Analiza zvo�nih lastnosti kompozitnih materialov
An Analysis of the Acoustic Properties of Composite Materials
Samo �ali - Uro� �nidari� - Janez Kopa�
Da bi izdelovalcu okrovov zvo�nikov olaj�ali izbiro optimalnega materiala, smo testirali razli�ne vrste plasti�nih materialov, oplemenitenih z drobno mletimi lesnimi delci, ter jih primerjali z aluminijem, MDF (srednje gosti komopziti), polistirenom drugega izdelovalca zvo�nikov (JVC) in ABS (akrilonitril -butadien - stiren). Vsi preizku�ani materiali so bili v obliki plo�� z izmerami 150*150 mm, njihova debelina pa je bila 2 mm. Ker so bile v ospredju testov zvo�ne lastnosti materialov, smo merili njihov relativni zvo�ni upor, relativno du�enje zvo�ne radiacije in faktor viskoznega du�enja. Prvi dve veli�ini sta izpeljani iz gostote in relativnega modula elasti�nosti, ki ju lahko dobimo iz meritev frekven�nega odziva prosto vpetih preizku�ancev. Rezultati ka�ejo, da se s pravilno izbiro drobno mletih lesnih delcev in plasti�ne osnove lahko pribli�amo materialu MDF, ki velja za zelo dobro izbiro pri izdelavi okrova zvo�nika. © 2004 Strojni�ki vestnik. Vse pravice pridr�ane. (Klju�ne besede: materiali kompozitni, lastnosti akusti�ne, analize modalne, metode presku�anja)
To help select the best material for loudspeaker boxes, we tested various types of polymer materials that are filled with fine, ground wood particles. In addition, we compared these materials with aluminium, MDF (medium-density fiberboard), polystyrene from another producer of loudspeaker boxes (JVC), and ABS (acrylnitril - butadiene - styrene). All the specimens were in the shape of square plates with dimensions 150X150 mm and the thickness 2 mm. Because the analysis was focused on the acoustic properties of the materials, we measured their relative sound-wave resistance, the relative damping of the sound radiation and the viscous-damping factor. The first two parameters are derived from the density and the relative modulus of elasticity, which can be obtained from measurements of the frequency response for free-supported specimens. The results show that a careful selection of fine, ground wood particles and polymer can give a satisfactory approximation to MDF, which is known as one of the best choices for the production of loudspeaker boxes.
© 2004 Journal of Mechanical Engineering. All rights reserved. (Keywords: composite materials, acoustic properties, modal analysis, testing methods)
Modalna analiza tankih �tirikotnih plo�� je razmeroma dobro raziskano podro�je (�1] do �4]). Poleg vpetja in oblike preizku�ancev na njihovo modalno obna�anje vsekakor vpliva tudi gradivo. Modalno obna�anje objekta pomeni amplitudo, du�enje in gibalne oblike pri posameznih frekvencah nihanja tega predmeta. Primerjava zvo�nih lastnosti razli�nih materialov pomeni torej primerjavo modalnega obna�anja preizku�ancev z enako obliko in vpetjem, pri �emer je spremenljivka vrsta materiala. V na�em primeru smo za dolo�anje zvo�nih lastnosti preizku�ancev uporabili metodo za prosto vpete, izotropne, tanke �tirikotne plo��e.
0 INTRODUCTION
The modal analysis of thin, square-shaped plates is relatively well investigated (�1] to �4]). Besides the specimens� shape and the type of support, their modal behaviour depends on the choice of material. The modal behaviour of an object means the amplitude, the damping and the modal shapes at certain frequen-cies of vibration (oscillation) for this object. A compari-son of the acoustic properties of different materials is therefore a comparison of specimens with equal shape and the some type of support, where the only variable is the material. In our case, for a definition of the acous-tic properties of the specimens, a method for free-sup-ported, isotropic and thin square-shaped plates was
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�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Podoben postopek je �e bil uspe�no uporabljen pri meritvah zvo�nih lastnosti izrazito izotropnega (ortotropnega) materiala, tj. lesa �5]. Zato predpostavljamo, da morebitna izotropnost preizku�anih materialov ni vplivala na kakovost analize, kar pa bo �e podrobneje razlo�eno. Cilj raziskave je bila metoda za merjenje zvo�nih lastnosti kvadratastih tankih plo�� ter kriterij za dolo�anje zvo�ne kakovosti razli�nih plasti�nih materialov z lesnimi vklju�ki, ki so namenjeni za velikoserijsko proizvodnjo brizganih okrovov za srednje kakovostne zvo�nike za poslu�anje glasbe.
Akusti�ne lastnosti materialov ne moremo definirati enopomensko. Za primer: med najbolj�e materiale za zvo�ne plo��e lesenih glasbil uvr��amo smreko, medtem ko se ta in podobne vrste lesa sploh ne uporabljajo pri gradnji okrovov za zvo�nike. Razlog je seveda v razli�nosti namena, ki ga imata zvo�na plo��a glasbila in okrov zvo�nika.
Modul elasti�nosti E in gostota r sta edini veli�ini, ki dolo�ata zvo�ni upor Z in du�enje zvo�nega sevanja J trdnih teles �6]:
applied. A similar approach was already applied during measurements of the acoustic properties of an extremely non-isotropic (orthotropic) material � wood �5]. It is as-sumed, therefore, that the probable non-isotropy of the tested materials did not affect the analysis, which will be explained in more detail later. The aim of this research was to devise method for measuring the acoustic properties of thin, square-shaped plates and a criterion for determin-ing the acoustic quality of different polymer materials with wood particles. These materials are intended for the large-scale production of middle-quality loudspeaker boxes using injection-moulding technology.
The acoustic properties of materials cannot be defined simply. For example, the best choice for the sound boards of wooden instruments is spruce, whereas this and similar types of wood are not used in the production of loudspeaker boxes. The reason for this lies in the different requirements of a sound board and a loudspeaker box.
The modulus of elasticity E and the density r are the only two variables that denote the sound-wave resistance Z and the damping of the sound radiation J of solids �6]:
Z = �r -E
r
Razlike v E in r se izra�ajo tudi v spremembi dinami�nega Youngovega modula. Ta modul izra�a razmerje togosti in specifi�ne te�e preizku�anca. Togost in gostota se lahko primerjata z E in r. Slika 1
Variations in E and r will also result in changes to the dynamic Young�s modulus. This modulus is defined as the ratio of the stiffness to the specific gravity of the speci-mens. The stiffness and the density can be compared to E
«4.
<" �
smreka, bor (najbolj�a kakovost)/spruce, pine (highest quality)
les za zvo�ne plo��e/wood for sound boards smreka, jelka/spruce, fir
javor, bukev, jesen/maple, beech, ash
vrba/willow
. kremenjak, aluminij, steklo/quartz, aluminium, glass
guma/rubber
srebro/silver
Z=Ȋ. (kg / m 2 s)
Sl. 1. Odvisnost du�enja zvo�nga sevanja (J) od zvo�nega upora (Z) za razli�ne vrste lesa in druga
gradiva �6] Fig. 1. Dependence of radiation damping (J) on sound wave resistance (Z) for several types of wood and
other materials �6]
�vmskmsmm
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
prikazuje odvisnost du�enja zvo�nega sevanja od zvo�nega upora za razli�ne vrste lesa in nekatera druga gradiva.
Prikazane odvisnosti potrjujejo, da sta pri zvo�nih plo��ah glasbil za�elena majhen zvo�ni upor in veliko du�enje zvo�nega sevanja. Z drugimi besedami, ve�ji dinami�ni Youngov modul zvo�ne plo��e je ugodnej�i.
Zvo�na plo��a glasbila mora namre� �im ve� prejete energije spremeniti v zvo�no energijo, izgube zaradi notranjega trenja morajo zato biti �im manj�e. Z drugimi besedami, pri �im manj�em faktorju viskoznega du�enja (definicija sledi) mora biti du�enje zvo�nega sevanja za zvo�ne plo��e glasbil �im ve�je. Faktor viskoznega du�enja 5 lahko izra�unamo na podlagi faktorja kakovosti Q iz ena�be, ki velja za malo du�ene sisteme �7]:
and r, respectively. Figure 1 shows the dependence of the damping of sound radiation on the sound-wave resistance for different wood species and other materials.
The presented relations confirm that in the sound boards of musical instruments, low sound-wave resistance and high damping of the sound radiation are desirable. In other words, a high rather than a low dynamic Young�s modulus of the sound boards is preferred.
The wooden resonant boards of musical instruments should translate most of the input energy into sound radiation. Therefore, losses due to internal friction are not desired. In other words, the factor of viscous damping (definition follows) should be as low as possi-ble, and the damping of the sound radiation should be as high as possible. The viscous-damping factor d can be calculated from the expression for the quality factor Q, which applies to low-damped systems �7]:
2d f2-f1
kjer je f lastna frekvenca modalnega na�ina, f1 in f2 pa pomenita frekvenci, kjer je amplituda frekven�nega vrha enaka P/2 (sl. 2).
Za primer resonan�ne frekvence s slike 2, ki pomeni lastni modalni na�in, je faktor viskoznega du�enja premo sorazmeren koeficientu viskoznega du�enja b in obratno sorazmeren zmno�ku modalne mase m in modalne togosti k �7]:
d =
where f0d is the natural frequency, and f1 and f2 are frequencies where the amplitude is P/2 (see Figure 2). In the case of the resonant frequency, which is presented in Figure 2, and which presumably indicates a natural mode, the factor of viscous damping is proportional to the coefficient of viscous damping b, and inversely proportional to the product of the modal mass m and the stiffness k �7]:
24k-
Podobno razmi�ljanje velja pri izbiri optimalnega materiala za okrove zvo�nikov, namenjenih za poslu�anje glasbe. V tem primeru mora okrov prepre�evati pojav izrazitih resonanc in odmevov, ki nastanejo zaradi izvira zvoka -vibracij membrane zvo�nika. To je logi�no, saj �elimo predvajati le signal, ki prihaja iz elektronskih komponent v mebrano zvo�nika, in to brez dodatnih negativnih vplivov, ki bi se utegnili pojaviti zaradi prisotnosti okrova. Po drugi strani okrov zvo�nika ne sme imeti pretiranih du�ilnih lastnosti, saj bi to pomenilo prevelike izgube zvo�ne
A similar way of thinking is applied when the selection of the best material for loudspeaker boxes is considered. In this case the box of a loudspeaker has to prevent the phenomenon of distinctive resonances and echoes that appear due to a sound source � vibra-tions of the loudspeaker diaphragm. Because we wish to produce only a signal from the electronic compo-nents into the loudspeaker diaphragm (without any additional and negative effects due to the loudspeaker box), this is logical. On the other hand, the loudspeaker box should not exhibit an excessive damping quality, because this would mean too high sound-energy losses.
Sl. 2. Definicija amplitude prvega resonan�nega vrha in faktorja viskoznega du�enja d Fig. 2. Definition of both amplitude of the first resonant peak and factor of viscous damping d
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�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
energije. To bi se lahko poznalo kot opazno zmanj�anje glasnosti ustvarjenega zvoka, kakor tudi preveliko du�enje vseh ali dolo�enih frekven�nih pasov. Potemtakem du�enje zvo�nega sevanja, ki pravzaprav pomeni zmo�nost sevanja zvoka v okolico, pri okrovu zvo�nikov ne sme biti preveliko, vsekakor pa mora biti bistveno manj �e kakor v primeru zvo�nih plo�� glasbil. Lahko bi rekli, da manj�anje velike stopnje du�enja zvo�nega sevanja, ki je zna�ilna za zvo�ne plo��e glasbil, pomeni izbolj�evanje zvo�nih lastnosti gradiva za okrov zvo�nika �8].
Z gotovostjo lahko trdimo, da majhen zvo�ni upor pomeni majhno zvo�no impedanco. To se ka�e v razmeroma hitrem odvajanju zvo�ne energije, torej premajhna zvo�na impedanca v primeru zvo�nih plo�� glasbil pomeni glasne, a kratko trajajo�e �ume brez glasbenega zna�aja �9]. V primeru okrova zvo�nika bi premajhen zvo�ni upor torej lahko pomenil interferenco tak�nih �umov z vibriranjem membrane zvo�nika, kar seveda ni za�eleno. Po drugi strani razmeroma velik zvo�ni upor pomeni preveliko zvo�no impedanco, torej pretirano po�asno odvajanje zvo�ne energije v okrov zvo�nika. To bi pomenilo mo�nost pojava stojnega valovanja in odmevov ustvarjenega zvoka znotraj okrova zvo�nika, kar vsekakor ne bi prispevalo h kakovosti zvoka.
Zvo�ni upor je glede na ena�bo (1) proporcionalen zmno�ku modula elasti�nosti in gostote materiala. Ta dva dolo�ata velikost modalne togosti k in mase m. Ob predpostavki, da je zvo�ni upor nespremenljiv, je vi�ji faktor viskoznega du�enja d posledica ve�jega koeficienta b (ena�bi (1) in (4)). Razmeroma veliko (majhno) du�enje 5 pomeni torej razmeroma velike (majhne) izgube zvo�ne energije znotraj okrova zvo�nika �8]. �e torej lahko govorimo o neki idealni vrednosti zvo�nega upora gradiva, potem za to vrednost obstaja tudi idealna vrednost faktorja viskoznega du�enja 5, ki ne sme biti ne previsoka, ne prenizka, torej mora biti optimalna. Nadalje, razmeroma velike vrednosti zvo�nega upora okrova zvo�nika pomenijo ob nespremenljivi vrednosti koeficienta b razmeroma majhne vrednosti faktorja viskoznega du�enja, kar pomeni mo�nost pojava odmeva in stojnega valovanja �9]. Po drugi strani pomeni razmeroma majhna vrednost zvo�nega upora pri nespremenljivem koeficientu b veliko vrednost faktorja S, s tem pa mo�nost prevelikih zgub zvo�ne energije od membrane zvo�nika v okrov (�8] in �9]).
Veliko okrovov zvo�nikov, med njimi tudi zelo kakovostni sistemi, je narejenih iz materiala MDF. Ta je v osnovi podoben iverni plo��i, le da gre pri MDF za bolj drobno mlete lesne delce. Primerjava strukture za MDF in klasi�no iverno plo��o je prikazana na sliki 3.
Z veliko gotovostjo lahko torej trdimo, da so zvo�ne lastnosti materiala MDF referen�ne
A consequence of this would be a significant decrease in the loudness, as well as too high damping of all, or only certain, frequency ranges. Therefore, the damping of sound radiation, which means the ability to radiate sound energy into the surroundings, must not be too high. In any case, it has to be significantly lower than in the case of the sound boards of musical instruments. One can say that a decrease of the relatively high level of damping of sound radiation, which is typical for musical instruments, indicates an improvement in the acous-tic properties of a material for loudspeaker boxes �8].
With great certainty one can say that a low value of sound-wave resistance means low sound im-pedance. This results in a relatively fast sound-en-ergy drain. Consequently, too low sound impedance of the sound boards results in loud and short-lasting noises without musical character �9]. Therefore, too low sound-wave resistance of a loudspeaker box could cause an interference of these noises with the loud-speaker diaphragm, which of course is not desired. On the other hand, a relatively high sound-wave resistance means too high acoustic impedance, which means that sound-energy drain into the loudspeaker box is too slow. This could result in the appearance of standing waves and echoes of the produced sound inside the loudspeaker box. Of course, this would not con-tribute to the sound quality in a positive way.
According to expression (1) the sound-wave resistance is proportional to the product of the modulus of elasticity and the material density. These two quanti-ties determine the magnitude of the modal stiffness k and the mass m. Considering that sound-wave resistance is a constant, an increase in the viscous-damping factor d is a consequence of an increase of coefficient b (see Equa-tions (1) and (4)). A relatively high (low) damping d there-fore means high (low) losses of sound energy inside the loudspeaker box �8]. If we are allowed to speak about an ideal magnitude of sound-wave resistance for a certain material, then for this value there is also an ideal factor of viscous damping d. This factor should be neither too high nor too low, i.e. it should be optimised. Next, if we assume that coefficient b is a constant, then a relatively high value of sound-wave resistance of the loudspeaker box will result in a relatively low factor of viscous damp-ing. This can lead to the appearance of echoes and standing waves �9]. On the other hand, a relatively low sound-wave resistance, at a constant coefficient b, means a high magnitude of factor d. This can significantly increase the losses of sound energy from the loudspeaker diaphragm into the box (�8] and �9]).
A lot of loudspeaker boxes, including high-quality systems, are made of MDF (medium-density fibreboard). In comparison to the particle board, MDF consists of smaller wood particles. Figure 3 shows a comparison between the MDF and the particle board structure.
With great certainty we can say that the acoustic properties of MDF are reference in terms of
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Sl. 3. Primerjava med MDF (levo) in iverne plo��e (desno) Fig.3. Comparison between MDF (left) and particle board (right)
lastnosti, �e imamo v mislih iskanje optimalnega materiala za okrov zvo�nikov. V nadaljevanju je predstavljena metoda za merjenje zvo�nih lastnosti kvadratastih tankih plo�� iz razli�nih materialov, predvsem kompozitov s plasti�no osnovo in z vklju�ki iz drobno mletih lesnih delcev. Pri tej metodi smo torej vrednosti veli�in (i) du�enje zvo�nega sevanja, (ii) zvo�ni upor in (iii) faktor viskoznega du�enja, ki so zna�ilni za MDF, ozna�ili za �elene vrednosti. Te so torej merilo za dolo�anje najbolj�e kombinacije plasti�ne osnove z drobno mletimi lesnimi delci kot polnilom.
1.1 Priprava preizku�ancev
Preizku�anci so bili kvadrataste plo��e z dimenzijo 150x150 mm. Oznake ter �tevilo preizku�ancev v vzorcu (n), njihova debelina (d), gostota (r) in sestava oziroma vrsta materiala so prikazani v preglednici 1.
Uporabljena sta bila dva tipa drobno mletih lesnih delcev. V primeru preizku�ancev z oznako vz4 so bili to razmeroma veliki delci iz mehkega lesa (smreka), v preostalih preizku�ancih pa razmeroma majhni delci iz trdega lesa (bukev). Kakor vidimo iz preglednice 1, se preizku�anci vz1 in vz4 lo�ijo samo po vsebini lesnih delcev. Razlika med preizku�anci vz2 in vz6 je v tipu polipropilena, sicer pa so masni dele�i vseh treh sestavnih komponent (pregl. 1) enaki. Enako velja za preizku�ance vz3 in vz5. Polistirenski preizku�anci vz9 so bili narejeni z injekcijskim brizganjem zdrobljenega okrova zvo�nikov proizvajalca JVC (tip XV THA35). Polistirenski preizku�anci vz10 so bili narejeni za primerjavo s preizku�anci vz9. Postopek izdelave vseh preizku�ancev s plasti�no osnovo je bilo iztiskanje, torej zvezna predelava plasti�nih mas, v katerem se polimerna talina potiska skozi orodje specifi�nega profilnega prereza. Material MDF je narejen iz drobno mletih lesnih delcev, ki so zlepljeni med seboj. Postopek lepljenja
the best material for loudspeaker boxes. A method for measuring the acoustic properties of square-shaped and thin plates made of various materials, especially of composites with a polymer matrix and fine, ground wood particles, is presented in the next section. In this method the values of (i) the damping of sound radiation, (ii) the sound-wave resistance, and (iii) the viscous-damping factor, which are typical for MDF are denoted as the desired values. Thus, these values present a criterion for determining the most suitable combination of a polymer material and fine wood particles as filler.
1.1 Specimens preparation
The specimens were square-shaped plates with dimensions of 150x150 mm. Denotations and the number of specimens in the group (n), their thickness (d), density (r), and composition or material type are presented in Table 1.
In the experiments two types of fine, ground wood particles were applied. For specimens vz4 this pulp consisted of relatively coarse particles of softwood (spruce), whereas for other specimens the fine, ground wood particles consisted of relatively small particles of hardwood (beech). As one can see from Table 1, the only difference between specimens vz1 and vz4 is in the content of wood particles. The difference between specimens vz2 and vz6 is in a type of polypropylene, whereas the mass portions of all three main components (see table 1) are the same. The same is true for specimens vz3 and vz5. Specimens based on polystyrene vz9 were made by injection moulding ground loudspeaker boxes JVC (type XV THA35). Specimens vz10 (also based on polystyrene) were used for a comparison with specimens vz9. All the specimens based on polymer were produced by extrusion, which means the continuous manufacturing of polymers, where a polymer melt is pushed through a die with a specific cross-section. MDF is made of fine, ground wood particles that are glued together. The process of gluing is performed at high
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�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Preglednica 1. Lastnosti preizkusancev Table 1. Properties of specimens
* kopolimer/copolymer **homopolimer/homopolymer
poteka pri visoki temperaturi in visokem tlaku. V primerjavi z borovim ali smrekovim lesom ima MDF obi�ajno pribli�no dvakrat manj�i modul elasti�nosti in pribli�no 70% ve�jo gostoto.
Preglednica 2 ka�e mehanske lastnosti preizku�ancev vz1 - vz6. Iz neenakih lastnosti v pre�ni in vzdol�ni smeri (glede na smer iztiskanja) vidimo, da so vsi ti preizku�anci anizotropni.
1.2 Meritve zvo�nih lastnosti preizkusancev
Slika 4 ka�e mesto meritve in merilno opremo. Vsi preizku�anci so bili vpeti na okoli 2 m dolgi elasti�ni vrvici, debeline okoli 0,3 mm. S tem zagotovimo najmanj�i vpliv vpetja na dinami�no obna�anje preizku�anca. Vzbujanje je bilo opravljeno s posebej izdelano napravo, katere glavni del je piezoelektri�ni merilnik vzbujevalnega impulza, ki je prikazan na sliki 5. Tipi�na oblika vzbujevalnega impulza in odzivnega signala je prav tako prikazana na sliki 5. Kakor vidimo, je amplitudna os vzbujevalnega signala prikazana brezrazse�no, saj prikazani vzbujevalnik ni umerjen v
temperature and pressure. In comparison to a fir or spruce wood, the MDF�s modulus of elasticity is approximately 100% smaller and its density is approximately 70% higher. The mechanical properties of specimens vz1 � vz6 are shown in Table 2. Based on unequal proper-ties in the transversal and longitudinal directions (ac-cording to the direction of extrusion) one can see that all the specimens are non-isotropic.
1.2 Measurements of the acoustic properties of the specimens
The measurement arrangement is shown in Figure 4. All the specimens were suspended on ap-proximately 2-m-long (0.3 mm in diameter) nylon line. This ensured a negligible effect of the specimen�s support on its dynamic behaviour. A special device with a piezoelectric sensor was used to excite the specimens, as shown in Figure 5. Typical shapes of the excitation and output signals are shown in Figure 5 as well. One can see that the amplitude axis of the input signal is presented on a dimensionless scale. The reason for this is that the excitation device was not calibrated for
stran 585
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Preglednica 2. Mehanske lastnosti preizkusancev vz1 do vz6 Table 2. Mechanical properties of specimens vz1 to vz6
* indeks te�enja/percolation index
zvo�no izolirana komora (mav�na obloga)/sound isolated chamber (gypsum)
mikrofon s predoja�evalnikom/ microphone with preamplifier
povratni gib/
return movement
t ' vzbujanje/
excitation
napajalnik B..l&Kjaer WB 1372/
power source Bruel&Kjaer WB 1372
oja�evalnik MIKOJ 01-95/ amplifier MIKOJ 01-95
merilna kartica National Instruments
AT-A2150C/
data acquisition board National Instruments
AT-A2150C
nabojni oja�evalnik/ charge amplifier
mikrofon/microphone: Bruel&Kjaer (tip/type 4188) predoja�evalnik/preamplifier: Bruel&Kjaer (tip/type 2671)
programska
oprema/software
X = 230 mm m
Sl. 4. Merilno mesto in oprema Fig. 4. Measurement place and arrangement
fizikalnih enotah. To seveda ne zmanj�a njegove uporabnosti, saj je kon�ni cilj meritev t.i. frekven�ni odziv preizku�ancev, pri katerem odzivni signal delimo z vzbujevalnim v frekven�nem podro�ju, torej merimo razmerje med odzivnim signalom ter vzbujevalnim signalom (mehanskim impulzom) �7]. Poleg tega nas niso zanimale absolutne vrednosti frekven�nih odzivov preizkusancev, temve� le njihova primerjava.
Akusti�ni odziv preizku�anca na mehansko impulzno motnjo (brezrazse�no), merjen s kondenzatorskim mikrofonom, ima enoto Pa, torej je amplituda zveznega frekven�nega odziva preizku�anca izra�ena v Pa �7]:
measurements in physical units. This does not affect its applicability because the aim of the measurements is the frequency response of the specimens, which is defined as the ratio of the output to the input signal �7]. In addition, we were interested in a comparison of the different frequency responses of the specimens rather than their absolute values.
The acoustic response of the specimen due to the mechanical impulse (on a dimensionless scale) which is measured with a condenser microphone is defined in Pascal units. Therefore, the amplitude of the continuous frequency-response function of a specimen is defined in Pascal units �7]:
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�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
vzbujevalna naprav(shematsko)/ excitation device(schematically)
elektromagnet
(povratni gib)/
electromagnet
(return movement)
piezoelektri�ni element/ piezoelectric element
* povratni gib/
�elezo/ magnetic iron
mesto vzbujanja/ place of excitation
return movement
4, vzbujanje/ excitation
os vrtenja/ axis of rotation
preizku�anec/specimen
ro�ica s piezoelektri�nim
elementom/
lever with piezoelectric element
t
vzbujevalni signal/ excitation signal
izhodni signal/ output signal
Sl. 5. Shematski prikaz naprave za vzbujanje preizkusancev, mesto vzbujanja ter oblika vzbujevalnega in
odzivnega signala Fig. 5. Schematic representation of excitation device, place of excitation, and both excitation and output
H(f)
kjer so: Gxy(f) povpre�ni kri�ni energijski spekter vhodnega in izhodnega signala, Gxx(f) povpre�ni energijski spekter vhodnega signala, f pa frekvenca z izmero Hz. �e nepovpre�ena spektra izra�unamo iz naslednjih ena�b �7]:
where Gxy (f) is an average cross power spectrum of both the input and output signals, Gxx (f) is an average power spectrum of an input signal, and f is the frequency with dimension Hz. Before the averag-ing both spectra from expression (4) are �7]:
Gxy( f ) = Sx( f )-Sy*( f ) G ( f ) = S( f )-S*( f )
kjer so: S (f) frekven�na slika vhodnega in S (f) izhodnega �asovnega signala, S *(f) in S *(f) pa njuni konjugirano kompleksni vrednosti y S tako definiranim frekven�nim odzivom se izognemo napaki zaradi navzo�nosti �uma. Pri meritvah frekven�nega odziva je pomembna koheren�na funkcija, ki je merilo za mo� izhodnega signala zaradi vhodnega signala. �e je koherenca 1, potem je bil ves izhodni signal povzro�en zaradi vhodnega, �e pa je 0, potem izhodni signal ni posledica vhodnega. Koheren�na funkcija f je �7]:
where Sx(f)and Sy(f)are frequency transformations of the input and output signals, respectively, and Sx*(f) and Sy*(f) are their complex conjugates. Such an approach ensures that a frequency-response func-tion is not influenced by the presence of noise. A criterion of the quality of the measurements is the coherence function, which indicates the power of the output signal due to the input signal. When this function is 1 then all the power of the output signal is a consequence of the input signal. When the coher-ence was 0 then the output signal was not caused by the input signal. The coherence function g2 is �7]:
stran 587
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Gxx(f)Gyy*(f)
kjer je Gyy(f) povpre�ni energijski spekter izhodnega signala, * pa ozna�uje kompleksno konjugacijo. Nepovpre�eni spekter G (f) izra�unamo analogno G (f) iz ena�be (6), tako da indeks nadomestimo z .
Dejansko so zaradi analogno/digitalne premene z merilno opremo zvezni spektri v izrazih (4) do (7) diskretni. Diskretni amplitudni frekven�ni spekter signala dobimo s hitro Fourierjevo preslikavo signala v �asovnem prostoru (�7] in �10]):
where Gyy (f ) is an average power spectrum of the output signal, and * indicates its complex conjuga-tion. A non-averaged spectrum Gyy(f) is calculated by analogy to Gxx(f) from expression (6), where the index x is replaced by y.
As a matter of fact, the continuous spectra in expressions (4) to (7) are discrete due to the ana-logue/digital conversion with the measurement equip-ment. The discrete amplitude spectrum of a signal is obtained with a fast Fourier transformation of this signal in a time domain (�7] and �10]):
FFT(s-Df)
� Xf(n-Dt)-e
- j2p sn / N
kjer so: s = 0, 1, 2 ... N/2, Df frekven�na lo�ljivost, T �as snemanja, N�tevilo diskretnih to�k, Dt �asovni korak med diskretnimi to�kami, f(n-Dt) diskretna vrednost signala v n-ti to�ki in j=-1. Frekvenca vzor�enja/je bila 8 kHz pri �tevilu diskretnih to�k N=4096. s FFT(s-Df) torej pomeni diskretno Fourierjevo preslikavo digitaliziranega diskretnega signala �asovne funkcije f(n-Dt). �e velja, da je z f(n-Dt) opisan vhodni signal v vzbujani predmet, katerega frekven�ni odziv merimo, velja tudi (�7] in �10]):
Sx(s-Df)*FFT(s-Df)
kjer je S(s-Df) frekven�na slika vhodnega signala v diskretni obliki. Podobno lahko re�emo tudi za izhodni signal. Torej, �e je z f(n-Dt) opisan izhodni signal iz vzbujanega predmeta, katerega frekven�ni odziv merimo, velja tudi (�7] in �10]):
where s = 0, 1, 2 ... N/2, Df is frequency resolution, T is time of signal recording, N is the number of discrete points, Dt is the time interval between these discrete points, f(n-Dt) is a discrete value of the signal in the n-th point, and j=>-1. The sampling frequency fs was 8 kHz and N was 4096. Thus, FFT(s-Df) indicates a discrete Fourier transformation of a digital discrete signal of a time-dependent function f(nDt). If f(n-Dt) describes the input signal into an object whose frequency response is measured, then the following is true (�7] and �10]):
� = f(n'Dt)-e
- j2p sn / N
where S (s-Df) is the frequency transformation of the input signal in a discrete form. Similarly, if f(n-Dt) describes the output signal from the excited object then the following is true (�7] and �10]):
Sy (s ■ Df) * FFT(s ■ Df ) = - £ f(n ■ Dt) ■ e
j2p sn / N
kjer je S (s-Df) frekven�na slika izhodnega signala v diskretni obliki.
Dvostranski amplitudni diskretni frekven�ni spekter je (�7] in �10]):
where S (s-Df) is a frequency transformation of the output signal in a discrete form.
The two-sided amplitude spectrum in a discrete form is (�7] and �10]):
\FFT(s ■ Df)\/N
N - �tevilo diskretnih to�k signala mora biti za natan�no Fourierjevo preslikavo 2n, n=1, 2 ... Izraz (11) predstavlja dvostranski spekter, po mno�enju z 2 pa dobimo amplitudni frekven�ni spekter, ki pomeni amplitude frekven�nih komponent signala. Frekven�ne komponente so na frekven�ni osi spektra med seboj oddaljene za Df (Hz). Zveza med frekven�no lo�ljivostjo in trajanjem signala je (�7] in �10]):
For a high-quality Fourier transformation the number of discrete points N has to be 2n, n=1, 2 ... Expression (11) represents a two-sided spectrum, however after multiplying it by a factor at 2 the result is an amplitude spectrum that represents the amplitudes of the fre-quency components of a signal. The frequency resolution between neighbouring components of this spectrum is Df (Hz). The relation between the frequency resolution and the time of recording is (�7] and �10]):
Df=1/T
2 jgnnat��llMliBilrSO |
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Zaradi poenostavitve naj za nadaljnjo analizo velja, da je vsak izmerjeni frekven�ni odziv obravnavan kot zvezni odziv H(f) iz ena�be (4), �etudi je dejansko nezvezen, torej odvisen od diskretnih vrednosti frekvence f z lo�ljivostjo Df.
Kakovosten analogno/digitalni pretvornik na merilni kartici z vgrajenim analognim filtrom za odstranitev visokih frekvenc je zagotovilo, da je bil Nyquistov pogoj (frekvenca vzor�enja vsaj 2-krat vi�ja od najvi�je frekvence v signalu) vedno izpolnjen. �as snemanja vstopnega sunka in akusti�nega odziva preizku�ancev je bil 0,512 s, torej je bila frekven�na lo�ljivost spektrov 1,953 Hz (ena�ba (12)). Snemanje impulza in rezultirajo�ega zvo�nega tlaka je bilo so�asno. Z obdelavo signala s programsko opremo je bilo poskrbljeno, da sta se oba, izhodni in vhodni signal, za�ela in kon�ala z amplitudo ni�. To prispeva h kakovosti frekven�ne analize, dokaz za to pa je bila vrednost koheren�ne funkcije med 0,95 in 1,0 za analizirano frekven�no obmo�je (prvega resonan�nega vrha) za vse materiale. Komponente frekven�nih spektrov so izra�ene v vrednostih kpk (korena povpre�ja kvadratov).
Za meritev morebitnih razlik v akusti�nem odzivu kvadratastih plo�� iz razli�nih materialov lahko uporabimo ena�bo (13). Ta povezuje frekvenco n-tega modalnega na�ina fn in mehanske lastnosti homogene, izotropne in prosto vpete kvadrataste plo��e �11]:
fn=Cn-t
kjer so C konstanta, odvisna od n-tega modalnega na�ina, t debelina plo��e, n Poissonovo razmerje in l dol�ina (�irina) plo��e. V analizi, ki bo prikazana, je bil analiziran prvi modalni na�in za vse plo��e. Ta modalni na�in ima najve�je pomike na sredini vzbujene plo��e, proti robovom pa so pomiki postopoma manj�i (podobno kakor pri trampolinu) (�11] in �12]). �e torej plo��o vzbudimo v sredini, vzbudimo prvi modalni na�in v najve�ji mo�ni meri Veli�ina fn je izmerjena, t, l, n in r so to�no ali vsaj pribli�no znane. Za prvi modalni na�in preizku�ancev iz ena�be (13) izhaja:
Due to a simplification let us denote that each measured frequency response function is analysed as a continuous response H(f) from expression (4), although in reality all the spectra were discontinuous, thus they were dependent on discrete values of frequency f with frequency resolution Df.
A high-quality analogue/digital converter on a data-acquisition board with an anti-aliasing filter ensured that the Nyquist criterion (the sampling frequency has to be at least two times higher than the highest frequency of interest in a signal) was fulfilled. The recording time of the input impulse and the specimen�s acoustic response was 0.512 sec. Thus, the frequency resolution of all spectra was 1.953 Hz (see expression (12)). The recording of the mechanical impulse and of the resulting sound pressure was performed simultaneously. Additional processing of both input and output signals was performed in order to set the amplitudes at their beginning and end to zero. This improves the quality of the frequency transformation, which was confirmed with a coherence function higher than 0.95 for the analysed frequency range (first resonant peak). The frequency-spectrum components are expressed in rms values.
To measure eventual differences in the acoustic response of square-shaped plates from various materials we can use expression (13). This includes the frequency of n-th mode/ and the mechanical properties of homogeneous, n isotropic and free-supported square-shaped plates �11]:
where C is a constant that depends on n-th mode, t is the plate n thickness, n is Poisson�s ratio, and l is the length (width) of the plate. In the analysis which follows, only the first mode of all specimens was analysed. The largest displacements for this mode are in the middle of the plate. The amplitudes of the displacements diminish towards the plate�s edges (like with trampoline) (�11] and �12]). Thus, when the specimen is excited in its geometrical centre, the first mode is excited as much as possible. The quantity fn is measured, and t, l, n and r are exactly or approximately known. Based on expression (13) for a first mode it follows:
Z zelo veliko verjetnostjo lahko re�emo, da je Poissonovo razmerje za aluminij 0,3, za MDF med 0,3 in 0,4, za preostale testirane materiale pa pribli�no 0,4 (�13] in �14]), vsekakor pa med 0,3 in 0,5. V nadaljevanju bosta tako pri analizi vseh materialov, razen aluminija, upo�tevani spodnja (0,3) in zgornja meja (0,5) Poissonovega razmerja
2 REZULTATI IN ANALIZA
Ker gre v nadaljevanju le za relativno primerjavo veli�in, lahko konstanto C1 in izmere
With great certainty we can say that Poisson�s ratio for aluminium is 0.3, for MDF between 0.3 and 0.4, and for other tested materials about 0.4; in any case between 0.3 and 0.5 (�13] and �14]). There-fore, in the following analysis the lower (0.3) and the upper (0.5) limit for Poisson�s ratio are considered.
2 RESULTS AND ANALYSIS
Because in the following analysis only a relative comparison of quantities is presented, it is reason-
gfin�OtJJlMlSCSD
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Preglednica 3. Rezultati meritev Table 3. Results of measurements
vseh veli�in iz ena�be (14) izvzamemo. Tako namesto veli�ine E dobimo brezrazse�ni modul elasti�nosti E l. Preglednica 3 prikazuje zbrane rezultate meritev za vse testirane plo��e. Prikazane so srednje vrednosti frekvenc in faktorjev viskoznega du�enja prvega modalnega na�ina ter relativnih modulov elasti�nosti.
�e v ena�bah (1) in (2) namesto veli�ine E upo�tevamo E l, in �e za gostoto ne upo�tevamo izmer, dobimo namesto veli�ine Z t.i. relativni zvo�ni upor Z l in namesto veli�ine J relativno du�enje zvo�nega sevanja Jr l. Obe relativni veli�ini in faktor viskoznega du�enja so za vse testirane materiale predstavljeni na slikah 6 oziroma 7. Zaradi zelo velike vrednosti relativnega zvo�nega upora za aluminij (vz8) so na sliki 8 �e enkrat prikazane vrednosti za vse nekovinske materiale.
able to exclude from Equation (14) the constant C1 and the dimensions of all quantities. Instead of quantity E we consequently obtain the dimensionless modulus of elasticity Erel. The results of the measurements of all the plates are presented in Table 3. More precisely, the mean values of the frequencies and the factors of viscous damping of the first mode, and the relative moduli of elasticity are presented.
Considering Erel instead of E and ignoring the units for density in Expressions (1) and (2), we obtain a relative sound-wave resistance Zrel instead of Z, and a relative damping of sound radiation Jrel instead of J. In addition to the viscous-damping factor, these two quan-tities are shown in Figures 6 and 7 for all the tested materials. Due to a high value of the relative sound-wave resistance for aluminium (vz8), Figure 8 shows Erel, Jrel and d for all the non-metal materials.
Sl. 6. Relativni zvo�ni upor in du�enje zvo�nega sevanja (vsi materiali) Fig. 6. Relative sound resistance and sound radiation damping (all materials)
2 jgnnat��llMliBilrSO |
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Sl. 7. Faktor viskoznega du�enja Fig. 7. Factor of viscous damping
3 -2,5 -
2 -1,5 -
1 -0,5
S- =
c
□ relativni zvo�ni upor (Poissonovo razmerje je 0,3)/relative sound resistance (Poisson's ratio is 0,3)
S relativni zvo�ni upor (Poissonovo razmerje je 0,5)/relative sound resistance (Poisson's ratio is 0,5)
■ relativno du�enje zvo�nega sevanja (Poissonovo razmerje je 0,3)/relative sound radiation damping (Poisson ratio is 0,3)
vz10 vz11
Sl. 8. Relativni zvo�ni upor in du�enje zvo�nega sevanja (vsi nekovinski materiali) Fig. 8. Relative sound resistance and sound radiation damping (all non-metal materials)
S slik 6 do 8 je razvidno, da imajo materiali vz4, vz9 in vz10 vse tri analizirane veli�ine (zvo�ni upor, du�enje zvo�nega sevanja in faktor viskoznega du�enja) podobne tistim za referen�ni material MDF. Pri tem velja, da imajo faktor viskoznega du�enja skoraj identi�en tistemu za MDF. Med temi tremi materiali je vz4 v splo�nem najbli�e MDF, saj ima ve�ji zvo�ni upor in du�enje zvo�nega sevanja kakor materiala vz9 in vz10. Iz pregl. 1 je razvidno, da ima samo material vz4 drobno mlete lesne delce iz mehkega lesa. Vpliv drobno mletih lesnih delcev (trdi oziroma mehki les) je razviden iz primerjave materialov vz1 in vz4.
One can see from Figures 6 to 8 that materials vz4, vz9 and vz10 indicate similar acoustic properties (sound-wave resistance, damping of sound radiation and viscous-damping factor) to the reference material MDF. In addition, their factor of viscous damping is almost identical to that of MDF. Among all three materials the closest to MDF is vz4, because it has a higher sound-wave resistance and a higher damping of sound radiation than materials vz9 and vz10. It is evident from Table 1 that only fine, ground wood particles in vz4 are made of softwood. The influence of fine, ground wood particles (hardwood, softwood) is evident from a com-parison between materials vz1 and vz4. It seems that
I isfinHi(s)bJ]�M]ifln;?n 04
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Kakor ka�e je ta vpliv precej zna�ilen, saj je faktor viskoznega du�enja za material vz1 za pribli�no 80% ve�ji od tistega za vz4, medtem ko sta za oba materiala zvo�ni upor in du�enje zvo�nega sevanja primerljiva.
Eden najbolj raz�irjenih materialov za tudi najbolj kakovostne okrove zvo�nikov je t.i. MDF. Zato smo zvo�ne lastnosti, ki jih ima ta material, definirali za referen�ne. Zvo�ne lastnosti gradiva so definirane z (i) zvo�nim uporom, (ii) du�enjem zvo�nega sevanja in (iii) faktorjem viskoznega du�enja. V primerjavi z zvo�nimi plo��ami glasbil, ki morajo �im ve� vibracijske energije sevati v okolico (pri �im manj�ih notranjih izgubah), je funkcija okrova zvo�nikov druga�na. Podrobneje, du�enje zvo�nega sevanja mora biti za zvo�ne plo��e glasbil razmeroma veliko, zvo�ni upor pa majhen. Energija tresenja membrane zvo�nika se mora pravilno absorbirati, pri tem pa stopnja absorpcije ne sme biti prevelika ali premajhna. Re�emo lahko torej, da mora biti zvo�ni upor materiala za okrov zvo�nikov razmeroma velik, da se ne oja�ujejo resonan�ne frekvence zvo�nika kot celote. Du�enje zvo�nega sevanja, torej sevanje zvoka v okolico, pa mora biti razmeroma majhno. Vendar bi previsok zvo�ni upor pomenil slabo prehajanje zvo�ne energije v okrov zvo�nika, kar pomeni veliko mo�nost pojava odmevov in stojnega valovanja. Lep primer tega je velika vrednost zvo�nega upora za aluminij (sl. 6). Ni si te�ko predstavljati, da bi se aluminijski okrov zvo�nika kazal v neza�elenih stranskih pojavih, na primer stojno valovanje in posledi�no resonan�na nihanja membrane zvo�nika. Logi�no je namre�, da namen okrova zvo�nika ni poudarjati dolo�enih frekvenc, temve� prav nasprotno, tak�ne pojave mora prepre�iti. Po drugi strani je majhen zvo�ni upor materiala povezan z razmeroma majhno zvo�no impedanco in hitrim odvajanjem zvo�ne energije v okrov zvo�nika, kar se lahko ka�e v kratko trajajo�ih, a izrazitih resonan�nih frekvencah okrova zvo�nika. To lahko pomeni velike izgube zvo�ne energije, sploh �e se razmeroma majhen zvo�ni upor pojavi v kombinaciji z razmeroma visokim faktorjem viskoznega du�enja (velike izgube zaradi notranjega trenja).
Smiselno je skleniti, da so zvo�ne lastnosti, ki smo jih izmerili na materialu MDF, optimalne. Tako lahko re�emo, da je material vz4 med vsemi testiranimi materiali najbolj primeren za okrove zvo�nikov. Ker sta oba, du�enje zvo�nega sevanja in zvo�ni upor za ta material nekoliko manj�a v primerjavi z MDF, se pojavi vpra�anje, kako obe veli�ini pove�ati, ne da bi bistveno spremenili faktor viskoznega du�enja, ki se zelo dobro ujema s tistim za MDF.
this influence is quite significant because the viscous-damping factor for material vz1 is approximately 80% higher than that one for vz4, whereas both the sound-wave resistance and the damping of sound radiation for these two materials are comparable.
3 CONCLUSION
One of the most common materials for loudspeaker boxes, including high-quality products, is MDF (medium-density fibreboard). Therefore, we denoted the acoustic prop-erties that are significant for this material as the reference properties. The acoustic properties of a material are defined by (i) the sound-wave resistance, (ii) the damping of sound radiation, and (iii) the viscous damping factor. In comparison to the sound boards of musical instruments, which should radiate their vibration energy into the surroundings as much as possible (in addition to minimal internal losses), the func-tion of loudspeaker boxes is different. More precisely, the damping of sound radiation for the sound boards of musical instruments has to be relatively high, and the sound-wave resistance should be low. The energy contained in the vibra-tions of a loudspeaker diaphragm has to be absorbed in a proper way, which means that the intensity of the absorption should be neither too high nor too low. We can say that the sound-wave resistance of a material for loudspeaker boxes has to be relatively high in order not to amplify the resonant frequencies of a whole loudspeaker. The damping of the sound radiation, and thus the radiating of sound into the surroundings, has to be relatively low. However, too high sound-wave resistance means an insufficient transition of acoustic energy into the loudspeaker box, which can result in phenomena like echoes and standing waves. A nice example of this is the high value of sound-wave resistance for aluminium (see Figure 6). It is not hard to understand that an aluminium loudspeaker box would re-sult in undesired effects like standing waves and, conse-quently, resonant vibrations of the loudspeaker diaphragm. It is logical that the purpose of a loudspeaker box is not to emphasize certain frequencies, but on the contrary, to pre-vent this phenomenon. On the other hand, too low sound-wave resistance correlates with a relatively low sound im-pedance and fast sound-energy drain into the loudspeaker box. This can result in short-lasting but distinctive reso-nant frequencies of the loudspeaker box. Finally, this can lead to high sound-energy losses, especially if relatively low sound-wave resistance appears together with a rela-tively high factor of viscous damping (high losses due to internal friction).
It is reasonable to conclude that the acoustic properties measured for MDF are the best ones. Thus, one can say that the most suitable material for loudspeaker boxes among the tested materials is vz4. Because both the damping of sound radiation and the sound-wave resistance for this material are slightly lower in compari-son to MDF, the question is how to increase these two parameters and not significantly affect the viscous-damp-ing factor that corresponds to that for MDF.
�ali S., �nidari� U., Kopa� J.: Analiza zvo�nih lastnosti - An Analysis of the Acoustic Properties
Zahvala
Prispevek je bil pripravljen s sodelovanjem projekta Eureka 2819: �Razvoj in ozna�ba okolju prijazne termoplastike� ter podjetja ECOPLAST iz Slovenije.
Aknowlegment
This paper was prepared with the coopera-tion with Eureka project 2819: �Development and characterisation of eco-friendly thermoplastics�, and the ECOPLAST factory in Slovenia.
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