Consider the mechanical system It consists of a rigid body of mass m that is free to rotate about a fixed point X O . The joint at X O does not permit the body to have a spin about e 1 . A vertical...

Consider the mechanical system It consists of a rigid body of mass m that is free to rotate about a fixed point X_O. The joint at X_O
does not permit the body to have a spin about e₁. A vertical gravitational force mgE₁
acts on the body. The inertia tensor of the body relative to its center of mass X¯ is

The position vector of the center of mass X¯ of the body relative to XO is ³0e1. To parameterize the rotation tensor of the body, we use a set of 1–3–1 Euler angles:

  For these angles

(a) Which orientations of the rigid body coincide with the singularities of the Euler angles?

(b) Derive expressions for the unconstrained potential energy U and unconstrained kinetic energy T of the rigid body.

(c) Show that the motion of the rigid body is subject to four constraints:

(d) Derive expressions for the constrained potential energy U and kinetic energy T of the rigid body.