Understanding Constrained Dynamics in Mechanical Systems: A Bead on a Wire Example
This article explores the principles of constrained dynamics by examining a bead on a wire. It highlights how the zero rate of change ensures constant constraints and discusses the limitations of solving equations with two unknowns. Additionally, it addresses the nature of constraint forces and their role in moving along legal velocities while ensuring work is not produced. The discussion incorporates numerical concerns related to damping to manage numerical drift. This serves as an essential guide for understanding complex constrained mechanical systems.
Understanding Constrained Dynamics in Mechanical Systems: A Bead on a Wire Example
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Presentation Transcript
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 Constraint force does not produce work
Constrained Dynamics General Recipe
Q q About Constraint Force… Constraint force Q-hat induced by Q to move q along legal velocity q-dot
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.
Numerical Concerns • To account for numerical drift, we often apply numerical damping as follows:
Q x Ex: Bead in Recipe
Example: Double Pendulum l1 p1 l2 p2
