Problem Statement
A force f acts on a rigid body that is initially at rest, causing it to accelerate with acceleration a. The body has inertia I, and it is attached to a revolute joint with rotation axis s, which constrains the body's motion. The constraint is maintained by a constraint force fc. The problem is to find a in terms of f.
Solution Details
The first equation is the equation of motion of the body. It expresses the relationship between the sum of the forces acting on the body and the resulting acceleration, assuming that the velocity is zero. The second equation expresses the constraint on the body's motion imposed by the joint. It states that the acceleration must be a scalar multiple of s. (α is a scalar.) The third equation states that the constraint force does no work.
Together, the first three equations state the problem mathematically. The second three then solve it in three easy steps: first eliminate the constraint force; then find an expression for α; then find an expresion for a in terms of f.
Why it Works
The quantities f, fc, s and a are all spatial vectors, and therefore provide a complete description of their respective physical quantities. So f describes not only the magnitude and direction of the applied force, but also the location of its line of action. Likewise, s describes both the direction and the location of the joint's rotation axis; a describes both the linear and angular acceleration of the body; and the spatial inertia matrix I provides a complete description of the body's inertia properties (mass, centre of mass, and rotational inertia).
By using spatial vectors, it is possible to dispense with a lot of the clutter that would have been necessary if the problem were to be posed and solved using traditional Euclidean (3D) vectors. The result is a substantial reduction in the quantity of algebra. A problem can be stated more succinctly and solved in fewer steps, arriving at a simpler expression of the solution. To find out more, visit Roy's teaching page.