Primary and super-harmonic resonances of Timoshenko pipes conveying high-speed fluid

Abstract

High-speed flow pipes suffer from severe vibration problems. When the fluid velocity is higher than the critical value, the straight equilibrium configuration of the pipe will lose stability. What follows is the supercritical vibration of the pipe near the non-trivial static equilibrium configuration. This paper attempts to reveal multiple resonance responses of forced vibration of pipes in the supercritical regime. Based on Timoshenko beam theory, the nonlinear coupled partial differential equations are deduced. The non-trivial static equilibrium configuration causes the parameters to vary with space variable. The approximate responses of the pipe are obtained and verified numerically. The results show that the flow velocity near the critical value is more prone to cause severe vibration. Unlike in the subcritical regime, there are third-order super-harmonic resonance and second-order super-harmonic resonance in the supercritical regime. So there are more resonance areas in the supercritical regime. High flow velocity or large external excitation can aggravate the difference between the Euler-Bernoulli model and the Timoshenko model, and the relative error of two models varies non-monotonically. Even for slender pipes, the difference between the two models is still very clear. Therefore, the Timoshenko model is more necessary to analyze the vibration of the high-speed pipe.