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Mat. Zametki, 2008, Volume 83, Issue 1, Pages 50–60 (Mi mz4335)  

This article is cited in 5 scientific papers (total in 5 papers)

Application of Maslov's Formula to Finding an Asymptotic Solution to an Elastic Deformation Problem

D. V. Kostin

Voronezh State University

Abstract: We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii–Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator $\mathscr A$ from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to $\mathscr A$. It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of $\mathscr A$ (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the form of its deflection is studied.

Keywords: elastic deformation, inhomogeneous elastic beam, bifurcation analysis, Maslov's projection-operator formula, energy functional, bifurcation diagram, orthogonal projection

DOI: https://doi.org/10.4213/mzm4335

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English version:
Mathematical Notes, 2008, 83:1, 48–56

Bibliographic databases:

UDC: 517.9
Received: 18.01.2007

Citation: D. V. Kostin, “Application of Maslov's Formula to Finding an Asymptotic Solution to an Elastic Deformation Problem”, Mat. Zametki, 83:1 (2008), 50–60; Math. Notes, 83:1 (2008), 48–56

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Malyugina M.A., “Lokalnye secheniya diskriminantnogo mnozhestva slabo neodnorodnoi uprugoi plastiny v usloviyakh narusheniya potentsialnosti”, Vestn. Voronezhskogo gos. un-ta. Ser. Fizika. Matematika, 2011, no. 2, 115–121  zmath  elib
    2. Dzhasim M.D., Karpova A.P., Kostin D.V., “Vetvlenie i optimizatsiya tsiklov pri nalichii kratnykh rezonansov”, Vestn. Voronezhskogo gos. un-ta. Ser. Fizika. Matematika, 2012, no. 1, 99–99  elib
    3. Kostin D.V., “K voprosu optimizatsii zakriticheskogo izgiba uprugoi lopatki turbiny”, Nasosy. turbiny. sistemy, 2012, no. 3, 67–72  elib
    4. Kostin D.V., “Initial Boundary Value Problems For Fuss-Winkler-Zimmermann and Swift-Hohenberg Nonlinear Equations of 4Th Order”, Mat. Vestn., 70:1 (2018), 26–39  mathscinet  isi
    5. Goy T., Negrych M., Savka I., “On Nonlocal Boundary Value Problem For the Equation of Motion of a Homogeneous Elastic Beam With Pinned-Pinned Ends”, Carpathian Math. Publ., 10:1 (2018), 105–113  crossref  mathscinet  zmath  isi
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