Page 97 - The Mechatronics Handbook

0066_Frame_C07 Page 7 Wednesday, January 9, 2002 3:39 PM

FIGURE 7.5 Sketch of a compound pendulum under gravity torques.

FIGURE 7.6 Sketch of a magnetically levitated flywheel on high-temperature superconducting bearings.
natural frequency

(
w = [ gm 2 L 2 – m 1 L 1 )/I] 1/2 (7.12)

For the simple pendulum m 1 = 0, and we have the classic pendulum relation in which the natural
frequency depends inversely on the square root of the length:

w = ( g/L 2 ) 1/2 (7.13)

Gyroscopic Motions
Spinning devices such as high speed motors in robot arms or turbines in aircraft engines or magnetically
levitated flywheels (Fig. 7.6) carry angular momentum, devoted by the vector H. Euler’s extension of
Newton’s laws says that a change in angular momentum must be accompanied by a force moment M,

M = H (7.14)
˙
In three-dimensional problems one can often have components of angular momentum about two dif-
ferent axes. This leads to a Coriolis acceleration that produces a gyroscopic moment even when the two
f
angular motions are steady. Consider the spinning motor with spin about an axis with unit vector e 1 and
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