An Improved Stiffness Model for Spur Gear with Surface Roughness Under Thermal Elastohydrodynamic Lubrication

Abstract

To investigate the contact performances and meshing characteristics of gears systematically, an improved comprehensive meshing stiffness model of spur gears with consideration of the tooth surface morphology, lubrication, friction, and thermal effects is presented based on the thermal elastohydrodynamic lubrication (TEHL) theory. The fractal feature of the tooth surface morphology is verified experimentally and characterized by the Weierstrass–Mandelbrot fractal function. Based on this, the rough contact stiffness, oil film stiffness, and thermal stiffness are introduced into the stiffness model. Comparisons between smooth and rough models are carried out, and the maximum temperature rise is increased by 24.7%. Subsequently, the influences of the torque, rotational speed, and fractal parameters on the oil film pressure and thickness, friction and temperature rise, and contact stiffness and comprehensive meshing stiffness are investigated. The results show that the oil film pressure and the maximum temperature rise increase by 125.18% and 69.08%, respectively, with an increasing torque from 20 N·m to 300 N·m. As the rotational speed is increased, the oil film thickness sharply increases, the rough peak contact area and friction reduce, and the stiffness fluctuation weakens. For fractal parameters, the oil film pressure and film thickness, friction, and temperature rise are nonlinear with changes in the fractal dimension D and fractal scale characteristic G. The results reveal that this work provides a more reasonable analysis for understanding the meshing characteristics and the design and processing of gears.

1. Introduction

Due to the widespread application of gears in aerospace, automobiles, ships, and other high-end mechanical equipment [1,2], gear systems often operate in complex and changing environments. In addition, the concave–convex microstructure and oil film on the tooth surface directly influence the load sharing, lubrication characteristics, wear, transmission efficiency, and thermal effects between the teeth, and then affect the gear-meshing characteristics and the contact properties at the meshing interface of the gears, which further influence the vibration and noise, meshing shock, safety and reliability, and service life of the gears through stiffness and error excitations. Hence, to understand the meshing characteristics of gears under multi-physical field coupling, the calculation of the comprehensive meshing stiffness is particularly important under thermal elastohydrodynamic lubrication and has become the focus of research in the field of gear systems in recent years, which has attracted more and more scholars to investigate this [3].

The elastic deformation and lubrication condition of gear teeth play a very important role in the gear-meshing process. The tooth stiffness and oil film stiffness directly affect the time-varying mesh stiffness (TVMS), which is one of the major factors that determine the vibration characteristics of gear systems. Hence, the elastohydrodynamic lubrication (EHL)/thermoelastohydrodynamic lubrication characteristics of gears have been studied thoroughly by domestic and foreign scholars in recent years. Wang [4,5] used a simplified multigrid method to obtain a complete numerical solution for the transient thermoelastohydrodynamic lubrication of gears. Yin [6] established a TEHL calculation model for double involute gears considering the characteristics of non-Newtonian fluid, and studied the lubrication characteristics. Zhou and his team members have carried out a lot of research on the lubrication properties of gears [7,8,9,10,11]. Firstly, based on the numerical calculation method, a non-Newtonian TEHL point contact model was proposed [7], and the lubrication properties under different parameters were analyzed. Then, to reduce the meshing impact in both normal and tangential directions, a new combined normal–tangential stiffness model [8] and oil film damping [9] were developed, respectively, and then the combined stiffness–damping [10] was proposed, which was investigated for the effects on mesh impact and gear nonlinear dynamics [11]. Based on this, the thermo-mechanical characteristics of gears in non-Newtonian transient micro-TEHL contacts were investigated [12]. Lu [13] proposed an analytical load distribution model suitable for EHL contact gears and researched the coupling effect between lubrication properties and load distribution. Xue [14,15] established a gear transient TEHL model based on dynamic loads, which was used to predict the scuffing load capacity of a spur gear system. Based on the six degrees of freedom (6-DOF) tribo-dynamics model of gear systems, Jian [16] investigated the TEHL properties of a gear system under vibrations and a static load. Comprehensively considering EHL, slice coupling, and extended tooth contact, Huang [17] developed an iterative model for the mesh stiffness of spur gears and studied the effects of EHL and slice coupling on meshing characteristics. Marques [18] developed an analytical model relying on the ISO 6336 maximum tooth stiffness and a parabolic single tooth stiffness per unit of single line length. To measure the meshing stiffness of gears, Refs [19,20,21] designed corresponding experimental methods to measure the gear tooth stiffness, which were verified through the finite element method (FEM). Considering the viscoelastic behavior of polymers, Chakroun [22] built a polymer–metal gear mesh stiffness model and researched the effect of the gear surface temperature rise. To reduce the gear-meshing impact, Pedrero [23,24] put forward analytical models for load sharing, TVMS, and transmission error with profile modification. Abruzzo [25] proposed a dynamic model to evaluate the meshing characteristic based on a two-phase approach, which could accurately analyze the contact forces and meshing stiffness. Chevrel-Fraux [26] developed a numerical–semi-analytical approach to compute the torsional stiffness of a planetary gear train reducer. Based on the teeth contact model, slicing method, and modified integral potential energy method, Anuradha [27] presented an analytical method for the TVMS of spiral bevel gears. Autiero [28] proposed a novel integrated approach combining the advanced mesh stiffness and partial EHL model, which was applied to evaluate the gear-meshing efficiency. Based on critical tribological parameters, Pei [29] developed a mixed EHL model, and researched the lubrication and friction characteristics under different working conditions. And, then, a mixed lubrication model combining a ball and a plane under pure impact motion was established [30], and the contact characteristics during the impact motion were investigated.

Due to different machining or surface strengthening processes, the tooth surface presents different roughness. As affected by the working environment of the gear, the tooth surface roughness and the oil film thickness were usually stayed on the same order of magnitude or an order of magnitude larger in the gear-meshing process, which made the contact status at the interface of gear meshing quite complicated, thereby affecting the lubrication performance, frictional characteristic, and meshing stability of gear systems. Moreover, the frequent tooth surface roughness contact may cause lubrication failures like stress concentration, micro-pitting, and film breakdown. Hence, it is necessary that we consider the effect of the tooth surface morphology in the analysis of the thermal elastohydrodynamic lubrication of gear. Akbarzadeh [31,32] presented a TEHL analysis for spur gears with the surface roughness using Johnson’s load-sharing concept. Shi [33] developed an integrated model with the analysis of the mixed lubrication state and dynamic-meshing performance for gears, and four typical machined roughness tooth surfaces were investigated. Sun [34] established a novel deterministic mesh stiffness-damping model of rough-surface gear considering the elastic–plastic contact and energy-dissipation mechanism, and investigated the model stability and multi-parameter sensitivity. Based on studying the lubrication characteristics of the smooth tooth surface, Zhou [35] proposed a novel TVMS model considering the asperity contact stiffness and oil film stiffness. Because the rough tooth surface was directly related to gear wear, Wang [36] developed a novel wear model for gears with rough surface contact, which was applied to the prediction of gear wear. Based on the load-sharing concept, Xiao [37] deduced the revised tooth surface contact stiffness combining the tooth surface roughness and oil film, which was investigated with key parameters. Considering the real rough surface topography and the 3D linear contact mixed EHL model, Cheng [38] presented a meshing stiffness calculation method under mixed lubrication. To obtain the gear tooth contact characteristics with different surface micro-topographies, an interface feature model and a tribo-dynamics coupling model of the gear system were proposed [39]. Considering the lubrication state, non-Newtonian shear characteristics, asperity deformation, and contact condition along the meshing path, Pei [40] built a comprehensive TVMS model, which provided an interdependence between the tooth surface contact and gear comprehensive meshing stiffness. Based on the modified contact deformation, lubricant film and EHL contact deformation, and the polynomial chaos expansion surrogate model, Tian [41] proposed a TVMS computing method for planetary gear, and investigated the influences of speed, torque, and rough surface. Gu [42] developed a TVMS calculation model of gears under mixed EHL based on load-sharing, which was investigated with the influence of temperature, speed, and load. Based on the improved gear body stiffness, Jiang [43] established a nonlinear dynamic model for the spur gear system considering random surface roughness. And the differences between the gear system with/without random surface roughness were compared. Huang [44] presented an improved multi-degree-of-freedom model considering the tooth surface roughness for employing a high-contact-ratio gear system, and established an indirect relationship between the surface roughness and dynamic responses. Wen [45] proposed a method for calculating the contact stress of rough tooth surfaces, and the influence of the surface topography on the contact stress of gears was investigated. You [46] presented a 3D rough surface thermoelastic–plastic lubrication-strain gradient analysis model, and investigated the impact of the thermal and micro-scale effects with various parameters.

Through the above literature survey, the rough tooth surface morphology has great influence on the meshing characteristics, which can directly change the dynamic characteristics of the gear. It can be seen that the tooth surface roughness has been researched relatively comprehensively, which is introduced into the gear-meshing stiffness calculation and the gear dynamic analysis under TEHL. Moreover, the existing research works on the TVMS of gears focus on improving the calculation of the gear tooth stiffness. However, the thermal stiffness caused by the thermal effect of the gear teeth is easily ignored, whereas the thermal effect directly affects the lubrication oil performance and tooth surface morphology. Conversely, the lubrication state and tooth surface roughness play a decisive role in tooth surface friction and heat production. Currently, the interaction has not received sufficient attention from researchers, and the systematic investigations on the comprehensive meshing stiffness with consideration of the tooth surface roughness, thermal effect, and lubrication are lacking. Given these limitations, the focus and innovations of the research are as follows: (1) a thermal elastohydrodynamic lubrication mathematical model with a comprehensive consideration of the gear surface morphology, thermal effect, and lubrication is established, which can precisely describe the gear-meshing process from a mathematical perspective; and (2) based on the comprehensive meshing stiffness (CMS) model, the lubrication performance, contact status, and meshing characteristic of gears are comprehensively analyzed under different parameters.

The remainder work of the research is organized as follows: Section 2 characterizes the tooth surface morphology by fractal theory. The TEHL model is deduced in Section 3. Section 4 establishes the comprehensive meshing stiffness considering the tooth surface roughness under TEHL. In Section 5, the lubrication performance, friction and temperature rise, and meshing characteristics of spur gears are investigated and compared with different parameters. Finally, the main conclusions are given.

2. Tooth Surface Morphological Characteristics

Due to the machining and post-processing of tooth surfaces (e.g., milling, grinding, and scraping surfaces) [34], the tooth surface inevitably exhibits a non-smooth feature. The measuring procedure of the tooth surface topography is illustrated in Figure 1(a₁–a₃). Two sample workpieces are selected in the middle position and edge position of the tooth surfaces, which are labeled a and b, and the workpiece surfaces are measured with an area covering 1 mm × 1 mm by a Newview 9000 3D non-contact surface morphometer (ZYGO, Middlefield, CT, USA) (the key parameters include RMS repeatability ≤ 0.008 nm, transverse resolution 0.34 um, spatial sampling 0.04 μm, sweep speed 171 μm/s, step measurement ≤ 0.1%, and linearity ≤ 0.1%). Moreover, the measured surface topographies are depicted in Figure 1(b₂,b₃,c₂,c₃) with an amplitude in the range of Sa ∈ [7.0, 12.0] μm. By comparing Figure 1(a₁,a₃,b₁,b₃), it can be noted that the gear surface is anisotropic (sample workpiece a: Sa = 4.533 μm (arithmetic mean height difference), Sq = 5.191 μm (root mean square height), and Sz = 20.235 μm (profile peak height), and sample workpiece b: Sa = 4.467 μm, Sq = 5.104 μm, and Sz = 19.129 μm), and the surface micro-texture shows certain directionality, which show that the topography has certain fractal characteristics [47]. Hence, the fractal model will be used to characterize the tooth surface topography and evaluate the local contact state instead of the actual tooth surface.

In this research, a modified Weierstrass–Mandelbrot (W_M) fractal function [48] is applied to characterize the 3D fractal surface topography of gears, which has the inherent capability of representing the surface at various length scales, different from those where the measurements were made. The rough surface random contour height W_M(x, y) can be expressed as follows:

$$W_M\left(x,y\right)=\left[\sqrt{{\displaystyle \frac{\ln \gamma }{M}}}\right]{\displaystyle \sum _{m=1}^{M}2\mathsf{\pi }{\left(\frac{2\mathsf{\pi }}{G}\right)}^{2-D}}{\displaystyle \sum _{n=-\infty }^{\infty }\left[1-\exp \left(\frac{2\mathsf{\pi }\sqrt{\left(x^2+y^2\right)}}{L}i{\gamma }^n\cos \left(\theta -\frac{\mathsf{\pi }m}{M}\right)\right)\right]\times \exp \left(i{\phi }_{m,n}\right){\left(\frac{2\mathsf{\pi }}{L}{\gamma }^n\right)}^{D-3}}.$$

(1)

where L represents the sample length, and D indicates the fractal dimension of the rough tooth surface, which is related to the density of asperities on the tooth surface. G is the characteristic scale parameter, which is related to the height amplitude of the asperity. x and y are the measure distance in the x- and y-direction, respectively. Moreover, other symbols can be found in Ref. [49].

To elucidate the effects of the fractal parameters D and G on the surface topography, Figure 2 and Figure 3 show the 3D surface topography with a different D and G [50]. The measurement length and width of the tooth surface are 1 mm, respectively. In Figure 2, the fractal dimension D remains the same (D = 2.4), and setting G varies from 1.02 × 10⁸ to 1.42 × 10⁸. It can be noticed that the variable G has generated remarkable differences in the bump peak of the fractal surface; this is to say, the bump peak of the surface increases obviously with the increasing G, which is consistent with the results in the other literature. Comparing the surface topography in Figure 3, it can be found that fractal dimension D determines the bump peak density and smoothness of the surface. As the D decreases and G increases (namely, as the gear surface roughness R_a increases), the tooth surface morphology displays a thicker shape, is less smooth, and has a higher hump. Therefore, W_M(x, y) can be used to simulate and characterize the real machined tooth surface morphology based on the anisotropic surface hypothesis, which can hardly be described by 2D models.

3. TEHL Model

3.1. TEHL Equations

During the meshing process of the gear, tooth surface contact is affected by the tooth surface morphology, tooth profile, working conditions, lubricating characteristics, temperature rise, and geometric dynamic characteristics. The gear tooth contact region contains both oil film and asperity contact under the thermal elastohydrodynamic lubrication region. The Reynolds equation for Newtonian fluid in the line contact TEHL condition including roughness can be written as follows [51]:

$${\displaystyle \frac{\partial }{\partial x}}\left[{\left({\displaystyle \frac{\rho }{\eta }}\right)}_{\mathrm{e}}h{\left(x,y,t\right)}^3{\displaystyle \frac{\partial p\left(x,y,t\right)}{\partial x}}\right]+{\displaystyle \frac{\partial }{\partial y}}\left[{\left({\displaystyle \frac{\rho }{\eta }}\right)}_{\mathrm{e}}h{\left(x,y,t\right)}^3{\displaystyle \frac{\partial p\left(x,y,t\right)}{\partial y}}\right]=12u_{\mathrm{r}}\left(t\right){\displaystyle \frac{\partial \left({\rho }^{*}h\left(x,y,t\right)\right)}{\partial x}}+12{\displaystyle \frac{\partial \left({\rho }_{\mathrm{e}}h\left(x,y,t\right)\right)}{\partial t}}.$$

(2)

in which u_r(t) is the entrainment velocity, and p, h, ρ, and η stand for the oil film pressure, oil film thickness, density, and viscosity of the lubricating oil, respectively. The boundary conditions of the Reynolds equation can be expressed as follows:

$$\begin{array}{l}p\left(x_{\mathrm{in}},y,t\right)=p\left(x_{\mathrm{out}},y,t\right)=p\left(x,y_{\mathrm{in}},t\right)=p\left(x,y_{\mathrm{out}},t\right)=0,\\ p\left(x,y,t\right)\ge 0, \left(x_{\mathrm{in}}<x<x_{\mathrm{out}}, y_{\mathrm{in}}<y<y_{\mathrm{out}} \right).\end{array}$$

(3)

The oil film thickness in TEHL consists of surface elastic deformation and surface roughness, which can be expressed as follows:

$$h\left(x,y,t\right)=h_0\left(t\right)+{\displaystyle \frac{x^2}{2R_{\mathrm{e}}\left(t\right)}}+{\displaystyle \frac{2}{\mathsf{\pi }{E}_{\mathrm{e}}}}{\displaystyle \underset{\mathrm{\Omega }}{\iint }\frac{p\left(\xi ,\varsigma ,t\right)}{\sqrt{{\left(x-\xi \right)}^2+{\left(y-\varsigma \right)}^2}}\mathrm{d}\xi \mathrm{d}\varsigma }+w_{\mathrm{m}}\left(x,y,t\right).$$

(4)

where E_e denotes the equivalent elastic modulus, Ω is the contact area, and $\xi \mathrm{and} \varsigma$ represent the pressures with calculating the deformation in the x and y coordinates, respectively. w_m(x, y, t) is the gear surface morphology (assume that the morphology of the driving and driven gears is identical).

Considering the thermal effect in the contact zone, the viscosity–pressure–temperature equation proposed by Roelands [52] and the density–pressure–temperature equation deduced by Dowson-Higginson [53] are governed by the following:

$$\rho \left(x,y,t\right)={\rho }_0\left\{\left.1+{\displaystyle \frac{0.6\times {10}^{-9}p\left(x,y,t\right)}{1+1.7\times {10}^{-9}p\left(x,y,t\right)}}-6.5\times {10}^{-3}\left[T\left(x,y,t\right)-T_0\right]\right\}\right..$$

(5)

$$\eta \left(x,y,t\right)={\eta }_0\exp \left\{\left.\left(\ln {\eta }_0+9.67\right)\left[{\left(1+5.1\times {10}^{9}p\left(x,y,t\right)\right)}^{z_0}\times {\left({\displaystyle \frac{T\left(x,y,t\right)-138}{T_0-138}}\right)}^{-s_0}-1\right]\right\}\right..$$

(6)

During the gear-meshing process, the pressure p(x, y, t) should meet the load equation

$${\displaystyle {\int }_{y_{\mathrm{in}}}^{y_{\mathrm{out}}}{\displaystyle {\int }_{x_{\mathrm{in}}}^{x_{\mathrm{out}}}p\left(x,y,t\right)\mathrm{d}x}}\mathrm{d}y=w\left(t\right).$$

(7)

where w(t) represents concentrated external loads.

Due to the shear and compression effect between the oil film layers, the gear teeth contact zone generates a certain amount of heat, which causes the oil film temperature to change. In order to investigate the thermal effect in the contact zone, the fluid energy equation can be written as follows:

$$c_{\mathrm{f}}\left[\rho {\displaystyle \frac{\partial T}{\partial t}}+\rho u_x{\displaystyle \frac{\partial T}{\partial x}}-\left({\displaystyle \frac{\partial }{\partial t}}{\displaystyle {\int }_0^z\rho \mathrm{d}{z}^{\prime }}+{\displaystyle \frac{\partial }{\partial x}}{\displaystyle {\int }_0^z\rho u_x\mathrm{d}{z}^{\prime }}\right){\displaystyle \frac{\partial T}{\partial z}}\right]-k_{\mathrm{f}}{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}=-{\displaystyle \frac{T}{\rho }}{\displaystyle \frac{\partial \rho }{\partial T}}\left(u_x{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial p}{\partial t}}\right)+{\eta }^{*}{\left({\displaystyle \frac{\partial u_x}{\partial z}}\right)}^2.$$

(8)

where k_f and c_f represent the thermal coefficient and specific heat capacity of the lubricating oil, T is the temperature of the lubricating oil, and other parameters’ meanings can be found in Ref [6]. Considering the convective heat transfer in the x and y directions, the solid energy equation can be written as follows [9]:

$$c_1{\rho }_1\left({\displaystyle \frac{\partial T}{\partial t}}+u_1{\displaystyle \frac{\partial T}{\partial x}}\right)=k_1{\displaystyle \frac{{\partial }^{2}T}{\partial z_1^2}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{c}_2{\rho }_2\left({\displaystyle \frac{\partial T}{\partial t}}+u_2{\displaystyle \frac{\partial T}{\partial x}}\right)=k_2{\displaystyle \frac{{\partial }^{2}T}{\partial z_2^2}}.$$

(9)

in which c_i, k_i, and ρ_i (i = 1, 2) are the specific heat capacity, thermal coefficient, and density of solids, respectively. Moreover, the boundary condition can be described as follows:

$$\begin{array}{l}T\left(x_{\mathrm{in}},y,z,t\right)=T_0, z\in \left[0,h\right],\\ T\left(x_{\mathrm{in}},y,z_1,t\right)=T_0 z_1\in \left[-d,0\right],\\ T\left(x_{\mathrm{in}},y,z_2,t\right)=T_0 z_2\in \left[0,d\right],\\ T\left(x,y,-d,t\right)=T\left(x,y,d,t\right)=T_0 x\in \left[x_{\mathrm{in}},x_{\mathrm{out}}\right].\end{array}$$

(10)

here d represents the calculated depth of the solid.

Assume that the temperature at the interface is continuous between the oil film and the solid, the heat flow on the oil–solid interface should be satisfied:

$${\left.k_{\mathrm{f}}{\displaystyle \frac{\partial T}{\partial z}}\right|}_{z=0}={\left.k_1{\displaystyle \frac{\partial T}{\partial z_1}}\right|}_{z_1=0},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\left.k_{\mathrm{f}}{\displaystyle \frac{\partial T}{\partial z}}\right|}_{z=h}={\left.k_2{\displaystyle \frac{\partial T}{\partial z_2}}\right|}_{z_2=0}.$$

(11)

In the following section, combined with multi-grid technology, the Gauss–Seidel low-relaxation iteration method and the chase method are applied to solve the Reynolds equation and energy equation, respectively.

3.2. Solution of Combined Friction Coefficient

According to the thermal elastohydrodynamic lubrication of the rough tooth surface, the friction between the teeth is mainly composed of hydrodynamic friction and asperity friction [19,20]. Combining the film thickness and viscosity of the lubricating oil, the hydrodynamic friction force can be written as follows:

$$F_{\mathrm{fo}}=2b_{\mathrm{H}}{B}_{\mathrm{H}}\eta {\displaystyle \frac{u_{\mathrm{s}}}{h}}.$$

(12)

where b_H is the half-width of Hertzian contact, B_H denotes the radius of curvature of the asperities, and u_s represents the sliding velocity.

Assume that all the asperities have the same friction characteristic, and the asperity friction force can be expressed as follows:

$$F_{\mathrm{fr}}=f_{\mathrm{r}}{\displaystyle \sum _{i=1}^N{p}_{\mathrm{a}}}{\mathrm{dA}}_{\mathrm{a}}=f_{\mathrm{r}}\cdot F_{\mathrm{r}}.$$

(13)

in which f_r is the friction coefficient between a single gear of asperities, p_a denotes the asperity pressure, and F_r represents the average positive pressure between asperities.

The combined friction coefficient can be written as follows [19,20]:

$$f={\displaystyle \frac{F_{\mathrm{fo}}+F_{\mathrm{fr}}}{F_{\mathrm{T}}}}.$$

(14)

here F_T is the transmitted force, which can be obtain in Ref. [19].

4. The Comprehensive Mesh Stiffness Modelling

4.1. Time-Varying Mesh Stiffness Model

Due to the tooth surface producing roughness by machining, the roughness and oil film in the contact zone exist simultaneously under THEL, and the contact stiffness is composed of the oil film stiffness and asperity stiffness. Hence, the comprehensive meshing stiffness k_cm of spur gears consists of the fillet-foundation stiffness k_f, tooth bending stiffness k_b, shear stiffness k_s, axial compressive stiffness k_a, and contact stiffness k_c (asperity stiffness k_r and oil film stiffness k_o). As shown in Figure 4, the equivalent contact model with the rough tooth surface and comprehensive meshing stiffness composition of spur gears is illustrated, which is written as follows:

$${\displaystyle \frac{1}{k_{\mathrm{cm}}}}={\displaystyle \frac{1}{k_{\mathrm{f}}}}+{\displaystyle \frac{1}{k_{\mathrm{a}}}}+{\displaystyle \frac{1}{k_{\mathrm{b}}}}+{\displaystyle \frac{1}{k_{\mathrm{s}}}}+{\displaystyle \frac{1}{k_{\mathrm{c}}}}.$$

(15)

in which the k_f, k_b, k_s, and k_a can be expressed as follows [42,54,55,56]:

$$k_{\mathrm{f}}={\displaystyle \frac{{\cos }^2\beta }{EL}}\left[L^{*}{\left({\displaystyle \frac{u_{\mathrm{f}}}{S_{\mathrm{f}}}}\right)}^2+M^{*}\left({\displaystyle \frac{u_{\mathrm{f}}}{S_{\mathrm{f}}}}\right)+P^{*}\left(1+Q^{*}{\tan }^2\beta \right)\right].$$

(16)

$${\displaystyle \frac{1}{k_{\mathrm{b}}}}={\displaystyle \frac{2U_{\mathrm{b}}}{F^2}}={\displaystyle {\int }_{\frac{\mathsf{\pi }}{2}}^{{\alpha }_0}\frac{{\left[\cos \beta \left(y_{\beta }-y_1\right)-x_{\beta }\sin \beta \right]}^2}{E{I}_{y_1}}\frac{\mathrm{d}{y}_1}{\mathrm{d}\gamma }}\mathrm{d}\gamma +{\displaystyle {\int }_{{\tau }_{\mathrm{c}}}^{\beta }\frac{{\left[\cos \beta \left(y_{\beta }-y_2\right)-x_{\beta }\sin \beta \right]}^2}{E{I}_{y_2}}\frac{\mathrm{d}{y}_2}{\mathrm{d}\tau }\mathrm{d}\tau }.$$

(17)

$${\displaystyle \frac{1}{k_{\mathrm{s}}}}={\displaystyle \frac{2U_{\mathrm{s}}}{F^2}}={\displaystyle {\int }_{\frac{\mathsf{\pi }}{2}}^{{\alpha }_0}\frac{1{.2\cos }^2\beta }{G{A}_{y_1}}\frac{\mathrm{d}{y}_1}{\mathrm{d}\gamma }}\mathrm{d}\gamma +{\displaystyle {\int }_{{\tau }_{\mathrm{c}}}^{\beta }\frac{1{.2\cos }^2\beta }{G{A}_{y_2}}\frac{\mathrm{d}{y}_2}{\mathrm{d}\tau }}\mathrm{d}\tau .$$

(18)

$${\displaystyle \frac{1}{k_{\mathrm{a}}}}={\displaystyle \frac{2U_{\mathrm{a}}}{F^2}}={\displaystyle {\int }_{\frac{\mathsf{\pi }}{2}}^{{\alpha }_0}\frac{1{.2\sin }^2\beta }{E{A}_{y_1}}\frac{\mathrm{d}{y}_1}{\mathrm{d}\gamma }}\mathrm{d}\gamma +{\displaystyle {\int }_{{\tau }_{\mathrm{c}}}^{\beta }\frac{1{.2\sin }^2\beta }{E{A}_{y_2}}\frac{\mathrm{d}{y}_2}{\mathrm{d}\tau }}\mathrm{d}\tau .$$

(19)

4.2. Contact Stiffness Model

Due to the different machining process of the tooth surface, the tooth surface micro-protrusions exceed the thickness of the oil film, which is considered to be in asperity peak contact. At this time, the load in the tooth contact zone is shared by the oil film and the asperity peak according to Johnson’s load-sharing concept [57] in mixed TEHL. Hence, the Hertzian contact stiffness is replaced by the rough contact stiffness and oil film stiffness, which is as follows:

$$k_{\mathrm{c}}=k_{\mathrm{r}}+k_{\mathrm{o}}.$$

(20)

According to Ref. [38], the total rough contact stiffness can be written as follows:

$$k_{\mathrm{r}}=2n{A}_{\mathrm{n}}E\sqrt{\beta }{\displaystyle \underset{h-l}{\overset{\infty }{\int }}\left[\sqrt{\left(z+l-h\right)}\cdot \frac{1}{\sqrt{2\mathsf{\pi }\sigma }}\cdot \exp \left(\frac{-z^2}{2{\sigma }^2}\right)\right]\mathrm{d}z}.$$

(21)

where A_n, n, E, β, and σ are the nominal contact area, asperity density, equivalent elastic modulus, equivalent curvature radius of asperity and standard deviation of the surface roughness, respectively. z/l are the distances between the asperity height/average height line of the rough surface and the mean height line of summits. In addition, other symbolic meanings can be found in Ref. [38].

The oil film stiffness is determined by the average pressure of the oil film thickness, which is given as follows [35,38]:

$$k_{\mathrm{o}}={\displaystyle \frac{B}{h}}{A}_{\mathrm{n}}\cdot \left(1-\gamma \right).$$

(22)

in which γ is the proportion of the asperities contact area in the entire domain, and B indicates the bulk modulus of the lubricant, which can be expressed as follows:

$$B=\left\{1-{\displaystyle \frac{1}{1+B_0^{\prime }}}\ln \left[1+{\displaystyle \frac{p}{B_0}}\left(1+B_0^{\prime }\right)\right]\right\}\cdot \left[B_0+p\left(1+B_0^{\prime }\right)\right].$$

(23)

where B₀ is the bulk modulus at ambient pressure. Moreover, the meanings and derivation process of the other parameters can be found in Refs. [35,38].

4.3. Thermal Stiffness Model

The gear teeth experiences thermal deformation in the meshing process as a result of the friction and temperature rise (due to the relatively thin oil film thickness, the central temperature rise is applied to replace the tooth surface temperature rise). Therefore, the thermal stiffness has an important effect on the stiffness characteristics of gears. To facilitate the calculation of tooth thermal deformation, it is assumed that the intermediate oil film temperature rise is approximately equal to the tooth surface temperature rise. Hence, the tooth profile deformation caused by tooth surface temperature rise σ_i (i = 1, 2) can be written as follows [58]:

$${\sigma }_i=-{\displaystyle \frac{T_{\mathrm{t}}\xi r_{\mathrm{b}i}(r_{\mathrm{b}i}+{\delta }_{\mathrm{b}i})}{2\left[r_{\mathrm{b}i}+{\delta }_{\mathrm{b}i}\cos {\alpha }_{\mathrm{a}i}+T_{\mathrm{t}}\xi r_{\mathrm{b}i}(1-\cos {\alpha }_{\mathrm{a}i})\right]}}\cdot \left[{\displaystyle \frac{L_i}{r_{\mathrm{b}i}}}-2\left(\mathrm{inv}{\alpha }_{\mathrm{a}i}-\mathrm{inv}\alpha \right)\right].$$

(24)

in which L_i (i = 1, 2) is the tooth thickness, α_ai represents the tip pressure angle, δ_bi is the heat deformation of the base circle with a steady state, ξ is the coefficient of linear expansion, and T_t denotes the tooth surface temperature difference, which can be written as follows:

$$\begin{array}{l}{T}_{\mathrm{t}}=T_{\mathrm{M}}+\mathrm{\Delta }T-T_0,\\ L_i={\displaystyle \frac{\mathsf{\pi }m}{2}}\cdot {\displaystyle \frac{r_{\mathrm{c}i}}{r_{\mathrm{p}i}}}-2r_{\mathrm{c}i}\left(\mathrm{inv}{\alpha }_{\mathrm{c}i}-\mathrm{inv}\alpha \right),\\ {\delta }_i=\lambda r_{\mathrm{b}i}{T}_{\mathrm{ro}i}+{\displaystyle \frac{\lambda \left(1+\mu \right)}{\left(1-\mu \right)}}\cdot {\displaystyle \frac{r_{\mathrm{b}i}\left[r_{\mathrm{b}i}^2(1-2\mu )-r_{\mathrm{o}i}^2\right]}{(r_{\mathrm{b}i}^2-r_{\mathrm{o}i}^2)}}\cdot \left(T_{\mathrm{rb}i}-T_{\mathrm{ro}i}\right).\end{array}$$

(25)

where r_ci (i = 1, 2) represents the distance from the meshing point to the center of the driving/driven gear, and T_roi and T_rbi are the temperatures of the driving/driven gear shaft and base cylinder, respectively.

The single tooth thermal stiffness k_Ti (i = 1, 2) and meshing thermal stiffness k_T caused by the tooth contact temperature are defined as follows:

$$k_{\mathrm{T}i}={\displaystyle \frac{F_{\mathrm{n}}}{{\sigma }_i}}, k_{\mathrm{T}}={\displaystyle \frac{k_{\mathrm{T}1}{k}_{\mathrm{T}2}}{k_{\mathrm{T}1}+k_{\mathrm{T}2}}}(i= 1, 2).$$

(26)

Based on the above analysis, Figure 5 shows a detailed flow chart of the process for solving the comprehensive meshing stiffness under TEHL.

5. Lubrication Performance and Meshing Characteristics Analysis

In this section, the influences of the torque, rotational speed, and fractal parameters on the contact properties and meshing characteristics with a rough tooth surface are investigated. The solution domain range is −4.0 < x < 1.5, the number of grids is 130, and the time step size is set 0.001. Moreover, the main parameters of the gear system are listed in

Table 1. The main parameters of the gear system.

Parameters	Value	Parameters	Value
Number of teeth z1, z2	24, 16	Density of gear ρ1, ρ2 (kg/m3)	7850
Module m (mm)	5.0	Viscosity of lubricant η (Pa·s)	0.03
Tooth width BH (mm)	10	Density of lubricant ρ (kg/m3)	860
Pressure angle α (°)	20	Pressure–viscosity coefficient (Pa−1)	2.0 × 10−8
Elasticity modulus E1, E2 (GPa)	210	Coefficient of heat conduction h1, h2 (J/ms°C)	46.47
Rotational speed n1 (r/min)	2000	Initial tooth surface temperature T0 (°C)	20
Torque T1 (N·m)	100	Poisson ratio ν1, ν2	0.3

.

5.1. Model Validation

In order to verify the influence of the tooth surface morphology on the gear lubrication properties, Figure 6 displays a comparison of the oil film pressure, film thickness, and central temperature rise between the smooth and rough tooth surfaces. It is noted that all figures exhibit obvious fluctuations along the line of action (LOA) due to the squeeze effect in single-to-double meshing points. Moreover, the maximum oil film pressure, temperature rise, and minimum oil film thickness occur in the single-teeth meshing zone, which is closely related to the load distribution on the tooth surface. Comparing Figure 6, it can be found that the rough and smooth lubrication models have big differences. For the rough tooth surface, the oil film pressure, film thickness, and temperature rise show obvious fluctuations, which are significantly affected by the elastic deformation of the local asperity and the random distribution, and the peak values of the pressure and temperature rise are obviously greater by 29.82% and 24.7% compared to that of a smooth tooth surface, while the film thickness displays the opposite trend (MP stands for meshing point). Figure 6g and h display the corresponding comparison results of a position. The distributions appear to show a clear 3D randomness, which can hardly be approximated by the simplified 2D lubrication model. Therefore, it is evident that the tooth surface morphology directly affects the critical behaviors of gears such as the contact stress, loading capacity, friction, meshing impact, etc., and the influence of the 3D tooth surface morphology must be considered during the meshing process of analyzing the gear contact state and comprehensive stiffness.

5.2. The Effect of Torque

Torque T₁, as an external excitation, has a significant effect on the lubrication state and meshing characteristic of gear contact. It is well-known that the increasing T₁ can increase the oil film pressure, actual contact area, and friction, and decrease the oil film thickness within a certain range, which further affect the gear tooth asperity contact and comprehensive meshing stiffness. Hence, the contact performance and stiffness characteristic of gears will be studied with T₁ = 20 N·m, 100 N·m, and 300 N·m in this section. Figure 7 shows the evolution of the oil film pressure and film thickness, friction and temperature rise, contact stiffness, and comprehensive meshing stiffness under different torques, which reveals notable differences in the results with a different T₁. At first glance, it is seen that the oil film pressure p and temperature rise ΔT exhibit a sudden increase, and film thickness h, contact stiffness k_c and comprehensive meshing stiffness k_cm present a decrease suddenly from double-to-single teeth meshing, and a reverse trend is observed from single-to-double teeth meshing along the line of action of the gears.

In the same position, due to the randomness of the asperity peak and the larger asperity peak deformation with a larger load, the fluctuation of the pressure in Figure 7(a₁–a₃) is becoming more and more obvious, and the second pressure-peak at the outlet of the contact zone increases significantly with the increasing T₁. In Figure 7(b₁–b₃), the oil film thickness shows the opposite trend compared to the oil film pressure, but the necking phenomenon of the oil film at the outlet becomes more and more obvious. The friction coefficient is closely related to the input torque, lubricant, and tooth surface morphology, which directly affects the thermal deformation and temperature rise. In the double-tooth meshing zone, the friction coefficient gradually decreases as the contact point moves toward the pitch point, and a sudden increase appears from double-to-single teeth meshing. In the single-tooth meshing zone, the friction coefficient presents a “V”-shaped structure with a low left and high right, and reaches a minimum value instead of being 0, which is caused by the local asperity contact. When the gear teeth enter the double-tooth meshing zone again, the friction coefficient suddenly decreases, and then gradually increases with the movement of the contact point. As T₁ increases from 20 N·m to 300 N·m, the h becomes thinner gradually, and the asperity peak contact probability and contact area increase, which lead to the friction coefficient increase. Meanwhile, the thermal elastohydrodynamic lubrication of the gear tooth contact state evolves to the mixed thermal elastohydrodynamic lubrication, and the meshing impact is weakened. It can be seen from Figure 7(d₁–d₃) that the ΔT is significantly affected by the asperity peak distribution, the temperature rise in the single-tooth meshing zone is obviously greater than that in the double-tooth meshing zone, and the maximum ΔT and the secondary peak temperature rise increase with the increasing torque. It can be found from Figure 7(e₁–e₃,f₁–f₃) that the k_c and k_cm increase significantly and the fluctuation reduces under different T₁, which is due to the larger contact load increasing the contact of the tooth surface asperity peak.

5.3. The Effect of Rotational Speed

Generally, the rotational speed n₁ directly affects the entrainment speed between the meshing teeth and further influences the oil film pressure and film thickness in the contact zone. Moreover, the rough contact, friction, and temperature rise are also significantly affected, which will change the lubrication performance and the stiffness characteristic of the gear. Therefore, Figure 8 demonstrates the variation in oil film pressure, thickness, friction coefficient, temperature rise, contact stiffness and comprehensive meshing stiffness under three rotational speeds (namely, n₁ = 500 r/min, 2000 r/min, and 3000 r/min), respectively. At a low rotational speed, the entrainment velocity between two tooth surfaces is small, which leads to a thinner oil film thickness and lower temperature rise, thereby further increasing the fluid viscosity and asperity peak contact probability. The lubrication state is poor, the oil film loading capacity is decreased, the local pressure and asperity peak contact zone are increased, and then the fluctuations of the contact state and meshing characteristic are exacerbated, which indicate that the gear teeth are in a mixed thermal elastohydrodynamic lubrication. Moreover, the maximum oil film pressure and the minimum film thickness occur at the local asperity peak, which may further lead to the stress concentration and film breakdown, and even lubrication failure. By now, the hydrodynamic friction is little but the asperity friction is large, and the decreased hydrodynamic friction dominates the effect of the increased asperity friction, which causes the friction coefficient to drop. Because the gear teeth are in the mixed TEHL state, it can be seen from Figure 8(e₁–e₃,f₁–f₃) that the contact stiffness and comprehensive stiffness are large with a thin oil film thickness. As n₁ increases, the film fluidity, film thickness, and loading capacity between the meshing teeth are increased, but the asperity peak contact probability and local oil film pressure/fluctuation are reduced. Due to the stable oil film between the teeth, the difference in friction coefficient is small, but the increasing hydrodynamic friction directly causes the increase in temperature rise and the secondary temperature rise peak changes significantly within a certain range. For k_c and k_cm, the thinner the film thickness is, the smaller the contact stiffness and comprehensive stiffness will be, indicating the n₁ has an obvious effect on the stiffness characteristic. Due to the randomness of the asperity peak contact, the analysis results exhibit a nonlinear increase with the increasing n₁, but the rate of increase decreases. Moreover, the fluctuation decreases in the alternating process of the single-double teeth, which helps to resist the meshing impact.

5.4. The Effect of Fractal Parameters

The varietal tooth surface morphology has a complex effect on the gear tooth contact performance. In engineering practice, the asperity peak density and amplitude are still the most intuitively used parameters to characterize the three-dimensional rough tooth surface morphology, which correspond to the fractal parameters D and G. Compared with the smooth tooth surface, the rough tooth surface will increase the friction between the teeth and reduce the efficiency of the gear transmission. Hence, in order to better analyze the influence of the tooth surface topography, the research focuses on the influence of parameters D and G on the contact performance and meshing characteristics. In this section, by setting D = 2.2, 2.6, 2.8 (G = 1.25 × 10⁸), and G = 1.02 × 10⁸, 1.30 × 10⁸, and 1.42 × 10⁸ (D = 2.4), the variation in oil film pressure, thickness, friction coefficient, temperature rise, contact stiffness, and comprehensive meshing stiffness under different fractal parameters are obtained and displayed in Figure 9 and Figure 10.

To evaluate the sensitivity of the asperity peak density, Figure 9 shows the distribution of the lubrication, friction, temperature rise, and stiffness with the different D, and the results show that the changes are closely related to the fractal dimension D. It can be noted from Figure 9(a₁–a₃) that, the denser tooth surface roughness peak is, the smaller the pressure fluctuation range is, and the larger the maximum pressure value is, but the smaller mean pressure value is; the uniform roughness peak distribution leads to a reduction in deformation. For the oil film thickness, due to the large contact ratio of the roughness peak, the distribution density in the contact zone somewhat increases, which causes the local film thickness to become thinner. Increasing D leads to more asperity peak contact on the tooth surface; the asperity friction will become the dominant part and the local friction coefficient increases nonlinearly but the fluctuation is remarkably reduced. Moreover, the overall change trend of the friction coefficient is almost constant. Compared to Figure 9(d₁–d₃), the fractal dimension D increases the maximum peak of the temperature rise significantly, but the fluctuation of the temperature rise decreases, which are mainly caused by the dense roughness peak distribution and the increasing contact probability. Figure 9(e₁–e₃,f₁–f₃) display the gear contact stiffness and comprehensive meshing stiffness, it can be noted that the increasing D obviously increases the k_c and k_cm. In the case of denser roughness, the asperity peak contact probability increases, and the asperity contact stiffness is larger than the film stiffness and plays a leading role. Moreover, the stiffness fluctuation decreases but randomness increases.

Figure 10 presents the contact performance and meshing characteristic distribution of the different characteristic scale parameter G. It can be observed from Figure 10(a₁–a₃,b₁–b₃) that, the greater tooth surface asperity peak is, the greater the pressure and film thickness fluctuation are, and the thinner the local film thickness in the contact zone is, but the lower the average pressure is. This is mainly because the increasing asperity peak reduces the contact area of the rough tooth surface, and the loading capacity of the tooth surface is weakened. Comparing Figure 10(c₁–c₃,d₁–d₃), it can be found that G has a remarkable influence on the tooth surface friction and temperature rise. With the increase in G, the asperity peak deformation increases and the asperity friction force becomes the dominant part; a slight increase in and violent fluctuation of the friction coefficient are observed. In Figure 10(d₁–d₃), the scale parameter G effect increases the maximum value and fluctuation of the temperature rise, but the decrease in the asperity peak amplifies the G effect on the average temperature rise and secondary temperature rise peak, which indicate the asperity peak has a significant thermal effect during lubrication. It can be found from Figure 10(e₁–e₃) that the contact stiffness is very close under the different G, but the fluctuation is intensified markedly, indicating the G has little effect on the contact stiffness. The influence of G on the k_cm is significant. Specifically, the larger the G is, the smaller the comprehensive meshing stiffness is, but the stronger the fluctuation will be.

In order to more intuitively exhibit the performances of the meshing tooth under the different T₁, n₁, D, and G, the variation rules are listed in

Table 2. The effect of different parameters on the variation rules.

Parameters	p	h	f	ΔT	kc	kcm	Fluctuation
T1 ↑	↑	↓	↑	↑	↑	↑	↑
n1 ↑	Local ↓	Local ↑	Local ↓	↑	↓	↓	↓
D ↑	Local ↓	Local ↓	↑	↑	↑	↑	↓
G ↑	Local ↓	Local ↑	Local ↑	Local ↑	↓	↓	↑

.

6. Conclusions

In this research, an improved comprehensive meshing stiffness model of the spur gear is developed, aiming at providing an interdependence between the rough tooth surface contact characteristics and gear comprehensive meshing stiffness, in which the tooth surface morphology, lubrication property, and thermal effect are considered; it can fully reflect the effects of the rough surface and lubrication, and accurately predict the lubrication performance and meshing characteristic of the gear. The analysis and verification of the machined surface morphology and the comparison with smooth and rough tooth surfaces are conducted. The influences of the torque, rotational speed, and fractal parameters on the contact performance and stiffness characteristics are investigated and discussed. The main research findings are drawn as follows:

(1)

The fractal characteristic of the tooth surface morphology is verified by experiment, which is described using the W_M fractal function. Compared to the lubrication performance of the smooth and rough tooth surfaces, the maximum values of the pressure and temperature rise are obviously greater by 29.82% and 24.7% compared to that of the smooth tooth surface, which aligns more closely with actual conditions.

(2)

Increasing T₁ can increase the oil film pressure and friction coefficient, and reduce the film thickness. Moreover, the rough peak contact probability and fluctuation increase significantly, which indicates the lubrication state becomes poorer.

(3)

The higher the n₁ is, the smaller the local pressure, friction coefficient, and stiffness will be, while, the thicker the film thickness is, the smaller the meshing stiffness will be, which helps to reduce the disturbance caused by roughness.

(4)

The fractal parameters have significant effects on the meshing characteristics. With a decreasing D and increasing G, the pressure and film thickness, friction, and temperature rise exhibit a nonlinear variation, but the contact stiffness and comprehensive meshing stiffness have clearly diminished, which lead to a poor lubrication state of the gear. The research work can provide certain guidance for the design and processing of gears.

In order to further improve the accuracy of the contact characteristics and meshing properties of the gear teeth, future research efforts will take into account the tooth surface temperature rise and the sensitivity of the tooth surface morphology parameters, and establish a more precise gear-meshing model, thereby providing technical support for the gear processing and dynamic analysis.