ME 6590 Multibody Dynamics
Velocities and Partial Velocities Using Absolute Coordinates
Mass Center Velocities and Partial Velocities
Consider the rigid body shown in the
diagram. Given that ( x1, x2 , x3 ) are the position
coordinates of the mass center G relative to the
inertial frame, then pG the position vector of G,
vG the velocity of G in R, and vG , x  the partial
velocity matrix for G may be written in matrix
form as
R
 x1 
 pG    x2  ,
x 
 3
 x1 
vG    x2  ,
x 
 3
  R vG
vG , x   
 x1
 vG
x2
R
1 0 0 

 vG


  0 1 0 
x3 
0 0 1 
R
Velocities of Other Points
For other points in the system, such as point P in the diagram above, we can write the
position vector in terms of the mass-center position vector as follows
 pP    pG    pP / G    pG   C  pP / G 
Differentiating this expression gives the velocity of P as
vP    pP    pG   C   pP / G   C  pP / G 
 vG   C   pP / G 
(1)
where  pP / G   0 because  pP / G  represents the coordinates of P relative to G in the
body-fixed system which are all constant.
To reduce Eq. (1) further, we must decide whether we will use inertial or body-fixed
components for the angular velocity vectors. The two results are
vP   vG   B C  pP / G 
vP   vG   C B  pP / G 
As before, the “primes” indicate components in the body-fixed system, and “no primes”
indicate components in the inertial system.
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