Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method

Amabili, Marco; Sarkar, Abhijit; Paı̈doussis, M.P.

doi:10.1016/j.jfluidstructs.2003.06.002

Cited by 100 publications

(73 citation statements)

References 22 publications

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“…The first interpretation regards the POD as the Karhunen-Loeve decomposition (KLD), and the second one considers that the POD consists of three methods: the KLD, the principal component analysis (PCA), and the singular value decomposition (SVD). In recent years, there have been many reported applications of the POD methods in engineering fields such as in studies of turbulence [9][10][11][12][13], vibration analysis [14][15][16][17][18], process identification [19][20][21][22], and control in chemical engineering [23][24][25][26][27][28][29][30][31][32]. In general, POD is a methodology that first identifies the most energetic modes in a time-dependent system and subsequently provides a means of obtaining a low-dimensional description of the system's dynamics where the low-dimensional system is obtained directly from the Galerkin projection of the governing equations on the empirical basis set (the POD modes).…”

Section: Proper Orthogonal Decomposition (Pod) and Galerkin Projectionmentioning

confidence: 99%

Model Reduction Using Proper Orthogonal Decomposition and Predictive Control of Distributed Reactor System

Márquez

Jos^eacute

Oviedo

et al. 2013

Journal of Control Science and Engineering

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This paper studies the application of proper orthogonal decomposition (POD) to reduce the order of distributed reactor models with axial and radial diffusion and the implementation of model predictive control (MPC) based on discrete-time linear time invariant (LTI) reduced-order models. In this paper, the control objective is to keep the operation of the reactor at a desired operating condition in spite of the disturbances in the feed flow. This operating condition is determined by means of an optimization algorithm that provides the optimal temperature and concentration profiles for the system. Around these optimal profiles, the nonlinear partial differential equations (PDEs), that model the reactor are linearized, and afterwards the linear PDEs are discretized in space giving as a result a high-order linear model. POD and Galerkin projection are used to derive the low-order linear model that captures the dominant dynamics of the PDEs, which are subsequently used for controller design. An MPC formulation is constructed on the basis of the low-order linear model. The proposed approach is tested through simulation, and it is shown that the results are good with regard to keep the operation of the reactor.

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Section: Proper Orthogonal Decomposition (Pod) and Galerkin Projectionmentioning

confidence: 99%

Model Reduction Using Proper Orthogonal Decomposition and Predictive Control of Distributed Reactor System

Márquez

Jos^eacute

Oviedo

et al. 2013

Journal of Control Science and Engineering

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“…Based on the convergence study conducted by Gonçalves et al [17] and by increasing the number of terms in (11), as will be shown in the following sections, convergence can be attained up to very large deflections (around two times the shell thickness) if all modes up to the 4th order are retained in (11), leading to the following modal expansion with eleven modes:…”

Section: Derivation Of the Reduced Order Modelmentioning

confidence: 99%

“…Rega and Troger [10], in an article that introduces a special issue of Non-linear Dynamics on reducedorder models, have analyzed the most common methods of dimension reduction in non-linear dynamics with emphasis on applications in mechanics. The POD method was used by Amabili et al [11,12] to study the non-linear vibrations of cylindrical shells. Amabili and Touzé [13,14] compared the efficiency of the proper orthogonal decomposition and the non-linear normal modes method to build reduced order models of a water-filled cylindrical shell.…”

Section: Introductionmentioning

confidence: 99%

An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells

2011

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Using Donnell non-linear shallow shell equations in terms of the displacements and the potential flow theory, this work presents a qualitatively accurate low dimensional model to study the nonlinear dynamic behavior and stability of a fluid-filled cylindrical shell under lateral pressure and axial loading. First, the reduced order model is derived taking into account the influence of the driven and companion modes. For this, a modal solution is obtained by a perturbation technique which satisfies exactly the in-plane equilibrium equations and all boundary, continuity, and symmetry conditions. Finally, the equation of motion in the transversal direction is discretized by the Galerkin method. The importance of each mode in the proposed modal expansion is studied using the proper orthogonal decomposition. The quality of the proposed model is corroborated by studying the convergence of frequency-amplitude relations, resonance curves, bifurcation diagrams, and time responses. The parametric analysis clarifies the influence of the lateral and axial loads on the non-linear vibrations and stability of the liquid-filled shell. Finally, the global response of the system is investigated in order to quantify the degree of safety of the shell in the presence of external perturbations through the use of bifurcation diagrams and basins of attraction. This allows one to evaluate the safety and dynamic integrity of the cylindrical shell in a dynamic environment.

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“…The use of modes for model reduction and parameter identification has been widely done in other contexts (e.g. [43][44][45][46][47][48][49]). …”

Section: Modal Reduction For Larger Systemsmentioning

confidence: 99%

Estimating Damping Parameters in Multi-Degree-of-Freedom Vibration Systems by Balancing Energy

Feeny

2007

Volume 1: 21st Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C

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A method of estimating damping parameters for multidegree-of-freedom vibration systems is outlined, involving a balance of dissipated and supplied energies over a cycle of periodic vibration. The power is formulated as the inner product between velocity and force terms, and integrated over a cycle. Conservative terms (mass and stiffness) drop out of the formulation. The displacement response and the input are measured, and the damping coefficients are estimated without knowledge of the mass and stiffness, which can be nonlinear, as illustrated in one example. The identification equations are also obtained with a modal reduction based on proper orthogonal decomposition. The method can be applied with a harmonic motion assumption, or by simple numerical integration. The method is illustrated with linear and quadratic damping, in simulations of a four degree-of-freedom system, and in a string with and without noise.

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Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method

Cited by 100 publications

References 22 publications

Model Reduction Using Proper Orthogonal Decomposition and Predictive Control of Distributed Reactor System

Model Reduction Using Proper Orthogonal Decomposition and Predictive Control of Distributed Reactor System

An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells

Estimating Damping Parameters in Multi-Degree-of-Freedom Vibration Systems by Balancing Energy

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