Skip to main content
SpringerLink
Log in

Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

A theoretical model for geometrically nonlinear vibration analysis of piezoelectrically actuated circular plates made of functionally grade material (FGM) is presented based on Kirchhoff’s-Love hypothesis with von-Karman type geometrical large nonlinear deformations. To determine the initial stress state and pre-vibration deformations of the smart plate a nonlinear static problem is solved followed by adding an incremental dynamic state to the pre-vibration state. The derived governing equations of the structure are solved by exact series expansion method combined with perturbation approach. The material properties of the FGM core plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. Control of the FGM plate’s nonlinear deflections and natural frequencies using high control voltages is studied and their nonlinear effects are evaluated. Numerical results for FG plates with various mixture of ceramic and metal are presented in dimensionless forms. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FGM plate on vibration characteristics of the smart structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Teymur, N. R. Chitkara, K. Yohngjo, J. Aboudi, M. J. Pindera and S. M. Arnold, Thermoelastic Theory for the Response of Materials Functionally Graded in Two Directions, Int. J. Solids Struct., 33 (1996) 931–966.

    Article  Google Scholar 

  2. M. Koizumi, The concept of FGM, Ceram. Trans. Funct. Grad. Mater. 34 (1993) 3–10.

    Google Scholar 

  3. T. Bailey and J. E. Hubbard, Distributed piezoelectric polymer active vibration control of a cantilever beam, J. Guid. Control Dyn., 8 (1985) 605–611.

    Article  MATH  Google Scholar 

  4. S. E. Miller and J. E. Hubbard, Observability of a Bernoulli-Euler beam using PVF2 as a distributed sensor, MIT Draper Laboratory Report, (1987).

  5. S. Dong and L. Tong, Vibration control of plates using discretely distributed piezoelectric quasimodal actuators/sensors, AIAA J., 39 (2001) 1766–1772.

    Article  Google Scholar 

  6. M. C. Ray, Optimal control of laminated shells with piezoelectric sensor and actuator layers, AIAA J., 41 (2003) 1151–1157.

    Article  Google Scholar 

  7. F. Peng, A. Ng and Y. R. Hu, Actuator placement optimization and adaptive vibration control of plate smart structures, J. Intell. Mater. Syst. Struct., 16 (2005) 263–271.

    Article  Google Scholar 

  8. W. J. Spencer, W. T. Corbett, L. R. Dominguez and B. D. Shafer, An electronically controlled piezoelectric insulin pump and valves, IEEE Trans. Sonics Ultrason., 25 (1978) 153–156.

    Google Scholar 

  9. S. Dong, X. Du, P. Bouchilloux and K. Uchino, Piezoelectric ring-morph actuation for valve application, J. Electroceram., 8 (2002) 155–161.

    Article  Google Scholar 

  10. C. Y. K. Chee, L. Tong and G. P. Steve, A review on the modeling of piezoelectric sensors and actuators incorporated in intelligent structures, J. Intell. Mater. Syst. Struct., 9 (1998) 3–19.

    Article  Google Scholar 

  11. L. Cao, S. Mantell and D. Polla, Design and simulation of an implantable medical drug delivery system using microelectromechanical systems technology, Sensors Actuators A, 94 (2001) 117–125.

    Article  Google Scholar 

  12. X. Chen, C. H. J. Fox and S. Mc William, Optimization of a cantilever microswitch with piezoelectric actuation, J. Intell. Mater. Syst. Struct., 15 (2004) 823–834.

    Article  Google Scholar 

  13. A. B. Dobrucki and P. Pruchnicki, Theory of piezoelectric axisymmetric bimorph Sensors, Actuators A 58 (1997) 203–212.

    Article  Google Scholar 

  14. C. J. Morris and F. K. Forster, Optimization of a circular piezoelectric bimorph for a micropump driver, J. Micromech. Microeng. 10 (2000) 459–465.

    Article  Google Scholar 

  15. S. Li and S. Chen, Analytical analysis of a circular PZT actuator for valveless micropump, Sensors Actuators A, 104 (2003) 151–161.

    Article  Google Scholar 

  16. E. F. Crawley and E. H. Anderson, Detailed models of piezoceramic actuation of beams, J. Intell. Mater. Syst. Struct., 1 (1990) 4–25.

    Article  Google Scholar 

  17. S. Timoshenko, Theory of Plates and Shells, Second Ed. McGraw-Hill, New York, (1940).

    MATH  Google Scholar 

  18. H. S. Tzou, Piezoelectric Shells-Distributed Sensing and Control of Continua, Kluwer Dordrecht, (1993).

    Google Scholar 

  19. H. S. Tzou and J. P. Zhong, Electromechanics and vibrations of piezoelectric shell distributed systems theory and applications ASME, J of Dynamic Systems, Measurements, and Control, 115(3) (1993) 506–517.

    Google Scholar 

  20. Y. Ootao and Y. Tanigawa, Control of transient thermoelastic displacement of a functionally graded rectangular plate bonded to a piezoelectric plate due to nonuniform heating, Acta Mech., 148 (2001) 17–33.

    Article  MATH  Google Scholar 

  21. J. N. Reddy and Z. Q. Cheng, Three-dimensional solutions of smart functionally graded plates, ASME J. Appl. Mech., 68 (2001) 234–241.

    Article  MATH  Google Scholar 

  22. B. L. Wang and N. Noda, Design of smart functionally graded thermo-piezoelectric composite structure, Smart Mater. Struct., 10 (2001) 189–193.

    Article  Google Scholar 

  23. X. Q. He, T. Y. Ng, S. Sivashanker and K. M. Liew, Active control of FGM plates with integrated piezoelectric sensors and actuators, Int. J. Solids Struct., 38 (2001) 1641–1655.

    Article  MATH  Google Scholar 

  24. K. M. Liew, J. Yang and S. Kitipornchai, Post buckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading, Int. J. Solids Struct., 40 (2003) 3869–3892.

    Article  MATH  Google Scholar 

  25. H. S. Shen, Post buckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings, Int. J. Solids Struct., 42 (2005) 6101–6612.

    MATH  Google Scholar 

  26. J. Yang, S. Kitipornchai and K. M. Liew, Nonlinear analysis of thermo-electro-mechanical behavior of shear deformable FGM plates with piezoelectric actuators, Int. J. Numer. Methods Eng., 59 (2004) 1605–1632.

    Article  MATH  Google Scholar 

  27. J. Yang, S. Kitipornchai and K. M. Liew, Large amplitude vibration of thermo-electric-mechanically stressed FGM laminated plates, Comput. Methods Appl. Mech. Eng., 192 (2003) 3861–3885.

    Article  MATH  Google Scholar 

  28. X. L. Huang and H. S. Shen, Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments, J. Sound Vib., 289 (2006) 25–53.

    Article  Google Scholar 

  29. F. Ebrahimi, A. Rastgoo and M. H. Kargarnovin, Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers, Journal of Mechanical Science and Technology, 22(6) (2008) 1056–1072.

    Article  Google Scholar 

  30. F. Ebrahimi and A. Rastgoo, Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers, Smart Mater. Struct., 17 (2008) No. 015044.

    Google Scholar 

  31. F. Ebrahimi and A. Rastgoo, An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory, Thin-Walled Structures, 46 (2008) 1402–1408.

    Article  Google Scholar 

  32. F. Ebrahimi, A. Rastgoo and A. A. Atai, theoretical analysis of smart moderately thick shear deformable annular functionally graded plate, European Journal of Mechanics — A/Solids, published online: 25 Dec. 2008.

  33. F. Ebrahimi and A. Rastgoo, Free Vibration Analysis of Smart FGM Plates, International Journal of Mechanical Systems Science and Engineering, 2(2) (2008) 94–99.

    Google Scholar 

  34. J. N. Reddy and G. N. Praveen, Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plate, Int. J. Solids Struct., 35 (1998) 4457–4476.

    Article  MATH  Google Scholar 

  35. X. L. Huang and H. S. Shen, Nonlinear vibration and dynamic response of functionally graded plates in thermal environment. International Journal of Solids and Structures, 41 (2004) 2403–2427.

    Article  MATH  Google Scholar 

  36. R. C. Wetherhold, S. Seelman and S. Wang, The use of functionally graded materials to eliminate or control thermal deformation, Compos. Sci. Technol., 56 (1996) 1099–1104.

    Article  Google Scholar 

  37. Y. Tanigawa, H. Morishita and S. Ogaki, Derivation of system of fundamental equations for a three dimensional thermoelastic field with nonhomogeneous material properties and its application to a semi infinite body, J. Therm. Stresses, 22 (1999) 689–711.

    Article  MathSciNet  Google Scholar 

  38. D. O. Brush and B. O. Almroth, Buckling of Bars Plates and Shells, McGraw-Hill, New York, (1975).

    MATH  Google Scholar 

  39. J. N. Reddy, Theory and Analysis of Elastic Plates, Taylor and Francis, Philadelphia, (1999).

    Google Scholar 

  40. G. Song, V. Sethi and H. N. Lic, Vibration control of civil structures using piezoceramic smart materials: A review, Engineering Structures, 28 (2006) 1513–1524.

    Article  Google Scholar 

  41. E. Efraim and M. Eisenberger, Exact vibration analysis of variable thickness thick annular isotropic and FGM plate, J. Sound Vib., 299 (2007) 720–738.

    Article  Google Scholar 

  42. C. Y. Chia, Nonlinear Analysis of Plates, McGraw-Hill, New York, (1980).

    Google Scholar 

  43. X. J. Zheng and Y. H. Zhou, Analytical formulas of solutions of geometrically nonlinear equations of axisymmetric plates and shallow shells, Acta Mech. Sinica, 6(1) (1990) 69–81.

    Article  MATH  Google Scholar 

  44. H. P. William, P. F. Brain and A. T. Sau, Numerical Recipes-the Art of Scientific Computing, Cambridge University Press, New York, (1986).

    MATH  Google Scholar 

  45. A. H. Nayfe and D. T. Mook, Nonlinear Oscillations, John Wiley, New York, (1979).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Tehran, Tehran, Iran

    Farzad Ebrahimi, Mohammad Hassan Naei & Abbas Rastgoo

  2. Mechanical Engineering Department, Faculty of Engineering and Technology, lmam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi

Authors
  1. Farzad Ebrahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Hassan Naei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abbas Rastgoo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farzad Ebrahimi.

Additional information

This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin

Farzad Ebrahimi received his B.S. and M.S. degree in Mechanical Engineering from University of Tehran, Iran. He is currently working on his Ph.D. thesis under the title of “Vibration analysis of smart functionally graded plates” at Smart Materials and Structures Lab in Faculty of Mechanical Engineering of the University of Tehran. His research interests include vibration analysis of plates and shells, smart materials and structures and functionally graded materials.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ebrahimi, F., Naei, M.H. & Rastgoo, A. Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation. J Mech Sci Technol 23, 2107–2124 (2009). https://doi.org/10.1007/s12206-009-0358-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-009-0358-8

Keywords

Search

Navigation