Exact Solution of Time History Response for Dynamic Systems With Arbitrary Viscous Damping Using Complex Modal Analysis

Abstract

A solution method for general, non-proportional damping time history response for piecewise linear loading is generalized to exact solutions which include piecewise quadratic loading. Comparisons are made to Trapezoidal and Simpson’s quadrature rules for approximating the time integral of the weighted generalized forcing function in the exact solution to the decoupled modal equations arising from state-space modal analysis of linear dynamic systems. Closed-form expressions for the weighting parameters in the quadrature formulas in terms of time-step size and complex eigenvalues are derived. The solution is obtained step-by-step from update formulas derived from the piecewise linear and quadratic interpolatory quadrature rules starting from the initial condition. An examination of error estimates for the different force interpolation methods shows convergence rates depend explicitly on the amount of damping in the system as measured by the real-part of the complex eigenvalues of the state-space modal equations and time-step size. Numerical results for a system with general, non-proportional damping, and driven by a continuous loading shows that for systems with light damping, update formulas for standard Trapezoidal and Simpson’s rule integration have comparable accuracy to the weighted piecewise linear and quadratic force interpolation update formulas, while for heavy damping, the update formulas from the weighted force interpolation quadrature rules are more accurate. Using a simple model representing a stiff system with general damping, we show that a two-step modal analysis using real-valued modal reduction followed by state-space modal analysis is shown to be an effective approach for rejecting spurious modes in the spatial discretization of a continuous system.