Forced Oscillation: Graph Peaks to Infinity Explained

[aaaa202]

So you've probably seen the graph for a forced oscillation that acts such that the frequency of the applied force almost equals the natural frequency of the harmonic oscillator. That graph peaks towards infinity. However I don't get why that is. Wouldnt it just peak towards the amplitude of the forced oscillation and the natural oscillations added?

 

[Philip Wood]

Your talk of infinity suggests that you're considering an undamped harmonic oscillator. You can get only limited insight from this model. A far more realistic case is a damped harmonic oscillator, with damping due to a resistive force F given by F = -kv.

In that case the oscillations at the natural frequency, which you do indeed get when you first apply the periodic 'driving force', die out (due to the damping!) leaving you with just those due to the driving force. These oscillations take place at the frequency of the driving force.

 

[Ken G]

To add to that correct and insightful answer, I think there may be one additional misconception that is showing up in the original question-- the "driving" is not an amplitude driving, it is a force driving. By that I mean, there is not some external mechanism that is trying to "drive" the oscillation to some given amplitude (like a hand grabbing it and shaking it a certain distance), it is some external force law. The force law has no idea what the amplitude of the oscillation will be, that depends on the system it is acting on (including things like mass and whatever damping there might be). And under some conditions (resonant driving, no damping), that can imply an infinite amplitude (which means the steady-state oscillation is never actually reached without including damping or something else that is being idealized).