Prediction of Coefficient of Restitution for Impact Elastoplastic Spheres Considering Finite Plate Thickness

Collisions between objects are a relatively common phenomenon in nature.Analyses of collision processes can greatly contribute to solving problems such as impact-rub faults and particle impacts.The coefficient of restitution is a critical parameter in the analysis of collision processes.Many experiments have shown that the coefficient of restitution is closely related to the plate thickness,and the smaller the plate thickness,the more inaccurate the coefficient of restitution predicted by the existing model,which seriously affects the process of collision analysis.To remedy this shortcoming,this paper proposes a plate thickness influence factor with the ratio of sphere diameter to plate thickness as the variable.The plate thickness influence factor can optimize the coefficient of restitution model to effectively predict the coefficient of restitution of impacting elastoplastic spheres with finite plate thickness.Finally,the validity of the new model is verified using a large amount of experimental data.

Modeling of softsphere normal collisions with characteristic of coefficient of restitution dependent on impact velocity

This letter presents a theoretical model of the normal （head-on） collisions between two soft spheres for predicting the experimental characteristic of the coefficient of restitution dependent on impact velocity. After the contact force law between the contacted spheres during a collision is phenomenologically formulated in terms of the compression or overlap displacement under considera- tion of an elastic-plastic loading and a plastic unloading subprocesses, the coefficient of restitution is gained by the dynamic equation of the contact process once an initial impact velocity is input. It is found that the theoretical predictions of the coefficient of restitution varying with the impact velocity are well in agreement with the existing experimental characteristics which are fitted by the explicit formula.

CFD Simulation of Particles Mixing and Segregation in a Rotary Coating Disk: Influence of Drag Models and Restitution Coefficient

In this paper, the dynamics of a transverse plane of a rotary coating disk of a binary mixture system comprising sand and urea particles were simulated using the two-fluid model along with the kinetic theory of granular flow in Fluent 19.1. Although some parameters relating to the material properties and size of the rotary coating disk have been researched, the effects of both drag force and restitution coefficient on the flow characteristics have yet to be examined. Thus, this paper numerically examines the effect of the inclusion of drag models and particle-particle restitution coefficients on particle dynamics in a rotary disk operating in the rolling regime of the granular bed. Three particle-particle drag models were considered: the Schiller-Naumann, Syamlal-O’Brien, and Gidaspow. The Syamlal-O’Brien and Gidaspow models were both able to successfully simulate particle segregation in a perfect rolling regime, whereas the Schiller-Naumann drag model appeared to be unable to predict the segregation of the particles and the rolling flow regime under the assumed conditions. Four different values of the restitution coefficient were also investigated: 0.7, 0.8, 0.9, and 0.95. The higher restitution values of 0.9 and 0.95 were found to substantially affect flow characteristics, ensuring suitable rolling regime behaviour for the bed during the rotational movement. The lower restitution coefficients of 0.7 and 0.8, on the other hand, indicated that irregular velocity vectors could be obtained in the active region of the granular bed.

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..

Investigation of Dynamic Behavior in 1D Non-uniform Granular System

We present a non-uniform granular system in one-dimensional case, whose granularity distribution has the fractal characteristic. The particles are subject to inelastic mutual collisions and obey Langevin equation between collisions. By Monte Carlo simulation we study the dynamic actions of the system. Far from the equilibrium, i.e., τ 〉〉 τe, the results of simulation indicate that the inhomogeneity of the system and the inelasticity of the particles have great influences on the dynamic properties of the system, and correspondingly the influence of the inhomogeneity is more significant.

Non-Gaussian and Clustering Behavior in One-Dimensional Polydisperse Granular Gas System

We present a one-dimensional dynamic model of polydisperse granular mixture with the fractal characteristic of the particle size distribution, in which the particles are subject to inelastic mutual collisions and are driven by Gaussian white noise. The inhomogeneity of the particle size distribution is described by a fractal dimension D. The stationary state that the mixture reaches is the result of the balance between energy dissipation and energy injection. By molecular dynamics simulations, we have mainly studied how the inhomogeneity of the particle size distribution and the inelasticity of collisions influence the velocity distribution and distribution of interparticle spacing in the steady-state. The simulation results indicate that, in the inelasticity case, the velocity distribution strongly deviates from the Gaussian one and the system has a strong spatial clustering. Thus the inhomogeneity and the inelasticity have great effacts on the velocity distribution and distribution of interparticle spacing. The quantitative information of the non-Gaussian velocity distribution and that of clustering are respectively represented.

Collision Statistics of Driven Polydisperse Granular Gases

We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension df. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension df, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with df, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior.

Global Pressure of One-Dimensional Polydisperse Granular Gases Driven by Gaussian White Noise

We study the global pressure of a one-dimensional polydisperse granular gases system for the first time, in which the size distribution of particles has the fractal characteristic and the inhomogeneity is described by a fractal dimension D. The particles are driven by Gaussian white noise and subject to inelastic mutual collisions. We define the global pressure P of the system as the impulse transferred across a surface in a unit of time, which has two contributions, one from the translational motion of particles and the other from the collisions. Explicit expression for the global pressure in the steady state is derived. By molecular dynamics simulations, we investigate how the inelasticity of collisions and the inhomogeneity of the particles influence the global pressure. The simulation results indicate that the restitution coefficient e and the fractal dimension D have significant effect on the pressure.

Equivalent Kelvin Impact Model for Seismic Pounding Analysis of Bridges

Based on Hertz contact theory, a method to determine the parameters of Kelvin impact model for seismic pounding analysis of bridges is proposed. The impact stiffness of Kelvin model is determined by the ratio of maximum impact force to maximum contact deformation, which is calculated based on Hertz contact theory with considering the vibration effect. The restitution coefficient which has great influence on the damping coefficient of Kelvin impact model is investigated by numerical analysis. Numerical results indicate that the impact stiffness of Kelvin impact model increases with the increment of the Hertz contact stiffness, approaching velocity or the length ratio of short to long girders. Vibration effect has remarkable influence on the impact stiffness and cannot be neglected. The restitution coefficient decreases when approaching velocity increases or the length ratio of short girder to long girder decreasing. The practical ranges of impact stiffness and restitution coefficient are obtained as 3 × 10^8--6 × 10^8 N/m and 0.6-3.95 respectively.

The collision experiment between rolling stones of different shapes and protective cushion in open-pit mines

Though gravel cushions are used worldwide in open-pit mines and railway slopes to control the impact of rolling stones,no universal technical standards have been put in place to guide engineers in their correct design,and few laboratory test results are available with which to characterize collisions between rolling stones and a gravel cushion.We carried out a large number of experiments in which rolling stones made of the same material but differently shaped were dropped from various heights onto cushions with various particle sizes and thicknesses.We investigated the characteristics of the resulting collisions,and the relationships between coefficients of restitution(CORs)of blocks with different shape and release height H,cushion thickness h and particle diameter d are obtained through linear fitting method.Orthogonal testing reveals the relative influence of block shape,release height,and the particle size and thickness of the cushion on the collision characteristics,which can assist engineers in designing a gravel cushion suitable to the distribution and weathering characteristics of rolling stones in a specific area.

Numerical Investigation of Particle Rebound Characteristics with Finite Element Method

In this work, investigation of particle rebound characteristics due to impact with surface of a target material is presented. The rebound of a spherical particle after impact on a planar surface was analyzed in detail. Specifically, the coefficient of restitution of the particle under various impact conditions was investigated numerically. This study has been conducted by carrying out a series of FEM-based (finite element method) simulations using ANSYS Autodyn software. First, a summary about the state of the art and the theoretical models for the elastic collisions were reviewed. Afterwards, the impact of an aluminum oxide particle on an aluminum alloy target surface was modeled. Using the Autodyn tool, the results were compared and validated by the experimental results of Gorham and Kharaz [1]. Selection of an appropriate equation of state (EOS) and a strength model for each material had a strong effect on the results. For both materials, the Shock EOS was applied for the final simulations. As the strength model, the Johnson-Cook and the elastic model were used, respectively. The agreement of the obtained numerical results with the experimental data confirmed that the proposed model can precisely predict the real behavior of the particle after the impact, when the material models are properly chosen. Furthermore, the effects of impact velocity and impact angle on the rebound characteristics of the particle were analyzed in detail. It was found that the selection of the exact value of friction coefficient has a drastic effect on the prediction of restitution coefficient values, especially the tangential restitution coefficient.

Measurement and analysis of restitution coefficient between maize seed and soil based on high-speed photography

The restitution coefficient is an important elementary physical parameter related to the research and development of agricultural machinery.The kinematic model of maize seed in the falling and impacting processes was developed to measure the restitution coefficient between maize seed and soil.A test bench for measuring the restitution coefficient was designed and built referred to the theory of mirror reflection.The velocities for impacting maize seed were measured and analyzed in a three-dimensional space via high-speed photography,and then restitution coefficients of in different impact conditions were obtained.On this basis,this study took flat dent seed and round seed as samples.Single factor tests were conducted to analyze the influences of these factors on the restitution coefficient.The impact angle,falling height,soil compaction,soil moisture,maize moisture content and different parts of seed were selected as test factors.The corresponding regression equations were obtained by analysis.The results showed that,as the impact angle was bigger than 25°,the restitution coefficient increased with the increase of impact angle.The restitution coefficient had a linear decreasing trend with the increase of falling height.As the soil compaction strength was 200-350 kPa,the restitution coefficient increased with the increase of soil compaction.As the soil compaction strength was larger than 350 kPa,the changing trend of the restitution coefficient was relatively stable.As the soil moisture content was 13.5%-18%,the restitution coefficient decreased with the increase of soil moisture.As the soil moisture content was 18%,the restitution coefficient was the minimum.As the maize moisture content was 11%-16%,the restitution coefficient decreased with the increase of maize moisture content.The rotational motion always occurred in falling process of flat dent seed and round seed.The probabilities of crown part and lateral part of maize seed impacting with soil were the highest,and the restitution coefficient between crown part and soil was higher than that of other parts in the same condition.

Experimental determination of restitution coefficient of garlic bulb based on high-speed photography

Restitution coefficient(RC)of garlic bulb is an important mechanical property that is required to establish the kinematics model of bulb collision and research the damage mechanism of bulb collision.In this study,kinetic equations of bulb collision were established based on Hertz's contact theory.The kinematics characteristics,elastoplastic deformation and contact damage during bulb collision were analyzed by using high-speed photography.The effects of bulb mass,moisture content,collision material,material thickness and release height on the RC were investigated by mixed orthogonal experiments and single-factor experiments.The results showed that the movement of bulb in the compression stage was translation,and the movement in the rebound stage was translation and rotation.During collision,the larger the rotational angular velocity of the bulb was,the smaller the measured RC would be.The contact damage of bulb included internal damage of the tissue,epidermis stretch and tear.The significance of effects of factors on RC decreased with the following sequence:collision material,release height,material thickness,bulb mass,and moisture content.Collision material,release height,material thickness,and bulb mass were significant factors.The RC between the bulb and Q235,nylon,and rubber decreased sequentially.The RC decreased with the increase of release height and bulb mass.The RC increased with the increase of material thickness of Q235,while it decreased with the increase of material thickness of rubber or Nylon.The determination coefficients of the regression equations between the significant factors and the RC were all greater than 0.96.The results will be helpful for damage mechanism analysis and design of garlic production equipment.

Cotton stalk restitution coefficient determination tests based on the binocular high-speed camera technology

The restitution coefficient(RC)of cotton stalks is an important elementary physical parameter that is required to establish the crushing mechanical model and research the film residue separation machinery.In this study,the calculation method of restitution coefficient considering the rotation motion of stalk-shaped agricultural materials was derived based on the principle of kinematics and the energy restitution coefficient method,and a test bench for measuring the RC was designed and built.The effects of collision material,moisture content,length,diameter,release height,and collision angle respectively on the RC were investigated by single-factor experiments and orthogonal experiments,and the regression models between influence factors and the RC were established.The results showed that Q235 showed the highest value of the RC,and it was followed by cotton stalks and soil lumps,sequentially.The RC of cotton stalks decreased with the increase of moisture content and diameter,while it increased at first and then decreased with the increase of length.As the release height was less than 500 mm,the RC increased with the increased release height.As the collision angle was less than 40°,the RC showed a linear increasing trend.The significance of the effects of factors on RC decreased with the following sequence:collision angle,length,release height,diameter,and moisture content.Length,collision angle,and release height were extremely significant.The contrast test results showed that the values based on Newton’s restitution coefficient method were smaller than that based on the energy restitution coefficient method.The verification test showed that the predicted rebound height of cotton stalks calculated based on the energy restitution coefficient method was closer to the actual rebound height,and the relative error was less than 5%.

Effect of coefficient of restitution in Euler-Euler CFD simulation of fluidized-bed hydrodynamics

Collision between particles plays an important role in determining the hydrodynamic characteristics of gas-solid flow in a fluidized bed. In the present work, earlier work （Loha, Chattopadhyay, ＆ Chatterjee, 2013） was extended to study the effect of the elasticity of particle collision on the hydrodynamic behavior of a bubbling fluidized bed filled with 530-~m particles. The Eulerian-Eulerian two-fluid model was used to simulate the hydrodynamics of the bubbling fluidized bed, where the solid-phase properties were calculated by applying the kinetic theory of granular flow. To investigate the effect of the elasticity of particle collision, different values of the coefficient of restitution were applied in the simulation and their effects were studied in detail. Simulations were performed for two different solid-phase wall boundary conditions. No bubble formation was observed for perfectly elastic collision. The bubble formation started as soon as the coefficient of restitution was set below 1.0, and the space occupied by bubbles in the bed increased with a decrease in the coefficient of restitution. Simulation results were also compared with experimental data available in the literature, and good agreement was found for coefficients of restitution of 0.95 and 0.99.

Coefficient of restitution for particles impacting on wet surfaces： An improved experimental approach

The coefficient of restitution is widely used to characterize the energy dissipation rate in numerical simulations involving particle collisions. The challenge in measuring the coefficient of restitution is the strong scatter seen in experimental data that results from varying particle properties, i.e. shape and surface roughness, and from imperfections in the experimental technique. To minimize this scattering, a novel experimental setup was developed based on two synchronized high-speed cameras capturing the collision behaviour of a particle in three dimensions. To measure the wet restitution coefficient, which describes particle impact in the presence of a liquid layer in the contact region, additional accuracy can be achieved by measuring the liquid layer thickness by a high-precision optical confocal sensor. The coefficient of restitution was measured for glass particles with two different diameters, at different relative velocities and liquid layer thicknesses, with a focus on small collision velocities and thin liquid layers, using both the improved （three dimensional） and the conventional （two dimensional） approaches to quantify the improvement of the new method＇s accuracy.

Energetic Coefficient of Restitution for a Planar Two-Body Oblique Collision with Friction

Although the coefficient of restitution was originally thought to be only a material property, the coefficient of restitution also depends upon initial conditions as well as on the frictional effect for oblique collisions. The objective of this paper is to demonstrate a method for obtaining the coefficient of restitution for oblique collisions and thereby to provide a theoretical guide for collision experiments. In this paper, we derive expressions for the energetic coefficient of restitution (e*) based on general normal contact deformation law, by which the value of e* can be obtained according to the initial conditions. An example shows that the results calculated by the derived expressions are reasonable.

Carbon dioxide desorption behavior of potassium carbonate supported on gamma-alumina solid sorbent in wet fluidized bed under steam atmosphere

Since the carbon dioxide(CO_(2))capture using solid sorbent is a reversible reaction,the solid sorbent can be regenerated by the desorption process.Therefore,the desorption process is one of the key important processes for the CO_(2)capture system.Traditionally,most of the literature studies focus on the desorption of solid sorbent under an N_(2)atmosphere.However,the desorption process of the solid sorbent is inappropriate in the real system because the system will need another process to separate CO_(2)and nitrogen(N_(2))after the desorption process.This study focused on the CO_(2)desorption of potassium carbonate supported on gamma-alumina(K2CO3/γ-Al_(2)O_(3))in a wet fluidized bed under a steam atmosphere by using the multiphase computational fluid dynamics(CFD)simulation.The effects of water thickness and dry restitution coefficient on CO_(2)desorption rate were investigated to provide a realistic particle collision behavior and to explore their effects on CO_(2)desorption phenomena.Moreover,the effect of steam velocity on the hydrodynamic behaviors of fluidization which on CO_(2)desorption rate was studied.The simulated results demonstrated that all the parameters,water thickness,dry restitution coefficient,and steam velocity had significantly affected system hydrodynamics and CO_(2)desorption rate in the wet fluidization desorption process.Furthermore,the effect of desorption temperature on CO_(2)desorption rate was evaluated for finding the appropriate temperature for CO_(2)desorption process of K2CO3/γ-Al_(2)O_(3).The results showed that the appropriate desorption temperature for CO_(2)desorption under steam atmosphere was the temperature over 150℃.

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.

Firmness measurement of peach by impact force response

The impact force response of a peach impacting on a metal flat-surface was considered as nondestructive determination of firmness. The objectives were to analyze the effect of firmness, drop height, fruit mass, and impact orientation on the impact force parameters, and to establish a relationship between the impact force parameter and firmness. The effect of fruit firmness, drop height and fruit mass on the impact force parameters (coefficient of restitution, percentage of energy absorbed, and coefficient of force-time) was evaluated. The study found that the coefficient of restitution, percentage of energy absorbed, and force-time impact coefficient were significantly affected by fruit ripeness, but not affected by drop height, impact position (fruit cheek), and mass. The percentage of absorbed energy increased with ripeness, while the force-time impact coefficient and coefficient of restitution decreased with ripeness. Relationships were obtained between the three impact characteristic parameters (force-time impact coefficient, coefficient of restitution, and percentage of energy absorbed) and peach firmness using a polynomial model (R2=0.932), S model (R2=0.910), and exponential model (R2=0.941), respectively.