Nonlinear vibration of plates by the hierarchical finite element and continuation methods

Ribeiro, Pedro; Petyt, M.

doi:10.1016/s0020-7403(98)00076-9

Cited by 79 publications

(64 citation statements)

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“…Lee and Ng [12] studied the nonlinear response of isotropic and composite plates using FEM and a modal reduction method in order to reduce the degrees of freedom. Ribeiro and Petyt [13,14] used FEM and the harmonic balance method to study rectangular plates subjected to harmonic excitations. They examined the influence of inplane and transverse boundary conditions on resonance and compared their results with experimental findings.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution

Babouskos¹,

Katsikadelis²

2010

TOMECHJ

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Abstract:The nonlinear dynamic response of thin plates made of linear viscoelastic material of fractional derivative type is investigated. The resulting governing equations are three coupled nonlinear fractional partial differential equations in terms of displacements. The solution is achieved using the AEM. According to this method the original equations are converted into three uncoupled linear equations, namely a biharmonic (linear thin plate) equation for the transverse deflection and two Poisson's (linear membrane) equations for the inplane deformation under time dependent fictitious loads. The resulting thus initial value problems for the fictitious loads is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term fractional differential equations. Several plates subjected to various loads and boundary conditions are analyzed and the influence of the viscoelastic character of the material is investigated. Without excluding other viscoelastic models, the viscoelastic material employed herein is described by the generalized Voigt model of fractional order derivative. The numerical results demonstrate the efficiency and validate the accuracy of the solution procedure. Emphasis is given to the resonance response of viscoelastic plates under harmonic excitation, where complicated phenomena, similar to those of Duffing equation occur.

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Section: Introductionmentioning

confidence: 99%

Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution

Babouskos¹,

Katsikadelis²

2010

TOMECHJ

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“…One approach of model reduction methods for structures is to first develop a full finite element model, then use a component mode synthesis approach to significantly reduce the number of degrees of freedom while incorporating the essential physics of the system including the nonlinearities [38,39,40,41]. Other approaches include the use of a Galerkin approximation based on the results of a finite element analysis [42] or a hierarchical finite element method in which the order of the approximating polynomial is increased while the mesh size is held constant, which allows for meshes with as little as one element for a plate [43,44]. These methods, however, can not be applied directly to nonlinear problems.…”

Section: Introductionmentioning

confidence: 99%

“…A number of approximate methods exist, however, that can accurately model von Karman plates and shells [46]. Approximate methods include finite elements, both spline finite strip methods [47] and hierarchical finite element methods [43,44], incremental harmonic balance [48], nonlinear normal modes [49], and assumed displacement fields with Taylor series expansions in the normal directions [50].…”

Section: Introductionmentioning

confidence: 99%

Reduced order modeling of fluid/structure interaction.

Barone¹,

Kalashnikova²,

Segalman³

et al. 2009

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This report describes work performed from October 2007 through September 2009 under the Sandia Laboratory Directed Research and Development project titled "Reduced Order Modeling of Fluid/Structure Interaction." This project addresses fundamental aspects of techniques for construction of predictive Reduced Order Models (ROMs). A ROM is defined as a model, derived from a sequence of high-fidelity simulations, that preserves the essential physics and predictive capability of the original simulations but at a much lower computational cost. Techniques are developed for construction of provably stable linear Galerkin projection ROMs for compressible fluid flow, including a method for enforcing boundary conditions that preserves numerical stability. A convergence proof and error estimates are given for this class of ROM, and the method is demonstrated on a series of model problems. A reduced order method, based on the method of quadratic components, for solving the von Karman nonlinear plate equations is developed and tested. This method is applied to the problem of nonlinear limit cycle oscillations encountered when the plate interacts with an adjacent supersonic flow. A stability-preserving method for coupling the linear fluid ROM with the structural dynamics model for the elastic plate is constructed and tested. Methods for constructing efficient ROMs for nonlinear fluid equations are developed and tested on a one-dimensional convectiondiffusion-reaction equation. These methods are combined with a symmetrization approach to construct a ROM technique for application to the compressible Navier-Stokes equations.

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“…A third-order shear deformation theory is used to model the displacements of the plate. Displacements and rotation of a general point in the mid-plane are discretised using a p-version finite element [16,17]. The plate is in the presence of a supersonic flow.…”

Section: Introductionmentioning

confidence: 99%

Flutter Analysis of Composite Laminates With Curvilinear Fibres

Akhavan¹,

Ribeiro²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. In this investigation, the authors intend to study the dynamic instability (flutter) of variable stiffness composite laminates (VSCLs) in the presence of supersonic flow. In the type of VSCL considered here, plies have curvilinear fibres and, consequently, the stiffness is variable in a macroscopic view. The plates considered are rectangular. In each ply, a reference fibre path, represented by a function of horizontal coordinate x, is defined. The reference fibre path orientation changes linearly from T 0 at the centre to T 1 at both vertical edges of the ply; the other fibre paths are defined shifting the reference path in direction y. The displacement and rotations are defined using a Third-order Shear Deformation Theory, and then they are discretised by a p-version finite element model that applies to VSCL plates. Piston theory is employed to model the aerodynamic force of the upstream flow in direction x. Flutter airspeeds are investigated for VSCL plates with different fibre angles. Changes of flutter speed are evaluated in two different regimes of steady and unsteady flow.

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Nonlinear vibration of plates by the hierarchical finite element and continuation methods

Cited by 79 publications

References 0 publications

Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution

Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution

Reduced order modeling of fluid/structure interaction.

Flutter Analysis of Composite Laminates With Curvilinear Fibres

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