Output Amplitude Gain Research Articles

Time-Delay Effects on the Collective Resonant Behavior in Two Coupled Fractional Oscillators with Frequency Fluctuations

In this study, we propose coupled time-delayed fractional oscillators with dichotomous fluctuating frequencies and investigate the collective resonant behavior. Firstly, we obtain the condition of complete synchronization between the average behavior of the two oscillators. Subsequently, we derive the precise analytical expression of the output amplitude gain. Based on the analytical results, we observe the collective resonant behavior of the coupled time-delayed system and further study its dependence on various system parameters. The observed results underscore that the coupling strength, fractional order, and time delay play significant roles in controlling the collective resonant behavior by facilitating the occurrence and optimizing the intensity. Finally, numerical simulations are also conducted and verify the accuracy of the analytical results.

Resonant behaviors of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler memory kernel

The dynamics of coupled oscillators and their collective behaviors are of great significance in various fields. This paper explores the resonant behaviors in the system composed of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler (M-L) memory kernel. The study begins by demonstrating that the two particles synchronize in the long-time limit, indicating that the mean-field behavior aligns with that of each individual particle. Accordingly, we derive an exact analytical expression for the steady-state output amplitude gain (OAG) of the mean field using the stochastic average method. Furthermore, the research investigates the diverse phenomena of generalized stochastic resonance (GSR) and examines how various system parameters influence GSR. To validate the theoretical analysis, numerical simulations are conducted to support and verify our findings.

The synchronization, stability and stochastic resonance of stochastic global coupled system

In this paper, we aim at a global coupled system influenced by a multiplicative dichotomous noise and an additive Gaussian white noise. Since the analysis of the synchronization and stability is important to a coupled system, and there is little general method to analyze them. We provide methods to derive the synchronization and stability of particles in the system. Guaranteed by the synchronization and stability, we obtain the analytic expression of each particle’s average displacement. To verify our results, we use stochastic Taylor expansion to derive numerical iterative expressions of the displacements of particles. As for numerical simulations, we find that the numerical solutions match the analytical solutions, and we display the synchronization speed of particles, the stable region of noise intensity, resonance areas and the output amplitude gain with respect to various parameters of the system. We find that the phenomenon of stochastic resonance occurs if and only if the parameters are situated in the resonance area.

Generalized stochastic resonance in a time-delay fractional oscillator with damping fluctuation and signal-modulated noise

In this study, we propose a time-delayed fractional oscillator (FO) subjected to damping fluctuation and signal-modulated noise, and investigate the generalized stochastic resonance (GSR) behaviors. By using (fractional) Shapiro–Loginov formula and Laplace transform, we obtain the first-order moment of system stationary state response, and the output amplitude gain (OAG). Based on the analytical results, it is observed that the GSR behaviors widely exist in the system, and they can be effectively controlled by the system parameters, including driving frequency, noise parameters, fractional order and time delay. It is also demonstrated that fractional order and time delay are two effective dimensions to regulate the GSR intensity under different parameter conditions, and the diversity of GSR behaviors mainly depends on the fractional order.

A coupled fractional-order system with fluctuating frequency and its application in bearing fault diagnosis

In this paper, a coupled fractional-order system with fluctuating frequency driven by different periodic signals under various damping strength is investigated. Firstly, based on the Shapiro-Loginov formula and Laplace transform method, the expressions for the output amplitude gain (OAG) of the two subsystems are derived and the resonant behaviors of particles are analyzed. The OAG exhibits various resonance behaviors in response to variations in system parameters, input signals and dichotomous noise, including parameter-induced stochastic resonance, bona-fide resonance and stochastic resonance. Especially, the average behavior of the two output signals is synchronized when two subsystems’ input signals and damping strengths are equal, which is verified in the numerical simulation. Finally, the proposed system is applied to the bearing fault diagnosis to evaluate its engineering application value. The results prove that the system is effective in diagnosing fault signals and has excellent performance.

Stochastic resonance of double fractional-order coupled oscillator with mass and damping fluctuations

In this study, the stochastic resonance phenomenon of a coupled double fractional-order harmonic oscillator with mass and damping fluctuation is investigated. Firstly, the Shapiro-Loginov formula and Laplace transform are used to obtain the analytical expression of the output amplitude gain of the system output. On this basis, aiming at the key factors involved in the model, including the coupling structure, fractional system, random fluctuation and external periodic force, the influence of coupling coefficient, double fractional order and driving frequency on the output amplitude gain (OAG) is analyzed, and reasonable physical explanations are provided. Secondly, numerical simulations are carried out to verify the accuracy of the theoretical solutions. The simulation results show that under certain conditions, the OAG of the system can appear stochastic resonance phenomenon with the above parameters, especially: (1) The OAG with the change of external drive frequency appears double peak, single peak and single valley stochastic resonance phenomenon, which does not appear under the same external disturbance with integer order and uncoupled conditions; (2) The order of double fractional derivative significantly affects the variation trend of OAG; (3) The coupling coefficient is not sensitive to the OAG.

Resonance behavior of fractional harmonic oscillator driven by exponentially correlated dichotomous noises

This paper investigates the resonance behaviors of a fractional-order harmonic oscillator driven by two exponentially correlated dichotomous noises, where the Caputo fractional derivative operator is applied to describe the power-law memory of the system. By using the stochastic averaging method and the Shapiro-Loginov formula, we derive the analytical expression of the output amplitude gain of the system, from which the existence and the correlation of noises are found to be keys for the occurrence of resonance. When either of the noises is absent or they are uncorrelated, the output amplitude gain is zero, indicating that the system is dissipative in this case. The numerical simulation shows that the system has rich resonance behaviors when noises are exponentially correlated. Three types of resonance, that is, the bona fide resonance, the classic stochastic resonance and the generalized stochastic resonance, are observed. And the effects of system parameters on these resonance behaviors are well discussed. Specifically, double-peak resonance and damping-coefficient–induced resonance are observed only in the fractional-order system rather than integer-order system.

Stochastic resonance of fractional-order coupled system excited by trichotomous noise

In order to describe the motion behavior of coupled particles with mass fluctuations in a viscous medium, we propose a corresponding model, namely a fractional-order coupled system excited by trichotomous noise. By using the Shapiro-Loginov formula and the Laplace transform, we find the statistical synchronization of the system, then obtain analytical expression of the system output amplitude gain. On this basis, this paper focuses on the key points, which are the coupled system, the fractional order system and the trichotomous noise, analyzes the influences of coupling coefficient, system order and noise steady-state probability on the generalized stochastic resonance phenomenon of system’s output amplitude gain, and gives some reasonable explanations. Specifically, first, as the coupling coefficient increases, the generalized stochastic resonance phenomenon of the output amplitude gain of the system first increases and then weakens until it converges. This phenomenon shows that the appropriate coupling strength can promote the generation of system resonance, thereby reflecting the importance of studying coupled systems. Second, with the order of the system increases, the generalized stochastic resonance phenomenon of the system’s output amplitude gain weakens gradually. When the system order value is 1, that is, when the system degenerates into an integer order system, the peak value of its output amplitude gain is smallest. This phenomenon shows that the fractional order system can obtain a larger output amplitude gain than the traditional integer order system. Third, the effect of the steady-state probability of noise on the output amplitude gain of the system changes with other related parameters. Under certain parameter conditions, trichotomous noise can not only make the output amplitude of the system larger than that of the system excited by dichotomous noise, but also change the resonance type of the system. Finally, the correctness of the above results is verified by numerical simulation.

Collective behaviors of two coupled harmonic oscillators driven by different frequency fluctuations with fractional damping

In a complex environment, the long-time memory and coupling effect are two important system characteristics that can be described by the fractional damping force and coupling force. This paper investigates the collective behaviors of two coupled fractional harmonic oscillators driven by different frequency fluctuations, including stability, synchronization and stochastic resonance (SR). Theoretically, the ‘synchronization condition’ and ‘stability condition’ of the system are derived. Comparative analysis shows that the latter is stricter than the former. Based on this, an analytical expression of the output amplitude gain is obtained. The numerical results show that when the stability condition is met, the average trajectories of two particles are both bounded and synchronous. Otherwise, they will diverge to infinity. Increasing ɛ (coupling strength) and decreasing α (fractional order) can both accelerate the synchronization speed. SR mainly occurs in the high-α or high-σ (noise amplitude) region, which means that SR emergence can be controlled by adjusting α or σ. The damping force, coupling force and frequency fluctuations compete with each other; thus, the SR intensity should be maximized by adjusting α, ɛ and σ simultaneously.

Cascaded Nonlinear Mass Fluctuation Stochastic Resonance System and Its Application in Bearing Fault Diagnosis

Stochastic resonance (SR) can realize bearing fault signal diagnosis by transferring noisy energy. In order to enhance the output response of the system and realize effective signal extraction, the nonlinear mass fluctuation SR system caused by nonlinear asymmetric dichotomous noise is cascaded to obtain the cascaded nonlinear mass fluctuation SR system. First, the output amplitude gain of the first-stage of the system is derived, and the influence of different parameters on it is explored; then the effects of different parameters of the cascaded system on the output amplitude gain and the output SNR are studied separately, which proves that the cascaded system can effectively double enhance the output response of the system; finally, the adaptive genetic algorithm is used to solve the difficulty of parameter adjustment, and the cascaded nonlinear mass fluctuation SR system is applied to the bearing fault diagnosis. The system proposed in this paper takes into account the effects of nonlinear asymmetric dichotomous noise and cascaded systems and performs waveform smoothing and double enhancement of the output signal. It can better extract fault signals and has effective engineering value.

Collective stochastic resonance behaviors of two coupled harmonic oscillators driven by dichotomous fluctuating frequency* *Project supported by the National Natural Science Foundation of China for the Youth (Grant Nos. 11501385 and 11801385).

The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated, including stability, synchronization, and stochastic resonance (SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain (OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.

The impact of memory effect on resonance behavior in a fractional oscillator with small time delay

We study stochastic resonance (SR) phenomenon in the fractional oscillator with time delay and damping fluctuation, analyze the impact of time delay and fractional damping as two memory ingredients on SR, and put forward firstly the concept of the robustness of GSR. By moment method, we obtain the analytical expression of the output amplitude gain and find that fluctuations in the output amplitude gain are non-monotonic. Using numerical simulations we verify the accuracy of the analytical results. We find (i) that the length of time delay and system order are parameters related to memory; (ii) that the output amplitude gain could attain a maximum by increasing driving frequency close to system frequency, and small time delay and system order contribute to enhance the resonance intensity; (iii) that the evolution of the output amplitude gain versus the noise intensity exhibits one-peak resonance, and small time delay can enhance the resonance intensity and the robustness of SR regarding to driving frequency and system frequency; (iv) that the evolution of the output amplitude gain versus the noise correlation rate presents one-peak resonance in the presence of small time delay, critical time delay is bigger with the increasing system order when noise intensity is fixed and critical time delay is smaller with the increasing noise intensity when system order is fixed.

Trichotomous noise induced resonance behavior of two coupled harmonic oscillators with fluctuating mass

This paper investigates the stochastic resonance phenomenon in two coupled harmonic oscillators with fluctuating mass. The fluctuation of the mass is modeled as a trichotomous noise. Applying the Shapiro-Loginov formula and the Laplace transform, the statistic synchronization between the two particles is obtained and defined. Then, the analytical expression of the output amplitude gain is derived. Based on the analytical result, the influence of the trichotomous noise’steady-state probability and the coupling strength on the resonance behaviors of the output amplitude gain are discussed. In the strong noise region, the output amplitude gain of the system will exhibit stochastic resonance with change of the steady state probability, so that the system output amplitude gain induced by trichotomous noise is greater than induced by dichotomous noise. Beside, the system output amplitude gain has resonance behavior with respect to the coupling strength too. Unlike the case of dichotomous noise, the system’s output amplitude gain can show double-peaks resonance when driven by trichotomous noise.

Stochastic resonance research on under-damped nonlinear frequency fluctuation for coupled fractional-order harmonic oscillators

Stochastic resonance on under-damped nonlinear frequency fluctuation for coupled fractional-order linear particles is investigated. First, the average motions of the two coupled Brownian particles obtained by the random average method and the Shapiro-Loginov formula are completely synchronized. Then, on the basis of synchronization, the mathematical formula of the system output amplitude gain of one of the particles is calculated. Numerical simulation results show that there exist various stochastic resonance phenomena with system and noise parameters changing, such as double-peak SR (DSR), bona-fide resonance (BFR), and parameter-induced SR (PSR). The nonlinear noise coefficient plays a significant role in the system. Finally, the accuracy of theoretical analysis is verified by Simulink numerical simulation.

Stochastic Resonance in Coupled Underdamped Harmonic Oscillators with Fluctuating Frequency Driven by Dichotomous Noise

In this paper, the resonance behaviors of two coupled harmonic oscillators with fluctuating frequency driven by an external periodic signal are investigated. Applying the Shapiro–Loginov formulas and the Laplace transform technique, the statistic synchronization between the two particles is observed and defined. Based on this finding, the analytical expression of the output amplitude gain (OAG) is derived. It is shown that the OAG is a non-monotonic function of the system parameters. The influences of the coupling strength as well as the other parameters on the resonance behaviors of OAG are discussed, including parameter-induced stochastic resonance, bona fide resonance and stochastic resonance. It is found that weak coupling can not only affect the resonance strength, but also change the resonance form. On the other hand, strong enough coupling can bind the particles together to move as a whole. In this case, the influence of the coupling will vanish. Furthermore, some numerical simulations are presented to verify the effectiveness of the theoretical analysis results.

Study on a Large-Scale Three-Dimensional Ultrasonic Plastic Welding Vibration System Based on a Quasi-Periodic Phononic Crystal Structure

The uniformity of amplitude distribution and amplitude gain are two main factors affecting the performance of ultrasonic welding vibration system. In order to improve the uniformity of amplitude distribution and amplitude gain of welding surface to enhance the performance of the vibration system, a new design method of a large-scale three-dimensional ultrasonic plastic welding vibration system based on a quasi-periodic phononic crystal structure is proposed. In this method, the composite horn combined with a conical section and a cylindrical section can effectively improve the output amplitude gain of the welding surface. In addition, the method forms a quasi-periodic phononic crystal structure by slotting in a large-scale three-dimensional tool head, and utilizes the band gap property of the structure to effectively suppress lateral vibration of the tool head and improve the amplitude distribution uniformity of the tool head’s welding surface. However, when the size of the tool head is relatively large, the quasi-periodic phononic crystal structure cannot suppress the lateral vibration very well. Therefore, the paper processes fan-shaped slopes on the output surface of the tool head which can further improve the uniformity of the amplitude distribution and amplitude gain. Finally, the simulation analysis and experiments show that the design method can optimize the large-scale three-dimensional ultrasonic plastic welding system, improve the uniformity of the vibration distribution and increase the output amplitude gain of the welding surface.

The resonance behavior in the fractional harmonic oscillator with time delay and fluctuating mass

We study stochastic resonance (SR) in a fractional harmonic oscillator with time delay and mass fluctuation. Using a moment equation, we derive the exact analytical expression of the output amplitude and find that fluctuations in the output amplitude are nonmonotonic. Using numerical simulations we verify the accuracy of the analytical result. We find (i) that the evolution of output amplitude versus the frequency of the external periodic force exhibits One-Peak SR and Two-Peaks SR, resonance region and resonance form largely depend on the noise correlation rate and noise intensity, (ii) that the evolution of output amplitude versus the noise intensity exhibits One-Peak SR and the increasing time delay lower the critical driving frequency, and (iii) that the evolution of output amplitude versus feedback gain performs diverse resonance forms, the increasing time delay widens the region for occurrence of SR and diversifies the resonance form.

Stochastic resonance in coupled fractional-order linear harmonic oscillators with damping fluctuation

The stochastic resonance (SR) in the coupled linear harmonic oscillators system containing the fractional-order damping with random dichotomous fluctuation is investigated in this paper. Firstly, by using the mathematical properties of the dichotomous noise, the Shapiro–Loginov formula and the stochastic averaging method, the first moment of the two particles are verified to be equal. Secondly, the analytic expression of the output amplitude gain (G) of the coupled oscillators system is obtained by the Laplace transform method. Thirdly, the numerical simulation results corresponding to the analytic expression show that the fractional-order, coupling coefficient, and damping coefficient all have a great influence on G of the system. Meanwhile, the double stochastic resonances (DSR), parameter-induced stochastic resonance (PSR) and bona-fide resonance (BFR) are discussed based on numerical simulation. Finally, the accuracy of the analytic expression of G has been verified by numerical simulation using the stochastic Euler algorithm.

The resonance behavior in two coupled harmonic oscillators with fluctuating mass

This paper studies the resonance behavior in two coupled harmonic oscillators with fluctuating mass. Firstly, the statistic synchronization between the two particles is obtained and defined. Then, the analytical expression of the output amplitude gain is derived by using the stochastic averaging method. Based on the analytical result and some corresponding numerical results, we analyze the coupling’s influence on the resonance behaviors of the output amplitude gain, including the parameter-induced stochastic resonance, the bona fide resonance, and the stochastic resonance. In weak coupling region, the coupling can not only enhance or weaken the resonance behaviors, but also change the resonance forms. In strong coupling region, the two particles are forced together by the coupling force and move synchronously; thus, the coupling’s influence will vanish. Finally, some numerical simulations are performed to provide a verification and an intuitive understanding of the theoretical results.

Stochastic resonance of two coupled fractional harmonic oscillators with fluctuating mass

The stochastic resonance (SR) of two coupled fractional harmonic oscillators with a dichotomous fluctuating mass is investigated. The complete synchronization between the average behaviors of the two oscillators is obtained, and the analytical expression of the output amplitude gain (OAG) is derived. The SR of the coupling system is analyzed based on the analytical results. Coupling strength and fractional order exert important influences on the SR behavior of OAG. They can not only control the emergence of SR, but also optimize the resonance intensity. Numerical simulations are also performed to provide an intuitive understanding of the theoretical results.