Duhamel Integral 的再推導

A New Look at the Duhamel Integral

一般對於Duhamel integral method 的推導都是先求得線性單自由度系統在單位脈衝作用下的位移反應函數，然後再基於任何的歷時載重皆可視為是一系列單位脈衝載重的組合，因而只需將各個單位脈衝載重的位移反應函數疊加起來即可獲得此線性單自由度系統在這歷時載重作用下的位移歷時反應。本文將從動量平衡運動方程式著手來推導單位脈衝位移反應函數。此推演方法非常簡單與一般求解一元二次微分方程式完全一樣，同時其所隱含的基本假設也非常淺顯易懂。因而可以非常簡單明瞭地求得線性單自由度系統之單位脈衝位移反應函數，並進而推導出Duhamel integral method。

In developing the general solution to an arbitrary external force for asingle degree of freedom system, the force is interpreted as a sequence ofimpulses of infinite duration and the response to this force is the sum of theresponse to each impulse. Each response can conveniently be written interms of the response of the system to a unit impulse. This procedure isgenerally known as the Duhamel integral. In this study, the response to aunit impulse will be derived based on a momentum equation of motion.This derivation is simple and is the same as the procedure to solve asecond-order differential equation. In addition, it is easy to capture thebasic assumptions of this derivation. Consequently, the response to a unitimpulse can be easily achieved, and then the Duhamel integral is derived.