Harmonic Oscillators with Maple
Copyright © 2002, 2003 by James F. Hurley, University of Connecticut Department of Mathematics, 196 Auditorium Road, Unit 3009, Storrs CT 06269-3009. All rights reserved.
1. Damped Oscillators.
Maple's plotting commands easily generate plots of solution functions for a damped harmonic oscillator with equation
The following routine plots an overdamped case, for which recall that
. In this plot, the graph is that of the solution function
Compare Figure 3.4.2 of Florin Diacu, Introduction to Differential Equations: Order and Chaos , Freeman, 2000.
> plot(0.2*exp(-t/2) + exp(-t/3), t = 0..6);
Next, let's plot a typical solution in the underdamped case, that is, one for which
Compare the plot with Figure 3.4.4 of Diacu .
> plot( exp(-t/8)*(sin(2*t) + cos(2*t)), t = 0..10);
Finally, it is just as easy to generate a plot of a solution in the critically damped case, for which . Compare the output of the following to Figure 3.4.3 of Section 3.4 of Diacu .
plot( (0.1 + t)*exp(-t), t = 0..6) ;
2. Forced Undamped Oscillations. The model for such a system is Equation (1) on p.113 of Diacu :
The derivation on pp. 113 and 114 of Section 3.4 shows that this problem has solution of the form x ( t ) =
A ( t ) sin , where
A ( t ) = .
The next Maple routine plots that solution function for the equation x ' ' + 4.81 x = 0.81 cos(2 t ), with initial conditions x (0) = x '(0) = 0 from Example 1 of Section 3.4 of Diacu . The plot illustrates the phenomenon of beats .
> plot (2*sin(t/10)*sin(21*t/10), t = 0..65) ;
The final routine graphs the solution function for a resonant system, namely the one from Example 2 of Section 3.4 of Diacu: x ' ' + 4.81 x = 0.81 cos(2.2 t ), with the same initial conditions as before: x '(0) = x (0) = 0. Note the steadily increasing amplitude of the oscillation.
> plot((0.81*t/4.4)*sin(2.2*t), t = 0..100);
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