How do conservative backbone curves perturb into forced responses? A Melnikov function analysis

Cenedese, Mattia; Haller, George

doi:10.1098/rspa.2019.0494

Cited by 33 publications

(52 citation statements)

References 68 publications

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“…As shown in the previous section, the limit value of the p þ -mode corresponds to a nonvanishing value of q 2 . Since the forced-damped solutions emanate from the backbone curves [5], this eventually creates the condition for the appearance of isolas.…”

Section: Isolasmentioning

confidence: 99%

Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

et al. 2021

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Quasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is investigated analytically and numerically in the specific case of a system of two coupled oscillators featuring a 1:2 internal resonance. More specifically, the locus of Neimark-Sacker points is analytically derived and its evolution with respect to the system parameters is highlighted. The backbone curves, solution of the conservative system, are first investigated, showing in particular the existence of two families of periodic orbits, denoted as parabolic modes. The behaviour of these modes, when the detuning between the eigenfrequencies of the system is varied, is underlined. The non-vanishing limit value, at the origin of one solution family, allows explaining the appearance of isolated solutions for the damped-forced system. The results are then applied to a Micro-Electro-Mechanical System-like shallow arch structure, to show how the analytical expression of the Neimark-Sacker boundary curve can be used for rapid prediction of the appearance of quasiperiodic regime, and thus frequency combs, in Micro-Electro-Mechanical System dynamics.

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Section: Isolasmentioning

confidence: 99%

Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

et al. 2021

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“…The classic HBM is used to solve dynamic equations in frequency domain referring to Eqs. (4,11,15). The nonlinear force in frequency domain is represented bỹ F nl (Q).…”

Section: Alternating Frequency/time Methodsmentioning

confidence: 99%

“…This crossing point is called "resonant shared solution" in [7]. Cenedese and Haller [4] provide a systematic mathematical analysis based on the Melnikov function, to identify the conditions under which conservative periodic orbits perturb into resonances of the forced response, thus defining the conditions of applicability of methods like the EBM. The EBM method in its original definition is applicable only to conservative nonlinear forces.…”

Section: Introductionmentioning

confidence: 99%

Comparison of different methodologies for the computation of damped nonlinear normal modes and resonance prediction of systems with non-conservative nonlinearities

et al. 2021

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The nonlinear modes of a non-conservative nonlinear system are sometimes referred to as damped nonlinear normal modes (dNNMs). Because of the non-conservative characteristics, the dNNMs are no longer periodic. To compute non-periodic dNNMs using classic methods for periodic problems, two concepts have been developed in the last two decades: complex nonlinear mode (CNM) and extended periodic motion concept (EPMC). A critical assessment of these two concepts applied to different types of non-conservative nonlinearities and industrial full-scale structures has not been thoroughly investigated yet. Furthermore, there exist two emerging techniques which aim at predicting the resonant solutions of a nonlinear forced response using the dNNMs: extended energy balance method (E-EBM) and nonlinear modal synthesis (NMS). A detailed assessment between these two techniques has been rarely attempted in the literature. Therefore, in this work, a comprehensive comparison between CNM and EPMC is provided through two illustrative systems and one engineering application. The EPMC with an alternative damping assumption is also derived and compared with the original EPMC and CNM. The advantages and limitations of the CNM and EPMC are critically discussed. In addition, the resonant solutions are predicted based on the dNNMs using both E-EBM and NMS. The accuracies of the predicted resonances are also discussed in detail.

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“…The harmonic balance method (HBM) with alternating frequency/time method [3] is used to solve Eqs. (1) and (4). In this method, the continuous properties in time domain Q are discretised based on Fourier series into frequency domain (Q c p + Q s p i) as shown in Eq.…”

Section: Nonlinear Modal Analysismentioning

confidence: 99%

“…The method relies on the fact that, for any steady-state response, the net energy transferred into a system over one period of vibration must be zero [8]. Moreover, the work of Cenedese and Haller [4] uses Melnikov function analysis that defines the mathematical conditions under which the EBM is valid.…”

Section: Introductionmentioning

confidence: 99%

An extended energy balance method for resonance prediction in forced response of systems with non-conservative nonlinearities using damped nonlinear normal mode

et al. 2020

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The dynamic analysis of systems with nonlinearities has become an important topic in many engineering fields. Apart from the forced response analyses, nonlinear modal analysis has been successfully extended to such non-conservative systems thanks to the definition of damped nonlinear normal modes. The energy balance method is a tool that permits to directly predict resonances for a conservative system with nonlinearities from its nonlinear modes. In this work, the energy balance method is extended to systems with non-conservative nonlinearities using the concept of the damped nonlinear normal mode and its application in a full-scale engineering structure. This extended method consists of a balance between the energy loss from the internal damping, the energy transferred from the external excitation and the energy exchanged with the non-conservative nonlinear force. The method assumes that the solution of the forced response at resonance bears resemblance to that of the damped nonlinear normal mode. A simplistic model and full-scale structure with dissipative nonlinearities and a simplistic model showing self-excited vibration are tested using the method. In each test case, resonances are predicted efficiently and the computed force-amplitude curves show a great agreement with the forced responses. In addition, the self-excited solutions and isolas in forced responses can be effectively

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How do conservative backbone curves perturb into forced responses? A Melnikov function analysis

Cited by 33 publications

References 68 publications

Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Comparison of different methodologies for the computation of damped nonlinear normal modes and resonance prediction of systems with non-conservative nonlinearities

An extended energy balance method for resonance prediction in forced response of systems with non-conservative nonlinearities using damped nonlinear normal mode

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