Applications of Centre Manifold Theory.

Carr, Jack

doi:10.21236/ada074052

Cited by 405 publications

(290 citation statements)

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“…[15][16][17]. This modern dynamical system foundation empowers us to systematically derive the novel and accurate models (2), (15) and (17) for the lateral momentum of fluids with nonlinear rheology.…”

Section: Resultsmentioning

confidence: 99%

“…Provided we can treat lateral derivatives ∂ x as a modifying influence, that is provided solutions vary slowly enough in x, centre manifold theorems (e.g. [15][16][17]) assure us three vitally important properties:…”

Section: Centre Manifold Theory Supports the Modellingmentioning

confidence: 99%

“…x that are neglected in (2) and (15). With the support of centre manifold theory, researchers may choose an approximate model that suits the parameter regime of their application.…”

Section: Power Law Fluidsmentioning

confidence: 99%

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The inertial dynamics of thin film flow of non-Newtonian fluids

Roberts

2008

Physics Letters A

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Consider the flow of a thin layer of non-Newtonian fluid over a solid surface. I model the case where the viscosity depends nonlinearly on the shear-rate; power law fluids are an important example, but the analysis here is for general nonlinear dependence. The modelling allows for large changes in film thickness provided the changes occur over a relatively large enough lateral length scale. Modifying the surface boundary condition for tangential stress forms an accessible foundation for the analysis where flow with constant shear is a neutral critical mode, in addition to a mode representing conservation of fluid. Perturbatively removing the modification then constructs a model for the coupled dynamics of the fluid depth and the lateral momentum. For example, the results model the dynamics of gravity currents of non-Newtonian fluids when the flow is not creeping.

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Section: Resultsmentioning

confidence: 99%

Section: Centre Manifold Theory Supports the Modellingmentioning

confidence: 99%

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The inertial dynamics of thin film flow of non-Newtonian fluids

Roberts

2008

Physics Letters A

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“…The z and z 3 terms vanish after integration on z in the case of a laminate with symmetric density with respect to the z-axis. In particular, for a laminate with the same density for all the layers and uniform thickness, the following simplified expression is obtained: The stress-strain relations for the kth orthotropic lamina of the shell, in the material principal coordinates (1,2,3), are obtained under the hypothesis…”

Section: First Step: Natural Modesmentioning

confidence: 99%

“…The most diffused ones are probably the nonlinear normal modes (under this name are classified the techniques based on the centre manifold theorem, the normal form theory and the inertial manifold) [1][2][3][4][5][6][7][8][9], including the most diffused version with asymptotic approach, the discretization of the equations of motion by using global (i.e. defined on the whole structure) admissible functions [10][11][12][13], the proper orthogonal decomposition method [14][15][16][17][18] and the natural mode discretization [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

Amabili

2013

Phil. Trans. R. Soc. A.

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Reduced-order models are essential to study nonlinear vibrations of structures and structural components. The natural mode discretization is based on a twostep analysis. In the first step, the natural modes of the structure are obtained. Because this is a linear analysis, the structure can be discretized with a very large number of degrees of freedom. Then, in the second step, a small number of these natural modes are used to discretize the nonlinear vibration problem with a huge reduction in the number of degrees of freedom. This study finds a recipe to select the natural modes that must be retained to study nonlinear vibrations of an angle-ply laminated circular cylindrical shell that the author has previously studied by using admissible functions defined on the whole structure, so that an accuracy analysis is performed. The higher-order shear deformation theory developed by Amabili and Reddy is used to model the shell.

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An I&I‐Based Observer to Solve the Output‐Feedback Synchronization Problem for a Class of Chaotic Systems

Aguilar-Ibáñez

García–Canseco²,

Martínez‐García

et al. 2017

Asian Journal of Control

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A novel robust observer, intended to solve the output-feedback chaos synchronization problem for the Master/Slave Configuration, is proposed here. Assuming that the given Master system belongs to a specific class of feedback-linearized systems, our solution is based on the well-known Immersion and Invariance (I&I) method. The proposed observer is devoted to the asymptotic estimation of the Master system's underlying dynamics, and its effectiveness is illustrated via computer-based simulations that involve both the so-called Duffing's oscillator and the Genesio & Tesi system.

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Applications of Centre Manifold Theory.

Cited by 405 publications

References 0 publications

The inertial dynamics of thin film flow of non-Newtonian fluids

The inertial dynamics of thin film flow of non-Newtonian fluids

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

An I&I‐Based Observer to Solve the Output‐Feedback Synchronization Problem for a Class of Chaotic Systems

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