Investigation of the dynamic stability of bending-torsional deformations of the pipeline

P. A. Vel'misov ; J. А. Tamarova ; Yu. V. Pokladova

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 1, pp. 72-81.

Résumé

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.

Détail

Keywords: differential equations, pipeline, deformation, dynamics, stability, Lyapunov functionals.
Mots-clés : flexural-torsional vibrations

Bibliographie

[1] V. I. Feodosev, “On vibration and stability of a pipeline containing flowing fluid”, Inzhenernyy sbornik, 10 (1951), 169–170 (In Russ.)

[2] A. A. Movchan, “On a problem of stability of a pipe when a fluid flows through it”, Applied Mathematics and Mechanics, 4 (1965), 760–762 (In Russ.)

[3] S. V. Chelomey, “On dynamic stability of elastic systems”, Dokl. of the Academy of Sciences of the SSR. Ser. “Mechanics”, 252:2 (1980), 307–310 (In Russ.) | MR

[4] V. A. Svetlitsky, Mechanics of Piping and Hoses: Problems of the interaction of rods with a fluid or air flow, Mashinostroyenie Publ., Moscow, 1982, 280 pp. (In Russ.)

[5] V. N. Vasina, “Parametric vibrations of a pipeline section with a flowing liquid”, Bulletin of the Moscow Power Engineering Institute, 1 (2007), 1–11 (In Russ.)

[6] V. V. Bolotin, V. P. Radin, V. P. Chirkov, A. V. Shchugorev, “Stability of a pipeline section with elastic support”, Rigid Body Mechanics, 1 (2009), 174–184 (In Russ.)

[7] E. Y. Kaya-Cekin, E. Aulisa, A. Ibragimov, P. Seshaiyer, “A stability estimate for fluid structure interaction problem with non-linear beam”, Discrete and Continuous Dynamical Systems. Supplement, 2009, 424–432 | MR

[8] R. T. Faal, D. Derakhshan, “Flow-induced vibration of pipeline on elastic support”, Procedia Engineering, 14 (2011), 2986–2993 | DOI

[9] S. V. Zefirov, A. V. Kochetkov, V. F. Ovchinnikov, A. O. Savikhin, L. V. Smirnov, A. V. Yaskelyin, “Numerical modeling of dynamic deformation of a spatial pipeline with a liquid under local shock loading”, Problems of Strength and Ductility, 75:2 (2013), 152–159 (In Russ.)

[10] E. Aulisa, A. Ibragimov, E. Y. Kaya-Cekin, “Fluid structure interaction problem with changing thickness beam and slightly compressible fluid”, Discrete and Continuous Dynamical Systems, Ser. S., 7:6 (2014), 1133—1148 | DOI | MR | Zbl

[11] K. Moditis, M. P. Paidoussis, J. Ratigan, “Dynamics of a partially confined, discharging, cantilever pipe with reverse external flow”, Journal of Fluids and Structures, 63 (2016), 120–139 | DOI

[12] K. Kontzialis, K. Moditis, M. P. Paidoussis, “Transient simulations of the fluid-structure interaction response of a partially confined pipe under axial flows in opposite directions”, Journal of Pressure Vessel Technology, Transactions of the ASME, 139:3 (2017), 031303 | DOI | MR

[13] D. B. Giacobbi, C. Semler, M. P. Paidoussis, “Dynamics of pipes conveying fluid of axially varying density”, Journal of Sound and Vibration, 473 (2020), 115202 | DOI

[14] A. R. Abdelbaki, M. P. Paidoussis, A. K. Misra, “A nonlinear model for a hanging cantilevered pipe discharging fluid with a partially-confined external flow”, International Journal of Non-Linear Mechanics, 118 (2020), 103290 | DOI

[15] P. A. Velmisov, A. A. Vasil'eva, E. P. Semenova, “Matematicheskoye modelirovaniye dinamiki uprugikh elementov pri aerogidrodinamicheskom vozdeystvii”, Mathematical modeling of physical, economic, technical, social systems and processes: Tr. 7 Mezhdunar. conf., Ulyanovsk, February 2-5, 2009, 68–70 (In Russ.)

[16] P. A. Velmisov, V. K. Manzhosov, Matematicheskoye modelirovaniye v zadachakh dinamiki vibroudarnykh i aerouprugikh sistem, UlGTU Publ., Ulyanovsk, 2014, 204 pp. (In Russ.)

[17] P. A. Velmisov, A. V. Korneev, “Matematicheskoye modelirovaniye v zadache o dinamicheskoy ustoychivosti truboprovoda”, Automation of control processes, 39:1 (2015), 63–73 (In Russ.) | DOI

[18] P. A. Velmisov, A. V. Gladun, “Ob upravlenii dinamikoy truboprovoda”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 18:4 (2016), 89–97 (In Russ.)

[19] A. V. Ankilov, P. A. Velmisov, Funktsionaly Lyapunova v nekotorykh zadachakh aerogidrouprugosti, UlGTU Publ., Ulyanovsk, 2019, 201 pp. (In Russ.)

[20] L. Collatz, Zadachi na sobstvennyye znacheniya, Nauka Publ., Moscow, 1968, 503 pp. (In Russ.)