Vibrational Resonance in an Overdamped System with a Fractional Order Potential Nonlinearity

Abstract

We investigate the vibrational resonance by the numerical simulation and theoretical analysis in an overdamped system with fractional order potential nonlinearities. The nonlinearity is a fractional power function with deflection, in which the response amplitude presents vibrational resonance phenomenon for any value of the fractional exponent. The response amplitude of vibrational resonance at low-frequency is deduced by the method of direct separation of slow and fast motions. The results derived from the theoretical analysis are in good agreement with those of numerical simulation. The response amplitude decreases with the increase of the fractional exponent for weak excitations. The amplitude of the high-frequency excitation can induce the vibrational resonance to achieve the optimal response amplitude. For the overdamped systems, the nonlinearity is the crucial and necessary condition to induce vibrational resonance. The response amplitude in the nonlinear system is usually not larger than that in the corresponding linear system. Hence, the nonlinearity is not a sufficient factor to amplify the response to the low-frequency excitation. Furthermore, the resonance may be also induced by only a single excitation acting on the nonlinear system. The theoretical analysis further proves the correctness of the numerical simulation. The results might be valuable in weak signal processing.

Nonlinear analysis of the semi-active particle damping vibration isolation system based on fractional-order theory

“…As the accuracy of G-L numerical solution has been certified in some references (Liu et al. , 2004; Yang et al. , 2018), in this section, we focus on verifying the correctness of the analytical solution through several calculation examples, and the main parameters used in these examples are introduced in Table 6:…”

Main sub-harmonic joint resonance of fractional quintic van der Pol-Duffing oscillator

“…For instance, Kwon DukSoo et al investigated the dual-frequency semi-elliptical shoe antenna employing Mathieu function for Femto-cell network [31]. Jianhua Yang et al, on the other hand, explored vibrational resonance in overdamped systems with fractional potential nonlinearity [32]. Li Jimeng et al's work focused on studying a novel adaptive parallel resonance system based on vibrational resonance and stochastic resonance cascade feedback model for its application in rolling bearing fault detection [33].…”

Effects of periodic parametric damping and amplitude-modulated signal on vibrational resonance and torus-doubling bifurcations occurrence in an asymmetric mixed Rayleigh-Liénard oscillator

“…The phenomenon of vibrational has received extensive research attention during the last two decades. Due to diverse potential applications of vibrational resonance in many areas of science and modern engineering [37][38][39][40][41], the phenomenon of vibrational resonance has been investigated theoretically and numerically on a diversity of dynamic systems such as asymmetric Duffing oscillator [42], systems with nonlinear damping [43][44][45][46], bistable systems [47][48][49], multistable systems [50,51], excitable systems [52], quintic oscillator [53,54], coupled oscillators [55,56], fractional order oscillators [57,58], biological nonlinear systems [59,60], position dependent mass oscillators [61], parametrically excited systems [61][62][63][64][65], oscillators chains and two-cell cubic autocatalytic reaction-diffusion model [66][67][68][69][70]. Moreover, the phenomenon of vibrational resonance has been explored experimentally [71][72][73][74].…”