THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

GENERAL MOTION

The kinematic analysis of a rigid body which has general three-dimensional motion is best accomplish with the aid of principles of relative motion.

 v  a A A    v  a B B     a v A A / / B B  v  a A A    v  a B B      r A  r A / / B B      r A / B 

Rotating Reference Axes

A more general formulation of the motion of a rigid body in space calls for the use of reference axes which rotate as well as translate. Reference axes whose origin is attached to the reference point B rotate with an absolute angular velocity which may be different from the absolute angular velocity  of the body.

  i     i  j       j  k      k

The expressions for the velocity and acceleration of point A become  a  v A A   a B B     r   r A A / / B B    v   rel      r  A / B   2     v rel   a rel Where,

v

rel relative to and

x-y-z

a

rel are the velocity and acceleration of point A measured by an observer attached to

x-y-z

.

 v  a rel rel    x   i i       y    j j     z k  z   k

We again note that  and may be different from the angular velocity body. We observe that, if x-y-z are rigidly to the body,  =  and v rel and a rel is the angular velocity of the axes are both zero.  of the

Example 1 The motor housing and its bracket rotate about the Z-axis at the constant rate  =3 rad/s. The motor shaft and disk have constant angular velocity of spin p=8 rad/s with respect to the motor housing in the direction shown. If g is constant at 30 o , determine the velocity and acceleration of Point A at the top of the disk and the angular acceleration a of the disk.

Example 2 The circular disk is spinning about its own axis (y-axis) at the constant rate p=10 p rad/s. Simultaneously, the frame is rotating about Z-axis at the constant rate  =4 p rad/s. Calculate the angular acceleration a of the disk and acceleration of Point A at the top of the disk. Axes x-y-z are attached to the frame, which has the momentary orientation shown with respect to the fixed axes X-Y-Z.