Duhamel's integral

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.

Going deeper

For a student

Duhamel's integral is the convolution on the half-line 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 with constant coefficients, and in the form of Duhamel's principle for linear evolution equations. It is a special case of variation of parameters. In structural dynamics it is used to compute responses to shocks and other non-periodic loads.