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Research Papers

Uncertainty Propagation in the Band Gap Structure of a 1D Array of Magnetically Coupled Oscillators

[+] Author and Article Information
Brian P. Bernard

Graduate Research Assistant
e-mail: brian.bernard@duke.edu

Benjamin A. M. Owens

Graduate Research Assistant
e-mail: bao4@duke.edu

Brian P. Mann

Associate Professor
Mem. ASME
e-mail: brian.mann@duke.edu
Dynamical Systems Laboratory,
Department of Mechanical Engineering,
Duke University,
Durham, NC 27708

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 27, 2012; final manuscript received November 15, 2012; published online June 6, 2013. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 135(4), 041005 (Jun 06, 2013) (7 pages) Paper No: VIB-12-1121; doi: 10.1115/1.4023821 History: Received April 27, 2012; Revised November 15, 2012

The propagation constant technique has previously been used to predict band gap regions in linear oscillator chains by solving an eigenvalue problem for frequency in terms of a wave number. This paper describes a method by which selected design parameters can be separated from the eigenvalue problem, allowing standard uncertainty propagation techniques to provide closed form solutions for the uncertainty in frequency. Examples are provided for different types of measurement or environmental uncertainty showing the varying robustness of a band gap region to changes in parameters of the same or different order. The system studied in this paper is comprised of repelling magnetic oscillators using a dipole model. Numerical simulation has been performed to confirm the accuracy of analytical solutions up to a certain level of base excitation amplitude after which nonlinear effects change the predicted band gap regions to low energy chaos.

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Figures

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Fig. 1

An example configuration showing n = 2 unit cells

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Fig. 2

The frequency band structure shows the plotted solutions to Eq. (14) for the case when m1 = 0.04 kg, m2 = 0.04 kg and shows that there is a propagation constant for all frequencies < 7 Hz

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Fig. 3

The frequency band structure shows the plotted solutions to Eq. (14) for the case when m1 = 0.04 kg, m2 = 0.02 kg. The shaded region illustrates the attenuation zone between frequencies 5–7 Hz where Eq. (14) has no propagation constant solutions.

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Fig. 4

The frequency band structure shows the plotted solutions to Eq. (14) for different mass ratios by holding m1 = 0.04 kg constant and varying m2. Shaded regions represent propagation zones and white space corresponds to attenuation zones.

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Fig. 5

The frequency band structure shows the plotted solutions to Eq. (14) for the case when m1 = 0.04 kg, m2 = 0.02 kg. Dotted lines represent uncertainty boundaries corresponding to ±10% of M, which significantly reduce the size of the shaded attenuation zone.

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Fig. 6

Propagation zones for the case when m1 = 0.04 kg, m2 = 0.02 kg as magnetization (M) suffers degradation

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Fig. 7

The frequency band structure shows the plotted solutions to Eq. (14) for the case when m1 = 0.04 kg, m2 = 0.02 kg. Dotted lines represent uncertainty boundaries corresponding to ±10% of m and V, which slightly reduce the size of the shaded attenuation zone.

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Fig. 8

Propagation zones for the case when m1 = 0.04 kg, m2 = 0.02 kg as mass and volume suffer degradation

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Fig. 9

Numerically simulated frequency response (relative to the displacement amplitude of the driven oscillator) of two oscillators in a 20 unit cell chain using the constants from Table 1 for the case when m1 = 0.04 kg, m2 = 0.02 kg and base excitation is (a) A = 0.0005 m, (b) A = 0.003 m, and (c) A = 0.01 m

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Fig. 10

Numerically simulated relative asymptotic kinetic energy of two oscillators in a 20 unit cell chain using the constants from Table 1 for the case when m1 = 0.04 kg, m2 = 0.02 kg and base excitation is (a) A = 0.01 m, (b) A = 0.025 m

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Fig. 11

Response of each oscillator of a 20 unit cell chain under base excitation of A = 0.01 m at ω = 3 Hz (propagation zone)

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Fig. 12

Response of each oscillator in a 20 unit cell chain under base excitation of A = 0.01 m at ω = 5.66 Hz (attenuation zone)

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Fig. 13

Response of each oscillator in a 20 unit cell chain under base excitation of A = 0.01 m at ω = 6.25 Hz (analytically predicted propagation zone, numerically predicted attenuation zone)

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