Normal Mode Localization for a Two Degrees-of-Freedom System With Quadratic and Cubic Non-Linearities

Bhattacharyya, Ranjan; Jain, Punit; Nair, Ashvini

doi:10.1006/jsvi.2001.3916

Cited by 12 publications

(6 citation statements)

References 16 publications

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“…This author also studied the transition between localization and non-localization as a function of the coupling/nonlinearities ratio; he found that localization was only possible for small values of this ratio and in cases where bifurcations appear as this ratio increases, thus leading to a disappearance of localized solutions. Localization phenomena have also been identified by many studies such as [8][9][10].…”

Section: Introductionmentioning

confidence: 86%

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Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

Grolet

Thouverez

2012

Journal of Sound and Vibration

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This work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity, through the use of the harmonic balance method (HBM). In order to study the influence of nonlinearity due to the large deflection of blades, a simplified model has been developed. This approach leads to a system of linearly coupled, second-order nonlinear differential equations, in which nonlinearity appears via cubic terms. Periodic solutions, in both the free and forced cases, are sought by applying HBM coupled with an arc-length continuation method. Solution stability has been investigated using Floquet's theorem. In addition to featuring similar and nonsimilar nonlinear modes, the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes. In the forced case, these nonlinear modes give rise to a complex dynamic behavior. Many bifurcations can take place, thus leading to strong or weak localization that may or may not be stable. In this study, special attention has been paid to the influence of excitation on dynamic responses. Several cases of excitation have been analyzed herein: localized excitation, and low-engine-order excitation. In the case of low-engine-order excitation, sensitivity of the response to a perturbation of this excitation type has been investigated, and it has been shown that for a localized, or sufficiently detuned excitation, several solutions can coexist, some of which are represented by closed curves in the Frequency-Amplitude domain. These various solutions overlap when increasing the force amplitude, leading to forced nonlinear localization. Because closed curves are not tied up with the basic nonlinear solution, they can easily be overlooked. In this study, they have been calculated using a sequential continuation with the force amplitude as a parameter.

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Section: Introductionmentioning

confidence: 86%

“…The expressions for U j , T j and W j have been obtained by replacing v by v j in Eqs. (16), (17), (10). The expressions of strain energy U, kinetic energy T, and work W for the complete system can then be given by…”

Section: Incorporation Of Cyclic Symmetrymentioning

confidence: 99%

Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

Grolet

Thouverez

2012

Journal of Sound and Vibration

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“…Nayfeh et al [20] investigated NNMs in clamped-clamped buckled beams and found that some modes could not interact at all, in spite of proper integer ratios between the associated frequencies, due to vanishing of the non-linear interaction coefficients in the normal forms. Bhattacharyya et al [21] studied the NNM localization of a discrete system with quadratic and cubic non-linearity only for the case of 1:1 internal resonance and found modes through which both masses pass at the same instant of time periodically through a non-zero point. Li et al [22] investigated NNMs and their bifurcation of a complex two degree of freedom system, and discussed the case of internal resonance and no internal resonance systematically.…”

Section: Introductionmentioning

confidence: 99%

Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes

Yan¹,

Wang²,

Zhang³

2011

Applied Mathematical Modelling

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a b s t r a c tThe aim of this paper is to show how the concept of nonlinear normal modes (NNMs) can be used to characterize the nonlinear dynamical behaviors of double-walled carbon nanotubes (DWCNTs). DWCNTs are modeled as double simply supported elastic beams with van der Waals (vdW) forces between the inner and outer walls. The multiscale method for deriving the approximate solutions of NNMs is applicable to the nonlinear systems. According to the procedure, the typical features -coaxial and noncoaxial vibrations of the system are exhibited in the literature. Moreover, the case of 1:3 internal resonance is discussed in detail, which can give rise to much more complex phenomenon for DWCNTs systems. Meanwhile, the amplitude-time curves of the nonlinear vibration with different initial conditions are presented, and the amplitude-frequency characteristic curves of the nonlinear vibration are also obtained.

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“…To determine the above integration constants, Equations (11) and (12) are substituted into Equations (7), and (13)- (16) are substituted into Equation (8). This step yields, after performing the corresponding integrations, the following expressions: …”

Section: Computation Of η I η Ii V I and V Ii Valuesmentioning

confidence: 99%

Lyapunov Equivalent Representation Form of Forced, Damped, Nonlinear, Two Degree-of-Freedom Systems

et al. 2018

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Featured Application: The qualitative and quantitative dynamic behavior of two degree-offreedom nonlinear systems can be studied by using their corresponding decoupled one degree of freedom Duffing type equivalent representation forms in the sense of Lyapunov, with the advantage of capturing amplitude-dependent nonlinear mode shapes.Abstract: The aim of this paper focuses on finding equivalent representation forms of forced, damped, two degree-of-freedom, nonlinear systems in the sense of Lyapunov by using a nonlinear transformation approach that provides decoupled, forced, damped, nonlinear equations of the Duffing type, under the assumption that the driving frequency and the external forces are equal in both systems. The values of Lyapunov characteristic exponents (LCEs), Lyapunov largest exponents (LLE), and time-amplitude and frequency-amplitude curves computed from numerical integration solutions, indicate that the decoupled Duffing-type equations are equivalent, in the sense of Lyapunov, to the original dynamic system, since both set of motion equations tend to have the same qualitative and quantitative behaviors.

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Normal Mode Localization for a Two Degrees-of-Freedom System With Quadratic and Cubic Non-Linearities

Cited by 12 publications

References 16 publications

Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes

Lyapunov Equivalent Representation Form of Forced, Damped, Nonlinear, Two Degree-of-Freedom Systems

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