Influence of hygro-thermal effects on the geometrically nonlinear free and forced vibrations of piezoelectric functional gradient beams with arbitrary number of concentrated masses

In this paper, the geometrically nonlinear free and forced vibrations of a beam made of functional gradient materials are analysed analytically, in which a piezoelectric actuator layer is integrated on the bottom and top faces. Moreover, the beam carrying concentrated masses is considered as a mass structure. Based on the Euler–Bernoulli beam theory and the nonlinear equilibrium equations are obtained using the Hamilton principle. An approximate multimodal approach in the vicinity of the principal mode based on a so-called second formulation applied to the relevant mode is used to solve the nonlinear equations. The results obtained from the proposed model are compared and validated with their counterparts from previous research in the literature. The good agreement between the current solution and equivalent data existing in the open literature proves the efficiency and accuracy of the method of analysis of the current solution. A detailed parametric study is conducted in this research to examine the influence of moisture change, temperature change, volume fraction, and the combined effect of positions, amplitudes and mass number on the nonlinear solutions of the vibration.