Abstract: Based on both numerical simulation and theoretical analysis, the linear and nonlinear vibration problems for free vibration of beams with functional gradient materials (FGM) are investigated. The mechanical behaviors of FGM beams under classical theory and first-order shear theory are solved analytically, the inherent frequencies of FGM beams under simple support and solid end restraint are obtained.The theory analyses different conditions that influence the inherent frequence of the FGM beams, including different boundary conditions, different beam theories, gradient index, slenderness ratio and so on. Regardless of classical beam theory or first-order shear theory, with the increase of gradient index, the inherent frequency of FGM beams decreases accordingly. The nonlinear inherent frequency of FGM beam under numerical simulation is obtained by ABAQUS simulation. The theoretical solutions are compared with the numerical solutions to verify each other and improve the mechanical model. The aim is to give linear and nonlinear solutions of the structural vibration response of beams with FGM under multiple theories, using analytical methods and numerical simulations.