Analytical solutions for twisting vibration of columns with nonlinear Rayleigh damping

The nonlinear Rayleigh damping is adopted in the governing equation, and 2-order quasi-linear PDE is solved. Assumed that both ends of the column are constrained without rotation. Substituting the assumed vibrational pattern function satisfying the boundary conditions into the PDE, and applying the orthognality of Fourier series, an infinite system of nonlinear 2-order ODEs is derived. A successive approximation method is adopted to obtain the analytical solution of the well known van der Pol equation for each time. The first two approximation solutions are obtained.