Oscillators

Hooke’s Law

The equation for the force that governs simple harmonic motion comes from Hooke’s law. In 1D that equation is famously written as,

\[\begin{split} \begin{align} F(x) &= -kx\\ U(x) &= \frac{1}{2}kx^2 \end{align} \end{split}\]

Simple Harmonic Oscillator

By Newton’s second law, the equation of motion for the simple harmonic oscillator (follows Hooke’s law directly) is,

\[\begin{split} \begin{align} m\ddot{x} &= -kx \nonumber\\ \ddot{x} &= -\frac{k}{m} x \end{align} \end{split}\]

Most are more used to using the natural frequency constant \(\omega \equiv \sqrt{\frac{k}{m}}\)

\[ \begin{align} \ddot{x} &= -\omega x \end{align} \]

This second order differential equation has the solutions,

\[\begin{split}\begin{gather} \boxed{x(t) = C_1e^{i\omega t} + C_2e^{-i\omega t}}\\ C_1 = C_2^*\\ \nonumber\\ \boxed{x(t) = B_1\cos(\omega t) + B_2\sin({\omega t})}\\ B_1 = C_1 + C_2 = x(0), \qquad B_2 = i(C_1 - C_2) = \frac{\dot x(0)}{\omega}\\ \nonumber\\ \boxed{x(t) = A\cos(\omega t - \delta)}\\ A = \sqrt{B_1^2 + B_2^2}, \qquad \delta = \tan\left(\frac{B_2}{B_1}\right)\\ \nonumber\\ \boxed{x(t) = \text{Re }Ce^{i\omega t}}\\ C = 2C_1\\ \nonumber\\ \boxed{x(t) = \text{Re}Ae^{i\omega t - \delta}} \end{gather}\end{split}\]

Multivariable Oscillators

Multivariable oscillators are simple only and a simple harmonic motion can be achieved if we consider Hooke’s law to be applied separately for each dimension. Say for \(N\)-dimensions, there are \(N\) differential equations each with possibly different amplitude \(A_i\) and different frequencies \(\omega_i\) (where \(i\) goes from \(x_1\) to \(x_2\)) then we may generalized the solution for each dimension to be,

\[ \begin{equation} \boxed{\boldsymbol{r} = \sum_{i=1}^N{A_i\cos(\omega_i t - \delta_i) \hat{x}_i}} \end{equation} \]

We may define a relative phase shift \(\delta_{ij} = \delta_j - \delta_i\) for \(j \ne i\). This is ususually useful in only one dimension since there exist only one \(\delta_{ij}\) for instance let \(\delta \equiv \delta_{yx}\):

\[\begin{split} \begin{gather} x = A_x\cos(\omega_x t)\\ y = A_y\cos(\omega_y t - \delta) \end{gather} \end{split}\]

Damped Oscillators

We consider an oscillator damped by a resistance force \(F_{f} = -b\dot x\) such that the differential equation becomes,

\[\begin{split} \begin{align} m\ddot x + b\dot x + kx = 0\\ \boxed{\ddot x + 2\beta\dot x + \omega^2 x = 0} \end{align} \end{split}\]

• $\beta\( : **Damping constant** where \)\beta \equiv b/2m$\

• \(\omega\) : As usual, the natural frequency \(\omega^2 = k/m\)

The solution to this equation is an exponential solved by the auxiliary equation method where,

\[\begin{split} \begin{gather} r_1 \equiv -\beta + \sqrt{\beta^2 - \omega^2}\\ r_2 \equiv -\beta - \sqrt{\beta^2 - \omega^2}\\ \boxed{x(t) = C_1e^{r_1 t} + C_2e^{r_2 t}} \end{gather} \end{split}\]

To visualize this solution we look at three cases seeing how \(\beta\) compares with \(\omega\).

Undamped Oscillation : For \(\beta = 0\) we expect to get the original harmonic oscillator

\[ \begin{gather} \boxed{x(t) = C_1e^{i \omega t} + C_2e^{-i \omega t}} \end{gather} \]

Underdamping Oscillation : For \(\beta < \omega\), the solution is a enveloped wave,

\[\begin{split} \begin{gather} \omega_1 \equiv \sqrt{\omega_0^2 - \beta^2}\\ x(t) = e^{-\beta t}\left(C_1e^{i\omega_1 t} + C_2^{-i\omega_1 t} \right)\\ \boxed{x(t) = Ae^{-\beta t}\cos(\omega_1 t - \delta)} \end{gather} \end{split}\]

Overdamping Oscillation : For \(\beta > \omega\), we observer a quickly dying non-periodical wave.

Critical Damping Oscillation : For \(\beta = \omega\), the solution is an exponential decay,

\[ \begin{equation} \boxed{x(t) = C_1e^{-\beta t} + C_2 t^{e^{-\beta t}}} \end{equation} \]

Damped Driven Oscillator

A driven oscillator adds in an external force call the driven force \(F_d(t)\). The damped driven oscillator is simply,

\[\begin{split} \begin{gather} F(t) = F_d(t) - b\dot x - kx\\ \boxed{\ddot x + 2\beta \dot x + \omega_0^2 = f(t)} \end{gather} \end{split}\]

• \(f(t)\) : The acceleration contributed by the external agent where \( f(t) = F_d(t)/m\)