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Research Papers

Sensitivity Enhancement of Cantilever-Based Sensors Using Feedback Delays

[+] Author and Article Information
Calvin Bradley

 Michelin Americas Research and Development Corporation, Greenville, SC 29605calvinbradley@gmail.com

Mohammed F. Daqaq1

Nonlinear Vibrations and Energy Harvesting Laboratory (NoVEHL), Department of Mechanical Engineering, Clemson University, Clemson, SC 29634mdaqaq@clemson.edu

Amin Bibo

Nonlinear Vibrations and Energy Harvesting Laboratory (NoVEHL), Department of Mechanical Engineering, Clemson University, Clemson, SC 29634abibo@clemson.edu

Nader Jalili

Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115n.jalili@neu.edu

Since delays are present in any feedback signal and their effect is especially apparent in systems with very high natural frequency such as those at the microscale; feedback signals without delays are hard to obtain making this approach even more feasible.

In general, neutral delay-differential equations need stronger conditions for stability, see Ref. 37 for further details.

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(4), 041014 (Aug 17, 2010) (9 pages) doi:10.1115/1.4001975 History: Received December 11, 2008; Revised June 04, 2010; Published August 17, 2010; Online August 17, 2010

This paper entails a novel sensitivity-enhancement mechanism for cantilever-based sensors. The enhancement scheme is based on exciting the sensor at the clamped end using a delayed-feedback signal obtained by measuring the tip deflection of the sensor. The gain and delay of the feedback signal are chosen such that the base excitations set the beam into stable limit-cycle oscillations as a result of a supercritical Hopf bifurcation of the trivial fixed points. The amplitude of these limit-cycles is shown to be ultrasensitive to parameter variations and, hence, can be utilized for the detection of minute changes in the resonant frequency of the sensor. The first part of the manuscript delves into the theoretical understanding of the proposed mechanism and the operation concept. Using the method of multiple scales, an approximate analytical solution for the steady-state limit-cycle amplitude near the stability boundaries is obtained. This solution is then utilized to provide a comprehensive understanding of the effect of small frequency variations on the limit-cycle amplitude and the sensitivity of these limit-cycles to different design parameters. Once a deep theoretical understanding is established, the manuscript provides an experimental study to investigate the proposed concept. Experimental results demonstrate orders of magnitude sensitivity enhancement over the traditional frequency-shift method.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A schematic of the cantilever beam subjected to delayed-feedback base excitations

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Figure 2

Stability pockets of Eq. 2. The shaded regions represent the gain-delay combinations leading to asymptotically stable cantilever response. The solid lines represent theoretical stability boundaries; the triangles (sweep up) and circles (sweep down) represent the experimental onset of the Hopf bifurcation points. This chart is obtained for μ=0.01 and the parameters listed in Table 1.

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Figure 3

Sketches of fixed points and their stability. (a) Nμ¯>0 and (b) Nμ¯<0.

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Figure 4

Variations of (a) μ¯N and (b) N along the stability boundary. Results are obtained for μ=0.01.

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Figure 5

Variation of the limit-cycle amplitude with κ for different values of δ

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Figure 6

Sensitivity of the limit-cycle amplitude to variations in γ̂ for different values of δ and κ=0.001

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Figure 7

Variation of the limit-cycle amplitude as κ is increased for different values of K̂

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Figure 8

Sensitivity of the limit-cycle amplitude to variations in γ̂ for different values of κ and δ=0.001

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Figure 9

Sensitivity of the limit-cycle amplitude to variations in the quality factor Q for different values of the critical gain K̂. Results are obtained for κ=0.003 and δ=0.001.

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Figure 10

Test fixture used for experimental investigation

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Figure 11

Variation of the limit-cycle amplitude with the gain K for time delay γ=0.4. The results are obtained experimentally for the cantilever beam under consideration.

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Figure 12

Variation of the limit-cycle amplitude with the gain K and the delay γ. The results are obtained experimentally for the cantilever beam under consideration. Theoretical curves (solid lines in the figure) were obtained using nonlinearity coefficients that are 20% higher than those obtained theoretically and listed in Table 1.

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Figure 13

Variation of the limit-cycle amplitude due to the addition of various tip masses to the cantilever beam for gain K=0.44 and delay γ=0.4

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Figure 14

Percentage drop in the limit-cycle amplitude and resonance frequency as function of the added mass. Results are obtained experimentally for a delay γ=0.4 and a gain K=0.44.

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Figure 15

Percentage drop in the limit-cycle amplitude as function of the added mass. Results are obtained experimentally for a delay γ=0.35 and a critical gain K̂≈0.53.

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