A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms

Waswa, Peter M.B.; Redkar, Sangram

doi:10.1007/s11071-019-05334-6

Cited by 12 publications

(12 citation statements)

References 18 publications

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“…3(a)-3(c), the red solid line indicates the temporal variation based on the STM,/ðtÞ, determined from the Normal Forms technique and the black dashed line represents the numerically integrated solution of Eq. (19). In Figs.…”

Section: System Temporal Variationsmentioning

confidence: 96%

“…Simultaneously, Eqs. (19) and (28) are numerically integrated with the same set of parameters and initial conditions. The phase plot comparison of the variation of the system states are shown in Fig.…”

Section: System Temporal Variationsmentioning

confidence: 99%

“…[16]. In the prior work, the authors have demonstrated an intuitive state augmentation enabling the direct application of the center manifold theory [17] and Normal Forms technique [18,19] toward reducing nonlinear systems. In this work, the authors have worked on simplifying the prior research, toward the dynamical analysis of linear time periodic system with the following objectives:…”

Section: Introductionmentioning

confidence: 99%

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Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

Subramanian

Redkar

2020

Journal of Computational and Nonlinear Dynamics

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In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique and apply them towards the investigation of stability bounds for linear time periodic systems. Though the Normal Forms technique has been predominantly used for the analysis of nonlinear equations, in this work the authors utilize it to transform a linear time periodic system to a time-invariant system, similar to the Lyapunov Floquet (L-F) transformation. The authors employ an intuitive state augmentation technique, modal transformation and near identity transformations to facilitate the application of time independent Normal Forms. This method provides a closed form analytical expression for the state transition matrix. Additionally, stability analysis is performed on the transformed system and the comparative results of dynamical characteristics and temporal variations of a simple linear Mathieu equation are also presented in this work.

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Section: System Temporal Variationsmentioning

confidence: 96%

Section: System Temporal Variationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

Subramanian

Redkar

2020

Journal of Computational and Nonlinear Dynamics

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“…The Equation 11could be subjected to a modal transformation and the Normal Forms technique could be applied to eliminate the non resonant nonlinear terms. A detailed implementation of the state augmentation along with normal forms technique, for nonlinear systems with periodic forcing terms, are discussed in detail in our prior work [46].…”

Section: State Augmentationmentioning

confidence: 99%

“…This theory was further applied to nonlinear systems with periodic coefficients by introducing a detuning parameter or a book-keeping parameter [44,45]. Though the use of book-keeping parameters appears to make the formulations using Normal Forms to be convoluted and cumbersome, a direct approach would ease the computations and provide comparable results [46,47].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of Suction Stabilized Floating Platforms

Subramanian

Dye

Redkar

2020

JMSE

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The occurrence of parametric resonance due to the time varying behavior of ocean waves could lead to catastrophic damages to offshore structures. A stable structure that could withstand the wave perturbations is quintessential to operate in such a harsh environment. In this work, the authors detail the relevance of a Suction Stabilized Float (SSF) or a Suction Stabilized Floating platform towards such an application. A generic design of a symmetrically shaped float structure along with its inherent stabilization behavior is discussed. Furthermore, the authors extend their prior research on this topic towards modelling the dynamics of SSF and perform stability analysis. The authors demonstrate the dynamical characteristics of SSF analytically using Floquet theory and Normal Forms technique, in this work. Additionally, the simulation results are verified and validated with the numerical methods.

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Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

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A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms

Cited by 12 publications

References 18 publications

Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

Dynamic Analysis of Suction Stabilized Floating Platforms

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

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