My Math Forum stability and unstability

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August 23rd, 2012, 02:53 PM   #1
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stability and unstability

Hello All
I wondering if any one can write the equation that proof the stability and instability of ball on concave and convex surface as shown in the figure below
this figure is always used to show the stability and instability . In the first case the ball is in neutral condition , and in the second case the ball is in
stable condition as if it moved from the equilibrium position it will back again to the first position (equilibrium position) or at least oscillate around it if there is no friction.
and in the third case the ball is unstable because any small perturbation will make the ball move a way from it's equilibrium position.
can any body proof that or have any reference about that, I think that one first has to write the differential equation that describe this (i think that the equation that describe that ball may be like that describe the pendulum ) and after that using any test for instability by adding small perturbation on the initial condition or using Lyapunov method.
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 image002.gif (2.0 KB, 427 views)

 August 24th, 2012, 12:47 AM #2 Senior Member   Joined: Aug 2012 Posts: 229 Thanks: 3 Re: stability and unstability Hey kernel. Can you use some kind of linearization of the system to show that once it goes above a certain point, it will "blow up" so to speak? I'm guessing that if you can show that the derivative will tend to stay the same magnitude (not zero) then you have shown that ball will always stray from its equilibrium position since that suggests that is moving monotonically away from the equilibrium position.
August 26th, 2012, 05:31 PM   #3
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Re: stability and unstability

Hello
Thanks for your replay , I think that if I write the equation of motion for the ball in the stable condition
i will get solution that is sinusoidal(like pendulum) but this is not indication for stability , have i to assume that there is
a small perturbation . I am not sure.
Attached Files
 pg_0002.pdf (67.9 KB, 9 views)

 August 26th, 2012, 05:37 PM #4 Senior Member   Joined: Aug 2012 Posts: 229 Thanks: 3 Re: stability and unstability Ohh sorry I thought you were talking about when an object was on a semi-circle but the other way around to what the PDF has posted, not the situation that is in your PDF.
 August 26th, 2012, 05:45 PM #5 Newbie   Joined: Aug 2012 Posts: 3 Thanks: 0 Re: stability and unstability Hello what in the pdf is the same as the first post , what in the pdf is just more illustration for the equation of motion . I am talking about ball movement on semi circle.

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