Abstract: In this study, the nonlinear Rayleigh damping is adopted which can depict the exciting during the initial stage of seismic response, and the decaying after the motion velocity reaches its maximum value in the neat stage. Since the governing equation is a nonlinear PDE, the vibration pattern function is assumed to fulfill all the boundary conditions as a Fourier series with coefficients to be functions of time variable only. By applying the orthogonality of Fourier series,this nonlinear PDE is deduced to be an infinite system of nonlinear ODEs. By successive approximation method, only one unknown function is taken in each computational process which is a nonhomogeneous van der Pol equation, and its first two asymptotic solutions are obtained.Finally, numerical results are given to show the bifurcation of the solutions.