The frequency of forced oscillations

[TheLil'Turkey]

So the frequency of an oscillator is always the same as the frequency of the force, if that force is a sinusoidal function of time. What's the best way to visualize why this is so? And also, why is the frequency of the oscillator in phase with the force if the force is below the resonance frequency of the oscillator, but 180 degrees out of phase with it if the frequency of the force is above the resonance frequency?

If the force has a constant frequency, but is not sinusoidal, does the oscillator also end up with that same frequency?

 

[sophiecentaur]

The (ideal) oscillator will only resonate at the fundamental of the impressed force. It is a filter so the resultant oscillation will be a band-passed version with lots of phase shift so the original waveform will be distorted in amplitude and phase. But no frequencies can change.

 

[Bobbywhy]

One excellent way to visualize the addition of forces to an oscillator: put your girlfriend on a playground swing. Push her once, wait for her to swing forward, then backward. When she just stops her backswing, apply a small push. You have just added energy in phase with the SHM oscillator. Now, imagine pushing her in the same direction, but when she is at the highest point of her forward swing. What happens then?

 

[TheLil'Turkey]

The (ideal) oscillator will only resonate at the fundamental of the impressed force. It is a filter so the resultant oscillation will be a band-passed version with lots of phase shift so the original waveform will be distorted in amplitude and phase. But no frequencies can change.

I didn't understand all that, but I'd like to know if the applied sinusoidal force is in phase with the natural restoring force of the oscillator if the frequency of the applied force is the resonance frequency.

One excellent way to visualize the addition of forces to an oscillator: put your girlfriend on a playground swing. Push her once, wait for her to swing forward, then backward. When she just stops her backswing, apply a small push. You have just added energy in phase with the SHM oscillator. Now, imagine pushing her in the same direction, but when she is at the highest point of her forward swing. What happens then?

I think that what would happen is her displacement would slowly increase with every push until the angle becomes so great that her frequency becomes equal to the frequency of my pushes. Her lowest position during these oscillations would be higher than the lowest possible position of the swing. Is that right?