university of florida thesis or dissertation formatting template

there is no relation between the two motions, the point does not return to its original position,
and its trajectory fills up the rectangle by its repeated passages. However, recognizable stationary
patterns emerge whenever the two motions are either of the same frequency or if the ratio of their
frequencies is a rational number and the initial phases are a simple fraction of 2π. 109
Historically, Lissajous figures are used in electrical and mechanical engineering. A signal
generator generates one signal of known frequency, and an oscilloscope compares the known
signal with an unknown by combining them at right angles to each other. If there is a small
difference between the frequencies, one of the patterns is maintained for a few oscillations. It
will gradually change to another pattern as the phase difference between the orthogonal
vibrations changes. If the frequencies are ω1 and ω2, then one motion gains ω1- ω2
periods/second on the other. Hence, the cycle of patterns repeats after 1/(ω1-ω2) seconds. Timing
the repetition cycle of the patterns accurately gives the difference in the two frequencies.
Lissajous figures used this way are very valuable in comparing frequencies, calibrating
frequency sources, and obtaining the natural frequencies of mechanical components.
Lissajous figures are also applied in aeroelasticity as an indication of flutter during flight
and wind tunnel testing. Pitch rate vs. pitch and plunge rate vs. plunge are plotted resulting in a
circular Lissajous figure. 91 This method provides a quantitative measure of the coupling between
the torsion and bending motions that occur as flutter is approached. 110
Wavelet Transform
The wavelet is originally introduced by French geophysicist Jean Morlet as a tool for
signal analysis in the applications for the seismic phenomena. It is now developed in various
fields of science and applied in practical engineering. The wavelet transform has two parameters,
one is the scale that corresponds to frequencies and the other is the position that corresponds to
time. A Fourier transform maps a signal into the frequency domain so information concerning
44