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Research Papers

Frequency Sweeping With Concurrent Parametric Amplification

[+] Author and Article Information
Nicholas J. Miller1

 Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824mille820@msu.edu

Steven W. Shaw

 Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824shawsw@msu.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 134(2), 021007 (Dec 30, 2011) (6 pages) doi:10.1115/1.4005363 History: Received October 04, 2010; Revised September 14, 2011; Published December 30, 2011; Online December 30, 2011

In this paper, we explore parametric amplification of multiple resonances in multidegree-of-freedom mechanical systems, and the use of frequency sweeping with a parametric pump to amplify several adjacent resonance peaks. We develop conditions under which it is possible to sweep with direct and parametric excitation to produce a sweep response with amplified effective quality factors for all resonances over a given frequency range. We determine gain and stability conditions and include analysis for potential problematic modal interactions. This technique makes it possible to improve the measurements of resonance locations in devices, for example, sensors that rely on tracking shifts in resonance peaks. The results are demonstrated on a model for a multi-analyte micro-electromechanical systems mass sensor.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Lumped mass model of the multi-analyte mass sensor of DeMartini [1]. The shuttle mass m is driven by direct excitation f and parametric excitation via the time-varying stiffness kb(t). The n smaller masses mi (<<m) represent sensing elements. The system is tuned such that the lowest mode is a bulk mode in which all masses move essentially in unison, and the next n modes have frequencies relatively close together with mode shapes having energy localized in each of the n elements. Resonance shifts of these modes correspond to changes in mass for the respective element.

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Figure 2

Bode plot for the example n = 4 model with resonances labeled A (bulk mode), B, C, D, and E (sensing modes)

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Figure 3

Stability boundary in normalized pump amplitude δ/kb 0 versus drive frequency parameter space. Gray regions are unstable, white regions are stable. Boundary “tongues” associated with primary resonances are labeled B–E, the others arise from various combination resonances.

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Figure 4

Stability boundary in frequency region of sensing resonances B–E, from Floquet theory and from perturbation analysis, demonstrating the utility of the analytical prediction

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Figure 5

Response amplitude (dB) versus drive frequency from numerical simulations of the unpumped and pumped systems, demonstrating parametric amplification of the resonances and the lack of combination resonances in the frequency range of interest

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Figure 6

Close up of resonance b of the pumped and unpumped systems, along with the analytical prediction of the amplitude from perturbation analysis. Amplitude in dB.

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