A method for calculating vertical dynamics of an electric vehicle considering integrated powertrain system

Abstract

The application discloses a kind of considering integrated powertrain system electric vehicle vertical dynamics calculation method, and its specific steps are as follows: 1) the model of driving motor radial unbalance magnetic pull is established;2) the dynamic model of two-stage gear transmission system is established;3) the model of road random excitation is established;4) the vertical dynamics model of electric vehicle considering integrated powertrain system is established;5) centralized drive electric vehicle vertical dynamics calculation.The beneficial effect is that the method integrates driving motor radial unbalance magnetic pull, transmission gear meshing excitation and road random excitation, forms wideband-multisource coupling excitation to simulate electric vehicle real vehicle working condition, which can improve the calculation accuracy of vibration, smoothness and component fatigue reliability related to electric vehicle vertical dynamics, provide technical support for electric vehicle vertical dynamics design, analysis and parameter optimization, and serve engineering application.

Description

Technical Field

[0001] This invention relates to the field of electric vehicle system dynamics, and in particular to a method for calculating the vertical dynamics of an electric vehicle considering an integrated power transmission system. Background Technology

[0002] To address the increasingly prominent contradiction between fuel supply and demand and environmental pollution, electric vehicles have been widely adopted worldwide. Vehicle system dynamics influences vehicle performance indicators such as vibration, ride comfort, operational stability, component fatigue reliability, road friendliness, and NVH (noise, vibration, and harshness), making it a hot research topic for automotive OEMs and related research institutions. Establishing accurate real-vehicle excitation is crucial for analyzing vertical dynamics. For the vertical system dynamics of electric vehicles, the main excitations are random road surface excitation and transmission system excitation.

[0003] Electric vehicles can be categorized into centralized drive and hub-distributed drive systems based on their drive mechanism. Current vertical dynamics calculation methods for the former only consider random road surface excitations, while some methods for the latter consider both road surface excitations and drive motor excitations. However, centralized drive and hub-distributed drive electric vehicles differ in structure and operating characteristics, making their vertical dynamics calculation methods not universally applicable. Furthermore, current electric vehicles primarily employ centralized drive, where the drive system excitation includes the combined effects of the drive motor and reducer, acting on the electric vehicle in conjunction with random road surface excitations, resulting in a wide-bandwidth, multi-source excitation characteristic. In summary, existing vertical dynamics calculation methods for centralized drive electric vehicles still face numerous limitations.

[0004] To address the aforementioned issues, this invention proposes a method for calculating the vertical dynamics of electric vehicles that considers an integrated powertrain system. This method calculates the impact of the powertrain system on the overall vehicle's vertical dynamics while simultaneously analyzing the effects of road conditions on the powertrain system's dynamics. This approach will promote the development of electric vehicle vertical dynamics calculation methods, improve the efficiency and reliability of electric vehicle vertical system dynamics calculations, provide theoretical support for electric vehicle design optimization, fill related technological gaps, and drive the development of system dynamics theory under broadband multi-source coupled excitation in electric vehicles, thus yielding significant social and economic benefits. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies and fill related technological gaps, this invention provides a method for calculating the vertical dynamics of electric vehicles considering integrated powertrain systems. The technical solution adopted by this invention to solve its technical problems is as follows:

[0006] A method for calculating the vertical dynamics of an electric vehicle considering an integrated powertrain system, characterized by the following steps:

[0007] Step (1): Establish a radial unbalanced magnetic pull model for the drive motor;

[0008] The radial unbalanced magnetic pull of the drive motor can be expressed as:

[0009] (1);

[0010] In equation (1), The inner radius of the stator; This refers to the effective length of the rotor. The amplitude of the synthesized magnetomotive force in the air gap; The air gap permeability; The angle between the point of minimum air gap and the x-axis; The Fourier coefficient is the permeability of the eccentric air gap. , is represented as:

[0011] (2);

[0012] In equation (2), The average air gap length when the stator and rotor are eccentric; The relative eccentricity; This refers to the rotor eccentricity.

[0013] Step (2): Establish a dynamic model of the two-stage gear transmission system;

[0014] The generalized displacement of a two-stage gear transmission system is defined as:

[0015] (3);

[0016] In equation (3), , , and These represent the translational displacement and rotational angular displacement of the first-stage driving gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the first-stage driven gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the second-stage driving gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the second-stage driven gear in the x, y, and z directions, respectively.

[0017] The dynamic equation of the two-stage gear transmission system is:

[0018] (4);

[0019] In equation (4), and These are the mass and moment of inertia of the first-stage driving gear, respectively. and These are the mass and moment of inertia of the first-stage driven gear, respectively. and These are the mass and moment of inertia of the second-stage driving gear, respectively. and These are the mass and moment of inertia of the second-stage driven gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the first-stage driving gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the first-stage driven gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the second-stage driving gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the second-stage driven gear, respectively. and These are the equivalent support damping in the x and z directions of the first-stage driving gear, respectively; and These are the equivalent support damping in the x and z directions of the first-stage driven gear, respectively; and These are the equivalent support damping in the x and z directions of the second-stage driving gear, respectively; and These are the equivalent support damping in the x and z directions for the second-stage driven gear, respectively; and The meshing stiffness of the first and second stage gear pairs are respectively defined by equation (5). and The meshing damping of the first and second stage gear pairs are respectively defined by equation (6). and These are the torsional stiffness and torsion of the intermediate shaft, respectively. and These are the pressure angles of the first and second stage gears, respectively. and These are the meshing helix angles of the first and second stage gears, respectively. and These are the base circle radii of the first-stage driving gear and driven gear, respectively. and These are the base circle radii of the second-stage driving gear and driven gear, respectively. and These are the input torque and the load torque, respectively. and These are the y-axis supporting forces exerted by the bearings on the first-stage driving gear and driven gear, respectively. and The bearing forces acting on the second stage driving gear and driven gear in the y direction are respectively defined as Equation (7). and The total deformation of the meshing gear pair projected onto the meshing line direction by the vibration of each direction is defined by equation (8). The torsional deformation of the intermediate shaft is defined by equation (10).

[0020] The meshing stiffness of the gear pair is:

[0021] (5);

[0022] In equation (5), and These are the average meshing stiffness values ​​of the first and second stage gear pairs, respectively. and These represent the fluctuation amplitude of the meshing stiffness of the first and second stage gear pairs, respectively. and These are the meshing angular frequencies of the first and second stage gear pairs, respectively. and These are the meshing phases of the first and second stage gear pairs, respectively;

[0023] The meshing damping of the gear pair is:

[0024] (6);

[0025] In equation (6), and The damping coefficients for the meshing of the first and second stage gear pairs;

[0026] The y-axis supporting force of the bearing on the gear:

[0027] (7);

[0028] In equation (7), Vertical vibration of the integrated drive system; and These are the y-axis equivalent support stiffnesses of the first-stage driving gear and driven gear, respectively. and These are the y-axis equivalent support stiffnesses of the second-stage driving gear and driven gear, respectively. and The y-axis equivalent support damping is respectively for the first-stage driving gear and driven gear; and The y-axis equivalent support damping is respectively for the second-stage driving gear and driven gear;

[0029] The total deformation of the meshing gear pair projected by the vibrations in all directions onto the meshing line is:

[0030] (8);

[0031] In equation (8), The gear pair transmission error is defined by equation (9).

[0032] (9);

[0033] In equation (9), and These represent the meshing transmission error amplitudes of the first and second stage gear pairs, respectively. and These are the meshing frequencies of the first and second stage gear pairs, respectively. and These are the meshing phase angles of the first and second stage gear pairs, respectively.

[0034] The torsional deformation of the intermediate shaft is as follows:

[0035] (10);

[0036] Step (3): Establish a road surface stochastic excitation model;

[0037] The road surface stochastic excitation model is defined as follows:

[0038] (11);

[0039] In equation (11), , , and These represent the random road surface excitations experienced by the left front, right front, left rear, and right rear wheels, respectively. For vehicle speed, This is the spatial cutoff frequency; For space reference frequency; This refers to the random road surface roughness coefficient. White noise with a power intensity of 0.5; The distance between the front and rear suspensions of a vehicle is divided by its speed. This is the distance between the left and right suspensions of the vehicle.

[0040] Step (4): Establish a vertical dynamics model of the electric vehicle considering the integrated drivetrain system;

[0041] The generalized displacement of the vertical dynamics system of an electric vehicle considering an integrated drivetrain system is defined as follows:

[0042] (12);

[0043] In equation (12), , , and These represent the vertical displacements of the unsprung mass of the left front, right front, left rear, and right rear suspensions, respectively. , and These are the vertical displacement of the sprung mass, the pitch angle displacement, and the roll angle displacement, respectively. , and These are the vertical displacement, pitch angle displacement, and roll angle displacement of the integrated drive system, respectively.

[0044] The vertical dynamic equation of an electric vehicle considering an integrated drivetrain system is:

[0045] (13);

[0046] In equation (13), , and These represent the sprung mass, pitch moment of inertia, and roll moment of inertia, respectively. , , and These represent the unsprung masses at the front left, front right, rear left, and rear right, respectively. , and These represent the mass, pitch moment of inertia, and roll moment of inertia of the drive system, respectively. , , and These are the suspension forces for the front left, front right, rear left, and rear right, respectively. , , and These are the forces on the front left, front right, rear left, and rear right tires, respectively. , and The front left, front right and rear suspension forces of the integrated drive system are defined by equation (14). and These are the distances from the sprung center of gravity to the front and rear suspensions, respectively. This is the distance from the sprung center of gravity to the left suspension. This is the distance from the sprung center of mass to the center of mass of the transmission system; and These are the distances from the center of gravity of the drivetrain to the front and rear suspensions, respectively.

[0047] (14);

[0048] In equation (14), , , and These are the suspension stiffnesses for the front left, front right, rear left, and rear right, respectively. , , and These are the front left, front right, rear left, and rear right suspension damping, respectively. , , and These are the stiffnesses of the front left, front right, rear left, and rear right tires, respectively. , , and These are the damping for the front left, front right, rear left, and rear right tires, respectively. and These are the lateral distances from the center of gravity of the integrated drivetrain system to the left and right front suspensions, respectively.

[0049] Step (5): Vertical dynamics calculation of centralized drive electric vehicle;

[0050] In numerical software, the radial unbalanced magnetic pull model of the drive motor described in step (1), the dynamic model of the two-stage gear transmission system described in step (2), the random excitation model of the road surface described in step (3), and the vertical dynamic model of the electric vehicle considering the integrated drive system described in step (4) are programmed and solved using numerical algorithms to output the motion displacement, velocity, and acceleration of the concentrated mass of all vehicles.

[0051] Compared with existing technologies, the beneficial effects of this invention are as follows: This method couples traditional vehicle vertical dynamics with power transmission system dynamics, forming a wide-frequency multi-source excitation under the conditions of radial unbalanced force of the drive motor, gear meshing excitation of the reducer, and random road surface excitation. This makes the vertical dynamics calculation results of centralized drive electric vehicles more accurate and more practically significant. Applying this method, while analyzing vehicle vertical dynamics, it is possible to study the dynamics of the power transmission system or the interaction between the two. This provides a method for calculating the vertical dynamics of electric vehicles that considers integrated power transmission systems, which has broad engineering application prospects. It can fill relevant technological gaps and generate significant social and economic benefits. Attached Figure Description

[0052] Figure 1 This is a flowchart illustrating the implementation of a method for calculating the vertical dynamics of an electric vehicle that considers an integrated power transmission system.

[0053] Figure 2 This is a schematic diagram of the cross-section of the drive motor;

[0054] Figure 3 It is a dynamic model of a two-stage gear transmission system;

[0055] Figure 4 It is a vertical dynamics model of an electric vehicle that considers the integrated drivetrain system;

[0056] Figure 5 This is a time-domain plot of the vertical vibration acceleration of the spring-loaded center of mass.

[0057] Figure 6 It is a frequency domain diagram of the vertical vibration acceleration of the spring-loaded center of mass;

[0058] Figure 7 This is a graph showing the relationship between the front suspension stiffness and the damping relative to the root mean square of the vertical vibration acceleration of the spring-loaded center of mass.

[0059] Figure 8 This is a graph showing the relationship between front suspension stiffness and damping and the root mean square of the vertical vibration acceleration of the center of mass of the integrated power transmission system. Detailed Implementation

[0060] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 8 The specific embodiments of the present invention will be described in detail below.

[0061] Reference Figure 1 A method for calculating the vertical dynamics of an electric vehicle considering an integrated powertrain system includes the following steps:

[0062] Step (1): Refer to Figure 2 Establish a radial unbalanced magnetic pull model for the drive motor;

[0063] The radial unbalanced magnetic pull of the drive motor can be expressed as:

[0064] (1);

[0065] In equation (1), R is the inner radius of the stator; L is the effective length of the rotor; F m The amplitude of the synthesized magnetomotive force in the air gap; The air gap permeability; This represents the angle between the point of minimum air gap and the x-axis; The Fourier coefficient is the permeability of the eccentric air gap. , is represented as:

[0066] (2);

[0067] In equation (2), The average air gap length when the stator and rotor are eccentric; The relative eccentricity; This refers to the rotor eccentricity.

[0068] Step (2): Refer to Figure 3 Establish a dynamic model of a two-stage gear transmission system;

[0069] The generalized displacement of a two-stage gear transmission system is defined as:

[0070] (3);

[0071] In equation (3), , , and These represent the translational displacement and rotational angular displacement of the first-stage driving gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the first-stage driven gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the second-stage driving gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the second-stage driven gear in the x, y, and z directions, respectively.

[0072] The dynamic equation of the two-stage gear transmission system is:

[0073] (4);

[0074] In equation (4), and These are the mass and moment of inertia of the first-stage driving gear, respectively. and These are the mass and moment of inertia of the first-stage driven gear, respectively. and These are the mass and moment of inertia of the second-stage driving gear, respectively. and These are the mass and moment of inertia of the second-stage driven gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the first-stage driving gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the first-stage driven gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the second-stage driving gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the second-stage driven gear, respectively. and These are the equivalent support damping in the x and z directions of the first-stage driving gear, respectively; and These are the equivalent support damping in the x and z directions of the first-stage driven gear, respectively; and These are the equivalent support damping in the x and z directions of the second-stage driving gear, respectively; and These are the equivalent support damping in the x and z directions for the second-stage driven gear, respectively; and The meshing stiffness of the first and second stage gear pairs are respectively defined by equation (5). and The meshing damping of the first and second stage gear pairs are respectively defined by equation (6). and These are the torsional stiffness and torsion of the intermediate shaft, respectively. and These are the pressure angles of the first and second stage gears, respectively. and These are the meshing helix angles of the first and second stage gears, respectively. and These are the base circle radii of the first-stage driving gear and driven gear, respectively. and These are the base circle radii of the second-stage driving gear and driven gear, respectively. and These are the input torque and the load torque, respectively. and These are the y-axis supporting forces exerted by the bearings on the first-stage driving gear and driven gear, respectively. and The bearing forces acting on the second stage driving gear and driven gear in the y direction are respectively defined as Equation (7). and The total deformation of the meshing gear pair projected onto the meshing line direction by the vibration of each direction is defined by equation (8). The torsional deformation of the intermediate shaft is defined by equation (10).

[0075] The meshing stiffness of the gear pair is:

[0076] (5);

[0077] In equation (5), and These are the average meshing stiffness values ​​of the first and second stage gear pairs, respectively. and These represent the fluctuation amplitude of the meshing stiffness of the first and second stage gear pairs, respectively. and These are the meshing angular frequencies of the first and second stage gear pairs, respectively. and These are the meshing phases of the first and second stage gear pairs, respectively;

[0078] The meshing damping of the gear pair is:

[0079] (6);

[0080] In equation (6), and The damping coefficients for the meshing of the first and second stage gear pairs;

[0081] The y-axis supporting force of the bearing on the gear:

[0082] (7);

[0083] In equation (7), Vertical vibration of the integrated drive system; and These are the y-axis equivalent support stiffnesses of the first-stage driving gear and driven gear, respectively. and These are the y-axis equivalent support stiffnesses of the second-stage driving gear and driven gear, respectively. and The y-axis equivalent support damping is respectively for the first-stage driving gear and driven gear; and The y-axis equivalent support damping is respectively for the second-stage driving gear and driven gear;

[0084] The total deformation of the meshing gear pair projected by the vibrations in all directions onto the meshing line is:

[0085] (8);

[0086] In equation (8), The gear pair transmission error is defined by equation (9).

[0087] (9);

[0088] In equation (9), and These represent the meshing transmission error amplitudes of the first and second stage gear pairs, respectively. and These are the meshing frequencies of the first and second stage gear pairs, respectively. and These are the meshing phase angles of the first and second stage gear pairs, respectively.

[0089] The torsional deformation of the intermediate shaft is as follows:

[0090] (10);

[0091] Step (3): Establish a road surface stochastic excitation model;

[0092] The road surface stochastic excitation model is defined as follows:

[0093] (11);

[0094] In equation (11), , , and These represent the random road surface excitations experienced by the left front, right front, left rear, and right rear wheels, respectively. For vehicle speed, This is the spatial cutoff frequency; For space reference frequency; This refers to the random road surface roughness coefficient. White noise with a power intensity of 0.5; The distance between the front and rear suspensions of a vehicle is divided by its speed. This is the distance between the left and right suspensions of the vehicle.

[0095] Step (4): Refer to Figure 4 Establish a vertical dynamics model for electric vehicles that considers the integrated drivetrain system;

[0096] The generalized displacement of the vertical dynamics system of an electric vehicle considering an integrated drivetrain system is defined as follows:

[0097] (12);

[0098] In equation (12), , , and These represent the vertical displacements of the unsprung mass of the left front, right front, left rear, and right rear suspensions, respectively. , and These are the vertical displacement of the sprung mass, the pitch angle displacement, and the roll angle displacement, respectively. , and These are the vertical displacement, pitch angle displacement, and roll angle displacement of the integrated drive system, respectively.

[0099] The vertical dynamic equation of an electric vehicle considering an integrated drivetrain system is:

[0100] (13);

[0101] In equation (13), , and These represent the sprung mass, pitch moment of inertia, and roll moment of inertia, respectively. , , and These represent the unsprung masses at the front left, front right, rear left, and rear right, respectively. , and These represent the mass, pitch moment of inertia, and roll moment of inertia of the drive system, respectively. , , and These are the suspension forces for the front left, front right, rear left, and rear right, respectively. , , and These are the forces on the front left, front right, rear left, and rear right tires, respectively. , and The front left, front right and rear suspension forces of the integrated drive system are defined by equation (14). and These are the distances from the sprung center of gravity to the front and rear suspensions, respectively. This is the distance from the sprung center of gravity to the left suspension. This is the distance from the sprung center of mass to the center of mass of the transmission system; and These are the distances from the center of gravity of the drivetrain to the front and rear suspensions, respectively.

[0102] (14);

[0103] In equation (14), , , and These are the suspension stiffnesses for the front left, front right, rear left, and rear right, respectively. , , and These are the front left, front right, rear left, and rear right suspension damping, respectively. , , and These are the stiffnesses of the front left, front right, rear left, and rear right tires, respectively. , , and These are the damping parameters for the front left, front right, rear left, and rear right tires, respectively. and These are the lateral distances from the center of gravity of the integrated drivetrain system to the left and right front suspensions, respectively.

[0104] Step (5): Vertical dynamics calculation of centralized drive electric vehicle;

[0105] In numerical software, the radial unbalanced magnetic pull model of the drive motor described in step (1), the dynamic model of the two-stage gear transmission system described in step (2), the random excitation model of the road surface described in step (3), and the vertical dynamic model of the electric vehicle considering the integrated drive system described in step (4) are programmed. The main parameters used in the model are shown in Table 1. The motion displacement, velocity and acceleration of all concentrated masses of the vehicle can be output by numerical algorithm.

[0106] Table 1: Main parameters of the model

[0107] R 0.065 m <![CDATA[β b1 , b b2 ]]> 0.4327, 0.3109 rad L 0.138 m <![CDATA[k e1 ,k e2 ]]> <![CDATA[6×10 8 ]]> N / m ε 0.05 — <![CDATA[k a1 ,k a2 ]]> <![CDATA[0.6×10 8 ]]> N / m <![CDATA[μ0]]> <![CDATA[4π×10 -7 ]]> — <![CDATA[E1, E2]]> <![CDATA[12×10 -6 ]]> m <![CDATA[F m ]]> 5900 A <![CDATA[m 1:4 ]]> 50 kg <![CDATA[k xp1 , k xp2 ]]> <![CDATA[4.667×10 8 ]]> N / m <![CDATA[m v ]]> 1000 kg <![CDATA[k xg1 , k xg2 ]]> <![CDATA[4.667×10 8 ]]> N / m <![CDATA[m t ]]> 110 kg <![CDATA[k yp1 , k yp2 ]]> <![CDATA[4.667×10 8 ]]> N / m <![CDATA[J v ]]> 1300 <![CDATA[kg·m 2 ]]> <![CDATA[k yg1 , k yg2 ]]> <![CDATA[4.667×10 8 ]]> N / m <![CDATA[J t ]]> 10 <![CDATA[kg·m 2 ]]> <![CDATA[k zp1 , k zp2 ]]> <![CDATA[4.667×10 8 ]]> N / m <![CDATA[I v ]]> 350 <![CDATA[kg·m 2 ]]> <![CDATA[k zg1 , k zg2 ]]> <![CDATA[4.667×10 8 ]]> N / m <![CDATA[I t ]]> 21 <![CDATA[kg·m 2 ]]> <![CDATA[ε m1 , eh m2 ]]> 0.01 — <![CDATA[l1]]> 1.3 m <![CDATA[m p1 , m p2 ]]> 0.367, 0.507 Kg <![CDATA[l2]]> 1.3 m <![CDATA[m g1 , m g2 ]]> 1.966, 6.461 Kg <![CDATA[l3]]> 0.8 m <![CDATA[J p1 , J g1 ]]> 0.011, 0.338 <![CDATA[kg·m 2 ]]> <![CDATA[l4]]> 1.3 m <![CDATA[J p2 , J g2 ]]> 0.016, 2.765 <![CDATA[kg·m 2 ]]> <![CDATA[l5, l6]]> 0.2 m <![CDATA[α1, α2]]> 0.3491, 0.3491 rad <![CDATA[l7, l8]]> 0.3 m

[0108] Vehicle ride comfort can typically be evaluated using the vertical vibration acceleration of the sprung center of mass. Figure 5 This is a time-domain plot of the vertical vibration acceleration of the spring-loaded center of mass. Figure 6 This is a frequency domain plot of the vertical vibration acceleration of the sprung center of mass. Further analysis can be conducted by adjusting the stiffness and damping of the suspension and drive system mounts to examine their impact on the vibration of the sprung center of mass and the drive system. Figure 7 This is a graph showing the relationship between the front suspension stiffness and the damping relative to the root mean square of the vertical vibration acceleration of the spring-loaded center of mass. Figure 8 This is a graph showing the relationship between front suspension stiffness and damping and the root mean square of the vertical vibration acceleration of the center of mass of the integrated power transmission system.

[0109] The above description is merely a preferred embodiment of the invention and does not constitute any limitation on the invention. Any modifications, alterations, or equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the protection scope of the invention.

Claims

1. A method for calculating the vertical dynamics of an electric vehicle considering an integrated powertrain system, characterized in that... Includes the following steps: Step (1): Establish a radial unbalanced magnetic pull model for the drive motor; The radial unbalanced magnetic pull of the drive motor can be expressed as: （1）； In equation (1), The inner radius of the stator; This refers to the effective length of the rotor. The amplitude of the synthesized magnetomotive force in the air gap; The air gap permeability; The angle between the point of minimum air gap and the x-axis; The Fourier coefficient is the permeability of the eccentric air gap. , is represented as: （2）； In equation (2), The average air gap length when the stator and rotor are eccentric; The relative eccentricity; This refers to the rotor eccentricity. Step (2): Establish a dynamic model of the two-stage gear transmission system; The generalized displacement of a two-stage gear transmission system is defined as: （3）； In equation (3), , , and These represent the translational displacement and rotational angular displacement of the first-stage driving gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the first-stage driven gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the second-stage driving gear in the x, y, and z directions, respectively. , , and These represent the translational displacement and rotational angular displacement of the second-stage driven gear in the x, y, and z directions, respectively. The dynamic equation of the two-stage gear transmission system is: （4）； In equation (4), and These are the mass and moment of inertia of the first-stage driving gear, respectively. and These are the mass and moment of inertia of the first-stage driven gear, respectively. and These are the mass and moment of inertia of the second-stage driving gear, respectively. and These are the mass and moment of inertia of the second-stage driven gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the first-stage driving gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the first-stage driven gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the second-stage driving gear, respectively. and These are the equivalent support stiffnesses in the x and z directions of the second-stage driven gear, respectively. and These are the equivalent support damping in the x and z directions of the first-stage driving gear, respectively; and These are the equivalent support damping in the x and z directions of the first-stage driven gear, respectively; and These are the equivalent support damping in the x and z directions of the second-stage driving gear, respectively; and These are the equivalent support damping in the x and z directions for the second-stage driven gear, respectively; and The meshing stiffness of the first and second stage gear pairs are respectively defined by equation (5). and The meshing damping of the first and second stage gear pairs are respectively defined by equation (6). and These are the torsional stiffness and torsion of the intermediate shaft, respectively. and These are the pressure angles of the first and second stage gears, respectively. and These are the meshing helix angles of the first and second stage gears, respectively. and These are the base circle radii of the first-stage driving gear and driven gear, respectively. and These are the base circle radii of the second-stage driving gear and driven gear, respectively. and These are the input torque and the load torque, respectively. and These are the y-axis supporting forces exerted by the bearings on the first-stage driving gear and driven gear, respectively. and The bearing forces acting on the second stage driving gear and driven gear in the y direction are respectively defined as Equation (7). and The total deformation of the meshing gear pair projected onto the meshing line direction by the vibration of each direction is defined by equation (8). The torsional deformation of the intermediate shaft is defined by equation (10). The meshing stiffness of the gear pair is: （5）； In equation (5), and These are the average meshing stiffness values ​​of the first and second stage gear pairs, respectively. and These represent the fluctuation amplitude of the meshing stiffness of the first and second stage gear pairs, respectively. and These are the meshing angular frequencies of the first and second stage gear pairs, respectively. and These are the meshing phases of the first and second stage gear pairs, respectively; The meshing damping of the gear pair is: （6）； In equation (6), and The damping coefficients for the meshing of the first and second stage gear pairs; The y-axis supporting force of the bearing on the gear: （7）； In equation (7), Vertical vibration of the integrated drive system; and These are the equivalent support stiffness in the y-direction for the first-stage driving gear and driven gear, respectively. and These are the equivalent support stiffnesses in the y-direction for the second-stage driving gear and driven gear, respectively. and These are the equivalent y-axis support damping for the first-stage driving gear and driven gear, respectively; and These are the equivalent y-axis support damping for the second-stage driving gear and driven gear, respectively; The total deformation of the meshing gear pair projected by the vibrations in all directions onto the meshing line is: （8）； In equation (8), The gear pair transmission error is defined by equation (9). （9）； In equation (9), and These represent the meshing transmission error amplitudes of the first and second stage gear pairs, respectively. and These are the meshing frequencies of the first and second stage gear pairs, respectively. and These are the meshing phase angles of the first and second stage gear pairs, respectively. The torsional deformation of the intermediate shaft is as follows: （10）； Step (3): Establish a road surface stochastic excitation model; The road surface stochastic excitation model is defined as follows: （11）； In equation (11), , , and These represent the random road surface excitations experienced by the left front, right front, left rear, and right rear wheels, respectively. For vehicle speed, This is the spatial cutoff frequency; For space reference frequency; This refers to the random road surface roughness coefficient. White noise with a power intensity of 0.5; The distance between the front and rear suspensions of a vehicle is divided by its speed. This is the distance between the left and right suspensions of the vehicle. Step (4): Establish a vertical dynamics model of the electric vehicle considering the integrated drivetrain system; The generalized displacement of the vertical dynamics system of an electric vehicle considering an integrated drivetrain system is defined as follows: （12）； In equation (12), , , and These represent the vertical displacements of the unsprung mass of the left front, right front, left rear, and right rear suspensions, respectively. , and These are the vertical displacement of the sprung mass, the pitch angle displacement, and the roll angle displacement, respectively. , and These are the vertical displacement, pitch angle displacement, and roll angle displacement of the integrated drive system, respectively. The vertical dynamic equation of an electric vehicle considering an integrated drivetrain system is: （13）； In equation (13), , and These represent the sprung mass, pitch moment of inertia, and roll moment of inertia, respectively. , , and These represent the unsprung masses at the front left, front right, rear left, and rear right, respectively. , and These represent the mass, pitch moment of inertia, and roll moment of inertia of the drive system, respectively. , , and These are the suspension forces for the front left, front right, rear left, and rear right, respectively. , , and These are the forces on the front left, front right, rear left, and rear right tires, respectively. , and The front left, front right and rear suspension forces of the integrated drive system are defined by equation (14). and These are the distances from the sprung center of gravity to the front and rear suspensions, respectively. This is the distance from the sprung center of gravity to the left suspension. This is the distance from the sprung center of mass to the center of mass of the transmission system; and These are the distances from the center of gravity of the drivetrain to the front and rear suspensions, respectively. （14）； In equation (14), , , and These are the suspension stiffnesses for the front left, front right, rear left, and rear right, respectively. , , and These are the front left, front right, rear left, and rear right suspension damping, respectively. , , and These are the stiffnesses of the front left, front right, rear left, and rear right tires, respectively. , , and These are the damping for the front left, front right, rear left, and rear right tires, respectively. and These are the lateral distances from the center of gravity of the integrated drivetrain system to the left and right front suspensions, respectively. Step (5): Vertical dynamics calculation of centralized drive electric vehicle; In numerical calculation software, the radial unbalanced magnetic pull model of the drive motor described in step (1), the dynamic model of the two-stage gear transmission system described in step (2), the random excitation model of the road surface described in step (3), and the vertical dynamic model of the electric vehicle considering the integrated drive system described in step (4) are programmed and solved by algorithms to output the motion displacement, velocity and acceleration of the concentrated mass of all vehicles.

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