Natural frequencies of a super-critical transporting Timoshenko beam

Super-critical transporting continua are often discussed in the context of an Euler beam model. Here Timoshenko beam theory is applied to study free vibration of high-speed transporting continua. Therefore, effects of rotary inertia and shear deformation on transverse vibration of super-critical transporting beams are discovered for the first time. The Galerkin method is applied to solve natural frequencies with the simply supported boundary conditions. Meanwhile, the weighted residual method (WRM) is employed to deal with the fixed ends. The natural frequencies of super-critical continua are verified by using the discrete Fourier transform (DFT). In the super-critical regime, the straight configuration of the beam loses its stability. The buckling configurations, due to the nonlinear stiffness, are deduced from the nonlinear static equilibrium equation. Furthermore, the governing equation for the transverse vibration of the super-critical transporting Timoshenko beam is derived based on the buckling shape. Time histories are calculated by using the finite difference method (FDM). Furthermore, natural frequencies of nonlinear free vibration are determined by using the DFT. In addition, the effect of nonlinear stiffness on the natural frequencies is discovered. On the other hand, a partially linearized equation is deduced by regarding buckling configurations and its spatial derivatives as constant coefficients. Then, natural frequencies are extracted by using the Galerkin method. The two different approaches are compared. Numerical results show that the rotary inertia and the shear deformation significantly affect vibration characteristics of the super-critical transporting beam for some configurations. Moreover, comparisons with an Euler-Bernoulli beam model reveal that the fundamental frequency of the Timoshenko beam is higher when the speed is slightly greater than the critical value.