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Transformation of sliding motion to rolling during spheres collision

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Abstract

In this research, we have investigated the three-dimensional elastic collision of two balls, based on friction in the tangential plane. Our aim is to offer analytical closed form relations for post collision parameters such as linear and angular velocities, collision time and tangential and normal impulse in three dimensions. To simplify the problem, stick regime is not considered. In other words, balls have a low tangential coefficient of restitution. Sliding, sliding then rolling, and rolling at the beginning of contact are three cases that can occur during impact which have been considered in our research. The normal interaction force is described by the Hertz contact force and dimensionless analysis is used for investigating normal interaction force; furthermore, Coulomb friction is considered during sliding. Experimental data for collisions show when sliding exists through the impact, tangential impulses can be taken as frictional impulses using the Coulomb law if the dynamic regime is not stick regime. To identify transformation of sliding motion to rolling or sticking during the impact process, linear and trigonometric functions are considered as an approximation for the normal interaction force. Afterwards, we have obtained the condition for the possibility of this transformation; moreover, we can estimate the duration of sliding and rolling or sticking. We have obtained an analytical solution for maximum force and deformation, collision time, impulses and post-collision linear and angular velocities in three dimensions.

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Abbreviations

R:

Ball radius (m)

m:

Ball mass (kg)

\({t}\,;\,\hat{t}\) :

Time(s); dimensionless time

r:

Position of each ball center

\({F}_{n} ; \hat{F}_{n}\) :

Normal force (N); dimensionless normal force

f :

Friction force (N)

V :

Linear velocities (m/s)

Y :

Young modulus (N/m\(^{2})\)

\({\xi }\,;\,\hat{\xi }\) :

Sum of the compressions of both spheres in the center of the contact area (m); dimensionless \({\xi }\)

\({\xi }_0 \) :

Characteristic length

\({\tau }_0 \) :

characteristic time

\({\mu }\) :

Coefficient of sliding friction

\(\vartheta \) :

Poisson ratio

\({\omega }\) :

Angular velocity (rad/s)

\(1\,;2\) :

Ball 1; ball 2

x; y; z :

Direction

rel :

Relative (relative linear and angular velocity)

tan :

Tangential (tangential velocity)

eff :

Effective (effective mass and radius)

i :

Variables at initial condition

\({f\,\mathrm{or}\,end}\) :

Variables at finial moment of impact

r :

Variables at instance of turning sliding into rolling

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Acknowledgements

The authors would like to express their gratitude to Sharif Research Office for supporting this research.

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Authors and Affiliations

  1. Nano Robotics Laboratory, School of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9465, Tehran, Iran

    Hossein Nejat Pishkenari, Hossein Kaviani Rad & Hamed Jafari Shad

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  1. Hossein Nejat Pishkenari
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  2. Hossein Kaviani Rad
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  3. Hamed Jafari Shad
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Correspondence to Hossein Nejat Pishkenari.

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Nejat Pishkenari, H., Kaviani Rad, H. & Jafari Shad, H. Transformation of sliding motion to rolling during spheres collision. Granular Matter 19, 70 (2017). https://doi.org/10.1007/s10035-017-0755-0

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