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Abstract
Nonlinear mathematical models are proposed that describe the dynamics of a pipeline with a fluid flowing in it: a) the model of bending-torsional vibrations with two degrees of freedom; b) the model describing flexural-torsional vibrations taking into account the nonlinearity of the bending moment and centrifugal force; c) the model that takes into account joint longitudinal, bending (transverse) and torsional vibrations. All proposed models are described by nonlinear partial differential equations for unknown strain functions. To describe the dynamics of a pipeline, the nonlinear theory of a rigid deformable body is used, which takes into account the transverse, tangential and longitudinal deformations of the pipeline. The dynamic stability of bending-torsional and longitudinal-flexural-torsional vibrations of the pipeline is investigated. The definitions of the stability of a deformable body adopted in this work correspond to the Lyapunov concept of stability of dynamical systems. The problem of studying dynamic stability, namely, stability according to initial data, is formulated as follows: at what values of the parameters characterizing the gas-body system, small deviations of the body from the equilibrium position at the initial moment of time will correspond to small deviations and at any moment of time. For the proposed models, positive definite functionals of the Lyapunov type are constructed, on the basis of which the dynamic stability of the pipeline is investigated. Sufficient stability conditions are obtained that impose restrictions on the parameters of a mechanical system.
Highlights
Nonlinear mathematical models are proposed that describe the dynamics of a pipeline with a fluid flowing in it: a) the model of bending-torsional vibrations with two degrees of freedom; b) the model describing flexural-torsional vibrations taking into account the nonlinearity of the bending moment and centrifugal force; c) the model that takes into account joint longitudinal, bending and torsional vibrations
All proposed models are described by nonlinear partial differential equations for unknown strain functions
To describe the dynamics of a pipeline, the nonlinear theory of a rigid deformable body is used, which takes into account the transverse, tangential and longitudinal deformations of the pipeline
Summary
Актуальной проблемой во многих отраслях техники является задача исследования динамики и устойчивости составных частей конструкций, устройств, приборов, установок различного назначения при аэрогидродинамическом воздействии. Д. Изучению и описанию колебаний трубопроводов, распространения волн в трубопроводах, содержащих газожидкостную среду, устойчивости упругих тел, взаимодействующих с потоком жидкости, посвящено большое количество теоретических и экспериментальных исследований, проведенных с середины прошлого века до сегодняшних дней и представленных, например, [1–14]. Контент доступен по лицензии Creative Commons Attribution 4.0 International License. Данной статьи по исследованию динамики и устойчивости упругих тел, взаимодействующих с потоком жидкости или газа, отметим статьи и монографии [15–19]. Для описания динамики трубопровода с протекающей в нем жидкостью предложены модельные уравнения, описывающие изгибно-крутильные колебания с двумя степенями свободы, уравнения, описывающие изгибно-крутильные колебания с учетом нелинейности изгибающего момента и центробежной силы, а также уравнения, описывающие совместные продольные, изгибные и крутильные колебания. Для исследования устойчивости используется аналитический метод, который основан на построении положительно определенных функционалов типа Ляпунова для указанных математических моделей
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