Parametric resonances of Timoshenko pipes conveying pulsating high-speed fluids

Parametric resonances of pipes caused by pulsating fluids have received much attention. However, the main concern is the pulsation of subcritical flow. Moreover, it is usually based on the Euler-Bernoulli pipe model. This paper focuses on revealing the characteristics of parametric resonances of the Timoshenko pipe with pulsation of supercritical high-speed fluids. Supercritical flow causes the linear instability of the pipe. The coupled partial differential equations with varying parameters are derived for governing the vibration of the pipe around the non-trivial static equilibrium configuration. A direct multi-scale method is developed to analytically obtain parametric resonance responses from coupled partial differential equations with varying parameters. For the first time, the nonlinear parametric response of the Timoshenko pipe is verified by using the finite difference method (FDM). Some interesting phenomena are demonstrated. For example, unlike parametric resonance at subcritical speeds, the steady-state response of velocity pulsation excitations at supercritical speeds is not monotonic with changes in some physical parameters. For another example, when pulsation occurs at supercritical speeds, the smaller steadystate response and the instability region can be obtained through the Timoshenko pipe model. In addition, the relationship between the instability threshold of the velocity pulsation amplitude and the slenderness ratio is non-monotonic, suggesting that the Euler-Bernoulli pipe model may simplify some vibration characteristics. Due to these different phenomena, the necessity of studying the velocity pulsation in the supercritical high-speed range and the necessity of the Timoshenko model are demonstrated. (C) 2020 Elsevier Ltd. All rights reserved.