238
6 Free Vibration of Multiple Degrees of Freedom System
Now, we obtain a unique value of rotation of the centre gear, whether we move
from the lower branch or the upper branch.
(f) Then, torques from the two branches are transferred to the single line part of
the system. Total torque becomes
T = (n
2
1 I 6 · 1 + n
2
2 I 5 θ 0 + n
2
1 I 4 θ 4 + n
2
2 I 2 θ 2 + I 3 θ 3 ) p
2
(6.131)
Substituting θ 4 = n 1 θ 3 and θ 2 = n 2 θ 0 into Eq. (6.131) yields
T = (n
2
1 I 6 · 1 + n
2
2 I 5 θ 0 + n
2
1 I 4 ¯
θ 3 + n
2
2 I 2 θ 0 + I 3 ¯
θ 3 ) p
2
(6.132)
(g) The rotation of disc I p is given by
θ 1 = θ 3 −
T
k 1
(6.133)
(h) Resulting torque at the far end is
T ext = T + I 1 p
2
θ 1
(6.134)
6.20 Reduction Methods for Dynamic Analysis
An analysis of the free vibration problem can be performed by solving the following
eigenvalue problem
( [K ] − p
2
[M]) {φ} = {0}
(6.135)
The direct method of generating [K ] and [M] matrices is suited only to problems
of smaller size. For bigger size problems, advantage is taken of the fact that [K ]
and [M] are symmetric and banded for structural problems. These matrices may
be stored in half-band and to save further storage space in skyline form. As most
eigenvalue solvers are increasingly expensive (takes time of solution proportional to
n
3 where n is the number of degrees of freedom), better methods have been sought
which would take less time. It is desirable to apply methods in which the size of these
matrices is reduced so that more economic solution of the eigenvalue is obtained.
Two such methods will be discussed here. They are Guyan reduction method of
dynamic analysis [13, 14] and the component mode synthesis method [15, 16].