Non-trivial Equilibrium Configuration Research Articles

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.

Parametric resonance for pipes conveying fluid in thermal environment

Parametric resonance is a kind of specific fluid-induced vibration for pipes conveying fluid. This paper focuses on revealing the qualitative characteristics of parametric resonances of the pipe with pulsating fluid speed in thermal environment, and compares the differences between the resonance characteristics in subcritical and supercritical regions. According to the generalized Hamilton's principle, the partial-differential-integral governing equation of a straight pipe is established. Under simply-supported boundary conditions, the non-trivial equilibrium configuration is obtained analytically. The governing equation of a supercritical pipe is derived by coordinate substitution on the basis of the new equilibrium configuration. The approximate analytical solution of parametric resonance for a pipe conveying fluid in a thermal environment is obtained by using the direct multi-scale method. The approximate analytical solution is verified to be reliable by using the Runge-Kutta method. The stability bounds of the parameters inducing parametric resonance are investigated via the introduced analytical method. The results show that the influence of temperature increment on sub-harmonic resonance is non-monotonic. In the subcritical region, when the temperature increment increases, the unstable region widens. This means that the system is more prone to parametric resonance, and the response amplitude decreases. In the supercritical region, when the temperature increment increases, the unstable bandwidth decreases, and the response amplitude increases. When viscous damping is increased or the average velocity is decreased, the sub-critical instability region decreases, and the supercritical instability region increases. With the increase in the pulsation velocity, the unstable bandwidth of subcritical and supercritical regions will increase, and parametric resonance is more likely to occur.

Static bifurcation and nonlinear vibration of pipes conveying fluid in thermal environment

In practical engineering, changes of environment temperature can significantly alter natural characteristics of a pipe system. In current work, the exact non-trivial equilibrium configuration of the pipe conveying fluid in a thermal environment is investigated for the first time, including the effect of temperature on natural frequencies and bending vibrations. Firstly, the partial-differential-integral governing equation for bending vibrating pipes subjected to changing temperature is established based on the Euler-Bernoulli beam theory, by means of the generalized Hamilton's principle. Exact solutions for non-trivial equilibrium configurations and critical flow velocities are obtained analytically. Stability of these configurations is discussed and reveals that just the first one is stable. The nonlinear resonance properties of the pipe conveying fluid in a thermal environment are analyzed by the harmonic balance method (HBM) on the basis of Galerkin truncation. The analytical results are verified by the differential quadrature element method (DQEM). It is found that an increasing temperature leads to a larger non-trivial equilibrium configuration and a lower critical flow velocity, with a consequent decrease in the stability of the system. High temperature and high flow velocity reduce the resonant frequency in the subcritical region. In the supercritical region, they increase the resonant frequency. Unlike pipes in the subcritical region, the response in the supercritical region has more resonant peaks and exhibits soft characteristics due to the quadratic nonlinearity. Besides, the asymmetry of the response caused by the zero drift is more pronounced at low temperatures or flow velocities. This study provides theoretical guidance for the vibration design of pipes conveying fluids in a thermal environment.

Free vibration analysis of Timoshenko pipes with fixed boundary conditions conveying high velocity fluid

Sever vibration will be induced due to the high flow velocity in the pipe. When the flow velocity exceeds the critical value, the static equilibrium configuration of the pipe loses its stability, and the vibration properties change accordingly. In this paper, the free vibration characteristics of the pipe with fixed-fixed ends are revealed in the supercritical regime. Based on the Timoshenko beam theory, the governing equations of the nonlinear vibration near the non-trivial static equilibrium configuration are established. The influences of system parameters on equilibrium configuration, critical velocity, and free vibration frequency is analyzed. The effects of supercritical velocity in different ranges on the natural frequencies are revealed. In addition, the comparison with the Euler-Bernoulli pipe model shows that the differences in critical velocity, equilibrium configuration, and frequency are still significant even the length-diameter ratio is large. The increase of the flow velocity reduces the difference of non-trivial static equilibrium configurations, but eventually aggravates the difference of natural frequencies. Within a certain supercritical velocity range, the vibration difference between the two pipe models is small, beyond this range, the vibration difference increases significantly.

Nonlinear forced vibrations of a slightly curved pipe conveying supercritical fluid

Vibrations of pipes caused by axially flowing fluids are very common in engineering applications. Due to material imperfections, guide misalignment, and improper supports, the installed pipes are prone to the initial curvature. Though small, the initial curvature can significantly change the dynamic characteristics of the slightly curved pipe system. This study investigates the non-linear forced vibration of a slightly curved pipe conveying supercritical fluid around the curved equilibrium, with the emphasis on amplitude–frequency responses around two asymmetric non-trivial equilibrium configurations. The governing equations for the forced vibration of a slightly curved pipe conveying supercritical fluids are derived using the generalized Hamilton principle. Then, the equations of motion are discretized into a set of coupled ordinary differential equations via the Galerkin truncation method and solved by the harmonic balance method combined with the pseudo-arc length technique. The approximate analytical results are verified by the numerical integration results. The obtained results demonstrate that the initial curvature has a significant effect on the dynamic characteristics of pipes conveying supercritical fluids, and can lead to significant differences in the dynamic response of the pipe system near different equilibrium configurations.

Critical velocity and supercritical natural frequencies of fluid-conveying pipes with retaining clips

Although retaining clips are widely used to constrain high-velocity fluid-conveying pipes, there are few studies on the influence of retaining clips on the supercritical vibration characteristics of the pipes. In this study, the vibration characteristics of the fluid-conveying pipe with retaining clip are firstly presented in the supercritical regime. According to the generalized Hamiltonian principle, the governing equation of the fluid-conveying pipe with the retaining clip is derived. When the flow velocity exceeds the critical velocity, the equilibrium configuration of the pipe becomes the non-trivial static equilibrium configuration. By coordinate transformation, the governing equation of the supercritical pipe around the non-trivial static equilibrium configuration is obtained. Emphasis is put on the effects of the clip stiffness and position on the critical flow velocity, equilibrium configuration and natural frequencies. It is found that variations in the retaining clip stiffness can cause different types of non-trivial equilibrium configurations. The natural frequencies will not be monotonic with the increasing clip stiffness in the supercritical flow. Furthermore, the two-span model of the pipe with the clip is obtained for verification. In conclusion, this work lays a foundation for the study of the dynamic response of the supercritical fluid-conveying pipe with retaining clip and can provide a reference for the analysis of the vibration characteristics of the clips and pipes system in engineering.

On the nonlinear effects of the mean wind force on the galloping onset in shallow cables

The critical aeroelastic behavior of horizontal, suspended, and shallow cables is analyzed via a continuous model accounting for both external and internal damping. Quasi-steady aerodynamic forces are considered, including their stationary contribution (mean wind force). This latter induces a rotation of the cable (steady swing) around the line connecting the suspension points, together with a deformation of the initial equilibrium profile under self-weight. First, by using perturbation methods, the nontrivial equilibrium configuration is analytically determined as a nonlinear function of the wind velocity. Then, the wind critical values at which bifurcations take place and the corresponding modal shapes are determined by solving a boundary value problem in the complex field. Numerical investigations are carried out to validate the perturbation solution. A preliminary nonlinear galloping analysis is also performed to verify the galloping onset in terms of non-trivial equilibrium path and critical modes. The nonlinear terms related to the fundamental path, from which bifurcations take place, play a key role revealing new insights. They are able to heavily influence the system bifurcation, making unstable configurations which instead would be aerodynamically stable without considering the effect of the mean wind force.

Parametric resonances of Timoshenko pipes conveying pulsating high-speed fluids

Parametric resonances of pipes caused by pulsating fluids have received much attention. However, the main concern is the pulsation of subcritical flow. Moreover, it is usually based on the Euler-Bernoulli pipe model. This paper focuses on revealing the characteristics of parametric resonances of the Timoshenko pipe with pulsation of supercritical high-speed fluids. Supercritical flow causes the linear instability of the pipe. The coupled partial differential equations with varying parameters are derived for governing the vibration of the pipe around the non-trivial static equilibrium configuration. A direct multi-scale method is developed to analytically obtain parametric resonance responses from coupled partial differential equations with varying parameters. For the first time, the nonlinear parametric response of the Timoshenko pipe is verified by using the finite difference method (FDM). Some interesting phenomena are demonstrated. For example, unlike parametric resonance at subcritical speeds, the steady-state response of velocity pulsation excitations at supercritical speeds is not monotonic with changes in some physical parameters. For another example, when pulsation occurs at supercritical speeds, the smaller steady-state response and the instability region can be obtained through the Timoshenko pipe model. In addition, the relationship between the instability threshold of the velocity pulsation amplitude and the slenderness ratio is non-monotonic, suggesting that the Euler-Bernoulli pipe model may simplify some vibration characteristics. Due to these different phenomena, the necessity of studying the velocity pulsation in the supercritical high-speed range and the necessity of the Timoshenko model are demonstrated.

Primary and super-harmonic resonances of Timoshenko pipes conveying high-speed fluid

High-speed flow pipes suffer from severe vibration problems. When the fluid velocity is higher than the critical value, the straight equilibrium configuration of the pipe will lose stability. What follows is the supercritical vibration of the pipe near the non-trivial static equilibrium configuration. This paper attempts to reveal multiple resonance responses of forced vibration of pipes in the supercritical regime. Based on Timoshenko beam theory, the nonlinear coupled partial differential equations are deduced. The non-trivial static equilibrium configuration causes the parameters to vary with space variable. The approximate responses of the pipe are obtained and verified numerically. The results show that the flow velocity near the critical value is more prone to cause severe vibration. Unlike in the subcritical regime, there are third-order super-harmonic resonance and second-order super-harmonic resonance in the supercritical regime. So there are more resonance areas in the supercritical regime. High flow velocity or large external excitation can aggravate the difference between the Euler-Bernoulli model and the Timoshenko model, and the relative error of two models varies non-monotonically. Even for slender pipes, the difference between the two models is still very clear. Therefore, the Timoshenko model is more necessary to analyze the vibration of the high-speed pipe.

Vibration around non-trivial equilibrium of a supercritical Timoshenko pipe conveying fluid

Vibration characteristics of pipes conveying fluid in the super-critical range are investigated by using Timoshenko beam theory for the first time. Generalized Hamiltonian principle is applied to derive the nonlinear transverse vibration governing equation. The non-trivial static equilibrium configuration and critical flow velocity of the pipe are analytically deduced. These analytical results are verified by using the finite difference method. Compared with Euler-Bernoulli flow pipeline, it is found that the equilibrium configuration of Timoshenko pipe is larger. In the supercritical regime, natural frequencies of the Timoshenko flow pipe are produced by the Galerkin truncation method. Numerical examples illustrate that vibration characteristics of the pipe are highly sensitive to length, thickness, shear modulus and velocity. The relative difference between the two pipe models is influenced by the velocity of the flow and is likely to exceed 100%. In general, this work found that the flow velocity makes the Timoshenko beam theory even more needed for researching vibration properties of pipes conveying fluid, especially at a high velocity.

Many-particle limits and non-convergence of dislocation wall pile-ups

The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repulsive force on each other, resulting in a nontrivial equilibrium configuration. This class of particle systems covers the setting of pile-ups of dislocation walls, which is an idealised setup for studying the microscopic origin of several dislocation density models in the literature. Such density models are used to construct constitutive relations in plasticity models.Our aim is to pass to the many-particle limit. The main challenge is the combination of the nonlocal nature of the interactions, the singularity of the interaction potential between particles of the same type, the non-convexity of the the interaction potential between particles of the opposite type, and the interplay between the length-scale of the domain with the length-scale of the decay of the interaction potential. Our main results are the Γ-convergence of the energy of the particle positions, the evolutionary convergence of the related gradient flows for sufficiently large, and the non-convergence of the gradient flows for sufficiently small.

Nonlinear frequency analysis of buckled nanobeams in the presence of longitudinal magnetic field

The nonlinear free vibration around the postbuckling configuration of nonlocal nanobeams with pinned–pinned boundary conditions are analytically investigated, considering the geometric nonlinearity arising from the mid-plane stretching and particularly the effect of a longitudinal magnetic field. When the applied axial force is small, the nanobeam is stable and its free vibration is around the straight equilibrium position. As the axial force becomes large and is beyond a certain critical value (critical axial force), however, either positive or negative non-trivial equilibrium configuration occurs and the free vibration is around the buckled configuration. The governing equation for the nanobeam before buckling exhibits a cubic nonlinearity while for a buckled nanobeam it contains both cubic and quadratic nonlinearities. Based on the nonlinear governing equations, approximate analytical solutions for the postbuckling configuration in terms of the applied axial force are evaluated, showing that the magnetic field parameter has a great effect on the critical axial force, buckled displacement and linear frequencies of the nanobeam system. More interestingly, the nonlinear frequency is found to be generally higher than the linear frequency in the case of prebuckling regime while it could be lower than the linear one in the case of postbuckling regime. The nonlinear frequency can be significantly influenced by the magnetic field, the nonlinear coefficient associated with the mid-plane stretching, and the initial vibration amplitude.

Forced vibration of axially moving beam with internal resonance in the supercritical regime

Local and global resonances under the condition of 3:1 internal resonance of a super-critically axially moving beam, subjected to a harmonic exciting force, are investigated in the present work. The governing equation is derived from the generalized Hamilton's principle and discreted into a multiple-degrees-of-freedom system by the Galerkin's method. In the super-critical regime, the axially moving beam becomes a bistable system with two symmetrical non-trivial equilibrium configurations. Based on the transformation around one of them, natural frequencies and the condition of internal resonance are obtained. By employing the method of multiple scales, resonances for first-two modes and harmonics under the condition of internal resonance are discussed analytically. Total displacement at the middle of the beam is composed by them and confirmed by direct numerical method. Internal resonance is found to have a big effect on the phase angle of and the amplitude. Coupling ship between the first-two modes is verified to be produced by the cubic nonlinearity and the 3:1 commensurability together. The effect of moving speed acting on the internal resonance is discussed and an energy transmission region is found. Different with the internal resonance in the sub-critical regime, most of the transferred energy is absorbed by the quadratic nonlinearity in the super-critical regime. The critical excitation of the local response is predicted by the analytical method and certified by simulations. The global response for the primary resonance has two stable focal points. However, the global response for the secondary resonance only has one stable focal point for the non-trivial equilibrium configuration is counteracted.

Global Dynamics of Pipes Conveying Pulsating Fluid in the Supercritical Regime

Global dynamics of supercritical pipes conveying pulsating fluid considering superharmonic resonance of the second mode with 1:2 internal resonance are investigated. The governing partial differential equations in the supercritical regime are obtained based on the nontrivial equilibrium configuration of the pipes conveying fluid and then transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motion to the near-integrable Hamiltonian standard form. The energy-phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are subsequently interpreted in terms of the physical motion of such gyroscopic system. Two types of nonlinear normal modal motion and the chaotic pattern conversion between the locked simple bidirectional traveling wave motion and the complex bidirectional traveling wave motion are discussed.

Homoclinic orbits and chaos of a supercritical composite panel with free-layer damping treatment in subsonic flow

Multi-pulse homoclinic orbits and chaotic dynamics of a supercritical composite panel with free layer damping treatment in subsonic flow are investigated considering one to two internal resonance. Inviscid potential flow theory is employed to exhibit the aerodynamic pressure and Kelvin’s model is used to describe the viscoelastic property of the free damping layer. By Hamilton’s principle, the governing equation of the composite panel in the subcritical regime is derived. In the supercritical regime, the buckling configuration is solved analytically and the PDE is obtained by introducing a displacement transformation for nontrivial equilibrium configuration. Then the governing equation in the first supercritical region is transformed into a discretized nonlinear gyroscopic system via assumed modes and then Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motions to the near-integrable Hamiltonian standard forms. The Energy-Phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are finally interpreted in terms of the physical motion of the gyroscopic continua and the dynamical mechanism of chaotic pattern conversion between the forward traveling wave motion and the complex bidirectional traveling wave motion are discussed.

Parametric resonance of a translating beam with pulsating axial speed in the super-critical regime

In the super-critical regime, parametric resonance of an axially translating beam is investigated for the first time. Resonances are found to be generated while the pulsated frequency is equal to not only two times of the natural frequency, but also just the natural frequency. By using the method of multiple scales, parametric resonance responses are solved out. Validity of the analysis method is certified by simulations. The quadratic nonlinearity caused by nontrivial equilibrium configuration is taken into consideration with the original cubic nonlinearity. A high-order multi-scaled perturbed solution with multi-modal shapes is introduced to ensure the accuracy of the results. The effects of the principal parameters are studied to certify the conditional existence of the parametric resonance. The parametric resonance in the sub-critical regime is used to demonstrate the difference between these two conditions. Comparison indicates that the translating beam at super-critical speed is more sensitive and the vibration is harder to be restrained.

Passive scalars, moving boundaries, and Newton's law of cooling

We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.

Warping and Ljapounov stability of non-trivial equilibria of non-symmetric open thin-walled beams

We investigate the effects of warping on the dynamic stability of non-trivial equilibrium configurations for non-symmetric open thin-walled beams. We use a direct one-dimensional model coarsely describing warping; the rest of the kinematics is exact. Dynamic derives from the balance of power; constitutive relations are non-linear, hyper-elastic, and distinguish the roles of the centroids and shear centres; inertial actions account for warping, too. By centred finite differences, the warping inertial action is found ineffective on the natural angular frequencies. Then, we follow non-trivial equilibrium paths and investigate their Ljapounov stability, by examining the small superposed oscillations. Results for generic, non-symmetric cross-sections are presented and discussed, showing the effects of warping and of coupling constitutive coefficients.

Global dynamics of an axially moving buckled beam

A parametric study for post-buckling analysis of an axially moving beam is conducted considering four different axial speeds in the supercritical regime. At critical speed, the trivial equilibrium configuration of this conservative system becomes unstable and the system diverges to a new non-trivial equilibrium configuration via a pitchfork bifurcation. Post-buckling analysis is conducted considering the system undergoing a transverse harmonic excitation. In order to obtain the equations of motion about the buckled state, first the equation of motion about the trivial equilibrium position is obtained and then transformed to the new coordinate, i.e. post-buckling configuration. The equations are then discretized using the Galerkin scheme, resulting in a set of nonlinear ordinary differential equations. Using direct time integration, the global dynamics of the system is obtained and shown by means of bifurcation diagrams of Poincaré maps. Other plots such as time traces, phase-plane diagrams, and Poincaré sections are also presented to analyze the dynamics more precisely.

Stability of non-trivial equilibrium paths of beams on a partially visco-elastic foundation

We consider a cantilever beam partially resting on a linear visco-elastic foundation of generalized Winkler type. The length and placement of the partial foundation are variable. The beam is subjected to a sub-tangential force at its unconstrained end. The stability of some of its non-trivial equilibrium configurations is investigated by a numerical procedure based on a finite differences technique. The critical boundaries of buckling and flutter are found; it turns out that the critical conditions for both static and dynamic instability depend on some physical parameters, and interactions between the boundaries of the domains of stability appear.