Free vibrations of a generally restrained rectangular plate with an internal line hinge
This paper deals with the study of free transverse vibrations of rectangular plates with an internal line hinge and elastically restrained boundaries. The equations of motion and its associated boundary and transition conditions are rigorously derived using Hamilton’s principle. The governing eigenvalue equation is solved employing a combination of the Ritz method and the Lagrange multipliers method. The deflections of the plate and the Lagrange multipliers are approximated by polynomials as coordinate functions. The developed algorithm allows obtaining approximate solutions for plates with different aspect ratios, boundary conditions, including edges elastically restrained by both translational and rotational springs, and arbitrary locations of the line hinge. Therefore, a unified algorithm has been implemented. Sets of parametric studies are performed and the results are given in graphical and tabular form.
Using close-coupling calculations of Burke and Moores for the scattering of electrons by ${\mathrm{Mg}}^{+}$ ions in the $3^{2}S_{\frac{1}{2}}$ and $3^{2}P_{\frac{1}{2},\frac{3}{2}}$ states, Baranger's expression for the impact approximation width of an isolated line is implemented for the components of the resonance doublet. These widths are extrapolated to below inelastic thresholds and averaged over elastic resonances according to theoretical threshold laws. In the experimental energy range, results compare reasonably with semiclassical approximations and with a semiempirical method involving effective Gaunt factors extrapolated to zero electron energy.