Some corner effects on the loss of selfadjointness and the non-excitation of vibration for thin plates and shells

Some models for axially moving orthotropic thin plates are investigated analytically via methods of complex analysis to derive estimates for critical plate velocities. Linearised Kirchhoff plate theory is used, and the energy forms of steady-state models are considered with homogeneous and inhomogeneous tension profiles in the cross direction of the plate. With the help of the energy forms, some limits for the divergence velocity of the plate are found analytically. In numerical examples, the derived lower limits for the divergence velocity are analysed for plates with small flexural rigidity.

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“…Bilinear energy forms, e.g. for stationary isotropic plates, have been discussed by Chen et al (1998).…”

Section: Axially Moving Orthotropic Platesmentioning

confidence: 99%

Estimates for Divergence Velocities of Axially Moving Orthotropic Thin Plates

Saksa

Jeronen

2014

Mechanics Based Design of Structures and Machines

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“…To show that the eigenvalues k of L 1 ðf Þ are non-negative, we proceed using general ideas from Chen et al (1998), who proved a similar result for an isotropic stationary plate. We introduce the bilinear form a(f, g) that corresponds to the strain energy of a plate (see Timoshenko and Woinowsky-Krieger, 1959, p. 377) …”

Section: Static Analysis Of Stability Loss Of a Platementioning

confidence: 99%

On the limit velocity and buckling phenomena of axially moving orthotropic membranes and plates

Баничук¹,

Jeronen²,

Kurki³

et al. 2011

International Journal of Solids and Structures

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a b s t r a c tIn this paper, we consider the static stability problems of axially moving orthotropic membranes and plates. The study is motivated by paper production processes, as paper has a fiber structure which can be described as orthotropic on the macroscopic level. The moving web is modeled as an axially moving orthotropic plate. The original dynamic plate problem is reduced to a two-dimensional spectral problem for static stability analysis, and solved using analytical techniques. As a result, the minimal eigenvalue and the corresponding buckling mode are found. It is observed that the buckling mode has a shape localized in the regions close to the free boundaries. The localization effect is demonstrated with the help of numerical examples. It is seen that the in-plane shear modulus affects the strength of this phenomenon. The behavior of the solution is investigated analytically. It is shown that the eigenvalues of the cross-sectional spectral problem are nonnegative. The analytical approach allows for a fast solver, which can then be used for applications such as statistical uncertainty and sensitivity analysis, real-time parameter space exploration, and finding optimal values for design parameters.

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“…In assuming regularity properties of solutions, we have taken account of [26] and the remarks in [16, pp. 6, 34-36], while cognizance is taken of the more recent work of Chen et al [4], which contains examples of plate problems in rectangular domains, showing that the "extraneous" solutions due to corner effects need not be singular, provided corner effects are properly treated [4, pp. 216, 238].…”

Section: Mathematical Setting For Pr Pmentioning

confidence: 99%

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

Dalsen

2002

Journal of Mathematical Analysis and Applications

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In this paper we consider a two-dimensional hybrid thermo-elastic structure consisting of a thermo-elastic plate which has a beam attached to its free end. We show that the initial-boundary-value problem for the interactive system of partial differential equations which take account of the mechanical strains/stresses and the thermal stresses in the plate and the beam, can be associated with a uniformly bounded evolution operator. It turns out that the interplay of parabolic dynamics due to the thermal effects in the hybrid structure and the hyperbolic dynamics associated with the elasticity of the structure yields analyticity for the entire system. This result yields solvability for the problem under optimal initial freedom of the displacement, velocity, and temperature in the plate and the beam, while uniform stability is readily available.  2002 Elsevier Science

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Some corner effects on the loss of selfadjointness and the non-excitation of vibration for thin plates and shells

Cited by 5 publications

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Estimates for Divergence Velocities of Axially Moving Orthotropic Thin Plates

Estimates for Divergence Velocities of Axially Moving Orthotropic Thin Plates

On the limit velocity and buckling phenomena of axially moving orthotropic membranes and plates

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

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