Nonlinear normal modes of a shallow arch with elastic constraints for two-to-one internal resonances

Yi, Zhuangpeng; Stanciulescu, Ilinca

doi:10.1007/s11071-015-2432-3

Cited by 33 publications

(12 citation statements)

References 53 publications

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“…Lately, Lakrad et al [19] analyzed via multiple scales the two-to-one internal resonance of a simply supported arch subjected to a spatially distributed harmonic load. Yi and Stanciulescu [20] analyzed the influence of different end constrain conditions to develop the nonlinear normal modes of an arch beam considering two-to-one internal resonance. Also, Yu and Chen [21], utilizing multiple scales, extended the work of Bi and Dai [18] and examined the one-to-one internal resonance of an arch beam and studied the stability by constructing the invariant manifold of the response equations.…”

Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

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Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

Alfosail

Hajjaj

Younis

2018

Journal of Computational and Nonlinear Dynamics

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We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

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Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

“…Next, we carry out the expansion to the next order to account for the cubic nonlinearity. Hence, we consider the particular solution of Eq (20),. which is written as…”

mentioning

confidence: 99%

Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

Alfosail

Hajjaj

Younis

2018

Journal of Computational and Nonlinear Dynamics

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“…Lee et al [7] studied free vibration of a rotating curved beam with elastically restrained roots. e two-to-one internal resonance between nonlinear normal modes of an elastically constrained shallow arch is examined [8]. In these cases, the assumption would be reasonable if the mass of the support is much larger than that of cable.…”

Section: Introductionmentioning

confidence: 99%

Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

Yang

Chen

et al. 2020

Shock and Vibration

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In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.

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“…A MEMS resonant mass sensor mainly realizes the detection by changing the resonant frequency and vibration amplitude of the structure caused by the adsorption of the elastic element of the sensor to the target analyzers [28]. Frequency shift tracking is the most common method [29].…”

Section: Introductionmentioning

confidence: 99%

Anti-Symmetric Mode Vibration of Electrostatically Actuated Clamped–Clamped Microbeams for Mass Sensing

Zhang

et al. 2019

Micromachines

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This paper details study of the anti-symmetric response to the symmetrical electrostatic excitation of a Micro-electro-mechanical-systems (MEMS) resonant mass sensor. Under higher order mode excitation, two nonlinear coupled flexural modes to describe MEMS mass sensors are obtained by using Hamilton’s principle and Galerkin method. Static analysis is introduced to investigate the effect of added mass on the natural frequency of the resonant sensor. Then, the perturbation method is applied to determine the response and stability of the system for small amplitude vibration. Through bifurcation analysis, the physical conditions of the anti-symmetric mode vibration are obtained. The corresponding stability analysis is carried out. Results show that the added mass can change the bifurcation behaviors of the anti-symmetric mode and affect the voltage and frequency of the bifurcation jump point. Typically, we propose a mass parameter identification method based on the dynamic jump motion of the anti-symmetric mode. Numerical studies are introduced to verify the validity of mass detection method. Finally, the influence of physical parameters on the sensitivity of mass sensor is analyzed. It is found that the DC voltage and mass adsorption position are critical to the sensitivity of the sensor. The results of this paper can be potentially useful in nonlinear mass sensors.

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Nonlinear normal modes of a shallow arch with elastic constraints for two-to-one internal resonances

Cited by 33 publications

References 53 publications

Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

Anti-Symmetric Mode Vibration of Electrostatically Actuated Clamped–Clamped Microbeams for Mass Sensing

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