Dynamic crushing of thin-walled spheres: An experimental study

Experimental studies on dynamic behavior of thin-walled spheres and sphere arrays in response to different impact velocity are presented. Ping pong balls are selected to study the collapse of thin-walled spheres. The tests were carried out by a modified Split Hopkinson Pressure Bar (SHPB) test system. The experimental results show that the deformation of thin-walled spherical shells depends on the impact velocity. The dynamic force in the range of small elastic deformation is larger than its quasi-static counterpart, but significantly below the latter after snap-through of the shell. The deformation and buckling mode are sensitive to the loading rate. It is noted that the strain rate effect of the materials and the inertia effect of the shell should be considered in the analysis of the shells response to dynamic loading.     

Soccer ball anisotropy modelling

The work presented in this paper details the development of a finite element (FE) model of a soccer ball, allowing for a greater understanding of the performance of soccer balls under dynamic conditions that are representative of play. The model consists of composite shell elements that include a hyperelastic strain energy potential equation to define the latex bladder layer and a plane stress orthotropic elastic material model to define the anisotropic woven fabric outer panels. The model was validated through a series of experimental impact tests whereby the ball was impacted normal to a rigid plate at an inbound velocity of approximately 34 ms⁻¹ (76 mph), with each impact recorded using high speed video (HSV) techniques. It was found that the combined effects of ball design and panel material anisotropy resulted in impact properties such as coefficient of restitution, contact time, deformation and the 2D shape taken up by the ball at maximum deformation, to vary with pre-impact ball orientation. The model showed good agreement with the measurements, and its ability to represent the effects of anisotropy in ball design.     

A sensing block method for measuring impact force generated at a contact part

A new method has been devised for measuring the impact force generated at the contact part in a collision. This new method, here called the sensing block method, is based on a simple structure which is made of a steel cylindrical block with a small cylindrical projection. The top of this projection serves as the contact plane in a collision. The impact force generated at the contact part can be measured using small strain gages mounted on the side surface of the projection. Through the calibration tests, the new method was confirmed to be more effective for direct measurement of impact force with relatively high sensitivity for both normal and oblique impacts. The obtained measurements are almost entirely free from disturbances caused by reflected wave interference. Using this method, an impact experiment on commercially available pure aluminum bars with different lengths and spherical ends was performed. From the experimental results, it was found that the generated maximum force gradually increased in proportion to the square root of potential energy, and that the generated mean stress on the contact part was constant and independent of the radius of curvature of the spherical end and impact velocity.     

Collision and rebound of ping pong balls on a rigid target

The collision and rebound behavior of ping pong balls impinging onto rigid target are studied. Three dimensionless dominant parameters are identified: (1) the ratio of the wall-thickness to the average radius of the ball; (2) the dimensionless initial velocity; and (3) the yield strain of the material.Depending on the dimensionless initial velocity, various collision and rebound behaviors of the ball are revealed: (1) When the initial velocity is low, the deformation of the ball remains purely elastic, for which the characteristic duration is theoretically obtained; (2) With the increase of initial velocity, the ball's cap begins to buckle and multiple impacts occur, leading to the increase of the restitution duration and the reduction of the coefficient of restitution (COR); (3) With the higher initial velocity, the ball's cap buckles permanently, leading to the disappearance of multiple impacts and a sudden drop of the restitution duration; consequently the COR decreases from 0.5 to 0.3; and (4) When the initial velocity is close to the material's yield velocity, the ball buckles into a non-axisymmetric mode.The simulation results are also compared with experimental ones. Furthermore, the effects of thickness-to-radius ratio, yield strain and coefficient of friction are also discussed.     

卧式行星球磨机最佳参数的数值模拟

采用颗粒离散元法模拟得到卧式行星磨磨筒内钢球平均接触力大小以及随填球率、钢球直径、磨筒半径和公转半径的变化规律,并与实验结果进行了比较分析。结果表明:行星磨的粉磨速率可以由平均接触力大小来确定,粉磨速率的对数是平均接触力线性函数,斜率为0.087。磨筒填球率的增加,导致钢球的平均接触力减小;随着钢球直径、磨筒半径、公转半径的增大,平均接触力呈线性增长,斜率分别为2.5,0.14,0.03,其中钢球直径对平均接触力的影响最大。     

滚珠丝杠副滚珠循环系统的动力学研究和仿真

研究滚珠丝杠副的滚珠循环系统中的碰撞接触现象,根据赫兹接触理论和经典碰撞理论,并考虑滚珠和导珠管的几何外形、丝杠的转动速度等因素,建立滚珠和导珠管碰撞接触的动力学模型。在此基础上分析了滚珠的入口速度、材料密度对碰撞力大小、接触时间的影响。应用多体动力学软件Adams,根据实际物理参数,创建了导珠管和滚珠的几何模型,对钢珠在导珠管中的运动情况进行仿真模拟,获得了碰撞过程中滚珠的运动轨迹、速度、加速度以及碰撞力的变化情况。通过对不同参数的物理模型理论计算和仿真,分析了影响滚珠循环系统动力学的主要因素。理论计算和仿真、试验结果吻合较好,这对于优化导珠管结构,研究开发新型滚珠丝杠副有重要的意义。     

考虑接触变形的梁受到球碰撞时弹塑性冲击荷载

使用有限元数值方法模拟了考虑接触变形时弹性、弹塑性梁分别受到弹性和弹塑性球横向撞击全过程,得到具有不同刚度球、不同刚度梁、不同质量比和不同撞击速度时梁受到的弹性和弹塑性冲击荷载。从中得到如下结论:对于弹性球撞击弹性梁:随着球-梁的相对刚度值的增大,冲击荷载峰值非线性增加,球和梁的接触时间减小。随着球速的增大,冲击荷载峰值随速度近似呈线性关系增长;接触时间降低,但不呈线性关系;冲击荷载最大值不一定出现在第一次接触的时段内。速度相同时,质量比越大,球和梁接触的时间越短;冲击荷载峰值越小。对于弹性和弹塑性球撞击弹塑性梁:梁的材料非线性性质对冲击荷载影响很大,弹塑性冲击荷载峰值比弹性荷载峰值低一个量级。冲击荷载的峰值和初速度成正比,但不是线性关系。质量比越大,冲击荷载峰值越小,且球和梁接触时间越短。     

Finite element analysis of contact–impact between a shell of revolution and a rigid wall

In this paper an attempt is made to use a shell theory that includes transverse normal stresses to study the contact–impact behaviour involving thick shells. The problem is solved using the finite element method. The contact criterion is imposed by means of displacement and force constraints. The Lagrangian multipliers technique is used to impose these constraints. The case of a spherical shell impacting a rigid wall with a constant velocity is considered for the analysis. It is shown that the method produces accurate results with less amount of computation than the usual three-dimensional analyses. The effects of the thickness of shell, velocity of impact and modulus of the material are studied.     

Experimental study and numerical simulation of the vertical bounce of a polymer ball over a wide temperature range

The dependence to temperature of the rebound of a solid polymer ball on a rigid slab is investigated. An acrylate polymer ball is brought to a wide range of temperatures, covering its glass to rubbery transition, and let fall on a granite slab while the coefficient of restitution, duration of contact, and force history are measured experimentally. The ball fabrication is controlled in the lab, allowing the mechanical characterization of the material by classic dynamic mechanical analysis. Finite element simulations of the rebound at various temperatures are run, considering the material as viscoelastic and as satisfying a WLF equation for its time–temperature superposition property. A comparison between the experiments and the simulations shows the strong link between viscoelasticity and time–temperature superposition properties of the material and the bounce characteristics of the ball.     

Equivalent normal stiffness of the ball in granular dynamics simulations

In the paper a concept of an equivalent normal ball stiffness based on averaging the work done by the nonlinear Hertz force during the impact is introduced and a numerical assessment is made on its efficiency in simulating collinear collisions. The systems of balls investigated, ranging from 2 to 30, are considered to be coupled and conservative. The energy and momentum conservation principles are used to assess the accuracy of simulation results. The effect of the time step for both linear and nonlinear models is investigated and it is shown that the linear model allows an increased time step compared to a nonlinear one while meeting energy and momentum conservation requirements with the same accuracy. In the paper also the pattern of break up for both models and different number of balls is investigated. It is found that for both models the pattern is the same: the balls are disconnected one at the time with constant rate and this rate does not depend on the number of balls. The financial assistance provided by the National Science and Engineering Research Council of Canada is gratefully acknowledged.     

Plane wave propagation in 2D and 3D monodisperse periodic granular media

Plane wave propagation in periodic ordered granular media comprising of elastic spherical particles is investigated. The spheres are under zero precompression and are assumed to interact via the Hertzian contact potential. Various two- and three-dimensional granular structures such as hexagonal packing (2D and 3D), face-centered cubic and body-centered cubic packings are considered in the present study, with the plane impact either normal or oblique to the granular system. For the normal impact case, 1D chains equivalent to the 2D and 3D structures are obtained. A universal relation between the wavefront speed and the force amplitude is derived, valid for all the granular structures studied. In the angular impact case, the shear component of the amplitude of the particle velocity is found to initially decay exponentially and further in a series of linear regimes. By employing simpler models, semi-analytical predictions are obtained for the decay of shearing effect.     

Dynamic simulation and energy absorption of tapered thin-walled tubes under oblique impact loading

In real-world impact loading situations the structure could be subjected to both axial and off-axis loads. Tapered thin-walled rectangular tubes have been considered desirable impact energy absorbers due to their ability to withstand oblique impact loads as effectively as axial loads. Despite this, relatively few studies have been reported on the response of such structures under oblique loading. The aim of this paper is to compare the energy absorption response of straight and tapered thin-walled rectangular tubes under oblique impact loading, for variations in the load angle, impact velocity and tube dimensions. It is found that the mean load and energy absorption decrease significantly as the angle of applied load increases. Nevertheless, tapering a rectangular tube enhances its energy absorption capacity under oblique loading. The outcome of the study is design information for the use of straight and tapered thin-walled rectangular tubes as energy absorbers in applications where oblique impact loading is expected.     

Experimental investigation of lift and drag on a ball rotated by an inclined water jet

Abstract

The lift and drag of rotating balls with Reynolds numbers from 4500 to 6500 were obtained using a ping‐pong ball impinged by a water jet with inclination angles ranging from θ = 3.43° to 11.3°. Results show that water strikes the ball surface and leaves from the wake region behind the ball, which spins more slowly when a large amount of water stays on the ball surface, causing low lift and low drag forces to be exerted on the ball. In contrast, the ball spins quickly when a small amount of water stays on the ball surface. This causes high lift force to be exerted on the ball, which verifies the Magnus effect of a rotating ball. The drag force increases nonlinearly with the angular speed of the ball because there is a branch flow out from the stretched wake behind the ball. Results also indicate that the lift force of the ball increases with jet exit speed, but the drag force does not change significantly with the jet exit speed. This reveals that the momentum change in the direction parallel to gravity is more significant than that in the direction normal to gravity.     

对称角铺设矩形复合材料叠层板受低速冲击时的应力波传及解层破坏分析

采用碰撞力分段模型和一阶剪切理论分析了给定初始速度的铁球与四边简支的复合材料叠层板中心发生碰撞的动力学行为,包括碰撞力及接触时间的变化规律、叠层板的振动响应、应力波传、表面沉陷等。并根据忽略厚度的界面模型假设及简化的Tsai Wu张量理论对复合材料叠层板的解层破坏进行了计算和分析,并给出了破坏区大小与铁球初始速度的关系。研究表明:碰撞力与铁球的初始速度成正比;复合材料叠层板中应力波传沿固定方向的相速度在各层内相同,拉伸应力波传速度沿纤维总体占优的方向大于其垂直的方向,而剪切应力波传速度则相反。即使在较低的初始碰撞速度下, 复合材料叠层板的解层破坏也很明显,并且破坏区域随初始碰撞速度的增大而不断扩展,其形状也会发生改变。      

Impact analysis of composite laminates with multiple delaminations

This paper concerns with the transient response of composite laminates with multiple delaminations subjected to low-velocity impact by a rigid ball. The finite element method based on the Mindlin plate theory is employed to describe the motion and deformation of the laminates. A Hertzian-type indentation law is adopted to calculate the impact force between the laminates and the rigid ball. To deal with the dynamic contacts between delaminated layers effectively, a modified Lagrange multiplier technique is employed. For multi-body dynamic contacts, the computation of contact force between delaminated layers is addressed. Numerical results provide much information for understanding the impact phenomenon of the composite laminates with multiple delaminations.     

Frictional viscoelastic based model for spherical particles collision

In this paper, collision of two balls in three dimensions with regard to energy loss and friction is studied. This investigation intends to propose the procedure which can determine the condition of pure sliding, sliding with rolling or sticking, and rolling or sticking at the beginning of contact. Furthermore, the other aim of this research is to suggest the closed-form relation for post-collision angular and linear velocities in which the possibility of three regimes of impact, pure sliding, sliding with rolling, and pure rolling, are considered. In this investigation, viscous Hertz contact force describes the normal interaction force. Moreover, stick regime is not taken into account and during sliding, Coulomb friction is considered as a tangential force, which is endorsed by collision experimental data in case the collision regime is not stick regime. Considering aforementioned issues along with employing momentum conservation, the condition for the possibility of transformation of sliding to sticking or rolling during collision, and furthermore, the analytical relation for post-collision velocities are achieved. Eventually, in order to demonstrate the procedure of solving two balls collision, numerical examples are presented.     

Experimental investigation of the kinematics of post-impact ice fragments

Hail is more hazardous for aircraft engines compared to rain and snow, mainly, because of its solid nature and high water content. In extreme cases it can lead to engine flame out. In order to avoid such situations, aero engines should be designed to withstand hail ingestion. For this purpose we have studied the post-impact characteristics of ice, such as particle velocity and directions of travel. To achieve this goal, a large experimental program has been undertaken, in which spherical ice specimens were projected against a rigid plate. Three specimen diameters (6.2, 12.9 and 27.5 mm) and four impact angles (20, 45, 67.2 and 90°) were considered, as well as a wide range of impact velocities (60-200 m/s). From this experimental work, we can conclude that the ice fragments formed after impact do not bounce back and that the post-impact ice trajectory angle is lower than 2°. This is in line with observations found in the literature. On the other hand, the ice fragments are mainly organised in a circular cloud, when observed in the target plane. The center of this cloud has the same velocity as the initial ice ball tangential impact velocity. Furthermore, the cloud radius expands with a rate proportional to the ice ball normal impact velocity. Finally, each fragment inside the cloud has a relative velocity which varies linearly with its distance from the cloud center. These experimental observations should be very helpful in developing models and simulations of hail ingestion by aircraft engines.     

Energy dissipation in collision of two balls covered by fine particles

A new fine particle impact damper (FPID) is composed of a spherical impactor and a small quantity of fine particles as damping agent. The model of energy dissipation in the collision between two balls covered by fine particles is necessary to investigate the mechanism and performance of FPID. In this study, a simplified model verified by FEA simulations is proposed to estimate the energy dissipation in collision between two balls covered by fine particles. In addition, the energy dissipation in the collision between two balls covered by fine particles is compared with that in the impact between two balls without fine particles, by means of theoretical predictions. FEA simulations are also carried out to discuss the effects of diameter ratio of particle to ball, particle material and particle amount on the energetic expression of the elastic–plastic loading (EPL) index (EPL_E). The results from the FEA simulations agree well with the estimations from the model proposed in this paper. It is concluded that the energy dissipation in the collision between two balls covered by fine particles can be predicted by classical collision models of two particles through the substitution of several parameters from balls; the plastic deformation of fine particles affixed on balls can exhaust much more energy than that of the two balls without particles, which is the reason for the good performance of FPID; the diameter ratio of particle to ball and the material of particles do not have significant effects on the EPL_E when the ratio is limited to the range of [1/200 – 1/10]. A correlation of the EPL_E and dimensionless initial relative velocity is also found for the collisions between two balls, which is independent not only of the particle size and material properties but also of the particles presence.     

Simulation of Adhesive–Dissipative Behavior of a Microparticle Under the Oblique Impact

The adhesive–dissipative behavior of a microparticle under the oblique impact is investigated numerically and the new discrete element method (DEM)-compatible interaction model is elaborated. The modeling approach is based on the Derjaguin–Muller–Toporov model of normal interaction for the adhesive elastic contact. Adhesion hysteresis is specified by the loss of the kinetic energy governed by the fixed amount of the adhesion work, required to separate two adhesive contacting surfaces. This effect is captured in the new interaction model by adding an additional dissipative force component to normal contact during unloading and detachment. The essential feature of this approach, differing from that of the viscous damping model, is that, according to the proposed method, the amount of the dissipated energy is not influenced by the actual initial velocity during the entire contact. The influence of adhesion on slip friction is reflected by considering the adhesive normal force components in the Coulomb's law of friction. The contribution of the adhesion-related dissipation is illustrated by a comparison of the behavior of the attractive–dissipative and attractive–non-dissipative models. The oblique impact of a microparticle on the plane surface at the intermediate impact angle is also investigated numerically. The link between adhesion and friction is supported by the numerical results.     

Transformation of sliding motion to rolling during spheres collision

In this research, we have investigated the three-dimensional elastic collision of two balls, based on friction in the tangential plane. Our aim is to offer analytical closed form relations for post collision parameters such as linear and angular velocities, collision time and tangential and normal impulse in three dimensions. To simplify the problem, stick regime is not considered. In other words, balls have a low tangential coefficient of restitution. Sliding, sliding then rolling, and rolling at the beginning of contact are three cases that can occur during impact which have been considered in our research. The normal interaction force is described by the Hertz contact force and dimensionless analysis is used for investigating normal interaction force; furthermore, Coulomb friction is considered during sliding. Experimental data for collisions show when sliding exists through the impact, tangential impulses can be taken as frictional impulses using the Coulomb law if the dynamic regime is not stick regime. To identify transformation of sliding motion to rolling or sticking during the impact process, linear and trigonometric functions are considered as an approximation for the normal interaction force. Afterwards, we have obtained the condition for the possibility of this transformation; moreover, we can estimate the duration of sliding and rolling or sticking. We have obtained an analytical solution for maximum force and deformation, collision time, impulses and post-collision linear and angular velocities in three dimensions.