Non-linear forced vibration analysis of piezoelectric functionally graded beams in thermal environment.

Document Type : Original Article


1 Ecole supérieure de technologie, Hassan II University of Casablanca, B.P 5366 Maarif, Casablanca, Maroc.

2 Laboratoire de Mécanique Productique et Génie Industriel, Ecole Supérieure de Technologie, Hassan II University of Casablanca, B.P.8012, Oasis, Casablanca, Morocco

3 Mohammed V University in Rabat, EMI-Rabat, LERSIM, B.P.765 Agdal, Rabat, Morocco


This work proposes a geometrically non-linear vibratory study of a functional gradation beam reinforced by surface-bonded piezoelectric fibers located on an arbitrary number of supports and subjected to excitation forces and thermoelectric changes. The non-linear formula is based on Hamilton's principle combined with spectral analysis and developed using Euler-Bernoulli's beam theory. In the case of a non-linear forced response, numerical results of a wide range of amplitudes are given based on the approximate multimodal method close to the predominant mode. In order to test the methods implemented in this study, examples are given and the results are very consistent with those of the literature. It should also be noted that the thermal charge, the electrical charge, the volume fraction of the structure, the thermal properties of the material, the harmonic force and the number of supports have a great influence on the forced non-linear dynamic response of the piezoelectrically functionally graded structure.


  1. Bakhshi Khaniki, H., Hosseini Hashemi, S. and Bakhshi Khaniki, M., “Free Vibration Analysis of Functionally Graded Materials Non-Uniform Beams”, International Journal of Engineering, Transactions C: Aspects, Vol. 29, No. 12, (2016), 1734-1740, DOI: 10.5829/idosi.ije.2016.29.12c.12.
  2. Tadi Beni, Z., Hosseini Ravandi, S. and Tadi Beni, Y., “Investigation of the Size Effect on the Nano-Beam Type Piezoelectric Low Power Energy Harvesting”, International Journal of Engineering, Transactions C: Aspects, Vol. 31, No. 9, (2018), 1585-1592, DOI: 10.5829/ije.2018.31.09c.15.
  3. Demir, Ç. and Civalek, Ö., "On the Analysis of Microbeams", International Journal of Engineering Science, Vol. 121, No., (2017), 14-33, DOI: 10.1016/j.ijengsci.2017.08.016.
  4. Habibi, B., Beni, Y.T., Mehralian, F., “Free Vibration of Magneto-Electro-Elastic Nanobeams Based on Modified Couple Stress Theory in Thermal Environment”, Mechanics of Advanced Materials and Structures, Vol. 26, No. 7, (2019), 601-613, DOI: 10.1080/15376494.2017.1410902.
  5. Samani, M.S.E. and Beni, Y.T, “Size Dependent Thermo-Mechanical Buckling of the Flexoelectric Nanobeam”, Materials Research Express, Vol. 5, No. 8, (2018), 085018, DOI: 10.1088/2053-1591/aad2ca.
  6. Tadi Beni, Y., “Size-Dependent Analysis of Piezoelectric Nanobeams Including Electro-Mechanical Coupling”, Mechanics Research Communications, Vol. 75, No., (2016), 67-80, DOI: 10.1016/j.mechrescom.2016.05.011.
  7. Tadi Beni, Y., “Size-Dependent Electromechanical Bending, Buckling, and Free Vibration Analysis of Functionally Graded Piezoelectric Nanobeams”, Journal of Intelligent Material Systems and Structures, Vol. 27, No. 16, (2016), 2199-2215, DOI: 10.1177/1045389X15624798.
  8. Tadi Beni, Z., Hosseini Ravandi, S., Tadi Beni, Y., “Size-Dependent Nonlinear Forced Vibration Analysis of Viscoelastic/Piezoelectric Nano-Beam”, Journal of Applied and Computational Mechanics, Vol., No., (2020), DOI: 10.22055/jacm.2020.32044.1958.
  9. Li, S.-r., Cheng, C.-j., “Free Vibration of Functionally Graded Material Beams with Surface-Bonded Piezoelectric Layers in Thermal Environment”, Applied Mathematics and Mechanics, Vol. 30, No. 8, (2009), 969-982, DOI: 10.1007/s10483-009-0803-7.
  10. Li, Y., Feng, W. and Cai, Z., “Bending and Free Vibration of Functionally Graded Piezoelectric Beam Based on Modified Strain Gradient Theory”, Composite Structures, Vol. 115, No., (2014), 41-50, DOI: 10.1016/j.compstruct.2014.04.005.
  11. Kiani, Y., Eslami, M., “Thermal Buckling Analysis of Functionally Graded Material Beams”, International Journal of Mechanics and Materials in Design, Vol. 6, No. 3, (2010), 229-238, DOI: 10.1007/s10999-010-9132-4.
  12. Rafiee, M., Yang, J. and Kitipornchai, S., “Large Amplitude Vibration of Carbon Nanotube Reinforced Functionally Graded Composite Beams with Piezoelectric Layers”, Composite Structures, Vol. 96, No., (2013), 716-725, DOI: 10.1016/j.compstruct.2012.10.005.
  13. Rafiee, M., Yang, J., Kitipornchai, S., “Thermal Bifurcation Buckling of Piezoelectric Carbon Nanotube Reinforced Composite Beams”, Computers & Mathematics with Applications, Vol. 66, No. 7, (2013), 1147-1160, DOI: 10.1016/j.camwa.2013.04.031.
  14. Yuan, M., “Compact and Efficient Active Vibro-Acoustic Control of a Smart Plate Structure”, International Journal of Engineering, Transactions B: Applications, Vol. 29, No. 8, (2016), 1068-1074,
  15. Tang, Y. and Ding, Q., “Nonlinear Vibration Analysis of a Bi-Directional Functionally Graded Beam under Hygro-Thermal Loads”, Composite Structures, Vol. 225, No., (2019), 111076, DOI: 10.1016/j.compstruct.2019.111076.
  16. Liu, C., Ke, L.-L., Yang, J., Kitipornchai, S., Wang, Y.-S., “Nonlinear Vibration of Piezoelectric Nanoplates Using Nonlocal Mindlin Plate Theory”, Mechanics of Advanced Materials and Structures, Vol. 25, No. 15-16, (2018), 1252-1264, DOI: 10.1080/15376494.2016.1149648.
  17. WANI, S.B., “Influence of Bi-Directional Fibreglass Grid Reinforcement on Drying Shrinkage and Mechanical Properties of Lightweight Foamed Concrete”, International Journal of Engineering, Transactions A: Basics, Vol. 34, No. 1, (2021), DOI: 10.5829/ije.2021.34.01a.02.
  18. Rahnavard, M., “Hot Corrosion Behavior of Functional Graded Material Thermal Barrier Coating (Research Note)”, International Journal of Engineering, Transactions A: Basics, Vol. 30, No. 1, (2017), 101-108, DOI: 10.5829/idosi.ije.2017.30.01a.13.
  19. Sherafatnia, K., Farrahi, G. and Faghidian, S.A., “Analytic Approach to Free Vibration and Buckling Analysis of Functionally Graded Beams with Edge Cracks Using Four Engineering Beam Theories”, International Journal of Engineering,Transactions C: Aspects Vol. 27, No. 6, (2014), 979-990. DOI:10.5829/idosi.ije.2014.27.06c.17
  20. Mirzavand, B. and Eslami, M.J., “Thermal Buckling of Simply Supported Piezoelectric Fgm Cylindrical Shells”, Journal of Thermal Stresses, Vol. 30, No. 11, (2007), 1117-1135, DOI: 10.1080/01495730701416036.
  21. Huang, X.-L., Shen, H.-S., “Vibration and Dynamic Response of Functionally Graded Plates with Piezoelectric Actuators in Thermal Environments”, Journal of Sound and Vibration, Vol. 289, No. 1-2, (2006), 25-53 DOI: 10.1016/j.jsv.2005.01.033.
  22. Zhang, D.-G. and Zhou, Y.-H., “A Theoretical Analysis of Fgm Thin Plates Based on Physical Neutral Surface”, Computational Materials Science, Vol. 44, No. 2, (2008), 716-720, DOI: 10.1016/j.commatsci.2008.05.016.
  23. Emam, S.A. and Nayfeh, A.H., “Postbuckling and Free Vibrations of Composite Beams”, Composite Structures, Vol. 88, No. 4, (2009), 636-642, DOI: 10.1016/j.compstruct.2008.06.006.
  24. Tian, Y., Fu, Y., Wang, Y., “Nonlinear Dynamic Response and Vibration Active Control of Piezoelectric Elasto-Plastic Laminated Plates with Damage”, Journal of Vibration, Vol. 15, No. 10, (2009), 1463-1492, DOI: 10.1177/1077546309103265.
  25. Kitipornchai, S., Ke, L., Yang, J., Xiang, Y., “Nonlinear Vibration of Edge Cracked Functionally Graded Timoshenko Beams”, Journal of Sound and Vibration, Vol. 324, No. 3-5, (2009), 962-982, DOI: 10.1016/j.jsv.2009.02.023.
  26. Maiz, S., Bambill, D.V., Rossit, C.A., Laura, P.A.A., “Transverse Vibration of Bernoulli–Euler Beams Carrying Point Masses and Taking into Account Their Rotatory Inertia: Exact Solution”, Journal of Sound and Vibration, Vol. 303, No. 3-5, (2007), 895-908, DOI: 10.1016/j.jsv.2006.12.028.
  27. Adri, A. and Benamar, R., “Linear and Geometrically Non-Linear Frequencies and Mode Shapes of Beams Carrying a Point Mass at Various Locations. An Analytical Approch and a Parametric Study”, Diagnostyka, Vol. 18, No. 2, (2017),
  28. Zhang, Z. and Zhang, C., “Mechanical Properties Analysis of Bilayer Euler-Bernoulli Beams Based on Elasticity Theory”, International Journal of Engineering, Transactions B: Applications, Vol. 33, No. 8, (2020), 1662-1667, DOI: 10.5829/ije.2020.33.08b.25.
  29. Ansari, R., Gholami, R. and Sahmani, S., “Free Vibration Analysis of Size-Dependent Functionally Graded Microbeams Based on the Strain Gradient Timoshenko Beam Theory”, Composite Structures, Vol. 94, No. 1, (2011), 221-228, DOI: 10.1016/j.compstruct.2011.06.024.
  30. Benamar, R., Bennouna, M., White, R., “The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of Thin Elastic Structures Part I: Simply Supported and Clamped-Clamped Beams”, Journal of Sound and Vibration, Vol. 149, No. 2, (1991), 179-195, DOI: 10.1016/0022-460X(91)90630-3.
  31. Azrar, L., Benamar, R., White, R., “A Semi-Analytical Approach to the Non-Linear Dynamic Response Problem of Beams at Large Vibration Amplitudes, Part Ii: Multimode Approach to the Steady State Forced Periodic Response”, Journal of Sound and Vibration, Vol. 255, No. 1, (2002), 1-41, DOI: 10.1006/jsvi.2000.3595.
  32. Fakhreddine, H., Adri, A., Chajdi, M., Rifai, S. and Benamar, R., “A Multimode Approach to Geometrically Non-Linear Forced Vibration of Beams Carrying Point Masses”, Diagnostyka, Vol. 21, No., (2020), DOI: 10.29354/diag/128603.
  33. El Kadiri, M., Benamar, R., White, R., “Improvement of the Semi-Analytical Method, for Determining the Geometrically Non-Linear Response of Thin Straight Structures. Part I: Application to Clamped–Clamped and Simply Supported–Clamped Beams”, Journal of Sound and Vibration, Vol. 249, No. 2, (2002), 263-305, DOI: 10.1006/jsvi.2001.3808.
  34. Rafiee, M., Mohammadi, M., Aragh, B.S. and Yaghoobi, H., “Nonlinear Free and Forced Thermo-Electro-Aero-Elastic Vibration and Dynamic Response of Piezoelectric Functionally Graded Laminated Composite Shells, Part I: Theory and Analytical Solutions”, Composite Structures, Vol. 103, No., (2013), 179-187 DOI: 10.1016/j.compstruct.2012.12.053.
  35. Reddy, J. and Chin, C., “Thermomechanical Analysis of Functionally Graded Cylinders and Plates”,Journal of Thermal Stresses, Vol. 21, No. 6, (1998), 593-626, DOI: 10.1080/01495739808956165.
  36. Babaee, A., Sadighi, M., Nikbakht, A. and Alimirzaei, S., “Generalized Differential Quadrature Nonlinear Buckling Analysis of Smart Sma/Fg Laminated Beam Resting on Nonlinear Elastic Medium under Thermal Loading”, Journal of Thermal Stresses, Vol. 41, No. 5, (2018), 583-607, DOI: 10.1080/01495739.2017.1408048.
  37. Fu, Y., Wang, J. and Mao, Y., “Nonlinear Analysis of Buckling, Free Vibration and Dynamic Stability for the Piezoelectric Functionally Graded Beams in Thermal Environment”, Applied Mathematical Modelling, vol. 36, no. 9, (2012), 4324–40, DOI: 10.1016/j.apm.2011.11.059
  38. Ke, L.-L., Yang, J. and Kitipornchai, S., “An Analytical Study on the Nonlinear Vibration of Functionally Graded Beams”, Meccanica, vol. 45, no. 6, (2010), 743–52, DOI: 10.1007/s11012-009-9276-1
  39. Shooshtari, A. and Rafiee, M., Nonlinear Forced Vibration Analysis of Clamped Functionally Graded Beams”, Acta Mechanica, vol. 221, no. 1, (2011), 23. DOI: 10.1007/s00707-011-0491-1
  41. Rubes, O., and Hadas, Z., “Designing, Modelling and Testing of Vibration Energy Harvester with Nonlinear Stiffness, Smart Sensors, Actuators, and MEMS VIII, Smart Sensors, Actuators, and MEMS VIII”, International Society for Optics and Photonics, vol. 10246, (2017), 102460W.