ISSN (print) 1995-2732
ISSN (online) 2412-9003


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This paper examines mechanisms the operating elements of which perform linear oscillations the frequencies of which are too high for the weights of the elements. This can create a considerable nonproductive mechanical reactive inertial power, which can by far exceed the productive (useful) dissipative power. A significantly higher energy efficiency can be achieved through neutralization of the mechanical reactive inertial power. Self-neutralization of the mechanical reactive inertial power can be possible in mechanisms with a constant equivalent moment of inertia due to displaced oscillation phases of the operating elements, the number of which may, in fact, be any. For a mechanism with two operating elements, the phase displacement is /2, whereas for a mechanism with three operating elements it is 2/3. Practical use of the mechanism with a constant equivalent moment of inertia with two big operating elements may be hindered because of the lack of balance. The degree to which a mechanism is balanced (or unbalanced) is directly related to the "degree" of central symmetry of the pattern formed by the cranks. The objective of this research is to identify the self-balancing conditions for mechanisms with a constant equivalent moment of inertia with four and three operating elements. It is shown that the star patterns formed by the cranks of mechanisms with a constant equivalent moment of inertia that have more than two operating elements are indeed centrally symmetric. Such mechanisms are balanced. It was found that, in a balanced mechanism with a constant equivalent moment of inertia, at least three operating elements perform linear oscillations.


Oscillations, phase, inertial power, self-balancing, central symmetry.

Igor P. Popov – Senior Lecturer

Kurgan State University, Kurgan, Russia. E-mail: ip.popow@yandex

Svetlana Yu. Kubareva – Senior Lecturer

Kurgan State University, Kurgan, Russia. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

1. Popov I.P., Chumakov V.G., Terentiev A.D. Drive power reduction in screen sizers. Nauchno-tekhnicheskie vedomosti Sankt-Peterburgskogo gosudarstvennogo politekhnicheskogo universiteta [St. Petersburg Polytechnic University Journal of Engineering Science and Technology], 2015, no. 2 (219), pp. 175-181. (In Russ.)

2. Popov I.P. Mechanical equivalents to reactive power. Vestnik Permskogo universiteta. Matematika. Mekhanika. Informatika [Bulletin of Perm University. Mathematics. Mechanics. Computer Science], 2015, no. 3(30), pp. 37–39. (In Russ.)

3. Popov I.P. Free vibration harmonics in systems with homogeneous elements. Prikladnaya matematika i mekhanika [Applied mathematics and mechanics], 2012, vol. 76, iss. 4, pp. 546–549. (In Russ.)

4. Popov I.P. Synthesis of an inert-inert oscillator. Prikladnaya matematika i voprosy upravleniya [Applied mathematics and control], 2017, no. 1, pp. 7–13. (In Russ.)

5. Popov I.P., Chumakov V.G., Davydova M.V., Popov D.P., Kubareva S.Yu. Ustroystvo dlya uravnoveshivaniya inertsyonnykh sil [Device for balancing inertia forces]. Patent RF, no. 2601891 RU, 2016.

6. Popov I.P. Oscillatory systems with homogeneous elements. Inzhenernaya fizika [Engineering physics], 2013, no. 3, pp. 52–56. (In Russ.)

7. Popov I.P. Modeling of a bi-inert oscillator. Prilozhenie matematiki v ekonomicheskikh i tekhnicheskikh issledovaniyakh: sb. nauch. tr. [Mathematics in economic and engineering studies: Research papers]. Ed. by V.S. Mkhitaryan. Magnitogorsk, 2017, pp. 188–192. (In Russ.)

8. Popov I.P. Sposob polucheniya mekhanicheskikh kolebaniy [Method of obtaining mechanical oscillations]. Patent RF, no. 2575763 RU, 2016.

9. Popov I.P. Oscillatory systems comprised of only inert or only elastic elements and the emergence of free harmonic oscillations. Vestnik Tomskogo gosudarstvennogo universiteta. Seria: Matematika i mekhanika [Tomsk State University Journal. Mathematics and mechanics], 2013, no. 1(21), pp. 95–103. (In Russ.)

10. Popov I.P., Shamarin E.O. Free mechanical harmonic oscillations with staggered phases. Vestnik Tikhookeanskogo gosudarstvennogo universiteta [Bulletin of PNU], 2013, no. 2(29), pp. 39–48. (In Russ.)