1:1 internal resonance in a two d.o.f. complete system: a comprehensive analysis and its possible exploitation for design

Clementi, Francesco; Lenci, Stefano; Rega, Giuseppe

doi:10.1007/s11012-020-01171-9

Cited by 28 publications

(12 citation statements)

References 49 publications

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“…In the whole four-dimensional phase space (including both velocities), we hence obtain a family of periodic orbits developing on a manifold such that its projection on ðq 10 ; q 20 Þ is a parabola. This solution is called parabolic mode, following also the designation given in [6,17,32] for the case of 1:1 resonance, where normal modes and elliptic modes are the two families of coupled solutions. More precisely, in the case p ¼ þ1, this solution is denoted as p þmode.…”

Section: Coupled Solutions: Parabolic Modes and Backbone Curvesmentioning

confidence: 99%

“…In recent years several occurrences of FCs have been documented in Micro Electro-Mechanical Systems (MEMS). These devices display a wide range of complex nonlinear phenomena, mainly due to their large quality factors Q, typically ranging from 10 3 to 10 6 . Indeed MEMS are often monolithic structures encapsulated in near vacuum packages and their dissipation is drastically reduced with respect to macrostructures.…”

mentioning

confidence: 99%

“…Indeed MEMS are often monolithic structures encapsulated in near vacuum packages and their dissipation is drastically reduced with respect to macrostructures. Internal resonance (IR) plays an important role in triggering more complex motions and facilitate energy transfer between modes and also the onset of quasiperiodicity [1,2,6,22,23,44,58,59]. Indeed in most cases, FCs in MEMS are associated with IR phenomena.…”

mentioning

confidence: 99%

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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: Coupled Solutions: Parabolic Modes and Backbone Curvesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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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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“…A micro-machined gyroscope system is a typical nonlinear vibration system with 1:1 internal resonance [ 8 , 10 , 12 ]. This 1:1 internal resonance is one of the important research directions of nonlinear dynamics [ 21 , 22 ]. It occurs in numerous mechanical systems, such as cables [ 23 ], beams [ 24 , 25 ], circular or square plates [ 26 , 27 ], etc.…”

Section: Introductionmentioning

confidence: 99%

“…It occurs in numerous mechanical systems, such as cables [ 23 ], beams [ 24 , 25 ], circular or square plates [ 26 , 27 ], etc. Many investigations, including those of periodic response [ 23 ], backbone curves [ 21 , 22 ], bifurcation characteristics [ 28 ], and chaos [ 29 ], have been done on this subject. The singularity theory is an effective method for bifurcation investigation.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis of a Micro-Machined Gyroscope with Nonlinear Stiffness and Electrostatic Forces

Zhang

2021

Micromachines

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The bifurcation of the periodic response of a micro-machined gyroscope with cubic supporting stiffness and fractional electrostatic forces is investigated. The pull-in phenomenon is analyzed to show that the system can have a stable periodic response when the detecting voltage is kept within a certain range. The method of averaging and the residue theorem are employed to give the averaging equations for the case of primary resonance and 1:1 internal resonance. Transition sets on the driving/detecting voltage plane that divide the parameter plane into 12 persistent regions and the corresponding bifurcation diagrams are obtained via the singularity theory. The results show that multiple solutions of the resonance curves appear with a large driving voltage and a small detecting voltage, which may lead to an uncertain output of the gyroscope. The effects of driving and detecting voltages on mechanical sensitivity and nonlinearity are analyzed for three persistent regions considering the operation requirements of the micro-machined gyroscope. The results indicate that in the region with a small driving voltage, the mechanical sensitivity is much smaller. In the other two regions, the variations in the mechanical sensitivity and nonlinearity are analogous. It is possible that the system has a maximum mechanical sensitivity and minimum nonlinearity for an appropriate range of detecting voltages.

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索-塔耦合结构面内动力学模型及其多重“1:1”内共振分析

Chen,

Li,

et al. 2023

Acta Mech. Sin.

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1:1 internal resonance in a two d.o.f. complete system: a comprehensive analysis and its possible exploitation for design

Cited by 28 publications

References 49 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

Bifurcation Analysis of a Micro-Machined Gyroscope with Nonlinear Stiffness and Electrostatic Forces

索-塔耦合结构面内动力学模型及其多重“1:1”内共振分析

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