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Research Papers: Elastohydrodynamic Lubrication

# Parametric Studies of Mechanical Power Loss for Helical Gear Pair Using a Thermal Elastohydrodynamic Lubrication ModelPUBLIC ACCESS

[+] Author and Article Information
Mingyong Liu

Hubei Agricultural Machinery Engineering
Hubei University of Technology,
Wuhan 430068, China
e-mail: lmy8508@hbut.edu.cn

Peidong Xu

Hubei Agricultural Machinery Engineering
Hubei University of Technology,
Wuhan 430068, China
e-mail: 463176132@qq.com

Chunai Yan

Dongfeng Passenger Vehicle Company,
Wuhan 430058, China
e-mail: 664874648@qq.com

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received January 17, 2018; final manuscript received June 21, 2018; published online August 20, 2018. Assoc. Editor: Yi Zhu.

J. Tribol 141(1), 011502 (Aug 20, 2018) (14 pages) Paper No: TRIB-18-1024; doi: 10.1115/1.4040723 History: Received January 17, 2018; Revised June 21, 2018

## Abstract

In this study, a comprehensive mechanical efficiency model based on the thermal elastohydrodynamic lubrication (TEHL) is developed for a helical gear pair. The tribological performance of the helical gear pair is evaluated in terms of the average film thickness, friction coefficient, mechanical power loss, mechanical efficiency, etc. The influence of basic design parameters, working conditions, thermal effect, and surface roughness are studied under various transmission ratios. Results show that the contribution of thermal effect on the tribological performance is remarkable. Meanwhile, the rolling power loss constitutes an important portion of the total mechanical power loss, especially around the meshing position where the pitch point is located in the middle of contact line and the full elastohydrodynamic lubrication (EHL) state with the friction coefficient less than 0.005. The proper increase of normal pressure angle and number of tooth can improve the tribological performance. The influence of helix angle on the mechanical efficiency is less significant. A positive addendum modification coefficient for pinion and a negative addendum modification coefficient for wheel are good for improving the mechanical efficiency. The results provide the tribological guidance for design of a helical gear pair in engineering.

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## Introduction

Gear drives are extensively applied in various industries, like automobiles, wind turbine gearbox, and marine transmissions. The machine elements such as gears and various bearings cause power loss in operation. The total power loss of the gearbox includes no-load and load-dependent power loss. The no-load power loss can be divided into windage loss, churning, and squeezing loss. The load-dependent loss depends on sliding friction loss and rolling friction loss. The power loss at the gear meshing interface represents a sizable portion of overall losses. Generally, the spur and helical gar pairs have a mechanical efficiency well over 99%. The power loss of a spur and helical gear pairs are quite small, but the multiplication of power losses in multistage gear drive might be significant. Meanwhile, the power losses amount to additional heat generation within the gearbox, several gear failures including surface pitting and teeth bonding are directly impacted by the temperature. Because of energy crisis and carbon emissions pressures, the mechanical transmission system efficiency has become a critical research topic in recent years.

In the past two decades, a number of numerical and experimental results on the mechanical power loss of spur and helical gear pairs have been published. In the aspect of theoretical research, the earlier studies investigated the efficiency of spur and helical gear pair by assuming a uniform friction coefficient or empirical friction formulae along the line of action (LOA). In the last decade, the elastohydrodynamic lubrication (EHL)-based friction coefficient has been widely used for the prediction of gear mechanical efficiency. Based on a simplified analytical friction model, Wu and Cheng [1] analyzed the power loss in spur gears due to the sliding and rolling friction. Haizuka et al. [2] studied the friction loss of helical gears in view of various helix angles, loads, and rotational speeds. Based on the experimental investigation, they also proposed the general formulas for the temperature of teeth and friction loss in view of gear design. Xu [3] had derived a novel friction coefficient formula by performing a multiple linear regression analysis to the massive EHL results and measured traction data. Then, the novel formula was applied for the efficiency prediction of both spur and helical gears. Heingartner and Mba [4] proposed a spur and helical gear efficiency models by various empirical formulas obtained from twin-disk tests. By discretizing a helical gear pair into a number of thin slices spur gears, Li et al. [5] studied the influence of design parameters and tooth surface modification on the mechanical power loss. Baglioni et al. [6] analyzed the spur gear efficiency variation versus the addendum modification through two different approaches for friction coefficient, a mean friction coefficient and a local friction coefficient formula proposed by Xu [3]. Douglas and Thite [7] established the spur gear efficiency model by comprehensive consideration of various empirical formulas. Marques et al. [8] proposed a four degrees-of-freedom dynamic model of spur and helical gear pairs accounting for friction. They evaluated the power loss for different gear geometries using constant and local time-varying friction coefficient. Liu et al. [9] developed a thermal EHL mode for a coated spur gear pair, and discussed the effect of elastic modulus and thickness of the coating material on the frictional power loss.

In terms of experimental research, Britton et al. [10] built a special four-gear rig and investigated the gear tooth frictional losses at loads and speeds representative of engineering practice. The results indicated that the superfinishing resulted in a reduction of friction and lower tooth surface temperature. Diab et al. [11] proposed a new traction law based on the friction measurements, and integrated in three-dimensional dynamic model of gears in order to discuss the effect of tooth friction and tip relief on the power loss. Petry-Johnson et al. [12] designed a power-circulating test to investigate the influence of two spur gear pairs, surface roughness, and lubricant on the power loss. Luis et al. [13] investigated the influence of toot profile and oil formulation on spur and helical gear power loss using the Forschungsstelle für Zahnräder und Getriebebau machine. Isaacson et al. [14] built a comprehensive experimental setup to evaluate the effective friction coefficient including the surface roughness and two different lubricants. Ziegltrum et al. [15] investigated the load-dependent power loss of different lubricants using a transient thermal elastohydrodynamic lubrication (TEHL) model for spur gear pair. By comparing with the measured value, the results revealed that mineral oil showed significantly higher friction coefficient and gear power loss than other ones. Andersson et al. [16] compared dip lubrication with spray lubrication regarding mesh efficiency and gear temperature using a back-to-back gear test rig.

As mentioned earlier, most of the published numerical results assumed a constant or empirical friction coefficient to calculate the efficiency of gear trains. In addition, most of the previous numerical and experimental results only discussed the limited design parameters on the mechanical power loss of gear pairs. Meanwhile, the gear mechanical efficiency model based on EHL is usually simulated as infinite line contact or a series of spur gear slices. The actual finite line contact model on the efficiency of helical gears is very limited.

In addition, the author and co-authors have already developed a non-Newtonian thermal EHL finite line contact model for helical gear pair and discussed the effect of design parameters and working conditions on the lubrication performance [1720]. In order to obtain more information on the mechanical power loss of helical gear pair, the present study employs the finite line thermal EHL model to investigate the effect of design parameters, working conditions, and surface roughness on the helical gear mechanical efficiency.

## Model Formulations

###### Geometrical and Kinematic Analysis.

According to the meshing theory of gear drives, a helical gear pair can be simulated as two cones with opposite direction at each engaging position. The half cone angle is the helix angle βb and the meshing model is shown in Fig. 1. The length of contact line KK' varies continuously during the meshing cycle, as shown in Fig. 2, while PP is the pitch line.

As shown in Fig. 2, the radius of curvature of a given mating tooth pair varies in time and in the direction of contact line KK'. The radius of curvature of arbitrary contact point C along the contact line can be defined asDisplay Formula

(1)${r1(y,t)=N1C¯=N1K¯(t)−(y−OK¯)⋅ sin βbr2(y,t)=N2C¯=N2K¯(t)+(y−OK¯)⋅ sin βb$

So, the surface velocities along the contact line can be expressed asDisplay Formula

(2)${u1(y,t)=r1(y,t)⋅ω1u2(y,t)=r2(y,t)⋅ω2$

where ω1 and ω2 are the angular velocities of gears 1 and 2, respectively.

Based on the geometry and kinematics analysis earlier, the parameters of gear EHL simulation along the contact line are given asDisplay Formula

(3)${Rx(y,t)=r1(y,t)⋅r2(y,t)/{[r1(y,t)+r2(y,t)]⋅ cos βb}ue(y,t)=[u1(y,t)+u2(y,t)]/2us(y,t)=u1(y,t)−u2(y,t)ξ(y,t)=us(y,t)/ue(y,t)$

where Rx is the equivalent radius of curvature. ue, us, and ξ are the entrainment velocity, sliding velocity, and slide-roll ratio, respectively.

###### Mixed Thermal Elastohydrodynamic Lubrication Model of a Helical Gear Pair.

The reduced Reynolds equation proposed by Hu and Zhu [21] is used to simulate the mixed EHL of helical gear pair. The nominal contact zone is divided into the hydrodynamic region and the asperity contact region. In the EHL region, the contact pressure is governed by the Reynolds equation accounting for non-Newtonian fluid behavior. It can be expressed asDisplay Formula

(4)$∂∂x(ρ12η*⋅h3⋅∂p∂x)+∂∂y(ρ12η*⋅h3⋅∂p∂y)=∂(ρueh)∂x+∂(ρh)∂t$

where p is the contact pressure, h is the film thickness, ρ is the density of lubricant, t is the time, and η* is the equivalent viscosity for non-Newtonian fluid.

The equivalent viscosity has been introduced in Eq. (4) to describe the non-Newtonian lubricant property. In the present study, the Eyring fluid has been adopted [22]. So, one can calculate the equivalent viscosity as follows:Display Formula

(5)$1η*=1η⋅τ0τx⋅sinh(τxτ0)$

where τ0 is the characteristic shear stress, τx is the surface shearing stress along the direction of entrainment velocity, and η is the viscosity of lubricant.

As mentioned in Ref. [21], the hydrodynamic pressure vanishes when the film thickness becomes zero. In this region, the asperities interact and the reduced Reynolds equation is employedDisplay Formula

(6)$∂(ρueh)∂x+∂(ρh)∂t=0$

The instantaneous lubricant film thickness at time t is defined asDisplay Formula

(7)$h(x,y,t)=h0(t)+x2/2Rx(y,t)+s1(x,y,t)+s2(x,y,t)+v(x,y,t)$

where h0(t) is the normal approach of two surfaces, s1 and s2 are the surfaces roughness, and v(x, y, t) is the surface elastic deformation, which is computed through the following integral:Display Formula

(8)$v(x,y,t)=2πE′∬Ωp(x′,y′,t′)(x−x′)2+(y−y′)2dx′dy′$

where E' is the equivalent elastic modulus and Ω is the computational domain.

The viscosity and density of lubricant apply the Roelands equation [23] and the Dowson–Higginson equation [24], respectively. These equations are as follows:Display Formula

(9)$η=η0 exp {(lnη0+9.67)[−1+(1+5.1×10−9p)z⋅(T−138T0−138)−s]}$
Display Formula
(10)$ρ=ρ0[1+0.6×10−9p1+1.7×10−9p−6.5×10−4(T−T0)]$

where η0 is the ambient viscosity of lubricant, ρ0 is the ambient density of lubricant, and T0 is the ambient temperature. $z=α/[5.1×10−9(lnη0+9.67)]$, $s=βT(T0−138)/(lnη0+9.67)$. Here, α and βT are the pressure–viscosity and the viscosity–temperature coefficient, respectively.

The applied load must be balanced by the contact pressure over entire solution domain. The equation is written asDisplay Formula

(11)$F(t)=Fm⋅ω(t)=∬Ωp(x,y,t)dxdy$

Here, Fm is the maximum load of tooth surface and ω(t) is the load ratio.

Due to the additional sliding movement along the teeth surfaces, the thermal effects should be considered. This effect can be modeled by incorporating a moving point hear source equation with the EHL governing equations to form a mixed TEHL model. Green's function, which is the temperature rise on the surface due to the moving point heat source, has been proposed by Carslaw and Jaeger [25]. Therefore, in consideration of the partition coefficient fi between the two teeth surfaces, the temperature rises on two surfaces at the specific position (x, y) from time 0 to t can be expressed as follows:Display Formula

(12)$ΔTi(x,y,t)=∫0t∬Ωfi(x′,y′,t′)⋅q(x′,y′,t′)4ρici[παi(t−t′)]1.5⋅ exp {−[(x−x′)−ui(t−t′)]2+(y−y′)24αi(t−t′)}dx′dy′dt′$

where, i = 1, 2 for gears 1 and 2. fi is the heat partition function and f1+f2=1. q is the heat flux density and can be obtained fromDisplay Formula

(13)$q(x,y,t)=μ(x,y,t)⋅p(x,y,t)⋅|us(y,t)|$

Here, μ(x, y, t) represents the instantaneous friction coefficient on the arbitrary contact point.

In the current study, we assume that the profile of temperature across the fluid film is close to the parabolic shape. This method has been widely adopted by the other research groups [26,27]. So, the temperature distribution across the film can be expressed asDisplay Formula

(14)$Tf(x,y,z,t)=(3T1+3T2−6Tm)(z/h)2+(6Tm−2T1−4T2)(z/h)+T2$

where T1 and T2 are the temperature of the teeth surfaces, and Tm is mean temperature across the film.

Then, according to Fourier's law, the heat flux on the two surfaces readsDisplay Formula

(15a)$f1(x,y,t)⋅q(x,y,t)=−kf⋅∂T∂z|z=h=kfh⋅(6Tm−4T1−2T2)$
Display Formula
(15b)$f2(x,y,t)⋅q(x,y,t)=kf⋅∂T∂z|z=0=kfh⋅(6Tm−2T1−4T2)$

By eliminating the mean film temperature Tm from Eqs. (15a) and (15b), the heat partition fi is determined by the following equation:Display Formula

(16)$ΔT2(x,y,t)−ΔT1(x,y,t)=q⋅h/2kf⋅(f1(x,y,t)−f2(x,y,t))$

where kf is the thermal conductivity of lubricant.

Then, the mean film temperature Tm can be calculated byDisplay Formula

(17)$Tm(x,y,t)=(2T1+T2)/3+f1⋅q⋅h/6kf$

###### Mechanical Power Loss of a Helical Gear Pair.

Both sliding and rolling actions at the gear mesh contact contribute to the mechanical power loss of a helical gear pair. In order to obtain the mechanical power loss of helical gear, the parameters such as surface shearing stress and friction coefficient should be determined first. The surface shearing stress between the teeth surfaces consists of the viscous shear within the hydrodynamic region and the contact friction within the asperity contact region. In the hydrodynamic region, the force equilibrium equation along the direction of entrainment velocity is given asDisplay Formula

(18)$∂τx−l∂z=∂p∂x$

An equivalent viscosity η* has been introduced in Eq. (5) to describe the non-Newtonian fluid. So, the viscous shear stress τx–l along the film thickness h is expressed asDisplay Formula

(19)$τx−l=η*∂u∂z$

Assuming no slip between the lubricant and teeth surfaces, i.e., u|z=0=u2 and u|z=h=u1. By integrating Eqs. (18) and (19) along the film thickness direction z, the lubricant velocity u can be obtained. So, the corresponding derivative of lubricant velocity along the film thickness direction z-axis is expressed asDisplay Formula

(20)$∂u∂z=ush+∂p∂x⋅1η*⋅(z−h2)$

Then, the viscous shear stress on the teeth surfaces can be defined asDisplay Formula

(21)$τx−l=η*∂u∂z=η*⋅ush+∂p∂x⋅(z−h2)$

The physical meaning of right-hand items of Eq. (21) is clear. The first and second terms are referred to as the sliding shear stress τs and rolling shear stress τr, respectively. So, the friction coefficient μl(x,y,t) in the hydrodynamic region is defined asDisplay Formula

(22)$μl(x,y,t)=|τx−l(x,y,t)|/p(x,y,t)$

In the asperity contact region, we assume that the friction coefficient is constant, μa(x,y,t) = 0.1. Then, the contact friction due to the asperity interaction is given asDisplay Formula

(23)$τx−a(x,y,t)=μa(x,y,t)⋅p(x,y,t)⋅us/|us|$

Therefore, the friction coefficient of tooth surface at a given time t can be written asDisplay Formula

(24)$μ(t)=|∬Ω[τx−l(x,y,t)+τx−a(x,y,t)]dxdy|/F(t)$

In the hydrodynamic region, the power loss at a certain time t is defined by integrating the product of viscous friction and the sliding velocity along the film thickness. It is as follows:Display Formula

(25)$PLl(t)=∬Ω∫0h[τx−l(x,y,t)⋅∂u∂z⋅dz]dxdy=∬Ωη*h⋅us2dxdy+∬Ωh312η*⋅(∂p∂x)2dxdy$

As mention earlier in Eq. (21), the first and second terms of the right-hand side of Eq. (25) are referred to as the sliding power loss PLs and rolling power loss PLr, respectively. Meanwhile, the corresponding power loss in the asperity contact region is given asDisplay Formula

(26)$PLa(t)=∬Ωτx−a(x,y,t)⋅usdxdy=∬Ωμa(x,y,t)⋅p(x,y,t)⋅|us|dxdy$

So, the total power loss at a certain time t over the entire meshing area of single tooth surface is found asDisplay Formula

(27)$PLm(t)=PLl(t)+PLa(t)$

In general, the contact ratio of helical gear pair is greater than one. It indicates that the helical gear pair has two or more teeth pairs in contact in part time during the whole meshing cycle. In the single tooth contact region, the overall gear power loss is determined by the power loss of the single tooth contact, as shown in Eq. (27). In the two or more teeth pairs contact region, the overall gear power loss is accumulated by the power loss per tooth pair. Hence, the overall helical gear mechanical power loss PL(t) and the instantaneous mechanical efficiency ηME are given asDisplay Formula

(28)$PL(t)=∑i=1N(t)[PLm(t)]i$
Display Formula
(29)$ηME(t)=1−PL(t)/(Tin⋅ω1)$

Here, N(t) is number of gear meshing teeth at time t.

## Numerical Method and Model Validation

The governing equations in Sec. 2 form differential–integral equations and are solved with the semi-system method [21]. The nominal Hertzian contact parameters pH and bH at the pitch point are introduced into the nondimension analysis. It should be noted that the dimensionless parameter along the y direction is chosen as the half of tooth width. So, in most calculation cases, the dimensionless calculation domain is chosen as X = [−2.5, 1.4] and Y = [−1.0, 1.0]. In order to avoid the numerical starvation, the calculation domain has been adjusted for some special cases. The elastic deformation and the thermal field are calculated through DC-FFT technique [28]. The iterative process is completed while the relative error of pressure is less than 10−6, the relative error of load is less than 10−5, and the relative error of thermal field is less than 10−6.

In order to validate the accuracy of the proposed model, the model predictions will be compared to the published data of Xu [3]. The parameters of helical gear and lubricant are listed in Table 1. The corresponding working condition is Tin = 500 N·m and ω1 = 4000 r/min. So, the nominal Hertzian contact parameters pH and bH are 0.79 GPa and 0.195 mm, respectively. It should be noted that the load variation is treated in the way used in Refs. [1720]. The corresponding instantaneous equivalent radius of curvature Rx, entrainment velocity ue, sliding velocity us, and slide-roll ratio ξ as a function of gear mesh position are shown in Fig. 3. Due to lack of the measured surface roughness used in Ref. [3], an alternative surface roughness is used in this study and the corresponding surface roughness in root-mean-square (RMS) is 0.1002 μm, as shown in Fig. 4. The comparisons of instantaneous average friction coefficient μav (μav = Average [μ(t)]i=N(t)) and mechanical efficiency ηME between the present study and Xu's results [3] of Fig. 4.5 are shown in Fig. 5.

It should be pointed out that the actual EHL analysis, experimental formula, and EHL-based formula results have been taken from Ref. [3]. Figure 5 reveals that the agreement between the current study and Xu's results is satisfactory. The minor difference between predicted value of the current model and Xu' results may be derived from these aspects. First, the finite line contact has been adopted in the current model, while a number of thin slices of spur gears as a line contact have been used to simulate the helical gear lubrication in Ref. [3]. Second, one possible reason for this difference could be the temperature models. However, based on the comparisons of Fig. 5, it can be stated that the proposed model can be applied to investigate the mechanical power loss of helical gear pair. Furthermore, in the calculation cases below, the whole meshing period along the LOA is discretized into 256 points. The computational grid covering the domain X × Y consists of 256 × 512 equal spaced nodes.

## Results and Discussion

###### Influence of Thermal Effect.

In order to discuss the influence of thermal effect on the mechanical power loss of helical gear pair, three typical helical gear pairs with different transmission ratios (tr) have been employed, such as tr = 0.5, tr = 1.0, and tr = 1.5. Except for the number of gear teeth z2, the other design parameters are the same as shown in Table 1. The input torque and speed are Tin = 500 N·m and ω1 = 4000 r/min. Meanwhile, the effect of surface roughness is not considered in this section.

Figure 6 shows the variation in the average film thickness, friction coefficient, and geometrical and kinematic parameters for three typical transmission ratios. The term ha represents the average film thickness in the central part of nominal Hertzian contact area. The subscripts iso and t represent the isothermal and thermal results, respectively. Figure 6 reveals that the variation of average film thickness is relatively smooth except for the engaging-in region. This phenomenon has been discussed in detail in Ref. [20]. Figure 6 shows that the influence of thermal effect on the friction coefficient is more significant than the average film thickness, especially in the larger slide-roll ratio region. Meanwhile, the overall variation trend of average film thickness is consistent with that of the equivalent radius of curvature and entrainment velocity, especially for tr = 0.5 and tr = 1.5.

In Fig. 7, the variation of the traction force and mechanical power loss along the tooth surface is shown for the same example cases as in Fig. 6. The subscripts s and r represent the sliding and the rolling traction force, respectively. The value of Fs and Fr is defined by integrating the sliding shear stress τs and the rolling shear stress τr in the whole computational domain. As shown in Figs. 7(a), 7(c), and 7(e), the distribution of sliding traction force acts like a sine function. Meanwhile, the rolling traction force is almost negligible compared to the sliding traction force. However, rolling power loss PLr generated by the rolling traction force constitutes an important portion of the total meshing power loss PLm, especially around the meshing position P where the pitch point is located at the middle of contact line KK'. For the current three helical gear pairs, neglecting the rolling power loss will result in the maximum 10.2%, 28.6%, and 43.6% underestimation of PLm for the tr = 0.5, tr = 1.0, and tr = 1.5, respectively. It is also observed from Fig. 7 that the influence of thermal effect on the traction force and mechanical power loss is significant. Meanwhile, the thermal effect results in the maximum 47.5%, 28.5%, and 17.8% of mechanical power loss error for tr = 0.5, tr = 1.0, and tr = 1.5, respectively.

For the example tr = 1.0, the corresponding contact line change and thermal EHL results at the meshing position P are illustrated in Figs. 8 and 9. As shown in Fig. 8, the meshing position P corresponds to the instantaneous contact line when the pitch point locates at the middle of the contact line KK'. The thermal EHL results have been shown in Fig. 9. Figure 9(c) shows that the temperature distribution at the central area is nearly equal to the ambient temperature as the local slide-roll ratio is being zero. At the meshing position P, the total surface traction force is almost zero due to the direction change of shear force, but the mechanical power loss is considerable. Figure 8 shows that the meshing positions A, B, C, and D are symmetric about the meshing position P for the unity-ratio example helical gear pair. For instance, at the meshing positions A and B, the pitch point is located at the end of contact line KK' and the length of contact line is maximum for this case. So, the surface traction force reaches its maximum level at the meshing positions A and B. Then, at the meshing positions C and D, the contact line reaches its maximum value along the whole meshing cycle. The total absolute value of slide-roll ratio of contact line at the meshing positions C and D is maximum compared to the each instantaneous along the LOA. For this reason, the mechanical power loss is the maximum value at the meshing positions C and D. This similar conclusion can be obtained in the example tr = 0.5 and tr = 1.5.

###### Influence of Input Torque and Speed.

In Sec. 4.1, the thermal effect on the lubrication performance and mechanical power loss has been studied under the specific working condition for three typical helical gear pairs. Then, this section tries to explore the lubrication performance and mechanical power loss under a wide range of working conditions. An example condition with a fixed transmission ratio is considered here. The input speed is varied from 250 to 5000 r/min. The input torque is varied between 125 and 1700 N·m. So, the corresponding Hertzian pressure pH and entrainment velocity uep at the pitch point are varied from 0.4 to 1.5 GPa and 0.67 to 13.4 m/s, respectively. In addition, the tribological performance parameters are time-varying along the LOA, such as ha, μ, μav, PL, and ηME. For the convenience of discussion, the mean value of tribological performance parameters under a specific working condition is defined asDisplay Formula

(30)$Δ¯=∫LOAΔds/LOA$

Here, LOA represents the transverse length of meshing along the line of action from B1 point to B2 point as shown in Fig. 2. Δ is the instantaneous value of tribological performance parameters, such as ha, μ, μav, PL, and ηME.

The effect of working conditions on the lubrication performance and mechanical power loss is shown in Fig. 10. Figure 10(a) shows that the input speed is still a major factor affecting the film thickness. As the input speed increases, the film thickness increases significantly. Figures 10(b) and 10(c) reveal that the friction coefficient is hardly affected by the input speed under light load, Tin < 280 N · m. This is because as the input torque is light, the film thickness is large enough and the helical gear pair lies in full-film lubrication regime. So, the corresponding friction coefficient is as low as μ < 0.005. Besides, as the input torque increases, friction coefficient increases gradually. The friction coefficient reaches its maximum μ ≈ 0.02 under the case of low speed and heavy-duty, as shown in Figs. 10(b) and 10(c).

As shown in Fig. 10, the contribution of sliding action to the total power loss is dominant during most of the working conditions, except for light-duty and high speed cases. As the input torque and input speed increases, the sliding power loss and the total power loss increase gradually and reach its maximum under a high speed and heavy load cases. It is seen from Fig. 10(f) that the input torque does not seem to have a great influence on the rolling power loss. Additionally, in Fig. 10(f), as the input speed goes up, the rolling power loss increases gradually. This phenomenon can be explained by Eq. (25). As the input speed increases, the sliding velocity us and the film thickness h increase simultaneously. According to the Dowson–Higginson empirical formula [29], the film thickness is proportional to 0.7 power of speed (H = 2.65G0.54U0.70/W0.13). So, the term of $us2/h$ goes up with the increase of the input speed. Furthermore, as the input torque increases, the equivalent viscosity increases. Hence, the effect of input speed and torque on the sliding power loss is obvious. It is also observed from Fig. 10(h) that the sliding power loss becomes increasingly dominant on the total power loss as the input speed and torque increase simultaneously. It should be pointed out that the rolling power loss is indeed not negligible, especially at the high speed and light load cases. The corresponding mechanical efficiency has been shown in Fig. 10(d). Figure 10(d) reveals that the mechanical power loss is less than 1% in the above working conditions. Meanwhile, the mechanical efficiency reaches its maximum under a low speed and light load case, and obtains its minimum under a low speed and heavy load cases.

###### Influence of Geometrical Parameters.

In general, the boundary conditions of design a gear pair usually include the gear center distance and input power. In order to isolate the effect of structure parameters, the gear center distance, the face width of gears, input torque, and rotational speed keep constant at a specific transmission ratio. Four basic design parameters, such as normal pressure αn, helix angle β, normal module mn, and addendum modification x1, are chosen to investigate with various transmission ratios. The normal pressure αn and helix angle β are varied between 10 deg and 30 deg. The normal module mn is varied from 1.4 to 3.0 mm. The addendum modification for the pinion is chosen as x1 = −0.5 to 0.5. The transmission ratio is chosen as tr = 0.36, 0.5, 0.8, 1.0, 2.0, and 4.0. In this section, all cases with a fixed input torque Tin = 500 N·m and input speed ω1 = 2000 r/min are considered. That is to say, the input power is constant Pin =1 04.7 kW. The thermal effect is considered in the simulations with ambient temperature T0 = 100 °C.

###### Influence of Normal Pressure Angle.

Figure 11 shows the effect of normal pressure angle on the lubrication performance and mechanical power loss. In this section, the helix angle, normal module, and addendum modification are kept at β = 25.232 deg, mn = 2.714 mm, and x1 = 0, respectively. Figure 11 shows that, as the normal pressure angle increases, the average film thickness increases as well as the friction coefficient decreases. The same trend is observed from the change of transmission ratio. This can be explained that, based on the gear meshing theory, as the normal pressure angle and transmission ratio go up, the equivalent radius and entrainment velocity go up. Figure 11(d) indicates that the mechanical efficiency can be improved by increasing the normal pressure angle.

Figures 11(e) and 11(f) show that the normal pressure angle has significant effect on the mechanical power loss, especially for the sliding power loss. The variation trend of overall mechanical power loss is consistent with the change of friction coefficient shown in Fig. 11(b). Figure 11(f) shows that, as the normal pressure angle increases, the rolling power loss increases gradually. With the increase of transmission ratio, the change of rolling power loss is not significant. This phenomenon can be explained by Eq. (25). It indicates that the contribution of film thickness to calculate the rolling power loss is dominant under the fixed working condition. So, the variation trend of rolling power loss is similar to that of film thickness drawn in Fig. 11(a). In additional, it is seen from Fig. 11(h) that the rolling power loss becomes increasingly dominant on the total power loss as the normal pressure angle and transmission ratio increase simultaneously. In most cases, the corresponding friction coefficient is less than 0.01 and the mechanical efficiency is larger than 0.9988.

###### Influence of Helix Angle.

In order to discuss the effect of helix angle on the lubrication performance and mechanical power loss, the normal pressure angle, tooth number of pinion, and addendum modification are kept at αn=18.224 deg, z1 = 50, and x1 = 0, respectively. Figure 12 shows the effect of helix angle on the lubrication performance and mechanical power loss. In general, the total contact ratio for helical gear pair can be increased conveniently by increasing the helix angle. In Fig. 12(a), the film thickness is increased by increasing the helix angle, but the trend is less significant, especially a helical gear pair with the transmission ratio tr less than 2. The variation of friction coefficient shown in Figs. 12(b) and 12(c) is opposite to that of film thickness. For a specific transmission ratio, Fig. 12(d) shows that the improvement of mechanical efficiency by increasing the helix angle is less significant. In most cases, the mechanical efficiency increases only 0.02% by increasing the helix angle from 10 deg to 30 deg for a specific transmission ratio. As we know, the larger helix angle will lead to an increase in axial force. Hence, the helix angle of helical gear should not be too large in the industrial design.

Figure 12(e) illustrates that the variation of sliding power loss is consistent with that of friction coefficient shown in Fig. 12(b). Figure 12(g) shows that increasing helix angle from 10 deg to 30 deg for a specific transmission ratio reduces mechanical power by only about 30%. As the total contact ratio increases conveniently by increasing the helix angle, the improvement of mechanical efficiency is limited. For all the example cases, Fig. 12(h) indicates that the contribution of sliding friction on the overall mechanical power loss is dominant.

###### Influence of Normal Module.

Figure 13 shows the effect of normal module on the lubrication performance and mechanical power loss for helical gear pair under several transmission ratios. In this situation, the other design parameters are fixed at αn=18.224 deg, β = 25.232 deg, and x1 = 0, respectively. With a fixed center distance, Fig. 13(a) reveals that, as the normal module increases, the average film thickness decreases and this trend is a less significant way. The corresponding friction coefficient has been shown in Figs. 13(b) and 13(c). The variation of friction coefficient is opposite to that of film thickness. The same conclusion has been obtained to discuss the effect of normal pressure angle and helix angle. Figure 13(d) illustrates that, as the normal module increases with the decrease in the corresponding tooth number, the mechanical efficiency decreases by about 0.1%. This means that the finer pitch gear has lower mechanical power losses as described in Refs. [1,5], and [12].

Figure 13(e) shows that the effect of normal module on the sliding power loss is similar to that of friction coefficient. It indicates that the sliding shear stress is the main component of the formation of friction coefficient and sliding power loss. In Fig. 13(f), it reveals that, as the normal module increases, the increasing trend of rolling power loss is significant. Figures 13(g) and 13(h) show that the contribution of sliding power loss on the overall mechanical power loss is dominant, especially a helical gear pair with the transmission ratio tr less than 2. In additional, Fig. 13(g) reveals that the overall mechanical power loss goes up by about 3.3 times when the normal module goes up from 1.4 to 3.0 mm. The change in trend is consistent with Ref. [5]. Besides, the corresponding mechanical efficiency is reduced by about 0.1%.

The addendum modification is often applied for improving the wear resistance and service life. The high displacement is chosen here to study the effect of addendum modification on the lubrication performance and mechanical power loss of the helical gear pair. That is to say, x1 + x2 = 0. The corresponding results have been shown in Fig. 14. In this section, the other design parameters are fixed at αn = 18.224 deg, β = 25.232 deg, and mn = 2.714 mm, respectively. Figures 14(a)14(c) show that the film thickness and the friction coefficient are less affected by the addendum modification under specific transmission ratio cases (tr < 1.0). For the transmission ratio tr > 1.0 cases, as the addendum modification increases, the film thickness increases and the friction coefficient decreases. This is due to the change of addendum modification, a different involute has been chosen as the tooth profile. Figure 14(d) indicates that, as the addendum modification increases, the mechanical efficiency increases first and then decreases when the addendum modification reaches x1≈0.2. This suggests that a positive addendum modification coefficient for pinion and a negative addendum modification coefficient for wheel are good for improving the mechanical efficiency.

Figure 14(f) shows that the effect of addendum modification on the rolling power loss is consistent with that of film thickness. The similar conclusion can been obtained by the other design parameters, such as normal pressure angle, helix angle, and normal module. Figures 14(g) and 14(h) illustrate that the contribution of rolling power loss on the overall mechanical power loss is enhanced when the transmission ratio increases. However, the proportion of sliding power loss in the overall mechanical power loss is more than 67.2% for all the example cases.

###### Influence of Surface Roughness.

In the study earlier, the ideal smooth tooth surface has been considered for the parameter sensitivity analysis of the mechanical power loss. In fact, gear teeth surfaces will form different roughness texture due to the diverse tooth surface processing methods. In order to discuss the effect of surface roughness on the mechanical power loss, the roughness profile shown in Fig. 4 is used for both teeth surfaces. The surface roughness with different RMS values is obtained by multiplying coefficient with the original roughness data. In the current simulation, the input torque and speed are kept at Tin = 500 N·m and ω1 = 2000 r/min, respectively. The transmission ratio is fixed at tr = 1.0. The surface roughness of gears is RMS = 0.02 and 0.05 μm.

The comparison results of lubrication performance and mechanical power loss between smooth surfaces and roughness teeth surfaces have been shown in Figs. 15 and 16. Due to the low RMS value, the whole meshing process belongs to a micro-EHL state under all simulation cases. Figure 15(a) shows that, with increases of the RMS value, the average film thickness increases correspondingly. This is due to the fact that the roughness valley holds much more lubrication in the micro-EHL state. As shown in Fig. 15(b), it is also shown that the effect of roughness on the friction coefficient is obvious, especially in the engaging-in and the engaging-out region. The same trend is also observed in Fig. 15(c) for the sliding traction force. Figure 15(d) shows that the effect of roughness on the rolling traction force is consistent during the whole meshing cycle. Figure 16 indicates that the mechanical efficiency is reduced by increasing the roughness RMS value. Additionally, in Fig. 16, the variations of the mechanical power loss along the LOA are shown for the same example cases as in Fig. 7.

## Conclusions

In this work, a thermo-EHL model is proposed to investigate the mechanical power loss of helical gear pair. The influence of basic design parameters, working condition, and surface roughness on the lubrication performance and mechanical power loss are evaluated. The conclusions can be made as follows:

1. (1)Results show that the contribution of thermal effect is indeed not negligible for the lubrication analysis and mechanical power loss prediction of helical gear pair, especially the calculation of friction coefficient and sliding power loss. It also says that the rolling power loss constitutes an important portion of the total meshing power loss, especially around the meshing position where the pitch point is located in the middle of contact line.
2. (2)Under different working conditions, the influence of input torque and speed on the lubrication performance is obvious difference. Such as the friction coefficient is hardly affected by the input speed under light load. As the input speed decreases and input torque increases, the friction coefficient increases and the corresponding mechanical efficiency is lowest within the range of simulated operating condition. Besides, the rolling power loss is mainly affected by the input speed. Moreover, the contribution of rolling power loss to the total power loss is not negligible, especially at the high speed and light load cases.
3. (3)The results indicate that the proper increase of normal pressure angle can improve the lubrication performance and enhance the mechanical efficiency. As the normal pressure angle increases, the contribution of rolling power loss to the total power loss is not negligible, especially at the larger transmission ratio. The influence of helix angle on the mechanical efficiency is less significant. As the normal module decreases with the increase in the corresponding tooth number, it is good for improving the lubrication performance and enhancing the mechanical efficiency. Meanwhile, as the normal module increases, the rolling power loss cannot be ignored. The proper addendum modification, a positive modification coefficient for pinion, and a negative modification coefficient for wheel are good for improving the mechanical efficiency.
4. (4)In the micro-EHL lubrication state, the average film thickness goes up with the increase of the surface roughness. The influence of surface roughness is remarkable on the friction coefficient, especially in the engaging-in and engaging-out region. Besides, the mechanical efficiency is reduced by increasing the roughness RMS value.

## Funding Data

• The National Key R&D Program of China (Grant No. 2017YFD0701105-03).

• The National Natural Science Foundation of China (Grant No. 51405142).

## Nomenclature

• B =

teeth width, mm

• c1, c2 =

specific heats of solids, J/kg·k

• E′ =

equivalent elastic module, GPa

• F(t), Fm =

• fi =

heat partition function, dimensionless

• h =

film thickness, μm

• KK′ =

contact line, dimensionless

• kf =

thermal conductivity of lubricant, W/m·k

• mn =

normal module, mm

• PP =

pitch line, dimensionless

• p =

pressure, GPa

• PLs, PLr =

sliding and rolling power loss, W

• PLm =

total power loss, W

• q =

heat flux density, W/m2

• r1, r2 =

• Rx =

• s1, s2 =

surface roughness, μm

• t =

time, s

• Ti, Tf =

temperature of two surfaces and oil film, K

• Tin =

input torque, N·m

• u1, u2 =

surface velocities, mm/s

• tr =

transmission ratio, dimensionless

• ue =

entrainment velocity, mm/s

• us =

sliding velocity, mm/s

• v =

surface elastic deformation, mm

• z1, z2 =

number of teeth, dimensionless

• x, y, z =

coordinate system, dimensionless

• α =

viscosity-pressure coefficient, GPa−1

• αn =

normal pressure angle, deg

• βb =

helix angle, deg

• βT =

viscosity-temperature coefficient, K−1

• ξ =

slide-roll ratio, dimensionless

• ρ0, ρ =

density of the lubricant, kg/m3

• η0, η =

viscosity of the lubricant, Pa·s

• η* =

equivalent viscosity, Pa·s

• ηME =

mechanical efficiency, dimensionless

• τ0 =

characteristic shear stress, MPa

• τx =

surface shearing stress, MPa

• μ, μl, μa =

friction coefficient, dimensionless

• ω1 =

input speed, r/min

## References

Wu, S. , and Cheng, H. S. , 1991, “A Friction Model of Partial-EHL Contacts and its Application to Power Loss in Spur Gears,” Tribol. Trans., 34(3), pp. 398–407.
Haizuka, S. , Naruse, C. , and Yamanaka, T. , 1999, “Study of Influence of Helix Angle on Friction Characteristics of Helical Gears,” Tribol. Trans., 42(3), pp. 570–580.
Xu, H. , 2005, “Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs,” Ph.D. thesis, The Ohio State University, Columbus, OH.
Heingartner, P. , and Mba, D. , 2003, “Determining Power Losses in the Helical Gear Mesh; Case Study,” ASME Paper No. DETC2003/PTG-48118.
Li, S. , Vaidyanathan, A. , Harianto, J. , and Kahraman, A. , 2009, “Influence of Design Parameters on Mechanical Power Losses of Helical Gear Pairs,” J. Adv. Mech. Des., Syst., Manuf., 3(2), pp. 146–158.
Baglioni, S. , Cianetti, F. , and Landi, L. , 2012, “Influence of the Addendum Modification on Spur Gear Efficiency,” Mech. Mach. Theory, 49, pp. 216–233.
Douglas, C. E. , and Thite, A. , 2015, “Effect of Lubricant Temperature and Type on Spur Gear Efficiency in Racing Engine Gearbox Across Full Engine Load and Speed Range,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 229(9), pp. 1095–1113.
Marques, P. M. T. , Martins, R. C. , and Seabra, J. H. O. , 2016, “Gear Dynamics and Power Loss,” Tribol. Int., 97, pp. 400–411.
Liu, H. , Zhu, C. , Zhang, Y. Y. , Wang, Z. , and Song, C. , 2016, “Tribological Evaluation of a Coated Spur Gear Pair,” Tribol. Int., 99, pp. 17–126.
Britton, R. D. , Elcoate, C. D. , Alanou, M. P. , Evans, H. P. , and Snidle, R. W. , 2000, “Effect of Surface Finish on Gear Tooth Friction,” ASME J. Tribol., 122(1), pp. 354–360.
Diab, Y. , Ville, F. , and Velex, P. , 2006, “Prediction of Power Losses Due to Tooth Friction in Gears,” Tribol. Trans., 49(2), pp. 260–270.
Petry-Johnson, T. T. , Kahraman, A. , Anderson, N. E. , and Chase, D. R. , 2008, “An Experimental Investigation of Spur Gear Efficiency,” ASME J. Mech. Des., 130(6), p. 062601.
Luis, M. , Ramiro, M. , Cristiano, L. , and Jorge, S. , 2010, “Influence of Tooth Profile and Oil Formulation on Gear Power Loss,” Tribol. Int., 43(10), pp. 1861–1871.
Isaacson, A. C. , Wagner, M. E. , Rao, S. B. , and Sroka, G. , 2016, “Impact of Surface Condition and Lubricant on Effective Gear Tooth Friction Coefficient,” American Gear Manufacturers Association Fall Technical Meeting 2016, Pittsburgh, PA, Oct. 2–4, pp. 41–47.
Ziegltrum, A. , Lohner, T. , and Stahl, K. , 2017, “TEHL Simulation on the Influence of Lubricants on Load-Dependent Gear Losses,” Tribol. Int., 113, pp. 252–261.
Andersson, M. , Sosa, M. , and Olofsson, U. , 2017, “Efficiency and Temperature of Spur Gears Using Spray Lubrication Compared to Dip Lubrication,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 231(11), pp. 1390–1396.
Zhu, C. , Liu, M. , Liu, H. , Xu, X. , and Liu, L. , 2013, “A Thermal Finite Line Contact EHL Model of a Helical Gear Pair,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 227(4), pp. 299–309.
Liu, M. , Zhu, C. , Liu, H. , Ding, H. , and Sun, Z. , 2014, “Effects of Working Conditions on TEHL Performance of a Helical Gear Pair With Non-Newtonian Fluids,” ASME J. Tribol., 136(2), p. 021502.
Liu, M. , Zhu, C. , Liu, H. , and Wu, C. , 2016, “Parametric Studies of Lubrication Performance of a Helical Gear Pair With Non-Newtonian Fluids,” J. Mech. Sci. Tech., 30(1), pp. 317–326.
Liu, M. , Liu, Y. , and Wu, C. , 2018, “The Transient and Thermal Effects on EHL Performance of a Helical Gear Pair,” Tribology, 13(3), pp. 81–90.
Hu, Y. , and Zhu, D. , 2000, “A Full Numerical Solution to the Mixed Lubrication in Point Contacts,” ASME J. Tribol., 122(1), pp. 1–9.
Eyring, H. , 1936, “Viscosity, Plasticity and Diffusion as Examples of Absolute Reaction Rates,” J. Chem. Phys., 4(4), pp. 283–291.
Roelands, C. J. A. , 1966, “Correlation Aspects of Viscosity-Temperature-Pressure Relationship of Lubricating Oils,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Dowson, D. , Higginson, G. R. , and Whitaker, A. V. , 1962, “Elasto-Hydrodynamic Lubrication: A Survey of Isothermal Solutions,” Archive J. Mech. Eng. Sci., 4(2), pp. 121–126.
Carslaw, H. S. , and Jaeger, J. C. , 1959, Conduction of Heat in Solids, 2nd ed., Oxford at the Clarendon Press, London.
Kim, H. J. , Ehret, P. , Dowson, D. , and Taylor, C. M. , 2001, “Thermal Elastohydrodynamic Analysis of Circular Contacts—Part 1: Newtonian Model,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 215(4), pp. 339–352.
Li, S. , Kahraman, A. , Anderson, N. , and Wedeven, L. D. , 2013, “A Model to Predict Scuffing Failures of a Ball-on-Disk Contact,” Tribol. Int., 60, pp. 233–245.
Liu, S. B. , Wang, Q. , and Liu, G. , 2000, “A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses,” Wear, 243(1–2), pp. 101–111.
Dowson, D. , 1995, “Elastohydrodynamic and Micro-Elastohydrodynamic Lubrication,” Wear, 190(2), pp. 125–138.
View article in PDF format.

## References

Wu, S. , and Cheng, H. S. , 1991, “A Friction Model of Partial-EHL Contacts and its Application to Power Loss in Spur Gears,” Tribol. Trans., 34(3), pp. 398–407.
Haizuka, S. , Naruse, C. , and Yamanaka, T. , 1999, “Study of Influence of Helix Angle on Friction Characteristics of Helical Gears,” Tribol. Trans., 42(3), pp. 570–580.
Xu, H. , 2005, “Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs,” Ph.D. thesis, The Ohio State University, Columbus, OH.
Heingartner, P. , and Mba, D. , 2003, “Determining Power Losses in the Helical Gear Mesh; Case Study,” ASME Paper No. DETC2003/PTG-48118.
Li, S. , Vaidyanathan, A. , Harianto, J. , and Kahraman, A. , 2009, “Influence of Design Parameters on Mechanical Power Losses of Helical Gear Pairs,” J. Adv. Mech. Des., Syst., Manuf., 3(2), pp. 146–158.
Baglioni, S. , Cianetti, F. , and Landi, L. , 2012, “Influence of the Addendum Modification on Spur Gear Efficiency,” Mech. Mach. Theory, 49, pp. 216–233.
Douglas, C. E. , and Thite, A. , 2015, “Effect of Lubricant Temperature and Type on Spur Gear Efficiency in Racing Engine Gearbox Across Full Engine Load and Speed Range,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 229(9), pp. 1095–1113.
Marques, P. M. T. , Martins, R. C. , and Seabra, J. H. O. , 2016, “Gear Dynamics and Power Loss,” Tribol. Int., 97, pp. 400–411.
Liu, H. , Zhu, C. , Zhang, Y. Y. , Wang, Z. , and Song, C. , 2016, “Tribological Evaluation of a Coated Spur Gear Pair,” Tribol. Int., 99, pp. 17–126.
Britton, R. D. , Elcoate, C. D. , Alanou, M. P. , Evans, H. P. , and Snidle, R. W. , 2000, “Effect of Surface Finish on Gear Tooth Friction,” ASME J. Tribol., 122(1), pp. 354–360.
Diab, Y. , Ville, F. , and Velex, P. , 2006, “Prediction of Power Losses Due to Tooth Friction in Gears,” Tribol. Trans., 49(2), pp. 260–270.
Petry-Johnson, T. T. , Kahraman, A. , Anderson, N. E. , and Chase, D. R. , 2008, “An Experimental Investigation of Spur Gear Efficiency,” ASME J. Mech. Des., 130(6), p. 062601.
Luis, M. , Ramiro, M. , Cristiano, L. , and Jorge, S. , 2010, “Influence of Tooth Profile and Oil Formulation on Gear Power Loss,” Tribol. Int., 43(10), pp. 1861–1871.
Isaacson, A. C. , Wagner, M. E. , Rao, S. B. , and Sroka, G. , 2016, “Impact of Surface Condition and Lubricant on Effective Gear Tooth Friction Coefficient,” American Gear Manufacturers Association Fall Technical Meeting 2016, Pittsburgh, PA, Oct. 2–4, pp. 41–47.
Ziegltrum, A. , Lohner, T. , and Stahl, K. , 2017, “TEHL Simulation on the Influence of Lubricants on Load-Dependent Gear Losses,” Tribol. Int., 113, pp. 252–261.
Andersson, M. , Sosa, M. , and Olofsson, U. , 2017, “Efficiency and Temperature of Spur Gears Using Spray Lubrication Compared to Dip Lubrication,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 231(11), pp. 1390–1396.
Zhu, C. , Liu, M. , Liu, H. , Xu, X. , and Liu, L. , 2013, “A Thermal Finite Line Contact EHL Model of a Helical Gear Pair,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 227(4), pp. 299–309.
Liu, M. , Zhu, C. , Liu, H. , Ding, H. , and Sun, Z. , 2014, “Effects of Working Conditions on TEHL Performance of a Helical Gear Pair With Non-Newtonian Fluids,” ASME J. Tribol., 136(2), p. 021502.
Liu, M. , Zhu, C. , Liu, H. , and Wu, C. , 2016, “Parametric Studies of Lubrication Performance of a Helical Gear Pair With Non-Newtonian Fluids,” J. Mech. Sci. Tech., 30(1), pp. 317–326.
Liu, M. , Liu, Y. , and Wu, C. , 2018, “The Transient and Thermal Effects on EHL Performance of a Helical Gear Pair,” Tribology, 13(3), pp. 81–90.
Hu, Y. , and Zhu, D. , 2000, “A Full Numerical Solution to the Mixed Lubrication in Point Contacts,” ASME J. Tribol., 122(1), pp. 1–9.
Eyring, H. , 1936, “Viscosity, Plasticity and Diffusion as Examples of Absolute Reaction Rates,” J. Chem. Phys., 4(4), pp. 283–291.
Roelands, C. J. A. , 1966, “Correlation Aspects of Viscosity-Temperature-Pressure Relationship of Lubricating Oils,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Dowson, D. , Higginson, G. R. , and Whitaker, A. V. , 1962, “Elasto-Hydrodynamic Lubrication: A Survey of Isothermal Solutions,” Archive J. Mech. Eng. Sci., 4(2), pp. 121–126.
Carslaw, H. S. , and Jaeger, J. C. , 1959, Conduction of Heat in Solids, 2nd ed., Oxford at the Clarendon Press, London.
Kim, H. J. , Ehret, P. , Dowson, D. , and Taylor, C. M. , 2001, “Thermal Elastohydrodynamic Analysis of Circular Contacts—Part 1: Newtonian Model,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 215(4), pp. 339–352.
Li, S. , Kahraman, A. , Anderson, N. , and Wedeven, L. D. , 2013, “A Model to Predict Scuffing Failures of a Ball-on-Disk Contact,” Tribol. Int., 60, pp. 233–245.
Liu, S. B. , Wang, Q. , and Liu, G. , 2000, “A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses,” Wear, 243(1–2), pp. 101–111.
Dowson, D. , 1995, “Elastohydrodynamic and Micro-Elastohydrodynamic Lubrication,” Wear, 190(2), pp. 125–138.

## Figures

Fig. 1

A pair of meshing helical gear pair

Fig. 2

The variation of contact line for helical gear pair along the LOA

Fig. 3

The variation of geometrical and kinematic parameters for helical gears in Ref. [3]

Fig. 4

The measured surface roughness profile in RMS = 0.1002 μm

Fig. 5

Comparison EHL results between current study and Xu's results [3]

Fig. 6

Variation of average film thickness, friction coefficient, and geometrical and kinematic parameters for three typical transmission ratios (tr = 0.5, tr = 1.0, and tr = 1.5)

Fig. 7

Variation of the traction force, mechanical power loss along the tooth surface for three typical transmission ratios (tr = 0.5, tr = 1.0, and tr = 1.5)

Fig. 8

Time varying contact line of helical gear pair with transmission ratio tr = 1.0

Fig. 9

Distribution of pressure, film thickness, oil film temperature, and shear stress for transmission ratio tr = 1.0 at meshing position P

Fig. 10

Effect of working conditions on the lubrication performance and mechanical power loss along the LOA

Fig. 11

Influence of normal pressure angle on the lubrication performance and mechanical power loss under several transmission ratios

Fig. 12

Influence of helix angle on the lubrication performance and mechanical power loss under several transmission ratios

Fig. 13

Influence of normal module on the lubrication performance and mechanical power loss under several transmission ratios

Fig. 14

Influence of addendum modification on the lubrication performance and mechanical power loss under several transmission ratios

Fig. 15

Effect of surface roughness on the average film thickness, friction coefficient and traction force along the LOA with the transmission ratio tr = 1.0

Fig. 16

Effect of surface roughness on the mechanical power loss and mechanical efficiency along the LOA with the transmission ratio tr = 1.0

## Tables

Table 1 Helical gear parameters and material properties of lubricant and solid

## Errata

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