Fly-wheel model exhibits the hither and thither motion of a bouncing ball

The to and fro motion of a bouncing ball on a flat surface is represented by a low-dimensional model. To describe the repeated reversals of the horizontal velocity of the ball, the elasticity of the ball has to be taken into account. We show that a simple fly-wheel model exhibits the observed hither and thither motion of elastic balls. The suggested model is capable of describing oblique impacts of spherical bodies, which can be important in many applications, including dynamical simulation of granular materials. We find that the behaviour of the bouncing fly-wheel is sensitive to the initial conditions, and the escape time plots are used to illustrate this observation.     

The dynamics of a bouncing ball with a sinusoidally vibrating table revisited

The dynamical behavior of a bouncing ball with a sinusoidally vibrating table is revisited in this paper. Based on the equation of motion of the ball, the mapping for period-1 motion is constructured and thereby allowing the stability and bifurcation conditions to be determined. Comparison with Holmes's solution [1] shows that our range of stable motion is wider, and through numerical simulations, our stability result is observed to be more accurate. The Poincaré mapping sections of the unstable period-1 motion indicate the existence of identical Smale horseshoe structures and fractals. For a better understanding of the stable and chaotic motions, plots of the physical motion of the bouncing ball superimposed on the vibration of the table are presented.     

含时滞反馈的楔式制动系统动力学分析

考虑含时滞反馈的影响,建立楔式制动系统动力学模型,运用多尺度方法对黏滑界面附近区域进行受迫主共振求解,分析时滞量、楔角与系统刚度对系统幅频响应的影响,应用Routh-Hurwitz判据分析系统稳定性的影响因素。基于解析解的分析表明:稳态幅值和稳定性边界都随时滞量发生周期性变化,周期内较大的时滞量引起鞍结分岔,并发展至不稳定多解;楔角和系统刚度增加引起主共振振幅增大,并扩大了不稳定区域。     

An analytical and numerical study of failure waves

Based on recent observations in shock experiments on glasses, a new failure process has been suggested for a certain type of brittle solids, in which a failure wave propagates through a solid at some distance behind the compressive stress wave near but below the Hugoniot elastic limit. Since the failure wave phenomenon is different from the usual inelastic shock waves, a combined analytical and numerical effort is made in this paper to explore the impact failure mechanisms associated with the failure wave. Based on the experimental data available, it appears that the physical picture of failure wave is related to local dilatation due to shear-induced microcracking. A mathematical argument then leads to the conclusion that the failure wave should be described by a diffusion equation instead of a wave equation, which is in line with the bifurcation analysis for localization problems. However, the occurrence of different governing equations in a single computational domain imposes both an analytical and a numerical challenge on the design of an efficient solution scheme. With the use of a partitioned-modeling approach, a simple solution procedure is proposed for failure wave problems, which is verified by the comparison with data.     

A NUMERICAL AND ANALYTICAL STUDY ON A TAIL-FLAPPING MODEL FOR FISH FAST C-START＊

The force production physics and the flow control mechanism of fish fast C-start are studied numerically and theoretically by using a tail-flapping model. The problem is simplified to a 2-D foil that rotates rapidly to and fro on one side about its fixed leading edge in water medium. The study involves the simulation of the flow by solving the two-dimensional unsteady incompressible Navier-Stokes equations and employing a theoretical analytic modeling approach. Firstly, reasonable thrust magnitude and its time history are obtained and checked by fitting predicted results coming from these two approaches. Next, the flow fields and vortex structures are given, and the propulsive mechanism is interpreted. The results show that the induction of vortex distributions near the trailing edge of the tail are important in the time-averaged thrust generation, though the added inertial effect plays an important role in producing an instant large thrust especially in the first stage. Furthermore, dynamic and energetic effects of some kinematic controlling factors are discussed. For enhancing the time-averaged thrust but keeping a favorable ratio of it to time-averaged input power within the limitations of muscle ability, it is recommended to have a larger deflection amplitude in a limited time interval and with no time delay between the to-and-fro strokes. The project supported by the CAS (KJCX-SW-L04)     

The mathematical phenomenological mapping in non-linear dynamics of spur gear pair and radial ball bearing due to the variable stiffness

In this paper the philosophy of mathematical phenomenological mapping has been applied to the non-linear dynamics of spur gears and radial ball bearings. The spur gear pair dynamics and rolling element bearing dynamics are analyzed separately, but with a tendency to reduce the both of the systems to the same mathematical model. The different reasonable assumptions are taken in every of these analyzes, but they do not have significant influence to the accuracy of the results. The systems are reduced to the single degree of freedom dynamics model. The total gear stiffness and ball bearing stiffness are recognized as the main influent factor of vibration behavior of these machine elements. Therefore, the special attention was paid to the new approach and procedure for stiffness solving and related problems. A single spur gear pair dynamics is solved and the results for total gear stiffness and vibration are shown. The conclusions emphasize the importance of described parallel analyzes in order to reduce the calculation time in solving different phenomena with usage of the principle of mathematical phenomenology.     

Conservation laws and solutions of a quantum drift-diffusion model for semiconductors

A non-linear system of partial differential equations describing a quantum drift-diffusion model for semiconductor devices is investigated by methods of group analysis. An infinite number of conservation laws associated with symmetries of the model are found. These conservation laws are used for representing the system of equations under consideration in the conservation form. Exact solutions provided by the method of conservation laws are discussed. These solutions are different from invariant solutions.     

Unsteady simple shear flow in a viscoplastic fluid: comparison between analytical and numerical solutions

In this paper, an unsteady flow of a viscoplastic fluid for simple shear flow geometry is solved numerically using two regularizing functions to overcome the discontinuity for zero shear rate of the Bingham constitutive law. The adopted models are the well-known Papanastasiou relation and one based on the error function. The numerical results are compared with the analytical solution of the same problem obtained by Sekimoto (J Non-Newton Fluid Mech 39:107–113, 1991). The analysis of the results emphasizes that the errors are much smaller in the yielded than in the unyielded region. The models approximate closer the ideal Bingham model as the regularization parameters increase. The differences between the models tend to vanish as the regularization parameters are at least greater than 10⁵.     

Numerical simulations of chaotic dynamics in a model of an elastic cable

The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities.The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.Part of this work was presented at the XVIIth Int. Congr. of Theor. and Appl. Mech., Grenoble, August 1988.     

An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam

Autoparametric interaction and the associated phenomenon of amplitude saturation are experimentally observed in a physical model of cable-and-beam structure. In this system, the horizontal beam is fixed at one end and supported at the other end by an inclined taut cable. The longitudinal axes of beam and cable are in a vertical plane. Three natural frequencies of the system are approximately of the ratio 1:1:2. This is a combination of two conditions that are very likely to occur in relatively long-span, multi-stay-cable bridges, namely, 1:1 tuning and 1:2 superharmonic tuning. While the beam is vertically excited with sufficiently large force near a primary resonance, the cable vibrates horizontally at half of excitation frequency. The beam also vibrates horizontally at half-frequency, as well as vertically. As the vertical excitation on the bean is further increased in amplitude, the vertical vibration amplitude gets saturated instead of increasing proportionately. A 3DOF analytical model of the structure is also derived, where the finite motion of the cable introduces geometric nonlinearities in quadratic and cubic forms. The system parameters having been carefully measured from the experimental model, steady-state solutions of the coupled nonlinear equations of motion are obtained, by the perturbation method of multiple time scales. Agreement between experimental observation and analytical prediction is very good, both qualitatively and quantitatively. Very good agreement is found also in the case of horizontal excitation of the beam, where effects of linear and nonlinear interaction are apparent.     

A model of two-velocity particles transport in a porous medium

A one-dimensional flow of suspension with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model of deep bed filtration which generalizes the known equations of mass balance and particle capture kinetics for a flow of fluid with identical particles is developed. The exact solution is evaluated at the filter inlet and on the concentration front of fast suspended and retained particles, asymptotic solutions are provided in certain vicinities of these lines. A global asymptotic solution to the problem with a small limit deposit is constructed. The asymptotics rapidly converges to the numerical solution.     

An analytical study of time-delayed control of friction-induced vibrations in a system with a dynamic friction model

We investigate the control of friction-induced vibrations in a system with a dynamic friction model which accounts for hysteresis in the friction characteristics. Linear time-delayed position feedback applied in a direction normal to the contacting surfaces has been employed for the purpose. Analysis shows that the uncontrolled system loses stability via. a subcritical Hopf bifurcation making it prone to large amplitude vibrations near the stability boundary. Our results show that the controller achieves the dual objective of quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. Consequently, the controlled system is globally stable in the linearly stable region and yields small amplitude vibrations if the stability boundary is crossed due to changes in operating conditions or system parameters. Criticality curve separating regions on the stability surface corresponding to subcritical and supercritical bifurcations is obtained analytically using the method of multiple scales (MMS). We have also identified a set of control parameters for which the system is stable for lower and higher relative velocities but vibrates for the intermediate ones. However, the bifurcation is always supercritical for these parameters resulting in low amplitude vibrations only.     

An analytical approximated solution and numerical simulations of a non-ideal system with saturation phenomenon

Nonlinear Dynamics - Nowadays, researches about non-ideal problems have been increased considerably in technical-scientific community. Nonlinear problems have been widely studied due to their...     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.     

Nonlinear dynamics and chaos of a finite-degree-of-freedom model of a buckled rod

The simple example of a mechanical system expressively exhibiting unpredictable and chaotic motions is a rod compressed by a supercritical force and subjected to a time-dependent transverse loading. Dynamics of this system can be analyzed either through modal analysis or through another lumped parameter modelling, for example, by discretization of the rod into an ensemble of segments. The paper is aimed to present the latter formulation of the problem and to discuss numerical results obtained in this framework.

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Sommario Un semplice esempio di sistema meccanico in grado di esibire in modo espressivo comportamenti dinamici non predicibili e caotici è rappresentato da una trave compressa in regime supercritico e soggetta ad un carico trasversale dipendente dal tempo. La dinamica di questo sistema può essere analizzata tramite approssimazioni modali, ovvero attraverso una modellazione a parametri concentrati, ad esempio discretizzando la trave in elementi rigidi con deformabilità localizzate. Il lavoro presenta quest'ultima formulazione del problema e ne discute i relativi risultati numerici.

     

New solvable problems in the dynamics of a rigid body about a fixed point in a potential field

We determine the general form of the potential of the problem of motion of a rigid body about a fixed point, which allows the angular velocity to remain permanently in a principal plane of inertia of the body. Explicit solution of the problem of motion is reduced to inversion of a single integral. A several-parameter generalization of the classical case due to Bobylev and Steklov is found. Special cases solvable in elliptic and ultraelliptic functions of time are discussed.