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By the theorem of Hamilton to the problem of bending vibra-
tions for an elastically restrained rotating nonuniform
Timo- shenko beam with attachments in this paper,the coupled
differen- tial governing equations of an nonuniform
Timoshenko beam are reduced into two complete forth-order
differential governing eq- uations with variable coefficients
in the flexural displacement or in the angle of rotation due
to bending ,respectively.The ex- plicit relation between the
flexural displacement and the angle of rotation due to bending
is established . The frequency equa- tions of the beam with a
general elastically restrained root are derived and expressed
in terms of the four normalized fundamen- tal solutions of the
associated governing differential equations . Consequently ,if
the geometric and material properties of the beam are in
polynomial form ,then by the method of Frobenius the exact
solution for the problem can be obtained. First, considering
the dimensionless length of a rotating beam shorter than the
radius of the rigid ring , as the rotating beam in all
compression due to the centrifugal force , it is ob- served
that setting angle , taper ratio and elastic constraint have
the influence on the natural frequencies. Second, an
nonuniform Timoshenko beam with the variance of length of
the rotating beam has the influence on the critical rotating
speed.Third,the variance of the tip mass given on which end of
a rotating nonuniform beam , it has the influence on the
natural frequencies. Finally, a rotating beam clamped at the
ri- gid ring which radius is shorter than the length of the
beam is partly in compression and partly in tension. Until now
only con- sidered , a uniform Bernoulli-Euler beam that clamped
at the ri- gid ring has a constant setting angle and rotating
speed.
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