A rigid-flexible coupling dynamics modeling method for a transmission system

Abstract

The application discloses a kind of rigid-flexible coupling dynamics modeling methods of transmission system, it includes the following steps: (1) the random wind speed prediction model is established, random wind speed is predicted based on double parameter Weibull model, determines input speed and input torque;(2) the time-varying mesh stiffness calculation model is established;(3) the calculation model of gear face friction force is established;(4) the bearing force calculation model is established;(5) the inner gear ring is discretized;(6) the meshing displacement and meshing force between three-stage gear pairs are calculated;(7) the planetary gear-bearing system dynamics vibration differential equation is established, and the vibration displacement curve of three-stage gear pair is obtained by solving;Beneficial effect is that the present rigid-flexible coupling gear-bearing system dynamics modeling related technical blank is filled by calculating random wind speed, time-varying mesh stiffness, gear face friction, time-varying support stiffness and gear ring flexible coupling dynamics equation.

Description

Technical Field

[0001] This invention relates to the field of gear system dynamics technology, and in particular to a method for dynamic modeling of rigid-flexible coupling in transmission systems. Background Technology

[0002] With the continuous development of science and technology, environmental issues are receiving increasing attention. Renewable green energy will gradually replace traditional fossil fuels, and wind energy is becoming the most popular renewable green energy source. Wind turbine generators operate in complex environments. Under the influence of dynamic excitation within the wind turbine gearbox transmission system and external random wind speed excitation, the gearbox is subjected to periodic alternating stress, which can easily lead to gear surface wear, cracks, and tooth root breakage, severely affecting the performance and service life of the wind turbine generator, and even causing serious safety accidents. Therefore, research on the dynamics of wind turbine gearbox systems is of great significance and engineering application value for the normal operation and safety assurance of wind turbine generators. Current technology lacks research on the nonlinear dynamics and vibration characteristics of rigid-flexible coupled gear-bearing systems.

[0003] To address the aforementioned issues, this invention discloses a dynamic modeling method for rigid-flexible coupling of transmission systems. This method considers external random wind speed excitation, establishes calculation models for time-varying meshing stiffness, tooth surface friction, and bearing force, derives expressions for gear meshing force and meshing displacement, thereby establishing a set of vibration differential equations for the planetary gear-bearing system, and analyzes the dynamic characteristics of the planetary gear-bearing system. This provides more accurate technical support for the theoretical design of wind turbine generators, promotes the development of engineering technology, and can generate significant social and economic benefits. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill the gaps in related technologies, this invention discloses a dynamic modeling method for rigid-flexible coupling of transmission systems. This method considers external random wind speed excitation, establishes calculation models for time-varying meshing stiffness, tooth surface friction, and bearing force, derives expressions for gear meshing force and meshing displacement, and thus establishes a set of vibration differential equations for planetary gear-bearing systems to analyze the dynamic characteristics of planetary gear-bearing systems.

[0005] The technical solution adopted by this invention to solve its technical problem is as follows: A method for modeling the dynamics of a rigid-flexible coupling transmission system, characterized by comprising the following steps:

[0006] Step (1): Use a two-parameter Weibull model to predict random wind speed and determine input rotational speed and input torque;

[0007] Step (2): Establish a time-varying meshing stiffness calculation model and use a first-order Fourier series to represent the meshing stiffness of the gear pair;

[0008] Step (3): Establish a tooth surface friction calculation model, calculate the tooth surface friction force and friction torque on each gear, and the expression for the friction force on the internal meshing planetary gear is as follows: The expression for the frictional torque is: The expression for the frictional force on the meshing micro-segment of the internal gear ring is: The expression for the frictional torque is: The expression for the frictional force on the external meshing planetary gear is: The expression for the frictional torque is: The expression for the frictional force on the sun gear is: The expression for the frictional torque on the sun gear is: The expression for the frictional force on herringbone gear 1 is: The frictional torque is The expression for the frictional force on the herringbone gear 2 is: The frictional torque is ;

[0009] in, These represent the radii of curvature of the end faces at the contact points of the internal meshing planetary gear, internal ring gear, external meshing planetary gear, sun gear, herringbone gear 1, and herringbone gear 2, respectively. These represent the overlap ratios of the low-speed, intermediate, and high-speed gear pairs, respectively. This refers to the frictional force experienced by a single tooth in the low-speed and intermediate-speed gear pairs. The frictional force on a single tooth of a high-speed gear pair is given by n, where n is the number of planetary gears and l is the distance from the tooth contact point to the front end of the tooth face.

[0010] Step (4): Establish the bearing force calculation model and determine the expression of the bearing force in the x, y, and z directions of the coordinate system as follows:

[0011] ;

[0012] in, This refers to the number of rolling elements in the bearing. This refers to the bearing's support stiffness. For the Heaviside function, This represents the deformation of the bearing rolling elements. The contact angle between the rolling element and the inner and outer rings after deformation is given. The index n is the rotation angle of the rolling element. When the selected bearing is a ball bearing, the index n = 3 / 2. When the selected bearing is a roller bearing, the index n = 10 / 9.

[0013] Step (5): Discretize the internal gear ring into M (M>100) rigid bodies and M equivalent springs of length 0. The relative displacements of the equivalent springs between adjacent gear ring segments along their respective x and y directions are:

[0014] ;

[0015] in, These are the micro segments of the gear ring. The vibration displacements along the x-axis, y-axis, and torsional direction, where 'a' is the distance from the center of mass of the gear ring micro-segment to the connecting spring. Let i be the phase angle of the gear ring micro-segment. For gear ring micro segments Phase angle;

[0016] Step (6): Calculate the meshing displacement and meshing force between the three-stage gear pairs. The expression for the meshing force is:

[0017] ;

[0018] in, These are the meshing stiffnesses of the low-speed, intermediate-speed, and high-speed gear pairs, respectively. These are the meshing damping parameters for the low-speed, intermediate-speed, and high-speed gear pairs, respectively. , , These are the backlash functions for the low-speed, intermediate, and high-speed gear pairs, respectively. It is half the backlash of the gear pairs in the low-speed, intermediate, and high-speed stages. The equivalent displacements of the low-speed, intermediate, and high-speed gear pairs are respectively expressed as follows:

[0019] ,

[0020] ,

[0021] ;

[0022] Among them, u 1i x 1i y 1i z 1i u2, x2, y2, z2, u3, x3, y3, z3, u4, x4, y4, z4, u5, x5, y5, z5, u6, x6, y6, z6 represent the torsional displacement and vibration displacement along the x, y, and z directions of the internal gear ring micro-segment, internal meshing planetary gear, external meshing planetary gear, sun gear, herringbone gear 1, and herringbone gear 2, respectively. , The meshing phase angle of the low-speed gear pair and the intermediate gear pair. The pressure angle of the high-speed gear pair end face. For the gear helix angle, For the transmission error of a three-stage gear pair, when When, in the formula " Take the "+", otherwise take the "-". " Let be the base circle radius of the internal gear ring. This is the distance from the centroid of the micro-segment of the internal gear ring to the center of the internal gear ring. The pressure angle of the low-speed gear pair end face is N, and N is the number of planetary gears.

[0023] Step (7): Establish the dynamic vibration differential equation of the planetary gear-bearing system; establish the coordinate system of the micro segment of the internal gear ring in a moving coordinate system that rotates synchronously with the internal gear ring with its micro segment's center of mass as the center. The degrees of freedom of the entire system are as follows:

[0024] ;

[0025] Where u, x, y, and z represent the degrees of freedom in the torsion, x-axis direction, y-axis direction, and z-axis direction, respectively. The subscripts 1i, 1b, 2, 2b, 3, 4, 3b, 5, 4b, 6, 5b, and 6b represent the gear ring micro-segment, planetary gear bearing 1, internal meshing planetary gear, planetary gear bearing 2, external meshing planetary gear, sun gear, sun gear bearing, herringbone gear 1, herringbone gear 1 bearing, herringbone gear 2, herringbone gear stage bearing 1, and herringbone gear stage bearing 2, respectively.

[0026] ,

[0027] ,

[0028] ,

[0029] ,

[0030] ,

[0031] ,

[0032] ,

[0033] ,

[0034] ,

[0035] ,

[0036] ,

[0037] ,

[0038] ;

[0039] in, , The input torque to the system is given by m, where g is the acceleration due to gravity, and m is the acceleration due to gravity. 1i m 1b , m 2b , m 3b m4, m5, m 4b m6, m 5b m 6b The masses of the following components are respectively: gear ring micro-segment, planetary gear bearing 1, internal meshing planetary gear, planetary gear bearing 2, external meshing planetary gear, sun gear bearing, sun gear, herringbone gear 1, herringbone gear 1 bearing, herringbone gear 2, herringbone gear stage bearing 1, and herringbone gear stage bearing 2, in K. pjx K pjy K pjz K sjx K sjy K sjz K 2jx K 2jy K 2jz Let K be the component of the support stiffness of each axis along the x, y, and z directions. risu K risx K risy K risz These represent the components of the torsional stiffness and support stiffness of the gear ring micro-segment along the x, y, and z directions, respectively. risu C risx C risy C risz These represent the components of the torsional stiffness and support stiffness of the gear ring micro-segment along the x, y, and z directions, respectively. pjx C pjy C pjz C sjx C sjy C sjz C 2jx C 2jy C 2jz These are the components of the support damping along the x, y, and z directions for each axis, respectively, C. pb1 C pb2 C sb C 1b C 21b C 22b The support damping of each bearing is F. 1bjx F 1bjy F 1bjz F 2bjx F 2bjy F 2bjz F 3bjx F 3bjy F 3bjzF 4bjx F 4bjy F 4bjz F 5bjx F 5bjy F 5bjz F 6bjx F 6bjy F 6bjz These are the components of the bearing force along the x, y, and z directions for each bearing, K. up K us K u2 K represents the torsional stiffness of each axis. e C is the equivalent spring stiffness of a micro-segment of the internal gear ring. up C us C u2 For the torsional damping of each axis, These are the base circle radii of the internal meshing planetary gear, the external meshing planetary gear, the sun gear, herringbone gear 1, and herringbone gear 2, respectively. , , The expression is:

[0040] ,

[0041] ,

[0042] ,

[0043] ;

[0044] in, The pressure angle of the gear pair. Let be the phase angle between the j-th internal meshing planetary gear and the internal gear ring. Let be the phase angle between the j-th external meshing planetary gear and the sun gear.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows: the proposed dynamic modeling method can more accurately reflect the dynamic characteristics of the rigid-flexible coupling of the wind turbine gearbox transmission system under random wind speed conditions, improve the dynamic theory system of gear-bearing system, and provide strong support for wind turbine design, vibration reduction, noise reduction and other aspects. Attached Figure Description

[0046] Figure 1 This is a flowchart of a dynamic modeling method for rigid-flexible coupling of a transmission system.

[0047] Figure 2 It is a three-dimensional model diagram of the transmission system;

[0048] Figure 3 This is a schematic diagram of a flexible gear ring;

[0049] Figure 4 It is a dynamic model of a planetary gear-bearing transmission system;

[0050] Figure 5 This is a vibration displacement curve of the three-stage gear pair in a wind turbine gearbox. Detailed Implementation

[0051] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 5 The specific embodiments of the present invention will be described in detail below.

[0052] like Figure 1 The diagram shows a flowchart of a dynamic modeling method for a wind turbine gearbox with a flexible gear ring, which includes the following steps:

[0053] Step (1): Predict random wind speed using a two-parameter Weibull model. The Weibull distribution model is as follows: The expressions for average wind speed and the power generated by the wind turbine through natural wind are:

[0054] ,

[0055] ;

[0056] Among them, shape parameters Dimensions ,function (a) can be based on empirical formulas calculate, This represents the average wind speed. The standard deviation of wind speed, The wind energy utilization coefficient, Let v be the air density, v be the natural wind speed, and r be the wind speed. a The radius of the wind turbine blade;

[0057] Step (2): Establish a time-varying meshing stiffness calculation model, and use a first-order Fourier series to represent the meshing stiffness of the gear pair:

[0058] ;

[0059] in, The average meshing stiffness of each gear pair. This represents the variation amplitude of the meshing stiffness of each gear pair. The meshing frequency of each gear pair. The phase angle of each gear pair;

[0060] Step (3): Establish a tooth surface friction calculation model, calculate the tooth surface friction force and friction torque on each gear, and the expression for the friction force on the internal meshing planetary gear is as follows: The expression for the frictional torque is: The expression for the frictional force on the meshing micro-segment of the internal gear ring is: The expression for the frictional torque is: The expression for the frictional force on the external meshing planetary gear is: The expression for the frictional torque is: The expression for the frictional force on the sun gear is: The expression for the frictional torque on the sun gear is: The expression for the frictional force on herringbone gear 1 is: The frictional torque is The expression for the frictional force on the herringbone gear 2 is: The frictional torque is ;

[0061] in, These represent the radii of curvature of the end faces at the contact points of the internal meshing planetary gear, internal ring gear, external meshing planetary gear, sun gear, herringbone gear 1, and herringbone gear 2, respectively. These represent the overlap ratios of the low-speed, intermediate, and high-speed gear pairs, respectively, where n is the number of planetary gears and l is the distance from the tooth contact point to the front end of the tooth face. The frictional forces acting on individual teeth of the low-speed, intermediate-speed, and high-speed gear pairs are respectively calculated using the following formulas:

[0062] ;

[0063] in, , , These are the end face overlap, axial overlap, and overlap of the three-stage gear pair, respectively. For gear width, For the gear helix angle, The base circle tooth pitch of the gear. Let the position of the contact line of the i-th pair of teeth on the gear end face be... ;

[0064] Step (4): Establish the bearing force calculation model and determine the expression of the bearing force in the coordinate system as follows:

[0065] ;

[0066] in, This refers to the number of rolling elements in the bearing. This refers to the bearing's support stiffness. For the Heaviside function, This represents the deformation of the bearing rolling elements. The contact angle between the rolling element and the inner and outer rings after deformation is calculated using the following formula:

[0067] ,

[0068] ,

[0069] ;

[0070] Where l is the center distance between the centers of curvature of the inner and outer raceways before the bearing is subjected to force. Let be the contact angle between the rolling element and the inner and outer rings before deformation, and let x, y, and z be the vibration displacements of the bearing rolling element along the x-axis, y-axis, and z-axis, respectively. For the angular deformation of the rolling element, The outer ring radius of the bearing. Where n is the rotation angle of the rolling element, and c is the clearance of the rolling bearing. When the selected bearing is a ball bearing, the exponent n = 3 / 2; when the selected bearing is a roller bearing, the exponent n = 10 / 9.

[0071] Step (5): Discretize the internal gear ring into M (M>100) rigid bodies and M equivalent springs of length 0. The relative displacements of the equivalent springs between adjacent gear ring segments along their respective x and y directions are:

[0072] ;

[0073] in, These are the micro segments of the gear ring. The vibration displacements along the x-axis, y-axis, and torsional direction, where 'a' is the distance from the center of mass of the gear ring micro-segment to the connecting spring. Let i be the phase angle of the gear ring micro-segment. For gear ring micro segments The phase angle;

[0074] Step (6): Calculate the meshing displacement and meshing force between the three-stage gear pairs. The expression for the meshing force is:

[0075] ;

[0076] in, These are the meshing stiffnesses of the low-speed, intermediate-speed, and high-speed gear pairs, respectively. These are the meshing damping parameters for the low-speed, intermediate-speed, and high-speed gear pairs, respectively. , , These are the backlash functions for the low-speed, intermediate, and high-speed gear pairs, respectively. It is half the backlash of the gear pairs in the low-speed, intermediate, and high-speed stages. The equivalent displacements of the low-speed, intermediate, and high-speed gear pairs are respectively expressed as follows:

[0077] ,

[0078] ,

[0079] ;

[0080] in, , The meshing phase angle of the low-speed gear pair and the intermediate gear pair. The pressure angle of the high-speed gear pair end face. For the gear helix angle, For the transmission error of a three-stage gear pair, when When, in the formula " Take the "+", otherwise take the "-". " Let be the base circle radius of the internal gear ring. This is the distance from the centroid of the micro-segment of the internal gear ring to the center of the internal gear ring. The pressure angle of the low-speed gear pair end face is N, and N is the number of planetary gears.

[0081] Step (7): Establish the dynamic vibration differential equation of the planetary gear-bearing system; establish the coordinate system of the micro segment of the internal gear ring in a moving coordinate system that rotates synchronously with the internal gear ring with its micro segment's center of mass as the center. The degrees of freedom of the entire system are as follows:

[0082]

[0083] Where u, x, y, and z represent the degrees of freedom in the torsion, x-axis direction, y-axis direction, and z-axis direction, respectively. The subscripts 1i, 1b, 2, 2b, 3, 4, 3b, 5, 4b, 6, 5b, and 6b represent the gear ring micro-segment, planetary gear bearing 1, internal meshing planetary gear, planetary gear bearing 2, external meshing planetary gear, sun gear, sun gear bearing, herringbone gear 1, herringbone gear bearing 1, herringbone gear 2, herringbone gear grade bearing 1, and herringbone gear grade bearing 2, respectively.

[0084] ,

[0085] ,

[0086] ,

[0087] ,

[0088] ,

[0089] ,

[0090] ,

[0091] ,

[0092] ,

[0093] ,

[0094] ,

[0095] ,

[0096] ;

[0097] in, , The input torque to the system is given by m, where g is the acceleration due to gravity, and m is the acceleration due to gravity. 1i m 1b , m 2b , m 3b m4, m5, m 4b m6, m 5b m 6b The masses of the following components are respectively: gear ring micro-segment, planetary gear bearing 1, internal meshing planetary gear, planetary gear bearing 2, external meshing planetary gear, sun gear bearing, sun gear, herringbone gear 1, herringbone gear 1 bearing, herringbone gear 2, herringbone gear stage bearing 1, and herringbone gear stage bearing 2, in K. pjx K pjy K pjz K sjx K sjy K sjz K 2jx K 2jy K 2jz Let K be the component of the support stiffness of each axis along the x, y, and z directions. risu K risx K risy K risz These represent the components of the torsional stiffness and support stiffness of the gear ring micro-segment along the x, y, and z directions, respectively. risu C risx C risy C risz These represent the components of the torsional stiffness and support stiffness of the gear ring micro-segment along the x, y, and z directions, respectively. pjx C pjy C pjz C sjx C sjy C sjz C 2jx C 2jy C 2jzThese are the components of the support damping along the x, y, and z directions for each axis, respectively, C. pb1 C pb2 C sb C 1b C 21b C 22b The support damping of each bearing is F. 1bjx F 1bjy F 1bjz F 2bjx F 2bjy F 2bjz F 3bjx F 3bjy F 3bjz F 4bjx F 4bjy F 4bjz F 5bjx F 5bjy F 5bjz F 6bjx F 6bjy F 6bjz These are the components of the bearing force along the x, y, and z directions for each bearing, K. up K us K u2 K represents the torsional stiffness of each axis. e C is the equivalent spring stiffness of a micro-segment of the internal gear ring. up C us C u2 For the torsional damping of each axis, These are the base circle radii of the internal meshing planetary gear, the external meshing planetary gear, the sun gear, herringbone gear 1, and herringbone gear 2, respectively. , , The expression is:

[0098] ,

[0099] ,

[0100] ,

[0101] ;

[0102] in, The pressure angle of the gear pair. Let be the phase angle between the j-th internal meshing planetary gear and the internal gear ring. Let be the phase angle between the j-th external meshing planetary gear and the sun gear.

[0103] In this example, the basic parameters of the wind turbine generator set are shown in Table 1:

[0104] Table 1 Basic Parameters of Wind Turbine Generator Sets

[0105] physical parameters numerical values physical parameters numerical values Wind turbine diameter (m) 171 Cut-in wind speed (m / s) 3 Number of leaves 3 Cut-off wind speed (m / s) 25 Rated power (MW) 5 Maximum wind energy utilization coefficient 0.48

[0106] The material parameters of the gear pair and bearing are shown in Tables 2 and 3, respectively.

[0107] Table 2 Gear Parameters of Wind Turbine Gearbox

[0108] parameter Internal gear ring Planetary Wheel 1 Planetary Wheel 2 Sun Gear Herringbone Gear 1 Herringbone Gear 2 Number of teeth 91 22 83 40 131 20 Module (mm) 28 28 16 16 10 10 Pressure angle (°) 20 20 20 20 15 15 Helix angle (°) 4.0 1.0 7.2 7.2 32.0 32.0 Width (mm) 560 560 410 410 365 365 Mass (kg) 5690 950 2320 440 5369 125

[0109] Table 3 Wind Turbine Gearbox Bearing Parameters

[0110] parameter pb1 pb2 sb 1b 21b 22b Number of rolling elements in a bearing 16 11 12 13 12 12 Bearing inner ring radius (mm) 160 300 100 350 40 40 Bearing outer ring radius (mm) 240 365 180 465 70 70 Bearing mass (kg) 47 121 50 295 2.2 2.2 Bearing support stiffness (N / m) <![CDATA[5×10 9 ]]> <![CDATA[8×10 9 ]]> <![CDATA[8×10 9 ]]> <![CDATA[5×10 9 ]]> <![CDATA[8×10 9 ]]> <![CDATA[2×10 9 ]]>

[0111] Among them, pb1, pb2, sb, 1b, 21b, and 22b represent planetary gear bearing 1, planetary gear bearing 2, sun gear bearing, herringbone gear bearing 1, herringbone gear grade bearing 1, and herringbone gear grade bearing 2, respectively.

[0112] Through the above steps, a dynamic model of the planetary gear-bearing transmission system is established as follows: Figure 3 As shown.

[0113] Figure 5 To obtain the vibration displacement curve of the three-stage gear pair, the data in the figure are all based on the natural frequency of the intermediate-stage gear pair. The time scale is used, and the displacement scale is half of the backlash of the intermediate gear pair. The displacement is then dimensionless, where... Figure 5 (a) shows the vibration displacement curve of the low-speed gear pair. Figure 5 (b) shows the vibration displacement curve of the intermediate stage gear pair. Figure 5 (c) shows the vibration displacement curve of the high-speed gear pair. It can be seen that under the same external excitation conditions, the vibration displacement curve of the three-stage gear pair exhibits periodic changes, and the vibration amplitude of the high-speed gear pair is the largest.

[0114] 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 modeling the dynamics of a rigid-flexible coupling transmission system, characterized in that, Includes the following steps: Step 1: Predict random wind speed using a two-parameter Weibull model to determine the input rotational speed and input torque. The Weibull distribution model is as follows: The power generated by the wind turbine through natural wind is expressed as: ； Among them, shape parameters Dimensions ,function (a) can be based on empirical formulas calculate, This represents the average wind speed. The standard deviation of wind speed, The wind energy utilization coefficient, Let v be the air density, v be the natural wind speed, and r be the wind speed. a The radius of the wind turbine blade; Step 2: Establish a time-varying meshing stiffness calculation model, and use a first-order Fourier series to represent the meshing stiffness of the gear pair: ； in, The average meshing stiffness of each gear pair. This represents the variation amplitude of the meshing stiffness of each gear pair. The meshing frequency of each gear pair. The phase angle of each gear pair; Step 3: Establish a tooth surface friction calculation model, calculate the tooth surface friction force and friction torque on each gear, and the expression for the friction force on the internal meshing planetary gear is as follows: The expression for the frictional torque is: The expression for the frictional force on the external meshing planetary gear is: The expression for the frictional torque is: The expression for the frictional force on the sun gear is: The expression for the frictional torque on the sun gear is: The expression for the frictional force on herringbone gear 1 is: The frictional torque is The expression for the frictional force on the herringbone gear 2 is: The frictional torque is ; in, These represent the radii of curvature of the end faces at the contact points of the internal meshing planetary gear, the external meshing planetary gear, the sun gear, herringbone gear 1, and herringbone gear 2, respectively. These represent the overlap ratios of the low-speed, intermediate, and high-speed gear pairs, respectively. This refers to the frictional force experienced by a single tooth in the low-speed and intermediate-speed gear pairs. The frictional force on a single tooth of a high-speed gear pair is given by n, where n is the number of planetary gears and l is the distance from the tooth contact point to the front end of the tooth face. Step 4: Establish the bearing force calculation model and determine the expression of the bearing force in the x, y, and z coordinate systems as follows: ； in, This refers to the number of rolling elements in the bearing. This refers to the bearing's support stiffness. For the Heaviside function, This represents the deformation of the bearing rolling elements. This refers to the contact angle between the rolling element and the inner and outer rings after deformation. The index n is the rotation angle of the rolling element. When the selected bearing is a ball bearing, the index n = 3 / 2. When the selected bearing is a roller bearing, the index n = 10 / 9. Step 5: Discretize the internal gear ring into M (M>100) rigid bodies and M equivalent springs of length 0. The relative displacements of the equivalent springs between adjacent gear ring segments along their respective x and y directions are: ； in, These are respectively the micro-segment i of the gear ring, Vibration displacement along the x-axis and y-axis, u 1i Let be the torsional displacement of the gear ring segment i, and let a be the distance from the center of mass of the gear ring segment i to the connecting spring. Let i be the phase angle of the gear ring micro-segment. For gear ring micro segments The phase angle; Step 6: Calculate the meshing displacement and meshing force between the three-stage gear pairs. The expression for the meshing force is: ； in, These are the meshing stiffnesses of the low-speed, intermediate-speed, and high-speed gear pairs, respectively. These are the meshing damping parameters for the low-speed, intermediate-speed, and high-speed gear pairs, respectively. , , These are the backlash functions for the low-speed, intermediate, and high-speed gear pairs, respectively. It is half the backlash of the gear pairs in the low-speed, intermediate, and high-speed stages. The equivalent displacements of the low-speed, intermediate, and high-speed gear pairs are respectively expressed as follows: ， ， ； Among them, u 1i x 1i y 1i z 1i u2, x2, y2, z2, u3, x3, y3, z3, u4, x4, y4, z4, u5, x5, y5, z5, u6, x6, y6, z6 represent the torsional displacement and vibration displacement along the x, y, and z directions of the gear ring micro-segment, the internal meshing planetary gear, the external meshing planetary gear, the sun gear, herringbone gear 1, and herringbone gear 2, respectively. , The meshing phase angle of the low-speed gear pair and the intermediate gear pair. The pressure angle of the high-speed gear pair end face. For the gear helix angle, For the transmission error of a three-stage gear pair, when When, in the formula " Take "+", otherwise take "". " Let be the base circle radius of the internal gear ring. This is the distance from the centroid of the micro-segment of the internal gear ring to the center of the internal gear ring. The pressure angle of the low-speed gear pair end face is N, and N is the number of planetary gears. Step 7: Establish the dynamic vibration differential equation of the planetary gear-bearing system; establish the coordinate system of the micro-segment of the internal gear ring in a moving coordinate system that rotates synchronously with the internal gear ring, with its micro-segment's center of mass as the center. The degrees of freedom of the entire system are as follows: ； Where u, x, y, and z represent the torsional displacement in the torsional direction, the vibration displacement in the x-axis direction, the y-axis direction, and the z-axis direction, respectively. The subscripts 1i, 1b, 2, 2b, 3, 4, 3b, 5, 4b, 6, 5b, and 6b represent the gear ring micro-segment, planetary gear bearing 1, internal meshing planetary gear, planetary gear bearing 2, external meshing planetary gear, sun gear, sun gear bearing, herringbone gear 1, herringbone gear 1 bearing, herringbone gear 2, herringbone gear grade bearing 1, and herringbone gear grade bearing 2, respectively. ， ， ， ， ， ， ， ， ， ， ， ， ； in, The input torque to the system is given by m, where g is the acceleration due to gravity, and m is the acceleration due to gravity. 1i m 1b , m 2b , m 3b m4, m5, m 4b m6, m 5b m 6b The masses of the following components are respectively: gear ring micro-segment, planetary gear bearing 1, internal meshing planetary gear, planetary gear bearing 2, external meshing planetary gear, sun gear bearing, sun gear, herringbone gear 1, herringbone gear 1 bearing, herringbone gear 2, herringbone gear stage bearing 1, and herringbone gear stage bearing 2, in K. e This represents the equivalent spring stiffness of a micro-segment of the internal gear ring. These are the base circle radii of the internal meshing planetary gear, the external meshing planetary gear, and the sun gear, respectively. , , The expression is: ， ， ， ； in, Let be the phase angle between the j-th internal meshing planetary gear and the internal gear ring. Let be the phase angle between the j-th external meshing planetary gear and the sun gear.

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