Effect of End Mass Rotation on the Dynamic Characteristics of a Cantilever Beam Using the Rayleigh-Ritz Principle

Abstract

The energy form of the Rayleigh-Ritz Principle was used to investigate the effect of rotating a mass at the free end on the dynamic characteristics of the cantilever beam. The complexity of boundary conditions caused by the end mass was avoided by applying the Rayleigh-Ritz principle. A weak form (energy form) Rayleigh Quotient for the beam with an end mass was derived. The stiffness and mass matrices of the system were obtained using 12 simple fixed-free (without end mass) eigenfunctions as admissible functions. Three different values of the end mass shape factor  (1, 2, and ), representing the mass moment of inertia of the end mass about the junction point, were used. The results showed that the influence of  on changing dynamic characteristics was significant in the second mode and above. The observed changes were 1.49%, 3.76% and 7.12% in the second, third, and fourth natural frequencies, respectively. A simple mathematical model was proposed to represent these changes. The end mass and its rotation transform the  mode shape of the cantilever into -mode shape of a fixed-pinned beam.