Stability And Internal Resonance Of Axially Moving Rayleigh Beam

The problem of vibration and stability of axially moving beam is very important in vibration mechanics, involving theoretical practical meaning. Natural frequencies of axial moving Rayleigh beam are investigated. The general Hamiltonian principle is developed to derive the transverse vibration equations of the axially moving Rayleigh beams. Under the simply and clamped supported boundary conditions, uses the differential quadrature method to calculate the subcritical natural frequencies of the axially moving Rayleigh beams. Numerical examples give the variations of the first and second natural frequencies versus mean velocities for various flexural rigidities and support rigidities, respectively. The results illustrate that natural frequency is bigger high orders than low orders and increases with flexural and support rigidities increases.Multi-scale method is used to solve the governing equation directly with solvability condition obtained. According to Routh-Hurwitz criterion stability boundary equation is derived. Therefore the instability conditions are presented under the simply and clamped supported boundary conditions. Numerical instances demonstrate effect of the stiffness, the support stiffness, the viscosity, and the mean axial speed on the stability region in principle resonance and summon resonance.Introducing nonlinear term to build a nonlinear model then infer solvability condition. The resolvable condition of parametric resonance incremental analysis method is derived, and the steady-state frequency response of sub-harmonic resonance and combination resonance of axially moving nonlinear viscoelastic Rayleigh beams is studied using stability condition of steady-state response from Routh-Hurwitz criterionResearch the viscoelastic Rayleigh beam in 3:1 internal resonance. According to the Routh-Hurwitz criterion, calculated the master type, Analyze the Rayleigh beam stability conditions and give the steady-state amplitude frequency response.