Dynamics of Constrained Mechanical Systems

Question 1

What are the two basic strategies that can be used to solve the MSD problem?

Answer 1

1) The problem is reduced to an equivalent set of Ordinary Differential Equations (ODEs), by eliminating the contraints and reducing the coordinates to the minimal set of truly independent ones (Minimal Coordinate Set, MCS).

2) The problem is directly solved in DAE form (Redundant Coordinate Set, RCS)

3) A third case, that of the Unconstrained Coordinate Set (UCS) is possible

Question 2

Describe the MCS.

Answer 2

Reduction to ODEs may not be straightforward and, except for special cases, might not be easy to implement automatically. However, the problem can then be integrated using explicit numerical schemes, which can be very efficient, but with a caveat:

Algorithmic stability is not guaranteed, so the choice of the time step is constrained by stability- rather than accuracy related considerations.

Question 3

Describe the RCS.

Answer 3

Integrating DAEs requires the use of numerical techniques with specific characteristics, namely unconditional stability and algorithmic dissipation, which can obly be provided by implicit integration schemes. The choice of the time-step can be dictated by accuracy-related considerations.

Question 4

Describe the UCS.

Answer 4

This strategy consists of preservign the structure of the unconstrained problem while indirectly enforcing the constraints. Formally, the problem is ODE. Thus, explicit integration schemes can be used with the same caveats indicated for the MCS strategy.