Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: 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.

Date of download: 10/27/2017 Copyright © ASME. All rights reserved. From: Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators J. Comput. Nonlinear Dynam. 2017;12(5):051011-051011-9. doi:10.1115/1.4036479 Figure Legend: The scheme for the VR at an arbitrary high frequency by the rescaled method