Page 165 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

152 Systems with Two or More Degrees of Freedom Chap. 5

Because the force acting on m 2 is
/c 2 2 — ^ 2 2 ^ 0
the absorber system ^ 2 ,^ 2 exerts a force equal and opposite to the disturbing
force. Thus, the size oi k 2 and m 2 depends on the allowable value of X 2.

5.7 CENTRIFUGAL PENDULUM VIBRATION ABSORBER

The vibration absorber of Sec. 5.6 is only effective at one frequency, o) = 0)22-
Also, with resonant frequencies on each side of (O22, the usefulness of the
spring-mass absorber is limited to a narrow frequency range.
For a rotating system such as an automobile engine, the exciting torques are
proportional to the rotational speed n, which can vary over a wide range. Thus, for
the absorber to be effective, its natural frequency must also be proportional to the
speed. The characteristics of the centrifugal pendulum are ideally suited for this
purpose.
Figure 5.7-1 shows the essentials of the centrifugal pendulum. It is a 2-DOF
nonlinear system; however, we limit the oscillations to small angles, thereby
reducing its complexity.
By placing the coordinates through point O' parallel and normal to r, line r
rotates with angular velocity (0 + </>). The acceleration of m is equal to the vector
sum of the acceleration of O' and the acceleration of m relative
to O'.

R6 sin (f) - RO^ cos (f) - r(^6 + (^) j/

+ ^RO cos (f) + Rd^ sin (f) -h r((9 + (5.7-1)
Because the moment about O' is zero, we have, from the 7-component of

Mq- = cos i:f) -h R6^ sin (/> + r(^6 + r = 0 (5.7-2)

r{9 + <^)

Figure 5.7-1. Centrifugal pendu­
lum.