1. Constrained Dynamics

2. Spring 20062 A Bead on a Wire x f

3. Spring 20063 A Bead on a Wire x f Zero rate-of- change ensures constant C!

4. Spring 20064 A Bead on a Wire x f

5. Spring 20065 A Bead on a Wire x f

6. Spring 20066 A Bead on a Wire x f However, one equation cannot solve two unknowns of f- hat!?

7. Spring 20067 A Bead on a Wire x f Constraint force does not produce work

8. Spring 20068 Constrained Dynamics General Recipe

9. Spring 20069 Constrained Dynamics

10. Spring 200610 About Constraint Force … Q q Constraint force Q-hat induced by Q to move q along legal velocity q-dot

11. Spring 200611 About Constraint Force … Q q 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. Spring 200612 Numerical Concerns To account for numerical drift, we often apply numerical damping as follows:

13. Spring 200613 Ex: Bead in Recipe x Q

14. Spring 200614 Example: Double Pendulum l1l1 l2l2 p1p1 p2p2