Fast solution method for fokker-planck equation of nonlinear dynamic response of vehicle suspension system

Abstract

The application discloses a fast solving method for Fokker-Planck equation of nonlinear dynamic response of a vehicle suspension system, and comprises the following steps: considering the nonlinear factors of the vehicle suspension, establishing a Fokker-Planck equation containing a probability density function of the vehicle suspension system by a motion control equation of the vehicle suspension system subjected to random excitation of a road surface; discretely processing the Fokker-Planck equation of the vehicle suspension system by using a finite element discrete method to obtain a global motion control equation; accelerating the solving of the global motion control equation by using an operator splitting method and a dimension reduction method to obtain the probability density function of displacement and velocity response of the vehicle suspension system; and integrating the probability density function of displacement and velocity of the vehicle suspension system in a state space to obtain the dynamic response of the vehicle system to the random excitation of the road surface. The application has engineering application guiding significance for structure design and damping control of the vehicle by fast solving the Fokker-Planck equation of the vehicle suspension to obtain the nonlinear dynamic response of the vehicle suspension system.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of solving method and technology of random dynamics response of vehicle suspension system, and particularly relates to a method for solving the nonlinear dynamics response of vehicle suspension system according to the transient solution of Fokker-Planck equation of vehicle suspension system under random road excitation. BACKGROUND

[0002] In the existing research, the vehicle suspension system is mostly simplified as a linear system, but in practice, the vehicle suspension is a nonlinear system, and the dynamics response of the vehicle suspension system under nonlinear factors needs to be considered. The time domain model of road excitation is often simulated and calculated using the superposition method of simple harmonic waves, 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 dynamics response of the vehicle cannot be expressed by a deterministic function of time and space coordinates, and is generally described and modeled by probability or statistical methods in random dynamics response analysis. In random vibration analysis, Fokker-Planck equation is a very important tool, and the probability density function of the dynamic response of a random dynamics system can be obtained by solving the Fokker-Planck equation.

[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 takes about 4 hours to solve the Fokker-Planck equation by using the finite element method under the premise of 100x100 finite element grid. Therefore, the Fokker-Planck equation of the vehicle suspension system is solved by combining the operator splitting and dimension reduction algorithm to quickly obtain the dynamic response of the speed and displacement of the vehicle suspension system. SUMMARY

[0004] The present application overcomes the above-mentioned shortcomings of the prior art and provides a fast solving method of Fokker-Planck equation for nonlinear dynamics response of vehicle suspension system.

[0005] The present application aims to quickly solve the Fokker-Planck equation of the vehicle suspension system under the influence of random road excitation. The purpose is to quickly solve the Fokker-Planck equation of the vehicle suspension system under a certain calculation accuracy by combining the operator splitting and dimension reduction algorithm with the finite element method, to obtain the complete probability density function of the system at any time, and to study the nonlinear dynamics response of the system through the probability density function.

[0006] The technical solution for achieving the object of the application is a fast solving method of Fokker-Planck equation of nonlinear dynamics response of a vehicle suspension system, comprising the following steps:

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

[0008] Step 2: the Fokker-Planck equation of the vehicle suspension system is discretized by using a finite element method, first, a weak solution form of the Fokker-Planck equation of the vehicle suspension system is obtained by using a double integral method, then a rectangular element is selected as an element shape and a bilinear interpolation function is used to obtain element coefficient matrices m and k, and a global motion control equation is obtained according to a finite element construction principle, the coefficient matrix of which is M and K;

[0009] Step 3: the coefficient matrix K in the global motion control equation is split into a convection term and a diffusion term by using an operator splitting method, and the eigenvalues and eigenvectors of the coefficient matrix M in the global motion control equation are solved, the eigenvectors are selected as effective reduced bases according to the size of the projection coefficient, and the global motion control equation is processed by using the operator splitting method and the dimension reduction method;

[0010] Step 4: given the parameters of the vehicle suspension system and the initial conditions, the processed global motion control equation is substituted and iteratively calculated to solve the Fokker-Planck equation of the vehicle suspension system, and the probability density function of the displacement and velocity of the vehicle suspension system evolving with time is obtained. The mean and mean square value responses of the displacement and velocity of the vehicle suspension system are obtained by integrating the probability density function.

[0011] The application is based on solving the Fokker-Planck equation of the vehicle suspension system to obtain the probability density function of the vehicle suspension system under random road excitation, and the nonlinear dynamics response of the vehicle suspension system is studied through the probability density function. Meanwhile, the Fokker-Planck equation of the vehicle suspension system is quickly solved by combining the operator splitting and dimension reduction method to quickly obtain the nonlinear dynamics response of the vehicle suspension system.

[0012] Compared with the prior art, the application has the following advantages:

[0013] (1) The nonlinear dynamics response of the system is obtained by solving the Fokker-Planck equation of the vehicle suspension system, which considers the nonlinear factors of the vehicle suspension and considers the road excitation as a Gaussian white noise, which is more consistent with the actual driving conditions of the vehicle on uneven roads.

[0014] (2) using operator splitting and dimension reduction method to calculate the Fokker-Planck equation of the vehicle suspension system, under the premise of meeting the numerical calculation accuracy, the calculation time can be effectively reduced, so as to quickly solve the nonlinear dynamic response of the vehicle suspension system, and help to realize the real-time control and optimization of the vehicle suspension system. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 is a flowchart of the method of the present application;

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

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

[0018] Figure 4 is a unit shape of the present application;

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

[0020] Figure 6 is a projection coefficient descending order diagram of the dimension reduction algorithm of the present application;

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

[0022] Fig. 8 is a contour plot of the probability density function of the vehicle suspension system of the present application evolving with time;

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

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

[0025] Figure 11 is a displacement and velocity evolution diagram of the vehicle suspension system of the present application with time. DETAILED DESCRIPTION

[0026] The technical solutions of the present application will be described in detail below with reference to the accompanying drawings.

[0027] The Fokker-Planck equation fast solving method for nonlinear dynamic response of the vehicle suspension system proposed by the present application specifically includes the following steps:

[0028] Step 1: The random excitation of the road surface is considered as an additive Gaussian white noise, and the nonlinear factor of the vehicle suspension is considered. The Fokker-Planck equation of the vehicle suspension system is obtained by the motion control equation of the vehicle suspension system.

[0029] A mechanical model of a quarter suspension system is established, as shown in Figure 2 where m s is the equivalent mass of the suspension, k is the equivalent damping of the suspension, and the nonlinear factor is considered, 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 of the ground to the suspension 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 the tire mass on the vehicle body system motion are ignored; (2) the equivalent mass m s of the suspension is the mass of the entire vehicle and the suspension distributed proportionally on a single wheel; (3) the quarter suspension under study is a single degree of freedom system, so the wheel and tire are assumed to be rigid bodies; and (4) the random excitation of the quarter suspension is assumed to be a Gaussian white noise.

[0031]

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

[0033]

[0034] where, w(t) is the random excitation of the road surface transmitted to the tire and then to the suspension, which 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 the mean value of the random variable, δ(·) represents the classical Dirac delta function, and D / π represents the amplitude of the constant bilateral spectral density function, which is the product of the road roughness coefficient and the vehicle speed:

[0038] D / π = A x V (5)

[0039] where A is the road roughness coefficient and V is the vehicle forward speed.

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

[0041]

[0042] where x1is the quarter suspension system displacement x s , x2is the quarter suspension system velocity

[0043] The Fokker-Planck equation satisfying the Markov process is derived by performing the random vibration analysis on the vehicle suspension system subjected to the random excitation of the road surface:

[0044]

[0045] p(x, t) is the probability density function, D is the excitation intensity of the Gaussian white noise, g1(x), 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] The specific expression of the Fokker-Planck equation corresponding to the vehicle suspension system is derived according to the formula (6) and the formula (7):

[0047]

[0048] Step 2: Discretize the Fokker-Planck equation of the vehicle suspension system using the finite element method, first, obtain the weak solution form of the Fokker-Planck equation of the vehicle suspension system by using the double integral method, then select a rectangular element as the element shape, use the bilinear interpolation function to obtain the element coefficient matrix m and k, and obtain the global motion control equation according to the finite element construction principle, the coefficient matrix of which is M and K;

[0049] First, derive the weak solution form of the equation, define the calculation domain Ω as shown in Figure 3 , 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 calculation domain Ω to obtain:

[0050]

[0051] By simplifying formula (9) using double partition integral, the weak solution form of Fokker-Planck equation of vehicle suspension system can be obtained:

[0052]

[0053] By using finite element method, matrix element shape as shown in Figure 4 is used. The equation (10) is discretized using bilinear interpolation function. Because the weight function w(x, t) and the probability density function p(x, t) are of the same form, they can be expressed by interpolation function as:

[0054]

[0055] Where, N r (x), N s (x) are the shape functions corresponding to the element nodes of the weight function w(x, t) and the probability density function p(x, t), and are the probability density function values at the element nodes, r, s = 1, 2, 3, 4, which give the specific form of bilinear interpolation function, where -1≤ξ≤1, -1≤η≤1.

[0056]

[0057]

[0058] Put formula (11) and formula (12) into formula (10) to get the element motion control equation of vehicle suspension system:

[0059]

[0060] Where the expressions of element coefficient matrix 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 element motion control equation of vehicle suspension system, using the finite element construction principle, according to the numbering of finite element grid, the global coefficient matrix M and K are constructed to get the global motion control equation:

[0065]

[0066] Step 3: using the operator splitting method, the coefficient matrix K in the global motion control equation is split into convection term and diffusion term, while solving the eigenvalues and eigenvectors of the coefficient matrix M in the global motion control equation, according to the size of the projection coefficient, the eigenvectors are selected as the effective reduced basis, and the global motion control equation is processed by using the operator splitting method and dimension reduction method;

[0067] The operator splitting method is applied to split the element coefficient matrix k into two parts k1 and k2. Formula (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 the global coefficient matrix K1 and K2 to obtain the global motion control equation:

[0072]

[0073] Next, the dimension reduction method is used 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 the part of the important eigenvectors to form the dimension reduction matrix T for dimension reduction solution, the form of the dimension reduction matrix T is as follows,

[0078] T=[Φ1,Φ2,…,Φ m ] (27)

[0079] At this time m<n, use the reduced matrix as the new coordinate system, and make the following conversion,

[0080]

[0081] Where is the coordinate representation of the probability density function in the new coordinate system. The coordinate conversion equation (28) is brought into the global motion control equation to obtain:

[0082]

[0083] Multiply both sides of the equation by T T , we get

[0084]

[0085] where

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

[0087]

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

[0089] Step 4: Given the parameters of the vehicle suspension system, given the initial conditions, use operator splitting and dimension reduction method to iteratively calculate the global motion control equation, solve the Fokker-Planck equation of the vehicle suspension system, and get the probability density function of the displacement and velocity of the vehicle suspension system with time evolution. Integrate the probability density function to get the mean and mean square value response of the displacement and velocity of the vehicle suspension system.

[0090] Solving the global motion control equation 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 connecting line of the four vertices of the solution region is 0. The initial condition uses a double probability density function to generate the probability density value of each initial node:

[0091]

[0092] Use operator splitting and dimension reduction method to iteratively calculate the global motion control equation, that is, calculate equation (33) to get 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 mean and mean square values of the displacement and velocity responses of the nonlinear vehicle suspension system can be obtained from the above formula, and thus the dynamic system state of the nonlinear vehicle suspension system can be analyzed.

[0099] Embodiment 2

[0100] This embodiment assumes that a car is driving forward at a speed of 20 m / s on a B-class road, and specific parameters of the vehicle suspension system are given. The Fokker-Planck equation of the vehicle suspension system is solved by using finite elements combined with operator splitting and dimension reduction method, so as to obtain the nonlinear dynamic response thereof.

[0101] The Fokker-Planck equation fast solving method for the nonlinear dynamic response of the vehicle suspension system of the present example specifically comprises the following steps:

[0102] Step 1: considering the road random excitation as an additive Gaussian white noise and considering the nonlinear factors of the vehicle suspension, the Fokker-Planck equation of the vehicle suspension system is obtained through the motion control equation of the vehicle suspension system, and then Step 2 is entered.

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

[0104]

[0105] Rewrite it into the form of state space

[0106]

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

[0108]

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

[0110] Table 1 Parameters of the vehicle suspension system

[0111]

[0112] Step 2: the Fokker-Planck equation of the vehicle suspension system is discretized by using the finite element method. First, the weak solution form of the Fokker-Planck equation of the vehicle suspension system is obtained by using the double integral method, then the rectangular element is selected as the element shape, the bilinear interpolation function is used to obtain the element coefficient matrix m and k, and the global motion control equation is obtained according to the finite element construction principle, and the coefficient matrix is M and K. Step 3 is entered.

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

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

[0115]

[0116] In the formula,

[0117] m = ∫∫NN T dx1dx2 (19)

[0118]

[0119] For easy calculation, g1(x) = x2 and 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: Using the operator splitting method, the coefficient matrix K in the global motion control equation is split into convection and diffusion terms, and the eigenvalues and eigenvectors of the coefficient matrix M in the global motion control equation are solved, and the eigenvectors are selected as effective reduced bases according to the size of the projection coefficient, and the global motion control equation is processed by using the operator splitting method and the dimension reduction method, and step 4 is entered;

[0123] Combined with the operator splitting method, the unit coefficient k is split into 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] The projection coefficient of the coefficient matrix M is solved, sorted by size, and the eigenvectors of the first 5000 projection coefficients are selected to form a dimension reduction matrix T, and the global motion control equation is reduced:

[0130]

[0131] Wherein, the dimension of coefficient matrix M, K1 and K2 in the full model is 10201x10201, and the dimension of coefficient matrix and after dimension reduction is 5000x5000, it can be seen that the dimension of equation to be solved is greatly reduced, and the calculation efficiency is obviously improved.

[0132] Step 4: Given the vehicle suspension system parameters and initial conditions, substitute the processed global motion control equation to perform iterative calculation, 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 evolving with 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 selected as the zero flux boundary condition, and the mean and variance of the initial condition are selected as μ x01 = μ x02 = 0.01, σ 01 = σ 02 = 1 / 100, and the parameters of the vehicle suspension system are shown in Table 1, and numerical simulation is performed.

[0134] To verify the accuracy of the numerical simulation results, the probability density function is integrated over its calculation domain, as shown in Figure 7, and the integral of the displacement and velocity probability density function in the state space is always equal to one, which indicates that the probability density function is convergent during the numerical simulation calculation, and the numerical simulation results obtained by using finite element combined with operator splitting and dimension reduction algorithm are correct.

[0135] Figure 8 shows the contour plot of the probability density function of the vehicle suspension system evolving with time, which illustrates the process of the vehicle suspension system from the initial condition to the steady state. According to the density of the contour at different times, it can be found that the vibration of the vehicle suspension system is relatively intense at the initial time, and the influence of the system stiffness and damping is more obvious; 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 of the vehicle suspension system fluctuate strongly in the first two seconds, and then gradually tend to be stable under the influence of the system stiffness and damping. Figure 11 The evolution of the displacement and velocity of the vehicle suspension system with time is shown, which can be directly observed from the initial condition to the steady state.

[0136] The embodiments of the present specification are merely a list of implementation forms of the inventive concept, and the protection scope of the present application should not be regarded as being limited to the specific forms stated in the embodiments, and the protection scope of the present application also encompasses equivalent technical means that can be thought of by those skilled in the art according to the inventive concept.

Claims

1. A fast solution method for Fokker-Planck equation of nonlinear dynamic response of vehicle suspension system, characterized in that, Comprising the following steps: Step 1: considering the road random excitation as an additive white Gaussian noise, considering the nonlinear factors of vehicle suspension, through the motion control equation of vehicle suspension system, the Fokker-Planck equation of vehicle suspension system is obtained; Step 2: the Fokker-Planck equation of vehicle suspension system is discretized by using finite element method, firstly, the weak solution form of the Fokker-Planck equation of vehicle suspension system is obtained by using double integral method, then the rectangular element is selected as the element shape, the bilinear interpolation function is used to obtain the element coefficient matrix m and k, and the global motion control equation is obtained according to the finite element construction principle, and the coefficient matrix is M and K; Step 3: the coefficient matrix K in the global motion control equation is split into convection term and diffusion term by using operator splitting method, and the eigenvalue and eigenvector of the coefficient matrix M in the global motion control equation are solved, the eigenvector is selected as the effective reduction base according to the size of the projection coefficient, and the global motion control equation is processed by using operator splitting method and dimension reduction method; Step 4: given the initial conditions under the given vehicle suspension system parameters, the processed global motion control equation is substituted into the global motion control equation to perform iterative calculation, the Fokker-Planck equation of vehicle suspension system is solved, and the probability density function of displacement and velocity of vehicle suspension system evolving with time is obtained; The probability density function is integrated to obtain the mean and mean square value responses of displacement and velocity of vehicle suspension system; Step 1 specifically comprises: Establish a mechanical model of the quarter suspension system, where m is in the system. s Let k be the equivalent mass of the suspension, and k be the equivalent damping of the suspension. Considering its nonlinearity, we have k = k s +αk s X 2 α is a nonlinear coefficient, c s For the equivalent damping of the suspension, x s (t) represents the vertical displacement of the suspension, and r is the random excitation transmitted from the ground to the suspension through the tires; The following assumptions are made for the quarter-car model: ① The influence of the suspension spring mass and the tire mass on the motion of the vehicle body system is ignored; ② The equivalent mass m s is the mass of the vehicle and the suspension mass distributed proportionally on the single wheel; ③ The quarter-car studied is a single degree of freedom system, so the wheel and tire are assumed to be rigid bodies; ④ The random excitation of the quarter-car is assumed to be Gaussian white noise; Let 2n = c s / m s , ξ = n / ω0, ω0 2 = k s / m s , n is the damping, ξ is the damping ratio, and ω0 is the natural frequency of the vehicle suspension system. Normalizing equation (1), we have: wherein w(t) is the random excitation from the road to the tire and from the tire to the suspension, considered as a white noise with Gaussian distribution, 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 calculating the mean value of random variable, δ(·) represents the classical Dirac delta function, and D / π represents the amplitude of the input constant double-sided spectral density function, which is the product of the road roughness coefficient and the vehicle driving speed: D / π=A×V (5) In the formula, A is the road roughness coefficient, and V is the speed of the vehicle driving forward; The motion control equation (2) of the quarter suspension system is converted into the form of state space: where x1 is a quarter suspension system displacement x s and x2 is a quarter suspension system velocity The random vibration analysis of the vehicle suspension system excited by the road random excitation is carried out, and the Fokker-Planck equation satisfying the Markov process is derived as: p(x, t) is the probability density function, D is the excitation intensity of Gaussian white noise, g1(x), 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 corresponding Fokker-Planck equation of the vehicle suspension system is derived as 2. The fast solution method of Fokker-Planck equation for nonlinear dynamics response of vehicle suspension system according to claim 1, characterized in that, Step 2 specifically comprises: Firstly, the weak solution form of the equation is derived, the calculation domain Ω is defined, which satisfies x1∈(a,b) and x2∈(c,d), all elements in equation (7) are multiplied by the weight function w of the same type as the probability density p, and then integrated on the calculation domain Ω to obtain: The weak solution form of the Fokker-Planck equation of the vehicle suspension system can be obtained by simplifying formula (9) by using double integral: By using the finite element method, the bilinear interpolation function is used to discretize equation (10) by using the rectangular element shape; because the weight function w(x,t) and the probability density function p(x,t) are of the same form, they can be expressed by using the interpolation function as: where N r (x), N s (x) are the shape functions corresponding to the element nodes, and are the values of the probability density function at the element nodes, r,s = 1,2,3,4, giving the specific form of the bilinear interpolation function, where -1≤ξ≤1, -1≤η≤1. Put formula (11) and formula (12) into formula (10), the unit motion control equation of the vehicle suspension system is obtained: Wherein the expression of unit coefficient matrix m and k is: 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, using the finite element construction principle, according to the numbering of the finite element grid, the global coefficient matrix M and K are constructed, and the global motion control equation is obtained:

3. The fast solution method of Fokker-Planck equation for nonlinear dynamics response of vehicle suspension system according to claim 1, characterized in that, Step 3 specifically includes: Using the operator splitting method, the unit coefficient matrix k is split into k1 and k2 two parts, formula (18) can be written as: Wherein k1 is called convection term, and k2 is called diffusion term: According to the finite element construction principle, assemble the global coefficient matrix K1 and K2, and obtain the global motion control equation: Next, use the dimension 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 that the eigenvector matrix of M is Φ, Φ = [Φ1, Φ2,..., Φ n ] (26) Φ n is the eigenvector corresponding to the eigenvalue Use the part of the important eigenvectors to form the dimension reduction matrix T for dimension reduction solution, and the form of the dimension reduction matrix T is as follows: T = [Φ1, Φ2,..., Φ m ] (27) At this time m < n, use the reduced matrix as the new coordinate system, and make the following conversion, where is the coordinate representation of the probability density function in the new coordinate system; bringing the coordinate transformation equation (28) into the global motion control equation gives Multiplying both sides of the equation by T simultaneously on the left T , we obtain: wherein The selection standard of the dimension reduction matrix T is to select the eigenvectors with larger projection coefficients, and the projection coefficient β is defined as: where p is an arbitrary vector, Φ i is an arbitrary chosen feature vector, and the projection coefficient represents the length of the projection of vector p onto the feature vector Φ i . The longer the projection, the more important the corresponding feature vector is proved to be.

4. The fast solution method of Fokker-Planck equation for nonlinear dynamics response of vehicle suspension system according to claim 1, characterized in that, Step 4 specifically includes: Solving the global motion control equation 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 connecting line of the four vertices of the solution region is 0; The initial condition is determined by generating the probability density value of each initial node from the double probability density function: Using the operator splitting and dimension reduction method to iteratively calculate the global motion control equation, that is, calculating equation (33), the probability density function p(x1, x2) of the vehicle suspension system is obtained; 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: E(x1) = ∫x1(∫p(x1, x2)dx2)dx1 (34) E(x2) = ∫x2(∫p(x1, x2)dx1)dx2 (35) From the above formula, the mean and mean square value of the nonlinear vehicle suspension system displacement and velocity response can be obtained, so the dynamic system state of the nonlinear vehicle suspension system can be analyzed.

Images