Consider a vibrating system described by the initial value problem u\'\' + 0:125

Question

Consider a vibrating system described by the initial value problem
u'' + 0:125u' + 4u = g(t); u(0) = 2; u'(0) = 0:
For each g(t) given below, we know that the solution will eventually behave like a pure oscillation,called the steady-state solution, since the homogeneous solution is a decaying oscillation. Estimate the amplitudes (rounded to one decimal place) of those steady-state oscillations by observing the graphs of the solutions as t gets larger. For which g(t) does the maximum amplitude occur and why?
(a) g(t) = 3 cos(0:5t)
(b) g(t) = 3 cos(1:5t)
(c) g(t) = 3 cos(2t)
(d) g(t) = 3 cos(2:5t)
(e) g(t) = 3 cos(3:5t)

I typed in the solutions in a program called mathematica put it is giving me the same graph for everyone and I was wondering if someone could help me out.

Explanation / Answer

Say g(t) = 3 cos(wt)
since we need steady state amplitudes, it is a particular solution and hence,
assume u(t)=uo*cos(wt), we need to get u0.
Substitute the assumed expression in the given differential equation and we get
uo as 3/((4-w2)+0.015625)).

Note that in obtaining the above value, we get a 90 degree phase difference of u''(or)u and u' and hence phasor sum has to calculated.

Now we can see as w increase 4-w2 decreases and overall value increases. Hence g(t) with highest w has the highest amplitude ie. (e) option. Note the difference is very less, hence may be you did not observe it. If there is a zoom function in your program try it to get the exact amplitude.

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