lecture_25 - MIT OpenCourseWare http://ocw.mit.edu 2.004...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . g r a v i t a t i o n a l f o r c e a c t s t o m o v e a d i s p l a c e d b a l l a w a y f r o m i t s e q u i l i b r i u m p o s i t i o n ( b ) a n u n s t a b l e s y s t e m ( a ) a s t a b l e s y s t e m ( c ) a n e u t r a l l y s t a b l e s y s t e m n o f o r c e s a c t t o m o v e a d i s p l a c e d b a l l g r a v i t a t i o n a l f o r c e a c t s t o r e s t o r e a d i s p l a c e d b a l l t o i t s e q u i l i b r i u m p o s i t i o n Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 25 1 Reading: Nise: Chap. 6 1 System Stability The figure above illustrates three stability conditions using a rolling ball on an undulating surface as a graphic example. Assume that the horizontal position and velocity of the ball are a pair of state variables describing this system. In a concave region of the surface, as shown in in (a) the base of the hollow is an equilibrium point. If the ball is displaced a small distance from this position and released it oscillates but ultimately returns to its rest position at the base as it loses energy due to frictional losses; this is therefore a stable equilibrium point. (Without energy dissipation the ball would roll back and forth forever and exhibit neutral, or marginal, stability.) On a convex portion of the surface, as shown in (b), the ball is in equilibrium if placed exactly at the top of the surface, but if it is displaced an infinitesimal distance to either side the net gravitational force acting on the ball causes it to roll down the surface and never to return to the equilibrium point. This equilibrium point is therefore unstable. If the ball is displaced along the at portion of the surface, as shown in (c) it neither moves away nor returns; the at portion represents a neutrally stable equilibrium...
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lecture_25 - MIT OpenCourseWare http://ocw.mit.edu 2.004...

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