Transcript THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
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.