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  "body": "\n## Statement\n\nLet $m \u003e 0$ and $c, k \\in \\R$, and let $F \\colon [0, \\infty) \\to \\R$ be continuous.\nLet $h$ be the solution of $m h'' + c h' + k h = 0$ with $h(0) = 0$ and $h'(0) = 1/m$.\nThen the solution of the [[initial value problem|0FXDW992DS]]\n$$m x'' + c x' + k x = F(t), \\qquad x(0) = 0, \\quad x'(0) = 0,$$\nis the convolution\n$$x(t) = \\int_0^t F(\\tau) \\, h(t - \\tau) \\, d\\tau, \\qquad t \\ge 0.$$\n\n## Remarks\n\nDifferentiate 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 $x''(t) = h'(0) F(t) + \\int_0^t F(\\tau) h''(t - \\tau) \\, d\\tau$.\nThen $m x'' + c x' + k x = m h'(0) F(t) = F(t)$, and $x(0) = x'(0) = 0$ because the integrals run over an interval of length $0$.\nThe function $h$ is the unit impulse response: the motion caused by a unit impulse at time $0$, which gives the mass the velocity $1/m$.\n\nExample: for an underdamped oscillator, with $\\omega_n = \\sqrt{k/m}$, $\\zeta = c / (2\\sqrt{km})$ and $0 \\le \\zeta \u003c 1$, put $\\omega_d = \\omega_n \\sqrt{1 - \\zeta^2}$.\nThen\n$$h(t) = \\frac{1}{m \\omega_d} e^{-\\zeta \\omega_n t} \\sin(\\omega_d t),$$\nand the response to $F$ is $x(t) = \\frac{1}{m \\omega_d} \\int_0^t F(\\tau) e^{-\\zeta \\omega_n (t - \\tau)} \\sin(\\omega_d (t - \\tau)) \\, d\\tau$.\nNonzero initial data are handled by adding the free motion with those data.\n\n## Explanations\n\n### L1 human\n\nA general force can be cut into many short kicks.\nEach kick sets the system moving in the same way, only scaled and delayed.\nBecause the system is linear, the total motion is the sum of all these delayed responses, and in the limit the sum becomes an integral.\n\n### L2 human\n\nDuhamel'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 $H(s) = 1/(m s^2 + c s + k)$.\nThe 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.\nIt is a special case of [[variation of parameters|0Y6TH5PQVS]].\nIn structural dynamics it is used to compute responses to shocks and other non-periodic loads.\n",
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