Background. We wish to study mechanical vibrations where the underlying D. E. is
m x'' + c x' + k x = f(t)

Computer Lab Work.

Exercise 1. Investigate the beat frequency of an undamped forced oscillation.
Consider the D. E. x''[t] + 100x[t] = 21 Cos[w t]with the I. C. x[0] = 0, x'[0] = 0.
Solve and plot the solution for .

The trigonometric identity 2 sin a sin b = cos(a-b) - cos(a+b) is used
to convert this solution to the form involving a product of sine functions.

Solve and plot the solution for w = 12

Solve and plot the solution for w = 11

Solve and plot the solution for w = 21/2

Solve and plot the solution for w = 41/4

Plot the maximum amplitude |A[w]| for 9 < w < 11.

Find the limit of the "slowly varying amplitude" function ,

use it to construct the limit of g[t,w].

Plot x[t,14], x[t,13], x[t,12], x[t,11], x[t,10.5], and f[t] on the same graph,
and observe that the functions are converging to f[t].

Exercise 2. Investigate the steady periodic solution of the D. E.
x''[t] + 4 x'[t] + 5x[t] = 40 Cos[3t].

Plot the steady state portion of the solution - Cos[3 t] + 3 Sin[3 t].

Solutions with the initial condition x'[0] = 0, x[0] = c for
c = -10, -9,..., 9, 10 are easily generated and plotted.
Display the various solutions and the steady state solution on the same graph.
Observe that they approach the steady periodic solution as increases.

Exercise 3. Investigate resonance of a damped forced oscillation.
Consider the D. E. m x''[t] + c x'[t] + k x[t] = F0 Cos[w t]

The transient portion of the solution is

and is not influenced by the choice of w.

The steady periodic function is influenced by the choice of w.
Find and plot the steady state function for w = 1, 1.5, 2, 2.5, and 2.95

Carry out the following investigation.

Find the maximum value of the amplitude P[w] for 0 < w < 6

An investigation of this graph with your cursor will reveal that the
maximum occurs somewhere near 2.95 ( actually at 2.95804 ).
Plot the solutions for w = 1, 1.5, 2, 2.5, and 2.95.

Solutions.

Return to the Differential Equations Project

Return to the Numerical Analysis Project

Return to the Complex Analysis Project

(c) John H. Mathews, 1998