Page 416 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.10 Geared System 403

Figure 12.9-3. Torsion-frequency
curve for the damped torsional sys­
tem of Fig. 12.9-2.

12.10 GEARED SYSTEM
Consider the geared torsional system of Fig. 12.10-1, where the speed ratio of shaft
2 to shaft 1 is n. The system can be reduced to an equivalent single shaft system as
follows.
With the speed of shaft 2 equal to O2 = nO^, the kinetic energy of the system
is
T = (12.10-1)
Thus, the equivalent inertia of disk 2 referred to shaft 1 is « ^ 2-
To determine the equivalent stiffness of shaft 2 referred to shaft 1, clamp
disks 1 and 2 and apply a torque to gear 1, rotating it through an angle 6^. Gear 2
will then rotate through the angle 02 = nS^, which will also be the twist in shaft 2.
The potential energy of the system is then
U = + {K2n^e\ ( 12.10-2)

and the equivalent stiffness of shaft 2 referred to shaft 1 is n^K2.

lU
n

n^Jr,

Figure 12.10-1. Geared system and
its equivalent single-shaft system.