Analytical method for steady state vibration of system with localized non-linearities using convolution integral and Galerkin method

The analytical method using transfer function or impulse response is very effective for analyzing non-linear systems with localized non-linearities. This is because the number of non-linear equations can be reduced to that of the equations with respect to points connected with the non-linear element. In the present paper, analytical method for the steady state vibration of non-linear system including subharmonic vibration is proposed by utilizing convolution integral and the impulse response. The Galerkin method is introduced to solve the non-linear equations formulated by the convolution integral, and then the steady state vibration is obtained. An advantage of the present method is that stability or instability of the steady state vibration can be discriminated by the transient analysis from convolution integral. The three-degree-of-freedom mass–spring system is shown as a numerical example and the proposed method is verified by comparing with the result by Runge–Kutta–Gill method.