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GENERAL MOTION
B/AB/ABA
B/ABA
rraa
rvv
The kinematic analysis of a rigid body which has general three-dimensional motion is best accomplish with the aid of principles of relative motion.
B/ABA
B/ABA
aaavvv
Rotating Reference Axes
kkjjii
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.
relrelB/AB/ABA
relB/ABA
av2rraa
vrvv
The expressions for the velocity and acceleration of point A become
kzjyixa
kzjyixv
rel
rel
Where, vrel and arel are the velocity and acceleration of point A measured relative to x-y-z by an observer attached to x-y-z.
We again note that is the angular velocity of the axes and may be different from the angular velocity of the body. We observe that, if x-y-z are rigidly to the body, = and vrel and arel are both zero.
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 is constant at 30o, determine the velocity and acceleration of Point A at the top of the disk and the angular acceleration of the disk.
Example 2
The circular disk is spinning about its own axis (y-axis) at the constant rate p=10 rad/s. Simultaneously, the frame is rotating about Z-axis at the constant rate =4 rad/s. Calculate the angular acceleration 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.
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