An analytical approach to geometrically nonlinear free and forced vibration of piezoelectric functional gradient beams resting on elastic foundations in thermal environments

Abstract

In this article, the geometrically nonlinear free and forced vibrations of a beam made of functional gradient materials are analytically analyzed, in which piezoelectric actuators are bound. Based on the Euler–Bernoulli beam theory and the nonlinear Von-Karman deformation field, the analytical solutions of the nonlinear free vibrations, and forced vibrations of a beam with two clamped ends and resting on an elastic foundation are predicted. Using Hamilton's principle and an approximate multimodal dominant mode method, the nonlinear governing equations are obtained. Due to the lack of forced vibration results of the functionally graded piezoelectric beam subjected to thermal loading and resting on elastic foundations, the results obtained by the current solution were compared with their counterparts. The good agreement between the current solution and equivalent data existing in the open literature proves the efficiency and accuracy of the current resolution analysis method. Finally, the new parametric studies covered in this research include the effects of several parameters such as elastic foundation, thermal loading, and thermal properties of constituent materials on the free and forced nonlinear vibration of the piezoelectric functional gradient beam.