# Pendulum

## (A Processing program illustrating vibration isolation)

Emil Schreiber

summer project
2009

Vibration isolation is an important challenge in many optical setups. To allow for high precision measurements, the optical components have to be kept very steady, even if the ground may shake, for example because of an experimenter walking by. One method to achieve this is to hang components like mirrors from long wires. The so formed pendulum works in a way that the mirror is hardly effected by any movements of it is suspension and stays in one place as desired. The applet below aims to explain this effect by demonstrating the behavior of a very simple pendulum. See also our Augmented Realty Pendulum sketch.

This applet has been built with Processing. Download the source code here: pendulum_src.zip

A pendulum, by definition, is a mass suspended from a string or (because a string is difficult to model) a stiff rod. The pendulum's bob is drawn towards the vertical position by gravity. If released from a non-vertical position, it starts to swing back and forth and will ultimately stop due to friction. Moving the suspension point, the bob will follow, but its inertia causes a slight delay depending on the velocity of the movement. For a periodic excitation different behaviors of the pendulum can be observed depending on the excitation frequency.

The simulation computes the occurring forces and the resulting movement of the pendulum using a numerical algorithm. The suspension point as well as the mass can be moved interactively to directly see the result. In addition an automatic movement of the suspension may be switched on, whose strength and speed are controlled by sliders. A second mass can be attached to the first to create a more complex double pendulum. Please note that even a simple pendulum can be a quite complex system to calculated accurately. An often used approximation as a so called harmonic oscillator is very exact for small movements of the pendulum, but loses its validity for large amplitudes. The numerical algorithm used in this simulation also has its limits and might give "unphysical" results when confronted with extreme parameters. More information about the physics involved can be found in this accompanying note (pdf file).

Things you can try:
• Grab the pendulum and move it. What happens if you move it slowly and what if you do it very fast?
• Switch on the automatic excitation by choosing a non-zero amplitude. Now slowly change the frequency and try to find the point where the pendulum swings highest. This is called the resonance frequency.
• How does the resonance frequency change with the amplitude or the length of the pendulum?
• Activate the double pendulum. Looking at the upper bob, you should now be able to find two resonance frequencies for which it starts to swing strongest.
• If your aim is to reduce any movement of the bob for a given excitation, which would be the best setup?
• Just for fun: Can you get the pendulum to constantly rotate? It is easiest with a short single pendulum but also possible for the double pendulum.

As you might have noticed playing with the simulation, the pendulum's bob can stay quite still if the suspension is moved fast, well above the resonance frequency. This suppression increases with the pendulum's length and it is even better for a double pendulum. In high precision laser interferometers quite complex systems are used, where the mirrors are part of a pendulum with several stages, each helping to minimize any motion of the mirrors. For slow movements of the suspension which are not suppressed by the pendulum's effect, additional active control systems can be used, so that the mirrors stay in place quite perfectly.