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Chattering and grazing in impact oscillators Chris Budd

Chattering and grazing in impact oscillators Chris Budd. Look at exceptional types of dynamics in piecewise-smooth systems. Hybrid systems. Maps. Key idea … The functions or one of their nth derivatives, differ when. Discontinuity set.

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Chattering and grazing in impact oscillators Chris Budd

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  1. Chattering and grazing in impact oscillators Chris Budd

  2. Look at exceptional types of dynamics in piecewise-smooth systems Hybrid systems Maps

  3. Key idea … The functions or one of their nth derivatives, differ when Discontinuity set Interesting discontinuity induced bifurcations occur when limit sets of the flow/map intersect the discontinuity set

  4. Impact oscillators: a canonical hybrid system u(t) g(t) obstacle

  5. ‘Standard’ dynamics Periodic dynamics Chaotic dynamics v Experimental Analytic u v u

  6. Grazing dynamics Grazing occurs when (periodic) orbits intersect the obstacle tanjentially v

  7. Chattering occurs when aninfinitenumber of impacts occur in afinite time v u u u v

  8. Poincare Maps

  9. Complex domains of attraction of periodic orbits

  10. Observe grazing bifurcations identical to the dynamics of thetwo-dimensional square-root map Transition to a periodic orbit Non-impacting orbit Period-adding

  11. Local analysis of a Poincare map associated with a grazing periodic orbit shows that this map has a locally square-root form, hence the observed period-adding and similar behaviour Poincare map associated with a grazing periodic orbit of a piecewise-smooth flow typically is smoother (eg. Locally order 3/2 or higher) giving more regular behaviour

  12. CONCLUSIONS • Piecewise-smooth systems have interesting dynamics • Some (but not all) of this dynamics can be understood and analysed • Many applications and much still to be discovered

  13. Parameter range for simple periodic orbits Fractions 1/n Fractions (n-1)/n

  14. Why are we interested in them? • Lots of important physical systems are piecewise-smooth: bouncing balls, Newton’s cradle, friction, rattle, switching, control systems, DC-DC converters, gear boxes … • Piecewise-smooth systems have behaviour which is quite different from smooth systems and is induced by the discontinuity: period adding • Much of this behaviour can be analysed, and new forms of discontinuity induced bifurcations can be studied: border collisions, grazing bifurcations, corner collisions.

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