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m,L

ME6441, Dr. Ferri. Example, Constrained Dynamics. Slender rod of mass m and length L slides without friction on inclined plane. B. m,L. C. A. q. b. b. Free-Body Diagram. q. C. b-q. N B. mg. N A. Force/Moment Balance. S F x. +. S F y. +. S M c. +. Kinematics. Combine.

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m,L

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  1. ME6441, Dr. Ferri Example, Constrained Dynamics Slender rod of mass m and length L slides without friction on inclined plane B m,L C A q b b Free-Body Diagram q C b-q NB mg NA

  2. Force/Moment Balance SFx + SFy + SMc + Kinematics

  3. Combine Differentiate or and

  4. Finally, get differential equation for q

  5. Lagrange’s Equation Approach: Recall

  6. Potential energy Virtual Work Calculate necessary derivatives

  7. Lagrange’s Equation: where

  8. B Alternate Formulation C NB q A yc NA b O xc

  9. Virtual displacements Virtual work

  10. Lagrange’s equations j=1,2,3 xc yc q

  11. B Alternate Formulation C q yc A b O xc In this formulation, we assume that the rod is free to move in a 2-D plane. Other than (conservative) gravity forces, no other forces are modeled.

  12. Constraint at A C A O

  13. Constraint at B B C O configuration constraint

  14. Constraint summary Constraint at A Thus, Constraint at B Thus,

  15. Lagrange’s Equations j=1,2,3 xc yc q Comparing with Newton-Euler equations

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