Coefficient of restitution

Game-Related Properties. Eot some activities, such as miming and wrestdng, the only consideration is the direct impact by the player. Eot others, eg, tennis, baseball, or soccer, the system must also provide acceptable bad-to-surface contact properties. Important bad-response properties on the artificial surface ate coefficients of restitution and friction, because these direedy determine the angle, speed, and spin of the bad. 

The coefficient of restitution is defined as the ratio of the vertical components of the impact and rebound velocities resulting when a bad is dropped or thrown onto a playing surface. The velocities or related rebound heights may be measured photographically. Criteria such as bad inflation pressure, air temperature, and other detads must be specified. 

In this case, an additional equation is required before the final velocities may be found. Thus, the coefficient of restitution e is defined as the ratio of the velocity of separation to the velocity of approach ... 

There are other, less commonly used, methods for measuring hardness. One is an impact method in which an indenter is dropped from a known height onto a specimen, and either the size of the indentation, or the coefficient of restitution, is measured. Another is the pendulum method in which a rocking pendulum is applied to a specimen surface. The damping of the pendulum s oscillations is a measure of the hardness. Still another is Moh s scratch method in which the ability of one specimen to scratch another is observed. These methods are described in various books (McColm, 1990), but only the... 

It is believed that this can be related to the differences in coefficient of restitution between the conveyed particles and the pipeline walls. On impact with the rubber, the particles will be decelerated, since the rubber will absorb much of the energy of impact. As a consequence, the particles will have to be re-accelerated back to their terminal velocity. The coefficient of restitution of the particles against the steel pipeline wall will be very much lower. This effect is clearly magnified by increase in velocity and explains why there is little difference between the two pipeline materials in low velocity dense phase conveying, but differ by 50% in high velocity dilute phase conveying. The results obtained with the barite were very similar. 

The rebounding velocities of the colliding spheres can be expressed in terms of the coefficient of restitution as... 

Equation (2.3) also provides a basis for the experimental determination of the coefficient of restitution. Consider the case where a ball at rest is dropped from a height h to a horizontal stationary massive rigid surface, rebounding back to a height of h . If we label the ball with the subscript 1 and the massive plane with 2, Eq. (2.3) can be rearranged to... 

The coefficient of restitution in an impact depends not only on the material properties of the colliding objects but also on their relative impact velocity. More discussion of the coefficient of restitution is given in 2.5.2. 

In order to determine the unknown velocities after collision uniquely, two more relationships for impact velocities need to be specified. For certain values of the coefficient of friction and the coefficient of restitution, simple expressions of impact velocities can be obtained. In a collision of two completely rough and inelastic spheres, where the coefficient of friction / reaches infinity and the coefficient of restitution e is equal to zero, the relative velocities must vanish. Therefore, we have... 

The recoverability or restitution of the kinetic energy during a normal collision between two objects can be represented by the coefficient of restitution defined by Eq. (2.3). Note that the coefficient of restitution cannot be used as a criterion to judge whether a collision is elastic or not unless the collision is solely considered as a normal collision. For example, the sliding at contact for the collision between two elastic spheres will make the collision inelastic while the value of the coefficient of restitution in this case is equal to 1. 

The preceding equation is only valid for a certain range of impact velocities. The lower limit comes from the fact that the coefficient of restitution has to be less than unity so that... 

Hence, the collisional rate of dissipation and the collisional stress tensor are related to the coefficient of restitution by Eq. (5.284) and Eq. (5.277). 

Of prime importance is the initial distribution of solids at the top of the heat-exchanging apparatus. Figure 48 shows the design of the bullet-head solids distributor. Solids fed from a nearly point source fall on a bullet-shaped target, from which they bounce off to land at some distance below on a fall-breaker baffle that either straightens the particles into essentially vertical paths or redistributes them by further deflection. The contour of the bullet head is calculated from the coefficient of restitution K between its material of construction and the solid particles concerned, and from the distance of solids feed point above it, h, and the distance to the fall-breaker baffle below, H. 

The effect of the particle movement on the fluid phase, and interactions between particles are neglected. This is justified because the number of particles in the freeboard is low. Particles are allowed to bounce off walls, where a coefficient of restitution of 0.8 is applied, which is based on the elasticity of sand and metal collisions but can be varied considerably (from 0.5 to 0.9) without affecting the entrainment results noticeably. 

When simulating the trajectories of dispersed phase particles, appropriate boundary conditions need to be specified. Inlet or outlet boundary conditions require no special attention. At impermeable walls, however, it is necessary to represent collisions between particles and wall. Particles can reflect from the wall via elastic or inelastic collisions. Suitable coefficients of restitution representing the fraction of momentum retained by a particle after a collision need to be specified at all the wall boundaries. In some cases, particles may stick to the wall or may remain very close to the wall after they collide with the wall. Special boundary conditions need to be developed to model these situations (see, for example, the schematic shown in Fig. 4.5). 

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

Lun, C.K.K. and Savage, S.B. (1986), The effect of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials, Acta Mechanica, 63, 15. 

Coefficient of restitution St Stokes number based on initial nuclei ... 

It is noticed that in-elastic collisions are characterized by the degree to which the relative speed is no longer conserved. For this reason the coefficient of restitution e in a collision is defined (i.e., with basis in (2.120)) as the ratio of the relative velocity after collision, divided by the relative velocity of approach [45] [69] ... 

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