Harmonic Oscillation

Simple Harmonic Oscillation

When the only force applying on an object is proportional to its position $$F = -kx$$ the equation of motion is $$m\frac{d^2x}{dt^2} = -kx$$ The general solution is $$B\cos (\omega t ) + C \sin (\omega t)$$ where $$\omega = \sqrt{\frac{k}{m}}$$

Damped Harmonic Oscillation

A damping force hinder an object's motion by exerting a force proportional to the object's velocity $$F_{damping} = -c\frac{dx}{dt}$$ the equation of motion is $$m\frac{d^2x}{dt^2} = -kx -c\frac{dx}{dt}$$ Let $$\gamma = \frac{c}{2m}$$ and $$\omega_0^2 = \frac{k}{m}$$ The equation of motion becomes $$\frac{d^2x}{dt^2} +2\gamma\frac{dx}{dt} + \omega_0^2 = 0$$ Let $$x = e^{\lambda t}$$ The equation of motion is $$(\lambda^2 + 2\gamma\lambda + \omega_0^2)e^{\lambda t} = 0$$ To have a non-trivial solution, $$\lambda^2 + 2\gamma\lambda + \omega_0^2 = 0$$ Solve for \(\lambda\) $$\lambda = -\gamma \pm \sqrt{\gamma^2-\omega_0^2}$$ The general solution is $$ \bbox[5px,border:2px solid #666] { x(t) = e^{\gamma t}\left[ B \exp\left(\sqrt{\gamma^2-\omega_0^2}t\right) + C\exp\left(-\sqrt{\gamma^2-\omega_0^2}t\right) \right] } $$

Underdamping

When $$\omega_0^2 > \gamma^2$$ Let $$\omega_d = \sqrt{\omega_0^2-\gamma^2}$$ The square root \(\sqrt{\gamma^2-\omega_0^2}\) is not real. The motion should be oscillation. The general solution is $$x(t) = e^{-\gamma t} (B e^{i\omega_d t} + C e^{-i\omega_d t}) = A e^{-\gamma t} \sin (\omega_d t + \phi)$$

Critical damping

When $$\omega_0^2 = \gamma^2$$ So $$\lambda = -\gamma$$ Let \(x(t) = f(t)e^{\gamma t}\). \begin{align} \frac{d^2x}{dt^2} +2\gamma\frac{dx}{dt} + \omega_0^2 &= 0 \\ \frac{d^2f}{dt^2} + (2\lambda + 2\gamma) \frac{df}{dt}e^{\lambda t} + (\lambda^2 + 2\lambda\gamma + \gamma^2)fe^{\lambda t} &= 0 \\ \frac{d^2f}{dt^2} &= 0 \\ f(t) &= At + D \end{align} The general solution is $$x(t) = (At + D)e^{-\gamma t}$$

Overdamping

When $$\omega_0^2 < \gamma^2$$ The general solution is $$x(t) = Be^{-(\gamma-q)t} + Ce^{-(\gamma + q)t}$$ where $$q = \sqrt{\gamma^2 - \omega_0^2}$$ The root is real so the motion is exponential rather than oscillation.