The paper deals with modeling and simulation of the connection of piezoelectric materials, FGM materials and LQR control, using state-space model, which is built by modal truncation MOR method. FEM equations of piezoelectric finite beam were implemented in FEM program MultiFEM written in Wolfram Language. Modal truncation MOR method, which uses results from modal analysis, is also implemented in program Wolfram Mathematica. Simple piezoelectric structure is analyzed in numerical experiments, where FEM results are compared with MOR results

Galik, Gabriel

Goga, Vladimir

Kutis, Vladimir

Murin, Justin

Paulech, Juraj

Scipedia

IX International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2021E. Oñate, M. Papadrakakis and B. Schrefler (Eds)PIEZOELECTRIC BEAM FINITE ELEMENT MODEL ANDITS REDUCTION TO STATE-SPACE MODELVLADIMÍR KUTIŠ∗, JURAJ PAULECH∗, GABRIEL GÁLIK∗,JUSTÍN MURÍN∗ AND VLADIMÍR GOGA∗∗Institute of Automotive Mechatronics,Faculty of Electrical Engineering and Information Technology,Slovak University of Technology in Bratislava,Ilkovičova 3, 812 19 Bratislava 1,Slovak Republic.e-mail: vladimir.kutis@stuba.sk, web page: http://uamt.fei.stuba.sk/web/Key words: Finite Element Method, Piezoelectric Analysis, State-Space Model, LQRControl, MORAbstract. The paper deals with modeling and simulation of the connection of piezo-electric materials, FGM materials and LQR control, using state-space model, which isbuilt by modal truncation MOR method. FEM equations of piezoelectric finite beamwere implemented in FEM program MultiFEM written in Wolfram Language. Modaltruncation MOR method, which uses results from modal analysis, is also implemented inprogram Wolfram Mathematica. Simple piezoelectric structure is analyzed in numericalexperiments, where FEM results are compared with MOR results.1 INTRODUCTIONCurrent mechatronic systems use a combination of smart materials [1] and moderncontrol [2]. Modern smart materials include magnetorheological materials, functionallygraded materials, piezoelectric materials, shape memory materials and many more [3].Modern control methods use a state description of the system’s behavior and these con-trol methods include linear quadratic controller (LQR) and linear quadratic Gaussiancontroller (LQG) [4]. To model a physical system, one of the most effective methods isthe finite element method [5], which can describe the analyzed system with high accu-racy. However, this accuracy will be reflected in an increase the number of degrees offreedom, which is undesirable from the control viewpoint. Therefore, the FEM modelmust be reduced to decrease the number of degrees of freedom. For this purpose, themodal truncation method, which is one of model order reduction (MOR) methods [6], isused.1Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r GogaThis paper deals with modeling and simulation of the connection of piezoelectric mate-rials, FGM materials and LQR control, using state-space model, which is built by modaltruncation MOR method. In the theoretical part of the paper, the FEM equations of apiezoelectric beam element with an FGM core is presented, then a reduced state-spacemodel is described, which uses the modal truncation method and at the end of the theo-retical part the LQR controller is described. The derived relations were implemented intothe FEM program MultiFEM, while the built-in functions of the Wolfram Mathematicaprogram [7] are used to formulate the state-space model. The functionality of the systemis presented on a simple example of a beam with an FGM core and piezoelectric layers,which is analyzed using the FEM program MultiFEM and then reduced and analyzed asa state-space model.2 PIEZOELECTRIC BEAM STRUCTUREFig. 1 shows a two-node beam finite element, the core of beam is made of FGMmaterial, piezoelectric layers with electrodes are placed on the outer surfaces of beamcore. The length of the beam finite element is Le, the width is be, the height FGM of theFigure 1: Beam element with FGM core and piezoelectric layers with electordescore is heFGM and the height of one piezoelectric layer is hep. This finite element has 10degrees of freedom – 6 mechanical and 4 electrical, which are shown in Fig. 2. Vector ofFigure 2: Piezoelectric beam element, degrees of freedom – left, element loads – rightelement unknowns is defined as[veφe]=[uei vei ϕei uej vej ϕej φeT1 φeT2 φeB1 φeB2]T(1)2Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r Gogaand vector of element loads is defined as[FeQe]=[F exi Feyi Mei Fexj Feyj Mej QeT1 QeT2 QeB1 QeB2]T(2)FEM equations for beam element with piezoelectric layers and FGM core for transientanalysis have classical form[Mevv 00 0] [v̈eφ̈e]+[Cevv 00 0] [v̇eφ̇e]+[Kevv KevφKeφv Keφφ] [veφe]=[FeQe](3)The piezoelectric system is described by the final FEM equations in the form[Mvv 00 0] [v̈φ̈]+[Cvv 00 0] [v̇φ̇]+[Kvv KvφKφv Kφφ] [vφ]=[FQ](4)This matrix equation (4) can be rewritten in the formMvvv̈ + Cvvv̇ + Kvvv + Kvφ (φ)sens = F − Kvφ (φ)actu (5)Kφvv + Kφφ (φ)sens = (Q)sens (6)Kφvv + Kφφ (φ)actu = (Q)actu (7)where (φ)sens represents measured electric potential of sensors and (φ)actu represents elec-tric potential of actuators. If electrodes on piezoelectric sensors are short circuit, i.e.(φ)sens = 0, then equations (5) and (6) have formMvvv̈ + Cvvv̇ + Kvvv = F − Kvφ (φ)actu (8)Kφvv = (Q)sens (9)If electrodes on piezoelectric sensors are open circuit, i.e. D = 0, then these equationshave formMvvv̈ + Cvvv̇ + Kvvv + Kvφ (φ)sens = F − Kvφ (φ)actu (10)Kφvv + Kφφ (φ)sens = 0 (11)3 STATE-SPACE MODELThe displacement of the nodal points expressed by the vector v can be expressed bythe eigenshapes Z of the system and the amplitudes x of these eigenshapes, i.e. v = Z.x.Using this transformation, equation (4) can be rewritten in form[MvvZ 00 0] [..x..φ]+[CvvZ 00 0] [ .x.φ]+[KvvZ KvφKφvZ Kφφ] [xφ]=[FQ](12)3Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r GogaUsing orthogonality properties of mode shapesZTMvvZ = diag(µk) ZTKvvZ = diag(µkω2k) ZTCvvZ = diag(2ξkµkωk) (13)equation (12) can be rewritten in form[µ 00 0] [..x..φ]+[2ξµω 00 0] [ .x.φ]+[µω2 ZTKvφKφvZ Kφφ] [xφ]=[ZTFQ](14)where µ, ω and ξ represent matrix of modal masses, matrix of modal frequencies andmatrix of modal damping ratios of structure, respectively. After some manipulation, wecan from equation (14) obtain state-space model[.x..x]=[0 I−ω2 −2ξω] [x.x]+[0 0µ−1ZT −µ−1ZTKvφ] [Fφ](15)4 CONTROL OF MOR MODELLinear quadratic controller is very suitable for control of flexible structural systems.Control law can be written as follows(φ)actu = −KLQRs (16)where s =[x.x]Tis state vector and KLQR is gain matrix, which is set up by minimizinga cost function JJ =12∫ ∞0(sTQs + uTRu)dt (17)Matrix Q is positive semidefinite matrix and R is positive matrix and they representweights of state and input vectors in the process of minimization of a cost function J .Vector u = (φ)actu represents input vector.5 NUMERICAL EXAMPLE5.1 Analyzed systemFig. 3 left shows the analyzed FGM beam structure on which the piezoelectric actuatorsand sensors are placed. The discretized FEM model is shown in Fig. 3 right. Effectivematerial properties of FGM beam are defined by material properties of constituents andtheir variations and Young’s modulus and density of considered FGM beam have formEFGM(x, y) =1.93 × 1019x3y2 − 1.93 × 1014x3 − 2.9 × 1018x2y2++ 2.9 × 1013x2 + 5.8 × 1015y2 + 2.55 × 1011 [Pa](18)ρFGM(x, y) =1.34667 × 1012x3y2 − 1.34667 × 107x3 − 2.02 × 1011x2y2++ 2.02 × 106x2 + 4.04 × 108y2 + 9200. [kg/m3](19)4Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r GogaFigure 3: Analyzed FGM beam system with piezoelectric actuators and sensors – left, discretized FEMmodel – rightDamping of structure is considered as Rayleigh damping with mass and stiffness constants3 × 10−5 s-1 and 3 × 10−5 s, respectively.Piezoelectric actuators and sensors are made from PZT5A piezoelectric material. PZT5Ais orthotropic material and has following material properties (direction of poling has index3): mechanical properties:– Young’s moduli: E1 = 61 GPa, E2 = 61 GPa, E3 = 53, 2 GPa– Poisson numbers: µ12 = 0.35, µ13 = 0.38, µ23 = 0.38– shear moduli: G12 = 22.6 GPa, G13 = 21.1 GPa, G23 = 21.1 GPa– density: 7750 kg/m3 piezoelectric properties: d31 = −171 × 10−12 C/N, d33 = 374 × 10−12 C/N, d15 =584 × 10−12 C/N, d24 = 584 × 10−12 C/N relative permittivity: εσ11 = 1728.8, εσ22 = 1728.8, εσ33 = 1694.9The goal of the analysis is to perform a static and dynamic FEM analysis of the beamstructure with different excitations and then create a MOR model and compare the resultsobtained from the MOR model and the FEM model. The analysis also includes a designof LQR control of the structure in order to minimize the vibration of the structure.5.2 FEM analysisStatic analysisStatic FEM analysis is considered as an initial analysis that defines the initial state forthe selected transient analysis. The deformed state is defined by the prescribed verticaldisplacement at node 8 with a value of −0.25 × 10−5m and at node 21 with a value of1 × 10−5 m – see Fig. 3. The deformed shape is shown in Fig. 4.5Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r GogaFigure 4: Static deformed shape of analyzed beam structureModal analysisThe goal of the modal analysis is to obtain the first 4 bending eigenfrequencies and thecorresponding eigenmodes of the analyzed beam structure. These eigenfrequencies andeigenmodes are subsequently used both in the creation of a reduced model of the systemand in the definition of the modal initial state of the system. The first four eigenshapesare shown in Fig. 5.Figure 5: The first 4 bending eigenmodesFirst four bending eigenfrequencies of analyzed systems are 188.4 Hz, 988.5 Hz, 5532 Hzand 9765 Hz.Harmonic analysisThe response of the analyzed structure to the harmonic vertical force acting at the endof the structure – node 21 is shown in Fig. 6. The amplitude of the harmonic force is6Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r Goga20 N. As can be seen from the harmonic analysis of piezoelectric system with damping,0 20 000 40 000 60 000 80 000100 00010-910-710-50.001displacement Ynode 10node 210 20 000 40 000 60 000 80 000100 00010-710-610-510-40.0010.010displacement Ynode 10node 21Figure 6: Response of system to harmonic vertical force, without considering damping – left, withdamping – rightthe dominant natural frequencies are the first four. For this reason, the first 4 bendingeigenshapes and eigenfrequencies of the analyzed system are used in the reduced model.Transient analysisIn the transient analysis, 3 different excitations were considered: excitation by initialconditions, excitation by external force, excitation by electric potential on actuators.Excitation by initial conditions: the initial conditions of the system are defined by the0.000 0.005 0.010 0.015 0.020 0.025-0.00001-5.×10-60.000005.×10-60.00001displacement Ynode 10node 210.000 0.005 0.010 0.015 0.020 0.025-5.×10-60.000005.×10-60.00001displacement Ynode 10node 210.000 0.005 0.010 0.015 0.020 0.025-1.5×10-9-1.×10-9-5.×10-1005.×10-101.×10-91.5×10-9charge Qelem. 7 Gb 1elem. 8 Gb 20.000 0.005 0.010 0.015 0.020 0.025-1.5×10-9-1.×10-9-5.×10-1005.×10-101.×10-91.5×10-9charge Qelem. 7 Gb 1elem. 8 Gb 2Figure 7: Excitation by initial conditions – displacements and the electric charge, without consideringdamping – left, with damping – rightstatic deformation, which was presented at the subsection Static analysis. Newmark’snumerical method was used to calculate the system response. The displacements of the7Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r Goganodes 10 and 21 and the electric charge on the electrodes on the elements 7 and 8 areshown in Fig. 7.Excitation by external force: the external force is located at the free end of the system -node 21. The force has a harmonic character with frequency 80 Hz and an amplitude of10 N. The displacement of nodes 10 and 21 as well as the electric charge on elements 7and 8 of damped system are shown in Fig. 8. The system had zero deformation and zero0.000 0.005 0.010 0.015 0.020 0.025-0.0015-0.0010-0.00050.00000.00050.00100.0015displacement Ynode 10node 210.000 0.005 0.010 0.015 0.020 0.025-8.×10-8-6.×10-8-4.×10-8-2.×10-802.×10-84.×10-86.×10-8charge Qelem. 7 Gb 1elem. 8 Gb 2Figure 8: Excitation by external force – displacements and the electric chargevelocity at the beginning of investigated time.Excitation by electric potential on actuators: piezoelectric actuators are located closerto the left side – elements 3 and 4 – see Fig. 3. The inner electrodes are grounded,an electric harmonic potential with an amplitude of 100 V and a frequency of 40 Hz isprescribed at the outer electrodes. The displacement of the nodes 10 and 21 as well asthe electric charge on the elements 7 and 8 of damped system are shown in Fig. 9. Also,0.000 0.005 0.010 0.015 0.020 0.025-2.×10-6-1.×10-601.×10-62.×10-6displacement Ynode 10node 210.000 0.005 0.010 0.015 0.020 0.025-2.×10-11-1.×10-1101.×10-112.×10-11charge Qelem. 7 Gb 1elem. 8 Gb 2Figure 9: Excitation by electric potential on actuators – displacements and the electric chargein this analysis the system had zero deformation and zero velocity at the beginning ofinvestigated time.5.3 MOR state-space modelBased on the modal analysis results, a reduced state space model described by equa-tion (15) was created using the first 4 bending eigenfrequencies and their corresponding8Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r Gogaeigenshapes. In the model, the damping of system was defined as in the FEM model. Inthe dynamic analysis of the reduced state-space model, all excitations which were used inthe transient FEM analysis, were considered. Initial deformations, external vertical forcelocated at the right end of the beam structure and electric potential on actuators had tobe transformed into modal space. Fig. 10 shows the vertical displacement of the node21, which was calculated from the full FEM model – blue line and from the reduced statespace model – brown rings, considering a system without damping as well as a systemwith damping. Fig. 11 shows time variation of the vertical displacement of the node 210.000 0.005 0.010 0.015 0.020 0.025-0.00001-5.×10-60.000005.×10-60.000010.000 0.005 0.010 0.015 0.020 0.025-5.×10-60.000005.×10-60.00001Figure 10: Vertical displacement of the node 21 calculated by FEM model – blue line and by MORmodel – brown rings, without considering damping – left, with damping – rightin the model with the damping under the load of the vertical force – the left figure, andunder the load of the electric potential on the piezoelectric actuator – the right figure.The blue line represents the results from the FEM analysis and the brown rings represent0.000 0.005 0.010 0.015 0.020 0.025-0.0015-0.0010-0.00050.00000.00050.00100.00150.000 0.005 0.010 0.015 0.020 0.025-2.×10-6-1.×10-601.×10-62.×10-6Figure 11: Vertical displacement of the node 21 calculated by FEM model – blue line and by MORmodel – brown rings, system loaded by vertical force – left, system loaded by electric potential on thepiezoelectric actuator – rightthe results from the reduced state-space model. As can be seen from Fig. 10 and 11, MOR9Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r Gogastate-space model very faithfully describes the behavior of the system and the state-spacemodel have significantly fewer degrees of freedom than the full FEM model.5.4 Control lawA reduced state space model was used to design the LQR controller, with a state weightof 50,000 and an input weight of 1. Based on the parameters thus selected, the gain matrixKLQR was calculated and then a feedback state-space model was built. Subsequently, thestate-space model with feedback was analyzed, considering the initial conditions definedby static FEM analysis and transformed into modal space. Comparison of the systemresponse, i.e. the vertical deflection of the node 21 for the system without control – blueline and for the system with control – brown line, is shown in Fig. 12 left. The electric0.000 0.005 0.010 0.015 0.020 0.025-5.×10-60.000005.×10-60.000010.000 0.005 0.010 0.015 0.020 0.025-4.×10-6-2.×10-60.000002.×10-64.×10-66.×10-68.×10-60.00001Figure 12: Comparison of the system response, blue line is response of the system without control andbrown line for the system with control – left, blue line is response of system with control computed byFEM model and brown rings is response of system with control computed by MOR model – rightpotential on the piezoelectric actuators generated from the linear quadratic controller wasdetermined – time variation of electric potential is shown in Fig. 13. This voltage was0.000 0.005 0.010 0.015 0.020 0.025-400-300-200-1000100200300Figure 13: The electric potential on the actuators generated from the linear quadratic controller10Vladimı́r Kutǐs, Juraj Paulech, Gabriel Gálik, Just́ın Muŕın and Vladimı́r Gogaapplied as a load to the full FEM model and a transient FEM analysis was performed.A comparison of the results from the FEM model and the MOR model when control isconsidered is shown in Fig. 12 right, where the blue line represents the displacement ofnode 21 from the FEM analysis and the brown rings from the MOR analysis.6 CONCLUSIONSThe paper presents finite beam element with core made from functionally graded ma-terial and piezoelectric patches placed on the outer surfaces. The derived FEM equa-tions were implemented in the FEM program MultiFEM. In this program, static, modal,harmonic and full transient analysis were performed. Using these analyzes a reducedstate-space model of the system was created. MOR state-space model was analyzed usingprogram Mathematica, and the results from the MOR model were compared with the re-sults from the FEM model. The comparison showed a very good agreement between thesetwo models. As a last step, linear quadratic regulator for a reduced state-space modelwas designed. The results from the MOR model with LQR controller were also comparedwith the FEM results, where the electric load of piezoelectric actuator was derived fromgain matrix and states of system.ACKNOWLEDGEMENTThis work was supported by the Slovak Grant Agency: VEGA 1/0081/18, VEGA1/0416/21, KEGA 011STU-4/2020 and APVV-19-0406.REFERENCES[1] Schwartz, M. Encyclopedia of Smart Materials. John Wiley & Sons, Inc., (2002).[2] Dorf, R. C. and Bishop, R. H. Modern Control Systems. Pearson, (2017).[3] Varadan, V. K., Vinoy, K. J. and Gopalakrishnan, S. Smart Material Systems andMEMS:Design and Development Methodologies. John Wiley & Sons, Inc., (2006).[4] Friedland, B. Control System Design An Introduction to State-Space Methods. DoverPublications, Inc., (2005).[5] Burnett, D.S. Finite Element Analysis: From Concepts to Applications. AddisonWesley Publishing Company, (1987).[6] Besselink, B. et al., A Comparison of Model Reduction Techniques from StructuralDynamics, Numerical Mathematics and Systems and Control. Journal of Sound andVibration (2013) 332:4403–4422.[7] Wolfram Research, Inc. Mathematica, Version 12.2. Publisher: Wolfram Research,Inc., (2020).11

PIEZOELECTRIC BEAM FINITE ELEMENT MODEL AND ITS REDUCTION TO STATE-SPACE MODEL

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PIEZOELECTRIC BEAM FINITE ELEMENT MODEL AND ITS REDUCTION TO STATE-SPACE MODEL

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