This exercise involves finding the solution of the problem for the coupled oscillators shown in Figure 7.17. The equation of motion in this case is
y”+
Ky
=
0, where
In the above matrix,
ij
is the spring constant for the spring connecting the
th and
th oscillator, and it is positive (these are the three smaller springs shown in Figure 7.17). The initial conditions are
(0) = (−1, 0, 2)
T
and
’(0) = (0, 0, 0)
T
.
(a) Assuming
introduce the velocity
i
(
) =
’
i
(). Letting
show that the oscillator problem can be rewritten as
z’ =
Az. What is
z(0)?
(b) If Euler’s method is used to solve the equation in part (a), what is the resulting finite difference equation? Your answer should be written in terms of
z
i
and
z
i
+1.
(c) If the trapezoidal method is used to solve the equation in part (a), what is the resulting finite difference equation? Your answer should be written in terms of
z
i
and
z
i
+1.
(d) If the RK4 method is used to solve the equation in part (a), what is the resulting finite difference equation? Your answer should be written in terms of
z
i
and
z
i
+1.
(e) Using one of the methods from (b)–(d), on the same axis, plot
1,
2,
3
for 0 ≤
≤ 10. Assume that
12
=
13
=
23
= 1/2. Make sure to state what method you used to solve the problem and why you picked that method. Also, explain why you believe your solutions are accurate approximations of the exact solutions.