Stability of axially accelerating viscoelastic Timoshenko beams: Recognition of longitudinally varying tensions

Abstract

Stability of axially accelerating viscoelastic Timoshenko beams is treated. The effects of longitudinally varying tensions due to the axial acceleration are focused in this paper, while the tension was approximatively assumed to be longitudinally uniform in previous works. The dependence of the tension on the finite axial support rigidity is also modeled. The governing equations and the accurate boundary conditions for coupled planar motion of the Timoshenko beam are established based on the generalized Hamilton principle and the Kelvin viscoelastic constitutive relation. The boundary conditions were approximate in previous studies. The method of multiple scales is employed to investigate stability in parametric vibration. The stability boundaries are derived from the solvability conditions and the Routh–Hurwitz criterion for principal and summation parametric resonances. Some numerical examples are presented to demonstrate the effects of the tension variation, the viscosity, the mean axial speed, the shear deformation coefficient, the rotary inertia coefficient, the stiffness parameter, and the pulley support parameter on the stability boundaries.