A modified wavelet approximation for multi-resolution AWCM in simulating nonlinear vibration of MDOF systems

Wavelet methods have been presented recently to solve the initial value problems of time-dependent ODEs. Before applying the current wavelet methods, a nonlinear vibration equation usually needs to be transformed to state equations, which may raise the storage and computation cost.In this paper, a modified form of wavelet approximation for the vibration solutions of linear/nonlinear MDOF systems is presented, based on which a multi-resolution AWCM (adaptive wavelet collocation method) is established for MDOF systems. The modified wavelet approximation’s wavelet coefficients explicitly include the boundary first-order time derivatives of the solutions, besides values of solutions on inner time locations of the time domain on which the solutions are defined. This can avoid to transform the dynamical equations of a linear/nonlinear vibrational MDOF system to state equations. Therefore, by using the modified wavelet approximation, the computation and storage cost required for wavelet collocation method may be reduced to at most 12 of that needed by those based on state equations; and the computation efficiency may be at least twice faster than wavelet collocation methods based on state equations, because the number of solution components that are needed to be computed is only 12 of the corresponding state equations. On the other hand, the implementation of initial conditions is straightforward by using the modified wavelet approximation proposed in this paper. The effectiveness of the modified wavelet approximation is tested by applying it to a linear and a nonlinear 10-DOF vibrational system.