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I have a system, where there are ball bearings on the pistons that clamp the metal plate with special dents for ball bearings. The system should be precise, because it is used for microscopy. It also should be as noiseless as possible. It also should be fast, so the impact at high velocity is inevitable.

I've thought of introducing some resin coating, but it will reduce the rigidity. Are there any solutions for this problem? Is there any strong relationship between sound and rigidity? I believe that there may be some rigid materials that somehow don't favor phonons.

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I thought of teflon coating. Have a look at this extensive list dynamiccoatingsinc.com/problemSolving.htm –  anna v Dec 15 '12 at 19:17
    
It's a helpful link. Still looks like most of the materials are soft –  neuronich Dec 15 '12 at 19:51
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Unfortunately, anything that reduces noise without slowing the system down is going to do it by sacrificing rigidity.

If it is within your engineering parameters, you might introduce a damper on the pistons that engages within a millimeter or so of the ultimate extension of the piston. By slowing the piston down before impact, you can reduce the noise, without compromising the rigidity of the surface. It would have to be just before impact, slowing over a very short distance, in order to keep your system from being too slow. (I'm thinking of something akin to the dampers that are used on kitchen drawers, that slow the drawer down just before the drawer slams shut.)

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That's a cool idea. Still it an interesting physical question, how to derive this relationship between soundness and rigidity. –  neuronich Dec 22 '12 at 2:06
    
If you're looking for more detail, perhaps this site will prove informative: kemt.fei.tuke.sk/Predmety/KEMT320_EA/_web/… Of course, your piston collision will produce a broad spectrum of frequencies, so you will always have to deal with some resonance. –  KDN Dec 22 '12 at 14:03
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The sounds waves will emanate at the point of impact and then echo off of various boundaries. You will want to engineer those boundaries to minimize or more likely direct the reflections and lead the energy to where it can be absorbed and converted to heat. I think the simplest way to do that is to make what the ball bearings collide with simply be as massive as possible. But it's not clear from your description what the geometric constraints are.

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I'm sorry for not mentioning all the constraints. Basically I can't change anything dramatically. I can't change size substantially. Ball bearings are quarter of inch in diameter now. I can potentially decrease their size but not increase. I can change the form of the dents. (They are conical right now. So sphere enters the cone) –  neuronich Dec 15 '12 at 19:45
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