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3 Forced Vibration of Single Degree of Freedom System
Fig. 3.9 Rotating shaft
The centre of rotation in Fig. 3.9 is O, S is the geometric centre, and G is the mass
centre. The elastic forces set up on the shaft are − kx and − ky. This is represented
by a complex quantity − kz, such that z = x + iy. The position of S relative to G
is e 0 e
iω t and position of G relative to O is z + r e
iω t . The equation of motion is as
follows
m
d
2
dt 2
z + r e
iω t
= −kz
(3.26)
or
m ¨
z + kz = mrω
2 e
i ω t
Substitution of the trial solution
z = Z e
i ω t
into Eq. (3.26) yields
Z =
mω
2 r
k − mω 2
(3.27)
or
Z
r
=
ω
2
/ p
2
1 −
ω 2
p 2