Modeling,Analysis And Simulation Of Nonlinear Vibration Of Timoshenko Pipe Conveying Fluid

Pipes conveying fluid are widely applied in aviation,aerospace,machinery and other fields.The research on nonlinear vibration of the pipe conveying fluid has an important application value.Meanwhile,as a typical gyro continuous system,the analysis of its vibration characteristics is of great theoretical significance.When the flow velocity is in the subcritical range,the pipe vibrates near the static equilibrium configuration.When the flow is in the supercritical field,the pipe vibrates near the non-trivial static equilibrium configuration.In this paper,based on Timoshenko beam theory and generalized Hamilton's principle,the nonlinear vibration model of the pipe conveying fluid is established.The analytical expressions of the critical velocity and non-trivial equilibrium configuration are derived.Multi-scale method,the harmonic balance method,finite difference method and Galerkin truncation method are used to study the free vibration,forced vibration and parametric vibration of the Timoshenko pipe conveying supercritical fluid.Comparing with the Euler-Bernoulli model,the differences of vibration characteristics predicted by the two models are given,and the necessity of the Timoshenko model in the supercritical field is determined.Firstly,considering the radial vibration,the nonlinear coupled vibration model of Timoshenko pipe conveying fluid is established.The coupled model is reduced to a partial differential model and an integro-partial differential model.The finite difference method is developed to numerically calculate the non-trivial static equilibrium configuration of the three pipe models.The differences among the three Timoshenko models are identified.The comparison between Timoshenko pipe model and Euler-Bernoulli pipe model shows that Timoshenko pipe will enter the supercritical field earlier and the non-trivial static equilibrium configuration will be larger.Secondly,the nonlinear free vibration model of Timoshenko pipe conveying supercritical fluid is established.Galerkin truncation method is used to analyze the dependence of supercritical natural frequency on relative parameters.Compared with Euler-Bernoulli pipe model,it is found that the natural frequency of Timoshenko pipe is larger in a certain supercritical velocity range.Besides,the transverse forced vibration model of Timoshenko pipe conveying fluid in the supercritical regime is established.The approximate analytical solution of the forced vibration response is obtained by using the harmonic balance method and verified numerically by the finite difference method.The results show that the influence of supercritical and subcritical velocity on the first order forced resonance response is completely opposite.In the supercritical regime,the second and third order superharmonic resonance responses can be excited.However,in the subcritical regime,only the third order superharmonic resonance response can be excited.Compared with Euler-Bernoulli pipe model,it is found that the vibration response of the Timoshenko pipe is always larger in the subcritical regime.However,in the supercritical regime,that of the Euler-Bernoulli pipe may be larger.At last,the nonlinear parametric vibration model of the Timoshenko pipe conveying supercritical pulsating fluid is established.Multi-scale method is applied to investigate approximately the sub-harmonic parametric resonance response and stability boundary.The results show that the stiffness coefficient and average velocity monotonously affect the stability boundary while non-monotonically affect the stable sub-harmonic parametric resonance response.