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On Translational Rectilinear Motion of a Rigid Body Carrying a Movable Inner Mass

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Abstract

We consider the motion of the mechanical system consisting of the case (a rigid body) and the inner mass (a material point). The inner mass circulates inside the rigid body on a circle centered at the center of mass of the rigid body. We suppose that the absolute value of the velocity of circular motion of the inner mass is constant. The rigid body moves translationally and rectilinearly on a flat horizontal surface with forces of viscous friction and dry Coulomb friction on it. The inner mass moves in the vertical plane.

We perform the full qualitative investigation of the dynamics of this system. We prove that there always exists a unique motion of the rigid body with periodic velocity. We study all possible types of such a periodic motion. We establish that for any initial velocity, the rigid body either reaches the periodic mode of motion in a finite time or asymptotically approaches to it depending on the parameters of the problem.

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References

  1. B. S. Bardin, “On shock-free jumps of the body carrying movable masses,” In: Proc. XVIII Int. Symp. “Dynamics of Vibroimpact Strongly Nonlinear Systems” (DYVIS-2015), pp. 42–49 (2015).

  2. B. S. Bardin and A. S. Panev, “On periodic motions of a body with movable inner mass on horizontal surface,” Tr. MAI, 84 (2015).

  3. B. Bardin and A. Panev, “On dynamics of a rigid body moving on a horizontal plane by means of motion of an internal particle,” Vibroeng. Procedia, 8, 135–141 (2016).

    Google Scholar 

  4. B. S. Bardin and A. S. Panev, “On the motion of a body with a moving internal mass on a rough horizontal plane,” Russ. J. Nonlin. Dyn., 14, No. 4, 519–542 (2018).

    MathSciNet  MATH  Google Scholar 

  5. B. S. Bardin and A. S. Panev, “On the motion of a rigid body with an internal moving point mass on a horizontal plane,” AIP Conf. Proc., 1959, 030002 (2018).

    Article  Google Scholar 

  6. G. G. Bil’chenko, “Influence of a movable load on the motion of the carrier,” In: Analytical Mechanics, Stability, and Control. Proc. XI Int. Chetaev Conf., pp. 37–44 (2017).

  7. N. N. Bolotnik and F. L. Chernous’ko, “Mobile robots controlled by movement of inner bodies,” Tr. Inst. Mat. i Mekh. UrO RAN, 16, No. 5, 213–222 (2010).

    Google Scholar 

  8. N. N. Bolotnik, A. M. Nunuparov, and V. G. Chashchukhin, “Capsular vibrational robot with electromagnetic gear and return spring: dynamics and motion control,” Izv. RAN. Teor. i Sist. Upravl., No. 6, 146–160 (2016).

    MATH  Google Scholar 

  9. N. N. Bolotnik and T. Yu. Figurina, “Optimal control of rectilinear motion of a solid body on rugged surface by means of transition of two inner masses,” Prikl. Mat. Mekh., 72, No. 2, 216–229 (2008).

    MathSciNet  MATH  Google Scholar 

  10. N. N. Bolotnik, T. Yu. Figurina, and F. L. Chernous’ko, “Analysis and optimization of motion of the body controlled by a movable inner mass,” Prikl. Mat. Mekh., 71, No. 1, 3–22 (2012).

    Google Scholar 

  11. N. N. Bolotnik, I. M. Zeydis, K. Tsimmermann, and S. F. Yatsun, “Dynamics of controllable motions of vibrational systems,” Izv. RAN. Teor. i Sist. Upravl., No. 5, 157–167 (2006).

    Google Scholar 

  12. F. L. Chernous’ko, “On the motion of body containing movable inner mass,” Dokl. RAN, 405, No. 1, 56–60 (2005).

    Google Scholar 

  13. F. L. Chernous’ko, “Analysis and optimization of the motion of the body controlled by the movable inner mass,” Prikl. Mat. Mekh., 70, No. 6, 915–941 (2006).

    MathSciNet  MATH  Google Scholar 

  14. F. L. Chernous’ko, “Motion of a body on a plane by means of movable inner masses,” Dokl. RAN, 470, No. 4, 406–410 (2016).

    Google Scholar 

  15. F. L. Chernous’ko, “Optimal motion control for a two-masses system,” Dokl. RAN, 480, No. 5, 528–532 (2018).

    Google Scholar 

  16. H. Fang and J. Xu, “Stick-slip effect in a vibration-driven system with dry friction: Sliding bifurcations and optimization,” J. Appl. Mech., 81, No. 5, 061001 (2014).

  17. T. Yu. Figurina, “Optimal motion control for the system of two bodies on an axis,” Izv. RAN. Teor. i Sist. Upravl., No. 2, 65–71 (2007).

    MathSciNet  Google Scholar 

  18. A. F. Filippov, “Differential equations with discontinuous right-hand side,” Mat. Sb., 51, No. 1, 99–128 (1960).

    MathSciNet  MATH  Google Scholar 

  19. A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side [in Russian], Nauka, Moscow (1985).

  20. M. V. Golitsyna, “Periodical mode of motion of the vibrational robot under constrains of control,” Prikl. Mat. Mekh., 82, No. 1, 3–15 (2018).

    MathSciNet  Google Scholar 

  21. M. V. Golitsyna and V. A. Samsonov, “Estimate of area of admissible parameters for the control system of the vibrational robot,” Izv. RAN. Teor. i Sist. Upravl., No. 2, 85–101 (2018).

    Google Scholar 

  22. A. P. Ivanov, Essentials of the Theory of Systems with Friction [in Russian], Izhevskiy in-t komp. issl., Izhevsk (2011).

  23. A. P. Ivanov and A. V. Sakharov, “Dynamics of the solid body with movable inner masses and the rotor on a rugged surface,” Nelin. Dinamika, 8, No. 4, 763–772 (2012).

    Article  Google Scholar 

  24. A. S. Panev, “On the motion of the solid body with movable inner mass on horizontal surface in a viscous media,” Tr. MAI, 98 (2018).

  25. N. A. Sobolev and K. S. Sorokin, “Experimental study of the model of vibrorobot with rotating masses,” Izv. RAN. Teor. i Sist. Upravl., No. 5, 161–170 (2007).

  26. K. S. Sorokin, “Motion of a mechanism on an inclined rugged plane by means of inner oscillating masses,” Izv. RAN. Teor. i Sist. Upravl., No. 6, 150–158 (2009).

  27. P. Vartholomeos and E. Papadopoulos, “Dynamics, design and simulation of a novel microrobotic platform employing vibration microactuators,” J. Dyn. Syst. Meas. Control. Trans. ASME, 128, No. 1, 122–133 (2006).

    Article  Google Scholar 

  28. P. Vartholomeos and E. Papadopoulos, “Analysis and experiments on the force capabilities of centripetal-force-actuated microrobotic platforms,” IEEE Trans. Robot., 24, 588–599 (2008).

    Article  Google Scholar 

  29. P. Vartholomeos, E. Papadopoulos, and K. Vlachos, “Analysis and motion control of a centrifugalforce microrobotic platform,” IEEE Trans. Automat. Sci. Eng., 10, 545–553 (2013).

    Article  Google Scholar 

  30. K. Vlachos, D. Papadimitriou, and E. Papadopoulos, “Vibration-driven microrobot positioning methodologies for nonholonomic constraint compensation,” Engineering, 1, 66–72 (2015).

    Article  Google Scholar 

  31. L. Yu. Volkova and S. F. Yatsun, “Modelling of flat controllable motion of three-mass vibrational system,” Izv. RAN. Teor. i Sist. Upravl., No. 6, 122–141 (2012).

  32. L. Yu. Volkova and S. F. Yatsun, “Study of regularities of motion of the jumping robot for different positions of mounting point of the leg,” Nelin. Dinamika, 9, No. 2, 327–342 (2013).

    Article  Google Scholar 

  33. Q. M. Wang, W. M. Zhang, and J. C. Ju, “Kinematics and dynamics analysis of a micro-robotic platform driven by inertial-force propulsion,” Appl. Mech. Mater., 733, 531–534 (2015).

    Article  Google Scholar 

  34. Z. Xiong and X. Jian, “Locomotion analysis of a vibration-driven system with three acceleration controlled internal masses,” Adv. Mech. Eng., 7, 1–12 (2015).

    Google Scholar 

  35. S. F. Yatsun, P. A. Bezmen, K. A. Sapronov, and S. B. Rublev, “Dynamics of the mobile vibrational robot with inner movable mass,” Izv. Kursk. Gos. Tekhn. Univ., 31, No. 2, 21–31 (2010).

    Google Scholar 

  36. S. F. Yatsun, I. V. Lupekhina, and K. A. Sapronov, “Modelling of the motion of the jumping vibrational micro-robot,” Izv. Kursk. Gos. Tekhn. Univ., 27, No. 2, 25–31 (2009).

    Google Scholar 

  37. S. F. Yatsun, V. Ya. Mishchenko, and D. I. Safarov, “Investigation of the motion of two-masses vibrational robot,” Izv. Vuzov. Ser. Mashin., No. 5, 32–42 (2006).

  38. S. F. Yatsun, A. V. Razin’kova, and A. N. Grankin, “Investigation of the motion of the vibrorobot with electromagnetic gear,” Izv. Vuzov. Ser. Mashin., No. 5, 53–64 (2007).

  39. S. F. Yatsun and L. Yu. Volkova, “Modelling of dynamical regimes of the vibrational robot moving on a surface with viscous friction,” Spetstekhn. i Svyaz’, No. 3, 25–29 (2012).

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), Moscow, Russia

    B. S. Bardin & A. S. Panev

  2. Mechanical Engineering Research Institute of the Russian Academy of Sciences, Moscow, Russia

    B. S. Bardin

Authors
  1. B. S. Bardin
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  2. A. S. Panev
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Correspondence to B. S. Bardin.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 65, No. 4, Proceedings of the S. M. Nikolskii Mathematical Institute of RUDN University, 2019.

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Bardin, B.S., Panev, A.S. On Translational Rectilinear Motion of a Rigid Body Carrying a Movable Inner Mass. J Math Sci 265, 728–762 (2022). https://doi.org/10.1007/s10958-022-06081-7

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  • DOI: https://doi.org/10.1007/s10958-022-06081-7

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