Kinematic Coefficient Of Restitution Research Articles

A parameter map of cavitation from impacts between solids immersed in water

Cavitation when two solids immersed in water collide has been studied theoretically and experimentally. A dimensionless parameter map showing a threshold line that delimits with good agreement the cavitation and non-cavitation zones was constructed. The threshold line was set by a cavitation number established with a lubrication model. Experiments were conducted using a device with a solid steel sphere colliding with a solid aluminum plane for different impact velocities and water column heights. When the impact forces are in the order of 100 to 1000 N, only cavitation generated by the rapid separation of solids in the liquid (CSSL) is observed; however, this type of cavitation along with cavitation due to tensile waves (TWC) are perceived for collision forces greater than 1000 N. In some events in which both types of cavitation (TWC-CSSL regime) appear, light emission also takes place. The kinematic coefficient of restitution and impact speed can be used to determine the cavitation onset, the transition from CSSL to TWC-CSSL regimes and the collision events in which photon production occurs. The obtained map will provide key information to develop potential applications as cavitation machining and surface treatment.

Investigating the Effect of Preimpact Energy Dissipation on Coefficient of Restitution regarding the Slope-Boulder Interaction

Coefficient of restitution is regarded as a dominating parameter in rockfall research. Generally, small-scale experiments were developed without considering interactions between boulder and slope. However, preimpact moving statuses are essential to evaluate rockfall behaviors. To reveal the effect of preimpact interactions on coefficient of restitution, energy dissipation considering initial velocity, surface type, and slope angle is executed based on medium-scale tests. The results show that (1) as the inclination of initial velocity, higher rebound height, and the declining normal coefficient of restitution occur, a determinable linear function could demonstrate relationships among energy dissipation and all coefficient of restitution; when initial velocity exceeds 5 m/s, the recovery ability shows and produces an increasing trend with respect to the variation of kinematic coefficient of restitution and kinetic energy coefficient of restitution. (2) As the surface material varies, slope hardness and rebound ability influence normal coefficient of restitution, and the surface roughness and rotation feature dominate tangential coefficient of restitution; considering preimpact slope and boulder interactions, four types of coefficient of restitution follow declining trend with different material sequence. (3) Slope angle affects normal coefficient of restitution, and tangential coefficient of restitution relatively descends 18% and inclines 10% when the angle ranges from 30° to 75°; regarding preimpact moving status, it differs from bounce times. The correlation between preimpact energy dissipation and four coefficients of restitution can be represented by the same decreasing linear function, when increasing the slope angle.

Laboratory study on the influencing factors and their control for the coefficient of restitution during rockfall impacts

Different opinions exist in the current research on the COR (coefficient of restitution) of rockfall, and these controversies can be attributed to the difference in the reduction degree of the real impact process during research on rockfall disasters. According to the previous research results, six main controlling factors affecting the COR were comprehensively determined. In this paper, experiments about the effects of the main controlling factors on the COR were conducted; a specially designed rockfall ejector was able to simulate the real rockfall impact process, from which the correlations between the main controlling factors and COR were obtained and then validated by theoretical derivation. By means of the proposed deviation theory between the impact contact point and the projection of mass center, the effects of the block shape on COR values were reasonably explicated. Synthetically, considering the relationships between the characteristic index of surface area distribution and the geometry with normal COR (Rn), a new index for quantifying the block shape, i.e., shape factor, was introduced, which shows a significant linear correlation of classification with Rn. Not equated to IH (inclined impact on a horizontal slope) or II (inclined impact on an inclined slope) in impact experiments of rockfall, the COR values of VI (vertical impact on an incline slope) and its evolving trend with impact angle are varied. COR values are not linearly related to the impact angle completely; kinematic COR (Rv), tangential COR (Rt), and Rn decrease with the increase of the impact angle and exhibit the power functions. The rotational speed (ω) of the block linearly increases with the increase of Rt; the effects of ω and shape on Rn are concomitant, and the ω of the non-round-shaped block is not always positively correlated with Rn, which may be the opposite when ω is small. The bouncing process is synthetically controlled by multiple influencing factors, and this research will provide an essential foundation for the follow-up study on the COR affected by multiple factors of interaction.

Rockfalls: influence of rock hardness on the trajectory of falling rock blocks.

Rockfalls pose significant hazard on human activities and infrastructure. The assessment of rockfall risk along roads and on other human activities is of great importance. Geological assessment can predict the outbreak of such events; explain the mechanism of occurrence and assist in the effective design of protection measures. The trajectory of a falling block is significantly affected by its response to the impact with the slope, since block kinematics properties mainly depend on the coefficient of restitution values. In this research a laboratory procedure based on the free fall of small blocks of different rock types and artificial materials was implemented in order to address the effect of their mechanical properties on the coefficient of restitution values. From the evaluation of testing results a correlation between kinematic coefficient of restitution and Schmidt rock hardness is proposed.

Image analysis measurements of particle coefficient of restitution for coal gasification applications

New robust Lagrangian computational fluid dynamic (CFD) models are powerful tools that can be used to study the behavior of a diverse population of coal particle sizes, densities, and mineral compositions in entrained gasifiers. By using this approach, the responses of the particles impacting the wall were characterized over a range of velocities (1 to 8m/s) and incident angles (90 to 20°). Within CFD models, the kinematic coefficient of restitution is the boundary condition defining the particle wall behavior. Four surfaces were studied to simulate the physical conditions of different entrained-flow gasification particle–surface collision scenarios: 1) a flat metal plate 2) a low viscosity silicon adhesive, 3) a high viscosity silicon adhesive, and 4) adhered particles on a flat metal plate with Young's modulus of elasticity ranging from 0.9 to 190GPa. Entrained flow and drop experiments were conducted with granular coke particles, polyethylene beads and polystyrene pellets. The particle normal and tangential coefficients of restitution were measured using high speed imaging and particle tracking. The measured coefficients of restitution were observed to have a strong dependence on the rebound angles for most of the data. Suitable algebraic expressions for the normal and the tangential component of the coefficient of restitution were developed based upon ANOVA analysis. These expressions quantify the effect of normalized Young's modulus, particle equancy, and relative velocity on the coefficient of restitution. The coefficient of restitution did not have a strong dependence on the particle velocity over the range considered as long as the velocity was above the critical velocity. However, strong correlations were found between the degree of equancy of the particles and the mean coefficient of restitution such that the coefficient of restitution decreased for smaller particle equancies. It was concluded that the degree of equancy and the normalized Young's modulus should be considered in applications such as gasification and other cases involving the impact of non-spherical particles and complex surfaces. Sliding was observed when particles impacted on oblique surfaces; however, the resulting effects were within the range of measurement uncertainties.

Geotechnical and kinematic parameters affecting the coefficients of restitution for rock fall analysis

This paper presents the results of laboratory and field tests regarding the determination of the parameters that influence the trajectory of falling blocks in rockfalls. Coefficients of restitution are considered to be critical parameters in rockfall modeling, but various definitions exist and there is no agreement on which one is more appropriate. In this research, the lamped-mass analysis method was used and coefficients of restitution were calculated using the kinematic definitions. The laboratory testing was performed using a number of common rocks found in Greece. The rock materials tested were characterized by different Schmidt hammer rebound hardness values while the mass of the falling blocks as well as the impact angle and the velocity were altered during the tests in order to estimate the effect of each parameter on the coefficients of restitution. The kinematic coefficient of restitution was compared to the normal coefficient of restitution and was found to be more appropriate when used in correlations. Impact angle was found to significantly affect the trajectory. Finally, a correlation between Schmidt hammer rebound hardness and kinematic coefficient of restitution is proposed whilst a method for estimating the normal coefficient of restitution is suggested including the impacts' angle effect.

Dynamic analysis of a flexible linkage mechanism with cracks and clearance

In this paper a planar mechanism with a flexible rod with cracks and slider clearance is investigated. The flexible rod is modeled with lumped masses. The equations of motion in which the influence of the cracks size, slider clearance and impact effects are taken into account, are developed. For the present study, the impact is modeled using a kinematic coefficient of restitution. The stability of the motion is studied using Lyapunov exponents. Numerical solution results show that clearance (that can be associated with fatigue and wear phenomena) and imperfect links change the dynamical behavior of the system. The effects of clearance combined with imperfect links should not be ignored when analyzing the dynamic performance and vibration characteristic of mechanical systems.

Rigid body impact with moment of rolling friction

In this paper, the impact between a rigid pendulum and rough surfaces is studied. The rolling friction moment and the coefficient of rolling friction are introduced, and an improved mathematical model of the planar impact with friction is presented. The influence of the moment of rolling friction on the energy dissipated by friction during the impact is analyzed. For a simple pendulum, using the energetic coefficient of restitution, more energy is dissipated for larger values of the coefficient of kinetic friction and contact radius, and for smaller values of the length of the beam. For a double pendulum using the kinematic coefficient of restitution, some energetically inconsistent results can be solved for some values of the coefficient of rolling friction.

Nonlinear dynamics of a flexible mechanism with impact

The nonlinear dynamics of a slider-crank mechanism with a flexible rod is considered in this study. The flexible rod is modeled with lumped masses and periodically impacted by an external flexible sphere. The impact is modeled using a kinematic coefficient of restitution. Nonlinear dynamics tools are applied to analyze the simulated data captured from the connecting rod of the mechanism. The chaotic behavior of the system is analyzed. The stability of the motion is studied using the Lyapunov exponents. The dependence between the Lyapunov exponents and the corresponding angular velocity of the driver link of the mechanism is investigated.

Influence of Reference Conditions on the Analysis of Impact-Induced Elastic Waves

Impact problems involving flexible bodies that undergo largerotations and translations were solved by using the floating frame ofreference approach. Analytically defined modes were selected to describedeformation. Different mode shapes can be used depending on the type ofattachment between the body frames of reference and the flexible bodies.Evaluating the kinematic coefficient of restitution during impactentails using the equivalent rigid body velocities of the flexiblebodies. In general, however, such velocities cannot be identified withreference velocities. When the colliding bodies can undergo large translations, freeattachments allow the use of mode shapes that possess zero totalmomentum. In this case, reference motion coincides with the equivalentrigid body motion of the flexible bodies and is not coupled to elasticmotion. When the colliding solids can rotate freely, the use of modeshapes with a zero angular momentum does not uncouple reference andelastic motion. However, if small deformation occurs during freerotation of the flexible body, the derivative of the reference angleremains roughly constant. Such a derivative can be interpreted as theequivalent angular velocity of the flexible body. When using a rigidattachment between the body frames of reference and the flexible bodies,reference motion does not coincide with the equivalent rigid bodyvelocity of the flexible bodies, which leads to non-constant referencevelocities when the bodies move freely. As a result, the equivalentrigid body velocities must be evaluated as linear combinations ofreference and elastic velocities. As shown in this paper, the use of free or rigid attachments asreference conditions, and their corresponding mode shapes allows theaccurate description of impact-induced elastic waves. However, theevaluation of elastic efforts with rigid attachments provides spuriouselastic forces at the attachment point. Such forces decrease as thenumber of modes increases.

Analytical Solutions of Contact Impact Problems

In this article, analytical solutions of the normal and tangential impact of half-spaces at low velocities are formulated. After a brief introduction, a classification of contact problems is proposed and some publications about impact are mentioned. The basic contact laws for monotonously increasing normal, tangential, and torsional contact have been found decades ago, while the general analytical solutions for tangential and torsional load histories, which are necessary for impact calculations, have only been found recently. Insertion of these contact laws into the equations of motion yields a system of nonlinear differential equations, which are uncoupled in the case of the oblique central impact. The normal solution of the uncoupled equations for Hertzian and some axisymmetric surfaces can be written as a hypergeometric function, which generalizes earlier solutions. Solutions in tangential direction for the compression phase and the restitution phase can be found for the case of complete adhesion. Finally, the kinematic coefficient of restitution is shown. This solutions may help to understand some open problems of the collision of rigid bodies, which are in contact on a small elastic domain, where a part of the kinetic energy is transformed into elastic energy and reconverted into kinetic energy in normal and tangential directions.