Constraint Moments and Constraint Forces in the Mechanics of Rigid Bodies

For those of you preparing to teach a course on Lagrangian Mechanics of Rigid Bodies in the forthcoming semester or quarter, you may be interested to read a new expository paper by Arun Srinivasa at Texas A&M and myself:

 A Simple Treatment of Constraint Forces and Constraint Moments in Rigid Body Dynamics, ASME Applied Mechanics Reviews, To Appear (2014).  

In this paper, we show how a prescription for constraint forces and constraint moments in rigid body mechanics can be developed that is readily related to the traditional treatment of generalized constraint forces in Lagrangian (analytical) mechanics. The prescription is illustrated with a range of examples. For myself as a student, I found the generalized constraint force very difficult to interpret physically and later as a teacher I found them difficult to motivate to students. We hope that our paper will be useful to both students and instructors alike.

A rapid summary of the prescription is as follows (with apologies in advance for formatting problems with the subscripts and superscripts). Consider a rigid body which is subject to a single kinematical constraint (e.g., a top sliding on a plane or a disk sliding on a plane). Let A be a material point on a rigid body and denote the velocity of A by v and the angular velocity of the rigid body by ω. We suppose that the constraint can be expressed in the form

f.v + h.ω + e  = 0   (*)

where the functions f, h, and e depend on the motion of the rigid body and time. Then a prescription for the constraint forces Fc and constraint moment Mc on the rigid body is

Fc =  λ f acting at A    (**)
Mc = λ h

If we choose coordinates q1,…q6 to describe the motion of the rigid body, then the constraint (*) can be expressed in the form

( ∑ Ak dqk/dt )+ b = 0                             (+)

and the usual prescription for the 6 components of the generalized constraint force Qc is

Qck =  λ Ak       (k=1,...,6)                                  (++)

We show the equivalence of  (**) and (++) and with the help of several examples note that (**) is far easier to interpret. It might be of interest to some readers for us to note that the tools we use to describe the constraint moment also find application in joint coordinate systems in biomechanics and representations for conservative moments.