Shaking Force Balancing of Mechanisms: An Overview
The balancing of mechanisms is a well-known problem in the field of mechanical engineering because the variable dynamic loads cause vibrations, as well as noise, wear and fatigue of the machines. A mechanical system with unbalance shaking force and shaking moment transmits substantial vibration to the frame. Therefore, the objective of the balancing is to cancel or reduce the variable dynamic reactions transmitted to the frame. The resolution of this problem consists in the balancing of the shaking force and shaking moment. It can be fully or partially, by internal mass redistribution via adding counterweights or by modification of the mechanism's architecture via adding auxiliary structures. The balancing problems are of continue interest to researchers. Several laboratories around the world are very active in this area and new results are published regularly. However, despite its ancient history, mechanism balancing theory continues to be developed and new approaches and solutions are constantly being reported. Various surveys have been published that disclose particularities of balancing methods. The author believes that this is an appropriate moment to present a state of the art of the shaking force balancing studies completed by new research results. This paper presents an overview of methods devoted to the shaking force balancing of mechanisms, as well as the historical aspects of the origins and the evolution of the balancing theory of mechanisms.
 Fischer O. Über die reduzierten Systeme und die Hauptpunkte der Glieder eines Gelenkmechanismus, Zeif. für Math. and Phys., 47, 429-466 (1902).
 Goryachkin V.P. The forces of inertia and their balancing (Russian). Collection of scientific works. Ed. "Kolos", Moscow, 283-418 (1914).
 Kreutzinger R. Über die Bewegung des Schwerpunktes beim Kurbelgetriebe, Getriebetechnik, 10 (9), 397-398 (1942).
 Yudin V.A. The balancing of machines and their stability. "Edition of Academy of Red Army", Moscow, 124p. (1941).
 Shchepetilnikov V.A. The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing. J. Mechanisms, 3, pp. 367-389 (1968).
 Shcepetilnikov V.A. The balancing of mechanisms with unsymmetrical links, Mech. and Mach. Theory, 10 (6), 461-466 (1975).
 Van der Wijk V. and Herder J.L. Synthesis method for linkages with center of mass at invariant link point - Pantograph based mechanisms. Mech. and Mach. Theory. 48, pp. 15-28 (2012).
 Van der Wijk V. and Herder J.L. The method of principal vectors for the synthesis of shaking moment balanced linkages. New Trends in Mechanism and Machine Science, Mechanisms and Machine Science Volume 7, 2013, pp 399-407.
 Van der Wijk V. Shaking-moment balancing of mechanisms with principal vectors and momentum, Front. Mech. Eng. 2013, 8(1), pp. 10–16.
 Grossley F.R. Dynamics in machines, New York, Roland Press (1954).
 Maxwell R.L. Kinematics and dynamics of machinery, Prentice-Hall, Englewood Cliffs, N.J. (1960).
 Smith M.R., Maunder L. Inertia forces in a four-bar linkage, Mechanical Engineering Science, 9 (3) 218-225 (1967).
 Talbourdet G.L., Shepler P.R. Mathematical solution of 4-bar linkages - IV. Balancing of linkages, Machine Design, n° 13, 73-77 (1941).
 Arakelian V. Shaking moment cancellation of self-balanced slider-crank mechanical systems by means of optimum mass redistribution. Journal of Mechanics Research Communications, Elsevier, Vol. 33, pp. 846-850 (2006).
 Artobolevckii I.I. Mechanism and machine theory, Ed. Naouka, Moscow, 644p. (1988).
 Davies T.H. The kinematics and design of linkages, balancing mechanisms and machines, Machine Design Eng., 40 (March), 40-51 (1968).
 Kamenski V.A. On the question of the balancing of plane linkages, Mechanisms, 3 (4), 303-322 (1968).
 Berkof R.S. Force balancing of a six-bar linkage. in. Proc. of the Fifth World Congress on Theory of Machines and Mechanisms, Montreal, Canada, July 8–13, 1979, pp.1082-1085 (1979).
 Doronin V.I., Pospelov A.I. Balanced slider-crank mechanism, Patent SU 1 627 769, 1991-02-15.
 Dresig H. Schwingungen mechanischer antriebssysteme: modellbildung, berechnung, analyse, synthese. Springer, 425p. (2001).
 Dresig H., Holzweißig F. Maschinendynamik, Springer, 2004, 526p.
 Filonov I.P., Petrikovetz I.P. Balancing device of lever mechanisms, Patent SU 1 296 762, 1987-03-15.
 Frolov K.V. Theory of mechanisms and machines, Ed. “Vishaya shkola”, Moscow, 1987, 496p.
 Turbin B.I, Koropetz A.A., Koropetz Z.A. The possibility of the shaking force balancing in the system with oscillating links. Russian Journal Mech. and Mach. Theory, n°7, pp.87-90 (1978).
 Van der Wijk V. and Herder J.L. Dynamic balancing of a single crank-double slider mechanism with symmetrically moving couplers. New Trends in Mechanism Science Mechanisms and Machine Science Volume 5, pp 413-420 (2010).
 Cormac P. A treatise on engine balance using exponentials. E.P. Dutton, New York (1923).
 Dalby W.E. The balancing of engines, Ed Arnnold, London (1923).
 Delagne G. Certaines propriétés générales d'équilibrage des machines à piston d'après la méthode des vecteurs tournants symétriques. C.R. Acad. Sci. 206(22), 1617-1618 (1938).
 Doucet E. Equilibrage dynamique des moteurs en ligne. Tech. Automobile et Arienne, v.37, pp. 30-31, 35-37, 55-56, 230-232 (1946).
 Kobayashi A. Analytical study of crank effort in reciprocating engines, Ryojun College Eng - Memoirs IV (3), 127-183 (1931).
 Root R.E. Dynamics of engine and shaft. John Wiley, New York (1932).
 Artobolevskii I.I., Edelshtein B.V. Methods of inertia calculation for mechanisms of agricultural machines (Russian), Moscow, Ed. Selkhozizdate (1935).
 Artobolevskii I.I. Methods of balancing of inertia forces in working machines with complex kinematic schemes (Russian), Moscow, Ed. Acad. Naouk SSSR (1938).
 Lanchester F.M. Engine balancing. Horseless Age, 33 (12-16), Mar. 25, Apr. 1,8,15, 22, pp. 494-498, 536-538, 571-572, 608-610, 644-646 (1914).
 Chiou S.T., Davies T.H. The ideal locations for the contra-rotating shafts of generalized Lanchester balancers. Proc. IMECH Engrs, Part C. Mechanical Engineering Sciences, Vol. 208, No. 1, pp. 29-37 (1994).
 Arakelian V., Makhsudyan N. Generalized Lanchester balancer. Mechanics Research Communications. 37(7), pp. 647-649 (2010).
 Kamenski V.A. On the problem of the number of counterweights in the balancing of plane linkages, Mechanisms, 3 (4), 323-333 (1968).
 Arakelian V., S. Briot. Simultaneous inertia force/moment balancing and torque compensation of slider-crank mechanisms. Mechanics Research Communications. 37(2), pp. 265-269 (2010).
 Chiou S.T., Davies T.H. Partial cancellation of shaking force harmonics by cam modification. Proc. IMECH Engrs, Part C. Mechanical Engineering Sciences, Vol. 211, pp. 253-263 (1997).
 Hilpert H. Weight balancing of precision mechanical instruments, Mechanisms, 3 (4), 289-302 (1968).
 Arakelian V. Balanced crank mechanism. Patent SU n°1802244. March 15, 1993.
 Arakelian V. Equilibrage dynamique complet des mécanismes, Mech. and Mach. Theory, 33 (4), 425-436 (1998).
 Arakelian V., Smith M. Shaking force and shaking moment balancing of mechanisms: an historical review with new examples. Transactions of the ASME. Journal of Mechanical Design, Vol. 127(2), pp. 334-339 (2005).
 Gheronimus Y.L. On the application of Chebychev's methods to the problem of balancing mechanisms, Mechanisms, 3 (4), 235-281 (1968).
 Gheronimus Y.L. An approximate method of calculating a counterweight for the balancing of vertical inertia forces, Mechanisms, 3(4), 283-288 (1968).
 Arakelian V. Synthèse dynamique des mécanismes basée sur les methodes d'approximation de la géométrie cinématique. Proc. of the Ninth World Congress on the Theory of Machines and Mechanisms, Italy, 1, 205-209 (1995).
 Arakelian V., An approximate method of calculating a counterweight for the optimum shaking force and shaking moment balancing of linkages. The 9th IFToMM International Conference on the Theory of Machines and Mechanisms, August 31 - September 3, 2004, Liberec, Czech Republic, pp. 41-46 (2004).
 Han C.-Y. Balancing of high speed machinery. ASME J. Eng. Ind. 89(1), pp. 111-117 (1967).
 Emöd I. Massenausgleich am Kurbelgertiebe von Sechszylinder-viertakt-V-motoren mit 6 Kurbeln und 60° Zylinderwinkeln, Period. Polytechn. Engng. 11 (3-4), 205-221 (1967).
 Gappoev T.T. Singularities of the balancing of the spatial mechanisms, Balancing of the machines and the apparatuses, Moscow, ed. Mechanical engineering, 243-251 (1979).
 Gappoev T.T., Tabouev D.B. Balancing of spatial mechanisms, Dynamics of machines, Moscow, Ed. Nauka, 50-56, (1980).
 Gappoev T.T., Salamonov M.S. Some problems of balancing of agricultural machines, Conference of the Soviet Union: Modern methods of balancing of the machines and the apparatuses, Moscow, 49-50 (1983).
 Innocenti C. Harmonic balancing of planar machinery by arbitrarily placed counterweighing shafts. In Proc. of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Las Vegas, Nevada, USA, September 4–7 (2007).
 Semenov M.V. The synthesis of balanced plane mechanisms, Mechanisms, 3 (4), 339-353 (1968).
 Stevensen E. N. Balancing of machines. J. Eng. Ind. 95(2), 650-656 (1973).
 Tsai L.W., Maki E.R. Planetary-gear-type second-harmonic balancers. J. of Mech., Trans, and Automation 111(4), 530-536 (1989).
 Urba A.L. Study of the elliptic harmonics and the possibility of their balancing by a counterweight, Collection of the scientific works of the Academy of agricultural science of Lithuania, 26 (3/28), 43-49 (1980).
 Urba A.L. Geometric disposition of the points with r.m.s. value of shaking moment of spatial mechanisms, Collection of the scientific works of the Academy of agricultural science of Lithuania, 27 (3/89) 40-49 (1981).
 Arakelian V., Smith M., Balancing method based on the properties of the Watt gear-slider mechanism, Proceefings of the International Symposium on Multibody Systems and Mechatronics: MUSME 2005, 6-9 March, Uberlandia, Brazil (2005).
 Tsai L.W. Oldham-coupling second-harmonic balancer. J. of Mech., Trans, and Automation 106(3), 285-290 (1984).
 Davies T.H., Niu G.H. On the retrospective balancing of installed planar mechanisms. Proc. IMECH Engrs, Part C. Mechanical Engineering Sciences, Vol. 208, pp. 39-45 (1994).
 Berkof R.S., Lowen G.G. A new method for completely force balancing simple linkages, TransASME, Eng. Ind., 91 B (1), 21-26 (1969).
 Bagci C. Shaking force balancing of planar linkages with force transmission irregularities using balancing idler loops. Mech. and Mach. Theory, 14, pp. 267-284 (1979).
 Balasubramanian S., Bagci C. Design equations for the complete shaking force balancing of 6R 6-bar and 6-bar slider-crank mechanisms. Mech. and Mach. Theory, 13, pp. 659-674 (1978).
 Berkof R.S., Lowen G.G., Tepper F.R. Balancing of linkages, Shock and Vibration Digest 9(6), 3-10 (1977).
 Elliot J.L., Tesar D. A general mass balancing method for complex planar mechanisms, Mech. and Mach. Theory, 17 (2), 153-172 (1982).
 Smith M.R. Dynamic analysis and balancing of linkages with interactive computer graphics, Computer Aided Design, 7(1), 15-19 (1975).
 Tepper F.R., Lowen G.G. General theorems concerning full force balancing of planar linkages by internal mass redistribution, Trans. ASME, J. Engin. Ind., 94B(3), 789-796 (1972).
 Walker M.J., Oldham K. A general theory of force balancing using counterweights. Mech. and Mach. Theory, 13, pp. 175-185 (1978).
 Briot, S., Arakelian, V., Le Baron, J.-P., 2012, “Shaking force minimization of high-speed robots via center of mass acceleration control”, Mech. Mach. Theory, Vol. 57, pp. 1–12.
 Arakelian, V., 2016, “Design of partially balanced 5R planar manipulators with reduced center of mass acceleration (RCMA)”, Robot Design, Dynamics and Control, Springer, pp. 113-122.
 Arakelian, V., Geng, J., Le Baron, J.-P., 2017, “Synthesis of balanced 3-RRR planar parallel manipulators”, In Proc. of the 19th International Conference on Robotics and Computer Integrated Manufacturing (ICRCIM'2017), Prague, Czech Republic, Vol. 4, pp. 37-43.
 Arakelian, V., Geng, J., Fomin A., 2018, “Inertia forces minimization in planar parallel manipulators via optimal control”, Journal of Machinery Manufacture and Reliability, Vol. 47, pp. 303-309.
 Geng, J., Arakelian, V., 2019, “Design of partially balanced planar 5R symmetrical parallel manipulators via an optimal motion planning”, Proceedings of ECCOMAS 2019, Advances in Mechanism and Machine Science, Springer, pp. 2211-2220.
 Geng, J., Arakelian, V., 2020, “Partial shaking force balancing of 3-RRR parallel manipulators by optimal acceleration control of the total center of mass”, Proceedings of IFToMM 2019, Springer, pp. 375–382.
 Arakelian, V., Geng, J., 2020, “Design of High-Speed Manipulators via Optimal Control of the Acceleration of the Total Mass Center”, Advanced Technologies in Robotics and Intelligent Systems, Springer, vol. 80, pp. 299-307.
 Acevedo, M., Orvañanos-Guerrero, M. T., Velázquez, R., and Arakelian, V., 2020, “An alternative method for shaking force balancing of the 3RRR PPM through acceleration control of the center of mass”, Journal of Applied Sciences, pp. 1-19.