Three-dimensional integral dry friction model for the motion of a rectangular body

Introduction. A three-dimensional dry friction model in the interaction of a rectangular body and a horizontal rough surface is considered. It is assumed that there is no separation of the body from the horizontal surface. The body motion occurs under the conditions of combined dynamics when, in addition to the longitudinal movement, the body participates in twisting.

Materials and Methods. Linear fractional Pade approximations are proposed, which replaced the cumbersome analytical expressions that most accurately describe the motion of bodies on rough surfaces. New mathematical models describing sliding and twisting of bodies with a rectangular base are proposed.

Results. Analytical expressions of the principal vector and moment of friction for rectangular contact areas are developed and scientifically established. A friction model that takes into account the relationship between sliding and twisting speeds, which provides finding solutions for Pade dependences, is developed. After numerical solution to the equations of motion, the dependences of the sliding speed and angular velocity on time were obtained and constructed. Graphs of the dependences of the friction forces and their moment on two parameters (angular velocity and slip velocity) were constructed, which enabled to compare the integral and normalized models of friction. The comparison results showed good agreement of the integral model and the model based on Pade approximations.

Discussion and Conclusions. The results obtained provide considering the dynamic coupling of components, which determines the force interaction of a rectangular body and a horizontal surface. These results can be used in mobile robotics. The analyzed motion of the body occurs through the motion control of a material point inside the body. Such mobile robots can be used when solving a wide class of problems: when creating autonomous robots for the exploration of outer space and planets; in the diagnosis and treatment in case of passing through complex structures of veins and arteries; in research under water, in places of large differential temperature; in underground operations.

1. Salimov MS, Ramzin NS. Dvizhenie tela na vibriruyushchei poverkhnosti v sluchae sukhogo treniya [Element movement on vibrating surface in case of dry friction]. Engineering and Automation Problems. 2019;4:100–(In Russ.)

2. Munitsyn LV. Vibrations of a Rigid Body with Cylindrical Surface on a Vibrating Foundation. Mech. Solids. 2017;52(6):675–685.

3. Voldrich J. Modelling of the three-dimensional friction contact of vibrating elastic bodies with rough surfaces. Appl. Comput. Mech. 2009;3(1):241–252.

4. Chowdhury MA, et al. Sliding friction of steel combinations. Open Mech. Eng. J. 2014;8(1):364–369.

5. Martinovs A, Gonca V. Descriptive model of sliding friction processes. Vide. Tehnologija. Resursi. =Environment. Technology. Resources. 2009;2:227–233.

6. Zhuravlev VF. O modeli sukhogo treniya v zadache kacheniya tverdykh tel [The model of dry fiction in the problem of the rolling of rigid bodies]. Journal of Applied Mathematics and Mechanics. 1998;62(5):762–767. (In Russ.)

7. Andronov VV, Zhuravlev VF. Sukhoe trenie v zadachakh mekhaniki [Dry friction in mechanical problems] Moscow. Izhevsk: R&C Dynamics; 2010. 184 p. (In Russ.)

8. Acary V, Brémond M, Huber O. On solving contact problems with coulomb friction: Formulations and numerical comparisons. In: Transactions of the European Network for Nonsmooth Dynamics on Advanced Topics in Nonsmooth Dynamics. Springer International Publishing; 2018. P. 375–457.

9. Haslinger J, Kučera R, Sassi T. A domain decomposition algorithm for contact problems with Coulomb’s friction. Lect. Notes Comput. Sci. Eng. 2014;98:889–897.

10. Kireenkov AA. Svyazannaya model' treniya skol'zheniya i vercheniya [Related model of sliding and twisting friction]. Doklady Akademii Nauk. 2011;441(6):750–755. (In Russ.)

11. Zhuravlev VF, Kireenkov AA. O razlozheniyakh Pade v zadache o dvumernom kulonovom trenii [Pade expansions in the two-dimensional model of Coulomb friction]. Mechanics of Solids. 2005;2:3–13. (In Russ.)

12. Santos AP, et al. Granular packings with sliding, rolling, and twisting friction. Phys. Rev. E. 2020;102(3):032903.

13. Kireenkov AA. Trekhmernye modeli treniya [Three-dimensional friction models]. Vestnik of Lobachevsky University of Nizhni Novgorod. 2011;4(2):174–176. (In Russ.)

14. Sakharov AV. Povorot tela bez vneshnikh dvizhitelei pri pomoshchi rotora A. V. [Rotation of a body without external propellers using rotor]. Trudy MFTI. 2014;6(2):80–91. (In Russ.)

15. Zhuravlev VF. Zakonomernosti treniya pri kombinatsii skol'zheniya i vercheniya [Friction laws in case of combination of slip and spin]. Mechanics of Solids. 2003;4:81–89. (In Russ.)

16. Kireenkov AA. Zakon Kulona v obobshchennoi differentsial'noi forme v zadachakh dinamiki tverdykh tel s kombinirovannoi kinematikoi [Coulomb law in generalized differential form in problems of dynamics of rigid bodies with combined kinematics]. Mechanics of Solids. 2010;2:15–26. (In Russ.)

17. Gluzman S, Yukalov VI. Self-similarly corrected Pade approximants for nonlinear equations. Int. J. Mod. Phys. B. 2020;33(29):1–23.

18. Baker GA, Graves-Morris P. Pade approximations. In: Encyclopedia of Mathematics and its Applications, Series no. 59, 2nd ed. Cambridge University Press; 1996. 764 p.

19. Kireenkov AA. Svyazannaya model' treniya skol'zheniya i kacheniya v dinamike tel na sherokhovatoi ploskosti [Combined model of sliding and rolling friction in dynamics of bodies on a rough plane]. Mechanics of Solids. 2008;3:116–131. (In Russ.)