MEAM 535 Advanced Dynamics

Fall 1999

Homework #4

Undergraduates need not solve problem #3 in this homework.
Extra credit will be given to those undergraduates who choose to do this problem.

1. The velocity of a particle with respect to an inertial frame is given as (3a1 + 4a2) in a Cartesian coordinate system with orthogonal basis unit vectors a1, a2, and a3.
   
   
   • Express this velocity vector in a rotated coordinate system whose unit basis vectors are given by b1, b2, and b3.
     b1 = -0.1470a1 - 0.1470a2 + 0.9781a3
     b2 = 0.2665a1 - 0.9582a2 - 0.1040a3
     b3 = 0.9525a1 + 0.2454a2 + 0.1801a3
     
     
   • Express the same velocity vector in another coordinate system whose unit basis vectors are given by c1, c2, and c3.
     c1 = 0.8192a1 + 0.5736a2
     c2 = 0.0872a1 + 0.9962a2
     c3 = a3
     
     
2. (a1, a2, and a3) is an inertial frame attached to the fixed body A, and (b1, b2, and b3) is a frame attached to a rotating body B. The two frames are initially concident. Now imagine that the body B rotates about the a3 at an angular rate of alpha rad/sec. Note that Kane defines the angular velocity vector of body B in body A as follows.
   
   Use this definition to obtain the angular velocity of body B in A.
   
   
3. In the class, we derived Kane's angular velocity vector by considering the derivative of a vector expressed in a moving frame. Now, do the reverse. That is, assume Kane's definition of the angular velocity vector (see above) and obtain the derivative of a vector p in frame A. Note that p is expressed in frame B. In other words, using the above equation of angular velocity vector, prove the following: