Theoretical study of the natural frequencies of a cantilever beam under dry friction

Abstract

An analytical method to obtain the natural frequencies of a cantilever beam under dry friction is described in this paper. The equations of motion of a cantilever beam under dry friction and an equivalent model are established using Lagrange equations of the first and second kinds, respectively. Then the equations expressing the relationship between exciting force and natural frequency are acquired according to the established equations. Lastly, based on the equivalent method of input energy between the two models, we deduce an analytical expression that is concerned with the natural frequencies of the cantilever beam under dry friction. The proposed method verified by numerical results reveals that the results between the analytical and numerical methods are in better consistency with each other. Besides, on account of the natural frequency strongly influenced by friction force, the model can be used as a feasible shock absorber for some structures over a wide frequency range. This is important because the analytical expression obtained by the proposed method can directly indicate the influence of the dry friction on the natural frequencies of a cantilever beam.