On the motion of a body with a moving internal mass on a rough horizontal plane. (English) Zbl 1419.70003

Summary: We consider a vibration-driven system which consists of a rigid body and an internal mass. The internal mass is a particle moving in a circle inside the body. The center of the circle is located at the mass center of the body and the absolute value of particle velocity is a constant. The body performs rectilinear motion on a horizontal plane, whereas the particle moves in a vertical plane. We suppose that dry friction acts between the plane and the body. We have investigated the dynamics of the above system in detail and given a full description of the body’s motion for any values of its initial velocity. In particular, it is shown that there always exists a periodic mode of motion. Depending on parameter values, one of three types of this periodic mode takes place. At any initial velocity the body either enters a periodic mode during a finite time interval or it asymptotically approaches the periodic mode.


70E18 Motion of a rigid body in contact with a solid surface
Full Text: DOI MNR


[1] Vartholomeos, P. and Papadopoulos, E., “Dynamics, Design and Simulation of a Novel Microrobotic Platform Employing Vibration Microactuators”, J. Dyn. Sys., Meas., Control, 128:1 (2005), 122-133 · doi:10.1115/1.2168472
[2] Vartholomeos, P. and Papadopoulos, E., “Analysis and Experiments on the Force Capabilities of Centripetal-Force-Actuated Microrobotic Platforms”, IEEE Trans. on Robotics, 24:3 (2008), 588-599 · doi:10.1109/TRO.2008.919298
[3] Vartholomeos, P., Papadopoulos, E., and Vlachos, K., “Analysis and Motion Control of a Centrifugal-Force Microrobotic Platform”, IEEE Trans. Autom. Sci. Eng., 10:3 (2013), 545-553 · doi:10.1109/TASE.2013.2248083
[4] Vlachos, K., Papadimitriou, D., and Papadopoulos, E., “Vibration-Driven Microrobot Positioning Methodologies for Nonholonomic Constraint Compensation”, Engineering, 1:1 (2015), 66-72 · doi:10.15302/J-ENG-2015016
[5] Jatsun, S. F., Mischenko, V. Ya., and Safarov, D. I., “Investigation of a Two-Mass Vibrating Motion of the Robot”, Izv. Vyssh. Uchebn. Zaved. Mashinostr., 2006, no. 5, 32-42 (Russian)
[6] Jatsun, S. F., Razin’kova, A. V., and Grankin, A. N., “Investigation of the Vibrating Motion of the Robot with Electromagnetic Drive”, Izv. Vyssh. Uchebn. Zaved. Mashinostr., 2006, no. 10, 53-64 (Russian)
[7] Yatsun, S. F., Lupekhina, I. V., and Sapronov, K. A., “Modeling of Motion of Hopping Vibration Driven Microrobot”, Izv. Kursk. Gos. Tekh. Univ., 2009, no. 2(27), 25-31 (Russian)
[8] Yatsun, S. F., Bezmen, P. A., Sapronov, K. A., and Rublev, S. B., “Dynamics of the Vibration Mobile Robot with Internal Movable Mass”, Izv. Kursk. Gos. Tekh. Univ., 2010, no. 2(31), 21a-31 (Russian)
[9] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2012, no. 6, 122-141 (Russian) · Zbl 1269.93010 · doi:10.1134/S1064230712060159
[10] Yatsun, S. F. and Volkova, L. Yu., “Simulation of Dynamic Modes of Vibration Robot Moving along the Surface of the Viscous Resistance”, Spectekhnika i svyaz, 2012, no. 3, 25-29 (Russian)
[11] Volkova, L. Yu. and Yatsun, S. F., “Studying of Regularities of Movement of the Jumping Robot at Various Positions of a Point of Fixing of a Foot”, Nelin. Dinam., 9:2 (2013), 327-342 (Russian) · doi:10.20537/nd1302009
[12] Wang, Q. M., Zhang, W. M., and Ju, J. Ch., “Kinematics and Dynamics Analysis of a Micro-Robotic Platform Driven by Inertial-Force Propulsion”, Engineering Decisions for Industrial Development, Applied Mechanics and Materials, 733, eds. J. Xu, P. Wang, Zh. Fang, TransTech, Stäfa, 2015, 531-534 · doi:10.4028/www.scientific.net/AMM.733.531
[13] Dokl. Ross. Akad. Nauk, 405:1 (2005), 56-60 (Russian) · doi:10.1134/1.2137795
[14] Prikl. Mat. Mekh., 70:6 (2006), 915-941 (Russian) · Zbl 1126.70334 · doi:10.1016/j.jappmathmech.2007.01.003
[15] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2006, no. 5, 157-167 (Russian) · Zbl 1263.93017 · doi:10.1134/S1064230706050145
[16] Prikl. Mat. Mekh., 72:2 (2008), 216-229 (Russian) · Zbl 1189.70033 · doi:10.1016/j.jappmathmech.2008.04.013
[17] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2007, no. 5, 161-170 (Russian) · Zbl 1308.93156 · doi:10.1134/S1064230707050140
[18] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2007, no. 2, 65-71 (Russian) · Zbl 1263.49006 · doi:10.1134/S1064230707020086
[19] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2009, no. 6, 150-158 (Russian) · Zbl 1211.93093 · doi:10.1134/S1064230709060136
[20] Ivanov, A. P. and Sakharov, A. V., “Dynamics of Rigid Body, Carrying Moving Masses and Rotor, on a Rough Plane”, Nelin. Dinam., 8:4 (2012), 763-772 (Russian) · doi:10.20537/nd1204006
[21] Zhan, X. and Xu, J., “Locomotion Analysis of a Vibration-Driven System with Three Acceleration Controlled Internal Masses”, Adv. Mech. Eng., 7:3 (2015), 12 · doi:10.1177/1687814015573766
[22] Dokl. Akad. Nauk, 470:4 (2016), 406-410 (Russian) · doi:10.1134/S1028335816100013
[23] Bardin, B. S., “On Non-Impact Jumps of a Body Carrying Movable Masses“, Proc. of the 18th Internat. Symp. “Dynamics of Vibroimpact (Strong Nonlinear) Systems” (Moscow, 2015), 42-49 (Russian)
[24] Bilchenko, G. G., “The Influence of Mobile Load on the Carrier Motion“, Proc. of the 11th Int. Chetaev Conf. “Analytical Mechanics, Stability and Control”, v. 1, Sect. 1, Analitical Dynamics, eds. S. N. Vassilyev et al., 37-44 (Russian)
[25] Fang, H. and Xu, J., “Stick-Slip Effect in a Vibration-Driven System with Dry Friction: Sliding Bifurcations and Optimization”, J. Appl. Mech., 81:5 (2013), 051001, 10 pp. · doi:10.1115/1.4025747
[26] Chernous’ko, F. L. and Bolotnik, N. N., “Mobile Robots Controlled by the Motion of Internal Bodies”, Tr. Inst. Mat. i Mekh. UrO RAN, 16:5 (2010), 213-222 (Russian)
[27] Prikl. Mat. Mekh., 76:1 (2012), 3-22 (Russian) · Zbl 1272.70036 · doi:10.1016/j.jappmathmech.2012.03.001
[28] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2016, no. 6, 146-160 (Russian) · Zbl 1384.93083 · doi:10.1134/S106423071605004X
[29] Golitsyna, M. V., “Periodic Mode of Movement of the Vibration-Driven Robot with Limited Control”, Prikl. Mat. Mekh., 82:1 (2018), 3-15 (Russian)
[30] Izv. Akad. Nauk. Teoriya i Sistemy Upravleniya, 2018, no. 2, 85-101 (Russian) · Zbl 1401.93141 · doi:10.1134/S1064230718020089
[31] Dokl. Akad. Nauk, 480:5 (2018), 528-532 (Russian) · Zbl 1400.49020 · doi:10.1134/S1064562418030195
[32] Bardin, B. S. and Panev, A. S., “On Periodic Motions of the Body with Movable Internal Mass over a Horizontal Surface”, Trudy MAI, 84 (2015), 5 (Russian)
[33] Bardin, B. S. and Panev, A. S., “On Dynamics of a Rigid Body Moving on a Horizontal Plane by Means of Motion of an Internal Particle”, Vibroeng. Procedia, 8 (2016), 135-141
[34] Panev, A. S., “On Motion of a Rigid Body with Mobile Internal Mass on a Horizontal Plane in a Viscous Medium”, Trudy MAI, 98 (2018), 2 (Russian) · Zbl 1419.70003
[35] Bardin, B. S. and Panev, A. S., “On the Motion of a Rigid Body with an Internal Moving Point Mass on a Horizontal Plane”, AIP Conf. Proc., 1959:1 (2018), 030002 · Zbl 1419.70003 · doi:10.1063/1.5034582
[36] Filippov, A. F., “Differential Equations with Discontinuous Right-Hand Side”, Mat. Sb. (N. S.), 51(93):1 (1960), 99-128 (Russian) · Zbl 0138.32204
[37] Filippov, A. F., Differential Equations with Discontinuous Righthand Sides, Math. Appl., 18, Springer, Dordrecht, 1988 · Zbl 0664.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.