Driven Harmonic Oscillator where Mass Hits Ground

[Nefertiti]

I started to ponder following problem. I have a driven, damped oscillator where the mass is free to vibrate in y-direction. If I put a wall or a ground near the mass, the mass touches it if the drive amplitude is larger than the distance to the ground. How does this change the normal dynamics. I would expect that the oscillation becomes pretty complex depending on forcing amplitude and spring parameters. Is it possible to derive the amplitude equation for given frequency and spring parameters?

 

[.Scott]

It would depend on how elastic the collisions were.

Assuming massless springs, rigid weights, etc:
If the collisions were completely inelastic, the results would be fairly simple because the new velocity (zero) would not depend on the incident velocity.
Generally, on collision the velocity of the oscillating weight would change by a factor of "a" in the y direction where 0>=a>=-1.

 

[Nefertiti]

Assuming inelastic collision, would it be possible to analytically derive the amount of energy transferred to the ground? Especially considering the spring parameters and the driving frequency.

 

[.Scott]

If you know the frequency, it should be easy to model the system.
You really haven't described the whole system. Are you pumping energy into this system? What is causing the "damping", friction in the spring?, a function of the velocity?
If it's inelastic and you are not going to add energy to the system, it will hit the floor once and then never again.

 

[Nefertiti]

I'm considering following scenario. The "ground" to which the spring is attached, is driven with n=C*Cos(wt). The natural frequency for this setup can be solved when there is no impact with the wall. Now I'm considering the case where there is an impact with the spring-mass and the wall. The impact should be inelastic but as the "ground" is driven, there is energy so that movement doesn't die out. The question is: is the amplitude-frequency plot altered by these non-elastic impacts?

 

[.Scott]

The natural frequency for this setup can be solved when there is no impact with the wall. Now I'm considering the case where there is an impact with the spring-mass and the wall. The impact should be inelastic but as the "ground" is driven, there is energy so that movement doesn't die out. The question is: is the amplitude-frequency plot altered by these non-elastic impacts?

If you look at the weight cycling at it's natural frequency, it will follow the sine function. If we consider the point when it is closest to the wall to be the 0-degree and 360-degree position, then what happens when it hits the wall will be that it will suddenly skip from a position such as 350 degrees to the 0 degree position.
Thus the new "natural frequency" will be higher and will be dependent on how much energy is added on each cycle.

If the w in the C cos(wt) is not changed, the collision will cause the weight to fall out of phase with N. So if the w is kept at the original natural frequency, a series of collisions (a series could be just one collision) will put the weight sufficiently out of phase that the weight will not efficiently increase the amplitude. Then the system will begin to recover - eventually bringing the weight into phase and increasing the amplitude thus setting it up for the next series of collisions.