A rigid-flexible coupling modeling method for gear transmission systems considering lubrication and friction

Abstract

This invention discloses a rigid-flexible coupling modeling method for gear transmission systems that considers lubrication and friction. This modeling method is designed for electric drive transmission systems and considers factors such as system lubrication characteristics, tooth surface roughness, friction, and flexible shafts. The specific steps are as follows: Step (1): Construct an elastohydrodynamic lubrication model considering tooth surface roughness; Step (2): Introduce time-varying friction coefficients under lubrication conditions and construct a time-varying meshing stiffness model for helical gears considering friction; Step (3): Calculate oil film stiffness, connect the time-varying meshing stiffness of the helical gears, and build an equivalent time-varying meshing stiffness model for the system; Step (4): Consider the flexible shaft, bearing nonlinear forces, etc., divide the system into node units, and establish a rigid-flexible coupling dynamic model of the system. The beneficial effect of this method is that it comprehensively considers the influence of lubrication, tooth surface roughness, shaft flexibility, and other factors on system dynamics, establishes a high-precision dynamic model of the system, and improves the accuracy of the model.

Description

Technical Field

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

[0002] Helical gear transmission systems are widely used in the new energy vehicle field due to their advantages such as high load-bearing capacity, precise transmission ratio, smooth operation, and high reliability. Under high-speed, heavy-load, and complex operating conditions, the actual dynamic behavior of the gear system depends not only on the elastic deformation of the structure itself but also on the coupled influence of the characteristics of the lubricating oil film between the tooth surfaces and the tribodynamic behavior. Therefore, establishing a rigid-flexible coupled dynamic model that can simultaneously reflect structural flexibility, lubrication state, and frictional excitation is an important theoretical foundation for accurately predicting the vibration and noise of gear systems and thus achieving high-performance gear transmission optimization design.

[0003] To address the aforementioned issues, this invention proposes a rigid-flexible coupling modeling method for gear transmission systems that considers lubrication and friction. Based on elastohydrodynamic lubrication theory, this method strongly couples and unifies the solutions for lubricating film pressure distribution, oil film thickness, time-varying friction coefficient, and rigid-flexible coupling dynamics model within the same framework. This achieves coordinated calculation of the elastohydrodynamic lubrication state, rough interface friction behavior, and structural flexible deformation during gear meshing, thereby improving the accuracy of the model solution. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill related technological gaps, this invention provides a rigid-flexible coupling modeling method for gear transmission systems considering lubrication and friction. The technical solution adopted by this invention to solve its technical problem is expressed as: a rigid-flexible coupling modeling method for gear transmission systems considering lubrication and friction, characterized in that:

[0005] Step (1): Construct an elastohydrodynamic lubrication model that takes into account tooth surface roughness;

[0006] By introducing a random tooth surface roughness function, a two-dimensional Reynolds equation system considering tooth surface roughness is established, and the variation law of lubricating film thickness and pressure is obtained by solving it.

[0007] During the entire meshing process, the contact point on the end face of the helical gear moves along the trajectory of the meshing line. Due to the helix angle of the helical gear, the length of the contact line between the gears is not constant during the meshing process. The meshing speed, radius of curvature, and contact load will change instantaneously. In order to accurately describe this series of processes, combined with the geometric characteristics of helical gear meshing, the helical gear meshing process can be equivalent to the rolling motion of two relatively moving cones.

[0008] Any meshing point on the contact line normal radius of curvature Represented as:

[0009] ;

[0010] Among them, subscript These are the driving and driven gears, respectively. The point of contact between gear teeth Length of the contact line of the gear pair The helix angle of the helical gear and the radius of the end face of the driving gear are given. With the radius of the driven gear end face It can be represented as:

[0011] ;

[0012] in, These are the tip circle radii of the driving and driven gears, respectively. These are the root circle radii of the driving and driven gears, respectively. This is the actual length of the meshing line. For the first Time elapsed from gear engagement The distance moved afterwards;

[0013] Equivalent radius of curvature of the meshing region of helical gears Represented as:

[0014] ;

[0015] Correspondingly, any point on the contact line Sliding speed along the tooth surface Represented as:

[0016] ;

[0017] in, These are the angular velocities of the driving and driven wheels, respectively. These are the base circle radii of the driving and driven gears, respectively. These are the pressure angles on the end faces of the driving and driven wheels, respectively.

[0018] and then, The suction speed at the point relative sliding speed , respectively represented as:

[0019] ;

[0020] Maximum contact pressure of gears Represented as:

[0021] ;

[0022] in, These are Poisson's ratio and elastic modulus of the gear, respectively, with subscripts. These represent the driving and driven gears, respectively. For gears at the meshing point The unit load at the contact line, assuming the load is uniformly distributed, is calculated using the formula for single-tooth load. ,in, The length of the contact wire. , For input torque;

[0023] Under lubrication conditions, the tooth contact during helical gear meshing is considered as line contact. In many past studies, the "end leakage effect" of lubrication has been neglected, and a one-dimensional lubrication model has been established, ignoring the changes in oil film thickness and pressure in the tooth width direction. This results in an inaccurate lubrication model. Therefore, it is necessary to establish a finite-length line contact elastohydrodynamic lubrication model that comprehensively considers the changes in oil film thickness and pressure in the tooth width direction. The general form of the Reynolds equation for finite-length line contact elastohydrodynamic lubrication can be expressed as:

[0024] ;

[0025] in, For the density of lubricating oil, The viscosity of the lubricating oil. These are oil film pressure and oil film thickness, respectively. These represent the directions of the entrainment velocity at the gear meshing point and the contact line of the gear teeth, respectively; neglecting the time-dependent variables in the equations... The resulting squeeze term, i.e., the second term on the right-hand side of the equation, can be expressed by the Reynolds equation as:

[0026] ;

[0027] The equation for the thickness of the elastohydrodynamic lubrication film of helical gears, considering tooth surface roughness, is expressed as:

[0028] ;

[0029] in, The thickness of the oil film at the center of the contact area. Any meshing point on the contact line of the helical gear The equivalent radius at that point, For the comprehensive elastic modulus of the contact surface, For the surface roughness of the contact tooth surface, Let be the pressure distribution function of the lubricating oil film in the meshing region. For lubrication of the area in contact with the tooth surface;

[0030] The viscous-pressure equation is expressed as:

[0031] ;

[0032] in, The viscosity-pressure coefficient, This refers to the initial viscosity of the lubricating oil.

[0033] The pressure-compression equation is expressed as:

[0034] ;

[0035] in, This refers to the initial viscosity of the lubricating oil.

[0036] The pressure boundary condition is expressed as:

[0037] ;

[0038] in, The lubricating oil film at and Location of the entrance and exit;

[0039] Load balance equations Represented as:

[0040] ;

[0041] Substitute the gear parameters and solve the Reynolds equations for finite-length line contact elastohydrodynamic lubrication simultaneously.

[0042] Step (2): Introduce the time-varying friction coefficient under lubrication conditions and construct a time-varying meshing stiffness model of helical gears that considers friction;

[0043] Calculate the time-varying meshing stiffness of helical gears considering time-varying friction coefficients, and the Hertzian contact stiffness of a single tooth. Represented as:

[0044] ;

[0045] Bending stiffness of a single tooth Represented as:

[0046] ;

[0047] in, It is half the angle of the base circle tooth profile. The engagement angle, The angle between the meshing force at the gear tooth engagement point and the direction perpendicular to the gear centerline is given. The coefficient of friction is time-varying. For direction function, For each slice of the gear tooth, consider the infinitesimal element along the tooth width direction;

[0048] single tooth shear stiffness Represented as:

[0049] ;

[0050] Axial compressive stiffness of a single gear tooth Represented as:

[0051] ;

[0052] Single gear tooth base meshing stiffness Represented as:

[0053] ;

[0054] in, For bending potential energy, The tooth thickness at the critical section of the tooth root. These parameters are used to calculate the contribution of the base material to the time-varying meshing stiffness of the entire gear.

[0055] Time-varying meshing stiffness of a single gear tooth Represented as:

[0056] ;

[0057] Among them, subscript They are the driving wheel and the driven wheel, respectively.

[0058] The overall time-varying meshing stiffness of the gear throughout the entire meshing process Represented as:

[0059] ;

[0060] in, This represents the number of tooth pairs that mesh simultaneously. The numbering of the meshing tooth pairs;

[0061] Step (3): Calculate the oil film stiffness, the time-varying meshing stiffness of the tandem helical gears, and build an equivalent time-varying meshing stiffness model of the system;

[0062] At a certain moment Lubrication contact area along the contact line length Segmented to length of If there are infinitesimal elements, then the oil film stiffness of each infinitesimal element is... It can be represented as:

[0063] ;

[0064] in, The length of the contact wire. For load increment, The load increment per unit line load. For pressure increment, This represents the increase in film thickness.

[0065] Overall oil film stiffness Represented as:

[0066] ;

[0067] Equivalent combined time-varying meshing stiffness considering the effect of lubricating oil film Represented as:

[0068] ;

[0069] Step (4): Considering the flexible rotating shaft, nonlinear force of the bearing, etc., divide the system into node elements and establish a rigid-flexible coupling dynamic model of the system;

[0070] To accurately establish the dynamic model of the gear system, the shaft was simulated using Timoshenko beam elements. The shaft was divided into multiple segments, with nodes marked at both ends of each segment. Each node considered 6 degrees of freedom. The shaft model... The generalized coordinates of the two nodes of the axis segment are represented as follows:

[0071] ;

[0072] in, and They are nodes along Displacement and rotation in direction, and They are nodes along Displacement and rotation in direction;

[0073] The finite element dynamic equations for the rotating shaft element are expressed as follows:

[0074] ;

[0075] in, These are the mass matrix, gyroscope matrix, stiffness matrix, and generalized force matrix of the rotating shaft element, respectively. This is the angular velocity matrix of the rotation axis element. for The first derivative, for The second derivative;

[0076] Located on the first pivot Expression of the bearing element support force at each node (5 degrees of freedom) Represented as:

[0077] ;

[0078] in, The bearing radial Support forces in two directions, This refers to the axial support force of the bearing. They are respectively Torque in both directions, The number of bearing balls, subscript Indicates the ball bearing number, Let j be the contact stiffness of the j-th ball. For the first Normal deformation of each ball bearing For the Heaviside function, The contact angle after loading. For the first The azimuth angle of each ball bearing. The radius of the trajectory of the inner raceway curvature center;

[0079] A coupled dynamic model of bending-torsion-shaft-swing is established for each pair of gear units on the rotating shaft using the lumped mass method. This model has the following 12 generalized degrees of freedom, denoted as follows:

[0080] ;

[0081] in, and They are the driving wheels along Displacement and rotation in direction, and They are the driving wheels along Displacement and rotation in direction;

[0082] The equation of motion for the rigid gear pair unit is expressed as:

[0083] ;

[0084] in, These are the mass matrix, gyroscope matrix, damping matrix, stiffness matrix, and generalized excitation force matrix of the gear meshing unit, respectively. The angular velocity matrix of the meshing unit. for The first derivative, for The second derivative;

[0085] Gear meshing unit mass matrix Represented as:

[0086] ;

[0087] in, These are the masses of the driving and driven gears, respectively. and The driving and driven gears are respectively wound around Moment of inertia of the shaft;

[0088] Gear meshing unit mass matrix Represented as:

[0089] ;

[0090] in, The projection vectors of displacements in each direction along the meshing line are expressed as:

[0091] ;

[0092] in, These are the pitch circle radii of the driving and driven gears, respectively. The helix angle of the base circle of a helical gear is defined as follows: right-hand helix is ​​positive, and left-hand helix is ​​negative. for The angle between the positive axis and the meshing surface is expressed as:

[0093] ;

[0094] in, The gear pair meshing angle, Line connecting the centers of the master and driven gears and Positive angle of the axis, For the angular velocity of the driving gear, when At that time, the driving gear rotates clockwise. At that time, the driving gear rotates counterclockwise;

[0095] Gear meshing unit damping matrix It can be represented as:

[0096] ;

[0097] in, The meshing damping of the gear pair is expressed as:

[0098] ;

[0099] in, The value range is [0.03, 0.17]. The moment of inertia of the master and driven gears;

[0100] Gear Generalized External Excitation Force Matrix , is represented as:

[0101] ;

[0102] in, The external torque matrix mainly contains the system's input torque and output torque, etc. The friction force matrix of the gear meshing unit. The meshing force matrix of the gear meshing unit. Represented as:

[0103] ;

[0104] in, The meshing force of the gear pair is expressed as:

[0105] ;

[0106] in, for The first derivative, The tooth flank clearance function is expressed as:

[0107] ;

[0108] in, It is half of the tooth flank clearance. The displacement of the gear along the direction of motion of the meshing line is expressed as:

[0109] ;

[0110] in, For the propagation error function;

[0111] External torque matrix Represented as:

[0112] ;

[0113] in, These are the torques of the driving and driven gears, respectively.

[0114] Friction Matrix Represented as:

[0115] ;

[0116] The differential equation of motion for the two-stage helical gear-shaft-bearing system is expressed as:

[0117] ;

[0118] in, These are the system's mass matrix, damping matrix, and stiffness matrix, respectively. For the generalized force matrix of the system, Let be the system's generalized displacement matrix. for The first derivative, for The second derivative, It can be represented as:

[0119] ;

[0120] in, Here is the mass matrix of the shaft and gear. For the damping matrix of the meshing gear, For the gyroscope matrix of the rotating shaft and meshing gears, Here is the angular velocity matrix of the shaft and meshing gears. These are the stiffness matrices for the shaft, bearing, and gear meshing stiffness matrix, respectively. These are the generalized external excitations of the shaft, meshing gears, and bearings, respectively.

[0121] Rayleigh damping Damping, which characterizes the system structure, can be expressed as:

[0122] ;

[0123] in, The coefficients related to system quality These are coefficients related to the system stiffness. For the system quality matrix, The system stiffness does not include gear meshing stiffness. and They are represented as follows:

[0124] ;

[0125] in, These correspond to the first and second order natural frequencies of the system, respectively. These are the damping ratios corresponding to the first and second natural frequencies, respectively. Attached Figure Description

[0126] Figure 1 This is a flowchart of a rigid-flexible coupling modeling method for gear transmission systems;

[0127] Figure 2 This is a schematic diagram of the equivalent transformation of the helical gear meshing process;

[0128] Figure 3 This is a schematic diagram of a flexible rotating shaft model;

[0129] Figure 4 This is a system node distribution diagram;

[0130] Figure 5 This is a time-domain diagram of gear vibration displacement. Detailed Implementation

[0131] 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.

[0132] like Figure 1 The diagram shown is a flowchart of a rigid-flexible coupling modeling method for gear transmission systems, which includes the following steps:

[0133] Step (1): Construct an elastohydrodynamic lubrication model that takes into account tooth surface roughness;

[0134] By introducing a random tooth surface roughness function, a two-dimensional Reynolds equation system considering tooth surface roughness is established, and the variation law of lubricating film thickness and pressure is obtained by solving it.

[0135] like Figure 2 The diagram shows the equivalent transformation of the helical gear meshing process. During the entire meshing process, the contact point on the gear end face moves along the trajectory of the meshing line. Due to the helix angle of the helical gear, the contact line length between the gears is not constant during the meshing process. The meshing speed, radius of curvature, and contact load will change instantaneously. In order to accurately describe this series of processes, combined with the geometric characteristics of helical gear meshing, the helical gear meshing process can be equivalent to the rolling motion of two relatively moving cones.

[0136] Any meshing point on the contact line normal radius of curvature Represented as:

[0137] ;

[0138] Among them, subscript These are the driving and driven gears, respectively. The point of contact between gear teeth Length of contact line of gear pair, β b The helix angle of the helical gear and the radius of the end face of the driving gear are given. With the radius of the driven gear end face It can be represented as:

[0139] ;

[0140] in, These are the tip circle radii of the driving and driven gears, respectively. These are the root circle radii of the driving and driven gears, respectively. This is the actual length of the meshing line. For the first Time elapsed from gear engagement The distance moved afterwards;

[0141] Equivalent radius of curvature of the meshing region of helical gears Represented as:

[0142] ;

[0143] Correspondingly, any point on the contact line Sliding speed along the tooth surface Represented as:

[0144] ;

[0145] in, These are the angular velocities of the driving and driven wheels, respectively. These are the base circle radii of the driving and driven gears, respectively. These are the pressure angles on the end faces of the driving and driven wheels, respectively.

[0146] and then, The suction speed at the point relative sliding speed , respectively represented as:

[0147] ;

[0148] Maximum contact pressure of gears Represented as:

[0149] ;

[0150] in, These are Poisson's ratio and elastic modulus of the gear, respectively, with subscripts. These represent the driving and driven gears, respectively. For gears at the meshing point The unit load at the contact line. Assuming the load is uniformly distributed along the contact line, the formula for calculating the single tooth load is as follows: ,in, The length of the contact wire. , For input torque;

[0151] Under lubrication conditions, the tooth contact during helical gear meshing is considered as line contact. In many past studies, the "end leakage effect" of lubrication has been neglected, and a one-dimensional lubrication model has been established, ignoring the changes in oil film thickness and pressure in the tooth width direction. This results in an inaccurate lubrication model. Therefore, it is necessary to establish a finite-length line contact elastohydrodynamic lubrication model that comprehensively considers the changes in oil film thickness and pressure in the tooth width direction. The general form of the Reynolds equation for finite-length line contact elastohydrodynamic lubrication can be expressed as:

[0152] ;

[0153] in, For the density of lubricating oil, The viscosity of the lubricating oil. These are oil film pressure and oil film thickness, respectively. These represent the directions of the entrainment velocity at the gear meshing point and the contact line of the gear teeth, respectively; neglecting the time-dependent variables in the equations... The resulting squeeze term, i.e., the second term on the right-hand side of the equation, can be expressed by the Reynolds equation as:

[0154] ;

[0155] The equation for the thickness of the elastohydrodynamic lubrication film of helical gears, considering tooth surface roughness, is expressed as:

[0156] ;

[0157] in, The thickness of the oil film at the center of the contact area. Any meshing point on the contact line of the helical gear The equivalent radius at that point, For the comprehensive elastic modulus of the contact surface, For the surface roughness of the contact tooth surface, Let be the pressure distribution function of the lubricating oil film in the meshing region. For lubrication of the area in contact with the tooth surface;

[0158] The viscous-pressure equation is expressed as:

[0159] ;

[0160] in, The viscosity-pressure coefficient, This refers to the initial viscosity of the lubricating oil.

[0161] The pressure-compression equation is expressed as:

[0162] ;

[0163] in, This refers to the initial viscosity of the lubricating oil.

[0164] The pressure boundary condition is expressed as:

[0165] ;

[0166] in, The lubricating oil film at and Location of the entrance and exit;

[0167] Load balance equations Represented as:

[0168] ;

[0169] Substitute the gear parameters and solve the Reynolds equations for finite-length line contact elastohydrodynamic lubrication simultaneously.

[0170] Step (2): Introduce the time-varying friction coefficient under lubrication conditions and construct a time-varying meshing stiffness model of helical gears that considers friction;

[0171] The time-varying meshing stiffness of helical gears considering time-varying friction coefficients and the Hertzian contact stiffness of individual teeth are calculated using the potential energy method and the slice method. Represented as:

[0172] ;

[0173] Bending stiffness of a single tooth Represented as:

[0174] ;

[0175] in, It is half the base circle tooth angle. It is the engagement angle. The angle between the meshing force at the gear tooth engagement point and the direction perpendicular to the gear centerline is given. The coefficient of friction is time-varying. For direction function, For each slice of the gear tooth, consider the infinitesimal element along the tooth width direction;

[0176] single tooth shear stiffness Represented as:

[0177] ;

[0178] Axial compressive stiffness of a single gear tooth Represented as:

[0179] ;

[0180] Single gear tooth base meshing stiffness Represented as:

[0181] ;

[0182] in, For bending potential energy, The tooth thickness at the critical section of the tooth root. These parameters are used to calculate the contribution of the base material to the time-varying meshing stiffness of the entire gear.

[0183] Time-varying meshing stiffness of a single gear tooth Represented as:

[0184] ;

[0185] Among them, subscript They are the driving wheel and the driven wheel, respectively.

[0186] The overall time-varying meshing stiffness of the gear throughout the entire meshing process Represented as:

[0187] ;

[0188] in, This represents the number of tooth pairs that mesh simultaneously. The numbering of the meshing tooth pairs;

[0189] Step (3): Calculate the oil film stiffness, the time-varying meshing stiffness of the tandem helical gears, and build an equivalent time-varying meshing stiffness model of the system;

[0190] At a certain moment Lubrication contact area along the contact line length Segmented to length of If there are infinitesimal elements, then the oil film stiffness of each infinitesimal element is... It can be represented as:

[0191] ;

[0192] in, The length of the contact wire. For load increment, The load increment per unit line load. For pressure increment, This represents the increase in film thickness.

[0193] Overall oil film stiffness Represented as:

[0194] ;

[0195] Equivalent combined time-varying meshing stiffness considering the effect of lubricating oil film Represented as:

[0196] ;

[0197] Step (4): Considering the flexible rotating shaft, nonlinear force of the bearing, etc., divide the system into node elements and establish a rigid-flexible coupling dynamic model of the system;

[0198] To accurately establish the dynamic model of the gear system, the shaft is simulated using Timoshenko beam elements, such as... Figure 3As shown, the rotation axis is divided into multiple segments, with nodes marked at both ends of each segment. Each node considers 6 degrees of freedom. The rotation axis model is... The generalized coordinates of the two nodes of the axis segment are represented as follows:

[0199] ;

[0200] in, and They are nodes along Displacement and rotation in direction, and They are nodes along Displacement and rotation in direction;

[0201] The finite element dynamic equations for the rotating shaft element are expressed as follows:

[0202] ;

[0203] in, These are the mass matrix, gyroscope matrix, stiffness matrix, and generalized force matrix of the rotating shaft element, respectively. This is the angular velocity matrix of the rotation axis element. for The first derivative, for The second derivative;

[0204] Located on the first pivot Expression of the bearing element support force at each node (5 degrees of freedom) Represented as:

[0205] ;

[0206] in, The bearing radial Support forces in two directions, This refers to the axial support force of the bearing. They are respectively Torque in both directions, The number of bearing balls, subscript Indicates the ball bearing number, For the first The contact stiffness of each ball bearing. For the first Normal deformation of each ball bearing For the Heaviside function, The contact angle after loading. For the first The azimuth angle of each ball bearing. The radius of the trajectory of the inner raceway curvature center;

[0207] A coupled dynamic model of bending-torsion-shaft-swing is established for each pair of gear units on the rotating shaft using the lumped mass method. This model has the following 12 generalized degrees of freedom, denoted as follows:

[0208] ;

[0209] in, and They are the driving wheels along Displacement and rotation in direction, and They are the driving wheels along Displacement and rotation in direction;

[0210] The equation of motion for the rigid gear pair unit is expressed as:

[0211] ;

[0212] in, These are the mass matrix, gyroscope matrix, damping matrix, stiffness matrix, and generalized excitation force matrix of the gear meshing unit, respectively. The angular velocity matrix of the meshing unit. for The first derivative, for The second derivative;

[0213] Gear meshing unit mass matrix Represented as:

[0214] ;

[0215] in, These are the masses of the driving and driven gears, respectively. and The driving and driven gears are respectively wound around Moment of inertia of the shaft;

[0216] Gear meshing unit mass matrix Represented as:

[0217] ;

[0218] in, The projection vectors of displacements in each direction along the meshing line are expressed as:

[0219] ;

[0220] in, These are the pitch circle radii of the driving and driven gears, respectively. The helix angle of the base circle of a helical gear is defined as follows: right-hand helix is ​​positive, and left-hand helix is ​​negative. for The angle between the positive axis and the meshing surface is expressed as:

[0221] ;

[0222] in, The gear pair meshing angle, Line connecting the centers of the master and driven gears and Positive angle of the axis, For the angular velocity of the driving gear, when At that time, the driving gear rotates clockwise. At that time, the driving gear rotates counterclockwise;

[0223] Gear meshing unit damping matrix It can be represented as:

[0224] ;

[0225] in, The meshing damping of the gear pair is expressed as:

[0226] ;

[0227] in, The value range is [0.03, 0.17]. The moment of inertia of the master and driven gears;

[0228] Gear Generalized External Excitation Force Matrix , is represented as:

[0229] ;

[0230] in, The external torque matrix mainly contains the system's input torque and output torque, etc. The friction force matrix of the gear meshing unit. This is the meshing force matrix of the gear meshing unit. Represented as:

[0231] ;

[0232] in, The meshing force of the gear pair is expressed as:

[0233] ;

[0234] in, for The first derivative, The tooth flank clearance function is expressed as:

[0235] ;

[0236] in, It is half of the tooth flank clearance. The displacement of the gear along the direction of motion of the meshing line is expressed as:

[0237] ;

[0238] in, For the propagation error function;

[0239] External torque matrix Represented as:

[0240] ;

[0241] in, These are the torques of the driving and driven gears, respectively.

[0242] Friction Matrix Represented as:

[0243] ;

[0244] according to Figure 4 The system is divided into node elements as shown. The kinematic differential equations of the second-order helical gear-shaft-bearing system are expressed as follows:

[0245] ;

[0246] in, These are the system's mass matrix, damping matrix, and stiffness matrix, respectively. For the generalized force matrix of the system, Let be the system's generalized displacement matrix. for The first derivative, for The second derivative, It can be represented as:

[0247] ;

[0248] in, Here is the mass matrix of the shaft and gear. For the damping matrix of the meshing gear, For the gyroscope matrix of the rotating shaft and meshing gears, Here is the angular velocity matrix of the shaft and meshing gears. These are the stiffness matrices for the shaft, bearing, and gear meshing stiffness matrix, respectively. These are the generalized external excitations of the shaft, meshing gears, and bearings, respectively.

[0249] Rayleigh damping Damping, which characterizes the system structure, can be expressed as:

[0250] ;

[0251] in, The coefficients related to system quality These are coefficients related to the system stiffness. For the system quality matrix, The system stiffness does not include gear meshing stiffness. and They are represented as follows:

[0252] ;

[0253] in, These correspond to the first and second order natural frequencies of the system, respectively. These are the damping ratios corresponding to the first and second natural frequencies, respectively.

[0254] Plot the vibration displacement time-domain diagrams of each gear and analyze the dynamic characteristics of the system. The vibration displacement time-domain diagram of the intermediate shaft drive gear along the x-direction is shown below. Figure 5 As shown.

[0255] 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 rigid-flexible coupling modeling method for gear transmission systems considering lubrication and friction, characterized in that: Step (1): Construct an elastohydrodynamic lubrication model that takes into account tooth surface roughness; By introducing a random tooth surface roughness function, a two-dimensional Reynolds equation system considering tooth surface roughness is established, and the variation law of lubricating film thickness and pressure is obtained by solving it. Step (2): Introduce the time-varying friction coefficient under lubrication conditions and construct a time-varying meshing stiffness model of helical gears that considers friction; Calculate the time-varying meshing stiffness and bending stiffness of a single tooth of a helical gear considering the time-varying friction coefficient. Represented as: ； in, It is half the base circle tooth angle. It is the engagement angle. The angle between the meshing force at the gear tooth engagement point and the direction perpendicular to the gear centerline is given. The coefficient of friction is time-varying. For direction function, For each slice of the gear tooth, consider the infinitesimal element along the tooth width direction; Single tooth shear stiffness Represented as: ； Axial compressive stiffness of a single gear tooth Represented as: ； Step (3): Calculate the oil film stiffness, the time-varying meshing stiffness of the tandem helical gears, and build an equivalent time-varying meshing stiffness model of the system; At a certain moment Lubrication contact area along the contact line length Segmented to length of If there are infinitesimal elements, then the oil film stiffness of each infinitesimal element is... It can be represented as: ； in, The length of the contact wire. For load increment, The load increment per unit line load. For pressure increment, This represents the increase in film thickness. Overall oil film stiffness Represented as: ； Step (4): Considering the flexible rotating shaft, nonlinear force of the bearing, etc., divide the system into node elements and establish a rigid-flexible coupling dynamic model of the system; The rotation axis is simulated using Timoshenko beam elements. The axis is divided into multiple segments, with nodes marked at both ends of each segment. Each node considers 6 degrees of freedom. The rotation axis model... The generalized coordinates of the two nodes of the axis segment are represented as follows: ； in, and They are nodes along Displacement and rotation in direction, and They are nodes along Displacement and rotation in direction; The finite element dynamic equations for the rotating shaft element are expressed as follows: ； in, These are the mass matrix, gyroscope matrix, stiffness matrix, and generalized force matrix of the rotating shaft element, respectively. This is the angular velocity matrix of the rotation axis element. for The first derivative, for The second derivative; A coupled dynamic model of bending-torsion-shaft-pendulum for each pair of gear units on the rotating shaft is established, with the following 12 generalized degrees of freedom, denoted as follows: ； in, and They are the driving wheels along Displacement and rotation in direction, and They are the driving wheels along Displacement and rotation in direction; The equation of motion for the rigid gear pair unit is expressed as: ； in, These are the mass matrix, gyroscope matrix, damping matrix, stiffness matrix, and generalized excitation force matrix of the gear meshing unit, respectively. The angular velocity matrix of the meshing unit. for The first derivative, for The second derivative; The differential equation of motion for the two-stage helical gear-shaft-bearing system is expressed as: ； in, These are the system's mass matrix, damping matrix, and stiffness matrix, respectively. For the generalized force matrix of the system, Let be the system's generalized displacement matrix. for The first derivative, for The second derivative, It can be represented as: ； in, Here is the mass matrix of the shaft and gear. For the damping matrix of the meshing gear, For the gyroscope matrix of the rotating shaft and meshing gears, Here is the angular velocity matrix of the shaft and meshing gears. These are the stiffness matrices for the shaft, bearing, and gear meshing stiffness matrix, respectively. These are the generalized external excitations of the shaft, meshing gears, and bearings, respectively.

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