On Some Closed-Form Solutions Of Duhamel’s Integral.

Abstract

In engineering analyses, the dynamic behavior of mechanical oscillators, or the electrical
performance of RLC circuits, is often modeled by linear, second order, inhomogeneous, ordinary
differential equations, with constant coefficients. Their general solution can be expressed in terms of
Duhamel's convolution integral, which involves the forcing function contained in the inhomogeneous
term of those equations. Depending on the complexity of this function, the integration may, or may
not, yield a closed-form solution. This article presents a general, and relatively compact, expression
for the recursive N th integration by parts of Duhamel’s integral. The obtained solution is based on the
use of second order recursive coefficients, which can be written in closed form. For forcing functions
with zero N th derivatives, the proposed expression is an exact and closed-form solution of the
original integral. This is the case for polynomial forcing functions of (N −1)th
degree. In this article,
due to space limitations, only final expressions are included, but their derivation process is
summarized. The summation format of the presented expressions allows for the proper identification
of all components contributing to the response. They are indicated as force-derivative and as initialforce-
derivative components. An example shows the use of the proposed exact, closed-form, solution.
It employs a forcing function defined as a quartic polynomial pulse. All different terms contributing to
either the displacement or velocity response are identified and analyzed. The proposed expressions
constitute ready tools for the solution of linear, second order differential equations subjected to
polynomial forcing functions.