Generalised Hertz — Hunt — Crossley Collision Model

The paper considers a nonlinear visco-elastoplastic model for a collinear collision between a body and a fixed barrier. The model combines the Hertz and Hunt --- Crossley collision models. We show that both elastic and perfectly inelastic collisions are possible, depending on the values of model parameters. We obtained the first integrals to the equations of motion for the deformation and restitution phases, as well as a solution by quadrature to the equations of motion describing the body. We determined the coefficient of restitution, the kinetic energy lost in the collision, and how they depend on the impact velocity and the constants of viscous and dry friction. For the case of an elastic collision, the coefficient of restitution is a monotonically decreasing function of impact velocity. As the impact velocity tends to zero, the coefficient of restitution tends to a certain maximum value that does not exceed unity, depends on the constant of dry friction only and decreases as it increases

The study was supported by RFBR (grant no. 16-01-00521)

References

[1] Goldsmit V. Udar. Teoreticheskie i fizicheskie svoystva soudaryaemykh tel [Collision. Theoretical and physical properties of colliding bodies]. Moscow, Stroyizdat Publ., 1965. 448 p.

[2] Panovko Ya.G. Vvedenie v teoriyu mekhanicheskogo udara [Introduction to mechanical shock theory]. Moscow, Nauka Publ., 1977. 224 p.

[3] Ivanov A.P. Dinamika sistem s mekhanicheskimi soudareniyami [Dynamics of systems with mechanical collisions]. Moscow, Mezhdunar. programma obrazovaniya Publ., 1997. 336 p.

[4] Lapshin V.V. A body collision with an obstacle. Inzhenernyy zhurnal: nauka i innovatsii [Engineering Journal: Science and Innovation], 2013, no. 12 (in Russ.). DOI: 10.18698/2308-6033-2013-12-1134

[5] Kochetkov A.V., Fedotov P.V. Some questions of the theory of blow. Naukovedenie, 2013, no. 5 (in Russ.). Available at: https://naukovedenie.ru/PDF/110tvn513.pdf

[6] Herts H. Uber die Beruhrung Fester Elastischer Korper. Journal Reine und Angewandte Mathematik, 1882, vol. 92, pp. 156–171.

[7] Gerts G. Printsipy mekhaniki, izlozhennye v novoy svyazi [Principles of mechanics stated in new relation]. Moscow, AN SSSR Publ., 1959. 387 p.

[8] Hunt K.H., Crossley F.R.E. Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech., 1975, vol. 42, iss. 2, pp. 440–445. DOI: 10.1115/1.3423596

[9] Dyagel R.V., Lapshin V.V. On a nonlinear viscoelastic model of Hunt — Crossley impact. Mech. Solids, 2011, vol. 46, iss. 5, pp. 798–806. DOI: 10.3103/S0025654411050141

[10] Lapshin V.V., Yurin E.A. Nonlinear elastoplastic model of collinear impact. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2016, no. 1, pp. 90–99 (in Russ.). DOI: 10.18698/1812-3368-2016-1-90-99

[11] Corless R.M., Gonnet G.H., Hare D.E.G., Jeffrey D.J., Knuth D.E. On the Lambert W function. Adv. Comput. Math., 1996, vol. 5, iss. 1, pp. 329–359. DOI: 10.1007/BF02124750

[12] Lapshin V.V. Mekhanika i upravlenie dvizheniem shagayushchikh mashin [Mechanics and control on walking machines motion]. Moscow, Bauman MSTU Publ., 2012. 200 p.