Numerical simulation of the industrial membrane nonlinear deformation

Article

Published: 27.01.2025

Authors: Podkopaev S.A., Gavrushin S.S.

Published in issue: #1(157)/2025

DOI: 10.18698/2308-6033-2025-1-2413

Category: Mechanics | Chapter: Theoretical Mchanics, Machine Dynamics

The paper indicates theoretical foundations of nonlinear deformation in the thin axisymmetric shells. It considers the membranes operational characteristics in various switching devices, valves and pressure sensors. The paper analyzes and summarizes the existing approaches to selecting the rational computation models of the industrial membranes. It presents the developed algorithm for nonlinear analysis and synthesis of the actuator designs based on the selected computation model. Types of the membrane supercritical nonlinear behavior, as well as a mathematical model to describe the processes of the axisymmetric shells nonlinear deformation, parameter discrete continuation and the technique of “changing the control parameters subspace” are considered. The supercritical behavior is studied using the example of a hinged spherical shell. A rational mathematical model is selected to describe nonlinear deformation of the flapping symmetric shells. The developed technique is implemented to solve practical problems in creating new and improving the existing design of the industrial membranes. A numerical algorithm is developed and implemented to study the processes of nonlinear deformation of a multiparameter system in the form of the author's program.

EDN CBEHNT

References
[1] Alfutov N.A. Osnovy rascheta na ustoychivost uprugikh sistem [Fundamentals of calculating the elastic systems stability]. Moscow, Mashinostroenie Publ., 1978, 312 p.
[2] Andreeva L.E. Uprugie elementy priborov [Elastic elements of devices]. Moscow, Mashinostroenie Publ., 1982, 456 p.
[3] Biderman V.L. Mekhanika tonkostennykh konstruktsiy. Statika [Mechanics of the thin-walled structures. Statics]. Moscow, Mashinostroenie Publ., 1977, 488 p.
[4] Valishvili N.V. Metody rascheta obolochek vrashcheniya na ETsVM [Methods of calculating rotation shells on a ECM]. Moscow, Mashinostroenie Publ., 1976, 278 p.
[5] Volmir A.S. Ustoychivost deformiruemykh sistem [Stability of the deformable systems]. Moscow, Fizmatgiz Publ., 1967, 984 p.
[6] Gavryushin S.S. Razrabotka metodov rascheta i proektirovaniya uprugikh obolochechenykh konstruktsiy pribornykh ustroystv: Dis. … d-ra tekhn. nauk [Development of methods for calculating and designing elastic shell structures of the instrument devices. Diss. … Dr. Sc. (Eng.)]. Moscow, 1994, 316 p.
[7] Gavryushin S.S., Baryshnikova O.O., Boriskin O.F. Chislennye metody v dinamike i prochnosti mashin [Numerical methods in dynamics and strength of the machines]. Moscow, BMSTU Publ., 2012, 492 p.
[8] Grigolyuk E.I., Lopanitsyn E.A. Konechnye progiby, ustoychivost i zakriticheskoe povedenie tonkikh pologikh obolochek [Finite deflections, stability and supercritical behavior of the thin flat shells]. Moscow, MSTU MAMI Publ., 2004, 162 p.
[9] Podkopaev S.A. Chislennoe modelirovanie zakriticheskogo nelineynogo deformirovaniya osesimmetricheskikh memmbran [Numerical simulation the post-buckling nonlinear deformation of axisymmetric membranes]. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2020, no. 1, pp. 64–87.
[10] Podkopaev S.A., Gavryushin S.S. Metodika issledovaniya zakriticheskogo povedeniya osesimmetrichnykh membran, ispol’zuyemykh v promyshlennom internete veshchey [Methodology for studying the supercritical behavior of axisymmetric membranes used in the industrial Internet of things]. Mezhvuzovskiy sbornik nauchnykh trudov: Matematicheskoe modelirovanie i eksperimentalnaya mekhanika deformiruemogo tverdogo tela [Interuniversity collection of scientific papers: Mathematical simulation and experimental mechanics of a deformable solid]. Issue 3. Tver, Tverskoy Gosudarstvennyi Tekhnicheskiy Universitet Publ., 2020, 108 p. https://doi.org/10.elibrary.ru/item.asp?id=29068499
[11] Podkopaev S.A. Gavryushin S.S., Nikolaeva A.S. Analiz protsessa nelineynogo deformirovaniya gofrirovannykh membran [Analysis of the process of nonlinear deformation of the corrugated membranes]. Mezhvuzovskiy sbornik nauchnykh trudov: Matematicheskoe modelirovanie i eksperimentalnaya mekhanika deformi-ruemogo tverdogo tela [Interuniversity collection of scientific papers: Mathematical simulation and experimental mechanics of a deformable solid]. Issue I. Tver, Tverskoy Gosudarstvennyi Tekhnicheskiy Universitet Publ., 2017, 152 p.
[12] Podkopaev S.A. Gavryushin S.S., Nikolaeva A.S., Podkopaeva T.B. Raschet rabochey kharakteristiki perspektivnykh konstruktsiy mikroaktyuatorov [Computation of performance characteristics of the promising microactuator designs]. Mezhvuzovskiy sbornik nauchnykh trudov: Matematicheskoe modelirovanie i eksperimentalnaya mekhanika deformiruemogo tverdogo tela [Interuniversity collection of scientific papers: Mathematical simulation and experimental mechanics of a deformable solid]. Issue I. Tver, Tverskoy Gosudarstvennyi Tekhnicheskiy Universitet Publ., 2017, 162 p.
[13] Reissner E. On axisymmetric deformations of thin shells of revolution. In: Proc. of Symp. In Appl. Math., Amer. Math. Soc., 1950, vol. 3, pp. 27–52.
[14] Podkopaev S.A., Gavryushin S.S., Podkopaeva T.B. Methods for studying the post-buckling behavior of axisymmetric membrane. In: The First International Symposium on Computer Science, Digital Economy and Intelligent Systems (CSDE-IS2019), October 4–6, 2019, Moscow, Russia.
[15] Feodosiev V.I. K raschetu khlopayushchey membrany [On calculation of a flapping membrane]. Prikladnaya matematika i mekhanika — Applied Mathematics and Mechanics, 1946, vol. 10, no. 2, pp. 295–306.
[16] Belhocine A. Exact analytical solution of boundary value problem in a form of an infinite hypergeometric series. International Journal of Mathematical Sciences and Computer Engineering (IJMSC), 2017, vol. 3, no. 1, pp. 28–37, https://doi.org/10.5815/ijmsc.2017.01.03
[17] Crisfield M.A. A fast incremental/iterative solution procedure that handles “snapthrought”. Calculations and Structures, 1981, vol. 13, no. 1, pp. 55–62.
[18] Podkopaev S.A. Chislennoe modelirovanie zakriticheskogo nelineynogo deformirovaniya osesimmetricheskikh memmbran [Numerical simulation the post-buckling nonlinear deformation of axisymmetric membranes]. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2020, no. 1, pp. 64–87.
[19] Feodosiev V.I. Uprugie elementy tochnogo priborostroeniya [Elastic elements of precision instrumentation]. Moscow, Oborongiz Publ., 1949.
[20] Ponomarev S.D., Biderman V.L., Likharev K.K. et al. Raschety na prochnost v mashinostroenii [Strength calculations in the mechanical engineering]. In 3 vols. Vol. 2. Moscow, Mashgiz Publ., 1958.
[21] Mescall J. Numerical solution of nonlinear equations for the rotation shell. AIAA J., 1966, vol. 4, no. 11, pp. 2041–2043.
[22] Gupta N.K. Experimental and numerical studies of dynamic axial compression of thin-walled spherical shells. International Journal of Impact Engineering, 2004, vol. 30, pp. 1225–1240.
[23] Riks E. The application of Newton’s method to the problem of elastic stability. J. Appl. Mech., 1972, vol. 39, pp. 1060–1065.