 | Physik M 513.805 (511.015) |
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| Physik M 513.805 (511.015) |
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Critically damped solution to a damped driven oscillatorA damped, driven oscillator is described by the equation, $$m\frac{d^2x}{dt^2}+b\frac{dx}{dt} + kx = F_0\cos(\omega t).$$If $b^2=4km$, the system is critically damped and the solution has the form, $$x(t) = C_1 \exp \left(-t/\tau\right)+C_2t \exp \left(-t/\tau\right)+\frac{F_0}{\rho}\cos (\omega t -\theta),$$where $$\rho = \sqrt{(k-m\omega^2)^2+\omega^2b^2},\qquad\tan\theta = \frac{\omega b}{k-m\omega^2},\qquad \tau=\frac{2m}{b},$$ $$C_1 = x_0-\frac{F_0}{\rho}\cos ( -\theta),\qquad C_2 = v_{x0} +\frac{C_1}{\tau} +\frac{\omega F_0}{\rho}\sin ( -\theta).$$Here $x_0$ is the position at $t=0$ and $v_{x0}$ is the velocity at $t=0$.
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