Resonance Phenomena by Solution of the ODE

The "periodic forcing function" can be any periodic function (i.e., a fourier series), but it is a bit simpler to analyze the effect of each mode separately.  Below, the forcing function will be assumed to be "index_17.gif"cos("index_18.gif"t)
Solve problems in terms of the mass and natural frequency--eliminate the spring constant in equations by defining it in terms of the mass and natural frequency.

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Mathematica can solve the nonhomogeneous ODE with a  forcing function at with an applied frequency:

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Consider the form of the above solution: The homogeneous part of the solution  (i..e, the part that depends on constants C[1] and C[2]) depends only on  the characteristic frequency, while the heterogeneous part includes the forcing and the characteristic frequency.   From a first glance at the denominator in the heterogeneous term, it appears the amplitudes are  maximized as ωapp → ωchar; and , it appears that,  as long as the viscous term η is non-zero, the solution is bounded. However, this is not the case as will be shown below.

Consider the behavior of the general solution at time t=0.  This will show that the homogeneous parts of the solution are needed to satisfy boundary conditions, even if the oscillator is initially at rest at zero displacement (i.e., y(0) = "index_23.gif"

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Therefore,  the C[1] and C[2] in the general solution, must pick up terms that depend on  ωapp and these must be included in considerations of the limit ωapp → ωchar.

Consider the particular case of an oscillator at rest at zero displacement:

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The resonant solution is the case: ωapp → ωchar

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To analyze the resonant behavior when the viscosity η is small, the solution is expanded about small viscosity to second order:
(Below, the function Map is used with a pure function to simplify each of the terms resulting from a series expansion (n.b. Normal converts the series form into a normal form).   Each term in the expansion is operated on in the following manner:
1) Convert any Exp[] into trignometric functions with ExpToTrig
2) Remove  fractional powers of containing powers of real numbers with PowerExpand
3) Simplify
by using a pure function as the first argument to Map,

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Of course, this leading behavior could have been obtained directly, viz

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but, from the direct solution, one cannot obtain the behavior as the viscosity becomes small

Considering the appearance of the resonant solution, there may be another special case when η is near 2 M ωchar.

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However, it appears that the only special feature is that, after very long times, the solution is a quarter-period out of phase.

Spikey Created with Wolfram Mathematica 6