Forced vibration with decreasing force amplitude

[abrandt]

I'm working on a problem that has forced vibration. The force, every time it is applied, is less than the previous impact. For clarity, the problem is dealing with a mining skip that is emptying. The inside of the skip is separated in sections using ribs but the top blocks still exert a force on the lower blocks. The impact of the material hitting the bottom of the skip (a 50 degree angle) is causing the skip to vibrate. This impact happens at distinct intervals with the time between impacts remaining roughly constant. The impact force is decreasing due to the decreasing total mass of the material inside the skip. I was wondering if you could point me in the right direction. Most of the problems i have dealt with deal specifically with a periodic force that is constant each time it is applied. I thought about adding a dampening force to diminish the force each time it is applied but wouldn't that just act to dampen the free vibration component? For background i am an engineering student who is working on a project with an external company.

 

[AlephZero]

You are right that the usual type of "forced response" analysis (modeled in the frequency domain) is about constant-amplitude response to a steady force.

Your situation seems more like a general transient dynamics problem. You could find the response of the system to an impulsive load (i.e. one impact). The response will be the sum of exponentially decaying vibrations in each of the structural modes that are excited. It's not obvious from your description whether the response would be mainly one vibration mode, or a combination of several modes.

If the system is linear, you can then find the response to a series of impulses of different sizes by linear superposition. That would be straightforward to do numerically, even with a spreadsheet if you don't want to do any programming.

You may want to do some sort of "worst case" analysis here, varying the times of the impulses in your model, since I'm guessing that what happens in real life is partly random.