Page 166 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 5.7 Centrifugal Pendulum Vibration Absorber 153

Assuming cf) to be small, we let cos (f) = 1 and sin cf>= cl) and arrive at the equation

for the pendulum:
(5.7-3)

If we assume the motion of the wheel to be a steady rotation n plus a small
sinusoidal oscillation of frequency co, we can write

0 = nt On sin cot

8 = n + codn cos cot = n (5.7-4)
6 = -0) On sin cot

Then Eq. (5.7-3) becomes

^

(j>+ = I^ sin ojt (5.7-3')
and we recognize the natural frequency of the pendulum to be

= n] (5.7-5)

and its steady-state solution to be

, (R -\-r)/r 2x1 •
cb = — sm cot (5.7-6)
^ -co^ + Rn^/r "
The same pendulum in a gravity field would have a natural frequency of yjg/r, so
it can be concluded that for the contrifugal pendulum, the gravity field is replaced
by the centrifugal field Rri^.
We next consider the torque exerted by the pendulum on the wheel. With the
y-component of equal to zero, the pendulum force is a tension along r, given by
m times the /-component of a^. By recognizing that the major term of ma^ is
-{ R + r)n^, the torque exerted by the pendulum on the wheel is

T = —m{R -h r)n^Rcl) (5.7-7)
Substituting for cf) from Eq. (5.7-6) into the last equation, we obtain

m{ R r) Rn^/r m(R -f r)
T= - co^0(. sin cot = —
Rn^/r - co^ 1 - rco^/Rn^
Because we can write the torque equation as T = Jq{\8, the pendulum behaves like