Forced Oscillations

Forced Oscillations

Apart from the force of the spring and damping force, an external force \(F(t)\), driving force, is acting on the object to force its motion $$m\ddot{x} + c\dot{x} + kx = F(t)$$ As this force drives the object to oscillate, it should be oscillating as well. Let $$F(t) = F_0 \cos \omega t $$

General Solution

The equation of motion $$m\ddot{x} + c\dot{x} + kx = F_0 \cos \omega t$$ Let \(\gamma = \frac{c}{2m}\) and \(\omega_0^2 = \frac{k}{m}\). $$\ddot{x} + 2\gamma\dot{x} + \omega_0^2x = \frac{F_0}{m} \cos \omega t$$ The general solution is $$x(t) = x_i(t) + x_h(t)$$ where \(x_i(t)\) is the solution with the driving force and \(x_h(t)\) is the solution without the driving force. \(x_h(t)\) has three solutions depending on whether \(\gamma\) is larger than, equal to or smaller than \(\omega_0\). \begin{align} x_h(t) &= Ae^{-\gamma t} \sin (\omega_d t + \omega_0 )\\ x_h(t) &= Ae^{-(\gamma-\sqrt{\gamma^2 - \omega_0^2})t} + Be^{-(\gamma+\sqrt{\gamma^2-\omega_0^2})t} \\ x_h(t) &= (At + B)e^{-\gamma t} \\ \end{align} They are called transient solutions because they eventually decay to zero.
The real part of $$m\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = \frac{F_0}{m}e^{i\omega t}$$ is the equation of motion, so solving this equation and the real part of the solution is the solution of equation of motion. Substitute \(x_i(t) = C e^{i\omega t} \) $$\left(-\omega^2 + 2i\gamma\omega + \omega^2\right)Ce^{i\omega t} = \frac{F_0}{m}e^{i\omega t} $$ So, we can solve for \(C\) \begin{align} C &= \frac{F_0/m}{-\omega^2 + 2i\gamma\omega + \omega_0^2} \\ &= \frac{F_0(\omega_0^2 - \omega^2 -2i\gamma\omega)}{m\left[(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2\right]} \\ &= Ae^{-i\phi} \end{align} where $$A = \frac{F_0}{m\sqrt{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}}$$ is the amplitude and $$\phi = \tan^{-1}\frac{2\gamma\omega}{\omega_0^2 - \omega^2}$$ So, $$x_i(t) = \Re (Ae^{i(\omega t - \phi)}) = A \cos(\omega t - \phi)$$ This is called the steady state solution. The general solution is \begin{align} x(t) &= x_h(t) + x_i(t) \\ &= A_he^{-\gamma t}\cos(\omega_d t + \phi_h) + \frac{F_0}{m\sqrt{(\omega_0^2 - \omega^2)^2-4\gamma^2\omega^2}}\cos(\omega t - \phi) \\ \end{align}

Resonance Frequency

When the frequency of the driving force maximize the amplitude, the frequency is called resonance frequency \begin{align} \frac{dA}{d\omega} &= 0 \\ \frac{F_0}{m}\left(-\frac{1}{2}\right)((\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2)^{\frac{3}{2}}(2(\omega_0^2 - \omega^2)(-2\omega) + 8 \gamma^2\omega) &=0 \\ \omega_0^2 - \omega^2 &= 2\gamma^2 \\ \omega &= \sqrt{\omega_0^2 - 2\gamma^2} \\ \end{align}

Work Done

The equation of motion is $$m\ddot{x} + c\dot{x} + kx = F_0 \cos \omega t$$ Multiply by \(\dot{x}\) on both sides \begin{align} m\ddot{x}\dot{x} + c\dot{x}\dot{x} + kx\dot{x} &= (F_0 \cos \omega t)\dot{x} \\ \frac{d}{dt}\left( \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2 \right) &= -c\dot{x}\dot{x} + (F_0\cos\omega t)\dot{x} \end{align} Left hand side the rate of change of the total internal energy while \(-b\dot{x}\dot{x}\) is the rate of work done by the damping force and \(\dot{x}F_0\cos\omega t\) is the rate of work done of the driving force on the object