Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

Abstract

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.