---
tag: 5KC6B0Z09Y
kind: statement
role: theorem
name: Duhamel's integral
msc: [34A30, 70J35, 44A35]
uses: [0QQRP17Z7S, FCKX8KFVGN, 0FXDW992DS]
about: [0QQRP17Z7S]
sources:
  - url: https://en.wikipedia.org/wiki/Duhamel%27s_integral
same_as:
  wikidata: Q845685
  wikipedia: Duhamel's integral
---

## Statement

Let $m > 0$ and $c, k \in \R$, and let $F \colon [0, \infty) \to \R$ be continuous.
Let $h$ be the solution of $m h'' + c h' + k h = 0$ with $h(0) = 0$ and $h'(0) = 1/m$.
Then the solution of the [[initial value problem|0FXDW992DS]]
$$m x'' + c x' + k x = F(t), \qquad x(0) = 0, \quad x'(0) = 0,$$
is the convolution
$$x(t) = \int_0^t F(\tau) \, h(t - \tau) \, d\tau, \qquad t \ge 0.$$

## Remarks

Differentiate under the integral sign: $x'(t) = h(0) F(t) + \int_0^t F(\tau) h'(t - \tau) \, d\tau = \int_0^t F(\tau) h'(t - \tau) \, d\tau$, and $x''(t) = h'(0) F(t) + \int_0^t F(\tau) h''(t - \tau) \, d\tau$.
Then $m x'' + c x' + k x = m h'(0) F(t) = F(t)$, and $x(0) = x'(0) = 0$ because the integrals run over an interval of length $0$.
The function $h$ is the unit impulse response: the motion caused by a unit impulse at time $0$, which gives the mass the velocity $1/m$.

Example: for an underdamped oscillator, with $\omega_n = \sqrt{k/m}$, $\zeta = c / (2\sqrt{km})$ and $0 \le \zeta < 1$, put $\omega_d = \omega_n \sqrt{1 - \zeta^2}$.
Then
$$h(t) = \frac{1}{m \omega_d} e^{-\zeta \omega_n t} \sin(\omega_d t),$$
and the response to $F$ is $x(t) = \frac{1}{m \omega_d} \int_0^t F(\tau) e^{-\zeta \omega_n (t - \tau)} \sin(\omega_d (t - \tau)) \, d\tau$.
Nonzero initial data are handled by adding the free motion with those data.

## Explanations

### L1 human

A general force can be cut into many short kicks.
Each kick sets the system moving in the same way, only scaled and delayed.
Because the system is linear, the total motion is the sum of all these delayed responses, and in the limit the sum becomes an integral.

### L2 human

Duhamel's integral is the [[convolution on the half-line|FCKX8KFVGN]] of the load with the impulse response, so the Laplace transform turns it into the product $X(s) = H(s) F(s)$ with $H(s) = 1/(m s^2 + c s + k)$.
The same idea works for any [[linear differential equation|0QQRP17Z7S]] with constant coefficients, and in the form of [[Duhamel's principle|Q28DM6ZCMF]] for linear evolution equations.
It is a special case of [[variation of parameters|0Y6TH5PQVS]].
In structural dynamics it is used to compute responses to shocks and other non-periodic loads.