Relative Velocity and Acceleration with Translating and Rotating Frames

 

 

Key Concepts:  Various frames of reference are used in kinematics.  One of the most important

ones is a fixed frame of reference such as one attached to earth (neglect rotation of the earth). 

Three other types include a translating frame of reference, a rotating frame of reference, and a

frame of reference that both translates and rotates.  Use of the appropriate frame of reference

may simplify the analysis for specific applications.

 

 

In a Nut Shell:  Start by attaching a translating and rotating frame of reference x1y1 to body 1 (rigid
link).  Examine the motion of a bug walking on body 1.  Let  rA  be the position vector of point A

with respect to the fixed frame, F, rB  be the position vector of bug B with respect to the fixed

frame, F, and rAB  be the position vector of  bug,  with respect to frame  x1y1 at any time t. 

                          

 

Start with vector addition.

                rB  =    rA  +  rAB    =    rA  +  rAB u       ,  u  is a unit vector directed from A to B

 

 

Now take the derivative of each vector with respect to time in the fixed frame of reference, F.

 

      drB /dt =   drA /dt  + d(rAB u) /dt  =  drA /dt  + d(rAB) /dt u + rAB  d(u)/dt

 

The result is:   drB /dt =  vB  is the velocity of point B with respect to the fixed frame, F

 

           drA /dt  = vA  is the velocity of point A with respect to the fixed frame, F

 

          d(rAB) /dt u  =  vB|1  is the velocity of B with respect to A in frame x1y1

 

          ω  =  dθ/dt k       where   ω  is the angular  velocity of body 1 in the fixed frame, F

 

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