Analysis of Diamond Cantilever Vibration
Purpose
In order to facilitate coherent bremsstrahlung radiation, the movement of the diamond radiator due to natural oscillation must be minimized. One possible way to mount the diamond involves supporting it from a single corner and leaving the other end free. The purpose of this work is to theoretically model the free vibrations of such a system to determine if it is a realistic solution to the problem of mounting the diamond.
Predictions
The diamond, when mounted from one corner, can be approximately modeled as a cantilever (a beam fixed at one end and free at the other) with non-uniform width. In order to determine the natural oscillatory motion of the diamond, I decided to develop a mathematical model for the motion of cantilevers with non-uniform width, and test that model with data from physical cantilevers.
Experiment
Fifteen test cantilevers of varying shapes have been cut from .032 inch thick sheet aluminum in the UConn metal-shop. (Thanks to Brendan Pratt.) Nine are of uniform width, while six have non-uniform width and are shaped like diamonds.
Step 1: Location of Resonance Frequencies Analysis was run in order to find the exact locations of the resonance frequencies for each individual cantilever. The cantilevers were mounted horizontally one at a time using a clamp to hold them fixed from one end. A microphone was set up near the tip of the cantilever, at an approximate distance of three centimeters away. The cantilever was then given a short, non-periodic impulse (a flick near the end) and allowed to settle into its natural oscillatory frequencies undisturbed. The microphone picked up the compressed air waves made by the cantilever’s vibration. This process was repeated three times for each cantilever. From this data the resonance frequencies were extracted using Fourier analysis. File:FourierFrequencies.pdf