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.     

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China     

Nonlinear reduced-order model for vertical sloshing by employing neural networks

The aim of this work is to provide a reduced-order model to describe the dissipative behavior of nonlinear vertical sloshing involving Rayleigh–Taylor instability by means of a feed forward neural network. A 1-degree-of-freedom system is taken into account as representative of fluid–structure interaction problem. Sloshing has been replaced by an equivalent mechanical model, namely a boxed-in bouncing ball with parameters suitably tuned with performed experiments. A large data set, consisting of a long simulation of the bouncing ball model with pseudo-periodic motion of the boundary condition spanning different values of oscillation amplitude and frequency, is used to train the neural network. The obtained neural network model has been included in a Simulink®  environment for closed-loop fluid–structure interaction simulations showing promising performances for perspective integration in complex structural system.

     

Large Motions of a Moored Floating Breakwater Modeled as an Impact Oscillator

Two-dimensional motions of a floating breakwater moored to thesea floor by two cables are considered. The breakwater is modeled bothas a point mass and as a rigid body. The mooring lines are assumed tohave no effect on the breakwater when they are slack, and to provide aninstantaneous impulsive force when they become taut, analogous to animpact oscillator or a ball bouncing on a rigid surface. The axialcomponent of the velocity is reduced at this instantaneous tautcondition. Fluid inertia and damping are not included, and the waveforces are assumed to be harmonic. A critical force is defined, and theeffects of the forcing frequency, the coefficient of restitution, andthe shape and size of the body on the critical force are examined.Trajectories of the motion are plotted and the impact velocities arecomputed and analyzed. Knowledge of the number and magnitude of theseimpacts is useful in assessing fatigue of the mooring lines.     

The bouncing motion appearing in a robotic system with unilateral constraint

The hopping or bouncing motion can be observed when robotic manipulators are sliding on a rough surface. Making clear the reason of generating such phenomenon is important for the control and dynamical analysis for mechanical systems. In particular, such phenomenon may be related to the problem of Painlevé paradox. By using LCP theory, a general criterion for identifying the bouncing motion appearing in a planar multibody system subject to single unilateral constraint is established, and found its application to a two-link robotic manipulator that comes in contact with a rough constantly moving belt. The admissible set in state space that can assure the manipulator keeping contact with the rough surface is investigated, and found which is influenced by the value of the friction coefficient and the configuration of the system. Painlevé paradox can cause either multiple solutions or non-existence of solutions in calculating contact force. Developing some methods to fill in the flaw is also important for perfecting the theory of rigid-body dynamics. The properties of the tangential impact relating to the inconsistent case of Painlevé paradox have been discovered in this paper, and a jump rule for determining the post-states after the tangential impact finishes is developed. Finally, the comprehensively numerical simulation for the two-link robotic manipulator is carried out, and its dynamical behaviors such as stick-slip, the bouncing motion due to the tangential impact at contact point or the external forces, are exhibited.     

Bifurcations of periodic solutions for plane mappings

In this paper,using some techniques,we prove that there exists the regular homoclinic point for Taylor mapping with 4相似文献   

A pseudo-stable structure in a completely invertible bouncer system

It is shown that a pseudo-stable structure of non-asymptotic convergence may exist in a completely invertible bouncing ball model. Visualization of the pattern of H-ranks helps to identify this structure. It appears that this structure is similar to the stable manifold of non-invertible nonlinear maps which govern the non-asymptotic convergence to unstable periodic orbits. But this convergence to the unstable repeller of the bouncing ball problem is only temporary since non-asymptotic convergence cannot exist in completely invertible maps. This nonlinear effect is exploited for temporary stabilization of unstable periodic orbits in completely reversible nonlinear maps.     

Numerical simulations of bouncing jets

Bouncing jets are fascinating phenomenon occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non‐Newtonian fluids when the jets fall in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier–Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is carried out with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing. Copyright © 2015 John Wiley & Sons, Ltd.     

研究柔性体撞击问题的子结构离散方法

本文用子结构方法研究刚性小球和均质柔性杆的纵向撞击以及和均质柔性梁的横向撞击问题,导出了用模态坐标表示的动力学方程,通过对刚性小球和柔性杆的纵向撞击的仿真教育处发现,在总单元数相同的情况下,取8个子结构比较合理,在此基础上对刚性小球和均质柔性梁的横向撞击的特性进行研究,发现撞击力在变化过程中会产生上下波动,当梁的弹性模量增加时,撞击力增大,撞击时间缩短。     

Modeling of droplet collisions using Smoothed Particle Hydrodynamics

We present an approach to model collisions of different droplets using Smoothed Particle Hydrodynamics (SPH). We consider bouncing and coalescence of two droplets. We only discretize the droplets neglecting the gaseous phase and consider a free surface at the boundaries. We use a modified continuum surface force model for the surface tension at a free surface. The transition between bouncing and coalescence is modeled using a critical Weber number and calculating the loss of kinetic energy during the collision to determine the point of coalescence. We demonstrate numerical convergence and analyze the error of the method for the transition of bouncing and coalescence. We show that the proposed approach is applicable to weakly-compressible SPH and incompressible SPH and compare binary collisions of Newtonian droplets with experimental results from the literature. Finally we apply the model to non-Newtonian droplets that show shear-thinning and shear-thickening behavior and discuss the differences to Newtonian droplets.     

On dual-grid level-set method for contact line modeling during impact of a droplet on hydrophobic and superhydrophobic surfaces

In this paper, a numerical methodology for modeling contact line motion in a dual-grid level-set method (DGLSM) – solved on a uniform grid for interface which is twice that for the flow equations – is presented. A quasi-dynamic contact angle model – based on experimental inputs – is implemented to model the dynamic wetting of a droplet, impacting on a hydrophobic or a superhydrophobic surface. High-speed visualization experiments are also presented for the impact of a water droplet on hydrophobic surfaces, with non-bouncing at smaller and bouncing at larger impact velocity. The experimental results for temporal variation of the droplet shapes, wetted-diameter and maximum height of the droplet match very well with the DGLSM based numerical results. The validation of the numerical results is also presented with already published experimental results, for the non-bouncing on a hydrophobic and bouncing on a superhydrophobic surface, at a constant impact velocity. Finally, a qualitative as well as quantitative performance of the DGLSM as compared to the traditional level set method (LSM) is presented by considering our experimental results. The accuracy of the partially refined DGLSM is close to that of the fine-grid based LSM, at a computation cost which is close to that of the coarse-grid based LSM. The DGLSM is demonstrated as an improved LSM for the computational multi-fluid dynamics (CMFD) simulations involving contact line motion.     

On the dynamics of a rolling ball actuated by internal point masses

The motion of a rolling ball actuated by internal point masses that move inside the ball’s frame of reference is considered. The equations of motion are derived by applying Euler–Poincaré’s symmetry reduction method in concert with Lagrange–d’Alembert’s principle, which accounts for the presence of the nonholonomic rolling constraint. As a particular example, we consider the case when the masses move along internal rails, or trajectories, of arbitrary shape and fixed within the ball’s frame of reference. Our system of equations can treat most possible methods of actuating the rolling ball with internal moving masses encountered in the literature, such as circular motion of the masses mimicking swinging pendula or straight line motion of the masses mimicking magnets sliding inside linear tubes embedded within a solenoid. Moreover, our method can model arbitrary rail shapes and an arbitrary number of rails such as several ellipses and/or figure eights, which may be important for future designs of rolling ball robots. For further analytical study, we also reduce the system to a single differential equation when the motion is planar, that is, considering the motion of the rolling disk actuated by internal point masses, in which case we show that the results obtained from the variational derivation coincide with those obtained from Newton’s second law. Finally, the equations of motion are solved numerically, illustrating a wealth of complex behaviors exhibited by the system’s dynamics. Our results are relevant to the dynamics of nonholonomic systems containing internal degrees of freedom and to further studies of control of such systems actuated by internal masses.     

Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2-cycles, grazing and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2-cycles in a corner-type bifurcation.     

浮力气泡对水平壁面的回弹动力学特性

黏性液体中的气泡浮升运动有趣而又复杂,而气泡与固壁边界的相互作用更是广泛存在于实际工程中.基于轴对称数值计算,模拟了浮力驱动下气泡在黏性液体中上升并与顶部水平固壁面碰撞、回弹的过程.采用考虑表面张力的不可压、变密度Navier-Stokes方程来描述气液两相流流动,并通过基于分级八叉树的有限体积法进行数值求解.为准确捕捉气泡在回弹过程中局部而迅速的拓扑变化,采用了动态自适应网格技术耦合流体体积法(volume of fluid,VOF)来重构气泡的形状. 从气泡对壁面的碰撞和回弹的基本现象入手,研究了伽利略数 Ga和接触速度$U_{a}$对气泡回弹动力学特性的影响, 分析了气泡碰撞过程中涡结构的变化.用回弹高度$H$、回弹周期$T$、长宽比{$A_{r}$}、浮升速度$U$、轴向位置$z$和回复系数$C_{r}$等参数来表征不同条件时气泡的运动和形状特性. 研究结果表明,气泡的回弹运动特性对 Ga十分敏感. Ga的增大可加剧气泡形变, 促进气泡的回弹运动, 增多回弹次数,增大回弹参数($T$和$H)$, 提升回复系数. 然而,接触速度并非决定气泡回弹动力学的控制参数, $U_{a}$的改变并不会改变回复系数.      

Solving Painlevé paradox: (P–R) sliding robot case

For a rigid body or a multibody system sliding on a rough surface, a range of uncertainty or non-uniqueness of solution could be found, which is termed: Painlevé paradox. Painlevé paradox is the reason of a wide range of undesired bouncing motions which are observed during sliding. As Painlevé paradox is a practical problem in case of multibody systems, this research work has investigated that paradox. In this research work, the condition leading to Painlevé paradox has been determined for a general multibody system. Investigating the motion of a prismatic–revolute (P–R), sliding robot has been conducted. In order to solve the paradox and find the motion, a tangential impact is assumed at the contact point. The impact model has been developed and the paradox, consequently, has been solved. Consequently, the kinematics of the motion has been specified.     

Periodic motion and bifurcations induced by the Painlevé paradox

In this paper we study the periodic motion and bifurcations of the Frictional Impact Oscillator, which consists of an object with normal and tangential degrees of freedom that comes in contact with a rigid surface. The Frictional Impact Oscillator contains the basic mechanism for a hopping phenomenon observed in many practical applications. We will show that the hopping or bouncing motion in this type of systems is closely related to the Painlevé paradox. A dynamical system exhibiting the Painlevé paradox has nonuniqueness and nonexistence of solutions in certain sliding modes. Furthermore, we will show that this type of systems can exhibit the Painlevé paradox for physically realistic values of the friction coefficient.     

垂直壁面附近上升单气泡的弹跳动力学研究

采用高速摄影技术结合阴影法,对静止水中垂直壁面附近上升单气泡运动进行实验研究,对比气泡尺度及气泡喷嘴与壁面之间的初始无量纲距离(S^*)对气泡上升运动特性的影响,分析气泡与壁面碰撞前后,壁面效应与气泡动力学机制及能量变化规律.结果表明,对于雷诺数Re≈580～1100,无量纲距离S^*<2～3时,气泡与壁面碰撞且气泡轨迹由无约束条件下的三维螺旋转变成二维之字形周期运动;当S^*> 2～3时,壁面效应减弱,有壁面约束的气泡运动与无约束气泡运动特性趋于一致.气泡与壁面碰撞前后,壁面效应导致横向速度峰值下降为原峰值的70%,垂直速度下降50%;气泡与壁面碰撞前,通过气泡中心与壁面距离(x/R)和修正的斯托克斯数相关式可预测垂直速度的变化规律.上升气泡与壁面碰撞过程中,气泡表面变形能量单向传输给气泡横向动能,使得可变形气泡能够保持相对恒定的弹跳运动.提出了气泡在与壁面反复弹跳时的平均阻力系数的预测模型,能够很好地描述实验数据反映出的对雷诺数Re、韦伯数We和奥特沃斯数Eo等各无量纲参数的标度规律.     

Computational study of bouncing and non-bouncing droplets impacting on superhydrophobic surfaces

We numerically investigate bouncing and non-bouncing of droplets during isothermal impact on superhydrophobic surfaces. An in-house, experimentally validated, finite element method-based computational model is employed to simulate the droplet impact dynamics and transient fluid flow within the droplet. The liquid–gas interface is tracked accurately in Lagrangian framework with dynamic wetting boundary condition at three-phase contact line. The interplay of kinetic, surface and gravitational energies is investigated via systematic variation of impact velocity and equilibrium contact angle. The numerical simulations demonstrate that the droplet bounces off the surface if the total droplet energy at the instance of maximum recoiling exceeds the initial surface and gravitational energy, otherwise not. The non-bouncing droplet is characterized by the oscillations on the free surface due to competition between the kinetic and surface energy. The droplet dimensions and shapes obtained at different times by the simulations are compared with the respective measurements available in the literature. Comparisons show good agreement of numerical data with measurements, and the computational model is able to reconstruct the bouncing and non-bouncing of the droplet as seen in the measurements. The simulated internal flow helps to understand the impact dynamics as well as the interplay of the associated energies during the bouncing and non-bouncing. A regime map is proposed to predict the bouncing and non-bouncing on a superhydrophobic surface with an equilibrium contact angle of 155°, using data of 86 simulations and the measurements available in the literature. We discuss the validity of the computational model for the wetting transition from Cassie to Wenzel state on micro- and nanostructured superhydrophobic surfaces. We demonstrate that the numerical simulation can serve as an important tool to quantify the internal flow, if the simulated droplet shapes match the respective measurements utilizing high-speed photography.     

DYNAMIC ANALYSIS OF ECCENTRIC WHEELS ALONG SLOPE FROM PURE ROLLING TO CONTINUOUS BOUNCING

研究了第十一届全国周培源大学生力学竞赛团体赛中轮子跳高试题的偏心圆轮方案,建立了无初速度释放的偏心圆轮沿斜面纯滚动到连续跳动的动力学理论模型,分析了弹跳现象的产生条件,研究了纯滚动到连续弹跳的动力学过程,给出了斜面倾角、初始释放角和摩擦系数等因素对跳跃行为的影响。     

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincarè maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.