which gives the frequency equation as

(n)

From equation (k), the frequency equation comes out to be

(o)

Noting equations (b) and (d), in the expanded form equation (o) can be written

(p)

For the numerical values of the present problem, equation (p) reduces to

(q)

From frequency equation (q), the following natural frequencies are obtained

It should be noted that for the present problem even four discs (polar mass moment of inertia) are present, however only three natural frequencies is obtained. This is due to the fact that gear pair is treated as a single polar mass moment of inertia, so effectively for the present problem only three generalized coordinates are sufficient to describe the motion.

For comparison with Example 6.11, let us put polar mass moment of inertia of the pinion and the gear to zero in equation (p), then we get

Which gives

and