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Research Papers

Dynamics of Globally and Dissipatively Coupled Resonators

[+] Author and Article Information
Andrew B. Sabater

School of Mechanical Engineering,
Birck Nanotechnology Center,
and Ray W. Herrick Laboratories,
Purdue University,
West Lafayette, IN 47907

Jeffrey F. Rhoads

School of Mechanical Engineering,
Birck Nanotechnology Center,
and Ray W. Herrick Laboratories,
Purdue University,
West Lafayette, IN 47907
e-mail: jfrhoads@purdue.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 21, 2013; final manuscript received November 19, 2014; published online January 19, 2015. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 137(2), 021016 (Apr 01, 2015) (9 pages) Paper No: VIB-13-1251; doi: 10.1115/1.4029226 History: Received July 21, 2013; Revised November 19, 2014; Online January 19, 2015

This work explores the dynamics of arrays of globally and dissipatively coupled resonators. These resonator arrays are shown to be capable of exhibiting seemingly new collective behaviors which are highly sensitive to the dispersion of the natural frequencies of the constituent resonators in the array, the intrinsic damping of the resonators in the array, and the magnitude of the global coupling coefficient that captures the strength of the dissipative coupling. These behaviors have been identified within the work as group attenuation, confined attenuation, and group resonance. Group and confined attenuation are associated with an absence of energy and are strongly dependent on the dispersion of the natural frequencies. In cases of moderate dissipative coupling, the effects of group and confined attenuation could be interpreted as frequency-dependent damping. In cases where the global coupling coefficient is large, group resonance is significant. This effect is synonymous with the resonances of the constituent resonators being shared and occurring at frequencies in between the isolated resonators' natural frequencies. Accordingly, one could view group resonance as the antithesis of localization, in that the localization of the modes of a conservatively coupled system with a finite dispersion of the constituent resonators' natural frequencies is most significant when the coupling is weak. The authors believe that collective behaviors, such as those described herein, have direct applicability in new single-input, single-output resonant mass sensors, and, with extension, a variety of other sensing and signal processing systems.

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Figures

Grahic Jump Location
Fig. 1

A scanning electron micrograph of an array of electromagnetically transduced microresonators. This device consists of two microcantilevers with Au/Cr wire traces deposited on the surface. When placed in close proximity to a magnetic field, these wire traces facilitate actuation and sensing. As is shown in the Appendix, this system can be characterized as an array of globally and dissipatively coupled resonators.

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Fig. 2

The responses of various arrays when σ = 0.001 and Q = 100. The rows of this figure correspond to N = 5, 25, and 1000, respectively, and the columns of this figure correspond to αQN = 1/50, 1/10, 1, and 5. These responses reveal that if an array of resonators is similar, or σQ is close to or less than 1, the product αQN can be used to characterize the array's response. Specifically, as αQN approaches 1, group attenuation, or the near uniform attenuation of the amplitude responses across the vast majority of the array, is observed.

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Fig. 3

The responses of various arrays when σ = 0.001 and Q = 1000. The rows and columns of this figure are laid out in the same manner as in Fig. 2. Since the resonators in this figure are less similar than the resonators in Fig. 2, a larger αQN is needed to observe group attenuation. In addition, Figs. 2 and 3 both show that as αQN exceeds 1, confined attenuation is observed, where the resonators near the middle of the set are attenuated more than the resonators near the fringes.

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Fig. 4

Various responses for N = 25 and Q = 100 when all but one resonator are similar. From top to bottom, the coupling coefficient is set to one-tenth and equal to [Q(N − 1)]−1 or the coupling coefficient used to characterize group attenuation for this set of resonators. These responses demonstrate that the relative attenuation of the response is confined to the resonators in the array that are similar.

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Fig. 5

Various responses for Q = 100 and N = 5 when the ln are selected such that the undamped natural frequencies are separated by 0.1. In Figs. 5(a)5(f), the coupling coefficient is equal to 0, 1, 5, 10, 20, and 100 times the inverse of the array quality factor, respectively. For small coupling coefficients, such as the cases in Figs. 5(a) and 5(b), the resonators only have a significant response near the uncoupled resonant frequencies. As the coupling coefficient increases, group resonance is observed, wherein individual resonances are lost and all of the resonators share resonances.

Grahic Jump Location
Fig. 6

A series of responses for Q = 10,000 and N = 5 when the ln are selected such that the undamped natural frequencies are separated by 1. In Figs. 6(a)6(f), the coupling coefficient is equal to 0, 0.1, 0.5, 1, 10 and 1000, respectively. These responses support the concept that the transition to group resonance is most strongly dependent on the coupling coefficient, and that group resonance is observed when α is greater than or equal to 1.

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