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Prototype Problems: Bifurcations of Different Kinds

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Stability of Axially Moving Materials

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 259))

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Abstract

In this chapter, we present some prototype bifurcation problems that arise in the mechanics of rigid and deformable structural elements. These problems are typical for engineering applications and characterize the approaches that can be applied in the investigation of stability. Some methods of bifurcation theory will be presented in the context of stability studies of the considered one-dimensional mechanical problems.

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References

  1. Banichuk NV (1978) Minimization of the weight of a wing with constraints on the velocity of divergence. Sci Notes Central Aerohydrodyn Inst (TsAGI) 9(5). in Russian

    Google Scholar 

  2. Clausen T (1851) About the shape of architectural columns. Bull Cl Phys-Math Acad Imp Sci St-Pétersbg 9:279–294

    Google Scholar 

  3. Lurie AI (1965) Application of the maximum principle to some simplest problems of mechanics. Res Works Leningr Polytechn Inst 252:34–46

    Google Scholar 

  4. Nikolai EL (1907) Lagrange’s problem on the best shape column. Notes of the St. Petersburg Polytechnical Institute 8

    Google Scholar 

  5. Chentsov NG (1936) Column of least weight. Res Articles, TsA GI 265:1–48

    Google Scholar 

  6. Keller JB (1960) The shape of the strongest column. Arch Rat Mech Anal 5(1):275–285

    Article  MathSciNet  Google Scholar 

  7. Banichuk N, Barsuk A (192b) On stability of torsioned elastic rods. Izv Akad Nauk S.S.S.R., M.T.T. 6:148–154

    Google Scholar 

  8. McIntosh SC, Easter FE (1968) Design of minimum-mass structures with specificial stiffness properties. AIAA J 016(7):962–964

    Google Scholar 

  9. Ashley H, McIntosh SC (1968) Applications of aeroelastic constraints on structural optimization. In: Proceedings of the 12th international congress of theoretical and applied mechanics, pp 100–113, Berlin, 1969. Springer. Stanford

    Chapter  Google Scholar 

  10. Jelicic ZD, Atanackovic TM (2006) On an optimization problem for elastic rods. Struct Multidiscip Optim 32(1):59–64

    Article  MathSciNet  Google Scholar 

  11. Banichuk NV, Makeev EV, Barsuk AA (2011g) Optimization of tension rods by the criterion of stability. Proc Nijegorod Univ 2(1):115–122 in Russian

    Google Scholar 

  12. Feodosjev VI (1965) On one problem of stability. Appl Math Mech 29(2). in Russian

    Google Scholar 

  13. Banichuk NV, Gura NM (1979b) On a dynamic problem of optimal design. Mech Deform Solid Bodies 41:20–24

    MathSciNet  Google Scholar 

  14. Bolotin VV (1961) Nonconservative problems of the theory of elastic stability. Fizmatgiz, Moscow. 339 pages, in Russian

    Google Scholar 

  15. Prasad SN, Hermann G (1969) The usefulness of adjoint systems in solving nonconservative problems of elastic continua. Int J Solids Struct 5

    Google Scholar 

  16. Plaut RH (1972) Determining the nature of instability in nonconservative problems. AIAA J 10(2)

    Article  Google Scholar 

  17. Banichuk NV, Mironov AA (1975) Optimization of vibration frequencies of an elastic plate in an ideal fluid. J Appl Math Mech 39(5):853–863. ISSN 0021-8928. https://doi.org/10.1016/0021-8928(75)90126-4. http://www.sciencedirect.com/science/article/pii/0021892875901264

    Article  MathSciNet  Google Scholar 

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Correspondence to Nikolay Banichuk .

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Banichuk, N., Barsuk, A., Jeronen, J., Tuovinen, T., Neittaanmäki, P. (2020). Prototype Problems: Bifurcations of Different Kinds. In: Stability of Axially Moving Materials. Solid Mechanics and Its Applications, vol 259. Springer, Cham. https://doi.org/10.1007/978-3-030-23803-2_1

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  • DOI: https://doi.org/10.1007/978-3-030-23803-2_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23802-5

  • Online ISBN: 978-3-030-23803-2

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