Vibration of a mass connected via preloaded spring

[sfensphan]

The setup: I have a mass (m1)connected to a much, much larger mass (m2) via a preloaded spring. They start out in contact because the preloaded spring holds them together. Now suppose the large mass is subject to vibrations, possibly at the resonance of the structure. Will the two masses separate? If they do, what is the maximum distance between the two?

Will the two masses separate?
My original chain of thought was that if the spring force is less than the inertial force of the small mass, then the two will separate.
F_spring = -k(Δx)
k = spring constant
Δx = displacement from spring resting state. Since there is a preload, this is non-zero

F_inertia = m1 * a
m1 = mass of the small mass
a = acceleration. This should be sinusoidal, since it's a vibration

I would then compare the two and see which one is larger. I would use the max value of F_inertia during this comparison.

If they do, what is the maximum distance between the two?
This is where I get a little lost. A static calculation would be to find where Δx is such that F_spring = F_inertia.
But what if the system vibrated at resonance frequency (which I believe would be sqrt(k/m1) for this case)?
How do I deal with the preload?
Is the Force body diagram such that the sinusoidal force acts directly on the m1?
Do I need to know how much the larger mass is displaced?

 

[drvrm]

I have a mass (m1)connected to a much, much larger mass (m2) via a preloaded spring. They start out in contact because the preloaded spring holds them together. Now suppose the large mass is subject to vibrations, possibly at the resonance of the structure. Will the two masses separate? If they do, what is the maximum distance between the two?

just as a help to your approach one can see the given reference-

actually ...In engineering practice, we are almost invariably interested in predicting the response of a structure or mechanical system to external forcing.

For example, we may need to predict the response of a bridge or tall building to wind loading, earthquakes, or ground vibrations due to traffic. Another typical problem you are likely to encounter is to isolate a sensitive system from vibrations. For example, the suspension of your car is designed to isolate a sensitive system (you) from bumps in the road.

Electron microscopes are another example of sensitive instruments that must be isolated from vibrations. Electron microscopes are designed to resolve features a few nanometers in size. If the specimen vibrates with amplitude of only a few nanometers, it will be impossible to see! Great care is taken to isolate this kind of instrument from vibrations. That is one reason they are almost always in the basement of a building: the basement vibrates much less than the floors above.

reference-http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_forced/vibrations_forced.htm

 

[sfensphan]

Thanks very much. From your link, I see that it is a base excitation case. Greatly appreciate you leading me in the right direction.