Generalized Hopf bifurcation of a non-smooth railway wheelset system

The strong coupling, multivariate, and nonlinear characteristics of the mathematical model of a doubly-fed induction generator (DFIG) make it challenging to analyze the bifurcation category of the DFIG. Therefore, in this study, we developed a method for analyzing the DFIG model via topologically equivalent dimensionality reduction in the neighborhood of bifurcation values based on the central manifold theorem and realized the discrimination of the Hopf bifurcation type of the DFIG. We established a complex frequency domain model of the DFIG and then calculated the critical open-loop gain point using the root trajectory method to obtain the relationship between the critical open-loop gain point and Hopf bifurcation. Thereafter, we performed topological equivalence dimensionality reduction in the neighborhood of the DFIG bifurcation values based on the central manifold theorem. Subsequently, the Hopf bifurcation type of the system was determined using the stability index (first Lyapunov exponent) of the Hopf bifurcation. Finally, a nonlinear time-domain simulation was used to verify that the four and two-dimensional DFIGs generate a stable limit cycle by supercritical Hopf bifurcation under the tuning parameter. The results indicated that the proposed method was feasible and effective in the neighborhood of bifurcation values in the state parameter space of DFIG, which provides a theoretical basis for the control of Hopf bifurcation of the DFIG.

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“…Let the smooth autonomous system be expressed as follows 21,22 :

\left\{\begin{array}{c}\displaystyle \frac{dx}{dt}={A}_{1}x+g(x,y)\\ \displaystyle \frac{dy}{dt}={A}_{2}y+h(x,y)\end{array}\right..

…”

Section: Equivalent Dimensionality Reduction Model Of Dfigmentioning

confidence: 99%

“…where A represents the linear part of the system and f x ( ) represents the nonlinear part. By introducing the nonsingular linear transformation matrix Q such that x = Qy, the linear part of Equation (20), that is, x Ax ̇= can be transformed into Equation (21).…”

Section: Dfig Topological Equivalence Dimensionality Reduction Modelmentioning

confidence: 99%

Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

Chen

Wei

et al. 2023

Energy Science & Engineering

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“…The nonlinear factors in the hunting motion mainly arise from the wheel-rail creepslip forces and the wheel-rail impacts. In studying the cone-tread wheel and small-amplitude hunting instability, wheel-rail contact geometry relationships based on equivalent conicity linearization are widely used [32][33][34][35][36]. In studies using worn-type wheels, scholars have used polynomial fi ing [37,38], least-squares fi ing [39], measured data [12], and software simulations [40][41][42] to obtain nonlinear wheel-rail contact geometry relationships.…”

Section: Introductionmentioning

confidence: 99%

The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System

Wang,

Ma,

Zhang

2024

Applied Sciences

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The complex service environment of railway vehicles leads to changes in the wheel–rail adhesion coefficient, and the decrease in critical speed may lead to hunting instability. This paper aims to reveal the diversity of periodic hunting motion patterns and the internal correlation relationship with wheel–rail impact velocities after the hunting instability of a bogie system. A nonlinear, non-smooth lateral dynamic model of a bogie system with 7 degrees of freedom is constructed. The wheel–rail contact relations and the piecewise smooth flange forces are the main nonlinear, non-smooth factors in the system. Based on Poincaré mapping and the two-parameter co-simulation theory, hunting motion modes and existence regions are obtained in the parameter plane consisting of running speed v and the wheel–rail adhesion coefficient μ. Three-dimensional cloud maps of the maximum lateral wheel–rail impact velocity are obtained, and the correlation with the hunting motion pattern is analyzed. The coexistence of periodic hunting motions is further revealed based on combined bifurcation diagrams and multi-initial value phase diagrams. The results show that grazing bifurcation causes the number of wheel–rail impacts to increase at a low-speed range. Periodic hunting motion with period number n = 1 has smaller lateral wheel–rail impact velocities, whereas chaotic motion induces more severe wheel–rail impacts. Subharmonic periodic hunting motion windows within the speed range of chaotic motion, pitchfork bifurcation, and jump bifurcation are the primary forms that induce the coexistence of periodic motion.

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“…Yan and Zeng 9 discussed the nonlinear governing motion equations of railway bogie and analyzed the nonlinear dynamical characteristics arising from the yaw damper and different wheel/rail contact relation related to the rolling radius and the contact angle. Miao et al 10 investigated the generalized Hopf bifurcation of a four‐dimension non‐smooth railway wheelset system. Li et al 11 investigated the bifurcation and chaotic behaviors of a wheelset system and discussed the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control method.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation, geometric constraint, chaos, and its control in a railway wheelset system

Cui

2022

Math Methods in App Sciences

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In this paper, an improved railway wheelset system is presented. The dynamical behaviors of the system are investigated, including dissipativity and invariance of the system, stability of zero-equilibrium point, and the bifurcation characteristics of railway wheelset system at zero equilibrium point. Furthermore, the chaotic behaviors of the motion of railway wheelset and the dynamical behaviors of the railway wheelset system under geometric constraint are studied.It shows that the railway wheelset system has complex dynamical phenomena owing to nonlinear factor and the railway wheelset system may have different complex dynamical behaviors with different nonlinear parameters. In addition, the motion of railway wheelset also has chaotic behaviors under geometric constraint. Finally, the chaos control of the railway wheelset system is achieved by using linear feedback control method. The numerical simulations are carried out in order to analyze the complex phenomena of the railway wheelset system.

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Generalized Hopf bifurcation of a non-smooth railway wheelset system

Cited by 9 publications

References 26 publications

Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System

Bifurcation, geometric constraint, chaos, and its control in a railway wheelset system

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