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.     

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简述了多刚体系统接触及碰撞动力学的研究方法，详细回顾了考虑库仑摩擦的多刚 体系统的能量不协调、Painlev\'{e}疑难及碰撞恢复系数等该领域中难点问题的研 究进展，探讨了解决该难点问题的研究思路，指出了未来该领域的研究方向.     

Non-linear behavior in a discretely forced oscillator

The simulated and experimental responses of a rigid-arm pendulum driven by an external impactor are considered. Here, impact occurs if the trajectory of a rotating impactor intersects that of the pendulum. Using the rotation rate of the impactor as the control parameter, experimental trials have demonstrated much of the dynamic behavior predicted by numerical simulations. The system exhibits chatter (i.e., multiple impacts within a single forcing period), sticking (i.e., contact between the pendulum and the impactor for non-negligible amounts of time), high-order periodicity, and behavior suggestive of chaos. A new convention for classifying periodic motions as well as insights regarding the nature of the coefficient of restitution (COR) in an experimental impacting system are also presented.     

Nonlinear dynamics of an elastic rod with frictional impact

A model is presented for the impact with friction of a flexible body in translation and rotation. This model consists of a system of nonlinear differential equations which considers the multiple collisions as well as frictional effects at the contacting end, and allows one to predict the rigid and elastic body motion after the impact. The kinetic energy is derived by utilizing a generalized velocity field theory for elastic solids. The model uses a dry coefficient of friction and a nonlinear contact force. We introduce a finite number of vibrational modes to take into account the vibrational behavior of the body during impact. The vibrations, the multiple collisions, and the angle of incidence angle, are found to be important factors for the kinematics of frictional impact. Analytical and experimental results were compared to establish the accuracy of the model.     

Role of particle stiffness and inter-particle sliding friction in milling of particles

Discrete element method （DEM） has been used to investigate the effects of particle elastic modulus and coefficient of inter-particle sliding friction on milling of mineral particles. An autogeneous mill of 600 mm diameter and 320 mm length with 14,500 particles has been selected for the simulation. Various mill performance parameters, for example, particle trajectories, collision frequency, collision energy and mill power have been evaluated to understand the effects of particle elastic modulus and inter-particle sliding friction during milling of particles. For the given model, it has been concluded that at high energy range, as the elastic modulus and particle sliding friction increase the energy dissipated among the particles increases. The collision frequency increases with the increase in elastic modulus, however, this trend is not clearly observed with increasing inter-particle sliding friction. The power draw of the mill increases with the increase in fraction of mill critical speed.     

Hysteresis losses in oscillatory systems

Hysteresis losses restrict the quality factors of oscillatory systems. We have considered the energy dissipation mechanism using as an example the contact spots of the ball supports of a pendulum tribometer with plane samples. The bending of a contact spot leads to a storage of the system׳s elastic energy. Measuring the quality factor of the pendulum at oscillations, owing to the inclination of the contact spot, makes it possible to determine the coefficient of hysteresis losses in materials and thin coatings without destroying their structures.     

具有摩擦的刚体碰撞

刚体碰撞是力学上的一个经典问题，但目前大都采用给定恢复系数进行分析的方法，本文则直接从两刚体碰撞时Newton第二定律出发，建立了计及摩擦的两刚体碰撞基本理论，并指出了以往研究通过给定恢复系数方法的错误之处。文中通过利用两刚体接触相对变位加速度与碰撞力的关系，定义了法向等效质量，对二维平面碰撞，通过一些简单的比较和定义给出了各种碰撞状态的分类判别法，并给出了碰撞力和冲量的解析式。指出法向等效质量依赖于碰撞状态，在反向滑动和停止滑动这两状态下的法向等效质量是变化的，因此导致其恢复系数是变化的，这样，恢复系数不仅与接触力(接触点情况)有关，而且还与两刚体的运动和动力参数有关，同时给出各种碰撞状态下的恢复系数计算式。     

Simulation of impact between rigid elements

The objective of this paper is to apply the concepts and theories of global impact based on a modern formulation in which the impact is considered with sliding, adherence or transition between sliding and adherence. The energy dissipated in the contact may be quantified by different definitions of the coefficient of restitution in Newton’s law and Poisson’s law. The method is applied to the impact between the damper (parabolic elements) and the platform of the blades in an airborne gas turbine. A numerical solution is calculated using a computer program constructed on the Matlab Platform.     

Angle of repose in the numerical modeling of ballast particles focusing on particle-dependent specifications: Parametric study

The discrete element method (DEM) is widely used in the realistic simulation of the shapes of particles. Researchers have considered the simplification of particle shapes owing to the high computational cost of such simulation. In this regard, the modeling of calibrated particles is a major challenge owing to the simultaneous effects of particle properties. The angle-of-repose test is a standard test method used to calibrate the bulk behavior of simulated particles. In the present study, the hollow-cylinder (slump) test was modeled for the verification of discrete element simulations. In this regard, a sensitivity analysis was conducted for all effective parameters, namely the static friction, rolling friction, restitution coefficient, sphericity, roundness, particle size distribution, and number of ballast particles. The results indicate that the rolling friction, roundness, number of particles, and size of particles are the most important parameters in the determination of the angle of repose (AOR). For particles in the range of ballast (20–60 mm), the effect of the number of particles on the angle of repose is reduced when the number is greater than 426. Additionally, it is concluded that angular particles can be replaced with sub-angular particles (R ≈ 0.2–0.45) with a higher rolling friction coefficient (μ_r > 0.14).     

Determining the angle of repose of sand under low-gravity conditions using discrete element method

This study is a comparative investigation of data, collected through experimental and numerical means, related to the flow of sand particles through a hopper under low-gravity conditions. During a parabolic airplane flight simulating low-gravity conditions, we determined effects of gravity on the angle of repose of sand pile particles by flowing dry sand from a hopper. The gravity effects on the angle of repose of the sand were negligible. Two-dimensional discrete element method (DEM) was used to simulate the angle of repose. Results were compared to observations made during the low-gravity experiments. Effects of varying parameters such as the friction coefficient and coefficient of rolling friction were determined by running various DEM simulations. Moreover, the effect of the elemental radius on the angle of repose was investigated using DEM. The angle of repose is influenced by certain changes in the friction coefficient and rolling friction values, but the elemental radius has only a negligible effect on the angle of repose within the range of variation. Results show that the DEM model used for this study might be applicable to determine terramechanical interactions under lunar surface gravity conditions, provided that parameters are adjusted and an extended period of simulation is achieved.     

一种滞弹簧耗能的新型离散元滚动阻力模型研究

颗粒间滚动阻力对颗粒体系的稳定性起着重要作用. 在传统的离散元法中, 滚动阻力模型通常由转动弹簧、转动黏壶和摩擦元件表达, 颗粒滚动动能由黏滞力(矩)和摩擦力做功耗散. 由于黏滞力(矩)与滚动速度相关, 临近静止状态的颗粒滚动速度变小, 动能耗散减弱, 传统的离散元模拟得到颗粒由滚动到静止耗费的时间比试验观测的结果要长. 为解决这一问题, 基于摩擦学理论分析了滚动阻力产生的材料滞弹性机理, 将其引入离散元滚动阻力模型, 提出了一种速度无关型动能耗散的滞弹簧, 给出了滞弹簧的弹性恢复力计算公式, 建立了一种新型的离散元滞弹性滚动阻力模型(HDEM). 为验证新型滚动阻力模型的正确性, 通过一个光学物理试验对单个圆形颗粒试件的自由滚动过程进行了测量, 将测量数据与新型的滞弹型离散元模型和传统离散元模型计算结果进行了对比. 结果显示, 基于滞弹性滚动阻力模型HDEM计算结果与试验数据吻合程度更高, 而且模拟得到的颗粒摆动频率更符合试验现象.      

Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law

This paper concerns the non-smooth dynamics of planar mechanical systems with isolated contact in the presence of Coulomb friction. Following Stronge [Impact Mechanics, Cambridge University Press, Cambridge, 2000], a set of closed-form analytic formulae is derived for a rigid-body impact law based on an energetic coefficient of restitution and a resolution of the impact phase into distinct segments of relative slip and stick. Thus, the impact behavior is consistent both with the assumption of Coulomb friction and with the dissipative nature of impacts. The analysis highlights the presence of boundaries between open regions of initial conditions and parameter values corresponding to distinct forms of the impact law and investigates the smoothness properties of the impact law across these boundaries. It is shown how discontinuities in the impact law are associated with discontinuity-induced bifurcations of periodic trajectories, including non-smooth folds and persistence scenarios. Numerical analysis of an example mechanical model is used to illustrate and validate the conclusions.     

与冲蚀有关的粘着与犁沟摩擦系数

提出一种与冲击磨损有关的有效摩擦系数，它被定义为冲击过程中，每一瞬间的切向有效摩擦力与法向载荷之比．有效摩擦力被表示为剪切项与犁沟项之组合．剪切项依赖于剪切金屑接合点的状态——滑移或滚动．犁沟项依赖于冲击物在冲击途中使靶材料发生移动的状态．此有效摩擦系数被应用于研究刚性球自由斜冲击延性靶问题．在忽略剪切项仅计及犁沟项的情况下，研究表明，Hutchings实验中的有效冲击摩擦系数是冲击过程与初始冲击角的函数，就时间平均而言，大于Hutchins为拟合计算与实验结果所取的摩擦系数常值0.05．有迹象表明，对于与固粒延性冲蚀有关的刚性球自由斜冲击延性靶问题，在一定条件下，有效摩擦系数主要由犁沟效应确定．     

Quantitative Impact Testing of Energy Dissipation at Surfaces

Impact testing with nanoscale spatial, force, and temporal resolution has been developed to address quantitatively the response of surfaces to impingement of local contact at elevated velocities. Here, an impact is generated by imparting energy to a pendulum carrying an indenter, which then swings towards a specimen surface. The pendulum displacement as a function of time x(t) is recorded, from which one can extract the maximum material penetration x _max , residual deformation x _r , and indentation durations t _in and t _out. In an inverse application one can use the x(t) response to extract material constants characterizing the impact deformation and extent of energy absorption, including material specific resistance coefficient C_in, coefficient of restitution e, and dynamic hardness H _imp . This approach also enables direct access to the ratio H/E, or resilience of the deformed material volume, at impact velocities of interest. The impact response of aluminum was studied for different contact velocities, and the mechanical response was found to correlate well with our one-dimensional contact model. Further experiments on annealed and work hardened gold showed that dynamic hardness H _imp scales with contact velocity and highlighted the importance of rate-dependent energy absorption mechanisms that can be captured by the proposed experimental approach.

STABILITY ANALYSIS OF REUSABLE LAUNCH VEHICLE LANDING STRUCTURE$^{\bf 1)}$

近年来,包括中国在内的诸多国家相继开展垂直起降重复使用火箭的研究,运载火箭在平台上垂直着陆时的着陆稳定性为实现运载火箭重复使用的关键问题. 由于在运载火箭设计初期结构设计尚未完成,不具有供着陆稳定性分析的详细的动力学模型,难以开展着陆过程动力学仿真,故对运载火箭着陆稳定性评估方法的研究尤为必要. 本文基于广义碰撞定律,对二维运动模式下运载火箭与着陆平台的多点碰撞过程进行了分析,切向采用库伦摩擦模型给出了切向运动学恢复系数的表达式. 本文首先通过机械能约束和接触碰撞中的单边约束给出了一般运动形式下广义运动学恢复系数的值域,再对两种典型运动模式,给出了该两种典型运动模式下广义运动学恢复系数的值域. 然后考虑着陆腿中缓冲器的作用,将运载火箭与平台的碰撞近似为完全非弹性碰撞,得到了其广义运动学恢复系数,并结合运动学分析和能量法提出了一种基于碰撞后速度的着陆稳定性的判别方法. 最后以某型运载火箭着陆样机的参数为例,分析了碰撞前速度、着陆腿跨距、摩擦系数对着陆稳定性的影响,结果表明,本文提出的稳定性判别方法较能量法更为精确,可以考虑触地速度、角速度、摩擦系数等参数间的耦合关系.     

等离子喷涂铁－镍－钴－碳化钨涂层制动摩擦特性的研究

重载制动摩擦磨损特性制约着刹车装置的刹车性能及其使用寿命．在ＭＭ－１０００试验机上，对等离子喷涂Ｆｅ－Ｎｉ－Ｃｏ－ＷＣ涂层的重载制动摩擦特性进行了试验研究．结果表明：在相同的试验条件下，等离子喷涂涂层分别与石棉摩擦材料和半金属摩擦材料对摩时的摩擦系数均比与基体３５ＣｒＭｏ钢对摩时的高，制动时间短，制动效率和耐磨性能都明显提高，制动力矩峰比小，刹车平稳，制动特性好；制动摩擦系数与制动压力、速度和转动惯量等参数密切相关，压力增大，摩擦系数减小，速度和转动惯量增大，石棉材料摩擦副的摩擦系数降低，半金属材料摩擦副的摩擦系数升高，而且半金属材料的摩擦系数的热稳定性比石棉摩擦材料的好     

重复使用火箭着陆结构稳定性分析

近年来,包括中国在内的诸多国家相继开展垂直起降重复使用火箭的研究,运载火箭在平台上垂直着陆时的着陆稳定性为实现运载火箭重复使用的关键问题. 由于在运载火箭设计初期结构设计尚未完成,不具有供着陆稳定性分析的详细的动力学模型,难以开展着陆过程动力学仿真,故对运载火箭着陆稳定性评估方法的研究尤为必要. 本文基于广义碰撞定律,对二维运动模式下运载火箭与着陆平台的多点碰撞过程进行了分析,切向采用库伦摩擦模型给出了切向运动学恢复系数的表达式. 本文首先通过机械能约束和接触碰撞中的单边约束给出了一般运动形式下广义运动学恢复系数的值域,再对两种典型运动模式,给出了该两种典型运动模式下广义运动学恢复系数的值域. 然后考虑着陆腿中缓冲器的作用,将运载火箭与平台的碰撞近似为完全非弹性碰撞,得到了其广义运动学恢复系数,并结合运动学分析和能量法提出了一种基于碰撞后速度的着陆稳定性的判别方法. 最后以某型运载火箭着陆样机的参数为例,分析了碰撞前速度、着陆腿跨距、摩擦系数对着陆稳定性的影响,结果表明,本文提出的稳定性判别方法较能量法更为精确,可以考虑触地速度、角速度、摩擦系数等参数间的耦合关系.      

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.