mathematical pendulum, moving suspension point, small own oscillations, frequency, Appel’s formalism, linearization


Background. The new dynamic problem, which is posed and solved in this article, is a theoretical generalization of the well-known classical problem of free oscillations of a mathematical pendulum. In the proposed setting, it is new and relevant, and can be successfully used in such fields of technology as vibration protection, vibration isolation and seismic protection of high-rise flexible structures, long power lines, long-span bridges and other large-sized supporting objects.

Objective. The aim of the work is to derive the equations of own oscillations of a new mathematical pendulum-absorber, to find a formula for determining the frequency of its small own oscillations and to establish those control parameters that allow you to tune the single-mass pendulum absorber to the frequency of the fundamental tone of the carrier object.

Methods. To achieve this goal, we used the methods of analytical mechanics, namely, the Appel’s formalism, as well as the linearization of the obtained differential equations.

Results. A mathematical model is constructed in the work that describes the own oscillations of a new-design mathematical pendulum with a movable (spring-loaded) suspension point with length L. The model is a system of differential equations obtained using the Appel’s formalism. Based on them, after linearization of nonlinear equations, a formula is established for the frequency of small own oscillations of a pendulum with a mobile suspension point.

Conclusions. It is shown that the frequency of own oscillations of the new mathematical pendulum coincides with the frequency of own oscillations of the classical mathematical pendulum with an equivalent suspension length, which is equal to gif.latex?L_%7Beq%7D%3DL+%5Cfrac%7Bmg%7D%7Bk%7D. In the case where the suspension point is fixed (k ® ¥), the frequency formula turns into a well-known formula for the frequency of small own oscillations of a classical mathematical pendulum gif.latex?%5Comega%20%3D%5Csqrt%7B%5Cfrac%7Bg%7D%7BL%7D%7D. If the stiffness coefficient of elastic elements tends to zero (k ® 0), then the frequency w of the damper also tends to zero. An important structural feature of the proposed pendulum is noted, consisting in the fact that due to the appropriate choice of the three control parameters of the pendulum (k, L, m), its frequency in magnitude can be made any in the range from zero to gif.latex?%5Csqrt%7B%5Cfrac%7Bg%7D%7BL%7D%7D.


V.P. Legeza, Vibration Protection of Dynamic Systems with Roller Dampers. Kyiv, Ukraine: Chetverta Khvylya, 2010, 280 p.

V.P. Legeza, Theory of Vibration Protection of Systems Using Isochronous Roller Dampers: Models, Methods, Dynamic Analysis, Technical Solutions. Saarbrucken, Germany: Lambert Academic Publishing, 2013, 108 p.

Z. Hu et al., “Non-linear model of the damping process in a system with a two-mass pendulum absorber”, Intellig. Syst. Applicat., Systems and Applications, vol. 11, no. 1, рр. 67–72, 2019. doi: 10.5815/ijisa.2019.01.07

V.P. Legeza et al., “Suppression of vibrations of wires and cables using a two-mass pendulum damper”, Elect. Syst. Netw., no. 2, pp. 7–13, 2016.

Zhengbing Hu et al., “Mathematical model of the damping process in a one system with a ball vibration absorber”, Int. J. Intellig. Syst. Applicat., vol. 10, no. 1, pp. 24–33, 2018. doi: 10.5815/ijisa.2018.01.04

J.P. Den Hartog, Mechanics. Mineola: Dover Publications, 464 p.

J.P. Den Hartog, Mechanical Vibrations. Crastre Press, 2008, 496 p.

B.G. Korenev and L.M. Reznikov, Dynamic Vibration Absorbers: Theory and Technical Applications. Chichester, UK: John Wiley and Sons Ltd., 1993, 368 p.

Dynamic Response of Lattice Towers and Guyed Masts. M.K.S. Madugula, Ed. Reston: SEI&ASCE, 2002, 264 p.

S.G. Kelly, Mechanical Vibrations: Theory and Applications. Cengage Learning, 2012, 672 p.

“Dynamics of Coupled Structures”, in Proc. the 33rd IMAC, A Conference and Exposition on Structural Dynamics, vol. 4, M. Allen et al., Eds. Springer International Publishing, 2015, 173 p. doi: 10.1007/978-3-319-15209-7

M. Geradin and D.J. Rixen, Mechanical Vibration: Theory and Applications to Structural Dynamic, 3rd ed. Chichester, UK: John Wiley and Sons Ltd., The Atrium, 2015, 616 p.

J.A. Karnovsky and E. Lebed, Theory of Vibration Protection. Switzerland: Springer International Publishing, 2016, 673 p.

J. Naprstek and C. Fischer, “Non-holonomic planar and spatial model of a ball-type tuned mass damping device”, in Proc. Engineering Mechanics-2017, Svratka, Czech Republic, May 15–18, 2017, pp. 698–701.

J. Naprstek and C. Fischer, “Forced movement of a ball in spherical cavity under kinematic excitation”, in Proc. Engineering Mechanics-2018, Svratka, Czech Republic, May 14–17, 2018, pp. 573–576. doi: 10.21495/91-8-573

H.W. Klein and W. Kaldenbach, “A new vibration damping facility for steel chimneys”, in Proc. Conf. Mechanics in Design, Trent University of Nottingham, UK, 1998, pp. 265–273.

T. Dahlberg, “On optimal use of the mass of a dynamic vibration absorber”, J. Sound and Vibrat., vol. 132, no. 3, pp. 518–522, 1989. doi: 10.1016/0022-460x(89)90645-7

B.V. Ostroumov, “Dynamic vibration damper in the form of an inverted pendulum with damping”, Bulletin of Higher Educational Institutions. Ser. Building, no. 9, pp. 36–38, 2002.

B.V. Ostroumov, “Calculation of a structure with a dynamic vibration damper”, Industrial and Civil Building, no. 5, pp. 18–22, 2003.

B. Diveyev, “Different type vibration absorbers design for beam-like structures”, in Proc. ICSV19, Vilnius, Lithuania, July 8–12, 2012, pp. 1499–1506.

B. Diveyev et al., “Influence of vibration protection object parameters and dynamic vibration damper on energy efficiency of vibration absorption”, Scientific Notes of Lutsk National Technical University, vol. 42, pp. 81–87, 2013.

I.A. Vikovich et al., “Application of different types of pendulum dynamic vibration dampers”, Bulletin of the National Transport University, vol. 29, no. 1, pp. 26–33, 2014.

B. Diveyev et al., “Different type vibration absorbers design for elongated console structures”, Scientific Notes of Lutsk National Technical University, vol. 41, pp. 10–16, 2014.

B.G. Korenev, “Dynamic vibration dampers”, in Proc. International Symposium Vibration Protection in Construction, Leningrad, SU, 1984, pp. 7–17.

B.G. Korenev and I.M. Rabinovich, Dynamic Calculation of Buildings and Structures: Designer’s Handbook. Moscow, SU: Stroyizdat, 1984, 304 p.

A.I. Lurie, Analitical Mechanics. Berlin, Heidelberg: Springer, 2002, 864 p.