Coefficient Of Restitution Values Research Articles

A low-cost method for quantifying bouncing ball dynamics with smartphone video

Abstract This study employed a low-cost method for quantifying bouncing ball dynamics with smartphone video. High-speed video analysis was used to investigate the bouncing behaviour of different balls, including table tennis, rubber, and tennis balls, on a table tennis surface. To examine the effect of surface material on bouncing dynamics, the table tennis ball was also tested on wood and tile surfaces. Key parameters analysed included bounce height decay, the coefficient of restitution (COR), and the bounce half-life. Additionally, we measured the total bouncing time and analysed the relationship between changes in gravitational potential energy, kinetic energy, and total mechanical energy (ME). Our results revealed a significant correlation between the COR and the decay rate of bounce height. Balls with higher COR values, indicative of lower energy loss per bounce, exhibited slower decay rates, resulting in longer bounce half-lives and extended total bounce times. For instance, the table tennis ball on a tile surface demonstrated a high COR of 0.93, a bounce half-life of 4.65 bounces, and a total bounce time of 8.22 ± 0.34 s. This suggests that material properties such as stiffness and elasticity of both the ball and the surface significantly influence the COR and decay rate, beyond just surface smoothness. Furthermore, our analysis of ME changes in the bouncing balls aligns with theoretical predictions, providing insights into the energy transformations occurring during bouncing. This improved understanding can also be a valuable tool for educators, helping teachers and students bridge the gap between theory and real-world observations, leading to a deeper grasp of the concepts.

Analysis of Restitution Coefficient Value on the Ball Using Tracker Software by Utilizing Video Based Laboratory

Performing experimental activities in a laboratory using the available equipment is not as straightforward as it may seem. In practice, conducting experiments can be challenging, and various factors can contribute to erroneous measurement results. One such factor is systematic errors, which occur due to inaccuracies in the measurement tools used. Additionally, error sources from external factors can also influence the measurement results. It is important to be aware of these potential errors and take appropriate measures to minimize their impact during experimental procedures. Tracker software is a valuable tool that can enhance the accuracy of experimental activities conducted in laboratories. This software assists in generating more precise data by tracking and analyzing various elements of the experiment. By utilizing tracker software, researchers can closely monitor the movement and behavior of objects, making it easier to measure and record data accurately. This technology plays a significant role in improving the reliability and efficiency of experimental procedures in the laboratory setting. From the study's objectives, it can be deduced that the restitution coefficient values obtained through manual data collection for the bekel ball are 𝑒 = 0.5136 ± 0.004214, and for the ping pong ball, it is 𝑒 = 0.4566 ± 0.01736. On the other hand, when using tracker software for data collection, the restitution coefficient value for the bekel ball is 𝑒 = 0.5397 ± 0.0008731, and for the ping pong ball, it is 𝑒 = 0.5115 ± 0.001154.

Comparison of collision dynamics of soccer balls with energy dissipation method

This study aimed to understand the collision characteristics of different soccer balls by determining energy dissipation (ED) during collisions. Soccer balls from five different brands were kicked towards a framed tempered glass. The kicks were performed with the balls inflated under three inner pressure conditions. The motion of each ball was recorded using two high-speed cameras at 6000 Hz. The contact surface area (CSA), ball indentation amount (IA), reaction forces (RF) and energetic coefficient of restitution (CoR) during contact were calculated via image processing methods. A new methodological approach was developed to determine the ED during the collision (EDDC). The accuracy of the estimated RF was validated using force plate data, and the CSA values validated the calculated IA. The EDDC were then reduced to a single value to compare with the CoR values obtained using the traditional formula. Differences between CoR and EDDC were statistically tested. One-way ANOVA tests were performed separately to compare the effect of the ball brand on the IA and EDDC for each inner pressure condition. The CoR and EDDC showed no significant difference, proving that the developed method is a valid measure of ED. At 1.0 bar inner pressure, IA and EDDC showed significant differences according to the brands. EDDC, depending on the ball quality, decreased at higher inner pressure values regardless of the brand. Consequently, the present study exhibited that ball quality becomes a critical factor in collision dynamics at high inner pressure values, considering that ball inner pressure may vary between 0.6 and 1.0 bar in official soccer matches.

Empirical investigation of parameter impacts on the three-dimensional coefficient of restitution of jujube at the time of harvest

To provide foundational data for parameter design and performance analysis of high-performance processing equipment, the collision model between jujube-parts and the parameter effects on coefficient of restitution (COR) were investigated. Jujube samples from the sandy lands of southern Xinjiang during the harvest season were utilized, with experimental factors including collision angle, falling height, collision material, moisture content, and collision parts. Single-factor tests and mixed orthogonal tests were conducted to explore the impact of parameter on COR. Utilizing the specular reflection principle, a 3D impact analysis of jujube was performed, and a kinematic model of the falling and impact process was established to determine the COR. Results indicate that the COR of Jun (Zizyphus jujuba cv. Junzao) is higher than that of Hui (Zizyphus jujuba cv. Huizao), and the ventral part exhibits a higher COR compared to the other four parts. Collisions with steel yield higher COR values compared to soil. Furthermore, for collisions with steel material, both Jun and Hui exhibit a decrease in COR with increasing falling height and moisture content, while the COR increases with an increase in collision angle. The influencing factors on COR follow the order of collision material > variety > moisture content > collision angle > falling height > collision part. These findings offer valuable insights for jujube processing research and for preventing mechanical damage to jujube fruits.

Exploring factors affecting the conversion of the coefficient of restitution to equivalent free fall drop height

AbstractThe coefficient of restitution (COR) is used to estimate the equivalent free fall drop height of packaging during handling and transportation. Various experimental studies were designed to explore factors affecting the conversion of the COR to equivalent free fall drop height. These factors include the number of drops, drop height, drop orientation, drop base, temperature and humidity on the COR. The results indicate that the number of drops and drop orientation are the two most influential factors affecting COR values. As the package undergoes more drops, the COR increases. Among the various orientations, the COR is largest for flat drops, followed by edge and corner orientations. Within flat drops, a smaller impact area results in a smaller COR. Manufacturing joints and the flats, edges and corners adjacent to manufacturing joints tend to yield smaller COR values. A COR correction matrix was developed to correct the conventional COR calculation. This matrix allows engineers to refine COR values based on the number of drops and drop orientations in the field. The matrix‐corrected equivalent free fall drop heights demonstrated an average accuracy of 96%. Compared to the conventional COR calculation method, it effectively reduces errors by a factor of 2–3.

Occurrence conditions of the reverse rotation of rockfall and its influence on the restitution coefficient

When rockfall occurs along dense rock slopes, the rotation direction of rockfall is not always downhill. Specifically, the rockfall may obtain a reverse rotation speed (RRS) after impact under certain conditions, the effect of which on the restitution coefficient (RC) cannot be ignored. According to the statistical results of the reverse rotation (RR) phenomena of blocks obtained from previous experiments, the occurrence of RR is correlated to the block shape, incident angle, and contact attitude. In this study, considering a typically shaped cubic block, the critical condition for the RR is preliminarily deduced. Based on the results, the influence of the RRS on the RC for four typically shaped blocks is examined using a customized device. Results show that the tangential RC (Rt) values of each block are not sensitive to the change in the RRS, the distribution is relatively concentrated and the values are high. Moreover, the normal RC (Rn) values are not sensitive to the RRS, and the distribution is relatively discrete. The RRS influences Rn; however, it is difficult to directly establish the relationship between them. To this end, considering the contact attitude and shape of the block, an integral variable, the impact coefficient (Ic), is proposed to determine the influence of RRS on Rn. Moreover, the impact-bounce behaviours of the block are categorized and analysed. For the block rebound following a single impact, Ic and Rn are positively and negatively correlated when the mass centre of the block (MC) is in front and behind the contact point (CP), respectively. For the block rebound following two successive impacts, with the increase in Ic, the Rn increases. These conclusions help clarify the mechanism of the influence of the RRS on RC and provide vital information and ideas for the development and optimization of a program to accurately predict rockfall trajectories.

Cohesive collisions of particles in liquid media studied by CFD-DEM, video tracking, and Positron Emission Particle Tracking

This paper investigates the cohesive collision of ice in an oil phase at temperatures ranging from −15.7 °C to −0.3 °C. The new information on the coefficient of restitution (COR) was obtained using three different velocity measurement methods: high-speed experimental video recording, Positron Emission Particle Tracking (PEPT), and numerical simulations. A new type of PEPT tracer was developed for the experiments. The COR values were in the interval 0.57...0.82, with a maximum at around −10 °C. The CFD-DEM coupled approach was applied to reproduce experiments with an ice particle drop and its collision with an inclined ice surface in a decane. The particle–wall interaction is modeled using commercial software, considering particle cohesion, particle size, and shape. CFD-DEM predicted the COR with an average deviation ∼10% from the experimental data. The numerical model’s results agree with the experiments, demonstrating that the CFD-DEM method is suitable for describing multiphase cohesive interactions.

Nonlinear dynamics investigation of contact force in cam–follower system using Lyapunov exponent parameter, power spectrum analysis, and poincare’ maps

ABSTRACT Non-linear-linear dynamics behaviour of the contact force is investigated for different cam angular speeds (N) and different internal dimensions of the follower guide (F.G.I.D.). The largest Lyapunov exponent parameter conception is used to detect whether the follower will detach from the cam or the follower will stay in permanent contact. Power spectrum analysis and Poincaré maps with a phase-plane diagram are added to detect the non-periodic motion of the contact force. The follower is moved with three degrees of freedom. Multi degrees of freedom (spring-damper-mass) systems are added at the end of the follower stem to suppress the non-linear dynamics behaviour of the contact force of the follower. SolidWorks software is used in the numerical simulation of the contact force. Contact force and follower displacement are determined at the different coefficient of restitution values. The friction and impact are considered between the cam and the follower and between the follower and its guide. The contact force is measured experimentally using OPTOTRACK/3020 by tracking the position of the follower through an infrared camera device. The value of the contact force has declined with the increase of cam angular speeds (N) and the internal dimension of the follower guide (F.G.I.D.). The peak of non-linear response of the follower displacement is reduced to 15%, 32%, 45%, and 62% after using multi degrees of freedom systems.

Numerical Study of Material Selection for Optimal Directed Energy Deposition Single Nozzle Powder Efficiency

Abstract The efficiency of powder-based directed energy deposition (DED) nozzle is dependent on its ability to direct the pneumatically conveyed powder into the meltpool. Computational fluid dynamics (CFD) with discrete phase modeling (DPM) has been used to investigate the optimization of DED nozzle geometry and DED parameter selection, however, the effect of material choice for nozzle fabrication has not been fully investigated. To explore the effects of the nozzle material on powder efficiency a CFD DPM model was created and analyzed in ansys fluent. Various nozzle materials were simulated using statistical models for the coefficient of restitution (COR) between the powder and nozzle wall from the literature. The results of the CFD DPM model aligned well with experimental data for a 316L stainless steel prototype nozzle. CFD DPM results indicated that using a material with a lower mean COR value improved the powder efficiency of the nozzle. Powder efficiency improved because the component of powder velocity normal to the direction of gas flow was reduced in nozzles made from materials with lower COR values, which in turn led to fewer impacts between the particles and the nozzle walls.

Effect of the Impact Coefficient of Restitution on the Nonlinear Dynamics Phenomenon of Flat-Faced Follower Mechanism with Clearance

The effect of impact coefficient of restitution on the nonlinear response of the follower is investigated at different follower guides’ clearances, different cam speeds and different followers’ offsets. The impact between the cam and the follower and between the follower and its guide is considered in the presence of coefficient of restitution and follower offset. The nonlinear dynamics phenomenon of the follower due to the impact coefficient of restitution is detected using the approach of largest Lyapunov exponent. Moreover, the chaotic phenomenon is detected using a phase-plane diagram. The numerical simulation of the nonlinear response of the follower is calculated using the SolidWorks program. The chaotic phenomenon in the cam follower system is increased with the increase of the impact coefficient of restitution value in which the potential energy of the follower has been decreased in the presence of follower offset. The chaotic motion of the follower response occurs due to the increase in cam speeds, follower’s offsets, follower guides’ clearances and impact coefficient of restitution.

The dynamics of bouncing, partially coalescing, liquid metal droplets in a viscous medium

Planar partial coalescence is a phenomenon in which a droplet at a free surface or interface between two fluids coalesces into the plane surface producing a smaller droplet rather than coalescing completely. This smaller, ‘daughter’ droplet will be driven towards the interface by gravity and capillary forces resulting in a cascade effect of progressively small daughter droplets until the Ohnesorge Number approaches $\sim$ 1 and the cascade terminates with a full coalescence event. This paper utilizes a room temperature liquid metal alloy composed of gallium, indium and tin to study partial coalescence in a viscous quiescent medium and observed bouncing of the coalescing droplets on the interface. We observed the event using high speed videography measuring effects such as the droplet to daughter droplet ratio, droplet velocities, droplet bounce heights and coefficients of restitution for the bouncing event. An existing model (Honey & Kavehpour, Phys. Rev. E, vol. 73, 2006) from our group was used, validated and expanded upon to include buoyancy effects to estimate the initial velocity of the droplet and we developed two new models for the droplet travel and maximum bounce height. The first utilizes the Stokes model for drag to moderate success while the second utilizes a model from Beard & Pruppacher (J. Atmos. Sci., vol. 26, 1969, pp. 1066–1072) and a fourth-order Runge–Kutta numerical integration scheme to predict the droplet velocity and position as functions of time. Additionally the coefficient of restitution was determined from the model using a shooting method technique in tandem with measured data to find a coefficient of restitution value of $A = 0.27 \pm 0.06$ . This ‘bouncing drop’ phenomenon continues in a quiescent viscous fluid to the sub-micron scale and was facilitated by the material properties of the liquid metal including the high density, moderate viscosity and particularly high interfacial tension.

Laboratory and discrete‐based numerical investigation on the collision problem of impactor‐block systems with soft‐porous and hard‐crystalline analog rocks

AbstractWe investigated experimentally the normal coefficient of restitution (COR) of impactors colliding on the surfaces of two different blocks; a soft‐porous block composed of plaster (dry impacts) and a hard‐crystalline rock composed of granite (dry and wet impacts). The experiments were performed in a range of low‐impact velocities and various particle types were used including perfectly spherical smooth glass beads and ceramic balls, rough glass beads as well as natural rough sand grains. Image processing was carried out to quantify the formed contours of craters caused by the surface damage of the soft blocks due to the impact. The results indicated very low COR values on the plaster block compared with the granite block as the energy was dissipated by means of surface plastic deformations on the plaster, however the contour crater images showed that the collision mechanisms depended on the roughness of the impactor. The behavior of impactor‐fluid‐block systems was dependent on both surface roughness and global morphology of the impactors. Discrete‐based (DEM) numerical simulations were performed to provide further insights into the behavior of the impactor‐block systems subjected to collision using the COR values from the experiments as the micro‐scale parameters and data from the literature as the macro‐scale parameters for the model calibration. The numerical output was used to observe the development of compression and tension force chain networks and how these involved during and after impact on different base blocks.

A Note on the Influence of Smectite Coating on the Coefficient of Restitution of Natural Sand Particles Impacting Granitic Blocks

The study of the collision behavior of solid objects has received a significant amount of research in various fields such as industrial applications of powders and grains, impacts of proppants and between proppant and rocks during hydraulic fracturing, and the study of debris flows and avalanches and the interactions of landslide materials with protective barriers. This problem has predominantly been studied through the coefficient of restitution (COR), which is computed from the dropping and rebound paths of particles; its value corresponds to 1 for perfectly elastic impacts and 0 for perfectly plastic impacts (i.e., at the collision there is no rebound of the particle). Often, the colliding particles (or particle–block systems) are not perfectly clean, and there is debris (or dust) on their surfaces, forming a coating, which is a highly possible scenario in the debris flows of natural particles and fragments; however, the topic of the influence of natural coatings on the surfaces of particles on the collision behavior of particle–block systems has been largely overlooked. Thus, the present study attempts to provide preliminary results with respect to the influence of natural coating on the surfaces of sand grains in the COR values of grain–block systems using a stiff granitic block as an analogue wall. Montmorillonite powder, which belongs to the smectite clay group, was used and a sample preparation method was standardized to provide a specific amount of clay coating on the surfaces of the sand grains. The results from the study showed a significant influence of the smectite coating in the COR values of the grain–block systems, which was predominantly attributed to the dissipation of energy at the collision moment because of the compression of the soft coating of microparticles. Additionally, the method of analysis for calculating the COR values based on one and two high-speed cameras was explored, as the impacts of natural grains involve deviations from the vertical, which influences the rebound paths. Thus, a sensitivity analysis was performed investigating the differences in the COR values in two-dimensional and three-dimensional analysis of the impact tests.

Influence of Drop Height on the Impact Characteristics of Futsal Ball Size Four

Futsal research on the ball characteristic has received little scientific attention. Recently modern football has been developed that afford excellent ball control and delivering. Difference constructions and sizes of ball deliver different characteristics of the ball. The purpose of this study is to examine Futsal ball size four deformations and coefficient of restitution (COR) during the impact. In this study, the Futsal ball was dropped vertically under three conditions. A subject performed five repetitive drops off the ball to impact a steel plate. The ball velocity was measured by SparkVue motion sensor attached on the top of the apparatus while the deformation was recorded using a high-speed video camera, that can up to 1,000 frames per second. From the experiment, the ball deformation and COR were measured. The results obtained showed the deformation and COR value vary with different heights for the same type of Futsal ball size four. It shows that the different heights of ball drops give different ball impact characteristics. It was found that the COR depended on the amount of the deformation of the ball.

The importance of parameter-dependent coefficient of restitution in discrete element method simulations

The coefficient of restitution (COR) is an important constant that represents the energy dissipation during contact between two objects. Simulation using the conventional discrete element method (DEM) involves a constant COR. This study presents a DEM simulation method that uses a parameter-dependent COR. The parameter-dependent COR was obtained from a collision incident between spherical particles and a plate surface using a drop-test apparatus. Glass and polypropylene beads of 3–6-mm diameter were used while acrylic and steel were used as the plate surfaces. The particle trajectories were captured by a high-speed camera and analyzed by an image analyzer. The COR was then correlated to a parameter-dependent COR function that depends on the material, impact velocity, and temperature. Free-fall DEM simulations using a constant COR and parameter-dependent COR were compared. The parameter-dependent COR approach obtained better agreement with experimental results than the constant-COR approach. The proposed concept could be applied for other material combinations with a wide range of operating conditions to obtain a database of parameter-dependent COR values for the simulation of solid handling applications.

Experimental Study on Rockfall Mechanism of Platy Rock on a Complex Slope

The rock fall trajectory and its mechanisms are the most difficult to predict, owing to the complexity of the slope and the Irregular shape of falling rocks. To acquire a better knowledge of the rock fall mechanism of platy rock and to investigate the influence of various impact parameters, a comprehensive physical model experimental study was undertaken based on 3D printing technology using a high-speed camera and specially developed block release system. Based on the experimental results, the effects of the slope angle on the stopping position, the instantaneous kinetic energy and collision position of platy rock block were analyzed. Meanwhile, the effects of movement forms of platy rock before and after collision on the normal coefficient of restitution and the tangent coefficient of restitution were discussed. It is observed that rock fall trajectory depends not only on slope material characteristics, slope angle but also on factors related to the platy block (weight, size and shape). The experimental results showed the value of restitution coefficient exceeding 1 has an important relation with the combination of various movement forms (including the flip motion) and the change of movement forms of platy rock before and after the collision. A new feasible experimental method for research and prevention of rock fall disaster was put forward. It would be important and helpful to the geo-hazard control work.

Experimental study on the coefficient of restitution of grain against block interfaces for natural and engineered materials

The coefficient of restitution (COR) is an important input parameter in the numerical simulation of granular flows, as it governs the travel distance, the lateral spreading, and the design of barriers. In this study, a new custom-built micromechanical impact loading apparatus is presented along with impact experiments on engineered and natural materials. The COR and energy loss of various grains and base block combinations are studied, including fairly regular-shaped Leighton Buzzard sand (LBS) grains as a natural soil and granite or rubber as base blocks, apart from the use of engineered materials for the grains (chrome steel balls, glass balls) and blocks (stainless steel, brass). The repeatability of the new micromechanical impact loading apparatus was checked by impacting chrome steel balls on stainless steel block. In all the test combinations, the higher and lower values of the COR are found for granite block (ranging between 0.75 and 0.95) and rubber block (ranging between 0.37 and 0.44) combinations, respectively. For the tested grain–block combinations, lower values of COR were observed for impacts between materials of low values of composite Young’s modulus. However, within the narrow range of composite surface roughness of the tested grain–block interfaces no particular trend was observed in the COR values. Compared to glass balls and chrome steel balls, greater scatter in the COR values is observed for natural sand grains. This is due to the variation of the elastic and morphological characteristics among individual LBS grains.

Effect of Grain Size and Surface Roughness on the Normal Coefficient of Restitution of Single Grains

The coefficient of restitution (COR) represents the fraction of pre-collision kinetic energy remained after the collision between two bodies. The COR parameter plays an important role in the discrete numerical analysis of granular flows or the design of protective barriers to reduce flow energy. This work investigated the COR for grain-block type impacts through comprehensive experiments using a custom-built impact loading apparatus. Glass balls of three different sizes were used as grains. The impact experiments were performed on three different types of materials as base blocks, namely brass, granite and rubber. Experiments on the brass block showed a decrease in COR values with increasing grain size. On the contrary, impacts on granite and rubber blocks showed an increase in COR values with increasing grain size. Additionally, the effect of surface roughness on the COR was investigated. It was revealed that the increase in surface roughness of either the grain or the block reduced the COR values due to the increased plastic deformations of surface asperities.

Multi-particle collisions in microgravity: Coefficient of restitution and sticking threshold for systems of mm-sized particles

Context. The current model of planet formation lacks a good understanding of the growth of dust particles inside the protoplanetary disk beyond mm sizes. A similar collisional regime exists in dense planetary rings. In order to investigate the low-velocity collisions between this type of particles, the NanoRocks experiment was flown on the International Space Station (ISS) between September 2014 and March 2016. We present the results of this experiment. Aims. The objectives of our data analysis are the quantification of the damping of energy in systems of multiple particles in the 0.1–1 mm size range while they are in the bouncing regime, and the study of the formation of clusters through sticking collisions between particles. Methods. We developed statistical methods for the analysis of the large quantity of collision data collected by the experiment. We measured the average motion of particles, the moment of clustering, and the cluster size formed. In addition, we ran simple numerical simulations in order to validate our measurements. Results. We computed the average coefficient of restitution (COR) of collisions and find values ranging from 0.55 for systems including a population of fine grains to 0.94 for systems of denser particles. We also measured the sticking threshold velocities and find values around 1 cm s−1, consistent with the current dust collision models based on independently collected experimental data. Conclusions. Our findings have the following implications that can be useful for the simulation of particles in PPDs and planetary rings: (1) The average COR of collisions between same-sized free-floating particles at low speeds (<2 cm s−1) is not dependent on the collision velocity; (2) The simplified approach of using a constant COR value will accurately reproduce the average behavior of a particle system during collisional cooling; (3) At speeds below 5 mm s−1, the influence of particle rotation becomes apparent on the collision behavior; (4) Current dust collision models predicting sticking thresholds are robust.

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.