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Research Papers

Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators

[+] Author and Article Information
J. H. Yang

School of Mechatronic Engineering,
China University of Mining and Technology,
Xuzhou 221116, China;
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109;
Jiangsu Key Laboratory of Mine Mechanical and
Electrical Equipment,
China University of Mining and Technology,
Xuzhou 221116, China
e-mail: jianhuayang@cumt.edu.cn

Miguel A. F. Sanjuán

Nonlinear Dynamics,
Chaos and Complex Systems Group,
Departamento de Física,
Universidad Rey Juan Carlos,
Tulipán s/n,
Móstoles 28933, Madrid, Spain;
Institute for Physical Science and Technology,
University of Maryland,
College Park, MA 20742

H. G. Liu

School of Mechatronic Engineering,
China University of Mining and Technology,
Xuzhou 221116, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 28, 2016; final manuscript received March 5, 2017; published online May 4, 2017. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 12(5), 051011 (May 04, 2017) (9 pages) Paper No: CND-16-1308; doi: 10.1115/1.4036479 History: Received June 28, 2016; Revised March 05, 2017

When the traditional vibrational resonance (VR) occurs in a nonlinear system, a weak character signal is enhanced by an appropriate high-frequency auxiliary signal. Here, for the harmonic character signal case, the frequency of the character signal is usually smaller than 1 rad/s. The frequency of the auxiliary signal is dozens of times of the frequency of the character signal. Moreover, in the real world, the characteristic information is usually indicated by a weak signal with a frequency in the range from several to thousands rad/s. For this case, the weak high-frequency signal cannot be enhanced by the traditional mechanism of VR, and as such, the application of VR in the engineering field could be restricted. In this work, by introducing a scale transformation, we transform high-frequency excitations in the original system to low-frequency excitations in a rescaled system. Then, we make VR to occur at the low frequency in the rescaled system, as usual. Meanwhile, the VR also occurs at the frequency of the character signal in the original system. As a result, the weak character signal with arbitrary high-frequency can be enhanced. To make the rescaled system in a general form, the VR is investigated in fractional-order Duffing oscillators. The form of the potential function, the fractional order, and the reduction scale are important factors for the strength of VR.

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Figures

Grahic Jump Location
Fig. 5

The time series of the system under different fractional-order values, the simulation parameters are f = 0.1, ω = 1500, β1=40, β2=5, a1=−1, b1=1

Grahic Jump Location
Fig. 4

The VR phenomenon occurs at the frequency ω for different fractional-order values. The simulation parameters are ω = 1500, β1=40, β2=5, a1=−1, b1=1, and in (a) α=0.6, f = 0.005, in (b) α=0.6, f = 0.1, in (c) α=1.0, f = 0.005, in (d) α=1.0, f = 0.1, in (e) α=1.4, f = 0.005, and in (f) α=1.4, f = 0.1. In each subplot, line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 3

The VR phenomenon occurs at the frequency ω for different fractional-order values. The simulation parameters are ω = 1500, f = 0.01, β1=40, a1=−1, b1=1, and in (a) α=0.6, β2=2, in (b) α=0.6, β2=5, in (c) α=1.0, β2=2, in (d) α=1.0, β2=5, in (e) α=1.4, β2=2, and in (f) α=1.4, β2=5. In each subplot, line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 2

The VR phenomenon occurs at the frequency ω for different fractional-order values. (a) The three-dimensional curve of the response amplitude Q obtained by the analytical predication. (b)–(f) The response amplitude versus the signal amplitude F for different factional-order values. The simulation parameters are ω = 1500, f = 0.01, β1=40, β2=4, a1=−1,b1=1. In (b)–(f), line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 1

The three-dimensional curve of the response amplitude Q in which there is no VR occurring at the frequency ω. The simulation parameters are ω = 1500, f = 0.01, β1=40, β = 1, a1=−1, b1=1.

Grahic Jump Location
Fig. 6

The VR phenomenon occurs at the frequency ω for different fractional-order values. The simulation parameters are f = 0.01, β1=40, β2=4, a1=−1, b1=1, and in (a) α=0.6, ω = 200, in (b) α=0.6, ω = 2000, in (c) α=1.0, ω = 200, in (d) α=1.0, ω = 2000, in (e) α=1.4, ω = 200, and in (f) α=1.4, ω = 2000. In each subplot, line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 9

The three-dimensional curve of the response amplitude Q. The simulation parameters are f = 0.01, ω = 1500, β1=40, β2=5, a1=−1, b1=1, and (a) δ=0.4, (b) δ=0.7, (c) δ=1.5, and (d) δ = 2.

Grahic Jump Location
Fig. 7

The VR phenomenon occurs at the frequency ω for different fractional-order values. The simulation parameters areω = 1500, f = 0.1, β2=4, a1=−1, b1=1, and in (a) α=0.6, β1=20, in (b) α=0.6, β1=50, in (c) α=1.0, β1=20, in (d) α=1.0, β1=50, in (e) α=1.4, β1=20, and in (f) α=1.4, β1=50. In each subplot, line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 8

There is no VR phenomenon at the frequency ω in monostable systems. (a) The three-dimensional curve of the response amplitude Q obtained by the analytical prediction. (b)–(f) The response amplitude versus the signal amplitude F for different factional-order values. The simulation parameters are ω = 1500, f = 0.01, β1=40, β2=4, a1=1, b1=1. In (b)–(f), line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 10

The VR phenomenon occurs at the frequency ω for different fractional-order values. The simulation parameters are f = 0.01, ω = 1500, β1=40, β2=5, a1=−1, b1=1, and in (a) α=0.5, δ=0.8, in (b) α=0.5, δ=1.2, in (c) α=1.0, δ=0.8, in (d) α=1.0, δ=1.2, in (e) α=1.5, δ=0.8, and in (f) α=1.5, δ=1.2. In each subplot, line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 11

The response amplitude versus the signal amplitude Ffor different factional-order values and different coefficients. The simulation parameters are ω = 1500, f = 0.01, β1=40,β2=5, a1=1, b1=1, and (a) α=0.6, δ=0.7, (b) α=0.6, δ=1.4, (c) α=1.0, δ=0.7, (d) α=1.0, δ=1.4, (e) α=1.5, δ=0.7, and (f) α=1.5, δ=1.4. In each subplot, line 1 is the analytical curve and line 2 is the numerical curve.

Grahic Jump Location
Fig. 12

The scheme for the VR at an arbitrary high frequency by the rescaled method

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