Research On Transverse Vibration Of Pipes Conveying Fluid On Pasternak Foundation

Pipes conveying fluid have been extensively applied in aeronautic and aerospace engineering,petrochemical industry,energy delivering,agriculture,water supply and sewerage,nuclear and our daily life.Due to the effect of fluid-structure interaction(FSI),pipes may even result in pipe rupture when the inner flow velocity is high or pulsation occurs.In addition,the transverse vibration of pipes conveying fluid is of the typical gyroscopic system,so the mechanism studies to the transverse vibration of the pipes conveying fluid on the elastic foundation are of great significance in engineering,economics,and theory.In this dissertation,the existing achievements in pipe FSI vibration are firstly summarized.Then the static and dynamic stability,parametric resonances,nonlinear vibration,and supercritical weakly forced vibration of the transverse vibration in pipes conveying fluid on Pasternak foundations model under the supporting condition on both sides are investigated,adopting the method of multiple scales,complex mode analysis,and the Galerkin discrete method.The main contents are as follows:1.The differential equations of transverse nonlinear vibration for the pipes conveying fluid on Pasternak foundations are established by Newton's method on the basis of considering Winkler foundation model and Pasternak foundation model.Furthermore,the differential equation and boundary conditions are non-dimensionalized.2.The mode function and the natural frequency of supported pipes on the Pasternak foundation are analyzed.As the basis of following chapters,the mode function and the natural frequencies of the linear free vibration are calculated by the complex mode analysis and Galerkin method.The influences of parameters such as foundation coefficient,mass ratio,and flow velocity under the supporting boundary condition on the system's static stabilities are discussed.The research reveals that the stabilities and the natural frequencies will decrease with the increase of flow velocity and tension force.The influence of mass ratio is not obvious.However,the effects of the foundation elastic coefficient and cutting rigidity on the static stabilities are shown to be great.3.Dynamic stabilities in transverse vibration of supported pipes on the Winkler or Pasternak foundation are analyzed,mainly considering the problem of parameter resonance in fluid-conveying Pipes caused by the pulsation of the inner flowing liquid;The dynamic stability of the pipe system conveying fluid on Pasternak foundations is laid special stress on.The stability boundaries of the parameter resonance in the pipe system are derived,adopting the method of direct multiple scales.The effects of the pulsating fluid flow velocity,damping coefficient,mass ratio,the linear,sheer foundation rigidities and other parameters of the parametric instability are discussed.Some numerical examples are presented to demonstrate the effects of related parameters on the stability boundaries.4.Dynamic response characteristics of the pipe system in the case of pulsating inner flow are analyzed by the method of multiple scales.The stabilities of trivial and nontrivial solutions are analyzed by the solvability conditions and the Routh-Hurwitz criterion.The result displays that when the pulsating inner flow frequency approaches the sum of any two natural frequencies or two times of a natural frequency of the autonomous linear generating system,summation parametric resonance or principal parametric resonance may occur if the fluctuation amplitude of flow velocity is large enough.The solutions of the response amplitude may bifurcate,which results in non-response,stable response or unstable response with them;when the parametric resonance of the nth mode occurs,the motion of this mode is dominant.The non-response instability range increases with the pulsating inner flow velocity;it will reduce with the increase of the mass ratio as well as the foundation elastic coefficient.5.Nonlinear weak forced vibration of a viscoelastic pipe conveying fluid on Pasternak foundation around the curved equilibrium resulting from the supercritical flow speed is investigated.The local dynamics behaviors of external and internal resonance are key research.The method of multiple scales and Galerkin method are developed to present the solvability condition of approximate solutions.Under super critical condition,the perturbed control equation is obtained by the transformation of the coordinate on non-zero equilibrium position to the transverse forced motion equations of the fluid conveying pipe on the Pasternak foundation.The first two primary resonance of vibration in the presence of two-to-one internal resonances are examined.The amplitude double-jumping,softening-spring,hardening-spring,hysteresis and saturation phenomenon are demonstrated.Especially,it is found that a softening spring is changed into a hardening spring when the velocity gradually approach to the neighborhood of internal resonance.