Artyunin A. I., Sumenkov O. Yu. Issledovaniye avtomaticheskoy balansirovki rotora v forme tsilindra, zhestko ustanovlennogo v korpuse na uprugikh oporakh [Research of the automatic balancing process of the cylindrical rotor, rigidly fixed in the housing on elastic supports]. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie [Modern Technologies. System Analysis. Modeling], 2019. Vol. 64, No. 4. Pp. 13–19. DOI: 10.26731/1813-9108.2019.4(64).13-19
The authors carried out the modeling process of the automatic balancing of an unstable rotor using four pendulums mounted in pairs on the rotating shaft on both sides of a cylindrical rotor. The rotor is rigidly fixed in the housing on elastic supports. Using the expressions for kinetic and potential energy, the Rayleigh dissipation function and the Lagrange equations of the second kind, differential equations are derived, describing the motion of our dynamic model both at acceleration and under stabilized conditions of the rotor motion. Numerical integration of the nonlinear differential equation system was implemented with the Runge-Kutta method of the forth order. But due to the insolvability of the top order equations with respect to top derivates, the calculation algorithm included matrix inversion at the each step of integration. Pendulum parameters and stability zones of the automatic balancing devices had been preliminarily calculated. The automatic balancing devices stability zones were artificially created by choosing the stiffnesses of the housing elastic supports so that the rotor operation speed was within the zone. The rotor and pendulum motion laws were derived, as well as laws of changing the angles of installation of the pendulums in relation to the imbalance. The authors demonstrated that in the selected zone the pendulums move apart and the rotor unbalance is compensated. The amplitudes of oscillations of the center of mass of the rotor with and without auto-balancer are compared. It has been established that the degree of compensation of the rotor imbalance by pendulums of the same mass and length substantially depends on the relative position of the center of the rotor mass, the center of the housing mass and the center of stiffness of the elastic supports of the housing and the distance between the pendulums in one pair. It is proved that automatic balancing can occur not only at rotational speeds higher than the critical rotor speed, but also in the resonance zones of the housing on elastic supports.
- Gusarov A.A. Balansirovka rotorov mashin [Balancing machine rotors]. Moscow: Nauka Publ., 2005. Book 2. 383 p.
- Spravochnik po balasirovke. Pod obshch. red. M.E. Levita [Balancing reference. In Levit M. E. (gen. ed.)] Moscow: Mashinostroenie Publ., 1992. 464 p.
- Nikiforov A.N. Sostoyanie problemy uravnoveshivaniya rotorov [The state of the problem of balancing rotors] Vestnik nauchno-tekhnicheskogo razvitiya [The Bulletin of scientific and technological development]. The Institute of Mechanical Engineering of the Russian Academy of Sciences, 2013. No.4 (68). Pp. 20-28.
- Artyunin A.I. Issledovanie dvizheniya rotora s avtobalansirom [Investigation of the movement of the rotor with auto-balancer]. Izvestiya vuzov [News of universities]. Mashinostroenie Publ., 1993. No.1., Pp. 10-15.
- Artyunin A.I., Alkhunsaev G.G., Serebrennikov K.V. Primenenie metoda razdeleniya dvizheniya dlya issledovaniya dinamiki rotornoi sistemy s gibkim rotorom i mayatnikovym avtobalansirom [Applying the motion separation method to study the dynamics of a rotor system with a flexible rotor and a pendulum auto-balancer]. Izvestiya vuzov [News of universities]. Mashinostroenie Publ., 2005. No. 9. Pp. 8-14.
- Bykov V.G. Avtobalansirovka zhestkogo rotora v vyazko-uprugikh ortotropnykh oporakh [Auto-balancing of a rigid rotor in visco-elastic orthotropic supports]. Vestnik SPbGU [The Bulletin of St. Petersburg State University], 2013. Ser.1. Vol. 2. Pp. 82-91.
- Kravchenko V.I. Romashchenko V.A. Ob avtomaticheskom uravnoveshivanii sharami [On automatic balancing with balls]. Teoriya mekhanizmov i mashin [Theory of mechanisms and machines]. Kharkov, 1985. Vol. 38. Pp. 69–74.
- Dubovik V.A., Pashkov E.N. Ustoichivost' statsionarnogo vrashcheniya neuravnoveshennogo rotora s zhidkim avtobalansirovochnym ustroistvom na gibkom valu [Stability of stationary rotation of an unbalanced rotor with a fluid self-balancing device on a flexible shaft] Izvestiya Tomskogo politekh. un-ta [The Bulletin of Tomsk Polytechnic University], 2007. Vol. 311. No. 2. Pp. 12-14.
- Ziyakaev G.R. Nekotorye voprosy dinamiki rotornykh sistem s mayatnikovymi avtobalansirovochnymi ustroistvami. Avtoreferat dissertatsii na soiskanie uchenoi stepeni kandidata tekhnicheskikh nauk [Some questions of the dynamics of rotary systems with pendulum automatic balancing devices. Author’s abstract of the Ph.D.(Engineering) diss.]. Tomsk Polytechnic University. Tomsk, 2009. Pp. 19.
- Filimonikhin G.B. Zrivnovazheniya i vibrozakhist rotoriv avtobalansirami z tverdimi koriguval'nimi vantazhami [Equilibrium and vibration protection of rotors with auto-balancers with solid corrective cargoes]. Ministry of Education and Science of Ukraine. Kirovograd. Nat. Techn. Un-ty Publ. Kirovograd, 2004. 352 p.
- Sperling L., Ryzhik B., Linz Ch., Duckstein H. Simulation of two plain balancing of a rigid rotor. Mathematics and Computers in Simulation, 2002, Vol. 58, No. 4-6, pp. 351-365.
- Rodrigues D.J., Champneys A.R., Friswell M.I. Automatic two-plane balancing for rigid rotors. International Journal of Non-Linear Mechanics. 2008, Vol. 43, Iss. 6, pp. 527-541.
- Rodrigues D.J., Champneys A.R., Friswell M.I., Wilson R.E. Two-plane automatic balancing: a symmetry breaking analysis. International Journal of Non-Linear Mechanics. 2011, Vol. 46, Iss. 9, pp. 1139-1154.
- Bolton J.N. Single-and dual-plane automatic balancing of an elastically mounted cylindrical rotor with considerations of coulomb friction and gravity. Dr. Diss. Blacksburg, Virginia, 2010. pp. 317.
- Nesterenko V.P. Avtomaticheskaya balansirovka rotorov priborov i mashin so mnogimi stepenyami svobody [Automatic balancing of the rotors of devices and machines with many degrees of freedom]. Tomsk: The publishing house of Tomsk University, 1985. 82 p.
- Dubovik V.A, Ziyakaev G.R. Osnovnoe dvizhenie dvukhmayatnikovogo avtobalansira na gibkom valu s uprugimi oporami [The main motion of a dual-swing auto-balancer on a flexible shaft with elastic supports]. Izvestiya Tomskogo politekh. un-ta, Matematika i mekhanika, Fizika [The Bulletin of Tomsk Polytechnic University. Mathematics and Mechanics, Physics], 2010. Vol. 317. No.2. Pp. 37-39.
- Artyunin A.I., Alkhunsayev G.G., Sushkeev Zh.B. Avtobalansirovka rotorov v korpuse na uprugikh oporakh [Auto-balancing of rotors in a housing on elastic supports]. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie [Modern technologies. System Analysis. Modeling]. Irkutsk, 2006. No. 6. Pp. 38-41.
- Artyunin A.I., Eliseev S.V., Sumenkov O.Yu. Determination of parameters and stability zones of pendulum auto-balancer of a rotor, installed in a housing on elastic supports. Proceedings of International Conference Advances in Engineering Research, 2018. Vol.158. Pp.25-29.
- Blekhman I.I. Vibratsionnaya mekhanika [Vibration mechanics]. Moscow: Fizmatgiz Publ., 1994. 400 p.