Mathematics for the Physical Sciences by Copley Leslie

Author:Copley, Leslie [Copley, Leslie]
Language: eng
Format: epub
Publisher: De Gruyter
Published: 2015-03-29T16:00:00+00:00

Assuming further that the series for x (t ) can be differentiated term by term the necessary number of times, we can substitute these two series plus

into the differential equation. Then, invoking orthogonality, we can equate the coefficients with the same exponential e inωt on both sides. The result is

or,

where = k /m is the natural frequency of the oscillator and λ = r /2m is the system’s damping factor. It only remains to determine the Fourier coefficients for f (t ) by applying

Thus, we obtain the steady state solution as a superposition of sinusoidal functions with frequencies that are integral multiples of 2π /T , T being the period of the driving force.

If the frequency of one of these functions is close to the natural frequency ω 0 of the system, a resonance effect occurs because of the near cancellation in the denominator of c n ; that function then becomes the dominant part of the system’s response to the imposed force. To offer a concrete illustration of this, let m = 1 (kg), r = 0.02 (kg/sec), and k = 25 (kg /sec 2 ), so that the equation of motion becomes

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