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Constrained Dynamics

Constrained Dynamics. f. x. A Bead on a Wire. f. x. A Bead on a Wire. Zero rate-of-change ensures constant C!. f. x. A Bead on a Wire. f. x. A Bead on a Wire. f. x. A Bead on a Wire. However, one equation cannot solve two unknowns of f-hat!?. f. A Bead on a Wire. x.

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Constrained Dynamics

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  1. Constrained Dynamics

  2. f x A Bead on a Wire

  3. f x A Bead on a Wire Zero rate-of-change ensures constant C!

  4. f x A Bead on a Wire

  5. f x A Bead on a Wire

  6. f x A Bead on a Wire However, one equation cannot solve two unknowns of f-hat!?

  7. f A Bead on a Wire x Constraint force does not produce work

  8. Constrained Dynamics General Recipe

  9. Constrained Dynamics

  10. Q q About Constraint Force… Constraint force Q-hat induced by Q to move q along legal velocity q-dot

  11. Q q About Constraint Force… Legal velocity q-dot is  to the hyperplane spanned by the gradients of all constraints. Hence, constraint force is a linear combination of all gradients.

  12. Numerical Concerns • To account for numerical drift, we often apply numerical damping as follows:

  13. Q x Ex: Bead in Recipe

  14. Example: Double Pendulum l1 p1 l2 p2

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