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.
A periodic vibrational resonance in the fractional-order bistable system
