Tim Scheffler

Coupled Oscillators

Here a single coupled oscillator consists of three mass points: two masses $m$ at $x_1$ and $x_3$ and a mass $2m$ at $x_2$ with $x_1 < x_2 < x_3$. The springs between the mass points shall have a spring constant $k$ and the unstretched length of the spring shall be $d$. This will give the Lagrangian: $$ L = \frac{m}{2} \dot{x}_1^2 + m \dot{x}_2^2 + \frac{m}{2} \dot{x}_3^2 - \frac{k}{2}\left(x_2 - x_1 - d \right)^2 - \frac{k}{2}\left(x_3 - x_2 - d \right)^2 \, . $$ The center of mass is $R = \frac{1}{4}(x_1 + 2x_2 + x_3)$ and we can separate the motion into two modes for the generalized coordinates: $$ \begin{align} Q & = x_3 - x_1 - 2d \\ q & = x_2 - \frac{x_1 + x_3}{2} \, . \end{align} $$ The original coordinates can be retrieved with $$ \begin{align*} x_2 &= R + q/2 \\ x_1 &= R - Q/2 - d - q/2 \\ x_3 &= R + Q/2 + d - q/2 \, . \end{align*} $$ With this we can write the Lagrangian as $$ L = \frac{1}{2}4m \dot{R}^2 + \frac{1}{2} \frac{m}{2} \dot{Q}^2 - \frac{1}{2} \frac{k}{2} Q^2 + \frac{1}{2} m \dot{q}^2 - \frac{1}{2} 2k q^2 \, . $$ The coordinates $q$ and $Q$ clearly describe two independent harmonic oscillations, the two modes.

Another way to look at this system is to see it as a linear chain with $N=3$ sites. The eigenfunctions (i.e. modes) for this system are standing waves with moving end points. The $N=3$ system can support two standing waves: the $\lambda_1 = 2 * L$ and $\lambda_2 = L$ waves, where $L$ is the length of the chain. The $Q$-mode is the $\lambda_1$ wave and the $Q$-mode is the $\lambda_2$ wave. Also note that the frequencies of the two modes are given by $\omega_2 = \sqrt{2k/m}$ and $\omega_1 = \sqrt{k/m}$ and so the dispersion $\omega / \lambda$ is not linear, because $\omega_2 / \omega_1 = \sqrt{2}$ but $\lambda_2 / \lambda_1 = 2$.

Below we see a simulation for the two modes. To the left there is the slower $Q$-mode with no movement of the middle mass. To the right we see the faster $q$-mode.

(Select the picture and press 's' or click the picture to start/stop the simulation.)

The animation is a processing molecular dynamics simulation using the velocity Verlet integration scheme. In the lower right corner the overall energy of the system is printed to give a simple test if the integration is correct (in that case the energy should not drift on a long time scale).

If you select the simulation canvas and press 'p' you can add a coupling between the inner mass points. With 'm' you can reduce the coupling again (holding down the shift key the steps will be larger). The coupling is a spring too with a spring constant as given by the value of "coupling". This spring is displayed as attached to the bottom of the two mass points just to distinguish it from the other springs.

With coupling between the two oscillators one can see the transfer of energy from one oscillator to the other and back.

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