This exercise involves finding the natural frequencies for the coupled oscillators in Figure 4.10. The equation of motion in this case is y ”+ Ky = 0 , where y = ( 1 ( ), 2 ( ), 3 ( )) T and In the...

This exercise involves finding the natural frequencies for the coupled oscillators in Figure 4.10. The equation of motion in this case is
y”+Ky
=
0, where
y
= (
₁(),

₂(),

₃())
^T

and

In the above matrix,


_ij

is the spring constant for the spring connecting the
th andth oscillator, and it is positive (these are the three smaller springs shown in Figure 4.10). Assuming
y
=
x


^Iωt
, where

I

=
, then the problem reduces to solving Kx
= λx, where λ =

².

(a) Show that
K
is positive definite.

(b) Using orthogonal iteration, compute the eigenvalues of
K
in the case of when


_ij

= 1/10. Also, state what stopping condition you used, and how many iteration steps were required.

(c) What is the eigenvalue for
K
when the oscillators are uncoupled? Note that this means that the


_ij

’s are zero.

(d) Using orthogonal iteration, compute the eigenvalues of
K
in the case of when


_ij

= 1/1000. You should use the same stopping condition as in part (b). How many iteration steps were required, and if significantly different than the number for part (b), explain why.