## Abstract

In the process of gear meshing, it is an inevitable trend to encounter failure cases such as contact friction thermal behavior and interface thermoelastic scuffing wear. As one of the cores influencing factors, the gear meshing contact interface micro-texture (CIMT) morphology significantly restricts the gear transmission system (GTS) dynamic characteristics. This subject intends to the contact characteristic model and interface friction dynamics coupling model of meshing gear pair with different CIMT morphologies. Considering the influence of gear meshing CIMT on the distribution type of hydrodynamic lubricating oil film, contact viscous damping, and frictional thermal load, the aforementioned models have involved time-varying meshing stiffness and static transmission error. Based on the proposed models, an example verification of meshed gear pair (MGP) is analyzed to reveal the influence of CIMT on the dynamic characteristics of GTS under a variety of micro-texture configurations and input branch power and rated speed/shaft torque conditions. Numerical simulation results indicate that the influence of CIMT on gear dynamic response (including meshing interface frictional thermal load, malicious damping, and impact vibration in the off-line direction of the action) is extremely restricted by the transient contact regularity of the meshing gear surface. Meshing gears dynamic characteristics (especially vibration and noise) can be obviously and effectively adjusted by setting a regular MGP with CIMT morphology instead of random gear surfaces.

## 1 Introduction

Involute cylindrical gears are often used in power rear transmission systems to satisfy the requirements for precise delivery index of motion, while having high power density and low friction loss. These cylindrical gears are often subjected to impact-type loads on them, which can naturally occur periodic oscillation due to the instantaneous meshed gear pair (MGP). The gear transmission systems (GTSs) have extensively focused on many cores’ industrial categories such as naval architecture, aerospace and ocean engineering, and wind power generation (roadbed and seabed).

The MGP contact interface with low speed and light load may have a thicker dynamic pressure oil film, and the meshing friction coefficient is small. The oil film pressure and temperature of the MGP interface with heavy-duty are relatively high, and the viscosity–temperature effect of the lubricating oil rapidly reduces the thickness of the interface oil film, which further leads to pitting corrosion and even thermal scuffing or wear on the meshing interface. The friction characteristics of the bearing seat of the recessed bushing manufactured by the processing technology and chemical etching technology are studied [1]. The load, oil varieties, groove size, depth, and shape are analyzed through experiments to examine the lubricating effect of the pit texture to explore their influence on the friction characteristics. The research results explained that the friction performance of the journal bearing is improved by designing the size of the pit texture reasonably. The sleeve has better friction properties. Bushings with etched pits on the entire circumference have better friction performance than bushings with grooves engraved on half of the circumference. Especially in hybrid thermal elastohydrodynamic lubrication (TEHL) systems, the secondary lubrication effect produced by pits is the main mechanism to improve the performance.

Based on this, how to improve the dynamic pressure lubrication effect of the meshing tooth surface (MTS) by changing the MGP with contact interface micro-texture (CIMT) morphology has become one of the core directions in the research field of gears [2]. With the gradual development of laser technology, the micro-texture processed by laser can significantly improve the TEHL performance of the contact surface [3]. After laser treatment, the textured MTS is customized with micron-level dynamic pressure lubrication structure grooves or pits that can store lubricating oil, which can significantly improve the compressive resistance of the MTS, thereby increasing the anti-scuffing load-carrying capacity of the MGP and providing pre-research solutions for the performance and response to extreme working conditions.

At present, the research of interface micro-texture mainly focuses on low-pair contact, focusing on the geometric shape design of micro-texture and the selection and setting of its size parameters. Researchers mostly recognize these surface micro-texture design shapes, such as circular pits, grooves, and convex hulls [4]. The theory and experimental feasibility of micro-pits of laser surface texturing technology is studied [5]. A regular surface texture with a micro-pit shape is formed on the contact interface of the friction machine, and bearing capacity, wear resistance, friction coefficient, and other aspects of the friction machine components have been significantly improved [6]. The research results show that in the case of full lubrication or hybrid TEHL, each micro-pit can be used as a miniature hydrodynamic lubrication bearing or as a miniature storage device for lubricant. The circular pits distributed at the intervals are processed by laser on the surface of the piston ring, and the influence of the diameter, spacing, and distribution position of the pits on the friction coefficient is analyzed, and the relevant influence parameters are optimized and analyzed, based on the joint solution of the Reynolds equation and the radial motion equation and then reveal the fluid dynamic pressure distribution law between the cylinder inner liner and piston ring [7]. With the aid of the material ablation process and the use of pulsating laser beams to create thousands of micro-pits arranged vertically and horizontally on the metal surface, tribological experiments are carried out in the range of 0.015–0.75 m/s through the disc needle device. The test results show that the laser-textured surface under the action of hydrodynamic lubrication can greatly reduce the friction coefficient, and the lower pit density is more conducive to the lubrication of the metal surface. In the case of higher moving speed, load, and higher viscosity oil, the effect of micro-texture on the metal surface is more obvious [8]. Based on a numerical simulation method of the virtual texture model, the influence of the bottom shape of the texture and the relative movement of the surface on the generated texture is studied. These textures are located on a triangularly distributed interaction surface and have the same density. The simulation results show that there is a thicker lubricating oil film related to the bottom configuration of the micro-element wedge and the micro-step hydrodynamic bearing [9]. The regularly distributed circular micro-textures are processed by laser on the stainless-steel surface, and the friction-reducing mechanism of micro-textures is studied with the numerical simulation method, which in turn reveals the law of oil film pressure change in the range of circular pit diameter from 20.0 μm to1000 μm, pit depth within 1.0–100 μm, and pit density within the range of 3–62% [10]. The analysis results show that the depth of the micro-texture is determined by the load. As the load increases, the depth of the micro-texture pits also increases accordingly to ensure a small contact interface friction force. The friction force of the micro-pit surface based on hydrodynamic lubrication is reduced by nearly 80% compared with the untextured surface.

Although the above-mentioned scholars experimentally verified the dynamic pressure lubrication effect of the micro-texture of the contact interface processed by laser and obtained the specific optimal micro-texture pit size, the basis for the selection of the micro-texture geometry and the optimization theory are not described in detail. The optimization form is mostly enumerated by experiments, facing many micro-textured geometric figures and their distribution patterns, which will make it difficult to obtain the most effective dynamic pressure lubrication prediction scheme. Based on this, some scholars have tried to establish a parameter model that optimizes the relationship between the micro-texture geometry and the friction performance, and develops aspects such as the micro-texture pit geometry, depth-to-diameter ratio, region occupancy, local texture and self-lubricating material filling, and other aspects [1113]. The micro-textured surface has a crucial correlation effect on the contact tribological performance, especially the micro-texture shape, micro-texture bottom profile, micro-texture direction, and micro-texture density are the most important key parameters affecting oil film thickness and friction coefficient [14]. The micro-textured triangle effect is the most obvious in reducing the friction [15]. Based on the combination of numerical analysis and experiment, the size and distribution of micro-texture can be optimized to a certain extent, and then the micro-texture contact interface structure can be innovatively designed. As well known, the micro-texture design method and design criteria are not unified, and the experimental data and its conclusions are quite different. And the experimental process and optimization analysis are too cumbersome. In view of the diversity and infinity of micro-texture shapes, it is impossible to determine which interface micro-texture form is most beneficial to reduce the contact friction and wear.

How to reduce the cases of micro-texture forms as much as possible and find the most superior micro-texture shape becomes the most critical issue. Many friction contact interfaces have been successfully micro-textured, such as turning tools, cylinder inner liners, dynamic and static sealing rings, MGPs, rolling/sliding bearings, oil–gas mixing/separation equipment, silicon-controlled wafer surface, and micro-nano mechatronics system of various intelligent machine tools. As shown in Fig. 1, the micro-texture of the contact interface shows its unique personality characteristics [1619]. Under different designed parameters and EHL lubrication conditions, the lubrication properties of conventional and textured spur gear pairs are experimentally disclosed. Three gear surface micro-textures (vertical oriented ovals, horizontal oriented ovals, and semicircular pits) are configured and analyzed by comparison. The results show the less damage of meshing tooth surface in the presence of interface micro-texture, the contact temperature rise of the gear surface is reduced, and the semicircular pits micro-element configuration with the vertical direction has the best effect.

Fig. 1
Fig. 1
Close modal

With the improvement of high-speed and heavy-duty GTS performance indicators, the dynamic responses of GTS have been extremely valued in the preliminary design and verification and optimization stages of MGPs [2022]. The transient contact characteristics of the MGP interfaces are one of the core related influencing indexes that restrict the GTS dynamic response. In particular, the CIMT morphology is an extremely interesting feature of the MGP surfaces. Various types of CIMT may lead to various dynamic responses of the GTS. A pit-type micro-textured surface with anti-friction and resistance-reducing, self-cleaning, and wear-resistant based on bionics is designed. The bionic micro-textured surface geometries studied in this simulation are quasi-circular, approximately square, linear, triangular, cross-hatched, and “S” shaped, and are shown in Fig. 2.

Fig. 2
Fig. 2
Close modal

Many monographs are related to the dynamic characteristics of the GTS [2326]. The transient mesh stiffness excitation injects a time-varying meshing parameter into the GTS dynamic equation, which is of great concern; it has constituted the intrinsic properties of the dynamics of MGPs, which in turn forms the characteristics and inherent properties of the GTS dynamics [2729]. The numerical methods for solving the time-varying meshing stiffness (TVMS) of MGPs are divided into two categories: the finite element method (FEM) [30] and the analytical method following the principle of potential energy [31]. The potential energy method occupies very little computational resources compared with the FEM, but it can still achieve extremely accurate numerical results comparable to the FEM analysis data. Some scholars have considered various stimulating factors, such as TVMS, MTS defects, and optimization analysis of gear tooth profile modification, etc., and have carried out research on the dynamic characteristics of related GTS [3234]. However, most researchers in academia are more inclined to study the CIMT when studying the interaction of MGP interface contact on the GTS dynamic characteristics, whereas the CIMT is abandoned on most levels. This can easily lead to misunderstandings about the effect of the surface roughness, the micro-asperity peak average (MAPA) is our primary concern, of meshing tooth as the CIMT morphology can also produce unexpected results in the gear dynamics. Herein, considering the appearance effect of CIMT’s micro-element body and then eliminating the adverse influence of MTS’ MAPA, this topic intends to study the dynamic characteristics of gears with the same contact interface MAPA but with different CIMT morphology. This is very likely to provide new ideas for improving the anti-thermoelastic scuffing load-bearing characteristics of GTS by controlling the micro-texture of the contact interface. Based on the fractal theory, the M–B rough surface elastoplastic contact fractal model is used to study the microscopic surface contact characteristics, and the relationship between the contact region of the asperities and the normal load is determined, and the fractal dimension is used to characterize the complexity of different surface profile features and irregularity, which describes the geometric characteristics of sections and curved surfaces with fractal geometry. This subject intends to use fractal graphics as the basis for the design of the micro-texture of the contact interface and analyze the changing trend between the fractal dimension and friction and wear of different fractal graphics, so as to obtain the best fractal micro-texture, which is the micro-texture of the MTS design and improve the contact interface lubrication and drag the reduction to provide new technology accumulation.

## 2 Gear Dynamic Model With CIMTs

In this sub-project, a gear friction dynamics model with CIMT is proposed, which mainly includes gear dynamics models with different CIMT evolutions and gear 3D TEHL transient meshing models to explore the correlation effect of CIMT on GTS dynamic characteristics. The GTS friction dynamics coupling equations with CIMTs are solved by an iterative loop between the earlier two models, as shown in Fig. 3. In the absence of iterative loop solution, first initialize the basic parameters of the MGP, such as the comprehensive error of gear tooth manufacturing, rated input torque, and its corresponding speed. The TVMS is analyzed according to the potential energy method. If it is not iterated, the starting values for viscous damping and sliding/rolling friction are set to. Considering the earlier excitations, the vibration equation of the MGP dynamics model can be solved by the Runge–Kutta analytical method. The sliding speed u1(t), u2(t) and curvature radius R(t) of the tooth surface at the meshing position are predicted by the GTS vibration motion analysis.

Fig. 3
Fig. 3
Close modal

In this preliminary study, an infinitely long cylindrical contact is assumed. A section of the textured micro-element computational domain for solving the fluid dynamics coincides with the x1 line, as shown in Fig. 4, where d is the height of the texture micro-element and l/a is the aspect ratio of the micro-element. In numerical simulations of lubricating contacts, the approximation errors caused by the classical asymptotic assumptions can be quite large, and the difference in scale leads to the solution of complex systems of equations. Assuming that the micro-scale is homogenous and periodic, based on the formal method of decoupling the macro-scale and micro-scale, a homogenized micro-elastohydrodynamic model is introduced, which considers the pressure and deformation that cannot be ignored at the micro-scale, and then the general applicability of the classical asymptotic homogenization method is extended.

Fig. 4
Fig. 4
Close modal

At the same time, the dynamic meshing force (DMF) FT(t) is deduced using the vibration response of the GTS and then substituted into the preset load model (distributed form) to derive the DMF FT(t) of the MGP. For the instantaneous values of u1(t), u2(t), R(t), FT(t), as well as the CIMT and lubricant properties, a typical semi-system method is used to analyze the control equations of the hybrid TEHL model [35]. The rolling friction Fr, the sliding friction Fs, and the viscous damping Cm of the gear meshing interface are subject to the convergent solution of the hybrid TEHL model, which is used to calculate the updated dynamic response of the MGP dynamic model proposed earlier.

An iterative cycle process is always executed until the DMF of the MGP meets the convergence criterion. The criterion for convergence of the iterative loop is as follows:
$∑γFpk(tγ)−Fpk−1(tγ)∑γFpk(tγ)
(1)
where γ represents the tooth pair mesh position in a meshing period, k is the number of steps in the loop iteration, $Fpk(tγ)$ denotes the calculated DMF at $tγ$ (instantaneous meshing time) calculated in the Kth loop iteration step, and the parameter err is a pre-defined convergence threshold, whose pre-determined value is set to 10−5 in the research issue. So, it can be deduced, whenever the relative error value between the DMFs analyzed in two consecutive steps is not greater than the pre-defined convergence threshold; once the iterative loop process stops, it is assumed that a stable solution is determined.

In the actual operation of GTS, the micro-appearance morphologies of MTS directly depend on the processing accuracy, manufacturing process, operation law, constituent materials, etc. The related discussion focuses on the three different MTS micro-appearance morphologies that exist in the actual processing and manufacturing. It is set in advance that these three cases all have the same micro-asperity peak Rq (root-mean-square (RMS) roughness value of MTS) and the same wavelength L; however, the cross sections of the two are not equal, and are marked by random permutation (case 1), sinusoidal distribution (case 2), and semi-ellipse designed configuration (case 3), as shown in Fig. 5. The associated influence of CIMT on the friction characteristics and dynamic responses of the MTS is further revealed.

Fig. 5
Fig. 5
Close modal

Figure 6 depicts a general spur gear dynamic model considering MTS friction. Herein, the representative parameters Ffp and Ffg describe separately the friction behavior of driving gear (pinion) and driven gear (bull gear). The transient meshing description of a spur meshing pair is modeled simultaneously by TVMS km(t) and viscous damping Cn. The support bearing of each gear is set to real-time simulation of equivalent support stiffness and damping indicated by the x and y directions; the relevant parameters are kpx, kgx, kpy, kgy and Cpx, Cgx, Cpy, Cgy. The driving/driven gears in the dynamic models are represented by the equivalent simplified representation of a rigid body whose mass mi(i = p, g), moment of inertia Ji, and radius are equal to the gear base circle radius ri. Assume that the rotational and translational movements of the two gears in the y direction are coupled along the line of action (LOA) by the spring damping unit. km represents the TVMS of the MGP, and Cn describes the equivalent meshing damping of the MGP. e is regarded as the static transmission error (STE), which mainly includes gear tooth elastic deformation and gear tooth manufacturing error under static load conditions. The rotational and translational degrees-of-freedom (DOFs) of the MGP in the x direction tend to be coupled in real-time in the off-line action (OLOA) direction. The control equations descriptions of the above gear dynamic model are expressed as follows:

$Jpθ¨p=Tp−Fp(t)rp−[∑i=1nFfp(t)Rp(t)]i$
(2)
$mpy¨p+cpyy˙p+kpyyp=−Fp(t)$
(3)
$mpx¨p+cpxx˙p+kpxxp=[∑i=1nFfp(t)]i$
(4)
$Jgθ¨g=Tg−Fg(t)rg−[∑i=1nFfg(t)Rg(t)]i$
(5)
$mgy¨g+cgyy˙g+kgyyg=Fg(t)$
(6)
$mgx¨g+cgxx˙g+kgxxg=−[∑i=1nFfg(t)]i$
(7)
where n represents the number of the MGPs; Ti denotes the external loads acting on the MGPs i ($i=p,g$, shown as the pinon and bull gear, respectively); $θ˙i$ and $θ¨i$ represent the torsional vibration velocity and acceleration of the MGPs i, respectively; $x˙i$ and $x¨i$ denote the translational vibration velocity and acceleration in the OLOA direction of the MGPs i, respectively; $y˙i$ and $y¨i$ represent the translational vibration velocity and acceleration in the LOA direction of the MGPs i, respectively; and Ri represents the radius scalar at the meshing position of the MGPs i. The frictional forces Ffp and Ffg of the driving/driven gears are described by the hybrid TEHL calculation model. The DMF Fp is expressed as follows:
$Fp=km(yp+rpθp−yg+rgθg−e)+Cn(y˙p+r˙pθ˙p−y˙g+r˙gθg−e˙)$
(8)
The tooth pair number in the meshing region presents a periodic time-varying law during the meshing transient process of the spur MGP. Consider a MGP with a normal contact ratio between 1 and 2, in which the number of meshing gear teeth alternates. The entire meshing region is set as a single tooth contact (STC) region and a double teeth contact (DTC) region. The overall gear DMF is expressed as the sum of the DMF of all MGPs in the meshing region:
$Fp=FT1+FT2$
(9)
where FT1 and FT2, respectively, denote the DMF of the first and second meshing gear teeth pairs, and FT2 = 0 represents the DMF when only one tooth pair is in meshing. According to the load distribution model between meshing gear teeth, the DMF of each MGP is derived as follows:
${FT1=k1δ1+c1δ˙1FT2=k2δ2+c2δ˙2$
(10)
where ki is simulated by the TVMS of the ith MGP $(i=1,2)$. ci represents the viscous damping of the ith MGP $(i=1,2)$, which is usually assumed to be an associated parameter dependent on ki. The calculation formula can be expressed as follows:
$ci=ki6(1−α)Vi((2α−1)2+3)$
(11)
where Vi represents the initial vibration velocity of the meshing gear teeth relative to the parameter $α=1−0.022Vi0.36$. In Eq. (11), δ and $δ˙i$, respectively, denote the relative displacement and velocity of the ith MGP $(i=1,2)$ along the LOA direction, which can be expressed as follows:
${δi=yp+rpθp−yg+rgθg−eiδ˙i=y˙p+rpθ˙p−y˙g+rgθ˙g−e˙i$
(12)
where ei indicates the displacement excitation caused by the tooth profile deviations (relative to the ideal involute tooth profile) for the ith MGP $(i=1,2)$ during a MGP meshing transient process, whereas $e˙i$ is shown as the derivative of the parameter ei, which is a quantitative characterization of the velocity excitation.
Fig. 6
Fig. 6
Close modal

## 3 Three-Dimensional TEHL Calculation Model of Gear Meshing Tooth Surface

Considering the comprehensive influence of factors such as MTS excitation load and slip speed, lubricating oil viscosity, and tooth surface MAPA, the lubrication conditions of MTS are divided into three situations: elastohydrodynamic, hybrid, and boundary lubrications. Among the aforementioned factors, the MTS excitation load is calculated by formula (11) to obtain the gear tooth DMF. The slip speed of the MGP is the difference between its linear speed in the OLOA direction, namely up(t) and ug(t), which is indicated as follows:
${up(t)=u¯p(t)−Rp(t)θ˙p(t)+x˙p(t)ug(t)=u¯g(t)−Rg(t)θ˙g(t)+x˙g(t)$
(13)
where ui(t) denotes the linear velocity associated with the nominal rotation of MGP $i(i=p,g)$ in the OLOA direction, which is defined as $ui(t)=−Rp(t)ωi$, herein ωi is its nominal rotational velocity. $Rp(t)θ˙p(t)$ and $xi(t)(i=p,g)$ represent the velocity vector components in the OLOA direction caused by rotation/translation vibration excitation.

Under hybrid TEHL conditions, the TEHL oil film and non-smooth contact in the gear meshing region co-exist. In the meshing region where there is no concave–convex contact, the MGPs are secluded by a layer of hydrodynamic TEHL oil film, thereby forming the full oil film lubrication. The typical feature of gear meshing is usually in line contact with the fluid hydrodynamic lubricating oil between the gear teeth participating in the meshing. Combined with the Reynolds equation, the expression of the commonly used 3D line contact TEHL model is as follows:

$∂∂x(ρ12η#h3∂p∂x)+∂∂y(ρ12η#h3∂p∂y)=μr∂ρh∂x+∂ρh∂t$
(14)
where x is regarded as the relative slip direction of the MGP and y is in the axial direction of the MGP; p and h represent the lubricating oil film thickness and pressure of the contact interface, respectively. ρ denotes the lubricating oil density, and the relative rolling/slipping speed μr is shown as an average instantaneous speed of the MGP, which can be expressed as $μr=1/2[μp(t)+μg(t)]$. η# stands for equivalent viscosity index. In view of the non-Newtonian features of the lubricating oil shear thinning effect, the following expression can be obtained from the Ree–Eying model:
$1η#=τ0ητmsinh(τmτ0)$
(15)
where τ0 is expressed as the reference value of the shear stress of the lubricating oil, τm is shown as the average value of the viscous shear stress of the lubricating oil, and η represents the viscosity of lubricating oil under low shear rate, which can be solved by Roelands’ formula.

The specific properties (rheological properties) for the hydrodynamic oil film at the contact interface are determined by the micro-texture morphology of the MTS. The thickness of the hydrodynamic oil film is extremely thin at the peak abrupt position of the uneven MTS micro-asperity, where the meshing gear tooth interface contact easily occurs. As a result, the original lubrication conditions are converted to hybrid TEHL. At a specific gear pair meshing position, due to the microscopic morphology of the contact interface of the MTS, unable to solve with the general form of Reynolds equation, and the thermoelastic contact theory needs to be sought.

In order to obtain a unified numerical solution for the contact region of the entire meshing interface, once the hydrodynamic lubricating film thickness is expressed as zero, all the hydrodynamic related parameter items in the Reynolds equation are regarded as closed state, which can satisfy the numerical solution of the initial contact pressure of the gear pair MTS. From the above, this can be equivalent to deriving the simplified form of Reynolds equation as:
${μr∂ρh∂x+∂ρh∂t=0h=0$
(16)
In view of the fact that Eq. (14) is regarded as another expression of Eq. (16), for the numerical solution of the TEHL problem, the pressure parameter automatically meets the non-intermittent conditions of the fluid dynamics boundary and the non-smooth contact region, and no additional settings are required. Based on this, the description of hybrid TEHL is analyzed by the same repeated process of iterative loop. The mathematical expression of the hydrodynamic oil film thickness in the TEHL region of the meshing gear interface can be expressed as follows:
$h(x,y,t)=h0(t)+x22Rx+V(x,y,t)+δ(x,y,t)$
(17)
where Rx represents the equivalent curvature radius of the MGPs at meshing point, which is set to $1/Rx=(1/Rp)+(1/Rg)$. δ indicates the geometric shape distribution parameter of the composite micro-topography of the two pairs of teeth surfaces at the meshing point of the driving and driven gears as shown in Fig. 7. V is regarded as the thermal elastic deformation parameter of the MTS and can be analyzed by the discrete convolution and fast Fourier transform (DC-FFT) method. Then its calculation expression is as follows:
$V(x,y,t)=2πE′∫∫Ωpf(ξ,s)+pc(ξ,s)(x−ξ)2+(y−s)2dξds$
(18)
where pf denotes the film pressure of the hydrodynamic oil, which can be achieved by analytic coupling (Eqs. (14), (17), and (18)). pc represents the non-smooth interface MAPA pressure, which is determined by the analytical coupling (Eqs. (16)(18)). ξ and s are, respectively, regarded as additional coordinates relative to the x and y axes. E′ represents the equivalent modulus of elasticity of the MGP and Ω denotes the calculation domain at meshing point of the MGP.
Fig. 7
Fig. 7
Close modal
Hybrid TEHL numerical solution problem is determined by the load conditions that have been set, and the solution pressure must meet the earlier mentioned pre-conditions for load balance. The objective optimization problem is a non-linear constrained optimization problem, which belongs to the framework of non-linear programming [36,37]. The load balancing equation is expressed as follows:
$∫∫Ωp(x,y,t)dxds=FT(t)$
(19)
when h ≠ 0, p denotes the film pressure of hydrodynamic oil, whereas when $h=0$, p is the pressure of non-smooth CIMT morphology. For the GTS, FT is the gear tooth DMF of a single MGP, solved by Eq. (11).
Consider that the friction of the gear meshing interface with TEHL conditions is caused by the hydrodynamic oil film viscous shear stress between the MGP. The aforementioned shear stress is caused by the combination of Poiseuille and Couette flows and changes linearly along the z direction (along the film thickness direction of the hydrodynamic oil), which is denoted as follows:
$τ(x,y,z,t)=η#(x,y,t)h(x,y,t)[μg(t)−μp(t)]+[z−12h(x,y,t)]∂p(x,y,t)∂x$
(20)
Since the meshing interface is not smooth and under the action of external excitation load, the peak position of the rough interface may have uneven and non-smooth contact, thereby forming a hybrid TEHL state. Therefore, the friction force of GTS consists of two parts. One is that there is a hydrodynamic oil film viscous shear stress between the MGP, and the other is that the non-indirect contact of rough peaks leads to the rupture of the oil film on the MGP interface, which weakens the lubrication effect and produces uneven friction thermoelastic behavior. According to the analysis process of hybrid TEHL, the intermediate oil film viscous shear force is regarded as the interface oil film contact friction force, and the real-time friction force of the MGP contact interface at any time can be shown as follows:
$Ff(t)=A∑im∑jn[τij(t)+μbpcij(t)]$
(21)
where M and N represent the grids number along the x and y directions in the numerical calculation domain, respectively. A denotes the grid division region unit, τij is the interfacial oil film equivalent viscous shear force at the middle layer (where z = 0.5h in Eq. (20)) on the grid node (i, j), pcij is the non-smooth interface contact pressure on the grid nodes (i, j), and the non-smooth interface contact friction coefficient μb is assumed to be 0.1. Equations (8) and (20) are substituted into Eq. (21), and the contact friction force at the MGP interface is derived from the following expression:
$Ffp(t)=Cm(t)[Rg(t)θ˙g(t)+x˙g(t)+Rp(t)θ˙p(t)−x˙p(t)]+Fs(t)+Fr(t)$
(22)
$Ffg(t)=−Ffp(t)$
(23)
The CIMT friction coefficient is expressed as follows:
$μfp(t)=Ffp(t)FT(t)$
(24)
Based on the film pressure p and film thickness h (convergent analytical value) of the hybrid TEHL model, the viscous damping Cm(t), sliding friction force Fs(t), and rolling friction force Fr(t) of the MGP are analyzed, which can be expressed as follows:
${Cm(t)=A∑im∑jn[η#(x,y,t)h(x,y,t)]Fs(t)=Cm(t)[μ¯g(t)−μ¯p(t)]+A∑im∑jn[μbpcij(t)]Fr(t)=A∑im∑jn[12h(xi,yj,t)∂p(xi,yj,t)∂x]$
(25)
From Eq. (25), it can be revealed that the lubricating oil viscous damping Cm(t) is proportional to its equivalent viscosity and inversely proportional to the hydrodynamic oil film thickness at the MGP contact interface. The rolling/sliding friction force Fs(t) increases with the increase of the viscous damping Cm, the relative slip speed μgμp of the MTS, and the non-smooth concave–convex contact pressure pcij. The rolling/sliding friction force Fr(t) and the oil film thickness h change in direct proportion to the oil film pressure gradient of the contact interface along the sliding direction of the MGP. The friction equation (23) is calculated from the numerical solution of the hybrid TEHL and substituted into the vibration equation (8). The frictional dynamic coupling equations of the meshing pair of GTS can be expressed as follows:
$Jpθ¨p−∑i=1n[Cm(t)Rp(t)]ix˙p+∑i=1n[Cm(t)Rp(t)]ix˙g+∑i=1n[Cm(t)Rp2(t)]iθ˙p+∑i=1n[Cm(t)Rp(t)Rg(t)]iθ˙g=Tp−Fp(t)rp−∑i=1n[Fs(t)Rp(t)+Fr(t)Rp(t)]i$
(26)
$mpy¨p+cpyy˙p+kpyyp=−Fp(t)$
(27)
$mpx¨p+(cpx+∑i=1n[Cm(t)]i)x˙p−∑i=1n[Cm(t)]ix˙g+kpxxp−∑i=1n[Cm(t)Rg(t)]iθ˙g−∑i=1n[Cm(t)Rp(t)]iθ˙p=−∑i=1n[Fs(t)+Fr(t)]i$
(28)
$Jgθ¨g+∑i=1n[Cm(t)Rg2(t)]iθ˙g+∑i=1n[Cm(t)Rp(t)Rg(t)]iθ˙p+∑i=1n[Cm(t)Rg(t)]ix˙g−∑i=1n[Cm(t)Rg(t)]ix˙p=Tg−Fp(t)rg−∑i=1n[Fs(t)Rg(t)+Fr(t)Rg(t)]i$
(29)
$mgx¨g+(cgx+∑i=1n[Cm(t)]i)x˙g−∑i=1n[Cm(t)]ix˙p+kgxxg−∑i=1n[Cm(t)Rp(t)]iθ˙p−∑i=1n[Cm(t)Rg(t)]iθ˙g=−∑i=1n[Fg(t)+Fr(t)]i$
(30)
$mgy¨g+cgyy˙g+kgyyg=Fp(t)$
(31)

From Eqs. (26)(31), it can be revealed that the rotational and translational DOFs are coupled in Eqs. (28) and (31) along the OLOA direction. Considering that the gear meshing interface frictional force exists in real-time, resulting in the interaction between the translational vibration and the torsional vibration of the GTS along the OLOA direction, the DMF is described as a function of displacement yi and velocity $y˙i$ along the LOA direction. The 6DOFs are coupled in Eqs. (26)(29) and are solved jointly by the DMF and frictional forces at the MGP interface. The torsional and all translational vibrations are in a state of interaction.

## 4 Simulation and Discussion

In this project, considering the same contact interface MAPA and hybrid TEHL conditions, the GTS friction dynamics models of different cases of CIMTs are established. The basic design parameters of the MGP (pinion and bull gear) in these cases are shown in Table 1, and the performance parameters of lubricating oil are reflected in Table 2. Under the experimental conditions of input torque 500 N m and speed 1000 r/min, the dynamic characteristics of GTS are studied.

Table 1

Basic design parameters of MGP

ParameterValue
Number of teeth, zp/zg27/33
Module, m (mm)4.5
Center distance, dc (mm)116
Pressure angle, α (deg)20
Face width, w (mm)45
Clearance coefficient, c*0.25
Addendum coefficient, $ha*$1
Tooth surface Rq value (μm)0.5
Elastic modulus, E (GPa)207
Density (ρp/ρg) (103 kg m−3)7.85/7.85
Poisson ratio, μ0.31
Mass (mp/mg) (kg)2.89/3.02
Moment of inertia (Jp/Jg) (10−3 kg m3)5.4/7.8
STE, e (μm)1
ParameterValue
Number of teeth, zp/zg27/33
Module, m (mm)4.5
Center distance, dc (mm)116
Pressure angle, α (deg)20
Face width, w (mm)45
Clearance coefficient, c*0.25
Addendum coefficient, $ha*$1
Tooth surface Rq value (μm)0.5
Elastic modulus, E (GPa)207
Density (ρp/ρg) (103 kg m−3)7.85/7.85
Poisson ratio, μ0.31
Mass (mp/mg) (kg)2.89/3.02
Moment of inertia (Jp/Jg) (10−3 kg m3)5.4/7.8
STE, e (μm)1
Table 2

Performance parameters of lubricating oil

ParameterValue
Reference shear stress, τ0 (MPa)18.2
Inlet viscosity, η0 (Pa s)0.10
Viscosity–pressure index, z00.71
Inlet density, ρ0 (kg m−3)886
ParameterValue
Reference shear stress, τ0 (MPa)18.2
Inlet viscosity, η0 (Pa s)0.10
Viscosity–pressure index, z00.71
Inlet density, ρ0 (kg m−3)886

The equivalent stiffnesses of supporting bearings of driving/driven gears in the x and y directions are expressed as kpx = kgx = 5.82 × 108 N/m and kpy = kgy = 5.82 × 108 N/m, and the equivalent dampings are cpx = cgx = 6.53 × 103 N/m and cpy = cgy = 6.53 × 103 N/m, respectively.

The TVMS of each MGP is solved by the potential energy method [7,38], and the result is shown in Fig. 8. Among them, the red line represents the gear meshing stiffness in four meshing cycles, whereas the dashed lines with blue and green denote the TVMS of gear pairs 1 and 2 in any meshing period. What needs to be emphasized here is that the TVMS of MGP is the sum of the meshing stiffness of all gear teeth pairs participating in the meshing behavior. The region where k2 = 0 is the STC region. If there is gear pair 2, no meshing in the region, and the total TVMS is significantly smaller than the DTC region as gear pair 2 is not in meshing state, and the total TVMS is significantly not higher than that in the DTC region.

Fig. 8
Fig. 8
Close modal

Substituting the total TVMS of the MGP into the dynamic friction coupling equations, the numerical simulation results of the vibration response of the GTS are derived. On this basis, the DMFs of the MGPs are determined by Eqs. (10) and (11). The DMFs are shown in Fig. 9. The pink solid line is regarded as the total DMF of the MGP, whereas the dashed lines with green and blue represent the DMF of the MGPs 1 and 2, respectively. What is concerned here is that the DMF of the MGP expresses the sum of the DMFs of all meshing gear teeth. Furthermore, it can be inferred that the following law of change is that the fluctuation range of the total DMF shows a trend of first decreasing and then increasing. One of the main reasons is that the meshing transient process is caused by the DTC region crossing into the STC region.

Fig. 9
Fig. 9
Close modal

### 4.1 Friction and Damping Between CIMTs in Three Cases.

The vibration response is solved based on the gear dynamics model to determine the transient DMF, the rolling/sliding velocity of the gear teeth interface, and the curvature radius at the meshing position. The aforementioned values are included in the hybrid TEHL model along with the microscopic morphology and lubricating oil characteristics at the MGP interface to evaluate the friction response, thickness, and pressure distribution of the lubricating oil film of the GTS. For the sake of comparison, a specific case of smooth gear tooth interface is provided, as shown in Figs. 10 and 11. Figure 10 shows the influence of the CIMT geometric distribution on the viscous damping as the wavelength L is 20 μm and the MAPA Rq (MTS roughness value) is 1.0 μm. The solid lines with red, green, and blue represent the viscous damping consistent with CIMT cases 1, 2, and 3, respectively, whereas the solid line with black denotes a smooth MTS. Table 3 shows the viscous damping values at the five specified points (A, B, C, D, and E represent the entry point, STC to DTC transition point, pitch point, DTC to STC transition point, and existing point) under the three conditions as mentioned earlier.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal
Table 3

Viscous damping values of five specified points in a meshing period under cases

CasesDamping (N s m−1)
ABCDE
151.7172.91.092 × 103113.991.03
261.6231.21.663 × 103154.2165.1
375.7281.61.339 × 103194.8174.2
CasesDamping (N s m−1)
ABCDE
151.7172.91.092 × 103113.991.03
261.6231.21.663 × 103154.2165.1
375.7281.61.339 × 103194.8174.2

From case 1 to case 3, a significant finding is that as CIMT becomes more regular and orderly, and the damping value becomes higher. The reason is that compared with case 1, the MTS micro-asperity peaks in case 3 are more evenly concentrated near the contact centerline, which leads to a closer connection between the MGP contact interfaces.

Therefore, the oil film thickness of case 3 is thinner than that of case 1. Furthermore, the smooth surface produces a higher viscous damping effect than the three rough MTS cases, which can be attributed to the thinnest hydrodynamic oil film produced between the smooth MTSs. Another noteworthy finding is that the damping value of the DTC region is higher than that of the STC region. This reason is attributed to the fact that the slip speed between the gear meshing interfaces of the DTC region is smaller than that of the STC region, which makes it easier to form an oil film. Figure 11 depicts the influence of the CIMT geometric distribution as the wavelength L is 20 μm and the MAPA Rq (MTS roughness value) of the MGP is 1 μm. The solid lines with red, green, and blue, respectively, describe the friction consistent with CIMT cases 1, 2, and 3, whereas the solid line with black represents a smooth MTS. It can be revealed that for the three rough MTS cases, the friction force of the DTC region is less than that of the STC region. As the CIMT geometric distribution cases 1, 2, and 3 gradually slowed down, the peak of the MTS micro-asperity decreased, which leads to the gradual weakening of the rough contact with the micro-textured interface. Therefore, the corresponding changes of friction force gradually show a decreasing trend from case 1 to case 3. Whether it is in the STC or DTC region, the friction generated by the smooth meshing gear tooth surface is the lowest, which is the “smooth” contact that is incorporated into the MTSs. Especially at the nodes, the friction between smooth MTSs is regarded as zero. The following reveals the correlation effect of the wavelength L (with a fixed Rq) under the three cases of CIMT on the friction and viscous damping of the gear MTS.

Figures 12(a1))–12(a3) reflect the friction trend of the three CIMT cases 1, 2, and 3 described in Fig. 12 at multiplex wavelengths L, where Figs. 12(b1)–12(b3) reveal the CIMT viscosity viscous damping absorption effect. The friction and damping of CIMT in the three cases are averaged within a meshing period, the correlation law of the CIMT case and wavelength on average friction force are analyzed, and the relative influence of average friction force and damping is explored. The average friction trend of different types of CIMT within a certain wavelength range is shown in Fig. 13(a). It can be seen from the above figures that as the wavelength is slight, the difference in average friction consistent with CIMT case 2 and case 3 is extremely small. As the wavelength is larger, the average friction of CIMT case 2 is not less than that of case 3. As the wavelength increases, the friction force (average value) of the CIMT case 1 drops slightly at a certain point (wavelength is about 30 μm), and then, due to the randomness of CIMT in case 1, it fluctuates sharply nearby. The average friction force of CIMT cases 2 and 3 suddenly decreased to a minimum (wavelength is approximately 14 μm), and followed the law of increasing with the increase of the wavelength.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

In turn, this can verify the key role of CIMT wavelength on the friction effect between MTSs. Figure 13(b) illustrates the changes of viscous damping (average value) under CIMT cases in multiplex wavelength ranges, and it can be seen that the average viscous damping is less affected by the CIMT wavelength. The viscous damping (average value) consistent with CIMT case 2 is approximately the same as that of case 3, but not less than that of case 1.

### 4.2 Dynamic Characteristics Under CIMT.

The proposed model considers the mutual influence of CIMT under different cases on the dynamic characteristics of MTS. Figure 14 reveals the correlation of the displacement response of driving gear MTS with CIMT wavelength (L = 20 μm) along the OLOA direction relative to the MAPA (roughness value RMS) and input torque in the three cases. It can be concluded that the MAPA decreases from CIMT case 1 to case 3, and the decrement increases as the torque increases. It is further learned that the r MAPA difference between any regular MTS (i.e., cases 2 or 3) and random MTS (i.e., case 1) is higher than the MAPA difference between two regular MTSs (i.e., cases 2 and 3) and the MAPA has relatively little influence on the dynamic response.

Fig. 14
Fig. 14
Close modal

### 4.3 Thermoelastic Hydrodynamic Lubrication Properties Under CIMT.

Based on the foregoing (as shown in Fig. 4) and taking into account the thermal effect, without considering the influence of body force and thermal radiation, and ignoring the heat conduction in the x and y directions, the energy equation of the lubricating medium at the meshing interface is expressed as follows:
$Cd[ρu∂t∂x+ρv∂t∂y−q∂t∂z]=kc∂2t∂z2−tp∂ρ∂t(u∂p∂x+v∂p∂y)+η[(∂u∂z)2+(∂v∂z)2]$
(32)
where the above four items are the convection heat transfer term, the heat conduction term, the pressure term, and the heat dissipation term. Cd is the specific heat capacity of the lubricating oil (J/(kg K)); kc is the thermal conductivity coefficient of the lubricating oil (W/(m2 K)); u and v are the flowrates of the lubricating oil along the x and y directions, respectively (m/s). q in the convection heat transfer term is obtained from the following equation [24,37]:
$q=∂∂x∫0zρudz′+∂∂y∫0zρvdz′$
(33)

The surface pressure, film thickness distribution, and temperature distribution of meshing teeth with interfacial micro-texture are solved by the multi-grid integration method. The meshing interface pressure calculation and thermoelastic deformation term calculation caused by the pressure adopt the multi-grid integration method, and the interface temperature calculation adopts the column-by-column scanning method. The grid is divided into five layers, the highest grid is divided into 512 nodes in the x direction and 512 nodes in the y direction, the temperature gradient in the oil film between the interfaces is larger, and the equidistant grid is used, and the number of nodes is ten. In the solid, the temperature gradient is larger near the solid–liquid interface, and the temperature change tends to be gentle at the distance away from the solid–liquid interface. Therefore, unequal spacing grids are used, the grid spacing is a proportional sequence, and the number of nodes in both solids is six. When the error of pressure before and after iteration is less than 1 × 10−4 and the relative error of load and temperature is less than 1 × 10−5, the iteration ends.

The distribution of the non-linear strong coupling field (film thickness, film pressure, and film temperature) at the micro-texture interface is solved by the combination of the energy equation and the Reynolds equation. The conclusions in Figs. 15 and 16 are as follows: the overall thickness of the oil film considering the thermal effect becomes thinner, especially the minimum film thickness, which decreases sharply after considering the thermal effect. After considering the thermal effect, the pressure fluctuation amplitude of the textured surface becomes larger, and the secondary pressure peak value increases significantly.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

By optimizing the parameters related to the CIMT, the synergistic regulation of improved lubrication increased load-carrying capacity. The Eyring model is used to analyze the influence of the cross-scale structure and distribution parameters of the meshing tooth surface micro-texture on the TEHL characteristics, and the optimal distribution interval of the micro-texture cross-scale structure parameters when the oil film properties of the MTS interface are in a good state under high-strength contact is obtained.

## 5 Conclusions

Time-varying meshing stiffness, STE, CIMT geometric distribution, and oil film shear effects are taken into consideration. According to the dynamic control equation of the 6DOFs MGP, the friction dynamic coupling model of the meshing interface of the MGP and the 3D TEHL contact model are established. The friction dynamics coupling model is solved by iterative loop until it converges. The coupling model that is proposed above is implemented to study the geometric distribution influence on MGP with CIMT morphology of the dynamic characteristics under MTS TEHL conditions. From this, the following conclusions are inferred:

1. As the CIMT changes from random permutation (case 1) to regular (sinusoidal distribution, case 2 and semi-ellipse designed configuration, case 3), the MTS viscous damping becomes higher and the smooth MTS has the highest damping value. Conversely, the frictional force decreases with the smoothness of the MTS.

2. For a regular MGP, the CIMT wavelength plays a vital role in the interface frictional contact effect between MTS, and the frictional force drops to an extreme value at a specific wavelength (the study case is about 14 μm).

3. The MTS dynamic response along the OLOA direction decreases as CIMT morphology becomes regular configuration (i.e., from case 1 to case 3), which means that the production of regular shape (rather than a random permutation shape) meshing gear tooth surface is more suitable for reducing the vibration response of the GTSs.

4. Optimized numerical value of CIMT scale parameters is analyzed, considering that the interfacial lubricating medium exhibits a strong non-Newtonian effect due to instantaneous pressure and thermal effects under heavy-duty meshing state of MTS with micro-texture characteristic.

The next focus of future work is to conduct an experimental research on the dynamic behavior of gear MTSs with multiplex types of CIMT morphology, so as to deepen the verification of relevant theoretical findings through the research of this sub-project.

## Acknowledgment

The authors would like to thank the Northeast Forestry University (NEFU), Heilongjiang Institute of Technology (HLJIT), and the Harbin Institute of Technology (HIT) for their support.

The research subject was supported by the Special Project for Double First-Class-Cultivation of Innovative Talents (Grant No. 000/41113102; Jiafu Ruan, NEFU), the Doctoral Research Start-Up Foundation Project of Heilongjiang Institute of Technology (Grant No. 2020BJ06; Yongmei Wang, HLJIT), the Natural Science Foundation Project of Heilongjiang Province (Grant No. LH2019E114; Baixue Fu, HLJIT), the Basic Scientific Research Business Expenses (Innovation Team Category) Project of Heilongjiang Institute of Engineering (Grant No. 2020CX02; Baixue Fu, HLJIT), the Special Scientific Research Funds for Forest Non-Profit Industry (Grant No. 201504508), the Youth Science Fund of Heilongjiang Institute of Technology (Grant No. 2015QJ02), and the Fundamental Research Funds for the Central Universities (Grant No. 2572016CB15).

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