Dynamics Analysis of the Steady and Transient States of a Nonlinear Piezoelectric Beam by a Finite Element Method

Document Type : Research Paper


Mechanical Engineering Department , Isfahan University of Technology


This paper presents a finite element formulation for the dynamics analysis of the steady and transient states of a nonlinear piezoelectric beam. A piezoelectric beam with damping is studied under harmonic excitation. A numerical method is used for this analysis. In the paper, the central difference formula of four order is used and compared with the central difference formula of two order in the time response of the structure. The NPBDA program is developed with Matlab software. In this program, the Newmark technique for dynamic analysis is used, the Newton-Raphson iterative and Simpson methods are used for the nonlinear solution. To verify the NPBDA results, the experimental results of Malatkar are used for the nonlinear vibration analysis of a beam without piezoelectric properties. Then, the piezoelectric effect on the frequency mode values and the time response are obtained. Afterwards, the modulation frequency in the nonlinear beam and the piezoelectric effect in this parameter are verified.


[1] Ha S.K., Keilers C., Chang F.K., 1992, Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators, American Institute of Aeronautics and Astronautics 30:780-772.
[2] Crawley E.F., Lazarus K.B., 1991, Induced strain actuation of isotropic and anisotropic plates, American Institute of Aeronautics and Astronautics 29: 951-944.
[3] Rao S.S., Sunar M., 1993, analysis of distributed thermo piezoelectric sensors and actuators in advanced intelligent structures, American Institute of Aeronautics and Astronautics 31:1286-1280.
[4] Mindlin R.D., 1974, Equations of high frequency vibrations of thermo piezoelectric crystal plates, International Journal of Solids Structure 10:637-625.
[5] Moetakef M.A., Lawrence K.L., Joshi S.P., Shiakolas P.S., 1995, Closed-Form expressions for higher order electro elastic tetrahedral elements, American Institute of Aeronautics and Astronautics 33:142-136.
[6] Moetakef M.A., Joshi S.P., Lawrence K.L., 1996, Elastic wave generation by piezoceramic patches, American Institute of Aeronautics and Astronautics 34:2117-2110.
[7] Suleman A., Goncalves M.A., 1995, Optimization issues in application of piezoelectric actuators in panel flutter control, IDMEC-Instituto Superior Tecnico, Departamento de Engenharia Mecanica 1096 Codex.
[8] Suleman A., Venkayya V.B., 1995, Flutter control of an adaptive laminated composite panel with piezoelectric layers, IDMEC-Instituto Superior Tecnico, Departamento de Engenharia Mecanica 1096 Codex.
[9] Zemcik R., Roifes R., Rose M., Tessmer J., 2007, High performance four-node shell element with piezoelectric coupling for the analysis of smart laminated structures, International Journal for Numerical Methods in Engineering 70:961-934.
[10] Lazarus A., Thomas O., Deu J.F., 2012, Finite element reduced order models for nonlinear vibration of piezoelectric layered beams with applications to NEMS, Finite Elements in Analysis and Design 49:51-35.
[11] Ghayour M., Jabbari M., 2013, The effect of support and concentrated mass on the performance of piezoelectric beam actuator and frequencies, 3rd International Conference on Acoustic and Viberation ( ISAV2013).
[12] Kogl M., Bucalem M.L., 2005, Analysis of smart laminates using piezoelectric MITC plate and shell elements, Computers and Structures 83:1163-1153.
[13] Piefort V., Preumont A., 2000, Finite element modeling of smart piezoelectric shell structures, 5th National Congress on Theoretical and Applied Mechanics.
[14] Delgado I., 2007, Nonlinear vibration of a cantilever beam, M.S in Mechanical Engineering.
[15] Malatkar P., 2003, Nonlinear vibrations of cantilever beams and plates, Ph.D. thesis, Virginia Polytechnic Institute and State University.
[16] Borse G.J., 1997, Numerical Methods with Matlab, PWS-Kent, Boston.
[17] Sebald G., Kuwano H., Guyomar D., 2011, Experimental duffing oscillator for broadband piezoelectric energy harvesting, Smart Materials and Structures 20: 1-10.
[18] Erturk A., Inman D.J., 2011, Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling, Journal of Sound and Vibration 330: 2339-2353.
[19] Friswell1 M.I., Faruque S.A., Bilgen O., Adhikari S., Lees A.W., Litak G., 2012, Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass, Journal of Intelligent Material Systems and Structures 23(13): 1505-1521.
[20] Bendigeri C., Tomar R., Basavaraju S., Arasukumar K., 2011, Detailed formulation and programming method for piezoelectric finite element, International Journal of Pure and Applied Sciences and Technology 7(1): 1-21.
[21] Cook R.D., 1995, Finite Element Analysis for Stress Analysis, John Wiley & Sons, New York.
[22] Clough R.W., Penzien J., 1975, Dynamics of Structures, John Wiley and Sons, NewYork.