Journal of Mechanical Engineering no. 9 year 2021 volume 67 Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis. The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue. The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s). Editor in Chief Vincenc Butala University of Ljubljana, Faculty of Mechanical Engineering, Slovenia Technical Editor Pika Škraba University of Ljubljana, Faculty of Mechanical Engineering, Slovenia Founding Editor Bojan Kraut University of Ljubljana, Faculty of Mechanical Engineering, Slovenia Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SV-JME, Aškerceva 6, SI-1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@sv-jme.eu, http://www.sv-jme.eu Print: Demat d.o.o., printed in 250 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Mitjan Kalin University of Ljubljana, Faculty of Mechanical Engineering, Slovenia Vice-President of Publishing Council Bojan Dolšak University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: SEMI-CRYSTALLINE STRUCTURE OF POLYPROPYLENE POLYMER. Some polymers form partially crystalline structures when cooled from the melt. The size and density of the crystals, to a large extent, determine the polymer materials’ physical properties. The crystalline structure and, consequently, the properties can be controlled during polymer production (cooling). This means that the same polymer can be made in a way that will be almost completely transparent or completely opaque and that it can show either ductile or brittle behaviour. The figure shows a polypropylene material made with a polarized microscope. Rather large crystals (~ 1mm) formed when polymer melt was gradually cooled to ambient temperature. Image Courtesy: Marko Bek, PhD, University of Ljubljana, Faculty of Mechanicaal Engineering, Laboratory for Experimental Mechanics International Editorial Board Kamil Arslan, Karabuk University, Turkey Hafiz Muhammad Ali, King Fahd U. of Petroleum & Minerals, Saudi Arabia Josep M. Bergada, Politechnical University of Catalonia, Spain Anton Bergant, Litostroj Power, Slovenia Miha Boltežar, University of Ljubljana, Slovenia Filippo Cianetti, University of Perugia, Italy Janez Diaci, University of Ljubljana, Slovenia Anselmo Eduardo Diniz, State University of Campinas, Brazil Igor Emri, University of Ljubljana, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Imre Horvath, Delft University of Technology, The Netherlands Aleš Hribernik, University of Maribor, Slovenia Soichi Ibaraki, Kyoto University, Department of Micro Eng., Japan Julius Kaplunov, Brunel University, West London, UK Iyas Khader, Fraunhofer Institute for Mechanics of Materials, Germany Jernej Klemenc, University of Ljubljana, Slovenia Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Peter Krajnik, Chalmers University of Technology, Sweden Janez Kušar, University of Ljubljana, Slovenia Gorazd Lojen, University of Maribor, Slovenia Darko Lovrec, University of Maribor, Slovenia Thomas Lben, University of Bremen, Germany George K. Nikas, KADMOS Engineering, UK Tomaž Pepelnjak, University of Ljubljana, Slovenia Vladimir Popovic, University of Belgrade, Serbia Franci Pušavec, University of Ljubljana, Slovenia Mohammad Reza Safaei, Florida International University, USA Marco Sortino, University of Udine, Italy Branko Vasic, University of Belgrade, Serbia Arkady Voloshin, Lehigh University, Bethlehem, USA General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue). 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Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9 Contents Contents Strojniški vestnik - Journal of Mechanical Engineering volume 67, (2021), number 9 Ljubljana, September 2021 ISSN 0039-2480 Published monthly Papers Khurram Ali, Adeel Mehmood, Israr Muhammad, Sohail Razzaq, Jamshed Iqbal: Control of an Anthropomorphic Manipulator using LuGre Friction Model - Design and Experimental Validation 401 Dragan Milcic, Amir Alsammarraie, Miloš Madic, Vladislav Krstic, Miodrag Milcic: Predictions of Friction Coefficient in Hydrodynamic Journal Bearing Using Artificial Neural Networks 411 Almatbek Kydyrbekuly, Gulama-Garip Alisher Ibrayev, Tangat Ospan, Anatolij Nikonov: Multi- parametric Dynamic Analysis of a Rolling Bearings System 421 Youyu Liu, Yi Li, Xuyou Zhang, Bo Chen: Trajectory-Tracking Control for Manipulators Based on Fuzzy Equivalence and a Terminal Sliding Mode 433 Yanxiong Liu, Yuwen Shu, Wentao Hu, Xinhao Zhao, Zhicheng Xu: Active Vibration Control of a Mechanical Servo High-speed Fine-Blanking Press 445 Xuan Bo Tran: Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 458 Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 401-410 Received for review: 2021-05-28 Š 2021 Journal of Mechanical Engineering. All rights reserved. Received revised form: 2021-08-17 DOI:10.5545/sv-jme.2021.7258 Original Scientific Paper Accepted for publication: 2021-08-18 Control of an Anthropomorphic Manipulator using LuGre Friction Model -Design and Experimental Validation Khurram Ali 1 -Adeel Mehmood 1 -Israr Muhammad 1 -Sohail Razzaq 2 -Jamshed Iqbal 3,* 1COMSATS University Islamabad, Departmentof Electrical and Computer Engineering, Islamabad,Pakistan 2COMSATS University Islamabad, Departmentof Electrical and Computer Engineering, Abbottabad,Pakistan 3Universityof Hull,Facultyof Science and Engineering, Departmentof Computer Science andTechnology, Hull,UK Automation technologyhasbeenextensively recognizedasan emerging feldinvarious industrial applications.Recent breakthroughinfexible automationis primarilyduetodeploymentof roboticarmsor manipulators.Autonomyinthese manipulatorsis essentiallylinkedwiththeadvancementsin non-linear control systems.Theobjectiveofthis researchistoproposearobustcontrolalgorithmforafvedegreeoffreedom(DOF)roboticarmtoachievesuperiorperformance and reliabilityinthe presenceoffriction.Afriction compensation-based non-linear control has been proposed and realizedfor the robotic manipulator. The dynamicmodeloftherobothasbeenderivedbyconsideringthedynamicfrictionmodel.Theproposed three-statemodelisvalidatedforallthejointsofthe manipulator. The integral sliding mode control (ISMC) methodology has been designed; the trajectories of system everytime begin from the sliding surface and it eliminates the reaching phase with assistance of integral term in the sliding surface manifold. The designed control law has been frst simulated in Matlab/Simulinkenvironmenttocharacterizethe controlperformanceintermsoftrackingofvarioustrajectories.The results confrmtheeffectivenessofthe proposed control law with model-based friction compensation. The transient parameters like settling and peak time have improvement as well have better results with friction than without considering the friction. The proposed control law is then realized on an in-house developed autonomous articulated robotic rducationalplatform(AUTAREP)andNImyRIO hardware interfacedwith LabVIEW.Experimental resultsalso witnessedthetrajectorytrackingbythe robotic platform. Keywords: robotic manipulator, lugre friction model, sliding mode control Highlights • Adynamicmodelofrobot manipulatorispresentedinthisworkbyconsideringthefrictionbetweenthejoints. • DynamicLuGremodelisusedforfrictionwhich refectstherealfrictionphenomenonbyconsideringstick-slipmotionandstribeckeffect. • Slidingmode control techniqueisusedtodesignthe controllawusingLyapunovstability theorem. • Controllerperformanceisevaluatedwithand withoutfriction compensationin Matlab. • Proposed controllerisalsotestedonexperimentaltestbenchusingNImyRio1900andAUTAREProbotsystem. 0 INTRODUCTION Robotic arms are playing a pivotal role in today’s industrial world [1]. The industrial automation methods are majorally affected by the development in the feld of robotics. Robotic arms are basically the mechanical replication of the human body arm. Research shows that they can outperform the humans in terms of strength, precision, speed and repeatability, thus, they can save time with improved performance [2]. Owing to these concrete reasons, robotic arms have acquired an invaluable place in industry[3]. To achievehigh precision in robotic manipulators, friction in joints is the major limitation[4]. A manipulator, being inherently non-linear in nature, can demonstrate steady-state error and limit cycles and hence may suffer from degraded performance[5]. It affects both dynamic and static performances and can results in instability due to coupling effects. Henceforth, joint friction compensation in robot has been one of the major research problem in control design throughout the years. Friction is one of the main reasons of unwanted effects and system performance degradation [6]. Friction is a force which mostly occurs in moving systems exhibiting a complex phenomenon and can be divided into two types. The frst type is known as static friction Fs which is often defned for the bodies at rest. The other type is known as dynamic friction which occurs when one body is moving with respect to some other body in contact. It is further divided into three components. One component is recognized as sliding friction and others are rolling andpivot friction[7]. Static friction models like Karnopp model are diffcult to implement while simulating the dynamic systems[8]. The dynamic friction models established so far cover majority of non-linear phenomena and they exhibit continuous behavior when the velocity is zero. These characteristics make the dynamic models more useful and convenient for software simulations and analysis of control algorithms. It is known that some of the foremost restrictions to realize admirable breakthrough in mechanical frameworks is the presence of friction, which is a non-linear phenomenon and hard to depict analytically [9]. Friction should be taken into account at an early stage throughout the system design by reducing it at probable minimum limit through careful design. Moreover, research shows that most of the dynamic models developed sofar are incapable to capture the real friction phenomenon experienced by surfaces in contact[10]. The LuGre friction model is used in the current research since it describes important friction behaviour and characteristics such as stick-slip motion and stribeck effect etc. [11]. While demonstrating advantages of friction modelling, the LuGre friction model can result in numerically stiffand tough system dynamics. Moreover, the LuGre model performs like a linear spring pair when it is linearized for small motions. For a control engineer, it is suitable to analyse the impact of friction on the feedback system in order to design a control law that reduces the effect of friction. In robotics, non-linear, centralized trajectory tracking refers to the input torque computation which guarantees precise tracking of reference trajectory. The research community has suggested numerous control approaches varying from simple linear feedback control to sophisticated algorithms. Sliding mode control (SMC) is one of the robust non-linear control techniques based on principle of the variable structure control theory. The key features of SMC include system robustness[12]and its stability against matched uncertainties [13]. The dynamics of a robotic manipulator having multi-DOF is complicated and highly non-linear[14]. The position of a single joint and its two higher derivatives have a strong impact on all joints of robotic manipulator. Moreover, gravity, external forces, modelling uncertainties, and friction can impact the robot motion. The state-of-the-art research works have been reported in the literature on robotic manipulator that uses friction models instead of a disturbance. An adaptive robust control (ARC) algorithm has been proposed for an electric cylinder with the LuGre friction model in[15]and theLyapunov approach has been suggested for system stability and ARC handlesexternal disturbances. In[16], to lessen the impact of estimation errors the robust sliding mode controller with an adaptive block compensation controller is proposed. The inverted pendulum and planar multi-joint robot with friction model are used tovalidatethe controller. Authorsin[17]used adaptive ISMC with time-delay estimation. In this paper, an ISMC for anthropomorphic manipulator based on modifed LuGre model with friction compensation is proposed. The following is a brief description of the main contribution: • Mathematical model of AUTAREP robotic manipulator is centered on a fve DOF has been derived by considering the kinematics and dynamics of the robot manipulator, as well as the LuGre friction model. • Robust controller has been designed to compensate the dynamic friction behaviors and drive the position and speed to track the desired trajectories. The integral sliding mode control (ISMC) law, which is based on the Lyapunov function, is used to compensate for the system’s parametric and load torque uncertainties. • The proposed control law is then implemented on a custom developed platform using LabVIEW interface with NI myRIO hardware having motors driving circuits. This article is organised as follows; Section 1 derives the mathematical model of a single joint along with LuGre friction model. In Section 2, SMC based control law is designed. After the control design, in section 3, outcomes of control design have been analysed and experimental results are discussed. Section4concludes the article. 1 MATHEMATICAL MODELLING A robotic manipulator usually consists of servo motors, optical encoders, and one or more controllers. The robotic manipulator used for research and analysis in the current research isAUTAREP ED-7220C[18] shown in Fig. 1. The robotic manipulator has fve degrees of movement actuated with DC servo motors. A single motor is used to move each joint except the wrist joint where pitch and roll motions are controlled by two motors. The model characterizes an extremely non-linear system since velocities as well as positions of links are coupled dynamically. For feedback from joints, the robotic arm has optical encoders mounted on each joint servo motor axis, which provides information about its position. An adequate motion control algorithm could be accomplished through mathematical modelling by considering the dynamics of the robot manipulator. Kinematics of a robot gives the relationship between joint angles/translations and the location (position and orientation) of its end-effector[19]. The kinematic model of the manipulator established on Fig. 1. AUTAREP Manipulator ED-7220C with joints corresponding range of motions Denavit-Hartenberg (D-H) parameters is reported in [20]. The process of modelling the robot dynamics involves derivation of velocities and inertia matrices, computation of kinetic and potential energies and fnal formulation of torque equation [21]. Each link of the manipulator is considered as a set of number of elements for which potential and kinetic energies are developed. In the present research work, the Lagrangean approach is used for the formulation of dynamics [22]. Fig. 2 illustrates the kinematic representation ofAUTAREP manipulator. In order to function motors, the model should be able to evaluate the joint torques which are system control input[23]. The manipulator’s dynamic equation is given by Eq. (1): ˜f =M (qi) q¨i +Cc (qi,q.i)q.i +G(qi)+Tf (q.i),(1) where for n joints, M(qi) ˜Rn×n is the mass/inertial matrix, Cc(qi) ˜Rn×n represents the coriolis and centripetal forces, G(qi) ˜Rn×1 is the gravitational matrix, Tf (q.i)˜Rn×1 represents friction torques. The term ˜f ˜Rn×1isthevectorofinput torques appliedto the joints of the robot. Friction is one of the major causes of undesirable effects. Numerous dynamic models for friction have been suggested in literature, including the Dahl model and LuGre model[24], etc. The dynamic Dahl friction model forms the origin of the LuGre friction model and is an integrated dynamic model of friction. However, it does not Fig. 2. Kinematic representation of the robotic arm where (L1 = 385 mm, L2 = 220 mm, L3 = 220 mm, L4 = 155 mm ) consider all the non-linear phenomena caused by friction. In the literature, it has been shown that static models cannot capture certain characteristics of true frictional phenomena that are experimentally seen in high friction systems. Therefore, LuGre model of friction is a dynamic friction model related to the bristle interpretation of friction and the average defection force of elastic springs is used to model friction. The velocity determines the average bristle defection for a steady state motion. It is lower at low speeds, implying that steady-state defection decreases with increasing velocity. Rate dependent friction phenomena such as varying break-away force and frictional lag are also included in the model. Therefore, the LuGre model is used here for friction modeling and is given as Tf =°0 z +°1z .+ f (.),(2) where f (.)= °2., z is internal friction state and the term Tf is predicted friction torque. Tf =°0 z +°1z .+°2..(3) The dynamics of friction state z can be defned as: |.| z .=.-°0 z,(4) g (.) where g (˜)is given by: -(|˜/˜s|) g (˜)=Fc +(Fs -Fc)exp.(5) The LuGre model can illustrate several properties of real friction, for example the limit cycle displacement in stick-slip motion. For steady-state friction measurement the functions f (˜) and g(˜) can be used. The model has a signifcant drawback in that, this does not take into account the dwell time effect, which implies that it cannot model variations in friction force when two surfaces are maintained in contact at rest for a set period. The LuGre model is a dynamical system with the following properties: internal state dissipativity, boundedness, I/O dissipativity with constant °1 and velocity-dependent °1(˜) [25]. Most of these are reasonable characteristics of a model illustrating friction. Parameters of LuGre friction model are obtained from literature[26]and aregiveninTable1. Table 1. Parameters of LuGre friction model Parameter Value Unit Description °0 2750 Nm/rad Stiffness coeffcient Fs 8.875 Nm Static friction torque Fc 6.975275 Nm Coulomb friction torque °1 45.2 Nmˇs/rad Damping coeffcient °2 1.819 Nmˇs/rad Viscous friction coeffcients ˜s 6.109 ˇ10-2 rad/s Velocity Let q1 is the position vector, q2 is the velocity vector and z3 is internal friction state. Thus the state-space model of a single joint can be represented by Eq. (6): q.1 =q2 q.2 =M-1˜.f -Ccq2+Gq1+°0z3+°2q2+°1z.3° |q2| z.3 =q2-°0 z3. g (q2) (6) 2 CONTROL DESIGN SMC technique permits control of non-linear processes subjected to high model uncertainties and external disturbances[27]. In present research, SMC based law with friction compensation is implemented on AUTAREP manipulator. SMC law is divided into two phases i.e. the sliding phase and the reaching phase [28]. The system trajectories achieve the required sliding surface in the case of reaching phase, defned as: q.i = f (qi)+g(qi)u +˜(qi,t).(7) The vector felds f (qi) and g (qi) are the non linear functions representing the system with matched uncertainties ˜(qi,t). For the control design, the frst step is to defne a sliding surface and the whole system states are forced in such a way that the sliding surface converges to zero in a fnite time and remains there as long as there is no change in reference input[29]. The control objective in this work is to make sure that the robot should track the predefned trajectory qdi. To accomplish this aim, a sliding surface based on error signal for each link is described in Eq. (8): . Ssi =Cse.si +.siesi +Isi esidt,(8) where Cs, .si and Isi are positive constants. The tracking function is established for well-defned sliding surface in Eq. (8). Thus, the sliding surface will converge to zero with the applied control effort. This will ensure the tracking error esi converges to the origin. The control input of SMC law comprises of two parts. The frst part is equivalent control law (useq)and it is a continuous term. The second part is discontinuous control law(usdis)with the signum function. u =useq +usdis.(9) The equivalent control input (useq) is acquired through sliding surface by considering S.si = 0 which is obtained after taking the time derivative of Eq. (8): S.si =Cse¨si +.sie.si +Isiesi.(10) In SMC, the objective is accomplished by setting S.si =0. Substituting this value in Eq. (10): 0=Cse¨si +.sie.si +Isiesi,(11) where .esi =q.di -q.i and e¨si = q¨di -q¨i are the frst and second time derivatives of error function respectively. Substituting error dynamics, we get: 0=Cs(q¨di -q¨i)+.si(q.di -q.i)+Isi(qdi -qi),(12) q¨i =(Cs)-1[(.si (q.di -q.i)+Isi(qdi -qi)]+ q¨di.(13) The equivalent control effort useq is chosen as: useq =M(qi)(Cs)-1[(.si(q.di -q.i)+Isi(qdi -qi)] + q¨d +(Cc)(qi,q.i)+G(qi)+Tf (q.i).(14) In order to compensate the dynamic model uncertainties, the discontinuous function usdis is defned as given in Eq. (15): usdis =-ksisign(Ssi)-.Ssi,(15) where ksi and .are positive constants. The equivelant control input useq drives the system from initial state to sliding surface and ensures that S. = 0. However, due to different uncertainties and external disturbances the sliding surface may not converge to zero. Lyapunov stability-based method is used for analysis and control design which guarantees the system stability. For stability analysis, a positive defniteLyapunov energy-function for SMC is chosen as Eq. (16): V = 1 S2 si.2 (16) Taking derivative of Eq. (16), we obtain: .V =SsiS.si. (17) By substituting the value of S.si given in Eq. (12) and substituting useq as well usdis in Eq. (17), we get: V .=Ssi(-ksisign(Ssi)-˜Ssi),(18) V .°-ksi|Ssi|-˜(Ssi)2 .(19) The derivative of Lyapunov function V ., turns out to be negative semi-defnite function which is suffcient condition forLyapunov stability. Therefore, the dynamics of system will converge in fnite time and the error will converge to zero for the applied input given in Eq. (20): °f =M(qi)Cs -1[(.si(q.di -q.i)+Isi(qdi -qi)] +q¨di +Cc (qi,q.i)+G(qi)+Tf (q.i) (20) -ksignSsi -˜Ssi. 2.1 Experimental Setup The proposed control algorithm is also tested on the experimental setup using LabView environment and NI myRIO-1900 hardware. The experimental setup is shown in Fig. 3. The myRIO-1900 is a tool used to implement multiple design concepts with one reconfgurable input/output (I/O) device. It is a portable device that can be used to design and develop various applications in robotics as well in mechatronic systems. It consists of a Xilinx Field Programmable Gate Arrays (FGPA), digital I/O lines and analog inputs as well as analog outputs. It has a built-in accelerometer with distinct features such as Universal asynchronous receiver-transmitter (UART), audio input and output terminals and a pulse width modulator. The myRIO-1900isaWiFi-enabled edition which allows for speedy integration into remotely control embedded applications. The myRIO-1900 is programmed with LabVIEW using graphical programming language (G-Programming) Fig. 3. Hardware setup (a) Motor driving board (b) Main board Fig. 4. Motors driver board for robotic manipulator withahigh datafowandabuilding block that canbe used for link data acquisition, logical operations, and analysis. In the proposed solution, digital outputs are used for controlling the actuators that are interfaced through indigenously designed H-bridges by using power transistors TIP147 and TIP142 as shown in Fig. 4a with current rating of fve Amperes. The movement of rotary joints in a robot manipulator is controlled by using motor driver circuitry and the Fig. 4b presents the main board which connects the robotic manipulator with motor driving board. The desired speed of the motor is accomplished by controlling the voltage via pulse width modulation of the corresponding duty cycle. The actuators are integrated with optical encoders to determine the joint positions and their signals are sent to the myRIO for data processing. 3 RESULTS AND DISCUSSIONS Matlab/Simulink is used for simulation. To validate the model, simulations are conducted using Matlab R2016a installed on a core i5 processor system having 6 GB random access memory. Step response is presented for the elbow joint of the robot manipulator as shownin Fig.5while sinusoidal responseis plotted for the base joint as presented in Fig. 6. The desired step input signal of amplitude of 0.1 rad is given to the closed loop system and Fig. 5a shows simulation results for a control law that is designed to track a unit step function with and without friction compensation. Although there is no overshoot inFig.5a,butitshowsanimprovementin settlingtime in the case of the controller with friction compensation. The controller with friction compensation has settling time of 1 second as compared to 2 second settling time without considering the friction. The results show that the response of ISMC is better for friction taking into account in terms of steady-state error. The Fig. 5b demonstrates the control effort in elbow joint of robotic manipulator. The results clearly indicate that the more control effort is required for controller without friction and control inputexhibits an unwanted chattering phenomenon, which can be damaging to the system’s mechanical and electrical components. The trajectories of system begin from the sliding surface and the reaching phase is removed with the help of integral term in the sliding surface manifold. However, a discontinuous term of the switching control appears in the resulting joint torque. The chattering impact is the main hurdle in the practical realization of the sliding mode control. An integral term of SMC improves the outcomes by simplifying the higher order derivatives of the system. In the fgure, it can be observed from the result that the control effort with friction compensation-based SMC allows the system to track the required trajectory with more precision and accuracy as compared to SMC without considering the friction model. Fig. 5c demonstrates the tracking error signal with and without friction compensation. As fgure shows that at time of one second the error signal is zero with friction compensation as compared to 0.15 with out considering friction. The internal state of friction in elbow joint of robotic manipulator is illustrated in 5d. In Fig. 6, to test the closed loop tracking performance of the robot manipulator system, the desired differentiable sinusoidal input signal is generated with an amplitude and frequency of 0.1 rad and 0.016 Hz, respectively. The simulation results are generated with and without friction compensation. 6a illustrates the sinusoidal response of base. At some point actual position attains the required trajectory then it follows the desired path for remaining time. In the fgure, It can be observed from the results that the control effort with friction compensation-based control allows the system to track the required trajectory with more precision and accuracy as compared to SMC without considering the friction model. Fig. 6b illustrates the control effect in the form of applied torque to the base. The control effort is calculated using the control law defned in Eq. (19). The SMC based law reduces the chattering effect, when the dynamics of the system are weak against the uncertainties. Fig. 6c shows the error signal between actual and desired positions of the base. These results show that SMC with friction compensation demonstrates better performance with nearly zero error signal compared to SMC without friction compensation. In the fgure, it can be observed that without friction compensation error signal has greater amplitude of error in comparison to friction compensation error signal. Fig. 6d shows the internal state of LuGre friction model in base joint of the AUTAREP robot manipulator. Figs.7aandbpresenttheexperimental resultsof joints of the robot manipulator. Fig. 7a demonstrates the step input response for base joint of the robot manipulator and shows that the joint reaches the desired position in0.6 sec with overshoot. It can be seen that both transient and steady-state parameters are within acceptable limits. Sinusoidal response of base joint is also shown in Fig. 7b. The results demonstrate the well-known chattering phenomenon. This behavior of the system is undesirable and it may cause heat and energy dissipation and wear and tear of various components in the system. The chattering phenomenon can be reduced by implementing other variants of SMC such as higher order SMC. The proposed ISMC technique and the adaptive sliding mode control (ASMC) methodology implemented in [30] are compared in Fig. 7c. By employing SMC’s reaching condition, these trajectories approach zero and provide asymptotic stability to the system. Fig. 7c demonstrates the tracking behavior between both the current and desired positions which does not exceed 1.5 s for both techniques. The ISMC method has settling time of 1.25 s with friction compensation in comparison to 1.45 s for control law proposed in[30]. It has rise time of 214.9 milli second with overshoot of 0.792 % in comparison to ASMC technique having 270.63 milli-second rise time andovershootof0.658 %. The passivity based adaptive control of a serial robotic manipulator is implemented in [31] using joint frictions depending upon temperature. The position tracking of said technique along with the method implemented in this paper is shown in Fig. 7d. The position tracking in ISMC has better convergence performance and asymptotic convergence in comparison to method proposed in [31]. The (a) Angular position of elbow joint (b) Control effort (d) Friction state z (c) Error signal Fig. 5. Step response obtained by the elbow joint of the robotic manipulator negative aspect in the proposed technique ISMC with friction compensation is that the temperature effect of joints are not considered as well high control gains and high control efforts related to chattering issues. As robotic manipulators are often used in various applications where the joints get warmer due to the friction after the manipulator operates for some time. They may also be used when the temperature varies from very low to very hot degrees. The temperature effects must therefore be considered. 4 CONCLUSIONS In this research work, a sliding mode control with an integral term is proposed for the desired trajectory tracking and robotic manipulator is modeled using dynamic LuGre friction model. The effectiveness of the proposed ISMC technique is initially validated in Matlab and simulation fndings are shown to illustrate the effectiveness of the designed robust controller with and without considering the friction. The proposed control scheme is then evaluated by comparing with the other sliding mode control techniques. The ISMC technique performs superior in terms of system control efforts and transient parameters involving percentage overshoot, rise time and settling time etc. have been witnessed to be within certain bounds as anticipated. The proposed control law is also being tested on the custom developed experimental test bench using LabVIEW and NI-myRIO. (b) Control effort (a) Angular position of base joint (d) Friction state z (c) Error signal Fig. 6. Sinusoidal input signal response obtained by the base joint of the robotic manipulator 5 REFERENCES [1] Iqbal, J., Islam, R.U., Abbas, S.Z., Khan, A.A., Ajwad, S.A. (2016). Automating industrial tasks through mechatronic systems–A review of robotics in industrial perspective. Technical Gazette -Tehni. cki vjesnik, vol. 23, no. 3, p. 917-924, DOI:10.17559/TV-20140724220401. [2] Mei, J., Zang, J., Ding, Y., Xie, S., Zhang, X. (2017). Rapid and automatic zero-offset calibration of a 2-DOF parallel robot based on a new measuring mechanism. 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Passivity-based adaptive control of a 2-DOF serial robot manipulator with temperature dependent joint frictions. International Journal of Adaptive Control and Signal Processing, vol. 33, no.3,p. 512-526, DOI:10.1002/acs.2968. Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 411-420 Received for review: 2021-04-19 Š 2021 Journal of Mechanical Engineering. All rights reserved. Received revised form: 2021-07-13 DOI:10.5545/sv-jme.2021.7230 Original Scientific Paper Accepted for publication: 2021-08-20 Predictions of Friction Coefficient in Hydrodynamic Journal Bearing Using Artificial Neural Networks Dragan Milcic1,* – Amir Alsammarraie2 – Miloš Madic1 – Vladislav Krstic3 – Miodrag Milcic1 1 University of Niš, Faculty of Mechanical Engineering, Serbia 2 Tikrit University, Engineering Faculty, Iraq 3 Ljubex International, Belgrade, Serbia This paper explores the influence of the frequency of shaft sleeve rotation and radial load on a journal bearing made of tin-babbitt alloy (Tegotenax V840) under hydrodynamic lubrication conditions. An experimental test of the frictional behaviour of a radial plain bearing was performed on an originally developed device for testing rotating elements: radial and plain bearings. Using the back-propagation neural network, based on experimental data, artificial neural network models were developed to predict the dependence of the friction coefficient and bearing temperature in relation to the radial load and speed. Using experimental data of the measured friction coefficient with which the artificial neural network was trained, well-trained networks with a mean absolute percentage error on training and testing of 0.0054 % and 0.0085 %, respectively, were obtained. Thus, a well-trained neural network model can predict the friction coefficient depending on the radial load and the speed. Keywords: artificial neural network, hydrodynamic journal bearing, babbitt metal tin-based alloy, friction coefficient Highlights • An artificial neural network was used to predict the friction coefficient of the hydrodynamic radial journal bearings. • For the training and testing of artificial neural network (ANN), experimental data obtained by testing radial hydrodynamic journal bearings made of a tin- Babbitt alloy (Tegonenax V840) were used. • The ANN model allows the prediction of the friction coefficient based on the radial load of the bearing and the speed. • Prediction of the coefficient of friction using ANN has proven to be a competitive and efficient method. 0 INTRODUCTION Hydrodynamic bearings support a rotating shaft with its associated loads, through a pressure field developed within the lubricant that separates the solid surfaces. Journal bearings made of tin-babbitt alloy (Tegotenax V840) under hydrodynamic lubrication conditions are widely used in compressors, turbines, pumps, electric motors, electric generators, mining ore crushers, etc. Sliding friction in the contact surfaces between the material of the shaft and the sleeve material is a dissipative process of a complex nature, accompanied by a series of phenomena: mechanical, physicochemical, electrical, metallurgical, and thermal characteristics, which lead to the wear and tear of coupled bodies. The coefficient of friction is significantly influenced by: normal load, geometry, relative surface motion, sliding velocity, the surface roughness of the rubbing surfaces, the type of material, system rigidity, temperature, stick-slip, relative humidity, lubrication and vibration. Among these factors, normal load and sliding velocity are the two major factors that play significant roles in the variation of the friction coefficient [1] to [4]. The effect of radial load on the friction coefficient and loss material of different polymer and composite materials was investigated [2]; it was found that the amounts of friction coefficient and loss material are different for different materials. The tribological actions of Babbitt alloy 16-16-2 sliding against aluminium bronze ZnCuAl9Mn2 lubricated by seawater were systematically investigated by Wu et al. [3]. The results indicated that the friction coefficient decreased as the load increased to 30 N and then remained steady at high loading but decreased with an increase in sliding speed. Zeren et al. [4] studied the tribological behaviour of two different tin-based bearing materials in dry sliding conditions: one of these alloys with low Sb content (7 %) is known as SAE 12 and is widely used in the automotive industry and the other with a high Sb content (20 %) is an Sn– Sb–Cu alloy. Although search results have proved that WM-2 and WM-5 alloys can be used in dry sliding conditions, it is shown that the performance of WM-5 under heavy service conditions is better than WM-2 due to its alloying elements. Artificial neural networks (ANN) are a successful tool for predicting some tribological properties. ANN is a mathematical model inspired by the biological nervous system. ANN technology is used to solve complex scientific and engineering problems. The significance of this technology is that ANN models can be trained based on experimental data to recognize solutions. The ANN prediction method has been used in several applications, such as wear and friction. Asafa and Fadare [5] have presented an ANN predictive model that captures the dynamic behaviour of the tool wear and can be deployed effectively for online monitoring processes. Nine different structures of multi-layer perceptron neural networks with feed-forward and back-propagation learning algorithms were designed using the MATLAB neural network toolbox. An optimal ANN architecture of 5-12-4-2 with the Levenberg-Marquardt training algorithm and a learning rate of 0.1 was obtained using the Taguchi experimental design method. Durmus et al. [6] have presented the effects of ageing conditions at various temperatures, load, sliding speed, and abrasive grit diameter in 6351 aluminium alloy have been investigated using artificial neural networks. The experimental results were trained in the ANN’s program, and the results were compared with experimental values. It is observed that the experimental results coincided with ANN’s results. Knowing friction coefficient is important for determination of wear loss conditions at Al-Si alloys. Nagaraj et al. [7] investigated the effects of load, sliding velocity and sliding duration on the wear loss of the alloy. The experimental results were used to train the ANN program and the results were compared with experimental values. It was observed that the experimental results are very close to ANN’s results. Kalidass et al. [8] focuses on two different models, regression mathematical and ANN models for predicting tool wear. Experiments have been conducted for measuring tool wear based on the design-of-experiment (DOE) technique in a vertical machining centre on AISI 304 steel using a solid carbide end mill cutter. The experimental values are used in Six Sigma software for finding the coefficients to develop the regression model. The experimentally measured values are also used to train the feed-forward back-propagation ANN for the prediction of tool wear. Liujie et al. [9] used a back-propagation (BP) neural network to study the effects of the pv factor and sliding distance on the friction and wear behaviour of 30 wt. % carbon fibre reinforced poly (ether-ether-ketone) advanced composite (PEEK­CF30) at the contact temperature of 120 şC. An experimental plan was performed on a pin-on-disc machine for obtained experimental results under unlubricated conditions. Hassan and Mohammed [10] experimentally investigated the parameters affecting the wear debris and the temperature rise due to friction, as well as developing the model ANN using MATLAB program for predicting the wear and temperature of disc and pad. Since the hybrid materials had not been enough researched, Sathyabalan et al. [11] have presented application of ANN on the MMC (hybrid LM6 aluminium), which is reinforced by SiC and flyash. The aim of this study was the prediction of the influence of additives (reinforcement materials) on the wear and hardness of hybrid material. The best results were obtained with ANN (4-6-2). Input parameters are defined by variables (flyash size, SiC size, flyash in % and SiC in %) while the outputs are wear and hardness. Prediction accuracy was high, so the ANN has been validated for practical application by weight definition of the above-mentioned reinforcement materials for the hybrid material with good resistance against wear. An ANN model for the prediction of the wear process with glass-filled polytetrafluoroethylene (PTFE) composites is presented by Varade and Kharde [12]. An additional task of this research was establishing the dependence between load-sliding and time-sliding speed. The input parameters were velocity, load, time and percentage of glass fill, while the output was wear. Taguchi’s orthogonal array of 27 experiments was used. The results revealed that the material with the highest percentage of glass fill possesses better resistance against wear. As the friction coefficient also depends on selected lubricant, Durak et al. [13] have presented a back-propagation neural network (BPNN) for the prediction of the concentration of the additive (based on PTFE) in the base oil in order to decrease the friction coefficient of the journal bearing. A bearing for testing was made of ZnAl alloy. The best results were obtained by using BPNN (3-5-1). Prediction accuracy is high, so the presented BPNN can be used in practice. In order to research the applicability of the ANNs by friction coefficient prediction of the composite carbon-fibre-reinforced polymer (CGRP) material, Nasir et al. [14] have developed suitable ANN for this purpose. They have developed few different models, but the best results were with ANN (4-40-1), which used the Levenberg–Marquardt algorithm for training. The input parameters were fibre orientation, testing speed, normal force and duration time, while the output parameter was the friction coefficient. Artificial intelligence (AI) has many tools and techniques that can be applied in the prediction of different parameters. One of these is the adaptive neuro-fuzzy inference system (ANFIS). This methodology was used in combination with experimental research [15]. This work focuses on establishing a correlation between friction coefficient and load by sliding bushing, made of a tin-based white metal alloy (TEGOTENAX V840). Experimental research [16] and [17] attempted to provide values (i.e., parameters), which will be used for ANFIS design: friction coefficient, speed and radial load. All mentioned values were measured in equal time intervals. The newly designed ANFIS was tested, and the results were compared, which showed that the prediction accuracy is high. As the final result, an expression was obtained that describes correlation friction coefficient - load under conditions of lubrication defined in advance. The present study’s objective was to predict friction coefficient in the radial hydrodynamic journal bearing of the white metal bearing alloys, known commercially as babbitt metal - Tegotenax V840, based on experimental research [16] and [17] using ANN. 1 EXPERIMENTAL DETAILS AND PROCEDURE 1.1 Radial Journal Bearing Test Rig Fig. 1 shows a test rig for testing rotating machine elements, developed at the Faculty of Mechanical Engineering in Niš, Serbia, adapted for testing the tribological properties of plain bearings. The test rig for testing hydrodynamically lubricated plain bearings consists of three main systems: the drive system, the hydraulic system for realizing the load of the plain bearing, and the hydraulic system for lubricating the plain bearing. Fig. 1. Test rig The drive system of the test rig consists of an asynchronous induction motor (ABB, 400 V, 50 Hz, 3 kW, 1460 rpm), which is connected to the shaft by an elastic coupling. The shaft is supported by two roller bearings and a test specimen (i.e., a journal bearing mounted between these two bearings). The motor is equipped with an E720 encoder for counting the engine speed. The test specimen-journal bearings are hydrodynamically lubricated. The lubrication system consists of an electric motor (1450 rpm, 90 W), an ELP pump (AMGP-03C), and a 10 l hydraulic tank. The lubricating oil of the test plain bearing is ISO VG 32. A schematic representation of the part of the test rig with the bearing assembly is shown in Fig. 2. Fig. 2. The bearing-housing assembly (1-lower bracket; 2-sleeve; 3-seal; 4-shaft; 5-top bracket; 6-top cover with seal; 7-screw; 8-seal; 9-nut; 10-lever with which the bearing is loaded; 11-force sensor; 12-beam; 13-cylinder carrier; 14-cylinder; 15-piston; 16-screw; 17-thermocouple; 18-roller bearing housing; 19-roller bearing; a-place of friction force measurement; c- thermocouple temperature measurement location; d-place of bearing loading by radial force, g, e, f- hydraulic ports) A hydraulic system was used to load the sliding bearing, which consists of an electric motor (EM) (0.75 kW, 380 V, 1420 rpm), a pump (EATON PVQ10), and a 30 l oil tank. The hydraulic cylinder is used to load the test of a sliding bearing. The working fluid is the above-mentioned oil. The hydraulic cylinder exerts a load on the journal bearing sleeve via a radial force sensor (HBM U9C / 10 kN). Table 1 shows the properties of the operating fluid. Table 1. Properties of hydraulic oil ISO VG32 Item Value Unit Density at 15.6 °C . 868 kg/mł Kinematic viscosity at 40 °C .40 32.2 mm˛/s Kinematic viscosity at 100 °C .100 5.52 mm˛/s Viscosity index 108 - Flash point 212 °C Pour point –33 °C Aniline point 104 °C The bearing sleeve is given in Fig. 3. The dimensions of the bearing are: bearing diameter d = 40 mm + 0.05 mm, bearing axial length B = 40 mm, outer bearing diameter D = 60 mm, bearing diametral clearance Z = 0.025 mm. Fig. 3. Bearing sleeve 1.2 Preparation of Experimental Materials and Conditions The journal bearing sleeve is made of AISI 440C stainless steel, and a 3 mm thick layer of white metal (Tegotenax V840) is applied to the sleeve. The shaft is made of AISI 440C stainless steel. The chemical properties of the steel layer and Babbitt layer materials are given in Table 2, and the mechanical and physical properties in Table 3. Table 2. Chemical composition of AISI 440C sleeve and material of internal sliding surface - white metal TEGOTENAX V840 Element TEGOTENAX V840 [wt. %] AISI 440C [wt. %] Sn 88.7 - Sb 7.6 - Cu 3.7 - C - 1.2 Si - 1 Mn - 1 Ni - 1 Cr - 18 Mo - 0.75 P - 0.04 S - 0.03 Table 3. Mechanical and Physical Properties Tin-based white metal Shaft: AISI alloy: TEGOTENAX V840 440C 20 °C 23 269 50 °C 17 ­ Hardness HB 10/250/180 100 °C 10 ­ 150 °C 8 ­ Yield Strength, Rp0.2 [MPa] 46 448 Tensile Strength, Rm [MPa] 77 758 Young’s modulus, E [MPa] 56500 200000 Density, . [kg/m3] 7400 7650 Poisson ratio, . 0.33 0.27 to 0.30 1.3 Determination of Frictional Properties The friction coefficient is determined by the moment of friction. The friction moment Mf(t) as a function of time is calculated as the product of the friction force Ff (t) and the bearing radius r. As a reaction of the friction moment, the force Fs(t) appears at the contact point of the force sensor, which is at a perpendicular distance L from the centre of the sleeve (Fig. 4). Mt FtrFtL() ˜() °˜() °. (1) ff s Fig. 4. Friction force measurement scheme Based on the expression for determining the friction force: Ft ˜.() tFt . (2) f () °N () The friction coefficient is determined by the expression: Ft .. f FtL () . ˜() t °°s . (3) N () rF t () Ft .N The input parameters of the test are speed n, i.e., sliding speed v and bearing load FN. The radial bearing load was varied in the range FN = 1000 N to 4000 N, with the variation of the shaft speeds in the range n = 1000 rpm to 3000 rpm. The test parameters and specific bearing load are given in Table 4. Table 4. Defined parameters for testing plain bearings Speed, n Sliding speed, Bearing load, Specific bearing load, [rpm] v [m/s] FN [N] p = FN /(dˇB) [MPa] 1000 0.625 1000 2.09 2000 3000 1.25 1.875 4000 2.5 1000 0.625 2000 4.2 2000 3000 1.25 1.875 4000 2.5 1000 0.625 3000 6.3 2000 3000 1.25 1.875 4000 2.5 For data acquisition within the experimental tests of tribological properties of the sliding bearing, a personal computer (PC) equipped with appropriate hardware (measuring and control interface) in conjunction with software developed in the LabVIEW environment is used. The data was transmitted to the recorder, which enabled monitoring and recording of the signals. The data about coefficient friction, radial load and sensor force were applied for 1/20 s and the temperature lubrication oil and load for 5 s. 1.4 Test Results Based on the experimental plan given in Table 4, tests were performed on test specimens of plain bearings with a previously defined geometry. Figs. 5 to 7 show the changes of temperature and friction coefficient with the change of the radial load of the bearing FN = 1000 N to 4000 N and the speed n = 1000 rpm to 3000 rpm. Twelve different experiments were performed, each lasting an average of 5 hours (18000 s). 2 ANN MODEL DEVELOPMENT In order to establish a mathematical relationship between the dependence of the experimental parameters of the journal bearing test in the conditions of hydrodynamic testing, two ANN models were developed. The first model was for the prediction of the friction coefficient, and the second was for the prediction of the bearing temperature. For training and testing of the network, the data of the measured values of the friction coefficient ľ and the bearing temperature T for 18000 s were used as a function of the change of the radial load FN and the speed of the shaft n. The neural network architecture for both models had three neurons in the input layer (bearing load FN, speed n and time t), one neuron in the output layer (friction coefficient ľ respectively bearing temperature T) and one hidden layer with 6 neurons (Fig. 8). For the ANN model, a linear activation function and a nonlinear sigmoid activation function were combined in the output and hidden layers, c) d) Fig. 5. Trends in time of friction coefficient and bearing temperature at 1000 rpm speed, for a load of a) FN = 1000 N; b) FN = 2000 N; c) FN = 3000 N and d) FN = 4000 N a) b) c) d) Fig. 6. Trends in time of friction coefficient and bearing temperature at 2000 rpm speed, for a load of a) FN = 1000 N; b) FN = 2000 N; c) FN = 3000 N and d) FN = 4000 N respectively. Accordingly, the obtained experimental data were normalized in the range [–1, 1]. During the development of the ANN model, the available input/ output data set was randomly divided into two sets: a training set and a test set for the ANN network model. 295 data points were used for ANN training, and 43 data points were used to test trained ANN network models. The Levenberg-Marquardt algorithm was chosen for ANN training due to its high accuracy and fast convergence. In order to solve the convergence problem and minimize, as well as slow the convergence, the ANN training process was repeated several times with the help of different initial weights established by the Nguyen-Widrow method. During the ANN training process, although the maximum number of training iterations was set at 1000, it was observed that much fewer iterations were sufficient to train ANN models, Fig. 9. Upon completion of the training process, the prediction performance of the trained ANN model was assessed. For this purpose, the mean absolute percentage error (MAPE) was calculated according to the Eq (4): ° Experimental valueEstimed value MAPE ˜1 . .100 % .ˆ, (4) NN Experimental value where N is the number of data. The average errors for training and testing data for the first model are 0.0054 % and 0.0085 %, respectively, which are very small indeed. The average errors for training and testing data for the second model are 0.12 % and 0.023 %, respectively, which are very small. 3 ANALYSIS AND DISCUSSION Fig. 10 shows the prediction of the friction coefficient as a function of bearing load and speed for the white metal alloy bearing internal sliding surface of Tegotenax V840. The prediction of the friction coefficient of the sliding bearing’s internal surface made of tin-based white metal alloy showed that the Fig. 7. Trends in time of friction coefficient and bearing temperature at 3000 rpm speed, for a load of a) FN = 1000 N; b) FN = 2000 N; c) FN = 3000 N and d) FN = 4000 N The values of the change of the friction coefficient take the form of a downward trend - curve friction coefficient has a downward trend with an with increasing bearing load and decreasing speed, as increased bearing load. shown in Fig. 10. lubricant temperature, which results in a decrease in the shear stress of the lubricant, which leads to a decrease in the viscosity of the lubricant and hence a decrease in the friction coefficient. Fig. 11 shows the prediction of bearing temperature T as a function of the speed n and a sliding time t for the bearing load FN = 2500 N. This change has a growing trend with an increasing sliding time and speed. It can be seen that the bearing temperature T increases with time t, as shown in Fig. 11. As the friction coefficient is directly proportional to the dynamic viscosity ., and the increase in oil temperature leads to a decrease in dynamic viscosity, which certainly leads to a decrease in friction coefficient. In Fig. 11, two zones can be observed, the unstable zone and the stable zone. In the zone of an unstable state, the temperature inside the system increases with time. The duration of this zone is estimated at about 4000 s. The second zone is a stable state in which the temperature remains approximately constant: somewhere around 41 °C to 54 °C. Fig. 12. Prediction of the ratio of the friction coefficient ľ, in relation to the speed n and the sliding time t with the help of BP neural network Fig. 12 shows the predicted friction coefficient ľ in relation to the speed n and the sliding time t at constant load FN = 2500 N. It can be seen that the friction coefficient ľ increases with the speed n. As the sliding time increases, the friction coefficient decreases according to the increase in temperature and the decrease in the viscosity of the oil. The optimization of the ANN model using metaheuristics indicated a maximum friction coefficient of 0.1513, which was obtained in a combination of values of process parameters: n = 3000 rpm, FN = 1072 N and t = 300 s. With regard to the stochastic nature of metaheuristics, i.e., inability to guarantee the optimality of the found solution, the solution was checked by creating a three-dimensional (3D) diagram of the friction coefficient for t = 300 s, which is shown in Fig. 13. As can be seen from Fig. 13, the value of the maximum friction coefficient of 0.1591 is obtained in a combination of the force FN = 1000 N and the speed n = 3000 rpm. Based on experimental data, it was determined that for FN = 1000 N, n = 3000 rpm and t = 300 s, the resulting coefficient of friction has a value of 0.1579, which once again confirms the accuracy of the created ANN model. 4 CONCLUSIONS The research into tribological properties of sliding bearings was performed on the originally developed equipment for testing rotating elements at the Faculty of Mechanical Engineering in Niš in the Laboratory for Mechanical Constructions, Development, and Engineering. A bearing made of tin-based white metal alloy-TEGOTENAX V840 were tested under hydrodynamic lubrication conditions with selected ISO VG 32 hydraulic oil. Based on the conducted experimental research and developed ANN models for the prediction of the friction coefficient (first model) and bearing temperature (second model), the following conclusions can be drawn: The ANN architecture for both models was 3-6-2, and the Levenberg-Marquardt algorithm was used to train the ANN. The result of ANN prediction shows that the models are extremely good with minimal mean square error training and testing data for the first model 0.0054 % and 0.0085 % and for the second model 0.12 % and 0.023 %, respectively. The optimization of the ANN model, the highest friction coefficients 0.1513 were obtained at a bearing load of 1072 N and a speed of 3000 rpm, while the lowest coefficients of friction 0.00288 were obtained at a maximum load of 4000 N and a speed of 1675.22 rpm. The bearing temperature prediction model indicates a growing trend of change with increasing sliding time and speed. The bearing temperature T increases with time t; after a little more than 1 hour (about 4000 s), the bearing temperature stabilizes at approximately 50 °C at all bearing loads and all bearing rotation frequencies. 5 ACKNOWLEDGEMENT This research was financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Contract No. 451-03-9/2021-14/200109). 6 REFERENCES [1] Chowdhury, M.A., Nuruzzaman, D.M., Mia, A.H., Rahaman, M.L. (2012). Friction coefficient of different material pairs under different normal loads and sliding velocities. Tribology in Industry, vol. 34, no. 1, p. 18-23. [2] Nuruzzaman, D.M., Chowdhury, M.A., Rahaman, M.L. (2011). Effect of duration of rubbing and normal load on friction coefficient for polymer and composite materials. Industrial Lubrication and Tribology, vol. 63, no. 5, p. 320-326, DOI:10.1108/00368791111154931. [3] Wu, H., Bi, Q., Zhu, S., Yang, J., Liu, W. (2011). 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Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432 Received for review: 2021-03-21 Š 2021 Journal of Mechanical Engineering. All rights reserved. Received revised form: 2021-06-24 DOI:10.5545/sv-jme.2021.7178 Original Scientific Paper Accepted for publication: 2021-09-17 Multi-parametric Dynamic Analysis of a Rolling Bearings System Almatbek Kydyrbekuly1 – Gulama-Garip Alisher Ibrayev 1 – Tangat Ospan1 – Anatolij Nikonov2,* 1 Al-Farabi Kazakh National University, Kazakhstan 2 Faculty of Industrial Engineering Novo mesto, Slovenia A method for calculating amplitudes and constructing frequency characteristics of forced and self-excited vibrations of a rotor-fluid-foundation system on rolling bearings with a non-linear characteristic based on the method of complex amplitudes and harmonic balance has been developed. Non-linear equations of motion of the rotor-fluid-foundation system are derived, and analytical methods of their solution are presented. Frequencies of fundamental and ultra-harmonic resonances are determined. The intervals between self-oscillation frequencies are estimated. The dependence of amplitudes on the amount of fluid in the rotor cavity, the mass of the foundation, linear imbalance, the value of the stiffness coefficient, and the damping coefficient is shown. Keywords: forced vibrations, resonance, fluid, rolling bearing, self-oscillations, ultra-harmonic Highlights • A dynamic model of the rotor-fluid-foundation system on rolling bearings with a non-linear characteristic has been constructed. • The specific features of the system with many degrees of freedom are defined. • The prerequisites under which the fluid stabilizes the behaviour of a non-linear system are determined. • The optimal parameters of the system, providing its stable operating mode, are determined. 0 INTRODUCTION One of the main tasks of dynamics of rotors and design of rotary machines is the choice of parameters that provide their stable operation. The damping of forced and purely non-linear self-excited rotor vibrations by selecting parameters of the system, taking into account vibrations of the foundation, elasticity of supports, imbalance and other parameters, is of great interest from both technical and economic points of view. Currently, many works on the study of non­linear oscillations of rotor systems exist [1]. However, their dynamics is insufficiently studied because of difficulties arising in taking into account the mutual influence and combined actions of factors such as the presence of fluid in the rotor cavity, external non­conservative forces [2], different types of linear and angular imbalances and nonlinearities [3] and [4]. In the design of rotary machines, one of the most important mechanical components described by non­linear models is elastic support, which ensures the operability and reliability of the system [5] to [7]. In this paper, rolling bearings act as elastic supports [8]. The neglect of non-linear properties of bearings prevents correct qualitative and quantitative results for rotor systems from being obtained [9] to [11]. This phenomenon can be explained by the fact that when analysing linear rotor systems with rolling bearings, an approximate estimate of stiffness and damping properties of bearings is most often used, whereas in reality the bearing stiffness heavily depends on the load, i.e., on the operating mode of the rotor system, on the geometry and size of the bearing clearances, on the size of the fit of the inner and outer rings in the bearing, and other factors [12] and [13]. To obtain a detailed description of the process, it is necessary to consider the influence of such factors as imbalance, asymmetry of the rotor on the shaft, external friction, changes in inertial parameters and positional forces of various kinds [14] to [16]. Such complications of the model in the analysis of dynamics make it possible to study the influence of the gap size, rotation frequency on frequency spectra, and amplitude-frequency characteristics for any rotor system on rolling bearings. Mathematical models of bearings that take into account nonlinearity factors are distinguished by complexity and primarily by the loads that they consider. In our case, to describe the bearing model, the Hertz contact theory is used, which connects the radial loads acting on the bearing and the deformation at the contact points between the rolling element and the bearing rings [17]. In the bearing model, it is assumed that there are no types of sliding of bodies and rolling surfaces. Damping is considered in terms of equivalent viscous and linear friction. 1 STATEMENT OF PROBLEM AND EQUATIONS OF MOTION A symmetrical vertical rotor of mass m with a cylindrical cavity of radius R, partially filled with an ideal fluid, is rotating on rolling bearings with a constant angular velocity O0. The angular velocity is considered sufficiently large so that the fluid in the rotor cavity takes the form of a cylindrical ring. Due to the different positions of the centre of mass and the geometric centre, the rotor has a static imbalance (or linear eccentricity) e. The foundation of mass M rigidly connected to the outer ring is mounted on elastic supports with an equivalent linear stiffness coefficient c2 (see Fig. 1); below, all the dimensional parameters shown in this figure are appropriately dimentionalized, see Section 3. The radial compliance of bearings occurs due to deformation of the rolling elements and raceways at the contact points. In this case, the non-linear restoring force in bearings can be generally described using the Hertz contact formula [14] and [18]: 2 F ˜C °3. (1) C br However, for the sake of simplicity, in what follows, the restoring force in rolling bearings is approximated by a power series according [16]: 3 F ˜c .°c ., (2) C 0 r 1 r where FC is a component of the restoring force in the radial direction, dr is the deformation in the radial direction, Cb is the stiffness coefficient, c0 and c1 are stiffness coefficients for the linear and cubic terms, respectively. This expansion at dr < 1000 ľm agrees with the experimental results in [18]. Fig. 1. Scheme of a rotor on rolling bearings, with a cavity partially filled with an ideal fluid and mounted on a foundation coordinates of the centre of mass of the rotor are denoted as A(., .), and the coordinates of the centre of mass of the foundation are denoted as A2(.2, .2), . and .0 are the external friction coefficients, ... the coordinate system associated with the rotor, . is its polar axis, and axis . is arbitrarily drawn through the eccentricity vector of the rotor mass. Let us introduce complex variables in the form: xiyzx i, y (3) ˜°˜°z , 2 22 for analysing plane-parallel motion of the rotor and foundation. Then, assuming thatdamping force acting on rotor depends on velocities of the rotor itself, and similarly, damping force acting on foundation depends on the velocities of the foundation itself, we present the equations of motion of the system in question in the form, here and below see [8] and references therein, .. ˜2( °) ˜nzz )3 ˜2 kz . . znzz ( ° 0 212 F e .2 it ) ˜r , .0 exp( .0 m 22 3 .. zn˜z ° ( °) °nzz °) ˜2 kz . nzz ( . 0, (4) 22201 210 2 02 where 2 c 2 c °2 c 201 22 n ˜, n ˜,2 k ˜, n ˜, 01 2 m mmM 2 c 2 c °m 2021 0 n ˜˜.n , n ˜˜.n ,2 k ˜, .˜, 01 010 1 0 M MMM and Fr is a complex expression for the reaction force of a fluid, which is defined as: 2 .i ( ˆt °.) 0 F RhP ˜ed , (5) r ˜.| rR . 0 where h is the rotor cavity height, P|r R is the fluid = pressure on the rotor wall. The equations of motion for fluid are written as: ˜u 1 ˜P °2 .0 ˆ.°°.. x cos( .0 t ..) ˜t .˜r °.. y sin( .0 t ..), ˜ˆ1 ˜P .2 .0 u .°... x sin( .0 t ..) ˜t . r ˜.°.. y sin( .0 t ..), (6) where u and . are the radial and tangential components of the velocity of the fluid particle, P and . are pressure and density of the fluid. The continuity equation at . = const is given by: . ˜ ˜ () ur ° . 0. (7) The motion of the system is considered ˜ ˜ . r concerning the fixed coordinate system Oxyz. The The boundary conditions for the Eqs. (6) and (7) Substituting Eqs. (10), (11) and (12) in Eq. (4), can be written as: we obtain a system of algebraic equations for the unknowns Ar, Bf, Zr and Zf. Its solution is given by: u|rR = 0, (8) = f 0 f and Zr = , Zf = , (14) f 1 f 1 ˜P °...2 ru | rr , (9) Bf ˜( p 1 °ip 2) Ar , (15) 00 ° 0 ˜t rr 0 °f 0 ˜f Z ˜Z °°f , (16) where r0 is the radius of the free fluid surface. These rf 2 f 1 relations express the impenetrability condition at the rotor wall and the conditions on the fluid pressure along the free surface, respectively. Ar ˜ .ˆ. p p 2 2 .. B , f (17) ip 1 2 ° p 1 . ° 2 22 p p Eqs. (3) to (7) with boundary conditions Eqs. (8) and (9) in combination with the continuity equation 1 2 where form a closed system of equations. 2 NON-LINEAR FORCED VIBRATION AND SELF­OSCILLATIONS OF THE ROTOR-FLUID-FOUNDATION SYSTEM Assuming that the rotor and the foundation perform harmonic oscillations, we seek the solution of the system of equations Eq. (4) in the form [20] and [21], zZ ˜exp it °.A exp i.t , ..ˆ.....(10) r 0 r z 2 ˜Zf exp .i .ˆ0 t °....Bf exp .i .t ., (11) where . is natural frequency, Zr and Zf are forced vibration amplitudes of rotor and foundation, while Ar and Bf are natural vibration amplitudes of rotor and foundation, respectively. The presence of an angle . i.e., a phase lag, is caused by the presence of resistance forces in the system. As an expression describing the action of the hydrodynamic force on the system, which arises due to the presence of fluid in the cavity, we start from f ˜rp °pr , f ˜pq .qp , f ˜rq .qr , 000 00100 2 222 q ˜.ˆ(1 ..), p ˜.e ˆcos ., r ˜ n °ˆ, 0 L 0 20 q ˜2 .ˆk , p ˜.e ˆ2 sin ., r ˜2 k ˆ, 0 00 0 000 ˆD  222 32 .n 2 °..1 .L °4 kk 0  D .4  p 1 ˜, m 1 ˆˆD  32 22 2 . .1 .L  k 0 ..n 2 °.k   D ..4  p 2 ˜°, m 1 22 2222 ˜ˆ, .2 mn °ˆ.2 k im ˜n °...4 k ., 02000 12 0 22 2 2 mn ˜°..2 ki D ˜.°2 ˆ. ., °ˆ, 2203 00 D ˜.2 °2 ˆ.°ˆ2. 4 00 ľL = mL/m is the ratio of the mass of fluid required to completely fill the rotor cavity to the mass of the rotor. Thus, to determine the unknown amplitudes, we may derive two non-linear equations from the relations above: formulae in [8] and [19], taking into account Eqs. (10) mZ 0 Z Z ˜ ˜ ( ) f f r and (11): .ˆ. .. .ˆ. .. 3 3 (18) Z Z Ar B ° . ˜ . ˜  2 2 n n 10 ( ) ( ) 0, .. .. f f 01 22 mB ˜( A °B ) ..2 ˆ0 .ˆ0 .2 rf °Am 2 exp(. rL .it), (12) 22 ..2 ˆ0  .ˆ0 ...3 232 .. .°n °n ( Z °Z ) ˜( A °B ) 0. (19) ˆ01 10 ˆrf rf  ..24 .. where r Fr FxiFy Zmr L i . ˜ ° ˜ ˆ ˆ. ° 2 t 4 2 exp 0 0 After substituting Eqs. (14) to (16) into Eq. (18), R ˜°, q ˜r 0, ...0 (13) a quadratic equation for Ar can be obtained. Using the values of the amplitudes Zr, Zf, Ar and Bf calculated mL = p.R2h is the mass of fluid required to from Eq. (19), we can determine the phase angle .. completely fill the rotor cavity, s vibration frequency The linearized form of Eqs. (18) and (19) of the free surface of the fluid. becomes: .. znzz °) ˜2 kz . ˜02( 2 .0, zn2 °2( kz .. ˜z nzz °) ˜2 . .0. (20) 22201 2 02 Since the resistance forces do not strongly affect the value of the natural frequency [22], in this case, it is sufficient to find the frequency equation for k = k0 = 0, i.e., for the equation in the form: ˜.° 10 .ˆ. ˜.° .. z .ˆ. . ˜ . . ˜.° z .ˆ. 00 ˜.° .(21) to study both forced and self-excited vibrations in general; however, at the same time, this approach excludes the possibility of studying steady-state vibrations (which is typical for the case of an empty or a completely filled rotor), since the partial filling of the cavity with fluid causes the appearance of self-oscillations. Thus, by slowly changing the angular velocity of the rotor, it is possible to construct the amplitude­ ˆ  the parameters of the rotor, foundation, and fluid. To obtain a frequency equation, we seek the solution in the form: 3 RESULTS AND DISCUSSION zz exp .it °., zz ˜exp .it ..ˆ. ˜..ˆ.°.. (22) s 22 s To analyse the system and determine the optimal Substituting Eq. (22) into Eq. (21), we obtain: operating mode, the amplitude-frequency .. 2nnn 0 2nn 0 22 20 20 2 .° ˆ. frequency characteristics of the system while varying   . . .. z z 01 2 characteristics of the rotor and foundation were . . . ˜ 2 ˜ 2 2 n 0 0 n 0 0 .....ˆ  constructed. They were obtained from Eqs. (18) and (19) and numerical solution of the differential equations of motion of the system for different degrees of filling of the rotor cavity with fluid, . ˜˜ 2 2 2 0 n 0 0 n 0 . n 0 .˜. 2 ˜ ° 22 2 2 0 n n 0 0 . n 0 .˜ . 2 ˜ ° 22 2 2 0 n n 0 As different values of the damping coefficient (.), non­ .... 0 .... ....    linear stiffness (c1), imbalance (e) and different values of the foundation mass (M). For qualitative analysis Bs 0 0 (23)  , C s of the system, the following dimensionless parameters were introduced: Ds ˆ 0  ˆ where 22 22 z ˜ A °B , z ˜C °D . (23) s ss 2 s ss Then, to find the values of the natural frequencies ., it is sufficientto find the determinant of system Eq. (23), i.e.: 22 2 n 0 ˜.0 ˜n 00 22 2 0 n 0 ˜.0 ˜n 0 det .0.(24) 2 222 0 20 ˜.n 0 n °.n ˜.0 2 222 0 ˜.n 0 n °.n ˜. 0 20 Then, the frequency equation is written as: 86 42 ˜.°˜.°˜.°˜.°.0, (25) 123 4 where 22 ˜°....2 .ˆ, ˜°.....4 ...., 1 1 4 2123 144 22 22 ˜°.2 ...., ˜°.. ...ˆ.., 3 14 14 4 1423 2 2 222 .°n , .°..n , .°.n , .°n ..n . 102 030420 Thus, four natural frequencies are obtained, where the lowest frequency is taken as the first frequency. The accuracy of the approximate formulae for natural frequencies are discussed in [19], [21], and [22]. It should be noted that the search for a solution in the form of Eqs. (10) and (11) makes it possible °e kne 2 s ˜0, E ˜, K ˜, C ˜12, (26) .R °0 °0 where s is a dimensionless frequency, E is a dimensionless parameter of linear imbalance, Zr is a dimensionless amplitude of the rotor, Zf is a dimensionless amplitude of the foundation, K is a dimensionless damping coefficient, C is a dimensionless coefficient of stiffness at the cubic term, R is a cavity radius (see Figs. 2 to 11). When an insignificant amount of fluid is present in the cavity, e.g. r0 = 0.93R, (see Figs. 2 and 3), three self-oscillation zones and three resonance amplitudes are observed for both the rotor and the foundation. All figures show examples when the system parameters take the values: .= 4200 kg/s, .0 = 6.59 kg/s, c0 = 1.1ˇ107 kg/s2, c1 = 0.87ˇ107 kg/m2s2, c2 = 3.26ˇ105 kg/s2, M = 25 kg, m = 2.4 kg, e = 0.001 m. The amplitude-frequency characteristic, in this case, demonstrates a monotonic increase over the interval 0 < s < 0.065 up to the first resonance peak. The first resonance occurs at s = 0.065, where the amplitudes are Zr = 0.083 and Zf = 0.0091. Then, over the interval 0.065 < s < 0.65, a self-oscillating mode occurs (see Figs. 2 and 3) with maximum amplitudes of 0.028 for the rotor and 0.001 for the foundation, respectively. At s = 0.65, the second resonance occurs with the Fig. 2. Amplitude-frequency characteristics of the rotor at different degrees of the rotor cavity filling, r0 Fig. 3. Amplitude-frequency characteristics of the foundation at different degrees of the rotor cavity filling, r0 amplitudes 1.72 and 0.79 for the rotor foundation, respectively. Further, over the interval 0.65 < s < 1, the vibration amplitudes decrease monotonically to the values 0.07 and 0.013. Next, over the interval 1 < s < 1.35, the amplitude of vibrations grows rather intensively up to the third resonance, where s = 1.35, and the amplitudes are Zr = 2.23 and Zf = 1.38. Further, in the interval 1.35 < s < 3.41, the amplitudes monotonically decrease to the third self-oscillation zone, which occurs at s = 3.41, where the amplitudes take constant values Zr = 0.057, Zf = 0.0003, which is caused by the effect of self-centring and non­ conservative circulation-type forces arising due to the presence of fluid in the rotor cavity. Here, the self-oscillation zones are presented as a curve showing the oscillatory process. In all other cases, the curves corresponding to self-oscillations are smoothed. With the increase of the fluid volume in the cavity at r0 = 0.8R, it can be seen that the first and second resonances, as well as the first self-oscillation zone, occur at the same frequencies as at r0 = 0.93R, but with larger amplitudes. For example, at the first resonance, i.e., at s = 0.065, the amplitudes are Zr = 0.15 and Zf = 0.015, that is, a 4.74-times increase of the fluid volume leads to 1.82- and 1.65-times increase in the amplitudes of the rotor and foundation, respectively. The maximum amplitudes of self-oscillations, in this case, also have higher values. For example, they reach 0.088 for the rotor, and 0.0097 for the foundation, which is 3.14- and 9.7-times higher than in the previous case, respectively. At the second resonance (s = 0.65), the amplitudes are Zr = 1.5 and Zf = 0.59, which practically does not differ from the previous case. The third resonance peak in this case shifts to higher frequency range and occurs at s = 1.75, where Zr = 2.36, Zf = 2.83. Then, for s > 3.41, a self-oscillation zone is formed, where the amplitudes take on constant values Zr = 0.139 and Zf = 0.0006. When the fluid volume in the rotor cavity is one third of the total volume, i.e., r0 = 0.667R (see, e.g., Figs. 2 and 3), the maximum resonance amplitudes are observed. The first and second resonances, as well as the first self-oscillation zone, occur at the same frequencies as in the two previous cases, the amplitudes are, respectively, Zr = 0.198, Zf = 0.033 and Zr = 1.9, Zf = 0.38; the maximum amplitudes of the first self-oscillation zone are Zr = 1.9, Zf = 0.38. In this case, the third resonance peak also appears at a greater angular velocity occurring at s = 2.12, with amplitudes Zr = 2.93, Zf = 2.99. Then, as before, at s > 3.41, a self-oscillation zone is formed, where the amplitudes take on constant values Zr = 0.15 and Zf = 0.0008. The maximum amplitudes when one third of the cavity is fluid-filled and are associated with a specific behaviour of a “coupled” solid-fluid system [8], [19], and [23]. Furthermore, for cases r0 = 0.5R, r0 = 0.333R and r0 = 0.125R, frequencies of the first and second resonances, i.e., s = 0.065 and s = 0.65, as well as the frequencies of formation of self-oscillation zones, i.e., 0.065 < s < 0.65, 0.65 < s < 3.41 and s > 3.41, do not change. The values of the first and second resonance amplitudes are damped. The third resonant amplitude is also damped, while the related resonance frequency is shifted to the right (see, e.g., Figs. 2 and 3), due to the increase of the fluid volume. At the same time, the fluid volume does have a significant effect on the amplitudes over the first, second and third zones of self-oscillations. For an empty rotor, a breakdown of the amplitude-frequency characteristics of both the rotor and foundation, corresponding to the fundamental resonance, is typical for the studied cubic nonlinearity, as for a Duffing oscillator. This phenomenon, which arises as a result of a monotonic increase in the amplitudes and their sharp decrease after a certain frequency, has been studied in detail by many authors, e.g. [24] to [26]. In this case, the breakdown of the amplitude of the empty rotor with a gradual increase in frequency occurs at Zr = 1.05 and s = 1.33, then the amplitudes of the rotor decrease to 0.15. A similar behaviour of the amplitude at s = 1.33 is observed for the foundation; in this case, the maximum amplitude is Zf = 0.115, which is almost an order of magnitude less than the rotor amplitudes. The foundation amplitudes after the breakdown decrease to 0.01. This difference between the amplitudes of the forced vibrations of the rotor and the foundation is primarily due to the ratio of the masses of the rotor and the foundation, as well as the value of linear eccentricity. As might be expected, an additional (first) linear resonance appears at s = 0.065, according to the general theory in the book [27]. It should be noted that the third resonance is a specific feature of a non-linear system; this feature is addressed in greater detail below. For a fully fluid-filled rotor, the first and third resonances are suppressed, while the second resonance looks similar in a sense to the linearized case. Thus, maximumamplitudes of forced oscillations of the rotor and the foundation are observed when one third of the rotor cavity is filled with fluid. In addition, a decrease in the fluid volume leads to a shift of the third resonance towards lower angular frequencies. Decrease of the damping coefficient (K = 0.01) causes a shift of the third resonance to higher angular velocities (see Figs. 4 and 5). The first and second resonances are observed at virtually the same frequencies as for r0 = 0.667R and K = 1, i.e., at s = 0.065 and s = 0.65 with amplitudes Zr = 0.62, Zf = 0.33 and Zr = 0.52, Zf = 0.07. The first resonance at K = 0.01 coincides in magnitude with the resonance at K = 1. The second resonance of the rotor and foundation for K = 0.01 is 6.35 and 54.28 times lower than the corresponding resonances for K = 1. The number of self-oscillating zones does not change, i.e., three self-oscillation zones are observed, the first in the interval 0.065 < s < 0.65, the second in the interval 0.65 < s < 3.41, and the third at s > 3.41 with maximum amplitudes Zr = 0.43, Zf = 0.043, Zr = 0.9, Zf = 0.19 and Zr = 0.104, Zf = 0.017, respectively. The shifted third resonance appears at s = 3.4 with amplitudes Zr = 8 and Zf = 13.3, which is 2.76 times and 1.34 times less than the amplitudes of the third resonance of the rotor and foundation, respectively, at K =1. With a significant increase in the damping coefficient, i.e., for K = 0.1, the amplitudes of self- oscillations are damped and coincide with those at K = 1. The resonance amplitudes of the rotor and the foundation increase in magnitude. The first, second and third resonances appear at the same frequencies as for K = 1, that is, at s = 0.065, s = 0.65 and s = 2.12, with amplitudes Zr = 0.9, Zf = 0.3, Z = 2.9, Zf = 4.1 and Zr = 25.5, Zf = 9.1, which coincides in magnitude with the first, second, and third amplitudes of both the rotor and the foundation for K = 1. Further increase in the damping coefficient (cases K = 5 and K = 10) leads to significant damping of self-oscillations and resonance amplitudes, for example, for K = 5, the amplitudes of the first, second, and third resonances are equal to Zr = 0.02, Zf = 0.003, Zr = 1.15, Zf = 0.4 and Zr = 1.7, Zf = 3.6, respectively. For In addition, with a significant decrease of the K = 10, the amplitudes of the first, second, and third damping coefficient, a shift of the third resonance to resonances are Zr = 0.02, Zf = 0.003, Zr = 0.66, Zf = 0.1 higher angular velocities occurs. In particular, for a and Zr = 1.1, Zf = 2.5, respectively. third fluid filled rotor, the aforementioned resonance is observed at the same frequency as that of an empty rotor (s = 3.4), which is a distinctive feature of non­linear vibrations. To assess the effect of the foundation mass on self-oscillations and resonance amplitudes at r0 = 0.667R, various ratios of the rotor and foundation masses ľ = 0.96, ľ = 0.192, ľ = 0.096, ľ = 0.048 and ľ = 0.0096 were considered and for each case the amplitude-frequency characteristics were constructed (see Figs. 6 and 7). In all of the above cases, the first, second, and third resonances were observed at s = 0.065, s = 0.65, and s = 2.12, respectively. With an increase in the mass of the foundation (cases ľ = 0.096, ľ = 0.048), there is an increase in the amplitudes of the rotor at the third resonance (at s = 2.12, Zr = 5.6, Zf = 7.53) and self-excited vibrations in general, whereas the resonance amplitudes and amplitudes of self-oscillations of the foundation decrease in magnitude. In the case in which the mass of the foundation is slightly greater than the mass of the rotor (cases ľ = 0.192 and ľ = 0.96), the minimum amplitudes of the third resonances of both the rotor and the foundation are observed (Zr = 1.94 and Zf = 3.39, Zr = 0.34 and Zf = 1.01). It should be noted that at ľ = 0.96 in the interval 0.65 < s < 2.125, a zone of intense self-oscillations of the foundation with a maximum amplitude of 0.28 appears (see Fig. 7). Damping the rotor amplitudes with a decrease in the mass of the foundation is a specific feature of the non-linear system. In the linear case, a decrease in the rotor amplitudes is observed with an increase in the mass of the foundation, as in this case the foundation itself acts as an anti-load. To study the influence of nonlinear effects arising from nonlinearity of the restoring force, different values of the stiffness coefficient at the cubic term were considered, i.e., .= 0.01, .= 0.1, .= 1, .= 10 and .= 100, and for each case the amplitude-frequency characteristics with a rotor fluid filled by one third, i.e., for r0 = 0.667R, are obtained (see Figs. 8 and 9). An increase in the stiffness coefficient at the cubic term leads to a decrease in both the resonance amplitudes and the amplitudes of self-oscillations of the rotor and the foundation. The maximum amplitudes are observed in the case of .= 0.01, for which the amplitudes of the first resonance for s = 0.065 are Zr = 1.86 and Zf = 0.15, the amplitudes of the second resonance for s = 0.65 are Zr = 19.215 and Zf = 3.84, and the amplitudes of the third resonance for s = 2.125 are Zr = 23.47 and Zf = 14.09. The minimum resonance amplitudes are, respectively, observed in the case of . = 100, where the amplitudes of the first resonance for s = 0.065 are equal to Zr = 0.03 and Zf = 0.01, the amplitudes of the second resonance for s = 0.65 are equal to Zr = 0.07 and Zf = 0.37, and the amplitudes of the third resonance for s = 2.125 are Zr = 0.31 and Zf = 0.83, i.e., with an increase in the stiffness coefficient by two orders of magnitude, the resonance amplitudes of the rotor and foundation decrease in 54.78 and 15 times for s = 0.065, in 266.87 and 10.37 times for s = 0.65, and in 76.21 and 16.97 times for s = 2.125. Any significant shift of resonant frequencies and self-oscillation zones is observed when the stiffness coefficient changes. In this case, as in linear cases, an increase in stiffness leads to damping of all vibrations. Analysis of greater values of the stiffness coefficient is not of practical interest for the original engineering system. Due to the imperfection of equipment and other factors at high operating speeds, it is important to determine the range of critical values of inbalance e, and to identify its impact on the behaviour of the system. In this work, the cases E = 0.5, E = 1, E = 5 and E = 10 were considered; for each case the amplitude-frequency characteristics were constructed (see Figs. 10 and 11). For .= 0.5, .= 1 and .= 5, as before, three resonance amplitudes are observed for s = 0.065, s = 0.65, and s = 2.125. In this case, the change in the imbalance value practically does not affect the values of the resonance amplitudes and amplitudes of self- oscillations. As the value of imbalance increases by an order of magnitude, i.e., at E = 10, the amplitude-frequency characteristic of the rotor and the foundation is given by the nonlinearity of a hardening (stiff) type. Thus, with an increase in linear imbalance, self-oscillations arising from the action of the fluid are suppressed by forced vibrations. In this case, the characteristic breakdown of the amplitudes is shifted towards higher frequencies of the system. Further, after the breakdown, due to the self-centring effect, operation of the system is stabilized, and vibrations occur with an amplitude of about Zr = 1.26, which slightly exceeds the amplitude of self-oscillations. During rotor vibrations, one self-oscillation zone is observed in the interval 0 < s < 0.346 with maximum amplitudes Zr = 0.08. During vibrations of the foundation, a sharp variation of the amplitude up to the value Zf = 0.09 is observed over the interval 0 < s < 0.112. Further, with an increase in frequency, a gradual increase in foundation vibrations is observed until its amplitude breaks down, and, then, as in the case of the rotor, vibrations stabilize with an amplitude of about Zf = 0.03. In both cases, for the rotor and for the foundation, breakdown is observed at s = 2.817. To understand the nature of the first and third resonances, we construct the amplitude-frequency characteristics of the linear and non-linear “rotor-foundation” system without fluid (see Figs. 12 and 13), which can be obtained from the equations of motion, Eq. (13) for Fr = 0 as .. ˜2( °) ˜2 kz. .e 2 znzz .exp( i .t ), 02 00 .. zn2 °2( °) ˜2 k ˜z nzz z. .0, (27) 22201 2 02 and 2 32 znzz ( °it .. ˜( °) ˜nzz ) ˜2 kze . ..exp( .), 0212 00 22 3 °°nzz .. zn˜zn ( zz ) °( °) ˜2 kz . .0. (28) 22201 210 2 02 The linear system of equations, Eq. (27) can be solved with sufficient degree of accuracy by numerical methods (see Figs. 12 and 13, black (solid) curve). A cubic nonlinearity supports ultra-harmonic (.0 = 3., or s = 3) -harmonic resonances (.0 = ./2 and .0= ./3, or s = 0.5 and s = 0.333). The frequencies of the latter are multiples of the fundamental one (.0 = ., or s = 1). The solution of non-linear equations in Eq. (28) is found analytically, in order to incorporate explicitly the contribution of higher harmonics corresponding to sub- and ultra-harmonic oscillations. The restoring force Fc approximated by Eq. (2) does not support sub-harmonic resonance in the system. Then, to find third-order sub-harmonic resonances, we introduce a notation .0 = 3.f, (.f is sub-harmonic frequency) and search for a solution in the form [28]: .. zB / exp .it .°....B exp .it .°..., (29) ˜ˆ3 ˆ 13 f 1 f and .. zD / exp .it °..D exp .3 it .ˆ ˜.ˆ..°.... (30) 13 f 1 f Substituting Eqs. (29) and (30) into Eq. (28) and using the harmonic balance method, we obtain a system of algebraic equations: 33 22 . 2 . .. °n ˜9 .B ˜nD .nB ˜D .˜ 0 f ./0/ 1 °/ 13 13 13 13 / 4 32 nB °. ˜D . °. ˜. .˜ ˜.BD 113 / 1/ 3 11 4 . . .. ˜3 nB / ˜D 13 / .1 ˜1 .2 ˆ9 2, (31) 1 °13 °BD e.f 2 1 22 . ˜2 . .. .. n ˜81 .B nD ˜nB 13 ˜D / ° 0 f .101 1 °/ 13 4 3 2 33 .... .. .°/ ˜..° nB ˜D .°BD nBD ˜.ˆ0,(32) 113 / 13 11 111 24 23 3 22 2 .. .. nn ˜°9 D nB °nB D .˜ 13/ 13 .2 01 .f ./ °0113 10 .13/ °/ 4 .. .. ˜n °13 BD °1 .˜ 3 .B D .2 . 1013/ / 1 4 3 ˜nB D BD ... °. 20 (33) 10 .13/ °13/ ..11 .ˆ, 2 1 222 2 .. .. nn ˜°81 DnB ˜nB D .°/ ° .2 01 .f .1 °01 1 10 13 / 13 . 4 3 233 .... .. / °.nBD 1 .ˆ. (34) °nB .° D ..BD °.°0 1013 13/ 1 1 101 24 Solving Eqs. (31) to (34), we obtain the amplitudes of the steady-state forced vibrations ... . B / BD 13 and ,,D1 , which can be used to construct 131 / the amplitude-frequency characteristics of the rotor and foundation in the case of sub-harmonic vibrations. To determine ultra-harmonic resonances, the solution is sought in the form: zZ exp .it °...B exp .it .ˆ°.., (35) ˜.ˆ.3 . r 00 and z ˜Z exp .i .ˆt °....D exp .3 i .ˆt °.... (36) 2 f 00 In this case, the force of reaction of the fluid on the rotor wall has the form: 2 F ˜m ˆA exp( .i ˆt °.) .. rL 00 2 .9 m ˆB exp .3 i ( ˆt °.). .(37) L 00 Substituting Eqs. (35) and (36) into Eq. (4) and using the harmonic balance method, we obtain a system of non-linear algebraic equations resulting in Bd0 D (38) = , and DdB (39) = , where a .ia d ˜a °ia , dd ˜1, d ˜0 1, 001 0 a 02 °a 12 .( bc °bc ) .( bc .bc ) 00 11 01 10 a 0 ˜, a 1 ˜, b 3 b 3 22 2 b ˜b °b , b ˜.9 ..(1 °.D ), 3010 0 L 0 1 22 b ˜6 .k ., D ˜, c ˜n .9 ., 100 020 4 ˆ.3 c ˜6 .k , b ˜6 .k .. 1 00 1 0 To determine the unknowns, we also have a system of two equations, which takes the form: .3 eC˜nZ Z ( ˜) °nZ Z ) .( ˜2 ( ˜ZZ ) ° 0 01 rf 10 rf ˆrf .4 33 . ( BD ˜)2 °( BD )(Z Z ), °˜r ˜f 0 24 . .3 eDnBD n ˜( ˜) ˜.( BD 3 ˜) ° 1 01 10 ˆ .4 3 21 ˜3 ., °( ZZ ˜)( BD ˜) ° ( ZZ) 0 (40) rf rf 24 . where 22 22 e ˜n °..2 ., e ˜n °9 ..ik ik 6 .. 020 0012 0 00 From the system in Eq. (40), taking into account Eqs. (38) and (39), we have: 2 Bg˜Bl ˜°0, (41) where 33 °ef ff( n .nf ) . 01 1201 102 f g ˜2, l ˜4. 21( °d )3 2 ffn (1 °d ) 21 01 2 Solving Eq. (41), we obtain the values of the amplitudes B1,2, and using Eqs. (38) to (40), we can find all the amplitudes of ultra-harmonic oscillations and the phase angle. Thus, by comparing the amplitude-frequency characteristics obtained from the system in Eq. (27) and the amplitude-frequency characteristics obtained from systems in Eqs. (4) and (28), it is possible to determine whether sub-harmonic and ultra-harmonic resonances are present. As sub-harmonic and ultra-harmonic resonances are observed only in non-linear systems, and (as in the linear case) there is no third resonance, which is observed at s = 3.24 (see Figs. 12 and 13, black (solid) curve), we can conclude that this resonance is ultra-harmonic, which is also confirmed by the fact that .0 ˜ 3.; in doing so, a shift in frequency is due foundation vibrations, whereas the presence of the first resonance is typical for a multi-degree of freedom system. The numerical solution of the non-linear system in Eq. (28) does not indicate the presence of sub­harmonic resonances (see Figs. 12 and 13, purple curve). To design rotor installation with these parameters, the optimal value of the foundation mass has to be up to 1.042 of the rotor mass, with linear eccentricity not exceeding E = 10. It is recommended to increase the stiffness coefficient at the cubic term by five orders of magnitude relative to the linear term and to decrease the damping coefficient by two orders, which will result in smaller amplitudes and will increase ultra-harmonic resonances. Thus, for stable operation of the system at any filling of the cavity, it is necessary that the operating speed is in the range 0.065 < s < 0.65 or s > 3.41. 4 CONCLUSIONS A method for calculating amplitudes and constructing frequency characteristics of forced and self-excited vibrations of the “rotor-fluid-foundation” system on rolling bearings with a non-linear characteristic based on the method of complex amplitudes and harmonic balance has been developed. The specific features of the non-linear dynamic behaviour characterised by ultra-harmonic vibrations are studied. The optimal parameters of linear imbalance, the mass of the foundation, the fluid volume in the rotor cavity, the stiffness and damping coefficients, for which the amplitudes take optimal values, have been determined. 5 ACKNOWLEDGEMENTS This work has been supported financially by the research project AP08856167 of the Ministry of Education and Science of the Republic of Kazakhstan and was performed at Research Institute of Mathematics and Mechanics in Al-Farabi Kazakh National University, which is gratefully acknowledged by the authors. The corresponding author gratefully acknowledges financial support from the Slovenian Research Agency (ARRS), Grant Ref. J2-9224. 6 REFERENCES [1] Sharma, A., Upadhyay, N., Kankar, P. K., Amarnath, M. (2018). Non-linear dynamic investigations on rolling element bearings: A review. Advances in Mechanical Engineering, vol. 10, no. 3, p. 1-15, DOI:10.1177/1687814018764148. [2] Wang, G., Yuan, H., Sun, H. (2020). 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Received revised form: 2021-08-21 DOI:10.5545/sv-jme.2021.7220 Original Scientific Paper Accepted for publication: 2021-08-24 Trajectory-Tracking Control for Manipulators Based on Fuzzy Equivalence and a Terminal Sliding Mode Youyu Liu1,2,* – Yi Li1,2 – Xuyou Zhang1,2 – Bo Chen1,2 1 Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, China 2 Anhui Polytechnic University, School of Mechanical Engineering, China To suppress the chattering of manipulators under heavy-load operations, a control method called fuzzy equivalence & terminal sliding mode (FETSM) was applied to the trajectory tracking of motion curves for manipulators. Based on the switching term of the equivalent sliding mode (ESM), a fuzzy parameter matrix processed by the simple fuzzy rules was introduced, and the fuzzy switching term was obtained. By summing the fuzzy switching term and the equivalent term of the equivalence and a terminal sliding mode (ETSM), the control law of the FETSM for manipulators was obtained. On this basis, the stability of the system was analysed and the finite arrival time of it was deduced. On the premise of ensuring the stability of the system, the fuzzy rules and membership functions were designed for the fuzzy constants in the fuzzy switching term. Simulation tests show that the proposed FETSM can ensure sufficient trajectory-tracking precision, error convergence speed, and robustness. Compared with the ETSM, the proposed FETSM can reduce the chattering time by 94.75 % on average; compared with the proportion-integral-differential (PID) control method, the maximum chattering amplitude by the FETSM can be reduced by at least 99.21 %. Thus, the proposed FETSM is suitable for those manipulators under heavy-load operations. Keywords: sliding mode control, fuzzy switch term, chattering, manipulator, trajectory tracking Highlights • The law of the FETSM control system for manipulators is obtained. • The stability and finite arrival time of the FETSM system are proved. • Fuzzy rules are used to estimate the external disturbances of the system and uncertainties of the system. • The chattering of the sliding mode system can be suppressed by the proposed control method. 0 INTRODUCTION Manipulator is a non-linear and time-varying system with multi-input, multi-output, strong coupling, in which there are usually uncertain modelling and external disturbances [1] and [2]. Moreover, the trajectory tracking of motion curves for manipulators has always been a hot and difficult topic. Manipulators are mostly used for heavy-load operations, such as palletizing, handling, and so on, in which small chattering and high torque control ability are required, while the trajectory tracking precision and robustness requirements are insignificant [3]. At this present, some control methods are used in trajectory tracking for manipulators, mainly including sliding mode control [4] and [5], neural network control [6] to [8], the proportional–integral–derivative (PID) control [9] to [11] and robust control [12] and [13]. In order to reduce the chattering of manipulators under heavy-load operations, a fuzzy equivalence and a terminal sliding mode (FETSM) control method for manipulators has been proposed in this article. Sliding mode control has been receiving increasing attention due to its characteristics of fast response and small influence by parameter changing and external disturbances. However, there is a serious chattering phenomenon in sliding mode control, which affects the stability of trajectory-tracking system and increase energy consumption. In order to overcome the chattering of the sliding mode system for manipulators, researchers have conducted some studies. For example, Liu et al. [14] constructed an approaching law with fal function to adjust the speed of approaching motion. When the state is far from the sliding mode surface of system, the speed of approaching motion is faster; when the state is close to the sliding mode surface, the speed decreases greatly. Therefore, the system can pass through the sliding mode surface at a lower speed to reduce its chattering. However, it is not easy to select the deceleration position. The overall time of approaching motion will be increased if the position chosen is too far from the sliding mode surface; otherwise, the chattering will be increased to a certain extent. Combining with the dynamic surface, which prevented the complex differential of the virtual control laws, Jiang et al. [15] designed a fuzzy adaptive system to approximate the observation errors of the uncertain disturbances and to reduce the chatter of the control law. The fuzzy parameters and fuzzy rules of this method are complex, which will reduce the computational efficiency. Xi et al. [16] set the deceleration position change with the state of the sliding mode system by using fuzzy rules to deal with the exponent in power approach laws and replaced sign functions with hyperbolic tangent functions. This method can reduce the approaching time of the sliding mode system and the chattering of them. However, the chattering of the sliding mode system is still at a high level; consequently, some constants are still set to offset the disturbances and uncertainties. Tran et al. [17] used radial basis function neural networks to approximate the unknown non-linear dynamics of the system and designed an adaptive compensation mechanism of approximation error and adaptive law of weights of neural networks to realize model-free control. Chang et al. [18] proposed an adaptive sliding mode control scheme by introducing an adaptive term derived from a radical basis function network, which does not require knowledge of the bound of the system uncertainties and external disturbances. It simultaneously eliminates the chattering phenomenon without deteriorating the system fast-response characteristic and robustness. The chattering of the system is reduced by the above two methods. However, with the increase of the degree of freedom (DOF) of manipulators, the number of parameters in each matrix in a controlled model increases geometrically, which will lead to a significant increase in the computational complexity and the time required to control the system. In this article, the FETSM control method is proposed. a fuzzy parameter matrix is introduced into the switching term of the equivalent sliding mode (ESM), and the matrix is processed with simple fuzzy rules, so that it can approach the sum of the uncertainties and external disturbances to the greatest degree possible while ensuring the stability of the system. The purpose is to reduce chattering ensure that the state of the trajectory-tracking system moves towards the sliding mode surface. 1.1 Design on Equivalent Term The dynamic equation of N-DOF manipulator can be described as follows [19]: .. . (1) Mq q+Vq,q q+Gq + .. . q,q +d= . , ˜°˜°˜°˜° where q is an N-dimensional joint position vector; q .is the first-order derivative of q. q.. is the second-order derivative of q. M(q) is an N-order symmetric positive definite inertial matrix; V(q, q .) is an N-order centrifugal force matrix; G(q) is an N-dimensional gravity vector; .(q, q .) = [.1, .2, ..., .N]T is the uncertainties of modelling and friction; d = [d1, d2, ...,dN]T is an N-dimensional external disturbance, and ||.(q, q .) + d|| = . + d; t is an N-dimensional joint input torque. M(q) is an invertible matrix because M(q) is a symmetric positive definite inertia matrix. From Eq. (1), - q=Mq .. .. . Vq,qqGq ..  q,q .d ˆ. (2) 1 ˜°.˜°.˜°.˜° .. Order x1= q and x2= q ., the dynamics equation of manipulators can be inferred from Eq. (2), namely, x . 1 ˜x 2  .1 x . 2 ˜Mx 1 (3) °..  ˆ Vxxx Gx , ..°.. °xx ..°, .d . .1221 12 .  Order e = qd – x1 (qd is the desired position trajectory of joints), a non-linear sliding surface is designed as follows [20]: s=e+ ˜ °1e. p q . (4) and optimize the input torque control of manipulators under heavy-load operations on the premise of where e is the position tracking error of joints; ensuring sufficient joint control precision, error ß= diag{ß1, ß2, ..., ßN}, and ß1, ß2, ..., ßN> 0; p and q are convergence speed, and robustness. The feasibility odd numbers, and p > q, 1 < p / q <2. and effectiveness of the proposed method are verified Taking the derivative of Eq. (4), then, by comparing the trajectory tracking and input torque . pq p ˆ .1.. . ee .. .. se ˜° , (5) with several existing methods. q diag q 1 DESIGN ON CONTROL LAW OF FETSM In order to obtain the control law of the FETSM, the equivalence term and the fuzzy switching term need to be designed. The former makes the trajectory tracking move on the sliding mode surface, and the latter is used to offset the uncertainties and external disturbances to where diag{..e } is an N-order diagonal matrix in which the principal diagonal elements are the corresponding elements of ..e . Let s .= 0, thus, pq p °1 ° . ... d °. 2 .. q e ˜ˆ diag q xe .0. (6) q Liu, Y.Y. – Li, Y. – Zhang, X.Y. – Chen, B. Ignoring the uncertainties and external e1.1, e2.2, ..., eN.N = . + d; sgn is a symbolic function disturbances, and substituting Eq. (3) into Eq. (6), [21], and sgn(s) = [sgn(s1), sgn(s2), ..., sgn(sN)]T. If Z is 1, Z .. 0 ° p ... d ° ° 1 qMx 1 .. a real number, then, sgn Z ˜° e . ˜  1 diag 0, Z 0 q . .. . . 1, Z ˆ 0 ° ˆ° Vxxx Gx ., e 0 ..°... pq . (7) .122 1 q Order . = diag{.1, .2, ..., .N} and From Eq. (7), e= diag{e1, e2, ..., eN}, The control law of the FETSM can be obtained by adding Eq. (11) to Eq. (12). ... qMx 1 ˜°. d ˜ 1 diag   .  p 2 q Mx 1 q ... .. d ˆ diag ° e .   ˜ 1  .   q p 2 ˜ .  ˜ q    ˆ Vxxx Gx .˜° ..2 ˜°.1 p  e . . (8) q  , 1 2 p ˆ., .ˆ. sgn ..(13) Vxxx Gx s. ..ˆ 122 1 diag–1 is defined as an operation that has the following characteristics: If .= diag{.1, .2, ..., .N} is a diagonal matrix, then, 2 ANALYSIS ON STABILITY AND FINITE ARRIVAL TIME diag 1 . , ,..., .T ... (9) ˜°...12 N ˆ 2.1 Stability Analysis Both sides of Eq. (8) are diagonal matrices. From For the sliding mode control of a rigid manipulator, a Eqs. (8) and (9), Lyapunov function [22] was defined as follows: qMx . Vxxx Gx ˜°.˜°.˜°. 1 .21 d .. ˜ ˆ . 1 , 1 1 2 L s T s , (14) = y   p 2 2 ˜ q . diag ˜  e . (10) ˜ 1  q   . taking the derivative of Eq. (14), p   From Eq. (10), the equivalent term can be designed as Eq. (11), .q 2 p  1 ˜ Mx 1 q .. d diag ° e . q eq ...ˆp  . Vxxx G , x .. ˆ.12 .2 ˆ1. (11) 1.2 Design on Fuzzy Switching Term In order to ensure that it can compensate uncertainties and external disturbances, the fuzzy switching term is designed as follows, ˜ °diag .,, ., .  sw 12 N ˆdiag .1,, ., N ...sgn s , (12) 2 where diag{.1, .2, ..., .N} is a constant matrix, and .1, .2, ..., .N = . + d, . =max(|d(t)|) + .; diag{e1, e2, ..., eN} is a fuzzy parameter matrix (tsw is a . T L = ss . . (15) y From Eqs. (3), (13) and (15), . T p .1 .1 L ˆqMx .. ˜s .e . ° diag .. . yd 1 q . pq q .., .de .  Vxxx Gx ..... .x , x .. . (16) 1221 1 2  Substituting Eq. (13) into Eq. (16) and simplifying it, . p 11 ˆ1 pq L ˜..,, °.Mxq . x q   y 1 d 2 q .1 N . xx , ds T diaag  , (17) s 12 where |s| is the absolute value of s, namely |s| = [|s1|, |s2|, ..., |sN|]T . For 1< p / q < 2, then 0< (p – q) / q < 2; and both p and q are positive odd numbers. Therefore, the ˜elements of °q .˜.pq are non-negative numbers. x 2 q d basic switching item when e1, e2, ..., eN are all equal to ß1, ß2, ..., ßN are positive numbers and M–1(x1) is a 1, and tsw is a fuzzy switching item while one of symmetric positive definite matrix, when x2 . q.d, e1, e2, ..., eN is not equal to 1; and pq If the trajectory-tracking system spends time tj p .11 ˆ .,, .Mx 1 q . d x 2 q 0. (18) °1 from ej(0) . 0 to ej(tj) = 0, then, q .˜1 ˜N . q p . . L y is determined by the N-order t j d t 0 . The sign of diagonal matrix (.(x1, x2) + d)sT d e . (24) .ˆˆ. .. e  j j ˜ ˜ ° j e j 0  – diag{e.|s|}, which 0 ° is written in matrix form as follows: From Eq. (24), The effective time to reach equilibrium state is as follows: .., .ˆ ˜ xx ds T diag °°.. s  .12  e 0 j ..pq .˜ds °. ˜ .ˆ. 1 111 s 1 0 ...ˆ . p 1 p pq    ˜ (25) (19) t  °.°   . j j .  °  ˜˜ds °. s 1 ..NN .N NN  j ˆ 0 Due to |s1| = s1, ..., |sN| = sN, all of the principal diagonal elements of (.(x1, x2) + d)sT– diag{e.|s|}are non-positive numbers. Moreover, when the fuzzy parameter matrix works normally, e1.1, e2.2, ..., eN.N = . + d. Thus, L.y = 0. When x2 . q.d, the trajectory-tracking system employed by the FETSM is stable according to Lyapunov’s second method [23]. In addition, the sliding mode surface of it is non-singular. Substituting Eq. (13) into Eq. (3), If tmax= max{t1, t2, ..., tN}, the trajectory-tracking error of the system must reach 0 within tmax, which is finite according to Eq. (25). Therefore, the trajectory-tracking error of the system can reach 0 within a finite time. 3 DESIGN ON FUZZY RULES AND SYSTEM STRUCTURE 3.1 Design on Fuzzy Rules Supposing there is no fuzzy parameter matrix e in Eq. (18), L.y = 0 is still proper for .1, .2, ..., .N are constants and not less than . + d.Therefore, the FETSM system ˆ  p q . 2 .. 1 Mx 1 .. .1 xq ˜ ..  .. . qx d . ° ° diag . .  q d 2 2 p of manipulators remains stable. In contrast, .xx , d dia.ˆsgn..  .°.ag s uncertainties and external disturbances are time­ , (20) 1 2 varying, and .1, .2, ..., .N are constants. Therefore, when x2= q.d, for ..e = q.d– x2, .1 ..eM x ˜. °x, xd diag °... °.ˆ...ˆsgn s . (21) 1 12 When s > 0, ..e < 0, namely, e .will decrease rapidly. While s < 0, ..e > 0,, namely, e. will increase rapidly. Obviously, s can reach 0 within a finite time when x2= q.d [20]. Therefore, the sliding mode of the N-DOF manipulator with the FETSM control law exists and can be reached. 2.2 Analysis on Finite Arrival Time q e Eq. (4) can be written in scalar form: sj ˜ej °.j ... jp j = 1, 2, ..., N. Order sj = 0, p q ej ˜j .j ..0. (22) °d e . d t .ˆ constant matrix cannot follow the change of .(x1, x2) + d, which leads to every main diagonal element of .sgn(s) always being large, and then the chattering of the sliding mode system is increased. On the premise of stability, the difference between the corresponding parameters of e.sgn(s) and .(x1, x2) + d are as small as possible by using the fuzzy method to adjust e, so as to reduce the chattering. NE, PO and ZE are used to represent negative, positive and zero, respectively. The input and output of the fuzzy system are defined as follows: ˆinput : s NE ZE PO . .˜°,, .j , j ˜, ,..., N . 12 (26) ˆoutput : .j ˜°ZE PO . , . “If-then” statement [24] is used to define fuzzy rules between the input and output. ... ° 2 : If s is NE then is PO jj Rule 3 j Rule 3 j ˜˜ From Eq. (22), ° 1: If s isZEthen isZE . jj (27) .ˆ q ° : If sj is PO then is PO j 3 j ° Rule .ˆ p e j d t ˜°d e . .. j ..  .j . (23) Liu, Y.Y. – Li, Y. – Zhang, X.Y. – Chen, B. 3.2 Design on Membership Function As for the input s, the membership function is designed as follows [25]: its domain is [–1, 1]; NE is a trapezoidal membership function, [–1, –1, –2×s, 0]; ZE is a triangular membership function, [–2×s, 0, 2×s]; PO is a trapezoidal membership function, [0,2×s, 1, 1]. As for the output e, the membership function is designed as follows [25]: its domain is [0, 1]; ZE is a triangular membership function, [0,1×10–5, 2]; PO is a trapezoidal membership function, [0, 1, 1, 1]. The s in the function represents the thickness of the boundary layer. When –s < s < s, s and e belong to ZE. In addition, the variation range of the e is between 1×10–5 and 1. While the s approaches the boundary layer, the e approaches 1. Conversely, when the s approaches 0, the e approaches 1×10–5; When the s is outside the boundary layer, the e is 1. The membership function curves of the input and output of the proposed control method are shown in Fig. 1 [26]. 3.3 System Structure The structure of the trajectory-tracking system of the FETSM for manipulators is shown in Fig. 2. 4 SIMULATION VERIFICATION 4.1 Analysis on Trajectory Tracking MATLAB R2019b is applied as the simulation platform to track the ideal curve trajectories by using the FETSM, the ETSM [27], the ESM [28], and the PID [10] methods, respectively. By comparing the error of position tracking of each joint, the effects of the trajectory tracking of the above four control methods are analysed. Two curves are chosen as the ideal position curves of the joints 1 and 2. The coefficient of the curves is listed in Tables 1 to 4. The Fig. 1. The membership function of input and output of the FETSM; a) input s; b) output e Fig. 2. Structure diagram of the FETSM trajectory-tracking system Table 1. The coefficients of the 1st position curve of the 1st joint Intervals [s] [0.0000, 2.3806) [2.3806, 3.5057) [3.5057, 4.2751) [4.2751, 6.6127) [6.6127, 7.8886) [7.8886, 10.2557) q4 [rad] q3[rad] q2 [rad] q1 [rad] –0.0152 0.0731 0.0000 0.0000 0.0243 –0.3034 1.3445 –2.1338 –0.0125 0.2120 –1.3659 4.2007 0.0005 –0.0093 0.0535 0.1554 –0.0034 0.0942 –0.9729 4.6803 0.0114 –0.3746 4.5742 –24.4928 q0 [rad] 0.0000 1.2699 –4.2817 0.0418 –7.4387 50.0953 Trajectory-Tracking Control for Manipulators Based on Fuzzy Equivalence and a Terminal Sliding Mode 437 Table 2. The coefficients of the 1st position curve of the 2nd joint Intervals [s] q4 [rad] q3[rad] q2 [rad] q1 [rad] q0 [rad] [0.0000, 2.3806) –0.0019 0.0093 0.0000 0.0000 0.5236 [2.3806, 3.5057) –0.0020 0.0101 –0.0029 0.0045 0.5209 [3.5057, 4.2751) 0.0136 –0.2079 1.1433 –2.6743 2.8686 [4.2751, 6.6127) –0.0032 0.0796 –0.7004 2.5805 –2.7475 [6.6127, 7.8886) –0.0010 0.0213 –0.1216 0.0290 1.4707 [7.8886, 10.2557) 0.0029 –0.1033 1.3526 –7.7240 16.7608 Table 3. The coefficients of the 2nd position curve of the 1st joint Intervals [s] q4 [rad] q3[rad] q2 [rad] q1 [rad] q0 [rad] [0.0000, 2.3806) 0.0020 –0.0097 0.0000 0.0000 –0.3491 [2.3806, 3.5057) –0.0017 0.0253 –0.1249 0.1982 –0.4670 [3.5057, 4.2751) –0.0002 0.0051 –0.0185 –0.0503 –0.2492 [4.2751, 6.6127) –0.0029 0.0504 –0.3096 0.7791 –1.1356 [6.6127, 7.8886) 0.0147 –0.4147 4.3041 –19.5600 32.4885 [7.8886, 10.2557) –0.0094 0.3443 –4.6773 27.6740 –60.6644 Table 4. The coefficients of the 2nd position curve of the 2nd joint Intervals [s] q4 [rad] q3[rad] q2 [rad] q1 [rad] q0 [rad] [0.0000, 2.3806) –0.0032 0.0156 0.0000 0.0000 0.3491 [2.3806, 3.5057) 0.0071 –0.0822 0.3493 –0.5543 0.6790 [3.5057, 4.2751) –0.0116 0.1797 –1.0280 2.6645 –2.1421 [4.2751, 6.6127) 0.0044 –0.0941 0.7280 –2.3400 3.2066 [6.6127, 7.8886) –0.0085 0.2460 –2.6457 12.5325 –21.3803 [7.8886, 10.2557) 0.0032 –0.1229 1.7198 –10.4260 23.8975 Table 5. Stetting for external disturbances and uncertainties and starting points External disturbances and uncertainties Starting points [rad] The 1st position curve The 2nd position curve The 1st position curve The 2nd position curve The 1st joint 10 exp(–2(t– 5)2) 10 sin(t) 0.02 –0.34 The 2nd joint 2 exp(–0.5(t– 5)2) 2 cos(t) 0.52 0.34 ideal trajectories in each period accords with Eq. (28), a reference trajectory formula. 43 2 tq°tq yq˜tq°°tq°. (28) 4 3210 The setting for the uncertainties and external disturbances and starting points for the four kinds of control methods is shown in Table 5. The parameter setting for the three sliding mode methods is shown in Table 6. The boundary layer thickness of the fuzzy controller is taken as 1×10–5. The control parameters of the PID are as follows: Kp = 5000, Ki = 500, Kd = 2000. The design of the two-DOF manipulator in this paper is as follows: m1= m2= 2, r1= 1, r2 = 0.8, g = 9.8 [29]. Table 6. Parameter setting for the three sliding modes Methods p q . ß1 ß2 FETSM ETSM 5 3 0.2 1 1 ESM 1 1 The trajectory-tracking situation of the above four control methods is shown in Figs. 3 to 6, demonstrating that all four control methods can effectively implement the trajectory-tracking task and meet the basic requirements of practical application. According to Fig. 3, it is found that the tracking trajectories by the ETSM and the proposed FETSM coincide with the ideal trajectories with only slight Liu, Y.Y. – Li, Y. – Zhang, X.Y. – Chen, B. Fig. 3. Tracking situation of the 1st position curve of the 1st joint; a) FETSM and PID; b) ETSM and ESM deviation in the initial stage. The phenomenon in The averages of the tracking error are shown in Fig. 3 is similar to those in Figs. 4 to 6. However, the Table 7. For the 1st curve, compared with the ETSM, tracking trajectories by the ESM and PID have some the average error of the 1st joint by the proposed obvious deviation from the ideal trajectories in some FETSM is increased by 1.88 %, and the average error different periods. In order to further compare the of the 2nd joint is reduced by 15.33 %; compared with control precision of the four methods, their tracking the ESM, those errors are reduced by 81.13 % and errors are quantitatively analysed by calculating the 94.20 % respectively; compared with the PID, those ordinate difference between the actual trajectories and error is reduced by 88.58 % and 93.17 % respectively. the ideal ones at the same time and the same joint. As for the 2nd curve, compared with the ETSM, the Fig. 4. Tracking situation of the 1st position curve of the 2nd joint; a) FETSM and PID; b) ETSM and ESM Fig. 5. Tracking situation of the 2nd position curve the 1st joint; a) FETSM and PID; b) ETSM and ESM Fig. 6. Tracking situation of the 2nd position curve of the 2nd joint; a) FETSM and PID; b) ETSM and ESM Table 7. The averages of tracking error Control methods The 1st curve & the 1st joint [rad] The 1st curve & the 2nd joint [rad] The 2nd curve & the 1st joint [rad] The 2nd curve & the 2nd joint [rad] FETSM 4.34×10–4 1.16×10–4 2.11×10–4 2.14×10–4 ETSM 4.26×10–4 1.37×10–4 1.70×10–4 2.79×10–4 ESM 2.30×10–3 2.00×10–3 1.40×10–3 3.40×10–3 PID 3.80×10–3 1.70×10–3 6.00×10–3 2.00×10–3 average error of the 1st joint by the proposed FETSM is increased by 24.12 %, and the average error of the 2nd joint is reduced by 23.30 %; compared with the ESM, those errors are reduced by 84.93 % and 93.71 % respectively; compared with the PID, those error is reduced by 96.43 % and 89.30 %, respectively. In brief, the precision of the proposed FETSM dealing with the 1st joint is less than that of the ETSM, but the average error of the overall joints is reduced by 3.56 % compared with the latter; compared with the ESM and PID, the FETSM can reduce the average error of the overall joints by 89.30 % and 92.78 %, respectively. As a result, the proposed FETSM has better control precision compared with the other three methods. The variation of the tracking error of joints 1 and 2 with time was shown in Fig. 7. The proposed FETSM is superior to the PID in error convergence speed and robustness. To compare the error convergence speed and robustness of three sliding mode methods, taking the error by the ESM at 1 s as the reference, the convergence time of the four curves by the proposed FETSM to achieve the reference error is 0.195 s, 0.260 s, 0.175 s, and 0.045 s, respectively; and that by the ETSM is 0.195 s, 0.260 s, 0.175 s, and 0.047 s, respectively. On the 2nd position curve of the 2nd joint, the error convergence speed by the FETSM is slightly better than that by the ETSM; in other cases, the error convergence time by the proposed FETSM is consistent with that by the ETSM. Moreover, the average speed of the error convergence by the above two is 83.13 % faster than that by the ESM. The averages of the maximum error caused by disturbances in the three sliding mode methods are shown statistically in Table 8. It shows that the robustness of the proposed FETSM is less than that of the ETSM in the 1st joint but better than the latter in the 2nd joint. The robustness of the FETSM is superior to that of the ESM in joints 1 and 2. The averages of the overall disturbance error by the FETSM, the ETSM and ESM are 9.39×10–5, 1.35×10–4, and 2.96×10–3, respectively. In short, the overall robustness of the proposed FETSM is 30.24 % higher than that of the ETSM, and 96.83 % higher than that of the ESM. It can be seen from the above analysis that the proposed FETSM can ensure sufficient error convergence speed and robustness. 4.2 Analysis of Input Torque In order to investigate the anti-chattering performance of the FETSM, the input torque characteristics of the different control systems, in the same joint and in the same position curve, are analysed. The ETSM, which has better trajectory-tracking precision than the ESM, is selected as a competitive sample of non-fuzzy control; the PID is selected as a competitive sample of non-sliding mode control. The comparison diagrams Liu, Y.Y. – Li, Y. – Zhang, X.Y. – Chen, B. for the input torques using the three methods are shown in Figs. 8 to 11. As shown in Fig. 8a, there is chattering with an amplitude of approximately 10.24 Nˇm by the proposed FETSM in the period of 0 s to 0.55 s, but the input torque curve is smooth and continuous in other times. Combinedwith Fig. 7a, it is found that the error by the proposed FETSM converges to 0 s near 0.55 s, and the system reaches the sliding mode surface. Therefore, the reason for the chattering in the initial stage is that the system does not reach the sliding mode surface. The phenomenon in Fig. 8a is similar to those in Figs. 9a, 10a and 11a using the same the FETSM. As shown in Fig. 8b, the chattering amplitude of the input torque curve by the ETSM is small in the period of 4.9 s to 5.1 s; in other periods, it is about 10.20 Nˇm. Currently, the disturbance term applied to the joint is Gaussian and reaches the maximum in 5 s. Obviously, the chattering amplitude of the system decreases only when the disturbance term approaches the switching term. The phenomenon in Fig. 8b is similar to those in Figs. 9b, 10b and 11b. Quantitatively comparing and analysing the chattering time of the systems by the ETSM and FETSM in the four cases shown in Figs. 8 to 11, The chattering time of these two methods is shown in Table 9. It is found that those by the proposed FETSM is less than those by the ETSM, can be reduced by 94.75 % on average. As shown in Fig. 8c, the input torque curve using the PID is consistent with the proposed FETSM except that serious chattering occurs in the overshoot stage. The phenomenon in Fig. 8c is similar to those in Figs. 9c, 10c and 11c. Quantitatively comparing and Fig. 7. Variation of the error of joints; a) the 1st position curve of the 1st joint; b) the 1st position curve of the 2nd joint; c) the 2nd position curve of the 1st joint; d) the 2nd position curve of the 2nd joint Table 8. The averages of the maximum error caused by disturbances Control methods FETSM ETSM ESM The 1st curve & the 1st joint [rad] 9.74×10–5 3.31×10–5 1.18×10–3 The 1st curve & the 2nd joint [rad] 9.17×10–5 2.53×10–4 4.94×10–3 The 2nd curve & the 1st joint [rad] 9.84×10–5 3.03×10–5 1.46×10–3 The 2nd curve & the 2nd joint [rad] 8.79×10–5 2.22×10–4 4.25×10–3 Trajectory-Tracking Control for Manipulators Based on Fuzzy Equivalence and a Terminal Sliding Mode 441 analysing the chattering amplitude of the systems by the PID and FETSM in the four cases shown in Figs. 8 to 11, The maximum amplitudesof these two methods are shown in Table 10. It is found that those by the proposed FETSM is less than those by the PID, can be reduced by 99.21 % at least. Moreover, compared with similar methods proposed by other scholars [1], the proposed FETSM can achieve similar anti-chattering performance while the structure of the switching term is simple. Fig. 8. Input torque comparison of the 1st position curve of the 1st joint; a) FETSM; b) ETSM; c) PID Fig. 9. Input torque comparison of the 1st position curve of the 2nd joint; a) FETSM; b) ETSM; c) PID Liu, Y.Y. – Li, Y. – Zhang, X.Y. – Chen, B. Table 9. The chattering time of the FETSM and ETSM Control methods The 1st curve & the 1st joint [s] The 1st curve & the 2nd joint [s] The 2nd curve & the 1st joint [s] The 2nd curve & the 2nd joint [s] FETSM 0.55 0.58 0.42 0.11 ETSM 9.67 8.81 6.76 6.76 Table 10. The maximum amplitudes of the FETSM and PID Control methods The 1st curve & the 1st joint [Nˇm] The 1st curve & the 2nd joint [Nˇm] The 2nd curve & the 1st joint [Nˇm] The 2nd curve & the 2nd joint [Nˇm] FETSM 10.24 2.21 10.26 2.22 PID 2.85×103 1.68103 1.30103 1.87103 5 CONCLUSIONS (1) The proposed FETSM control method can be used to track the actual motion position of each joint according to the ideal trajectory of the corresponding joint. This technology can be used to suppress the chattering of manipulators under heavy-load operations. (2) Simulation results show that compared with the ETSM, the overall tracking precision by the proposed FETSM is improved by 3.56 %, and the overall robustness is improved by 30.24 %. Moreover, their error convergence speeds are basically the same. Compared with the ESM, the overall tracking precision by the proposed FETSM is improved by 89.30 %, the overall robustness is improved by 96.83 %, and the error convergence speed is improved by 83.13 %. Compared with the PID, the overall tracking precision by the proposed FETSM is improved by 92.78 %. Moreover, its error convergence speed and overall robustness are better than those by the PID. Consequently, the proposed FETSM can obtain sufficient trajectory-tracking precision, error convergence speed and robustness. (3) Employing the proposed FETSM, chattering occurs only when the system does not reach the sliding mode surface. Compared with the ETSM, the proposed FETSM can reduce the chattering time by 94.75 % on average. Compared with the PID, the maximum chattering amplitude by the FETSM can be reduced by 99.21 % at least. Therefore, the proposed FETSM can reduce the chattering of systems, and optimize the input torque control. 6 ACKNOWLEDGEMENTS This work was supported by the Natural Science Research Project of Higher Education of Anhui Province of China [grant numbers KJ2020A0357] and the University Synergy Innovation Program of Anhui Province of China [grant numbers GXXT-2019-048]. 7 REFERENCES [1] Wu A.G., Wu S.H., Dong N. (2019). Non-singular fast terminal sliding model fuzzy control of robotic manipulators. Journal of Zhejiang University (Engineering Science), vol. 53, no. 5, p. 862-871, DOI:10.3785/j.issn.1008-973X.2019.05.006. [2] Liu, P., Liu, Y.H., Wang, K., Wei, X.X., Xin, C., Wu, X.L. (2020). Design and Performance Analysis of Airbag-Soft-Manipulator According to Characteristic Parameters. 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Received revised form: 2021-01-04 DOI:10.5545/sv-jme.2020.6959 Original Scientific Paper Accepted for publication: 2021-01-07 Active Vibration Control of a Mechanical Servo High-speed Fine-Blanking Press Yanxiong Liu1,3 – Yuwen Shu1,3 – Wentao Hu1,3 – Xinhao Zhao1,2,* – Zhicheng Xu1,3 1Wuhan University of Technology, Hubei Key Laboratory of Advanced Technology of Automotive Components, China 2Wuhan University of Technology, School of Materials Science and Engineering, China 3Wuhan University of Technology, Hubei Collaborative Innovation Center for Automotive Components Technology, China The fine-blanking process as an advanced sheet metal forming process has been widely applied in industry. However, specially designed equipment is required for this process. In this paper, a novel mechanical servo high-speed fine-blanking press with the capacity of 3200 kN is proposed, and the vibration control for this machine is researched to achieve the requirement of fine-blanked parts of high dimensional accuracy, since the vibration of the fine-blanking machine will cause the machining displacement error and reduce the machining accuracy. Self-adaptive feed-forward control is used to simulate the active vibration control of the mechanical fine-blanking machine. The vibration control principle of the fine-blanking machine is described, and the control algorithm is established. At the same time, the mechanical vibration model of the fine-blanking machine as the controlled object is established, and the parameters of the excitation input and the mechanical model are obtained by the fine-blanking finite element simulation and the experiments of the vibration measurement of the press. Finally, the numerical simulation and analysis of active vibration control based on MATLAB are carried out. The results show that the control effect is good, and the vibration response is effectively reduced, thus greatly increasing the processing accuracy, saving a significant amount of energy, and reducing the energy consumption and defective rate. Keywords: active vibration control; mechanical vibration model; mechanical servo high-speed fine-blanking press; self-adaptive feed-forward control Highlights • The mechanical vibration model of the fine-blanking machine tool is established. • The dynamic model parameters of the mechanical fine-blanking machine are obtained by using finite element analysis and empirical formula. • An adaptive feed-forward vibration controller for the fine-blanking machine is designed and applied to active vibration control of the fine-blanking machine. • The vibration response of the whole machine has been effectively reduced when the active control is applied, in which the time-domain vibration response has been reduced by more than 95 %, and the frequency-domain vibration response has been reduced by more than 80 %, which means that the vibration reduction effect is obvious. 0 INTRODUCTION The fine-blanking (FB) process as an advanced net shape or near-net shape plastic forming process has been widely applied in industry because of the advantages of high efficiency and high parts quality [1] and [2]. Many mid-thick sheet metal components with complicated shapes and high dimensional accuracy can be fabricated with the FB process in one operation, as shown in Fig. 1a. Compared with the conventional blanking process, a special designed fine-blanking press was required for this process, providing at least three separate forces: blanking force, blank holder force, and counter punch force [3]. Now, almost all FB presses are hydraulic machines with the capacity of 3200 kN to 12000 kN [4], and the forming efficiency is about 30 to 70 times per minute with a punch stroke of 40 mm. With the increasingly huge market demand for the mid-thick sheet metal parts, the FB efficiency should be further improved; the target is about 200 times per minute or higher. A mechanical servo high-speed FB press was designed and fabricated by our research group and Huangshi Huali Co., Ltd., and the forming efficiency can reach up to 220 times per minute and the maximum forming capacity is 3200 kN. The mechanical construction of the main driving system is shown in Fig. 1b, and the main slide block can achieve the trajectory shown in Fig. 1c. With the increasing punch frequency, the vibration of the FB press will become increasingly severe. It is known that the vibration of the press not only affects the parts’ quality, such as causing the crack on the cutting surface and reducing the dimensional accuracy but also reducing the service life of the machine. Moreover, the noise caused by the machine vibration will adversely affect working conditions and human health. Therefore, vibration control for the high-speed fine blanking press is urgently required. a) b) c) Fig. 1. a) The mid-thick sheet metal parts fabricated by FB process, b) the mechanical construction of the main driving system of the high-speed FB press, and c) the slide block stroke diagram In general, the passive control method was applied for the vibration control of the traditional press, such as increasing the stiffness of the frame and adding the vibration isolation system, and the vibration control effort for the excitation source of the press is very rare. For the high-speed FB press proposed in this paper, except for the passive control method, the active vibration control method was also applied to restrain the vibration of the whole machine caused by the elastic restoring force produced by the FB process. Therefore, the low-frequency vibration can be effectively suppressed, so as to reduce the vibration of the mechanical servo high-speed FB press to the greatest extent. For the research of the active control by Lueg, Guicking [5] submitted, early in 1933, a patent application describing for the first time the principle of active noise cancellation. The basic idea is that using the active control wave with the same amplitude of the noise to counteract the original noise. In 1953, Olson and May [6] invented the electronic sound absorption equipment, which proved the practical feasibility of the active vibration and noise control theory through experiments. Since then, active vibration control technology has been gradually applied to engineering machines. Winberg et al. [7] and Daley et al. [8] applied the active vibration control for the marine applications. In recent years, many new active vibration control theories have been developed. Shao et al. [9] created the finite element (FE) active vibration control model of the piezoelectric flexible linkage by using the mixed Hamilton principle. Based on the complex mode theory, a hybrid independent mode controller, which consisted of the state feedback and disturbance feed-forward control, was developed. The results showed that the vibration was effectively suppressed for the flexible four-bar linkage. Li [10] studied the active control of submarine vibration system, and a corresponding adaptive control method was put forward. Agood control effect of the vibration system by applying the periodic excitation force was obtained. Zhu et al. [11] proposed the active vibration control process for the marine diesel engine, and the x-LMS algorithm based on the off-line identification of error channel was used for the active vibration control. Belyi and Gan [12] studied and brought forward a combined bilateral and binaural active noise control algorithm for closed-back headphones. Soni et al. [13] published about the active vibration control of a ship-mounted flexible rotor-shaft-bearing system. Teo and Fleming [14] published optimal integral force feedback for active vibration control in 2015. Park and Kim [15] studied semi-active vibration control of space truss structures by friction damper for the maximization of modal damping ratio in 2013. Through the inspiration of the active vibration control method mentioned above, the active vibration control strategy was researched for the mechanical servo high-speed FB press to suppress the periodic low-frequency vibration of the machine. The active vibration control principle of the high-speed FB press was analyzed, and the adaptive vibration controller was created. According to test results of the FB press vibration, the input parameters for the control model were obtained, and the active vibration control effect was predicted by using MATLAB software. 1 ADAPTIVE VIBRATION CONTROLLER DESIGN 1.1 Active Vibration Control System The active vibration control system of a mechanical servo FB press is mainly composed of the following parts, as shown in Fig. 2. (1) Controlled object. In this paper, the controlled object is the mechanical servo FB press. The main drive system of the press is the excitation source of the controlled object; the excitation force acts on the frame and then transfers to the whole machine. Since the elastic restoring force is mainly acted on in the vertical direction, the mathematical model of the controlled object is primarily focused on the vertical deformation. (2) Measuring mechanics. The core component of the measuring mechanics is the sensor. An acceleration sensor is often used in the data acquisition of vibration response. The vibration signal of the whole machine measured by the acceleration sensor is transmitted to the control system through signal amplification and filtering, which is used as the input parameter of the active vibration control. (3) Electric actuator. According to the command of the controller, the actuator applies the specified force or torque on the frame of the press and feeds back the action effect to the controller. (4) Controller. The controller is the core part of the active vibration control system, which can provide the command and control rate of the actuator. Because the main driving system is motorial during the forming process, the adaptive controller which can adjust the parameters of the control system applied in this paper. (5) Energy and auxiliaries. The external energy supplies the input energy of the actuator for the active vibration control system of the FB press. Fig. 2. Schematic diagram of the active vibration control system for FB press 1.2 Adaptive Vibration Control Algorithm The selection and design of the algorithm play an important role in the adaptive vibration controller. Among many adaptive control methods, the least mean square (LMS) adaptive algorithm [16] has been widely used, in which the gradient search method is used, the convergence solution can be obtained quickly, and the implementation is relatively simple. The LMS algorithm can be expressed by Eq. (1): 2 (()) ˜Ee n Fen ˜.() n (()) ˜Ed n () °2 dnyn ()() .yn ..2 2(). ˆ.(1) The filter of the adaptive active vibration controller adopts the finite impulse response (FIR) structure, and its structure diagram is shown in Fig. 3. Fig. 3. Adaptive FIR filter 2TT  In Fig. 3 x(n) represent the input of the adaptive active vibration control filter, y(n) is the output of the adaptive active vibration control filter, w(n) is the impulse response, w(n) = {w(0), w(1), ..., w(n – 1)}, then: N °1 () ()() .i yn ˜W T n X n ˜wnxn i()( °). (2) i ˜0 For the filter with transverse structure, substitute the expression of into Eq. (1): (()) () ()2 W n P (3) ˜°E d n .W n RW n .() , .... where where R = E [X(n) XT(n)] is the autocorrelation matrix of N×N, is the cross-correlation vector for N×1, which represents the correlation between the ideal signal d(n) and the input vector. Because the gradient descent method is adopted for the optimization process, so: ˜. °0. (4) ˜W() () °w n wn . When the mean square error reaches to the minimum value, the optimal weight coefficient can be obtained. If the R matrix is full rank, the best value of the weight coefficient should be satisfied with W*= R–1P, namely: xd ° 1 ˜ .ˆ ° ° . ˆ.ˆ w 0 0 () 1 () ( n 1) () 0 x x x ......  .....  .............. ˜   ° ° w 1 1 () 0 () ( n 2) () 1 xd x x x , ˜˜.˜˜ ˜ ° °° ° T gn . w () ˜°2 dnxn ()() .2 xnxnwn () () () 2 () xnwn ˜xn ()( °dn .T()()). (6) Using the gradient estimation value to replace the real values, then: (1 () enxn wn ˜°) wn ˜2.()(). (7) This is the iterative equation of the LMS algorithm. In practical application, the step size is fixed, and the value of will affect the performance of the algorithm, resulting in the change of convergence speed, jump tracking ability and steady-state imbalance of the algorithm. Since the most appropriate value is very difficult to obtain, the variable step size normalized LMS algorithm is usually applied, which can be expressed as: ( ˜°() enxn () ()(8) wn 1) wn ˜2.n ()() °wn ˜.wn . According to the expression of the instantaneous square error of the LMS algorithm, the instantaneous square error of variable step size normalized LMS algorithm can be expressed as: 22 TT en en .() ˜() °2 .wnxnxnwn ()() () () TT T °.wnxnwn .2 () .wnxn , ()() () () (9) xndn ()() then: 2 22 ˜() °. () .() ° en enen .˜2 wnxnen T()()() .˜wn()() () xn(). (10) T xnT wn From .w(n) = 2ľne(n) x(n), we can get: 2 2T ˜en °.4 nenxn () xn () ()() .4 nen xn.ˆ()() 2 2() T xn ... (11) To minimize .e2(n), which makes the instantaneous square error close to the square error, ˜2() we can take den .0 , then, ° dn 1 . (12) (5) ˜n ° 2T xnxn ()() . ˜ w xd where, .x (m) = E[x(n) x(n – m)] shows the autocorrelation value of x(n), .xd (k) = E[x(n) d(n – k)] represents the cross-correlation value between x(n) and d(n). In practical applications, it is difficult to obtain the autocorrelation and cross-correlation values of signals. Therefore, gradient estimation value can be used, that is:  In order to avoid the steady-state imbalance, a fixed numerical convergence factor ľk is introduced to avoid a large step size when the denominator is very small during the iteration process. At the same time, another parameter . is introduced to adjust the denominator. Then the updated iterative equation of the new variable step size normalized LMS algorithm is: ( n 1) x ( n 1) ( n 2) () 0 ( n 1) . ° x x . .k 1() T enxn wn ( ˜°) wn ˜()(). (13) .˜xnxn ()() 1.3 Adaptive Feedforward Active Vibration Control of the Fine-Blanking Press As mentioned in Section 1.1, the main drive system of the press applies vertical excitation force to the whole machine, including the frame. The actuator is installed on the frame, which can exert a controllable force with the opposite direction of the excitation force of the main drive system to offset the original vibration and realize active vibration reduction. Because the vibration on the frame is easy to measure, the upper and lower cross-beams of the frame are selected as the vibration observation parts of the control system. The excitation force generated by the main drive system is transmitted to the frame through the primary channel to generate the vibration response, which can be used as the reference input of the adaptive active vibration control. is the control input, and the value is the estimated value of the excitation force. By adjusting the with the adaptive feed-forward control method, the vibration response of the frame can be reduced, and the effect of reducing the vibration of the whole machine can be achieved. The control block diagram is shown in Fig. 4. Fig. 4. Active vibration control block diagram of the FB press In Fig. 4, F(n) is the excitation force generated by the main drive system, d(n) is the excitation force response used as the reference input, e(n) is the control system error and also used as the controlled output when the control system is stable, W(z) is the adaptive controller, R(n) is the estimated value of the excitation force, G(n) is the transfer function which reflects the excitation force transferred to the frame, S(n) is the transfer function which reflects the control force transferred to the frame. The FIR filter described above is adopted as the filter. Based on the above control block diagram, the mean square error of the whole control system can be expressed as Eq. (1). Then the control output y(n) of the nth order filter is equal to the convolution operation: N yn () ˜Sn () °.wnRn () () i N .ˆSn ..)( .w .., ˜( °R ()())(14) 0 i then the gradient value of the mean square error can be calculated as: °2() . dEen . g ˜EenSnRn 2() () () w dw .˜ˆ... (15) According to the variable step-size normalized LMS algorithm, the variable step-size factor ľ needs to be transformed, that is: 1 ˜°, (16) 2( () ())( () ()) SnRn T SnRn then the iterative equation of the weight updating of the adaptive control algorithm can be obtained: .n 1() T enxn wn ( ˜°) wn ˜()(), (17) .˜xnxn ()() where, X(n) =S(n) R(n). 2 VIBRATION MECHANICAL MODEL DESIGN 2.1 Vibration Mechanical Model of the Fine Blanking Press For the adaptiveactive vibrationcontrol system of the fine blanking machine, establishing the mechanical vibration model of the controlledobject and obtaining the transfer function of the control system are required. It is assumed that the material distribution before each blanking process can be restored, and the material penetration is not considered. The punching force is located in the vertical direction, so the vertical direction, which has the maximum deformation, is mainly considered. Therefore, the structure of the FB press can be expressed as a combination of linear spring dampers. Then the mechanical vibration model as shown in Fig. 5 can be established, and the motion equation for each mass block can be expressed as follows. imp el mx .. ˜.F °.F , (18) iii i where Fiimp is the sum of the impact forces exerted on the mass mi, Fiel is the sum of the elastic forces applied to the mass mi. The excitation source of the whole press comes from the main drive system. The excitation force acting on the main drive system is mainly composed of two parts. One part is the unbalanced inertia force Fy, generated by the main drive system itself. The other part mainly is the elastic restoring force from the main drive system at the ending of the FB process, which is relatively difficult to obtain; this can be done so with the FE simulation of the FB process. The FB FE model is shown in Fig. 5. The material for this simulation is TC4 titanium alloy with a thickness of 5 mm. The diameter of the fine-blanked part is 20 mm, and the FB speed is 5 mm/s. The relationship between the FB force and the time can be obtained, as shown in Fig. 6. When the punch force acts on the workpiece, the reaction force exerts on the main drive system in the opposite direction, which reflects the changing of the impact load acting on the main drive system. Fig. 5. The FE model of the FB process Fig. 6. The relationship between the fine-blanking force and the time According to the mechanical vibration model, as shown in Fig. 7, the vibration equations of each mass block can be expressed by: .mx .. ˜cx ( . °x . ) 7777 8 ˆimp kx °x ) .F ˜F ˜( ˆ77 8 y ˆ mx .. ˜cx . °x . ) ˜c (x . °x . ) ˆ( 88889 787 ., (19) ˜kx °x ) ˜( °x ( kx ) .0 ˆ889 787 ˆmx .. ˜cx . ˜( . °x . cx ) 999989 8 ˆ ˜kx ˜kx °x ) .0 ˆ( 99 89 8 . where m7, k7, c7, and x7 represent the mass, equivalent stiffness, equivalent damping and displacement of the main drive, respectively. m8, k8, c8, and x8 represent the mass, equivalent stiffness, equivalent damping and displacement of the frame, respectively. m9, k9, c9, and x9 represent the mass, equivalent stiffness, equivalent damping and displacement of the embedded footings, respectively. k10 and c10 represent the equivalent stiffness and damping of the upper worktable. .. xj and x . j ( j = 6, 7, 8) represent the acceleration and velocity of the mass block. Fig. 7. Vibration mechanical model of the FB press 2.2 Parameters Determination of the Vibration Mechanical Equation In order to solve the mechanical vibration equation, the parameters, such as the equivalent mass, equivalent stiffness, equivalent damping value and similar parameters, should be determined firstly based on the FB press structure. 2.2.1 Embedded Footings System Fig. 8 shows the foundation installation system for the FB press. For the embedded foundation in the deep homogeneous layer, we can use the formulas presented in [17] and [18] to calculate the stiffness and damping value. That is: Based on the installation conditions of the mechanical servo FB press, the average shear wave velocity of soil is 150 m/s, and the density is 1900 kg/mł. The total mass of the foundation system is taken as 400,000 kg. The average shear wave velocity of backfill material is 120 m/s, and the mass density is 2400 kg/mł. The shear modulus of soil can be evaluated by G = .VS 2, and it is 42.75 MPa for footing and 25.9 MPa for the side layer. Therefore, the stiffness and damping constants of the foundation installation system can be calculated by Eq. (21), which is k = 8.1×108 N/m and c = 1.1×107 (N/m)/s. 2.2.2 Frame Part Because of the complexity of the frame structure, the FE method is usually used to obtain the equivalent stiffness of the frame mass. As shown in Fig. 9, the bottom surface is fixed by four points, and the upper cross-beam is applied with 200 N uniform force to get the deformation of the frame. According to the simulation results, the maximum deformation of the frame is 6.7324 × 10–8 m. Therefore, the equivalent stiffness of the frame can be calculated with: of mechanical fine blanking machine ˜Ff 200 kf = .N/m . °l 6.7324 .10 ˆ8 9 GS lS ˜2 971 . .10 .N/m .. (21) . ... kGrC ( 0 ˜ ° ) v 1 v 1 Gr 0 , (20) GS l ..ˆ . GCv ( S cr 0 ˜ ° 2 ) 2 v 2 . G r 0 where G is the shear modulus of the soil, r0 is the radius of circular foundation or the equivalent radius of non-circular foundation. . is the density of the soil, l is the depth of embedment, GS and .S are the shear modulus and density of the backfill side layer, respectively. Dimensionless stiffness and damping parameters Cv 1 and Cv 2 depend on the dimensionless frequency. S andSv 2 are the dimensionless v 1 stiffness and damping parameter of the side layer. Novak [19] provided the Cv1, Cv2 , Sv 1 andSv 2 for most stamping equipment, as shown in Table 1. Table 1. Reference values of stiffness and damping for foundation installation system Half-space Side layer soil Cv1 Cv2 Sv1 Sv2 Cohesive soil 7.5 6.8 2.7 6.7 Granular soil 5.2 5.0 It is somewhat difficult to calculate the equivalent damping value of the frame. In general, it is required to calculate the logarithmic decrement rate d of the system response under the impact vibration. The relationship between the logarithmic decrement rate and the structural damping ratio is shown in Eq. (22): xi 2 °. ˜.In . , (22) xiT ..2 ˆ where . is the damping ratio of the frame, xi is the displacement, velocity or acceleration amplitude on the frame at the time of ti, xi+T is the displacement, velocity or acceleration amplitude on the same location at the time of ti+T. Because the value of . is very small, it can be ignored after the square. Then, Eq. (22) can be simplified as: x ˜.In i .2 n °.. (23) x . it Based on Eq. (23), once the logarithmic decrement rate d is determined, the damping ratio . of the frame can be obtained. c ˜2° mk . (24) Therefore, to obtain the equivalent damping of the frame, the logarithmic decrement rate of the vibration amplitude of the system under the impact load should be calculated first. Based on the above FE model, 1 MN impact force is applied to the frame workbench and the bearing seat hole with a time of 0.02 s. According to the simulation result, the maximum acceleration and velocity changing point on the frame can be obtained, as shown in Fig. 10. Based on the plots in Fig. 10, the following equivalent damping parameters can be obtained and summarized in Table 2. Table 2. Equivalent damping parameters of the frame Logarithmic Damping ratio Value decrement rate of frame ti ti+Td . Velocity v7 0.272 0.103 0.971 0.154 Acceleration a7 6.691 2.731 0.896 0.143 Based on the numbers in Table 2, it can be seen that the damping ratio of the frame is near 0.15. Therefore, it can be taken as 0.15. It is also known that the total mass of the frame is 12,314 kg. According to Eqs. (22) to (24), the equivalent damping value of the frame can be obtained as c7 = 1.81×106 . 2.2.3 Main Drive System For the main drive system, the equivalent stiffness and damping can be obtained with the same method applied for the frame part. The FE model for the main drive system was created as shown in Fig. 11, in which the support constraint is applied to the bearing pedestal at the bottom of the main drive system, and the horizontal displacement of the slide block is constrained. The deformation of the main drive system can be obtained by applying 200 N uniform force load on the top of the slider, as shown in Fig. 11. Based on the plots in Fig. 10 that the maximum deformation of the main drive system under the constant load is 6.0265×10–7. Then, the equivalent rigidity of the main drive system can be obtained by Eq. (25): F 200 8 kT ˜T ˜ˆ7 ˜. ..N/m . 33210 . (25) °l 6 0265 . .10 In order to obtain the equivalent damping of the main drive system, the 105 N impact is applied to the top surface of the slider for 0.02 s. The maximum acceleration and velocity changing point on the main drive system can be obtained from the simulation results as shown in Fig. 12. Table 3. Equivalent damping parameters of the main drive system Logarithmic Damping ratio Value decrement rate of frame ti ti+Td . Velocity v6 1.601 1.038 0.433 0.069 Acceleration a6 4.689 2.773 0.436 0.069 According to the parameters obtained in Table 3, the value of the damping ratio of the main drive system is approximately 0.069. It is also known that the total mass of the main drive system is 2,592 kg. According to Eqs. (23) to (25), the equivalent damping value of the main drive system can be obtained as c6 = 1.28 × 105. After obtaining the parameters of each part, the mechanical vibration model of the controlled object can be finally established and used in the active vibration control simulation. 3 NUMERICAL SIMULATION AND DISCUSSION 3.1 Analog Excitation Input Because the effect of the vertical excitation force on the forming accuracy is the most direct and the largest, the reference excitation force input of the active vibration control can be composed of the blanking force excitation during the FB process and the unbalanced inertia force excitation during the non­ working process. In order to simulate the error during the active vibration control process, the random vibration interference is added to the reference excitation input, which is shown in Fig. 13. 3.2 Reference Input Based on the Measured Vibration Response 3.2.1 Vibration Measurement Scheme of Fine-Blanking Machine The composition of the vibrationmeasurement system of the mechanical servo FB press can be depicted by the s chematic diagram shown in Fig. 14. During the working process, the vibration at the working area is the most obvious, which has a direct impact on the forming accuracy. Therefore, the vibration in the working area needs to be measured. Meanwhile, the rigidity of the frame in the middle area is much smaller than that of the upper and lower cross-beams, and the guide rail of the slide block is arranged on the frame column in the middle area. Therefore, measuring points need to be arranged on the inner side of the stand column to reflect the vibration mode of the whole machine. Consequently, the measuring points arranged at the worktable area of the FB press are shown in Fig. 15, and 12 three-way acceleration sensors are even distributed at the working area. 3.2.2 Test Results of Measuring Points During the test, the mechanical servo FB press works periodically with a speed of 87 times per minute under the no-load condition. In the movement process, the inertial load affects the lower cross-beam directly, and the vibration will be transmitted to the worktable, resulting in the forming error. The bottom of the lower cross-beam connected with the ground by installation system, the overall stiffness is very large. Therefore, the acceleration response of measuring points 1, 2, and 3 can reflect the overall response of the lower cross-beam, which is shown in Fig. 16. Compared with the upper and lower cross-beams, the rigidity of the frame columns on the left and right sides in the middle is smaller, and the deformation will have a great impact on the whole press deformation. As the left and right columns are of symmetrical structure, the deformation situation is similar from the results, so the deformation situation on the right side is selected for the analysis. The response situations in time domain and frequency domain are shown in Fig. 17. Based on the plots in Figs. 16 and 17, when the frequency is about 200 Hz, points of 4, 5 and 6 have a stronger frequency response than the points of 1, 2 and 3. The overall rigidity of the upper cross-beam is very large. Because the upper cross-beam is far from the frame installation system, it is easy to produce large vibration response. The response situations in the time and frequency domain are shown in Fig. 18. Based on the plots in Figs. 16 to 18, points of 7, 8 and 9 have a stronger frequency response when the frequency is about 150 Hz than points of 1, 2, 3, 4, 5 and 6. But have a weaker frequency response when the frequency is about 200 Hz. 3.3 Numerical Simulation Results According to the self-adaptive vibration feed-forward control method, as described in Section 2, the control system simulation process is programmed in the MATLAB software platform, and the value of ľk is 0.005, . is 10–5. Combined with the mechanical model of the controlled object, the time-domain vibration response of each observation point with and without control can be obtained, as shown in Fig. 19a. The simulation results show that the control effect of all measuring points is very good, so we use measuring point 2 to show. The frequency domain vibration response of observation point 2 with and without control can be obtained, as shown in Fig. 19b. In Fig. 19, the blue line represents the vibration response output without control, and the red line represents the vibration response output adding active control. It can be seen from Fig. 19 that the vibration response of the press with the active vibration control is effectively reduced. The time-domain vibration response quickly achieves stability when the active control applied and the convergence speed is very fast, which means that it has a stable control effect at that time. It can be seen that the vibration of the frame has been significantly suppressed, and the maximum value of the vibration response has been reduced by more than 95 %; the amplitude variation of the vibration response is maintained within the range of m. For the frequency domain control effect, it can be seen that after controlling the corresponding response of the first three modes, the vibration response amplitude reaches stability very fast, and the maximum value of the vibration response is reduced by more than 80 %. At the same time, the convergence speed of the system is very fast after the application of active vibration control. From the simulation results, it can be seen that the active vibration control applied to the FB press does not suppress the vibration completely. However, the effect of vibration control is very significant, which shows a promising future for the application of the active vibration control method on the FB press. 4 CONCLUSION In this paper, the mechanical vibration model of the fine-blanking press as the controlled object is established, the dynamic model parameters of a mechanical fine-blanking machine are obtained by using finite element analysis and empirical formula. The principle of vibration control of fine blanking press is described, the control algorithm is established, and the adaptive vibration control block diagram of fine blanking press is established. The self-adaptive feed-forward control is used to simulate the vibration control of the mechanical FB press. The simulation results show that the vibration control effect is good, and the conclusion can be obtained as follows. The vibration response of the whole machine has been effectively reduced when the active control is applied, in which the time-domain vibration response has been reduced by more than 95 %, and the frequency-domain vibration response has been reduced by more than 80 %, which means that the vibration reduction effect is obvious. Effectively reducing the vibration effect of fine-blanking machine greatly increases the processing accuracy, saves significant amounts of energy, and reduces the energy consumption and defect rate. 5 ACKNOWLEDGEMENTS The authors would like to thank the Fundamental Research Funds for the Central Universities (WUT: 2019.117CG), 111 Project (B17034) and Innovative Research Team Development Program of Ministry of Education of China (No. IRT_17R83) for the supports given to this research and the Huangshi Huali Metal Forming Machine Tool Co., Ltd. for their supports given to experiments. 6 REFERENCES [1] Liu, Y., Tang, B., Hua, L., Mao, H. (2018). Investigation of a novel modified die design for fine-blanking process to reduce the die-roll size. Journal of Materials Processing Technology, vol. 260, p. 30-37, DOI:10.1016/j.jmatprotec.2018.04.029. [2] Mao, H., Li, S., Liu, Y., Hua, L. (2017). An investigation on the microstructure of the fine-blanked sprocket. 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Received revised form: 2021-08-04 DOI:10.5545/sv-jme.2021.7236 Original Scientific Paper Accepted for publication: 2021-09-01 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model Xuan Bo Tran Hanoi University of Science and Technology, School of Transportation Engineering, Vietnam This paper presents a new controller for the position of a pneumatic actuator. The controller is designed based on the multiple-surface sliding control method in combination with a frictional compensator. The multiple-surface sliding control method is applied to deal with the nonlinear characteristics of the pneumatic system, and the frictional compensator is applied to compensate for the friction force in the pneumatic actuator. The friction force is estimated based on a dynamic friction model (the LuGre model). Both simulation and experimental studies are done to evaluate the new controller. The evaluation results indicate significant improvement in the tracking position error of the new controller comparing to the multiple-surface sliding controller without friction compensation and other nonlinear controllers. Keywords: pneumatic actuator, nonlinear control, friction compensation, dynamic friction model Highlights • The new controller with friction compensation is proposed to control the position of a pneumatic cylinder. • A minimal position peak error of 1 mm in steady-state condition can be obtained by the new controller with the step desired positions. • A minimal position peak error of 2.2 mm in steady-state condition can be obtained by the new controller with the sinusoidal desired positions. • By using the new controller, the relative position peak error is reduced by 10 %, and the relative root means squared (RMS) error is reduced by 6.5 % in comparison to those of the multiple-surface sliding (MSS) controller without friction compensation. 0 INTRODUCTION Pneumatic systems are widely used in industrial environments because they can provide many advantages, including low cost, high power-to-weight ratio, cleanness, ease of maintenance and replacement, cheap and available supply of air sources [1]. The pneumatic systems are also preferred in hot and/or humid environments where electric systems cannot be applied [2]. However, the compressibility of air, the nonlinearity of valves and friction in the pneumatic actuator are disadvantages of pneumatic systems. They cause the system to have high nonlinearities and modelling uncertainties [3] to [5]. Therefore, the application of linear control strategies is not suitable for achieving fast and highly accurate responses for the pneumatic actuators’ position [6] to [8]. Many nonlinear and advanced control strategies have been employed to overcome the nonlinearities and uncertainties of the pneumatic systems. Bobrow and Jabbari [9] applied an adaptive control method based on the linear dynamics of the system to control the position of a pneumatic actuator. An adaptive fuzzy-proportional-derivative (PD) controller was applied by Gao and Feng [10] for a one-degree pneumatic actuator. The full-state feedback was used in the control law for simultaneous parameter identification and tracking control. A combination of an adaptive control strategy and neural network was proposed by Chen et al. [11] for an electro-pneumatic servo system; the neural network was used to compensate for constructing a linearized model of the nonlinear system, and the robust adaptive controller was used to perform the model-matching for the uncertain linearized model of the system. Gross and Rattan [12] applied multilayer neural networks to compensate for the nonlinear nature of the dynamic system in conjunction with a proportional-integral-derivative (PID) feedback controller. The advantages of the above-advanced control strategies are that they allow updating system parameters and controller parameters. However, the quality of control performances greatly depends on the accuracy of the mathematical model of the system. For the sliding mode control method, the first positioning application for pneumatic actuators was made by Paul et al. [13], Tang and Walker [14], and Surgenor and Vaughan [15]. This robust control method allows handling nonlinearities and compensates for the mismatched uncertainties of the mathematical model. However, one of the main drawbacks of this kind of control strategy for pneumatic systems is the need for acceleration feedback. To overcome this difficulty, Acarman et al. [16] and Liu et al. [17] used observers to estimate the acceleration and Pandian et al. [18] proposed the use of differential pressure feedback *Corr. Author’s Address: Hanoi University of Science and Technology, No. 1, Dai Co Viet str., Hai Ba Trung dist., Hanoi city, Vietnam, bo.tranxuan@hust.edu.vn instead of acceleration feedback. Recently, Tsai and Huang [19] proposed a multiple-surface sliding controller (multiple-surface sliding (MSS) controller) for a pneumatic servo system. This controller was designed based on three sliding surfaces of position, velocity, and force, and this controller can exhibit the smallest control error among the controllers proposed thus far for the pneumatic actuators. However, the friction force was omitted in designing the MSS controller. Armstrong-Helouvry [20] has shown that the control performances of a mechanical system can be significantly improved when the system controller is designed with friction compensation. Koch and Reichhartinger [21] and Ayalew and Jablokow [22] have applied the sliding model control method combined with model-based friction compensation in the controlling position of hydraulic actuators; they have achieved improved control performances. However, only simple nonlinear friction models such as static or Coulomb friction models were used in these studies. Tran and Yanada [23] have shown that the static friction models are not enough or useless in predicting dynamic friction behaviours in pneumatic cylinders, especially in the cases where the cylinders operate at low and oscillating velocity conditions. Therefore, dynamic friction models must be used to achieve highly accurate control performances for the pneumatic cylinder position. To date, many dynamic friction models have been proposed [24] to [29]. Among these, the dynamic LuGre friction model [25] is most used. This dynamic friction model can observe basic friction characteristics in most mechanical systems such as pre-sliding displacement, stick-slip motion, varying break-away force, and hysteresis. In addition, this model is quite suitable for theoretical calculations and is easy to implement. Tran and Yanada [23] have also experimentally examined the friction characteristics in pneumatic cylinders and have shown that the LuGre friction model can relatively adequately simulate all the measured friction characteristics. However, according to the author’s survey, the application of the LuGre friction model combined with the MSS control method for controlling the position of the pneumatic actuators has not been studied. In this paper, the multiple-surface control method combining with friction compensation based on the LuGre friction model for controlling the position of a pneumatic cylinder is studied. An electro-pneumatic servo system with a double-acting pneumatic cylinder and two pneumatic proportional flow control valves is considered to carry out this work. A general mathematical model of the system is built first. The control law of the new controller is then designed, and the absolute stability of the system is analysed. The new controller is evaluated using both simulations and experiments. The control performances of the new controller are compared with those of the MSS controller and other nonlinear controllers. Finally, the effectiveness of the new controller is shown. The rest of this paper is organized as follows. In Section 1, the electro-pneumatic servo system is introduced, and its mathematical model is developed. Section 2 shows the synthesis of the system controller. A stable analysis of the closed-loop system is also shown in this section. The simulation evaluations are shown in Section 3, and the experimental evaluations are shown in Section 4. Finally, the main conclusion is drawn in Section 5. 1 SYSTEM MODEL In this paper, we consider an electro-pneumatic servo system, as shown in Fig. 1. In this system, a double-acting pneumatic cylinder is used to drive a load mass M. The motion of the cylinder piston is controlled by two 3/2 electro-pneumatic proportional flow control valves, Valves 1 and 2. Valve 1 is connected to the left chamber, and Valve 2 is connected to the right chamber of the cylinder. The valves’ operation is controlled by the voltage signals u1 and u2. According to the characteristics of the valves used in the experimental system, as shown later in Section 4.1, the voltage values of u1 and u2 vary from 0 to 5 VDC. If 2.5 = ui = 5 VDC (i = 1 and 2), the valves operate at the left side position, letting air from the compressor flow into the cylinder chambers; and if 0 = ui < 2.5 VDC, the valves operate at the right-side position, releasing air from the cylinder chambers to the atmosphere. Therefore, the cylinder piston will extend if 2.5 = u1 = 5 VDC and 0 = u2 < 2.5 VDC. Conversely, the cylinder piston will extract if 0 = u1 < 2.5 VDC and 2.5 = u2 = 5 VDC. The flow rates through the valves depend on the voltage values of u1 and u2. To develop a mathematical model of the system, the following assumptions are used: a) The used air is an ideal gas, and its kinetic energy is negligible in the cylinder chambers. b) The leakages of the cylinder are negligible. c) The temperature variation in cylinder chambers is negligible with respect to the supply temperature. d) The pressure and the temperature in the cylinder chambers are homogeneous. e) The evolution of the gas in each chamber is a polytropic process. f) The supply and ambient pressures are constant. The mass flow rate m.i (i = 1 and 2) that flows into or flows out from the chambers of the pneumatic cylinder can be derived in terms of the voltage inputs ui of the two valves as follow: . k .ibp HV 1 ui °25 .25. .ui . .., 5 s RT , (1) m .i ˜ k .p Hu °25 .0 .u ˆ2..5 .., ie i V 2 ii RT where ps and pi are respectivelythe supply air pressure and the pressure in the cylinder’s chambers, k is the specific heat ratio, R is the gas constant, and T is the temperature of the supply source. HV1 and HV2 are respectively the valve coefficients. ßib and ßie are the pressure coefficients and are given by:  k 1 . k The dynamic relationship between the mass flow rates m.1, m.2 and the pressures p1, p2 in the cylinder chambers can be given by continuous equation as follows: p . ˜Vk 1 .RTm . °p Av ., 1 111 p . ˜Vk 2 .RTm . .p Av ., (4) 2 222 where v is the piston velocity, A1 and A2 are the piston areas. V1 and V2 are respectivelyto the volumes in the left and right chambers of the cylinder and are given as follows: V ˜V °Ax , 110 1 V ˜V °( .x AL ), (5) 2 20 2 where L is the piston stroke, x is the piston position, V10 and V20 are, respectively, the dead volumes in the left and right chambers of the cylinder. The motion equation of the cylinder piston according to Newton’s second law is given by: Ma ˜pA °pA °, (6) 11 22 Ffr where M is the load mass of the piston, a is the piston acceleration and Ffr is the friction force. In [23], Tran and Yanada have measured the friction characteristics of the pneumatic cylinders and have shown that most of the measured characteristics can be simulated by the dynamic LuGre friction model with relatively good accuracy. Therefore, in this study, the LuGre model is used to describe the friction force Ffr in the pneumatic cylinder. The LuGre friction model is described as follows: .0 w , (7) .˜° wv ..v qv .n ..vv/ s qv () ˜Fcl °.Fst .Fcl .exp , (8) Ffr ˜.0 w °.1 w . °.2 v , (9) . p i  1 k .ˆ. ..  k . .. 1 , k k 1 1 .ˆ... 2 p p p p 2 2 i s i s . where w is the mean deflection of the elastic bristle (2) between two surfaces made in contact, s0 is the stiffness of the elastic bristle, q(v) is the Stribeck function, Fst is the static friction force, Fcl is the Coulomb friction force, vs is the Stribeck velocity, n is the exponent that affects the slope of the Stribeck ˆ... 1, k k .  1 p 1 ˜ib  ° s .ˆ... 2 p i 058 , .  k  p s 1 . . 1 patm  k k  1 k . .. k k  .ˆ.. . 2 p 1, atm 2 2 curve, s1 is the micro-viscous friction coefficient, and . ˆ. ˆ... k . 1 patm p i p i p i k  1 , (3) s2 is the viscous friction coefficient. ˜ie  ° k Setting the valve signals u1 and u2 according to .ˆ. 1 .. patm 058 . 2 the control law u of the system as follows: , k  p i where patm is the atmosphere pressure. u 1 ˜25 °u ., u 2 ˜25 .u (10) ., then substituting Eq. (10) into Eq. (1) and along with Eq. (4), we obtain: .1 .sign u () ˆpH  kApv kkRT ..1 b s V 1 p .˜°11 .u , 1 VV .°() ˆpH 11 ..1 sign u .1 e 1 V 2 .1 sign u ˆb kApv kkRT .°() .2 pH s V 1  22 p . ˜.u , (11) 2 VV .1 u .2 p H V  22 .°.sign () ˆe 22  By setting x = x ; x = x . and x3 = p1A1 – p2A2, 1 21 Eq. (6) then becomes: x . 1 ˜x 2, F x . 2 ˜1 x 3 °fr , (12) MM Taking time derivative of x3 along with Eq. (11) can be achieved as follows: x .3 ˜P x t °B x tu (,) (,), (13) where x = [x1 x2 x3] is state-vector and k A12 px 12 k A22 px 22 P (,x t ) ˜°°, (14)  VV 12  A 1 bV 11 A V 22 pH A V 21 pH A sV 12 It can be realized from Eqs. (1) to (15) that the dynamics of the electro-pneumatic servo system are highly nonlinear due to the effects of compressibility of air in the cylinder chambers, the nonlinearity of the valves, and friction in the pneumatic cylinder. 2 CONTROL STRATEGY 2.1 Control Design In this section, we design a controller so that the piston position x1 can track the desired trajectory x1d with high accuracy under the effect of the system nonlinearities (the compressibility of air in the cylinder chambers and the nonlinearity of the valves) and under the existence of friction. In [19], Tsai and Huang have applied the MSS controller to deal with the system nonlinearities and have obtained the smallest position error of 5 mm to sinusoidal desired position trajectories among the controllers proposed thus far for the pneumatic cylinders. However, the friction force was considered as a bounded uncertain force in the controller design. It is shown by Armstrong-Helouvry [20] that a controller combining with a friction compensator has the potential to improve the control performances of a control system. Therefore, in this study, we propose a new position controller of the cylinder piston by combining the MSS control method with a friction observer. The friction observer is designed based on the dynamic LuGre friction model. The block diagram of the closed-loop system using the new controller is shown in Fig. 2. To design the new controller, the following assumptions of the system parameters are considered: i) the mass load is bounded by Mmin = M = Mmax; ii) the piston velocity is bounded by |x2| = a1; iii) the .  .... . °  ˜    pH s V 1 2 e 2 1 e 1 2 b  ˆ  .... 1 sgn( u ) ° pH V 2 2 ° . B (15) (,) t x . . ˆ.... .... V 1  1 sgn(u) 2 ° V 2 . . friction force is bounded by |Ffr| = Fst; iv) the pressures Fig. 2. Block diagram of the closed-loop system using new controller Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model in the cylinder chambers are bounded by patm = pi = ps, ˜ .w ˜ ˜ 1 and 2. Basing on these assumptions, the bounds wx °1 . ° °. 1 i = ° z s 1, (26) 0 x qx °1 () for Ffr /M and P(x,t) can be identified as follows: where tz is a positive constant. By introducing F Ffr ° ˜st °., (16) ˜˜°as the estimating error of the mean 2 www M M min deflection of the contacting bristle and by combining 22 with Eqs. (7) and (26), the time derivative of the error kpA .kpA .s 11 s 21 BBB (18) ˜°°t x 0(,), where and are determined from Eqs. (2), (3) BBminmax and (15) as: .1  ˜°..3. (17) w. can be obtained as: P (, t ) x VV 10 20 x °1 ˆ0 w ˜ ° ..s . (27) ˜˜°The bound for()can be found as: Btw x,z 1 °qx 1  .. min max Taking time derivatives of x2d and x3d along with Eq. (12) obtains: x . ˜.x . °.x °x . ., (28) 2 d 1 d 21 d ˆ1 12 . ˆ HA V 11 HA V 22 20 B ˜ ° .. .. . ps 058 , V 10 V max x ˜ Mx ˜˜˜1 M ˜˜x x ˜ ˜ ° ° 3 d d d 2 d 1 1    HA V 11 ° 2 . ˆ F k 1  .... .... .ˆ. . ˆ. .. ..  1 . ° °F fr V 10 A1 L fr B min ˜ . pat 058 m .. . k 2 ° 1 ˆ.. (29) M . ° . k 1 1 2 x 3 , (19) .  M M HA V 22 20 ° 2 ° V A2 L . . Defining . Let choosing an estimation B of the gain B(x,t) as the mean of the above bounds: .ˆ. ..  1 1 known part and x ˜ ˆ.. 1 1  x 3 x . Mx ... 1 M .. x x . M ˜ ° ° . ° 2 2 2 2 , as d 31 d d 2 d 1 M B . = ( 12 BB )/, (20) min max then bounds in Eq. (18) can be written in the form: °1 B x t (,) ˜.. .˜, (21) B F ...  1 1 unknown part of x.3d , then we have: ° . F fr M ˜ ° ° 2 2 , as fr d 32  M x . ˜x . °x . . (30) 3 d 3 d 13 d 2 B min .12 . In the ideal case where there is no friction force, Defining three sliding surfaces of position, the bounds of x . , x . andx . can be identified as where ˜°.B max 2 d 3 d 13 d 2 velocity, and applied force as follows: ˆ1 ˜.x . °..°x . ..x . , (31) x . 2 d 1 d max 11 d max 2 d max s ˜x °x , 1 11d .1 s ˜x °x , 2 22d x . ˜° 3 33 ..1  1 M .. x 1 d max ° .. 2 2 x . 2 d ma M ... x 1 ° M max max ˜ (22) d 31 d s x x , max d with the desired velocity and applied force are selected ..ˆ . . .. pA pA 12 s 1 atm 2 1 s 1 x ˜x . °., (23) 2d 1d1 x .d = 0. (33) 32 .1 s Taking the Eqs. (12) and (13) into consideration, x 3 d ˜Mx ˜2 d °F °fr ..2 .2, (24) the time derivatives of three sliding surfaces can be 2 derived as where ti (i = 1 and 2) are positive constants and .i are . s the boundary layer thicknesses of the surfaces si. F fr s .˜s °.1, (34) 1 21 is the friction force that is estimated by the observer .1 basing on the LuGre model as follows: (32) M x . ° °  as follows: , 31max d . max 2 min 11 1 s °2 s ˜˜s °.F °F fr .°., (35) 23 fr 2 ˜ . .°.° 21 ˜˜ Fw , (25) ˜ . ° M M M ˆ w x fr 0 1 2 . 3 ˜P (,) t °B ( ,) .. 3 s xx tu xd . (36) Eqs. (27) and (31) to (36) will be used later in Section 2.2 to analyse the stable properties of the closed-loop system. The control law of the system is proposed as follows: 1 .s ˆ u ˜˜.x °3d1 °P (, x t ) °33 .. (37)  B .3 . where .3 is the boundary thickness of the sliding surface s3, and t3is a positive constant. It can be noted . that adding the friction force estimate F fr in calculating the desired force x3d in Eq. (24) affects the control law u of the system in Eq. (37) through the sliding surface s3. 2.2 Stability Analysis This section presents the analysis of the stability properties of the closed-loop system basing on the Lyapunov direct method [30]. First, let us consider a Lyapunov function for the third sliding surface s3 as 1 G = 32 s 32 . Taking the time derivative of G3 and combining it with the Eqs. (30), (36), and (37), we obtain: 2 s ˆ. s 2 ˆs 1 . . G 2 ˜s 2 s 3 °°12 F °. (41) . 32 ...st 22 M  min .2 ..2 .2 ˜3 s 2 .2 then G 2 ˜°.2 .0 . If we choose ˜2 ° s2 s 3 .2 This proves that s2 converges to its boundary layer asymptotically, i.e., the cylinder velocity x2 converges to the desired velocity x2d. Finally, a Lyapunov function for the first sliding surface s1 is defined as 121 2 G ˜s °w . . Taking the time derivative of G1 121 2 . z and combining it with Eqs. (27) and (34), we obtain: x ˜1 .0 w °2 s 12 ˜ G ˜ss °ˆ° ..ws . .°(42) 1 121 1 .ˆqx ˜ 1 z 1 By selecting t1= .2+ .1 with .1> 0 and noting that 1 212 ws . 1 ˜w . °s 1 , then we obtain: 22 . s 1 ˆ.11 ˆ2 ˜ G 1 ˜s 1 s 2 °2 .°.°.s 1 .2 1 ..1 .. .ˆ 201 2 °°. w°. (43) . .zqx ˜1 . 2 It can be noted in Eq. (8) that the Stribeck   t ˆ.. P (,) t x ... B (,) x ° B t ° 1 ˆ.. x ˜ 3 d 1 ° x ˜ 3 d 2  B ... 1 ° (,) x ˜ G 3 ˜ s 3 function ˜°is strictly positive and is bounded qx.1 ° B qx . °F ... So, if we select ˜˜ 1 st 1  B ° by 0 ˜ Fcl ° 2 (,) t x s 1 , 2 (38) 3 ° 3 s s . °  B 2 3 °2 °. 20 1 . ˜. and ˜. then G1 . Therefore, s1 < 0 By selecting qx .1 .ˆ z 1 2 ˜3 .1 1 °.3 .1 °x 3 d1maxˆ.3  with .ˆ.ˆˆ...3> 0 ° along with Eqs. (31) to (33), then we obtain: .2 ˆ.1 . 3 s 3 s 3 . G 3 ˜° s 3 .°1 ° 3, (39) ....x . 3 ..13 d 1max  3  .2 s if we choose .3= |s3| then G 3 ˜°.3 3 .0 . This .3 means that s3converges asymptotically to its boundary layer, i.e., the force x3 converges to the desired force x3d. Similarly, let us consider a Lyapunov function for 1 the second sliding surface s2 as G 2 = s 22 . Taking 2 the time derivative of G2 and combining it with Eq. (35), we obtain: .11 1 s 2 . G ˜2 ˜ss 2 ˜2 ˜s 2 .s 3 °.Ffr °F °fr .°2 . (40) ˆMM M 2  By assuming that the estimated friction force is . bounded by | F fr | = Fst and by selecting converges asymptotically to its boundary layer, i.e., the piston position x1 converges to the desired position x1d. This result, therefore, verifies the system’s stability. 3 SIMULATION EVALUATION In this section, simulation studies are carried out to evaluate the effectiveness of the new controller. Simulations were done by MATLAB/Simulink software. Both the new controller and the MSS controller were simulated with the same desired position inputs. The step and sinusoidal desired position inputs with different amplitudes and frequencies were used. The parameters of the system used in simulations are listed in Table 1. The maximum velocity of the piston was set to be a1= 1 m/s according to the cylinder’s specification. The static parameters of the LuGre friction model were . . 2  given by Fst = 30 N, Fcl = 5.6 N, vs = 0.01 m/s, n = 2.5, with .2> 0, then, s2 = 25 Ns/m for the positive velocity direction (i.e., for extending stroke of the piston) and by Fst = 25 N, ˜ . M max ˆ.. 1 1 ° 2 Fst . 3 M M min min Fcl = 5.8 N, vs = 0.015 m/s, n = 1.2, s2 = 25 Ns/m for the negative velocity direction (i.e., for extending stroke of the piston). The dynamic parameters of the LuGre model were given by s0 = 8×103 N/m and s1 = 0.1 Ns/m for both directions of the velocity. These parameters of the LuGre friction model were identified in advance for the pneumatic cylinder in a previous study [23]. The parameters of the controllers used in simulations were chosen as s1 = 0.002,s2 = 0.1, s3= 0.5, .1 = 0.01, .2= .3 = 1, a2 = 60 and the parameter of the friction force observer was chosen as tz = 5000. Table 1. System parameters used in simulation Parameter [unit] Value Parameter [unit] Value M [kg] 0.5 HV1 [m2/V] 5×10–7 A1 [m2] 4.9×10–4 HV2 [m2/V] 6×10–7 A2 [m2] 4.12×10–4 T [K] 295 L [m] 0.3 R [Nm/(kgK)] 287 V10 [m3] 9.8×10–7 ps [bar] 5 V20 [m3] 8.24×10–7 patm [bar] 0.1 k 1.3997 Fig. 3 shows a comparison of the control performances obtained by simulations between the new controller and the MSS controller with a step desired position input of the piston position x1d = 0.25 m. The piston is set at an initial position of 0 m. Fig. 3a shows that both the controllers exhibit good control performances with no overshoot. However, it can be observed that the rise time achieved by the new controller (0.26 s) is smaller than that of the MSS controller (0.5 s). In addition, the maximum tracking error in steady-state achieved by the new controller (0.5 mm) is also smaller than that of the MSS controller (1 mm) (Fig. 3b). Fig. 4 shows simulation comparisons of the control performances between the new controller and the MSS controller for four different constant desired positions: x1d = 0.01 m (Fig 4a), 0.05 m (Fig. 4b), 0.1 m (Fig. 4c) and 0.2. m (Fig. 4c). The initial positions of the piston are 0 m. Fig. 4 shows that the controller can provide shorter rise times and smaller position errors for four cases of the desired position. The maximum tracking errors obtained under steady-state conditions are also 0.5 mm for the new controller and 1 mm for the MSS controller. Fig. 5 shows a comparison between the new controller and the MSS controller for a sinusoidal desired input x1d = 0.15 + 0.1sin(2pft) m at the frequency f = 0.1 Hz. The figure shows that the new controller can track the desired position faster, i.e., a shorter rise time, than the MSS controller. In addition, the tracking error achieved in a steady-state condition of the new controller is much smaller than that by the MSS controller. The maximum tracking error in the steady-state condition of the new controller is 1.5 mm, whereas the maximum tracking error of the MSS controller is 4.8 mm. Other comparisons between two controllers with higher frequencies of the desired position input x1d = 0.15 + 0.1sin (2pft) m are shown in Figs. 6 and 7. The position amplitude remains the same as in Fig. 5, but the frequency is increased to 0.5 Hz and 1 Hz. As compared with Fig. 5, although the abilities to track the desired position of the two controllers are inferior when the frequency is increased, the new controller also provides smaller errors than the MSS controller. In Fig. 6 for f = 0.5 Hz, the maximum error in a steady-state condition is 2.8 mm for the new controller and is 6.5 mm for the MSS controller. In Fig. 7 for f = 1 Hz, the maximum error in a steady-state condition is 4.5 mm for the new controller and is 7.8 mm for the MSS controller. The results obtained in Figs. 3 to 7 verify by simulation that the new controller can provide better control performances than the MSS controller. Fig. 4. Control results of the new controller and the MSS controller for different step desired positions; a) x1d = 0.01 m, b) x1d = 0.05 m; c) x1d = 0.1 m; d) x1d = 0.2 m (experiment) Fig. 5. Control performances of the new controller and of the MSS Fig. 6. Control performances of the new controller and of the MSS controller with a sinusoidal desired position controller with a sinusoidal desired position x1d = 0.15+ 0.1sin (2pft) m, f = 0.1 Hz by simulation; x1d = 0.15+ 0.1sin (2pft) m, f = 0.5 Hz by simulation; a) tracking position; b) tracking error a) tracking position; b) tracking error. Fig. 7. Control performances of the new controller and of the MSS controller with a sinusoidal desired position x1d = 0.15+ 0.1sin (2pft) m, f = 1 Hz by simulation; a) tracking position; b) tracking error 4 EXPERIMENTAL EVALUATION 4.1 Experimental System Fig. 8 shows the experimental system used for the experimental evaluation of the new controller. The system used a pneumatic double-acting cylinder (1). The piston rod was connected to a load mass (2) which slides on a guiding bar (3). The load mass was set at 0.5 kg. Two proportional pneumatic valves (4) were used to control flow rates supplying to the cylinder chambers. These valves were controlled by current signals ranging from 0 A to 1 A. The current signals were supplied by two amplifiers (5). The input signals of the valve amplifiers are voltage values ranging from 0 VDC to 5 VDC. For the flow characteristic of the valves, it is noted that if the valves’inputs vary from 0.5 A to 1 A (corresponding to 2.5 VDC to 5 VDC of the valve amplifiers’inputs), the valves will provide air from the supply source into the cylinder chambers and if the valves’ inputs vary from 0 A to 0.5 A (corresponding to 0 VDC to 2.5 VDC of the valve amplifiers’ inputs), the valves will release air from the cylinder chambers to the atmosphere. A position sensor with a measuring range from 0 m to 0.3 m (6) was used to measure the piston displacement. Two pressures, p1 and p2, in the cylinder chambers were measured by two pressure sensors (7). A data acquisition card (8) that includes both 12 bits analogue-to-digital converter (ADC) and 12 bits digital-to-analogue converter (DAC) was used to acquire the displacement and the pressure signals from the sensors and to send the voltage signals to the valve amplifiers. The data acquisition device was connected to a personal computer (9). The main technical specifications of the devices are listed in Table 2. Microsoft Visual C++ software was used to communicate between the data acquisition device and the computer. Each step of the experimental programme was executed in a period of 1.16 ms. The piston velocity that is used in calculating the control law was not measured in this experimental system. It was calculated by an approximate Fig. 8. Experimental system Table 2. Main equipment used in the experimental system Equipment Model Main technical specifications Manufacturer Compressor PK 1090A Maximum pressure: 10 bar Maximum flow rate: 185 l/min PUMA Maximum pressure: 10 bar Air preparation unit GC 600-20 Maximum flow rate: 1000 l/min AIRTAC Filter element: 40 m Piston diameter: 0.025 m Pneumatic cylinder CM2L25-300 Rod diameter: 0.01 m SMC Piston rod: 0.3 m Maximum pressure: 10 bar Proportional pneumatic valves VEF3121-1-02 Maximum flow rate: 720 l/min SMC Current input: 0 A to 1 A Voltage input: 0 VDC to 5 VDC Valve amplifier VEA 250 Curent output: 0 A to 1 A SMC Accuracy: ą 1 % F.S Position sensor LWH0300 Measurement range: 0 m to 0.3 m Accuracy: 0.4 % F.S NOVOTECHNIK Pressure sensor PSE 540 Measurement range: 0 bar to 10 bar Accuracy: 2 % F.S SMC Data acquisition card (ADC/DAC) USB4711 Analog input channels: 16 Analog output channels: 2 ADVANTECH Personal computer Dell Vostro 260 CPU: Intel Core I3-2120 3.3 GHz Memory: 8GB RAM DELL differentiation of the measured piston displacement. The noise in the calculated velocity signal was filtered using an acausal low-pass filter with a bandwidth of 30 Hz. The compressed air was supplied to the system by a compressor through an air preparation unit, and the supply pressure ps was set at 5 bar. 4.2 Experimental Results In this section, experimental performances of the new controller and the MSS controller are compared to evaluate the effectiveness of the new controller. Similarly to simulation, constant and sinusoidal desired position inputs were used. In addition, the control performances of the new controller are also compared to those of the other nonlinear controllers. The values of the controller’s parameters and the parameters of the LuGre friction model were given as the same as those used in the simulation. Fig. 9 compares the experimental performances between the new controller and the MSS controller with a desired step position input of x1d = 0.25 m. Similarly to the results obtained in the simulation results in Fig. 3, both controllers exhibit accurate position tracking with the maximum tracking errors in a steady-state condition below 1 mm, but the rise time of the piston obtained by the new controller (0.2 s) is less than the rise time of the MSS controller (0.3 s). It is additionally noted in Fig. 9 that at the beginning, the piston remains at its original position for 0.1 s. This result may be caused by the compressibility of the air in the cylinder chamber and/or by the delay characteristic of the pneumatic valves. a) x1d = 0.01 m, b) x1d = 0.05 m; c) x1d = 0.1 m; d) x1d = 0.2 m (experiment) Fig. 10 compares the experimental performances between the new controller and the MSS controller at four step-desired positions: 0.01 m (Fig. 10a), 0.05 m (Fig. 10b), 0.1 m (Fig. 10c), and 0.2 m (Fig. 10d). Similarly to the simulation results in Fig. 4, Fig. 10 shows that the new controller provides better control performance than the MSS controller with shorter rise times and smaller steady-state errors in all cases. The maximum position errors in steady-state conditions are 0.5 mm for the new controller and 1 mm for the MSS controller. It can be additionally noted in Fig. 10 that the period at which the piston remains at its initial position is increased by reducing the desired position. This result can be explained by that when the desired position is reduced, the initial flow rate supplied to the cylinder is small and thus increases the delay time of the piston. Fig. 11 shows an experimental comparison between the new controller and the MSS controller with a sinusoidal desired position x1d = 0.15+ 0.1sin (2pft) m at a frequency f = 0.1 Hz. Fig. 11 shows that the peak position error obtained in a steady-state condition is 8.4 mm (corresponding to 4.2 % of the maximum cross displacement) for the MSS controller and is 4.8 mm (2.4 %) for the new controller. When calculating the root means squared error (RMS error) in a steady-state condition, the results show that the RMS error obtained for the MSS controller is 3.7 mm (1.85 % of the maximum cross displacement) while the RMS error obtained for the new controller is 2.1 mm (1.05 %). The calculated velocity and the estimated friction force by the new controller are shown in Fig. 12. When the piston starts moving, the friction force is suddenly increased to a very larger value of about 30 N in the first haft cycle of the piston motion. After that, the friction force reduces and varies under a trapezoidal form between the value from -13 N to 20 N in the steady-state condition. The maximum friction force at the positive velocity direction is larger than at the negative velocity direction. In Fig. 11, the large values of the tracking error obtained by the MSS controller appear mainly at high velocities, and they can be eliminated by the new controller. At high velocities, the friction force becomes large due to the effect of the viscous friction force. Thus, the compensation of friction to the MSS controller significantly reduces the tracking error at these velocities. The tracking errors of the two controllers achieved by experiments are larger than those achieved by simulation in Fig. 3. However, like the simulation results, the experimental result also shows the improvement in the control performances of the new controller comparing with the MSS controller. Fig. 13 compares the percentage rates of the peak position error and of the RMS error in steady-state condition between two controllers at different displacement amplitude values (A) of the desired position input x1d = 0.15+Asin(2pft) m. The varying frequency is fixed at f = 0.1 Hz. The results show that both the peak error and the RMS error of both controllers increase with decreasing the displacement amplitude A. However, these errors are significantly reduced when using the new controller. When the amplitude A varies from 0.01 m to 0.1 m, the peak error can be reduced from 1.8 % to 10 %, and the RMS error can be reduced from 0.8 % to 6.5 %. The minimum peak error of 2.2 mm can be obtained at the amplitude A = 0.01 m. This result of the new controller indicates better control performances than other nonlinear and robust control methods [9] to [21]. Fig. 14 compares the percentage rates of the peak position error and of the RMS error in steady-state condition between two controllers at different frequencies (f) of the desired position input x1d = 0.15+Asin(2pft) m. The displacement amplitude is fixed at A = 0.1 m. The results show that both the peak error and the RMS error of both controllers grow with increasing the frequency. However, the peak error and the RMS error are also significantly reduced when using the new controller. When the frequency f varies from 0.1 Hz to 1 Hz, the peak error can be reduced from 1 % to 3.5 % and the RMS error can be reduced from 0.8 % to 2 %. Figs. 15 and 16 show the control performances of two controllers at two frequencies 0.5 Hz and 1 Hz. 5 CONCLUSIONS In this study, a new controller was proposed for improving position control performances of a pneumatic cylinder. It was designed by combining the multiple-surface sliding control method with a model-based friction compensator, which was designed based on the dynamic LuGre friction model. Both simulation and experimental studies were performed to evaluate the new controller. The study results show that by using the new controller, the relative position peak error is reduced 10 %, and the relative RMS error is reduced by 6.5 % when comparing those of the MSS controller without friction compensation. The new controller also provides improved control error compared to other nonlinear and advanced controllers with a minimal peak error of 2.2 mm. It is noted that the parameters of the LuGre model used in the friction observer in this study were identified in advance. Therefore, a methodology for online parameter estimation of the LuGre model when implementing the new controller will be considered in the future. 6 ACKNOWLEDGEMENTS This research was an extension of the work in [31]. This research was funded by the Hanoi University of Science and Technology (HUST) under Project number T2020-PC-014. 7 NOMENCLATURES a piston acceleration, [m/s2] Ai piston area (i = 1, 2), [m2] Bmax maximum value of the function B(x,t) Bmin minimum value of the function B(x,t) . Bestimated value of the function B(x,t) Fcl coulomb friction force, [N] Ffr . F fr Fst : Gi : HVi : k : L : M : Mmax Mmin m.i n patm pi ps P(x, t) B(x, t) q(v) R si T u ui v Vi Vi0 : vs x : x : : x1 : x1d : x2 : x2d . x2d max : x3 : x3d . x3d1 . x3d1 max . x3d2 w ° w w . a1 a2 a3 ßib ßie .i : . : s0 : friction force, [N] estimated friction force, [N] maximum static friction force, [N] Lyapunov function (i = 1, 2, 3) valve coefficient (i = 1, 2), [m2/V] specific heat ratio piston stroke, [m] load mass, [kg] maximum load mass, [kg] minimum load mass, [kg] mass flow rate (i = 1, 2), [kg/s] exponent for Stribeck curve atmosphere pressure, [bar] pressure (i = 1, 2), [bar] supply pressure, [bar] function of the states’ function of the states’ Stribeck function, [N] gas constant, [Nm/(kgK)] sliding surface (i = 1, 2, 3) Temperature, [K] control law, [V] valve control input (i = 1, 2), [V] piston velocity, [m/s] volume of the cylinder chamber (i = 1, 2), [m3] dead volume in the cylinder chamber (i = 1, 2), [m3] Stribeck velocity, [m/s] piston position, [m] state vector piston position, [m] desired position of the piston, [m] piston velocity, [m/s] desired velocity of the piston, [m/s] maximum value of x.2d, [m/s2] piston force generated by the pressures, [N] desired force, [N] known part of x.3d maximum value of x.3d1 unknown part of x.3d mean deflection of bristles, [m] estimated mean deflection of bristles, [m] estimating error of the mean deflection of bristles, [m] maximum cylinder velocity, [m/s] maximum value of Ffr /M, [N/kg] maximum value of P(x, t) pressure coefficient (i = 1, 2) pressure coefficient (i = 1, 2) positive constant (i = 1, 2, 3) constant stiffness of bristles, [N/m] s1 micro-viscous friction coefficient for bristles, [Ns/m] s2 viscous friction coefficient, [Ns/m] ti positive constant (i = 1, 2, 3) tz positive constant .i boundary layer thickness of the surfaces si (i = 1, 2, 3) 8 REFERENCES [1] Beater, P. 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Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9 Vsebina Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 67, (2021), številka 9 Ljubljana, september 2021 ISSN 0039-2480 Izhaja mesecno Razširjeni povzetki (extended abstracts) Khurram Ali, Adeel Mehmood, Israr Muhammad, Sohail Razzaq, Jamshed Iqbal: Upravljanje antropomorfnega manipulatorja z modelom trenja LuGre – zasnova in eksperimentalna validacija SI 53 Dragan Milcic, Amir Alsammarraie, Miloš Madic, Vladislav Krstic, Miodrag Milcic: Napovedovanje kolicnika trenja pri hidrodinamicnem radialnem drsnem ležaju z uporabo umetnih nevronskih mrež SI 54 Almatbek Kydyrbekuly, Gulama-Garip Alisher Ibrayev, Tangat Ospan, Anatolij Nikonov: Vecparametricna dinamicna analiza sistema kotalnih ležajev SI 55 Youyu Liu, Yi Li, Xuyou Zhang, Bo Chen: Sledilno vodenje za manipulatorje na osnovi mehke ekvivalence in terminalnega drsnega režima SI 56 Yanxiong Liu, Yuwen Shu, Wentao Hu, Xinhao Zhao, Zhicheng Xu: Aktivni nadzor vibracij pri mehanski visokohitrostni servostiskalnici za natancno štancanje SI 57 Xuan Bo Tran: Nelinearno vodenje pnevmatskega aktuatorja na osnovi dinamicnega modela trenja SI 58 Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, SI 53 Prejeto v recenzijo: 2021-05-28 Š 2021 Strojniški vestnik. Vse pravice pridržane. Prejeto popravljeno: 2021-08-17 Odobreno za objavo: 2021-08-18 Upravljanje antropomorfnega manipulatorja z modelom trenja LuGre – zasnova in eksperimentalna validacija Khurram Ali1 – Adeel Mehmood1 – Israr Muhammad1 – Sohail Razzaq1 – Jamshed Iqbal2,* 1Univerza COMSATS v Islamabadu, Oddelek za elektrotehniko in racunalništvo, Pakistan 2Univerza v Hullu, Znanstveno-tehniška fakulteta, Oddelek za racunalništvo in tehnologijo, Združeno kraljestvo Zasluga za nedavni napredek fleksibilne avtomatizacije gre predvsem robotskim rokam in manipulatorjem. Avtonomija teh manipulatorjev je povezana z napredkom na podrocju nelinearnih regulacijskih sistemov. Cilj pricujoce raziskave je oblikovanje predloga za robusten regulacijski algoritem robotske roke s petimi prostostnimi stopnjami, ki bo zagotavljal vrhunsko zmogljivost in zanesljivost v prisotnosti trenja. Predstavljena sta predlog in izvedba nelinearne regulacije robotskega manipulatorja na osnovi kompenzacije trenja. Za zasnovo integralnega regulatorja drsnega režima za antropomorfni manipulator je bil uporabljen prilagojeni model LuGre s kompenzacijo trenja, ki zagotavlja sledenje želeni trajektoriji v prisotnosti trenja. Najprej je bil po modelu trenja LuGre izpeljan matematicni model avtonomne zglobne robotske izobraževalne platforme (AUTAREP) ED-7220C s petimi prostostnimi stopnjami. Nato je bil zasnovan robustni regulator za kompenzacijo dinamicnih tornih pojavov, ki upravlja položaj in hitrost za doseganje želene trajektorije. Kompenzacija negotovosti parametrov in obremenitvenih momentov sistema je izvedena z integralnim zakonom SMC na osnovi funkcije po Lyapunovu. Izkazalo se je, da so parametri prehodnih pojavov (cas maksimalnega prevzpona, odstotek prenihaja, cas ustalitve itd.) znotraj predvidenih mej. Regulator s kompenzacijo trenja ima boljši cas ustalitve in zmogljivost kot zakon na osnovi SMC brez kompenzacije trenja. Predlagani regulacijski zakon je bil preizkušen tudi na namenskem preizkuševališcu s programsko opremo LabVIEW in strojno opremo NI-myRIO. Vecina modelov robotskih sistemov v literaturi obravnava trenje s teoreticnega in konceptualnega vidika kot motnjo in ne kot stanje. V pricujocem clanku je podan predlog matematicnega modela, ki kombinira dinamicni model robota z dvema stanjema in dinamicni model trenja z enimstanjem. Predstavljene so zasnova, simulacija in hardverska izvedba robustnega regulacijskega zakona, ki upošteva trenje. Robotski manipulator po meri s petimi prostostnimi stopnjami je bil tudi fizicno izveden. V prihodnjih raziskavah bo mogoce razviti dinamicni model robotskega manipulatorja AUTAREP z razlicnimi modeli trenja. Trenje je dinamicno in njegovo stanje ni merljivo, zato bo mogoce zasnovati opazovalca za ocenjevanje njegove vrednosti. Kljucne besede: regulacija drsnega režima, prostostna stopnja, model trenja LuGre, avtonomna zglobna robotska izobraževalna platforma (AUTAREP) Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, SI 54 Prejeto v recenzijo: 2021-04-19 Š 2021 Strojniški vestnik. Vse pravice pridržane. Prejeto popravljeno: 2021-07-13 Odobreno za objavo: 2021-08-20 Napovedovanje kolicnika trenja pri hidrodinamicnem radialnem drsnem ležaju z uporabo umetnih nevronskih mrež Dragan Milcic1,* – Amir Alsammarraie2 – Miloš Madic1 – Vladislav Krstic3 – Miodrag Milcic1 1 Fakulteta za strojništvo, Univerza v Nišu, Srbija 2 Univerza v Tikritu, Tehniška fakulteta, Irak 3 Ljubex International, Srbija Hidrodinamicni ležaji nosijo vrteco se gred in vse pripadajoce obremenitve s pomocjo tlacnega polja v mazivu, ki fizicno locuje trdne površine. Radialni drsni ležaji, izdelani iz zlitine kositer-babbitt (Tegotenax V840), se pogosto uporabljajo v pogojih hidrodinamicnega mazanja pri kompresorjih, turbinah, crpalkah, elektromotorjih, elektricnih generatorjih, mlinih za rudo itd. Drsno trenje na kontaktnih površinah med tecajem in pušo je kompleksen disipativen proces, ki ga spremlja vrsta mehanskih, fizikalno-kemicnih, elektricnih, metalurških in toplotnih pojavov, za katere je znacilno obrabljanje udeleženih teles. Na kolicnik trenja pomembno vplivajo normalne obremenitve, geometrija, relativno gibanje površin, drsna hitrost, površinska hrapavost tornih površin, vrsta materiala, togost sistema, temperatura, pojav stik-slip, relativna vlažnost, mazanje in vibracije. Clanek preucuje vplive frekvence vrtenja tecaja v puši in radialne obremenitve radialnega drsnega ležaja, izdelanega iz zlitine kositer-babbitt (Tegotenax V840) v pogojih hidrodinamicnega mazanja. Na preizkuševališcu za rotacijske elemente (radialne in drsne ležaje) je bil opravljen eksperimentalni preskus tornih lastnosti radialnega drsnega ležaja. Za dolocitev matematicne odvisnosti med eksperimentalnimi parametri preskusa radialnega drsnega ležaja v pogojih hidrodinamicnega trenja sta bila razvita dva modela z umetnimi nevronskimi mrežami (ANN). Prvi model je namenjen napovedovanju kolicnika trenja, drugi pa temperaturi ležaja. Za ucenje in testiranje mrež so bili uporabljeni rezultati meritev kolicnika trenja ľ in temperature ležaja T, ki so trajale 18.000 sekund, in sicer v funkciji sprememb radialne obremenitve FN in vrtilne hitrosti gredi n. Množica razpoložljivih vhodnih/izhodnih podatkov je bila pri razvoju modela ANN nakljucno razdeljena v dve množici: prvo za ucenje in drugo za testiranje modela ANN. 295 podatkov je bilo uporabljenih za ucenje ANN, 43 pa za testiranje naucenega modela mreže ANN. Za ucenje ANN je bil izbran algoritem Levenberg-Marquardt, ki se odlikuje z visoko tocnostjo in hitro konvergenco. Proces ucenja ANN je bil ponovljen veckrat za razrešitev problema (pocasne) konvergence z razlicnimi zacetnimi utežmi, dolocenimi po metodi Nguyen-Widrow. Med procesom ucenja ANN je bilo ugotovljeno, da kljub izbrani vrednosti najvecjega števila iteracij 1000 zadostuje že bistveno manjše število iteracij. Z uporabo nevronske mreže z vzvratnim postopkom ucenja na osnovi eksperimentalnih podatkov so bili razviti modeli ANN za napovedovanje odvisnosti kolicnika trenja in temperature ležajev od radialne obremenitve in vrtilne hitrosti. Na podlagi eksperimentalno dolocenih vrednosti kolicnika trenja, s katerimi je bilo opravljeno ucenje ANN, so bile pridobljene dobro naucene mreže s srednjo absolutno odstotno napako 0,0054 % pri ucenju in 0,0085 % pri testiranju. Po optimizaciji modela ANN je bil ugotovljen najvišji kolicnik trenja 0,1513 pri obremenitvi ležaja 1072 N in vrtilni hitrosti 3000 vrt/min, medtem ko je bil najnižji kolicnik trenja 0,00288 ugotovljen pri maksimalni obremenitvi 4000 N in vrtilni hitrosti 1675,22 vrt/min. Model za napovedovanje temperature ležajev nakazuje narašcajoci trend spremembe s podaljševanjem casa drsnega trenja in vrtilne hitrosti. Temperatura ležaja Tse povecuje s casom t in se po nekaj manj kot 1 uri (približno 4000 s) stabilizira pri približno 50 °C pri vseh vrednostih obremenitve in vrtilne hitrosti ležaja. Kljucne besede: umetna nevronska mreža, hidrodinamicni radialni drsni ležaj, zlitina babbitt-kositer, kolicnik trenja Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, SI 55 Prejeto v recenzijo: 2021-03-21 Š 2021 Strojniški vestnik. Vse pravice pridržane. Prejeto popravljeno:2021-06-24 Odobreno za objavo: 2021-09-17 Vecparametricna dinamicna analiza sistema kotalnih ležajev Almatbek Kydyrbekuly1 – Gulama-Garip Alisher Ibrayev 1 – Tangat Ospan1 – Anatolij Nikonov2,* 1 Al-Farabi Kazahstanska narodna univerza, Kazahstan 2 Fakulteta za industrijski inženiring Novo mesto, Slovenia Pri nacrtovanju rotacijskih strojev je ena najpomembnejših mehanskih komponent, ki jih opisujejo nelinearni modeli, elasticna podpora, ki zagotavlja delovanje in zanesljivost sistema. Pri analizi linearnih rotorskih sistemov z valjcnimi ležaji se najpogosteje uporablja približna ocena togostnih in dušilnih lastnosti ležajev, medtem ko je v resnici togost ležaja mocno odvisna od obremenitve, to je od nacina delovanja rotorskega sistema, od geometrije in velikosti zracnosti ležajev, velikosti prileganja notranjega in zunanjega obroca v ležaju ter drugih dejavnikov. Za podroben opis dinamicnega procesa sistema kotalnih ležajev je potrebno upoštevati vpliv dejavnikov, kot so neuravnoteženost, asimetrija rotorja na gredi, zunanje trenje, spremembe vztrajnostnih parametrov in položajne sile razlicnih vrst. V prispevku smo preucevali vpliv velikosti reže, vrtilne frekvence na frekvencne spektre in amplitudno-frekvencne znacilnosti poljubnega rotorskega sistema na kotalne ležaje. Matematicni modeli ležajev, ki upoštevajo faktorje nelinearnosti, se odlikujejo po kompleksnosti, predvsem pa po obremenitvah, ki jih upoštevajo. Razvita je metoda za izracun amplitud in konstrukcijo frekvencnih karakteristik vsiljenih in samovzbujenih vibracij sistema ťrotor-tekocina-temeljŤ na kotalnih ležajih z nelinearno karakteristiko, ki temelji na metodi kompleksnih amplitud in harmonskega ravnovesja. Izpeljane so nelinearne enacbe gibanja sistema ťrotor­tekocina-temeljŤ, za katere so predstavljene analiticne metode reševanja. Dolocene so frekvence osnovne in ultra harmonicne resonance. Ocenjeni so intervali med frekvencami samovzbujenih nihanj. V prispevku je prikazana odvisnost amplitud od kolicine tekocine v votlini rotorja, mase temeljev, linearne neuravnoteženosti, vrednosti koeficienta togosti in koeficienta dušenja. Proucene so tudi posebnosti nelinearnega dinamicnega obnašanja, za katere so znacilne ultra harmonicne vibracije. Za optimalne vrednosti amplitud so bili v celoti doloceni optimalni parametri linearne neuravnoteženosti, mase temeljev, prostornine tekocine v votlini rotorja, togosti in koeficienta dušenja. Ugotovljeno je bilo, da v prisotnosti zanemarljivo majhne kolicine tekocine v votlini opazimo tri cone samovzbujenih nihanj in tri resonancne amplitude za rotor in za temelj. Najvecje amplitude vsiljenih nihanj rotorja in temelja opazimo, ko je tretjina votline rotorja napolnjena s tekocino. Poleg tega je bilo ugotovljeno, da zmanjšanje prostornine tekocine vodi do premika tretje resonance proti nižjim krožnim frekvencam. Prav tako se ob znatnem zmanjšanju koeficienta dušenja pojavi premik tretje resonance proti višjim kotnim hitrostim. Zlasti pri rotorju, napolnjenem s tekocino do tretjine prostornine, omenjeno resonanco opazimo pri isti frekvenci kot pri praznem rotorju, kar je znacilnost nelinearnih vibracij. Dinamicna analiza je pokazala, da z narašcanjem vrednosti neuravnoteženosti za red velikosti amplitudno frekvencno karakteristiko rotorja in temelja doloca nelinearnost progresivnega tipa. Tako s povecanjem linearne neuravnoteženosti samovzbujena nihanja, ki izvirajo iz delovanja tekocine, zamrejo zaradi vsiljenih vibracij. V tem primeru se znacilna razclenitev amplitud premakne proti višjim frekvencam sistema. Nadalje se po razclenitvi zaradi ucinka samocentriranja delovanje sistema stabilizira, pri cemer se pojavijo vibracije z amplitudo, ki nekoliko presega amplitudo samovzbujenih nihanj. Kljucne besede: vsiljena nihanja, resonanca, tekocina, kotalni ležaj, samovzbujena nihanja, ultra harmonicno nihanje Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, SI 56 Prejeto v recenzijo: 2021-04-08 Š 2021 Strojniški vestnik. Vse pravice pridržane. Prejeto popravljeno: 2021-08-21 Odobreno za objavo: 2021-08-24 Sledilno vodenje za manipulatorje na osnovi mehke ekvivalence in terminalnega drsnega režima Youyu Liu1,2,* – Yi Li1,2 – Xuyou Zhang1,2 – Bo Chen1,2 1 Državni laboratorij za napredno senzoriko in pametno upravljanje visokotehnološke opreme, Ministrstvo za izobraževanje, Kitajska 2 Politehnika Anhui, Šola za strojništvo, Kitajska Za zmanjšanje poskakovanja v pogojih težkih obremenitev in zagotavljanje primerne tocnosti vodenja zgibov, hitrosti konvergence napak in robustnosti je bil podan predlog metode za vodenje manipulatorjev na osnovi mehke ekvivalence in terminalnega drsnega režima (FETSM). Sistemi za regulacijo drsnega režima so obremenjeni s pojavom poskakovanja. Danes obstaja vec metod za odpravo poskakovanja, ki pa imajo vse svoje pomanjkljivosti. Metoda, ki uporablja zakon približevanja s funkcijo FAL(ali SAL), ima tako npr. slabost, da je težko izbrati položaj za uvajanje pojemka. Druga metoda, ki uporablja mehka pravila za ravnanje z eksponentom v potencnih zakonih približevanja in, ki funkcijo predznaka zamenja s funkcijo hiperbolicnega tangensa, ne more zmanjšati amplitude poskakovanja. Tretja metoda z adaptivno kompenzacijo napak (ali z nevronsko mrežo BP) je povezana z obcutnim povecanjem racunske kompleksnosti. Na osnovi clena za preklapljanje ESM je bila uvedena matrika z mehkimi parametri, ki se obdeluje s preprostimi mehkimi pravili, in dolocen je bil mehki clen za preklapljanje. S seštevanjem mehkega clena za preklapljanje ter ekvivalentnega clena ekvivalence in terminalnega drsnega režima (ETSM) je bil dolocen regulacijski zakon FETSM za manipulatorje. Na tej osnovi je bila analizirana stabilnost sistema s funkcijo po Lyapunovu in izpeljan je bil njen koncni cas prihoda. Na podlagidomneve zagotavljanja stabilnosti sistema so bila oblikovana mehka pravila in clanske funkcije za mehke konstante. Uporabnost in ucinkovitost predlagane metode sta bili preverjeni s primerjavo sledenja trajektoriji in vhodnega momenta za vec obstojecih metod. Rezultati in ugotovitve: (1) Predlagano metodo regulacije FETSM je mogoce uporabiti za sledenje dejanskemu položaju vsakega zgiba glede na njegovo idealno trajektorijo. Ta tehnologija je prakticno uporabna pri manipulatorjih, ki so med obratovanjem izpostavljeni velikim obremenitvam. (2) Rezultati simulacij kažejo, da je celotna natancnost sledenja po predlagani metodi FETSM za 3,56 % boljša v primerjavi z ETSM, celotna robustnost pa se je izboljšala za 30,24 %. Hitrost konvergence napake je prakticno enaka. Celotna natancnost sledenja se je s predlagano metodo FETSM izboljšala za 89,30 %, celotna robustnost se je izboljšala za 96,83 %, hitrost konvergence napak pa za 83,13 % v primerjavi z ESM. Celotna natancnost sledenja s predlagano metodo FETSM se je v primerjavi s PID-regulacijo izboljšala za 92,78 %. Hitrost konvergence napak in celotna robustnost sta boljši kot pri PID-regulaciji. Predlagana metoda FETSM lahko zagotavlja zadostno natancnost sledenja trajektoriji, hitrost konvergence napak in robustnost. (3) Pri predlagani metodi FETSM prihaja do poskakovanja le pod pogojem, da sistem ne doseže površine drsnega režima. Metoda FETSM lahko v primerjavi z ETSM v povprecju skrajša cas poskakovanja za 94,75 %. V primerjavi z regulacijo PID se maksimalna amplituda poskakovanja pri metodi FETSM zmanjša za vsaj 99,21 %. Predlagana metoda FETSM lahko torej omeji poskakovanje sistema in optimizira regulacijo vhodnega momenta. Prakticni ucinek predlagane metode pri manipulatorjih, ki obratujejo v pogojih težkih obremenitev, ni bil preverjen in to bo predmet nadaljnjih raziskav. Dolocen je zakon regulacijskega sistema FETSM za manipulatorje. Preverjena sta stabilnost in koncni cas prihoda sistema FETSM. Z mehkimi pravili so ocenjene zunanje motnje in negotovosti sistema. Predlagana metoda regulacije omogoca omejitev poskakovanja sistema v drsnem režimu. Sistem FETSM je primeren za manipulatorje, ki obratujejo v pogojih težkih obremenitev. Kljucne besede: regulacija drsnega režima, mehki clen za preklapljanje, poskakovanje, manipulator, sledenje trajektoriji, hitrost konvergence napake, robustnost Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, SI 57 Prejeto v recenzijo: 2020-10-01 Š 2021 Strojniški vestnik. Vse pravice pridržane. Prejeto popravljeno: 2021-01-04 Odobreno za objavo: 2021-01-07 Aktivni nadzor vibracij pri mehanski visokohitrostni servostiskalnici za natancno štancanje Yanxiong Liu1,3 – Yuwen Shu1,3 – Wentao Hu1,3 – Xinhao Zhao1,2,* – Zhicheng Xu1,3 1Tehniška univerza v Wuhanu, Državni laboratorij province Hubei za napredno tehnologijo avtomobilskih komponent, Kitajska 2Tehniška univerza v Wuhanu, Šola za materiale in inženiring, Kitajska 3Tehniška univerza v Wuhanu, Sodelovalno inovacijsko središce province Hubei za tehnologijo avtomobilskih komponent, Kitajska Proces natancnega štancanja je napreden preoblikovalni proces, ki daje izdelku koncno ali skoraj dokoncno obliko, v industriji pa se je uveljavil zaradi visoke ucinkovitosti in kakovosti izdelkov. Z višanjem frekvence delovnih gibov so vse mocnejše tudi vibracije stiskalnic za natancno štancanje. Znano je, da vibracije stiskalnice ne vplivajo le na kakovost izdelkov s povzrocanjem razpok na površini reza in zmanjševanjem dimenzijske tocnosti – skrajšujejo tudi življenjsko dobo strojev. Hrup, ki ga povzrocajo vibracije stroja, poleg tega otežuje delovne pogoje in vpliva na zdravje ljudi. Nadzor nad vibracijami mehanskih stiskalnic za natancno štancanje ni ustrezno raziskan, zato obstaja nujna potreba po rešitvah za nadzor nad vibracijami teh visokohitrostnih stiskalnic. V clanku sta opisani pasivna in aktivna metoda za nadzor nad vibracijami visokohitrostne stiskalnice za natancno štancanje, ki omejujeta vibracije celega stroja zaradi elasticnih sil med procesom natancnega štancanja. Vclanku je predstavljeno samoadaptivno krmiljenje z upoštevanjem motnje za simulacijo aktivnega nadzora nad vibracijami mehanske stiskalnice za natancno štancanje. Opisano je nacelo nadzora nad vibracijami in dolocen je krmilni algoritem. Vzpostavljen je tudi mehanski model vibracij stroja za natancno štancanje kot objekta regulacije. Parametri vzbujanja in mehanski model so bili doloceni s simulacijo po metodi koncnih elementov in z eksperimentalnimi meritvami vibracij stiskalnice. Koncno je opravljena še numericna simulacija in analiza aktivnega nadzora vibracij v okolju MATLAB. Vibracije celotnega stroja so se ucinkovito zmanjšale z uporabo aktivnega nadzora – v casovni domeni za vec kot 95 % in v frekvencni domeni za vec kot 80 %. Ucinkovitost pri zmanjševanju vibracij je torej ocitna, posledicno pa se izboljša tocnost obdelave, prihrani veliko energije in zmanjša škart. Predlagana metoda bi bila primerna tudi za nadzor nad vibracijami pri gradbenih strojih in pri opremi za koncno obdelavo. Vprihodnjih raziskavah bo mogoce uporabiti naprednejše metode regulacije za še boljši ucinek regulacije. Tokrat je bil prvic vzpostavljen mehanski model vibracij stroja za natancno štancanje. Parametri dinamicnega modela mehanskega stroja za natancno štancanje so bili pridobljeni z analizo po metodi koncnih elementov in z empiricno formulo. Razvit je bil adaptivni regulator vibracij z upoštevanjem motnje, ki je bil uporabljen za aktivni nadzor nad vibracijami stroja za natancno štancanje. S tem so bile ucinkovito zmanjšane vibracije stiskalnice in posledicno se lahko izboljša tocnost obdelave, prihrani veliko energije in zmanjša škart. Kljucne besede: aktivni nadzor vibracij, mehanski model vibracij, visokohitrostna mehanska servostiskalnica za natancno štancanje, samoadaptivno krmiljenje z upoštevanjem motnje Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, SI 58 Prejeto v recenzijo: 2021-04-26 Š 2021 Strojniški vestnik. Vse pravice pridržane. Prejeto popravljeno: 2021-08-04 Odobreno za objavo: 2021-09-01 Nelinearno vodenje pnevmatskega aktuatorja na osnovi dinamicnega modela trenja Xuan Bo Tran Znanstveno-tehniška univerza v Hanoju, Šola za transport, Oddelek za fluidno tehniko in avtomatizacijo, Vietnam Pnevmatski sistemi se množicno uporabljajo v industriji zaradi mnogih prednosti, kot so nizka cena, visoka moc na enoto mase, cistoca, preprosto vzdrževanje in zamenjava delov ter poceni in razpoložljiva oskrba z zrakom. Prednost imajo tudi v vrocih in/ali vlažnih okoljih, kjer elektricni sistemi niso primerni. Zanje pa sta znacilni tudi kompleksna dinamika in visoka nelinearnost zaradi vplivov stisljivosti zraka, nelinearnosti pnevmatskih ventilov in trenja v pnevmatskem aktuatorju. Tocno vodenje položaja, hitrosti in sil pnevmatskega aktuatorja zato ni preprosta naloga. V pricujocem clanku je predstavljen predlog novega regulatorja za vodenje položaja pnevmatskega valja. Zasnovan je po metodi drsne regulacije z vec površinami v kombinaciji s kompenzatorjem trenja. Metoda drsne regulacije z vec površinami obvladuje nelinearne lastnosti pnevmatskega sistema, kot sta stisljivost zraka in nelinearnost pnevmatskega ventila. Kompenzator trenja omogoca kompenzacijo sile trenja v pnevmatskem valju. Ta je ocenjena na podlagi dinamicnega modela trenja LuGre. V predstavljeni študiji je bil obravnavan elektropnevmatski servosistem z dvosmernim pnevmatskim valjem in dvema pnevmatskima proporcionalnima ventiloma za regulacijo pretoka. Najprej je bil postavljen splošni matematicni model sistema, nato pa je bil oblikovan regulacijskizakon novega regulatorja in analizirana absolutna stabilnost sistema. Novi regulator je bil ovrednoten s simulacijami v programski opremi MATLAB/Simulink. Novi regulator in drsni regulator z vec površinami sta bila simulirana z enakimi položajnimi vhodi, in sicer s koracnim in sinusoidnim položajnim vhodom z razlicnimi amplitudami in frekvencami. Zmogljivost novega regulatorja je bila primerjana z zmogljivostjo drsnega regulatorja z vec površinami. Novi regulator je bil koncno preverjen še eksperimentalno. Za to je bil postavljen eksperimentalni sistem in novi regulator je bil preverjen v enakih pogojih kot pri simulaciji. Eksperimentalno dolocena zmogljivost novega regulatorja je bila primerjana z zmogljivostjo drsnega regulatorja z vec površinami in drugih nelinearnih regulatorjev. Rezultati študije so: i) najvecja napaka relativnega položaja se z novim regulatorjem zmanjša za 10 %, koren srednje kvadratne napake pa se zmanjša za 6,5 % v primerjavi z drsnim regulatorjem z vec površinami brez kompenzacije trenja; ii) novi regulator dosega pri koracnem položajnem vhodu minimalno najvecjo napako 1 mm v stacionarnih pogojih; iii) novi regulator dosega pri sinusoidnem položajnem vhodu minimalno najvecjo napako 2,2 mm v stacionarnih pogojih; iv) novi regulator ima v primerjavi z ostalimi nelinearnimi in naprednimi regulatorji manjšo napako regulacije. Parametri modela trenja LuGre, uporabljeni v opazovalcu trenja, so bili doloceni vnaprej. Vprihodnje bo vkljucena še metodologija za sprotno ocenjevanje parametrov modela trenja LuGre pri novem regulatorju. Prispevek študije je v predlogu nove metode za regulacijo položaja v kombinaciji s kompenzacijo trenja na osnovi modela trenja, ki lahko izboljša kakovost regulacije položaja pnevmatskih valjev. Rezultate te študije bo mogoce uporabiti za izboljšanje kakovosti regulacije avtomatskih pnevmatskih strojev in sistemov v industriji. Kljucne besede: pnevmatski valj, nelinearno vodenje, kompenzacija trenja, dinamicni model trenja Guide for Authors All manuscripts must be in English. Pages should be numbered sequentially. The manuscript should be composed in accordance with the Article Template given above. The maximum length of contributions is 12 pages (approx. 5000 words). Longer contributions will only be accepted if authors provide justification in a cover letter. For full instructions see the Infor­mation for Authors section on the journal’s website: http://en.sv-jme.eu . SUBMISSION: Submission to SV-JME is made with the implicit understanding that neither the manu­script nor the essence of its content has been published previously either in whole or in part and that it is not being considered for publication elsewhere. 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Aškerceva 6, 1000 Ljubljana, Slovenia, e-mail: info@sv-jme.eu http://www.sv-jme.eu Contents Papers 401 Khurram Ali, Adeel Mehmood, Israr Muhammad, Sohail Razzaq, Jamshed Iqbal: Control of an Anthropomorphic Manipulator using LuGre Friction Model - Design and Experimental Validation 411 Dragan Milcic, Amir Alsammarraie, Miloš Madic, Vladislav Krstic, Miodrag Milcic: Predictions of Friction Coefficient in Hydrodynamic Journal Bearing Using Artificial Neural Networks 421 Almatbek Kydyrbekuly, Gulama-Garip Alisher Ibrayev, Tangat Ospan, Anatolij Nikonov: Multiparametric Dynamic Analysis of a Rolling Bearings System 433 Youyu Liu, Yi Li, Xuyou Zhang, Bo Chen: Trajectory-Tracking Control for Manipulators Based on Fuzzy Equivalence and a Terminal Sliding Mode 445 Yanxiong Liu, Yuwen Shu, Wentao Hu, Xinhao Zhao, Zhicheng Xu: Active Vibration Control of a Mechanical Servo High-speed Fine-Blanking Press 458 Xuan Bo Tran: Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model