In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

This paper presents a multi-objective optimization approach for a fractional-order control system for electricmagnetic suspension (EMS) type magnetic levitation (maglev) models, aiming to optimize the non-linear magnetic buoyancy control parameters. Firstly, a multi-body dynamics model of an EMS-type maglev vehicle with a fractional-order PID control system is developed. To fulfill the multi-objective requirements and optimize the fractional order parameters, an improved multiobjective optimization algorithm is proposed. The Sperling index and the magnitude of the vertical suspension gap variation are employed as the optimization objectives. Then, the aforementioned multi-objective optimization algorithm is applied to the constructed multi-body dynamics model to perform multiobjective optimization of the control parameters. Finally, the Pareto front solution is calculated and the optimal control parameters are obtained. The results demonstrate that the optimized control parameters can significantly reduce the vertical suspension gap and Sperling index during operation, with the reduction in the vertical suspension gap amplitude and its standard deviation being 10∼73%, and the reduction in the Sperling index being 8∼13%.

The electromagnet levitation control system is the core of maglev trains, but various external disturbances make the design of the controller difficult. In addition, multiple electromagnet maglev systems are coupled, and the independent control mode lacks the necessary synchronization coordination mechanism. This study proposes a linear expanded state observer (LESO)–backstepping control method with double cross coupling of air gap and velocity for the levitation frame of electromagnetic suspension maglev vehicles to eliminate external disturbances and unknown influences, thus improving the synchronization performance of each point on the levitation frame. A virtual controller is designed with the backstepping control strategy, and the unknown disturbances in the controller are compensated for by LESO. A single-electromagnet control strategy is derived. Then, the synchronization error among the levitation points is used as an additional feedback signal for the input of the four levitation point controllers to demonstrate the closed-loop stability of the system under the proposed strategy. Results demonstrate the tracking effect of the observer and the superiority of the proposed method in improving the synchronization and tracking performance of the levitation frame points.

In this paper, the necessary condition for the chaotic motion of a Duffing oscillator with the fractional‐order derivative under harmonic excitation is investigated. The necessary condition for the chaos in the sense of Smale horseshoes is established based on the Melnikov method, and then the chaotic threshold curve is obtained. The largest Lyapunov exponents are provided, and some other typical numerical simulation results, including the time histories, frequency spectrograms, phase portraits, and Poincare maps, are presented and compared. From the analysis of the numerical simulation results, it could be found that, near the chaotic threshold curve, the system generates chaos via the period‐doubling bifurcation, from single periodic motion to period‐2 motion and period‐4 motion to chaotic motion. The effects of fractional‐order parameters, the stiffness coefficient, and the damping coefficient on the threshold value of the chaotic motion are analytically discussed. The results show that the coefficient of the fractional‐order derivative has great effect on the threshold value of the chaotic motion, while the order of the fractional‐order derivative has less. The analysis results reveal some new phenomena, and it could be useful for designing or controlling dynamic systems with the fractional‐order derivative.

At present, there are few studies considering both nonlinear and fractional characteristics of suspension in vehicle systems. In this paper, a fractional nonlinear model of a quarter vehicle with two-degree-of-freedom (2-DOF) is innovatively proposed to describe the suspension system containing the viscoelastic material metal rubber. Given the lack of a general calculation scheme for the multi-degree-of-freedom fractional-order incremental harmonic balance method (IHBM), a general calculation scheme for the 2-DOF incremental harmonic balance method for nonlinear systems with fractional order is derived. The nonlinear dynamical properties of the presented model are acquired using this method. The accuracy of the proposed method is verified through a comparison with the power series expansion method. Afterward, the effects of the various parameters on the dynamic performance are analyzed. The vibration peak value of the fractional-order model is significantly higher than that of the integer-order model (IOM). Therefore, the suspension parameters should be designed with a margin when using IOM.

The chaotic behavior of a nonlinear vehicle suspension system with a fractional-order derivative is considered. A hysteretic nonlinear model of vehicle suspension with a fractional-order derivative term is established and the analytically necessary condition for heterotopic chaos is derived based on the Melnikov method. The largest Lyapunov exponents are compared. Then, the necessary condition is numerically verified by various simulation factors. The possibility of chaotic motion in vehicles should be higher for larger amplitude of road excitation. It is found that the coefficient and order of the fractional differential term, the stiffness coefficient, and the damping coefficient of the system all affect the necessary condition, and analysis on the effects of these parameters is presented individually. It has been shown that the larger the coefficient of the fractional differential term, or the stiffness and damping coefficients of the system, the lower the possibility of chaos in the system. Meanwhile, without considering the fractional order, the integer order suspension model obviously reduces the actual area where chaos may occur, so designing suspensions according to the fractional order model can avoid chaos more accurately than the integer order model.