The displacement resonance in forced oscillations
In every oscillating system there is dissipation of mechanical energy, which results that the motion of the mass-spring system described in the previous sections dies out. If the oscillation are to be maintained, energy must be supplied to the system. In this section we shall assume that the system is acted on by a periodic driving force. Suppose that the mass-spring system is subjected to a periodic force ![`*`(F[0], `*`(cos(`*`(omega, `*`(t)))))](images/paper_180.gif)
, where ![F[0]](images/paper_181.gif)
is the maximum value of the applied force and 
is its frequency. The equation of motion is then
![`+`(`*`(m, `*`(Student:-MultivariateCalculus:-diff(x(t), t, t))), `*`(r, `*`(Student:-MultivariateCalculus:-diff(x(t), t))), `*`(k, `*`(x(t)))) = `*`(F[0], `*`(cos(`*`(omega, `*`(t)))))](images/paper_183.gif)
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The transfer function of the mass-spring system is given by:
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It is easy to show that the steady-state frequency response to an input ![`*`(F[0], `*`(cos(`*`(omega, `*`(t)))))](images/paper_197.gif)
becomes
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![x(t) = `*`(F[0], `*`(abs(('T')(`*`(I, `*`(omega)))), `*`(cos(`+`(`*`(omega, `*`(t)), arg(('T')(`*`(I, `*`(omega)))))))))](images/paper_198.gif) |
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The steady-state system response is also a cosine having the same frequency 
as the input.
And the amplitude of this response is ![F[0]](images/paper_203.gif)
. The variation in both the magnitude 
and argument 
as the frequency 
of the input cosine is varied constitute the frequency response of the system, as the following example shows.
Let us first solve the differential equation using Maples dsolve.
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The displacement 
of a mass-spring system undergoing forced oscillations plotted against the time with 
The graph shows that the transient solution dies out as 
increases. The transfer function of the system is given by:
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With 
and 
, we get
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There is some frequency called the resonance frequency at which the amplitude 
becomes a maximum. This resonance frequency can be recognized in many vibrating systems unless the damping force 
is too large.
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With zero damping force 
the resonance frequency is 
The phase angle 
is given by
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Substituting the values 
and 
gives the steady-state response
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As 
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![plot([f(2, t), xs(t)], t = 30 .. 39, thickness = [1, 2], legend = [typeset(f(2, t) = input), typeset(xs(t) = '`*`(steady_state, `*`(response))')], labels = [t, x(t)], labelfont = [TIMES, BOLD, 12])](images/paper_251.gif)
![plot([f(2, t), xs(t)], t = 30 .. 39, thickness = [1, 2], legend = [typeset(f(2, t) = input), typeset(xs(t) = '`*`(steady_state, `*`(response))')], labels = [t, x(t)], labelfont = [TIMES, BOLD, 12])](images/paper_252.gif) |
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![text2 := proc (i) options operator, arrow; plots:-textplot([2, Tabs(i, 2), convert(r = i, string)], align = ABOVE, font = [TIMES, BOLD, 12]) end proc; -1](images/paper_257.gif) |
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![plots:-display(seq(plt2(i), i = [.1, .25, .5, .75, 1, 1.41]), labels = [omega, A], labelfont = [TIMES, BOLD, 12])](images/paper_258.gif) |
Phase shift between steady-state response and input displacement.
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![plots:-display(seq(plt2(`+`(`*`(0.5e-1, `*`(i)))), i = 1 .. 28), insequence = true, labels = [omega, A], labelfont = [TIMES, BOLD, 12])](images/paper_260.gif) |
Animation
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![text3 := proc (i) options operator, arrow; plots:-textplot([2.4, phi(i, 2.4), convert(r = i, string)], align = {ABOVE, RIGHT}, font = [TIMES, BOLD, 12]) end proc; -1](images/paper_263.gif) |
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![plots:-display(seq(plt3(i), i = [.1, .25, .5, .75, 1, 1.41]), labels = [omega, phi], labelfont = [TIMES, BOLD, 12])](images/paper_265.gif) |
Variation of the phase shift 
against the angular velocity 
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![plots:-display(seq(plt3(`+`(`*`(0.5e-1, `*`(i)))), i = 1 .. 28), labelfont = [TIMES, BOLD, 12], insequence = true)](images/paper_269.gif) |
Animation
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The displacement of the mass-spring system undergoing forced oscillations with damping constant 
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The displacement of the mass-spring system undergoing forced oscillations with damping constant 
![x(t) = `*`(F[0], `*`(abs(T(`*`(I, `*`(omega)))), `*`(cos(`+`(`*`(omega, `*`(t)), arg(T(`*`(I, `*`(omega)))))))))](images/paper_199.gif){kind=small}
![x(t) = `/`(`*`(F[0], `*`(cos(`+`(`*`(omega, `*`(t)), arg(`/`(1, `*`(`+`(`-`(`*`(m, `*`(`^`(omega, 2)))), `*`(I, `*`(r, `*`(omega))), k)))))))), `*`(abs(`+`(`-`(`*`(m, `*`(`^`(omega, 2)))), `*`(I, `*`(...](images/paper_201.gif)
