Helical gear meshing stiffness calculation method considering friction influence under mixed elastohydrodynamic lubrication

Abstract

The invention provides a helical gear meshing stiffness calculation method considering friction influence under mixed elastohydrodynamic lubrication, and belongs to the field of gear transmission. The method comprises the following steps: firstly, calculating variable contact line length and entrainment speed of the helical gear during meshing, then calculating helical gear mixed elastohydrodynamic lubrication friction acting force based on shear stress under the action of oil film extrusion, and then considering fluid extrusion action and helical gear friction action of mixed elastohydrodynamic lubrication, nonlinear contact, matrix rigidity correction and meshing extension influence. The time-varying meshing stiffness of the bevel gear is calculated based on a bevel gear integral potential energy method, meanwhile, the meshing oil film stiffness of the bevel gear under the action of mixed elastohydrodynamic lubrication is calculated, and finally, the time-varying meshing stiffness and the oil film stiffness are combined to solve the comprehensive time-varying meshing stiffness. According to the method, the fluid extrusion effect of mixed elastohydrodynamic lubrication and the bevel gear friction effect in bevel gear meshing under high speed and heavy load are completely considered, the analytical model for solving the time-varying meshing stiffness of the bevel gear pair is improved, the calculation speed is high, and the calculation precision is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of gear transmission, and in particular relates to a method for calculating the meshing stiffness of helical gears taking into account the influence of friction under mixed elastohydrodynamic lubrication. Background Art

[0002] In the context of the transition to new energy, high-speed precision electric drive reducers are core basic components of engineering machinery and heavy vehicles, and their performance is directly related to the energy efficiency and reliability of the entire system. Gear transmission systems have inherent advantages such as high transmission accuracy, excellent efficiency, and long service life, as well as the support of a relatively complete international standard system. They have become key transmission components for electric drive reducers under high-speed and heavy-load conditions. However, with the deepening of gear technology research, the existing theoretical system has gradually exposed several technical bottlenecks that need to be broken through when dealing with extreme working conditions such as high speed and heavy load: in terms of dynamic characteristics, the insufficient calculation accuracy of time-varying meshing stiffness leads to deviations in the prediction of the system vibration response; in terms of lubrication mechanism, the dynamic coupling effect of the friction coefficient under the mixed elastohydrodynamic lubrication state has not been effectively characterized.

[0003] Current research has gradually deepened from spur gears to helical gears, mainly focusing on improving the calculation method of time-varying meshing stiffness of helical gear pairs. The main calculation methods are: (1) Empirical formula method: The calculation is simple, but the calculation accuracy is low, and it is only applicable to the conceptual design stage; (2) Analytical method: Although it can consider basic factors such as shaft deformation, it has certain limitations for studying oil-injection lubricated gear pairs under high-speed and heavy-load conditions. This limitation is mainly reflected in the fact that it does not consider the friction of gears under high-speed and heavy-load conditions and the influence of mixed elastohydrodynamic lubrication on the time-varying meshing stiffness of gears; (3) Finite element method: It obtains accurate solutions through three-dimensional contact simulation, but the modeling process is complicated, and the calculation time increases exponentially with the number of meshing cycles. In addition, the simulation of friction under mixed elastohydrodynamic lubrication is too complicated and difficult to calculate quickly; (4) Analytical-finite element hybrid method: It combines the advantages of the two methods, but the modeling is also complicated and the calculation efficiency is low. It also does not fully consider the friction under mixed elastohydrodynamic lubrication. The friction coefficient in its calculation process still relies on empirical formulas, resulting in a large error in meshing stiffness. Summary of the Invention

[0004] In view of the fact that the existing method for calculating the time-varying mesh stiffness of helical gears does not fully consider the influence of friction and mixed elastic lubrication of helical gear meshing under high speed and heavy load on the time-varying mesh stiffness, the present invention provides a method for calculating the mesh stiffness of helical gears considering the influence of friction under mixed elastic-hydrodynamic lubrication.

[0005] To achieve the above objectives, the technical solutions provided by the present invention are:

[0006] A method for calculating the mesh stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication is provided, which includes the following steps:

[0007] Step 1: Calculate the length of the contact line when the helical gears are meshing based on the basic parameters of the helical gear pair;

[0008] Step 2: Determine the entrainment speed of the helical gear meshing according to the basic parameters of the helical gear pair and the length of the contact line during meshing;

[0009] Step 3, based on the basic parameters of the lubricating oil of the oil-injection lubricated helical gear pair and the determined entrainment velocity, a hybrid elastohydrodynamic lubrication model for the helical gear is constructed based on the oil film squeezing effect;

[0010] Step 4: According to the constructed hybrid elastohydrodynamic lubrication model, the time-varying friction coefficient of the tooth surface during the meshing process of the helical gear under hybrid elastohydrodynamic lubrication is solved;

[0011] Step 5: The teeth of the helical gear pair are evenly decomposed into multiple independent thin-slice spur gear pairs along the tooth width direction. Based on the obtained time-varying contact line length and tooth surface time-varying friction coefficient of the helical gear meshing, and considering the effects of friction, oil film squeezing, nonlinear contact, modified matrix stiffness, and extended meshing, the time-varying mesh stiffness of each thin-slice spur gear pair is calculated and summed to obtain the time-varying mesh stiffness of the helical gear pair.

[0012] Step 6, calculate the time-varying oil film stiffness of the helical gear meshing under mixed elastohydrodynamic lubrication;

[0013] Step 7: Calculate the comprehensive time-varying mesh stiffness of the helical gear pair under the action of mixed elastohydrodynamic lubrication based on the obtained time-varying mesh stiffness and time-varying oil film stiffness.

[0014] Furthermore, step 1 includes the following sub-steps:

[0015] Step 1.1, determine the width f of the helical gear meshing plane:

[0016]

[0017] Where r a1 is the radius of the top circle of the driving gear tooth, r a2 is the radius of the top circle of the driven wheel, r1 is the pitch circle radius of the driving wheel, r2 is the pitch circle radius of the driven wheel, α t is the end pressure angle;

[0018] Step 1.2: Determine the length of the helical gear meshing contact line:

[0019]

[0020] Where point P is the node of the meshing plane, B represents the tooth width, B1B2 is the actual meshing line, and β b is the helix angle of the base circle of the helical gear, B lis the projection length of the contact line in the tooth width direction, which is used to represent the width of the effective meshing tooth. s represents the independent variable, which is the distance from the intersection of the contact line or its extended line and the action line to the theoretical meshing point. L is the length of the helical gear meshing contact line.

[0021] Furthermore, step 2 includes the following sub-steps:

[0022] Step 2.1, determine the equivalent radius of curvature of the helical gear pair meshing:

[0023] When s≤PB1,

[0024]

[0025] When s≥PB1,

[0026]

[0027] Where R x1 and R x2 are the equivalent curvature radii of the driving and driven wheels respectively, R is the comprehensive curvature radius of any point on the contact line in the normal section of the gear teeth, N1 and N2 are the two endpoints of the ideal meshing line;

[0028] Step 2.2, calculate the entrainment speed u of the helical gear pair e :

[0029] u1=ω1R x1

[0030] u2=ω2R x2

[0031]

[0032] Where u1 and u2 are the linear velocities of the driving wheel and the driven wheel respectively, and ω1 and ω2 are the angular velocities of the driving and driven wheels respectively.

[0033] Furthermore, in step 3, the Reynolds equation under the oil film squeeze effect of the helical gear under mixed elastohydrodynamic lubrication is constructed:

[0034]

[0035] Where u e (t) is the entrainment velocity, η e (x, y, t) is the equivalent viscosity of non-Newtonian fluid lubrication, h(x, y, t) is the oil film thickness, ρ(x, y, t) is the lubricating oil density equation, p(x, y, t) is the mixed elastohydrodynamic lubrication pressure distribution, and (x, y, t) represents different direction and time components.

[0036] Furthermore, step 4 includes the following sub-steps:

[0037] Step 4.1, solve the hybrid elastohydrodynamic lubrication Reynolds equation according to the multigrid method to obtain the tooth surface contact friction force:

[0038]

[0039] F f =∫∫ Ω τdxdy

[0040] Where τ is the shear stress, η is the viscosity equation of the lubricating oil, u is the fluid velocity equation, z represents the fluid lubrication contact direction, and F f is the tooth contact friction force, x and y represent the contact direction of fluid lubrication;

[0041] Step 4.2: Determine the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication based on the tooth surface contact friction force:

[0042] μ(t)=F f / w

[0043] Where μ(t) is the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication, and w is the tooth surface load.

[0044] Furthermore, step 5 includes the following sub-steps:

[0045] Step 5.1, calculate the Hertzian contact stiffness, bending stiffness, axial compression stiffness, shear stiffness and matrix stiffness of the meshing stiffness of a single thin-plate spur gear pair:

[0046]

[0047] u f =r b2 (cosα1+sinα1tanα1-1)

[0048] S f =2r b2 α2

[0049] Where k h is the Hertzian contact stiffness, k a is the bending stiffness, k b is the axial compression stiffness, k s is the shear stiffness, k fis the gear matrix stiffness, E is the comprehensive elastic modulus, L is the length of the helical gear contact line, v is the Poisson's ratio, d is the length of the cantilever beam assuming the gear is along the gear rolling direction, x is the independent variable parameter, dx is the differential increment in the cantilever beam length direction, dL is the unit differential length along the contact line, α1 is the gear base tooth angle, α2 is half of the gear base tooth angle, μ is the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication, h is the distance from a point on the tooth profile curve to the tooth centerline, β is the helical angle of the helical gear, G is the gear shear modulus, A x1 and A x2 are the cross-sectional areas of the cantilever beam at the distances x1 and x2 from the tooth root circle, I x1 and I x2 are the moments of inertia of the cross sections at distances x1 and x2 from the tooth root circle, u f and S f is the tooth profile parameter, r b2 is the base circle radius of the driven wheel, M, P, Q are fitting coefficients;

[0050] Step 5.2, calculate the mesh stiffness of the unit spur gear pair based on the bending stiffness, axial compression stiffness, shear stiffness and Hertz contact stiffness in the mesh stiffness of the unit spur gear pair, and sum them to obtain the time-varying mesh stiffness of the helical gear pair:

[0051]

[0052] Where k t is the time-varying meshing stiffness of the helical gear pair, N represents the total number of thin-spur gear pairs, is the unit gear pair meshing stiffness, The unit bending stiffness of the master and driven wheels are respectively, and k b The formula for calculation is: The unit shear stiffness of the master and driven wheels are respectively, and k s The formula for calculation is: The unit compression stiffness of the master and driven wheels are respectively, and k a The formula for calculation is: is the unit contact stiffness, through k h The formula for calculating λ is p ,λ g are the correction coefficients of the master and driven wheel matrix respectively, are the corrected unit matrix stiffness of the master and driven wheels respectively.

[0053] Furthermore, in step 6, the oil film stiffness under mixed elastohydrodynamic lubrication is calculated as follows:

[0054]

[0055] Where k ois the oil film stiffness, B is the tooth width, Δp is the EMAL pressure increment, Δh is the EMAL film thickness increment, dX is the mesh size along the gear rolling direction, and n is the number of nodes in the EMAL contact domain.

[0056] Furthermore, in step 7, the comprehensive time-varying meshing stiffness of the helical gear pair under the action of mixed elastohydrodynamic lubrication is calculated as follows:

[0057]

[0058] Where k is the comprehensive meshing stiffness of the helical gears, and I represents the number of gear pairs that are engaged simultaneously when the helical gears are meshing.

[0059] Furthermore, in step 5.2, the master and slave wheel matrix correction coefficient λ p ,λ g The calculation is as follows:

[0060] λ j =1+r kfj

[0061] r kfj =(k fAi -k fBi )×100% / k fBi

[0062] In the formula, the value of i is p and g, r kfj k is the change in base stiffness when double teeth mesh relative to that when single teeth mesh, fAi is the double tooth meshing stiffness of the i-th thin spur gear at the meshing position, k fBi is the meshing stiffness of the single tooth area of the i-th thin-slice spur gear at the meshing position.

[0063] The advantages of the present invention are:

[0064] The present invention proposes a method for calculating the meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication. First, the time-varying contact line length and entrainment velocity of the helical gear meshing are calculated. Then, the friction force of the helical gear under mixed elastohydrodynamic lubrication is calculated based on the shear stress under the action of the oil film extrusion. Then, the fluid extrusion effect of the mixed elastohydrodynamic lubrication and the friction effect of the helical gear, nonlinear contact, modified matrix stiffness and extended meshing are considered. The time-varying meshing stiffness of the helical gear is calculated based on the principle of the cumulative integral potential energy method of the helical gear. At the same time, the oil film stiffness of the helical gear meshing under the action of mixed elastohydrodynamic lubrication is calculated. Finally, the time-varying meshing stiffness and the oil film stiffness are combined to solve the comprehensive time-varying meshing stiffness of the helical gear pair. Therefore, the present invention fully takes into account the fluid squeezing effect of mixed elastohydrodynamic lubrication and the friction effect of helical gears in the meshing of helical gears under high speed and heavy load. Not only are the fluid squeezing effect of mixed elastohydrodynamic lubrication and the friction effect of helical gears, nonlinear contact, modified matrix stiffness and extended meshing effects considered when calculating the time-varying meshing stiffness of helical gears, but the oil film stiffness of the helical gear meshing under the action of mixed elastohydrodynamic lubrication is also solved and used together with the time-varying meshing stiffness to calculate the comprehensive time-varying meshing stiffness of the helical gear pair. The analytical model for solving the comprehensive time-varying meshing stiffness of the helical gear pair is improved, the calculation speed is fast, and the calculation accuracy of solving the comprehensive time-varying meshing stiffness of the helical gear pair is improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] The above and / or other features and advantages of the present invention will become more readily understood through the following description with reference to the accompanying drawings, which are not drawn to scale and in which some features are exaggerated or minimized to show details of particular components.

[0066] Figure 1 is a flow chart of a method for calculating meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication according to the present invention;

[0067] Figure 2 Schematic diagram of the driving wheel engagement angle in the present invention;

[0068] Figure 3 This is a schematic diagram of the decomposition and slicing of the helical gear in the present invention;

[0069] Figure 4 Schematic diagram of the hybrid elastohydrodynamic lubrication effect on helical gears in the present invention;

[0070] Figure 5 The figure is a comparison diagram of the comprehensive time-varying meshing stiffness of the helical gear pair obtained by using the method of the present invention and the existing method in an example. DETAILED DESCRIPTION

[0071] The present invention will be described in detail below with reference to the accompanying drawings by means of exemplary embodiments of the present invention. It should be noted that the following detailed description of the present invention is only for the purpose of illustration and is not intended to limit the present invention.

[0072] The present invention provides a method for calculating the meshing stiffness of helical gears taking into account the influence of friction under mixed elastohydrodynamic lubrication (HEL). The method is used to fully take into account the fluid extrusion effect of HEL and the friction effect of helical gears, thereby improving the calculation accuracy and efficiency of the time-varying meshing stiffness of helical gear pairs under extreme working conditions such as high speed and heavy load.

[0073] Reference Figure 1 The method for calculating the meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication provided by the present invention comprises the following steps:

[0074] Step S1, calculating the time-varying contact line length of the helical gears according to the basic parameters of the helical gear pair;

[0075] Step S2, determining the entrainment speed of the helical gear meshing according to the basic parameters of the helical gear pair and the length of the contact line varying during meshing;

[0076] Step S3, constructing a hybrid elastohydrodynamic lubrication model for the helical gear based on the oil film squeezing effect according to the basic parameters of the lubricating oil of the oil-injection lubricated helical gear pair and the determined entrainment velocity;

[0077] Step S4, solving the time-varying friction coefficient of the tooth surface during the meshing process of the helical gear under hybrid elastohydrodynamic lubrication according to the constructed hybrid elastohydrodynamic lubrication model;

[0078] Step S5: The teeth of the helical gear pair are evenly decomposed into multiple independent thin-slice spur gear pairs along the tooth width direction. Based on the obtained time-varying contact line length and tooth surface time-varying friction coefficient of the helical gear meshing, and taking into account the effects of friction, oil film squeezing, nonlinear contact, modified matrix stiffness, and extended meshing, the time-varying mesh stiffness of each thin-slice spur gear pair is calculated and summed to obtain the time-varying mesh stiffness of the helical gear pair.

[0079] Step S6, calculating the time-varying oil film stiffness of the helical gear meshing under mixed elastohydrodynamic lubrication;

[0080] Step S7: Calculate the comprehensive time-varying meshing stiffness of the helical gear pair under the action of mixed elastohydrodynamic lubrication according to the obtained time-varying meshing stiffness and time-varying oil film stiffness.

[0081] like Figure 2 As shown, in each step of the present invention, the basic parameters of the helical gear pair include: number of teeth, elastic modulus, Poisson's ratio, inner hole radius, module, tooth width, pressure angle, helix angle, tooth top height coefficient and top clearance coefficient. Step S1 analyzes the meshing kinematics of the helical gear pair and calculates the length of the contact line of the helical gear pair meshing. The time-varying nature of the contact line is one of the important characteristics of helical gear transmission. Changes in the number of contact lines will cause changes in stiffness and tooth surface friction. In some embodiments, step S1 includes the following sub-steps:

[0082] Step S1.1, calculate the basic parameters of the involute meshing of the helical gear:

[0083]

[0084] Where, α max and α min are the starting and ending angles of the driving wheel respectively, NA is the distance from the end point of the ideal meshing line to the meshing point, NA=NP-AP=r b1 tanα0-r b2 ·(tanα a2 -tanα0), NP and AP are the distances from the endpoint of the ideal meshing line to the node and the distance from the meshing point to the node, respectively;

[0085] r1=mz1 / cosβ

[0086] r2=mz2 / cosβ

[0087] r b1 =r1 cosα0

[0088] r b2 =r2 cosα0

[0089] r a1 =r1+h a m

[0090] r a2 =r2+h a m

[0091] α a2 =arccos(r b2 / r a2 )

[0092] Where r1 is the pitch circle radius of the driving wheel, r2 is the pitch circle radius of the driven wheel, m is the module, z1 is the number of teeth of the driving wheel, z2 is the number of teeth of the driven wheel, β is the helical angle of the helical gear, and r b1 is the base circle radius of the driving wheel, r b2 is the base circle radius of the driven wheel, α0 is the pressure angle, r a1 is the radius of the top circle of the driving gear tooth, r a2 h is the radius of the driven gear tooth top circle, a is the tooth top height, α a2 is the pressure angle corresponding to the meshing point of the driven gear tooth top;

[0093] Step S1.2, determine the width f of the meshing plane of the helical gears. The width of the meshing plane is formed by the meshing line and the addendum circles of the driving and driven gears:

[0094]

[0095] Where r a1 is the radius of the top circle of the driving gear tooth, r a2 is the radius of the top circle of the driven wheel, r1 is the pitch circle radius of the driving wheel, r2 is the pitch circle radius of the driven wheel, α t is the end pressure angle;

[0096] Step S1.3: Determine the length of the helical gear meshing contact line:

[0097]

[0098] Where point P is the node of the meshing plane, B represents the tooth width, B1B2 is the actual meshing line, and β b is the helix angle of the base circle of the helical gear, B l is the projection length of the contact line in the tooth width direction, which is used to represent the width of the effective meshing tooth. s represents the independent variable, which is the distance from the intersection of the contact line or its extended line and the action line to the theoretical meshing point. L is the length of the helical gear meshing contact line.

[0099] Reference Figure 4 In step S2, the entrainment speed of the helical gear meshing can be calculated by the following sub-steps:

[0100] Step S2.1, determine the equivalent radius of curvature of the meshing helical gear pair:

[0101] When s≤PB1,

[0102]

[0103] When s≥PB1,

[0104]

[0105]

[0106] Where R x1 and R x2 are the equivalent curvature radii of the driving and driven wheels respectively, R is the comprehensive curvature radius of any point on the contact line in the normal section of the gear teeth, N1 and N2 are the two endpoints of the ideal meshing line;

[0107] Step S2.2, calculate the entrainment speed u of the helical gear pair meshing e :

[0108] u1=ω1R x1

[0109] u2=ω2R x2

[0110]

[0111] Where u1 and u2 are the linear velocities of the driving wheel and the driven wheel respectively, and ω1 and ω2 are the angular velocities of the driving and driven wheels respectively.

[0112] In step S3 of the present invention, the basic parameters of the lubricant for the high-speed, heavy-load, oil-injection lubricated helical gear pair include: lubricant brand, initial lubricant viscosity, initial lubricant density, lubricant thermal conductivity, lubricant specific heat, lubricant solid density, lubricant viscosity-temperature coefficient, lubricant viscosity-pressure coefficient, lubricant comprehensive elastic modulus, lubricant material parameters, ambient temperature, and lubricant solid specific heat. In a specific embodiment of the present invention, this step constructs the Reynolds equation for the oil film squeeze effect under mixed elastohydrodynamic lubrication of helical gears as follows:

[0113]

[0114] Where u e (t) is the entrainment velocity, η e (x, y, t) is the equivalent viscosity of non-Newtonian fluid lubrication, h(x, y, t) is the oil film thickness, ρ(x, y, t) is the lubricating oil density equation, p(x, y, t) is the mixed elastohydrodynamic lubrication pressure distribution, and (x, y, t) represents different direction and time components.

[0115] Step S4 can solve the hybrid elastohydrodynamic lubrication Reynolds equation using a multigrid method. In hybrid elastohydrodynamic lubrication, due to the squeezing effect of the helical gear oil film, the elastic hydrodynamic contact stiffness requires remodeling and analysis of the elastic contact body's stiffness. Therefore, the calculation of parameters such as shear stress and film thickness is of great significance, providing more accurate calculation results for the friction analysis under hybrid elastohydrodynamic lubrication. This step can specifically include the following substeps:

[0116] Step S4.1, solve the hybrid elastohydrodynamic lubrication Reynolds equation according to the multigrid method to obtain the tooth surface contact friction force:

[0117]

[0118] F f =∫∫ Ω τdxdy

[0119] Where τ is the shear stress, η is the viscosity equation of the lubricating oil, u is the fluid velocity equation, z represents the fluid lubrication contact direction, and F f is the tooth contact friction force, x and y represent the contact direction of fluid lubrication;

[0120] Step S4.2, based on the tooth surface contact friction force, determine the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication:

[0121] μ(t)=F f / w

[0122] Where μ(t) is the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication, and w is the tooth surface load.

[0123] For step S5, due to the characteristics of helical gears, when solving the stiffness, the helical gears are often calculated based on the cumulative integral potential energy method, which regards the helical gears as spur gears of a certain width, such as Figure 3 As shown, the helical gear is decomposed into multiple spur gear slices of size dz along the tooth width B. The gear mesh stiffness is calculated for each spur gear slice, and then the mesh stiffness of the helical gear pair is calculated by summing the two. Therefore, the Hertzian contact stiffness, bending stiffness, axial compression stiffness, shear stiffness, and matrix stiffness of the gear pair are first calculated separately. When considering mixed elastohydrodynamic lubrication, the friction and extrusion effects under mixed elastohydrodynamic lubrication affect the bending stiffness, axial compression stiffness, shear stiffness, and matrix stiffness, thereby affecting the overall mesh stiffness of the gear pair. This can include the following sub-steps:

[0124] Step S5.1, calculate the Hertzian contact stiffness, bending stiffness, axial compression stiffness, shear stiffness and matrix stiffness of the meshing stiffness of a single thin-film spur gear pair:

[0125]

[0126] u f =r b2 (cosα1+sinα1tanα1-1)

[0127] S f =2r b2 α2

[0128] Where k h is the Hertzian contact stiffness, k a is the bending stiffness, k b is the axial compression stiffness, k s is the shear stiffness, k f is the gear matrix stiffness, E is the comprehensive elastic modulus, L is the length of the helical gear contact line, v is the Poisson's ratio, d is the length of the cantilever beam assuming the gear is along the gear rolling direction, x is the independent variable parameter, dx is the differential increment in the cantilever beam length direction, dL is the unit differential length along the contact line, α1 is the gear base tooth angle, α2 is half of the gear base tooth angle, μ is the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication, h is the distance from a point on the tooth profile curve to the tooth centerline, β is the helical angle of the helical gear, G is the gear shear modulus, A x1 and A x2 are the cross-sectional areas of the cantilever beam at the distances x1 and x2 from the tooth root circle, I x1 and I x2 are the moments of inertia of the cross sections at distances x1 and x2 from the tooth root circle, u f and Sf is the tooth profile parameter, r b2 is the base circle radius of the driven wheel, M, P, Q are fitting coefficients;

[0129] In step S5.2, the time-varying mesh stiffness of the unit spur gear pair involved in the meshing is obtained by using a gear pair meshing characteristic analysis method that considers the fluid squeezing effect of mixed elastohydrodynamic lubrication, the friction effect of the helical gears, nonlinear contact, finite element correction of the matrix stiffness, and the effect of extended meshing. The time-varying mesh stiffness of the helical gear pair is obtained by cumulative summation:

[0130]

[0131] Where k t is the time-varying meshing stiffness of the helical gear pair, N represents the total number of thin-spur gear pairs, is the unit gear pair meshing stiffness, The unit bending stiffness of the master and driven wheels are respectively, and k b The formula for calculation is: The unit shear stiffness of the master and driven wheels are respectively, and k s The formula for calculation is: The unit compression stiffness of the master and driven wheels are respectively, and k a The formula for calculation is: is the unit contact stiffness, through k h The formula for calculating λ is p ,λ g are the correction coefficients of the master and driven wheel matrix respectively, are the corrected unit matrix stiffness of the master and driven wheels respectively.

[0132] Among them, the correction coefficient λ of the master and driven wheel matrix p ,λ g The calculation is as follows:

[0133] λ j =1+r kfj

[0134] r kfj =(k fAi -k fBi )×100% / k fBi

[0135] In the formula, the value of j is p and g, λ p is the base correction coefficient of each thin-slice spur gear driving wheel, λ g is the base correction coefficient of each thin spur gear driven wheel, r kfj k is the change in base stiffness when double teeth mesh relative to that when single teeth mesh, fAi is the double tooth meshing stiffness of the i-th thin spur gear at the meshing position, kfBi is the meshing stiffness of the single tooth area of the i-th thin-slice spur gear at the meshing position.

[0136] In step S6, the oil film stiffness under mixed elastohydrodynamic lubrication is calculated as follows:

[0137]

[0138] Where k o is the oil film stiffness, B is the tooth width, Δp is the EMAL pressure increment, Δh is the EMAL film thickness increment, dX is the mesh size along the gear rolling direction, and n is the number of nodes in the EMAL contact domain.

[0139] Finally, in step S7, the comprehensive time-varying meshing stiffness of the helical gear pair under the action of mixed elastohydrodynamic lubrication is calculated using the following formula:

[0140]

[0141] Where k is the comprehensive meshing stiffness of the helical gears, and I represents the number of gear pairs that are engaged simultaneously when the helical gears are meshing.

[0142] As described above, the present invention fully considers the fluid squeezing effect of mixed elastohydrodynamic lubrication and the friction effect of helical gears in the meshing of helical gears under high speed and heavy load. Not only are the fluid squeezing effect of mixed elastohydrodynamic lubrication and the friction effect of helical gears, nonlinear contact, modified matrix stiffness and extended meshing effects considered when calculating the time-varying meshing stiffness of helical gears, but the oil film stiffness of the helical gear meshing under the action of mixed elastohydrodynamic lubrication is also solved and used together with the time-varying meshing stiffness to calculate the comprehensive time-varying meshing stiffness of the helical gear pair. The analytical model for solving the comprehensive time-varying meshing stiffness of the helical gear pair is improved, the calculation speed is fast, and the calculation accuracy of solving the comprehensive time-varying meshing stiffness of the helical gear pair is also improved.

[0143] Next, the method for calculating the meshing stiffness of helical gears taking into account the influence of friction under mixed elastohydrodynamic lubrication provided by the present invention is further explained with reference to examples.

[0144] In this example, the basic parameters of the helical gear pair are shown in Table 1, and the basic parameters of the lubricating oil are shown in Table 2:

[0145] Table 1 Basic parameters of helical gear pairs

[0146]

[0147] Table 2 Basic parameters of mixed elastohydrodynamic lubrication oil

[0148]

[0149]

[0150] When the helical angle is 27°, the time-varying meshing stiffness diagram of the helical gear obtained by using the algorithm of the present invention and the existing finite element method is as follows: Figure 5 As shown, the blue dashed line represents the meshing stiffness of helical gears under mixed elastohydrodynamic lubrication (HEL), calculated using the method of the present invention. Under high-speed and heavy-load conditions, the extrusion of the lubricating oil film of the helical gears will exert an extrusion effect on the gear pair, and tooth surface friction will also have a significant impact on the dynamic characteristics of the helical gear meshing. Therefore, the present invention incorporates the dynamic characteristics of helical gears under HEL into the calculation of the meshing stiffness of the helical gears, fully considering the friction effects of the helical gears under high-speed and heavy-load conditions, and more closely approximating the actual lubrication and friction state of the gear pair meshing, thereby improving the accuracy of the calculation of the meshing stiffness of the helical gears. Furthermore, compared to the inefficient finite element method, the present invention can quickly calculate the meshing stiffness of helical gear pairs under HEL with different parameters, greatly improving the calculation efficiency. This figure compares the accuracy of the time-varying comprehensive meshing stiffness of the helical gear pair obtained by the algorithm of the present invention with the time-varying meshing stiffness of the helical gear pair obtained by the existing finite element method. It can be seen that the helical gear meshing stiffness curve of the present invention conforms to the meshing law of the gear pair, and the error in size with the meshing stiffness calculated by the finite element method is within a reasonable range, which verifies the effectiveness of the time-varying meshing stiffness calculation method of the helical gear pair proposed in the present invention.

[0151] Finally, it should be noted that the features mentioned and / or illustrated in the above description of the exemplary embodiments of the present invention may be incorporated into one or more other embodiments in the same or similar manner, combined with features in other embodiments, or substituted for corresponding features in other implementations. The technical solutions obtained by such combination or substitution shall also be deemed to be included in the scope of protection of the present invention.

Claims

1. A method for calculating the meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication, characterized in that: The following steps are involved: Step 1: Calculate the length of the contact line when the helical gears are meshing based on the basic parameters of the helical gear pair; Step 2: Determine the entrainment speed of the helical gear meshing according to the basic parameters of the helical gear pair and the length of the contact line during meshing; Step 3, based on the basic parameters of the lubricating oil of the oil-injection lubricated helical gear pair and the determined entrainment velocity, a hybrid elastohydrodynamic lubrication model for the helical gear is constructed based on the oil film squeezing effect; Step 4: According to the constructed hybrid elastohydrodynamic lubrication model, the time-varying friction coefficient of the tooth surface during the meshing process of the helical gear under hybrid elastohydrodynamic lubrication is solved; Step 5: The teeth of the helical gear pair are evenly decomposed into multiple independent thin-slice spur gear pairs along the tooth width direction. Based on the obtained time-varying contact line length and tooth surface time-varying friction coefficient of the helical gear meshing, and considering the effects of friction, oil film squeezing, nonlinear contact, modified matrix stiffness, and extended meshing, the time-varying mesh stiffness of each thin-slice spur gear pair is calculated and summed to obtain the time-varying mesh stiffness of the helical gear pair. Step 6, calculate the time-varying oil film stiffness of the helical gear meshing under mixed elastohydrodynamic lubrication; Step 7: Calculate the comprehensive time-varying mesh stiffness of the helical gear pair under the action of mixed elastohydrodynamic lubrication based on the obtained time-varying mesh stiffness and time-varying oil film stiffness.

2. The method for calculating meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication according to claim 1, characterized in that: Step 1 includes the following sub-steps: Step 1.1, determine the width f of the helical gear meshing plane: Where r a1 is the radius of the top circle of the driving gear tooth, r a2 is the radius of the top circle of the driven wheel, r1 is the pitch circle radius of the driving wheel, r2 is the pitch circle radius of the driven wheel, α t is the end pressure angle; Step 1.2: Determine the length of the helical gear meshing contact line: Where point P is the node of the meshing plane, B represents the tooth width, B1B2 is the actual meshing line, and β b is the helix angle of the base circle of the helical gear, B l is the projection length of the contact line in the tooth width direction, which is used to represent the width of the effective meshing tooth. s represents the independent variable, which is the distance from the intersection of the contact line or its extended line and the action line to the theoretical meshing point. L is the length of the helical gear meshing contact line.

3. The method for calculating meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication according to claim 2, characterized in that: Step 2 includes the following sub-steps: Step 2.1, determine the equivalent radius of curvature of the helical gear pair meshing: When s≤PB1, When s≥PB1, Where R x1 and R x2 are the equivalent curvature radii of the driving and driven wheels respectively, R is the comprehensive curvature radius of any point on the contact line in the normal section of the gear teeth, N1 and N2 are the two endpoints of the ideal meshing line; Step 2.2, calculate the entrainment speed u of the helical gear pair e : u1=ω1R x1 u2=ω2R x2 Where u1 and u2 are the linear velocities of the driving wheel and the driven wheel respectively, and ω1 and ω2 are the angular velocities of the driving and driven wheels respectively.

4. The method for calculating meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication according to claim 1, characterized in that: In step 3, the Reynolds equation under the oil film squeeze effect of mixed elastohydrodynamic lubrication of helical gears is constructed: Where u e (t) is the entrainment velocity, η e (x, y, t) is the equivalent viscosity of non-Newtonian fluid lubrication, h(x, y, t) is the oil film thickness, ρ(x, y, t) is the lubricating oil density equation, p(x, y, t) is the mixed elastohydrodynamic lubrication pressure distribution, and (x, y, t) represents different direction and time components.

5. The method for calculating meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication according to claim 4, characterized in that: Step 4 includes the following sub-steps: Step 4.1, solve the hybrid elastohydrodynamic lubrication Reynolds equation according to the multigrid method to obtain the tooth surface contact friction force: F f =∫∫ Ω τdxdy Where τ is the shear stress, η is the viscosity equation of the lubricating oil, u is the fluid velocity equation, z represents the fluid lubrication contact direction, and F f is the tooth contact friction force, x and y represent the contact direction of fluid lubrication; Step 4.2: Determine the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication based on the tooth surface contact friction force: μ(t)=F f / w Where μ(t) is the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication, and w is the tooth surface load.

6. The method for calculating meshing stiffness of helical gears considering the influence of friction under mixed elastohydrodynamic lubrication according to claim 1, characterized in that: Step 5 includes the following sub-steps: Step 5.1, calculate the Hertzian contact stiffness, bending stiffness, axial compression stiffness, shear stiffness and matrix stiffness of the meshing stiffness of a single thin-plate spur gear pair: u f =r b2 (cosα1+sinα1tanα1-1) S f ＝2r b2 α2 Where k h is the Hertzian contact stiffness, k a is the bending stiffness, k b is the axial compression stiffness, k s is the shear stiffness, k f is the gear matrix stiffness, E is the comprehensive elastic modulus, L is the length of the helical gear contact line, v is the Poisson's ratio, d is the length of the cantilever beam assuming the gear is along the gear rolling direction, x is the independent variable parameter, dx is the differential increment in the cantilever beam length direction, dL is the unit differential length along the contact line, α1 is the gear base tooth angle, α2 is half of the gear base tooth angle, μ is the time-varying friction coefficient of the tooth surface under mixed elastohydrodynamic lubrication, h is the distance from a point on the tooth profile curve to the tooth centerline, β is the helical angle of the helical gear, G is the gear shear modulus, A x1 and A x2 are the cross-sectional areas of the cantilever beam at the distances x1 and x2 from the tooth root circle, I x1 and I x2 are the moments of inertia of the cross sections at distances x1 and x2 from the tooth root circle, u f and S f is the tooth profile parameter, r b2 is the base circle radius of the driven wheel, M, P, Q are fitting coefficients; Step 5.2, calculate the mesh stiffness of the unit spur gear pair based on the bending stiffness, axial compression stiffness, shear stiffness and Hertz contact stiffness in the mesh stiffness of the unit spur gear pair, and sum them to obtain the time-varying mesh stiffness of the helical gear pair: Where k t is the time-varying meshing stiffness of the helical gear pair, N represents the total number of thin-spur gear pairs, is the unit gear pair meshing stiffness, The unit bending stiffness of the master and driven wheels are respectively, and k b The formula for calculation is: The unit shear stiffness of the master and driven wheels are respectively, and k s The formula for calculation is: The unit compression stiffness of the master and slave wheels are respectively, and k a The formula for calculation is: is the unit contact stiffness, through k h The formula for calculation is: p ,λ g are the correction coefficients of the master and driven wheel matrix respectively, are the corrected unit matrix stiffness of the master and driven wheels respectively.

7. The method for calculating meshing stiffness of helical gears considering the influence of friction under hybrid elastohydrodynamic lubrication according to claim 6, characterized in that: In step 6, the oil film stiffness under mixed elastohydrodynamic lubrication is calculated as follows: Where k o is the oil film stiffness, B is the tooth width, Δp is the EMAL pressure increment, Δh is the EMAL film thickness increment, dX is the mesh size along the gear rolling direction, and n is the number of nodes in the EMAL contact domain.

8. The method for calculating meshing stiffness of helical gears considering the influence of friction under hybrid elastohydrodynamic lubrication according to claim 7, characterized in that: In step 7, the comprehensive time-varying mesh stiffness of the helical gear pair under mixed elastohydrodynamic lubrication is calculated as follows: Where k is the comprehensive meshing stiffness of the helical gears, and I represents the number of gear pairs that are engaged simultaneously when the helical gears are meshing.

9. The method for calculating meshing stiffness of helical gears considering the influence of friction under hybrid elastohydrodynamic lubrication according to claim 6, characterized in that: In step 5.2, the master and slave wheel matrix correction coefficient λ p ,λ g The calculation is as follows: l j =1+r kfj r kfj =(k fAi -k fBi )×100% / k fBi In the formula, the value of i is p and g, r kfj k is the change in base stiffness when double teeth mesh relative to that when single teeth mesh, fAi is the double tooth meshing stiffness of the i-th thin spur gear at the meshing position, k fBi is the meshing stiffness of the single tooth area of the i-th thin-slice spur gear at the meshing position.

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