INVESTIGATION OF THE MATHEMATICAL MODEL OF A SINGLE PENDULUM UNDER THE ACTION OF THE FOLLOWER FORCE
Abstract
One of the main constructive elements of roadmakers, railway bridge supports and structures is a compressed rod, to the end of which a follower force is applied. Recently, the most frequently used model of such rod in the form of an inverted mathematical pendulum under the influence of an asymmetric follower force. Asymmetry is due to the simultaneous presence of both angular and linear eccentricities. The work is devoted to the study of vertical and non-vertical states of equilibrium of a single pendulum. The reduced mathematical model of a single inverted mathematical pendulum is generalized, since it takes into account both the angular eccentricity and the linear eccentricity of the follower force. In addition, the coefficients of influence allow to consider all types of elastic elements (rigid, soft or linear). In this case, both elements can have characteristics of the same type or of different types. For direct integration of the differential equation of the pendulum motion, and also the decoupling of the corresponding Cauchy problem, the authors use the method of extending the parameter of the outstanding Japanese scientist Y.A. Shinohara. Varying of the angular eccentricity of the follower force at zero linear eccentricity results in the inverted pendulum having one or three non-vertical equilibrium positions. The type of characteristics of the elastic elements affects the maximum possible deviation from the vertical, at which the pendulum will be in a state of equilibrium. Analysis of the results of computer simulation shows that the orientation of the follower force for fixed values of other parameters of the pendulum has a significant effect on the configuration of the equilibrium curve
Downloads
References
Lobas, L. H. (1998). Bifurkatsii statsyonarnukh sostoianyi i peryodycheskikh dvizhenyi konechnomernukh dinamicheskikh sistem s prosteishei symmetryei. Prikladnaya mekhanika, 1, 3–29.
Lobas, L. H., Lobas, L. L. (2004). Bifurkatsii, ustoychivost' i katastrofy sostoyaniy ravnovesiya dvoynogo mayatnika pod vozdeystviem asimmetrichnoy sledyashhey sily. Izvestiya RAN. Mekhanyka tverdoho tela, 4, 139–149.
Lobas, L. H., Lobas, L. L. (2002). Vliyanie orientatsii sledyashhey sily na oblasti ustoychivosti verkhnego polozheniya ravnovesiya perevernutogo dvoynogo mayatnika. Problemu upravlenyia i informatiki, 6, 26–33.
Lobas, L. H., Lobas, L. L. (2002). Modelirovanie dinamicheskoho povedeniia odnomernoho upruhoho tela pri vozdeistvii asymmetrichnoi slediashchei sily. Elektronnoe modelirovanie, 6, 19–31.
Jin, J.-D., Matsuzaki, Y. (1988). Bifurcations in a two-degree-of-freedom elastic system with follower forces. Journal of Sound and Vibration, 126 (2), 265–277. doi: 10.1016/0022-460x(88)90241-6
Koval’chuk, V. V., Lobas, V. L. (2004). Divergent Bifurcations of a Double Pendulum under the Action of an Asymmetric Follower Force. International Applied Mechanics, 40 (7), 821–828. doi: 10.1023/b:inam.0000046227.50540.17
Lobas, L. G. (2005). Dynamic Behavior of Multilink Pendulums under Follower Forces. International Applied Mechanics, 41 (6), 587–613. doi: 10.1007/s10778-005-0128-y
Lobas, L. G. (2005). Generalized Mathematical Model of an Inverted Multilink Pendulum with Follower Force. International Applied Mechanics, 41 (5), 566–572. doi: 10.1007/s10778-005-0125-1
Lobas, L. G., Koval’chuk, V. V. (2005). Influence of the Nonlinearity of the Elastic Elements on the Stability of a Double Pendulum with Follower Force in the Critical Case. International Applied Mechanics, 41 (4), 455–461. doi: 10.1007/s10778-005-0110-8
Lobas, V. L. (2005). Influence of the Nonlinear Characteristics of Elastic Elements on the Bifurcations of Equilibrium States of a Double Pendulum with Follower Force. International Applied Mechanics, 41 (2), 197–202. doi: 10.1007/s10778-005-0077-5
Shinohara, Y. (1972). A geometric method for the numerical solution of nonlinear equations and its application to nonlinear oscillations. Publications of the Research Institute for Mathematical Sciences, 8 (1), 13–42. doi: 10.2977/prims/1195193225
Copyright (c) 2017 Olha Bambura, Oleksii Rezydent

This work is licensed under a Creative Commons Attribution 4.0 International License.
Our journal abides by the Creative Commons CC BY copyright rights and permissions for open access journals.
Authors, who are published in this journal, agree to the following conditions:
1. The authors reserve the right to authorship of the work and pass the first publication right of this work to the journal under the terms of a Creative Commons CC BY, which allows others to freely distribute the published research with the obligatory reference to the authors of the original work and the first publication of the work in this journal.
2. The authors have the right to conclude separate supplement agreements that relate to non-exclusive work distribution in the form in which it has been published by the journal (for example, to upload the work to the online storage of the journal or publish it as part of a monograph), provided that the reference to the first publication of the work in this journal is included.