Face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and micro-texture coupling

Abstract

The invention discloses a face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling, which comprises the following steps: step 1, based on a gear dynamics theory, constructing an eight-degree-of-freedom face gear-rotor system dynamics model; 2, establishing a finite element analysis model of the micro-texture face gear transmission pair, solving time-varying meshing stiffness and coupling the time-varying meshing stiffness to a system kinetic equation; 3, coupling a micro-texture item representing the film thickness equation with the film thickness equation of mixed thermal elastohydrodynamic lubrication; 4, coupling the mixing and thermal influence with the elastohydrodynamic lubrication basic equation, and establishing a mixed thermal elastohydrodynamic lubrication model; 5, carrying out iterative solution on the mixed thermal elastohydrodynamic lubrication and micro-texture coupling equation, outputting the tooth surface friction force and the friction coefficient of a solution domain, and inputting the tooth surface friction force and the friction coefficient as parameters into a system kinetic equation; and 6, solving the coupled system kinetic equation by using a Runge-Kutta numerical integration method to obtain the kinetic response of the face gear transmission system.

Description

Technical Field

[0001] The present invention belongs to the technical field of face gear transmission, and specifically provides a face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling. Background Art

[0002] Face gear transmissions, with their high contact ratio, excellent stability, and ability to handle high speeds and heavy loads, are increasingly becoming a leading transmission method in the aviation field. These gear transmissions meet the requirements of extreme operating conditions, such as aviation transmissions. However, face gears operating under high-speed and heavy-load conditions are prone to high tooth surface temperatures and greater elastic deformation during long-term friction, which can lead to pitting and bonding failures, as well as significant vibration and noise. Therefore, predicting, analyzing, and improving the performance of face gears operating under extreme conditions is a hot topic and a challenging area of ​​research in the field of gear transmissions.

[0003] Existing research indicates that surface microtexturing can create a hydrodynamic lubrication effect during surface flow, significantly increasing the oil film's carrying capacity while reducing the friction coefficient. This method, which improves lubrication performance by modifying surface topography, has already achieved relatively mature theoretical applications in machining tools, sliding bearings, washers, and other objects, but research on gears is still in its infancy. Texturing the tooth surfaces of face gears can significantly improve the lubrication performance between contacting tooth surfaces and reduce friction excitation on the tooth surfaces. This method is expected to enhance the dynamic performance of face gear transmissions and extend their applicability and service life under extreme conditions of high speed and heavy loads. Summary of the Invention

[0004] The dynamic analysis methods for face gears under extreme conditions of high speed and heavy load are very complex. High-precision predictive analysis methods often need to consider the effects of mixed thermal elastohydrodynamic lubrication (HELHL). Research on the dynamic performance of face gear transmissions due to tooth surface microtexturing is still immature, and there is a lack of a mechanism for the coupling effect of surface morphology changes caused by microtexturing on tooth surface lubrication and system dynamic parameters. In view of this, the present invention integrates the theories of microtexture and lubrication, and microtexture and dynamics in the field of face gear transmission dynamics, and proposes a face gear dynamic analysis method that considers the coupling of HELHL and microtexture. This method achieves a high-precision predictive analysis of the dynamic performance of gear transmissions under extreme conditions by considering multiple factors.

[0005] In order to achieve the above object, the present invention provides the following technical solutions:

[0006] A face gear dynamics analysis method considering the coupling of hybrid thermal elastohydrodynamic lubrication and microtexture includes the following steps:

[0007] Step 1: Based on gear dynamics theory, construct an eight-degree-of-freedom face gear-rotor system dynamics model including the prime mover, spur gears, face gears, and load, and derive the system dynamics equations;

[0008] Step 2: Establish a finite element analysis model of the micro-textured gear transmission pair using the finite element method, solve the time-varying meshing stiffness containing micro-texture, and couple the time-varying meshing stiffness into the system dynamics equation;

[0009] Step 3: Construct the micro-texture geometry design and the mathematical expression of the micro-texture region of the flow domain to be solved, characterize the micro-texture term of the film thickness equation and couple it with the film thickness equation of mixed thermoelastic hydrodynamic lubrication;

[0010] Step 4: Based on the EHL theory, a mixed lubrication characterization method is defined. The mixing and thermal effects are coupled with the basic EHL equations. A mixed thermal EHL model is established, which includes the point contact two-dimensional EHL Reynolds equation, the viscosity-temperature-viscosity equation, the density-temperature-density-pressure equation, and the energy equation.

[0011] Step 5: Using the finite difference method to iteratively solve the hybrid thermoelastic-hydrodynamic lubrication and microtexture coupling equation, output the tooth surface friction force and friction coefficient in the solution domain, and input them as parameters into the system dynamics equation;

[0012] Step 6: Use the Runge-Kutta numerical integration method to solve the coupled system dynamic equations and obtain the dynamic response of the face gear transmission system.

[0013] Furthermore, in step 1, the eight degrees of freedom include four torsional angular displacements of the prime mover, spur gear, face gear, and load, and three lateral displacements and one axial displacement of the spur gear and face gear. The generalized displacement matrix is ​​expressed as:

[0014] {δ}={x p ,y p ,θ p ,θ e ,y g ,z g ,θ g ,θ l} T

[0015] Where: {δ} is the eight-degree-of-freedom generalized displacement matrix; Figure 2 In the Cartesian coordinate system defined in p is the vibration displacement of the cylindrical gear in the x-axis direction; y p is the vibration displacement of the cylindrical gear in the y-axis direction; θ p is the rotational displacement of the cylindrical gear around its z-axis; θ eis the rotational displacement of the prime mover around its z-axis; g is the vibration displacement of the face gear in the y-axis direction; z g is the vibration displacement of the face gear in the z-axis direction; θ g is the rotational displacement of the face gear around its z-axis; θ l is the rotational displacement of the load around its z-axis.

[0016] The dynamic model of the eight-degree-of-freedom face gear-rotor system is expressed as:

[0017]

[0018] Where: m i (i=p,g) is the concentrated mass of the driving wheel and the driven wheel; I i (i=e,p,g,l) is the moment of inertia of the prime mover, driving wheel, driven wheel and load; c i ,k i (i=sp,sg) is the torsional damping and torsional stiffness of the transmission shaft in the torsional direction; r i (i=mp,mg,fp) is the rotation radius of the dynamic meshing force and tooth surface friction around the rotation center of the driving wheel and the driven wheel; c i ,k i (i=px,py,gy,gz) is the comprehensive equivalent damping and stiffness of the transmission shaft and supporting bearing for the gear in the lateral vibration direction; F mi ,F fi (i=px,py,gy,gz) are the two-dimensional components of the dynamic meshing force and tooth surface friction force in the Cartesian coordinate system of the driving wheel and the Cartesian coordinate system of the driven wheel respectively; T i (i=e,l) are the driving torque and load torque.

[0019] Furthermore, in the step 2, the finite element method is used to simulate the time-varying meshing stiffness of the gear teeth during the surface gear transmission process. The time-varying meshing stiffness is expressed as:

[0020]

[0021] Where: Δθ i (i=p,g) is the rotation angle of the driving and driven wheels under load; Δθ′ i (i=p,g) is the rotation angle of the driving and driven wheels under no-load conditions; the discrete rotation angle data is obtained through the face gear dynamics simulation results; T l is the load moment; R g and R p are the base circle radius of face gear and cylindrical gear respectively;

[0022] The time-varying meshing stiffness of the face gear is fitted to the discrete data results using the Fourier series expansion method, and the following is obtained:

[0023]

[0024] Where: i is the order of the Fourier series expansion; ω is the gear meshing frequency, which is related to the number of gear teeth z and the speed n, and k ai 、k bi (i=1→∞) is the sine and cosine amplitude constant.

[0025] Furthermore, in step 3, the two-dimensional film thickness equation of the micro-textured tooth surface point contact considering elastic deformation is expressed as:

[0026]

[0027] Where: h0 is the central oil film thickness, the specific size of which is determined by the load balance condition; R is the equivalent radius, which is related to the curvature radii R1 and R2 of the two spheres used for point contact analysis, and The curvature radii R1 and R2 are obtained through the three-dimensional model or simulation of the face gear transmission pair; λ(x, y) is the micro-texture term, which is characterized by constructing a corresponding mathematical model based on the specific geometric shape of the micro-texture; δ(x, y) is the elastic deformation term, which is obtained through the surface elastic deformation formula under normal load in elastic mechanics.

[0028] Furthermore, in step 4, the elastohydrodynamic lubrication theory is: the geometric surface at the ideal smooth plane position is taken as the tolerance base surface, a certain section of the actual rough surface is taken as the tolerance baseline, and there is a deviation from the tolerance baseline in the height direction and the offset is The geometric line is the tolerance limit, and the part of the actual rough surface that exceeds the upper tolerance limit is considered to be in a non-elastohydrodynamic lubrication state, while the part within the tolerance limit is considered to be in an elastohydrodynamic lubrication state. Generally, the value can be taken as the calculated oil film thickness h.

[0029] Furthermore, in step 4, the point contact two-dimensional elastohydrodynamic lubrication Reynolds equation based on face gear transmission is expressed as:

[0030]

[0031] Where: ρ is the density of lubricating oil; η is the viscosity of lubricating oil; p is the oil film pressure; h is the thickness of lubricating oil film; is the average velocity of the fluid between the contacting tooth surfaces in the x-axis direction, which is determined by the upper boundary velocity u x1 With the lower boundary velocity u x2 Decide: is the average velocity of the fluid in the y-axis direction, given by the upper boundary velocity u y1 With the lower boundary velocity u y2 Decide:

[0032] The viscosity-temperature-viscosity-pressure equation is expressed as:

[0033]

[0034] Where: η0 is the viscosity of the lubricating oil at zero pressure and initial temperature T0; p0 is the pressure viscosity coefficient; z is the viscosity-pressure formula coefficient; T is the current temperature;

[0035] The density-temperature-pressure equation is expressed as:

[0036]

[0037] Where: ρ0 is the density of the lubricating oil at zero pressure and initial temperature T0; D is the density-temperature coefficient;

[0038] The energy equation is expressed as:

[0039]

[0040] Where: c p is the constant-pressure heat capacity of the lubricating oil; k is the heat conductivity coefficient of the lubricating oil; T(x,y,0) and T(x,y,h) are the upper and lower boundary temperature conditions of the flow domain to be calculated, which are used to solve the energy equation Item; u, v, and ω are the flow velocities of the fluid in the x-axis, y-axis, and z-axis directions respectively.

[0041] Furthermore, in step 5, the tooth surface friction force under the mixed lubrication state and the friction coefficient under the lubrication state are expressed as:

[0042]

[0043] Among them: F T is the load, which can be calculated by the load balance equation; ne is the non-elastohydrodynamic lubrication proportional coefficient, which is determined by the characterization method of elastohydrodynamic and non-elastohydrodynamic mixed lubrication; μ ne is the tooth surface friction coefficient in the non-elastohydrodynamic lubrication state; τ is the shear stress in the elastohydrodynamic lubrication state; the shear stress in the elastohydrodynamic lubrication state is expressed as:

[0044]

[0045] Where: u2―u1 is the velocity difference between the upper and lower boundaries of the basin.

[0046] The beneficial effects of the present invention are:

[0047] The present invention considers the face gear dynamics analysis method coupled with hybrid thermal elastohydrodynamic lubrication and microtexture. The multi-factor coupling analysis method significantly improves the prediction accuracy and reliability of the face gear dynamics performance under extreme working conditions, including:

[0048] (1) Multi-physics coupling modeling improves analysis accuracy

[0049] By coupling the hybrid thermoelastic-hydrodynamic lubrication model with the time-varying mesh stiffness of microtexture to the dynamic model of the eight-degree-of-freedom face gear-rotor system, and by coupling the microtexture term of the film thickness equation with the film thickness equation of hybrid thermoelastic-hydrodynamic lubrication, the synergistic effects of lubrication state, thermal effects, elastic deformation, and surface microtexture are simultaneously incorporated into the face gear dynamic analysis for the first time. This multi-physics coupling modeling effectively overcomes the limitations of traditional methods that only consider a single factor in isolation (such as lubrication alone or stiffness change alone), and solves the problem of "unclear coupling mechanism between microtexture, lubrication, and dynamics" mentioned in the background technology. It significantly improves the prediction accuracy of tooth surface contact meshing force, oil film thickness, and friction excitation, thereby enabling the prediction and analysis method to further accurately adjust and optimize system parameter design and reduce the risk of failure such as tooth surface pitting and bonding.

[0050] (2) Microtexture optimizes lubrication performance and reduces friction excitation

[0051] By constructing a mathematical model of microtexture, a controllable fluid dynamic pressure effect is introduced in the tooth surface contact area; combined with the friction force calculation under mixed lubrication conditions, the contribution of microtexture to the improvement of oil film bearing capacity and the reduction of non-elastohydrodynamic lubrication ratio can be quantified, which can effectively reduce the tooth surface friction coefficient, thereby reducing system vibration and noise.

[0052] (3) Efficient numerical methods to simulate complex working conditions

[0053] A multi-scale solution strategy combining the finite difference method and Runge-Kutta numerical integration is adopted to significantly improve the computational efficiency while ensuring the stability of the solution of the mixed thermoelastic-hydrodynamic lubrication equation and the dynamic equation.

[0054] (4) Extending service life and expanding application scenarios

[0055] The output system dynamic response (such as vibration amplitude, vibration signal frequency component, and dynamic meshing force) can directly guide the collaborative design of microtexture parameters (such as array spacing and aspect ratio) and lubrication conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] In order to make the purpose, technical solutions and beneficial effects of the present invention more clear, the present invention provides the following drawings for illustration:

[0057] Figure 1This is a flow chart of the face gear dynamics analysis method considering the coupling of hybrid thermal elastohydrodynamic lubrication and microtexture in the present invention;

[0058] Figure 2 It is a structural diagram of the dynamic model of the eight-degree-of-freedom face gear-rotor system;

[0059] Figure 3 Schematic diagram of the simulation of time-varying mesh stiffness of micro-textured gears;

[0060] Figure 4 It is the geometric model of diamond-shaped microtexture;

[0061] Figure 5 It is a characterization method for mixed elastohydrodynamic and non-elastohydrodynamic lubrication. DETAILED DESCRIPTION

[0062] The present invention will be further described below with reference to the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it. However, the embodiments are not intended to limit the present invention.

[0063] like Figure 1 As shown, the present embodiment considers the face gear dynamics analysis method of hybrid thermal elastohydrodynamic lubrication and microtexture coupling, and includes the following steps.

[0064] Step 1: Based on gear dynamics theory, construct an eight-degree-of-freedom face gear-rotor system dynamics model including the prime mover, spur gears, face gears, and load, and derive the system dynamics equations.

[0065] In this embodiment, the lumped parameter method is used to construct the eight-degree-of-freedom face gear-rotor system dynamic model, and the prime mover (drive motor), spur gear, face gear and load are regarded as four elements with rotational inertia, such as Figure 2 As shown. The factors considered in the system dynamics model include the time-varying meshing stiffness of the gear teeth, meshing damping, static transmission error, torsional damping and torsional stiffness of the transmission shaft, damping and stiffness of the supporting bearings, and tooth surface friction, among which the damping and stiffness of the input and output transmission shafts and the gear supporting bearings are regarded as a certain equivalent damping and equivalent stiffness, respectively. The eight generalized displacements in the system are: 4 torsional angular displacements of the prime mover, spur gears, face gears and loads, and 3 lateral displacements and 1 axial displacement of spur gears and face gears. That is, in this embodiment, the eight degrees of freedom include the 4 torsional angular displacements of the prime mover, spur gears, face gears and loads, as well as the 3 lateral displacements and 1 axial displacement of spur gears and face gears. Its generalized displacement matrix is ​​expressed as:

[0066] {δ}={x p ,y p ,θ p ,θe ,y g ,z g ,θ g ,θ l} T

[0067] Where: {δ} is the eight-degree-of-freedom generalized displacement matrix; Figure 2 In the Cartesian coordinate system defined in p is the vibration displacement of the cylindrical gear in the x-axis direction; y p is the vibration displacement of the cylindrical gear in the y-axis direction; θ p is the rotational displacement of the cylindrical gear around its z-axis; θ e is the rotational displacement of the prime mover around its z-axis; g is the vibration displacement of the face gear in the y-axis direction; z g is the vibration displacement of the face gear in the z-axis direction; θ g is the rotational displacement of the face gear around its z-axis; θ l is the rotational displacement of the load around its z-axis.

[0068] Based on this dynamic model, the system meshing coupling dynamic equation is established according to Newton's law. The dynamic model of the eight-degree-of-freedom face gear-rotor system is expressed as:

[0069]

[0070]

[0071] Where: m i (i=p,g) is the concentrated mass of the driving wheel and the driven wheel; I i (i=e,p,g,l) is the moment of inertia of the prime mover, driving wheel, driven wheel and load; c i ,k i (i=sp,sg) is the torsional damping and torsional stiffness of the transmission shaft in the torsional direction; r i (i=mp,mg,fp) is the rotation radius of the dynamic meshing force and tooth surface friction around the rotation center of the driving wheel and the driven wheel; c i ,k i (i=px,py,gy,gz) is the comprehensive equivalent damping and stiffness of the transmission shaft and supporting bearing for the gear in the lateral vibration direction; F mi ,F fi (i=px,py,gy,gz) are the two-dimensional components of the dynamic meshing force and tooth surface friction force in the Cartesian coordinate system of the driving wheel and the Cartesian coordinate system of the driven wheel respectively; T i (i=e,l) are the driving torque and load torque.

[0072] Step 2: Establish a finite element analysis model of the micro-textured gear transmission pair using the finite element method, solve the time-varying meshing stiffness containing micro-texture, and couple the time-varying meshing stiffness into the system dynamics equation.

[0073] Based on the above face gear dynamics equation, the microtexture factor is coupled with the face gear dynamics. The coupling method considering the microtexture is as follows: Since the structural changes caused by the micro-textured tooth surface have an influence on the face gear transmission that cannot be ignored, from the perspective of dynamic excitation, an important and significant influence of the micro-textured tooth surface on the face gear dynamics is the "time-varying mesh stiffness". Therefore, the effect of microtexture on the face gear dynamics can be characterized by the "time-varying mesh stiffness".

[0074] Given that face gear transmission is more complex than ordinary gear transmission, its time-varying meshing stiffness is not suitable for direct solution using the relevant calculation formulas for ordinary gear meshing stiffness. This embodiment uses the finite element method to simulate the time-varying meshing stiffness of the gear teeth in the face gear transmission process, and then considers its periodic characteristics and uses the Fourier series expansion method to fit the discrete data results. Specifically, the time-varying meshing stiffness can be calculated using the following formula:

[0075]

[0076] Where: Δθ i (i=p,g) is the rotation angle of the driving and driven wheels under load; Δθ′ i (i=p,g) is the rotation angle of the driving and driven wheels under no-load condition; T l is the load moment; R g and R p are the base circle radius of the face gear and cylindrical gear respectively. The discrete rotation angle data is obtained through the face gear dynamic simulation results.

[0077] The time-varying meshing stiffness of the face gear is fitted to the discrete data results using the Fourier series expansion method, and the following is obtained:

[0078]

[0079] Where: i is the order of the Fourier series expansion; ω is the gear meshing frequency, which is related to the number of gear teeth z and the speed n, and k ai 、k bi (i=1→∞) is the sine and cosine amplitude constant.

[0080] The finite element method is used to simulate the micro-textured face gear transmission pair model, and the "diamond-shaped micro-texture" is coupled into the face gear transmission pair. The micro-texture is added near the contact trace of the transmission pair. The time-varying meshing stiffness of the micro-textured face gear can be obtained by the above formula. The simulation method of the "time-varying meshing stiffness" of the micro-textured face gear is as follows: Figure 3 shown.

[0081] Step 3: Construct the micro-texture geometry design and the mathematical expression of the micro-texture area of ​​the flow domain to be solved, characterize the micro-texture term of the film thickness equation and couple it with the film thickness equation of mixed thermoelastic hydrodynamic lubrication.

[0082] The coupling between microtexture and elastohydrodynamic lubrication is achieved through the microtexture term in the film thickness equation and the representation of the microtexture mathematical model. The key to constructing the film thickness equation for microtexture tooth surfaces that considers elastic deformation is to add the elastic deformation term and the microtexture term to the two-dimensional film thickness equation for point contact. The two-dimensional film thickness equation for microtexture tooth surfaces that considers elastic deformation can be expressed as:

[0083]

[0084] Where: h0 is the central oil film thickness, the specific size of which is determined by the load balance condition; R is the equivalent radius, which is related to the curvature radii R1 and R2 of the two spheres used for point contact analysis, and The curvature radii R1 and R2 are obtained through the three-dimensional model or simulation of the face gear transmission pair; λ(x, y) is the micro-texture term, which is characterized by constructing a corresponding mathematical model based on the specific geometric shape of the micro-texture; δ(x, y) is the elastic deformation term, which is obtained through the surface elastic deformation formula under normal load in elastic mechanics.

[0085] Specifically, the elastic deformation term δ(x,y) is expressed as:

[0086]

[0087] Where: E is the comprehensive elastic modulus of the two contact surfaces, which is related to the Poisson's ratio ν1, ν2 and elastic modulus E1, E2 of the contact surface material. The relationship is: Ω is the elastic deformation integration area, (s, t) and (x, y) are the position coordinates of the load application point and the position coordinates of the point to be calculated respectively; p(s,) is the two-dimensional distribution function of the load.

[0088] The film thickness equation for a micro-textured tooth surface needs to additionally consider the changes in the original two-dimensional film thickness at the elastic deformation point contact caused by the micro-texture region. Therefore, a micro-texture term is added to the actual film thickness equation. This embodiment proposes an asymmetric "diamond-shaped" micro-texture and uses it as an example to express the micro-texture term in the film thickness equation for a flow domain containing this micro-texture. The main structural parameters of the "diamond-shaped" micro-texture include: major axis parameter a, minor axis parameter b, transition parameter d, transverse center spacing parameter l, and longitudinal center spacing parameter e. Figure 4 The geometric structure of the "diamond-shaped" microtexture and the aspect ratio of And it contains 6 rectangular oil film flow domains Ω to be calculated with this micro-texture. The micro-texture term λ(x,y) of the film thickness of this flow domain can be expressed as:

[0089]

[0090] in: The non-microtexture region is represented by the difference between the whole flow area and the microtexture region. t The mathematical expression is:

[0091]

[0092] Where: e0 and l0 are the vertical distances between the micro-texture tip near the origin of the coordinate system and the x-axis and y-axis; the "diamond-shaped" micro-texture is composed of a triangular half and an elliptical half, y1 and y2 represent the vertical range of the triangular half and the elliptical half of the micro-texture area respectively; x m1 and x m2 They represent the lower bound of the current x value range, x m1 =l0+kl,x m2 =l0+d+kl; k is the linear array coefficient of the micro-texture in the flow domain. For the rectangular oil film flow domain Ω in this embodiment, k=0, 1, 2.

[0093] Step 4: Based on the elastohydrodynamic lubrication theory, a mixed lubrication characterization method is defined, and the mixing and thermal effects are coupled with the basic elastohydrodynamic lubrication equations to establish a mixed thermal elastohydrodynamic lubrication model that includes the point contact two-dimensional elastohydrodynamic lubrication Reynolds equation, viscosity-temperature-viscosity-pressure equation, density-temperature-density-pressure equation, and energy equation.

[0094] Due to the manufacturing errors of the gear tooth surface, the high roughness and sharp surface transition peaks easily lead to the destruction of the oil film, resulting in an elastohydrodynamic-non-elastohydrodynamic mixed lubrication state. The elastohydrodynamic lubrication theory defined in this embodiment is: to characterize the elastohydrodynamic and non-elastohydrodynamic mixed lubrication state of the contact interface of the face gear pair, the geometric surface in the ideal smooth plane position is the tolerance base surface, a certain section of the actual rough surface is the tolerance baseline, and there is a deviation from the tolerance baseline in the height direction and the offset (or tolerance value) is The geometric line is the tolerance limit. The part of the actual rough surface that exceeds the upper tolerance limit is considered to be in a "non-elastohydrodynamic lubrication" state, while the part within the tolerance limit is considered to be in a "elastohydrodynamic lubrication" state. The offset Generally, the calculated oil film thickness h can be used as the value. Characterization methods of elastohydrodynamic and non-elastohydrodynamic mixed lubrication are as follows: Figure 5 shown.

[0095] For this elastohydrodynamic and non-elastohydrodynamic mixed lubrication characterization method, the actual rough surface can be obtained by measuring the tooth surface through the white light interferometry experimental method of the optical profilometer, or it can be simulated using the mathematical method of probability distribution.

[0096] (1) Reynolds equation for two-dimensional elastohydrodynamic lubrication of point contacts

[0097] The contact form of a cylindrical gear-face gear transmission pair is "point contact," and a contact ellipse exists during the transmission process. Typically, the composite velocity of the lubricating oil forms a certain angle with the major axis of the ellipse, meaning that the composite velocity is not collinear with the major axis of the contact ellipse. Therefore, if the major axis of the contact ellipse is defined as the x-axis, when considering the two-dimensional elastohydrodynamic lubrication of "point contact," it is important to note that the velocity has components in both the x-axis and y-axis directions. The Reynolds equation for two-dimensional elastohydrodynamic lubrication of "point contact" face gear transmission can be expressed as:

[0098]

[0099] Where: ρ is the density of lubricating oil; η is the viscosity of lubricating oil; p is the oil film pressure; h is the thickness of lubricating oil film; is the average velocity of the fluid between the contacting tooth surfaces in the x-axis direction, which is determined by the upper boundary velocity u x1 With the lower boundary velocity u x2 Decide: is the average velocity of the fluid in the y-axis direction, given by the upper boundary velocity u y1 With the lower boundary velocity u y2 Decide:

[0100] (2) Load balance equation

[0101] The correctness of the numerical solution of elastohydrodynamic pressure needs to be verified by the load-pressure balance equation. In theory, the pressure of the area being sought should be consistent with the load on the area. For point contact, the load-pressure balance equation can be expressed as:

[0102] F T ―∫∫ Ω p(x,y)dxdy=0

[0103] Among them: F Tis the point contact load; p(x,y) is the point contact pressure.

[0104] (3) Viscosity-temperature-viscosity-pressure equation

[0105] For face gear transmissions operating under extreme conditions of high speed and heavy load, the tooth contact area is often exposed to high pressure and high temperature. As the temperature and pressure of the lubricating oil change, the viscosity and density of the lubricating oil will also change accordingly. Especially for face gear transmissions operating under high speed and heavy load conditions, the oil's viscosity-pressure characteristics, viscosity-temperature characteristics, density-pressure characteristics, and density-temperature characteristics will be important factors affecting the elastohydrodynamic lubrication performance of the lubricating oil between the teeth.

[0106] Roelands viscosity-pressure and viscosity-temperature coupling equations can be expressed as:

[0107]

[0108] Where: η0 is the viscosity of the lubricating oil at zero pressure and initial temperature T0; p0 is the pressure-viscosity coefficient; z is the coefficient of the viscosity-pressure equation; T is the current temperature in Kelvin (SI units), which can be solved using the energy equation; and p is the pressure in GPa. In this embodiment, T0 = 303 K (30°C).

[0109] (4) Density-temperature-pressure equation

[0110] The density pressure and density temperature coupling equation can be expressed as:

[0111]

[0112] Where: ρ0 is the density of the lubricating oil at zero pressure and initial temperature T0; D is the density-temperature coefficient.

[0113] (5) Energy equation

[0114] The effect of heat on lubrication is mainly reflected in the change in the viscosity and density of the lubricant due to temperature. The solution to the temperature distribution relies on the energy equation. For the tooth surface of a point contact gear, the energy equation can be expressed as:

[0115]

[0116]

[0117] Where: c p is the constant-pressure heat capacity of the lubricating oil; k is the heat conductivity coefficient of the lubricating oil; T(x,y,0) and T(x,y,h) are the upper and lower boundary temperature conditions of the flow domain to be calculated, which are used to solve the energy equation Item; u, v, and ω are the flow velocities of the fluid in the x-axis, y-axis, and z-axis directions respectively.

[0118] Step 5: Use the finite difference method to iteratively solve the hybrid thermoelastic-hydrodynamic lubrication and microtexture coupling equations, output the tooth surface friction force and friction coefficient in the solution domain, and input them as parameters into the system dynamics equation.

[0119] By integrating the above hybrid TEHL theory with microtexture characterization methods, a series of equations for the coupling of hybrid TEHL and microtexture can be derived. Finite difference methods are then used to iteratively solve these equations to obtain the input parameters required to calculate the tooth surface friction force and friction coefficient. Hybrid TEHL and face gear dynamics are coupled through the tooth surface friction force and friction coefficient. The calculation method for tooth surface friction force and friction coefficient under hybrid lubrication is as follows:

[0120]

[0121] Among them: F T is the load, which can be calculated by the load balance equation; ne is the non-elastohydrodynamic lubrication proportional coefficient, which is determined by the characterization method of elastohydrodynamic and non-elastohydrodynamic mixed lubrication; μ ne is the tooth surface friction coefficient in the non-elastohydrodynamic lubrication state; τ is the shear stress in the elastohydrodynamic lubrication state; the shear stress in the elastohydrodynamic lubrication state is expressed as:

[0122]

[0123] Where: u2―u1 is the velocity difference between the upper and lower boundaries of the basin.

[0124] The calculated tooth surface friction force and friction coefficient can be used as parameter input for the subsequent face gear-rotor system dynamic equations, completing the coupling of "hybrid thermal elastohydrodynamic lubrication" and "face gear dynamics".

[0125] Step 6: Use the Runge-Kutta numerical integration method to solve the coupled system dynamics equations and obtain the dynamic response of the face gear transmission system. Specifically, substitute the calculation formulas for the parameters in the dynamics equations back into the original system dynamics equations and use the Runge-Kutta numerical integration method to solve the face gear-rotor system dynamics equations that take into account the multi-factor coupling to obtain the system's dynamic response.

[0126] The above embodiments are merely preferred embodiments for the purpose of fully illustrating the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are within the scope of protection of the present invention. The scope of protection of the present invention shall be subject to the claims.

Claims

1. A face gear dynamics analysis method considering the coupling of hybrid thermal elastohydrodynamic lubrication and microtexture, characterized by: The steps include: Step 1: Based on gear dynamics theory, construct an eight-degree-of-freedom face gear-rotor system dynamics model including the prime mover, spur gears, face gears, and load, and derive the system dynamics equations; Step 2: Establish a finite element analysis model of the micro-textured gear transmission pair using the finite element method, solve the time-varying meshing stiffness containing micro-texture, and couple the time-varying meshing stiffness into the system dynamics equation; Step 3: Construct the micro-texture geometry design and the mathematical expression of the micro-texture region of the flow domain to be solved, characterize the micro-texture term of the film thickness equation and couple it with the film thickness equation of mixed thermoelastic hydrodynamic lubrication; Step 4: Based on the EHL theory, a mixed lubrication characterization method is defined. The mixing and thermal effects are coupled with the basic EHL equations. A mixed thermal EHL model is established, which includes the point contact two-dimensional EHL Reynolds equation, the viscosity-temperature-viscosity equation, the density-temperature-density-pressure equation, and the energy equation. Step 5: Using the finite difference method to iteratively solve the hybrid thermoelastic-hydrodynamic lubrication and microtexture coupling equation, output the tooth surface friction force and friction coefficient in the solution domain, and input them as parameters into the system dynamics equation; Step 6: Use the Runge-Kutta numerical integration method to solve the coupled system dynamic equations and obtain the dynamic response of the face gear transmission system.

2. The face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling according to claim 1 is characterized by: In step 1, the eight degrees of freedom include four torsional angular displacements of the prime mover, spur gear, face gear, and load, and three lateral displacements and one axial displacement of the spur gear and face gear. The generalized displacement matrix is ​​expressed as: {δ}={x p ,y p ,i p ,i e ,y g ,z g ,i g ,i l } T Where: {δ} is the eight-degree-of-freedom generalized displacement matrix; in the Cartesian coordinate system defined in Figure 2, x p is the vibration displacement of the cylindrical gear in the x-axis direction; y p is the vibration displacement of the cylindrical gear in the y-axis direction; θ p is the rotational displacement of the cylindrical gear around its z-axis; θ e is the rotational displacement of the prime mover around its z-axis; g is the vibration displacement of the face gear in the y-axis direction; z g is the vibration displacement of the face gear in the z-axis direction; θ g is the rotational displacement of the face gear around its z-axis; θ l is the load's rotational displacement around its z-axis; The dynamic model of the eight-degree-of-freedom face gear-rotor system is expressed as: Where: m i (i=p,g) is the concentrated mass of the driving wheel and the driven wheel; I i (i=e,p,g,l) is the moment of inertia of the prime mover, driving wheel, driven wheel and load; c i ,k i (i=sp,sg) is the torsional damping and torsional stiffness of the transmission shaft in the torsional direction; r i (i=mp,mg,fp) is the rotation radius of the dynamic meshing force and tooth surface friction around the rotation center of the driving wheel and the driven wheel; c i ,k i (i=px,py,gy,gz) is the comprehensive equivalent damping and stiffness of the transmission shaft and supporting bearing for the gear in the lateral vibration direction; F mi ,F fi (i=px,py,gy,gz) are the two-dimensional components of the dynamic meshing force and tooth surface friction force in the Cartesian coordinate system of the driving wheel and the Cartesian coordinate system of the driven wheel respectively; T i (i=e,l) are the driving torque and load torque.

3. The face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling according to claim 1 is characterized by: In the second step, the finite element method is used to simulate the time-varying meshing stiffness of the gear teeth during the surface gear transmission process. The time-varying meshing stiffness is expressed as: Where: Δθ i (i=p,g) is the rotation angle of the driving and driven wheels under load; Δθ′ i (i=p,g) is the rotation angle of the driving and driven wheels under no-load conditions; the discrete rotation angle data is obtained through the face gear dynamics simulation results; T l is the load moment; R g and T are the base circle radii of the face gear and cylindrical gear respectively; The time-varying meshing stiffness of the face gear is fitted to the discrete data results using the Fourier series expansion method, and the following is obtained: Where: i is the order of the Fourier series expansion; ω is the gear meshing frequency, which is related to the number of gear teeth z and the speed n, and k ai 、k bi (i=1→∞) is the sine and cosine amplitude constant.

4. The face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling according to claim 1 is characterized by: In step 3, the two-dimensional film thickness equation of the micro-textured tooth surface point contact considering elastic deformation is expressed as: Where: h0 is the central oil film thickness, the specific size of which is determined by the load balance condition; R is the equivalent radius, which is related to the curvature radii R1 and R2 of the two spheres used for point contact analysis, and The curvature radii R1 and R2 are obtained through the three-dimensional model or simulation of the face gear transmission pair; λ(x, y) is the micro-texture term, which is characterized by constructing a corresponding mathematical model based on the specific geometric shape of the micro-texture; δ(x, y) is the elastic deformation term, which is obtained through the surface elastic deformation formula under normal load in elastic mechanics.

5. The face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling according to claim 1 is characterized by: In step 4, the elastohydrodynamic lubrication theory is: the geometric surface in the ideal smooth plane position is the tolerance base surface, a certain section of the actual rough surface is the tolerance baseline, and there is a deviation from the tolerance baseline in the height direction and the offset is The geometric line is the tolerance limit, and the part of the actual rough surface that exceeds the upper tolerance limit is considered to be in a non-elastohydrodynamic lubrication state, while the part within the tolerance limit is considered to be in an elastohydrodynamic lubrication state. Generally, the value can be taken as the calculated oil film thickness h.

6. The face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling according to claim 1 is characterized by: In step 4, the Reynolds equation for point contact two-dimensional elastohydrodynamic lubrication based on face gear transmission is expressed as: Where: ρ is the density of the lubricating oil; η is the viscosity of the lubricating oil; p is the oil film pressure; h is the thickness of the lubricating oil film; is the average velocity of the fluid between the contacting tooth surfaces in the x-axis direction, which is determined by the upper boundary velocity u x1 With the lower boundary velocity u x2 Decide: is the average velocity of the fluid in the y-axis direction, given by the upper boundary velocity u y1 With the lower boundary velocity u y2 Decide: The viscosity-temperature-viscosity-pressure equation is expressed as: Where: η0 is the viscosity of the lubricating oil at zero pressure and initial temperature T0; p0 is the pressure viscosity coefficient; z is the viscosity-pressure formula coefficient; T is the current temperature; The density-temperature-pressure equation is expressed as: Where: ρ0 is the density of the lubricating oil at zero pressure and initial temperature T0; D is the density-temperature coefficient; The energy equation is expressed as: Where: c p is the constant-pressure heat capacity of the lubricating oil; k is the heat conductivity coefficient of the lubricating oil; T(x,y,0) and T(x,y,h) are the upper and lower boundary temperature conditions of the flow domain to be calculated, which are used to solve the energy equation Item; u, v, and ω are the flow velocities of the fluid in the x-axis, y-axis, and z-axis directions respectively.

7. The face gear dynamics analysis method considering hybrid thermal elastohydrodynamic lubrication and microtexture coupling according to claim 1 is characterized by: In step 5, the tooth surface friction force under mixed lubrication state and the friction coefficient under lubrication state are expressed as: Among them: F T is the load, which can be calculated by the load balance equation; ne is the non-elastohydrodynamic lubrication proportional coefficient, which is determined by the characterization method of elastohydrodynamic and non-elastohydrodynamic mixed lubrication; μ ne is the tooth surface friction coefficient in the non-elastohydrodynamic lubrication state; τ is the shear stress in the elastohydrodynamic lubrication state; the shear stress in the elastohydrodynamic lubrication state is expressed as: Where: u2―u1 is the velocity difference between the upper and lower boundaries of the basin.

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