Exact Solution of Time History Response for Dynamic Systems with Coupled Damping using Complex Mode Superposition

An exact solution method for general, non-proportional damping time history response for piece-wise linear loading proposed by Dickens is generalized to piece-wise quadratic loading. Comparisons are made to Trapezoidal and Simpson\u27s 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. The time integral of the forcing function is recognized as a weighted integral with complex exponential and the general update formulas are derived using polynomial interpolation to the forcing function. 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 obtained from the piecewise linear and quadratic interpolatory quadrature rules starting from the initial conditions. Linear approximation for loading within a time-step used by Dickens is shown to be a special case of the quadrature rules with linear interpolation. The solution methods are exact for piecewise linear and quadratic loading with or without initial conditions and are computationally efficient with low memory for time-history response of linear dynamic systems including general non-proportional viscous damping. 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\u27s 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, 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