How Does Damping Affect the Resonant Amplitude of a Driven Pendulum?

[stanli121]

Homework Statement

Given a simple pendulum with a mass on the end and a massless string. The support point for the pendulum is moved laterally with an amplitude D at the resonant frequency. The damping is from the air and is considered viscous i.e. not turbulent. The difference between the resonant frequency and the frequency in the presence of damping is negligible. What is the amplitude of the oscillation in the steady state.

Homework Equations

The Attempt at a Solution

I am trying to consider this conceptually. Because the damping is light, can I assume that the damping coefficient -> 0 and that the damping term in general cancels out? In that event, would the steady state amplitude end up being the same as the drive amplitude, D? I'm getting hung up trying to understand how the damping affects this situation because I'm told the resonant frequency is the same as the frequency in the presence of damping. Any thanks is much appreciated.

 

[JesseC]

Without damping oscillations would grow infinitely big... for a steady state to be achieved there must be some damping. The driving force is putting energy into the system, and if its not getting lost anywhere then what would happen?

 

[stanli121]

That's one of the dilemmas I encountered -- what happens to a driven, undamped oscillator. The amplitude can't grow larger than the length of the string so the other possibility seems to be a period that tends to infinity. I can't seem to grasp the fundamentals of this situation.

 

[JesseC]

To arrive at a quantitative answer the problem would require some values for the pendulum parameters and an equation relating them to the amplitude of oscillation. Such an equation does exist! I can't for the life of me remember it off by heart, but a quick search on wikipedia about damped driven harmonic oscillators might help you out.

Another problem I can see is that at large amplitude oscillations the small angle approximation will break down and the problem will become non-linear!