TEHL analysis of VH-CATT cylindrical gear transmission in elliptical contact considering time-varying parameters

Abstract

The lubrication characteristics of a cylindrical gear transmission with a variable hyperbolic circular-arc-tooth-trace (VH-CATT) are important theoretical bases for the tribology design of the gear and its fatigue life prediction. Existing lubrication characteristics studies of it are based on simplified linear contact models, which are limited to one specific meshing position only. Therefore, the mathematical models for tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) are established to calculate most time-varying parameters in a meshing period, such as the kinematic parameters, spatial geometric parameters, and contact parameters, are calculated. Accordingly, a thermo-elastohydrodynamic lubrication (TEHL) model of an elliptical contact of this gear transmission is presented. Using all the proposed mathematical models, the TEHL results with smooth and rough tooth surfaces in a meshing period are analyzed. The obtained results reveal that these models can be used to predict the distribution of the film thickness, pressure, and temperature rise on the smooth and rough tooth surfaces of VH-CATT cylindrical gear transmission in each contact area of a meshing period. And it shows that the time-varying parameters have great influences on lubrication performance. Furthermore, the lubrication mechanism in a meshing period is investigated systematically in VH-CATT cylindrical gear transmission.

Introduction

VH-CATT cylindrical gear is a new type of gear developed based on the Gleason spiral bevel gear, as shown in Figure 1. Compared with traditional spur and helical gears, VH-CATT cylindrical gears have a strong bearing capacity, higher transmission efficiency, lower installation precision requirements, and no axial force.^1,2 Because of its excellent performance, VH-CATT cylindrical gears have significant application prospects. In recent years, the exploration of VH-CATT cylindrical gear transmission progressed considerably. Tseng and Tsay ³ and Wu et al. ⁴ deduced the full tooth surface equation for VH-CATT cylindrical gears with transition tooth surface, and Zhao et al. ⁵ established an accurate three-dimensional mathematical model. Then, Wu et al. ⁶ studied the effect of machine tool errors on the tooth surface errors of this gear. Generally, poor lubrication is the main factor leading to frictional wear of gears. In a meshing period, most time-varying parameters, such as the gear geometry, speed, contact geometry, and load at each meshing moment, constantly vary. Concurrently, with the influences of pressure, temperature field, and tooth surface roughness in the lubricated contact area, the surface elastic deformation, lubricating oil viscosity and density will significantly change. Therefore, it is necessary to develop a TEHL model of VH-CATT cylindrical gears considering most time-varying parameters to study the lubricant behavior in the meshing process.

Figure 1.

Model of VH-CATT cylindrical gear.

In 1916, for the first time, Martin ⁷ used a simplified model to apply the Reynolds equation for analyzing gear lubrication. With the maturity of gear technology and the elastohydrodynamic lubrication (EHL) theory as well as significant improvement in numerical algorithms and computer performance, the application to EHL theory has spanned a century and is still providing innovations. For instances, Xu and Kahraman ⁸ and Liu et al. ⁹ developed the EHL-line and TEHL-line contact model of a hypoid bevel gear and helical gear transmission, respectively. It should be noted that these studies mainly focused on theoretical conjugate tooth surfaces. In practical industrial applications, owing to the deviation caused by gear assembly and machining errors, elliptical contacts are formed on gear contact traces instead of theoretical line contacts. Therefore, a highly realistic elliptical contact was introduced into the EHL model of hypoid gears ¹⁰ and the applicability of this model was subsequently expanded to non-Newtonian fluid. ¹¹ To achieve further closeness to a real scenario, the present study examines the lubrication mechanism of VH-CATT cylindrical gears with elliptical contact.

Following the determination of the effects of tooth surface roughness and thermal effect factor on gear lubrication performance, many scholars have progressively conducted EHL model studies with these on various gears. After Bobach et al. ¹² introduced the roughness into the TEHL of an involute spur gear, Xue et al. ¹³ further considered load changes in the TEHL analysis of this gear. The lubrication performance of helical gears was studied by using a shear-thinning TEHL model incorporating ultimate shear stress. ¹⁴ For planetary gear, the model of EHL was established, which consider the effects of roughness. ¹⁵ After Pu et al. ¹⁶ established the EHL model considering roughness of a spiral bevel gear, Zhang et al. ¹⁷ predicted the coefficient of friction and tooth surface wear of this gear using TEHL technology. After establishing a TEHL model covering the full film, mixed, and boundary lubrication, Simon presented the methods to improve the lubrication performance of hypoid gear based on multi-objective optimization, ¹⁸ optimization of manufacture parameters, ¹⁹ and machine tool setting parameters. ²⁰ Therefore, when analyzing the lubrication performances of VH-CATT cylindrical gears, the effects of the tooth surface roughness and the temperature rise cannot be ignored.

Most recently, numerous scholars have considered the time-varying parameters in the meshing process when analyzing the lubrication performance of gear transmission. For instance, the effect of load and speed were introduced into the TEHL model of a helical gear. ²¹ In addition, Wang et al. ²² presented a TEHL model of a spiral bevel gear, considering of the velocity vector and the surface roughness. Considering certain time-varying parameters along the meshing trace of a spiral bevel gear, the progressive mesh densification was introduced in the numerical calculation of a EHL model ²³ and the behavior of dynamical wear, ²⁴ friction, temperature, and contact fatigue ²⁵ was subsequently discussed. Then, an EHL analysis of spiral bevel gears was conducted after considering the contact point migration due to the deformation under a load. ²⁶ These studies serve as important references for the establishment of an accurate TEHL model of VH-CATT gear transmission.

The coupling of kinematic, spatial geometric, and contact parameters increases the complexity of lubrication analyses of VH-CATT cylindrical gears. However, existing studies on the EHL ²⁷ and TEHL ²⁸ of VH-CATT cylindrical gears are based on simplified linear contact models, which are limited to a specific meshing position. These studies ignore not only the time-varying parameters along the contact trace during gear meshing but also the tooth surface roughness. Therefore, this paper proposes a method for comprehensively calculating the time-varying parameters on a contact trace in a meshing cycle, which is developed using TCA and LTCA models. Subsequently, a TEHL model of VH-CATT cylindrical gears is established, and these time-varying parameters are extracted as the input parameters of the TEHL model. After the numerical calculation, the film thickness, pressure, and temperature rise distributions of a VH-CATT cylindrical gear pair under the effects of the time-varying parameters in a meshing period along the contact trace are analyzed in detail.

Contact analysis of VH-CATT cylindrical gears

To simulate the meshing of a gear pair under real working conditions, TCA and LTCA mathematical models based on the tooth surface equation of VH-CATT cylindrical gear transmission are established. The contact trace as well as time-varying parameters such as the main curvature, entrainment speed, slide-roll ratio, contact ellipse, load distribution, and maximum Hertzian contact pressure in the meshing process, obtained via TCA and LTCA in a meshing period are used as the input parameters for the TEHL model.

Tooth surface equation

VH-CATT cylindrical gears are machined using a double-edged milling cutter fixed on a rotating cutter head. The machining principle is shown in Figure 2.

Figure 2.

Principle of rotating cutter-head milling.

The tooth surface equation ⁵ of the VH-CATT cylindrical gears is written as equation (1).

Where the upper (lower) symbols of ∓ and ± represent the convex (concave) tooth surface of the gears machined by the inner (outer) cutting edge; α is the pressure angle; θ is the cutter rotation angle, φ is the workpiece rotation angle; m is the gear module; R is the radius of the rotating cutter head (radius of the tooth trace), R₁ is the pitch radius.

$$\{\begin{array}{c}{x}_0=[-(R\mp \frac{\pi }{4}m\pm u_0\sin \alpha )\cos \theta +R+R_1\phi ]\hfill \\ \multicolumn{1}{c}{\cos \phi +(u_0\cos \alpha -R_1)\sin \phi ,}\\ \multicolumn{1}{c}{y_0=[(R\mp \frac{\pi }{4}m\pm u_0\sin \alpha )\cos \theta -R-R_1\phi ]}\\ \multicolumn{1}{c}{\sin \phi +(u_0\cos \alpha -R_1)\cos \phi ,}\\ \multicolumn{1}{c}{z_0=(R\mp \frac{\mathrm{\pi }}{4}m\pm u_0\sin \alpha )\sin \theta ,}\\ \multicolumn{1}{c}{u_0=\mp \sin \alpha \frac{\cos \theta (R\mp \frac{\mathrm{\pi }}{4}m)-(R_1\phi +R)}{\cos \theta }}\end{array}$$

(1)

TCA model

As shown in Figure 3, the spatial coordinate system of the VH-CATT cylindrical gear pair is established based on the spatial configuration during meshing. o_fx_fy_fz_f is the fixed coordinate system; o_px_py_pz_p and o_gx_gy_gz_g are the workpiece dynamic coordinate systems fixed with driving and driven gears, respectively; o_mx_my_mz_m and o_nx_ny_nz_n are the intermediate coordinate systems in the coordinate transformation; l is the center distance; ϕ_p and ϕ_g are the meshing angles of the driving and driven gear, respectively.

Figure 3.

Coordinate systems used for movement of driving and driven tooth surfaces.

It is assumed that Σ_p (the concave tooth surface) in o_px_py_pz_p represents the family of surfaces r _p and that its unit normal vector is n _p . Similarly, Σ_g (the convex tooth surface) in o_gx_gy_gz_g denotes the family of surfaces r _g , and its unit normal vector is n _g . In o_fx_fy_fz_f, the vector and unit normal vector of Σ_p and Σ_g are denoted as ( r _fp , n _fp ) and ( r _fg , n _fg ), respectively.

Thus, the TCA mathematical model of the VH-CATT cylindrical gear pair in the meshing process is expressed as:

$$\{\begin{array}{c}{\mathit{r}}_{\mathrm{fp}}({\phi }_p,{\theta }_p,{\varphi }_p)-{\mathit{r}}_{\mathrm{fg}}({\phi }_g,{\theta }_g,{\varphi }_g)=0\hfill \\ \multicolumn{1}{c}{{\mathit{n}}_{\mathrm{fp}}({\phi }_p,{\theta }_p,{\varphi }_p)-{\mathit{n}}_{\mathrm{fg}}({\phi }_g,{\theta }_g,{\varphi }_g)=0}\end{array}$$

(2)

The transformation of Σ_p into o_fx_fy_fz_f is completed by:

$$\{{\Sigma }_p\}:\{\begin{array}{c}{\mathit{r}}_{\mathrm{fp}}={\mathit{M}}_{\mathrm{fp}}{\mathit{r}}_p\hfill \\ \multicolumn{1}{c}{{\mathit{n}}_{\mathrm{fp}}={\mathit{L}}_{\mathrm{fp}}{\mathit{n}}_p}\end{array}$$

(3)

Where M (4 × 4) is the transformation matrix of r _p from o_px_py_pz_p to o_fx_fy_fz_f, and L _fp (3 × 3) is the transformation matrix corresponding to n _p of Σ_p.

The transformation of Σ_g into o_fx_fy_fz_f is completed by

$$\{{\Sigma }_g\}:\{\begin{array}{c}{\mathit{r}}_{\mathrm{fg}}={\mathit{M}}_{\mathrm{fn}}{\mathit{M}}_{\mathrm{nm}}{\mathit{M}}_{\mathrm{mg}}{\mathit{r}}_g\hfill \\ \multicolumn{1}{c}{{\mathit{n}}_{\mathrm{fg}}={\mathit{L}}_{\mathrm{fn}}{\mathit{L}}_{\mathrm{nm}}{\mathit{L}}_{\mathrm{mg}}{\mathit{n}}_g}\end{array}$$

(4)

Where M _mg (4 × 4), M _nm (4 × 4), and M _fn (4 × 4) are the transformation matrices of r _g from o_gx_gy_gz_g to o_fx_fy_fz_f, and L _mg (3 × 3), L _nm (3 × 3), and L _fn (3 × 3) are the transformation matrices corresponding to n _g of Σ_g.

Based on the existence theorem of implicit function, ²⁹ five independent equations with five unknown parameters are derived from equation (2), as follows:

$$\{\begin{array}{c}{f}_i[{\phi }_p\left({\varphi }_p\right),{\theta }_p\left({\varphi }_p\right),{\phi }_g\left({\varphi }_p\right),{\theta }_g\left({\varphi }_p\right),{\varphi }_g\left({\varphi }_p\right)]=0\hfill \\ \multicolumn{1}{c}{f_i\in C}\end{array},(i=1,2,3,4,5)$$

(5)

Taking the design parameters of the VH-CATT cylindrical gear pair listed in Table 1 as a case, all data reflecting the gear meshing status are solved through numerical iterations, the results as shown in Figure 4.

Table 1.

Design parameters.

Parameters	Numerical value
Module of gear pair (mm)	8
Number of driving gear teeth	21
Number of driven gear teeth	29
Pressure angle (°)	20
Tooth width (mm)	80
Radius of the tooth trace (mm)	500

Figure 4.

Contact trace in (a) coordinate system o_px_py_pz_p and (b) coordinate system o_gx_gy_gz_g.

The curvature of a VH-CATT cylindrical gear is an important spatial geometric parameter affecting its lubrication behavior. As shown in Figure 5, the principal orientations of Σ_p and Σ_g are t ^a _p , t ^b _p , t ^a _g , and t ^b _g , respectively. Correspondingly, the principal curvatures are K ^a _p , K ^b _p , K ^a _g , and K ^b _g .

Figure 5.

Curvature relationship of tooth surface.

Thus, the mathematical expression of the main curvature of the VH-CATT cylindrical gears can be derived as:

$$\{\begin{array}{c}{K}_i^a=K_{\mathrm{iA}}+\sqrt{{K_{\mathrm{iA}}}^2-K_{\mathrm{iG}}}\hfill \\ \multicolumn{1}{c}{K_i^b=K_{\mathrm{iA}}-\sqrt{{K_{\mathrm{iA}}}^2-K_{\mathrm{iG}}}}\end{array},(i=p,g)$$

(6)

Where K_iG is the Gaussian curvature and K_iA is the average curvature of the tooth surface, which can be deduced by the surface theory. ³⁰

Generally, velocity parameters have a major influence on the TEHL analysis of gear transmission. Therefore, the entrainment velocity, u _e , the absolute velocities, v _fp and v _fg , and the slide-roll ratio, S, of the current meshing point are introduced into the TEHL model as velocity parameters. The v _fp and v _fg , of the driving and driven gears, respectively, are obtained using the transmission characteristics of the VH-CATT cylindrical gear pair:

$$\{\begin{array}{c}{\mathit{v}}_{fp}={\omega }_{fp}\times {\mathit{r}}_{fp}=-{\omega }_p(y_{fp}+R_p)\mathit{i}+{\omega }_p{x}_{fp}\mathit{j}\\ {\mathit{v}}_{fg}={\omega }_{fg}\times {\mathit{r}}_{fg}={\omega }_g(y_{fg}-R_g)\mathit{i}-{\omega }_g{x}_{fg}\mathit{j}\end{array}$$

(7)

Where ω _fp and ω _fg are the angular velocities of the meshing point, and R_p and R_g are the radii of the pitch circles.

The sliding velocity, v _r , at the meshing point can be calculated as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{v}}_r={\mathit{v}}_{\mathrm{fp}}-{\mathit{v}}_{\mathrm{fg}}=-[{\omega }_p(y_{\mathrm{fp}}+R_p)+{\omega }_g(y_{\mathrm{fg}}-R_g)]\mathit{i}}\\ \multicolumn{1}{c}{+({\omega }_p{x}_{\mathrm{fp}}+{\omega }_g{x}_{\mathrm{fg}})\mathit{j}}\end{array}$$

(8)

The method for calculation u _e at any meshing point is as follows:

$${\mathit{u}}_e=\frac{{\mathit{v}}_{\mathrm{fp}}·{\mathit{l}}_r·{\mathit{l}}_r+{\mathit{v}}_{\mathrm{fg}}·{\mathit{l}}_r·{\mathit{l}}_r}{2}$$

(9)

Where, l _r is the unit vector of the v _r .

Thus, the calculation method of S is as follows:

$$S=\frac{2\left|{\mathit{v}}_r\right|}{\left|{\mathit{u}}_e\right|}$$

(10)

LTCA model

Generally, the real working conditions of gear pair can be simulated using LTCA model. The LTCA model of the VH-CATT cylindrical gear transmission is established based on the method proposed in Klima et al. ³¹ and Jiang and Fang, ³² which is illustrated in Figure 6.

Figure 6.

LTCA model of VH-CATT cylindrical gears.

In a meshing period, the VH-CATT cylindrical gear pair is in a contact state at a specific time, assuming that the contact load between Σ_p and Σ_g is distributed along the main direction of the Hertz contact ellipse. In Figure 6, the tooth curve is the intersection of the tooth surface and the normal plane along the relative main direction, ik (k = I, II) is a contact point, and jk (k = I, II) is a discrete point along the relative main direction. The gaps and contact loads of the contact points on each tooth surface in the meshing satisfy the following conditions:

$$\{\begin{array}{c}{\mathit{F}}_k{\mathit{f}}_k+{\mathit{w}}_k=\mathit{Z}+{\mathit{d}}_k,(k=I,\mathrm{II})\hfill \\ \multicolumn{1}{c}{\sum _{j=1}^n{f}_{\mathrm{jI}}+\sum _{j=1}^n{f}_{\mathrm{jII}}=F}\\ \multicolumn{1}{c}{d_{\mathrm{jk}}=0,(\mathrm{if}{f}_{\mathrm{jk}}>0)}\\ \multicolumn{1}{c}{d_{\mathrm{jk}}>0,(\mathrm{if}{f}_{\mathrm{jk}}=0)}\end{array}$$

(11)

Where F _k (n × n) is the assembled flexibility matrix determined using the finite element method, f _k (1 × n) and d _k (1 × n) are column matrixes composed of elements [f_1k, f_2k, …, f_nk] and [d_1k, d_2k, …, d_nk], respectively, w _k (1 × n) is a column matrix that contains the initial tooth gaps, and Z (1 × n) represents the approach of the tooth surface along the action line direction. f_jk (j = 1, 2, …, n) is the contact load at point j on tooth k, F is the total load, and d_jk (j = 1, 2, …, n) is the final tooth gap at point j on tooth k.

Thus, the maximum Hertzian contact pressure, p_h, in the contact area can be deduced as follows:

$$p_h=\frac{3F}{2\mathrm{\pi }\mathrm{ab}}$$

(12)

Where a (or b) is the minor (or major) semi-axis of the Hertz contact ellipse.

The a and b are expressed as follows:

$$\{\begin{array}{c}a={\alpha }_{\mathrm{ab}}\sqrt[3]{\frac{3F}{4A}(\frac{1-{\mu }_p^2}{E_p}+\frac{1-{\mu }_g^2}{E_g})}\hfill \\ \multicolumn{1}{c}{b={\beta }_{\mathrm{ab}}\sqrt[3]{\frac{3F}{4A}(\frac{1-{\mu }_p^2}{E_p}+\frac{1-{\mu }_g^2}{E_g})}}\end{array}$$

(13)

Where E_p and E_g are the elastic moduli of the gear material, μ_p and μ_g are Poisson’s ratios of the gear material, and α_ab, A, and β_ab are coefficients.

Basic equations of the TEHL model

Referring to the model proposed in Zhu, ³³ the Reynolds equation of the VH-CATT cylindrical gear pair with an elliptical contact is as follows:

$$\frac{\partial }{\partial x}\left(\frac{\rho h^3}{\eta }\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{\rho h^3}{\eta }\frac{\partial p}{\partial y}\right)=12\left|{\mathit{u}}_e\right|\frac{\partial \left(\rho h\right)}{\partial x}+12\frac{\partial \left(\rho h\right)}{\partial t}$$

(14)

Where η is the viscosity of the lubricant, ρ is the density of the lubricant, h is the film thickness, p is the film pressure, and t is the meshing time.

Accordingly, the nondimensional Reynolds equation is

$$\frac{\partial }{\partial X}\left({\epsilon }_X\frac{\partial P}{\partial X}\right)+\frac{\partial }{\partial Y}\left({\epsilon }_Y\frac{\partial P}{\partial Y}\right)=\frac{\partial \left(\overline{\rho }H\right)}{\partial X}+\frac{\partial \left(\overline{\rho }H\right)}{\partial \overline{t}}$$

(15)

Where

$${\epsilon }_X=\frac{a}{12\left|{\mathit{u}}_e\right|}\frac{p_h}{{\eta }_0}\left(\frac{\overline{\rho }{H}^3}{\overline{\eta }}\right),{\epsilon }_Y=\frac{a^3}{12b^2\left|{\mathit{u}}_e\right|}\frac{p_h}{{\eta }_0}\left(\frac{\overline{\rho }{H}^3}{\overline{\eta }}\right)$$

(16)

Where η₀ is the initial viscosity of the lubricant.

The boundary conditions of equation (15) are

$$\begin{array}{c}\multicolumn{1}{c}{P(X_{\mathrm{in}},Y)=P(X_{\mathrm{out}},Y)=0,P(X,Y_{\mathrm{in}})=P(X,Y_{\mathrm{out}})=0,}\\ \multicolumn{1}{c}{\frac{\partial P}{\partial X}=\frac{\partial P}{\partial Y}=0,X_{\mathrm{in}}\le X\le X_{\mathrm{out}},Y_{\mathrm{in}}\le Y\le Y_{\mathrm{out}}}\end{array}$$

(17)

To ensure that the TEHL model replicates a real scenario, the dynamic surface topography of the VH-CATT cylindrical gears is introduced into the film thickness equation in the form of the three-dimensional random roughness distribution. The film thickness equation is as follows:

$$h=h_0\left(t\right)+\frac{x^2}{2R_x}+\frac{y^2}{2R_y}+{\delta }_p(x,y,t)+{\delta }_g(x,y,t)+V(x,y,t)$$

(18)

Where h₀ is the initial central film thickness; R_x and R_y are the equivalent radii of the curvature at the current meshing point in the directions of the tooth profile and tooth trace, respectively, which are derived from equation (6); x²/2R_x and y²/2R_y are the transient geometries before elastic deformation; δ_p and δ_g are the three-dimensional random roughness of Σ_p and Σ_g, respectively. δ_p and δ_g are not introduced into equation (18) for smooth tooth surfaces.

The local contact deformation, V, is calculated using the Boussinesq integral equation:

$$V(x,y,t)=\frac{2}{\pi E\prime }\int {\int }_{\mathrm{\Omega }}\frac{p(\xi ,\psi )}{\sqrt{{(x-\xi )}^2+{(y-\psi )}^2}}d\xi d\psi$$

(19)

Where $E\prime$ is the integrated elastic modulus.

Therefore, the nondimensional film thickness equation is:

$$\begin{array}{c}\multicolumn{1}{c}{H(X,Y,\overline{t})=H_0\left(\overline{t}\right)+\frac{X^2}{2}+e_k\frac{Y^2}{2}+{\overline{\delta }}_p(X,Y,\overline{t})}\\ \multicolumn{1}{c}{+{\overline{\delta }}_g(X,Y,\overline{t})+L{\int }_{X_{\mathrm{in}}}^{X_{\mathrm{out}}}{\int }_{Y_{\mathrm{in}}}^{Y_{\mathrm{out}}}}\\ \multicolumn{1}{c}{\frac{P(\overline{\xi },\overline{\psi })}{\sqrt{(X-\overline{\xi })+(Y-\overline{\psi })}}d\overline{\xi }d\overline{\psi }}\end{array}$$

(20)

Where e_k is the ellipticity, and e_k = R_x/R_y. L is the contact deformation coefficient of the contact ellipse, and L = 3WR_x/a²bE′.

Considering the influence of temperature change on the lubrication of VH-CATT cylindrical gears, the three-dimensional energy ³⁴ is introduced into the TEHL model based on the theory of moving heat over a semi-infinite solid. The three-dimensional energy equation in the x, y, and z directions (z direction represents the film thickness direction) is as follows:

$$\begin{array}{c}\multicolumn{1}{c}{c_a\rho (u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}-w\frac{\partial T}{\partial z})=\kappa (\frac{{\partial }^{2}T}{\partial z^2}+\frac{{\partial }^{2}T}{\partial y^2})}\\ \multicolumn{1}{c}{-\frac{T}{\rho }\frac{\partial \rho }{\partial T}(u\frac{\partial p}{\partial x}+v\frac{\partial p}{\partial y})+\eta [{\left(\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}\right)}^2]}\end{array}$$

(21)

Where C_a and κ are the isobaric heat capacity and thermal conductivity of the lubricant, respectively, and u, v, and w are the flow rates along the x, y, and z directions, respectively.

To obtain the solution of the energy equation, the velocity field must be determined first. When the pressure distribution and the film thickness are known and the lubricant viscosity changes along the z direction, the flow velocities, u, v, and w, in the x, y, and z directions, respectively, are expressed as:

$$\{\begin{array}{c}\begin{array}{c}u=\left|{\mathit{v}}_{\mathrm{fg}}\right|+\frac{\partial p}{\partial x}({\int }_0^z\frac{z}{\eta }\mathrm{dz}-\frac{{\int }_0^h\frac{z}{\eta }\mathrm{dz}}{{\int }_0^h\frac{1}{\eta }\mathrm{dz}}{\int }_0^z\frac{1}{\eta }\mathrm{dz})+\frac{\left|{\mathit{v}}_{\mathrm{fp}}\right|-\left|{\mathit{v}}_{\mathrm{fg}}\right|}{{\int }_0^h\frac{1}{\eta }\mathrm{dz}}{\int }_0^z\frac{1}{\eta }\mathrm{dz}\hfill \\ \multicolumn{1}{c}{v=\frac{\partial p}{\partial y}({\int }_0^z\frac{z}{\eta }\mathrm{dz}-\frac{{\int }_0^h\frac{z}{\eta }\mathrm{dz}}{{\int }_0^h\frac{1}{\eta }\mathrm{dz}}{\int }_0^z\frac{1}{\eta }\mathrm{dz})}\\ \multicolumn{1}{c}{w=-\frac{1}{\rho }{\int }_0^z[\frac{\partial \left(\rho u\right)}{\partial x}+\frac{\partial \left(\rho v\right)}{\partial z}]\mathrm{dz}}\end{array}\hfill \end{array}$$

(22)

The fundamental equation for calculating the temperatures of the upper and lower tooth surfaces in the meshing is as follows:

$$\{\begin{array}{c}T\left(x,y,0\right)=T_0+\kappa {(\pi {\rho }_p{c}_p\left|{\mathit{v}}_{fp}\right|{\kappa }_p)}^{-0.5}{\int }_{-\infty }^x\frac{\partial T}{\partial z}\frac{ds}{{\left(x-s\right)}^{0.5}}\\ T\left(x,y,h\right)=T_0+\kappa {(\pi {\rho }_g{c}_g\left|{\mathit{v}}_{fg}\right|{\kappa }_g)}^{-0.5}{\int }_{-\infty }^x\frac{\partial T}{\partial z}\frac{ds}{{\left(x-s\right)}^{0.5}}\end{array}$$

(23)

Where T₀ is the initial temperature, (ρ_p, ρ_g), (c_p, c_g), and (κ_p, κ_g) are the density, specific heat capacity, and thermal conductivity of the tooth surface material of Σ_p and Σ_g, respectively.

Viscosity is used as the dependent variable of pressure and temperature to reflect their relationship. The Roelands viscosity–pressure–temperature equation is used to calculate the viscosity:

$$\eta ={\eta }_0\exp \left\{(\ln {\eta }_0+9.67)[{(1+\frac{p}{p_0})}^{Z_{\eta }}{\left(\frac{T-138}{T_0-138}\right)}^{-1.1}-1]\right\}$$

(24)

Where Z_η is the coefficient of the equation, Z_η= α₀/5.1(lnη₀+ 9.67), and α₀ is the viscosity-pressure coefficient.

The Dowson–Higginson density–pressure–temperature equation is selected to reflect the specific effects of pressure and temperature on density, and the equation is as follows:

$$\rho ={\rho }_0[1+\frac{0.6p}{1+1.7p}+D(T-T_0)]$$

(25)

Where ρ₀ and D are the thermal expansion rate and initial density of the lubricant, respectively.

The load is balanced by the integrating the pressure over the entire solution region. The load-balancing equation is

$${\int }_{x_{\mathrm{in}}}^{x_{\mathrm{out}}}{\int }_{y_{\mathrm{in}}}^{y_{\mathrm{out}}}p(x,y,t)\mathrm{dxdy}=F\left(t\right)$$

(26)

Numerical computation method

Figure 7 shows the program flow of above models. The Reynolds equation and the energy equation are discretized iteratively by combining the central and forward difference methods. The Jacobi bipolar iteration and the Gauss–Seidel iteration methods is used in high-pressure and low-pressure regions, respectively. To prevent divergence in the iteration process, the relaxation factor, c₁, of the correcting pressure iteration is limited to 0.005–0.01, and the correction factor, c₂, of initial central film thickness is limited to 0.001–0.01. The convergence precisions of the pressure and the temperature are $\left|\frac{p_{i,j}^{\mathrm{New}}-p_{i,j}^{\mathrm{Old}}}{p_{i,j}^{\mathrm{New}}}\right|{10}^{-5}$ and $\left|\frac{T_{i,j}^{\mathrm{New}}-T_{i,j}^{\mathrm{Old}}}{T_{i,j}^{\mathrm{New}}}\right|{10}^{-5}$ for smooth surface solutions, respectively, and ≤10⁻⁴ for rough surface solutions. The solving domain is defined as −2.0a ≤ x ≤ 2.0a, −2.0b ≤ y ≤ 2.0b. The computing grid covering this domain consists of 299 ×299 equidistant nodes. The number of layers in the direction of the film thickness is 5. To get numerical solution of approximate actual condition, the roughness of both friction surfaces, is moved with the meshing time.^23,25 Since VH-CATT cylindrical gear has similar precision grade and working condition with the spiral bevel gear, the roughness is set as 0.3 μm according to the value in Pu et al. ¹⁶ and Wang et al. ²³ One of the three-dimensional rough surfaces is shown in Figure 8.

Figure 7.

Calculation flowchart.

Figure 8.

Three-dimensional rough surface profiles.

Model validation

In order to verify the present TEHL model of VH-CATT cylindrical gear transmission, the film thickness distribution solved by the present model is compared with the data solved by Wang et al. ²² and measured by Omasta et al. ³⁵ To ensure that the conditions for validation are identical, the same input parameters, solving domain, and the computing grid are obtained via from Wang et al. ²² Table 2 and Figure 9 show that the central film thickness and film thickness distribution calculated by present model are substantially consistent with numerical simulation ²² and measured result ³⁵ at same working condition. Thus, the TEHL model has a certain computational accuracy, and the error is small compared with the results in the literature.

Table 2.

Central film thickness comparison.

h-cen (nm)	Data sources
388	Calculated by present TEHL model
392	Calculated by Wang et al. 22
395	Measured by Omasta et al. 35

Figure 9.

Comparison of film thickness distributions calculated by (a) present model, (b) Wang et al., ²² and (c) measured by Omasta et al. ³⁵

Results and analysis

Results of TCA and LTCA

The design parameters of the VH-CATT cylindrical gear pair used for the TCA and the LTCA are listed in Table 1. In addition, the input rotational speed, n_in, of the driving gear is constant at 500 rpm, and the input torque of the driven gear is set as 232 Nm. The analysis results for the time-varying parameters along the contact trace in the VH-CATT cylindrical gears are generalized in Figure 10.

Figure 10.

The (a) composite curvature radii along directions of tooth profile and tooth trace, (b) sliding velocities of driving gear (red line) and driven gear (blue line), (c) entrainment velocities, (d) slide-roll ratio, (e) gaps between contacting teeth, (f) load distributions and maximum Hertz contact pressures, and (g) contact ellipses on tooth surface.

As shown in Figure 10(a), R_x reaches a maximum near the pitch point and exhibits a parabola-like variation with the meshing process. In comparison, R_y gradually increases linearly with the meshing time. As shown in Figure 10(b), the | v _fp | increases gradually while | v _fg | decrease, and reaches a same velocity near the pitch point. According to the calculation results in Figure 10(c) and (d), | u _e | and S gradually increase with the meshing time, and the value of S is 0 near the pitch point. Figure 10(e) gives the gaps between contacting teeth along the contact trace in a meshing period. As shown in Figure 10(f), both F and p_h have a sharp rise and fall in the middle of meshing period. Because of the transformation of the meshing process between single-tooth or double-teeth meshing, the area size of contact ellipse abruptly increases and decreases when the meshing process first enters the single-tooth and double-teeth meshing areas, respectively. A certain proportion of the transient contact ellipses is uniformly extracted from the path of contact and is projected on the tooth surface of driving gear, as shown in Figure 10(g).

Note that the rotation angle increment, Δϕ, of the gear is converted to the corresponding time increment, Δt, based on the input rotational speed, n_in, that is, Δt = Δϕ/n_in. In general, the meshing process of the studied gear pair alternately involves single or double teeth. Moreover, the values of some time-varying parameters between the single-tooth and double-teeth meshing areas rise and fall sharply. Therefore, a total of nine time nodes, t_i (i = 1, 2, 3, …, 9), are selected as typical examples for the analysis of the lubrication characteristics in a meshing period as shown in Figure 10(f). Note that the entire gear meshing process is disaggregated into three stages, which are the first meshing stage (t_1 − t₃), the second meshing stage (t_4 − t₆), and the third meshing stage (t_7 − t₉), respectively.

Results of TEHL with smooth tooth surfaces

The time-varying parameters in Figure 10 and the lubrication parameters in Table 3 are taken as the input parameters of the TEHL model. Figures 11 to 13 give the numerical calculation results of the film thickness h, pressure p, and temperature rise, T, along the X- and Y-axes in a complete meshing period.

Table 3.

Material parameters of gear pair and lubricant.

Lubricant	Numerical value	Gear pair	Numerical value
Initial viscosity (Pa s)	0.09	Initial temperature (K)	303
Initial density (kg/m3)	846	Thermal conductivity	46
Specific heat (J/kg/K)	2000	Specific heat (J/kg/K)	470
Thermal conductivity (W/m K)	0.14	Density (kg/m3)	7850
Thermal expansivity (K−1)	−0.00065	Elastic modulus (Pa)	2.07 × 1011
Viscosity–pressure coefficient (Pa−1)	2.3 × 10−8	Poisson’s ratio	0.3

Figure 11.

Film thickness distributions along X- and Y-axis directions of contact ellipse in meshing period.

Figure 12.

Pressure distributions along X- and Y-axis directions of contact ellipse in meshing period.

Figure 13.

Temperature rise distributions along X- and Y-axis direction of contact ellipse in meshing period.

h, p, and T, exhibit different variation trends in different meshing stages, as observed from Figures 11 to 13. The R_x, R_y, | u _e | gradually increase, and p_h decreases sharply in t₁ − t₃, so h increases significantly, whereas p and T decrease rapidly with t. In addition, S increase gradually while T decreases rapidly, enabling the viscosity of the lubrication to be recovered. Thus, the lubrication performance of the VH-CATT cylindrical gears in t₁ − t₃ is significantly improved with the meshing process. The sensitivity of h and p of numerical calculation results to the time-varying parameters in t₄ − t₆ is similar to that in t₁ − t₃. However, S reaches 0 near the pitch point (it’s near t₅), so T manifest as changing trend of “V” shape in this stage. T in Δ(t₄ − t₅) increases more than the decrement in Δ(t₅ − t₆), and this trend can be explained by the variation in p_h in this stage, which is shown in Figure 10. Notably, in t₇ − t₉, the variation laws of h, p, and T are different from those in t₁ − t₃ and t₄ − t₆. h and p continue to increase (or decrease) slightly during Δ(t₇ − t₈). Correspondingly, because the contact ellipse and R_x continue to decrease and p_h exhibits a slight increasing trend in Δ(t₈ − t₉), the variation trends of h and p are opposite to those in the previous period. In addition, with the increment of S (in Figure 10) in this stage, effect of lubricant viscosity on T gradually less than thermal expansion, so T increases continuously.

To evaluate the lubrication performance in the different meshing stages more intuitively, the contour lines of h, p, and T under smooth tooth surfaces are plotted in Figure 14.

Figure 14.

The contours of (a) film thickness h, (b) pressure p, and (c) temperature rise T in meshing period.

Figure 14 shows the significant differences in h, p, and T at different meshing moments, which are caused by the variations in the time-varying parameters at each moment. Therefore, the positions of the minimum film thickness, maximum pressure, and maximum temperature rise deviate with the meshing time, and at each meshing instant, a prominent squeezing effect occurs along the flow direction (X-axis) of the lubricant. During different meshing stages, that is, when the VH-CATT cylindrical gear pair operates under the transmission condition of a single-tooth and double-teeth alternately carrying the load, the h, p, and T change abruptly in the calculation area. This is because of the rapid variations in the load and the contact ellipse in a short time.

Based on the TEHL results for the VH-CATT cylindrical gears considering smooth tooth surfaces, Figure 15 gives the variations in the minimum film thickness h-min, center film thickness h-cen, maximum pressure p-max, and maximum temperature rise T-max in a meshing period. Here, define λ as the increment in the TEHL results at the current moment compared with that in the previous one, that is, λ = 100(C_r −P_r)/P_r%, where P_r and C_r represent the TEHL values at the previous and current meshing moments, respectively.

Figure 15.

The (a) minimum film thickness, (b) central film thickness (c) maximum pressure, and (d) maximum temperature rise.

First, Figure 15 shows that the lubrication condition is the least favorable at the instant when the gears initially mesh, when the occurrence of friction and wear failure are the most probable. The lubrication condition gradually improves as the meshing progresses. Second, because the time-varying parameters change significantly in Δ(t₁ − t₂), λ increases or decreases rapidly in this period. The first peak values (0.63 ms) of the four λ curves are 12.49%, 8.02%, −7.61%, and −7.10%, respectively. Concurrently, it is noted that the λ values clearly fluctuate at 3.74 and 5.71 ms caused by the switch between single-tooth and double-teeth load. Finally, when t is 4.62 ms, the gear pair is located near the pitch point, and S is 0 at this time. Thus, h-min,p-max, and T-max have good extreme values of obvious lubrication near the pitch point, which are 0.8036 μm, 0.6948 GPa, and 38.6125°C.

Results of TEHL with rough tooth surfaces

To predict the lubrication characteristics more accurately in a meshing period of a real machined surface, this section introduces the three-dimensional surface roughness based on the input parameters of smooth TEHL. The TEHL results considering rough tooth surfaces for the VH-CATT cylindrical gears are summarized in Figures 16 to 18.

Figure 16.

Film thickness distributions along X- and Y-axis directions of contact ellipse in meshing period.

Figure 17.

Pressure distributions along X- and Y-axis directions of contact ellipse in meshing period.

Figure 18.

Temperature rise distributions along X- and Y-axis directions of contact ellipse in meshing period.

Figures 16 to 18 show that the TEHL numerical simulation results for rough tooth surfaces are significantly different from those of smooth tooth surfaces, which are shown in Figures 11 to 13. Under the effect of roughness, although the variation laws of the TEHL results for smooth tooth surfaces still apply in each meshing stage for rough tooth surfaces, the latter exhibit clear oscillations. The temperature rise between the different meshing stages for rough tooth surfaces exhibits numerous divergent peaks, as seen in Figure 18. In a meshing period, the smaller film thickness and larger pressure are, the higher temperature is. Therefore, the first meshing stage is most likely subjected to the influence of direct contact of the rough peaks by comparing the peaks of temperature at different meshing stages.

To facilitate direct observation of the lubrication performance of the VH-CATT gear pair under thermal and roughness effects, Figure 19 presents contour maps.

Figure 19.

The contours of (a) film thickness h, (b) pressure p, and (c) temperature rise T in meshing period.

In Figure 19(a), the squeezing effect caused by the extrusion of lubrication at each meshing instant is still evident; however, the shape of the film fluctuates owing to the roughness of the gear surface. The pressure in Figure 19(b) exhibits abrupt changes between different meshing stages. A large ratio of the contact load to the contact ellipse implies a high viscosity of lubrication and a strong influence of the roughness on the pressure. In Figure 19(c), the influence of roughness on the lubrication performance of the VH-CATT cylindrical gears can be manifested by the number of dark red spots in the contact ellipse. Note that the there are some dark red spots randomly distributed on the right side of the contact ellipse at t₈ and t₉ in red rectangular box of Figure 19(c). The reasons for this phenomenon are the random distribution of roughness and the poor lubrication performance in t₇ − t₉, so it is still possible to damage the oil film by the roughness of the tooth surface.

Conclusions

In this study, the TEHL model of VH-CATT cylindrical gear transmission in elliptical contact has been established by considering the effects of time-varying parameters. The numerical method for calculating the time-varying parameters of this gear transmission in a meshing period has been proposed. After introducing the time-varying parameters in a meshing period, the lubrication characteristics considering smooth and rough tooth surfaces have been investigated. Based on numerical simulation, the following conclusions can be drawn:

1. The kinematic parameters, spatial geometric parameters, and contact parameters of VH-CATT cylindrical gear transmission constantly vary in a meshing period. To obtain these time-varying parameters, the TCA and LTCA models are established. Considering the effects of temperature and time-varying parameters for the analysis of the lubrication characteristics, a TEHL model of the VH-CATT cylindrical gears in elliptical contact is presented.

2. The TEHL results for smooth tooth surfaces show that the lubrication characteristics of the VH-CATT cylindrical gears are directly affected by time-varying parameters. The lubrication characteristics in the entire meshing process fluctuate because the load distribution and Hertz contact ellipse vary significantly. The lubrication performance at the beginning of the first meshing stage—when the film thickness is the smallest, the pressure is the largest, and the temperature rise is highest—is the worst. However, the performance improves rapidly as the meshing progresses. The second meshing stage presents the best lubrication performance. The lubrication performance of the third meshing stage is better than that of the first meshing stage and inferior to that of the second meshing stage.

3. The TEHL results for rough tooth surfaces show that the lubrication characteristics obtained after considering roughness are closer to the actual lubrication characteristics. The overall trends of the variations in the lubrication performance in a meshing period for rough tooth surfaces are similar to those observed for smooth tooth surfaces. However, the positions of the minimum film thickness, maximum pressure, and maximum temperature rise shift are more stochastic than the corresponding positions for smooth tooth surfaces. The stage most likely subjected to the influence of direct contact due to rough peaks is the first meshing stage, followed by the third meshing stage.