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Unformatted text preview: realistic model would allow the
carts to move in all three directions. This would require new variables yi and
zi such that
yi (t) = the y -component of displacement of the i-th weight,
zi (t) = the z -component of displacement of the i-th weight.
60 20 40 60 80 100 -0.02
-0.14 Figure 2.9: Shape of the fundamental mode (n=100).
If we were to set x1
x x = 2 ,
x y = 2 ,
z z = 2 ,
zn then an argument similar to the one given above would show that the vectors
x, y and z would satisfy the matrix diﬀerential equations
m and each of these could be solved just as before.
Using a similar approach, we could consider even more complicated systems
consisting of many masses and springs connected in a two- or three-dimensional
array. 61 Chapter 3 Fourier Series
3.1 Fourier series The theory of Fourier series and the Fourier transform is concerned with dividing
a function into a superposition of sines and cosines, its components of various
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This document was uploaded on 01/12/2014.
- Winter '14