Nonlinear Vibrations Of Pipes Conveying Fluid Subjected To Impacting Constraints

Pipe conveying fluid was a kind of slender structures,which was mostly used in engineering.As time goes by,the fixed motion constraints may be destabilized by the flow-induced vibration of the pipe systems,which may lead to a gap between the fluid conveying pipe and motion constraints.This kind of loose constraints,or in another words,impacting constraints,may have significant influences on the vibration behaviors of the fluid conveying pipes,which may also lead to complicated nonlinear impacting behaviors of the pipe systems.Therefore,it is of great importance in exploring the nonlinear impacting dynamics of the constrained pipe systems both theoretically and experimentally.In this paper,the nonlinear impacting vibrations of a fluid conveying pipe subjected to loose constraints are investigated,mainly focusing on the chaotic oscillations,as well as planar and non-planar dynamics,discovering a series new nonlinear responses of the system,which have not been observed before.The explorations of this work would be helpful in the effective designing of pipe systems.The main features of the present paper are organized as follows:1.The two-dimensional(2-D)mathematical model and nonlinear equations of motion are constructed for a simply supported pipe conveying pulsating fluid subjected to distributed impacting constraints.Particular attention is paid on the buckling and nonlinear forced vibration behaviors of the pipe system under various mean flow velocities and pulsating frequencies,unfolding some new phenomena and mechanisms of the constrained pipe systems.During the theoretical modeling,the distributed impacting constraints are depicted as a wall at both sides of the pipe.The impacting force is modeled by a modified trilinear spring.In the numerical process,a matrix transforming method is proposed,aiming at decoupling the nonlinear terms in the governing equations,which would cut the cost of a huge number of numerical integration greatly.Results show that,when the pulsating frequency is zero,i.e.,a constant internal flow velocity,the pipe would lose stability by buckling.With the increasing of pulsating frequency,the pipe is capable of displaying both periodic and chaotic motions.Several types of the distribution of impacting forces and deflections of the pipe are calculated when the pipe is in contact with the impacting constraints.2.The 2-D nonlinear equations are constructed for a cantilevered pipe conveying fluid with the existence of distributed impacting constraints.The possible nonlinear vibrations and forced vibrations of the system under constant and pulsating fluid are explored.Compared with the case of a simply supported pipe,non-extensible condition of a cantilevered pipe is considered in this system.The nonlinear terms in the governing equations are complicated,including multiple integration and variable limit integration.In the numerical process,the matrix transforming method proposed in this work can be adopted,which could improve the efficiency of calculating by decoupling the large number of nonlinear terms.Interesting results have been obtained with distributed impacting constraints modeled either by a cubic spring or a modified trilinear spring,comparing with the case of a point impacting constraint.Responses of a distributed cubic spring model are singleness,mainly performing period-1 vibration;for some particular velocity ranges,the system may exhibit another type of period-1 motion with two or three extreme values.For a modified trilinear spring model,the pipe undergoes a transformation from periodic oscillation to chaotic oscillation,and again periodic oscillation.Period-1 motion occurs at the onset of critical velocity.Multi-period motion happens in a wide range of flow velocities.The nonlinear forced vibrations are influenced greatly by the mean flow velocity,pulsating amplitude and pulsating frequencies.In the range of flow velocities in this paper,the system may also show chaotic vibrations.The instability regions of the cantilevered pipe conveying pulsating fluid are calculated using the Floquet theory,discovering the relationships of the resonances with mean flow velocity,pulsating amplitude and pulsating frequency.3.The three-dimensional impacting dynamics of a cantilevered pipe conveying fluid subjected to a point constraint is investigated,aiming at discussing the nonlinear impacting dynamics of two types of impacting constraints,one is two parallel bars(TPBs)and the other is tube support plate(TSP).For TPBs,friction effects are considered,illustrating the transforming between planar and non-planar vibrations.Results show that,for the asymmetry of the TPBs in the two coordinates,responses get large distinguishes,showing quasi-periodic oscillations with large lateral displacement but small amplitude,and small lateral displacement but large amplitude.The friction between the pipe and constraints changes the direction of the planar motion at large flow velocities;for small flow velocities,the friction effects could be neglected.For the case of TSP,for the symmetry of the structure and the nonlinear equations,the behaviors in the two coordinate directions are similar,both showing periodic,quasi-periodic as well as chaotic oscillations.For some particular flow velocities,the pipe may stick to the impacting constraints.The vectorial resultant of the responses in the two coordinate directions leads to the planar or non-planar vibrations.The other system parameters have significant influences on the vibration styles.From the above,the detailed theoretical and numerical exploring of weak impacting dynamics of a fluid conveying pipe impacting systems reveals the behaviors of bifurcation and the route to chaos of engineering structures.Some new aspects of the nonlinear vibrations and stabilities of the impacting systems are discussed for the first time.The present work is of great innovation in the field fluid-structure interaction and nonlinear dynamics.The results and conclusions in this paper are of great significance in the optimization of pipe/impacting systems.