Influence of the boundary relaxation on the free vibration of rotating composite laminated Timoshenko beams

In this paper, the influence of boundary relaxation on the free vibration characteristics of a rotating composite laminated Timoshenko beam is studied. Based on the first order shear theory, the theoretical model of the rotating composite laminated Timoshenko beam is established, in which the Possion's effect is also considered. The relaxed boundary conditions of the beam are simulated using a set of artificial springs. By adjusting the stiffness of the springs different extent relaxation boundary conditions of the beam can be obtained. Relaxation parameters are introduced to evaluate the extent of boundary relaxation. A uniformed formula for the centrifugal force of the rotating beam with relaxed boundary conditions is deduced. By adopting the Rayleigh-Ritz method together with the artificial spring technique the solution for the free vibration characteristics of the rotating beam with relaxed boundary is carried out. The results achieved by the present approach are compared with the results obtained by the finite element method and those in the existing literature. Some new results about the influences of boundary relaxation on the first order natural frequency and mode shape of the rotating beam are presented. The influences of parameters, such as the rotating speed, the fiber orientation angles, the slenderness ratios, and the hub ratios on the free vibration behaviors of the beam with relaxed boundary conditions are discussed.