Abstract
Direct analytical solutions, i.e., analytical solutions without any reference to the theory of differential equations, are given for the vibration frequencies and modes of beams with discontinuities in moment and shear. The calculated displacement, slope, moment and shear are given. In every case, these data are compared to corresponding data from the rigorously exact solution to the differential equation subjected to the rigorously imposed boundary conditions. The results show that, contrary to the approximation that would be obtained from the Rayleigh-Ritz method when there are discontinuities in the higher derivatives, the direct solution converges to the exact solution as defined by the mathematician: i.e., the solution that satisfies the differential equation of equilibrium throughout the domain and the conditions on the boundary of the domain.
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