Methods for dimension reduction and their application in nonlinear dynamics

Steindl, Alois; Tröger, H.

doi:10.1016/s0020-7683(00)00157-8

Cited by 102 publications

(95 citation statements)

References 20 publications

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“…Many different methods have been proposed in the past, one of the most popular being the Proper Orthogonal Decomposition (POD) method [3][4][5]29]. In this paper, the NNMs of the unforced structure are tested as basis functions for reducing the dynamics for shell models with harmonic forcing.…”

Section: Discussionmentioning

confidence: 99%

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Touzé

Amabili

Thomas

2008

Computer Methods in Applied Mechanics and Engineering

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. Reduced-order models for large-amplitude vibrations of shells including in-plane inertia. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 197 (21-24) AbstractNon-linear normal modes (NNMs) are used in order to derive reduced-order models for large amplitude, geometrically non-linear vibrations of thin shells. The main objective of the paper is to compare the accuracy of different truncations, using linear and non-linear modes, in order to predict the response of shells structures subjected to harmonic excitation. For an exhaustive comparison, three different shell problems have been selected: (i) a doubly curved shallow shell, simply supported on a rectangular base; (ii) a circular cylindrical panel with simply supported, in-plane free edges; and (iii) a simply supported, closed circular cylindrical shell. In each case, the models are derived by using refined shell theories for expressing the strain-displacement relationship. As a consequence, in-plane inertia is retained in the formulation. Reduction to one or two NNMs shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases, and this limitation is extended to two times the thickness for two of the selected models.

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Section: Discussionmentioning

confidence: 99%

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Touzé

Amabili

Thomas

2008

Computer Methods in Applied Mechanics and Engineering

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“…As a result, improving the reduced dynamics requires adding more modes/basis vectors (and hence dimensions) to the reduced space, even though there may be no geometric reason to do so. Methods that attempt to describe the nonlinear inverse are called nonlinear Galerkin methods, or approximate inertial manifolds [32,43].…”

Section: Galerkin Projectionmentioning

confidence: 99%

Dimensionality reduction of dynamical systems with parameters

Welshman¹,

Brooke²

2013

AIP Conference Proceedings

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“…1 It turns out that while C is self-adjoint, K is not self-adjoint because of the presence of the follower force µ. The adjoint (or left) eigenvalue problem therefore reads:…”

Section: Right and Left Eigenvalue Problemsmentioning

confidence: 99%

“…The three displacements, however, are not independent because of the internal constraints: sin ϑ = u ; ε := (1 + w ) 2 + u 2 − 1 = 0 (1) expressing shear-undeformability and inextensibility, respectively. The curvature κ(s) is assumed as the (unique) strain measure; from Equation (1 1 ) it follows that…”

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confidence: 99%