# 杜哈梅积分

## 概要介绍

### 问题背景

${\displaystyle m{\frac {d^{2}x(t)}{dt^{2}}}+c{\frac {dx(t)}{dt}}+kx(t)=p(t)}$

${\displaystyle h(t)={\begin{cases}{\frac {1}{m\omega _{d}}}e^{-\varsigma \omega _{n}t}\sin \omega _{d}t,&t>0\\0,&t<0\end{cases}}}$

${\displaystyle h(t-\tau )={\frac {1}{m\omega _{d}}}e^{-\varsigma \omega _{n}(t-\tau )}\sin[\omega _{d}(t-\tau )]}$${\displaystyle t\geq \tau }$

### 结论导出

${\displaystyle p(t)\approx \sum {p(\tau )\cdot \Delta \tau \cdot \delta }(t-\tau )}$

${\displaystyle x(t)\approx \sum {p(\tau )\cdot \Delta \tau \cdot h}(t-\tau )}$

${\displaystyle \Delta \tau \to 0}$时，连续求和转化为积分，此时上面的等式是严格成立的

${\displaystyle x(t)=\int _{0}^{t}{p(\tau )h(t-\tau )d\tau }}$

h(t-τ)的表达式代入即得杜哈梅积分的一般形式：

${\displaystyle x(t)={\frac {1}{m\omega _{d}}}\int _{0}^{t}{p(\tau )e^{-\varsigma \omega _{n}(t-\tau )}\sin[\omega _{d}(t-\tau )]d\tau }}$

## 参考文献

• 倪振华 编著，《振动力学》，西安交通大学出版社，西安，1990
• R. W. Clough, J. Penzien, Dynamics of Structures, Mc-Graw Hill Inc., New York, 1975.（中文版：R.W.克拉夫，J.彭津 著，王光远等 译，《结构动力学》，科学出版社，北京，1981）
• Anil K. Chopra, Dynamics of Structures - Theory and applications to Earthquake Engineering, Pearson Education Asia Limited and Tsinghua University Press, Beijing, 2001
• Leonard Meirovitch, Elements of Vibration Analysis, Mc-Graw Hill Inc., Singapore, 1986