An Improved Algorithm for Calculating Friction Force and Torque in Involute Helical Gears

Abstract

Time varying frictional force and torque are one of the main exciting sources of vibration in helical gears. This paper presents an approach to determine the friction force and torque in involute helical gears considering nonuniform load distribution along contact lines. An analytical load distribution model is employed and extended to obtain the load per unit of length along contact lines. Friction force and torque models under nonuniform assumption are derived. Comparisons of the determined friction force and torque with the results from uniform assumption are made. In addition, the differences between constant friction coefficient and varying coefficient are revealed. Moreover, two typical design cases of helical gears are studied. Results show that the fluctuations of friction force and torque under uniform assumption are more significant than those under nonuniform assumption in sample I for a single tooth, but less significant for the sum of those of the three teeth, while in sample II, the fluctuations under uniform assumption are less significant than those under nonuniform assumption. The friction coefficient induced difference is negligible compared with the difference induced by load distribution assumptions.

1. Introduction

Gears have been widely used in many fields, such as heavy-duty machines, helicopters, marine ships, and machine tools. The noise and vibration significantly affect the efficiency and reliability in these applications. Besides the time-varying mesh stiffness, damping, static transmission error, and backlash, the time-varying friction force and torque between meshing tooth pairs are also important exciting sources of the noise and vibration in gear pairs.

A number of literature works concerning the effect of friction on dynamics of spur gears could be found. The finite element method was employed in [1] to model the elastic properties of spur gear pair and to analyze the effects of friction. It was concluded that the input and output torque did not remain constant when frictional effects were taken into account, and a time-varying dynamic transmission error was induced. The influence of tooth friction on gear dynamics, where average mesh stiffness per unit of contact length, uniform normal load distributions, and constant coefficient of friction have been assumed, has been investigated and the results revealed the potentially significant contribution of tooth friction to gear vibration and noise. Sliding friction accounts for most of the losses at low-medium speeds while rolling friction becomes more significant at high speed [2, 3].

The authors of [4, 5] presented the procedure to obtain the friction force and torque. The author of [4] has presented a 16-DOF dynamic model for a spur gear pair by incorporating friction effects between meshing teeth. The coefficient of friction was assumed to vary linearly with shaft angle, and the friction force is calculated from the normal meshing force comprising both mesh stiffness and damping values. A linear time-varying and a nonlinear time-varying dynamics for spur gear systems were modeled to investigate the effects introduced by friction, with different considerations of coefficient of friction. And then the authors gave the semianalytical solution and numerical solution, respectively [5]. The authors of [6, 7] presented an improved MDOF spur gear pair model with alternate sliding friction formulations, and the simulations indicated small differences when different sliding friction formulations were employed.

Compared with spur gear, the length of contact line for a tooth in helical gear varies with different meshing positions of pinion and gear, which will excite vibrations especially in helical gear system [8]; the load distribution along the contact line is not uniform, which is different from that of spur gear pairs. It was shown in [9] that the fluctuations of friction force and torque induced more significant effects than the effects introduced by the values of friction force and torque in gear system. The author of [10] proposed a new analytical model for helical gears that characterized the contact plane dynamics and captured the velocity reversal at the pitch line due to sliding friction, with the assumption of constant mesh stiffness per unit length along contact line. A 12-DOF dynamic model with the inclusion of friction force and torque was established, and a simplified 6-DOF model was proposed for dynamics analysis. Kar and Mohanty [11, 12] presented an algorithm for determination of time-varying frictional force and torque at meshing teeth and the bearings in a helical gear system, which was an extension of the method proposed in [4] for spur gears. One of the assumptions in their work is the uniform load distribution along contact lines.

The objective of this paper is to reveal the differences between uniform and nonuniform load distribution along contact lines while calculating friction force and torque for helical gears. The approach is an extension of the work presented by [11] for helical gears. Firstly, the variation of contact line is presented. Then an analytical load distribution model for involute helical gears proposed by Pedrero et al. [13] is employed and extended to get the normal load along contact lines. With the assumption of constant friction coefficient, both friction force and torque at each contact point are calculated and compared with the results from uniform load distribution assumption. Moreover, another typical design sample is studied and the results are presented. The influences of different friction coefficient models on the resultant force and torque are also investigated.

2. Mathematical Model

2.1. Time-Varying Length of Contact Lines

Figure 1 shows the meshing schematic of an involute helical gear pair. The contact lines are formed by the engaged teeth of pinion and gear, and all lie in the inner common tangent plane of the two base cylindrical surfaces of pinion and gear. The angle between the gear axis and instantaneous contact line is 𝛽_𝑏. Typically, the total contact ratio 𝜀 of helical gears is 2<𝜀=𝜀_𝛼+𝜀_𝛽<3, which means that there are two or three teeth in simultaneous contact.

Figure 1 

Meshing of a helical gear pair.

The following are denoted:𝑑_𝛼=rem𝜀_𝛼,𝑑,1_𝛽=rem𝜀_𝛽,1, (1a)(1b)where the function rem(𝜀_𝛼,1) means the decimal fraction of transverse contact ratio 𝜀_𝛼, and rem(𝜀_𝛽,1) means the decimal fraction of axial contact ratio 𝜀_𝛽. Two typical design samples will be investigated as follows: (𝑑_𝛼+𝑑_𝛽<1): the sum of residuals of transverse and axial contact ratios is smaller than one; (𝑑_𝛼+𝑑_𝛽>1): the sum of residuals of transverse and axial contact ratios is larger than one.

The variations of contact lines for both samples are shown in Figure 2.

(a)

(b)

Figure 2 

Contact line variations: (a) sample I, (b) sample II.

For a single tooth, the contact line increases gradually before it reaches the maximum. Then the length of contact line remains constant for a while, after which the contact line decreases gradually as the pinion or gear further rotates until finishing meshing. In Figure 2(a), the length reaches its maximum value 𝑓cosec𝛽_𝑏 at point C and keeps the value till point D. In Figure 2(b), the contact line keeps the maximum value 𝑏sec𝛽_𝑏 during the mesh period from point D to point C.

The individual and total lengths of contact lines for three teeth in one transverse pitch travel are expressed as follows.(1)Sample I (𝑑_𝛼+𝑑_𝛽<1)𝐿₀(𝑡)=𝑟_𝑏𝜔_𝑝𝑡sin𝛽_𝑏𝐿,0≤𝑡<𝑇,₁(𝑡)=𝑝_𝑡+𝑟_𝑏𝜔_𝑝𝑡sin𝛽_𝑏,0≤𝑡<𝑡₁₁=𝑓sin𝛽_𝑏,𝑡₁₁≤𝑡<𝑡₁₂=𝑏⋅tan𝛽_𝑏+𝑓−𝑝_𝑡−𝑟_𝑏𝜔_𝑝𝑡sin𝛽_𝑏,𝑡₁₂𝐿≤𝑡<𝑇,₂(𝑡)=𝑏⋅tan𝛽_𝑏−𝑟_𝑏𝜔_𝑝𝑡+2𝑝_𝑡−𝑓sin𝛽_𝑏,0≤𝑡<𝑡₂₁=0,𝑡₂₁𝐿(𝑡)≤𝑡<𝑇=𝐿₁(𝑡)+𝐿₂(𝑡)+𝐿₃(𝑡). (2)(2)Sample II (𝑑_𝛼+𝑑_𝛽>1)𝐿₀(𝑡)=𝑟_𝑏𝜔_𝑝𝑡sin𝛽_𝑏,0≤𝑡<𝑡₀=𝑏cos𝛽_𝑏,𝑡₀𝐿≤𝑡<𝑇,₁(𝑡)=𝑏cos𝛽_𝑏,0≤𝑡<𝑡₁₁=𝑏tan𝛽_𝑏−𝑟_𝑏𝜔_𝑝𝑡−𝜀_𝛼−1𝑝_𝑡sin𝛽_𝑏,𝑡₁₁𝐿≤𝑡<𝑇₂(𝑡)=𝑏tan𝛽_𝑏−𝑟_𝑏𝜔_𝑝𝑡+2−𝜀_𝛼𝑝_𝑡sin𝛽_𝑏,0≤𝑡<𝑡₂₁=0,𝑡₂₁𝐿(𝑡)≤𝑡<𝑇,=𝐿₁(𝑡)+𝐿₂(𝑡)+𝐿₃(𝑡), (3)where 𝑣=𝑟_𝑏𝜔_𝑝,𝑡₀=𝑏tan𝛽_𝑏/𝑣,𝑡₁₁=(𝑓−𝑝_𝑡)/𝑣,𝑡₂₁=(𝑓+𝑏⋅tan𝛽_𝑏−2𝑝_𝑡)/𝑣,𝑡₁₂=𝑏⋅tan𝛽_𝑏−𝑝_𝑡𝑣𝑝,𝑇=^𝑡𝑣. (4)

2.2. Load per Unit of Length along Contact Lines

Though uniform load distribution is assumed in most of the literature, the load per unit of length varies along contact lines. In this paper, to get the real load distribution in involute helical gears, an analytical load distribution model is employed.

The expression is given by 𝑓𝜁,𝜁₀=𝜀_𝛽cos𝛽_𝑏𝑏𝑣(𝜁)𝐼_𝑣𝜁₀𝐹, (5) where function 𝑓(𝜁,𝜁₀) represents the load per unit of length applied to the point of contact line corresponding to 𝜁 at the moment in which the reference transverse section is in contact at the point 𝜁₀ and 𝐹 is the applied load. More details could be found in [13].

In this paper, to get the load per unit of length along contact lines, each contact line is divided into segments along the axis of gear and the expression for the normal load applied on the 𝑖th segment of 𝑘th tooth at 𝑗th meshing instant is written here as 𝑓_𝑘𝑖_𝑘=𝜀,𝑗^𝛽cos𝛽_𝑏𝑏𝑣𝜁_𝑖𝑘𝑡_𝑗𝐼_𝑣𝜁_0,𝑘𝑡_𝑗𝐹,𝑖_𝑘=1,2,…,𝑁_𝑗,𝑘,𝑘=0,1,2,𝑗=1,2,…,𝑁_𝑡. (6) As for time-varying length of contact lines, 𝑁_𝑗,𝑘 is the number of segments for each tooth in contact, varying with meshing positions.

Parameters in (6) are given by 𝑣𝜁_𝑖𝑘𝑡_𝑗=cos𝑏₀𝜁_𝑖𝑘𝑡_𝑗−𝜀_𝛼2,𝑏₀=12𝜀1+^𝛼2²−1^−1/2,𝜁_𝑖𝑘𝑡_𝑗=𝜉_𝑖𝑘𝑡_𝑗−𝜉_inn,𝜉_inn=𝑧2𝜋𝑟²_𝑎𝑟²_𝑏−1−𝜀_𝛼,𝜉_𝑖𝑘𝑡_𝑗=𝑧2𝜋𝑟²_{𝑖𝑘,𝑗}𝑟²_𝑏𝐼−1,_𝑣𝜁_0,𝑘𝑡_𝑗=1𝑏₀^𝐸𝛾_𝑛=0sin𝑏₀𝜁_𝑛,sup−𝜀_𝛼2−sin𝑏₀𝜁_𝑛,inf−𝜀_𝛼2,𝜁_𝑛,sup=𝜁_0,𝑘𝑡_𝑗+𝑛+𝜀_𝛼−min𝜁_0,𝑘𝑡_𝑗𝜁+𝑛,0−max_0,𝑘𝑡_𝑗+𝑛,𝜀_𝛼,𝜁_𝑛,inf=𝜁_0,𝑘𝑡_𝑗+𝑛−𝜀_𝛽+𝜀_𝛼−min𝜁_0,𝑘𝑡_𝑗+𝑛−𝜀_𝛽𝜁,0−max_0,𝑘𝑡_𝑗+𝑛−𝜀_𝛽,𝜀_𝛼. (7)𝜉_inn denotes the profile parameter corresponding to the inner point of contact of the pinion, and 𝑟_{𝑖𝑘,𝑗} is the distance from the applied force point to the center of gear or pinion in the transverse plane shown in Figure 3.

Figure 3 

Geometrical parameter of an involute tooth.

The expressions for parameters 𝜁_0,𝑘(𝑡_𝑗) and 𝜁_𝑖𝑘(𝑡_𝑗) for three teeth in a mesh cycle are given by 𝜁_0,0𝑡_𝑗=0,𝑡_𝑗≤𝑡₀,𝑧⋅𝜔_𝑝𝑡2𝜋^𝑗−𝑡₀,𝑡₀<𝑡_𝑗𝜁≤𝑇,_0,1𝑡_𝑗=𝑧2𝜋𝑟_𝑏𝑝𝑝_𝑡−𝑏⋅tan𝛽_𝑏+𝑣⋅𝑡_𝑗,𝜁_0,2𝑡_𝑗=𝑧2𝜋𝑟_𝑏𝑝2𝑝_𝑡−𝑏⋅tan𝛽_𝑏+𝑣⋅𝑡_𝑗,𝑡_𝑗≤𝑡₂,0,𝑡₂<𝑡_𝑗𝜁≤𝑇,_𝑖𝑘𝑡_𝑗=𝑧2𝜋𝑟_𝑏𝑝𝑅_𝑒𝑖_𝑘,𝑗−𝐴𝐵, (8) respectively, where 𝑅_𝑒(𝑖_𝑘,𝑗) is the radii of curvature given in the following section.

2.3. Friction Force

The direction of friction force is perpendicular to the line of action in the transverse plane parallel to the cross section of the gear and pinion, and perpendicular to the contact lines in the plane of action. The force is assumed to be positive when the contact point on the contact line is above the pitch line and negative otherwise. Hence, the friction force of 𝑖th segment of 𝑘th tooth of pinion at 𝑗th meshing instant could be given by 𝐹_𝑘𝑖_𝑘,𝑗=𝜇⋅𝑓_𝑘𝑖_𝑘,𝑗⋅Δ𝑙, (9) where 𝜇 is the constant coefficient of friction, and Δ𝑙  is the length of a segment. The friction force of 𝑘th tooth at 𝑗th meshing instant is expressed as 𝐹₀𝑡_𝑗=−𝜇⋅^𝑁00_𝑖0=1𝑓₀𝑖₀,𝑗⋅Δ𝑙0≤𝑡_𝑗<𝑓−𝑓₃𝑣=−𝜇⋅^𝑁01_𝑖0=1𝑓₀𝑖₀,𝑗−^𝑁0,𝑗_{𝑖0=𝑁01+1}𝑓₀𝑖₀,𝑗⋅Δ𝑙𝑓−𝑓₃𝑣≤𝑡_𝑗𝐹<𝑇,₁𝑡_𝑗=−𝜇⋅^𝑁11_𝑖1=1𝑓₁𝑖₁,𝑗−^𝑁1,𝑗_{𝑖1=𝑁11+1}𝑓₁𝑖₁,𝑗⋅Δ𝑙0≤𝑡_𝑗𝐹<𝑇,₂𝑡_𝑗=−𝜇⋅^𝑁21_𝑖2=1𝑓₂𝑖₂,𝑗−^𝑁2,𝑗_{𝑖2=𝑁21+1}𝑓₂𝑖₂,𝑗⋅Δ𝑙0≤𝑡_𝑗<𝑇, (10) where the function round (𝑥) gets the integer nearest to 𝑥. And 𝑁₀₀𝐿=round¹,𝑁Δ𝑙₀₁𝐿=round¹¹,𝑁Δ𝑙₁₁𝐿=round²¹,𝑁Δ𝑙₂₁𝐿=round³¹.Δ𝑙 (11) The contact line segments divided by pitch line in sample I, shown in Figure 4, are given by 𝐿₁₂=𝑓−𝑓₃sin𝛽_𝑏,𝐿₁₁=𝐿₁−𝐿₁₂,𝐿₂₂=𝑓−𝑓₃sin𝛽_𝑏,0≤𝑣𝑡<𝑓+𝑓₁−𝑝_𝑡,𝐿₂₁=𝐿₂−𝐿₂₂,𝐿₂₁=𝑓₃sin𝛽_𝑏,𝑓+𝑓₁−𝑝_𝑡≤𝑣𝑡<𝑝_𝑡,𝐿₂₂=𝐿₂−𝐿₂₁,𝐿₃₁=𝑓₃sin𝛽_𝑏,0≤𝑣𝑡<𝑓−𝑓₃+𝑓₄,𝐿₃₂=𝐿₃−𝐿₃₁,𝐿₃₁=𝐿₃𝑓−𝑓₃+𝑓₄≤𝑣𝑡<𝑝_𝑡. (12) Similarly, the contact line segments in sample II are derived as 𝐿₁₁=𝑣⋅𝑡_𝑗−𝑓−𝑓₃cosec𝛽_𝑏,𝐿₁₂=𝐿₁−𝐿₁₁,𝐿₂₁=𝑝_𝑡−𝑓−𝑓₃+𝑣⋅𝑡_𝑗cosec𝛽_𝑏,0≤𝑡_𝑗<𝑏⋅tan𝛽_𝑏−𝑓₃+𝑓−𝑝_𝑡𝑣,𝐿₂₁=𝐿₂,𝑏⋅tan𝛽_𝑏−𝑓₃+𝑓−𝑝_𝑡𝑣≤𝑡_𝑗𝐿<𝑇,₂₂=𝐿₂−𝐿₂₁,𝐿₃₁=𝐿₃𝐿,0≤𝑡<𝑇,₃₂=𝐿₃−𝐿₃₁. (13) Accordingly, the total friction force for one transverse pitch travel is obtained as 𝐹_𝑓𝑡_𝑗=^floor(𝜀)_𝑘=0𝐹_𝑘𝑡_𝑗,0≤𝑡_𝑗<𝑇. (14) Function floor (𝜀) gets the maximum integer smaller than 𝜀.

Figure 4 

Schematic of friction force direction.

2.4. Frictional Torque

The formulations for friction torque of sample I under uniform load assumption could be found in [14], while the formulations for each tooth in sample II under uniform assumption are derived as 𝑇_𝑓0𝐹=𝜇𝐿𝐿₁𝐿_AC−𝑓−𝑣𝑡20≤𝑣𝑡<𝑏tan𝛽_𝑏𝐹=𝜇𝐿𝐿₁𝐿_AC−𝑓−𝑣𝑡−𝑏tan𝛽_𝑏+𝑏tan𝛽_𝑏2,𝑏tan𝛽_𝑏≤𝑣𝑡<𝑓−𝑓₃𝐹=𝜇𝐿𝐿₁₂𝐿_AC−𝑓−𝑓₃−𝑣𝑡−𝑏tan𝛽_𝑏2+𝑓₃𝐹−𝜇𝐿𝐿₁₁AC−𝑓₃−𝑣𝑡−𝑓−𝑓₃2,𝑓−𝑓₃≤𝑣𝑡<𝑝_𝑡,𝑇_𝑓1𝐹=𝜇𝐿𝐿₂₂𝐿_AC−𝑓−𝑓₃−𝑝_𝑡−𝑏tan𝛽_𝑏−𝑣𝑡2+𝑓₃𝐹−𝜇𝐿𝐿₂₁𝐿_AC−𝑓₃−𝑝_𝑡−𝑓−𝑓₃+𝑣𝑡2,0≤𝑣𝑡<𝑏tan𝛽_𝑏−𝑝_𝑡−𝑓−𝑓₃𝐹=−𝜇𝐿𝐿₂₁×𝐿_AC−𝑓−𝑏tan𝛽_𝑏2+𝑣𝑡+𝑝_𝑡−𝑏tan𝛽_𝑏,𝑏tan𝛽_𝑏−𝑝_𝑡−𝑓−𝑓₃≤𝑣𝑡<𝑓−𝑝_𝑡𝐹=−𝜇𝐿𝐿₂₁𝐿_AC−𝑏tan𝛽_𝑏−𝑣𝑡−𝑓−𝑝_𝑡2,𝑓−𝑝_𝑡≤𝑣𝑡<𝑝_𝑡,𝑇_𝑓2𝐹=−𝜇𝐿𝐿₃𝐿_AC−𝑓−𝑝_𝑡−𝑏tan𝛽_𝑏−𝑝_𝑡−𝑣𝑡2,0≤𝑣𝑡<𝑓+𝑏tan𝛽_𝑏−2𝑝_𝑡=0,𝑓+𝑏tan𝛽_𝑏−2𝑝_𝑡≤𝑣𝑡<𝑝_𝑡. (15)

The following are the derivations of friction torque under nonuniform load assumption. The friction moment arm is equal to the radii of curvature in entrainment direction at each contact point, as shown in dashed lines in Figure 5. According to the involute profile of a tooth, the radii of curvature of ith segment of kth tooth pair of pinion at jth meshing instant in the transverse plane could be written as 𝑅_𝑒𝑖₀,𝑗=𝐿_AB+𝑖₀⋅𝐿₀𝑡_𝑗⋅sin𝛽_𝑏𝑁_𝑗,0,0≤𝑡_𝑗<𝑡₀=𝐿_AB+𝑣⋅𝑡_𝑗−𝑏⋅tan𝛽_𝑏+𝑖₀⋅𝐿₀𝑡_𝑗⋅sin𝛽_𝑏𝑁_𝑗,0,𝑡₀≤𝑡_𝑗𝑅<𝑇,_𝑒𝑖₁,𝑗=𝐿_AB+𝑝_𝑡−𝑏⋅tan𝛽_𝑏+𝑣⋅𝑡_𝑗+𝑖₁⋅𝐿₁𝑡_𝑗⋅sin𝛽_𝑏𝑁_𝑗,1,0≤𝑡_𝑗𝑅<𝑇,_𝑒𝑖₂,𝑗=𝐿_AB+2⋅𝑝_𝑡−𝑏⋅tan𝛽_𝑏+𝑣⋅𝑡_𝑗+𝑖₂⋅𝐿₂𝑡_𝑗⋅sin𝛽_𝑏𝑁_𝑗,2,0≤𝑡_𝑗<𝑡₂=0,𝑡₂≤𝑡_𝑗<𝑇. (16) Accordingly, the frictional torque arm for pinion is given by 𝐿_𝑓𝑖_𝑘,𝑗=𝑅_𝑒𝑖_𝑘,𝑗. (17)

Figure 5 

Frictional torque arm.

The sign convention of the frictional torque is considered as positive when facilitating the rotation and negative otherwise. The friction torque at ith segment of kth tooth pair of pinion at jth meshing instant is calculated as 𝑇₀𝑡_𝑗=𝜇⋅Δ𝑙⋅^𝑁00_𝑖0=1𝑓₀𝑖₀,𝑗⋅𝐿_𝑓𝑖₀,𝑗,0≤𝑡_𝑗<𝑓−𝑓₃𝑣=𝜇⋅Δ𝑙⋅^𝑁01_𝑖0=1𝑓₀𝑖₀,𝑗⋅𝐿_𝑓𝑖₀−,𝑗_{𝑁0,𝑗𝑖0=𝑁01+1}𝑓₀𝑖₀,𝑗⋅𝐿_𝑓𝑖₀,𝑗,𝑓−𝑓₃𝑣≤𝑡_𝑗𝑇<𝑇,₁𝑡_𝑗=𝜇⋅Δ𝑙⋅^𝑁11_𝑖1=1𝑓₁𝑖₁,𝑗⋅𝐿_𝑓𝑖₁−,𝑗_{𝑁1,𝑗𝑖1=𝑁11+1}𝑓₁𝑖₁,𝑗⋅𝐿_𝑓𝑖₁,𝑗,0≤𝑡_𝑗𝑇<𝑇,₂𝑡_𝑗=𝜇⋅Δ𝑙⋅^𝑁21_𝑖2=1𝑓₂𝑖₂,𝑗⋅𝐿_𝑓𝑖₂−,𝑗_{𝑁2,𝑗𝑖2=𝑁21+1}𝑓₂𝑖₂,𝑗⋅𝐿_𝑓𝑖₂,𝑗,0≤𝑡_𝑗<𝑇. (18) Hence, the total friction torque in three teeth for one transverse pitch travel is given by 𝑇_𝑓𝑡_𝑗=^floor(𝜀)_𝑘=0𝑇_𝑘𝑡_𝑗,0≤𝑡_𝑗<𝑇. (19)

3. Results and Discussion

3.1. Model Verification

The load distribution model used in this paper is verified by comparing the results from [13] with the current simulation. The normalized load per unit of length obtained from the finite element analysis, integration of elastic potential energy, and the current model is shown in Figure 6, respectively. The reference transverse section is chosen at 𝜉₀=𝜉_𝑎.

Figure 6 

Load distribution model verification: (a) integration of elastic potential energy; (b) finite element analysis; (c) current simulation.

As could be seen from Figure 6, a good agreement is obtained.

3.2. Time-Varying Length of Contact Lines

The simulation parameters used in sample I and II are shown in Tables 1 and 2, respectively. The mesh cycle for one transverse pitch travel is divided into 50 time steps, and 𝑁_𝑡=150. The maximum value of 𝑁_𝑗,𝑘 is 62 in sample I and 60 in sample II corresponding to the maximum length of contact line in both cases.

Table 1 

Specifications of helical gear pair for sample I.

Table 2 

Specifications of helical gear for sample II.

The time-varying length of contact line for a single tooth in sample I is shown in Figure 7(a). There is a good agreement with the function formulated by Velex and Sainsot [3].

(a)

(b)

(c)

(d)

Figure 7 

Time-varying contact lines for (a) single tooth in sample I; (b) three teeth in sample I; (c) single tooth in sample II; (d) three teeth in sample II.

The maximum value of the contact line is  𝑓cosec𝛽_𝑏, while the maximum length of contact line in sample II is 𝑏sec𝛽_𝑏, as shown in Figure 7(c). The variations of total contact lengths for one transverse pitch travel in both cases are shown in Figures 7(b) and 7(d), respectively. As could be seen from the figure, the time intervals for minimum total length of contact lines differ from each other.

3.3. Load Distribution

In [10], uniform load distribution is assumed, while the load per unit of length varies with different position along contact line.

The load distributions for a single tooth of both samples in a complete cycle are shown in Figures 8(a) and 8(b). As could be seen from the figure, the x-axis represents the coordinate along instantaneous contact line, while the y-axis represents meshing time for a single tooth in three transverse pitch travels. And z-axis means the load per unit of length along contact line. Obviously, the distributions in two design cases are different. To make the difference clear, the load distributions during the time interval that contact lines reach maximum values and keep constant are also presented, as shown in Figures 8(c) and 8(d). It is indicated that the maximum load per unit of length in sample I lies in the middle part of a contact line, while that in sample II lies in one of the faces of gear.

(a)

(b)

(c)

(d)

Figure 8 

Load distribution for a single tooth: (a) in a mesh cycle of sample I; (b) in a mesh cycle of sample II; (c) in time interval 𝑡 (63) ~t (75) of sample I; (d) in time interval 𝑡 (31) ∼𝑡 (75) of sample II.

3.4. Friction Force and Torque

Friction force and torque for both a single tooth and three teeth are shown in Figure 9. As could be seen from the figure, the friction force is not sinusoidal though periodic, similar to the case in spur gear [15]. The dashed lines represent the friction forces obtained under uniform load distribution assumption, while the solid lines represent the results from nonuniform assumption.

(a)

(b)

(c)

(d)

Figure 9 

Friction force and torque in sample I: (a) friction force of a single tooth; (b) friction force of three teeth; (c) friction torque of a single tooth; (d) friction torque of three teeth.

For a single tooth, the force difference under two assumptions is trivial before the time when contact line reaches its maximum value. However, the difference becomes significant during the left time in a whole meshing cycle. Different from those of a single tooth, there is always a significant difference between the total friction forces under two assumptions for a transverse pitch travel. The statistical values of Figure 9 are shown in Table 3. It shows that the fluctuation under uniform assumption is more significant than that under nonuniform assumption.

Table 3 

Statistical values of Figure 9.

Friction torques for a single tooth and three teeth are shown in Figures 9(c) and 9(d), respectively. Similar to friction force, torque fluctuation under uniform assumption is more significant than that under nonuniform assumption. To be noted, as could be seen from Figure 9(d), the peak value of friction torque under nonuniform assumption is positive, while that under uniform assumption is negative. Those differences may result in a different dynamic response in helical gear systems.

Results are shown in Figure 10 and statistical values of the figure are shown in Table 4. Obviously, the fluctuations of both friction force and torque under uniform assumption are less significant than those under nonuniform assumption, which is contrary to sample I. The short constant force interval appearing in the first sample does not come out in this one.

Table 4 

Statistical values of Figure 10.

(a)

(b)

(c)

(d)

Figure 10 

Results for sample II: (a) friction force for a single tooth; (b) friction force for three teeth; (c) friction torque for a single tooth; (d) friction torque for three teeth.

3.5. Results from Time-Varying Friction Coefficient

To investigate the influence of time-varying friction coefficient on the resultant friction force and torque, a friction coefficient formula proposed by [14] is employed as 𝜇_𝑘𝑖_𝑘,𝑗=0.0127×1.131.13−𝑆_avglog₁₀29700𝑓_𝑘𝑖_𝑘,𝑗𝜂𝑣_𝑠𝑖_𝑘,𝑗𝑣²_𝑒𝑖_𝑘,,𝑗 (20) where 𝑆_avg is the averaged surface roughness, and 𝜂is the dynamic viscosity of the oil entering the gear contact. 𝑣_𝑠(𝑖_𝑘,𝑗) and 𝑣_𝑒(𝑖_𝑘,𝑗) represent the sliding velocity and the entraining velocity of the ith segment of kth tooth pair at jth meshing instant, respectively, as𝑣_𝑠𝑖_𝑘,𝑗=𝑣_𝑝𝑖_𝑘,𝑗−𝑣_𝑔𝑖_𝑘,𝑣,𝑗_𝑒𝑖_𝑘,𝑗=𝑣_𝑝𝑖_𝑘,𝑗+𝑣_𝑔𝑖_𝑘,𝑣,𝑗_𝑝𝑖_𝑘,𝑗=𝜔_𝑝𝑅_𝑝𝑒𝑖_𝑘,𝑣,𝑗_𝑔𝑖_𝑘,𝑗=𝜔_𝑔𝑅_𝑔𝑒𝑖_𝑘,𝑗, (21a)(21b)(21c)(21d) where 𝑅_𝑝𝑒(𝑖_𝑘,𝑗) is the radii of curvature at each segment of pinion as given by (16), and 𝑅_𝑔𝑒(𝑖_𝑘,𝑗) is the radii of curvature at corresponding contact segment of gear and is given by 𝑅_𝑔𝑒𝑖_𝑘,𝑗=𝑟_𝑏+𝑅_𝑏tan𝛼_𝑡−𝑅_𝑝𝑒𝑖_𝑘,𝑗. (22) Here, only sample II is taken as an example. The legends NV, NC, UV, and UC in Figure 11 denote the results from nonuniform load distribution and varying friction coefficient assumption; nonuniform load distribution and constant friction coefficient assumption; uniform load distribution and varying friction coefficient assumption; uniform load distribution and constant friction coefficient assumption, respectively. The constant friction coefficient is obtained by averaging the time-varying coefficients calculated by (20).

(a)

(b)

(c)

(d)

Figure 11 

Results for sample II with varying friction coefficient: (a) friction force of a single tooth; (b) friction force of three teeth; (c) friction torque of a single tooth; (d) friction torque of three teeth.

As could be seen from Figures 11(a) and 11(b), the friction forces under NV assumption and NC assumption are very close to each other; similarly, the force difference between UV assumption and UC assumption is negligible. However, it is obvious that there is a nonnegligible difference between uniform and nonuniform load distribution assumption, which means that the force fluctuation under nonuniform assumption is more significant than that under uniform assumption. A similar conclusion could be made on frictional torque, as shown in Figures 11(c) and 11(d).

4. Conclusion

This paper proposed an improved algorithm to calculate the friction force and torque in helical gear pairs, considering the nonuniform load distribution along contact lines. Two design samples are studied. In sample I, the maximum load per unit of length lies in the middle area of a contact line, while that in sample II lies in one of the faces of gear. The fluctuations of friction force and torque under uniform assumption in the first sample for a single tooth is more significant than those under nonuniform assumption, while the sum of friction force and torque for three teeth are less significant. On the contrary, the fluctuations of both friction force and torque under uniform assumption in the second sample are less significant than those under nonuniform assumption for both a single tooth and the sum of the three teeth. Compared with the difference induced by uniform and nonuniform load distribution assumptions, the differences induced by different friction coefficient models are negligible. Those differences may result in a different dynamics response in a geared system. In addition, both friction coefficient and applied load are linearly proportional to the difference between the fluctuations of friction force and torque obtained under two assumptions. The work presented here provides a way for further dynamics analysis for gear systems.

Nomenclature

𝑏:	Gear width
𝑏0:	Auxiliary parameter
𝑑𝛼:	Decimal fraction of 𝜀𝛼
𝑑𝛽:	Decimal fraction of 𝜀𝛽
𝐸𝛾:	Integer part of ε
𝑓:	Length of contact zone
𝑓1,𝑓2,𝑓3,𝑓4:	Length parameters in contact zone shown in Figure 2
𝑓𝑘(𝑖𝑘,𝑗):	Load per unit of length
𝐹:	Applied load
𝐹𝑘:	Friction force of kth tooth
𝐹𝑓:	Total friction force of three teeth
𝐹uni:	Force from uniform assumption
𝐹non:	Force from nonuniform assumption
𝐼𝑣:	Sum of inverse unitary potential
𝐿:	Total length of three contact lines
𝐿𝑓:	Frictional torque arm
𝐿𝑖:	Length of 𝑖th contact line
𝐿𝑖𝑗:	𝑗th segment on 𝑖th contact line
Δ𝑙:	Length of each segment
𝑁𝑗,𝑘:	Number of segments of 𝑘th tooth at 𝑗th instant
𝑁𝑡:	Total time steps
𝑁00,𝑁01,𝑁11,𝑁21:	Number of segments on line segments
𝑝𝑡:	Pitch in transverse plane
𝑝𝑏𝑡:	Base pitch in transverse plane
𝑟:	Reference radius shown in Figure 3
𝑟𝑎1:	Radius of addendum circle
𝑟𝑏1:	Radius of base circle
𝑅𝑏:	Radius of base circle of gear
𝑅𝑒(𝑖𝑘,𝑗):	Radius of curvature at each segment
𝑡:	Meshing time
𝑡11,𝑡12,𝑡0,𝑡2,𝑡21:	Time instant corresponding to meshing position
𝑇:	Meshing period for a pitch travel
𝑇𝑓:	Frictional torque of three teeth
𝑇𝑘:	Frictional torque of 𝑘th tooth
𝑣:	Linear speed along line of action
𝑣(𝜁):	Inverse unitary potential
𝑧:	Number of teeth
𝛽𝑏:	Base helix angle
𝜀:	Total contact ratio of helical gears
𝜀𝛼:	Transverse contact ratio
𝜀𝛽:	Axial contact ratio
𝜁:	Auxiliary parameter
𝜁0:	Auxiliary parameter
𝜇:	Friction coefficient
𝜉:	Profile parameter
𝜉inn:	Profile parameter corresponding to the inner contact point of pinion
𝜔:	Rotational speed of gears
𝜔𝑝:	Rotational speed of pinion.

Acknowledgment

This work was supported by the National Science and Technology Major Projects of China (Grant no. 2012ZX04012-031).