Boundary Conditions and Mode Jumping in the von Kármán Equations

Holder, E.J.; Schaeffer, David G.

doi:10.1137/0515034

Cited by 30 publications

(15 citation statements)

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“…The conditions (6), (4b) and (6), (5) correspond to the clamped sides x = 0, l 1 and simply supported ends y = 0, l 2 . Holder and Schaeffer in [12] considered (6) when the plate is compressed only on the clamped sides i.e., L = ∂ 2 /∂ x 2 . This situation was also investigated numerically by Chien, Gong and Mei in [6], Chien, Chang and Mei in [5] and others.…”

Section: The Nonlinear Mechanical Model and Boundary Conditionsmentioning

confidence: 99%

“…This implies nonstability of the primary solution branches through further bifurcation. This phenomenon was investigated extensively by Holder, Schaeffer and Golubitsky ( [12,15]). Later their theoretical results for the partially clamped plate, subjected to compression along its two sides were verified numerically by Chien, Gong and Mei in [6].…”

Section: Introductionmentioning

confidence: 97%

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The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

Muradova

2007

Adv Comput Math

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In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin's approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton's iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented.

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Section: The Nonlinear Mechanical Model and Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

Muradova

2007

Adv Comput Math

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“…Numerous works have been devoted to the study of bifurcation in the von Kármán problems (see for instance [1], [6], [7], [8], [10], [11], [16], [18] and [21]). However, these investigations do not concern the situation when the plate is fixed to the elastic foundation.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation

Janczewska

2001

Ann. Polon. Math.

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Abstract. We investigate bifurcation in the solution set of the von Kármán equations on a disk Ω ⊂ R 2 with two positive parameters α and β. The equations describe the behaviour of an elastic thin round plate lying on an elastic base under the action of a compressing force. The method of analysis is based on reducing the problem to an operator equation in real Banach spaces with a nonlinear Fredholm map F of index zero (to be defined later) that depends on the parameters α and β. Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if (0, p) is a bifurcation point then dim Ker F x (0, p) > 0. Next, we give a full description of the kernel of the Fréchet derivative of F . We study in detail the situation when the dimension of the kernel is one. We prove that (0, p) is a bifurcation point by the use of the Lyapunov-Schmidt finite-dimensional reduction and the Crandall-Rabinowitz theorem. For a one-dimensional bifurcation point, analysing the Lyapunov-Schmidt branching equation we determine the number of families of solutions, their directions and asymptotic behaviour (shapes).

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“…The dimensionality of the plate is reduced to two by appropriate coupling of the loads with the thickness of the plate, see e.g., Ciarlet [10] and Antman [4, Chapter XIV], Holder/Schaeffer [16] and Schaeffer/Golubitsky [18]. The classical boundary conditions (1.2) are hard to be realized experimentally.…”

Section: Introductionmentioning

confidence: 99%

“…However, they simplify significantly mathematical analysis of the bifurcation scenario of the solutions (cf. Holder/Schaeffer [16], Szilard [20], Stein [19], Schaeffer/Golubitsky [18]). Moreover, in [18] Schaeffer/Golubitsky showed that physically the boundary conditions…”

Section: Introductionmentioning

confidence: 99%

Symmetry and scaling properties of the von Kármán equations

Chien

Kuo

Mei

1998

Z. angew. Math. Phys.

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We study symmetries in the von Kármán equations with simply supported boundary conditions on rectangular domains. By embedding this fourth order plate problem into a space of periodic functions, we obtain hidden symmetries and scaling properties in its solution manifold. These properties are exploited for efficient numerical approximation of the solution branches at the bifurcation points, and are demonstrated with numerical examples.Mathematics Subject Classification (1991). 35B32, 73H05, 65N25, 65N06.

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Boundary Conditions and Mode Jumping in the von Kármán Equations

Cited by 30 publications

References 10 publications

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation

Symmetry and scaling properties of the von Kármán equations

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