Fokker-Planck equation rapid solving method for nonlinear dynamic response of vehicle suspension system

Abstract

A Fokker-Planck equation rapid solving method for nonlinear dynamic response of a vehicle suspension system comprises the steps that nonlinear factors of the vehicle suspension are considered, and a Fokker-Planck equation containing a probability density function of the vehicle suspension system is established through a vehicle suspension system motion control equation randomly excited by a road surface; the method comprises the following steps of: performing discrete processing on a Fokker-Planck equation of a vehicle suspension system by utilizing a finite element discrete method to obtain a global motion control equation; solving the global motion control equation in an accelerated manner by using an operator splitting method and a dimension reduction method to obtain a probability density function of displacement and speed response of the vehicle suspension system; and in the state space, carrying out integration on the probability density function of the displacement and the speed of the vehicle suspension system to obtain the dynamic response of the vehicle system to the road surface random excitation. According to the method, the nonlinear dynamic response of the vehicle suspension system is obtained by rapidly solving the Fokker-Planck equation of the vehicle suspension, and the method has engineering application guiding significance for structural design and vibration reduction control of the vehicle.

Description

Technical Field

[0001] The present invention belongs to the field of random dynamic response solution method and technology of vehicle suspension system, and specifically relates to a method for solving the nonlinear dynamic response of the vehicle suspension system based on the transient solution of the Fokker-Planck equation of the vehicle suspension system under random road excitation. Background Art

[0002] In most existing studies, the vehicle suspension system is simplified as a linear system. In reality, the vehicle suspension is a nonlinear system, and the dynamic response of the vehicle suspension system under nonlinear factors needs to be considered. The time domain model of road excitation often uses the simple harmonic superposition method for simulation calculation, which cannot accurately describe the time domain information of road excitation. Considering it as Gaussian white noise can more accurately express the time domain information of road excitation. The nonlinear dynamic response of the vehicle cannot generally be expressed by deterministic functions of time and space coordinates. In random dynamic response analysis, it is usually described and modeled by probability or statistics. In random vibration analysis, the Fokker-Planck equation is a very important tool. By solving the Fokker-Planck equation, the probability density function of the dynamic response of the random dynamic system can be obtained.

[0003] Therefore, the probability density function of the dynamic response of the vehicle suspension system can be obtained by solving the Fokker-Planck equation, and the dynamic response of the vehicle suspension system can be obtained by integrating the probability density function. However, it is very time-consuming to solve the Fokker-Planck equation using the finite element method. It takes about 4 hours to calculate when the finite element grid is 100×100. Therefore, it is proposed to accelerate the solution of the Fokker-Planck equation of the vehicle suspension system by combining the finite element with the operator splitting and the dimensionality reduction algorithm to quickly obtain the dynamic response of the velocity and displacement of the vehicle suspension system. Summary of the invention

[0004] The present invention aims to overcome the above-mentioned shortcomings of the prior art and provide a fast solution method for the Fokker-Planck equation of the nonlinear dynamic response of a vehicle suspension system.

[0005] The purpose of the present invention is to quickly solve the Fokker-Planck equation of a vehicle suspension system under the influence of random road excitation. The purpose is to use the finite element method combined with operator splitting and dimensionality reduction algorithms to quickly solve the Fokker-Planck equation of the vehicle suspension system under a certain calculation accuracy, obtain the complete probability density function of the system at any time, and study the nonlinear dynamic response of the system through the probability density function.

[0006] The technical solution to achieve the purpose of the present invention is: a fast solution method for the Fokker-Planck equation of the nonlinear dynamic response of the vehicle suspension system, comprising the following steps:

[0007] Step 1: Consider the random excitation of the road surface as an additive Gaussian white noise, consider the nonlinear factors of the vehicle suspension, and obtain the Fokker-Planck equation of the vehicle suspension system through the motion control equation of the vehicle suspension system;

[0008] Step 2: Use the finite element method to discretize the Fokker-Planck equation of the vehicle suspension system. First, use the double partial integration method to obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system. Then, select the rectangular unit as the unit shape and use the bilinear interpolation function to obtain the unit coefficient matrix m and k. According to the finite element construction principle, the global motion control equation is obtained, and its coefficient matrix is ​​M and K.

[0009] Step 3: Use the operator splitting method to split the coefficient matrix K in the global motion control equation into convection terms and diffusion terms, and solve the eigenvalues ​​and eigenvectors of the coefficient matrix M in the global motion control equation at the same time. Select the eigenvector as the effective reduction basis according to the size of the projection coefficient, and use the operator splitting method and dimensionality reduction method to process the global motion control equation;

[0010] Step 4: Given the vehicle suspension system parameters and initial conditions, substitute the processed global motion control equations for iterative calculations, solve the Fokker-Planck equation of the vehicle suspension system, and obtain the probability density function of the displacement and velocity of the vehicle suspension system over time. Integrate the probability density function to obtain the mean and mean square value response of the displacement and velocity of the vehicle suspension system.

[0011] The present invention is based on solving the Fokker-Planck equation of the vehicle suspension system, obtains the probability density function of the vehicle suspension system under random road excitation, and studies the nonlinear dynamic response of the vehicle suspension system through the probability density function. At the same time, the Fokker-Planck equation of the vehicle suspension system is quickly solved by combining the operator splitting and dimensionality reduction method to quickly obtain the nonlinear dynamic response of the vehicle suspension system.

[0012] Compared with the prior art, the present invention has the following significant advantages:

[0013] (1) The nonlinear dynamic response of the system is obtained by solving the Fokker-Planck equation of the vehicle suspension system. This method takes into account the nonlinear factors of the vehicle suspension. At the same time, the road excitation is considered as Gaussian white noise, which is more in line with the actual driving conditions of the car on uneven roads.

[0014] (2) The Fokker-Planck equations of the vehicle suspension system are calculated using operator splitting and dimensionality reduction methods. This can effectively reduce the calculation time while meeting the numerical calculation accuracy, thereby quickly solving the nonlinear dynamic response of the vehicle suspension system, which helps to achieve real-time control and optimization of the vehicle suspension system. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 is a flow chart of the method of the present invention;

[0016] Figure 2 is a model diagram of a quarter vehicle suspension system of the present invention;

[0017] Figure 3 is the finite element calculation domain of the present invention;

[0018] Figure 4 It is the unit shape of the present invention;

[0019] Figure 5 is the projection coefficient of the present invention;

[0020] Figure 6 It is a descending graph of projection coefficients of the dimensionality reduction algorithm of the present invention;

[0021] Figure 7(a) to Figure 7(b) FIG. 7( a ) is an integral diagram of the probability density function of the edge of the vehicle suspension of the present invention; FIG. 7( a ) is an integral diagram of the probability density function of the displacement; FIG. 7( b ) is an integral diagram of the probability density function of the velocity;

[0022] FIG8 is a contour plot of the probability density function of the vehicle suspension system of the present invention evolving over time;

[0023] FIG. 9( a ) is a non-stationary response of the displacement mean of the vehicle suspension system of the present invention; FIG. 9( b ) is a non-stationary response of the velocity mean of the vehicle suspension system of the present invention;

[0024] FIG. 10( a ) is a non-stationary response of the mean square displacement value of the vehicle suspension system of the present invention; FIG. 10( b ) is a non-stationary response of the mean square velocity value of the vehicle suspension system of the present invention;

[0025] Fig.11 It is a graph showing the displacement and velocity evolution over time of the vehicle suspension system of the present invention. DETAILED DESCRIPTION

[0026] The technical solution of the present invention is described in detail below with reference to the accompanying drawings.

[0027] The invention provides a fast solution method for the Fokker-Planck equation of the nonlinear dynamic response of a vehicle suspension system, which specifically comprises the following steps:

[0028] Step 1: Consider the random excitation of the road surface as an additive Gaussian white noise, consider the nonlinear factors of the vehicle suspension, and obtain the Fokker-Planck equation of the vehicle suspension system through the motion control equation of the vehicle suspension system;

[0029] Establish a mechanical model of the quarter suspension system, such as Figure 2 As shown, where m s is the equivalent mass of the suspension, k is the equivalent damping of the suspension, and considering its nonlinear factors, k = k s +αk s X 2 , α is the nonlinear coefficient, c s is the equivalent damping of the suspension, x s (t) is the vertical displacement of the suspension, and the random excitation transmitted to the suspension by the ground through the tire is r.

[0030] The following assumptions are made for the quarter suspension system model: (1) the effects of the suspension spring mass and tire mass on the body system motion are ignored; (2) the suspension equivalent mass m s is the mass of the entire vehicle and the suspension distributed proportionally on a single wheel; (3) the quarter suspension under study is considered to be a single-degree-of-freedom system, so the wheel and tire are assumed to be rigid bodies; (4) the random excitation of the quarter suspension is assumed to be Gaussian white noise.

[0031]

[0032] Let 2n = c s / m s ,ξ=n / ω0,ω0 2 =k s / m s , n is damping, ξ is damping ratio, ω0 is the natural frequency of vehicle suspension system, and equation (1) is normalized to be:

[0033]

[0034] in, w(t) is the random excitation transmitted from the road surface to the tire and then to the suspension. It is considered as a Gaussian distributed white noise, defined by its first two moments:

[0035] E[w(t)]=0 (3)

[0036] E[w(t)w(t+t′)]=2Dδ(t′) (4)

[0037] In the equation, E[·] represents the function of finding the mean of random variables, δ(·) represents the classical Dirac delta function, and D / π represents the amplitude of the input constant bilateral spectral density function, which is the product of the road roughness coefficient and the vehicle speed:

[0038] D / π=A×V (5)

[0039] Where A is the road roughness coefficient, and V is the forward speed of the vehicle.

[0040] Convert the motion control equation (2) of the quarter suspension system into the state space form:

[0041]

[0042] Where x1 is the displacement x of one quarter of the suspension system s , x2 is one quarter of the suspension system speed

[0043] The random vibration analysis of the vehicle suspension system subjected to random road excitation is carried out, and the Fokker-Planck equation that satisfies the Markov process is derived as follows:

[0044]

[0045] p(x,t) is the probability density function, D is the excitation intensity of Gaussian white noise, g1(x) and g2(x) are the system state equations. For the vehicle suspension system, g1(x) = x2, g2(x) = -2ξω0x2-ω0 2 x1-αω0 2 x s 3 .

[0046] According to formula (6) and formula (7), the specific expression of the Fokker-Planck equation corresponding to the vehicle suspension system is derived as follows:

[0047]

[0048] Step 2: Use the finite element method to discretize the Fokker-Planck equation of the vehicle suspension system. First, use the double partial integration method to obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system. Then, select the rectangular unit as the unit shape and use the bilinear interpolation function to obtain the unit coefficient matrix m and k. According to the finite element construction principle, the global motion control equation is obtained, and its coefficient matrix is ​​M and K.

[0049] First, the weak solution form of the equation is derived and the computational domain Ω is defined as Figure 3 As shown, satisfying x1∈(a,b) and x2∈(c,d), multiply all elements in equation (7) by the weight function w of the same type as the probability density p, and then integrate over the computational domain Ω to obtain:

[0050]

[0051] By simplifying formula (9) using double partial integration, we can obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system:

[0052]

[0053] Using the finite element method, we use matrix elements with the shape Figure 4 As shown. Use the bilinear interpolation function to discretize equation (10). Because the weight function w(x, t) and the probability density function p(x, t) have the same form, they can be expressed by the interpolation function as follows:

[0054]

[0055] Among them, N r (x), N s (x) is the weight function w(x, t) and the probability density function p(x, t) is the shape function corresponding to the unit node, and is the probability density function value at the unit node, r,s=1,2,3,4, and the specific form of the bilinear interpolation function is given, where -1≤ξ≤1,-1≤η≤1.

[0056]

[0057]

[0058] Substituting formula (11) and formula (12) into formula (10), we can obtain the unit motion control equation of the vehicle suspension system:

[0059]

[0060] The expressions of the unit coefficient matrices m and k are:

[0061] m=∫∫NN T dx1dx2 (19)

[0062]

[0063] N=[N1(x1,x2),N2(x1,x2),N3(x1,x2),N4(x1,x2)] is the global interpolation function matrix.

[0064] Using the unit motion control equation of the vehicle suspension system and the finite element construction principle, the global coefficient matrices M and K are constructed according to the numbering of the finite element grids to obtain the global motion control equation:

[0065]

[0066] Step 3: Using the operator splitting method, split the coefficient matrix K in the global motion control equation into a convection term and a diffusion term. At the same time, solve the eigenvalues and eigenvectors of the coefficient matrix M in the global motion control equation. Select the eigenvectors as the effective reduced basis according to the magnitude of the projection coefficients, and use the operator splitting method and the dimensionality reduction method to process the global motion control equation;

[0067] Apply the operator splitting method to split the element coefficient matrix k into two parts, k1 and k2. Equation (18) can be written as

[0068]

[0069] where k1 is called the convection term and k2 is called the diffusion term.

[0070]

[0071] Similarly, according to the finite element construction principle, assemble them into the global coefficient matrices K1 and K2 to obtain the global motion control equation:

[0072]

[0073] Next, use the dimensionality reduction method to reduce the dimension of the system to be solved, reduce the calculation difficulty, and thus reduce the calculation time.

[0074] Given the element coefficient matrix M, assume that the eigenvector matrix of M is Φ,

[0075] Φ = [Φ1, Φ2, …, Φ n (26)

[0076] Φ n is the eigenvector corresponding to the eigenvalue,

[0077] Use some important eigenvectors to form a reduced matrix T for dimensionality reduction solution. The form of the reduced matrix T is as follows,

[0078] T = [Φ1, Φ2, …, Φ m (27)

[0079] At this time, m < n. When using the reduced matrix as the new coordinate system, make the following transformation,

[0080]

[0081] where is the coordinate representation of the probability density function in the new coordinate system. Substitute the coordinate transformation equation (28) into the global motion control equation to obtain:

[0082]

[0083] Multiply both sides of the equation by T on the left T ,get:

[0084]

[0085] in

[0086] The selection criteria of the dimension reduction matrix T is to select the eigenvector with a larger projection coefficient, such as Figure 5 As shown, the projection coefficient β is defined as

[0087]

[0088] Where p is an arbitrary vector, Φ i is an arbitrarily selected eigenvector. The physical meaning of the projection coefficient can be seen from the figure, which represents the vector p in the eigenvector Φ i The length of the projection on the eigenvector. The longer the projection length, the more important the corresponding eigenvector is.

[0089] Step 4: Given the vehicle suspension system parameters and initial conditions, the global motion control equations are iteratively calculated using the operator splitting and dimensionality reduction method to solve the Fokker-Planck equation of the vehicle suspension system and obtain the probability density function of the displacement and velocity of the vehicle suspension system over time. The probability density function is integrated to obtain the mean and mean square value response of the displacement and velocity of the vehicle suspension system.

[0090] Solving the global motion control equations for iterative calculations requires defining initial conditions and boundary conditions. The system boundary generally uses zero flux boundary conditions, that is, the probability density function of all boundary nodes on the four vertex lines of the solution area is 0. The initial condition uses a double probability density function to generate the probability density value of each initial node:

[0091]

[0092] The global motion control equations are iteratively calculated using operator splitting and dimensionality reduction methods, i.e., equation (33) is calculated to obtain the probability density function p(x1, x2) of the vehicle suspension system.

[0093]

[0094] Using the probability density function of the vehicle suspension system, the nonlinear dynamic response of the vehicle suspension system is obtained, and the formula is:

[0095] E(x1)=∫x1(∫p(x1,x2)dx2)dx1 (34)

[0096] E(x2)=∫x2(∫p(x1,x2)dx1)dx2 (35)

[0097]

[0098] The above formulas can be used to obtain the mean and mean square values ​​of the displacement and velocity responses of the nonlinear vehicle suspension system, so the dynamic system state of the nonlinear vehicle suspension system can be analyzed.

[0099] Example 2

[0100] This embodiment assumes that a car is traveling forward at a speed of 20 m / s on a Class B road. Given the specific parameters of the vehicle suspension system, the Fokker-Planck equation of the vehicle suspension system is solved by using finite element combined with operator splitting and dimensionality reduction methods to obtain its nonlinear dynamic response.

[0101] In this example, a fast solution method for the Fokker-Planck equation of the nonlinear dynamic response of a vehicle suspension system includes the following steps:

[0102] Step 1: Consider the random excitation of the road surface as an additive Gaussian white noise, consider the nonlinear factors of the vehicle suspension, and obtain the Fokker-Planck equation of the vehicle suspension system through the motion control equation of the vehicle suspension system, and then proceed to step 2;

[0103] The motion control equation of the vehicle suspension system is:

[0104]

[0105] Rewrite it in state space form

[0106]

[0107] The Fokker-Planck equation of the vehicle suspension system is obtained as

[0108]

[0109] The vehicle suspension system parameters and initial conditions are shown in Table 1.

[0110] Table 1 Parameters of vehicle suspension system

[0111]

[0112] Step 2: Use the finite element method to discretize the Fokker-Planck equation of the vehicle suspension system. First, use the double partial integration method to obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system. Then, select the rectangular unit as the unit shape and use the bilinear interpolation function to obtain the unit coefficient matrix m and k. According to the finite element construction principle, the global motion control equation is obtained, and its coefficient matrix is ​​M and K. Go to step 3;

[0113] The calculation domain is divided by matrix units. The calculation domain of the vehicle suspension system state space is -1≤x1≤1, -1≤x2≤1, and the calculation domain is divided into a 100×100 rectangular grid.

[0114] The unit motion control equation of the vehicle suspension system is obtained using the finite element discretization method:

[0115]

[0116] In the formula,

[0117] m=∫∫NN T dx1dx2 (19)

[0118]

[0119] For ease of calculation, g1(x)=x2, g2(x)=-2ξω0x2-ω0 2 x1-αω0 2 x s 3 .

[0120] According to the grid division principle, the corresponding global motion control equation is obtained

[0121]

[0122] Step 3: Use the operator splitting method to split the coefficient matrix K in the global motion control equation into convection terms and diffusion terms, and solve the eigenvalues ​​and eigenvectors of the coefficient matrix M in the global motion control equation at the same time. Select the eigenvector as the effective reduction basis according to the size of the projection coefficient, and use the operator splitting method and dimensionality reduction method to process the global motion control equation, and then go to step 4;

[0123] Combined with the operator splitting method, the unit coefficient k is split into two terms k1 and k2. Formula (21) can be written as

[0124]

[0125] The specific expressions of k1 and k2 are as follows

[0126]

[0127] The global motion control equation can be written as:

[0128]

[0129] Solve the projection coefficients of the coefficient matrix M, sort them by size, select the first 5000 eigenvectors of the projection coefficients to form a dimension reduction matrix T, and reduce the order of the global motion control equation:

[0130]

[0131] Among them, the dimensions of the coefficient matrices M, K1 and K2 in the full model are 10201×10201, and the coefficient matrices after dimensionality reduction are and The dimension is 5000×5000. It can be seen that the dimension of the equation that needs to be solved is greatly reduced, and the calculation efficiency is significantly improved.

[0132] Step 4: Given the vehicle suspension system parameters and initial conditions, substitute the processed global motion control equations for iterative calculations, solve the Fokker-Planck equation of the vehicle suspension system, and obtain the probability density function of the displacement and velocity of the vehicle suspension system over time. Integrate the probability density function to obtain the mean square value response of the displacement and velocity of the vehicle suspension system.

[0133] The boundary condition is zero flux boundary condition, and the mean and variance of the initial condition are μ x01 =μ x02 =0.01,σ 01 =σ 02 =1 / 100, the parameters of the vehicle suspension system are shown in Table 1, and numerical simulation is carried out.

[0134] In order to verify the accuracy of the numerical simulation results, the probability density function of the vehicle suspension displacement and velocity is integrated over its computational domain by utilizing the normalization of the probability density function, that is, its integral is always equal to one within the computational domain. As shown in Figure 7, in the state space, the integrals of the probability density functions of displacement and velocity are always equal to one, indicating that the probability density function converges during the numerical simulation calculation. This indicates that the numerical simulation results using the finite element combined with operator splitting and dimensionality reduction algorithm are correct.

[0135] Figure 8 shows the contour map of the probability density function of the vehicle suspension system evolving over time, illustrating the process of the vehicle suspension system from initial conditions to steady state. According to the density of the contour lines at different times, it can be found that the vibration of the vehicle suspension system changes violently at the initial moment, and is more obviously affected by the stiffness and damping of the system. Figures 9 and 10 show the mean and mean square value of the displacement and velocity of the vehicle suspension system, respectively, and the nonlinear dynamic response of the vehicle suspension system is obtained. It can be seen that the displacement and velocity fluctuations of the vehicle suspension system are relatively strong in the first two seconds, and then the displacement and velocity responses gradually become stable under the influence of the stiffness and damping of the system. Fig.11 The evolution of the displacement and velocity of the vehicle suspension system over time is shown, which allows us to intuitively see the process of the system evolving from initial conditions until it reaches a steady state.

[0136] The contents described in the embodiments of this specification are merely an enumeration of the implementation forms of the inventive concept. The protection scope of the present invention should not be regarded as limited to the specific forms described in the embodiments. The protection scope of the present invention also extends to equivalent technical means that can be conceived by those skilled in the art based on the inventive concept.

Claims

1. A fast solution method for the Fokker-Planck equation of the nonlinear dynamic response of a vehicle suspension system, characterized in that: The steps include: Step 1: Consider the random excitation of the road surface as an additive Gaussian white noise, consider the nonlinear factors of the vehicle suspension, and obtain the Fokker-Planck equation of the vehicle suspension system through the motion control equation of the vehicle suspension system; Step 2: Use the finite element method to discretize the Fokker-Planck equation of the vehicle suspension system. First, use the double partial integration method to obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system. Then, select the rectangular unit as the unit shape and use the bilinear interpolation function to obtain the unit coefficient matrix m and k. According to the finite element construction principle, the global motion control equation is obtained, and its coefficient matrix is ​​M and K. Step 3: Use the operator splitting method to split the coefficient matrix K in the global motion control equation into convection terms and diffusion terms, and solve the eigenvalues ​​and eigenvectors of the coefficient matrix M in the global motion control equation at the same time. Select the eigenvector as the effective reduction basis according to the size of the projection coefficient, and use the operator splitting method and dimensionality reduction method to process the global motion control equation; Step 4: Given the vehicle suspension system parameters and initial conditions, substitute the processed global motion control equations for iterative calculations, solve the Fokker-Planck equation of the vehicle suspension system, and obtain the probability density function of the displacement and velocity of the vehicle suspension system over time; The probability density function is integrated to obtain the mean and mean square responses of the displacement and velocity of the vehicle suspension system.

2. The Fokker-Planck equation fast solution method for the nonlinear dynamic response of a vehicle suspension system according to claim 1, characterized in that: Step 1 specifically includes: Establish the mechanical model of the quarter suspension system. s is the equivalent mass of the suspension, k is the equivalent damping of the suspension, and considering its nonlinear factors, k = k s +αk s X 2 , α is the nonlinear coefficient, c s is the equivalent damping of the suspension, x s (t) is the vertical displacement of the suspension, and the random excitation transmitted to the suspension by the ground through the tire is r; The following assumptions are made for the quarter suspension system model: (1) the effects of the suspension spring mass and tire mass on the body system motion are ignored; (2) the suspension equivalent mass m s is the mass of the entire vehicle and the suspension distributed proportionally on a single wheel; (3) the quarter suspension under study is considered to be a single-degree-of-freedom system, so the wheel and tire are assumed to be rigid bodies; (4) the random excitation of the quarter suspension is assumed to be Gaussian white noise. Let 2n = c s / m s ,ξ=n / ω0,ω0 2 =k s / m s , n is damping, ξ is damping ratio, ω0 is the natural frequency of vehicle suspension system, and equation (1) is normalized to be: in, w(t) is the random excitation transmitted from the road surface to the tire and then to the suspension. It is considered as a Gaussian distributed white noise, defined by its first two moments: E[w(t)]=0 (3) E[w(t)w(t+t ′ )]=2Dδ(t ′ ) (4) In the equation, E[·] represents the function of finding the mean of random variables, δ(·) represents the classical Dirac delta function, and D / π represents the amplitude of the input constant bilateral spectral density function, which is the product of the road roughness coefficient and the vehicle speed: D / π=A×V (5) Where A is the road roughness coefficient, and V is the forward speed of the vehicle; Convert the motion control equation (2) of the quarter suspension system into the state space form: Where x1 is the displacement x of one quarter of the suspension system s , x2 is one quarter of the suspension system speed The random vibration analysis of the vehicle suspension system subjected to random road excitation is carried out, and the Fokker-Planck equation that satisfies the Markov process is derived as follows: p(x,t) is the probability density function, D is the excitation intensity of Gaussian white noise, g1(x) and g2(x) are the system state equations. For the vehicle suspension system, g1(x) = x2, g2(x) = -2ξω0x2-ω0 2 x1-αω0 2 x s 3 ; According to formula (6) and formula (7), the specific expression of the Fokker-Planck equation corresponding to the vehicle suspension system is derived as follows:

3. The Fokker-Planck equation fast solution method for the nonlinear dynamic response of a vehicle suspension system according to claim 1, characterized in that: Step 2 specifically includes: First, we derive the weak solution form of the equation and define the computational domain Ω, which satisfies x1∈(a,b) and x2∈(c,d). Multiply all elements in equation (7) by the weight function w of the same type as the probability density p, and then integrate over the computational domain Ω to obtain: By simplifying formula (9) using double partial integration, we can obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system: Using the finite element method, using rectangular unit shapes, and using bilinear interpolation functions, equation (10) is discretized; because the weight function w(x, t) and the probability density function p(x, t) have the same form, they can be expressed by the interpolation function as follows: Among them, N r (x), N s (x) is the weight function w(x, t) and the probability density function p(x, t) is the shape function corresponding to the unit node, and is the probability density function value at the unit node, r,s=1,2,3,4, and the specific form of the bilinear interpolation function is given, where -1≤ξ≤1,-1≤η≤1. Substitute Equation (11) and Equation (12) into Equation (10) to obtain the unit motion control equation of the vehicle suspension system: The expressions of the unit coefficient matrices m and k are: m=∫∫NN T dx1dx2 (19) n = [n1(x1, x2), N2(x1, x2), N3(x1, x2), N4(x1, x2)] is the global interpolation function matrix. Using the unit motion control equation of the vehicle suspension system, based on the finite element construction principle, according to the numbering of the finite element mesh, construct the global coefficient matrices M and K to obtain the global motion control equation:

4. The Fokker-Planck equation fast solution method for the nonlinear dynamic response of a vehicle suspension system according to claim 1, characterized in that: Step 3 specifically includes: Apply the operator splitting method to split the unit coefficient matrix k into two parts, k1 and k2. Equation (18) can be written as: where k1 is called the convection term and k2 is called the diffusion term: Similarly, according to the finite element construction principle, assemble them into the global coefficient matrices K1 and K2 to obtain the global motion control equation: Next, use the dimensionality reduction method to reduce the dimension of the system to be solved, reduce the calculation difficulty, and thus reduce the calculation time. Given the unit coefficient matrix M, assume the eigenvector matrix of M is Φ, Φ=[Φ1,Φ2,…,Φ n ] (26) Φ n is the eigenvector corresponding to the eigenvalue, Use some important eigenvectors to form the dimensionality reduction matrix T for dimensionality reduction solution. The form of the dimensionality reduction matrix T is as follows: T=[Φ1,Φ2,…,Φ m ] (27) At this time, m < n. When using the reduced matrix as the new coordinate system, make the following transformation, in is the coordinate representation of the probability density function in the new coordinate system; substituting the coordinate transformation equation (28) into the global motion control equation yields: Multiply both sides of the equation by T on the left T ,get: in The selection criterion for the dimensionality reduction matrix T is to select the eigenvectors with larger projection coefficients. The projection coefficient β is defined as: Where p is an arbitrary vector, Φ i is an arbitrarily selected eigenvector, and the projection coefficient represents the vector p on the eigenvector Φ i The length of the projection on the eigenvector. The longer the projection length, the more important the corresponding eigenvector is.

5. The Fokker-Planck equation fast solution method for the nonlinear dynamic response of a vehicle suspension system according to claim 1, characterized in that: Step 4 specifically includes: To solve the global motion control equation for iterative calculation, initial conditions and boundary conditions need to be defined; generally, the zero-flux boundary condition is used for the system boundary, that is, the probability density function of all boundary nodes on the four vertex connection lines of the solution region is 0; the initial conditions are determined by generating the probability density value of each initial node from the double probability density function: Use the operator splitting and dimensionality reduction method to perform iterative calculation on the global motion control equation, that is, calculate Equation (33) to obtain the probability density function p(x1, x2) of the vehicle suspension system; Using the probability density function of the vehicle suspension system, obtain the nonlinear dynamic response of the vehicle suspension system. The formula is: E(x1) = ∫x1(∫p(x1, x2)dx2)dx1 (34) E(x2) = ∫x2(∫p(x1, x2)dx1)dx2 (35) From the above formulas, the mean and mean square values of the displacement and velocity responses of the nonlinear vehicle suspension system can be obtained, so the dynamic system state of the nonlinear vehicle suspension system can be analyzed.

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