A Modified Newmark Scheme for Simulating Dynamical Behavior of MDOF Nonlinear Systems

Abstract

The Newmark integration scheme, originally developed for simulating responses of linear dynamical systems, is modified in this paper to effectively model nonlinearities through introduction of the incremental displacements and the effective mass, damping and tiffness matrices. Based on the results of comparisons with the analytical solutions and the Runge-Kutta (RK) method for well-known nonlinear oscillators, the proposed scheme is found to capture accurately dynamical behaviors of nonlinear systems such as mplitude-dependent stiffening or softening natural frequencies, jump phenomena, superharmonic resonances, sub-harmonic resonances, and even chaos and bifurcations. The proposed scheme is more efficient than the RK method for large scale nonlinear ystems with some type of sparsity for which a targeted algebraic equations solver can be employed to speed up the solution for each time step.