A theoretical model of collision between soft-spheres with Hertz elastic loading and nonlinear plastic unloading

This paper presents a theoretical model on the normal (head-on) collision between soft-spheres on the basis of elastic loading of the Hertz contact for compression process and a nonlinear plastic unloading for restitution one, in which the parameters all are determined in terms of the material and geometric ones of the spheres, and the behaviors of perfect elastic, inelastic, and perfect plastic collisions appeared in the classical mechanics are fully described once a value of coefficient of restitution is specified in the region of 0 ≤ ε ≤ 1. After an empirical formula of the coefficient of restitution dependent on the impact velocity is suggested to fit the existing experimental measurements by means of the least square method, the predictions of the dependency and the collision duration are in well quantitative agreement with their experimental measurements. It is found that the measurable quantities are dependent on both the impact velocity and the parameters of spheres. Following this model, finally, an approach to determine the spring coefficient in the linear viscoelastic model of the collision is also displayed. These results obtained here will be significantly beneficial for the applications where a collision model is requested in the simulations of relevant grain flows and impact dynamics etc..     

Outer synchronization of small-world networks by a second-order sliding mode controller

The current work presents an experimentally validated analytical model for low-velocity impact between a sphere and a plate. The model accounts for plastic deformation as well as flexural vibrations. The elastic phases are modeled with a nonlinear Hertzian contact model, and the plastic phase is linearized with a non-homogeneous expression. The results are compared against newly carried out experiments. The model well captures the effect of plate thickness-to-sphere diameter ratio, impact velocity, and material properties. The model’s generalized framework allows consideration for various expressions of contact parameters, critical velocity, and residual indentation. Moreover, the proposed methodology can be easily incorporated into particle-based or discrete element modeling approaches for granular flows to evaluate the real-time coefficient of restitution as opposed to assuming the constant value beforehand. Simplified relations are provided to assist in evaluating the coefficient of restitution.     

Predicting the coefficient of restitution of impacting elastic-perfectly plastic spheres

The current work presents a different methodology for modeling the impact between elasto-plastic spheres. Recent finite element results modeling the static deformation of an elasto-plastic sphere are used in conjunction with equations for the variation of kinetic energy to obtain predictions for the coefficient of restitution. A model is also needed to predict the residual deformation of the sphere during rebound, or unloading, of which several are available and compared in this work. The model predicts that a significant amount of energy will be dissipated in the form of plastic deformation such that as the speed at initial impact increases, the coefficient of restitution decreases. This work also derives a new equation for the initial critical speed which causes initial plastic deformation in the sphere that is different than that shown in previously derived equations and is strongly dependant on Poisson’s Ratio. For impacts occurring above this speed, the coefficient of restitution will be less than a value of one. This work also compares the predictions between several models that make significantly different predictions. The results of the current model also compare well with some existing experimental data. Empirical fits to the results are provided for use as a tool to predict the coefficient of restitution.     

An analytical elastic-perfectly plastic contact model

A new formulation for elastic-perfectly plastic contact in the normal direction between two round surfaces that is solely based on material properties and contact geometries is developed. The problem is formulated as three separate domains: the elastic regime, mixed elastic–plastic behavior, and unconstrained (fully plastic) flow. Solutions for the force–displacement relationship in the elastic regime follow from Hertz’s classical solution. In the fully plastic regime, two well supported assumptions are made: that there is a uniform pressure distribution and there is a linear force–deflection relationship. The force–displacement relationship in the intermediate, mixed elastic–plastic regime is approximated by enforcing continuity between the elastic and fully plastic regimes. Transitions between the three regimes are determined based on empirical quantities: the von Mises yield criterion is used to determine the initiation of mixed elastic–plastic deformation, and Brinell’s hardness for the onset of unconstrained flow. Unloading from each of these three regimes is modeled as an elastic process with different radii of curvature based on the regime in which the maximum force occurred. Simulation results explore the relationship between the impact velocity and coefficient of restitution. Further comparisons are made between the model, experimental results found in the literature, and other existing elastic–plastic models. The new model is well supported by the experimental measurements of compliance curves for elastic–plastic materials and of coefficients of restitution from impact studies, and in elastic-perfectly plastic regimes is demonstrated to be more accurate than existing models found in the literature.     

A nonlinear visco-elastoplastic impact model and the coefficient of restitution

A visco-elastoplastic model for the impact between a compact body and a composite target is presented. The model is a combination of a nonlinear contact law that includes energy loss due to plastic deformation and a viscous element that accounts for energy losses due to wave propagation and/or damping. The governing nonlinear equations are solved numerically to obtain the response. A piecewise linear version of the model is also presented, which facilitates analytical solution. The model predictions are compared to those of the well-known and commonly used Hunt–Crossley model. The effects of the various impact parameters, such as impactor mass, velocity, plasticity, and damping, on the impact response and coefficient of restitution are investigated. The model appears to be suitable for a wide range of impact situations, with parameters that are well defined and easily calculated or measured. Furthermore, the resulting coefficient of restitution is shown to be a function of impact velocity and damping, as confirmed by published experimental data.     

单自由度冲击扰动下模拟磁头／磁盘系统接触回复特性的实验研究

采用自制的试验装置研究了单自由度冲击扰动下磁头 /磁盘系统的接触回复特性 ,分析了不同载荷下球形模拟磁头跳跃振动回复系数和接触时间与输入速度之间的关系 .结果表明 :当载荷较小时 ,回复系数随着输入速度的减小而增加 ,而当载荷较大时 ,回复系数随着输入速度的减小而减小 .在不同载荷条件下 ,接触时间均随输入速度的减小而缩短 ,但小载荷下接触时间缩短的幅度较大 .球形模拟磁头的回复特性和接触时间与载荷、输入速度及润滑膜特性有关 ,增大载荷和选择合适的润滑剂均可抑制跳跃振动     

Influence of liquid layers on energy absorption during particle impact

The influence of the thickness of a covering liquid layer and its viscosity as well as the impact velocity on energy loss during the normal impact on a flat steel wall of spherical granules with a liquid layer was studied. Free-fall experiments were performed to obtain the restitution coefficient of elastic-plastic γ- Al2O3 granules by impact on the liquid layer, using aqueous solutions of hydroxypropyl methylcellulose with different concentrations for variation of viscosity （1-300 mPa s）, In the presence of a liquid layer, increase of liquid viscosity decreases the restitution coefficient and the minimum thickness of the liquid layer at which the granule sticks to the wall. The measured restitution coefficients were compared with experiments performed without liquid layer. In contrast to the dry restitution coefficient, due to viscous losses at lower impact velocity, higher energy dissipation was obtained, A rational explanation for the effects obtained was given by results of numerically solved force and energy balances for a granule impact on a liquid layer on the wall. The model takes into account forces acting on the granule including viscous, surface tension, capillary, contact, drag, buoyancy and gravitational forces. Good agreement between simulations and experiments has been achieved.     

Measurements of impact properties of small, nearly spherical particles

The authors report impact properties for collisions of small, nearly spherical particles that present interesting experimental challenges. They consider difficulties arising with surface reflectivity, slight asphericity, surface damage and collisions with particles affixed to a rigid plate. To measure these impact properties, the authors refine the experimental technique of Foersteret al. To permit straightforward incorporation in rapid granular theories, the impacts are described with three coefficients. The first is the Newtonian coefficient of normal restitution. The second represents the frictional properties of the contact surfaces. The last characterizes the restitution of the tangential component of the contact point velocity for impacts that involve negligible sliding.     

Dynamic Deformation Behavior of a Golf Ball during Normal Impact

The normal impact between a golf ball and a rigid steel target was studied to examine ball deformation and the contact force during the impact. Using high-speed video images, the normal and tangential compression ratios of the ball were measured to analyze the ball deformation quantitatively. In addition, the inbound and rebound ball velocities, contact time, and coefficient of restitution were determined as basic parameters of the impact. As the inbound ball velocity increased, the maximum normal compression ratio increased while the maximum tangential compression ratio, contact time and coefficient of restitution decreased. The ball center displacements during the impact were measured to determine the ball center velocity and acceleration, and the contact force was calculated by the product of the mass and acceleration. The contact force increased almost linearly with the inbound ball velocity, and its relationship agreed well quantitatively with the results from a load-cell, and also agreed well qualitatively with Hertz contact theory.     

Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider

A two-degree-of-freedom plastic impact oscillator with a frictional slider is considered. Dynamics of the plastic impact oscillator are analyzed by a three-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the impact Poincaré map. The piecewise property of the map is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of the map is generated via the grazing contact of two masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularity and parameter variation on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams of before-impact velocity as a function of the excitation frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-N single-impact motions of the plastic impact oscillator are found to exhibit extensive and systematic characteristics. Dynamics of the impact oscillator, in the elastic impact case, is also analyzed. This type of impact is modelled by using the conditions of conservation of momentum and an instantaneous coefficient of restitution rule. The differences in periodic-impact motions and bifurcations are found by making a comparison between dynamic behaviors of the plastic and elastic impact oscillators with a frictional slider. The best progression of the plastic impact oscillator is found to occur in period-1 single-impact sticking motion with large impact velocity. The largest progression of the elastic impact oscillator occurs in period-1 multi-impact motion. The simulative results show that the plastic impact feature for the impact-progressive oscillator is of a considerable importance in minimizing adverse effects such as high noise levels, wear and tear caused by impacts.     

Analysis of repeated impacts on a steel rod with visco-plastic material behavior

In machine dynamics impacts are usually common phenomena, resulting from collisions of moving bodies. Even low velocity impacts might produce high stresses in the contact region, which result in inelastic deformation. Thereby, visco-plastic materials, such as steel, show a significant increase of the yield stress with the strain rate. In machine dynamics repeated collisions occur, resulting in repeated impacts on a previously deformed contact area. Then, inelastic deformation and the resulting residual stresses produced by previous impacts have an influence on the behavior of the following impacts. Thus, the impact behavior varies with the number of impacts. This paper presents a numerical and experimental evaluation of repeated impacts with identical impact velocity up to 3 m/s, whereby the deformation history of the contact area, due to previous impacts, is included. The approach is applied to longitudinal impacts of an elastic steel sphere on a steel rod with distinct visco-plastic material behavior which is identified by Split Hopkinson Pressure Bar tests. A Finite Element analysis and experimental verification using two Laser-Doppler-Vibrometers are performed. It is shown that for an accurate impact simulation the FE model must include the visco-plastic material behavior of the steel. Further it is found that the maximal contact force, the rebound velocity and the coefficient of restitution increase with the number of impacts, while the contact duration decreases with the number of impacts. After several impacts these quantities show saturation to a constant value, indicating no significant additional inelastic deformation in the later impacts. Further, the residual stress distribution, the maximal von Mises stress distribution and the local deformation at the contact point are evaluated and a characteristic force-deformation diagram is obtained. Finally, an analysis is performed to describe the relation between maximal force and remaining crater at the contact point.     

Rigid body impact with moment of rolling friction

In this paper, the impact between a rigid pendulum and rough surfaces is studied. The rolling friction moment and the coefficient of rolling friction are introduced, and an improved mathematical model of the planar impact with friction is presented. The influence of the moment of rolling friction on the energy dissipated by friction during the impact is analyzed. For a simple pendulum, using the energetic coefficient of restitution, more energy is dissipated for larger values of the coefficient of kinetic friction and contact radius, and for smaller values of the length of the beam. For a double pendulum using the kinematic coefficient of restitution, some energetically inconsistent results can be solved for some values of the coefficient of rolling friction.     

Viscoplastic analysis for direct impact of sports balls

A coefficient of restitution (COR) is used to represent viscoplastic dissipation of energy in the contact region of colliding bodies [6]. A model has been developed that distinguishes between rate-independent plastic and rate-dependent viscous energy dissipation during impact. A nondimensional contact force shape factor for compression α is introduced which can be measured experimentally. The measured α and COR are used to separate the energy dissipated during collision into part due to plastic deformation and another part due to viscous dissipation. In comparison to the viscoelastic compliance model, the viscoplastic compliance model approximates the force profile accurately, especially in terms of the maximum force that occurs during impact.     

Continuous contact force models for impact analysis in multibody systems

One method for predicting the impact response of a multibody system is based on the assumption that the impacting bodies undergo local deformations and the contact forces are continuous. In a continuous analysis, the integration of the system equations of motion is carried out during the period of contact; therefore, a model for evaluating the contact forces is required. In this paper, two such contact force models are presented, both Hertzian in nature and based upon the direct-central impact of two solid particles.At low impact velocities, the energy dissipation during impact can be represented by material damping. A model is constructed based on the general trend of the Hertz contact law in conjuction with a hysteresis damping function. The unknown parameters are determined in terms of a given coefficient of restitution and the impact velocity. When local plasticity effects are the dominant factor accounting for the dissipation of energy at high impact velocities, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are again evaluated. The two particle models are generalized to an impact analysis between two bodies of a multibody system.     

Study on impact phenomenon between a sphere and a layered structure

As part of a general investigation of impact-printing phenomenon, a series of impact experiments were performed in which the print action was simulated by a freely falling sphere impacting with a horizontal stack composed of ribbon, paper and carbon paper, backed by a steel platen. The principal quantity measured was the coefficient of restitution of the system, which is related to the energy lost during the collision process. In certain cases, a piezoelectric transducer was used to support the stack so that information about contact time and impact force could be obtained. The behavior of the coefficient of restitution as a function of stack thickness was determined, not only for the case of ribbon, paper and carbon paper, but for various thicknesses of other materials also supported by a hard-steel platen. The latter experiments were performed in order to obtain a better understanding of the nature of the collision process. In general, similar behavior was found for all materials and energies tested. The empirical results were compared with theoretical models for energy-loss mechanisms and it was concluded that the major energy-loss mechanisms were associated with plastic and viscoelastic phenomenon, rather than with traveling waves.     

多体系统碰撞动力学中接触力模型的研究进展

接触碰撞行为作为大自然与多体系统中的常见现象,其接触力模型对于多体系统的碰撞行为机理研究与性能预测至关重要.静态弹塑性接触模型与考虑能量耗散的连续接触力模型是研究接触碰撞行为的两类不同方法,在多体系统碰撞动力学中存在诸多共性与差异.本文分别从上述两类接触模型的发展历程入手,详细介绍了两类模型的区别与联系.首先,根据阻尼项分母中是否含有初始碰撞速度将连续接触力模型分为黏性接触力模型与迟滞接触力模型,讨论了能量指数与Hertz接触刚度之间的关系,阐述了现有连续接触力模型在计算弹塑性材料接触碰撞行为时存在的问题.其次,着重介绍了分段连续的准静态弹塑性接触力模型(可连续从完全弹性转换到完全塑性接触阶段),分析了利用此类弹塑性接触力模型计算碰撞行为的技术特点.同时,以恢复系数为桥梁和借助线性化的弹塑性接触刚度,避免了Hertz刚度对弹塑性接触刚度的计算误差,根据碰撞前后多体系统的能量与动能守恒推导了弹塑性接触模型等效的迟滞阻尼因子.探索了连续接触力模型与准静态弹塑性接触力模型之间的内在联系,数值计算结果定量说明了人为阻尼项代表的能量耗散与弹塑性接触力模型中加卸载路径代表的能量耗散具有等效性.另外...     

Impact Oscillator with Hertz's Model of Contact

Dynamic properties of simple mechanical 1 DOF system containing soft stop is described and analyzed. The proposed general dynamical impact model respects the nonlinearity of the restoring contact force between solid bodies as function of deformation and velocity. It describes the real behavior of impacting system more exact than the piecewise linear model or the Kelvin–Voigt model and even model based on application of constant coefficient of restitution. Free and forced vibrations of system with Hertz's contact are investigated and domains of various types of impact motion, response curves and phase-plane trajectories are presented.     

The coefficient of normal restitution for the Hertzian contact of two rough spheres colliding in a viscous fluid

We use previous theoretical results for the added mass, history and lubrication forces between two spheres colliding in a fluid with viscosity ν to investigate the effect of viscous dissipation on the coefficient of restitution during contact. We assume that the mechanical interaction is governed by Hertzian mechanical contact of small duration τ and that the minimum approach distance between particles is approximately equal to the height σ of surface micro-asperities. A non-dimensionalization of the equation of motion indicates that the contact dynamics is governed by two parameters – the ratio ϵ of the surface roughness σ and the sphere radius a, and a contact Stokes number defined as St_c = σ²/ντ. An asymptotic solution of the equation of motion in the limit of small ϵ/St_c is used to obtain an explicit expression for the coefficient of restitution during contact and the latter is compared with estimates based on numerical solutions of the non-linear equation of motion.     

Elastic–plastic contact force history and response characteristics of circular plate subjected to impact by a projectile

A new elastic–plastic impact–contact model is proposed in this paper. By adopting the principle of minimum acceleration for elastic–plastic continue at finite deformation, and with the aid of finite difference method, the proposed model is applied in the problem of dynamic response of a clamped thin circular plate subjected to a projectile impact centrally. The impact force history and response characteristics of the target plate is studied in detail. The theoretical predictions of the impact force and plate deflection are in good agreements with those of LDA experimental data. Linear expressions of the maximum impact force/transverse deflection versus impact velocity are given on the basis of the theoretical results. The project supported by the National Natural Science Foundation of China (10532020).     

Impact of a bead on a rotating wall

We study experimentally the impact of a plastic bead on a rotating wall made of steel (velocity Ω; radial position x₀). The results show that the restitution coefficient is directly function of the impact velocity x₀Ω and is invariant by changing frame reference. The influence of the height of release of the particle on its angular velocity after impact is also studied. We observe an increase of the angular velocity with height followed by a saturation. We propose an interpretation for this evolution considering that the particle may roll without sliding during all the impact. This physical feature is not always taken into account in existing models of impact between rigid bodies. To cite this article: F. Rioual et al., C. R. Mecanique 336 (2008).