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I am trying to get my head around the behaviour and the effectivity of the the filaments movement of a toothbrush inside your mouth. I want to know the mechanical formulas and/or properties that I need to consider in order to understand how the filament moves and bends.

The ultimate goal is to know how much it will move and bend over the time and the relation of the parameters.

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    $\begingroup$ This is going to involve mostly buckling deformation - to begin, you should modify your free body diagram to have a spring reaction moment at the base. Page 9 of web.mit.edu/16.unified/www/SPRING/materials/Lectures/… should get you close to seeing the setup I'm describing. $\endgroup$ – Mark Apr 11 '17 at 16:17
  • $\begingroup$ i don't see a buckling problem. You don't push straight down.. I do see however the interaction/friction among the filaments has to be taken into account. $\endgroup$ – agentp Oct 10 '17 at 16:44
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The best approach to model this, since we will pass well beyond small-deflection beam theory, is to use the elastica. When you plug in the various sloped constraints of the tooth, and compare to the "soft" moment and force resistance of the individual filament, the elastica will yield the most accurate results. Having said that, it's a heavily numerical method model, and subsequently has minimal actual useful results. If you're attempting to define measurable quantities (such as how hard should you push the brush on your mouth?), the use of standard buckling theory will be the best model, but realize that all of your deflections will be invalid after the filament buckles (which will be a very small force).

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