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The governing equations and the associated boundary conditions
for three problems of nonuniform Timoshenko beams are derived
by using the Hamilton''s principle. They are:(1) the free
vibrations of nonuniform Timoshenko beams subjected to axial
forces,(2) the nonconservative stability of nonuniform
Timoshenko beams subjected to partially tangential follower
forces, (3) the pure bending vibrations of rotating nonuniform
Timoshenko beams. In each subject, the two coupled
differential equations are decoupled into one complete fourth-
order differential equations with variable coefficients in
the angle of rotation due to bending or in the flexuarl
displacement, respectively. The explicit relation between
the flexural displacement and the angle of rotation due to
bending is established. The frequency equations of the beam
are derived and expressed in terms of four normalized
fundamental solutions of the associated reduced governing
differential equation. Consequently, if the geometric and
material properties of the beam are in polynomial forms,
then the exact solutions for the problem can be
constructed by the method of Frobenius. But its rate of
convergence for the solution will be slow for some problem
with special variable coefficients. The modified method of
Frobenius by which the rate of convergence for the solution is
fast,is constructed. However, if the coefficients of the system
are not in polynomial form, a semi-closed method is
established. Moreover, the mechanism of tensional instability
of beam is studied. And the neccesary and sufficient
condition at which the phenomenuous of tensional
instability happens, is discovered. Finally, the influence
of some parameters on the vibrations and the
nonconservative stability of nonuniform Timoshenko beams is
studied.
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