On the dynamic behaviour of slender beams with elastic ends carrying a concentrated mass

The dynamic behaviour of a slender beam carrying a concentrated mass at an arbitrary abscissa is examined. The beam is supposed to be elastically restrained against the rotation and the translation at both the ends, so that it is possible to study all the common boundary conditions. First, the exact solution is calculated, by solving the differential equations of motion and by imposing the corresponding boundary conditions. The resulting frequency equation is numerically solved. Subsequently, various approximate results are given, using the optimized Rayleigh-Schmidt approach with trigonometric and static shape functions, so that some comparison becomes possible. Finally, an application of the Morrow method allows us to obtain a lower bound to the true results.