ε 2-Order Normal Form Analysis for a Two-Degree-of-Freedom Nonlinear Coupled Oscillator

Liu, Xuanang; Wagg, DJ

doi:10.1007/978-3-030-34713-0_3

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…It should be noted that the direct normal form method relies on the nonlinear terms being small in the sense that they should be significantly smaller than the ω 2 ni values. However, in this example, the nonlinear coefficients are of the same order as the ω 2 ni values, and yet despite this, it is not the determining factor for the discrepancy observed by Breunung & Haller [24]-see [43] for further details. Now specific values are chosen to compute the analytical forced response curves defined by, equation (4.16), and backbone curves, equation (4.17), of the example mass-spring system.…”

Section: An Order-reduced Variant Of the Nonlinear Transformmentioning

confidence: 68%

“…So for example, when the comparison was carried out by Breunung & Haller [24], the ε 1 -order approximation was used, and as a result, the softening nonlinear effects were not captured appropriately. As we show in the results below, (and for the full system in [43]) this is corrected by the addition of the ε 2 terms. It should be noted that the direct normal form method relies on the nonlinear terms being small in the sense that they should be significantly smaller than the ω 2 ni values.…”

Section: An Order-reduced Variant Of the Nonlinear Transformmentioning

confidence: 99%

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Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

Liu

Wagg

2019

Proc. R. Soc. A.

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In this paper, we describe a direct normal form decomposition for systems of coupled nonlinear oscillators. We demonstrate how the order of the system can be reduced during this type of normal form transformation process. Two specific examples are considered to demonstrate particular challenges that can occur in this type of analysis. The first is a 2 d.f. system with both quadratic and cubic nonlinearities, where there is no internal resonance, but the nonlinear terms are not necessarily ε 1 -order small. To obtain an accurate solution, the direct normal form expansion is extended to ε 2 -order to capture the nonlinear dynamic behaviour, while simultaneously reducing the order of the system from 2 to 1 d.f. The second example is a thin plate with nonlinearities that are ε 1 -order small, but with an internal resonance in the set of ordinary differential equations used to model the low-frequency vibration response of the system. In this case, we show how a direct normal form transformation can be applied to further reduce the order of the system while simultaneously obtaining the normal form, which is used as a model for the internal resonance. The results are verified by comparison with numerically computed results using a continuation software.

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Section: An Order-reduced Variant Of the Nonlinear Transformmentioning

confidence: 68%

Section: An Order-reduced Variant Of the Nonlinear Transformmentioning

confidence: 99%

Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

Liu

Wagg

2019

Proc. R. Soc. A.

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“…In order to compare with our proposed symbolic computation method, the same problem has been solved in conservative case (unforced-undamped case), and the conservative backbone curve is computed using Eq. (18) and Table (4 as the value of  is positive, hardening behavior is clearly seen, in contrary, if  is negative softening behavior will be noticed. Furthermore, as the figure shows the relation between natural frequency and amplitude, the backbone curves do not perfectly coincide with the manifolds, and this is due to the presence of damping.…”

Section: Non-resonant Duffing Oscillator With Cubic Nonlinearitymentioning

confidence: 91%

Computing Backbone Curves for Nonlinear Oscillators With Higher Order Polynomial Stiffness Terms

Nasir¹,

Sims²,

Wagg³

2020

XI International Conference on Structural Dynamics

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Single-degree-of-freedom (SDOF) nonlinear oscillators are widely used for modelling systems with just one degree-of-freedom in addition to single mode approximations to structural elements such as beams and cables, as well as other multi-degree-of-freedom (MDOF) applications. In this work, an investigation of the behavior of SDOF nonlinear oscillators is carried out using the method of direct normal forms. So far, this method has only been considered as a theoretical technique used for solving limited nonlinear dynamical systems in which low orders of nonlinearities appear, involving quadratic and cubic nonlinearities. In this work, thanks to the implementation of symbolic computations, the method of direct normal forms is generalized for solving nonlinear SDOF systems with any order of polynomial (or geometric) weak nonlinearities. Using this new approach, the effect of any higher order nonlinear term, or any combination of nonlinear terms can be investigated. Backbone curve relations are obtained for a selection of example systems representing both hardening and softening systems, and the results are verified by comparing the approximate analytical solutions to numerical solutions generated using COCO numerical continuation toolbox in Matlab.

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ε 2-Order Normal Form Analysis for a Two-Degree-of-Freedom Nonlinear Coupled Oscillator

Abstract: This is a repository copy of ε^2-Order normal form analysis for a two-degree-of-freedom nonlinear coupled oscillator.

Cited by 2 publications

References 14 publications

Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

Computing Backbone Curves for Nonlinear Oscillators With Higher Order Polynomial Stiffness Terms

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