Quasi-periodic Motions Research Articles

Modeling and dynamics simulation of spur gear system incorporating the effect of lubrication condition and input shaft crack

PurposeThe paper is devoted to presenting a systematic investigation on the mechanical model and nonlinear dynamic characteristics of spur gear system with and without input shaft crack.Design/methodology/approachConsidering the backlash, load-distribution, time-varying meshing stiffness and sliding friction, the modelling of a 5DOF gear system is proposed. Likewise, stiffness and damping models under elastohydrodynamic lubrication are developed, and sliding friction between gear pair is also outlined. In particular, a cracked input shaft which affects the support stiffness is presented, and breathing crack in keyway is adopted. On this basis, the dynamic responses of a gear system with and without input shaft crack are examined using numerical method, and some classical response diagrams are given, illustrating the effect of the important parameters on the gear system.FindingsDynamic simulation demonstrates that there exist periodic, quasi-periodic and chaotic motions in the gear system, and rational speed of the gear pair has noteworthy effects on vibration characteristic. Besides, comparison between healthy and cracked condition of input shaft indicates that occurring of crack convert periodic motion to quasi-periodic or chaotic motion.Originality/valueThe results give an understanding of the operating conditions under which undesirable dynamic behavior occurs, and provide some useful information to design and diagnose such gear system with crack fault.

Asymmetric vibrations and chaos in spherical caps under uniform time-varying pressure fields

This paper presents a study on nonlinear asymmetric vibrations in shallow spherical caps under pressure loading. The Novozhilov’s nonlinear shell theory is used for modeling the structural strains. A reduced-order model is developed through the Rayleigh–Ritz method and Lagrange equations. The equations of motion are numerically integrated using an implicit solver. The bifurcation scenario is addressed by varying the external excitation frequency. The occurrence of asymmetric vibrations related to quasiperiodic and chaotic motion is shown through the analysis of time histories, spectra, Poincaré maps, and phase planes.

Research on the influence of system parameters on the electromechanical dynamics of a large wind turbine drivetrain

A wind turbine (WT) is a complex system integrating mechanical and electrical controls, and WTs can be coupled and interact with each other. This paper established an electromechanical model of a WT drivetrain and explored the influence of the system parameters of the WT drivetrain on the dynamic response and current harmonics. The results show that the introduction of a generator produces an electromagnetic frequency component in the mechanical system and that the mechanical parameters directly determine whether the generator undergoes periodic motion, quasiperiodic motion or chaotic motion, which affects the power generation efficiency of a WT. The research results can be used as a reference for the mechanical design of WT drivetrains and the selection of generators.

Nonlinear Forced Vibration of Spinning Pipes Conveying Fluid Under Lateral Harmonic Excitation

As a crucial offshore drilling equipment, the drill strings often encounter instability due to various external excitations. In order to deeply understand the dynamical mechanism of such issues, in this paper, the working drill strings are modeled as a spinning pipe conveying fluid, and the forced vibration characteristics of such system under an equivalent lateral excitation are studied analytically and numerically. The system considered is essentially under triple excitations: the fluid–structure interaction (FSI), spinning motion and external force. The viscoelastic material and geometrical nonlinearity due to extensible pipe axis are taken into account. The governing equation is derived by the Hamilton principle and then treated by the numerical simulation. The Lissajous trajectory and whirling motion shape of the pipe are obtained, in which the subcritical and supercritical cases are both discussed. It is revealed that the system conducts periodic or quasi-periodic motion at subcritical state, while presents chaotic motion at supercritical flow velocity. Perturbation analysis is also carried out to investigate the nonlinear frequency–amplitude features of the system. Comprehensive parametric studies are performed and the flow velocity, spinning speed, nonlinearity, excitation and damping are all found to have significant effects on the moving trajectory and frequency response of the pipe.

Dynamic analysis and stability prediction of nonlinear feed system coupled with flexible workpiece

The workpiece is prone to self-excited vibration when turning a slender shaft and the flexibility cannot be ignored in this condition. In this paper, a new flexible tool-workpiece system is developed to investigate the nonlinear dynamics and chatter stability for in a turning process. The piecewise-nonlinear restoring function of the tool system is derived by calculating axial deformations and forces of the ball screw feed system. With the variation of time delay, the dynamic behaviors of the cutting system are analyzed by numerical method, which manifests as periodic, quasi-periodic, and chaotic motions. Through determining the critical chip width under different spindle speed, the stability lobe diagram (SLD) of nonlinear system is obtained to distinguish stable and unstable states. In addition, the dynamic characteristics and SLD of nonlinear system are compared with linear system. An experiment is carried out to observe the vibration phenomenon under stable and unstable turning conditions and verify the accuracy of stability prediction.

Three-dimensional quasiperiodic torsional flows in rotating spherical fluids at very low Prandtl numbers

The aim of this study is to determine through numerical simulations the extent and robustness of the three-dimensional torsional dynamics of the thermal convection in rotating spherical fluids at very low Prandtl numbers. It is known that the kinetic energy of the periodic axisymmetric flows propagates latitudinally on the surface of the sphere. Here, it is shown that when the axisymmetry is broken at a secondary Hopf bifurcation, the flow starts to drift in the azimuthal direction giving rise to a quasiperiodic motion that propagates the energy in latitude and longitude. The double direction of propagation gives rise to a meandering path of the kinetic energy, which is still concentrated on the surface, but highly localized. Several new stable states of convection with different symmetries have been identified in a large range of Rayleigh numbers, all of them retaining the torsional motion of the basic velocity field. Particular attention is paid to their dependence on the Rayleigh number and on the values of the frequencies, of the mean zonal flow, and of the kinetic energy of the fluid.

Effects of journal static eccentricity on dynamic responses of a rotor system under base motions using FDM inertia model

When an oil-lubricated rotor-bearing system is subjected to base motions, it will inevitably be affected by oil-film forces, base excitations, and inertial forces. The alignment state cannot be guaranteed over time, resulting in a static eccentricity. An inertia model for open-ended SFD incorporated three-dimensional fluid velocities and pressure distribution represented by the Reynolds number. In addition to viscous pressure, the inertial pressure distribution was also solved by finite difference method (FDM), and the oil-film forces were determined by numerical integration. Using transient modal integration method, dynamic responses of a rotor-SFD-support system considering oil-film inertia were simulated. The effects of oil-film inertia, static eccentricity, and base motions on transient responses and bifurcation behavior were discussed. The results indicate that for large eccentricity ratio, the oil-film inertia increases the reaction forces. For medium and large static eccentricity, the journal performs quasi-periodic motions. Under base harmonic translation, a series of sub- and super-harmonic frequencies kΩ±Ωz are stimulated along with the multiples of rotational frequency kΩ and base harmonic frequency Ωz. The bifurcation motion of journal with static eccentricity fluctuates seriously in resonance. Overall, nonlinear dynamic responses of rotor-damping-support systems subjected to base motions can be evaluated during dynamics design and vibration suppression.

Dynamic Characteristics Analysis of Gear-Bearing System Considering Dynamic Wear with Flash Temperature

The influence of the dynamic wear model considering the tooth contact flash temperature on the dynamic characteristics of a gear-bearing system is studied. Firstly, the meshing stiffness model, based on flash temperature theory, is established. Then, the changing of tooth surface temperature and meshing stiffness in the process of gear meshing is analyzed. Next, the initial tooth surface wear is calculated based on the Archard theory, and the dynamic wear model of the system is established. Finally, the effects of initial wear, friction factors, and damping ratio on the system response are studied. The results show that with the increase of fractal dimension D, the uncertainty and the fluctuation amplitude of backlash decrease, and the meshing force decreases. Therefore, the initial tooth surface wear is reduced, and the stability of the system response with a dynamic wear model is improved; with the increase of the friction coefficient, the tooth surface flash temperature rises, and the root mean square value of the vibration displacement of the system amplifies, which indicates that the system tends to be unstable; with the increase of damping ratio, the system changes from unstable quasi-periodic and chaotic motion to the stable periodic motion. The increase of damping accelerates the energy loss of the system and makes the system prone to be stable.

Theoretical and Numerical Analysis of Coupled Axial-Torsional Nonlinear Vibration of Drill Strings

Based on a lumped parameter model with two degrees of freedom, the periodic response of the coupled axial-torsional nonlinear vibration of drill strings is studied by HB-AFT (harmonic balance and alternating frequency/time domain) method and numerical simulations. The amplitude-frequency characteristic curves of axial relative displacement and torsional relative angular velocity are given to reveal the mechanism of bit bounce and stick-slip motion. The stability of periodic response is analyzed by Floquet theory, and the boundary conditions of bifurcation of periodic response are given when parameters such as nominal drilling pressure, angular velocity of turntable, and formation stiffness are varied. The results show that the amplitude of the periodic response of the system precipitates a spontaneous jump and Hopf bifurcation may occur when the angular velocity of the turntable is varied. The variation of parameters may lead to the complex dynamic behavior of the system, such as period-doubling motion, quasiperiodic motion, and chaos. Bit bounce and stick-slip phenomenon can be effectively suppressed by varying the angular velocity of turntable and nominal drilling pressure.

Solitonic, quasi-periodic, super nonlinear and chaotic behaviors of a dispersive extended nonlinear Schrödinger equation in an optical fiber

Various dynamic behaviors of the dispersive extended nonlinear Schrödinger equation (NLSE) are studied in this paper. For this, a new Φ6−model expansion method is used to examine the solitary waves of the considered model. The main objective of this approach is to provide a medium by involving various parameters for Jacobian elliptic solutions. It is also noted that certain bright–dark optical solitons are given by the NLSE, as well as bell-shaped bright optical solitons. By applying Galilean transformation to evaluate the nonlinear and super nonlinear periodic behaviors in nonlinear optics, the system is transformed into a planar dynamical system. By using phase portrait analysis, we first investigate the presence of a set of novel optical solitons of the NLSE. The dynamic motions for the nonlinear, super nonlinear, periodic, quasi-periodic and chaotic motions are studied using various parametric conditions in the presence of external periodic force. Bifurcation analysis and sensitivity analysis for NLSE propagation is represented using phase plane analysis via phase plots of the dynamic structure. Further, via time series plots, we present analytical forms of the solitary waves.

Analytical Solution and Quasi-Periodic Behavior of a Charged Dilaton Black Hole

With the vast breakthrough brought by the Event Horizon Telescope, the theoretical analysis of various black holes has become more critical than ever. In this paper, the second-order asymptotic analytical solution of the charged dilaton black hole flow in the spinodal region is constructed from the perspective of dynamics by using the two-timing scale method. Through a numerical comparison with the original charged dilaton black hole system, it is found that the constructed analytical solution is highly consistent with the numerical solution. In addition, several quasi-periodic motions of the charged dilaton black hole flow are numerically obtained under different groups of irrational frequency ratios, and the phase portraits of the black hole flow with sufficiently small thermal parameter perturbation display good stability. Finally, the final evolution state of black hole flow over time is studied according to the obtained analytical solution. The results show that the smaller the integral constant of the system, the greater the periodicity of the black hole flow.

Response and stability analysis of a two-spool aero-engine rotor system undergoing multi-disk rub-impact

The main aim of this paper is to propose a numerical procedure for capturing the nonlinear dynamic characteristics of a two-spool aero-engine rotor system undergoing multi-disk rub-impact. In aero-engines, the possibility for the multi-disk rub-impact is high during the fan blade-out (FBO) event and subsequent windmilling action. It intensifies the nonlinear effects on the rotor vibrations and leads to certain undesired circumstances in the engine. The dual-rotor model consists of multi-stage compressors and single-stage turbines that undergo rubbing whenever their deflection exceeds the clearance. The dynamic model of the dual-rotor system is constructed using the tapered Timoshenko beam elements, rigid disks and rolling contact bearings. A proper model reduction technique based on component mode synthesis coupled with the Craig-Bampton substructuring is utilized to reduce the size of the finite element model. A semi-analytic technique called the approximate time variational method is employed to investigate the steady-state response of the system under multi-disk rub impact. Based on the proposed method, the response characteristics of the model are obtained and are verified with the numerical integration results. Compared to the single-disk rub-impact, the nonlinearities are intensified and significant variations are observed in the response characteristics and stability of the system. Period-5, quasi-periodic, and dry friction backward whirl motions are observed in the response for different values of the system parameters. During quasi-periodic motion, some unknown fractional components such as 0.716ω1, 0.766ω1, 0.916ω1 and 0.964ω1 are appeared in the response. Moreover, the dry friction backward whirl happened with a very large amplitude and it contains a superharmonic frequency component in the response.

Quantitative orbit classification of the planar restricted three-body problem with application to the motion of a satellite around Jupiter

In this work we numerically investigate the planar circular restricted three-body problem and apply this model to the Sun-Jupiter-particle problem where the particle is orbiting Jupiter. Our aim is to complement the qualitative mapping technique with quantitative techniques to trace a given trajectory in phase space. Qualitatively, this problem can be investigated using the Poincaré surface of sections mapping technique. Here we compute such maps for various Jacobian energies. While the computation of classic Poincaré surface of sections are useful to qualitatively classify phase-space regions with periodic, quasi-periodic, or chaotic motion, the method is inadequate to describe escape and/or collision orbits. To mitigate this shortcoming we complement such maps with the calculations of quantitative dynamical maps based on the Smaller-Alignment Index (SALI) technique. This allows for a complete assessment of the global dynamics of the problem resulting in a detailed classification of orbits of differing dynamical character. We revealed the network of simple periodic orbits around Jupiter, along with their linear stability. As a highlight, we identified a region of flipping orbits that are not detected with Poincaré surface of sections. We outline and discuss various assumptions. After a short review of the underlying model and applied numerical techniques, we present and discuss results from this work.

Molecular dynamics simulation of synchronization of a driven particle

Synchronization plays an important role in many physical processes. We discuss synchronization in a molecular dynamics simulation of a single particle moving through a viscous liquid while being driven across a washboard potential energy landscape. Our results show many dynamical patterns as the landscape and driving force are altered. For certain conditions, the particle's velocity and location are synchronized or phase-locked and form closed orbits in phase space. Quasi-periodic motion is common, for which the dynamical center of motion shifts the phase space orbit. By isolating synchronized motion in simulations and table-top experiments, we can study complex natural behaviors important to many physical processes.

Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser.

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known; however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

Dynamic Responses of the Aero-Engine Rotor System to Bird Strike on Fan Blades at Different Rotational Speeds

To study the effect of a bird striking engine fan on the rotor system, a low-pressure rotor system dynamic model based on a real aero-engine structure was established. Dynamic equations were derived considering the case of the bird strike force which transferred to the rotor system. The bird strike force was obtained from the bird strike process simulation in LS-DYNA, where a smoothed particle hydrodynamics (SPH) mallard model was constructed using a computed tomography (CT) scanner, and finite element method (FEM) was used to simulate the bird strike on an actual fan model. The dynamic equations were solved using the Newmark-β method. The effect of rotational speeds on the rotor system dynamics after bird strike was investigated and discussed. Results show that the maximum bird impact force can reach 104 kN at 3772 r/min. Impact time is only 0.06 s, but the bird strike on fan blades lead to a transient shock on the rotor system. Under the action of transient shocks, the rotor system displacement in the horizontal and vertical directions increase sharply, and the closer the mass point is to the fan, the more it is affected; the vibration amplitude at the fan will increase 15 times within 0.1 s of the bird strike and will gradually decrease with the effect of damping. The dynamics of the rotor system changes from a stable single periodic motion to a complex irregular quasi-periodic motion after a bird strike, and the strike force excites the first-order vibrational mode of the rotor system. This phenomenon occurs at all speeds when bird strikes occur. Bird strikes will cause resonance in the rotor system, which may cause damage to the engine. It was also seen that the bird strike force, and hence the effects on the rotor system, increases as the engine rotational speed increases; the peak force is larger and the number of peaks has increased. The impact force at 3772 r/min is 99.5 kN higher than at 836 r/min, and three additional peaks emerged. This effect is more reflected in the amplitude, and the overall vibration characteristics do not change. Combining the bird strike with the rotor dynamics calculation, the dynamic response of the aero-engine rotor system to bird strike is studied at different flight stages, which is of guiding significance for power evaluation of aero engines after bird strike.

Nonlinear dynamics of fluid-conveying composite pipes subjected to time-varying axial tension in sub- and super-critical regimes

In the present work, nonlinear dynamics of a simply-supported fluid-conveying composite pipe subjected to axial tension in sub- and super-critical regimes are investigated. The tension acting on the pipe is assumed to comprise an average component and a harmonically changing disturbance component. More specifically, the nonlinear equation of motion is derived based on the Euler-Bernoulli beam theory considering the von Kármán nonlinearity for large deformation assumption, which is then truncated via the Galerkin's approach. In addition, the Kelvin-Voigt model is adopted on the pipe materials with visco-elastic structural damping. Afterwards, parametric studies are performed to evaluate the effects of the axial tension and fiber orientation angle on pre- and post-buckling frequencies, critical fluid velocities and static buckling displacements. The global dynamic behaviors of the pipe are studied through the forms of bifurcation diagrams, phase portraits, Poincaré maps, power spectral density (PSD) diagrams and linear instability maps, in both sub- and super-critical regimes. The results show that, based on symmetric layers, pipes with smaller fiber orientation angles have larger critical divergence fluid velocities but smaller static buckling displacements. Some interesting dynamical phenomena, including periodic, multi-periodic, quasi-periodic and chaotic motions, can be observed with different fiber orientation angles. Moreover, the system shows ‘soft spring’ characteristic in super-critical regime, compared with the ‘hard spring’ characteristic in sub-critical case.

Time-Varying Shadows of Quasi-Periodic Motion Across Sections of the Flow Within Nearly Time-Periodic Three-Body Dynamics

Time-Varying Shadows of Quasi-Periodic Motion Across Sections of the Flow Within Nearly Time-Periodic Three-Body Dynamics

Geometrically exact dynamics of cantilevered pipes conveying fluid

In this paper, the global dynamics of a hanging fluid conveying cantilevered pipe with a concentrated mass attached at the free end is investigated. The problem is of interest not only for engineering applications, but also because it displays interesting and often surprising dynamical behaviour. The widely used nonlinear models based on the transverse motion of the pipe are not able to accurately capture the dynamical behaviour of the system at very high flow velocities. Thus, a high-dimensional geometrically-exact model is developed for the first time, utilising Hamilton’s principle together with the Galerkin modal decomposition technique. Extensive numerical simulations are conducted to investigate the influence of key system parameters. It is shown that at sufficiently high flow velocities past the first instability (Hopf bifurcation), the system undergoes multiple bifurcations with extremely large oscillation amplitudes and rotations, beyond the validity of third-order nonlinear models proposed to-date. In the presence of an additional tip mass, quasi-periodic and chaotic motions are observed; additionally, it is shown that for such cases an exact model is absolutely essential for capturing the pipe dynamics even at relatively small flow velocities beyond the first instability.

Probing the phase space of coupled oscillators with Koopman analysis.

With the development of probing and computing technology, the study of complex systems has become a necessity in various science and engineering problems, which may be treated efficiently with Koopman operator theory based on observed time series. In the current paper, combined with a singular value decomposition (SVD) of the constructed Hankel matrix, Koopman analysis is applied to a system of coupled oscillators. The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures with periodic, quasiperiodic, or chaotic motion. By checking the amplitude of the principal modes along a straight line in the phase space, cusps of different sizes on the magnitude profiles are identified whenever a qualitative change of dynamics takes place. There seems to be no obstacle to extend the current analysis to high-dimensional nonlinear systems with intricate orbit structures.