Velocity-dependent Restitution Coefficient Research Articles

Gravity effects in mass-spring-damper models of inelastic collisions

A ball bouncing on a rigid surface is modeled as a mass-spring-damper system. We consider the effect of including or neglecting the force of gravity, extending previous work that shows that including gravity yields a velocity-dependent coefficient of restitution (COR). This velocity dependence is most pronounced at low-impact velocities and high damping. Previously-published models differ in defining the termination of the collision, with some referencing the ball’s position and others noting when the contact force becomes zero. We propose a new model that combines aspects of these approaches. The various models are compared in their predictions for the COR and collision duration, and are compared to experimental data from a cart on an inclined track bouncing repeatedly on a spring. While the new model shows some improvement over the prior collision termination conditions, the inclusion of gravity is the more important consideration in modeling repeated bouncing.

A nonresonant triboelectric-electromagnetic energy harvester via a vibro-impact mechanism for low-frequency multi-directional excitations

A majority of current studies about non-resonant energy harvesters for low-frequency energy are based on pure experiment without theoretical modeling. In this work, a novel hybrid triboelectric-electromagnetic harvester in form of a rolling magnet inside a casing is proposed for scavenging low-frequency energy from multi-directional excitations, which includes a number of sub triboelectric energy harvesters (sub-TEHs) and two sub electromagnetic energy harvesters (sub-EEHs). Both the mechanical model and electrical model are established for a horizontal excitation, a rocking excitation and a vertical excitation. A velocity-dependent coefficient of restitution for impact and an impact-velocity-dependent charge density are obtained via experiment, which improve the theoretical model and result in better agreement between theoretical and experimental results. Structural response and electric outputs under different types of external excitations are investigated theoretically. It is found that the harvester has a good sensitivity to small-angle ultra-low-frequency rocking excitations, which is beneficial to low-frequency energy harvesting. When tested on a shake table at 4 Hz, a sub-TEH and a sub-EEH can generate peak powers of 212.1 µW and 57.8 mW, respectively. An electronic clock is able to work continuously when powered by the harvester shaken by hand. The harvester is also tested inside a handbag carried by a walking human for biomechanical energy and on a floating structure for wave energy, which exhibits good performance. This work presents a novel design with a capability of harvesting low-frequency energy under different external excitations, which contributes to the development and applications of triboelectric and electromagnetic energy harvesting.

Inelastic bouncing of a spherical ball in the presence of quadratic drag with application to sports balls

The bouncing motion of a spherical ball following its repeated inelastic impacts with a horizontal flat surface is analyzed. The effect of air resistance on the motion of the ball is accounted for by using the quadratic drag model. The effects of inelastic impacts are accounted for by using the coefficient of restitution, which is assumed to remain constant during repeated impacts. Also presented is an extension of the analysis allowing for a velocity-dependent coefficient of restitution. Closed-form expressions are derived for the velocity, position, maximum height, duration, and dissipated energy during each cycle of motion. The decrease of successive rebound heights in the presence of air resistance is more rapid for higher values of the launch velocity, because the drag force is stronger and acts longer. Air resistance can significantly affect the value of the coefficient of restitution determined in a dropping ball test. For a given number of rebounds, the energy dissipated by inelastic impacts is greater than the energy dissipated by air resistance, if the launch velocity is sufficiently small. The opposite is true for greater values of the launch velocity. The derived formulas are applied to analyze the bouncing motion of a ping pong ball, tennis ball, handball, and a basketball.

Modification of the spherical particle spring-damping contact model from contact velocity dependent restitution coefficients

The most common granular flow Discrete Element Method (DEM) simulation contact model is the spring-damping model, which typically uses constant restitution coefficients to estimate the model damping coefficient. The predicted rebound velocities are thus pre-determined by the ad hoc restitution coefficient setting. Considering the fact that the coefficient of restitution is contact velocity-dependent and event driven, we advance the spring-damping model using three contact velocity-dependent restitution coefficient approaches. We show that the spring-damping model using a new restitution coefficient approach derived from the model of Walton & Braun (1986) gives good rebound velocity predictions over a range of contact velocities for four spherical particles of different hardness.

Role of cohesion in the formation of kink wave fronts in vibrofluidized granular materials

The formation of kink wave fronts (KWFs) in a quasi-two-dimensional granular system is investigated numerically with a focus on the role of cohesive interactions between individual particles. The cohesive particle–particle interaction is achieved through tuning the velocity-dependent coefficient of restitution, based on an analytical model introduced recently (Gollwitzer et al., 2012; Müller and Huang, 2016). A comparison with experimental results (Butzhammer et al., 2015) indicates that the threshold for the emergence of traveling KWFs matches the regime in which the center of mass height of the granular layers fluctuates with a period that triples the vibration period. Further comparisons between dry and wet granular dynamics reveal that KWFs are more pronounced in wet granular layers because of enhanced collective motion induced by cohesion.

NON-FICKIAN DIFFUSION IN TIME–SPACE FLUCTUATING DIFFUSIVITY LANDSCAPES: FROM SUPERFAST TO ULTRASLOW

In this study, the local time and space structural derivative diffusion model is introduced to describe non-Fickian diffusion in heterogeneous media from ultraslow to superfast. Its equivalent heterogeneous diffusion process (HDP) model with time and space-dependent diffusion coefficient is also given. The connection and difference between the proposed model and the existing HDP models are compared. Different types of diffusion processes can be captured by selecting proper structural functions in the structural derivative model, which is determined based on the concentration of tracers, mean squared displacement (MSD) and diffusion coefficient of the heterogeneous media. The dynamics of the free cooling granular gases with constant restitution coefficient and velocity-dependent restitution coefficient is investigated to test the proposed model. The results show that the diffusion law of the granular particles is not uniform for the short and long-time scales, but one uniform diffusion law derived from the proposed model can well fit the data in different time scales. The patterns of the time-dependent diffusion coefficient are also simulated to deeply understand the complicated non-Fickian diffusion in complex system.

Internal and surface waves in vibrofluidized granular materials: Role of cohesion.

Wave phenomena in vibrofluidized dry and partially wet granular materials confined in a quasi-two-dimensional geometry are investigated with numerical simulations considering individual particles as hard spheres. Short-ranged cohesive interactions arising from the formation of liquid bridges between adjacent particles are modeled by changing the velocity-dependent coefficient of restitution. Such a change effectively suppresses the formation of surface waves, in agreement with previous experimental observations. The difference in pattern creation arises from the suppressed momentum transfer due to wetting and it can be quantitatively understood from an analysis of binary impacts.

Impact velocity-dependent restitution coefficient using a coupled Eulerian fluid phase-Eulerian solid phase-Lagrangian discrete particles phase model in gas-monodisperse particles internally circulating fluidized bed

A Coupled Eulerian fluid phase-Eulerian solid phase-Lagrangian discrete particles phase model (CEEL) is proposed to study of the hydrodynamics of gas and monodisperse particles in an internally circulating fluidized bed (ICFB). The impact velocity-dependent coefficient of restitution (CoR) used in the kinetic theory of granular flow (KTGF) is determined from the dynamic information of Lagrangian discrete particle phase using discrete element method (DEM). The interphase momentum exchange coefficient between Eulerian fluid phase and Eulerian solid phase gives back to the Lagrangian discrete particle phase. Numerical simulations show that the impact velocity-dependent CoR decreases with the increase of solid volume fraction and impact velocity of discrete particles. The predicted velocities and circulation mass fluxes of Eulerian solid phase and Lagrangian discrete particles phase are close to each other in the ICFB. The granular temperature and rotational granular temperature of Eulerian solid phase and Lagrangian discrete particles phase are predicted. Present CEEL model provides all fields of continuous fluid phase, Eulerian solid phase and the motion of discrete particles in the ICFB.

Incorporation of velocity-dependent restitution coefficient and particle surface friction into kinetic theory for modeling granular flow cooling.

Kinetic theory (KT) has been successfully used to model rapid granular flows in which particle interactions are frictionless and near elastic. However, it fails when particle interactions become frictional and inelastic. For example, the KT is not able to accurately predict the free cooling process of a vibrated granular medium that consists of inelastic frictional particles under microgravity. The main reason that the classical KT fails to model these flows is due to its inability to account for the particle surface friction and its inelastic behavior, which are the two most important factors that need be considered in modeling collisional granular flows. In this study, we have modified the KT model that is able to incorporate these two factors. The inelasticity of a particle is considered by establishing a velocity-dependent expression for the restitution coefficient based on many experimental studies found in the literature, and the particle friction effect is included by using a tangential restitution coefficient that is related to the particle friction coefficient. Theoretical predictions of the free cooling process by the classical KT and the improved KT are compared with the experimental results from a study conducted on an airplane undergoing parabolic flights without the influence of gravity [Y. Grasselli, G. Bossis, and G. Goutallier, Europhys. Lett. 86, 60007 (2009)10.1209/0295-5075/86/60007]. Our results show that both the velocity-dependent restitution coefficient and the particle surface friction are important in predicting the free cooling process of granular flows; the modified KT model that integrates these two factors is able to improve the simulation results and leads to better agreement with the experimental results.

An Improved Mathematical Model of Galton Board With Velocity-Dependent Restitution

A Galton board is an instrument invented in 1873 by Francis Galton (1822–1911). It is a box with a glass front and many horizontal nails or pins embedded in the back and a funnel. Galton and many modern statisticians claimed that a lead ball descending to the bottom of the Galton board would display random walk. In this study, a new mathematical model of Galton board is developed, to further improve three very recently proposed models. The novel contribution of this paper is the introduction of the velocity-dependent coefficient of restitution. The developed model is then analyzed using symbolic dynamics. The results of the symbolic dynamics analysis prove that the developed Galton board model does not behave the way Galton envisaged.

A coupled Eulerian fluid phase-Eulerian solids phase-Lagrangian discrete particles hybrid model applied to gas-solids bubbling fluidized beds

Because the traditional Eulerian-Eulerian approach and Lagrangian-Eulerian approach are essentially the description of dense gas-solid two-phase flow in two reference frames, it is natural to expect that these approaches are related. A coupled Eulerian fluid phase-Eulerian solids phase-Lagrangian discrete particle phase (CEEL) hybrid approach is proposed where transport equations of fluid phase and Eulerian solids phase are solved in an Eulerian framework and the discrete particle phase is represented as Lagrangian framework. In this CEEL model, the solid phase is described by Eulerian solids phase and Lagrangian discrete particle phase. The ghost phase is introduced to represent either Eulerian solids phase or discrete particle phase as the solid phase. The Eulerian solids phase properties are modeled by means of kinetic theory of granular flow with an impact velocity-dependent restitution coefficient as functions of granular temperature. The collisions of discrete particles are handled using a discrete element method. Simulations are carried out in a dense gas-solid bubbling fluidized bed. The predicted solids flux agrees with the experimental data. Such approaches to closure model building enforce the macroscopic conservation principles while attempting to accommodate local variations in momentum exchange due to bubble formation mechanisms in gas-solid fluidized beds.

Response regimes in equivalent mechanical model of strongly nonlinear liquid sloshing

Equivalent mechanical model of liquid sloshing in partially-filled cylindrical vessel is treated in the cases of free oscillations and of horizontal base excitation. The model is designed to cover both regimes of linear and essentially nonlinear sloshing. The latter regime involves hydraulic impacts applied to the walls of the vessel. These hydraulic impacts are commonly simulated with the help of high-power potential and dissipation functions. For analytical treatment, we substitute this traditional approach by consideration of the impacts with velocity-dependent restitution coefficient. The resulting model is similar to recently explored vibro-impact nonlinear energy sink (VI NES) attached to externally forced linear oscillator. This similarity allowed exploration of possible response regimes. Steady-state and chaotic strongly modulated responses are encountered. Besides, we simulated the responses to horizontal excitation with addition of Gaussian white noise, and related them to reduced dynamics of the system on a slow invariant manifold (SIM). All analytical predictions are in good agreement with direct numerical simulations of the initial reduced-order model.

Granular temperature with discrete element method simulation in a bubbling fluidized bed

The Discrete Element Method (DEM) plays an important role in understanding and modeling the kinetic characteristics in granular systems. A soft-sphere method with a linear spring–dashpot model was used in the simulation of a bubbling fluidized bed. The time-averaged granular temperature and vertical particle velocity at different heights were numerically studied and compared to experimental measurements of Müller. The influence of a velocity-dependent coefficient of restitution and three drag models were also investigated in this work. Good agreement was found between the DEM simulation and Müller’s experiment, especially using the DiFelice drag model. The variable coefficient of restitution, with a sufficiently high yielding relative velocity, gives a granular temperature that is a little lower compared to that of a constant coefficient of restitution, while it predicts a more intense velocity fluctuation, with a lower yielding relative velocity. By comparing the granular temperature in the vertical direction and in the transverse direction, a strong anisotropy is found in the bed.

Simulations of granular gravitational collapse

A freely cooling granular gas in a gravitational field undergoes a collapse to a multicontact state in a finite time. Previous theoretical [D. Volfson et al., Phys. Rev. E 73, 061305 (2006)] and experimental work [R. Son et al., Phys. Rev. E 78, 041302 (2008)] have obtained contradictory results about the rate of energy loss before the gravitational collapse. Here we use a molecular dynamics simulation in an attempt to recreate the experimental and theoretical results to resolve the discrepancy. We are able to nearly match the experimental results, and find that to reproduce the power law predicted in the theory we need a nearly elastic system with a constant coefficient of restitution greater than 0.993. For the more realistic velocity-dependent coefficient of restitution, there does not appear to be a power-law decay and the transition from granular gas to granular solid is smooth, making it difficult to define a time of collapse.

Granular gas of viscoelastic particles in a homogeneous cooling state

Kinetic properties of a granular gas of viscoelastic particles in a homogeneous cooling state are studied analytically and numerically. We employ the most recent expression for the velocity-dependent restitution coefficient for colliding viscoelastic particles, which allows us to describe systems with large inelasticity. In contrast to previous studies, the third coefficient a 3 of the Sonine polynomials expansion of the velocity distribution function is taken into account. We observe a complicated evolution of this coefficient. Moreover, we find that a 3 is always of the same order of magnitude as the leading second Sonine coefficient a 2 ; this contradicts the existing hypothesis that the subsequent Sonine coefficients a 2 , a 3 … , are of an ascending order of a small parameter, characterizing particles inelasticity. We analyze evolution of the high-energy tail of the velocity distribution function. In particular, we study the time dependence of the tail amplitude and of the threshold velocity, which demarcates the main part of the velocity distribution and the high-energy part. We also study evolution of the self-diffusion coefficient D and explore the impact of the third Sonine coefficient on the self-diffusion. Our analytical predictions for the third Sonine coefficient, threshold velocity and the self-diffusion coefficient are in a good agreement with the numerical finding.

Equivalence of the freely cooling granular gas to the sticky gas

A freely cooling granular gas with a velocity-dependent restitution coefficient is studied in one dimension. The restitution coefficient becomes near elastic when the relative velocity of the colliding particles is less than a velocity scale delta . Different statistical quantities, namely, density distribution, occupied and empty cluster length distributions, and spatial density and velocity correlation functions, are obtained using event driven molecular dynamic simulations. We compare these with the corresponding quantities of the sticky gas (inelastic gas with zero coefficient of restitution). We find that in the inhomogeneous cooling regime, for times smaller than a crossover time t{1} , where t{1} approximately delta;{-1} , the behavior of the granular gas is equivalent to that of the sticky gas. When delta-->0 , then t{1}-->infinity and, hence, the results support an earlier claim that the freely cooling inelastic gas is described by the inviscid Burgers equation. For a real granular gas with finite delta , the existence of the time scale t{1} shows that, for large times, the granular gas is not described by the inviscid Burgers equation.

Simulations of dense granular gases without gravity with impact-velocity-dependent restitution coefficient

We report two-dimensional simulations of strongly vibrated granular materials without gravity. The coefficient of restitution depends on the impact velocity between particles by taking into account both the viscoelastic and plastic deformations of particles, occurring at low and high velocities respectively. Use of this model of restitution coefficient leads to new unexpected behaviors. When the number of particles N is large, a loose cluster appears near the fixed wall, opposite the vibrating wall. The pressure exerted on the walls becomes independent of N, and linear in the vibration velocity V, quite as the granular temperature. The collision frequency at the vibrating wall becomes independent of both N and V, whereas at the fixed wall, it is linear in both N and V. These behaviors arise because the velocity-dependent restitution coefficient introduces a new time scale related to the collision velocity near the cross over from viscoelastic to plastic deformation.

Statistical properties of granular gas under microgravity one-dimensional inelastic hard rod system

We have studied numerically statistical properties of granular gas in a one-dimensional inelastic (viscoelastic) hard rod model under microgravity, which is designed to the mimic experimental granular-vibrated beds by introducing a velocity-dependent restitution coefficient (VDRC). Our systematic simulations show that various macroscopic properties of this model are quantitatively different from a linear combination of the previous simulations based on the constant restitution coefficient (CRC). The present results are significantly important to study a vibration response and dynamics of granular gas especially in microgravity experiment.

Voronoi neighbor statistics of homogeneously sheared inelastic hard disks and hard spheres

In this work we extend our earlier Voronoi neighbor analysis [J. Chem. Phys. 123, 074502 (2005)] to homogeneously sheared inelastic hard-particle structures, the simplest model for rapid granular matter. The pair distribution function is partitioned into the nth neighbor coordination number (Cn), and the nth neighbor position distribution [fn(r)]. The distribution of the number of Voronoi faces (Pn) is also considered since C1 is its mean. We report the Cn and Pn for the homogeneously sheared inelastic hard-disk and hard-sphere structures. These statistics are sensitive to shear ordering transition, the nonequilibrium analogue of the freezing transition. In the near-elastic limit, the sheared fluid statistics approach that of the thermodynamic fluid. On shear ordering, due to the onset of order, the Cn for sheared structures approach that of the thermodynamic solid phase. The suppression of nucleation by the homogeneous shear is evident in these statistics. As inelasticity increases, the shear ordering packing fraction increases. Due to the topological instability of the isotropically perturbed face-centered cubic lattice, polyhedra with faces 12 to 18, with a mean at 14, coexist even in the regular close packed limit for the thermodynamic hard-sphere solid. In shear ordered inelastic hard-sphere structures there is a high incidence of 14-faceted polyhedra and a consequent depletion of polyhedra with faces 12, 13, 15-18, due to the formation of body-centered-tetragonal (bct) structures. These bct structures leave a body-centered-cubic-like signature in the Cn and Pn data. On shear ordering, close-packed layers slide past each other. However, with a velocity-dependent coefficient of restitution, at a critical shear rate these layers get disordered or amorphized. We find that the critical shear rate for amorphization is inversely proportional to the particle diameter, as compared to the inverse square scaling observed in dense colloidal suspensions.

Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models

The revised Enskog approximation for a fluid of hard spheres which lose energy upon collision is discussed for the case that the energy is lost from the normal component of the velocity at collision but is otherwise arbitrary. Granular fluids with a velocity-dependent coefficient of restitution are an important special case covered by this model. A normal solution to the Enskog equation is developed using the Chapman-Enskog expansion. The lowest order solution describes the general homogeneous cooling state and a generating function formalism is introduced for the determination of the distribution function. The first order solution, evaluated in the lowest Sonine approximation, provides estimates for the transport coefficients for the Navier-Stokes hydrodynamic description. All calculations are performed in an arbitrary number of dimensions.