Page 169 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

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

By assuming the solution to be in the form
6 =
(5.8-2)
(f) = (f>ge‘
where 6q and (/>„ are complex amplitudes, their substitution into the differential
equations results in
. CO) ICO) _ M q
-F i - ^0 y 00 j

and
^ .CO) \ , ICO) ^
(5.8-3)

By eliminating (/>(, between the tv/o equations, the expression for the amplitude
of the pulley becomes

(O^J^ —icco

(5.8-4)
- Ju)^)] +ic(o[o)^Jj- ( K-Jco^)]
Letting 0)1 = K/J and /i, = Jj/J, the critical damping is

Cc c = —2/o)„ = 2Uo)„

The amplitude equation then becomes

K6 M (w/w„) + 4^

A/n
(5.8-5)

which indicates that \Kd^/M^^\ is a function of three parameters, ¡jl, and (o)/o)^).
This rather complicated equation lends itself to the following simple inter­
pretation. If f = 0 (zero damping), we have an undamped single-DOF system with
resonant frequency of coj = ^ K /J . A plot of |/C0q/ M q| vs. the frequency ratio will
approach oo at this frequency. If = oo, the damper mass and the wheel will move
together as a single mass, and again we have an undamped single-DOF system, but
with a lower natural frequency of ^ K /{ J + .
Thus, like the Lanchester damper of the previous section, there is an
optimum damping ¿'qfor which the peak amplitude is a minimum, as shown in Fig.

5.8-3. The result can be presented as a plot of the peak values as a function of