Detecting internal resonances during model reduction

Nicolaidou, Evangelia; Hill, Thomas L.; Neild, Simon A

doi:10.1098/rspa.2021.0215

Cited by 9 publications

(6 citation statements)

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“…Finally, we note that, in line with approaches using a static condensation procedure [8], the proposed method requires that the natural frequencies of the reduced and condensed modes are well-separated, which allows the coupling between them to be approximated as quasi-static, while incorrect results can be obtained when this assumption does not hold. Limitations are also likely to be encountered when the non-conservative forces cause the quasi-static coupling approximation to break down [31].…”

Section: Discussionmentioning

confidence: 99%

Nonlinear mapping of non-conservative forces for reduced-order modelling

2022

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Non-intrusive or indirect reduced-order modelling strategies, such as the implicit condensation and expansion method, are applicable to geometrically nonlinear structures modelled using commercial finite-element (FE) software. Traditionally, the non-conservative forces acting on the structure are reduced via a linear projection onto the space spanned by the reduced modeshapes. As such, only the forces acting directly on these reduced modes can be captured, while any energy gained or dissipated by the statically condensed modes is neglected. This can lead to significant inaccuracies in the reduced-order model (ROM) predictions, which is demonstrated here using a 2-degrees-of-freedom (DOF) oscillator, and an FE model of a cantilever beam. It is shown that the non-conservative forces acting on the statically condensed modes can be captured using a nonlinear mapping of the physical DOFs into the reduced coordinates. This introduces additional terms in the reduced equations of motion, which we describe as force compensation . Excellent agreement is observed between the forced response curves of the full-order models and those of our proposed ROMs, both for the oscillator as well as the cantilever beam under different external excitation conditions (i.e. a constant-direction force and a follower force).

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Section: Discussionmentioning

confidence: 99%

Nonlinear mapping of non-conservative forces for reduced-order modelling

2022

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“…This model was proposed in [46] to predict internal resonances, between qr and h, during reduced-order modelling. As with discussions in §3, the internally resonant component, h, may be assumed to be small when compared with the dominant mode, qr.…”

Section: Effect Of Symmetry Breaking On Internal Resonancesmentioning

confidence: 99%

“…While the internal resonances between qr and h is captured by equation (4.3 b ). This is equivalent to considering internal resonances in the neighbourhood of the primary NNM branch; for details of this derivation please see [46].…”

Section: Effect Of Symmetry Breaking On Internal Resonancesmentioning

confidence: 99%

Existence and location of internal resonance of two-mode nonlinear conservative oscillators

2022

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Internal resonances can be widely observed in nonlinear systems; even a simple nonlinear system can exhibit intricate internal resonances when vibrating at large amplitudes. In this study, the existence and locations of internal resonances of a general two-mode system with an arbitrary eigenfrequency ratio are considered. This is achieved by first considering the symmetric case, where the internal resonances are found to be approximately captured by the Mathieu equation. It is shown that the bifurcations can exist in pairs; and, for each pair, the bifurcated solution branches capture modal interactions with the same commensurate frequency relationship but different phase relationships. To determine the existence and locations of internal resonances, the divergence and convergence for correlated bifurcation pairs are then considered. Lastly, the internal resonances in asymmetric cases are analytically derived, where the asymmetry induced bifurcation splitting is captured by a non-homogeneous extended Mathieu equation. This work explores the mechanism underpinning internal resonances, and explains their topological features, such as which internal resonances are observed as amplitude increases. A graphical method is also proposed for efficient determination of the existence and locations of internal resonances.

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“…However, it is not trivial to identify these dynamically significant modes without a prior computation. To overcome this limitation, Nicolaidou et al introduce time-dependent terms to represent the error of the quasi-static coupling relationship to detect the dynamic interaction in the system efficiently [44]. Monitoring these errors allows for a modal selection strategy for developing a multi-mode ROM, since the mode with a large error is deemed as dynamically significant and should be added to the reduction basis.…”

Section: Introductionmentioning

confidence: 99%

Ensuring the Accuracy of FE-based Nonlinear Dynamic Reduced-order Models

Xiao

Hill

Neild

2023

Preprint

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Numerous powerful methods exist for developing Reduced-order Models (ROMs) using Finite Element (FE) models. Ensuring the accuracy of these ROMs is essential; however, the validation using dynamic responses is expensive. In this work, we propose a method to ensure the accuracy of ROMs without extra dynamic FE simulations. It has been shown that the well-established Implicit Condensation and Expansion (ICE) method can produce an accurate ROM when the FE model's static behaviours are captured accurately. However, this is achieved via a fitting procedure, which may be sensitive to the selection of load cases and ROM's order, especially in the multi-mode case. To alleviate this difficulty, we define an error metric that can evaluate the ROM's fitting error efficiently within the displacement range, specified by a given energy level. Based on the fitting result, the proposed method provides a strategy to enrich the static dataset, i.e. additional load cases are found until the ROM's accuracy reaches the required level. Extending this to the higher-order and multi-mode cases, some extra constraints are incorporated into the standard fitting procedure to make the proposed method more robust. A clamped-clamped beam is utilised to validate the proposed method, and the results show that the method can robustly ensure the accuracy of the static fitting of ROMs.

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Detecting internal resonances during model reduction

Cited by 9 publications

References 50 publications

Nonlinear mapping of non-conservative forces for reduced-order modelling

Nonlinear mapping of non-conservative forces for reduced-order modelling

Existence and location of internal resonance of two-mode nonlinear conservative oscillators

Ensuring the Accuracy of FE-based Nonlinear Dynamic Reduced-order Models

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