Question

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?

Answer 1

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.