letnik/volume 52 - št ./no. IS/06 - stn./pp. 7B3- Ljubljana, dec./Dec. 200B, zvezek/issue 5QO STROJNIŠKI VESTNIK JOURNAL OF MECHANICAL ENGINEERING ^«TROJNlS^ cena BOD SIT 9 770039"248001 ISSN DD33-24BO Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 783 Vsebina - Contents Vsebina - Contents Strojniški vestnik - Journal of Mechanical Engineering letnik - volume 52, (2006), številka - number 12 Ljubljana, december - December 2006 ISSN 0039-2480 Izhaja mesečno - Published monthly Uvodnik Editorial Alujevič A.: 500. številka (zvezek) Strojniškega Alujevič A.: 500th issue of the Journal of Mechani-vestnika 784 cal Engineering Razprave Papers Jakomin, M., Kosel, F., Batista, M., Kosel, T.: Preskok Jakomin, M., Kosel F., Batista, M., Kosel, T.: Snap-through sistema plitve osnosimetrične bimetalne of the System for a Shallow Axially symmetric lupine z uporabo nelinearne teorije 785 Bimetallic Shell using Non-linear Theory Banovec, P., Kozelj, D., Šantl, S., Steinman, F.: Izbira Banovec, P, Kozelj, D., Šantl, S., Steinman, F.: merilnih mest v vodovodnih sistemih z Sampling Design for Water Distribution genetskimi algoritmi 817 System Models by Genetic Algorithms Herakovič, N., Noe, D.: Analiza delovanja pnevmatiČnega Herakovič, N, Noe, D.: Analysis of the Operation of ventila s predkrmilnim piezoventilom 835 Pilot-Stage Piezo-Actuator Valves Podržaj, P., Kariž, Z.: Programljivi logični krmilniki na Podržaj, P, Kariž, Z.: Programmable Logic Controllers temelju rešitve algebraične Riccatijeve enačbe 852 Based on the Algebraic Riccati Equation Solution Nikolič, D., Vujadinovič, R., Iida, N.: Raziskava Nikolič, D., Vujadinovič, R, Iida, N.: Experimental učinkov različnih stopenj vračanja izpušnih Study of the Effects of Different Exhaust Gas plinov na temperaturo plamena in nastanek Recirculation Ratios on the Flame Temperature saj pri uporabi dizelskega goriva z različnimi and Soot Formation when Using Diesel Fuels T90 temperaturami destilacije 863 With Different T90 Distillation Temperatures Strokovna literatura Professional Literature Ocene knjig 873 Book Reviews Osebne vesti Personal Events Zoisovo in Puhovi priznanji 875 Zois and Puch Awards Častni doktorat - prof. dr. Manfred Geiger 876 Honors Doctor of Science - Prof. Dr. Manfred Geiger Prof. dr. Viljem Kralj - petinsedemdesetletnik 877 Prof. Dr. Viljem Kralj - 75 years Magisterij, specializacije in diplome 879 Master’s, Specialisation’s and Diploma Degrees Vsebina 2006 880 Contents 2006 Navodila avtorjem 885 Instructions for Authors Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 784 Uvodnik - Editorial 500. številka (zvezek) Strojniškega vestnika - 500th issue of the Journal of Mechanical Engineering Kmalu bo minilo 52 let od izida 1. številke Strojniškega vestnika, kar smo proslavili 24.3.2005. Najprej je izhajal 4-krat na leto, kasneje 6-krat na leto in sedaj 1-krat mesečno. Poletna številka je iz praktičnih vzrokov dvojna. Tako se je nabralo 500 številk (zvezkov) Strojniškega vestnika. V zadnjih nekaj letih so tri tematske izdaje (energetika, konstrukterstvo, tehnologija) in preostale, ki so mešane, saj avtorjev po končani presoji (doma in v tujini) ne želimo pustiti predolgo čakati. Ker je tujih avtorjev vedno več, je Izdajateljski svet sklenil, da znanstvene prispevke tujih avtorjev lahko tiskamo samo v angleškem jeziku (s slovenskim povzetkom). Pomemben dejavnik izhajanja Strojniškega vestnika je denarna pomoč Javne agencije za raziskovalno dejavnost Republike Slovenije, ki nam je za naslednja leta (2006 do 2008) odobrila približno polovico potrebnih sredstev. Preostalo zbiramo iz prispevkov tematskih številk, vedno bolj redkih oglasov, naročnin ter pokrivanja primanjkljaja s strani izdajateljev (predvsem obeh fakultet). V naslednjem letu čaka slovensko strojništvo izvedba Bolonjske prenove univerzitetnih učnih programov. Uredništvo Strojniškega vestnika pričakuje od vodstev obeh Fakultet za strojništvo v Ljubljani in Mariboru, da po odobritvi novih učnih programov na Svetu za visoko šolstvo Slovenije z objavo teh programov ovrže dvome, da gre le za preobleko dosedanjih učnih programov v nove kalupe, temveč za nov in drugačen način izobraževanja. Urednik prof.dr. Andro AlujeviČ Very soon it will be 52 years since the first issue of the Journal of Mechanical Engineering was published, an event we already celebrated on 24 March 2005. The journal was originally published 4 times a year, later this was increased to 6 times a year, and now the journal is published every month; although for practical reasons a double issue appears in the summer. In total there have been 500 issues. In recent years, three thematic issues - power, design, and technology - have been printed, the remaining issues being composed of mixed papers, thus enabling the publication of papers as soon as possible after an external and internal review procedure. Due to the abundance of foreign papers, the Publishing Council decided to allow the presentation of these papers in English only (with a translated Slovenian summary). An important contribution to publishing the Journal of Mechanical Engineering is that made by the Slovenian Research Agency. The agency will be providing approximately one half of the required funding for the period 2006 to 2008. The rest of the costs are covered by fees for thematic issues, occasional adverts, subscriptions and contributions from the Founders and Publishers (mainly the two faculties). In the coming year the national mechanical engineering teaching is to be subject to an assessment based on the Bologna agreement. The Editorial Board of the Journal of Mechanical Engineering expects that the two faculties of mechanical engineering, in Ljubljana and in Maribor, after the approval by the National Higher Education Board, will publish new programmes relating to this new and different way of providing education in Slovenia. Editor Prof. Dr. Andro AlujeviČ 784 Uvodnik - Editorial Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 UDK - UDC 624.074.4:539.3/.4 Izvirni znanstveni članek - Original scientific paper (1.01) Preskok sistema plitve osnosimetrične bimetalne lupine z uporabo nelinearne teorije Snap-through of the System for a Shallow Axially symmetric Bimetallic Shell using Non-linear Theory Marko Jakomin1 - Franc Kosel2 - Milan Batista3 - Tadej Kosel2 (1Primorski inštitut za naravoslovne in tehnične vede, Koper; 2Fakulteta za strojništvo, Ljubljana; 3Fakulteta za pomorstvo in promet, Portorož) V prispevku obravnavamo napetostne, deformacijske in stabilnostne razmere pri tankih, osnosimetričnih plitvih bimetalnih lupinah. Po teoriji drugega reda, ki upošteva ravnotežje sil na deformiranem telesu, podajamo model z matematičnim opisom geometrije sistema, premikov, napetosti in termoelastičnih deformacij. Enačbe temeljijo na teoriji velikih premikov. Kot primer predstavljamo rezultate za krogelne lupine, ki jih aproksimiramo s parabolično funkcijo. Poleg prosto položenih lupin obravnavamo tudi vrtljivo in konzolno vpete lupine ter lupine, ki so poleg segrevanja obremenjene tudi s silo v temenu. Deformacijsko krivuljo in temperaturo preskoka računamo numerično z nelinearno strelsko metodo. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: lupine bimetalne, obremenitve toplotne, preskok sistema, teorija velikih premikov) The paper deals with the stresses, strains and buckling conditions in thin, axially symmetric, shallow, bimetallic shells. Based on third-order theory, which takes into account the equilibrium state of the forces and moments that are acting on the deformed system, the paper presents a model with a mathematical description of the geometry of the system, the stresses, the thermoelastic strains and the displacements. The mathematical formulation is based on the theory of large displacements. As an example, the results for spherical, shallow shells are shown, approximated by a parabolic function. Besides simple roller-supported shells, also simple bearing-supported shells and clamped shells are discussed. The shells are loaded with temperature and/or a concentrated load acting at the top. The displacement state and the snap-through temperature are calculated numerically using a non-linear method. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: bimetallic shell, thermal loads, stability, snap-through of the system, large displacement theory) 0 UVOD Pospešen razvoj strojnih znanosti v zadnjih stoletjih je omogočil izdelavo različnih naprav, od razmeroma preprostih mehanizmov pa vse do zelo zapletenih strojnih naprav, ki jih človeštvo uporablja v tehnično tehnološkem postopku preoblikovanja materialnih dobrin. Čeprav so sodobne naprave po obliki, namenu in zgradbi med seboj zelo različne, pa se zaradi pomembnosti nemotenega in zanesljivega delovanja ter njihove vrednosti izraža zahteva po njihovi zaščiti pred različnimi preobremenitvami. Posebej pri strojih, ki spreminjajo eno obliko energije v drugo ter se pri tem segrevajo, je potrebno poskrbeti za zanesljivo zaščito pred toplotno 0 INTRODUCTION The development of mechanical sciences over the centuries has enabled the production of various devices, from relatively simple devices to very complex mechanical appliances, which are used in the technical-technological process of remodelling the material goods. Although modern devices differ greatly in shape, purpose and structure, they – for the sake of their unobstructed and reliable functioning – all require protection from various forms of overloading. This is particularly so for machines that transform one type of energy into another, warming up during the process, thus requiring real protection against excessive temperature over-loading. For this 785 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 preobremenitvijo. S tem namenom v naprave vgrajujemo elemente, ki opravljajo funkcijo “toplotne varovalke”, tako da stroj izklopijo takoj, ko posamezen del doseže največjo še dopustno temperaturo. Zaradi zanesljivosti delovanja so se pri zaščiti pred toplotno preobremenitvijo uveljavili črtni in ploskovni bimetalni konstrukcijski elementi, katerih delovanje sloni na znanem fizikalnem dejstvu, da se telesa s povečevanjem temperature raztezajo. Idealno homogena telesa se širijo in krčijo izotropno. V primeru bimetalnih teles, ki so izdelana iz dveh materialov z različnima temperaturnima razteznostnima koeficientoma, pa deformacije zaradi temperaturnih sprememb ne bodo več izotropne. V prispevku želimo poiskati in matematično formulirati funkcijsko zvezo med temperaturo, napetostjo in premiki bimetala. Ta zveza je med drugim odvisna tudi od geometrijskih značilk bimetala, saj se, npr. črtni bimetalni konstrukcijski elementi v primerjavi s ploskovnimi, na temperaturne spremembe različno odzivajo. Za prakso so predvsem pomembne razlike v stabilnostnih razmerah. Plitve bimetalne lupine imajo lastnost, da pri določeni temperaturi pridejo v indiferentno stanje, kar vodi v pojav, ki je v literaturi znan pod pojmom preskok sistema. 1 GEOMETRIJA SISTEMA Na sliki 1 je predstavljena osrednja ploskev tanke bimetalne vrtilno simetrične lupine. Osnosimetrična oblika lupine nastane z vrtenjem neke funkcije okoli ordinatne osi. Obliko nedeformirane osnosimetrične lupine v Lagrangevem koordinatnem sistemu torej določa funkcija y = y(x). Zaradi spremembe temperature se lupina deformira v novo obliko, ki jo določa funkcija Y = Y(X) v Eulerjevem koordinatnem sistemu. Tudi ta oblika je osnosimetrična v predpostavljenem homogenem temperaturnem polju, zaradi česar se mehanske veličine v odvisnosti od kota j ne spreminjajo. Vektor premika u v naravnem koordinatnem sistemu (y,j,ry ) poljubne točke P na osrednji ploskvi določa točko P’. Torej: purpose, elements that function as “heat cut-out” devices are installed, disconnecting the appliance as soon as a particular part reaches the maximum permissible temperature. Because of their reliability in functioning as protection from heat over-loading, linear and plane bimetallic structural elements are often used in these devices. Their working is based on the known physical fact that bodies expand with increasing temperature. Ideally, however, homogeneous objects expand and shrink in an isotropic manner. In the case of bimetallic bodies, which are made of two materials with different linear expansion coefficients, however, the deformations due to temperature changes will no longer be isotropic. In this paper we try to find and mathematically formulate the functional connection between the temperature, the strain and the displacements of a bimetal. This connection, among other things, also depends on the bimetal’s geometrical characteristics as, for instance, the linear bimetallic structural elements in comparison with the plane elements react differently to temperature changes. For practical purposes, above all, the differences in the stability conditions are important. Shallow bimetallic shells have the property that at a certain temperature they change to an indifferent state, which leads to a phenomenon known in the literature as a “snap-through of a system”. 1 GEOMETRY OF THE SYSTEM Fig. 1 shows the flexible, middle plane, of a thin, bimetallic rotationally symmetrical shell. The axially symmetric form of the shell is a result of the rotation of a function around the ordinate axis. The shape of undeformed axially symmetric shells in the Lagrange coordinate system is thus determined by the function y = y(x). Because of the change in temperature, the shell will deform into new shape, determined by the function Y = Y(X) in the Euler coordinate system. This shape is axially symmetric too, providing that at each point the bimetallic shell is exposed to the same temperature change. The axially symmetric bimetallic shell within a homogeneous temperature field, however, represents an axially symmetric loading example, and because of this the physical magnitudes depending on the angle j remain unchanged. The displacement of an optional point P on the middle plane of the shell to the point P’ is determined by the r displacement vector u in the natural coordinate system. Therefore: ,j,ry) = (u,v,w) 786 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Ker je problem osnosimetrične narave, v = 0, lahko tudi pišemo: The displacement state is axially symmetric, so we can also write: r u = (u,0,w) Osrednja ploskev lupine je določena z enačbo r = r (v), zaradi cesar je vektor premika u funkcija kota V u = u (ip). Opazujmo premike na tanki bimetalni osnosimetrični lupini v homogenem temperaturnem polju (sl. 1), ki je v temenu obremenjena s silo F. Zaradi temperaturne obremenitve se točka P, ki ima na nedeformirani lupini lego P(x,y(x)), premakne v lego P’ s koordinatama P’(X,Y(X)). Premik točke P v točko P’ določimo z enačbo (1), ki jo v ravnini (^r) lahko tudi pišemo u = (u,w). Zveza med Eulerjevim (X,Y(X)) in Lagrange-vim (x,y(x)) koordinatnim sistemom je (sl. 1): X = (1). The middle plan is determined by the equation ry = ry(y). Consequently, the displacement vector r rr u is the function of angle y: u = u (y) . Let us observe the displacement state of a thin bimetallic axially symmetric shell in a homogeneous temperature r field, Fig. 1, loaded at the top with the force Fk . Because of the temperature load, the point P(x,y(x)) on the undeformed shell will move to the point P’(X,Y(X)). The shifting of point P to the point P’ is determined by Equation (1), which in the plane r (y,ry) can also be written as u = (u, w) . The relationship between the Euler (X,Y(X)) and Lagrange (x,y(x)) coordinates is, Fig. 1: >y,Y ty,Y Sl. 1. Osnosimetrična lupina v homogenem temperaturnem polju in zveza med Eulerjevim in Lagrangevim koordinatnim sistemom Fig. 1. The axi-symmetric shell in the homogeneous temperature field and the connection between Euler and Lagrange coordinate system Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 787 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 kjer je: iz česar izhaja: r u (x,y) = where: cosy sin y siny -cos y following that: X = x +w siny +u cosy Y = y -w cosy +u siny (2) (3). Nadalje določimo ukrivljenost nedeformirane The curvature of the undeformed shell in the lupine v meridialni smeri y in obročni smeri j. meridian y and circular j direction is determined. An Infinitezimalno majhen del loka na krivulji je: infinitesimally small part of the arc of the curve in V obročni smeri: the circular direction is: dsj = r -siny-dj = x-dj v meridialni smeri : and in the meridian direction: dsy = ry •dy Geometrija sistema na sliki 1 določa naslednja The geometry of the system in Fig. 1 leads to razmerja: the following relations: y =arctan(y/) 1 ky == y " kj== r ry V(1 + y'2)3 sinarctan(y ) 1 kjer je s ky in k označena ukrivljenost lupine v meridialni oziroma obročni smeri, z eno oziroma dvema črticama pa prvi oziroma drugi odvod po Lagrangevi koordinati x: 1 (4) (5) (6), where ky and kj are the curvatures of the shell in the meridian and circular directions, respectively, while the one and two apostrophes, respectively, mark the first and second derivative with respect to the independent variable x: dx ( ) = ( )', T2( ) = ( )" Zaradi deformacije lupine se kot y spremeni v kot y, ukrivljenost lupine v meridialni smeri r oziroma obročni smeri r smeri pa v ry oziroma rj. Veljajo naslednje zveze, določene z geometrijo sistema na sliki 1: dx Due to the deformation of a shell the angle y y changes into the angle . The curvatures of the deformed shell are ry and rj in the meridian and circular directions, respectively. From the geometry of the system in Fig. 1 the following relations are determined: _ - • - - 1 Y X = r siny, Y = tany, ky= — = , & 1 siny sinarctanY 1 == = X & Y X X Z eno piko oziroma dvema pikama nad koordinato Y(X) sta označena prvi oziroma drugi odvod po Eulerjevi koordinati X: dY Y& = in/and Y& & dX One and two points above the coordinate Y(X) mark the first and the second derivatives, respectively, with respect to the variable X d2Y dX (7). 788 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Ker je koordinata Y zaradi enačbe (2) posredno funkcija Lagrangeve koordinate x, torej Y = Y(X) = Y[X(x)] izračunamo oba odvoda v enačbah (7) z uporabo t.i. verižnega pravila. Po vstavitvi teh odvodov v enačbo za koordinato X in enačbi za ukrivljenost lupine v deformiranem stanju sledi: Due to Equation (2) the coordinate Y is indirectly a function of the variable x , i.e., Y = Y(X) = Y[X(x)]. The derivatives in Equations (7) can be calculated by means of the so-called Chain Rule. After inserting into the Equation for the coordinate X and the Equations for the curvature of the shell in the undeformed state, it follows: y = arctan {Y' X} Y' X" Y' Y' X X2 + Yl (8) (9) (10). Diferencial dolDine dsy v meridialni smeri deformirane lupine je: The differential of the length dsy in the meridian direction of the deformed shell is: dsy JdX2 + dY2 = dxX2 + Y' ryd (11). 2 DOLOČITEV TENZORJEV DEFORMACIJ IN NAPETOSTI TER VEKTORJA PREMIKA Deformacijo lupine podamo z elementi vektorja premika na osrednji oziroma primerjalni ploskvi, in sicer s premikom u v meridialni ter s premikom w v prečni smeri. Osrednja ploskev se zaradi upogibnih napetosti ne deformira. Elemente deformacijskega tenzorja v krivočrtnem pravokotnem koordinatnem sistemu določimo po pravilih za spremembo tenzorjev iz enega v drugi koordinatni sistem ali pa z neposredno postavitvijo na podlagi deformiranega stanja elementarnega dela lupine, kjer upoštevamo premike na ukrivljeni ploskvi ([1] do [3]): du dy/ dw dz 2 DEFINING THE STRAIN AND STRESS TENSORS AS WELL AS THE DISPLACEMENT VECTOR The shell’s deformation state is shown by the displacement vector in the middle, i.e., the reference plane, and that, by the displacement of u in the meridian, and the displacement of w in the radial direction. Due to bending, the middle plane does not deform. The elements of the strain tensor in the curvilinear orthogonal coordinate system are determined by rules for tensor transformation from one into another coordinate system or by direct forming on the basis of the deformed state of the elementary part of the shell, where displacements on the curved plane are taken into account ([1] to [3]): 1 2ry2 g yj g yr dw dy dv 1 +--------- dj sin y dv 1 du dy ry dj -u du du dy/ 2r2 dw dy / 1 tanyJ r [ tan y/ ------cot y = 0 rjsiny ry dw 1 -u dw 1 ry dr dy ry dy ry (12) (13) (14) (15) (16) Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 789 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 -v dv \ dr V enačbah od (12) do (17) smo upoštevali predpostavko, da je lupina tanka in osnosimetrična tudi v deformiranem stanju, kakor sledi: du dw V enačbi (13) smo za pravokotno specifično deformacijo s v meridialni smeri upoštevali tudi nelinearni člen. Izkazalo se je namreč, da je upoštevanje nelinearnega člena in s tem teorije velikih premikov nujno za pravilnost rezultatov. Pri tem smo upoštevali, da je v nelinearnem členu komponenta u v primerjavi s komponento w zanemarljivo majhna. Z uvedbo krajevne koordinate z, z izhodiščem na osrednji ploskvi lupine, so zaradi ukrivljenosti lupine elementi tenzorja specifičnih deformacij tudi funkcija koordinate z: +—-----------= 0 (17). dp r ¦ sin tp In Equations (12) to (17) we have assumed that the shell is also thin and axially symmetric in the deformed state, as follows: ±( ) = 0, v = 0 (18). dp According to the third-order theory we also take into consideration the non-linear term in for the strain s in the meridian direction. Furthermore, we take into account that the displacement u in the nonlinear term is negligible compared with the displacement w. By introducing the local coordinate z with the centre of origin in the middle plane of the shell, the elements of the strain tensor are, due to the shell’s curvature, also a function of the coordinate z, as follows: L. = in eyz 1 + L* 1 1 r V ) = ey +z and = ej +z { 1 ( 1 r 1 1 r ?) zz 1+ 1+ ryry ejz zz 1+ 1+ = Li,+z-^ z-T y(19) j(20), kjer smo upoštevali, da je pri tankih lupinah: where it has been taken into account that for thin shells: @ @y@j@0 Deformacijski in napetostni tenzor sta: ryr The stress and strain tensors are: L, = Ll 0 ez yr sl 0 tz yr 0 ejz 0 , sij = 0 sjz 0 ipr 0 0 ipr 0 0 kjer je zveza med deformacijskim in napetostnim tenzorjem [3]: (21), where the relationship between the stress and strain tensors [3] is: E 1-m2 E 1-m2 (22) (23) (24) Z oznako T je v enačbah (22) in (23) označena sprememba temperature glede na primerjalno temperaturo T, pri kateri je napetostno stanje v lupini povsod enako nič: where T in Equations (22) and (23) denotes the change in temperature with respect to the reference temperature T0 at which the stress state in the shell is throughout equal to zero: syz (T0,y,z)= sjz (T0,y,z)= tyz r (T0,y,z)= 0 790 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 3 SILE, MOMENTI IN MEHANSKO RAVNOVESJE 3 EQUILIBRIUM STATE OF THE FORCES AND MOMENTS Slika 2 prikazuje elementarno majhen del Fig. 2 shows an elementary small part of the bimetalne lupine z napetostmi, ki se pojavijo na bimetallic shell, with stresses that appear on the prereznih ploskvah lupine. Zaradi napetostnega cross-section of the shell’s plane. The forces dNy, stanja, delujejo na ploskvah elementa ABCD sile dNy, dNj, dTy and the bending moments dMy, dMj on the dNj in dTyr ter upogibna momenta dMy in dMj: element ABCD planes are: dNy = J syz -h1 r +z dT dNj = J s -h1 r+z ry +z h2 yr yr f -h rj r +z dMy = J zs -h1 dMj =- 2 zs -h r siny ¦ dz ¦ dj = ny ¦ r sin y ¦ dj = nyX ¦ dj ry-dz-dy = njry ¦ dy = njdL rj sin y ¦ dz -dj = tyrrj sin y ¦ dj = tyrX ¦ dj r sin y ¦ dz ¦ dj = my ¦ r sin y ¦ dj = myX ¦ dj ry +z ry -dz-dy = m ¦ ry ¦ dy = m dL (25) (26) (27) (28) (29) kjer so n , n in t enotske sile in m , m enotska where n , n , ty denote unit forces and m , m unit mo- momenta, ki se pr yr tankih lupinah poenostavijo: ments, which are simplified in the case of thin shells: ny = 2 sydz h nj = 2 sjz¦ dz -h 2 my = J zs -h1 ¦ dz (30) (31) (32) (33) fzcr'-dz (34). j J j Po definiciji je osrednja ali primerjalna -h According to the definition, the middle, i.e., the ploskev tista ploskev na oddaljenosti h od reference plane is the one that lies at the distance h1 spodnjega roba lupine, ki se zaradi upogibnih from the lower edge of a shell. Because of the bending napetosti ne deformira. Lego osrednje ali primerjalne stresses the middle plane does not deform. The position ploskve dobimo iz pogoja ravnovesja notranjih osnih of the middle plane is obtained from the condition that sil, ki se pojavijo zaradi delovanja upogibnih takes into account that all inner forces, as a result of napetosti: the bending moment, must be in an equilibrium state: Jsyz-dA= 11 E1eyzxj- dz+ 2 E2eyzxj-dz = 0 (35). A -h1 d1 -h1 Specifično deformacijo ez računamo po The normal strain ez is calculated using enačbi (19), ki v primeru tanke lupine, ob upoštevanju, Equation (19), which, in the case of a thin shell and da je specifični raztezek s osrednje ploskve enak taking into account that the normal strain s of the nič, preide v obliko: y plane is equal to zero, takes the form: Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 791 Osrednja ravnina The middle plane Sl. 2. Element na deformirani lupini Fig. 2. Element on deformed shell z my+ dmy ny+ dny Y(X) ny + dny dmy Sl. 3. Enotske sile in momenti na elementu deformirane lupine Fig. 3. Unit forces and moments in the element of deformed shell 792 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 11 (36). Po integraciji enačbe (35) z uporabo enačbe After integration of Equation (35) and by using (36) je lega h1 osrednje ploskve : Equation (36), the position h1 of the middle plane is: E3+2E2SA+E2S22 (37). 2 E1d1+E2d2 () Iz pogoja h + h = d se določi tudi vrednost From the condition h1 + h = d the value for h h2. Ker je element na sliki 2 v ravnovesju, lahko is determined. Because the element in Fig. 2 is in an zapišemo enačbe ravnovesja sil in momentov equilibrium state we can write Equations of (sl. 3). equilibrium of the forces and moments, Fig. 3. Ravnovesje sil v meridialni smeri y: The equilibrium Equation of the forces in the meridian direction y: (dNy +d(dNy))-dNy cosdy -dTy sindy -2dN dip sin-----cos 2 y in v prečni smeri: and in the radial direction: (dTy +d(dTyr ))-dTyr cosdy +dNy sindy +2dNj dj sin-----sin 2 y dy dy (38). (39). Ravnovesna enačba v obročni smeri je zaradi Due to axially symmetric deformation, the osnosimetričnega napetostno-deformacijskega equilibrium Equation in the circular direction is equal stanja enaka nič. Zapišemo še upogibno momentno to zero. The equilibrium Equation for the bending ravnovesje: moments is: (dMy +d (dMy))-dMy -2dMj sin dj [2) dT,costp-dX Upoštevajmo, da je: ter dT^ sin^ • dY - dn^ sin^ -dX + dn^ cos^ • dY + 2dn,fi sin Note that: sin( dip) @ dtp, sin(dip) @ dip, cos dtp @ 1 dj dY (40). d(dN y Tako imamo po ureditvi: d(dN ) and -------- dtp = dn^X ¦ dip + ndX ¦ dip dtp d(dT, ) = ( dTr ) d^j = dt X¦ dip + t.dX¦ dip dtp d(dM,)= ( d-----) dtp = dm.X ¦ dip + mdX ¦ dip dtp After rearranging: ( naX) -nL'cos^-t^X^) =0 ( taX) + nvL' sin ^ + n^X ¦ (a) = 0 ( m^X ) - m^L' + n^X ( Y' cos tp - sin tp) - t^X ( Y' sin tp + cos tp) = 0 (41) (42) (43). 4 REŠEVANJE ENAČB ZA PRIMER PLITVIH LUPIN 4 SOLUTION OF THE EQUATIONS OF A SHALLOW SHELL Mešani sistem enačb (2), (3), (8) do (10), (13), The mixed system of Equations (2), (3), (8) to (14), (16), (19), (20), (22) do (24), (30) do (34) in (41) (10), (13), (14), (16), (19), (20), (22) to (24), (30) to (34) Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 793 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 do (43) z neznankami crz , ¦ y @ sin y @ tan y = y , cos y @ 1, y @ 0 Y' y < 1 =>¦ y @siny @ tan y = —j @ Y', cosy @ 1, Y 2 @ 0 Eulerjevi koordinati (X,Y) v enačbi (2) sta s Considering these simplifications, Euler’s co- temi poenostavitvami sedaj: ordinates (X,Y) in Equation (2) are now: X = x + wy' + u, Y = y-w + uy' (48). Iz enačb (5), (6), (9) in (10) izhaja: From Equations (5), (6), (9) and (10) it follows that: — @ y, — @ —, —@Y @y -w , — @ — @--------- ter zato: ry r x ry and thus: r x 11 ±y r y. = 1 - 1 @ -w/ 45 r r x tp tp in iz enačb (13) in (14): and from Equations (13) and (14): 11 1 1 / i\2 = y w + u + (w ey 2v ) (49) (50) (51) 794 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 e =-(y'w + u) (52). * x V enačbah (48) upoštevamo tudi, da je premik In Equations (48) we have taken into account u majhen v primerjavi s premikom w, ta pa je majhen that the displacement u is small in comparison with v primerjavi z Lagrangevo koordinato x, tako da sta the displacement w, and the latter is small in com- Eulerjevi koordinati X in Y približno: parison with the Lagrange coordinate x, so that the Euler coordinates X and Y are approximately: X @ x (53) Y@y-w (54). Ravnovesne enačbe (41) do (43) so sedaj: Thus the equilibrium Equations (41) to (43) are: nx'-n,-trxY" = 0 (55) ( rx ) +nY' + n,xY" = 0 (56) (mi,x)-mv-t^x = 0 (57). Pomnožimo sedaj enačbo (55) z Y’ ter ji Let us now multiply Equation (55) with Y’ and prištejmo enačbo (56). Tako imamo: add Equation (56). Thus, we have: (n^x) Y' + n^xY" + (t+rx) - t^xY'Y" = 0 Zadnji člen v enačbi zanemarimo, saj je po We disregard the last term in the equation, predpostavki: assuming that: Y'Y" = -(Y'2)'@0 Tako imamo po ureditvi zvezo: In this way we obtain the relationship: (trx )^-( nxY")' ter po integriranju: and after integration: c x tr=-nY + : (58), kjer je c integracijska stalnica, ki je odvisna od zunanje where c is a constant of integration, which depends on sile Fk v temenu lupine. Zvezo (58) vstavimo v enačbi the outer force Fk in the top of the shell. The (56) in (57): connection (58) is inserted into Equations (56) and (57): ( n*x ) -n =0 (59) ( mi>x ) -mv+ni>xY'-c = 0 (60). Sedaj imamo sistem enajstih enačb (44) do Now we obtained a system of 11 Equations (44) (47), (49) do (52), (54), (59) in (60) z enajstimi to (47), (49) to (52), (54), (59) and (60) and with eleven neznankami ny , n, m , m, u, w, e , ej, Y,? ,? . Naprej variables ny , n, m , m, u, w, ey , ej, Y,Y, Y . The next postopamo takole. Izrazimo premik u iz enačbe (52), steps are as follows. The displacement u is j expressed ga odvajamo po koordinati x ter vstavimo v enačbo from Equation (52). The derivative of the displacement (51). Nastane zveza: u with respect to the variable x is then obtained and inserted into Equation (51). In this way we get: (xe) +-(w')2=e<,+w'y' (61). Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 795 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Iz enačb (44) in (45) izrazimo ey in e ter ju vstavimo v enačbo (61). T in T nadomestimo z enačbama (49) in (59), enotsko silo n pa s prvim členom v enačbi (59). S tem dobi enačba (61) obliko: From Equations (44) and (45) are expressed ey and e, which are inserted into Equation (61). Next/ and Y are substituted by Equations (49) and (59), while the unit force n is substituted by the first term in Equation (59). Thus, Equation takes the form: -(x3n) A2 -A2 A y'w' AB -AB A (62). Sedaj v enačbo (60) vstavimo enačbi (46) in (47) ter spet izrazimo specifični deformaciji ey in e z enotskima silama n , n in razlikama ukrivljenosti T in T . Tako kako/prej nadomestimo T in T z enačbama (49) in (50), enotsko silo n pa s prvim členom v enačbi (59). Če iz enačbe (61) izrazimo še n' postane enačba (60): Now, Equations (46) and (47) are inserted into Equation (60). The normal strains ey and e are expressed by the unit forces ny, n and the differences in curvature Y andY . Also, T and Y are substituted by Equations (49) and (59), and the unit force n by the first term in Equation (59). If from Equation (61) is expressed too, Equation (60) becomes [4]: (y AB -AB A y'w' wi AC -B2 A (63). Prvotni sistem, ki ga je sestavljalo 21 enačb in prav toliko neznank, smo naposled prevedli v sistem dveh diferencialnih enačb (62) in (63), ki opisujeta najbolj splošen primer, ko sta m1 * m2 in & ^ <52. Enačbi (62) in (63) poenostavimo, če imata obe plasti lupine enak Poissonov količnik^ = M2, ker je_ takrat: A = mA; B = mB; C = mC; AB-AB = 0. Zato sta v tem primeru enačbi (62) in (63): The initial system, originally consisting of 21 equations and as many variables, has been finally converted into a system of two differential equations, (62) and (63), which outline the commonest example where m1 ^ m2 and S1 ^ S2. Equations (62) and (63) can be simplified if both layers of a shell have an equal Poisson’s coefficient M1 = ^because then: A = mA; B= mB; C = mC; AB - AB = 0 . Hence, in this case Equations (62) and (63) are: x 3 / = A(1-m in )\y'w'--(w')\ and xn^ (y - w) AC -B2 A (64) (65). Z uvedbo brezrazsežne vodoravne koordinate %. By introducing a dimensionless horizontal coordinate c: (66) in Wittrickovih funkcij G, F0 in F: no = -Y' = xa prevedemo problem v brezrazsežno obliko [6]. Iz enačbe (66) izhajajo namreč razmerja: and Wittrick’s functions G, F0 and F: (67) (68) (69) A2 - A2 the problem is converted into the dimensionless form [6]. Because of the introduction of a dimensionless coordinate ^we can write: G(c) a2 (AC-B2 A j F0(c) J2(AC-B2) a2 V A2-A2 F(c) J2(AC-B2) 796 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 x = ac, — = ~c, Ox a 'dc .dx, 2 c, „2 2 a ox a zato so posamezni členi v enačbah (64) in (65) po therefore, the individual terms in Equations (64) and spremembi: (65), after the transformation, are: ( xn) = 3x + x2 = 4c dx ii i( i i) c y w = y y - Y = — AC -B 2 x a '2 ( AC-B2 ) A d2 dc (c-G(c)) [A2(1-^2)) xw" + 1 wy] =x ( y'-Y')" + ( y'-Y')' - x1( y'-Y') = 4a c2 (F0(c)-F(c))F0(c) d2 2 AC-B2 A2(1-fj,2 2 c-(F0(c)-F(c) (y-w)' =Y' Diferencialni enačbi (64) in (65) v brezrazsežni obliki sta tako: Thus, the differential equations (64) and (65) in the dimensionless form are: 4(c(F-F0)) =FG 4(cG)" =F02-F2 c-a2 A2 1-S c AC-B2\2(AC-B2 (70) (71), pri čemer smo z dvema črticama označili drugi odvod where two apostrophes mark the second derivative po koordinati Z: with respect to the coordinate Z: — ( ) = ( )', d2 ( ) = ( )ff dc dc Odvisni brezrazsežni spremenljivki, po The dependable, dimensionless variables katerih rešujemo diferencialni enačbi (70) in (71), with which we are solving the differential equations sta torej oblikovna funkcija trenutne oblike (70) and (71) are thus the formative function F() of bimetalne lupine F(Z) ter napetostna funkcija the present form of the bimetallic shell and the stress G(x). Integracijsko stalnico c v enačbi (71) function G(Z). The constant of integration c in določimo z upoštevanjem ravnovesja sil na robu Equation (71) is determined by taking into account lupine. the equilibrium of forces at the edge of a shell. Če je lupina prosto položena, je sila podpore If the shell is simply roller supported, the force V na robu lupine nasprotno usmerjena in po r vrednosti enaka sili Fk v temenu lupine (sl. 4): V at the edge of a shell is pointing in the opposite Fk r direction and is equal to the force Fk in the top of the shell, Fig. 4: 1 Ve Sl. 4. Obremenitev prosto poloUene lupine s silo Fk Fig. 4. Loading of simply-roller supported shell with the force F Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 797 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 rrr r Fk +V = Fk +(2pa)Ve = 0 (72), kjer je z V označena enotska navpična sila na robu lupine. Enačba (72) je v komponentni obliki: r where Ve denotes the vertical unit force at the edge of the shell. Equation (72) in component form is: F tyr +ny dY dx 2pa = 0, -tyr dY dx +n =0 (73). Rešitev sistema enačb (73) je potem, ko upoštevamo tudi plitvost: When also taking into account the shallowness, the solution of the system of Equations (73) is: ny(a)= ta yr() 2pa dx 2pa (74) (75). Vstavimo enačbi (74) in (75) v enačbo (58) ter izrazimo stalnico c: Če na lupino ne deluje zunanja sila Fk je integracijska stalnica c v enačbi (76) enaka nič, diferencialni enačbi (70) in (71) pa se poenostavita v obliko, ki jo je v [6] zapisal W. H. Wittrick: Equations (74) and (75) are inserted into Equation (58) and the constant c is expressed by: -Fk 2p (76). r If the external force Fk does not act upon the shell, the integrating constant c in Equation (76) is equal to zero, whilst the differential Equations (70) and (71) are simplified into a form, according to W. H. Wittrick in [6]: 4(cG)"=F02-F2 4(c(F-F0))"=FG (77) (78). 5 ANALIZA RAZMER PRI KROGELNIH LUPINAH V enačbah (70) in (71) se pojavlja oblikovna funkcija začetne oblike lupine F(X). Ta je odvisna od funkcije y = y(x), ki opisuje osrednjo ploskev začetne, nedeformirane oblike lupine. V primeru krogelnih lupin, katerih teme je postavljeno v izhodišče koordinatnega sistema, je ta funkcija v posredni obliki: x2 + y2 = 2yR. Zaradi plitvosti lupine zanemarimo drugi člen ter dobimo po zamenjavi spremenljivke x v brezrazsežno spremenljivko^: 5 ANALYSIS OF THE CIRCUMSTANCES IN SPHERICAL SHELLS In Equations (70) and (71) occurs the formative function F0(c) of the initial shape of the shell. It depends on the function y = y(x), which describes the middle plane of the initial, undeformed shape of the shell. However, in the case of spherical shells, whose crowns are set at the beginning of the coordinate system, this function is in the implicit form x2 + y2 = 2yR. Because of the shallowness of the shell, the second term is neglected, so that after the substitution of the variable x into the dimensionless variable c, we obtain: x ac = 2R 2R h0c (79). Oblikovna funkcija začetne oblike lupine je po enačbi (68): According to Equation (68) the formative function of the initial shape of the shell is: 2hA 1-m2 2 AC-B2 konst (80). Če upoštevamo, da je začetna oblikovna funkcija F0 nespremenljiva, sta diferencialni enačbi When taking into account that the initial formative function F0 is a constant, the differential 798 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 (70) in (71): Equations (70) and (71) are: 4G"x + 8G' 4F"x + 8F' = FG- F02 -F2 A2 1-t? X AC-B2\2(AC-B2 (81) (82). Diferencialni enačbi (81) in (82) najlažje rešimo za primer, ko lupina ni obremenjena z zunanjo silo F , ker je takrat v enačbi (82) stalnica c enaka nič. k e je lupina prosto položena, so napetosti in momenti na njenem robu enaki nič, v temenu lupine pa imajo napetosti in momenti le končne vrednosti. Za napetostno funkcijo G(c) veljata torej robna pogoja: The easiest way to solve Equations (81) and (82) is in the case where the shell is not loaded by an external force Fk because then the constant c in Equation (82) is equal to zero. If the shell is simply roller supported, the stresses and moments at its edges are equal to zero, while the stresses and moments in the top of the shell have only limited values. For the stress function G(c) the following boundary conditions thus hold: G(1)=0, G(0)^oo Robna pogoja za oblikovno funkcijo F(c) dobimo iz enačbe (46), ki jo zapišemo v brezrazsežni obliki. Iz enačbe (44) in (45) izrazimo ey in ej ter vrednosti vstavimo v enačbo (46) za enotski moment m . Spremembo ukrivljenosti T in T nadomestimo z enačbama (49) in (50) prvi in drugi odvod premika w pa izrazimo z odvodom enačbe (54). Če nadalje spet vzamemo enak Poissonov koeficient za obe plasti bimetalne lupine se enotski moment my izraža: (83). The boundary conditions for the formative function F(c) are obtained from Equation (46), written in dimensionless form. From Equation (44) we express ey and insert the value into Equation (46) for the unit moment my . The curvature differences Y andY are then replaced by Equations (49) and (50), while j the first and second derivatives of the displacement w are expressed by the derivative of Equation (54). In the case of an equal Poisson’s ratio for both layers of the bimetallic shell, the unit moment m runs as follows: C 2C A(1-f) (F0(1 + fi) - F(X)(1 + ^) -2XF'(X)- kadar velja tudi: d1 =d2 in the special case where: in/and E1 = E2 = E QT(84), (85). Ker je na robu lupine poleg enotske sile n ničen tudi enotski moment m , izrazimo iz (84) povezavo med temperaturo lupine T in trenutno oblikovno funkcijo F: At the edge of the shell, besides the unit force ny, also the unit moment my is equal to zero. So, the relationship between the shell’s temperature T and the present formative function F can be derived from Equation (84): F F(1) 1+ m F (1) (86), kjer je t brezrazsežna funkcija temperature, T pa nespremenljiva: where t is the dimensionless function of the temperature and Tm is a constant: 2h0(1+m)(AC -B2) Tm = 2 (87). a (AQ -BP) Za oblikovno funkcijo F sta torej robna Thus, the boundary conditions for the pogoja: formative function F are: F F(1) 1 +m F (1) F(0) (88). Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 799 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Kadar je bimetalna lupina takšna, da veljajo enačbe (85), potem je iz enačbe (37): h = h = d1 = d2 = d/2. Če slednje upoštevamo pri izračunu stalnic A,A, B, B, C, C, P in Q, se enačbi (80) in (87) za nespremenljivi F0 in T poenostavita: F 2y/6-h 1 T 2d Let us take into consideration the case where Equations (85) are fulfilled. Consequently, from Equation (37) it follows that h1 = h2 = d1 = d2 = d/2. If what was stated above is considered in the calculation of the constants A, A , B, B , C, C , P and Q, Equations (80) and (87) for the constants F0 and Tm are simplified: 2d2F (89). 3R(a1-a2) 3V6-a21~ Za numerično reševanje problema pa robna pogoja G(0) ^ oo in F(0) ^ oo nista primerna, zato uvedemo novi spremenljivki g(c) in f(c): f = c-F, Po spremembi diferencialnih enačb (81), (82) in robnih pogojev (83) in (88) ter upoštevanju, da sta funkciji G in F v c = 0 omejeni, imamo naslednji problem robnih vrednosti: 4g = F02-g(0) = g(1) = f(0) = 0, t = 1 f c2 For a numerical solution of the problem, however, the boundary conditions G(0) ^ oo andF(0) ^ oo are not suitable; hence new variables g(c) and f(c) have been introduced: g = c-G (90). After transforming the differential equations (81) and (82), the boundary conditions (83) and (88), and taking into consideration that functions G and F in c = 0 are limited, we obtain the following boundary-value problem: f-g 2 4f" = (2f'(1) f(1)•(1-m )) (91). Prevedemo ga v sistem navadnih diferencialnih enačb prvega reda: y1 = y2, y2 y1(0) = y1(1) = y3(0) če uvedemo zamenjavo: F2 2 c , t = 1- F0(1 + m) It is then converted into a system of ordinary differential equations of the first order: 4 c2 (2y4(1)-y3(1)-(1-)) y3 =y4, y4 (92), dg dc g = y1, g = — = y2, Sistem enačb (92) s temperaturo t v območju 0 < t < 2 smo rešili z uporabo nelinearne strelske metode. Izbrali smo približni vrednosti y2 (1) in y4 (1) ter izračunali približne vrednosti funkcij g in f po običajni enokoračni metodi Runge Kutta 4. reda. Določanje natančnejših vrednosti y2 (1) in y4 (1) je potekalo po Newtonovi metodi reševanja nelinearnih enačb. Ker je sistem enačb (92) v točki c = 0 singularen, smo odstopanje približnih vrednosti g in/od danih robnih pogojev v c = 0 računali v točki c = c = 1010. Deformacijo lupine smo zapisali z razmerjem x med trenutno višino deformirane lupine in začetno višino nedeformirane lupine: F0(1 + m) when the substitution is introduced: f f i df = ž/3, = — = y4 The system of Equations (92) with a temperature t in the interval 0 < t < 2 was solved using the non-linear shooting method. We took approximate values for y2 (1) and y4(1) and calculated rough values for the functions g and f using the classical one-step Runge Kutta method of the fourth order. For defining more exact values of y2 (1) and y4(1), the Newton method of solving non-linear equations was used. And since the system of Equations (92) at the point c = 0 is singular, the digressions from the approximate values g and f from the given boundary conditions in c = 0 were calculated at the point c = c = 10-10. The shell’s deformation was recorded with the ratio x between the present height of the deformed shell and the initial height of the undeformed shell: h h Y (1) y(1) 1 1 1 -f-f(c)-dc F0c0 c (93). 800 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Integral v (93) smo računali numerično. Zaradi singularnosti v = 0 je integracija potekala od = 1 do X= Z0. Tako smo izračunali razmerje višin Lpri različnih temperaturah t. Problem robnih vrednosti (91) lahko rešimo tudi po metodi, ki jo je v [6] predlagal W. H. Wittrick. Ker sta funkciji g in f v točki x= 0 singularni, ju v okolici te točke zapišemo v obliki potenčne vrste, tako da zadostimo robnim pogojem v = 0: The integral (93) was solved numerically. Due to the singularity in^= 0 the integration occurred from = 1 to x=X0. Consequently, we have calculated the values of ratio L at various temperatures t. The boundary value problem (91) could be solved too using the method proposed by W. H. Wittrick in [6]. Because the functions g and f at the point x = 0 are singular, they have been written in the form of power series about the point = 0 in such a way as to meet the boundary conditions in = 0: CO J2gnXn+1, f = J2fnXn+1 (94). n=0 n=0 Po vstavitvi potenčnih vrst (94) v diferencialni enačbi robnega problema (91) ter primerjavi koeficientov pri enakih potencah spremenljivke X na obeh straneh enačb, dobimo razmerja med koeficienti: F0-f0 After inserting the power series (94) into the boundary-value problem (91) and comparing the coefficients in equal exponents of the variable c on both sides of the equations, we obtain the relations between the coefficients: g1 f1 g2 -2f0f1 g4 g3= 4 -2f0f3-2f1f2 242 -f1 f0g1 +f1g0 = 24 f0g2 +f1g1 +f2g0 f4 , f2 f3 48 f0g3 +f1g2 +f2g1 +f3g0 (95). itn./etc. Sistem enačb (95) je rekurziven. Pri izbranem g0 in f0 so določeni vsi nadaljnji koeficienti g in fn funkcije g in f. Koeficienta g0 in f0 seveda izberemo tako, da funkciji g in f na robu lupine v * = 1 zadoščata robnima pogojema v enačbi (91). Za izbrani koeficient g0 smo izbrali neko začetno vrednost koeficienta f0 ter v območju 0 < v < 0 05 določili člene potenčne vrste funkcij g in f. S potenčnima vrstama smo izračunali funkcijske vrednosti g(0,05), g’(0,05), f(0,05) in f’(0,05) ter s temi začetnimi vrednostmi numerično izračunali vrednosti funkcij g in f v območju 0,05 < x < 1 . Ker je izbrana začetna vrednost koeficienta f0 le približek, funkcija g v točki x = 1 odstopa od robnega pogoja v enačbi (91). Po Newtonovi metodi smo zato določili novo, natančnejšo vrednost koeficienta f0, ter postopek ponovili tolikokrat, da je postala absolutna vrednost funkcije g(1) natančna do vnaprej predpisane vrednosti. Prednost tega numeričnega postopka pred prej opisanim je v tem, da z Newtonovo metodo ali bisekcijo računamo vrednost samo ene neznanke, in sicer vrednost koeficienta f0 oziroma y3 (0,05), medtem ko je treba pri strelski metodi izračunati vrednosti dveh The system of Equations (95) is recursive. By selecting g0 and f0 all further coefficients g and fn of the function g and f have been determined. Clearly the coefficients g0 and f0 are selected in such a way that the functions g and f at the edge of a shell in^= 1 meet the boundary conditions in Equation (91). For the selected coefficient g0 we have taken an initial value for the coefficient f0 and in the interval 0 < y < 0 05 determined the elements of the power series’ , for functions g and f. Then we have calculated, using power series, the functional values g(0,05), g’(0,05), f (0,05) and f’(0,05). These initial values were used to compute the numerical values of the functions g and f in the interval 0,05 < y < 1. Because the selected initial value for the coefficient f0 is only an approximation, the function g at the point^= 1 deviates from the boundary condition in Equation (91). Using Newton’s method we have then determined a new, more exact value for the coefficient f0 and kept repeating the procedure until the absolute value of the function g(1) was as precise as set up initially. The advantage of this numerical procedure over the one described before is that by using Newton’s method or bisection we calculate the value of only one variable y3(0,05), while with the shooting method it is necessary to calculate the values of two variables, y2(1) Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 801 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 FQ = 12 -0.5 Sl. 5. Funkcija t = t(0 za primer osnosimetrine lupine s F = 12 in M = 1/3, ki izkazuje pojav preskoka sistema med točkama AB ob segrevanju in točkama CD ob ohlajanju Fig. 5. The function t = t(® as an example of axi-symmetric shell for F0 = 12 and M = 1/3 expressing the phenomenon of a snap-through system between points AB in the process of heating up and the points CD in the process of cooling spremenljivk y2 (1) in y4 (1). Po obeh numeričnih postopkih smo dobili enake rezultate. Slika 5 prikazuje razmere v lupini s Poissonovim koeficientom M = 1/3 in začetno oblikovno funkcijo F0 = 12. Graf funkcije brezrazsežne temperature t v odvisnosti od razmerja višin 4 predstavlja stabilnostne razmere ob temperaturnem obremenjevanju bimetalne lupine . V začetnem, temperaturno neobremenjenem stanju t = 0, v točki O(1,0) je razmerje višin ^enako ena. S povečevanjem brezrazsežne temperature t se to razmerje zmanjšuje. Kakor je razvidno s slike 5, je območje na krivulji med točko O in točko A(p1, tp1) = A(0,360;1,195), kjer ima funkcija t(^ lokalni vrh, območje stabilnega ravnovesja. Do preskoka lupine bo torej prišlo v točki A pri temperaturi tp1 = 1,195, ker je korak med točko A in točko C(^2, A = C(-0,360;0,805), kjer ima funkcija lokalni dol, območje nestabilnega ravnovesja. Po preskoku bo lupina zavzela novo ravnovesno lego v točki B(-0,752;1,195) pri temperaturi t = 1,195. Pri nadaljnjem segrevanju lupine razmerje L še naprej zmanjšuje. Pri ohlajanju lupine imamo nasproten pojav in v točki C pri temperaturi tp2 = 0,805, ponoven preskok. Tokrat lupina preskoči v ravnovesno lego v točki D(0,752;0,805) pri temperaturi t = 0,805. S and y4(1). Nevertheless, with both of these numerical methods we obtained the same results. Fig. 5 shows the condition for the shell with Poisson’s ratio m = 1/3 and the initial formative function F0 = 12. The graph of the function of dimensionless temperature t depending on the ratio of heights x represents the stability circumstances during the shell’s temperature load. In the initial state with temperature t = 0, at point O(1,0) the ratio of heights x is equal to one. By increasing the dimensionless temperature t this ratio is decreasing. As shown in Fig. 5, the portion of the curvature between point O and point A(xp1, tp1) = A(0.360,1.195) where the function t(x) has the local maximum is the range of stable equilibrium. Hence, the snap-through of the shell will happen at point A at temperature tp1 = 1.195 because the interval between point A and point C(xp2, t p 2 ) = C(-0.360,0.805)where the function has the local minimum is the range of unstable equilibrium. After the snap-through, the shell will take a new equilibrium state at point B(-0.752,1.195) at temperature t = 1.195. In the course of the shell’s further heating up, the ratio of heights x continues to decrease. However, in the cooling down of the shell the reverse situation occurs and at point C at temperature tp2 = 0.805, another snap-through happens. This time the shell snaps into the equilibrium state at point 802 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Sl. 6. Stabilnostne razmere pri lupinah z različnimi vrednostmi funkcije F0 in M = 1/3 Fig. 6. Snap-through behaviour for shells with various values of function F0 and M = 1/3 Preglednica 1. Temperatura in lega preskoka lupine za različne vrednosti funkcije F0 in M = 1/3 Table 1. Temperature and position of the shell's snap-through for different values of function F0 and M= 1/3 F0 8, 93 10 12 14 16 S.T.1 tp1 =1 xp1 =0 tp1 = 1,0 43 xp1 = 0, 248 tp1 = 1,195 xp1 = 0, 361 tp1 = 1, 381 xp1 = 0, 411 tp1 = 1, 567 xp1 = 0, 445 S.T.2 tp2 =1 xp2 =0 t p2 = 0.957 xp2 = -0, 248 tp2 = 0, 805 xp2 = -0, 361 tp2 = 0, 619 x p2 = -0, 41 1 tp2 = 0, 433 xp2 = -0,445 ponovnim segrevanjem lupine do temperature prvega preskoka tp = 1,195 lahko celoten krog preskokov lupine ponovimo. Stabilnostne razmere pri lupinah z drugačnimi vrednostmi začetne oblikovne funkcije F0 so prikazane na sliki 6, preglednične vrednosti za temperaturo preskoka tp in razmerje višin L v trenutku preskoka lupine pa so zapisane p v preglednici 1. Kritična vrednost začetne oblikovne funkcije F0 izpod katere preskok lupine ni mogoč, znaša za lupine s Poissonovim količnikom H= 1/3, F0 = Fk = 8,93. Krivulja s F0 = 0 ponazarja razmere pri okrogli bimetalni plošči. Potek krivulje za brezrazsežno temperaturo t v odvisnosti od razmerja višin ^ je asimetrična glede na premico F0 = 0. D(0.752,0.805) at the temperature t = 0.805. By heating the shell up to the temperature of the first snap-through tp1 = 1.195, the whole cycle of the shell’s snaps is repeated. The snap-through behaviour of the shells with different values of the initial formative function F0 is shown in Fig. 6, while the tabulated values of the snap-through temperature tp and the ratio of height xp at the moment of the shell’s snap-through are presented in Table 1. The critical value of the initial formative function F0 under which the shell’s snap-through is not possible amounts to F0 = Fkr = 8.93 for the shells with Poisson’s ratio m = 1/3.The curve with F0 = 0 shows the conditions for the round bimetallic plate. The curve’s line for dimensionless temperature t relative to ratio x, is asymmetrical with respect to the straight line F0 = 0. Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 803 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Na sliki 7 sta prikazani krivulji za temperaturo prvega preskoka tp 1 in razmerje višin L v odvisnosti od začetne oblikovne funkcije F0. p1 Posledica temperaturne obremenitve prosto položene lupine je tudi vodoravni premik, določen z razliko med Eulerjevo in Lagrangevo koordinato X-x. Enačba (48) za Eulerjevo koordinato X je, potem ko izrazimo komponento u vektorja premika u iz enačbe (52), specifično deformacijo s pa iz enačb (44) in (45): Fig. 7 shows the curvatures for the temperature of the first snap-through tp1 and the ratio of heights xp1 depending on the initial formative function F0. The consequence of the temperature loading of the simply roller-supported shell is also a horizontal displacement, determined by the difference between the Euler and Lagrange coordinates X-x. After expressing the component u of the dis-r placement vector u from Equation (52) and the normal strain ej from Equations (44) and (45), Equation (48) for the Euler coordinate X is: X = aVx mny PT A(1-m2) A(1+ m) (96). Enotsko silo n izrazimo iz enačbe (67), enotsko silo n pa z zvezo med obema enotskima silama iz enačbe (59). Vodoravni premik X(,t) je torej pri danih parametrih lupine odvisen od napetostne funkcije G(et): The unit force ny is expressed by Equation (67) and the unit force nj by the connection between both unit forces by means of Equation (59). Hence, the horizontal displacement X(c,t), for the given parameters of the shell, depends on the stress function G(c,t): X(X,t) = ay/X C(2G'x + G(1-fi)) P-t-Tm (97). a2A(1-n2) A(1 + fi) Za primer smo izračunali brezrazsežni As an example, we have calculated the di- vodoravni premik (X(a) - a)/a v odvisnosti od mensionless horizontal displacement (X(a) - a)/a temperature t za lupino s parametri : depending on the temperature t for the shell with the parameters: F =12, h0= 0,78mm, a=15mm, <5=0,3mm, a1 = 3,41 • 10-5 / K, a2 = 1,41 • 10-5 / K Kakor je razvidno s slike 8, se vodoravni premik bimetalne lupine s temperaturo veča. Največji vodoravni premik je v trenutku preskoka v As shown in Fig. 8, the horizontal displacement of the bimetallic shell increases with temperature. The biggest horizontal displacement is at the beginning of z, "C 1.5 J_-^-~""""" 1.25 _ 0.75 0.5 _^L.____—------- 0.25 ?; 10 13 14 .1 IS Sl. 7. Temperatura in lega preskoka v odvisnosti od oblikovne funkcije F0 za lupine z m = 1/3 Fig. 7. Temperature and snap-through position depending on the formative function F0 for the shells with m = 1/3 804 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 F0=V2: h, = 0,78mm, a=16mm, <5=0:3mm. Ql = 3.41 ¦ 1CT° / K. a, = 1,41 ¦ 10"s / K 0.003 - Sl. 8. Vodoravni premik na robu lupine v odvisnosti od temperature t za prosto položeno lupino Fig. 8. Horizontal displacement at the edge of the simply-roller supported shell relative to temperature t FO = Br93: 10; 12: 14; 16 0.005 ¦ 3 . u j 4 ¦ : .o: 3 3.032 ¦ 3.031 F0=1L Sl. 9. Vodoravni premik na robu lupine za prosto položene lupine različnih vrednosti začetne oblikovne funkcije F0 Fig. 9. Horizontal displacement at the edge of the simply-roller supported shell of various values of the initial formative function F0 točki A. Po preskoku zavzame lupina novo the snap-through process, at the point A. After the snap- ravnovesno lego, vodoravni premik pa preide v točko B. Na sliki 9 je prikazan vodoravni premik na robu lupine za lupine z debelino d = 0,3 mm, tlorisnim polmerom a = 15 mm, razteznostnima koeficientoma a1 = 3,41 . 10-5 /K in a2 = 1,41 . 10-5 /K ter začetno oblikovno funkcijo F0, kar izhaja s slike. S povečevanjem začetne višine h0 in s tem through, the shell assumes a new equilibrium position, while the horizontal displacement passes to the point B. Fig. 9 shows the horizontal displacement at the edge of the shell with thickness d = 0.3 mm and horizontal radius a = 15 mm, coefficients of linear temperature .. expansion a1 = 3.41 10-5 /K and a2 = 1.41 10-5 /K and initial formative function F0, as follows from the graphical presentation. By increasing the initial height h0 and Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 805 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 funkcije F0 poleg temperature preskoka se zvečuje tudi vodoravni premik na robu lupine v x = a. Obravnavali bomo tudi preskok sistema lupine z vrtljivim vodoravno nepomično vpetim robom. Takšen je primer, če lupino vstavimo v valj ter tako preprečimo širjenje lupine v smeri osi X. V tem primeru vpetja je premer lupine a med temperaturnim obremenjevanjem stalen: a(T) = a = konst. Veljata robna pogoja: s (1) = 0; m(1) = 0. y Specifična deformacija s ima v primeru, ko sta debelini slojev enaki d1 = d2 = d/2 naslednjo obliko: subsequently the function F0, as well as an increase in the snap-through temperature the horizontal displacement at the edge of the shell at x = a is increased too. The next example we will discuss is the snap-through of the simply bearing-supported shell. This is the case when the shell is inserted into a cylinder, preventing in this way the expansion of the shell in the direction of the X axis. In this type of support, the radius of the shell a remains constant: a(T) = a = const. during the temperature loading. The boundary conditions are: ej(1) = 0; my(1) = 0. The normal strain ej has, in the case where the thicknesses of layers are equal, the following form: L, = (98). (ny-n)+PT(1-//) Ker se ny in n v (98) izražata z enačbama (67) Because n and n in (98) are expressed by in (59), zapišemo problem robnih vrednosti za vrtljivo Equations (67) and (59) we can write the boundary- vodoravno nepomično vpeto lupino: value problem for simply bearing-supported shells: 4g F2 f2 f 4 fll = ' g g(0) = f(0) = 0, g'(1)-g(1)[ c 1+m -72/3-F0(a1+a2)1-~ (99). t =1- F0(1 + M) Sistem enačb (99) smo rešili numerično s predhodno opisano nelinearno strelsko metodo. (2f'(1)-f(1).(1-M)) The system of Equations (99) was solved numerically using the above-described non-linear 15mm, 1/3. č = 0. Snm 1.75 n2=3.41xl{TtKr1 afcl.aixiar^KT1 a2=1.41»;l(r5K-1 :[k; Sl. 10. Razmerje višin $v odvisnosti od temperature T za vrtljivo vodoravno nepomično vpeto lupino Fig. 10. Ratio of heights L relative to the temperature T for simply bearing-supported shell 806 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Izbrali smo približni začetni vrednosti za funkciji g in f v točki x = 1 ter problem začetnih vrednosti rešili po metodi Runge Kutta 4. reda. Bolj natančne funkcijske vrednosti g(1) in f(1) smo spet računali z Newtonovo metodo za reševanje nelinearnih enačb. Na sliki 10 so predstavljene razmere pri temperaturnem obremenjevanju vrtljivo vodoravno nepomično vpete lupine za številčni primer F = 12; a = 15 mm;^1/3; d = 0,3 mm. Pri takšnem vpetju bimetalne lupine se razmerje višin L s povečevanjem temperature povečuje. Zaradi segrevanja se lupina razteza. Ker je raztezanje lupine v vodoravni smeri onemogočeno, se lupina razteza v smeri navpičnice. S temperaturo T se povečuje trenutna višina lupine Y in s tem razmerje LKer funkcijamT) nima lokalnega ekstrema sklepamo, da vrtljivo vpeta lupina za ta primer nima preskoka. Podobne razmere opažamo tudi pri enoslojnih lupinah oziroma lupinah z nespremenljivim koeficientom linearnega temperaturnega raztezka a1 = a2. Kakor je razvidno s slike 10, je deformacija bimetalne lupine s temperaturnim raztezkom slojev a1 > a2 nekje med deformacijama enoslojnih lupin z nespremenljivim temperaturnim raztezkom a1 in a2. Deformacijske krivulje, ki prikazujejo obliko lupine glede na temperaturno obremenitev, so razvidne s slike 11. shooting method. The approximate initial values for the functions g and f at the point c = 1 were selected and the problem of the initial values solved by the Runge Kutta method of the fourth order. More precise functional values g(1) and f(1) were again calculated by means of Newton’s method for solving non-linear equations. Fig. 10 shows the conditions in the temperature loading of simply bearing-supported shells for the numerical sample F0 = 12; a = 15 mm; m = 1/3; d = 0.3 mm. In this example of the bimetallic shell, the ratio x increases with increasing temperature. Because of the heating, the shell is expanding. And since the expansion of a shell in the horizontal direction is not possible, the shell is expanding in the vertical direction. Along with the temperature T the height of the shell Y is also expanding, and with this the ratio x. Since the function x(T) does not have a local extreme we can conclude that shells with a simple bearing-support do not have a snap-through. Similar results were also observed in the single-layer shells with a constant coefficient of linear temperature expansion a1 = a2. As seen from Fig. 10, the deformation of the bimetallic shell with a temperature expansion of layers a1 > a2 is somewhere between the deformations of single layer shells with constant temperature expansion a1 and a2. The deformation states of the shell relative to the temperature are shown in Fig. 11. Sl. 11. Deformacijske krivulje za vrtljivo vodoravno nepomično vpeto lupino s F = 12 in M = 1/3 Fig. 11. Displacement states for simply bearing-supported shell with F0 = 12 and M = 1/3 Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 807 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Sl. 12. Razmerje višin x v odvisnosti od temperature T za vrtljivo vodoravno nepomično vpeto lupino z m = 1/3, (a + a2)/(a1 - a) = 1 in F = 1,86 Fig. 12. Ratio of heights x relative to temperature T for simply bearing-supported shell with m = 1/3, (a1 + a2)/(a1 - a) = 1 and F0 = 1.86 Sl. 13. Razmerje višin x v odvisnosti od temperature T za vrtljivo vodoravno nepomično vpeto lupino z m = 1/3, (a + a2)/(a1 - a) = 1 in F = 1,87 Fig. 13. Ratio of heights x relative to the temperature T for simply bearing-supported shell with m = 1/3, (a1 + a2)/(a1 - a) = 1 in F0 = 1.87 Analizirali smo tudi razmere pri zelo plitvih 2 We have also analyzed the conditions in very lupinah z majhno začetno višino h0. Izkazalo se je, da se pri takšnih lupinah razmerje višin x(T) z višanjem temperature T zmanjšuje, če je le začetna oblikovna funkcija F0 dovolj majhna glede na shallow shells with a small initial height h0. It turned out that in such shells the ratio x(T) decreased with increasing temperature T only if the initial formative function F0 was small enough compared to the ratio 808 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 razmerje (a1 + a2)/(a1 - a2) v sistemu enačb (99). Največja vrednost za začetno oblikovno funkcijo, pri kateri se razmerje višin RT) z višanjem temperature Tše vedno zmanjšuje, znaša F = 1,86 (sl. 12). Takrat ima razmerje med vsoto in razliko razteznostnih koeficientov najmanjšo mogočo vrednost (a1 + a2)/(a1 - a2) = 1. Če vrednost začetne oblikovne funkcije F0 le malenkostno povečamo, se bo razmerje višin RT) z višanjem temperature T zvečalo, kar je razvidno s slike 13. V primeru konzolno vpete lupine sta robna pogoja: s (1) = 0; F(1) =/(1) = konst. Funkciji Y’ in y’ izrazimo iz enačb (69) in (68). Po krajšanju enakih členov je robni pogoj: F(1) = F0. Problem robnih vrednosti za konzolno vpeto lupino je s tem: 4g F2 (a1 + a2)/(a1 - a2) in the system of equations (99). The highest value for the initial formative function, in which the ratio of heights x(T) with increasing temperature T is still decreasing, is F0 = 1.86, Fig. 12. In this case the relation between the sum and the difference of the expansion factor has the smallest possible value (a1 + a2)/(a1 - a2) = 1. If we only slightly increase the value of initial formative function F0, the ratio x(T) will increase with the increase of the temperature T, as is clear from Fig. 13. In the case of the clamped shell, the boundary conditions are: ej(1) = 0; Y’(1) = y’(1) = const. The functions Y’ and y’ are expressed from Equations (69) and (68). After reducing the same terms, the boundary condition is: F(1) = F0. Then, the boundary-value problem for the clamped shell is: f 2 4f" g(0) = f(0) = 0, f(1) = F0, g'(1)-g(1) Tokrat smo izbrali približni vrednosti za g(1) in f’(1). Ob vsakem koraku smo preverili, kolikšno je odstopanje od robnih pogojev v točki x = 0 ter z Newtonovo metodo izračunali boljša približka za g(1) in f’(1). Rezultati so razvidni s slik 14 in 15 in so podobni razmeram pri vrtljivo vodoravno nepomično vpeti lupini. Razlika v stopnji deformacije različno vpetih lupin z enakimi snovno geometrijskimi značilkami je razvidna s slik 16 in 17. S C0 smo označili obliko obeh lupin v nedeformiranem stanju, s C1 obliko za vrtljivo vodoravno nepomično vpeto lupino in s C2 obliko za konzolno vpeto lupino. Pri tem pomenita zgornji deformacijski krivulji obliko lupin pri temperaturi t = 1, spodnji krivulji pa obliko pri temperaturi t = -1. Kakor je razvidno s slike 16 je razlika v deformaciji različno vpetih lupin očitnejša šele pri večji temperaturni obremenitvi. Pri konzolno vpeti lupini se razmerje višin RT) z višanjem temperature T povečuje ne glede na vrednost začetne oblikovne funkcije F0. Oglejmo si še razmere pri analizi preskoka sistema pri prosto položeni lupini, ki je v temenu obremenjena z zunanjo silo Fk. Ta obremenitveni primer opisujeta diferencialni enačbi (81) in (82). Iz primerjave enačb (74) in (75) izhaja, da je normalna enotska sila np v primerjavi s strižno enotsko silo V majhna, zato jo zanemarimo: ny (1) @ 0 . f g 2 c )1 (100). We chose approximate values for g(1) and f’(1). In each step we checked how far the digression from the boundary conditions can go at point c = 0 and by means of Newton’s method, better proximities for g(1) and f’(1) are computed The results can be seen in Figures 14 and 15; they are similar to the conditions found in the simple bearing-supported shell. Displacement states for different kinds of boundary conditions of the shells of the same material and geometrical characteristics can be seen in Figures 16 and 17. C0 denotes the shape of both shells in the undeformed state, C1 denotes the simple bearing-supported shell and C2 the clamped shell. In such case the upper displacement curves represent the shell’s shape at temperature t = 1 and the lower ones the shape at the temperature t = -1. As is clear from Figures 16 and 17, the difference in the displacement of the various boundary conditions of the shells is clearly expressed only in the case of higher temperature loads. In the clamped shells, the ratio of heights x(T) is increasing with increasing temperature T, regardless of the value of the initial formative function F0. The object of discussion is also the snap-through phenomenon in the simple roller-supported shell, wrhich is additionally loaded with an external force Fk in its top. This loading example is outlined by the differential equations (81) and (82). When comparing Equations (74) and (75) it follows that the normal unit force nj compared to the tangential force ty is small, hence it can be neglected: ny(1) @ 0 . Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 809 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Fli = 12, a = lEcnrc, ji = 1/3, 5 = 0.3mm r ol^.^IxlOT^ItL Sl. 14. Razmerje višin L v odvisnosti od temperature T za konzolno vpeto lupino Fig. 14. Ratio of heights {relative to the temperature T for clamped shell Sl. 15. Deformacijske krivulje za konzolno vpeto lupino s F0 = 12 in m = 1/3 Fig. 15. Displacement states for clamped shell with F0 = 12 and m = 1/3 Če ponovno upoštevamo, da sta funkciji G in F v točki = 0 omejeni, zapišemo problem robnih vrednosti: g(0) = g(1) = f(0)=0 t = 1 Considering that functions G and F at point c = 0 are limited, the boundary-value problem can be expressed as: 2 2 X 1 F0(1 +m) 2f'(1)-f(1).(1-/*)) (101), 810 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 F0=12, H=l/3, ; 1.4 1.2 -^ci 3- -: —0 . 5 0.8 0.6 o.4 0.5 i Sl. 16. Razmerje višin {pri vrtljivo vodoravno nepomično vpeti lupini (C1 ) in konzolno vpeti lupini (C2 ) v odvisnosti od temperaturnega obremenjevanja za lupine s F = 12 in M = 1/3 Fig. 16. Ratio of heights t in simply bearing-supported shell (C1 ) and clamped shell (C2 ) depending on temperature loading for shells with F0 = 12 and M = 1/3 ro = 12, f*= 1/3 St/a L-i ^VToi' ¦cl / 0.06 i = t) T = l f/ s °-04 c&^ ff / 0-03 T = -L if/ _^~~------"___Q - °^ If/J^^' 0-01 '--' j% Sl. 17. Deformacijske krivulje pri vrtljivo vodoravno nepomično vpeti lupini (C ) in konzolno vpeti lupini (C 2 ) za temperaturne vrednosti t = 0, t = -1, t = 1 Fig. 17. Displacement states in simply bearing-supported shell (CJ and clamped supported shell (C2 ) for . temperature values t = 0, t = -1, t = 1 kjer je fk, po analogiji z brezrazsežnotemperaturo t, where f is a dimensionless force, and FM is a con- brezrazsežna sila, FM pa nespremenljiva: stant: Fk ~ fk' FM ; F 2tt 2C3 E pd4 a2VA(1-M2) a26j6(1-M2) 3 Reševanje sistema enačb (101) je potekalo To solve the system of Equations we use the po prej opisani numerični metodi. Zaradi nazornosti above-described numerical method. For the sake of Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 811 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 Preglednica 2. Temperatura in lega preskoka lupine za različne vrednosti sile Fk Table 2. Temperature and shell's snap-through position for different force values Fk Fk[N] 0N 10N 20N 30N Fkr = 34,44N 40N S.T.1 tp1 = 1,19 5 x1 = 0, 361 tp1 = 0.879 xp1 = 0.366 t p1 = 0.535 xp1 = 0, 360 t p1 = 0.166 xp1 = 0, 335 tp1 =0 xp1 = 0, 312 t p1 = -0, 214 xp1 = 0, 269 S.T.2 tp2 = 0, 805 x p2 = -0, 361 tp2 = 0.510 xp2 = -0.348 t p2 = 0.231 xp2 = -0, 327 t p2 = -0, 04 0 xp2 = -0, 293 t p2 = -0.159 xp2 = -0, 2 74 t p2 = -0, 3 08 x p2 = -0, 24 0 Sl. 18. Stabilnostne razmere pri različnih vrednostih sile Fk Fig. 18. Snap-through behaviour for different force values Fk predstavljamo grafične in preglednične rezultate za clarity, the graphical and tabular results for the shell lupino z naslednjimi podatki: are shown below, with the following data: F 12, h0 = 0,78mm, a = 15mm, d = 0.3mm, E = 1, 7 • 105 MPa (102). S slike 18 je razvidno, da se temperatura preskoka lupine tp znižuje z večanjem sile Fk. Pri sili Fk = 0 smo za temperaturo tp in lego xp preskoka lupine dobili enake rezultate, kakor smo jih predhodno izračunali za samo temperaturno obremenjeno lupino. Temperaturo preskoka tp 1 za vmesne vrednosti sile Fk v območju [0,40] smo določili z interpolacijskim polinomom 4. stopnje (sl. 19): From Fig.18 it follows that the snap-through temperature tp is decreasing with increasing force Fk. For the force Fk = 0 we get the same results for the temperature tp and position xp of shell’s snap-through as those previously calculated for the shell that was loaded only by temperature. The snap-through temperature tp1 for the intermediate values of the force Fk in the interval [0,40] was determined by the interpolating polynomial of the fourth degree, Fig. 19: tp1 (Fk) = 1, 195 - 3,03 • 10-2Fk -1,045 • 10-4F2 - 2,25 • 10-6Fk3 + 4,583 • 10-8F4 (103). S polinomom (103) smo določili kritično silo Fkr, pri kateri lupina preskoči pri temperaturi t = 0: Using the polynomial (103) the critical force Fkr, at which the shell snaps-through at temperature t = 0, is determined: 812 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 F0=12, a=15mn, 5=0. arms, E=lr VxlO-^/nmi,2 ^=1/3 T 1.2 1 o.e 0.6 0.4 0.2 "^ 0.2 10 L '. 30 J [K] r Sl. 19. Temperatura preskoka tp1 v odvisnosti od zunanje sile Fk r Fig. 19. Snap-through temperature tp1 depending on the external force Fk Pri kritični sili Fk = 34,44 N lupina preskoči, ne da bi jo bilo potrebno segrevati. Če se po preskoku lupine sila zmanjša, se zmanjša tudi razmerje višin x. Kadar je lupina dovolj plitva, se po prenehanju sile vrne v izhodiščno lego. Obravnavana lupina je že takšna, saj je pri mehansko neobremenjeni lupini Fk = 0 možno samo eno ravnovesno stanje pri temperaturi t = 0 in sicer pri razmerju višin x = 1. Da bi lupina po prenehanju kritične sile F ne preskočila v izhodiščno lego, mora krivulja za temperaturo t v odvisnosti od razmerja višin x sekati negativni del abscisne osi ali se je vsaj dotakne. Samo v tem primeru sta mogoči dve stabilni ravnovesni stanji in razmerje višin ^ 1 pri temperaturi t = 0. Približno najmanjšo vrednost začetne oblikovne funkcije F0 i izračunamo z interpolacijskim polinomom n Interpoliramo preglednico 1, in sicer temperaturo tp2 drugega preskoka (S.T.2) z začetno oblikovno funkcijo F0 v območju [8,93 < F0 < 16]: At the critical force Fk = 34.44 N the shell snaps-through without being heated up. If after the shell’s snap-through the force decreases, the ratio x is decreased too. When the shell is shallow enough, on removal of the force it returns to the initial position. The shell we are discussing is of this type, because only one equilibrium state is possible at temperature t = 0 at x = 1 if there is no external force in the top of the shell. To prevent the shell snapping-through into the initial position on removal of the critical force Fkr the curve for the temperature t depending on the ratio x should be crossing the negative part of the abscissa axis or at least touch it. Only in this way are two stable equilibrium conditions and a ratio of heights L * 1 possible at the temperature t = 0. As an approximation, the lowest value of the initial formative function F0 is calculated using the interpolating polynomial In Table 2 we interpolate the temperature tp2 of the second snap-through (S.T.2) with the initial formative function F0 in the interval [8,93 1 za T < 0, razen pri vrtljivo vodoravno nepomično vpeti lupini z majhno začetno oblikovno funkcijo F0. Če na lupino poleg spremembe temperature T deluje tudi sila Fk v temenu lupine, se pojavi preskok lupine pri nižji temperaturir. Pri dovolj veliki sili Fk lupina preskoči, ne da bi p jo bilo treba dodatno segrevati. Velikost kritične sile Fkr je odvisna od snovno geometrijskih lastnosti lupine v enačbah In thin-walled, shallow bimetallic shells a snap-through into a new equilibrium state occurs when a certain temperature is reached. The snap-through temperature Tp depends on the material and the geometrical properties of the shell. As a special case, the stability conditions for spherical shells whose two layers have equal thickness d1 = d2 = d/2 and the same Poisson’s ratio m1 = m2 = m were analyzed. From Equation (89) it follows that the position of a snap-through xp, in shells with equal thickness d and Poisson’s number m, depends only on the initial value of the shell’s height h0. The curvature of the shell 1/R and the difference in the coefficients of the linear expansion a1 - a2 affect only the snap-through temperature Tp and have no influence on the snap-through xp position. In shallow, single-layer shells with a constant coefficient of the linear temperature expansion a(z) = a = const. the ratio of heights x(T) remains constant, regardless of the temperature loading. With increasing temperature T the shell’s horizontal radius a increases while the vertical component of the deformation Y at the edge of the shell always remains the same: = konst (104). From Equation (104) it follows that single-layer shells have no snap-through. Also, very thin bimetallic shells with the initial formative function F0 < 8.93 and Poisson’s ratio m = 1/3 have no snap-through. In order that snap-through can occur in bimetallic shells, it is necessary, besides a high enough temperature, to ensure that the edge of the shell can freely expand in the horizontal direction. The greatest horizontal displacement occurs at the edge of a shell at the moment of the shell’s snap-through. Bimetallic shells with a greater initial height h0 have, for the same material and geometrical characteristics, a greater horizontal displacement. Clamped bimetallic shells cannot expand in the horizontal direction. Because the expansion in the horizontal direction of such shell is prevented, the shell is expanding vertically. Consequently, the ratio of heights x(T) increases along with the temperature increase x(T) >1 for T < 0, except in simple bearing-supported shells with a small initial formative function F0. When in addition to the temperature T, also a force Fk is acting on the shell’s top, the snap-through occurs at a lower temperature tp. When the force Fkr is high enough, the shell snaps-through without the need to be additionally loaded with temperature. The value of the critical force Fkr depends on the material and the geometrical =y(a) 814 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 (102). Po prenehanju kritične sile se lupina vrne v začetno ravnovesno lego, razen pri manj plitvih lupinah, pri katerih je vrednost začetne oblikovne funkcije F0 i > 21 66 za lupine z značilkami v enačbah (102) n properties of the shell in Equations (102). When the force is removed, the shell returns to its initial equilibrium state, except in the case of shallower shells, whose value of initial formative function is F0 i > 21 66 for shells with the characteristics in EquationT(102). tlorisni polmer nedeformirane lupine Youngov elastični modul materiala 1 in 2 začetna oblikovna nedeformirana funkcija trenutna oblikovna deformirana funkcija sila v temenu lupine kritična sila v temenu lupine členi potenčne vrste trenutne oblikovne funkcije napetostna funkcija členi potenčne vrste trenutne napetostne funkcije začetna višina nedeformirane lupine razdalji osrednje ploskve od spodnje (h1) oziroma zgornje (h2) ploskve bimetalne lupine ukrivljenosti nedeformirane lupine ukrivljenosti deformirane lupine notranja momenta v lupini notranja enotska momenta v lupini notranji sili v smeri normale v lupini notranji enotski sili v smeri normale na lupino polmer nedeformirane krogelne lupine polmera ukrivljenosti nedeformirane lupine polmera ukrivljenosti deformirane lupine dol Qna na nedeformirani lupini dol Qna na deformirani lupini primerjalna temperatura temperatura, temperatura preskoka notranja striaa sila na lupini notranja enotska striaa sila v lupini vektor premika členi vektorja premika sila podpore, enotska sila podpore Lagrangev koordinatni sistem Eulerjev koordinatni sistem temperaturna koeficienta dolinskega raztezka materiala 1 in 2 7 OZNAKE 7 SYMBOLS a E1,E2 F0 (c) F (c) Fk F kr fi G (c) gi h0 h1,h2 ky,kj ky,kj My,Mj Ny,Nj ny,nj R s s T0 T,Tp T yr t yr r u u,v,w V,Ve (x,y,z) (X,Y,Z ) horizontal radius of undeformed shell Young’s elastic modulus of materials 1 and 2 initial formative function present formative function force acting at the top of a shell critical force acting at the top of a shell elements of power series of the present formative function stress function elements of power series of the stress function initial height of the undeformed shell middle plane distance from the lower (h1) and upper (h2) plane of the bimetallic shell the curvatures of the undeformed shell the curvatures of the deformed shell internal moments in the shell internal unit moments in the shell internal forces in the direction normal to the shell internal unit forces in the direction normal to the shell radius of the undeformed spherical shell radii of the undeformed shell radii of the deformed shell the length on the undeformed shell the length on the deformed shell reference temperature temperature, snap-through temperature internal tangential force in the shell internal unit tangential force in the shell displacement vector elements of the displacement vector reaction force, unit reaction force Lagrange coordinates Euler coordinates linear temperature expansion coefficients of material 1 and material 2 Preskok sistema plitve osnosimetrične bimetalne lupine - Snap-through of the System for a Shallow Axially 815 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 785-816 debelina lupine debelini slojev iz materiala 1 in 2 deformacijski tenzor Poissonovi števili za material 1 in 2 napetostni tenzor brezrazse Cna temperatura, temperatura preskoka brezrazse Cna neodvisna spremenljivka razliki ukrivljenosti kota na nedeformirani lupini kot na deformirani lupini [1] [2] [3] [4] [5] [6] d d1,d2 e ij m1,m2 s ij c y shell’s thickness thickness of layers made of material 1 and 2 strain tensor Poisson’s ratios of materials 1 and 2 stress tensor dimensionless temperature, snap-through temperature dimensionless independent variable curvature differences angles of the undeformed shell angles of the deformed shell 8 LITERATURA 8 REFERENCES Timoshenko S. (1959) Theory of plates and shells. McGraw-Hill Book, New York. Alfutov N. A. (2000) Stability of elastic structures. Springer-Verlag. Reddy J. N. (1999) Theory and analysis of elastic plates. Taylor & Francis. Drole R., Kosel F. (1993) Stability analysis of shallow axi-symmetric bimetallic shells in a homogeneous temperature field. The 14th Canadian Congress of Applied Mechanics, Kingston, Ontario, June 1993. Drole R., Kosel F. (1994) Analysis of stress-strain state in shallow spherical bimetallic shells by non-linear theory, Z. angew. Math. Mech. Wittrick. W. H. (1953) Stability of bimetallic disk, The Quarterly Journal of Mechanics and Applied Mathematics. Naslovi avtorjev: dr. Marko Jakomin Univerza na Primorskem Primorski inštitut za naravoslovne in tehnične vede, Pristaniška cesta 14 6000 Koper markojakomin@pint.upr.si prof. dr. Franc Kosel doc. dr. Tadej Kosel Univerza v Ljubljani Fakulteta za strojništvo Aškerčeva 6 1000 Ljubljana franc.kosel@fs.uni-lj tadej.kosel@fs.uni-lj prof. dr. Milan Batista Univerza v Ljubljani Fakulteta za pomorstvo in promet Pot pomorščakov 4 6320 Portorož milan.batista@fpp.edu Prejeto: Received: 12.5.2005 Sprejeto: Accepted: Authors’ Addresses:Dr. Marko Jakomin University of Primorska Primorska Institute for Natural Sciences and Technology Pristaniška cesta 14 6000 Koper , Slovenia marko.jakomin@pint.upr.si Prof. Dr. Franc Kosel Doc. Dr. Tadej Kosel University of Ljubljana Faculty of Mechanical Eng. Aškerčeva 6 1000 Ljubljana, Slovenia franc.kosel@fs.uni-lj tadej.kosel@fs.uni-lj Prof. Dr. Milan Batista University of Ljubljana Faculty of Maritime studies and Transport Pot pomorščakov 4 6320 Portorož, Slovenia milan.batista@fpp.edu Odprto za diskusijo: 1 leto 22.6.2006 Open for discussion: 1 year 816 Jakomin M. - Kosel F. - Batista M. - Kosel T. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 UDK - UDC 628.1:53.08 Izvirni znanstveni članek - Original scientific paper (1.01) Izbira merilnih mest v vodovodnih sistemih z genetskimi algoritmi Sampling Design for Water Distribution System Models by Genetic Algorithms Primož Banovec - Daniel Kozelj - Sašo Šantl - Franci Steinman (Fakulteta za gradbeništvo in geodezijo, Ljubljana) Obravnavana je izbira merilnih mest za umerjanje hidravličnih modelov vodovodnih sistemov. Izbira merilnih mest za umerjanje hidravličnih modelov je podana kot optimizacijski problem, ki je sestavljen iz dveh normaliziranih ciljnih funkcij. S prvo ciljno funkcijo se povečuje natančnost umerjanja parametrov hidravličnega modela, z drugo pa zmanjšuje število potrebnih merilnih mest. Predstavljeno je reševanje optimizacijskega problema z uporabo genetskih algoritmov. Overitev in uporaba razvitega optimizacijskega modela (imenovan IMMe) je bila opravljena na hidravličnem modelu teoretičnega vodovodnega sistema "Anytown", ki je namenjen kot primerjalni model za testiranje različnih raziskav pri hidravličnem modeliranju. Overjeni model IMMe je bil uporabljen na stvarnem vodovodnem sistemu mesta Sežane. Uporaba genetskih algoritmov se je v obeh primerih izkazala za zelo učinkovito optimizacijsko orodje pri izjemno kombinatornih optimizacijskih problemih. Razvita modela umerjanja in izbire merilnih mest omogočata vzpostavitev kar se da natančnih hidravličnih modelov vodovodnih sistemov, ki bodo v prihodnosti ključnega pomena za zagotavljanje gospodarnosti in učinkovitosti oskrbe s pitno vodo. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: hidravlika, vodovodi, modeliranje, optimiranje, merilna mesta, algoritmi genetski) In this paper we discuss sampling design for the calibration of water distribution system hydraulic models. The sampling design for the calibration of water distribution-system models is formulated as an optimisation problem consisting of two normalised objective functions. The first objective function is used to increase the calibration accuracy of the model parameters, and the second one is used to reduce the number of necessary measurement locations. The optimisation problem was solved by using genetic algorithms. The verification and application of the developed optimisation model (called IMMe) were carried out on the artificial water distribution system of Anytown, which serves as a reference model for testing various researches in hydraulic modelling. The verified IMMe model was applied to a real water-distribution system in the town of Sežana. For both water distribution models, the use of genetic algorithms proved very efficient with extremely combinatorial optimisation problems. The developed calibration and sampling design allow very accurate hydraulic modelling of the water distribution systems, which is of key importance for ensuring the economy and efficiency of drinking-water supplies in the future. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: hydraulic water distribution systems, modelling, optimisation, measurement locations, genetic algorithms) 0 UVOD 0 INTRODUCTION Hidravlično modeliranje vodovodnih sistemov (v nadaljevanju VS) je znanstveno zelo dobro razvita panoga, ki rabi kot učinkovito orodje za podporo pri odločanju, upravljanju, načrtovanju in obnovi VS. Podpora informacijske tehnologije pri vodenju katastra komunalnih naprav in zbiranju podatkov ter meritev omogoča učinkovito The hydraulic modelling of water distribution systems (hereinafter referred to as WDS) is a scientifically well-developed discipline serving as an efficient decision-support tool for the management, development and rehabilitation of WDS. The information technology supports the asset management of the WDS and the data acquisition allows the effi- 817 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 vzpostavitev hidravličnih modelov in njihovo vključitev v analize, vsakdanje obratovanje in načrtovanje ter v končni fazi tudi izvajanje potrebnih ukrepov. Informacijski sistem se uporablja kot vir razpoložljivih podatkov o fizičnih značilkah VS, ki izhajajo iz dejansko znanih podatkov oziroma meritev in pa iz ocen posameznih hidravličnih parametrov, če količine niso natančno znane. S temi podatki je mogoče postaviti “surov” hidravlični model, katerega obsežnost je treba prilagoditi upravljavsko obvladljivim razmeram. Hidravlični modeli istega VS se lahko med seboj razlikujejo glede na namen analize, ki bo izvedena s tem modelom. Namen modela je bistvenega pomena, saj določa stopnjo natančnosti modela in njegove poenostavitve. Poenostavitev hidravličnega modela se imenuje ogrodenje (skeletiranje), cilj katerega je, iz modela izključiti elemente, ki niso bistveni za njegovo hidravlično enakost z dejanskim dogajanjem v VS. Po postavitvi takega modela se najprej izvede postopek grobega umerjanja oziroma makrokalibracije (sl. 1), cilj katerega je prilagajanje parametrov hidravličnega modela, dokler delovanje hidravličnega modela VS ne izkazuje ujemanja s sedanjimi meritvami z odstopanjem največ 30 odstotkov [1]. Makrokalibracija je usmerjena v odpravljanje posameznih virov odstopanj, to so napačni podatki o topologiji omrežja, nastavitev tlačnih con, parametrov elementov VS, napak merilnih naprav in odčitavanja meritev kakor tudi odpravljanje človeških napak. Običajno z makrokalibracijo hidravlični model postane grob približek stvarnega VS z določeno stopnjo natančnosti. Hidravlični model je namenjen podpori pri odločanju na tehničnem, ekonomskem in pravnem področju, kar terja, da je hidravlični model natančneje umerjen. Kot uvod v postopek natančnejšega umerjanja oziroma mikrokalibracije (v INFORMACIJSKI SISTEM GIS podpora HIDRAVLIČNO MODELIRANJE <.......?¦ OGRODENJE IN OCENA KALIBRACIJSKIH PARAMETROV A 1 ' ' IZVEDBA / STORITEV -I MAKROKALIBRACIJA t > ' ANALIZA IN NAČRTOVANJE 1 MIKROKA LIBRACIJA cient building of hydraulic models and their integration into analysis, daily operation and planning, and finally also the carrying out of the necessary measures. The information system serves as a source of available data on the physical characteristics of WDS deriving from actually known data or measurements as well as from the evaluations of individual hydraulic parameters if the exact quantities are not known. These data allow us to build a rough hydraulic model, the size of which has to be adapted to a manageable extent. Hydraulic models of the same WDS can vary substantially in their configuration with regard to the purpose of the model and the analysis to be performed with it. The model's purpose is essential since it determines the accuracy level to be applied. The model's adaptation involves a procedure of skeletonization, i.e., simplification of the model, the aim of which is to exclude those elements that are not essential to represent the model's hydraulic behaviour according to the real WDS. With such a model, the procedure of macro-calibration is first carried out (Fig. 1), with the adjustment of certain model parameters to achieve correspondence with the hydraulic behaviour of WDS, i.e., the system variables should not exceed 30 percent, based on the existing measurements [1]. Macro-calibration strives to eliminate the sources of differences, such as incorrect data on network topology, pressure-zone settings, the parameters of network elements, the errors of the measurement equipment and readings, as well as human error. Normally, macro-calibration turns the hydraulic model into a rough approximation of the real WDS, with a certain level of accuracy. Decision making should tackled with technical, economic and legal issues, so the hydraulic model has to be calibrated more precisely. As an introduction to the so-called micro-calibration (hereinafter referred to as INFORMATION SYSTEM GIS support HYDRAULIC MODELLING <.......* SKELETONIZATION AND CALIBRATION PARAMETER ESTIMATION A i MEASURES / SERVICE -1 MACRO CALIBRATION 1 T ANALYSIS AND DESIGN <— ¦ MICRO CALIBRATION Sl. 1. Upravljanje vodovodnih sistemov s poudarkom na modeliranju Fig. 1. Water distribution system management with the emphasis on modelling 818 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 nadaljevanju umerjanje) se uporablja postopek izbire merilnih mest, ki zagotavlja zbiranje pomembnih vzorcev meritev. Izbira merilnih mest za potrebe umerjanja je osredotočena na zbiranje meritev za razpoznavanje značilk in parametrov hidravličnega modela VS, odkrivanje virov težav pri vsakodnevnem obratovanju VS in za določene raziskovalne potrebe. Namen pričujočega prispevka je določitev merilnih mest in zbiranje meritev za natančno razpoznavanje, tj. umerjanje, fizičnih parametrov hidravličnih modelov VS. 1 UMERJANJE HIDRAVLIČNIH MODELOV VS kjer so: E - vrednost ciljne funkcije umerjanja; y*ijk -meritve hidravličnih veličin na VS; yijk - napovedi modela za (i=1,...N; j=1,...NM; k=1,...NV); NL -število obtežnih primerov; NM -število merilnih mest; N - tip hidravlične veličine (tj. tlaki, pretoki ...); ai -utežni koeficient Na področju umerjanja VS je bilo razvitih več postopkov in metod, ki jih v splošnem delimo na metodo poskus - napaka ter izrecne in posredne metode. Razvite posredne metode umerjanja so metode, ki so formulirane in se rešujejo kot optimizacijski problemi in so se izkazale za zelo učinkovite ([2] in [3]). Njihova ciljna funkcija je običajno izražena v obliki, ki omogoča zmanjšanje razlik med meritvami in napovedmi oziroma rezultati modela (en. 1). Razvoj optimizacijskih modelov za umerjanje je v veliki meri pripomogel k povečani natančnosti hidravličnih modelov VS. Merila natančnosti napovedovanja hidravličnih veličin so podana v različnih smernicah, ki določajo meje "calibration") level, a sampling design programme would provide representative samples of measurements for the calibration. Sampling design is focused on collecting measurements for the determination of WDS model properties, on the detection of problems in the daily operation of WDS, as well as covering certain research needs. The aim of the presented paper is to determine the measurement locations in which the measurements allow accurate identification, i.e., the calibration of the physical parameters of WDS hydraulic models. 1 CALIBRATING HYDRAULIC MODELS OF WDS where E is the value of the objective function; y*ijk are measurements of the system variables; yijk are model predictions of the system variables for (i=1,...NL; j=1,.. NM; k=1,-•-NV); NL is the number of loading conditions; NM is the number of measurement locations; NV is the type of system variables (e.g. pressure, flow); and ai are a weighting coefficients. In the area of WDS model calibration, several approaches and methods were developed, which, in general, comprise the trial-and-error method and the explicit and implicit methods. The developed implicit methods of calibration, formulated and solved as optimisation problems, have proven their effectiveness ([2] and [3]). Their objective function is expressed in a form that allows minimisation of the differences between the measurements and predictions of the model (Eq. 1). The developments of optimisation models for calibration have to a great extent contributed to the increased accuracy of WDS models. The levels of accuracy according to the purpose of the WDS model Postopek umerjanja se izvaja zaradi The calibration process is conducted to vzpostavitve zaupanja v napovedi hidravličnega provide confidence in the predictions of the WDS modela. Poleg zanesljivosti napovedovanja daje model. Besides the reliability of the predictions, umerjanje tudi natančen vpogled v delovanje VS, calibration offers a deep insight into the WDS operation, kar omogoča tudi vrednotenje občutljivosti modela which enables an assessment of the model's sensitivity na spremembe posameznih fizikalnih in/ali to changes in individual physical and/or non-physical nefizikalnih parametrov. Umerjanje je torej postopek parameters. The calibration of hydraulic models is thus določitve posameznih neznanih parametrov a procedure of determining individual unknown model hidravličnega modela, s katerim se zmanjšajo razlike parameters, which minimises the differences of med meritvami opravljenimi na stvarnem VS, in predicted and measured system variables. The napovedmi oz. rezultati hidravličnega modela. calibration problem is formulated to minimise the differ- Nastavljen problem umerjanja se lahko rešuje z ences of predicted and measured system variables, and zapisom ciljne funkcije v naslednji obliki: its general objective function is expressed as: \nm nl Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 819 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 odstopanj med meritvami in rezultati modela, vezane na namene analiz hidravličnih modelov [4]. Za uspešno umerjanje je treba najprej zbrati niz meritev, ki se pridobijo z načrtnim zbiranjem podatkov hidravličnih veličin pri obratovanju na pomembnih mestih na VS. Kakovost informacij, pridobljenih iz meritev, ima velik vpliv na natančnost umerjanja hidravličnega modela, večje število informacij pa daje v končni fazi tudi večjo stopnjo zaupanja v model. 2 IZBIRA MERILNIH MEST ZA UMERJANJE HIDRAVLIČNIH MODELOV VS Izbira merilnih mest se v praksi pogosto izvaja po načelu “ad hoc”, pri katerem se izdelovalec hidravličnega modela s pomočjo svojih strokovnih ocen odloči in predlaga kraje merilnih mest. Glavni poudarek pri zagotovitvi kakovosti in uporabnosti rezultatov izbire merilnih mest obsega določevanje [5]: (a) katere hidravlične veličine opazovati, (b) kje opazovati, (c) kdaj opazovati in (d) v kakšnih okoliščinah oz. obtežnih primerih opazovati. Odgovore na vprašanja pod točko (a), (c) in (d) mora predhodno podati inženir s svojim strokovnim znanjem. Glavni poudarek optimizacije izbire merilnih mest je osredotočen na vprašanje (b), ki je najzahtevnejše in obsega določevanje najboljših krajev merilnih mest. Izkazalo se je, da je občutljivost merilnih mest bistvenega pomena za določevanje najboljših merilnih mest. Za določevanje občutljivosti hidravličnih veličin merilnih mest glede na spremembe parametrov hidravličnega modela se uporablja t.i. Jacobijeva matrika. Koeficienti Jacobijeve matrike (i,j) so parcialni odvodi izrazov hidravličnih veličin glede na parametre modela in jih je mogoče določiti z numerično aproksimacijo, metodo končnih razlik [3]: are expressed as the differences between the measurements and the model's predictions [4]. For a successful calibration, a set of measurements has to be gathered, first by means of systematic sampling of the hydraulic quantities on a WDS. Since the relationship between the quality and quantity of a measurement is not predefined and self-evident, much attention needs to be paid to evaluating the informational content and its impact on the calibration accuracy. 2 SAMPLING DESIGN FOR THE CALIBRATION OF WDS MODELS Sampling design is often carried out according to the ad-hoc principle, where a designer of a WDS model determines and proposes the measurement location on the basis of his or her expertise. To ensure the quality and relevance of the measurements, the main emphasis in sampling design involves determining [5]: (a) which model parameters to observe, (b) where to observe them, (c) when to observe them and (d) in what circumstances or loading conditions to observe them. While the answers to questions under points (a), (c) and (d) have to be provided in advance by an engineer using his or her expert knowledge, the main emphasis of sampling design is focused on question (b), which in turn is the most demanding and involves a determination of the optimal measurement locations. It has been proven that model parameter sensitivities are essential for the determination of optimal measurement locations. To determine the optimal measurement locations with regard to model sensitivities, the so-called Jacobian matrix is used. The coefficients (i,j) of the Jacobian matrix are partial derivatives of the model predictions with regard to the model parameters and can be determined by applying numerical approximations, e.g., the finite-difference method [3]: dyi daj = yi (a*j ) - yi(aj ) a*j -aj vsak/for each (i=1,..., NO; j=1,..., Na) (2), kjer so: d y - sprememba hidravlične veličine i;Qa -sprememba parametra umerjanja hidravličnega modela j; yi(a*) - vrednost hidravlične veličine pri izhodiščni vrednosti parametra umerjanja; yi(a) -vrednost hidravlične veličine pri spremembi vrednosti parametra umerjanja; a*j - vrednost izhodiščnega parametra umerjanja; aj - vrednost spremenjenega parametra umerjanja; NO - število meritev in Na - število parametrov umerjanja. Za where d y is the change of the hydraulic quantity i; d a is the change of the hydraulic parameter of the model j; yi(a*) is the value of the hydraulic quantity at the initial calibration parameter value; yi(a) is the value of the hydraulic quantity at the change of the calibration parameter value; a*j is the value of the initial calibration parameter value; aj is the value of the hydraulic quantity at the changed calibration parameter value; NO is the number of observations and Na is the number of 820 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 rešitev izraza (2) uporabimo metodo končnih razlik, za katero je potrebno Na+1 hidravličnih simulacij, pri čemer je dodatna simufacija pridobljena s predhodno oceno vrednosti parametrov modela. Bolje umerjeni parametri hidravličnega modela dajejo natančnejše vrednosti koeficientov Jacobijeve matrike. Iz tega izhaja, da je smiselno izvajati iterativni postopek umerjanja in izbire merilnih mest (sl. 1) in to do sprejemljive natančnosti glede na namen hidravličnega modela [6]. Optimizacijski problem izbire merilnih mest je obravnavan z vidika izpolnjevanja dveh ciljnih funkcij, ki bosta podrobneje obravnavani v nadaljevanju. 2.1 Ciljna funkcija natančnosti umerjanja Z večanjem števila merilnih mest se povečuje tudi natančnost umerjanja parametrov hidravličnega modela. Metoda, ki je uporabljena za vrednotenje prve ciljne funkcije natančnosti umerjanja hidravličnega modela, je povzeta s področja teorije regresijske analize in je bila uspešno uporabljena pri razvoju modelov za umerjanje in izbiro merilnih mest modelov podtalnice [7]. Kasneje se je ta metoda uspešno uporabila tudi na področju izbire merilnih mest za izvedbo umerjanja hidravličnih modelov VS. S prvo ciljno funkcijo se vrednoti prostornina območja zaupanja parametrov. V nasprotju z določevanjem intervalov zaupanja se prostornina območja zaupanja parametrov ukvarja z verjetnostjo, da so iskane vrednosti N parametrov hkrati v določenem N - razsežnem območju. Prostornina območja zaupanja je sorazmerna kvadratnemu korenu determinante kovariančne matrike parametrov hidravličnega modela in se ga matematično lahko izrazi takole [6]: kjer sta: Cov - kovariančna matrika parametrov hidravličnega modela; det() - determinanta matrike. Kovariančna matrika iz enačbe (3) je podana z izrazom: kjer so: s2 - varianca regresijske napake; J - Jacobijeva matrika z elementi matrike Qy/Qaj (i=1,...,N ; j=1,...,N); W - simetrična, pozitivno definirana matrika utežnih koeficientov; JT - transponirana Jacobijeva matrika. calibration parameters. Eq. (2) is solved using the finite-difference method with Na+1 hydraulic simulations, where the additional simulation is obtained for the initial values, i.e., by a preliminary parameter-value assessment. Better calibrated model parameters give more precise values of the Jacobian matrix coefficients. Consequently, it is reasonable to carry out an iterative procedure of calibration and sampling design (Fig. 1) to reach an acceptable accuracy level with regard to the hydraulic model's purpose [6]. The optimisation problem of the sampling design has two objectives and will be further discussed in the next sections. 2.1 The Objective Function of Calibration Accuracy An increase in the number of measurement locations will consequently increase the confidence level or the accuracy of the obtained parameter assessments. The method used to evaluate the first objective function of the calibration accuracy derives from the regression analysis theory and was successfully used in the development of models for the calibration and sampling design of groundwater models [7]. Later on it was also successfully used in the area of WDS hydraulic models. The first objective function is to evaluate the parameter's confidence region volume. In contrast to the determination of confidence intervals, the volume of a parameter's confidence region deals with the probability that the sought values of Na parameters can at the same time be found in a certain Na dimensional area. The confidence region volume is proportional to the square root of the determinant of the covariance matrix of hydraulic model parameters and can be mathematically expressed as follows [6]: where Cova is the covariance matrix of the hydraulic model parameters; and det() is the matrix determinant. The covariance matrix from Eq. (3) is expressed as follows: where s2 is the regression error variance; J is the Jacobian matrix with the matrix elements d y/d aj (i=1,.. .,N ; j=1,.. .,N ); W is the symmetrical, positively defined matrix of weight coefficients; and JT is the transposed Jacobian matrix. (detCova 1 (3), Cova=s2-(JTWJ )1 (4), Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 821 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 Pri oblikovanju prve ciljne funkcije se lahko brez večje izgube splošnosti izrazov v enačbi (4) predpostavi, da veljajo tipične domneve regresijske analize: natančnost meritev je neodvisna od kraja na VS; vsa merilna oprema, ki bo uporabljena pri zbiranju meritev, ima podobno natančnost; in samo tlaki so predmet obravnave pri optimizaciji izbire merilnih mest. Posledično se lahko matriko utežnih koeficientov W približno izrazi kot diagonalno matriko, katere diagonalni elementi zavzemajo vrednosti 1/a2H, kjer je Na detCov- (8), where Cova-1F is the full inverse covariance matrix taking into account all the potential measurement locations [5]. The determinant of the inverse covari-ance matrix is calculated by means of a numerical method for defining the lower and upper triangular matrix (LU Decomposition) [8]. After the decomposition of the matrices, the determinant is a product of the diagonal elements of the upper triangular matrix. 2.2 The Objective Function of Measurement Costs The second objective function is optimising the costs related to measurements performed on a system. Considered here are the following: (a) capital or investment costs and (b) variable or operational costs. It is difficult to determine general equations to estimate the measurement costs; therefore, the procedure is simplified by the assumption that the main measurement costs for each measurement performed are related only to the operational part of the total costs. The objective function can thus be expressed with regard to the number of measurement locations. Its purpose is to minimize the measurement costs and indirectly also the number of measurement locations. It is desirable to use the second objective function in a normalised form [5]: min F =N max F =1 N NF (9), kjer sta: N - število merilnih mest na sistemu; NF -število vseh možnih merilnih mest v sistemu. Iz izraza (9) je razvidno, da tako oblikovana druga ciljna funkcija teži k čim manjšemu številu merilnih mest. 2.3 Vzpostavitev optimizacijskega problema Ker sta si zgoraj navedeni ciljni funkciji protislovni, je treba najti rešitev, ki bo v največji meri zadovoljevala tako prvo kakor drugo ciljno funkcijo. Prva ciljna funkcija bo najboljša, če bodo meritve zbrane na vseh možnih merilnih mestih, medtem ko bo druga najboljša, če bo število merilnih mest čim manjše, saj bodo s tem povezani tudi najnižji stroški. Optimizacijski problem je sestavljen iz naslednjih delov: (a) ciljne funkcije, (b) optimizacijskih spremenljivk oziroma neznank in (c) robnih pogojev. Formulacija optimizacijskega problema za izbiro merilnih mest je sestavljena iz dveh ciljnih funkcij, opredeljenih v zgornjih where N is the number of measurement locations in the system; NF is the number of all potential measurement locations in the system. Eq. (9) shows that the thus formulated second objective function tends towards the lowest possible number of measurement locations. 2.3 Definition of the Optimisation Problem Since the above-mentioned objective functions are mutually contradictory, a solution has to be found that will, to the largest extent, satisfy both the objective functions. The first objective function will be optimal if measurements are collected at all possible measurement locations, while the second will be optimal if the number of measurement locations is zero, since this will incur the lowest costs. The optimisation problem involves the following parts: (a) the objective function; (b) the optimisation variables or unknowns; and (c) the constraints. The optimisation problem formulation for the sampling Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 823 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 odstavkih. Normalizirana izraza (8) in (9), ki sta oblikovana na način, da se njuni vrednosti zveča, sta združena v en izraz: design consists of two objective functions defined in the previous paragraphs. The normalised Eq. (8) and Eq. (9), formulated so as to maximise their values, are joined into a single expression: maxi?= H>>-^ Pe (10), N>Nm 0 , Nmm 0); N - največje Število merilnih mest v sistemu (N < NF); NF - število vseh možnih merilnih mest v s ax stemu; pc1 in pc2 - stroškovna koeficienta kazenske funkcije. Optimizacijska orodja se uporabljajo za reševanje zahtevnih in obsežnih optimizacijskih problemov. Genetski algoritmi (v nadaljevanju GA) so razred nelinearnih, prilagodljivih in hevrističnih metod za iskanje in optimizacijske probleme in spadajo v skupino razvojnih algoritmov. GA posnemajo naravno načelo reprodukcije, mutacije in naravne izbire za zagotovitev preživetja najbolj sposobnega oziroma najboljšega osebka, tj. rešitve. Načelo GA in njegova uporaba pri optimizaciji VS je podano v [9] do [11]. GA so optimizacijska metoda, ki se posebej dobro izkaže pri reševanju velikih in zapletenih problemov z mnogimi lokalnimi ekstremi in s katero se skoraj vedno najdejo rešitve blizu najboljše [12]. Ta lastnost izvira iz dejstva, da GA raziskujejo prostor rešitev s skupnostjo (populacijo) kromosomov, tj. možnimi rešitvami, ki so naključno razpršene po celotnem prostoru rešitev. Ena od prednosti GA je tudi ta, da zahtevajo le vrednotenje ciljne funkcije za optimizacijski postopek brez zahtevnih numeričnih opravil ter možnost uporabe tako where E is the common objective function; wi are the weight coefficients to determine the importance of an individual objective function; p is the norm order; Fi is the i-th objective function; and Pe is the penalty function. The weight coefficients w1 and w2 are selected so that their sum is equal to 1. The values assumed by the joint objective function E are within the interval (0, 1), if this does not exceed the constraints. The penalty function is used to introduce explicit constraints into an optimisation problem and will represent the maximum and minimum values of the number of measurement locations: N 0); Nmax is the maximum number of measurement locations on the system (Nmax < NF); NF is the number of all the potential measurement locations in the system; and pc1 and pc2 are the cost coefficients of the penalty function. Optimisation tools are used for solving demanding and comprehensive optimisation problems. Genetic algorithms (hereinafter referred to as GAs) are a class of nonlinear, adaptive and heuristic methods for search and optimisation problems and belong to the group of evolutionary algorithms. GAs mimic nature's principles of reproduction, mutation and natural selection to ensure the survival of the fittest and therefore the best individual, i.e., the best solution. The principle of GAs and their application to WDS optimisation are given by [9] to [11]. GAs are an optimisation method especially useful in solving large and complex problems with many local minimums and maximums that almost always provides a solution close to the optimum one [12]. This feature arises from the fact that GAs explore the solution space with a population of chromosomes, i.e., possible solutions randomly distributed across the entire solution space. One of the advantages of GAs is that they require only the evaluation of the objective function for the optimisation procedure, without demanding numerical operations, as well as 824 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 diskretnih kakor tudi zveznih spremenljivk v postopku optimizacije [13]. 3 OVERITEV MODELA ZA IZBOR MERILNIH MEST NA VS ANYTOWN Za uporabo in overitev prikazane metode je bil izdelan optimizacijski model za najboljšo izbiro merilnih mest z uporabo GA ki je poimenovan IMMe. Samo preizkušanje pa je bilo izvedeno na hidravličnem modelu “Anytown“, ki ga je prvi definiral Thomas M. Walski [14]. Omenjeni model je bil izbran zaradi njegove razširjenosti na področju preizkušanja različnih orodij za analizo VS in zasnove, ki vključuje tudi določene značilnosti dejanskih VS. Optimizacijski model izbire merilnih mest je namenjen določevanju merilnih mest, ki bodo sporočali najbolj še informacije v fazi umerjanja izbranih parametrov hidravličnega modela. Koeficienti hrapavosti cevi po Hazen-Williamsu so bili izbrani kot parametri umerjanja. Čeprav je v hidravličnem modelu “Anytown“ število parametrov enako številu cevi (tj. 34 cevi), se lahko število parametrov znatno zmanjša prek združevanja koeficientov hrapavosti, s čimer se je določilo pet skupin (N = 5). Umerjanje parametrov hidravličnih modelov tj. koeficientov hrapavosti cevi, zahteva kakovostne meritve hidravličnih veličin, ki bodo v največji meri podajale zadostno količino informacij za učinkovito umerjanje [15]. Optimizacijski problem je bil oblikovan na način, da določa najboljša merilna mesta za pet neodvisnih ustaljenih hidravličnih simulacij, tj. štirje neodvisni požarni odvzemi vode in ob normalni porabi. Ker je skupno število vozlišč N = 16, je največje število meritev NO = 80 vrstic (16 vozlišč krat 5 obtežnih primerov). Izhajajoč iz teoretičnega ozadja izbire merilnih mest, se tlačna občutljivost vseh možnih merilnih mest zapiše v Jacobijevi matriki, ki ima NO = 80 vrstic in N = 5 stolpcev. Optimizacija ciljne funkcije (enačba (10)) z GA se je posebno dobro izkazala v izjemno kombinatornih problemih, kamor spadata tako umerjanje kakor tudi izbira merilnih mest [12]. Uporabljeni so bili Simple GA v povezavi z binarnim kodiranjem kromosomov, sestavljenih iz 16 genov (16 možnih merilnih mest). Populacija 100 kromosomov je bila začeta in iz nje so se izbirali starševski kromosomi z uporabo t.i. “ruletnega kolesa” ter verjetnostjo križanja 0,75 in verjetnostjo pojava mutacije 0,07. Vzpostavitev nove generacije the possibility of using both discrete and continuous variables in the optimisation procedure [13]. 3 SAMPLING DESIGN MODEL VERIFICATION FOR AN ARTIFICIAL WDS IN ANYTOWN For the application and verification of the presented method, an optimisation model (called IMMe) for the optimal selection of measurement locations using GAs was developed. The verification process was carried out on the Anytown hydraulic model, which was first defined by Thomas M. Walski [14]. His model is seen as a benchmarking model for testing various tools of WDS analysis and implies realistic concepts of real WDS characteristics. The sampling design optimisation model was applied to determine the measurement locations that will provide the most information for the model's parameter calibration. Pipe-roughness coefficients according to Hazen-Williams were selected as the calibration parameters. Although the Anytown model consists of only 34 individual pipe-roughness coefficients, their number was seriously reduced by introducing pipe grouping criteria, which resulted in five groups (N = 5). Calibrating the model parameters, i.e., the pipe roughness coefficients, requires high-quality measurements of system variables, which will, to the largest extent possible, provide sufficient information for an efficient calibration [15]. The sampling design optimisation problem was formulated to determine optimal measurement locations for five independent steady-state loading conditions, i.e., four independent fire flow tests and a normal operation loading condition. Since the total number of nodes is N = 16, the maximum number of observations is NO max 80 rows (16 nodes x 5 loading conditions). Following the theoretical background of the sampling design, the pressure sensitivity of the possible measurement locations is expressed by the Jacobian matrix consisting of NO = 80 rows and N = 5 columns. The optimisation of the objective function (Eq. (10)) by the GA thus proved to be successful in extremely combinatorial problems, like sampling design and calibration [12]. Simple GAs were applied and solutions were presented in binary coding for the chromosome composed of 16 genes (16 possible measurement locations). A population of 100 chromosomes was initialised to perform an evolution chromosome based on a roulette-wheel parent selection with a probability of 0.75 and a mutation probability of 0.07. Full generation replacement with elitism was Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 825 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 potomcev iz starševskih kromosomov je potekala s popolno zamenjavo populacije skupaj s funkcijo odličnosti [13]. Poleg opisanega primera Simple GA so bili za potrebe overitve modela in njegove uporabe na primeru VS Sežana uporabljeni tudi ustaljeni GA s celoštevilčnim kodiranjem kromosomov. Ostali parametri GA so ostali enaki kakor v zgornjem primeru. Pri reševanju optimizacijskega problema izbire merilnih mest so bili uporabljeni tako Simple kakor ustaljeni GA. Rezultati optimizacijskega postopka so prikazani na sliki 2, na kateri je prikazana prva ciljna funkcija F1, tj. natančnosti umerjanja, v odvisnosti od števila merilnih mest. Na splošno se lahko podata dve ugotovitvi, tj. vključevanje dodatnih merilnih mest povečuje natančnost umerjanja in krivulja ciljne funkcije F1 spremeni nagib. V pričujočem primeru se nagib spremeni med 4. in 5. merilnim mestom, kar pomeni, da je uvajanje nadaljnjih merilnih mest manj učinkovito. Rezultati na sliki 2 so prikazani skupaj s primerjavo z rezultati modela Kapelan idr. [5], ki so povzeti iz literature in so namenjeni postopku verifikacije. Opravljena overitev modela za izbiro merilnih mest v obeh primerih uporabe posameznih tipov GA izkazuje enake rezultate. Ker je obsežnost optimizacijskega problema na dejanskem VS Sežane prevelika za uporabo Simple GA z binarnim kodiranjem, bodo uporabljeni le ustaljeni GA v kombinaciji s celoštevilčnim kodiranjem kromosomov. applied to create a new generation of offspring chromosomes [13]. Besides the described example of the Simple GA, also a Steady-state GA with integer coding were used for model verification for its later use on the Sežana WDS example. The other GA parameter settings remained the same as those above. The optimisation problem of sampling design was solved by using both Simple and Steady-state GAs. The results of the optimisation process are presented in Fig. 2, where the first objective function F1, i.e., the objective function of the calibration accuracy, is presented in relation to the number of measurement locations. In general, two characteristics can be perceived, i.e., the inclusion of additional measurement locations results in a higher calibration accuracy, and the curve of the objective function F has a change of slope. In the presented case the slope changes between the 4th and 5th measurement locations, which indicates that the introduction of further measurement locations would be far less efficient. The results in Fig. 2 are also compared to the sampling design model of Kapelan et al. [5] by literature review for model verification purposes. The performed verification of the sampling design model shows the same results in both cases of the use of individual types of GA. Since the application of the optimisation problem to the real Sežana WDS is considered to be too large for solving the problem by the use of the Simple GA with a binary coding, only the steady-state GA in combination with integer coding will be applied to the aforementioned case study. 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Število merilnih mest N / Number of measuring locations N Sl. 2. Rezultati modela izbire merilnih mest IMMe in primerjava s Kapelan idr. [5] Fig. 2. Results of "IMMe" model and a comparison to Kapelan et al. "CAO1" model [5] 826 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 4 PRIMER VS SEŽANA Overjena metoda izbire merilnih mest je bila uporabljena na dejanskem VS mesta Sežana, da bi vzpostavili niza merilnih mest na VS za zbiranje kakovostnih meritev za razpoznavo parametrov umerjanja. Uporaba optimizacijskega modela z uporabo genetskih algoritmov je bila izvedena za določevanje merilnih mest tlakov za umerjanje koeficientov hrapavosti cevovodov pod neodvisnimi požarnimi odvzemi vode, tj. odpiranje hidrantov VS mesta Sežana je na Primorskem in oskrbuje približno 5.500 prebivalcev, povprečna poraba pa je ocenjena na 16,55 l/s. Popoln hidravlični model VS je bil vzpostavljen z uporabo ustreznih podatkov geografskega informacijskega sistema (GIS) in računalniško podprtega načrtovanja upravljalca Kraški vodovod Sežana. Postavljeni model sestavljajo 1 vodni vir, 6 vodnih zbiralnikov, 4črpalke, 696 vozlišč, 738 cevi in 1 zniževalni ventil tlakov. Osrednji oz. mestni del VS je oskrbovan prek dveh vodnih zbiralnikov, ta dva sta v bližini prenosnega cevovoda, ki dovaja vodo naprej na obalno področje. Popoln hidravlični model je bil poenostavljen, tako da je ogrodni model sestavljalo le še 132 vozlišč in 173 cevi. Oba hidravlična modela sta bila grobo umerjena z meritvami pretoka v cevovodih in ravni vode v vodohranih. 4.1 Oblikovanje problema izbire merilnih mest Koeficienti hrapavosti cevovodov ogrodnega hidravličnega modela so bili razvrščeni po “ameriškem” kriteriju združevanja [2]. Končno število skupin koeficientov hrapavosti je bilo 48, kar pa bi posledično privedlo do izjemno velikega števila merilnih mest, potrebnega za rešitev inverznega problema določevanja parametrov, tj. umerjanja [15]. Nanašajoč se na ciljno funkcijo stroškov izvajanja meritev, lahko ugotovimo, da je množično zbiranje meritev zelo drag postopek. Zato je bilo število skupin koeficientov hrapavosti zmanjšamo z uporabo ohlapnejših kriterijev, vezanih samo na podatke materiala in starosti cevovodov, ki so bili najbolj nezanesljivi. Tako je bilo določenih 16 skupin hrapavosti (Na = 16), kar naj bi omogočalo vzpostavitev dobro določenega problema umerjanja. Za potrebe določitve občutljivostnih koeficientov za postavitev Jacobijeve matrike so bile določene predhodne ocene koeficientov hrapavosti za vsako skupino na podlagi strokovnih ocen in ustreznih preglednic. 4 CASE STUDY OF SEŽANA The verified IMMe sampling design model was applied to the real-life WDS of the town of Sežana, to find an optimal set of measurement locations on the WDS to collect quality measurements for the identification of model parameters. To identify the pressure measurement locations for the calibration of the pipe roughness values, the multiple independent fire flow loading conditions, i.e., hydrant flushing, were assessed and GAs were applied to perform the optimisation task. The WDS of Sežana is located in the coastal region of Slovenia, supplying a population of around 5,500 residents, while the average residential and industrial demand is estimated to be 16.55 l/s. A full-pipe WDS model was established using the relevant GIS and CAD data, resulting in a WDS model consisting of 1 reservoir, 6 tanks, 4 pumps, 696 junctions, 738 pipes and 1 pressure reducing valve. The inner part of the WDS is supplied through two tanks, which are located near the transmission main, delivering water to the coastal region. For the purposes of the sampling design analysis the full pipe model was skeletonized to a model of 132 junctions and 173 pipes. Both, full pipe and skeletonized WDS models were macro-calibrated for demands using flow and tank level measurements. 4.1 Formulation of the Sampling-Design Problem Using the sampling design approach on the skeletonized WDS model, pipe roughness values were grouped using the "American" grouping criterion [2]. The resulting number of pipe roughness groups is 48, which would consequently lead to a high number of measurement locations for the inverse problem of parameter identification, i.e., the calibration [15]. According to the second objective function, measurement collection is a costly process. Therefore, the number of pipe roughness groups was reduced according to loosened criteria of the pipe material and age, which were unreliable and sparse. A total of 16 pipe roughness groups (N = 16) were created, which should ennsure a well-posed calibration problem. The pipe roughness values for each group were previously estimated using expert judgment and relevant pipe roughness tables for the purpose of sensitivity analysis and Jacobian matrix determination. 827 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 Poleg krajev tlačnih meritev je bilo treba določiti tudi kraje odpiranja hidrantov, število mogočih lokacij je 95 (NLd, = 95) od skupaj 132 vozlišč. Izvedena je bila občutl ax ivostna analiza krajev odpiranja hidrantov, kjer so vozlišča najbolj občutljiva na povečanje hitrosti v ceveh [16]. Skupaj je bilo določenih 14 krajev odpiranja hidrantov (NL d = 14), pri katerih se je reševal problem izbire merilnih mest za umerjanje koeficientov hrapavosti. Optimizacijski problem izbire merilnih mest je bil oblikovan skladno s ciljem določitve najboljših krajev merilnih mest za umerjanje 16 skupin hrapavosti cevovodov (N = 16). Kraji zbiranja tlačnih meritev na sistemu so bile omejene na 95 (N = 95) mogočih krajev merilnih mest, kjer bodo ma zbrane meritve pri 14 (NL d = 14) neodvisnih ustaljenih obtežnih primerih. Izvedena je bila analiza občutljivosti tlakov v vozliščih za vse mogoče kraje merilnih mest in vse obtežne primere za določitev tlačnih občutljivosti za 16 skupin hrapavosti cevovodov. Postavljena je bila ti. “cela” Jacobijeva matrika za vsa mogoča merilna mesta z NO = 1330 vrsticami (95 vozlišč x 14 obtežnih primerov) in N = 16 stolpcev (16 skupin koeficientov hrapavosti cevovodov). Ker je Jacobijeva matrika edinstvena matrika za definiran problem izbire merilnih mest VS, jo je treba le enkrat izračunati pred optimizacijskim postopkom [5], medtem ko so vse rešitve optimizacijskega postopka izpeljane iz polne Jacobijeve matrike. Rešitve postopka izbire merilnih mest so bile pridobljene z ustaljenim GA skupaj s celoštevilčnim kodiranjem kromosomov, ki predstavljajo indekse krajev merilnih mest. Kodiranje s celimi števili je dobilo prednost pred binarnim kodiranjem, ker je primernejše za reševanje izbire merilnih mest na velikih realnih VS [5]. Možnost ponavljanja ista merilna mesta v eni in isti rešitvi je bila odpravljena z izbrisom vsakega dodatnega kraja merilnega mesta iz te rešitve, če bi se le-ta ponavljala. V optimizacijskem postopku GA je bila uporabljena skupnost 200 kromosomov z možnostjo križanja v dveh točkah kromosoma z verjetnostjo 0,85 in verjetnostjo mutacije v genu 0,07. Za določevanje funkcije sposobnosti so bile uporabljene naslednje vrednosti utežnih koeficientov ciljnih funkcij w1 = 1 in w = 0 ter potenca p = 1. Večkratni zagoni GA so bili uporabljeni v optimizacijskem postopku izbire merilnih mest. Besides the determination of pressure measurement locations, hydrant flushing locations needed to be identified. A total number of 95 (NL d, = 95) out of 132 junctions were identified. A sensitivity analysis was performed to identify the hydrant flushing locations, i.e., junctions at which pipes were most sensitive to increased velocities [16]. A total of 14 locations (NL d = 14) were determined, at which the sampling design problem for the calibration of the pipe roughness values was solved. The sampling design problem was formulated in order to obtain the optimal measurement locations for identifying a total number of 16 pipe roughness groups (N = 16). The pressure data collection locations were reduced to 95 (N = 95) possible measurement locations, where data could be obtained from 14 (NL d = 14) independent steady-state loading conditions. A pressure sensitivity analysis was performed for all the possible measurement locations and all the loading conditions to obtain the pressure sensitivities of the 16 pipe roughness groups. A full Jacobian matrix was written with NO = 1330 rows (95 nodes x 14 loading conditions) and Na = 16 columns (16 pipe roughness groups). Since the Jacobian matrix is a unique matrix for a defined sampling design problem of a WDS it has to be calculated only once prior to the optimisation process [5], while the estimations on sampling design solutions are derived from the full Jacobian matrix. The sampling design solutions were obtained using a Steady-State GA together with integer coding of the chromosomes, which in turn represent the IDs of the measurement locations. Integer coding was preferred to binary coding since integer coding is more suitable for large, real-life sampling design problems [5]. The possibility of repeated measurement locations in one sampling design solution was handled with the exclusion of all the additional measurement devices at one and the same measurement location. In the GA search process a population of 200 chromosomes was used with a two-point crossover probability of 0.85 and a gene mutation probability of 0.07. The weighting coefficients w1 = 1 and w2 = 0, and an exponent of p = 1 were used to evaluate the fitness function. Trivial solutions (with N = 0 and N = 95) were excluded and multiple GA runs were performed to identify non-trivial sampling design solutions. 828 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 r...........> -IMMe Število merilnih mest N / Number of measurement locations N Sl. 3. Rezultati izbora merilnih mest modela IMMe v VS Sežana Fig. 3. Results of IMMe sampling design model on WDS of Sežana 4.2 Rezultati izbire merilnih mest VS Sežana Pri izbiri merilnih mest se določa razmerje med prvo in ciljno funkcijo z uporabo utežnih koeficientov w1 in w2. Ker je razmerje utežnih koeficientov težko natančno določiti, se lahko v postopku uporabi stalno število iskanih merilnih mest. Zato je bil izbran utežni koeficient w = 0. Z uporabo celoštevilčnega kodiranja vseh mogočih merilnih mest (indeksacija) in z določitvijo dolžine kromosoma z želenim številom merilnih mest je možno točno določiti najboljša merilna mesta. Rezultat takšnega postopka je diagram na sliki 3, ki pomeni natančnost umerjanja parametrov (prva ciljna funkcija) v odvisnosti od določenega števila merilnih mest (druga ciljna funkcija). Na sliki 3, ki prikazuje natančnost umerjanja parametrov, tj. prvo ciljno funkcijo, je razvidno strmo naraščanje funkcije F1 do 13 merilnih mest. Zatem se z dodajanjem števila merilnih mest rast funkcije F1 upočasni, kar pomeni, da vsako dodano merilno mesto prispeva manj “informacije” k natančnosti umerjanja parametrov. Optimizacijski model podaja potrebno pomoč pri določevanju razporeditve merilnih mest po VS. Rezultati s slike 3 so lahko pomoč v dveh primerih odločanja: (a) če je vnaprej izbrana natančnost umerjanja parametrov, ki jo želimo doseči v samem postopku umerjanja, lahko ugotovimo koliko merilnih mest je potrebnih; (b) ocenimo kakšno natančnost umerjanja lahko pričakujemo pri izbranem številu 4.2 Results of the Sampling Design Case Study -WDS of Sežana The sampling design problem deals with the trade-off between the first and the second objective functions using the weighting coefficients w1 and w2. Since such a trade-off and the corresponding weights are hard to correctly identify, it is preferable to apply the search process for a fixed number of measurement locations; therefore, w2 = 0. Using integer coding of all the measurement locations and a chromosome length equal to the number of desired measurement locations, it is possible to correctly identify a sampling design solution within those constraints. The result of such a search process is a diagram of the calibration parameter accuracy (the first objective function) against the number of measurement locations (the second objective function), and is shown Fig. 3. Fig. 3 shows that the calibration parameter accuracy, i.e., the objective function 1, rapidly increases until it reaches a total of 13 measurement locations. Later, the slope rises more slowly, indicating that every additional measurement location adds less "information" to the calibration parameter accuracy. The optimisation model provides the necessary support in determining how the selected number of measurement locations is to be distributed across the WDS. There are two ways to interpret the results from Fig. 3: (1) one can determine in advance what level of calibration parameter accuracy is desirable for the calibration process, and through that the number of measurement locations can be identified; (2) the level of calibration parameter accuracy can be estimated from a certain Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 829 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 Preglednica 1. Rezultati izbire merilnih mest za primer VS Sežane Table 1. Results of sampling design for WDS of Sežana Ciljna funkcija Objective function ID vozlišč / Node ID F 1 N 13 41 42 45 49 50 52 55 67 68 70 83 90 97 98 99 112 114 117 118 119 132 0.502 10 1 111 0.556 11 0.608 12 1 0.646 13 11 0.671 14 0.693 0.712 15 16 111 111 111 111 111 11 1 11 11 111111 11 11 111 1 11 111 1 1 111 111 111 111 1 111 0.731 17 1111 0.748 1111 0.764 0.777 0.789 19 20 21 111 11 1 111 1 111 111111 11111 1111 1111111 11111 1 11 111 111 11 1 11111 11111111111111111 1 111 111 111 111 0.800 22 11111111111111111 11 111 merilnih mest oziroma razpoložljivi merilni opremi. Končni rezultat je določitev najboljših merilnih mest z uporabo GA. Optimizacijski rezultati s slike 3 so ločeno podani v preglednici 1, kjer so prikazana najboljša vozlišča za izbrano število merilnih mest. V preglednici 1 je podan le delež celotnih rešitev optimizacije in predstavlja rešitve za 10 do 22 merilnih mest, tj. določitev vozlišč, kjer bo nameščena merilna oprema. Metodologija izbire merilnih mest modela IMMe sloni na vrednotenju tlačnih občutljivosti na spremembe parametrov umerjanja, zato je primerno postaviti vprašanje tlačnih občutljivosti in merilnih mest [8]. Vsota tlačnih občutljivosti posameznih merilnih mest, ki je podana na sliki 4, prikazuje, da večja ko je tlačna občutljivost vozlišča, bolj je vozlišče primerno, da postane merilno mesto. Kot primer bo obravnavano vozlišče 99, ki ima veliko tlačno občutljivost pri spremembi parametrov umerjanja in je blizu odvzema požarne vode, kar povzroča povečane pretoke in s tem tudi hidravlične izgube, poleg tega pa je tudi oddaljeno od vodnih virov. Izsledki, ki izhajajo iz predhodnih ugotovitev, so v skladu z ugotovitvami drugih raziskovalcev ([5] in [6]), ki trdijo, da naj se tlačne meritve zbirajo blizu vozlišč z veliko porabo (odjemom) in naj bodo na obrobju VS, daleč proč od vodnih virov. Določevanje najboljših N merilnih mest z uporabo kriterija občutljivosti posameznih merilnih mest ni vedno tudi najboljše za določevanje skupine N-1 merilnih mest [6]. Upoštevanje načela nalaganja v tem primeru lahko vodi v napačne razlage, če upoštevamo povezavo med parametri umerjanja ob izračunu prve ciljne funkcije. number of available measurement locations, e.g., the measurement equipment that can be installed. The final result is an optimal set of measurement locations obtained within the GA search process. The optimisation results in Fig. 3 are directly related to Table 1, where optimal sets of measurement location IDs are given. Table 1 is just a segment of the entire sampling design results and presents solutions from 10 to 22 measurement locations, i.e., the table determines the node IDs where the measurement devices are to be placed. The sampling design methodology of the IMMe model is based on quantifying the pressure sensitivities to the calibration parameter changes; therefore, the pressure sensitivities and measurement location selection should be addressed [8]. Fig. 4 shows briefly the sums of the pressure sensitivities at individual measurement locations, just to indicate that the higher the node sensitivity the better the node is suited to being selected as a measurement location. The case of node 99, with the high pressure sensitivity for all the calibration parameters changes, situated near the fire flow discharges, which causes high flows and thus large hydraulic losses, is also quite remote from the water sources. This behaviour is also in agreement with the general findings by different researchers ([5] and [6]) stating that pressure measurements should be collected near the high consumption nodes and on the margin of the WDS far away from the water sources. Determination of an optimal set of N measurement locations by using the sensitivity criterion for a certain measurement location is not always optimal for determining the set of N-1 measurement locations [6]. Using the superposition principle could be misleading if the correlation between the calibration parameters becomes significant when calculating the first objective function. 1 1 1 1 1 1 1 1 1 1 1 830 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 a i * S, 1-5 -t 1 " , l n - .l n n n Sl. 4. Vsota tlačnih občutljivosti vozlišč Fig. 4. Sum of pressure sensitivities Sl. 5. Najbolj«e lokacije merilnih mest za primer 13 merilnih mest Fig. 5. Optimal locations for 13 measurement locations Izjavo lahko podkrepimo kar z rezultati izbire najboljših merilnih mest iz preglednice 1 in tlačnih občutljivosti s slike 4. Zanimiva primerjava z vidika tlačnih občutljivosti je med vozliščema 99 in 112. S This case can be underlined by examining the sampling design results from Table 1 and the pressure sensitivities from Fig. 4. From the point of view of the pressure sensitivity of the measurement Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 831 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 slike 4 je razvidno, da izkazuje vozlišče 99 precej večjo tlačno občutljivost kakor vozlišče 112, medtem ko je mogoče iz preglednice 1 razbrati, da je vozlišče 112 najboljše merilno mesto v primeru N = 12, medtem ko se vozlišče 99 ne uvršča v najboljši niz merilnih mest. Načelo nalaganja merilnih mest se izkaže za napačno, saj se vozlišče 112 ne pojavlja v nobenem od naslednjih petih naborov najboljših merilnih mest. Obstaja več razlogov za omenjene razlike, ki pa imajo odločilen vpliv na izbiro merilnih mest: (a) razvrščanje parametrov umerjanja, (b) hidravlične razmere, ki jih povzročijo požarni preizkusi in povečane hidravlične izgube, (c) bližina vozlišča 112 vozlišču s povečano porabo in (d) vpliv topologije omrežja. Pri razlagi primera N = 12 merilnih mest prispevajo vozlišča 112 in preostala izmed najboljših merilnih mest večjo natančnost umerjanja parametrov, kakor to izkazujejo njihove posamične tlačne občutljivosti. Opazovana lastnost izhaja iz povezave med koeficienti Jacobijeve matrike, ki določa strukturo najboljših merilnih mest glede na največjo natančnost umerjanja parametrov hidravličnega modela. Prostorska porazdelitev najboljših 13 merilnih mest, podanega z vozlišč ID, je prikazana na sliki 5. Od 14 leg povečanega odvzema je bilo za namestitev merilne opreme izbranih samo pet vozlišč, medtem ko je preostalih 8 razporejenih drugod po VS Sežane. Z upoštevanjem dejstva, da največ “informacije” podajo prav vozlišča s povečano porabo, obravnavani primer dokazuje, da to le delno drži. Glede na dejstvo, da so se koeficienti hrapavosti cevi razvrstiti glede na podane kriterije, so najboljša merilna mesta določena na krajih, kjer bodo meritve zagotavljale največ “informacij” za čim večjo natančnost umerjanja koeficientov hrapavosti cevi. 5 SKLEPI Razvoj, overitev in uporaba računalniškega optimizacijskega modela IMMe za najboljšo izbiro merilnih mest z uporabo GA so bili uspešni. Metodologija izbire merilnih mest upošteva lastnosti občutljivostne matrike, ki je določena s strukturo parametrov umerjanja, hidravličnimi razmerami ob zbiranju meritev, hidravličnimi veličinami, ki se merijo, in kraji zbiranja meritev. Optimizacijski problem izbire merilnih mest je določen z dvema kriterijema, in sicer največje natančnosti umerjanja in najmanjšega števila merilnih mest. Uporaba GA v postopku izbora locations, a comparison of the nodes 112 and 99 is interesting. As can be seen from Fig. 4., node 99 has a much higher pressure sensitivity than node 112, while Table 1 in turn shows that node 112 presents the optimal measurement location in the case of N = 12, when node 99 is not selected. Differences to the superposition principle arise because node 112 does not occur in any of the following five sets of optimal measurement locations. There could be several reasons for the mentioned differences, all having a decisive impact on the sampling design: (a) calibration parameter grouping, (b) hydraulic conditions caused by the fire flow tests and its increased head losses, (c) proximity of node 112 to a node with increased consumption, (d) influence of network topology. When considering N = 12 measurement locations, node 112 and others from the optimal set contribute to a higher calibration parameter accuracy due to the first objective function, regardless of the individual pressure sensitivities. This property can be observed because of the correlation of the coefficients of the sensitivity matrix, which indicates that the optimal measurement locations are those which contribute the most to the identification of the calibration parameters. The resulting node IDs for the 13 most informative measurement locations from Table 1 are presented in Fig. 5. Out of 14 hydrant flushings only 5 measurement locations correspond to the same location, while the remaining 8 are located elsewhere throughout the WDS. When referring to the fact that measurement data provides the most "information" at locations of high discharge, the presented case study shows this is only partly true. Optimal measurement locations were placed at positions where measurements provide most "information" for calibrating those parameters, i.e., the calibration grouped pipe roughness coefficients. 5 CONCLUSIONS The development, verification and application of the optimisation model called IMMe for optimal sampling design by using GAs were successful. The sampling design methodology takes into account the properties of a sensitivity matrix determined by the structure of the calibration parameters, the hydraulic conditions during measurement collection, the hydraulic quantities measured, and the measurement locations. The optimisation problem of the sampling design is determined by two criteria, i.e., on the maximisation of the calibration accuracy and on the 832 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 najboljših merilnih mest hidravličnega modela Anytown se je izkazala za zelo učinkovito. Po opravljeni overitvi je bil model IMMe uporabljen na dejanskem VS Sežana, kjer so bila določena najboljša merilna mesta za umerjanje koeficientov hrapavosti cevovodov. Nadaljnje raziskovalno delo naj bi bilo usmerjeno v določitev ustreznejšega zapisa druge ciljne funkcije, ki bi bila izražena oziroma odvisna neposredno od stroškov izvajanja meritev, hkrati pa tudi v vrednotenje uporabe metodologij na dejanskih VS. Zahvala Rezultati raziskave so delno financirani v okviru raziskovalnega programa P2-180 in raziskovalnega projekta L2-6468, za katerega je sredstva zagotovila Agencija za raziskovalno dejavnost Republike Slovenije, s tesnim sodelovanjem University of Exeter, VB. minimisation of the number of measurement locations. The use of GAs in the sampling design procedure on the artificial hydraulic model of Anytown proved very efficient. After verification, the IMMe model was applied to a real WDS of Sežana, where optimal measurement locations were obtained for the pipe roughness calibration. Further research work is focused on a more appropriate expression for the second objective function, which would be directly dependent on the measurement costs and in parallel research, on other assessments of the sampling on the WDS. Acknowledgments The obtained results are partially supported by the research programme P2-180 and research project L2-6468-0792 by the Slovenian Research Agency, with the close collaboration of the University of Exeter, UK. 6 LITERATURA 6 REFERENCES [I] Ormsbee, L.E., Lingireddy, S. (1997) Calibrating hydraulic network models, Journal AWWA, Volume 89, Number 2, 1997, p. 42-50. [2] Walters, G.A., Savic, DA., Morley, M., de Schaetzen, W. (1998) Calibration of water distribution network models using genetic algorithms, Proc. 7th International Conference on Hydraulic Engineering Software - Hydrosoft 98, Southampton, UK, p. 131-140. [3] Lansey, K.E., El-Shorbagy, W., Ahmed, I., Araujo, J., Haan, C.T (2001) Calibration assessment and data collection for water distribution networks, Journal of Hydraulic Engineering, Volume 127, Number 4, 2001, p. 270-279. [4] ECAC (1999) Calibration guidelines for water distribution modelling, Proc. AWWA 1999 Imtech Conference, Seatlle, USA. [5] Kapelan, Z., Savic, DA., Walters, GA. (2003) Multiobjective sampling design for water distribution model calibration, Journal of Water Resources Planning and Management, Volume 129, Number 6, 2003, p. 466-479. [6] Bush, CA., Uber, J. (1998) Sampling design methods for water distribution model calibration, Journal of Water Resources Planning and Management, Volume 124, Number 6, 1998, p. 334-344. [7] Knopman, D.S., Voss, C.I. (1989) Multiobjective sampling design for parameter estimation and model discrimination in groundwater solute transport, Water Resources Research, Volume 25, Number 10, 1989, p. 2245-2258. [8] Kozelj, D., Šantl, S., Steinman, F., Banovec, P. (2005) Sampling design for calibration of water distribution system models by genetic algorithms, Proceedings of the Eight International Conference on Computing and Control for the Water Industry, University of Exeter, UK, p. 173-178. [9] Goldberg, D.E. (1989) Genetic algorithms in search, optimization and machine learning, Addison-Wesley Publishing Company, USA. [10] Steinman, F, Banovec, P., Šantl, S. (2001) Genetic-algorithms-supported planning of water supply systems, Journal of Mechanical Engineering, Volume 47, Number 6, 2001, p. 263-279. [II] Keedwell, E., Khu, S.T (2005) A hybrid genetic algorithm for the design of water distribution networks, Engineering Applications of Artificial Intelligence, Volume 18, Number 4, June 2005, p. 461-472. Izbor merilnih mest na vodovodnih sistemih - Sampling Design for Water Distribution System 833 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 817-834 [12] Simpson, A.R., Dandy, C.G. (1994) Genetic algorithms compared to other techniques for pipe optimization, Journal of Water Resources Planning and Management, Volume 120, Number 4, 1994, p. 423-443. [13] Wall, M., Galib (1996) A C++ library of genetic algorithm components, Mechanical Engineering Department, Massachusetts Institute of Technology, USA. [14] Walski, T.M. et al. (1987) Battle of the network models: Epilogue, Journal of Water Resources Planning and Management, Volume 113, Number 2, 1987, p. 191-203. [15] Mallick, K.N., Ahmed, I., Tickle, K.S. Lansey, K.E. (2002) Determining pipe groupings for water distribution network, Journal of Water Resources Planning and Management, Volume 128, Number 2, 2002, p. 130-139. [16] Meier, R.W., Barkdoll, B.D. (2000) Sampling design for network model calibration using genetic algorithms, Journal of Water Resources Planning and Management, Volume 126, Number 4, 2000, p. 245-250. Naslov avtorjev: doc. dr. Primož Banovec Authors’ Address: Doc. Dr. Primož Banovec Daniel Kozelj Daniel Kozelj Sašo Šantl prof. dr. Franci Steinman Prof. Dr. Franci Steinman Univerza v Ljubljani University of Ljubljana Fakulteta za gradbeništvo in Faculty of Civil and Geodetic geodezijo Engineering Hajdrihova 28 Hajdrihova 28 1000 Ljubljana 1000 Ljubljana, Slovenia primoz.banovec@fgg.uni-lj.si primoz.banovec@fgg.uni-lj.si daniel.kozelj@fgg.uni-lj.si daniel.kozelj@fgg.uni-lj.si saso.santl@evj-kabel.net saso.santl@evj-kabel.net fsteinma@fgg.uni-lj.si fsteinma@fgg.uni-lj.si Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 5.5.2005 Received: Accepted: 25.10.2006 Open for discussion: 1 year 834 Banovec P. - Kozelj D. - Šantl S. - Steinman F. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 UDK - UDC 62-33:621.5 Izvirni znanstveni članek - Original scientific paper (1.01) Analiza delovanja pnevmatičnega ventila s predkrmilnim piezoventilom Analysis of the Operation of Pilot-Stage Piezo-Actuator Valves Niko Herakovič- Dragica Noe (Fakulteta za strojništvo, Ljubljana) Zaradi vse večjih zahtev po boljših dinamičnih lastnostih in manjši porabi energije pnevmatičnih ventilov, je treba nenehno izboljševati sedanje elektromagnetne izvršilnike, obenem pa raziskati možnosti uporabe alternativnih aktuatorjev Odkar so piezoelektrični izvršilniki, predvsem upogibni, pridobili na preprostosti oz. cenenosti, hitrosti delovanja in zanesljivosti, so v zadnjih letih pritegnili veliko pozornosti tudi za uporabo pri krmiljenju pnevmatičnih ventilov. V okviru predstavljene raziskave je bil razvit predkrmilni ventil z dvema upogibnima piezoelementoma ter prigrajen standardnemu pnevmatičnemu ventilu velikosti ISO 3, 5/2. V prispevku so prikazane bistvene značilnosti piezoelementov, možnosti uporabe le-teh za krmiljeneje pnevmatičnih ventilov in na koncu analiza ventila s predkrmilnim piezoelementom. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: ventili pnevmatični, analize delovanja, predkrmiljenje ventilov, elementi piezoelektrični) Due to the ever-growing demands for better dynamic properties and lower energy consumption for pneumatic valves, constant efforts to improve existing electromagnetic actuators are required, while at the same time new possibilities for alternative actuators are being searched for. Since piezoelectric actuators, especially the bender type, have become low-cost, easy to use, faster and more reliable in operation, they have also attracted a lot of attention in pneumatic control-valve applications. The presented study reports on the experimental development of a pilot-stage valve with two bending piezo-elements attached to a standard pneumatic valve of size ISO 3, 5/2. The most important characteristics of piezo-actuators and the possibilities for their use in pneumatic control valves are discussed, and the features of the developed pilot-stage piezo-actuator valve are presented. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: pneumatic valves, operation analysis, piezo-controlled hydraulic pilot stage, piezo-elements) 0 UVOD 0 INTRODUCTION Pri razvoju ventilov se srečujemo z zahtevami po čim manjših izmerah, dobrih dinamičnih značilkah ter čim manjšo porabo električne energije. Pogosto se zahteva izpolnjevanje vseh treh meril. Tehnične izboljšave pri razvoju pnevmatičnih ventilov pa bo mogoče doseči tako z optimizacijo konstrukcije kakor tudi z novimi načini krmiljenja in uporabo novih materialov [1]. Raziskave na področju razvoja predkrmilnih pnevmatičnih ventilov so pokazale, da je uporaba elektromagnetnih izvršilnikov, kot predkrmilnih ventilov, omejena in je treba iskati nove rešitve. Ugotoviti je mogoče, da dosedanja uporaba piezoelektričnih pogonov v razvoju In the design and development of valves, the requirements that have to be met are the smallest possible dimensions, good dynamic characteristics and the lowest possible electric power consumption. Quite often these three criteria have to be fulfilled. Technical improvements in the development of pneumatic valves can be achieved by optimizing the design as well as introducing new principles of pneumatic control and new materials [1]. Research in the field of pilot-stage pneumatic valves has shown that the use of electromagnetic actuators in the pilot-stage has its own restrictions and that new solutions have to be sought. On the other hand, it can be established that in the development of fluid-power components and 835 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 fluidnotehničnih komponent in drugih mehatronskih sistemov kaže obetavne rezultate ([2] do [6]). Razvoj piezoelektričnih izvršilnikov zahteva dobro poznavanje piezotehnike, postopka izdelave piezoelementov ter možnosti, ki jih posamezne izvedbe piezoelementov ponujajo. Pomembna sta tako raztezek in sile, oblike in izmere piezoelementov pa so lahko prilagojene vsakokratni uporabi. V predkrmilnih ventilih so bili v predhodnih raziskavah uporabljeni predvsem upogibni konzolni piezoelementi ([6],[10] in [18]). Namen naše raziskave pa je analizirati možnosti uporabe mostičnega piezoelementa [1]. Zato je bil okviru raziskave standardni izvedbi pnevmatičnega potnega ventila, namesto magnetnih premikal, prigrajen predkrmilni ventil z mostičnim piezoizvršilnikom, rezultate delovanja oz. značilke ventila pa smo primerjali z elektromagnetnim predkrmilnim ventilom. 1 IZHODIŠČA ZA RAZVOJ VENTILA S PREDKRMILNIM PIEZOVENTILOM Pnevmatični posredno krmiljeni ventili so zgrajeni iz glavnega ventila, ki usmerja zrak od napajanja k porabniku ter predkrmilnega ventila (pk1 in pk2), ki preklaplja oziroma krmili glavni ventil - gv (sl. 1). Naloga predkrmilnega ventila je spremeniti električno energijo v mehansko delo, oziroma premik zapirnega elementa, ki usmerja krmilni zrak na krmilni priključek glavnega ventila. Na ta način so izmere izvršilnikov predkrmilnega ventila manjše, manjša pa je tudi poraba moči. V preteklosti je bilo razvitih več načinov pogonov, ki električni signal spremenijo v silo ali premik. Naloga pogona je povezovati električni krmilni obtok s pnevmatičnim, oziroma spremeniti električno energijo v energijo other mechatronic systems the use of piezoelectric actuators has become very promising ([2] to [6]). The development of piezoelectric actuators requires a good knowledge of piezo technology, manufacturing procedures and the possibilities offered by their particular designs. Factors that are important to consider are the extension and the force, while the shapes and sizes of piezo elements can be adapted to each particular application. In pilot-stage valve applications, previous researchers have mainly used bending cantilever piezo-elements ([6],[10] and [18]). The aim of our study was to analyze the use of another possible type, i.e., a bending crossbow piezo-element [1]. For this purpose, a pilot-stage valve with a crossbow piezo-actuator was built to a standard pneumatic directional valve, and the results and the characteristics of the thus obtained valve were compared to an electromagnetic pilot-stage valve. 1 CONSIDERATIONS IN THE DESIGN OF PILOT-STAGE PIEZO-ACTUATOR VALVE Pneumatic pilot-stage valves consist of the main valve that directs the air from the supply source to the consumer and a pilot-stage component (pk1 and pk2) that switches on/off the main valve (gv), Fig 1. The task of the pilot-stage valve is to convert the electrical energy into mechanical work or displacement of the closing element, which then directs the controlling air onto the control element of the main valve. This kind of design makes it possible for the dimensions of pilot-stage actuators to be smaller and the power consumption to be lower. In the past, a number of actuator principles for converting an electrical signal into a force or motion have been developed. The task of the actuator is to connect the electrical control circuit with the pneumatic circuit, or, in other words, to convert the electri- Sl. 1. Elektromagnetni posredno krmiljeni pnevmatični 5/2 potni ventil [17] Fig. 1. Electromagnetic pilot-stage directional pneumatic valve 5/2 [17] 836 HerakovičN. - Noe D. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 w. *¦==.:, w,. -J EW [ wm. I id \ w a) Elektromagnetni izvršilniki Electromagnetic actuators b) Električni izvršilniki Electric actuators Sl. 2. Spreminjanje električne energije v mehansko Fig. 2. Conversion of electrical energy into mechanical energy stisnjenega zraka. Po načinu delovanja so elektromehanski pretvorniki (EP) razdeljeni v elektromagnetne in električne izvršilnike. Elektromagnetni najprej električno energijo spremenijo v magnetno energijo in nato v mehansko, pri električnih pa je ta sprememba neposredna (sl. 2 a in b). Vprašanje je, kakšne možnosti ima elektromagnetni izvršilnik v smislu skrajševanja preklopnih časov, zmanjševanja izmer ter zmanjševanja porabe moči. V fizikalnem smislu nastanejo, zaradi magnetne indukcije in vrtinčnih tokov pri vklopu električnega toka, zakasnitve pri preklopu. Preklopni časi so tem krajši, čim manjše so mase gibljivih delov in čim krajši je gib jedra tuljave. Pri najboljši izvedbi elektromagnetnih premikal so preklopni časi v območju 500 ms, za običajne izvedbe ventilov pa je čas preklopa med 10 in 20 ms. Pomembna za uporabnike pa je tudi ponovljivost preklapljanja v vsej dobi trajanja v pričakovanem okolju. Posebni ventili lahko dosežejo ponovljivost tudi pri 100 milijonih gibov ([1], [3], [6] in [7]). Naslednji ukrepi, ki jih je mogoče zaslediti pri razvoju elektromagnetnih predkrmilnih ventilov, so povezani z zmanjševanjem prostornine pnevmatičnih komponent pri enakih ali celo povečanih pretokih stisnjenega zraka. Trenutno se je zmanjševanje ustavilo pri 10 mm. Nadaljnje zmanjševanje izmer onemogočajo elektromagnetni izvršilniki. Izdelava manjših ventilov je zaradi zahtevnih tehnoloških postopkov manjših elektromagnetov ekonomsko neupravičena. Napredek je mogoče iskati v smeri izvršilnikov na podlagi silicija. Trenutno je tako imenovana mikropnevmatika, pri kateri so izmere pnevmatičnih predkrmilnih ventilov samo nekaj mm, na začetku svojega razvoja. Problem, ki nastane pri zmanjševanju izmer, se kaže dandanes v nesorazmerju pnevmatičnega dela s priključki za zaznavala, pri cal energy into the energy of compressed air. In terms of operation, electro-mechanical transducers (EP) can be divided into electromagnetic and electrical actuators. The electromagnetic actuators convert electrical energy first into magnetic energy and then into mechanical energy, whereas the electrical actuators transform it directly (Fig 2 a and b). The potentials of the electromagnetic actuators are questionable when factors such as shorter switchover times, smaller dimensions and lower power consumption are considered. In the physical sense, due to magnetic induction and eddy currents during switching the current on, delays in switchover occur: the shorter the switchover times, the lower are the masses of the moving parts and the shorter is the movement of the core of the coil. In optimal electromagnetic-actuator design solutions, the switchover times are in the range of 500 ms, while for standard valve designs the switchover time ranges between 10 to 20 ms. The switchover repeatability during the life time in a foreseen environment is also important for customers. Special valves may reach a repeatability as high as 100 million strokes ([1], [3], [6] and [7]). Some efforts in the development of electromagnetic pilot-stage valves are directed into reducing the volume of the pneumatic component at equal or even higher compressed air-flow rates. So far the reductions have stopped at 10 mm3. A further scale-down of dimensions is not possible because of the electromagnetic actuators. Due to demanding technological procedures the manufacturing of smaller valves is not economically justifiable. However, an advancement could be looked for in the direction of making actuators based on silicon. Currently, the so-called micro-pneumatics, where the dimensions of pneumatic pilot-stage valves reach just a few mm, is still in its early stages. The problem in scaling down the dimensions is in the disproportion between the pneumatic part and the sensor connectors, where the pneumatic part can be Analiza delovanja pnevmatičnega ventila - Analysis of the Operation of Pilot-Stage 837 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 Sl. 3. Izvedbe predkrmilnih ventilov, a - elektromagnetni, b - z dvoslojnim upogibnim piezoelementom, c - z mostičnim piezoelementom - izvedba z odbojno šobo, upe - upogibni piezoelement Fig. 3. Designs of pilot-stage valves a - electromagnetic, b - two-layer bending piezo-element, c - crossbow piezo-element - flapper nozzle system, upe - bending piezo-element katerih je pnevmatični del bistveno manjših izmer od significantly smaller than the connectors. At the priključkov za zaznavala. Za zdaj tu še ni ustreznih current stage of development there are still no rešitev (M12 priključek je sedaj še nujnost) [6] . suitable solutions available (an M12 connection is Več razlogov narekuje razvoj učinkovitih still a necessity) [6]. elektromehanskih pretvornikov, ki se uporabljajo v There is a growing need to develop efficient vrsti elektronskih naprav in nenazadnje tudi v electro-mechanical transducers that can be used in elec- pnevmatiki. Eden izmed njih je tudi zahteva po tronic devices and fluid power systems. One reason for energetsko varčnih komponentah in je posledica this is the need for energy-saving as a general trend in zahtev po varčevanju energije v vseh vrstah uporab. all kinds of applications. Some of the most recent Razviti so piezoelektrični, elektrouporni, developments are piezoelectric, magnetoresistive, magnetouporni, spominski elementi, elektromagnetni electromagnetic and shape-memory elements [1]. There itn. [1]. Za velike dinamične zahteve je piezoelektrični is no doubt that for high-dynamic-load applications the izvršilnik prav gotovo najprimernejši. Prednost piezoelectric actuators are the best choice. The piezoelektričnih materialov je v tem, da spreminjajo advantage of piezoelectric materials lies in the fact that električno energijo v mehansko in nasprotno skoraj they convert electrical energy into mechanical and vice brez izgub (sl. 2). versa, almost without any losses (Fig 2). Med različnimi izvedbami električnih Among the various kinds of electric actuator, izvršilnikov imajo piezoelektrični največje možnosti the piezoelectric actuators have the best chance to beza široko in ekonomsko upravičeno uporabo ([4], come widespread and economically justifiable. ([4], [6] in [7]). Ker so v pnevmatiki preklopne sile, v [6] and [7]). Since in pneumatic systems the switchover primerjavi s hidravliko majhne, so primerni za forces are low compared to hydraulic systems, the uporabo predvsem upogibni piezoizvršilniki oziroma bending piezo-actuators are the ones that are especially elementi. Razvoj piezo materialov in pogonov je v suitable for pneumatic applications. Recently, there zadnjih letih močno napredoval, tako da je mogoče has been considerable progress in the development of z uporabo novih materialov in tehnologij izdelati piezo-materials and piezo-actuators so that with the upogibne piezoizvršilnike, primerne za krmilne sile use of new materials and technologies it is possible to tudi do 100 N ali v posebnih izvedbah do 1kN [15]. manufacture bending piezo-actuators suitable for V pnevmatiki so se do sedaj v največji meri control forces up to 100 N, or in cases of special design, uporabljali predvsem konzolni upogibni even 1 kN [15]. In pneumatic systems, cantilever piezoelementi, vendar imajo tudi mostični bending piezo-actuators are mostly used, though the piezoelementi primerne značilnosti za uporabo pri crossbow piezo-actuators also possess properties krmiljenju potnih ventilov (sl. 3). suitable for use in directional control valves (Fig 3). 2 UPOGIBNI PIEZOIZVRŠILNIKI 2 BENDING PIEZO-ACTUATORS Piezopojav je lastnost številnih materialov, ki The piezo effect is a feature of numerous jih najdemo v naravi. Njihova poglavitna značilnost materials found in nature. Their basic characteristic is 838 Herakovič N. - Noe D. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 je ta, da se na kristalu pri mehanski deformaciji, zaradi tlačne ali natezne sile, pojavi električni naboj. Ta pojav je imenovan tudi neposredni piezopojav. Nasprotni pojav, ko se kristal pod vplivom električnega polja deformira - skrči ali raztegne, pa se imenuje inverzni ali povratni piezopojav in je uporaben za premikala oziroma pogone. 2.1 Osnovna načela delovanja Za razumevanje povratnega piezopojava, ki sta ga opredelila v letu 1881 Curie in Lippman ([8] in [9]) je treba razumeti električno polarnost, katere povzročitelji so električni dipoli. Pri tem je električni dipol primerljiv z magnetnim, torej s tako imenovanim elementarnim magnetom vendar v električnem smislu. Dipol je mogoče prikazati z negativnim (- q) in pozitivnim nabojem (+q ) in puščico, ki je usmerjena od negativnega k pozitivnemu dipolu. Opisati ga je mogoče z momentom dipola, z vektorjem p, ki je definiran kot zmnožek naboja q in medsebojne oddaljenosti l in je podan v enoti dolžine (sl. 4 a). Električna polariteta je lahko naravna ali pa nastane pri delovanju električnega polja na nepolarizirane atome - spontana polarizacija. Polarizacija je lahko trajna ali pa po prenehanju delovanja električnega polja izgine. Za razvoj piezopogonov je zanimiva trajna ali permanentna polarnost. Dipol se v močnem električnem polju E usmeri glede na smer polja. Pri tem deluje na pozitivni del dipola sila F v smeri električnega polja in sila F2 v nasprotni smeri (sl. 4 b). Dvojica sil se izrazi kot vrtilni moment M, ki je enak zmnožku potenciala p in električnega polja E (M= p . E). that on a crystal subjected to mechanical deformation, as a result of compressive and tensile forces, an electrical charge is created. This effect is also known as the direct piezo effect. The opposite phenomenon, when under the influence of an electric field the crystal deforms, i.e., it contracts or extends, is called the reverse piezo effect. This is used in actuators. 2.1 Basic principles of operation The reverse piezo-effect, first described in 1881 by Curie and Lippman ([8] and [9]), can be explained by the electrical polarity caused by electric dipoles. Here, the electrical dipole is comparable to the magnetic one, i.e., to the so-called elementary magnet, but in the electrical sense. A dipole can be presented with a negative (- q) and positive charge (+ q) and an arrow directed from the negative to the positive dipole. It can be defined by the dipole moment, vector p, defined as the product of the charge q and the distance l between them, Fig 4a. Electrical polarity can be natural or can occur when an electrical field acts on non-polarized atoms, i.e., spontaneous polarization. Polarization can be permanent or can disappear after the electrical field stops acting. It is the permanent polarization that is of interest in the development of piezo-actuators. In a strong electric field, E, a dipole aligns with the direction of the electric field. In this way only the positive part of the dipole is acted on by force F1 in the direction of the electrical field, and F2 in the reverse direction (Fig 4b). The force couple expresses itself as a rotating moment M equal to the product of . the potential, p, and the electrical field E (M = p E). Sl. 4. Dipol (a) in dipol v električnem polju (b) Fig. 4. Dipole (a) and dipole in the electric field (b) Analiza delovanja pnevmatičnega ventila - Analysis of the Operation of Pilot-Stage 839 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 Za dipole v dielektriku velja, da so naključno usmerjeni in je material navzven nevtralen. S postavitvijo materiala v električno polje se dipoli ustrezno usmerijo in na površini dielektrika nastane površinski naboj. Kadar govorimo o dipolih v trdnih telesih, mislimo s tem polarizirane elementarne kristalne mreže. Najbolj znan piezomaterial je kremen, vendar piezopojav pri njem ni dovolj izrazit, sile so majhne in tako ni primeren za krmiljenje pnevmatičnih ventilov. Mnogo primernejši materiali so različne feroelektrične keramike, med njimi barijev titanat (BaTiO3), svinčev titanat, svinčev cirkonat in svinčev cirkonat-titanat (PZT). Fizikalne lastnosti PZT piezokeramike so odvisne od mešanice cirkonija in titanata in dodanih drugih materialov, ki so specifični za posamezne izdelovalce piezoelementov. Kristaliti PZT so pred polarizacijo centrosimetrično kubični (izotropno), po polarizaciji pa kažejo tetragonalno simetrijo (anizotropna sestava), in sicer v pogojih, ko vrednost temperature ne presega Curie-jeve temperature [9]. Nad to temperaturo kristali izgubijo piezoelektrične lastnosti. V piezomaterialu, ki je zgrajen iz več osnovnih kristalov, so dipoli umerjeni naključno. Vendar se v materialu najdejo skupine kristalov oziroma dipolov, ki so enako usmerjene. Take skupine so označene kot območja, ki v bistvu ustrezajo makroskopskemu dipolu, saj so znotraj polarizirane. Ker so ta območja ponovno naključno usmerjena, kristal na zunaj ne kaže polarnosti. Če se tak material postavi v močno električno polje nekaj kV na mm dolžine, se območja pričnejo usmerjati, posamezne skupine jeder kristala dobijo enako usmerjenost in jedra rastejo na račun drugih. Z zamrznitvijo polariziranega stanja dobimo piezoelektrični kristal s polarizirano osjo. Pod Curie- Dipoles in a dielectric fluid have a random sense of direction, and outwardly the material acts as neutral. By placing the material into an electric field, the dipoles align accordingly, and on the surface of the dielectric a surface charge is produced. Dipoles in reference solids are displayed as polarized elementary crystals in the crystal lattice. A is commonly known, quartz is a piezo-material; however, in quartz the piezo effect is not distinct enough, the forces induced are small, which makes it inappropriate for use in pneumatic valves. Much more suitable materials include some ferromagnetic ceramics, among them barium-titanate (BaTiO3), lead-titanate, lead-zirconate and lead-zirconate-titanate (PZT). The physical properties of PZT piezo-ceramics depend on the proportion of zirconium and titanium and their additions specific for each particular manufacturer of piezo-elements. Prior to polarization PZT crystallites are axisymmetrical cubes (isotropic), after the polarization, however, they display a tetragonal symmetry (anisotropic structure), and this is in conditions when the temperature does not exceed the Curie temperature [9]. Above this temperature the crystals lose their piezoelectric properties. In a piezo material made from several basic crystals, the dipoles are directed randomly. However, in the material it is possible to find groups of crystals that have the same direction. These groups are marked as domains and actually correspond to the macroscopic dipole because they are polarized on the inside. Since these domains again are randomly directed, on the outside the crystal does not display any polarity. If such materials are subjected to a strong electric field of a few kV per mm of length, the domains start aligning themselves, and particular groups in the crystal core become equally directed and these cores start growing at the expense of others. By freezing such a polarized state, a piezoelectric crystal with a polarized axis is obtained. Below the Curie temperature and on removal Sl. 5. Predstavitev posameznih kristalnih zrn in njihovih območij Fig 5. Presentation of crystal grains and their domains 840 HerakovičN. - Noe D. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 Sl. 6. Deformacija kristalne rešetke pri postavitvi v električno polje Fig. 6. Deformation of crystal lattice when subjected to electric field jevo temperaturo in po odstranitvi električnega polja se dipoli sicer nekoliko razbremenijo, vendar v večini ostanejo v dani usmerjenosti (sl. 5). Pri ponovni postavitvi piezoelektrične keramike v električno polje se polarizacija območij ponovno ojači in kristal se deformira, oziroma razširi vzdolžosi polarizacije (sl. 6). Deformacija kristala Dl je odvisna od jakosti električnega polja E oziroma napetosti U in piezo modula dij, kjer je i smer električnega polja v kartezičnem koordinatnem sistemu in j smer deformacije za dano polarizacijo kristala (sl. 7). of the electric field, the dipoles discharge slightly, but mainly remain in the given direction (Fig 5). By putting the piezoelectric ceramic back into the electric field, the polarization of the domains intensifies again and the crystal deforms or extends along the polarization axis (Fig 6). The deformation of the crystal depends on the intensity of the electric field, E, or the voltage, U, and piezo-module, dij, where i denotes the direction of the electric field in the Cartesian coordinate system and j the direction of deformation for a given polarization of the crystal (Fig 7). Dl = d-E-l = d-l-U/l = d-U (1). Glede na smer delovanja električnega polja in smer polarizacije se kristal lahko deformira v smeri polarizacije ali pravokotno na polarizacijo. Tako predstavlja piezomodul d33 vzdolžno deformacijo kristala v smeri osi z, ko je smer električnega polja enaka smeri prvotne polarizacije, v osi z, piezomodul d31 pa prečno deformacijo glede na smer električnega polja, ko je smer polarizacije v osi z in deformacija With respect to the working direction of the electric field and the direction of polarization, the crystal may deform in the direction of polarization of perpendicularly to it. Thus, the piezo module, d33, represents the longitudinal deformation of the crystal in the direction of the z-axis, when the sense of direction of the electrical field is the same as the direction of the original polarization in the z- axis, and the piezo-module d31 presents a transverse deformation with respect to the direction of Sl. 7. Opredelitev indeksov piezomodula Fig. 7. Determination of the subscripts of a piezo module Analiza delovanja pnevmatičnega ventila - Analysis of the Operation of Pilot-Stage 841 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 pravokotno na os z. Za polikristalno keramiko PZT je vrednost modula d3 med vrednostjo 50 in 765 pm/V [11]. Pri izbiri piezoizvršilnikov, primernih za pogon pri pnevmatičnih ventilih, je poleg največjega raztezka aktuatorja Dl treba poznati še največjo dopustno silo F, ki jo lahko piezoizvršilnik doseže, ter točko delovanja glede na najboljšo silo in raztezek. Največja sila, imenovana tudi omejevalna sila, ki se vzpostavi v piezoelementu, ko je ta pod napetostjo in je vpet med dve aboslutno togi površini, ki ne dopuščata raztezka - Dl/l=0. Sila je odvisna od vzmetne stalnice kristala in deformacije: the electrical field, when the sense of direction of the polarization is in the z-axis and the deformation is perpendicular to the z-axis. For PZT polycrystalline ceramics the value of the module d33 ranges between 50 to 765 pm/V [11]. When selecting piezo-actuators suitable for pneumatic valves, besides the maximum elongation of the actuator, we also need to know the maximum allowable force F a piezo-actuator can reach, and the point of action with respect to the optimal force and extension. The maximum force, called the blocking force, is the force occurring in the piezo element when the latter is under stress and is fixed between two absolutely rigid surfaces allowing no extension, i.e., Dl/l = 0. The force depends on the spring constant of the crystal and the deformation. F= k-dij-U (2), kjer je k - vzmetna stalnica piezo kristala oziroma količnik togosti, ki je odvisen od vrste parametrov. Primer delovnega diagrama piezoizvršilnika je podan na sliki 8. S tem diagramom, ki je specifičen za vsak piezoizvršilnik, je mogoče preprosto razbrati delovne točke piezoizvršilnika med obema mejnima vrednostima sile in raztezka. Najboljša točka delovanja vsakega piezoizvršilnika je na polovici zveznice med največjo silo in največjim upogibom oz. raztezkom (sl. 8). Pri ogrevanju piezomaterialov nad Curie-jevo temperaturo T polarizacija kristalov piezoizvršilnika izgine. Podatek je pomemben prav za uporabo v where kT is the spring constant of the piezo-crystal or the coefficient of stiffness, depending on a number of parameters. Figure 8 shows in a graph the optimal operating point of a piezo-actuator. From this force-deflection graph, which is specific for each piezo-actuator, we can easily see the operating points of piezo-actuators lying between the limit values of force and extension. The optimal operating point of each piezo-actuator is in the middle of the connecting line between the maximum force and the maximum deflection or extension (Fig. 8). When piezo-materials are heated above the Curie temperature, Tc, the polarization of the crystals disappears. This data is important for high- 11 _ (blocking force) (for <"" omejevalna sila cd ^v* (directionaf voltage enlargement) '« \ j smer večanja napetost i yC (optimal operating point) \ > /\^najugodnejša točka delovanja pThere is also a flapper (op) distanced from the adjustable nozzle by a distance h and fixed to the crossbow piezoelement. The actuator is fed by compressed air with a pressure p , which flows out through both nozzles into the atmosphere, but also into the control chamber of the pneumatic valve. When the flapper is approaching the pressure pk at the outlet gradually increases and when the flapper is returning the pressure decreases. Here, it was necessary to consider that the piezo-actuator has to enable air inflow to one side of the directional valve and air relief on the other side (Fig 11) and provide a large enough pressure for piloting the main valve as well as a suitable switchover frequency. To determine the optimal diameter ratio of both nozzles (d. / d.) it was necessary to analyze the operation of the n pilot-stage piezo-actuators. For this purpose a test rig was set up ([12] and [13]) to measure and observe the histories of the pressure pk and the switchover force of the main valve. The measurements provided a basis for the determination of the necessary flapper displacement, h, and the necessary force, F, that have to be provided by the piezo-element to achieve complete air relief in the control chamber of the main valve, on the one side, and a large enough switchover Sl. 11. Shematični prikaz sistema soba - odbojna plošča (a) in glavni ventil s predkrmilnima piezoventiloma, ki delujeta po načelu soba - odbojna plošča, v prerezu (b) Fig. 11. a) Schematic of the nozzle-flapper system; b) the main valve with the pilot-stage piezo-actuators, flapper/nozzle design, cross section 844 HerakovičN. - Noe D. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 Sl. 12. Potek odvisnosti sile F od odmika odbojne plošče in premera nastavljive sobe Fig. 12. Force, F, versus flapper displacement for various adjustable nozzle diameters v krmilni celici na drugi strani bata glavnega ventila. Na sliki 12 so prikazani rezultati meritev tlaka pk oz. sile F v odvisnosti od odmika odbojne plošče h od nastavljive šobe pri različnih razmerjih premerov obehšob (kšin nš). Poteki sile F pri različnih premerih nastavljive šobe so značilni in kažejo odvisnost sile F od odmika odbojne plošče h. Rezultati kažejo, da se sila F prične zmanjševati pri odmiku odbojne plošče h= 0,1 mm (sl. 12). Hitrost zmanjševanja sile se z večanjem premera šobe veča. Sila F se pri odmiku h=0,4 mm zmanjša na vrednost 0,5 N, neodvisno od premera nastavljive šobe. Iz tega je mogoče sklepati, da v krmilni celici ostaja še vedno tlak v vrednosti okrog 1 bar. Ugotovimo tudi, da je največja sila pri premeru nastavljive šobe 3 mm pod 4,5 N. Pri manjših premerih šob se sila tlaka na odbojno ploščo zmanjšuje. Na potek preklapljanja ventila neposredno vplivata obe šobi v predkrmilnem ventilu ter odmik zaslonke h. Raziskava je pokazala, da na delovanje pnevmatičnega potnega ventila značilno vpliva razmerje med premerom nespremenljive šobe in premerom spremenljive šobe. Izpolnjen mora biti pogoj, da je d. < d.. Številne meritve so pokazale, da je za dani primer najugodnejše delovanje piezopremikala pri premeru d. = 1,6 mm in d. = 2,5 mm do dn. = 3 mm ([12] in [13]). Za določitev piezoelementa in njegovih značilk je treba opredeliti še razmere pri praznjenju in polnjenju krmilnih celic glavnega ventila v povezavi s premikom odbojne plošče h in sile F s katero deluje curek zraka pri iztoku iz d. na mostični piezoelement. force or pressure in the control chamber on the other side of the piston of the main valve. Fig 12 shows the measured results of the pressure, pk, or force, F, versus the flapper displacement, h, from the adjustable nozzle at various diameter ratios of both nozzles (kš and nš). The successive changes of the force, F, at various diameters of the adjustable nozzle are characteristic and show the dependence of force, F, on the flapper displacement, h. The results show that the force, F, starts falling at the flapper displacement h = 0.1 mm (Fig 12). The rate at which the force is falling increases with increasing nozzle diameter. At the displacement h = 0.4 mm, the force, F , falls to 0.5 N, irrespective of the diameter of the adjustable nozzle. From this we can conclude that in the control chamber there still remains a pressure of about 1 bar. It can also be seen that the maximum force at the adjustable nozzle diameter 3 mm is below 4.5 N. At smaller nozzle diameters the pressure acting on the flapper is falling. The switchover of the valve is directly dependent on the sizes of both nozzles in the pilot-stage valve and the flapper displacement, h. Our research has shown that the operation of a pneumatic directional valve is largely influenced by the ratio between the constant nozzle diameter and the adjustable nozzle diameter. For a successful operation the condition d < d. has to be fulfilled. Numerous measurements have shown that for the studied case, the actuator operates best at a diameter d = 1,6 mm and d. = 2,5 mm to dn. = 3 mm ([12] in [13]). nSTo determine the piezo element characteristics, it is also necessary to measure the relationships between the supply and relief of the main chamber and the flapper displacement, h, and force, F, with which the air jet acts at the nozzle outlet (dn") onto the crossbow piezo element. Analiza delovanja pnevmatičnega ventila - Analysis of the Operation of Pilot-Stage 845 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 Sl. 13. Pnevmatični ventil ISO 3, 5/2 s predkrmilnim piezoventilom Fig 13. Pneumatic valve ISO 3, 5/2 with a pilot-stage piezo-actuator. Dana meritev je temelj za izbiro piezoelementa in določitev njegove najboljše delovne točke. Sila piezoelementa mora biti večja od sile tlaka. Temelj za izbiro piezoelementa sta tako potreben gib h oziroma deformacija piezoelementa in sila, ki jo mora piezoelement zagotavljati. Sklepati je mogoče, da mora upogibni piezoelement omogočati gib med 0,1 in 0,4 mm in zagotavljati silo, večjo od 5 N, če želimo doseči zanesljivo delovanje ventila. Na podlagi rezultatov uvodne raziskave je bil izbran piezoelement [16] z največjim upogibom Dx = 250 mm, z največjo silo F = 35 N pri U = 200 V enosmerne napetosti. Na sliki 13 je prikazan prirejen glavni potni ventil velikosti ISO 3, 5/2 s prigrajenim predkrmilnim piezoventilom. Raziskava možnosti uporabe piezoizvršilnikov z upogibnimi elementi je pokazala, da je za krmiljenje 5/2 potnega ventila mogoče uporabiti na novo razviti predkrmilni ventil z upogibnim piezoelementom, ki deluje po načelu šoba - odbojna plošča. V predkrmilnem ventilu sta uporabljena dva mostična piezoelementa (sl. 3c, 9b in 13) ([16] in [17]). Vod napajalnega tlaka p je speljan iz glavnega napajalnega priključka glavnega potnega ventila. Napajalni zrak tako teče skozi nespremenljivo šobo do spremenljive šobe. V mirujoči legi sta oba piezoelementa neobremenjena, zrak teče iz obeh šob v ozračje. Pri vklopu enega izmed predkrmilnih ventilov se piezoizvršilnik upogne in se premakne proti šobi za premik h. Zaradi oviranja iztekanja zraka v krmilnem prostoru se tlak pk zviša. Pri ustrezno veliki sili tlaka v krmilnem prostoru se ventil premakne v novo lego. 4 RAZULTATI MERITEV IN ANALIZA 4.1 Preizkuševalce This measurement serves as the basis for the selection of the piezo-element and the determination of its optimal operating point. The force of the piezo-element has to be greater than the force resulting from the pressure. The basis for the selection is thus the necessary displacement, h, or the deformation of the piezo-element and the force it has to ensure. It is possible to conclude that a bending piezo-element has to enable a deflection of between 0.1 to 0.4 mm and ensure a force greater than 5 N for reliable operation. On the basis of the results of our preliminary research, a piezo-element was selected with a maximum deflection of Dx = 250 mm, maximum force F = 35 N at U = 200 V DC. Fig 13 shows the main directional valve of size ISO 3, 5/2 with a pilot-stage piezo-actuator. The research into the possibilities of using piezo-actuators with bending elements has shown that for controlling a 5/2 directional valve it is possible to use the discussed crossbow piezo-element with a flapper/nozzle system. In the pilot-stage two such crossbow elements were used (Fig 3c, 9b and 13) ([16] in [17]). The supply pressure, pn, is led from the main power supply on the main directional valve. The supply air flows through the constant nozzle to the adjustable nozzle. At standstill both piezo-elements are unloaded and the air flows from both nozzles into the atmosphere. When switching on one of the pilot-stage valves, the piezo-actuator bends and moves towards the nozzle by a displacement h. Due to greater blockage of the air outflow, the pressure, pk, in the chamber rises. When the pressure in the chamber is big enough, the valve is switched over into a new position. 4 ANALYSIS OF MEASUREMENT RESULTS 4.1 Test rig Za potrebe izvedbe meritev značilk ventila je bilo postavljeno preizkuševališče, kar je prikazano A test rig was set up to carry out the measurements of the valve’s characteristics. A photograph of the rig is 846 HerakovičN. - Noe D. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 a) b) Krmilnik optičnega zaznavala / Optical sensor controller \ Merilni ojačevalnik / Amplifier HBM Piezoizvršilnik / Piezoactuator O,™zo - Elektromagnet / Electromagnet Ojačevalnik/ Amplifier 5V/24V Optično zaznavalo / Optical sensor Tlačno zaznavalo / Pressure sensor 2 x Ventil / Valve ¦^- Večfunkcijska kartica / -*j Multifunction card NI 6024E Osebni računalnik / PC LabVIEW Sl. 14. a) Preizkuševalce, b) shema merilne verige Fig. 14. a) A photograph of the test rig b) block diagram of the measuring chain na sliki 14a. Verižna shema merilne verige je prikazana na sliki 14b. Poteki tlakov v obeh celicah so bili merjeni s piezomerilniki tlakov ([13] in [14]), lega gibljivega dela glavnega ventila pa s svetlobnimi mejnimi stikali, ki omogočajo zaznavanje le končnih leg bata ventila, medtem ko vmesnih leg ni mogoče meriti. Potek preizkušanja je bil voden in rezultati so vrednoteni v okolju LabVIEW. 4.2 Analiza delovanja Temelj za uspešen razvoj pnevmatičnega ventila s piezoventilom je potrebni potek tlaka v obeh krmilnih celicah pri različnih premerih nespremenljive shown in Fig 14a, while a block diagram of the measuring chain is presented in Fig. 14b. The histories of the pressures in both chambers were measured with piezo pressure gauges [13, 14], and the position of the moving part of the main valve with light limit switches, sensing only the final positions of the valve piston, while the intermediate positions could not be measured. The testing procedure was guided and the results evaluated in the LabVIEW environment. 4.2 Analysis of operation The basis for the successful development of a pneumatic valve with a piezo-actuator is to be able to produce the required continuous changes of pres- Analiza delovanja pnevmatičnega ventila - Analysis of the Operation of Pilot-Stage 847 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 in spremenljive šobe pri napajalnem tlaku p = 6 bar. Rezultati meritev potekov tlakov pri običajnem elektromagnetnem ventilu so prikazani na sliki 15. Poteki tlakov so značilni in kažejo, da se v krmilni celici pri preklopu tlak pk na napajalni (tlačni) strani najprej hitro zviša, nato pa zniža in ob koncu preklopa ventila ponovno zviša na vrednost, ki je enaka napajalnemu tlaku. Sočasno se na drugi, odzračeni strani, tlak p hitro zniža in ob koncu preklopa doseže vrednost pod pkD = 1 bar, praktično pkD = 0 bar. Iz diagrama je mogoče določiti tudi čas zakasnitve po vklopu predkrmilnega ventila, čas preklopa predkrmilnega ventila t , čas preklopa glavnega ventila t ter skupni čas preklopa *. Čas preklopa določa frekvenco ventila. p Na velikost potrebne sile za preklop in s tem krmilnega tlaka - pkL, vplivajo masa gibljivega dela ventila m, sila trenja F ter višina tlaka na nasprotni premakn večja od smeri gib N tlakov v predkrmi p = 5 bar hitro zvi ponovno nasprotn v celici sure in both chambers at different diameters of the constant and adjustable nozzles at the supply pressure pn = 6 bar. The results of the measurements of pressure history in the standard electromagnetic valve are shown in Fig 15. The pressure histories are characteristic and show that on switchover in the control chamber the pressure, pkL, on the supply side (pressure side) first rises steeply and then falls, and then at the end of the switchover rises again to a value equal to the supply pressure. Simultaneously, on the other air-relief side, the pressure, pkD, falls and at the end of the switchover reaches a value below pkD = 1bar, practically pkD = 0 bar. From the graph we can also determine the switchover time of the pilot-stage valve, tpv, the switchover time of the main valve, tgv, and total switchover time, tp. This defines the frequency of the valve. The magnitude of the force necessary for switching-over, i.e., the control pressure, pkL, depends on the mass of the moving part of the valve, m, friction force, F tr, alve, pkD. w position r that the ction. ments of he devel-e supply e typical. eeply and again to ite side is the value 6,0 4,0 2,0 0 0 Krmilni impulz / Start tz 0,05 tpv 0,1 tgv tp 0,15 0,2 Čas /Time [s] 0,5 0 848 1 6,0 4,0 2,0 Krmilni impulz / Start impulise Ž pkL Končna Ilega desno Right end position i 1 Končna lega levo l—Left end position 1 i/ pkD^ 0,5 0,1 0,2 0,3 0,4 Čas /Time [s] 0 Sl. 16. Odvisnost posameznih časov od višine tlaka pri ventilu s piezopogonom Fig. 16. Characteristic times versus pressure values in a piezo-actuator valve pkD = venti siste pona bata zakas (sl. 1 časi napaj napa prekl Mejno število preklopov prikazanega ventila je največje število gibov, ki jih gibljivi del ventila lahko opravi ob največji amplitudi, to pomeni, da se popolnoma premakne iz začetne v končno lego in nazaj. Za opazovani ventil s piezopogonom je mejna frekvenca 4,35 Hz in je 5 bar, e with e rela-times al into nd the lay an times, wever, es not The limiting number of switchovers of the developed piezo-valve represents the highest number of strokes that the moving part of the valve can perform at maximum amplitude, i.e., the fully completed motion from the initial into the final position and back. The piezo-drive valve under observation had a limit frequency of 140 120 100 80 60 40 20 0 pr ek lop LEV O / switch to lef t -•-------preklop DESNO / switch to right -*------gib LEVO / movement left A------gib DESNO / movement right #------zakasnitev LEVO / delay left H------zakasnitrev DESNO / delay right 2,5 3 3,5 4 4,5 5 5,5 6 6,5 Tlak / Pressure [bar] Sl. 17. Odvisnost posameznih značilnih časov ventila pri različnih napajalnih tlakih Fig 17. Characteristic times of the valve versus supply-pressure values 1 0 0 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 bila določena iz odziva ventila na vhodni pravokotni signal. Seveda na mejno frekvenco poleg pogonskega dela vplivata še masa gibljivega dela ventila, trenje med gibljivimi in negibljivimi deli ventila ter pretočne in tlačne sile v ventilu. 5 SKLEP Razvoj pnevmatičnih komponent v svetu terja nenehno optimiranje sedanjih in tudi razvoj novih komponent. Eno izmed možnosti razvoja novih ventilov in optimiranja sedanjih je vsekakor uporaba drugačnih, predvsem piezoizvršilnikov za krmiljenje ventilov. Na podlagi teoretičnih izhodišč in meritev pnevmatičnega ventila je mogoče sklepati, da uporaba upogibnih piezoelementov v predkrmilnih potnih ventilih ponuja vrsto prednosti pred elektomagnetnimi izvršilniki. Poleg konzolnih upogibnih izvršilnikov je mogoče uporabiti za preklapljanje ventila tudi mostično izvedbo v sistemu šoba odbojna plošča, kar je bilo izvedeno tudi v tej raziskavi. Razviti predkrmilni izvršilnik je prigrajen k običajnemu potnemu ventilu velikosti ISO 3, 5/2, kar omogoča tudi primerjave med elektromagnetnim in piezoizvršilnikom. Meritve so pokazale, da ima prototipna izvedba ventila preklopne frekvence v mejah, kakršne imajo primerjalni elektromagnetni ventili. Mejne frekvence so dovolj velike tudi za hitre delovne kroge. Ustrezne konstrukcijske rešitve in integracija napajalne elektronike bo v nadaljevanju omogočila gradnjo manjših in hitrejših ventilov. 4.35 Hz, which was determined from the valve response to the rectangular signal. It is, of course, understood that, apart from the driving part, the limit frequency is influenced by the mass of the moving part of the valve, the friction between the moving and inert parts of the valve, the flow force and the pressure in the valve. 5 CONCLUSION Trends in the development of fluid power systems dictate continuous efforts to optimise the existing components and develop new ones. One of the possible developments in this area is the use of alternative piezo-actuators for piloting valves. On the basis of a theoretical study and experimental measurements it is possible to conclude that the use of bending piezo elements in the pilot-stage of directional valves offers a number of advantages over electromagnetic actuators. Besides the cantilever configuration, also the crossbow flapper nozzle design can be successfully used, which was also realized in this research. The developed pilot-stage actuator is a unit built to the standard directional valve ISO 3 5/2, offering a comparison between the electromagnetic and piezo-actuators. The measurements have shown that the prototype design has switchover frequencies ranging within the limits that are also characteristic for electromagnetic valves. The limit frequencies are high enough to allow for rapid working cycles. In future, better design solutions and the integration of power-supply electronics will enable smaller valve sizes and faster operation. 6 LITERATURA 6 REFERENCES [1] HerakovičN., Noe D. (1998) Nova generacija energetsko varčnih pnevmatičnih ventilov. Ventil, letnik 4, št. 4, str. 210-214. [2] O+P-Gesprachsrunde (2002) Aktoren fur Ventile und ihre Peripherie, O+P Olhydraulik und Pneumatik, 46 (2002) Nr. 11-12, Str. 640-689 [3] HerakovičN. (1999) Die Untersuchung des Nutzung des Piezoeffektes zur Ansteuerung fluidtechnischer Ventile, Verlag Meinz, Aachen. [4] Matten N., Ohmenhauser M. (1994) Stetigwirkendes Wegeventil mit piezoaktuatorischem Stellelement. Olhydraulik und Pneumatik, Vol. 38, Nr. 6, Seite 350-355. [5] HerakovičN. (1995) Piezoaktorbetatigung fur ein einstufiges hochdynamisches Servoventil. Olhydraulik und Pneumatik, Vol. 39, Nr. 8, Seite 601-605. [6] Murenhoff H. (2003) Trends in Ventilentwicklung, O+P Olhydraulik und Pneumatik, Nr. 47 (2003), Str. 226-241. 850 HerakovičN. - Noe D. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 835-851 [7] Wennmacher G, Yamada H. (1993) Prototyp eines dynamischen Schnellschaltventils mit piezoelektrischer Ansteuerung. Olhydraulik und Pneumatik, Vol. 37, Nr. 10, Seite 794-797. [8] N.N. Physik Instrumente (PI) Katalog. [9] N.N. Piezooxide (PXE), Eigenschaften und Aufwendungen, Valvo, Philips 1.Auflage, Juli 1988. [10] Loesel M. (2003) Klein- und Versteuerventile, Steuerung neutraler und aggressiver Fluide, O+P Olhydraulik und Pneumatik, 47 (2003) Nr. 5, Str. 360-362. [11] http://users.telenet.be/educypedia/electronics/sensorspiezo.htm [12] Kezele N. (2001) Določitev karakteristik pnevmatičnih potnih ventilov. Diplomska naloga, Fakulteta za strojništvo, Univerza v Ljubljani, Ljubljana. [13] Slamnik G. (2002) Razvoj krmiljenja pnevmatičnega ventila s piezoaktuatorjem. Diplomska naloga, Fakulteta za strojništvo, Univerza v Ljubljani, Ljubljana. [14] Fabjan R. (2003) Določitev dinamičnih karakteristik pnevmatičnih potnih ventilov. Diplomska naloga, Fakulteta za strojništvo, Univerza v Ljubljani, Ljubljana. [15] www.face-int.com/pdf/whtpaper.pdf [16] Podatki podjetja Ceramtec [17] Katalog podjetja TIO [18] Hoerbiger katalog Naslov avtorjev: doc. dr. Niko Herakovič prof. dr. Dragica Noe Univerza v Ljubljani Fakulteta za strojništvo Aškerčeva 6 1000 Ljubljana niko.herakovic@fs.uni-lj.si dragica.noe@fs.uni-lj.si Authors’ Address: Doc. Dr. Niko Herakovič Prof. Dr. Dragica Noe University of Ljubljana Faculty of Mechanical Eng. Aškerčeva 6 1000 Ljubljana, Slovenia niko.herakovic@fs.uni-lj.si dragica.noe@fs.uni-lj.si Prejeto: Received: 7.3.2006 Sprejeto: Accepted: 22.6.2006 Odprto za diskusijo: 1 leto Open for discussion: 1 year Analiza delovanja pnevmatičnega ventila - Analysis of the Operation of Pilot-Stage 851 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 UDK - UDC 681.511:517.9 Izvirni znanstveni članek - Original scientific paper (1.01) Programljivi logični krmilniki na temelju rešitve algebraične Riccatijeve enačbe Programmable Logic Controllers Based on the Algebraic Riccati Equation Solution Primož Podržaj - Zoran Kariž (Fakulteta za strojništvo, Ljubljana) V prispevku je predstavljena sinteza podoptimalnega krmilnika, ki sloni na rešitvi algebraične Riccatijeve enačbe (ARE). Predstavljen je numerični postopek, s katerim lahko pridemo do te rešitve. V praksi se parametri krmiljenega sistema mnogokrat precej razlikujejo od tistih v ARE. V tem primeru sta vprašljivi optimalnost in celo stabilnost krmilnega sistema. Zelo uporabno bi torej bilo, če bi lahko zasnovali prilagodljivi linearni podoptimalni krmilnik. Tak krmilnik bi bil zmožen odkriti spremembe v parametrih sistema in prirediti parametre. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: sistemi prilagodljivi, enačbe Riccatijeve, krmilniki logični, krmilniki programljivi, algoritmi numerični) This paper deals with the synthesis of a suboptimal controller, based on the solution of the algebraic Riccati equation (ARE). The numerical procedure for obtaining the solution is presented. In applications the controlled system parameters often differ from the ones used in the ARE. In this case the optimality of the control system and even its stability are questionable. Therefore, it would be very useful to design an adaptive linear suboptimal controller. Such a controller should be able to detect changes in the system parameters and adjust its parameters. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: adaptive systems, Riccati equations, programmable logic controllers, numerical algorithms) 0 UVOD Linearni časovno neodvisni sistem (LČN) z diferencialno enačbo stanj: 0 INTRODUCTION A linear time-invariant (LTI) system with the state differential equation: x&(t) = Ax(t) + Bu(t) (1), prikazan na sliki 1 je optimalen, če mu je dodana povratna zveza: shown in Figure 1 is said to be optimal if the feedback: u(t) = -F(t)x(t) u B —H ¦) x J x '+ A Sl. 1. LČN sistem Fig. 1. LTI system 852 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 tako, da je integralski kriterij: is added in such a way that the following cost functional: t1 J [xT (t)R1 (t)x(t) + uT (t)R2 (t)u(t)] -dt + xT (t1 )P1x(t1 ) najmanjši. Pri tem so: • R1(t) in R2 (t) pozitivno definitni simetrični matriki za t = t = t1, . P pozitivno poldefinitna simetrična matrika. Pokazati se da [1], da je treba za minimizacijo funkcionala rešiti naslednjo matrično Riccatijevo enačbo (MRE): is minimized. This means that . R1(t) and R2(t) are positive definite symmetrical matrices for t0 = t = t1 . P is a positive semi-definite symmetrical matrix It can be shown [1], that in order to minimize the cost functional the following matrix Riccati equation (MRE) has to be solved: & P(t) = R1 - P(t)BR2- 1BT P(t) + P(t)A+ AT (t)P Če rešitev MRE konvergira, potem je limita: If the solution of the MRE converges, then the limiting solution: P = limP(t) rešitev naslednje algebraične Riccatijeve enačbe (ARE): is the solution of the following algebraic Riccati equation (ARE): R - PBR-1BT P + PA+ AT P = 0 Krmilnik, ki je zasnovan na temelju te rešitve, se imenuje podoptimalni krmilnik. 1 OPIS PROBLEMA Podoptimalni krmilniki se veliko uporabljajo. Dejstvo pa je, da se lastnosti krmiljenega objekta, dane z matriko A, spreminjajo s časom zaradi staranja in spremenljivih delovnih razmer, čeprav se predpostavlja, da so časovno neodvisne. Matrika A je lahko časovno neodvisna, če opazujemo tipični čas prehodne funkcije krmilnega sistema, toda je časovno odvisna med dobo trajanja sistema. Če se dejanska matrika stanja AR razlikuje od tiste, ki smo jo uporabili v ARE (matrika A) za D, se lahko pojavijo težave. AR = A+D Zaradi tega se izgubi podoptimalnost sistema. V mnogih primerih je vprašljiva celo njegova stabilnost. Zaželeno bi torej bilo, da se zgradi krmilnik, ki bi lahko: . določil dejansko matriko stanja AR, . rešil ARE na podlagi dejanske matrike stanja AR. The controller based on this solution is called a suboptimal controller. 1 PROBLEM FORMULATION Suboptimal controllers are frequently used. However, it is a fact that the properties of the controlled system given in matrix A change over time, due to ageing and different working conditions etc., even though they are assumed to be time-invariant. The matrix A may be time-invariant if we consider a typical transient response time of a system, but it is time variant during the lifetime of the system. Many problems can be encountered if the real system matrix, AR, which differs from the one used in the ARE (matrix A) by the amount D, is introduced. d ¦¦¦ d Consequently, the system’s suboptimal performance is lost. Even its stability may be questionable in many cases. Therefore, it is desirable to design a controller capable of: . determining the real system matrix, A, . solving the ARE considering the real system matrix, AR, Programljivi logični krmilniki - Programmable Logic Controllers 853 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 2 DOLOČEVANJE DEJANSKE MATRIKE STANJA AR Celoten postopek določevanja dejanske matrike stanja A R je razdeljen na dva koraka, in sicer v odvisnosti od tega ali matrika stanja AR ni poznana (prvi korak), ali pa je vsaj približno poznana (drugi korak). 2.1 Prvi korak Če v enačbi (1) uporabimo dejansko matriko stanja v prvem koraku, dobimo: 2 DETERMINING THE REAL SYSTEM MATRIX, AR The whole procedure for determining the real system matrix, AR, is divided into two steps, depending on whether the system matrix, AR, is not known at all (first step) or is approximately known (second step). 2.1 First step When the real system matrix, AR1, is used in the first step, Equation (1) implies the following equality: x&(t) - Bu(t) = AR1x(t) (2). Matrika stanja AR1 pomeni linearno preoblikovanje vektorskega prostora 5R n v vektorski prostor M n in je enolično določena z baznimi vektorji prostora W. Torej je za določitev elementov sistemske matrike treba najti n linearno neodvisnih vektorjev stanj in njihove slike. Predpostavimo, da lahko merimo vse komponente vektorja stanj x in da je med fazo branja vhodni signal signal u enak 0 (sl. 2). Uporabljena je bila tudi predpostavka, da je mogoče izračunati odvode vseh komponent vektorja stanja. Postopek, ki ga lahko uporabimo v ta namen, je opisan na primer v [8] in [9]. Levo stran enačbe (2) lahko torej določimo za vsak čas t. Recimo, da krmilnik med fazo branja (u=0) za nek čas vzorčenja t prebere x(0+), x(T), x(2T) in tako naprej, dokler ne najde n linearno neodvisnih vektorjev stanj x1 , x2 , ... , x . Potem lahko oblikujemo naslednji matriki: The system matrix, AR1, represents a linear trans-formation of the space 9T into the space 9T and is uniquely determined by the images of the vectors on the basis of 9T. Therefore, in order to determine the elements of the system matrix we only need to find n linearly independent state vectors x and their images. Let us suppose that all the components of the state vector x can be measured, and that during the acquisition phase the control signal u equals 0 (Fig. 2). Another supposition, that the derivatives of all the components of the state vector x can be calculated, was also made. The procedure, which might be used in this case, is given for example in [8] and [9]. Therefore the left-hand side of Equation (2) can be determined at any time t. Now, let us say that the controller reads x(0+), x(T ), x(2T) and so on during the acquisition phase (u=0) for a specified sampling time TS, until n linearly independent state vectors x1 , x2 , ... , x are found. Then, the following matrices can be formed: X& =[x&1 x&2 L x&n] ; X=[x1 x2 L xn] Dejansko matriko stanja AR1 lahko torej določimo iz naslednje enačbe: x f x • J '------------- A R1«—' Sl. 2. Prvi korak, faza branja Fig. 2. First-step acquisition phase Therefore, the real system matrix, AR1, can be determined by the following equation: B ~+ "^ » r x "V A R1 -F1 Sl. 3. Prvi korak, delovna faza Fig. 3. First-step working phase 854 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 B +^ x —+—* J x AB2 -F, Sl. 4. Drugi korak, faza branja Fig. 4. Second-step acquisition phase B +^> x J x \ i '+ A R2 -F2 Sl. 5. Drugi korak, delovna faza Fig. 5. Second-step working phase X-X- Z določeno matriko stanja A 1 lahko pridemo do rešitve Riccatijeve enačbe. Njena rešitev P1 se nato uporabi za določitev ustrezne povratne matrike F1. S tem se faza branja konča. Sistem brez povratne zveze je spremenjen v sistem s povratno zvezo, prikazan na sliki 3. Ta postopek je treba izvesti le prvič, ko uporabimo tak tip krmilnika na sistemu, o katerem ne vemo ničesar, razen razsežnosti vektorja stanj. Če je matrika stanja AR (A R1) vsaj približno poznana, potem je pametneje uporabiti povratno matriko F1, ki ustreza tej matriki stanja in nadaljevati z drugim korakom. 2.2 Drugi korak Ko je matrika stanja določena (ali pa je uporabljena približna), je treba analizirati sistem s povratno zvezo, prikazan na sliki 4. Med fazo branja je uporabljena povratna matrika F iz prvega koraka. Dejanska matrika stanja je tekoča in ima zato indeks 2. Sistem opišemo z diferencialno enačbo: After the system matrix, AR1, is determined, the Riccatti equation can be solved. Its solution P1 is then used to find the appropriate feedback matrix, F1. At this moment the acquisition phase is ended and the working phase starts. The open-loop system is changed to the closed-loop system shown in Figure 3. But this procedure only needs to be implemented the first time we use this type of controller on a system we know nothing about, except the number of states. If the system matrix, AR ( A R1), is known approximately, then it is better to use the feedback matrix F1 associated with this system matrix and proceed immediately to the second step. 2.2 Second step Once the system matrix has been determined (or an approximate matrix has been used), the closed-loop system shown in Figure 4 should be analyzed. During the acquisition phase the feedback matrix F1 from the first step is used. The real system matrix, however, is current and therefore has index 2. The system is described by the differential equation: x&(t) = ( AR2 + BF1 )x(t) Krmilnik mora tako kot v prvem koraku prebrati x(0+), x(T), x(2T) in tako naprej. Nato sestavimo matriki X in X. Realno sistemsko matriko v drugem koraku AR2 lahko po končani fazi branja v drugem koraku določimo na podlagi naslednje enačbe: The controller has to read x(0+), x(TS), x(2TS) and so on, as in the first iteration. Then the matrices X and X& can be formed. The real system matrix in the second step, AR2, can then be determined after the acquisition phase of the second step is ended, according to the following equation: & XX-1 - BF Ko je nova povratna matrika F2 določena, se lahko začne delovna faza v drugem koraku. Končni blokovni diagram je prikazan na sliki 5. Drugi korak se nato iterativno ponavlja. After the new feedback matrix F2 is determined the working phase of the second step can begin. The associated block diagram is shown in Figure 5. The second step is then iteratively continued. Programljivi logični krmilniki - Programmable Logic Controllers 855 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 3 REŠEVANJE ARE Uporabljeni sta bili dve metodi reševanja ARE, in sicer: . Metoda z uporabo enačb Ljapunova, .Potterjeva metoda. 3.1 Metoda z uporabo enačb Ljapunova Ta metoda se je izkazala za dokaj počasno in zato ne bo opisana. Podroben opis je podan v [6] in [7]. 3.2 Potterjeva metoda Ta metoda je bila uporabljena za reševanje ARE. Sestavimo matriko M reda 2x2 [1]: M A -R Matriko R1 lahko zapišemo v obliki: 3 SOLUTION OF THE ARE Two methods for solving the ARE were considered: . a method using Lyapunov equations . Potter’s method 3.1 Method using Lyapunov equations This method turned out to be rather slow and will therefore not be described. A detailed description can be found in [6] and [7]. 3.2 Potter’s method This method was used in order to solve the ARE. Let us consider the following 2x2 matrix M [1]: -B-R-1-BT The matrix R1 can be expressed as: Če je matrična dvojica: . (A,B) ustaljiva in . (A,H) pregledljiva, R=H-HT If the matrix pair: .(A,B) is stabilizable . (A,H) is observable Im Im Im Im Re Re Re Re Sl. 6. Mogoče razporeditve lastnih vrednosti matrike reda 6x6 Fig. 6. Possible eigenvalue locations of a 6x6 matrix 856 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 potem se vse lastne vrednosti matrike M pojavljajo v dvojicah (Ai,-A). Nekaj mogočih primerov je prikazanih na sliki 6. Naj bo lastni vektor ali korenski vektor yi matrike M, ki ustreza lastni vrednosti Xi z negativno realno komponento, oblike: Potem se rešitev ARE dobi z [5]: yi P =[v ,L,v then all the eigenvalues of matrix Mr appear in pairs (li,-li). Some possible examples are shown in Figure 6. Let the eigenvector or generalized eigenvector yi of the matrix Mr corresponding to li, which has a negative real part, be: Then, the solution of the ARE is given by [5]: [] u ,L,u -1 3.2.1 Izračun lastnih vrednosti in lastnih vektorjev Da pridemo do rešitve ARE, je najprej treba dobiti lastne vektorje matrike M. Ta postopek se največkrat izvede v štirih korakih, in sicer: .z balansiranjem matrike [4], . s preoblikovanjem matrike v Hessenbergovo [3], . s QR algoritmom [3], . z izračunom lastnih vektorjev [3]. 4 ZGRADBA SISTEMA 4.1 Zgradba krmilnika Uporabljen je bil programljivi logični krmilnik SIEMENS S7-300 s funkcijskim modulom FM 356-4 in dodatnimi vhodno izhodnimi enotami (slika 7). 1. napajalnik / power supply 2. CPU 314 IFM 3. FM 356-4 funkcijski modul FM 356-4 function module 4. vhodno-izhodne enote interface modules 5. osebni računalnik PC 6. vmesnik za osebni računalnik PC adapter 7. spominska kartica memory card 8. programirna naprava programming device (Js 3.2.1 Calculation of the eigenvalues and eigenvectors In order to solve the ARE we must find the eigenvectors of the M. This procedure is usually done in four steps, by: r • matrix balancing [4], . reducing a matrix to the Hessenberg form [3], . performing the QR algorithm [3], . computing the eigenvectors [3]. 4 SYSTEM DESIGN 4.1 Controller design As an application of the introduced method the controller was built using the SIEMENS S7-300 programmable logic controller along with the FM 356-4 function module and several interface (input and output) modules (Figure 7). 3) (3) 1 7^ CD @ • j © S f (D / Sl. 7. Programljivi logični krmilnik z osebnim računalnikom Fig. 7. PLC with supporting PC Programljivi logični krmilniki - Programmable Logic Controllers 857 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 Spominska kartica EPROM se uporablja za vgradnjo in zagon operacijskega sistema ter za nalaganje uporabniškega programa. Za vgradnjo opravilnega sistema na spominsko kartico je potrebna posebna programirna naprava. Uporabniški program pa se lahko spreminja in shranjuje z uporabo osebnega računalnika in vmesnika zanj, ki je priključen na osrednjo procesno enoto (CPU). Program se naloži v CPU, ki je prek vodila (BUS) povezan s funkcijskim modulom (FM). 4.2 Modeliranje krmiljenega sistema A flash EPROM memory card is used to install and boot the operating system and to load the user software. A special programming device is needed to install the operating system on the memory card. The user software, on the other hand, can be modified and saved with the PC along with the appropriate PC adapter, which is connected to the CPU. The software is downloaded to the CPU, which is connected to the function module (FM) via a backplane BUS. 4.2 Controlled system modelling Za simulacijo krmiljenega objekta je bil A Meda 41TC iterative analogue computer uporabljen iterativni analogni računalnik Meda 41TC. was used to simulate the controlled system. 5 REZULTATI Kot primer je prikazan sistem z matrikama: 5 RESULTS As an example, a system with the following matrices: in začetnim pogojem: Matriki R1 in R2 v integralskem kriteriju sta: 114 231 425 x(0) 312 1 21 212 (3) and the initial condition: Odgovor prilagodljivega podoptimalnega krmilnika (tri komponente vektorja stanj) v prvem koraku za Ts=0,1 je prikazan na sliki 8. Dejanska matrika stanja A je bila na analognem računalniku namerno modelirana tako, da je sistem brez povratne zveze nestabilen. Zaradi tega je na sliki 8 jasno vidna faza branja. Rezultate lahko primerjamo z odgovorom linearnega podoptimalnega krmilnika, katerega povratna matrika F ustreza matriki AR (sl. 9). Očitno je, da je v primeru, ko je povratna matrika F v neprilagodljivem krmilniku točna, le-ta boljši od podoptimalnega. Do sprememb pa pride, če se matrika stanja A, za katero je izračunana povratna matrika F, le malo razlikuje od dejanske sistemske matrike AR. Recimo, da je dejanska matrika stanja AR enaka kakor v enačbi (3), povratna matrika neprilagodljivega krmilnika pa je izračunana za matriko A, definirano v enačbi (4). was used. The matrices R1 and R2 in the cost functional were: ; R2=[1] The response of an adaptive suboptimal controller (three components of the state vector) during the first step for Ts=0.1 is shown in Fig. 8. The real system matrix, AR, was deliberately modelled on an analogue computer in such a way that the open-loop system is unstable. Therefore, the acquisition phase can clearly be seen in Fig. 8. These results can be compared to the response of a linear suboptimal controller whose feedback matrix F is associated with the matrix AR (Fig. 9). It is obvious that an adaptive controller does not perform as well as a non-adaptive controller if the feedback matrix in the non-adaptive controller is accurate. The situation changes, however, if the system matrix A with which the feedback matrix of an non-adaptive controller is associated differs from the real system matrix, AR, only slightly. Let us say that the real system matrix, AR, is the same as in Eq. (3), but the non-adaptive controller ’s feedback matrix is associated with the matrix A defined in Eq. (4). 858 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 xi (V) \ 1 iti ~hh" 1\ ^ !"Txi f I \ | j~x^| f vvJ I / j ] j '"¦ čas / time (s) Sl. 8. Odgovor prilagodljivega linearnega podoptimalnega krmilnika Fig. 8. The response of an adaptive linear suboptimal controller xi (V) ________ l~x~l ¦VI i \ < \ \._____. 1 \ Ixl 'ft V .T-------- /VA4- - -\ * xTI _______X^_J^r: ¦^-—1------------------- čas / time (s) Sl. 9. Idealni odgovor neprilagodljivega linearnega podoptimalnega krmilnika Fig. 9. Ideal non-adaptive linear suboptimal controller response Odgovor neprilagodljivega krmilnika za ta primer je prikazan na sliki 10. Sklepamo lahko, da se obnašanje neprilagodljivega linearnega podoptimalnega krmilnika že pri majhnih spremembah parametrov sistema hitro poslabšuje. Kriterijski integral za vse tri primere (prilagodljivi podoptimalni (AS), idealni neprilagodljivi (IN), realni neprilagodljivi (RN)) je prikazan na sliki 11. 1,1 4,1 3,1 1,1 2 4,8 (4). The response of the non-adaptive linear controller for such a case is shown in Figure 10. We can conclude that with only a slight change in the system parameters the performance of the non-adaptive linear suboptimal controller deteriorates rapidly. The cost functionals for all three cases (adaptive suboptimal (AS), ideal non-adaptive (IN), real non-adaptive (RN)) are shown in Figure 11. 60 50 40 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 9 10 60 50 40 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 9 10 Programljivi logični krmilniki - Programmable Logic Controllers 859 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 xi (V) L ------- J ± _L i 1 "^. ---]---f--- ¦-. r r r i^ ~™ 1 - — -1 1 —. 1 ----------—_ 1 ------- ------- ------- 01 23456789 10 čas / time (s) Sl. 10. Realni odgovor neprilagodljivega linearnega podoptimalnega krmilnika Fig. 10. Real non-adaptive linear suboptimal controller response 1 Kriterijski integral 1 l S | jf-~ i RN ------ )' ¦\- - r/ X s|aS1 Tf / -\- -----1------ —[- ------- ------ \ —r~A— ' ''' fINl / / čas / time (s) Sl. 11. Primerjava kriterijskih integralov Fig. 11. Comparison of the cost functionals Jasno je, da se obnašanje prilagodljivega linearnega podoptimalnega krmilnika spreminja v povezavi s časom vzorčenja T, še posebej, če je sistem brez povratne zveze nestabilen, kajti med fazo branja v prvem koraku le-te ni. Slika 12 prikazuje kriterijske funkcionale za različne čase vzorčenja (od T=0,025s do T=0,125s s korakom 0,025s). Jasno je, da je treba za izboljšano obnašanje zmanjšati čas vzorčenja TS. To je mogoče v teoriji in do neke mere tudi na analognem računalniku. V praksi pa lahko pride do problemov, ker se prebrani vektorji stanj zaradi It is to be expected that the performance of an adaptive linear suboptimal controller varies with sampling time, TS, especially if the open-loop system is unstable, because during the acquisition phase of the first step, there is no feedback. Figure 12 shows cost functionals depending on different sampling times (from TS=0.025s to TS=0.125s with a 0.025s step). It is obvious that in order to improve the performance of the controller we must decrease the sampling time, TS. This can be easily done in theory and to some extent on the analogue computer. In practice, on the other hand, problems may occur because the acquired 60 50 40 30 20 0 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 860 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 Kriterijski integral i V i TS = 0,125 s — s*- i i i i i i i V^ i \ TS = 0,025 s čas / time (s) Sl. 12. Primerjava časov vzorčenja Fig. 12. Cost functional versus sampling time šumov ne izražajo v pravi dejanski sistemski matriki AR. Ti učinki se povečujejo, ko gremo s časom vzorčenja proti 0. 6 SKLEP Jasno so bile prikazane prednosti prilagodljivega linearnega podoptimalnega krmilnika pred neprilagodljivim. Te prednosti se povečujejo s povečanimi odstopanji v parametrih sistema. V tem primeru prilagoldjivi krmilnik ohrani svoje zmožnosti. Obnašanje neprilagodljivega krmilnika pa se hitro poslabša. Če matrike stanja ne poznamo, imamo med fazo branja v prvem koraku sistem brez povratne zveze. Obnašanje sistema je torej odvisno od trajanja te faze, le-to pa od časa vzorčenja. Nadaljnje raziskave je torej treba osredotočiti na določevanje optimalnega časa vzorčenja za različne jakosti šuma. state vectors during the acquisition phase do not result in the correct real system matrix, AR, due to the noise and other disturbances. These effects are magnified as the sampling time approaches 0. 6 CONCLUSION The advantages of an adaptive linear suboptimal controller over a non-adaptive controller have been clearly demonstrated. These advantages increase when deviations in the system parameters become larger. In this situation, the adaptive linear suboptimal controller maintains its capability. The performance of a non-adaptive controller, on the other hand, deteriorates rapidly. However, if the approximate system matrix is not known in advance, we are basically operating the open-loop system during the first-step acquisition phase. The overall performance is therefore dependent on the duration of this phase, which further depends on the sampling time. Further research should, therefore, focus on a determination of the optimal sampling time with regard to the presence of different levels of noise. 7LITERATURA 7REFERENCES [1] H. Kwaakernak, R. Sivan (1972) Linear optimal control systems, John Wiley & Sons Inc. [2] F. L. Lewis (1988) Optimal control, John Wiley & Sons Inc. [3] David M. Young, Robert Todd Gregory (1973) A survey of numerical mathematics, Addison—Wesley Publishing Company. [4] J. H. Wilkinson, C. Reinsch (1973) Die Grundlehren der mathematischen Wissenschaften: Linear Algebra, Springer Verlag. x 10 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Programljivi logični krmilniki - Programmable Logic Controllers 861 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 [5] Katsuhisa Furuta, Akira Sano, Derek Atherton (1988) State variable methods in automatic control, John Wiley & Sons Inc. [6] E. J. Davison, F. T. Man (1968) The numerical solution of AQ+QA-C, IEEE Transactions on Automatic Control, August, str.: 448-449. [7] David L. Kleinman (1968) On an iterative technique for Riccati equation computations, IEEE Transactions on Automatic Control, February, str.: 114-115. [8] B.A.BeceKepcKHH, B.B.HapauueB (1987) CncreMH aBTOMaranecKoro ynpaBJiefflia c MHKpo3BM, «Haym» [9] H.H.HBameHKO (1978) ABTOMaranecKoe perynnpoBafflie, «MauiuHocmpoenue» Naslov avtorjev: dr. Primož Podržaj Authors’ Address: Dr. PrimožPodržaj doc. dr. Zoran Kariž Doc. Dr. Zoran Kariž Univerza v Ljubljani University of Ljubljana Fakulteta za strojništvo Faculty of Mechanical Eng. Aškerčeva 6 Aškerčeva 6 1000 Ljubljana 1000 Ljubljana, Slovenia primoz.podrzaj@fs.uni-lj.si primoz.podrzaj@fs.uni-lj.si zoran.kariz@fs.uni-lj.si zoran.kariz@fs.uni-lj.si Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 12.4.2005 25.10.2006 Received: Accepted: Open for discussion: 1 year 862 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 UDK - UDC 621.436:621.43.06 Izvirni znanstveni članek - Original scientific paper (1.01) Raziskava učinkov različnih stopenj vračanja izpušnih plinov na temperaturo plamena in nastanek saj pri uporabi dizelskega goriva z različnimi T90 temperaturami destilacije Experimental Study of the Effects of Different Exhaust Gas Recirculation Ratios on the Flame Temperature and Soot Formation when Using Diesel Fuels With Different T90 Distillation Temperatures Danilo Nikolič1 - Radoje Vujadinovič1 - Norimasa Iida2 (1University of Montenegro, Podgorica, Montenegro; 2 KEIO University, Yokohama, Japan) V študiji, opisani v tem prispevku, smo preizkušali krmiljenje dušikovega oksida (NOx) in nastanek saj. Kot razredčilo, pri simulaciji kroženja izpušnih plinov, smo uporabili ogljikov dioksid (CO 2 ) vsebnosti 4,3%, 9,5 in 14,3%, kar pomeni vsebnosti kisika (O2 ) 20%, 19% in 18%. V nadaljevanju smo uporabili tri različna dizelska goriva z različnimi T90 temperaturami destilacije. Lastnosti goriva smo zavarovali pred vplivi vsebnosti aromatov, žvepla in cetanskega števila. Za simulacijo dizelskega zgorevanja smo uporabili hitrokompresijski motor z enim cilindrom. Postopek vžiga in zgorevanje pri vbrizgavanju dizelskega goriva smo opazovali s pomočjo hitrega neposrednega fotografiranja. Temperaturo plamena (kazalnik nastanka NO) in faktor KL (kazalnik vsebnosti saj v vbrizgu dizelskega goriva) smo analizirali z uporabo dvobarvne metode. Preizkus je pokazal, da se s povečanjem vsebnosti CO2 v dovodu zmanjšata najvišja temperatura plamena in nastajanje saj. Prav tako so rezultati pokazali, da pri vsebnosti CO2 = 4,3% v dovodu, T90 temperatura destilacije nima posebnega vpliva na najvišjo temperaturo plamena in nastanek saj. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: gorivo dizelsko, plini izpušni, nastanek saj, temperatura destilacije) In this paper the diesel in-cylinder control of nitrogen oxide (NOx) and soot formation was tested. Carbon dioxide (CO2 ) was used as a diluent to simulate the exhaust-gas recirculation (EGR) process at ratios of 4.3%, 9.5% and 14.3%, thus making oxygen (O2 ) concentrations of 20%, 19% and 18% respectively. In addition, three diesel fuels with different T90 distillation temperatures were used. The fuel parameters were isolated from the influence of the aromatics content, sulfur content, and cetane number. A single-cylinder rapid compression machine (RCM) was used to simulate the diesel-type combustion. The ignition and combustion processes of the diesel-fuel spray were observed using high-speed direct photography. The flame temperature (an indication of NO formation) and KL factor (an indication of the soot concentration inside the diesel-fuel spray) were analyzed using the two-color method. The study demonstrated that with an increase of the CO2 concentration in the intake charge, the maximum flame temperature and the soot formation decrease. Also, when there was a CO =4.3% concentration in the intake charge, the results showed no significant influence of the diesel-fuel T90 distillation temperature on the maximum flame temperature and the soot formation. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: diesel fuels, exhaust gas recirculation (EGR), distillation temperature, rapid compression machine) 0 INTRODUCTION The environmental impact of motor vehicles is of great concern worldwide. In particular, the con- tribution to atmospheric pollution from motor vehicles points to the need to reduce vehicular emissions. Relative to the spark-ignited internal combustion engine, the diesel engine emits large quantities 863 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 of particulate matter (PM) and nitrogen oxide (NOx). It is difficult to reduce both of these pollutants at the same time because of their trade-off. Diesel emission-reduction strategies can be divided into two main types: in-cylinder control and after-treatment. In-cylinder control implies changes to the design of the engine and diesel-fuel reformulation, with the former providing more opportunities to reduce emissions [1]. Exhaust-gas recirculation (EGR) is an effective in-cylinder technique that reduces NOx because it lowers the maximum flame temperature. But the application of EGR can also adversely affect the quality of the lubricating oil, the engine durability and produce higher unburned hydrocarbon (HC) and PM exhaust emissions, resulting from the lower oxygen (O2) concentration. The influence of EGR on exhaust emissions can be efficiently simulated by the addition of carbon dioxide (CO2) to the intake charge. Ladommatos et al. [2] identified three major effects of introducing CO2 into the intake charge of a diesel engine: the dilution effect, the chemical effect, and the thermal effect. The dilution effect, which represents the reduction in O2 and nitrogen (N2) fractions in the intake charge due to the replacement with CO2, was shown to be the most significant, having an influence on both the combustion process and exhaust emissions. With the increase of CO2 in the intake charge, the ignition delay periods increase ([3] to [7]), the NOx formation decreases ([4] to [7]), while soot formation in some cases increases ([8] and [5]) and in other cases decreases ([4] and [6]). Many studies have been carried out to assess the effect of the fuel properties on diesel emissions. These studies showed that the fuel properties, such as the cetane number, the aromatic content and type, the distillation temperature, the density, and the viscosity, are the most influential on the combustion process and exhaust emissions. Many papers ([9] to [13]) have investigated the influence of the T90 Driving Pistone Piston Cylinder Injection Nozzle 1 Bore : 145mm Stroke : 692mm Compression Ratio : 15.5 Combustion Chamber f 145 x 48 mm Fig. 1. The Rapid Compression Machine distillation temperature of diesel fuels on exhaust emissions. Most of them reported an increase of NOx and PM emissions with the increase in the T90 distillation temperature. The aim of this study was to show the combined effects of the EGR and T90 distillation temperatures of diesel fuel on the formation of NOx and soot inside the combustion chamber. A rapid compression machine (RCM) was used to simulate diesel combustion, having a single fuel-spray injection in the high-temperature and high-pressure atmosphere of the surrounding gas. The RCM is capable of a diesel-fuel spray-combustion investigation with minimized influences of some parameters specific to a high-speed diesel engine. The ignition and the combustion processes of diesel-fuel spray were observed using high-speed direct photography. The flame temperature (an indication of NO formation) and KL factor (an indication of soot formation) were analyzed with the two-color method. The two-color method is based on the continuous radiation of soot particles during the burning diesel-fuel spray. 1 TEST EQUIPMENT AND CONDITIONS The RCM, Figure 1, used in this study for the simulation of the diesel-combustion process is a duplicated single-diesel-type compression cycle after the combustion process, which is carried out in the environment of a constant volume with high temperature and high pressure. The RCM is a pancake-type combustion chamber with a diameter of 145 mm and a thickness of 48 mm, as illustrated in Figure 2. The single fuel spray from one nozzle hole was injected straight down and did not collide with the walls of the combustion chamber. The piston stroke during compression is 692mm. The ratio and the time of compression are 15.5 and 200ms, respectively. Fuel Injection Nozzle Pressure Pick up '(strain gage) Pressure Pick up (piezo) Optical Probe Fig. 2. The RCM combustion chamber 864 Nikolič D. - Vujadinovič R. - Iida N. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 Fuel Injection Combustion Nozzle -^Chamber Safety Enclosure High Speed Camera Mirror - Fig. 3. Optical arrangement The in-cylinder pressure inside the RCM combustion chamber was measured with a piezoelectric sensor. Observations and optical measurements were performed via the quartz windows installed on both the piston and the cylinder head. The image of the flame shape passed through the quartz window was reflected from two plane mirrors (one in the safety box and the other outside) and was caught by the high-speed camera. The speed of the high-speed camera is 5000 flashes per second (FPS) (Model: NAC 16 HD; Shutter constant: 5). The film used was Kodak VISION 500T 7279 (16 mm). Figure 3 shows the optical arrangement of the high-speed photography. The image of a halogen lamp, with a known luminous temperature, was also recorded on each frame as a standard light source for the two-color method analysis. This lamp was positioned at the optical distance from the camera that was equal to the fuel-spray flame. This study involved high-speed direct photography of the luminous flames, a combustion analysis Table 1. Composition and state of the intake charge 400 350 300 r T10 T1 > / T9 K yy ----- 250 200 0 20 40 60 80 90100 Evaporated Fraction % Fig. 4. Distillation curves of the test fuels from pressure diagrams, and flame-temperature and KL-factor analyses by the two-color method applied to the color image of the luminous flames. CO2 was used as a diluent to simulate the EGR process at the ratios of 4.3%, 9.5% and 14.3%, thus making O2 concentrations of 20%, 19% and 18%, respectively. Table 1 shows the composition and the state of the intake charge (gas). Table 2 shows the experimental conditions. In the experiment, three JCAP (Japanese Clean Air Project) fuels were used, T1, T9, and T10, each with a different T90 distillation temperature. The influence of the T90 was isolated from the influences of the aromatics content (0%), the sulfur content (0%), and the cetane number. Some types of n-paraffins and i-paraffins were mixed to keep the cetane number constant. Table 3 shows the main properties of the test fuels, while Figure 4 shows their distillation curves. Figure 5 shows the chromatogram data analyzed by Miwa [14]. The fuels with a higher T90 contain heavier elements. Air + CO2 O2 vol % CO2 vol % N2 vol % Ar vol % Tin K T0 K Pin MPa P0 MPa 21 0 78 1 353 905+5 0.1 3.0+0.1 20 4.3 74.2 0.95 353 895+5 0.1 3.0+0.1 19 9.5 70.5 0.91 353 885+5 0.1 3.0+0.1 18 14.3 66.8 0.86 353 875+5 0.1 3.0+0.1 Table 2. Experimental conditions Max. Inj. Pressure Pinj 70 MPa Nozzle Hole Diameters d 0.18 mm Fuel-Injection Period Tinj 4.0+0.1 MPa Injection Equipment JERK type Fuel Pump Injection-valve Opening pressure Pop 23 MPa Raziskava učinkov različnih stopenj - Experimental Study of the Effects 865 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 Table 3. Properties of the test fuels Fuel properties __________ 0.7880___________0.7852___________0.7916 Viscosity@30°C | mm2/s | 3.630 | 3.322 | 4.946 Density @15°C unit g/cm3 T1 T9 T10 Cetane Number 48.8 48.6 48.5 Distillation IBP °C 213.5 213.0 211.0 10%____________^C____________229.5____________2305____________233.0 50%___________^C__________249.0___________247.5___________254.5 90%____________^C____________307.0____________275.0____________387.5 EP °C 326.0 295.5 396.5 Sulfur mass % Mean Molecular Weight Lower Calorific Value C/H Ratio J/g 206 n-Paraffin i-Paraffin (atom/ atom) % (v/v) 43910 202 0.476 43920 219 Naphthene % (v/v) 36 0.479 43970 Total-Aromatics % (v/v) 57 35 0.479 % (v/v) 7 59 24 0 6 70 0 6 0 Fuel T9 Fuel T1 5 10 15 20 25 30 Retention Time min Fig. 5. Chromatograms of the test fuels Figure 6 shows the time histories of the in-cylinder pressure and temperature when only air was compressed. The fuel injection was set to begin at 60 ms after the RCM drive piston reached its top position. After the in-cylinder pressure reaches its maximum at the top dead center (TDC) and constant-volume conditions begin, the in-cylinder pressure gradually declines due to heat losses and gas leakages. To and po are the in-cylinder temperature and pressure at the time of the fuel injection. 1.1 Definition of the ignition delay time and the characteristic combustion times Figure 7 shows a typical example of the in-cylinder pressure rise at the time of the fuel injection and during combustion, and the fuel-injection pressure and the needle-lift histories. When the fuel was injected, mixture cooling occurred and the in-cylinder pressure decreased more rapidly. The time from the start of the fuel injection to the start of the heat release was defined as the ignition delay time 866 Nikolič D. - Vujadinovič R. - Iida N. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 I Fuel Injection Timing *—----------------------¦ 1200 Compression T0=908 + 5K Dura tion P0=3.0±0.1MPa Air Tin=353K Pin=0.1MPa 1000 800 600 400 200 0 100 200 300 400 500 Time after Compression Start t ms Fig. 6. Typical in-cylinder pressure and temperature histories for the RCM 0 tid t10 t50 t90 Fuel Inj. Period tinj h-----------H tpmax Time after Fuel Injection Start t Fig. 7. Model of in-cylinder pressure, fuel-injection pressure and needle lift tid, while the time from the start of the fuel injection to the peak of in-cylinder pressure was defined as t. Pmax The in-cylinder pressure increase was defined as DPmax, and the times to reach 10%, 50% and 90% of the peak pressures (DPmax) were defined as t10, t50 and t90, respectively. Because the equivalence ratio was rather small, the specific heat during combustion was assumed to be nearly constant, and thus the burnt-fuel fraction and the pressure increase were considered almost proportional. Therefore, t10, t50 and t90 could be considered as 10%, 50% and 90% of the burned-fuel timings. The time from the start of the fuel injection to the first appearance of the luminous flame was defined as tfa, and the time of the luminous flame’s disappearance was defined as tfd. CO2 = 0 %vol. O2 =21 % vol. T = 905±5 K CO2 = 4.3 %vol. O2 =20 % vol. T = 895±5 K CO2 = 9.5 %vol. O2 = 19 % vol. T =885±5 K CO2 =14.3 %vol O2 = 18 % vol. T = 875±5 K 2 RESULTS AND DISCUSSION Figure 8 shows examples of two-dimensional direct photograph images of the T10 fuel-spray combustion for all the test conditions. With the increase of the CO2 concentration in the intake charge, the luminous flame area and the intensity of the flame decrease. A similar trend appears for the T1 and T9 fuels. Figure 9 shows pressure diagrams for all the test conditions and the test fuels. During compression and at the time of the fuel injection, the pressure of the intake charge (gas mixture) decreases as the CO2 concentration increases, maintaining lower pressures during the whole combustion period. This pressure decrease with the increase of the CO2 concentration in the intake charge is the reason for 3ms 4ms 5ms 6ms 7ms Time after fuel injection start, t ms Fig. 8. Examples of direct photograph images of diesel-fuel spray combustion (fuel T10) Raziskava učinkov različnih stopenj - Experimental Study of the Effects 867 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 Fuel injection period Fuel injection period 500 8 10 12 Time after fuel injection start t, ms Fig. 9. In-cylinder pressure histories the changed composition of the combustion air and could be associated with the increase of the specific heat of such a mixture. Figure 10 shows the rate-of-heat-release (RHR) diagrams for all the test conditions and the test fuels. With the increase of the CO2 concentration in the intake charge the high peaks of the RHR increase and are delayed. This is caused by the increased ignition delay, which means that at the time of the ignition there is more fuel available in the cylinder, well mixed, and with a faster burning rate after the ignition. Figure 11 shows the relationships between the ignition delays, the luminous-flame appearance times, the luminous-flame periods and the combustion periods with characteristic times. The ignition-delay periods are longer with the increase of CO2 concentrations. This is caused by the decrease of O2 concentrations in the intake charge -2 0 2 4 6 8 10 12 Time after fuel injection start t, ms Fig. 10. Rate of the heat-release-time histories as well as the decrease of the intake gas temperature and pressure at the moment of the diesel-fuel injection. This fact makes the periods of fuel/gas-mixture preparation longer. The luminous-flame appearance times (the start of soot radiation) follow the same trend as the ignition delay periods. The luminous-flame periods and combustion periods decrease with the increase of the CO2 concentration in the intake charge. This could be due to a prolonged ignition delay, the decrease of the O2 concentrations from 21% to 18%, the lower in-cylinder temperatures and pressures, which all contribute to incomplete combustion. Figure 12 shows an example of two-dimensional images of the flame temperature and the KL value distribution inside the diesel-fuel spray flame determined by the two-color method. In the case of CO2=14.3%, due to the low luminosity of the flame, the two-color method was not applicable. 868 Nikolič D. - Vujadinovič R. - Iida N. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 Luminous Flame Period tfa t10 t50 t90 0 tid ¦^--------------------------------------------------------------------------------------------------------------------------------- Combustion Period Injection period tpmax CO =14.3% CO =9.5% T90 CO2=4.3% CO2=0% T1 CO =14.3% CO =9.5% T9 T90=275°C CO =4.3% CO =0% T10 02 6 8 1012 T i me af t e r f ue l i nj ectio n start, ms Fig. 11. Ignition delays, characteristic combustion periods and luminous-flame periods Figure 13 (at the end of the paper) shows the time histories of the luminous-flame area and the flame-temperature distribution inside the fuel-spray flames. The flame temperatures obtained from the image analysis using the two-color method were hierarchies at 100K intervals, starting from 1750K. There are decreases in the luminous-flame areas and in the high-temperature areas with the increase of the CO2 concentrations in the intake charge. Figure 13 also shows the area-averaged flame temperature, which is defined by the ratio of SAiTi / SAi, in which Ti is the median value of each hierarchy, Ai is the area having temperature Ti, and SAi is the total flame area. The maximum values of this temperature decrease with the increase of the CO2 concentration in the intake charge. This decrease is more Flame Temp. Flame Temp. K KL Value 2425 «0.375 2175 0.283 Fuel T10; CO2 =0%,; t =5ms Fig. 12. Examples of direct photograph images of luminous fuel spray flame, temperature and KL factor distribution in the fuel spray flame obvious for CO2 concentrations of 4.3% and higher. This decrease is a consequence of the O2 concentration decrease in the combustion chamber and the increase of the inert gases, the decrease of the intake-charge temperatures and the pressures at the time of the fuel injection as well as during the whole combustion period. Regarding the T90 distillation temperature of the diesel fuel, there is a decrease in the maximum area-averaged flame temperature as well as the duration of high temperatures with a decrease of the T90 distillation temperature in the cases of CO2=0% and CO2=9.5%. In the case of CO2=4.3% there is no significant difference in the maximum area-averaged flame temperatures as well as in the duration of the high temperatures. Figure 14 shows the time history of the area-integrated KL value. The KL value is a multiple of the absorption coefficient K, which is nearly proportional to the soot-particle number density in the flame (Beers law) and the optical path length L in the soot region. The KL value is an index of the total number of soot particles along the optical path. The area-integrated KL value SAi(KL)i was defined as the product of the median value of each hierarchy (KL)i and its area, Ai. With the increase of the CO2 concentration in the intake charge, soot starts forming later due to increased ignition delays. The maximum area-integrated KL values decrease with the increase of the CO2 concentration. It was expected that there would be more soot formed with the increase of CO2 concentration due to a lower O2 concentration. Lower values of the area-integrated KL value with the increase of CO2 concentration could be the result of a more homogeneous mixture at the time of the ignition due to more time being available for mixture Raziskava učinkov različnih stopenj - Experimental Study of the Effects 869 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 To = 905 ± 5K O2 = 21% CO2 =0% To = 895 ± 5K O2 = 20% CO2 =4.3% To = 885 ± 5K O2 = 19% CO2 =9.5% 2300 2000 2300 2000 2300 Flame Temp. 2350K 2250K 2150K 2350K 2250K 2150K 2050K 1950K 1850K 1750K 2000 1700 80 80 T i me af te r inj e c tio n start, ms Fig. 13. Time histories of temperature distribution inside diesel fuel spray flame and of area averaged flame temperature Fuel injection period 1000 800 600 400 200 0 800 600 400 200 0 800 600 400 200 0 CO T1 T90=307°C 0 1 2 3 4 5 6 7 8 910 Time after injection start, ms Fig. 14. Time history of the area-integrated KL values preparation, due to lower temperatures inside combustion chamber at the time of the ignition, as well during the whole combustion period, and due to the fact that there is still enough air in such a big combustion chamber, which supports soot oxidation. Iida SAE950213 and Mitchell SAE932798 published similar results. The period of the KL existence decreases with the increase of CO2 concentration in the intake charge, except for the fuel T9 with the lowest T90 distillation temperature. The KL existence period could be related to the soot exhaust emissions from a real diesel engine. By lowering the intake-gas temperature it is possible to use higher EGR rates, which will significantly lower NOx and, at the same time, not influence significantly the soot-formation and soot-extinction periods. Regarding the effect of the fuel on soot formation and emission, for cases of CO2=0% and 9.5%, the soot-extinction periods decrease with the decrease of the T90 distillation temperature. For the case of CO2=4.3%, there is a slight influence of the T90 distillation temperature on the soot-formation and soot-existence periods. This means that by using this CO2 concentration there is a possibility to use heavier fuels for the same soot exhaust emission. 870 Nikolič D. - Vujadinovič R. - Iida N. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 3 CONCLUSION In this study exhaust gas recirculation (EGR) was simulated by the introduction of carbon dioxide (CO2) into the intake air for four different concentrations, 0%, 4.3%, 9.5%, and 14.3%, using three diesel fuels with different T90 distillation temperatures of 275°C, 307°C, and 387.5°C, in order to determine the influence on the nitrogen oxide (NOx) and soot formation inside the combustion chamber of a rapid compression machine (RCM). The main conclusions from this study are as follows: Maximum flame temperature (NOx formation) decreases with CO2 concentration increase in the intake charge. Soot formation and existence periods decrease with CO2 concentration increase in the intake charge. The fuel with the lowest T90 distillation temperature, T9, showed a good soot-NOx trade-off in the case without any CO2 addition to the intake charge. The fuel with the highest T90 distillation temperature, T10, showed a good soot-NOx trade-off in the case of CO2=4.3% in the intake charge. 4 REFERENCES [I] K.Mitchell (2000) Effects of fuel properties and source on emission from five different heavy duty Diesel engines, SAE Paper No. 2000-01-2890. [2] N.Ladommatos, S.M.Abdelhalim, H.Zhao and Z.Hu (1996) The dilution, chemical and thermal effects of exhaust gas recirculation on Diesel engine emission – Part1: Effects of reducing inlet charge Oxygen, SAE Paper No. 961165. [3] D. Nikolic, K. Wakimoto, S. Takahashi and N. Iida (2001) Effect of nozzle diameter and EGR ratio on the flame temperature and soot formation for various fuels, SAE Paper No. 2001-01-1939. [4] D.L.Mitchell, J.A.Pinson and T.A.Litzinger (1993) The effects of simulated EGR via intake air dilution on combustion in an optically accessible DI Diesel engine, SAE Paper No. 932798. [5] N.Ladommatos, S.M.Abdelhalim, H.Zhao and Z.Hu (1996) The dilution, chemical and thermal effects of exhaust gas recirculation on Diesel engine emission – Part2: Effects of Carbon Dioxide, SAE Paper No. 961167. [6] N.Iida (1996) Surrounding gas effects on soot formation and extinction – Observations of Diesel spray combustion using a rapid compression machine, SAE Paper No. 960603. [7] S.Ropke, G.W.Schweimer and T.S.Strauss. NOx formation in Diesel engines for various fuels and intake gases, SAE Paper No. 950213. [8] A.M.Kreso, J.H.Johnson, L.D.Gratz, S.T.Bagley and D.G.Leddy (1998) A study of the effects of exhaust gas recirculation on heavy duty Diesel engine emissions, SAE Paper No. 981422. [9] K.Nakakita, S.Takasu, H.Ban, T.Ogawa, H.Naruse, Y.Tsukasaki and L.I.Yeh (1998) Effect of hydrocarbon molecular on Diesel exhaust emissions – Part 1: Comparison of combustion and exhaust emission characteristics among representative Diesel fuels, SAE Paper No. 982494. [10] N.Miyamoto, H. Ogawa, M.Shibuya, K.Arai, and O.Esmilaire (1994) Influence of the Hydrocarbon fuels on Diesel exhaust emissions, SAE Paper No. 940676. [II] M.Tamanouchi, H.Morihisa, S.Yamada, J.Iida, T.Sasaki and H.Sue (1997) Effects of fuel properties on exhaust emissions for Diesel engines with and without oxidation catalyst and high pressure injection, SAE Paper No. 970758. [12] W.S.Neill, W. L.Chippior, O.L.Gulder, J.Cooley, E.K.Richardson, K.Mitchell and C.Fairbridge (2000) Influence of fuel aromatics type on the particulate matter and NOx emissions of a H.D. Diesel engine, SAE Paper No. 2000-01-1856. [13] K.Tsurutani, Y.Takei, Y. Fujimoto, J. Matsudaira and M. Kumamoto (1995). The effects of fuel properties and oxygenates on Diesel exhaust emissions, SAE Paper No. 952349. [14] Consignment research report concerning influence of Diesel engine technology and fuel technology on exhaust emission, Petroleum Energy Center PEC-1999JC-25 (1999). • • • • Raziskava učinkov različnih stopenj - Experimental Study of the Effects 871 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 863-872 Authors’ Addresses: Prof. Dr. Danilo Nikolič Doc. Dr. Radoje Vujadinovič University of Montenegro Cetinjski put bb 81000 Podgorica, Montenegro dannikol@cg.yu Prof. Dr. Norimasa Iida KEIO University 3-14-1 Hiyoshi, Kohoku-ku Yokohama 223-8522, Japan iida@sd.keio.ac.jp Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 13.3.2006 25.10.2006 Received: Accepted: Open for discussion: 1 year 872 Nikolič D. - Vujadinovič R. - Iida N. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 873-874 Strokovna literatura - Professional Literature Strokovna literatura - Professional Literature Ocene knjig - Book Reviews Dušan Ješič MERNA TEHNIKA Zal.: Mašinski fakultet, Banja Luka, 2004 393 strani, 487 slik, format 17 x 23,8 cm Nenehni razvoj znanosti, tehnike in tehnologije, razvoj novih kakovostnejših materialov in izdelkov terjajo razvoj novih metod in tehnik merjenja, merilno tehniko pa postavljajo kot posebno znanstveno panogo, ki obsega zelo široka in prepletena znanja. Zavedajoč se vseh razsežnosti problematike merilne tehnike avtor pričujočo knjigo, ki je pisana kot učbenik za študente strojništva, razdeli pogojno na štiri celote: osnove merjenja, merjenje geometrijskih veličin, merjenje temperature in hidromehanskih veličin ter merjenje hrupa in vibracij. Bolj podrobno pa je knjiga razdeljena na 19 poglavij, katerih vsebina je podana slikovno izredno pregledno in strokovno prepričljivo brez posebnih poglobljenih teoretičnih izvajanj in analiz. 1. Osnove merjenja: Obravnava uvod v metrologijo, nadzor kakovosti pri izdelavi izdelkov, osnove tehničnega merjenja ter klasifikacijo in izbiromerilnih in nadzornih instrumentov. 2. Merilni sistemi in zaznavala: Zajema osnove, naloge, klasifikacijo ter strukturo merilnih sistemov. Podaja podrobno delitev in opis vseh vrst zaznaval, ki se dandanes uporabljajo v merilni tehniki. 3. Industrijska merila za dolžine: Obravnava enonamenska merila (merilne kladice) in večnamenska merila. 4. Optična merilna tehnika: Podaja ABBE-jev merilnik dolžin, merilne stroje in merilne mikroskope. 5. Interferentna merilna tehnika: Obravnava problematiko merjenja z ravnim vzporednim steklom, svetlobne interferometre ter laserske merilne sisteme. 6. Merjenje in nadzor kotov, konusov in nagibov: Podaja primerjalne metode merjenja in nadzora ter trigonometrične in goniometrične metode merjenja kotov. 7. Merjenje in nadzor navojev: Obravnava nadzor navojev in problematiko merjenja značilnih parametrov navojev. 8. Merjenje in nadzor zobnikov: Zajema osnove in napake posameznih metod nadzora zobnikov, zahteven nadzor zobnikov, merjenje specifičnih parametrov profila ozobij, nadzor sledi nošenja ter nadzor šumnosti zobnikov. 9. Nadzor makrogeometrije površin: Definira makrogeometrična odstopanja površin, metode nadzora oblike, lege ter ravnosti površin. 10. Nadzor mikrogeometrije - hrapavosti površin: Poudarek je na parametrih hrapavosti, na kakovostnih (primerjalnih) ter kolikostnih (brezdotikalnih ali dotikalnih) metodah. 11. Koordinatni merilni stroji (KMS): Zajema osnovne pojme, načelo delovanja in področja uporabe, programiranje KMS, strukturo in klasifikacijo KMS ter smeri razvoja na področju KMS. 12. Avtomatizacija merjenja in nadzora: Definirani so dejavni in posredni merilni sistemi, pnevmatski in fotoelektrični merilni sistemi, merilno-nadzorni roboti, prilagodljivi metrološki sistemi (moduli, celice, središča in postaje). 13. Merjenje temperature: Posebej obdelane vsebine, ki obravnavajo razteznostne plinske, manometrske ter bimetalne termometre. Večji poudarek je na popisu električnih merilnih zaznaval ali termometrov, kakor so uporovni termometri, termopari, sevalni ali optični termometri. Prikazane so posebnosti in omejitve pripadajočih merilnih metod. 14. Merjenje tlaka: Obsega kratek pregled temeljnih pojmov in njihovih definicij, kapljevinske merilnike tlaka, batne manometre, tlačne tehtnice, manometre z elastičnimi elementi ter kratek popis posebnih merilnih metod merjenja tlaka. 15. Merjenje količine in gostote: Zajema podpoglavja o: določanju in merjenju prostornin trdnih teles, kapljevin in plinov, merjenje višine gladin kapljevin, merjenje mase in gostote trdnih teles, kapljevin ter plinov. Pri tem so na kratko podane posebnosti posameznih merilnih metod. 16. Merenje hitrosti in pretoka tekočin: Podane so osnove za: standardizirane merilne metode za umerjanje merilnikov pretoka; merilnike prostorninskega toka kapljevin in plinov; turbinske merilnike prostorninskega toka; 873 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 873-874 merjenje pretoka in hitrosti tekočin po dušilnih metodah; induktivne merilnike pretoka: rotametre; merilnike krajevnih hitrosti toka tekočin (anemometre, LDA, Pitot-Prandtlovo merilno sondo); merjenje prostorninskega toka kapljevin v odprtih pretočnih kanalih ter kratek oris nekaterih sodobnih za merjenje pretoka, npr.: ultrazvočni, vrtinčni in termični merilni postopki. 17. Merjenje vizkoznosti: Kratek pregled uveljavljenih merilnikov ter merilnih metod za merjenje viskoznosti. 18. Merjenje vlažnosti: Pregled izbranih merilnih metod za merjenje vlažnosti plinastih in sipkih ali prašnatih snovi. Posebej so na kratko popisane tehnike določanja ali merjenja vlažnosti, ki temeljijo na rosiščni higrometriji, psihrometriji ter kapacitivni higrometriji. 19. Merjenje hrupa in vibracij: Nekatere teoretične osnove s področja tehnične akustike in dinamike nihanja, ki se nanašajo na merjenje hrupa ter vibracij. Knjiga naj bi bila namenjena ne samo študentom strojništva temveč tudi strokovnjakom v praksi za lažje spoznavanje in razumevanje merilne tehnike ter razvoja na področju merilnih naprav in sistemov, ki se uporabljajo tako v kovinsko-predelovalni industriji kakor tudi v tehniki postopkov in storitvenih dejavnostih. prof. dr. Mirko Sokovič prof. dr. Ivo Bajsic 874 Strokovna literatura - Professional Literature Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 875-879 Osebne vesti - Personal Events Osebne vesti - Personal Events Zoisovo priznanje za pomembne dosežke na področju strojništva in tribologije Doktor Mitjan Kalin je izredni profesor na Fakulteti za strojništvo Univerze v Ljubljani. V fakultetnem Centru za tribologijo in tehnično diagnostiko, ki je pomembna raziskovalna enota predvsem v mednarodnem prostoru, raziskuje pojave mehanskega trenja. Posebej pomembna so njegova raziskovanja triboloških pojavov na področju biomedicine. Raziskoval je pojave obrabe naravnih zob zaradi obrabe hidroksiapatita, ki je poglavitna sestavina zobne sklenine. Ugotovil je, da se obraba zob poveča tudi do 20-krat, če se pojavijo že manjša odstopanja od idealno poliranih površin. Pojav je popisal z ustreznim mehanizmom obrabe, v katerega je vključil tudi tribokemijske pojave, to je kemijske spremembe, ki spremljajo trenje in obrabo. Z različnimi tribološkimi sistemi je doktor Kalin v raziskavah dokazoval potrebo po poglobljenem razumevanju pojavov. Zato je razvil mehanizme mejnih nano-filmov na primerih mazanja keramike z vodo. Tribološki mehanizmi, ki jih je raziskal, so pomembna podlaga za oblikovanje mejnih filmov keramičnih materialov v stikih s kovinskimi kompoziti pri izjemnih temperaturah. Raziskuje tribološki mehanizem trdih DLC prevlek po mazanju z olji. Ugotovil je, da ni mogoče doseči želene "inertnosti" dodatkov v oljih tribološkega sistema, in dokazal, da tudi v tem primeru prihaja do značilnega tribokemijskega pojava. V zadnjih treh letih je profesor Mitjan Kalin objavil kar 15 člankov v mednarodnih revijah, tri sestavke v monografijah ter bil soavtor dveh ameriških patentov, nadgrajenih iz slovenskih. Njegovo mednarodno vpetost dokazujeta tudi citiranost objavljenih del ter sodelovanje pri soorganiziranju mednarodnih konferenc v ZDA in več evropskih državah. Puhovo priznanje za izume in razvojne dosežke na področju laserskih sistemov za tridimenzionalno merjenje oblike teles Raziskovalci Fakultete za strojništvo Univerze v Ljubljani: dr. Matija Jezeršek, dr. Drago Bračun in izr. prof. dr. Janez Diaci so razvili več izvirnih optičnih merilnih sistemov, ki na principu aktivne triangulacije zaznavajo in merijo obliko poljubnih teles. Tovrstne naprave v podjetju Alpina omogočajo izdelavo obutve, prilagojene kupcu. V podjetju Goodyear laserski sistemi dinamično testirajo gumijaste izdelke in zračne vzmeti. V Kliničnem centru zdravnikom omogočajo spremljati celjenje kožnih razjed. Laserske sisteme so uporabili tudi pri nadzoru varilnih postopkov in ugotavljanju stopnje obrabe elektrodnih konic pri uporovnem točkovnem varjenju. Slovenski laserski sistemi za trirazsežno merjenje oblik telesa se v primerjavi z evropskimi odlikujejo s hitrostjo, natančnostjo, enostavnostjo uporabe in nizko ceno. Dr. Matija Jezeršek, dr. Drago Bračun in prof. dr. Janez Diaci so svoje razvojne dosežke predstavili tudi v obliki člankov, referatov, patentov in patentnih prijav. Puhovo priznanje za izume in razvojne dosežke na področju numeričnega modeliranja materialov in procesov Raziskovalno in razvojno delo profesorja Božidarja Šarlerja, zaposlenega na Univerzi v Novi Gorici, obsega posodobitev kontinuirnega ulivanja aluminijevih zlitin in jekel ter informatizacijo livnih naprav. Optimizacijo značilnosti postopkov je dosegel z vključitvijo zaznaval in simulacijskih sistemov v proizvodne postopke. S tem se je izboljšala kakovost obstoječih izdelkov, povečala storilnost livnih naprav, pospešil razvoj izdelkov, zmanjšal izmet in povečala varnost postopkov. Te 875 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 875-879 uporabne dosežke, vrhunske tudi v mednarodnem merilu, je profesor Šarler prenesel v podjetja Acroni, Štore Steel, Impol in Talum. Profesor Šarler sodeluje v številnih evropskih projektih. Posebej ga odlikuje dejstvo, da je v evropske projekte uspel vključiti tudi slovenska podjetja. Z objavljenimi članki, vabljenimi predavanji in članstvom v uredniških odborih revij pa izstopa tudi po znanstveni odličnosti. Častni doktorat znanosti prof.dr. Manfredu Geigerju - dolgoletnemu članu mednarodnega svetovalnega odbora Strojniškega vestnika Univerza v Ljubljani vsak prvi teden v decembru proglasi za svoj teden, kajti tretjega decembra leta 1919 je bilo na novoustanovljeni Univerzi v Ljubljani opravljeno prvo predavanje. V tem tednu se na sedežu Univerze pa tudi pri njenih članicah zvrsti veliko dogodkov, med katerimi vsekakor zaseda prvo mesto svečana seja Senata Univerze v Ljubljani, kajti tedaj so proglašeni tudi njeni novi častni doktorji znanosti. Med letošnjima nagrajencema je tudi profesor doktor Manfred Geiger z Univerze Erlangen -Niirnberg. Častni doktor prve slovenske univerze je leta 1974 doktoriral na Tehniški univerzi v Stuttgartu pri profesorju Langeju, enemu od soustanoviteljev nove znanstvene panoge plastomehanskih postopkov. Leta 1977 je dr. Geiger odšel v industrijo, v podjetju Trumpf je bil odgovoren za razvoj laserskega stroja za razrez pločevin, kar mu vsi priznavajo kot inovacijo v svetovnem merilu. Dr. Geiger je bil nato glavni vodja razvoja in raziskav pri podjetju PEBRA, kjer so pod njegovim vodstvom razvili prve z vlakni ojačane plastične dele za avtomobilske karoserije. Profesor Geiger je leta 1982 prešel v akademske vode ko je bil od Univerze Erlangen-Niirnberg povabljen in pooblaščen za ustanovitev nove Katedre za izdelovalne tehnologije. Katedra je danes v Nemčiji pa tudi v svetu priznana kot ena od vodilnih pedagoških in znanstveno-raziskovalnih institucij za svoje področje. Katedra ima trenutno 59 raziskovalcev in tehniškega osebja in okoli 55 študentov v dopolnilnem delovnem razmerju. Med najpomembnejše merljive uspehe profesorja bi lahko našteli: - Njegova skrb za znanstveni podmladek in raziskovalno delo se kaže v 89 doktorskih disertacijah. - Profesor Geiger je ustanovitelj treh podjetij oziroma tehnoloških centrov: Bayerisches Laser-zentrum, Erlanger Lasertechnik in Laserequipment, kjer dela skupaj okoli 80 raziskovalcev. Profesor Geiger je prve stike s Fakulteto za strojništvo navezal že kot doktorand leta 1965, ko se je na Tehniški univerzi v Stuttgartu srečal s tedanjim docentom Francem Golograncem, ki je na tej šoli prav tako pripravljal svojo doktorsko disertacijo. Naslednja leta se je to sodelovanje in prijateljevanje razširilo še na tedanjega asistenta Karla Kuzmana, ki je na tej ustanovi občasno pomagal docentu Golograncu pri eksperimentalnem delu pri disertaciji. Ko je docent Gologranc promoviral pri prof. Langeju, so se stiki ohranjali s pogostimi obiski v obe smeri, pri katerih so slovenski partnerji pridobivali dragocene informacije o smereh razvoja preoblikovalnih tehnologij. Večletni prijateljski stiki so se še okrepili, ko je dr. Geiger postal profesor na Univerzi v Erlangenu. Med najpomembnejša dokazila o nesebični pomoči prof. dr. Geigerja pri razvoju nove znanstvenoraziskovalne panoge na Fakulteti za strojništvo bi lahko navedli: - Koordinator TEMPUS-Projekta "Advanced Manufacturing Technology, Engineering Economy an CIM-Oriented Techniques in Metal forming" 1991 - 1994; - Član mednarodnega svetovalnega odbora Strojniškega vestnika (več ko 10 let); - Pooblaščenec in koordinator Pogodbe o sodelavi med Univerzama Erlangen in Ljubljana; - Darilo laserskega obdelovalnega sistema Katedri 876 Osebne vesti - Personal Events Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 875-879 za izdelovalne tehnologije in sisteme, Fakultete za strojništvo v Ljubljani; - Bivanja do- in podiplomskih študentov, raziskovalcev v Erlangenu (nekaj mesecev, leto in več): Tomaž Pepelnjak, Andrej Horvat, Igor Komel, Uroš Flere, Edvard Govekar, Karl Kuzman; - Profesor Geiger je skupaj z raziskovalci FS predstavil 17 del v uglednih revijah ali znanstvenih srečanjih. Področje sinergij bi bilo mogoče tisto, s katerim bi najbolje predstavili profesorjev lik. Ne išče sinergij le med tehnološkimi procesi, še bolj pomembne so sinergije med ljudmi, in to: med raziskovalci iz akademske sfere ter strokovnjaki iz industrijskega okolja, sinergije med mladimi in starimi, med predstavniki različnih narodov, med predstavniki različnih kultur. To priznanje ni bilo le priznanje učitelju, temveč je pokazalo, kako zna najstarejša slovenska Univerza ceniti ljudi, ki so toliko svoje energije in znanja vložili v napredek več sinergijsko povezanih znanstvenih tehniških panog in hkrati ta znanja uspešno prenesli v gospodarska okolja z vrhunsko tehnologijo. prof. dr. Karl Kuzman dekan Fakultete za strojništvo Prof. dr. Viljem Kralj - petinsedemdesetletnik Dne 31.7.2006 je dopolnil 75 let dr. mag. Viljem Kralj, univ. dipl. inž. elektrotehnike, upokojeni redni profesor Fakultete za strojništvo Univerze v Ljubljani. Rodil se je v Dugem Ratu pri Splitu, kot prvi od petih otrok materi Zlati in očetu Rudolfu, kjer je oče delal pri izgradnji tovarne karbida, ki jo je gradila francoska delniška družba Le Dalmatien d.d. v sodelovanju s prav tako francoskim podjetjem LAir Liquide. Ko otroška igrivost še ni prerasla v fantovske podvige, so očeta že čakale nove službene zadolžitve. Takrat se je družina Kraljevih preselila v Novi Sad, od koder pa jih je 1941 leta madžarska okupacijska oblast izgnala. Izgon se je končal v Mariboru, kjer si je oče našel zaposlitev v tamkajšnjihŽelezniških delavnicah. Podobno kakor mladostno odraščanje je bilo pestro in bogato tudi šolanje. Osnovno šolo je začel obiskovati v Novem Sadu, nadaljeval in končal pa jo je v Mariboru, kjer je dokončal tudi pet razredov klasične gimnazije, maturiral pa je na Srednji hidrometeorološki šoli v Beogradu. Po maturi se je zaposlil kot klimatolog in sinoptik na Hidrometeorološkem zavodu SR Slovenije v Ljubljani, in se leta 1950 vpisal na Naravoslovno matematično fakulteto ter 23.1.1957 na Tehniški fakulteti diplomiral iz elektrotehnike - šibki tok. Prva inženirska zaposlitev je bila v tovarni Telekomunikacije v Ljubljani, kjer je sodeloval pri razvoju električnih elementov in sistemov ter bil vodja merilnice. Leta 1959 ga je življenjska pot privedla na novoustanovljeni Zavod za varjenje (danes Institut za varilstvo), kjer se je dejavno vključil v strokovno-razvojno, znanstveno-raziskovalno in pedagoško delo. Od tod je leta 1962 bil preko ASTEF (francoske fundacije za tehnično pomoč državam v razvoju) poslan tudi na varilsko specializacijo v Paris. Kot raziskovalec in vodja raziskav je delal na razvoju varilnih strojev in naprav ter opreme za varjenje, tako za zavod kot tudi za druga podjetja. Na tem mestu je težko predstaviti vse dosežke tega oddelka, navedimo le najodmevnejše: plazemski rezalnik PR-100, naprava za plazemsko mikro varjenje PV-20, naprava za plamensko rezanje z ročnim, magnetnim ali fotoelektronskim vodenjem, naprava za avtomatsko privarjanje v proizvodnji malih kompresorjev v tovarni Iskra v Kranju, sistem za avtomatizirano vodenje varilne proge pri uporovnem varjenju streh avtobusov v Avtomontaži, v Ljubljani. Razvil je napravi za obločno varjenje v zaščiti nevtralnih plinov TIGVAR in MIGVAR. Za novo ustanovljeno podjetje VARSTROJ v Lendavi so takrat na Zavodu pripravili tehnično dokumentacijo za industrijsko proizvodnjo domala celotnega proizvodnega programa strojev in naprav: prenosni varilni transformator za obločno varjenje tipa Furlan, stroj za uporovno varjenje Furlan TT 1 in TT 2, ob tem pa še povsem na novo razvili stroj za sočelno obžigalno varjenje ter namizno izvedbo stroja za plamensko rezanje. Rezultati tega razvojno-raziskovalnega dela so bili predstavljeni tudi širši strokovni javnosti doma in po svetu, ob rednih prispevkih v strokovnem glasilu Varilna tehnika tudi v berlinskem SchweiBtechnik, pa tudi v strokovnem glasilu vsesovjetskega inštituta za znanost in tehniko, pri Akademiji nauk SSSR, kjer je bil podan skrajšan prikaz njegovega referata z naslovom "Ekonomska primerjava plazemskega varjenja z Osebne vesti - Personal Events 877 različnimi plini in plinskimi mešanicami", ki ga je pred tem predstavil na Mednarodnem posvetovanju o toplotnem rezanju v Halleju, NDR (Ekspres informacije-Svarka, Moskva 1973/No. 39, str. 26-28). Že med zaposlitvijo na Zavodu za varjenje je na Fakulteti za elektrotehniko opravil podiplomski magistrski študij s področja avtomatike in ga 1969 uspešno končal. Začete raziskave in spoznanja je poglobil in razširil v doktorski disertaciji z naslovom: Raziskave gibov roke človeka elektrovarilca pri ročnem obločnem varjenju z metodami biokibernetike, ki jo je izdelal pod mentorstvom akademika prof. dr Alojza Vodovnika in somentorstvom akademika prof. dr. Ludvika Gyergyeka in jo 1973 leta tudi uspešno zagovarjal. Rezultati raziskav na tem področju so bili resnično svetovno odmevni. Prvič je bil namreč popisan in izmerjen gib varilčeve roke, narejena avto in križna korelacija med gibom varilčeve roke in varilnimi parametri, dokazana je bila uporaba varilčevega otipa, zastavljene so bile raziskave za študij kinematike paličaste elektrode. Rezultati teh raziskav so bili predstavljeni na več Letnih skupščinah Mednarodnega instituta za varjenje IIS/ IIW, v študijski skupini za fiziko obloka št. 212 in objavljeni v več znanstvenih člankih na domačih in mednarodnih konferencah ter recenziranih strokovnih revijah. Z gotovostjo se lahko trdi, da če je slovenska in jugoslovanska varilska stroka v čem dosegla svetovni vrh, ga je prav gotovo z rezultati raziskav s področja biokibernetike varilca. Teze in zamisli teh raziskav so bile namreč sprejete tudi v raziskovalni program pri IIS/IIW. V 80 letih so bile te raziskave eno od temeljnih področij za raziskovalne programe v komisiji II-B- za obločno varjenje. Prof. Kralj je kot habilitiran docent prišel na Fakulteto za strojništvo leta 1975, v času izredne rasti gospodarstva, velikega povpraševanja po inženirskih kadrih, med njimi tudi po takšnih z varilsko usmeritvijo, ki so jo prav v tistem času udejanjali na Fakulteti za strojništvo. Tu je bil aprila 1979 izvoljen za izrednega profesorja in junija 1984 za rednega profesorja. Ob že uvedenih vsebinah predavanj in laboratorijskih vaj za predmete Varjenje, Tehnika spajanja in Varilska tehnologija, je zaoral ledino pri uvedbi predavanj in laboratorijskih vaj za ostale varilske predmete: Varilni stroji in naprave, Oprema za varilne procese in varstvo pri delu, ter Fizikalno-kemijske osnove varilnih procesov (skupaj s prof. Prosencem). Tako je leta 1985, ob uvedbi nove visokošolske usmeritve na Fakulteti za strojništvo v Ljubljani, prvič v Sloveniji bil uveden redni univerzitetni študij varjenja. V okviru podiplomskegaštudija pa je pripravil program predavanj za Posebne načine toplotnega varjenja in rezanja, ki sta ga s pokojnim profesorjem Francem Roethlom zaokrožila v predmet Posebni načini obdelave. Ob neposrednem pedagoškem delu ne gre prezreti njegovega mentorstva številnim diplomantom na vseh stopnjah in različnih programih dodiplomskega in podiplomskega študija. Če se odmisli listo z Zavoda, se je samo na Fakulteti za strojništvo nabralo zajetno številko preko 300 mentorstev. Ena od temeljnih značilnosti njegovega dela s kolegi kot s študenti je bila in ostaja izjemna neposrednost, odprtost in strokovna verodostojnost. Kar opazen del vsebinsko bogatih in strokovno poglobljenih predavanj je predstavil tudi v izdanih učbenikih: Kontrola brez porušitve in Novejši varilni postopki, oboje Zavod za varjenje, 1959; Varjenje v zaščiti CO2 (soavtor z Ivanom Limplom), Društvo za varilno tehniko Slovenije, 1971; Točkovno uporovno varjenje (poglavje: Stroji in naprave za uporovno točkovno varjenje), Institut za varilstvo, 1992; Krautov strojniški priročnik (dodano na novo napisano poglavje o varjenju), Littera picta. Kot eden od dveh predstavnikov Jugoslavije (med 30-imi iz celotnega sveta) je sodeloval pri pripravi knjige Physics of welding (je tudi naveden), ki jo je v izdaji Mednarodnega Instituta za varjenje, 1984 založil Pergamon Press. Na področju znanstvene raziskovalne dejavnosti v mednarodnih okvirih izstopajo njegovi prispevki pri Mednarodnem Institutu za varjenje. Od leta 1959, ko je kot opazovalec prvič sodeloval na 12. letni skupščini IIW-ja, pa do 56. letne skupščine 2001 v Ljubljani, je deloval v domala vseh komisijah kot ekspert, imenovani delegat ali delegat Jugoslavije oz. Slovenije. Bil je član Organizacijskega odbora in član Upravnega odbora v Mednarodnem inštitutu za varjenje in tam predstavil preko 30. znanstvenih dokumentov. Na javni seji 47. letne skupščine Mednarodnega inštituta za varjenje leta 1982 v Ljubljani je sopredsedoval bloku predavanj z naslovom "Value of the new Welding Processes in Respect of Energy, Economy and Technology" in imel sam predavanje z naslovom "Welding and Allied Processes". Za udeležence te Letne skupščine je bil, skupaj s pokojnim doc. dr. Andrejem Grželjem in podpisanima, tudi eden od organizatorjev in vodičev ogleda laboratorijev na Fakulteti za strojništvo. Znanstvenoraziskovalno delo na Fakulteti za strojništvo je bilo usmerjeno predvsem v temeljne raziskave varjenja v okviru večletnih raziskovalnih projektov, ki so bili financirani s strani Raziskovalne skupnosti Slovenije, oziroma Ministrstva za znanost. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 875-879 Te raziskave se lahko strne v tri zaokrožene sklope, Od leta 1976 je častni član Društva za varilno in sicer: Študij varilnih procesov, njihove dinamike tehniko Slovenije. Za zasluge pri razvoju varilne in matematično modeliranje ročnega obločnega tehnike je dobil pisno priznanje Društva za varilno varjenja. Študij kinematike paličaste taljive elektrode tehniko SR Slovenije, za sodelovanje na strokovnem pri obločnem varjenju in Študij varjenja in navarjanja področju je leta 1984 prejel tudi pisno priznanje z dvojno in trojno elektrodo pod praškom in v Društva za tehniku zavarivanja Hrvatske. Za zaščitnih plinih. dolgoletno sodelovanje pri razvoju družine varilnih Na tem mestu je nemogoče zajeti vsega izvorov mu je bilo podeljeno posebno priznanje s njegovega družbenega delovanja. Na Zavodu za strani Iskre - Industrije za avtomatiko. varjenje je bil član in predsednik Sveta, ter član Če se znanstveno-raziskovalnemu in Upravnega odbora, Znanstvenega sveta in Odbora izobraževalno vzgojnemu opusu dodajo njegove za razvoj in raziskave. Na Fakulteti za strojništvo je bil obče človeške dobrine: široko splošno izobraženost, član in predsednik Odbora za znanstveno raziskovalno znanje tujih jezikov (ob tekočem znanju delo in Odbora za vzgojno-izobraževalno delo, terčlan srbohrvaščine in nemščine še znanje ruščine, Odbora za koordinacijo v samoupravni interesni angleščine in francoščine), poštenost in doslednost skupnosti RSS. V letih 1988 do 1993 je bil predsednik pri delu, ugotavljamo, da njegov prispevek (še) ni bil Katedre za tehnologijo materialov med leti 1990 in 1996 ustrezno umeščen. Vendar: "Tisto kar je vredno, s vodja Laboratorija za varjenje in med leti 1991 in 1993 časom pridobiva na vrednosti, kar ni vredno z vsakim prodekan za pedagoško delo. dnem izgublja na svoji vrednosti". Pri starosti 65 let, z izpolnjenimi 40 leti delovne Ob kar visokem življenjskem jubileju, se dobe, se je septembra 1996, po tedaj veljavnem prijatelji in sodelavci s hvaležnostjo spominjamo zakonu upokojil, ohranil pa je stike tako s Fakulteto slavljenca ter mu želimo predvsem trdnega zdravja, za strojništvo kot z Institutom za varilstvo. Na saj za zdravo-razumsko skeptičnost in hkrati Fakulteti za strojništvo je še zmeraj udeležen kot optimizem in vedrino zna najbolje poskrbeti kar mentor ali somentor v dodiplomskem in sam. podiplomskem študiju, na Institutu za varilstvo pa kot predavatelj na tečajih specialističnega programa prof.dr. Viktor Prosenc IWE/IWT, z mednarodno priznanimi diplomami. doc.dr. Ivan Polajnar Doktorata in diplome - Doctor’s and Diploma Degrees DOKTORATA Na Fakulteti za strojništvo Univerze v Mariboru Na Fakulteti za strojništvo Univerze v sta z uspehom zagovarjala svoji doktorski disertaciji: Ljubljani so pridobili naziv diplomirani inženir dne 14. novembra 2006: mag. Aleš Belšak, strojništva: z naslovom: "Časovno-frekvenčna analiza stanja dne 9. novembra 2006: Matija JEREB, Blaž zobniških gonil"; KOREN, Damjan PIRNAR, Ivan SIMONIČ, Matej dne 17. novembra 2006: mag. Bojan Vohar, ŽEŽE; z naslovom: "Optimizacija manipulatorja z dne 10. novembra 2006: Rajko ARKAR upoštevanjem elastičnih deformacij". Tomaž DREMELJ, Aleš KLINAR, Darko ROGELJ, S tem sta navedena kandidata dosegla Janez ŠKERL, Rok VOZEL; akademsko stopnjo doktorja znanosti. dne 13. novembra 2006: Franc PERKO, Jure RAJŠELJ, Matej STAVANJA, Marko UMEK. DIPLOMIRALI SO Na Fakulteti za strojništvo Univerze v Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir Ljubljani so pridobili naziv univerzitetni diplomirani strojništva: inženir strojništva: dne 30. novembra 2006: Danijel CAJNKO, dne 23. novembra 2006: Miha BAZNIK, Bine ČARE, Roland GOMBOŠI, Ermin LUJINOVIČ, Gregor GRGIČ, Uroš TRDAN, Luka ŽAGAR. Tomaž OBAD. Osebne vesti - Personal Events 879 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 880-884 Vsebina 2006 - Contents 2006 Vsebina 2006 - Contents 2006 Uvodnik Editorial Fajdiga M. 2 Fajdiga M. Tuma M.: Boris Černigoj (1915 - 2006) 208 Tuma M.: Boris Černigoj (1915 - 2006) Kopač J. 704 Kopač J. Alujevič A.: 500. številka (zvezek) Strojniškega Alujevič A.: 500th issue of the Journal of Mechani- vestnika 784 cal Engineering Razprave Papers Hribernik, A.: Vpliv biodizla na vbrizgavanje goriva, Hribernik, A.: The Influence of Biodiesel Fuel on the zgorevanje, nastanek emisij in značilnice Injection, Combustion, Emissions and dizelskega motorja z neposrednim vbrizgom 3 Performance of a Direct-Injected Diesel Engine Katrašnik, T.: Izboljšan algoritem za simulacijo turbine Katrašnik, T.: A Novel Algorithm for the Simulation avtomobilskega turbopolnilnika 15 of an Automotive Turbocharger Turbine Magister, T, Krulec, R., Batista, M., Bogdanovič, L.: Magister, T, Krulec, R., Batista, M., Bogdanovič, L.: Meritve voznikovega odzivnega časa 26 Measuring a Driver’s Reaction Time Okorn, I., Bešter, T., Orbanič, P, Fajdiga, M.: Okorn, I., Bešter, T, Orbanič, P, Fajdiga, M.: The Primerjalna analiza preračuna prednje preme Difference Between a Front-Axle Stress po metodi končnih elementov in standardu Calculation Using the Finite-Element Method DIN743 41 and the Same Calculation According to DIN743 Wagner, A., Bučar, T, Fajdiga, M.: Vpliv različnih Wagner, A., Bučar, T, Fajdiga, M.: The Effects of parametrov na trdnostni preračun toplotno Different Analysis Parameters on the obremenjenega žarometa 52 Calculated Temperature of a Loaded Headlamp Nagode, M., Fajdiga, M.: Modeliranje naključnih Nagode, M., Fajdiga, M.: Thermo-Mechanical termomehanskih napetostno-deformacijskih Modelling of Stochastic Stress-Strain stanj 74 States Vesenjak, M., Ren, Z., Miillerschon, H., Matthaei, Vesenjak, M., Ren, Z., Miillerschon, H., Matthaei, S.: S.: Računalniško modeliranje gibanja goriva Computational Modelling of Fuel Motion and in njegov vpliv na konstrukcijo rezervoarja 85 Its Interaction with the Reservoir Structure Borovinšek, M., Vesenjak, M., Ulbin, M., Ren, Z.: Borovinšek, M., Vesenjak, M., Ulbin, M., Ren, Z.: Simulacija naleta tovornega vozila ob cestno Simulating the Impact of a Truck on a Road- varnostno ograjo 101 Safety Barrier Furlan, M., Rebec, R, Černigoj, A., Čelič, D., Čermelj, Furlan, M., Rebec, R., Černigoj, A., Čelič, D., Čermelj, P, Boltežar, M.: Vibroakustično modeliranje P, Boltežar, M.: VVibro-Acoustic Modelling alternatorja 112 of an Alternator Slavic, J., Nastran, M., Boltežar, M.: Modeliranje in Slavič, J., Nastan, M., Boltežar, M.: Modeling and analyzing analiza dinamike ščetke elektromotorja 126 the dynamics of an electric-motor brush Leskovar, M., Mavko, B.: Simuliranje preizkusa Leskovar, M., Mavko, B.: Simulation of the Phebus težke nesreče Phebus FPT1 s programom FPT1 Severe Accident Experiment with the MELCOR 142 MELCOR Computer Code Taskesen, A., Mendi, F., Kisioglu, Y., Kulekci, M. K: Taskesen, A., Mendi, F., Kisioglu, Y., Kulekci, M. K: Analiza deformacij vrtal po analitični metodi Deformation Analysis of Boring Bars Using in s končnimi elementi 161 Analytical and Finite Element Approaches Musizza, B., Petrovčič, J., Tinta, D., Tavčar, J., Dolanc, Musizza, B., Petrovčič, J., Tinta, D., Tavčar, J., Dolanc, G., Koblar, J., Juričič, D.: Izvedba sistema za G., Koblar, J., Juričič, D.: Implementation of a avtomatsko končno kontrolo kakovosti System for the Automatic End-Quality elektromotorjev za sesalnike 170 Assessment of Vacuum-Cleaner Motors Emri, I., Cvelbar, R.: Uporaba gladilnih funkcij za Emri, I., Cvelbar, R.: Using Spline Functions to glajenje podatkov podanih v diskretni obliki 181 Smooth Discrete Data 880 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 880-884 Markič, M., Likar, B.: Regionalni vidiki inoviranja kot Markič, M., Likar, B.: Regional Aspects of Innovation osnova konkurenčnosti podjetja znotraj as a Cornerstone of the Competitiveness of a države in EU 195 Company within the State and the EU Vujica Herzog, N., Polajnar, A., Pižmoht, P.: Vujica Herzog, N., Polajnar, A, Pižmoht, P.: Perform- Merjenje izvedbe pri prenovi poslovnih ance Measurement in Business Process Re- tokov 210 Engineering Spruogis, B., Turla, V.: Dušenje torzijskih vibracij in Spruogis, B., Turla, V.: A Damper of Torsional Vibrations raziskava učinkovitosti 225 and an Investigation of Its Efficiency Kravčenkiene, V., Aleksa, A., Ragulskis, M., Kravčenkiene, V., Aleksa, A., Ragulskis, M., Maskeliunas, R: Izbira gladilne spremenljivke Maskeliunas, R.: Choice of Smoothing z optičnimi napetostnimi nanosi 237 Parameter Using Photo-Elastic Coatings Pšunder, I.: Obvladovanje delovne zastarelosti Pšunder, I.: Managing the Functional Obsolescence of strojev in opreme: količnik ozdravljivosti Machinery and Equipment: the Quotient of delovne zastarelosti kot osnova za Curability of Functional Obsolescence as a Basis odločitve o obnovi ali zamenjavi strojev in for Decisions Made about the Renovation or opreme 242 Replacement of Machinery and Equipment Drev, D, Vrhovšek, D., Panjan, J.: Raziskave možnosti Drev, D., Vrhovšek, D., Panjan, J.: Using Porous uporabe porozne keramike kot podstave ali Ceramics as a Substrate or Filter Media filtrirne snovi pri čiščenju odpadnih vod 250 During the Cleaning of Sewage Lerher, T., Potrč, I.: Načrtovanje in optimiranje Lerher, T., Potrč, I.: The Design and Optimization of avtomatiziranih regalnih skladiščnih sistemov 268 Automated Storage and Retrieval Systems Leskovar, M., Končar, B., Cizelj, L.: Simuliranje Leskovar, M., Končar, B., Cizelj, L.: Simulation of a eksplozije pare v reaktorski votlini s splošnim Reactor Cavity Steam Explosion with a General programom za računsko dinamiko tekočin 292 Purpose Computational Fluid Dynamics Code Kulvietiene, R., Kulvietis, G., Tumasoniene, I.: Kulvietiene, R., Kulvietis, G, Tumasoniene, I.: A Simbolno-številčno analiziranje nihanj Symbolic-Numeric Vibrations Analysis of sistemov z velikim številom stopenj prostosti 309 Systems with Many Degrees of Freedom Štubna, I., Trnik, A.: Popravni količniki za izračun Štubna, I., Trnik, A.: Correction Coefficients for Cal- Youngovega modula iz resonančnega culating the Young’s Modulus from the Reso- upogibnega nihanja 317 nant Flexural Vibration Drev, D., Panjan, J.: Teoretične in eksperimentalne Drev, D., Panjan, J.: Theoretical and Experimental osnove za izdelavo mehanskih izolacijskih Foundations for the Manufacturing of pen 323 Mechanical Insulation Foams Kljenak, I., Mavko, B., Škerlavaj, A.: Modeliranje Kljenak, I., Mavko, B., Škerlavaj, A.: Modelling of razslojene atmosfere v eksperimentalni the Stratified Atmosphere in a Nuclear Power napravi jedrske elektrarne s popisom z Plant Experimental Facility with a Lumped- zgoščenimi parametri 340 Parameter Description Tollazzi, T, Lerher, T, Šraml, M.: Analiza vpliva Tollazzi, T, Lerher, T, Šraml, M.: An Analysis of the prometnega toka pešcev na prepustno Influence of Pedestrians’ Traffic Flow on the zmožnost krožišča z uporabo diskretnih Capacity of a Roundabout Using the Discrete simulacij 359 Simulation Method Vidmar, P, Petelin, S.: Model požara ob prometni Vidmar, P, Petelin, S.: Model of an Accident-Induced nesreči v bližini jedrske elektrarne 380 Fire around a Nuclear Power Plant Jovanovič, J., Krivokapič, Z.: Uporaba neobičajnih Jovanovič, J., Krivokapič, Z.: The Application of an nevronskih mrež za ovrednotenje okoljskih Atypical Neural Network when Quantifying vidikov modeliranja 392 the Modeling of Environmental Aspects Basic, I., Crnjac, P.: Ocena vpliva vhodnih parametrov Bašič, I., Crnjac, P.: The Estimated Influence of the v analizi toplotnega prehodnega pojava pod Input Parameters in the analysis of the PTS tlakom v reaktorski posodi Nuklearne elektrarne in the Core of the PWR Krško NPP in the Krško v primeru majhne izlivne nezgode 404 Case of the SB LOCA Hengl, T, Jurišič, M., Martinič, I.: Ocena natančnosti Hengl, T, Jurišič, M., Martinič, I.: An Accuracy Vsebina 2006 - Contents 2006 881 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 880-884 satelitske navigacije pri upravljanju naravnih Assessment of Satellite Navigation in virov 419 Natural-Resource Management Krivokapič Z., Zogovič, V., Spaič, O.: Spremljanje Krivokapič Z., Zogovič, V., Spaič, O.: Using Neural obrabe vijačnega svedra (S390) z uporabo Networks to Follow the Wear of a S390 Twist nevronskih mrež 437 Drill Monno, M., Ravasio, C: Vpliv spreminjanja tlaka na Monno, M., Ravasio, C.: The Effect of Pressure Fluctuations rezalno zmožnost vodnega curka 443 on the Cutting Ability of Pure Water Jet Biskup, C, Hover, M., Versemann, R, Bach, Fr.-W., Biskup, C, Hover, M., Versemann, R, Bach Fr.-W., Kr6mer, Kromer, S., Kirsch, L., Andreae, A., Pude, F, S., Kirsch, L., Andreae, A, Pude, F, Schmolke, S.: Schmolke, S.: Merjenje generacije toplote pri Heat Generation During Abrasive Water-Jet rezanju kosti z abrazivnim vodnim curkom 451 Osteotomies Measured by Thermocouples Uhlmann, E., Hollan, R., El Mernissi A.: Uhlmann, E., Hollan, R, El Mernissi A: Hybrid Dry- Hibridno obstreljevanje s suhim ledom Ice Blasting Laser Processing: Nd-YAG-Laser- in laserjem 458 assisted Dry-Ice Blasting for De-Coating Werth, H, Hiller, W, Luetge, C, Koerner, J.-P, Pude, Werth, H, Hiller, W, Luetge, C, Koerner, J.-P, Pude, F, Lefevre, I., Lefevre, R.: Učinkovitost F., Lefevre, I., Lefevre, R.: Cutting rezanja in dosegljiva kakovost pri uporabi Performance and Obtainable Quality when abrazivnega vodnega curka pri tlaku 6000 bar 463 Applying 6000-Bar Abrasive Water-Jets Djinovič, Z, Tomič, M., Vujanič, A., Pavelka, R., Mitič, Djinovič, Z., Tomič, M., Vujanič, A., Pavelka, R., Mitič, S., Vujanič, D, Cordes, M.: Preučevanje in razvoj S., Vujanič, D., Cordes, M.: Investigation and stekleno-vlaknenega vibrometra primernega za Development of a Fiber-Optic Vibrometer for popolno vsadne slušne pripomočke 470 Use in Totally Implantable Hearing Aids Vujovič, A., Krivokapič, Z.: Uvajanje informacijskih Vujovič, A., Krivokapič, Z.: Implementation of tehnologij z namenom izboljšanja sistema za Information Technology for the Purpose of upravljanje s kakovostjo 477 Quality Management System Improvement Herrero, A., Igor Goenaga, I., Azcarate, S., Uriarte, L., Herrero, A., Igor Goenaga, I., Azcarate, S., Uriarte, L., Ivanov, A., Rees, A., Wenzel, C, Miiller, C: Ivanov, A., Rees, A., Wenzel, C, Miiller, C: Mehanska mikroobdelava s frezanjem, žično Mechanical Micro-Machining Using Milling, erozijo, potopno erozijo in diamantnim Wire EDM, Die-Sinking EDM and Diamond struženjem 484 Turning GeiBdorfer, S., Putz, A, Engel, U: Mikropreoblikovanje GeiBdorfer, S., Putz, A., Engel, U.: Microforming - - Trenutno stanje in prihodnje zahteve 495 Current Status and Future Demands Museau, M., Masclet, C.: Integrirano oblikovanje izdelkov Museau, M., Masclet, C: Integrated Design of s področja drobnih elektromehanskih izdelkov 506 Mycro-Electro-Mechanical Systems Levy, P., Knowles, D., Stagg, C, Junkar, M.: Dvanajst Levy, P., Knowles, D., Stagg, C, Junkar, M.: The "znakov umiranja" rastočega proizvodnega Twelve "Death Signs" for a Growing podjetja 515 Manufacturing Company Dragoi, G, Cotet, C. E, Rosu, L., Rosu, S. M.: Naloga Dragoi, G, Cotet, C. E., Rosu, L., Rosu, S. M.: The navideznih omrežij v navideznem podjetju 526 Role of Virtual Networks in a Virtual Enterprise Karpiriski, A.: Uvod v diagnostiko postopka Karpiriski, A.: An Introduction to the Diagnosis of razslojevanja kompozitnih materialov pri the Delamination Process for Glass/Epoxy rezanju z visokotlačnim abrazivnim vodnim Composites During High-Pressure Abrasive curkom 532 Water-Jet Cutting Bach, Fr.-W., Louis, H., Versemann, R., Schenk, A.: Bach, Fr.-W., Louis, H., Versemann, R, Schenk, A.: Karakterizacije postopka čiščenja z vodnim Characterization of a Pure Water-Jet Cleaning curkom - simulacija postopka 539 Process - Process Simulation Hadar, A., Nica, M. N, Constantinescu, I. N, Dan Hadar, A., Nica, M. N, Constantinescu, I. N, Dan Pastama, S.: Konstrukcijska in geometrijska Pastama, S.: The Constructive and Geometrical optimizacija spojev, narejenih iz laminiranih Optimization of the Junctions in Structures kompozitnih materialov 546 Made from Laminated Composite Materials Šukšta, M., Bazaras, Ž., Leonavičius, M., Šukšta, M., Bazaras, Ž., Leonavičius, M., 882 Vsebina 2006 - Contents 2006 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 880-884 Krenevičius, A., Stupak, S., Petraitis, G.: Krenevičius, A., Stupak, S., Petraitis, G.: The Trdnost litegaželeza in normaliziranega litega Strength of as Cast Iron and Normalized Cast železa pri izmenični obremenitvi 558 Iron Subjected to Cyclic Loading Vidrih, B., Dolinar, M., Medved, S.: Povezava modela Vidrih, B., Dolinar, M., Medved, S.: The Connection podnebnih sprememb z modelom toplotnega Between the Climate Change Model and a odziva stavb - primer Slovenije Building's Thermal Response Model: A Case 568 of Slovenia He, J.-H., Zhang, L.-N.: Variacijski način He, J.-H., Zhang, L.-N.: A Variational Approach to določitve enačbe mazanja ne-newtonske the Establishment of a Lubrication Equation tanke plasti 587 for a Non-Newtonian Thin Film Luo, Y.-X., Huang, H.-Z., Fan, X.: Splošna siva Luo, Y.-X, Huang, H.-Z., Fan, X.: The Universal Grey prenosno matrična metoda in uporaba v Transfer Matrix Method and Its Application in izračunu naravnih frekvenc sistemov 592 Calculating the Natural Frequencies of Systems Rotar, F, Sluga, A.: Razvoj generične strukture in Rotar, F, Sluga, A: Development of the Generic Structure programskih modulov elementarnega and Programming Modules of an Elementary delovnega sistema porazdeljenega tiskanja 599 Work System for Distributed Printing Drev, D.: Izdelava politetrafluoretilenskih membran Drev, D: Producing Polytetrafluorethylene Membranes in njihovo laminiranje na tekstilne podloge 613 and Laminating Them on Textile Backings Žagar, B., Nardin, B., Glojek A., Križaj, D: Prilagodljivi Žagar, B., Nardin, B., Glojek A., Križaj, D.: An Adaptive sistem za hlajenje orodij za brizganje plastike System for Cooling Injection-Moulding s pomočjo termoelektričnih modulov 630 Moulds Via Thermoelectric Modules Radovanovič, M.: Nekatere možnosti določitve Radovanovič, M.: Some Possibilities for Determining rezalnih podatkov pri laserskem odrezovanju 645 Cutting Data when using Laser Cutting Barzdaitis, V, Bogdevicius, M.: Dinamično vedenje Barzdaitis, V, Bogdevicius, M.: The Dynamic vrtilnega sistema turbine 653 Behavior of a Turbine Rotating System Augustaitis, V K.,Šešok, N, Iljin, I.: Metoda raziskave Augustaitis, V. K., Sheshok, N, Iljin, I.: A Method nelinearnih upogibov valjev pri rotacijskem for Investigating the Nonlinear Bends of the ofsetnem tiskarskem stroju 662 Cylinders of a Web Offset Printing Station Filipovič, D., Krička, T: Energijska analiza pridelave Filipovič, D., Krička, T: An Energy Analysis of oljne ogrščice za potrebe proizvodnje biodizla Rapeseed Production for Biodiesel in na Hrvaškem 680 Croatia Pušavec, F, Krajnik, P, Kopač, J.: Odrezovanje Pušavec, F., Krajnik, P, Kopač, J.: High-Speed mehkih materialov z velikimi hitrostmi 706 Cutting of Soft Materials Stoič, A., Lucič, M., Kopač, J.: Vrednotenje stabilnosti Stoič, A., Lucič, M., Kopač, J.: Evaluation of the pri struženju v trdo 723 Stability During Hard Turning Dolinšek, S., Panjan, P., Syvanen, T, Ramovš, J.: Dolinšek, S., Panjan, P., Syvanen, T, Ramovš, J.: Lasersko sintranje orodja za tlačno litje Laser-Sintered Tools for the Die-casting of aluminija 738 Aluminum Cedilnik, M., Sokovič, M., Jurkovič, J.: Umerjanje in Cedilnik, M., Sokovič, M., Jurkovič, J.: Calibration preverjanje geometrijske natančnosti and Checking the Geometrical Accuracy of a računalniško krmiljenih obdelovalnih strojev 752 CNC Machine-Tool Antič, A., Hodolič, J., Sokovič, M.: Razvoj sistema Antič, A., Hodolič, J., Sokovič, M.: Development of za nadzor obrabe orodja pri struženju na a Neural-Networks Tool-Wear Monitoring temelju nevronskih mrež 763 System for a Turning Process Jakomin, M., Kosel, F, Batista, M., Kosel, T: Preskok Jakomin, M., Kosel F., Batista, M., Kosel, T.: Snap-through sistema plitve osnosimetrične bimetalne of the System for a Shallow Axially symmetric lupine z uporabo nelinearne teorije 785 Bimetallic Shell using Non-linear Theory Banovec, P., Kozelj, D., Šantl, S., Steinman, F: Izbira Banovec, P, Kozelj, D., Šantl, S., Steinman, F.: merilnih mest v vodovodnih sistemih z Sampling Design for Water Distribution genetskimi algoritmi 817 System Models by Genetic Algorithms Herakovič, N, Noe, D: Analiza delovanja pnevmatiČnega Herakovič, N., Noe, D.: Analysis of the Operation of Vsebina 2006 - Contents 2006 883 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 880-884 ventila s predkrmilnim piezoventilom 835 Podržaj, P., Kariž, Z.: Programljivi logični krmilniki na temelju rešitve algebraične Riccatijeve enačbe 852 Nikolič, D., Vujadinovič, R., Iida, N.: Raziskava učinkov različnih stopenj vračanja izpušnih plinov na temperaturo plamena in nastanek saj pri uporabi dizelskega goriva z različnimi T90 temperaturami destilacije 863 Pilot-Stage Piezo-Actuator Valves Podržaj, P, Kariž, Z.: Programmable Logic Controllers Based on the Algebraic Riccati Equation Solution Nikolič, D., Vujadinovič, R., Iida, N.: Experimental Study of the Effects of Different Exhaust Gas Recirculation Ratios on the Flame Temperature and Soot Formation when Using Diesel Fuels With Different T90 Distillation Temperatures Poročila Reports Odmevi iz varilske stroke 693 Welding News Strokovna literatura Professional Literatura Ocene knjig Book Reviews Iz revij From Journals Osebne vesti Personal Events Profdr. Jože Puhar - 80 letnik 66 ProfDr. Jože Puhar - 80th Anniversary Prešernove nagrade Fakultete za strojništvo Univerze 2005 Preseren’s Awards of Faculty of Mechanical v Ljubljani za leto 2005 67 Engineering of University of Ljubljana Prof. dr. Branko Gašperšič (1935-2006) 625 Prof. Dr. Branko Gašperšič (1935-2006) 80 let dr. Jožeta Hlebanje, zaslužnega profesorja 80th Anniversary of Dr. Jože Hlebanja, Professor Univerze v Ljubljani 697 Emeritus of the University of Ljubljana Bunshahova nagrada 699 Bunshah Award Akademik profesor dr. Janez Peklenik 80-letnik, 80th Anniversary of Professor Dr. Janez Peklenik, Academician and Professor Emeritus of the 777 University of Ljubljana Zoisovo in Puhovi priznanji 875 Zois and Puch Awards Častni doktorat - prof. dr. Manfred Geiger 876 Honors Doctor of Science - Prof. Dr. Manfred Geiger Prof. dr. Viljem Kralj - petinsedemdesetletnik 877 Prof. Dr. Viljem Kralj - 75 years Doktorati, magisteriji, specializacije in diplome Doctor‘s, Master’s, Specialisation’s and Diploma Degrees Pisma uredništvu Navodila avtorjem 626 Letters to the Editorial Board Instructions for Authors 884 Vsebina 2006 - Contents 2006 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 885-886 Navodila avtorjem - Instructions for Authors Navodila avtorjem - Instructions for Authors Članki morajo vsebovati: - naslov, povzetek, besedilo članka in podnaslove slik v slovenskem in angleškem jeziku, - dvojezične preglednice in slike (diagrami, risbe ali fotografije), - seznam literature in - podatke o avtorjih. Strojniški vestnik izhaja od leta 1992 v dveh jezikih, tj. v slovenščini in angleščini, zato je obvezen prevod v angleščino. Obe besedili morata biti strokovno in jezikovno med seboj usklajeni. Članki naj bodo kratki in naj obsegajo približno 8 strani. Izjemoma so strokovni članki, na željo avtorja, lahko tudi samo v slovenščini, vsebovati pa morajo angleški povzetek. Za članke iz tujine (v primeru, da so vsi avtorji tujci) morajo prevod v slovenščino priskrbeti avtorji. Prevajanje lahko proti plačilu organizira uredništvo. Če je članek ocenjen kot znanstveni, je lahko objavljen tudi samo v angleščini s slovenskim povzetkom, ki ga pripravi uredništvo. VSEBINA ČLANKA Članek naj bo napisan v naslednji obliki: - Naslov, ki primerno opisuje vsebino članka. - Povzetek, ki naj bo skrajšana oblika članka in naj ne presega 250 besed. Povzetek mora vsebovati osnove, jedro in cilje raziskave, uporabljeno metodologijo dela,povzetek rezulatov in osnovne sklepe. - Uvod, v katerem naj bo pregled novej šega stanja in zadostne informacije za razumevanje ter pregled rezultatov dela, predstavljenih v članku. - Teorija. - Eksperimentalni del, ki naj vsebuje podatke o postavitvi preskusa in metode, uporabljene pri pridobitvi rezultatov. - Rezultati, ki naj bodo jasno prikazani, po potrebi v obliki slik in preglednic. - Razprava, v kateri naj bodo prikazane povezave in posplošitve, uporabljene za pridobitev rezultatov. Prikazana naj bo tudi pomembnost rezultatov in primerjava s poprej objavljenimi deli. (Zaradi narave posameznih raziskav so lahko rezultati in razprava, za jasnost in preprostejše bralčevo razumevanje, združeni v eno poglavje.) - Sklepi, v katerih naj bo prikazan en ali več sklepov, ki izhajajo iz rezultatov in razprave. - Literatura, ki mora biti v besedilu oštevilčena zaporedno in označena z oglatimi oklepaji [1] ter na koncu članka zbrana v seznamu literature. Vse opombe naj bodo označene z uporabo dvignjene številke1. OBLIKA ČLANKA Besedilo članka naj bo pripravljeno v urejevalnilku Microsoft Word. Članek nam dostavite v elektronski obliki. Ne uporabljajte urejevalnika LaTeX, saj program, s katerim pripravljamo Strojniški vestnik, ne uporablja njegovega formata. Enačbe naj bodo v besedilu postavljene v ločene vrstice in na desnem robu označene s tekočo številko v okroglih oklepajih Papers submitted for publication should comprise: - Title, Abstract, Main Body of Text and Figure Captions in Slovene and English, - Bilingual Tables and Figures (graphs, drawings or photographs), - List of references and - Information about the authors. Since 1992, the Journal of Mechanical Engineering has been published bilingually, in Slovenian and English. The two texts must be compatible both in terms of technical content and language. Papers should be as short as possible and should on average comprise 8 pages. In exceptional cases, at the request of the authors, speciality papers may be written only in Slovene, but must include an English abstract. For papers from abroad (in case that none of authors is Slovene) authors should provide Slovenian translation. Translation could be organised by editorial, but the authors have to pay for it. If the paper is reviewed as scientific, it can be published only in English language with Slovenian abstract, that is prepared by the editorial board. THE FORMAT OF THE PAPER The paper should be written in the following format: - A Title, which adequately describes the content of the paper. - An Abstract, which should be viewed as a mini version of the paper and should not exceed 250 words. The Abstract should state the principal objectives and the scope of the investigation, the methodology employed, summarize the results and state the principal conclusions. - An Introduction, which should provide a review of recent literature and sufficient background information to allow the results of the paper to be understood and evaluated. - A Theory - An Experimental section, which should provide details of the experimental set-up and the methods used for obtaining the results. - A Results section, which should clearly and concisely present the data using figures and tables where appropriate. - A Discussion section, which should describe the relationships and generalisations shown by the results and discuss the significance of the results making comparisons with previously published work. (Because of the nature of some studies it may be appropriate to combine the Results and Discussion sections into a single section to improve the clarity and make it easier for the reader.) - Conclusions, which should present one or more conclusions that have been drawn from the results and subsequent discussion. - References, which must be numbered consecutively in the text using square brackets [1] and collected together in a reference list at the end of the paper. Any footnotes should be indicated by the use of a superscript1. THE LAYOUT OF THE TEXT Texts should be written in Microsoft Word format. Paper must be submitted in electronic version. Do not use a LaTeX text editor, since this is not compatible with the publishing procedure of the Journal of Mechanical Engineering. Equations should be on a separate line in the main body of the text and marked on the right-hand side of the page with numbers in round brackets. 885 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 885-886 Enote in okrajšave V besedilu, preglednicah in slikah uporabljajte le standardne označbe in okrajšave SI. Simbole fizikalnih veličin v besedilu pišite poševno (kurzivno), (npr. v, T, n itn.). Simbole enot, ki sestojijo iz črk, pa pokončno (npr. ms1, K, min, mm itn.). Vse okrajšave naj bodo, ko se prvič pojavijo, napisane v celoti v slovenskem jeziku, npr. časovno spremenljiva geometrija (ČSG). Slike Slike morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot sl. 1, sl. 2 itn. Posnete naj bodo v ločljivosti, primerni za tisk, v kateremkoli od razširjenih formatov, npr. BMP, JPG, GIF. Diagrami in risbe morajo biti pripravljeni v vektorskem formatu. Pri označevanju osi v diagramih, kadar je le mogoče, uporabite označbe veličin (npr. t, v, m itn.), da ni potrebno dvojezično označevanje. V diagramih z več krivuljami, mora biti vsaka krivulja označena. Pomen oznake mora biti pojasnjen v podnapisu slike. Vse označbe na slikah morajo biti dvojezične Preglednice Preglednice morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot preglednica 1, preglednica 2 itn. V preglednicah ne uporabljajte izpisanih imen veličin, ampak samo ustrezne simbole, da se izognemo dvojezični podvojitvi imen. K fizikalnim veličinam, npr. t (pisano poševno), pripišite enote (pisano pokončno) v novo vrsto brez oklepajev. Vsi podnaslovi preglednic morajo biti dvojezični. Seznam literature Vsa literatura mora biti navedena v seznamu na koncu članka v prikazani obliki po vrsti za revije, zbornike in knjige: [ 1 ] A. Wagner, I. Bajsič, M. Fajdiga (2004) Measurement of the surface-temperature field in a fog lamp using resistance-based temperature detectors, Stroj. vestn. 2(2004), pp. 72-79. [2] Vesenjak, M., Ren Z. (2003) Dinamična simulacija deformiranja cestne varnostne ograje pri naletu vozila. Kuhljevi dnevi ’03, Zreče, 25.-26. september 2003. [3] Muhs, D. et al. (2003) Roloff/Matek Maschinenelemente - Tabellen, 16. Auflage. Vieweg Verlag, Wiesbaden. Podatki o avtorjih Članku priložite tudi podatke o avtorjih: imena, nazive, popolne poštne naslove in naslove elektronske pošte. SPREJEM ČLANKOV IN AVTORSKE PRAVICE Uredništvo Strojniškega vestnika si pridržuje pravico do odločanja o sprejemu članka za objavo, strokovno oceno recenzentov in morebitnem predlogu za krajšanje ali izpopolnitev ter terminološke in jezikovne korekture. Avtor mora predložiti pisno izjavo, da je besedilo njegovo izvirno delo in ni bilo v dani obliki še nikjer objavljeno. Z objavo preidejo avtorske pravice na Strojniški vestnik. Pri morebitnih kasnejših objavah mora biti SV naveden kot vir. Units and abbreviations Only standard SI symbols and abbreviations should be used in the text, tables and figures. Symbols for physical quantities in the text should be written in italics (e.g. v, T, n, etc.). Symbols for units that consist of letters should be in plain text (e.g. ms1, K, min, mm, etc.). All abbreviations should be spelt out in full on first appearance, e.g., variable time geometry (VTG). Figures Figures must be cited in consecutive numerical order in the text and referred to in both the text and the caption as Fig. 1, Fig. 2, etc. Pictures may be saved in resolution good enough for printing in any common format, e.g. BMP, GIF, JPG. However, graphs and line drawings sholud be prepared as vector images. When labelling axes, physical quantities, e.g. t, v, m, etc. should be used whenever possible to minimise the need to label the axes in two languages. Multi-curve graphs should have individual curves marked with a symbol, the meaning of the symbol should be explained in the figure caption. All figure captions must be bilingual Tables Tables must be cited in consecutive numerical order in the text and referred to in both the text and the caption as Table 1, Table 2, etc. The use of names for quantities in tables should be avoided if possible: corresponding symbols are preferred to minimise the need to use both Slovenian and English names. In addition to the physical quantity, e.g. t (in italics), units (normal text), should be added in new line without brackets. All table captions must be bilingual. The list of references References should be collected at the end of the paper in the following styles for journals, proceedings and books, respectively: [1] A. Wagner, I. Bajsič, M. Fajdiga (2004) Measurement of the surface-temperature field in a fog lamp using resistance-based temperature detectors, Stroj. vestn. 2(2004), pp. 72-79. [2] Vesenjak, M., Ren Z. (2003) Dinamična simulacija deformiranja cestne varnostne ograje pri naletu vozila. Kuhljevi dnevi ’03, Zreče, 25.-26. september 2003. [3] Muhs, D. et al. (2003) Roloff/Matek Maschinenelemente - Tabellen, 16. Auflage. Vieweg Verlag, Wiesbaden. Author information The information about the authors should be enclosed with the paper: names, complete postal and e-mail addresses. ACCEPTANCE OF PAPERS AND COPYRIGHT The Editorial Committee of the Journal of Mechanical Engineering reserves the right to decide whether a paper is acceptable for publication, obtain professional reviews for submitted papers, and if necessary, require changes to the content, length or language. Authors must also enclose a written statement that the paper is original unpublished work, and not under consideration for publication elsewhere. On publication, copyright for the paper shall pass to the Journal of Mechanical Engineering. The JME must be stated as a source in all later publications. 886 Navodila avtorjem - Instructions for Authors