If I have an over-damped mechanical system that is excited with a sinusoidal motion. That sinusoidal motion starts with a determined frequency then increases frequency over time. Of course, it is known that there will be a phase shift between the driving force and the motion of the hanging mass.
My question is, how to figure out phase lag of mass motion in relation to driving force?
A damped harmonic oscillator with a sinusoidal driving force is represented by the equation
where (b
is the damping coefficient,
) and
is the resonant frequency of the oscillator. The particular
solution to this equation can be determined by taking the imaginary
part of the solution to
If you assume* the solution takes the form
and plug that in, you get
and
Solving for the phase difference gives
This depends on the frequency of the driving force and the resonant frequency of the oscillator, but not on the amplitude of the driving force.
You can express this in terms of the dimensionless variable
as
and if you graph it,
(graph generated by Wolfram Alpha) you'll see how the response
of the oscillator jumps from leading to lagging when
(at
), that is, when the
driving and resonant frequencies are equal.
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