Study On Stability And Critical Flow Velocity Of Cantilever Pipeline Conveying Fluid With Elastic Support

In this paper, based on using the two kinds of research methods which are theoretical analysis (Ritz-Galerkin method) and numerical analysis (DQM method), the stability and critical flow velocity of pipeline conveying fluid with elastic support and cantilever were studied under the role of unchanging flow.First of all, the analytical expressions of frequency equation and eigenfunctions of bending vibration beam with the elastic support and cantilever have been derived, at the same time the expressions of vibration of beam which are composed from the superimposed of all the main vibration were given. And the first two-order eigenvalues of the beam have been calculated by use of numerical methods (dichotomy) when stiffness coefficient of elasticity and the bending rigidity of the beam are differently, the eigenvalues are influenced by the bending stiffness of the beam have been analyzed. In addition, through giving the variation curve that the eigenvalues of beam are changing along with the stiffness coefficient of elasticity at each location of the intermediate fixed support, the eigenvalues are influenced by stiffness coefficient of elasticity have been analyzed.And then, the differential equation of the pipeline's motion with elastic support and cantilever were derived under the role of unchanging flow. The differential equation of the motion after which was dealed by dimensionless was discretized by Ritz-Galerkin method based on the natural frequency and eigenfunctions of pipeline which were regarded as the similar natural frequency and eigenfunctions of beam with the same boundary conditions, thus getting the first-order state equations after the appropriate transformation. According to the characteristic equation of the first-order state equations, and the occurred conditions of dynamic flutter and static instability of pipeline conveying fluid with elastic support and cantilever were obtained by using qualitative methods of vibration theory under the role of unchanging flow(the conditions of the stability were under zero balanced state).Promptly, the conditions of critical flow velocity were meeted with which pipeline conveying fluid was dynamic flutter and static instability state. According to these conditions, the effect of linear stiffness coefficient of elasticity, mass ratio, fluid pressure which was the main physical parameters on critical instability flow velocity was analyzed and discussed based on the eigenequation. In addition, the dynamics of pipeline conveying fluid with elastic support and cantilever were by using the method of numerical simulations based on fourth-order Runge-Kutta method.Finally, the differential equation of the motion and boundary conditions of pipeline conveying fluid with elastic support and cantilever were deal by using DQM method, the unstable form and critical flow velocity of pipeline conveying fluid with elastic support and cantilever were obtained by numerical analysis, to check the results were obtained based on theoretical analysis.