Resonance and natural oscillations

[Cheman]

Why does everything have a natural frequency at which it oscillates when struck by a single force and then left to oscillate? Does everything only have one? Does it vary depending on what the original force to cause it to oscillate was?

Also, why does force imposed at natural frequency casue the amplitude to build up? (ie - cause resonance?)

Thanks in advance.

 

[quasar987]

Why does everything have a natural frequency at which it oscillates when struck by a single force and then left to oscillate?

Not everything has a natural frequency. For an object to have a natural frequency of oscillation, it has to be submitted to a very special kind of force (and I'm not talking about the force that "struck" the object before leaving it to oscillate, I'm talking about the force of the spring or the analogous). It has to be submited to a force that has a stable equilibrium position, i.e. there must be a position in space at which the object experiences no force and also such that when the object is displaced to the left, the force pushes it to the right, and when displaced to the right, the force pushes it to the left. For any such force applied to an object, if we consider small enough displacements of the object, the movement will be to a good approximation what we call simple harmonic, i.e. described to a good degree of accuracy by

$$x(t) = Acos(\omega t)$$

where A is the amplitude of oscillation and $\omega$ is the angular frequency of oscillation, or what we call the natual frequency.

Does everything only have one?

The natural frequency of an object does not depend on the object itself as much as it depends on the force acting on it. It is possible that an object be submited to a force that allows multiple stable equilibrium points. In this case, the object has a natural frequency that depends "around" which equilibrium point it is oscillating.

Does it vary depending on what the original force to cause it to oscillate was?

No.

Also, why does force imposed at natural frequency casue the amplitude to build up? (ie - cause resonance?)

I don't know of a simple answer to that.

 

[krab]

Why does everything have a natural frequency at which it oscillates when struck by a single force and then left to oscillate?

It has to do with the fact that nothing is truly rigid. Objects are held together by forces. These, together with inertia are sufficient to cause a natural frequency.

Does everything only have one?

No. The simplest example is a guitar string. It has many resonances. The special thing about them though is that all its natural frequencies are separated by musical intervals. For an arbitrary object, the natural frequencies are not related in this way.

Does it vary depending on what the original force to cause it to oscillate was?

All such oscillations eventually become nonlinear at sufficiently large amplitude. That means eventually it will get out of step with a constant frequency driver.

Also, why does force imposed at natural frequency casue the amplitude to build up? (ie - cause resonance?)

It's like asking why does an object continue to speed up with a constant applied force. But in the oscillating case, you only apply the force every time it is moving in the right direction, so the motion gets faster and faster in the same way.

 

[Cheman]

If you had a swing or something on a swing force example, if your initial force to set it into vibration was, say, 7N then it would oscillate at its natural frequency. (as long as the force was only exerted once.) Would it also oscillate at natural frequency is this force was 4N, or 10N or 25N?

Thanks in advance.

 

[quasar987]

I can only speak about the case where the friction and the amplitude of oscillation are small, but is such condition, yes it will oscillate at the same frequency no matter the force you use to set it in motion.

Normally, in textbooks, this fact is embodied in the sentence "The frequency of oscillation of an harmonic oscillator is independant of the amplitude of oscillation.".

 

[Claude Bile]

Also, why does force imposed at natural frequency cause the amplitude to build up? (ie - cause resonance?)

At resonance, the restoring force is in phase with the driving force. As one moves away from the resonance frequency (commonly called detuning), the amplitude begins to drop as the restoring force and driving force begin to move out of phase.

Claude.