Nonlinear dynamics of a circular curved cantilevered pipe conveying pulsating fluid based on the geometrically exact model

Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine,the investigations on the mechanical responses of the pipes have attracted considerable attention.The fluid-structure interaction(FSI)between the pipe with a curved shape and the time-varying internal fluid flow brings a great challenge to the revelation of the dynamical behaviors of flexible pipes,especially when the pipe is highly flexible and usually undergoes large deformations.In this work,the geometrically exact model(GEM)for a curved cantilevered pipe conveying pulsating fluid is developed based on the extended Hamilton's principle.The stability of the curved pipe with three different subtended angles is examined with the consideration of steady fluid flow.Specific attention is concentrated on the large-deformation resonance of circular pipes conveying pulsating fluid,which is often encountered in practical engineering.By constructing bifurcation diagrams,oscillating shapes,phase portraits,time traces,and Poincarémaps,the dynamic responses of the curved pipe under various system parameters are revealed.The mean flow velocity of the pulsating fluid is chosen to be either subcritical or supercritical.The numerical results show that the curved pipe conveying pulsating fluid can exhibit rich dynamical behaviors,including periodic and quasi-periodic motions.It is also found that the preferred instability type of a cantilevered curved pipe conveying steady fluid is mainly in the flutter of the second mode.For a moderate value of the mass ratio,however,a third-mode flutter may occur,which is quite different from that of a straight pipe system.

Nonlinear Dynamics of Viscoelastic Pipe Conveying Pulsating Fluid Subjected to Base Excitation

Based on the Euler-Bernoulli beam theory and Kelvin-Voigt model,a nonlinear model for the transverse vibration of a pipe under the combined action of base motion and pulsating internal flow is established.The governing partial differential equation is transformed into a nonlinear system of fourth-order ordinary differential equations by using the generalized integral transform technique(GITT).The effects of the combined excitation of base motion and pulsating internal flow on the nonlinear dynamic behavior of the pipe are investigated using a bifurcation diagram,phase trajectory diagram,power spectrum diagram,time-domain diagram,and Poincare map.The results show that the base excitation amplitude and frequency significantly affect the dynamic behavior of the pipe system.Some new resonance phenomena can be observed,such as the period-1 motion under the base excitation or the pulsating internal flow alone becomes the multi-periodic motion,quasi-periodic motion or even chaotic motion due to the combined excitation action.

THE DYNAMIC BEHAVIORS OF VISCOELASTIC PIPE CONVEYING FLUID WITH THE KELVIN MODEL

Based on the differential constitutive relationship of linearviscoelastic material, a solid-liquid coupling vibration equation forviscoelastic pipe conveying fluid is derived by the D'Alembert'sprinciple. The critical flow velocities and natural frequencies ofthe cantilever pipe conveying fluid with the Kelvin model (flutterinstability) are calculated with the modified finite differencemethod in the form of the recurrence for- mula. The curves betweenthe complex frequencies of the first, second and third mode and flowvelocity of the pipe are plotted. On the basis of the numericalcalculation results, the dynamic behaviors and stability of the pipeare discussed. It should be pointed out that the delay time ofviscoelastic material with the Kelvin model has a remarkable effecton the dynamic characteristics and stability behaviors of thecantilevered pipe conveying fluid, which is a gyroscopicnon-conservative system.

In-plane forced vibration of curved pipe conveying fluid by Green function method

The Green function method （GFM） is utilized to analyze the in-plane forced vibration of curved pipe conveying fluid, where the randomicity and distribution of the external excitation and the added mass and damping ratio are considered. The Laplace transform is used, and the Green functions with various boundary conditions are obtained subsequently. Numerical calculations are performed to validate the present solutions, and the effects of some key parameters on both tangential and radial displacements are further investigated. The forced vibration problems with linear and nonlinear motion constraints are also discussed briefly. The method can be radiated to study other forms of forced vibration problems related with pipes or more extensive issues.

FLOW-INDUCED INTERNAL RESONANCES AND MODE EXCHANGE IN HORIZONTAL CANTILEVERED PIPE CONVEYING FLUID (Ⅱ)

The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3：1, 2：1 and 1：1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3：1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.

Stability and Chaotic Vibrations of a Pipe Conveying Fluid under Harmonic Excitation

The stability and chaotic vibrations of a pipe conveying fluid with both ends fixed, excited by the harmonic motion of its supporting base in a direction normal to the pipe span, were investigated with the aid of modern numerical techniques,involving the phase portrait,Lyapunov exponent and Poincare map tc. The nonlinear differential equations of motion of the system were derived by considering the additional axial force due to the lateral motion of the pipe. Attention was concentrated on the effect of forcing frequency and flow velocity on the dynamics of the system. It is shown that chaotic motions can occur in this system in a certain region of parameter space,and it is also found that three types of routes to chaos exist in the system:(i)period doubling bifurcations;(ii)quasi periodic motions;and (iii)intermittent chaos.

Effects of supported angle on stability and dynamical bifurcations of cantilevered pipe conveying fluid

The effects of the supported angle on the stability and dynamical bifurcations of an inclined cantilevered pipe conveying fluid are investigated. First, a theoretical model of the pipe is developed through the force balance and stress-strain relationship. Second, the response surfaces, stability, and critical lines of the typical hanging system （H-S） and standing system （S-S） are discussed based on the modal analysis. Last, the bifurcation diagrams of the pipe are presented for different supported angles. It is shown that pipes will undergo a series of bifurcation processes and show rich dynamic phenomena such as buckling, Hopf bifurcation, period-doubling bifurcation, chaotic motion, and divergence motion.

Dynamical Stability of Cantilevered Pipe Conveying Fluid with Inerter-Based Dynamic Vibration Absorber

Cantilevered pipe conveying fluid may become unstable and flutter instability would occur when the velocity of the fluid flow in the pipe exceeds a critical value.In the present study,the theoretical model of a cantilevered fluid-conveying pipe attached by an inerter-based dynamic vibration absorber(IDVA)is proposed and the stability of this dynamical system is explored.Based on linear governing equations of the pipe and the IDVA,the effects of damping coefficient,weight,inerter,location and spring stiffness of the IDVAon the critical flow velocities of the pipe system is examined.It is shown that the stability of the pipe may be significantly affected by the IDVA.In many cases,the stability of the cantilevered pipe can be enhanced by designing the parameter values of the IDVA.By solving nonlinear governing equations of the dynamical system,the nonlinear oscillations of the pipe with IDVA for sufficiently high flow velocity beyond the critical value are determined,showing that the oscillation amplitudes of the pipe can also be suppressed to some extent with a suitable design of the IDVA.

Dynamic analysis and regulation of the flexible pipe conveying fluid with a hard-magnetic soft segment

The recently developed hard-magnetic soft(HMS)materials can play a significant role in the actuation and control of medical devices,soft robots,flexible electronics,etc.To regulate the mechanical behaviors of the cantilevered pipe conveying fluid,the present work introduces a segment made of the HMS material located somewhere along the pipe length.Based on the absolute node coordinate formulation(ANCF),the governing equations of the pipe conveying fluid with an HMS segment are derived by the generalized Lagrange equation.By solving the derived equations with numerical methods,the static deformation,linear vibration characteristic,and nonlinear dynamic response of the pipe are analyzed.The result of the static deformation of the pipe shows that when the HMS segment is located in the middle of the pipe,the downstream portion of the pipe centerline will keep a straight shape,providing that the pipe is stable with a relatively low flow velocity.Therefore,it is possible to precisely regulate the ejection direction of the fluid flow by changing the magnetic and fluid parameters.It is also found that the intensity and direction of the external magnetic field greatly affect the stability and dynamic response of the pipe with an HMS segment.In most cases,the magnetic actuation increases the critical flow velocity for the flutter instability of the pipe system and suppresses the vibration amplitude of the pipe.

Nonlinear Dynamic Analysis of A Viscoelastic Pipe Conveying Fluid

Incremental harmonic balance method (IHBM) is applied to geometrically nonlinear vibration analysis of a simply supported pipe conveying fluid. the material of which is viscoelastic of the Kelvin- Voigt type. Some unstable phenomena - the appearance of the peak or jumps in the pipe's amplitude frequency curves, which are considered to be of importance to this kind of structure, are presented in the numerical results, and the influence of several parameters of the system on the dynamic characteristic of the pipe are also studied. It is believed that this is the first attempt to search for the periodic solution of the pipe and its intrinsic property with such a method.

Utilization of nonlinear vibrations of soft pipe conveying fluid for driving underwater bio-inspired robot

Creatures with longer bodies in nature like snakes and eels moving in water commonly generate a large swaying of their bodies or tails,with the purpose of producing significant frictions and collisions between body and fluid to provide the power of consecutive forward force.This swaying can be idealized by considering oscillations of a soft beam immersed in water when waves of vibration travel down at a constant speed.The present study employs a kind of large deformations induced by nonlinear vibrations of a soft pipe conveying fluid to design an underwater bio-inspired snake robot that consists of a rigid head and a soft tail.When the head is fixed,experiments show that a second mode vibration of the tail in water occurs as the internal flow velocity is beyond a critical value.Then the corresponding theoretical model based on the absolute nodal coordinate formulation(ANCF)is established to describe nonlinear vibrations of the tail.As the head is free,the theoretical modeling is combined with the computational fluid dynamics(CFD)analysis to construct a fluid-structure interaction(FSI)simulation model.The swimming speed and swaying shape of the snake robot are obtained through the FSI simulation model.They are in good agreement with experimental results.Most importantly,it is demonstrated that the propulsion speed can be improved by 21%for the robot with vibrations of the tail compared with that without oscillations in the pure jet mode.This research provides a new thought to design driving devices by using nonlinear flow-induced vibrations.

RESEARCH ON SOLID-LIQUID COUPLING DYNAMICSOF PIPE CONVEYING FLUID

On the basis of Hamilton principle. the equation of sonlid-liquid coupling vibration of pipe conveying fluid is deduced. An asymmetrical sonlid-liquid coupling damp matrix and a symmetrical solid-liquid coupling Stiffness matrix are obtained. Using QR method , pipe’s nature frequencies are calculated. The curves of the first four orders of natural frequency-flow velocity of pipe waw given .The influence of flowing velocity ,pressure, solid-liquid coupling damp and solid-liquid coupling stiffness on natural frequency are discussed respectively.The dynamic respondence of the pipes for stepload with different flow velocity are calculated by Newmark method .It is found that,with the flow velocity increased, the nature frequency of the pipes reduced, increased,reduced again and so on.

Flow-Induced Vibration of A Nonlinearly Restrained Curved Pipe Conveying Fluid

Investigated in this study is the flow induced vibration of a nonlinearly restrained curved pipe conveying fluid. The nonlinear equation of motion is derived by equilibrium of forces on microelement of the system under consideration. The spatial coordinate of the system is discretized by DQM (differential quadrature method). On the basis of the boundary conditions, the dynamic equation is solved by the Newton Raphson iteration method. The numerical solutions reveal several complex dynamic motions for the variation of the fluid velocity parameter, such as limit cycle motion, buckling and so on. The result obtained also shows that the sub parameter regions corresponding to the several motions may change with the variation of some parameters of the curved pipe. The present study supplies a new reference for investigating the nonlinear dynamic response of some other structures.

HOPF BIFURCATION OF A NONLINEAR RESTRAINED CURVED PIPE CONVEYING FLUID BY DIFFERENTIAL QUADRATURE METHOD

This paper proposes a new method for investigating the Hopf bifurcation of a curved pipe conveying fluid with nonlinear spring support.The nonlinear equation of motion is derived by forces equilibrium on microelement of the system under consideration.The spatial coordinate of the system is discretized by the differential quadrature method and then the dynamic equation is solved by the Newton-Raphson method.The numerical solutions show that the inner fluid velocity of the Hopf bifurcation point of the curved pipe varies with different values of the parameter, nonlinear spring stiffness.Based on this,the cycle and divergent motions are both found to exist at specific fluid flow velocities with a given value of the nonlinear spring stiffness.The results are useful for further study of the nonlinear dynamic mechanism of the curved fluid conveying pipe.

Parametric resonance of axially functionally graded pipes conveying pulsating fluid

Based on the generalized Hamilton's principle,the nonlinear governing equation of an axially functionally graded(AFG)pipe is established.The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam.Via the direct multi-scale method,the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method(DQEM).The influence of Young's modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions.In general,the pipe in the supercritical region is more sensitive to the pulsating excitation.The nonlinearity changes from hard to soft,and the non-trivial equilibrium configuration introduces more frequency components to the vibration.Besides,the increasing Young's modulus gradient improves the critical pulsating flow velocity of the parametric resonance,and further enhances the stability of the system.In addition,when the temperature increases along the axial direction,reducing the gradient parameter can enhance the response asymmetry.This work further complements the theoretical analysis of pipes conveying pulsating fluid.

Three-dimensional dynamics of supported pipes conveying fluid

This paper deals with the three-dimensional dynamics and postbuckling behavior of flexible supported pipes conveying fluid, considering flow velocities lower and higher than the critical value at which the buckling instability occurs. In the case of low flow velocity, the pipe is stable with a straight equilibrium position and the dynamics of the system can be examined using linear theory. When the flow velocity is beyond the critical value, any motions of the pipe could be around the postbuckling configuration(non-zero equilibrium position) rather than the straight equilibrium position, so nonlinear theory is required. The nonlinear equations of perturbed motions around the postbuckling configuration are derived and solved with the aid of Galerkin discretization. It is found, for a given flow velocity,that the first-mode frequency for in-plane motions is quite different from that for out-of-plane motions. However, the second-or third-mode frequencies for in-plane motions are approximately equal to their counterparts for out-of-plane motions, keeping almost constant values with increasing flow velocity. Moreover, the orientation angle of the postbuckling configuration plane for a buckled pipe can be significantly affected by initial conditions, displaying new features which have not been observed in the same pipe system factitiously supposed to deform in a single plane.

FORCED VIBRATIONS WITH INTERNAL RESONANCE OF A PIPE CONVEYING FLUID UNDER EXTERNAL PERIODIC EXCITATION

Applying the multidimensional Lindstedt-Poincare （MDLP） method, we study the forced vibrations with internal resonance of a clamped-clamped pipe conveying fluid under ex- ternal periodic excitation. The frequency-amplitude response curves of the first-mode resonance with internal resonance are obtained and its characteristics are discussed; moreover, the motions of the first two modes are also analyzed in detail. The present results reveal rich and complex dynamic behaviors caused by internal resonance and that some of the internal resonances are de- cided by the excitation amplitude. The MDLP method is also proved to be a simple and efficient technique to deal with nonlinear dynamics.

VIBRATION AND STABILITY OF VERTICAL UPWARD-FLUID-CONVEYING PIPE IMMERSED IN RIGID CYLINDRICAL CHANNEL

A theoretical model is developed for the vibration and stability of a vertical pipe subjected concurrently to two dependent axial flows. The external fluid, after exiting the outer annular region between the pipe and a rigid cylindrical channel, is conveyed upwards inside the pipe. This configuration thus resembles of a pipe that aspirating fluid. The equation of planar mo- tion is solved by means of the differential quadrature method （DQM）. Calculations are conducted for a slender drill-string-like and a bench-top-size system, for different confinement conditions of the outer annular channel. It is shown that the vibrations of these two systems are closely related to the degree of confinement of the outer annular channel. For a drill-string-like system with narrow annuli, buckling instability may occur in the second and third modes. For a bench-top-size system, however, both buckling and flutter may occur in the lowest three modes. The form of instability depends on the annuli size.

Extremely large-amplitude oscillation of soft pipes conveying fluid under gravity

In this work,the nonlinear behaviors of soft cantilevered pipes containing internal fluid flow are studied based on a geometrically exact model,with particular focus on the mechanism of large-amplitude oscillations of the pipe under gravity.Four key parameters,including the flow velocity,the mass ratio,the gravity parameter,and the inclination angle between the pipe length and the gravity direction,are considered to affect the static and dynamic behaviors of the soft pipe.The stability analyses show that,provided that the inclination angle is not equal to π,the soft pipe is stable at a low flow velocity and becomes unstable via flutter once the flow velocity is beyond a critical value.As the inclination angle is equal to π,the pipe experiences,in turn,buckling instability,regaining stability,and flutter instability with the increase in the flow velocity.Interestingly,the stability of the pipe can be either enhanced or weakened by varying the gravity parameter,mainly dependent on the value of the inclination angle.In the nonlinear dynamic analysis,it is demonstrated that the post-flutter amplitude of the soft pipe can be extremely large in the form of limit-cycle oscillations.Besides,the oscillating shapes for various inclination angles are provided to display interesting dynamical behaviors of the inclined soft pipe conveying fluid.

ANALYSIS OF COUPLED-MODE FLUTTER OF PIPES CONVEYING FLUID ON THE ELASTIC FOUNDATION

The governing equation of solid-liquid couple vibration of pipe conveying fluid on the elastic foundation was derived. The critical velocity and complex frequency of pipe conveying fluid on Winkler elastic foundation and two-parameter foundation were calculated by po,ver series method. Compared,with pipe without considering elastic foundation, the numerical results show that elastic foundation can increase the critical flow velocity of static instability and dynamic instability of pipe. And the increase of foundation parameters may increase the critical flow velocity of static instability and dynamic instability of pipe, thereby delays the occurrence of divergence and flutter instability of pipe. For higher mass ratio beta, in the combination of certain foundation parameters, pipe behaves the phenomenon of restabilization and redivergence after the occurrence of static instability, and then coupled-mode flutter takes place.