INVESTIGATION OF THE MATHEMATICAL MODEL OF A SINGLE PENDULUM UNDER THE ACTION OF THE FOLLOWER FORCE

Olha Bambura, Oleksii Rezydent

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

Keywords


single pendulum; equilibrium states; follower force; bifurcations; mathematical model; orientation parameter; catastrophe; phase space; mechanical system; eccentricity

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References


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DOI: http://dx.doi.org/10.21303/2585-6847.2017.00472

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ISSN 2585-6847 (Online), ISSN 2585-6839 (Print)