Nonlinear reduced-order modelling for limit-cycle oscillation analysis

Gai, G.; Timme, Sebastian

doi:10.1007/s11071-015-2544-9

Cited by 11 publications

(4 citation statements)

References 29 publications

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“…Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types arising in engineering applications. Furthermore, many non-linear aeroelastic analysis studies adopt a state-space representation for the unsteady aerodynamics, which is possible using reduced order models, as in [37], for example. The method presented here would be anticipated to be applicable in these cases also.…”

Section: Discussionmentioning

confidence: 99%

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Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Lewis

2019

International Journal of Non-Linear Mechanics

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This paper presents analytical higher order approximations to limit cycles of an autonomous multidegree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill's Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs. The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation. Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.

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Section: Discussionmentioning

confidence: 99%

“…The determinantal equation (37) will have infinitely many solutions of the form Ž ± . In principle, Equation (37) could be solved approximately by taking a finite number of rows and columns in the determinant.…”

Section: Stability Analysismentioning

confidence: 99%

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Lewis

2019

International Journal of Non-Linear Mechanics

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“…The fourth step is to reduce Eq. (8) to its first-order center manifold [32,33,36,37]. For a system encompassing only third order nonlinearities, this corresponds to neglect variables not related to the bifurcation, i.e.…”

Section: Criticality Analysismentioning

confidence: 99%

Flutter Control of a Two-Degrees-of-Freedom Airfoil Using a Nonlinear Tuned Vibration Absorber

Malher

Touzé

Doaré

et al. 2017

Journal of Computational and Nonlinear Dynamics

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International audienceThe influence of a Nonlinear Tuned Vibration Absorber (NLTVA) on the airfoil flutter is investigated. In particular, its effect on the instability threshold and the potential subcriticality of the bifurcation is analyzed. For that purpose, the airfoil is modeled using the classical pitch and plunge aeroelastic model together with a linear approach for the aerodynamic loads. Large amplitude motions of the airfoil are taken into account with nonlinear restoring forces for the pitch and plunge degrees of freedom. The two cases of a hardening and a softening spring behavior are investigated. The influence of each NLTVA parameter is studied and an optimum tuning of these parameters is found. The study reveals the ability of the NLTVA to shift the instability, avoid its possible subcriticality and reduce the LCOs amplitude

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“…However, the solution of full-models with many degrees of freedom entails a large computational cost. In order to reduce this cost, reduced-order models (ROMs) are introduced and, very often, researchers used a limited number of DOFs -generally six to eight [2][3][4][5][6][7][8][9] -in the analysis of nonlinear flutter of plates. The main goal of this study is quantifying the degree of approximation of ROMs with respect to its associated FOM, in plates subjected to supersonic airflow.…”

Section: Introductionmentioning

confidence: 99%

Approximations By Reduced-Order Models for Nonlinear Flutter of Variable Stiffness Composite Plates

Akhavan¹,

Ribeiro²

2019

SAMPE 2019 - Charlotte, NC

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In this investigation, we analyze errors due to using reduced-order models instead of full-order models in the examination of nonlinear flutter of variable stiffness composite laminates (VSCLs). These plates can be made, e.g., by Automated Tow Placement Machines, using composite laminates with curvilinear fibers; in our particular case, the orientation angle of the reference curvilinear fiber path changes linearly from T0 at the left edge to T1 at the right edge of the plate. A Third-order Shear Deformation Theory (TSDT) is used to model the laminate and a p-version finite element is applied to discretize the displacements and rotations. The plates are subjected to a supersonic airflow of which the aerodynamic pressure is approximated using linear Piston theory. The equations of motion of the full-order model of the self-excited vibrational system are formed using the principle of virtual displacements. In order to reduce the number of degrees-of-freedom of the full-order model, static condensation and/or a modal summation method are used. The equations of motion of the reduced-order and full-order models are solved using Newmark method to study the dynamic responses, focusing on limit cycle oscillations (LCOs). Approximation errors are discussed for LCO amplitudes of VSCL plates with various curvilinear fiber paths.

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Nonlinear reduced-order modelling for limit-cycle oscillation analysis

Cited by 11 publications

References 29 publications

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Flutter Control of a Two-Degrees-of-Freedom Airfoil Using a Nonlinear Tuned Vibration Absorber

Approximations By Reduced-Order Models for Nonlinear Flutter of Variable Stiffness Composite Plates

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