Generalization of Duhamel's integral to multi-degree-of-freedom systems

Abstract

This article presents a generalization to multi-degree-of-freedom (MDOF) systems of the classical convolution formula for single-degree-of-freedom (SDOF) systems that is widely known as Duhamel's integral. Remarkably, the extended convolution formula for a MDOF system is ultimately cast directly in terms of the system stiffness, mass and damping matrices in a manner that fully parallels the SDOF case, and without resorting to real or complex modal superposition. Still, although the proof based on the use of the Laplace transform might suggest that the obtained formulae are applicable to any type of viscous damping matrix, a more careful evaluation demonstrates that they contain a rather subtle constraint that implies that damping is of the classical type. The free vibration elicited by non-vanishing initial conditions is also considered. As a result, we ultimately obtain the general solution for MDOF systems with arbitrary dynamic loads and initial conditions. For undamped systems, the equivalence of the classical modal analysis with the direct approach proposed herein is demonstrated analytically. Special closed-form solutions, singular solutions and numerical examples are also investigated to a limited extent so as to further illustrate the application of the proposed formulae in the context of practical problems.