Random double-time-delay model for railway wheel set system and establishment method thereof

Abstract

The invention discloses a random double-time-delay model for a railway wheel set system and an establishment method thereof, relates to the technical field of wheel set systems, and aims to consider few model structures for time-delay displacement feedback control on a series of suspension in the research of the wheel set system and to better conform to the practical significance. A dynamical model of a double-time-lag wheel set system under Gaussian white noise parametric excitation needs to be established. The invention provides a random double-time-delay model for a railway wheelset system and an establishment method thereof, and the method comprises the following steps: S1, establishing a wheelset system motion equation, and obtaining a balance point system equation according to the state conversion of the wheelset system at a balance point; s2, using a central manifold theorem and a random average method to carry out dimension reduction processing on the wheel set system motion equation to obtain a random double-time-delay model; and S3, analyzing the random double-time-delay model by using the maximum Lyapunov index and the singular boundary theory, and judging the random stability of the random double-time-delay model.

Description

Technical Field

[0001] The present invention relates to the technical field of wheel-set systems, and particularly relates to a stochastic double-delay model for a railway wheel-set system and a method for establishing the same. Background Art

[0002] With the rapid development of trains, the increase in operating speed has brought a series of safety problems. In particular, the hunting motion, which is an inherent property during train operation, is one of the important factors affecting train operation safety. During the process of the wheel-set advancing along the railway track, affected by external random factors, self-excited vibration damping motion with both lateral displacement and yawing occurs. To a certain extent, the hunting motion has good guiding properties for the advancement of the wheel-set. However, when the amplitude is large, the edge of the wheel-set collides with the railway track, easily leading to instability and causing the train to derail. Therefore, it is necessary to study the hunting motion of the wheel-set system, which helps to optimize the parameters of train design and improve the safety performance of the train.

[0003] There are many non-linear factors during the operation of the vehicle system, including non-linear constraints of wheel-rail geometry, wheel-rail creep relationship, suspension components, etc. In practice, the wheel-set system under random parametric excitation can better reflect the actual situation during train operation. The dynamic model of the train includes a randomly variable wheel-rail interaction. In the research on the wheel-set system, there are few model structures considering the time-delay displacement feedback control on the primary suspension. In order to be more in line with the actual meaning, it is necessary to establish a dynamic model of a double-delay wheel-set system under Gaussian white noise parametric excitation, analyze the influence of time delay on the hunting stability and bifurcation characteristics of the wheel-set system, which helps to understand the complex dynamic behavior of the wheel-set during high-speed operation. Summary of the Invention

[0004] Aiming at the above existing problems, the present invention aims to provide a stochastic double-delay model for a railway wheel-set system and a method for establishing the same, establish a dynamic model of a double-delay wheel-set system under Gaussian white noise parametric excitation, analyze the influence of time delay on the hunting stability and bifurcation characteristics of the wheel-set system, which helps to understand the complex dynamic behavior of the wheel-set during high-speed operation.

[0005] The main idea of the technical solution adopted by the present invention: taking double delay as a parameter, exploring the stochastic stability and bifurcation behavior of the wheel-set system under Gaussian white noise random parametric excitation. First, use the center manifold theorem and stochastic averaging method to reduce the dimension of the system, and analyze the stochastic stability of the system by using the maximum Lyapunov exponent and singular boundary theory; secondly, judge the conditions and types of stochastic bifurcation through the three-exponent method and the joint probability density function diagram, and explore the influence of double delay on the critical speed of stochastic P-bifurcation; finally, draw the single-parameter bifurcation diagram and double-parameter bifurcation diagram of the lateral displacement of the wheel-set system, and explore the influence of time delay on the critical speed of system instability.

[0006] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A random double-delay model for a railway wheel set system and a method for establishing the same, comprising the following steps:

[0008] S1. Establish the motion equation of the wheel set system, and obtain the equilibrium point system equation according to the state transformation at the equilibrium point;

[0009] S2. Use the center manifold theorem and the stochastic averaging method to reduce the dimension of the motion equation of the wheel set system to obtain a random double-delay model;

[0010] S3. Analyze the random double-delay model by using the maximum Lyapunov exponent and the singular boundary theory to judge the stochastic stability of the random double-delay model;

[0011] S4. According to the analysis of the random double-delay model in step S3, obtain the conditions satisfying the three-exponent method, and further use the three-exponent method to analyze the conditions and types of stochastic bifurcation of the system;

[0012] S5. According to the analysis of the random double-delay model in step S3, draw a joint probability density function graph, and observe the change of its topological structure to explore the influence of double delay on the occurrence of stochastic bifurcation of the system.

[0013] S6. By drawing a single-parameter bifurcation diagram of the lateral displacement of the random double-delay model, analyze the influence of delay on the critical speed of system instability; draw a two-parameter bifurcation diagram of the random double-delay model to analyze the influence on the dynamic behavior of the system.

[0014] Through the above technical solutions, further, in the said S1, according to Newton's second law, the motion equation of the wheel set system is established, and the motion equation of the wheel set system is as follows:

[0015]

[0016] Where: y is the lateral displacement of the wheel set, ψ is the yaw angle of the wheel set, v is the running speed of the wheel set, τ1 and τ2 are the lateral suspension delay and the longitudinal suspension delay respectively, ξ(t) is a Gaussian white noise with a mean of 0 and an intensity of 2D, m is the mass of the wheel set, I z is the yaw moment of inertia of the wheel set, I y is the self-rotation moment of inertia of the wheel set, W is the wheel set load, r0 is the nominal rolling circle radius, l is half of the lateral span of the primary suspension, a is half of the lateral span of the left and right rolling circles, f 11 is the longitudinal creep coefficient, f 22 is the lateral creep coefficient, f 23 is the lateral spin creep coefficient, f 33 is the spin creep coefficient, k yis the primary lateral stiffness, kx is the primary longitudinal stiffness, c y is the primary lateral damping, c x is the primary longitudinal damping, λ e is the equivalent taper, α1 is the non - linear wheel - rail contact control parameter, α2 is the non - linear wheel - rail contact control parameter, β1 is the stochastic parametric excitation control parameter of the lateral stiffness, β2 is the stochastic parametric excitation control parameter of the longitudinal stiffness, β3 is the stochastic parametric excitation control parameter of the equivalent taper, D is the noise intensity coefficient; is the lateral wheel - set displacement velocity, is the lateral wheel - set displacement acceleration, is the yaw angular velocity of the wheel - set, is the yaw angular acceleration of the wheel - set.

[0017] Through the above technical solutions, further: In the said S1, assume that the equilibrium point of formula (1) is Let Then the equilibrium point system equation is obtained as:

[0018]

[0019] where X=(x1, x2, x3, x4) T

[0020]

[0021]

[0022] ε is the equivalent substitution parameter.

[0023] Through the above technical solutions, further: In the said S2, apply the center - manifold theorem to transform the equilibrium point system equation into a finite - dimensional ordinary differential equation, and the equation about the amplitude A(t) and the phase is:

[0024]

[0025] where, A(t) is the amplitude, is the phase, ε is the equivalent substitution parameter; ξ(t) is a Gaussian white noise with a mean of 0, an intensity of 2D, and a constant power - spectral density K; ω is the imaginary part of the eigenvalue λ = iω; A3, B3 are the elements in the basis function before normalization; J 12 , J 22 , J 14 , J 24 are the elements in the normalized basis function.

[0026] With the above technical solutions, further: in S2, according to the stochastic averaging method, the stochastic double-delay model can be obtained as: dA = m(A)dt + σ(A)dB(t) (8)

[0027] where m(A) = μ1A + μ2A 3 + μ3A 5 , σσ T (A) = μ4A 2

[0028] where:

[0029]

[0030] where K is the power spectral density of Gaussian white noise, and B(t) is the Wiener increment process.

[0031] With the above technical solutions, further, in S3, according to the definition of Lyapunov exponents, the following expression is obtained:

[0032] where λ is the Lyapunov exponent, ||A|| represents the definition of the norm, m′(0) is the initial value of the first derivative of the drift coefficient, and σ′(0) is the initial value of the first derivative of the diffusion coefficient.

[0033] With the above technical solutions, further: according to the singular boundary theory, the diffusion exponent, drift exponent, and characteristic value at A = 0 are respectively:

[0034] where α L is the diffusion exponent at A = 0, β L is the drift exponent at A = 0, c L is the characteristic value at A = 0.

[0035] The diffusion exponent, drift exponent, and characteristic value at A = +∞ are respectively:

[0036]

[0037] where α R is the diffusion exponent at A = +∞, β R is the drift exponent at A = +∞, c R is the characteristic value at A = +∞.

[0038] With the above technical solutions, further: the joint probability density function obtained in S4 is the following expression.

[0039]

[0040] where P stis the joint probability density function corresponding to the random double - time - delay model, C is the normalization constant, and u1 and u2 are the polar coordinate transformations with respect to the amplitude A(t) and the phase and

[0041] The beneficial effects of the present invention are as follows: Taking double - time - delay as a parameter, the random stability and bifurcation behavior of the wheel - set system under Gaussian white - noise random - parameter excitation are explored. First, the center - manifold theorem and the stochastic averaging method are used to reduce the dimension of the system, and the random stability of the system is analyzed by using the maximum Lyapunov exponent and the singular - boundary theory; Second, the conditions and types of stochastic bifurcation are judged by the three - exponent method and the joint - probability - density - function diagram, and the influence of double - time - delay on the critical speed of stochastic P - bifurcation is discussed; Finally, the single - parameter bifurcation diagram and the double - parameter bifurcation diagram of the lateral displacement of the wheel - set system are drawn to explore the influence of time - delay on the critical speed of system instability.

[0042] The present invention clearly points out the significant influence of time - delay on the random stability and bifurcation behavior of the wheel - set system, especially that the critical speed of stochastic P - bifurcation decreases with the increase of time - delay. The different states of the lateral displacement of the system are analyzed in detail, providing a certain theoretical basis for the safe operation of trains and parameter design. The accuracy of the occurrence of stochastic P - bifurcation of the model is verified by Monte Carlo simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is the schematic structural diagram of the dynamic model of the wheel - set system;

[0044] Figure 2 is the influence of time - delays τ1, τ2 on the characteristic quantity μ of the wheel - set system, where (a) is the schematic diagram of the influence of time - delay τ1 on the characteristic quantity μ of the wheel - set system; (b) is the schematic diagram of the influence of time - delay τ2 on the characteristic quantity μ of the wheel - set system;

[0045] Figure 3 is the three - dimensional diagram and the equipotential diagram of the characteristic quantity μ in the parameter space τ1 - v, where (a) is the three - dimensional diagram of the characteristic quantity μ in the parameter space τ1 - v; (b) is the equipotential diagram of the characteristic quantity μ in the parameter space τ1 - v;

[0046] Figure 4 is the three - dimensional diagram and the equipotential diagram of the characteristic quantity μ in the parameter space τ2 - v, where (a) is the three - dimensional diagram of the characteristic quantity μ in the parameter space τ2 - v; (b) is the equipotential diagram of the characteristic quantity μ in the parameter space τ2 - v;

[0047] Figure 5 is the schematic diagram of the influence of time - delay τ1 and time - delay τ2 on stochastic P - bifurcation, where (a) is the schematic diagram of the influence of time - delay τ1 on stochastic P - bifurcation, and (b) is the schematic diagram of the influence of time - delay τ2 on stochastic P - bifurcation;

[0048] Figure 6The joint probability density function \(P\) with time delays \(\tau_1=\tau_2 = 0.23\) and different velocities \(v\) st and the schematic diagram of the stationary probability density function \(P(A)\); where: (a) \(v = 70m / s\), (b) \(v = 120m / s\), (c) \(v = 70m / s\), (d) \(v = 120m / s\);

[0049] Figure 7 The joint probability density function \(P\) with different time delays \(\tau_1\) when \(\tau_2 = 0.23\) and \(v = 72\) st and the schematic diagram of the stationary probability density function \(P(A)\); where: (a) \(\tau_1 = 0.17\), (b) \(\tau_1 = 0.27\), (c) \(\tau_1 = 0.17\), (d) \(\tau_1 = 0.27\);

[0050] Figure 8 The joint probability density function \(P\) with different time delays \(\tau_2\) when \(\tau_1 = 0.82\) and \(v = 68\) st and the schematic diagram of the stationary probability density function \(P(A)\); where: (a) \(\tau_2 = 0.79\), (b) \(\tau_2 = 0.815\), (c) \(\tau_2 = 0.79\), (d) \(\tau_2 = 0.815\);

[0051] Figure 9 The bifurcation diagrams of the wheel - set system with \(\varepsilon = 0.1\), \(D = 0.01\), initial displacement \((0.001,0,0,0)\) and different time delays \(\tau_1,\tau_2\); where: (a) \(\tau_1 = 0.001\), \(\tau_2 = 0.001\), (b) \(\tau_1 = 0.001\), \(\tau_2 = 0.01\), (c) \(\tau_1 = 0.01\), \(\tau_2 = 0.001\), (d) \(\tau_1 = 0.01\), \(\tau_2 = 0.01\);

[0052] Figure 10 The time - series diagrams of the wheel - set system with different time delays \(\tau_1,\tau_2\); where: (a) \(\tau_1 = 0.001\), \(\tau_2 = 0.001\), (b) \(\tau_1 = 0.001\), \(\tau_2 = 0.01\), (c) \(\tau_1 = 0.01\), \(\tau_2 = 0.001\), (d) \(\tau_1 = 0.01\), \(\tau_2 = 0.01\);

[0053] Figure 11 The phase diagrams of the wheel - set system with different running velocities \(v\); where: (a) \(v = 70m / s\), (b) \(v = 84m / s\), (c) \(v = 85m / s\), (d) \(v = 120m / s\);

[0054] Figure 12 For \(v-\lambda\) e The two - parameter bifurcation diagrams on the plane, where (a) \(\tau_1=\tau_2 = 0.1\) (b) \(\tau_1=\tau_2 = 1\) (c) \(\tau_1=\tau_2 = 2\);

[0055] Figure 13It is a two-parameter bifurcation diagram on the v-β3 plane, where (a) τ1 = τ2 = 0.1, (b) τ1 = τ2 = 1, and (c) τ1 = τ2 = 2. Detailed implementation manners

[0056] To make the objectives, technical solutions, and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are some, but not all, of the embodiments of the present application. Usually, the components of the embodiments of the present application described and illustrated herein can be arranged and designed in various different configurations.

[0057] The inventors' research found that there are many non-linear factors during the operation of the vehicle system, including non-linear constraints of wheel-rail geometry, wheel-rail creep relationship, suspension elements, etc. In practice, the wheel pair system under random parametric excitation can better reflect the actual situation during train operation. The dynamic model of the train includes a randomly variable wheel-rail interaction. However, in the research on the wheel pair system, there are few model structures considering the time-delay displacement feedback control on the primary suspension. To be more in line with the actual meaning, it is necessary to establish a dynamic model of a two-time-delay wheel pair system under Gaussian white noise parametric excitation, analyze the influence of time delay on the hunting stability and bifurcation characteristics of the wheel pair system, which helps to understand the complex dynamic behavior of the wheel pair during high-speed operation.

[0058] Based on the above findings, the present application proposes a random two-time-delay model for a railway wheel pair system and its establishment method. Taking the two-time delay as a parameter, it explores the random stability and bifurcation behavior of the wheel pair system under Gaussian white noise random parametric excitation. First, the center manifold theorem and the stochastic averaging method are used to reduce the dimension of the system, and the maximum Lyapunov exponent and the singular boundary theory are used to analyze the random stability of the system; secondly, the three-exponent method and the joint probability density function diagram are used to judge the conditions and types of random bifurcation, and explore the influence of the two-time delay on the critical speed of random P-bifurcation; finally, the single-parameter bifurcation diagram and the two-parameter bifurcation diagram of the lateral displacement of the wheel pair system are drawn to explore the influence of time delay on the critical speed of system instability.

[0059] Embodiment 1

[0060] Refer to Figure 1 , the present application discloses a random two-time-delay model for a railway wheel pair system and its establishment method. The wheel pair system is a key part of the railway vehicle dynamic system, which includes a suspension system and wheel pairs connected together by an axle. The wheel pairs and the bogie are connected by a primary suspension system. As Figure 1As shown in the figure, without considering the coupling relationship between the wheel set and the bogie frame, the gyroscopic effect and the primary suspension damping are introduced into the wheel set system. Considering the random effects of the equivalent conicity and the suspension stiffness, they are regarded as stochastic processes with Gaussian white noise parametric excitation. In addition, the influence of time delay on the primary lateral and longitudinal suspensions is considered. The non-linear wheel-rail contact force adopts Kalker's linear creep theory. The influence of the wheel-rail normal force is reflected by the gravity stiffness and the gravity angular stiffness of the wheel set.

[0061] According to Newton's second law, the motion equations of the wheel set system are established and obtained as follows:

[0062]

[0063] where, ξ(t) is Gaussian white noise with a mean of 0 and an intensity of 2D, y is the lateral displacement of the wheel set, ψ is the yaw angle of the wheel set, v is the running speed of the wheel set, τ1 and τ2 are the time delays of the primary lateral suspension and the longitudinal suspension respectively, is the lateral displacement speed of the wheel set, is the lateral acceleration of the wheel set, is the yaw angular velocity of the wheel set, is the yaw angular acceleration of the wheel set. Other parameters are shown in Table 1.

[0064] Table 1 Main parameters of the wheel set system

[0065]

[0066]

[0067] Assume that the equilibrium point of system (1) is Let Then system (1) is transformed into the following form;

[0068]

[0069] where X = (x1, x2, x3, x4) T

[0070]

[0071]

[0072] where, ε is the equivalent substitution parameter.

[0073] The characteristic equation of the linear part of system (2) is as follows:

[0074]

[0075] When the speed v reaches the linear critical speed v cWhen τ1, τ2 = 0, the characteristic equation (3) has a pair of complex conjugate eigenvalues ±iω with real part 0 at the equilibrium point, and the real parts of the other eigenvalues are also 0. When τ1, τ2 = 0, the characteristic equation (3) can be written as follows:

[0076]

[0077] By comparing with the terms of equation (3), we can obtain

[0078]

[0079] When τ1≠0, τ2≠0, substituting the eigenvalue λ = iω into the characteristic equation (3), and differentiating both sides of equation (3) with respect to the velocity v, we can obtain the transversality condition as follows: where

[0080]

[0081] When v = v c , the system (2) undergoes a stochastic Hopf bifurcation near the equilibrium point.

[0082] Next, we analyze the dynamics of the system (2) near v = v c , and analyze the snake-like stability during the operation. The solution space of the system (2) is infinite-dimensional. Mapping the interval [-τ, 0] to R 2 , the system (2) can be written as

[0083]

[0084] where L(φ(θ), v c ) and Δf(φ(θ), v c , ε) are the linear and nonlinear parts of the system (2) respectively

[0085] Define the bilinear operator as follows:

[0086]

[0087] where ψ j2 , ψ j4 are the 2nd and 4th elements in ψ j respectively, and φ k1 , φ k3 are the 1st and 3rd elements in φ k respectively.

[0088] The space generated by all the eigenvalues of the characteristic equation (3) It is divided into two parts. A space P is generated by the complex conjugate pure imaginary eigenvalues ±iω of the characteristic equation (3), and another space Q is generated by the remaining eigenvalues. Next, the infinite-dimensional system (2) is reduced to a finite-dimensional ordinary differential equation by using the center manifold theorem. The center manifold is tangent to the two-dimensional subspace P generated by the conjugate pure imaginary eigenvalues ±iω. According to the conjugate pure imaginary eigenvalues λ = ±iω, the basis functions of P ∈ C in the characteristic equation (3) and those in its corresponding adjoint equation are obtained of the basis functions

[0089]

[0090] where:

[0091] Substituting the basis functions (5) into the bilinear expression (4), we get

[0092]

[0093] Normalize the basis function Ψ(s)

[0094]

[0095] to obtain

[0096]

[0097] where

[0098] J 11 (s) = (a 11 a 22 -a 12 a 21 ) -1 (a 22 cosωs - a 12 sinωs)

[0099] J 12 (s) = -(a 11 a 22 -a 12 a 21 ) -1 ω(a 22 sinωs + a 12 cosωs)

[0100] J 13 (s) = (a 11 a 22 -a 12 a 21 ) -1 [a 22 (A3cosωs - B3sinωs) - a 12(A3sinωs + B3cosωs)]

[0101] J 14 (s) = (a 11 a 22 -a 12 a 21 ) -1 [a 22 (A4cosωs - B4sinωs) - a 12 (A4sinωs + B4cosωs)]

[0102] J 21 (s) = (a 11 a 22 -a 12 a 21 ) -1 (-a 21 cosωs + a 11 sinωs)

[0103] J 22 (s) = (a 11 a 22 -a 12 a 21 ) -1 ω(a 21 sinωs + a 11 cosωs)

[0104] J 23 (s) = (a 11 a 22 -a 12 a 21 ) -1 [-a 21 (A3cosωs - B3sinωs) + a 11 (A3sinωs + B3cosωs)]

[0105] J 24 (s) = (a 11 a 22 -a 12 a 21 ) -1 [-a 21 (A4cosωs - B4sinωs) + a 11 (A4sinωs + B4cosωs)]

[0106] Assume the unique solution of system (2) can be expressed as:

[0107]

[0108] where φ(θ) = φP (θ) + φ Q (θ), under the semigroup J(t, β) and its infinitesimal generator A(θ, v) are respectively the projections of x t (φ(θ), v, ε) on two subspaces P and Q. According to the relation A(θ, P)Φ(θ) = Φ(θ)B, we get B = [(0, ω) T , (-ω, 0) T , for the central manifold the transformation φ(θ) = Φ(θ)B ∈ C and the transformation For θ = t, the approximate expression is calculated as follows:

[0109] Let A4 = b2, B4 = b3. On the central manifold M V ∈ C, the system (2) is as follows

[0110]

[0111] Introduce the polar coordinate transformation: Transform the system (2) into an equation about the amplitude A(t) and the phase of

[0112]

[0113] According to the stochastic averaging method, we can obtain the stochastic differential equation

[0114] dA = m(A)dt + σ(A)dB(t) (8)

[0115] where m(A) and σ(A) are respectively the drift coefficient and the diffusion coefficient of the system (7)

[0116] m(A) = μ1A + μ2A 3 + μ3A 5 , σσ T (A) = μ4A 2

[0117] where

[0118]

[0119] The local stability of the system (2) can be obtained by analyzing the corresponding stochastic differential equation. According to the definition of the Lyapunov exponent, we get

[0120]

[0121] Therefore, it is obtained that when λ < 0, the wheel-set system (2) is locally stochastically asymptotically stable in the sense of probability; when λ > 0, the wheel-set system (2) is locally stochastically asymptotically unstable in the sense of probability; when λ = 0, the wheel-set system (2) will undergo stochastic D-bifurcation near this point.

[0122] According to the singular boundary theory, the global stability of the wheel-set system (2) can be analyzed. Since σ(0) = 0 and m(0) = 0, the left boundary A = 0 is the first type of singular boundary. By calculation, the diffusion index, drift index, and characteristic value at A = 0 can be obtained.

[0123]

[0124] The right boundary A = +∞ is the second type of singular boundary at infinity. By calculation, the diffusion index, drift index, and characteristic value at A = +∞ can be obtained.

[0125]

[0126] When m(-∞) > 0 or m(+∞) < 0, the right boundary A = +∞ is an entrance boundary, satisfying the conditions of the three-index method. It can be obtained that when c L -α L = -1, the system undergoes stochastic D-bifurcation; when c L = α L the system undergoes stochastic P-bifurcation.

[0127] At this time, the properties of the system only depend on the left boundary A = 0. When the characteristic value c L = 1, the speed of the wheel-set system reaches the critical value. Through analysis, it can be obtained that when the speed v < v c the wheel-set system (2) is globally stable in the sense of probability; when the speed v > v c the wheel-set system (2) loses stability in the sense of probability.

[0128] One-dimensional The FPK equation corresponding to the stochastic differential equation (8) is

[0129]

[0130] After integrating and solving the FPK equation (9), the stationary probability density function can be obtained.

[0131]

[0132] Let Combined with equation (10), the joint probability density function can be obtained as follows

[0133]

[0134] Among them, the characteristic quantity μ = c L -α L , which is used to describe the bifurcation characteristics of the wheel set system. At this time, combining the definition of the Lyapunov exponent, the conclusion is obtained that the wheel set system undergoes stochastic D bifurcation when μ = -1 and stochastic P bifurcation when μ = 0.

[0135] Embodiment 2

[0136] Reference Figure 2 - Figure 13 , considering the influence of time delay on the primary lateral and longitudinal suspensions, the topological structure of the wheel set system has undergone an essential change. The critical speed at which the wheel set system undergoes stochastic P bifurcation is no longer a fixed value but changes with the change of time delay. Therefore, it is necessary to discuss the influence of time delays τ1 and τ2 on the occurrence of stochastic P bifurcation in the wheel set system.

[0137] From Figure 2 , it can be seen that as the speed v increases, the characteristic quantity μ shows non-linear growth. The growth rate is slower before reaching the critical speed and faster after reaching the critical speed. When the characteristic quantity μ grows to 0, the wheel set system undergoes stochastic P bifurcation. The influence of the two time delays τ1 and τ2 on the occurrence of stochastic P bifurcation in the wheel set system is similar. As the time delay τ1 and the time delay τ2 increase, the characteristic quantity curve μ moves to the left, and the corresponding critical speed gradually decreases. When the speed v is fixed, the characteristic quantity increases with the increase of the time delay τ1 and the time delay τ2, especially near the critical speed, which will cause the wheel set system to undergo stochastic P bifurcation and exhibit hunting instability.

[0138] As Figure 3 - Figure 4 shown, in order to explore the change process of stochastic P bifurcation of the double-time-delay wheel set system, taking τ1 - v and τ2 - v as variables respectively, the three-dimensional diagram and the equipotential diagram of the characteristic quantity μ are plotted.

[0139] Taking the case where the time delay τ1 and the speed v are variables as an example, from Figure 3 , it can be seen that the characteristic quantity μ increases with the increase of the time delay τ1 and the speed v, and the growth rate gradually becomes larger. When the two increase to a certain value, the characteristic quantity curve crosses the critical line μ = 0 of stochastic P bifurcation, and the value of the characteristic quantity μ changes from negative to positive, that is, the system undergoes stochastic P bifurcation. At this time, the stable solution of the wheel set system changes from probabilistic stability to probabilistic instability, and gradually generates hunting instability in the probabilistic sense. In Figure 3 , there is a non-linear relationship between the critical speed and the time delay τ1, and the critical speed decreases with the decrease of the time delay τ1.

[0140] From Figure 5It can be seen that the curve in the figure represents the critical line μ = 0 for the system to undergo stochastic P-bifurcation. The critical line divides the entire parameter space into two parts, namely the stochastic D-bifurcation region and the stochastic P-bifurcation region from left to right. At this time, the equilibrium solution probability of the wheel-set system is stable in the left region and unstable in the right region. The critical line for the wheel-set system to undergo stochastic P-bifurcation gradually moves to the left as the time delays τ1 and τ2 increase, and the critical speed also decreases accordingly. The parameter space for the system to undergo stochastic P-bifurcation gradually becomes larger.

[0141] Next, in order to analyze the influence of each parameter variable on the operating state of the wheel-set system before and after stochastic P-bifurcation, the joint probability density function graph and the stationary probability density function graph are plotted with the speed v, time delays τ1 and τ2 as variables respectively, and other parameters are shown in Table 1.

[0142] When the time delays τ1 = τ2 = 0.23 are selected and the speed v is used as the variable, the stochastic P-bifurcation is analyzed by observing the shape change of the joint probability density function. Figure 6 It can be seen that as the speed v increases, the joint probability density function of the wheel-set system gradually changes from a "single-peak" shape to a "crater" shape, and the topological structure changes, which indicates that the wheel-set system has undergone stochastic P-bifurcation, and the equilibrium solution of the system changes from probability stability to probability instability. In Figure 6 Figure (a), when the speed v = 70, the joint probability density function is in a "single-peak" shape, and the maximum value is obtained at the equilibrium point. The phase diagram forms a limit cycle region that gradually spreads out centered on the equilibrium point, and the chromaticity value at the equilibrium point is the largest. By observing Figure 6 Figure (c), it can be seen that the function value of the stationary probability density function P(A) gradually decreases until it approaches 0 far from the equilibrium point. This phenomenon indicates that the wheel-set system is stable in the sense of probability at the equilibrium point, that is, the probability of the wheel-set system approaches the equilibrium point A = 0 after running for a period of time, and the system behavior shows unstable non-limit cycle random vibration in the sense of probability 1. From Figure 6 Figure (a) to Figure 6 Figure (b), the function shape changes to a "crater" shape, and the peak value is significantly smaller than that in Figure 6 Figure (a). On the phase diagram, a limit cycle band with the largest chromaticity value is formed at the non-equilibrium point, which indicates that the system is stable in the sense of probability at the limit cycle band and unstable in the sense of probability at the equilibrium point. Combining Figure 6 Figure (d), it can be seen that the stationary probability density function takes the minimum value at the origin A = 0, and the wheel-set system has undergone stochastic P-bifurcation. This indicates that the wheel-set system is unstable in the sense of probability at this time, and snake-like instability motion has occurred in the actual situation. As the speed increases, the snake-like instability motion becomes more obvious, eventually leading to system collapse and train derailment. In Figure 6 Figure (c) and Figure 6In Figure (d), the blue curve represents the analytical solution of the function, and the red dotted line represents the numerical solution of the Monte Carlo simulation function. The two are in agreement, indicating the correctness of the theoretical derivation.

[0143] When τ2 = 0.23 and v = 72 are selected, the probability density function graph with the time delay τ1 as the variable is as Figure 11 shown. When τ1 = 0.82 and v = 68 are selected, the probability density function graph with the time delay τ2 as the variable is as Figure 12 shown. The effects of the time delays τ1 and τ2 on the operating state of the wheel-set system before and after random P-bifurcation are similar. Here, the case where τ1 is the variable is taken as an example for analysis.

[0144] From Figure 7 Figure (a), it can be seen that when the time delay τ1 = 0.17, the function graph is in a "single-peak" shape and reaches the maximum value at the equilibrium point. In Figure 7 Figure (c), the probability density function gradually decreases with the increase of the amplitude A until it approaches 0, indicating that the wheel-set system is stable in the sense of probability at the equilibrium point. From Figure 7 Figure (a) to Figure 7 Figure (b), the joint probability density function of the wheel-set system changes from a "single-peak" shape to a "crater" shape, and the peak value of the function also decreases. In Figure 7 , the stationary probability density function takes the minimum value of 0 at A = 0, and the peak value of the function will not be obtained at the equilibrium point but at a non-equilibrium point. At this time, the topological structure of the wheel-set system changes, and random P-bifurcation occurs. The system will no longer converge probabilistically to the equilibrium point but converge probabilistically to a new steady-state position. This indicates that the wheel-set system has exhibited hunting instability, which is not conducive to the safe operation of the train. In the actual train parameter design, the influence of time delay on the primary suspension of the train should be considered to improve the safety performance of the train. Figure 7 , Figure 8 The Monte Carlo simulation in

[0145] verified the correctness of the theory.

[0146] Select ε = 0.1, D = 0.01, the initial displacement is (0.001, 0, 0, 0), and other parameters are shown in Table 1. From Figure 9It can be seen that the threshold between the wheel-set system period and chaos gradually decreases with the increase of time delays τ1 and τ2, which indicates that the chaotic phenomenon of the system is more likely to occur with the increase of time delays τ1 and τ2, and the stability of the system also gradually decreases. When the speed exceeds the critical bifurcation speed, chaotic phenomenon and hunting instability will occur in the system. In actual situations, the lateral displacement of the train wheel-set is affected by time delays τ1 and τ2. The larger the time delays τ1 and τ2 are, the smaller the critical bifurcation speed of the wheel-set is, and the greater the possibility of hunting instability movement is, and vice versa.

[0147] As Figure 10 shown, to further reflect the response dynamics of the wheel-set system, time series diagrams of the lateral displacement of the wheel-set system at different running speeds are plotted through numerical simulation to analyze its bifurcation behavior. The following takes the situation in the (a) diagram of Figure 9 as an example for specific elaboration. It can be seen from the (a) diagram of Figure 10 that when the running speed V < 84.8, with the increase of running time, the lateral displacement of the wheel-set system gradually tends to the equilibrium point. When the speed V > 84.8, the lateral displacement no longer tends to the equilibrium point with the increase of running time but shows periodic amplitude motion. In the phase diagram, it can be seen from the (c) diagram of Figure 11 that when the speed V = 85, the trajectory line of the system approaches the limit cycle with a smaller amplitude rather than the stable equilibrium point. At this time, since the amplitude of the limit cycle does not exceed the flange clearance (about 9 mm), the system will not derail and can still operate normally. When the speed continues to increase, the amplitude of the limit cycle will quickly approach and exceed the value of the flange clearance. It can be seen from the (d) diagram of Figure 11 that when the speed V = 120, the trajectory line quickly approaches the stable limit cycle, and the amplitude at this time is significantly larger than the value of the flange clearance, and the wheel-set system is very likely to derail. In addition, Figure 11 the limit cycle in the (d) diagram shows the characteristic of blurred edges, and it is composed of many limit cycles that probably exist, which is essentially different from the phase diagram of the deterministic system without perturbation terms, and the latter shows a single closed-loop trajectory.

[0148] To more deeply explore the influence of various parameters and time delays on the wheel-set system, as Figure 12 , Figure 13 shown, the bifurcation diagram of the lateral displacement y in the double-parameter plane is plotted. The color bar on the right side of the diagram represents different oscillation periods. For example, yellow represents the period-1 oscillation state, blue represents the period-2 oscillation state, and so on. The double-parameter bifurcation diagram about the speed v and the equivalent conicity λ e is as Figure 12 shown. It can be observed from the (a) diagram of Figure 12 that in the speed v and the equivalent conicity λ eIn the region with lower values, the system is in a mixed oscillation state where cycles 1 and cycles 2 and 3 alternate. At velocity v and equivalent taper λ e In the region with higher values, the overall behavior of the system shows an oscillation state where cycles 2 and cycles 3 alternate, interspersed with oscillations with higher cycle numbers such as cycle 10 and cycle 15. In Figure 12 the (b) figure of Figure 12 and the (c) figure of e in the region with lower values of velocity v and equivalent taper λ, as the time delay increases, the image gradually becomes chaotic and the system exhibits more complex dynamic behavior. At velocity v and equivalent taper λ e in the region with higher values, the influence of the time delay on the system is smaller and the system behavior maintains the original periodic oscillation mode.

[0149] In addition, the double-parameter bifurcation diagram plotted with velocity v and control parameter β3 as the parameter plane is as shown in Figure 13 In Figure 13 the (a) figure, it can be observed that as the velocity v increases, oscillations with lower periods alternate, while the influence of the parameter β3 on the bifurcation behavior of the system is smaller. Observing Figure 13 the (b) figure of Figure 13 and the (c) figure of

[0150] it can be seen that as the time delay changes, the oscillation period number will increase or decrease. The number of cycle 1 oscillation states of the system gradually increases and the velocity range for generating cycle 1 oscillations also changes, and the mixing phenomenon of different oscillation periods is more complex. The above phenomena indicate that by appropriately adjusting the time delay, relatively stable and regular periodic oscillation behavior can be induced in the system, providing a certain reference basis for the safety and design of trains.

[0151] 1. The judgment conditions and types of stochastic stability and stochastic bifurcation of the wheel-set system are theoretically given. Based on the three-exponential method, the stochastic stability of the system is determined by the left boundary, and the occurrence of stochastic bifurcation is reflected by the characteristic quantity μ. The characteristic quantities μ = -1 and μ = 0 correspond to the occurrence of stochastic D-bifurcation and stochastic P-bifurcation of the system respectively, which is convenient for theoretically exploring the influence law of time delays τ1 and τ2 on the hunting motion of the system.

[0152] 2. The change of time delays τ1 and τ2 causes the qualitative behavior of the system to change, and the critical velocity of the wheel-set system for stochastic P-bifurcation also changes accordingly. The action of time delays τ1 and τ2 causes the critical velocity to drift, and the critical velocity of stochastic P-bifurcation decreases significantly as time delays τ1 and τ2 increase, and the wheel-set system changes from probabilistic stability to probabilistic instability at the equilibrium point.

[0153] 3. Under the action of time delays τ1 and τ2, the lateral displacement of the wheel set system has different dynamic behaviors, such as periodic and chaotic states. And the speed threshold between period and chaos gradually decreases as the time delays τ1 and τ2 increase. Once the chaotic phenomenon appears in the wheel set system, it is theoretically considered unstable and is not conducive to the operation of the train. The periodic oscillation behavior of the wheel set system under the action of time delay is analyzed through the double-parameter bifurcation diagram, which provides a certain theoretical basis for the safe operation and parameter design of the train.

[0154] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. What is described in the above embodiments and the specification only illustrates the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of the present invention claimed is defined by the appended claims and their equivalents.

Claims

1. A method for establishing a random double-time-delay model for a railway wheelset system, characterized in that: The following steps are involved: S1. Establish the motion equation of the wheelset system, and obtain the equilibrium point system equation according to its state transformation at the equilibrium point; S2. Use the center manifold theorem and the random average method to reduce the dimension of the wheelset system motion equation and obtain a random double-delay model; S3. Analyze the random double-delay model using the maximum Lyapunov exponent and singular boundary theory to determine the random stability of the random double-delay model; S4, according to the analysis of the random double-delay model in step S3, the conditions satisfying the three-exponential method are obtained, so as to further use the three-exponential method to analyze the conditions and types of random bifurcation of the system; S5. According to the analysis of the random double-delay model in step S3, a joint probability density function diagram is drawn, and the change of its topological structure is observed to explore the influence of the double-delay on the random bifurcation of the system; S6. Analyze the influence of time delay on the critical speed of system instability by drawing a single-parameter bifurcation diagram of the lateral displacement of the random double-time-delay model; Draw a two-parameter bifurcation diagram of the random double-time-delay model to analyze the influence of the system's dynamic behavior.

2. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 1, characterized in that: In S1, the motion equation of the wheelset system is established according to Newton's second law, and the motion equation of the wheelset system is as follows: Where: y is the lateral displacement of the wheelset, ψ is the yaw angle of the wheelset, v is the travel speed of the wheelset, τ1 and τ2 are the primary lateral suspension delay and longitudinal suspension delay, ξ(t) is a Gaussian white noise with a mean of 0 and an intensity of 2D, m is the wheelset mass, I z is the moment of inertia of the wheelset, I y is the rotational inertia of the wheelset, W is the wheelset load, r0 is the nominal rolling circle radius, l is half of the lateral span of the primary suspension, a is half of the lateral span of the left and right rolling circles, and f 11 is the longitudinal creep coefficient, f 22 is the transverse creep coefficient, f 23 is the transverse spin creep coefficient, f 33 is the spin creep coefficient, k y is the lateral stiffness, k x is the longitudinal stiffness, c y is the lateral damping, c x is the longitudinal damping, λ e is the equivalent taper, α1 is the nonlinear wheel-rail contact control parameter, α2 is the nonlinear wheel-rail contact control parameter, β1 is the lateral stiffness random parameter excitation control parameter, β2 is the longitudinal stiffness random parameter excitation control parameter, β3 is the equivalent taper random parameter excitation control parameter, and D is the noise intensity coefficient; is the wheelset lateral speed, is the wheelset lateral acceleration, is the wheelset angular velocity, is the wheelset angular acceleration.

3. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 2, characterized in that: In S1, it is assumed that the equilibrium point of formula (1) is make Then the equilibrium point system equation is: Where: X = (x1, x2, x3, x4) T ε is the equivalent substitution parameter.

4. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 3, characterized in that: In S2, the central manifold theorem is used to transform the equilibrium point system equation into a finite-dimensional ordinary differential equation, and the amplitude A(t) and phase The equation is: Where A(t) is the amplitude, is the phase, ε is the equivalent substitution parameter; ξ(t) is a Gaussian white noise with a mean of 0, an intensity of 2D, and a power spectrum density of a constant K; ω is the imaginary part of the eigenvalue λ=iω; A3, B3 are the elements of the basis function before normalization; J 12 ,J 22 ,J 14 ,J 24 are the elements in the normalized basis function.

5. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 4, characterized in that: In S2, the random double-delay model can be obtained according to the random averaging method. The stochastic differential equation is: dA=m(A)dt+σ(A)dB(t) (8) where, m(A) = μ1A + μ2A 3 + μ3A 5 , σσ T (A) = μ4A 2 in: Where: K is the power spectral density of Gaussian white noise, and B(t) is the Weiner increment process.

6. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 5, characterized in that: In S3, according to the definition of Lyapunov exponent, the following expression is obtained: Where λ is the Lyapunov exponent, ||A|| denotes the definition of the mode, m′(0) is the initial value of the first-order derivative of the drift coefficient, and σ′(0) is the initial value of the first-order derivative of the diffusion coefficient.

7. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 6, characterized in that: According to the singular boundary theory, the diffusion index, drift index and characteristic value when A = 0 are obtained as follows: Among them, α L is the diffusion index when A=0, β L is the drift index when A=0, c L is the characteristic value when A=0; The diffusion index, drift index and characteristic value when A=+∞ are: Among them, α R is the diffusion index when A=+∞, β R is the drift index when A=+∞, c R is the characteristic value when A=+∞.

8. The method for establishing a random double-time-delay model for a railway wheelset system according to claim 7, characterized in that: The joint probability density function obtained in S4 is as follows: Among them, P st is the joint probability density function corresponding to the random double-delay model, C is the normalization constant, u1 and u2 are the amplitude A(t) and phase Polar coordinate transformation.

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