Transverse and longitudinal dynamic modeling of bimorph piezoelectric actuators with investigating the effect of vibrational modes

Document Type : Invited by Davoud Younesian


1 Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

2 New Technologies Research Centre, Amirkabir University of Technology, Tehran, Iran


Bimorph piezoelectric cantilevered (BPC) actuators have recently received a great deal of attention in a variety of micro-electromechanical systems (MEMS) applications. Dynamic modeling of such actuators needs to be improved in order to enhance the control performance. Previous works have usually taken transverse vibration into account without considering longitudinal vibration. This paper presents a comprehensive modeling for a set of transverse and longitudinal vibration equations for piezoelectric cantilevered actuators. In addition, dynamic behavior and exact non-minimum phase region along BPC is derived by analyzing first three vibrational modes. A simulation study is propounded to better analyze the system dynamic behavior. Finally, an experimental setup is developed to verify the proposed dynamic model. The modal frequency response of the system for the first three modes, obtained from the proposed model, is compared with those obtained from the experiment and a good consistency between them confirms the validity of the proposed dynamic model.


  • The set of transverse and longitudinal vibration equations have been derived and discretized for a BPC actuator.
  • The dynamic behavior of BPC actuator has been analyzed based on vibration equations for first three vibration modes.
  • The exact non-minimum phase region has been acquired along the actuator’s length.
  • The proposed dynamic behavior has been simulated to identify the model parameters and finally it has been verified via the experimental results.


Main Subjects

[1] M. Motamedi, M.T. Ahmadian, G. Vossoughi, S.M. Rezaei, M. Zareinejad, Adaptive sliding mode control of a piezo-actuated bilateral teleoperated micromanipulation system, Precision Engineering, 35 (2011) 309-317.
[2] T. Müller, A. Kugi, G. Bachmaier, M. Gerlich, Modelling and identification of a piezoelectrically driven fuel injection control valve, Mathematical and Computer Modelling of Dynamical Systems, 16 (2010) 285-305.
[3] S. Shim, M.G. Kim, K. Jo, Y.S. Kang, B. Lee, S. Yang, S.-M. Shin, J.-H. Lee, Dynamic characterization of human breast cancer cells using a piezoresistive microcantilever, Journal of biomechanical engineering, 132 (2010) 104501.
[4] Q. Xu, Precision position/force interaction control of a piezoelectric multimorph microgripper for microassembly, IEEE Transactions on Automation Science and Engineering, 10 (2013) 503-514.
[5] S. Bashash, R. Saeidpourazar, N. Jalili, Development, analysis and control of a high-speed laser-free atomic force microscope, Review of Scientific Instruments, 81 (2010) 023707.
[6] A. Salehi-Khojin, S. Bashash, N. Jalili, Modeling and experimental vibration analysis of nanomechanical cantilever active probes, Journal of Micromechanics and Microengineering, 18 (2008).
[7] N. Garcia, Theory of scanning tunneling microscopy and spectroscopy: resolution, image and field states, and thin oxide layers, IBM journal of research and development, 30 (1986) 533-542.
[8] M. Mohammadpour, M. Dardel, M.H. Ghasemi, M.H. Pashaei, Nonlinear energy harvesting through a multimodal electro-mechanical system, Journal of Theoretical and Applied Vibration and Acoustics, 1 (2015) 73-84.
[9] R. Toscano, I.A. Ivan, Robust structured controllers for piezoelectric microactuators, ISA transactions, 53 (2014) 1857-1864.
[10] W.M. Chen, T.S. Liu, Modeling and experimental validation of new two degree-of-freedom piezoelectric actuators, Mechatronics, 23 (2013) 1163-1170.
[11] I.A. Ivan, M. Rakotondrabe, P. Lutz, N. Chaillet, Quasistatic displacement self-sensing method for cantilevered piezoelectric actuators, Review of Scientific instruments, 80 (2009) 065102.
[12] O. Bilgen, M.A. Karami, D.J. Inman, M.I. Friswell, The actuation characterization of cantilevered unimorph beams with single crystal piezoelectric materials, Smart Materials and Structures, 20 (2011) 055024.
[13] S.-N. Chen, G.-J. Wang, M.-C. Chien, Analytical modeling of piezoelectric vibration-induced micro power generator, Mechatronics, 16 (2006) 379-387.
[14] G. Wang, Analysis of bimorph piezoelectric beam energy harvesters using Timoshenko and Euler–Bernoulli beam theory, Journal of Intelligent Material Systems and Structures, 24 (2013) 226-239.
[15] S. Peng, X. Zheng, J. Sun, Y. Zhang, L. Zhou, J. Zhao, S. Deng, M. Cao, W. Xiong, K. Peng, Modeling of a micro-cantilevered piezo-actuator considering the buffer layer and electrodes, Journal of Micromechanics and Microengineering, 22 (2012) 065005.
[16] S. Gorthi, A. Mohanty, A. Chatterjee, Cantilever beam electrostatic MEMS actuators beyond pull-in, Journal of Micromechanics and Microengineering, 16 (2006) 1800.
[17] M. Vagia, A frequency independent approximation and a sliding mode control scheme for a system of a micro-cantilever beam, ISA transactions, 51 (2012) 325-332.
[18] J. Yi, S. Chang, Y. Shen, Disturbance-observer-based hysteresis compensation for piezoelectric actuators, IEEE/Asme transactions on mechatronics, 14 (2009) 456-464.
[19] P.-P. Chao, P.-Y. Liao, M.-Y. Tsai, C.-T. Lin, Robust control design for precision positioning of a generic piezoelectric system with consideration of microscopic hysteresis effects, Microsystem technologies, 17 (2011) 1009-1023.
[20] H. Ghafarirad, S.M. Rezaei, A.A.D. Sarhan, M. Zareinejad, Continuous dynamic modelling of bimorph piezoelectric cantilevered actuators considering hysteresis effect and dynamic behaviour analysis, Mathematical and Computer Modelling of Dynamical Systems, 21 (2015) 130-152.
[21] A.A. Tahmasebi Moradi, S. Ziaei-Rad, R. Tikani, H.R. Mirdamadi, A finite element model for extension and shear modes of piezo-laminated beams based on von Karman's nonlinear displacement-strain relation, Journal of Theoretical and Applied Vibration and Acoustics, 2 (2016) 35-64.
[22] N. Jalili, Piezoelectric-Based Systems Modeling, in:  Piezoelectric-Based Vibration Control, Springer, 2010, pp. 183-232.
[23] S.S. Rao, Vibration of continuous systems, John Wiley & Sons, 2007.
[24] B. Engquist, A.-K. Tornberg, R. Tsai, Discretization of Dirac delta functions in level set methods, Journal of Computational Physics, 207 (2005) 28-51.
[25] A. Erturk, Assumed-modes modeling of piezoelectric energy harvesters: Euler–Bernoulli, Rayleigh, and Timoshenko models with axial deformations, Computers & Structures, 106 (2012) 214-227.
[26] S. Basak, A. Raman, S.V. Garimella, Dynamic response optimization of piezoelectrically excited thin resonant beams, Journal of vibration and acoustics, 127 (2005) 18-27.
[27] S. Yu, S. He, W. Li, Theoretical and experimental studies of beam bimorph piezoelectric power harvesters, Journal of Mechanics of Materials and Structures, 5 (2010) 427-445.
[28] A. Erturk, D.J. Inman, A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters, Journal of vibration and acoustics, 130 (2008) 041002.
[29] A. Erturk, D.J. Inman, On mechanical modeling of cantilevered piezoelectric vibration energy harvesters, Journal of Intelligent Material Systems and Structures, 19 (2008) 1311-1325.
[30] H.K. Khalil, Nonlinear Systems. 3rd Prentice-Hall, Upper Saddle River, NJ, (2002).
[31] K. Ogata, Modern control engineering 5th edition. Lugar: Upper Saddle River, New Jersey 07458, Ed: Prentice Hall, (2009) 55.
[32] S.C. Stanton, A. Erturk, B.P. Mann, D.J. Inman, Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification, Journal of Applied Physics, 108 (2010) 074903.
[33] S.C. Stanton, A. Erturk, B.P. Mann, D.J. Inman, Resonant manifestation of intrinsic nonlinearity within electroelastic micropower generators, Applied Physics Letters, 97 (2010) 254101.
[34] M. Daqaq, N. Jalili, S.N. Mahmoodi, Nonlinear Dynamics of A Piezoelectrically-actuated Microcantilever Sensor, in:  ENOC-2008, Saint Petersburg, Russia, 2008.