其他摘要：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.
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