On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

Givois, Arthur; Grolet, Aurélien; Thomas, Olivier; Deü, Jean-François

doi:10.1007/s11071-019-05021-6

Cited by 74 publications

(126 citation statements)

References 64 publications

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“…Geometrically nonlinear effects appear generally in thin structures such as beams, plates and shells, when the amplitude of the vibration is of the order of the thickness [26,37]. The von Kármán family of models for beams, plates and shells allows one to derive explicit partial differential equations (PDE) [1,8,18,37,39], showing clearly that a coupling between bending and longitudinal motions causes a non-linear restoring force of polynomial type in the equations of motion. This geometric nonlinearity is then at the root of complex behaviours, that also need dedicated computational strategies in order to derive quantitative predictions.…”

Section: Introductionmentioning

confidence: 99%

“…In its first version as described in [23], it allows computation of the nonlinear coupling coefficients of the discretized problem in the modal basis from a series of static computations with prescribed modal displacements. It has then been used in a number of contexts [8,19,21,22,28], and is generally connected to the modal basis. However one has to understand that per se, STEP is just an evaluation technique, a computational nonintrusive method, that can be used with other inputs than those from the modal basis.…”

Section: Introductionmentioning

confidence: 99%

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Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements

et al. 2020

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Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the modal expansion is observed when applying the method with 3D elements, because of nonlinear couplings occurring with very high frequency modes involving 3D thickness deformations. Focusing on the case of flat structures, we first show by computing all the modes of the structure that a converged solution can be exhibited by using either static condensation or normal form theory. We then show that static modal derivatives provide the same solution with fewer calculations. Finally, we propose a modified STEP, where the prescribed displacements are imposed solely on specific degrees of freedom of the structure, and show that this adjustment also provides efficiently a converged solution.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements

et al. 2020

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“…In this paper we propose a method that utilizes only the information from the state matrix to obtain simultaneously the coefficients of the higher order terms in (7). Similar idea was introduced in [13] and used in [14] for studying geometric nonlinearities of second order mechanical systems with the help of finite element method. To the author's best knowledge, this is the first time the method is being formulated in order to study power system in first order.…”

Section: The Proposed Methodsmentioning

confidence: 99%

“…The essence of the negative part is to create two equations in order to solve simultaneously. By prescribing as above and solving for linear and nonlinear static solutions, P NL is obtained from (14). Then from (6) and (12) we can write set of linear equations as…”

Section: The Proposed Methodsmentioning

confidence: 99%

A Novel Method for Accelerating the Analysis of Nonlinear Behaviour of Power Grids using Normal Form Technique

Ugwuanyi

Kestelyn

Thomas

et al. 2019

2019 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe)

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Today's power systems are strongly nonlinear and are becoming more complex with the large penetration of power-electronics interfaced generators. Conventional Linear Modal Analysis does not adequately study such a system with complex nonlinear behavior. Inclusion of higher-order terms in small-signal (modal) analysis associated with the Normal Form theory proposes a nonlinear modal analysis as an efficient way to improve the analysis. However, heavy computations involved make Normal Form method tedious, and impracticable for large power system application. In this paper, we present an efficient and speedy approach for obtaining the required nonlinear coefficients of the nonlinear equations modelling of a power system, without actually going through all the usual high order differentiation involved in Taylor series expansion. The method uses eigenvectors to excite the system modes independently which lead to formulation of linear equations whose solution gives the needed coefficients. The proposed method is demonstrated on the conventional IEEE 9-bus system and 68-bus New England/New York system.

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“…This method is simple and is based on prescribing the linear eigenvectors as unknown field in the initial nonlinear system, which leads to solving linear-only equations to obtain the coefficients of the nonlinear modal model. It was introduced in the mechanical engineering field in [38] and widely applied since, to compute the coefficients of nonlinear modal reduced order models of mechanical structures discretized by a finiteelements method [39], [40]. It is particularly attractive since a standard commercial finite-elements code can be used in a so called non-intrusive way.…”

Section: Introductionmentioning

confidence: 99%

A New Fast Track to Nonlinear Modal Analysis of Power System Using Normal Form

Ugwuanyi

Kestelyn

Thomas

et al. 2020

IEEE Trans. Power Syst.

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The inclusion of higher-order terms in small-signal (modal) analysis augments the information provided by linear analysis and enables better dynamic characteristic studies on the power system. This can be done by applying Normal Form theory to simplify the higher order terms. However, it requires the preliminary expansion of the nonlinear system on the normal mode basis, which is impracticable with standard methods when considering large scale systems. In this paper, we present an efficient numerical method for accelerating those computations, by avoiding the usual Taylor expansion. Our computations are based on prescribing the linear eigenvectors as unknown field in the initial nonlinear system, which leads to solving linear-only equations to obtain the coefficients of the nonlinear modal model. In this way, actual Taylor expansion and associated higher order Hessian matrices are avoided, making the computation of the nonlinear model up to third order and nonlinear modal analysis fast and achievable in a convenient computational time. The proposed method is demonstrated on a single-machine-infinitebus (SMIB) system and applied to IEEE 3-Machine, IEEE 16-Machine and IEEE 50-Machine systems.

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On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

Cited by 74 publications

References 64 publications

Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements

Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements

A Novel Method for Accelerating the Analysis of Nonlinear Behaviour of Power Grids using Normal Form Technique

A New Fast Track to Nonlinear Modal Analysis of Power System Using Normal Form

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