Freely vibrating beams
An operator based formulation is used to show the completeness of the eigenvectors of a non-uniform, axially loaded, transversely vibrating Euler-Bernoulli beam having eccentric masses and supported by off-set linear springs. This result generalizes the classical expansion theorem for a beam having conventional end conditions. Furthermore, the effect of truncating a series approximation of the initial deflection is investigated for the first time. New asymptotic forms of the eigenvalues and eigenvectors are determined which are themselves often sufficiently accurate for high frequency calculations. A numerical procedure normally needs to be used for a transversely vibrating Euler-Bernoulli beam having complicated interior and end conditions because closed form solutions (including their asymptotic forms) are mostly beyond reach. The Rayleigh-Ritz approximate procedure has been applied widely to self-adjoint problems in structural dynamics. However, the numerical convergence of the Rayleigh-Ritz procedure deteriorates significantly if Gibbs phenomenon occurs. In this thesis, generalized force mode functions are suggested as one means of avoiding this effect. The convergence rate of the eigenvalue approximations resulting from the use of such functions is determined for a discontinuous, freely vibrating Euler-Bernoulli beam. Moreover, the pointwise convergence of the deriv tives that correspond to the practically important bending moment and shear force is examined for the first time. Then, a numerical example is given to corroborate the new theory. Non-self-adjoint systems are encountered when viscous damping forces or a gyroscopic effect exists. The generalized force mode functions method is extended to accommodate a spinning Timoshenko beam having a stepped cross-section. Numerical data suggests that this approach can very accurately approximate the backward and forward precession frequencies, bending moment and shear force.