MEAM 535 Advanced Dynamics

Fall 1999

Summary of Week #4

1. Vectorial mechanics vs. variational mechanics

2. Newtonian (vectorial) mechanics vs. analytical (variational) mechanics

3. The concept of degrees of freedom (dof)
   "Minimum number of (independent) scalar quantities needed to completely describe the configuration of the system".

4. Configuration space

5. Generalized coordinates

6. Grubler's formula
   Jointed rigid-body systems = mechanisms
   Derivation of Grubler's formula for dof of mechanisms
   Some examples

7. Constraints
   
   • Holonomic (integrable) constraints
     holo = whole, integral
   • Scleronomic (time-independent)
     scleros = hard, inflexible, independent
   • Rheonomic (time-dependent)
     rheo = flexible, changing
   • nonholonomic (non-integrable
   • Pfaffian differentiable form
   • Unilateral vs. bi-lateral constraints
     Inequality and equality constraints

8. Implication of non-holonomic constrains
   Accessibility is not limited.
   Velocity degrees of freedom is reduced.

9. Conditions for integrability and exact differential of the Pfaffian form
   • Conditions for three dimensions
   • Relationship to Stokes theorem
   • Frobenius theorem for conditions in multiple dimensions

10. Example of a rheonomic, holonomic system

11. Examples of nonholonomic systems

12. Constraint forces

13. Workless constrains
    Scleronomic, holonomic constraints are workless constraints.

14. Virtual displacements
    Infinitesimal
    Not real
    Obey constraints
    Slightly perturbed imaginary configuration of the systems that obeys constraints