# Nonlinear Optimization Method for Transmission Error of Hypoid Gear Machined by the Duplex Helical Method

Authors: Shunxing Wu; Hong-Zhi Yan; Rengui Bi; Zhiyong Wang; Pengfei Zhu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this study, synchronous cutting of concave and convex surfaces for hypoid gear was achieved using a duplex helical method. Precise, nonlinear optimization of the transmission error driven by machine tool parameters was performed to reduce the vibration noise of the gear pair. First, the transmission error curve and contact path of the tooth surface of the initial pinion were solved using tooth contact analysis. Second, according to the preset parabolic transmission error curve, the initial gear was used to generate the target pinion, which coincided with the contact path of the initial pinion. Finally, a deviation correction model of the discrete points, corresponding to the contact paths on the concave and convex surfaces of the target and initial pinions, was established. This model was solved using the Levenberg–Marquard algorithm with the trust region strategy, to obtain optimized machine tool parameters. Synchronous optimization of the transmission errors of concave and convex surfaces of the pinion was achieved by correcting the deviations of the contact points. The effectiveness of the proposed method was verified by a numerical example and by performing a contact area rolling test.


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## Body

## 1. Introduction

Owing to the strong bearing capacity and high structural reliability of hypoid gear pairs, they are widely used in transmission devices such as automobiles and helicopters. A hypoid gear is normally produced using a traditional cutting method (i.e., the five-cut method). In this approach, the vibration and noise can be controlled easily because the concave and convex surfaces are cut and modified separately [1, 2]. Currently, the five-cut method is gradually being replaced by the duplex helical method. The duplex helical method is advantageous because of its higher machining efficiency and superior quality consistency compared to those of the five-cut method [3]. However, because the duplex helical method involves simultaneous processing of concave and convex surfaces, it is difficult to control the vibration and noise of the gear pair [4]. Owing to the increasing demand for comfort and reliability in cars, higher requirements are simultaneously being imposed for the vibration noise and life span of gear pairs. The vibration noise and life span of a gear pair are directly affected by the transmission error. Therefore, it is crucial to reduce the transmission error of gear pairs while ensuring efficient cutting in the duplex helical method [5, 6].Presetting and optimizing the transmission error curve of a gear pair are effective ways to reduce vibration noise while improving service life. Researchers have investigated spiral bevel gear pairs cut using the five-cut method. Based on the local synthesis method, Litvin et al. [7] and Simon [8] reduced the transmission error amplitude of a gear pair by presetting the three second-order contact parameters of the transmission error curve. Taking the preset parameters of the local synthesis method as optimization variables, Wang et al. [9] and Wang et al. [10] optimized the load transmission error amplitude through a genetic algorithm and kriging algorithm, respectively. Considering that the local synthesis method can only control the contact performance near the reference point, Liu and Fan [11] proposed optimizing the transmission error amplitude based on the local synthesis method by taking the modified coefficient as the design variable. Zhuo et al. [5] optimized the transmission error amplitude through the genetic algorithm, which is constrained by the composite shape method. This optimization was accomplished using the normal curvature and short-range torsion of the tooth surface as the controlling parameters. By expressing the tooth surface as a higher-order polynomial, Artoni et al. [12] used the coefficient of the higher-order polynomial as the optimization variable. This approach was employed to optimize the loaded transmission error amplitude and the maximum contact stress. By expressing the machine tool parameters as a sixth-order polynomial on the cradle angle, Astoul et al. [13] reduced the amplitude of the transmission error by optimizing the polynomial coefficients.Based on research, the transmission error amplitude can be reduced by optimizing the cutter head and machine tool parameters related to the transmission error. However, the transmission error amplitude cannot be obtained accurately. The target pinion tooth surface is obtained by presetting the transmission error on the pinion tooth surface, where it is a conjugate with the initial gear. The cutter head and machine tool parameters of the initial pinion can be modified by minimizing the deviation between the initial and the target pinions, which is a feasible method of obtaining the transmission error amplitude accurately. This modification process can be attributed to the least-squares optimization problem, which is a hot topic in current research. Krenzer [14] first proposed a linear regression correction method for machine tool parameters using a spiral bevel gear with minimum tooth surface deviation. Based on this research, Litvin et al. [15] and Stadtfeld [16] proposed minimizing tooth surface deviation as a nonlinear least squares optimization problem, and the accuracy of the machine tool parameters improved. Subsequently, to improve the stability and accuracy of the solution further, a sensitivity matrix describing the variations of the machine tool parameters relative to the tooth surface deviation was applied. The sequential quadratic programming algorithm [17, 18] and singular value decomposition algorithm [19] have also been used to correct the machine tool parameters. However, for the problem of excessive ill-health, the Hessian matrix has ill-health. In other words, the matrix is not positive, which makes accurate and stable solutions of nonlinear equations difficult to obtain. Therefore, the Levenberg–Marquard method [20, 21] is proposed to solve the correction value of the machine tool parameters. The core of the Levenberg–Marquard algorithm involves introducing a damping coefficient greater than zero to ensure that the Hessian matrix obtained in each iteration step is positive definite and reversible. Consequently, the iteration can be performed with high precision.As previously stated, this approach optimizes the machining parameters related to the transmission error to reduce the transmission error amplitude or to correct the tooth surface deviation to obtain greater accurate transmission error amplitude. Limited research has been conducted on the duplex helical method. Unlike the five-cut method, the duplex helical method simultaneously cuts concave and convex surfaces. It is more difficult to obtain the transmission error amplitude accurately for a gear pair cut by the duplex helical method. Therefore, the high-precision optimization of the transmission error of a hypoid gear pair cut by the duplex helical method was investigated in this study.


## 2. Tooth Surface Modeling of Hypoid Gear Pairs Machined by the Duplex Helical Method

To optimize the transmission error of a gear pair, the machining mathematical model of a gear pair should first be established. In this study, the gear was machined by a nongenerated method and the pinion was machined by a duplex helical method. The detailed machining mathematical model of a gear is given in reference [7]. R2 (ug, θg) and n2 (ug, θg) represent the tooth surface equation and unit normal vector of a gear in the gear coordinate system S2, respectively, where (ug, θg) are the Gaussian coordinates of the gear tooth surface. The detailed process of cutting a pinion is described below.The coordinate systems applied for pinion generation are shown in Figure1. Sm1, Sc, and Sd are coordinate systems rigidly fixed to the cutting machine. Sp is a coordinate system rigidly fixed to the head cutter and rotates around the axis zp. Sb and S1 are the movable coordinate systems rigidly fixed to the cradle and pinion, respectively. Sa is the reference coordinate system of the tilt angle and swivel angle. When the pinion is cut by the duplex helical method, the cradle rotates around the axis zm1 while moving as a helix along the axis zm1. Simultaneously, the pinion makes a rotary motion around the axis xd. The tooth surface of the pinion is cut into an envelope of the tool curved family through the generating movement. As shown in Figure 1, the machine tool settings q1, sr1, i, j, e1, xb1, γm1, xg1, and hl represent the center roll position, radial distance, tilt angle, swivel angle, work offset, sliding base, machine root angle, the machine center to the cross point, and the velocity coefficient of helical motion, respectively. φ is the rotation angle of the pinion, φ • mb1 is the rotation angle of the cradle, and mb1 is the rolling ratio of the pinion.Figure 1
Coordinate systems applied for pinion generation.The tooth surface of the pinion is generated by the cutter head with a straight blade. In theSp coordinate system, the vector function, rp, of the cutting cone formed by the head cutter and its unit normal vector, np, can be expressed as follows [7]:(1)rp=rc1∓upsinαpcosθprc1∓upsinαpsinθp−spcosαp1,np=cosαpcosθpcosαpsinθp∓sinαp,where sp and θp are the Gaussian coordinates of the pinion tooth surface; αp is the blade angle of the cutter head, and rc1 is the point radius of the cutter head.The upper and lower signs in equation (1) correspond to the convex and concave surfaces of the pinion. They can form a mesh with the concave and convex surfaces of the gear, respectively. According to the coordinate systems applied for pinion generation shown in Figure 1, rp and np are converted into the S1 coordinate system. As a result, the tooth surface equation r1 and its unit normal vector n1 of the pinion in the S1 coordinate system can be obtained as follows:(2)r1up,θp,φ=M1prpup,θp,n1up,θp,φ=L1pnpup,θp,f1=n1⋅∂r11:3∂φ,where f1 is the meshing equation when generating pinion [7], matrix M1p represents the transformation matrix from the coordinate system Sp to S1, and matrix L1p is the third-order submatrix of matrix M1p. M1p can be expressed as(3)M1p=10000cosφ−sinφ00sinφcosφ00000⋅cosγm10sinγm1−xbsinγm1−xg1010e1−sinγm10cosγm10000−xbcosγ1⋅cosqsinq00−sinqcosq0000100001⋅−sinj−cosj0sr1cosj−sinj0000100001⋅cosi0sini00100−sini0cosi00001,where q = q1 + f1• mb1. In addition, when the pinion is generated by the duplex helical method, the cradle has an axial helical motion. Thus, xb can be expressed as(4)xb=xb0−hl•φ•mb1.


## 3. Solving the Contact Point of the Target Tooth Surface while considering the Transmission Error

### 3.1. Solving the Contact Path

The coordinate system of gear pair meshing is shown in Figure2, which establishes the tooth contact analysis (TCA). Sh is a fixed coordinate system. Sd is an auxiliary coordinate system. S1 is a moving coordinate system fixed to the pinion and rotates around the xd-axis at an angular velocity ω(1). S2 is a moving coordinate system fixed to the gear and rotates around the xh-axis at an angular velocity ω(2). ψ1 and ψ2 are the rotation angles of the rotary motion of the pinion and gear, respectively.Figure 2
Coordinate system of gear pair meshing.According to the coordinate system of gear pair meshing shown in Figure2, the tooth surface equation r1 and unit normal vector n1 of the pinion are converted into the meshing coordinate system Sh to obtain rh1 and nh1:(5)rh1up,θp,φ,ψ1=Mh1r1up,θp,φ,nh1up,θp,φ,ψ1=Lh1n1up,θp,φ,where matrix Mh1 represents the transformation matrix from the S1 to Sh and matrix Lh1 is the third-order submatrix of matrix Mh1.Similarly, the tooth surface equationr2 and its unit normal vector n2 of the gear are converted to the meshing coordinate system Sh to get rh2 and nh2:(6)rh2ug,θg,ψ2=Mh2r2ug,θg,nh2ug,θg,ψ2=Lh2n2ug,θg,where matrix Mh2 is the transformation matrix from the S2 to Sh and matrix Lh1 is the third-order submatrix of matrix Mh1.In the gear pair meshing process of a gear pair, the two tooth surfaces should have the same radial vector and unit normal vector at any meshing point; hence, the TCA equation can be constructed [22]:(7)rh1up,θp,φ,ψ1=rh2up,θp,ψ2,nh1up,θp,φ,ψ1=nh2up,θp,ψ2.The transmission error curve and contact path can be obtained by solving equation (7). The tooth surfaces of the pinion and gear obtained by the initial machine tool parameters are defined as the initial pinion tooth surface Σ(0) and initial gear tooth surface, respectively. The radial vector rt0 and unit normal vector nt0 of the discrete point of the contact path on the initial pinion tooth surface Σ(0) can be obtained by the TCA. Here, t represents the number of discrete points on the contact path, and t = 1, 2, ..., m. The radial vectors rt0 of the discrete points of the contact path are converted into the coordinate system of the rotating projection surface using equation (8). Consequently, the contact points (Lt, Rt), which correspond to rt0, are obtained as follows:(8)Lt=rt01,Rt=rt022+rt032.


### 3.2. Solution of the Contact Point of the Target Tooth Surface Driven by the Transmission Error

The transmission error △ψ2 (ψ1) of the gear pair is defined as the difference between the actual rotation angle ψ2 and theoretical rotation angle (z1/z2)•ψ1 of the gear when the pinion rotates through an angle ψ1 [23]:(9)Δψ2ψ1=ψ2−ψ20−z1z2•ψ1−ψ10,where ψ10 is the initial rotation angle of the pinion, ψ20 is the initial rotation angle of the gear, Z1 is the number of pinions, and Z2 is the number of gears.As shown in Figure3, the second-order parabola transmission error curve is predicted on the concave and convex surface of the initial pinion to obtain the target pinion tooth surface. The contact point of the target pinion tooth surface will coincide with the contact point of the initial pinion (Lt, Rt). Given the coordinate value (δte, π/z1) of the meshing transition point of the parabolic transmission error curve, the second-order parabolic transmission error curve can be expressed as(10)Δψ2ψ1=−δte•z12π2•ψ1−ψ102,where δte is the amplitude of the meshing transition point of the transmission error curve. Combining equation (9) and equation (10), the motion relationship between the gear and the pinion can be obtained:(11)ψ2=ψ20+z1z2•ψ1−ψ10−δte•z12π2•ψ1−ψ102.Figure 3
Preset second-order parabola transmission error curve.Combining equation (8) with equation (11), the tooth surface equation r2 of the initial gear can be converted into the pinion coordinate system S1 according to the meshing coordinate system as shown in Figure 2. Consequently, the radial vector rt∗ and unit normal vector nt∗ of the target pinion tooth surface Σ∗ corresponding to the contact point (Lt, Rt) of the initial pinion tooth surface point can be obtained can be obtained using the following equations:(12)rt∗=M12r2ug,θg,nt∗=L12n2ug,θg,ft∗ug,θg,ψ2=nt∗•∂rt∗1:3∂ψ2,ψ2=ψ20+z1z2ψ1−ψ10−δteψ1−ψ102,rt∗x=Lt,rt∗y2+rt∗z2=Rt,where f1∗ is the meshing equation in the coordinate system of gear pair meshing [24]. Matrix M12 represents the transformation matrix from S2 to S1. Meanwhile, matrix L12 is the third-order submatrix of matrix M12. Matrix M12 can be expressed as(13)M12=10000cosψ1−sinψ100sinψ1cosψ100000•cosΓ0sinΓ0010E−sinΓ0cosΓ00000•10000−cosψ2sinψ200−sinψ2−cosψ200001.


## 3.1. Solving the Contact Path

The coordinate system of gear pair meshing is shown in Figure2, which establishes the tooth contact analysis (TCA). Sh is a fixed coordinate system. Sd is an auxiliary coordinate system. S1 is a moving coordinate system fixed to the pinion and rotates around the xd-axis at an angular velocity ω(1). S2 is a moving coordinate system fixed to the gear and rotates around the xh-axis at an angular velocity ω(2). ψ1 and ψ2 are the rotation angles of the rotary motion of the pinion and gear, respectively.Figure 2
Coordinate system of gear pair meshing.According to the coordinate system of gear pair meshing shown in Figure2, the tooth surface equation r1 and unit normal vector n1 of the pinion are converted into the meshing coordinate system Sh to obtain rh1 and nh1:(5)rh1up,θp,φ,ψ1=Mh1r1up,θp,φ,nh1up,θp,φ,ψ1=Lh1n1up,θp,φ,where matrix Mh1 represents the transformation matrix from the S1 to Sh and matrix Lh1 is the third-order submatrix of matrix Mh1.Similarly, the tooth surface equationr2 and its unit normal vector n2 of the gear are converted to the meshing coordinate system Sh to get rh2 and nh2:(6)rh2ug,θg,ψ2=Mh2r2ug,θg,nh2ug,θg,ψ2=Lh2n2ug,θg,where matrix Mh2 is the transformation matrix from the S2 to Sh and matrix Lh1 is the third-order submatrix of matrix Mh1.In the gear pair meshing process of a gear pair, the two tooth surfaces should have the same radial vector and unit normal vector at any meshing point; hence, the TCA equation can be constructed [22]:(7)rh1up,θp,φ,ψ1=rh2up,θp,ψ2,nh1up,θp,φ,ψ1=nh2up,θp,ψ2.The transmission error curve and contact path can be obtained by solving equation (7). The tooth surfaces of the pinion and gear obtained by the initial machine tool parameters are defined as the initial pinion tooth surface Σ(0) and initial gear tooth surface, respectively. The radial vector rt0 and unit normal vector nt0 of the discrete point of the contact path on the initial pinion tooth surface Σ(0) can be obtained by the TCA. Here, t represents the number of discrete points on the contact path, and t = 1, 2, ..., m. The radial vectors rt0 of the discrete points of the contact path are converted into the coordinate system of the rotating projection surface using equation (8). Consequently, the contact points (Lt, Rt), which correspond to rt0, are obtained as follows:(8)Lt=rt01,Rt=rt022+rt032.


## 3.2. Solution of the Contact Point of the Target Tooth Surface Driven by the Transmission Error

The transmission error △ψ2 (ψ1) of the gear pair is defined as the difference between the actual rotation angle ψ2 and theoretical rotation angle (z1/z2)•ψ1 of the gear when the pinion rotates through an angle ψ1 [23]:(9)Δψ2ψ1=ψ2−ψ20−z1z2•ψ1−ψ10,where ψ10 is the initial rotation angle of the pinion, ψ20 is the initial rotation angle of the gear, Z1 is the number of pinions, and Z2 is the number of gears.As shown in Figure3, the second-order parabola transmission error curve is predicted on the concave and convex surface of the initial pinion to obtain the target pinion tooth surface. The contact point of the target pinion tooth surface will coincide with the contact point of the initial pinion (Lt, Rt). Given the coordinate value (δte, π/z1) of the meshing transition point of the parabolic transmission error curve, the second-order parabolic transmission error curve can be expressed as(10)Δψ2ψ1=−δte•z12π2•ψ1−ψ102,where δte is the amplitude of the meshing transition point of the transmission error curve. Combining equation (9) and equation (10), the motion relationship between the gear and the pinion can be obtained:(11)ψ2=ψ20+z1z2•ψ1−ψ10−δte•z12π2•ψ1−ψ102.Figure 3
Preset second-order parabola transmission error curve.Combining equation (8) with equation (11), the tooth surface equation r2 of the initial gear can be converted into the pinion coordinate system S1 according to the meshing coordinate system as shown in Figure 2. Consequently, the radial vector rt∗ and unit normal vector nt∗ of the target pinion tooth surface Σ∗ corresponding to the contact point (Lt, Rt) of the initial pinion tooth surface point can be obtained can be obtained using the following equations:(12)rt∗=M12r2ug,θg,nt∗=L12n2ug,θg,ft∗ug,θg,ψ2=nt∗•∂rt∗1:3∂ψ2,ψ2=ψ20+z1z2ψ1−ψ10−δteψ1−ψ102,rt∗x=Lt,rt∗y2+rt∗z2=Rt,where f1∗ is the meshing equation in the coordinate system of gear pair meshing [24]. Matrix M12 represents the transformation matrix from S2 to S1. Meanwhile, matrix L12 is the third-order submatrix of matrix M12. Matrix M12 can be expressed as(13)M12=10000cosψ1−sinψ100sinψ1cosψ100000•cosΓ0sinΓ0010E−sinΓ0cosΓ00000•10000−cosψ2sinψ200−sinψ2−cosψ200001.


## 4. Nonlinear Accurate Optimization of Transmission Error

The flow chart of the nonlinear optimization method for the transmission error of hypoid gears machined by the duplex helical method is shown in Figure4.Figure 4
Flow chart of the nonlinear optimization method for the transmission error of hypoid gears machined by the duplex helical method.


### 4.1. Optimization Model

Ease-off is defined as the deviation of the corresponding contact point between the target and the initial tooth surfaces [25]. In the past, the optimization of the transmission error of the spiral bevel gear cut by the five-cut method has considered the deviation of the discrete points of the entire tooth surface [6, 9, 20]. For the simultaneous machining of concave and convex surfaces, the duplex helical method is applied. A method of correcting only the ease-off of the contact point between the target pinion tooth surface Σ∗ and initial pinion tooth surface Σ(0) is proposed. The proposed method is more practical and effective than correcting the deviation of the discrete points of the entire tooth surface. By correcting the machine parameters, the ease-off of the contact point can be minimized to optimize the transmission error. The radial vector rt0 and its unit normal vector nt0 of the contact point on the initial pinion tooth surface are expressed as(14)rt0=rt0εt,xnt0=nt0εt,x,t=1,2,…,m,where et = (up, θp, φ) and x = [q1, sr1, mb1, γm1, xb1, xg1, e1, hl, i, j], which is a vector containing 10 machine tool parameters. Meanwhile, t (t = 1, 2, …, m) can be expressed as the t-th contact point.The deviation correction model for contact points is shown in Figure5. rt0 and rt∗ represent the position vectors of the contact point on the initial pinion tooth surface Σ(0) and target pinion tooth surface Σ∗, respectively. ht is the deviation vector of the corresponding contact point. Thus, the position vector rt∗ of the contact point of the target pinion tooth surface Σ∗ can be expressed as(15)rt∗=rt0εt,x+nt0εt,xht.Figure 5
Deviation correction model for contact points.The Gaussian coordinate parametersup and θp of the initial tooth surface are located along the tangential plane of the tooth surface. Therefore, parameter εt (up, θp, φ) can be eliminated as follows:(16)rt∗−rt0εt,x•∂rt0εt,x∂up=0,rt∗−rt0εt,x•∂rt0εt,x∂θp=0,fεt,x=0,where f (εt, x) is the meshing equation for cutting the initial pinion tooth surface.Therefore, the deviation vector,ht, of the tooth surface can be expressed as(17)ht=rt∗−rt0εtx,xnt0εtx,x.Leth (x) = ht (x) be the nonlinear least squares optimization model for the synchronous optimization of the transmission errors. The concave and convex surfaces of the pinion are established by taking the machine tool parameters as design variables and the minimum sum of the residual squares of h (x) as the objective function:(18)x∗=argminfxwithfx=12hxThx.


### 4.2. Selection of Machine Tool Parameters

The machining parameters of the generated tooth surface include the head-cutter and machine tool parameters. The cutter head parameters include the tooth profile angleαp and point radius rc1. When αp and rc1 are used as optimization variables, the geometry of the cutter head needs to be changed, that is, the cutting edge of the cutter head must be ground. However, grinding the cutting edge is time consuming, expensive, and difficult. Therefore, αp and rc1 are not used as optimization variables. Instead, optimization variables are only selected from machine tool parameters. In addition, when selecting optimization variables, the coupling between machine tool parameters is considered; machine tool parameters that strongly influence the normal deviation of the tooth surface are preferred. If the optimization result does not satisfy the requirements, the optimization variables are sequentially increased according to the influence degree of the machine tool parameters on the normal deviation of the tooth surface. The degree of influence of the machine tool parameters on the normal deviation of the tooth surface is defined by the sensitivity coefficient. Assume that the machine tool parameter change is cp and the sum of the absolute values of the normal deviations that cause discrete points on the tooth surface is sd; then, the sensitivity coefficient is cp/sd. We set the angle parameter change amount to 0.05°, the linear parameter error to 0.05 mm, and the roll ratio error to 0.005. Then, the sensitivity coefficients of the machine tool parameters to the tooth surface normal deviation are listed as in Table 1.Table 1
Sensitivity coefficient of machine tool parameters to normal deviation of tooth surface.
Machine tool parametersSensitivity coefficientTilt anglei (°)271Swivel anglej (°)102Radial distancesr1 (mm)9Center roll positionq1 (°)1.2Work offsete1 (mm)4.9Machine center to backxg1 (mm)2.3Sliding basexb1 (mm)1.7Ratio of rollmb1116Machine root angleγm1 (°)291Velocity coefficient of helical motionhl (mm/rad)2.4


### 4.3. Optimization Algorithm

Presently, the nonlinear least squares problem is typically solved using an iterative algorithm. However, when an iterative algorithm is used to solve equation (18), the solution of the equation set is unstable due to the ill-conditions of the coefficient matrix. Thus, it is difficult to solve the accurate and stable corrections of the machine tool parameters [17–19]. To ensure the accuracy of the solution, the Levenberg–Marquard algorithm with a trust region strategy is used to solve equation (18) to ensure the positive definiteness of the Hessian matrix at each step in the iterative process. This approach ensures an accurate and efficient iteration for each step. Using the quadratic approximation function qk (x), the solution of the objective function f (x) of equation (18) can be transformed into a solution to the subproblem for the following trust region:(19)sk=argminqks=fxk+gks+12sTGks,s.t.s≤Δk,where k is the number of iterations and fk = f (xk). Furthermore, s = x−xk, which is the iterative step and is the variable to be determined. gk and Gk are the Jacobian and Hessian matrices of the objective function fk (x) at xk, respectively. h (xk) is the ease-off vector of the tooth surface. Additionally, gk = h (xk)TJ (xk) and Gk = J (xk)TJ (xk). △k is the radius of the trust region. When Δk changes, the solution of equation (19) is a spatial curve, which is the optimal curve shown in Figure 6. To ensure the positive characterization of Gk, the solution of equation (19) is equivalent to solving the following equation:(20)sk=sμk=−Gk+μkI−1gkT,where µk ≥ 0 is the damping coefficient equivalent to the positive parameter. The damping coefficient is used to ensure the regularity of Gk in each iteration and I is the identity matrix. According to equation (20), skGN=−Gk−1gkT is the Gauss–Newton iteration step when µk = 0. skCP=−1/μkgkT is a small step in the direction of steepest descent when µk is large. Therefore, the Levenberg–Marquard algorithm with a trust region strategy focuses on two strategies. The first strategy involves finding the iterative step size, s (uk), for the given trust region radius △k, which is the solution of equation (20). The second strategy involves updating △k.Figure 6
Tangent single dogleg method.To solve equation (20), the single dogleg method [26] and double dogleg method [27] are used. Figure 6 shows the tangent single dogleg method. When solving equation (20) by the tangent single dogleg method, the optimal curve s (△k) is replaced with the polyline sy (△k). The polyline sy (△k) can be obtained through the following steps. First, the Gauss–Newton point sGN in the Gauss–Newton direction is taken as the starting point to form the tangent line of the optimal curve s (△k). Second, the tangent line is projected onto the plane spanned by the fastest descending direction −gkT and the Gauss–Newton direction −Gk−1gkT; the projected line has an intersection point skZP with −gkT. Finally, sy (△k) can be obtained by connecting the initial point xk, skZP and skGN. Compared with the single dogleg method, the tangent single dogleg method yields sy (△k) closer to the Gauss–Newton direction, in other words, closer to the optimal curve. Thus, the tangent single dogleg method is better with convergence.When the Gauss–Newton stepskQN is within the trust region radius △k, that is, if skQN≤Δk, then the iterative step is the optimal solution of equation (20), which is skQN. When skQN>Δk, the tangent single dogleg method can be used to determine the approximate iteration step size. The tangent simple dogleg path sy (△k) consists of two line segments, which replaces the optimal curve s (△k), as shown in Figure 6. Equation (20) is the parametric equation of the optimal curve; by applying uk, this makes ||s|| = △k. The tangent of the curve at any point is(21)sk′=Gk+μkI−1Gk+μkI−1gkT,μ≥0,and when considering µk = 0, the tangent of the point skQN is(22)sk′|μ=0=Gk−1Gk−1gkT.Then, the equation represented by lineskZPskQN is(23)y=Gk−1gkT+αGk−1Gk−1gkT,α∈R.The line is projected onto the plane formed by−gkT,Gk−1gkT, and let A=−gkT,−Gk−1gkT. Then, P = A (ATA)−1AT is the projection matrix of the z-dimensional vector onto the A plane. If the steepest descending direction −gkT is not parallel to the Gaussian iteration direction −Gk−1gkT, then P−Gk−1gkT+αGk−1Gk−1gkT must have an intersection with −gkT. Thus, the solution of the following equation must be unique:(24)β−gkT=Gk−1gkT+αPGk−1Gk−1gkT.Letd=Gk−1gkT and e=PGk−1Gk−1gkT. Then, equation (24) can be transformed into(25)−βgkT=−d+αe.IfP < z, then the first and p-th components of vector gkT, d and e, respectively, are used to establish the system of equations:(26)−βgkT1=−d1+αe1,−βgkTp=−dp+αep.IfgkT1ep−gkTpe1≠0, then equation (26) has a solution:(27)β=dpe1−d1epgkT1ep−gkTpe1.Then, pointskZP=−βgkT can be determined. Therefore, the solution sk of equation (20) is the iteration step where the trust region can be expressed as follows:(28)sk=ΔkskZP.skZP,Δk≤skZP,skQN,Δk≥skQN,skZP+τskQN−skZD,skZP≤Δk≤skQN,τmadesk=Δk.The second problem involves updating the trust region radius △k, after controlling the iterative step by the tangent single dogleg method. In other words, we update the damping parameter µk. When the trust region algorithm is used, the damping parameter µk, can be updated by the increment ρk of the ratio of the objective function f (x) and the quadratic approximation function qk (x):(29)ρk=ΔfxΔqkx=fxk+sk−fxkqksk−qk0.Each iterative process of the Levenberg–Marquard algorithm with trust region strategy is achieved through the following steps. First,µk is given an initial value. Second, sk can be calculated according to the value of the previous iteration step. Afterwards, µk is adjusted according to the value of ρk. Finally, sk is calculated according to the adjusted µk. The update rules of parameter µk are as follows [28]:(30)μk+1=0.1μk,ρk>0.75,μk,0.25≤ρk≤0.75,10μk,ρk<0.25.


## 4.1. Optimization Model

Ease-off is defined as the deviation of the corresponding contact point between the target and the initial tooth surfaces [25]. In the past, the optimization of the transmission error of the spiral bevel gear cut by the five-cut method has considered the deviation of the discrete points of the entire tooth surface [6, 9, 20]. For the simultaneous machining of concave and convex surfaces, the duplex helical method is applied. A method of correcting only the ease-off of the contact point between the target pinion tooth surface Σ∗ and initial pinion tooth surface Σ(0) is proposed. The proposed method is more practical and effective than correcting the deviation of the discrete points of the entire tooth surface. By correcting the machine parameters, the ease-off of the contact point can be minimized to optimize the transmission error. The radial vector rt0 and its unit normal vector nt0 of the contact point on the initial pinion tooth surface are expressed as(14)rt0=rt0εt,xnt0=nt0εt,x,t=1,2,…,m,where et = (up, θp, φ) and x = [q1, sr1, mb1, γm1, xb1, xg1, e1, hl, i, j], which is a vector containing 10 machine tool parameters. Meanwhile, t (t = 1, 2, …, m) can be expressed as the t-th contact point.The deviation correction model for contact points is shown in Figure5. rt0 and rt∗ represent the position vectors of the contact point on the initial pinion tooth surface Σ(0) and target pinion tooth surface Σ∗, respectively. ht is the deviation vector of the corresponding contact point. Thus, the position vector rt∗ of the contact point of the target pinion tooth surface Σ∗ can be expressed as(15)rt∗=rt0εt,x+nt0εt,xht.Figure 5
Deviation correction model for contact points.The Gaussian coordinate parametersup and θp of the initial tooth surface are located along the tangential plane of the tooth surface. Therefore, parameter εt (up, θp, φ) can be eliminated as follows:(16)rt∗−rt0εt,x•∂rt0εt,x∂up=0,rt∗−rt0εt,x•∂rt0εt,x∂θp=0,fεt,x=0,where f (εt, x) is the meshing equation for cutting the initial pinion tooth surface.Therefore, the deviation vector,ht, of the tooth surface can be expressed as(17)ht=rt∗−rt0εtx,xnt0εtx,x.Leth (x) = ht (x) be the nonlinear least squares optimization model for the synchronous optimization of the transmission errors. The concave and convex surfaces of the pinion are established by taking the machine tool parameters as design variables and the minimum sum of the residual squares of h (x) as the objective function:(18)x∗=argminfxwithfx=12hxThx.


## 4.2. Selection of Machine Tool Parameters

The machining parameters of the generated tooth surface include the head-cutter and machine tool parameters. The cutter head parameters include the tooth profile angleαp and point radius rc1. When αp and rc1 are used as optimization variables, the geometry of the cutter head needs to be changed, that is, the cutting edge of the cutter head must be ground. However, grinding the cutting edge is time consuming, expensive, and difficult. Therefore, αp and rc1 are not used as optimization variables. Instead, optimization variables are only selected from machine tool parameters. In addition, when selecting optimization variables, the coupling between machine tool parameters is considered; machine tool parameters that strongly influence the normal deviation of the tooth surface are preferred. If the optimization result does not satisfy the requirements, the optimization variables are sequentially increased according to the influence degree of the machine tool parameters on the normal deviation of the tooth surface. The degree of influence of the machine tool parameters on the normal deviation of the tooth surface is defined by the sensitivity coefficient. Assume that the machine tool parameter change is cp and the sum of the absolute values of the normal deviations that cause discrete points on the tooth surface is sd; then, the sensitivity coefficient is cp/sd. We set the angle parameter change amount to 0.05°, the linear parameter error to 0.05 mm, and the roll ratio error to 0.005. Then, the sensitivity coefficients of the machine tool parameters to the tooth surface normal deviation are listed as in Table 1.Table 1
Sensitivity coefficient of machine tool parameters to normal deviation of tooth surface.
Machine tool parametersSensitivity coefficientTilt anglei (°)271Swivel anglej (°)102Radial distancesr1 (mm)9Center roll positionq1 (°)1.2Work offsete1 (mm)4.9Machine center to backxg1 (mm)2.3Sliding basexb1 (mm)1.7Ratio of rollmb1116Machine root angleγm1 (°)291Velocity coefficient of helical motionhl (mm/rad)2.4


## 4.3. Optimization Algorithm

Presently, the nonlinear least squares problem is typically solved using an iterative algorithm. However, when an iterative algorithm is used to solve equation (18), the solution of the equation set is unstable due to the ill-conditions of the coefficient matrix. Thus, it is difficult to solve the accurate and stable corrections of the machine tool parameters [17–19]. To ensure the accuracy of the solution, the Levenberg–Marquard algorithm with a trust region strategy is used to solve equation (18) to ensure the positive definiteness of the Hessian matrix at each step in the iterative process. This approach ensures an accurate and efficient iteration for each step. Using the quadratic approximation function qk (x), the solution of the objective function f (x) of equation (18) can be transformed into a solution to the subproblem for the following trust region:(19)sk=argminqks=fxk+gks+12sTGks,s.t.s≤Δk,where k is the number of iterations and fk = f (xk). Furthermore, s = x−xk, which is the iterative step and is the variable to be determined. gk and Gk are the Jacobian and Hessian matrices of the objective function fk (x) at xk, respectively. h (xk) is the ease-off vector of the tooth surface. Additionally, gk = h (xk)TJ (xk) and Gk = J (xk)TJ (xk). △k is the radius of the trust region. When Δk changes, the solution of equation (19) is a spatial curve, which is the optimal curve shown in Figure 6. To ensure the positive characterization of Gk, the solution of equation (19) is equivalent to solving the following equation:(20)sk=sμk=−Gk+μkI−1gkT,where µk ≥ 0 is the damping coefficient equivalent to the positive parameter. The damping coefficient is used to ensure the regularity of Gk in each iteration and I is the identity matrix. According to equation (20), skGN=−Gk−1gkT is the Gauss–Newton iteration step when µk = 0. skCP=−1/μkgkT is a small step in the direction of steepest descent when µk is large. Therefore, the Levenberg–Marquard algorithm with a trust region strategy focuses on two strategies. The first strategy involves finding the iterative step size, s (uk), for the given trust region radius △k, which is the solution of equation (20). The second strategy involves updating △k.Figure 6
Tangent single dogleg method.To solve equation (20), the single dogleg method [26] and double dogleg method [27] are used. Figure 6 shows the tangent single dogleg method. When solving equation (20) by the tangent single dogleg method, the optimal curve s (△k) is replaced with the polyline sy (△k). The polyline sy (△k) can be obtained through the following steps. First, the Gauss–Newton point sGN in the Gauss–Newton direction is taken as the starting point to form the tangent line of the optimal curve s (△k). Second, the tangent line is projected onto the plane spanned by the fastest descending direction −gkT and the Gauss–Newton direction −Gk−1gkT; the projected line has an intersection point skZP with −gkT. Finally, sy (△k) can be obtained by connecting the initial point xk, skZP and skGN. Compared with the single dogleg method, the tangent single dogleg method yields sy (△k) closer to the Gauss–Newton direction, in other words, closer to the optimal curve. Thus, the tangent single dogleg method is better with convergence.When the Gauss–Newton stepskQN is within the trust region radius △k, that is, if skQN≤Δk, then the iterative step is the optimal solution of equation (20), which is skQN. When skQN>Δk, the tangent single dogleg method can be used to determine the approximate iteration step size. The tangent simple dogleg path sy (△k) consists of two line segments, which replaces the optimal curve s (△k), as shown in Figure 6. Equation (20) is the parametric equation of the optimal curve; by applying uk, this makes ||s|| = △k. The tangent of the curve at any point is(21)sk′=Gk+μkI−1Gk+μkI−1gkT,μ≥0,and when considering µk = 0, the tangent of the point skQN is(22)sk′|μ=0=Gk−1Gk−1gkT.Then, the equation represented by lineskZPskQN is(23)y=Gk−1gkT+αGk−1Gk−1gkT,α∈R.The line is projected onto the plane formed by−gkT,Gk−1gkT, and let A=−gkT,−Gk−1gkT. Then, P = A (ATA)−1AT is the projection matrix of the z-dimensional vector onto the A plane. If the steepest descending direction −gkT is not parallel to the Gaussian iteration direction −Gk−1gkT, then P−Gk−1gkT+αGk−1Gk−1gkT must have an intersection with −gkT. Thus, the solution of the following equation must be unique:(24)β−gkT=Gk−1gkT+αPGk−1Gk−1gkT.Letd=Gk−1gkT and e=PGk−1Gk−1gkT. Then, equation (24) can be transformed into(25)−βgkT=−d+αe.IfP < z, then the first and p-th components of vector gkT, d and e, respectively, are used to establish the system of equations:(26)−βgkT1=−d1+αe1,−βgkTp=−dp+αep.IfgkT1ep−gkTpe1≠0, then equation (26) has a solution:(27)β=dpe1−d1epgkT1ep−gkTpe1.Then, pointskZP=−βgkT can be determined. Therefore, the solution sk of equation (20) is the iteration step where the trust region can be expressed as follows:(28)sk=ΔkskZP.skZP,Δk≤skZP,skQN,Δk≥skQN,skZP+τskQN−skZD,skZP≤Δk≤skQN,τmadesk=Δk.The second problem involves updating the trust region radius △k, after controlling the iterative step by the tangent single dogleg method. In other words, we update the damping parameter µk. When the trust region algorithm is used, the damping parameter µk, can be updated by the increment ρk of the ratio of the objective function f (x) and the quadratic approximation function qk (x):(29)ρk=ΔfxΔqkx=fxk+sk−fxkqksk−qk0.Each iterative process of the Levenberg–Marquard algorithm with trust region strategy is achieved through the following steps. First,µk is given an initial value. Second, sk can be calculated according to the value of the previous iteration step. Afterwards, µk is adjusted according to the value of ρk. Finally, sk is calculated according to the adjusted µk. The update rules of parameter µk are as follows [28]:(30)μk+1=0.1μk,ρk>0.75,μk,0.25≤ρk≤0.75,10μk,ρk<0.25.


## 5. Numerical Example

### 5.1. Nonlinear Optimization of Transmission Error

In this study, we used a pair of hypoid gear pairs machined by the forming method for the gear and the duplex helical method for the pinion as an example. Table2 lists the main geometric parameters of the hypoid gear pair; Table 3 lists the parameters of the preset second-order parabolic transmission error curve; and Table 4 is the initial head-cutter parameters and machine tool parameters of the hypoid gear pair.Table 2
Main geometric parameters of the hypoid gear pairs.
GearPinionNumber of teeth438Module6.861Face width (mm)4144.691Mean cone distance (mm)130.762128.323Pinion offset (mm)25.4Mean spiral angle (°)33.845Hand of spiralRightLeftHand of spiral (°)90Pitch angle (°)77.21512.549Face angle of blank (°)78.12416.154Root angle (°)73.54711.655Table 3
Parameters of the preset second-order parabolic transmission error curve.
ConcaveConvexAmplitude of meshing transition point (″)56Table 4
Initial head-cutter and machine tool parameters of the hypoid gear pair.
GearPinionBlade angleap (°)ConcaveConvexConcaveConvex20252025Point radiusrc1 (mm)115.824112.776114.966112.206Tilt anglei (°)015.572Swivel anglej (°)0−35.332Radial distancesr1 (mm)116.573117.385Center roll positionq1 (°)60.46565.626Work offsete1 (mm)028.568Machine center to backxg1 (mm)3.1416−0.147Sliding basexb1 (mm)016.637Ratio of rollmb105.326Machine root angleγm1 (°)72.2366−3.631Velocity coefficient of helical motionhl (mm/rad)00.766The target pinion can be obtained using the two constraints of the preset transmission error curve and the coincidence of the contact line with the initial pinion. According to the flowchart shown in Figure4, the Levenberg–Marquard algorithm with trust region strategy is used to optimize the transmission error and converges after 35 iterations. The deviation of the concave surface is reduced from a maximum of −13 um to a maximum of −1.3 um; and the deviation of the convex surface is reduced from a maximum of 12.9 μm to a maximum of 1.5 μm. In addition, the sequential quadratic programming algorithm [17, 18] was also used to optimize the transmission error; and the results were compared with those obtained using the Levenberg–Marquardt algorithm with the trust region strategy. After optimization using the sequential quadratic programming algorithm, the deviation of the concave surface is reduced from a maximum of −13 μm to a maximum of −5.6 μm; and the deviation of the convex surface is reduced from a maximum of 12.9 um to a maximum of 4.5 um. The comparison between the Levenberg–Marquard algorithm with trust region strategy and the sequential quadratic programming algorithm shows that the Levenberg–Marquard algorithm with trust region strategy has a better optimization effect. The optimized machine tool parameters of the pinion are listed in Table 5. TCA was applied to the hypoid gear pair that corresponding to the initial and optimized machine tool parameters. The optimized transmission error curve obtained by the Levenberg–Marquardt algorithm with trust region strategy was defined as optimized transmission error curve 1, whereas that obtained by the sequential quadratic programming algorithm was defined as optimized transmission error curve 2. The transmission error curve obtained from the initial machine tool parameters was defined as the initial transmission error curve. The transmission error curve obtained from the initial machine tool parameters is was defined as the initial transmission error curve. Figure 7 shows the initial and preset curves, as well as optimized transmission error curves 1 and 2, for the concave surfaces of the pinion. Figure 8 shows the initial and preset curves, as well as optimized transmission error curves 1 and 2, for the convex surfaces of the pinion. Lines ○, ●, □, and ∆ in Figures 7 and 8 correspond to the initial curve, preset curve, optimized transmission error curve 1, and optimized transmission error curve 2, respectively.Table 5
Optimized machine tool parameters of the pinion.
Levenberg–Marquard algorithm with trust region strategySequential quadratic programming algorithmTilt anglei (°)14.64115.049Swivel anglej (°)−36.249−36.018Radial distancesr1 (mm)117.968117.721Center roll positionq1 (°)65.62665.626Work offsete1 (mm)29.10028.880Machine center to backxg1 (mm)−0.138−0.117Sliding basexb1 (mm)15.93816.303Ratio of rollmb15.3765.358Machine root angleγm1 (°)−2.671−3.077Velocity coefficient of helical motionhl (mm/rad)0.5160.595Figure 7
Initial curve, preset curve, optimized transmission error curve 1, and optimized transmission error curve 2 for the concave surfaces of the pinion.Figure 8
Initial curve, preset curve, optimized transmission error curve 1, and optimized transmission error curve 2 for the convex surfaces of the pinion.Furthermore, Figure7, Figure 8, and Table 3 show that the Levenberg–Marquard algorithm with trust region strategy is better than sequential quadratic programming algorithm in optimizing transmission error. Therefore, the following content only discusses the optimization effect of the Levenberg–Marquard algorithm on the transmission error. Figures 7 and 8 show that the optimized transmission error curve 1 is consistent with the preset transmission error curve. Further analysis of the transmission error curve was conducted, and the results are summarized in Table 6. When comparing with the preset values, it was found that the amplitudes of the meshing transition points of the concave and convex surfaces of the optimized pinion differ by 0.1″ and 0.2″, i.e., 2% and 3.3%, respectively. The amplitudes of the maximum transmission error of the concave and convex surfaces of the optimized pinion differ by 0.1″ and 0.5″, i.e., 1.2% and 4.6%, respectively. When comparing with the initial values, the maximum amplitude and the amplitude of the meshing transformation point of optimized transmission error curve 1 are reduced. When performing further analysis of the transmission errors in Table 6, they were compared with the initial transmission error curve. The amplitudes of the meshing transition points of the concave and convex surfaces of the optimized pinion are reduced by 7.2″and 4.5″, i.e., 59.5% and 43.7%, respectively. The amplitudes of the maximum transmission error for the concave and convex surfaces of the optimized pinion are reduced by 11.3″ and 7.6″, i.e., 57.1% and 42.2%, respectively. In summary, optimized transmission error curve 1 is consistent with the preset transmission error curve. Furthermore, the amplitudes of the meshing transition point and maximum transmission error are significantly reduced in comparison to the initial transmission error curve. These characteristics verify the effectiveness of the proposed method, which involves the precise, nonlinear optimization of the hypoid gear transmission error driven by the duplex helical method.Table 6
Maximum amplitude and amplitude of the meshing transition point for the transmission error.
Amplitude of meshing transition pointAmplitude of maximum transmission errorConcaveConvexConcaveConvexInitial (″)12.110.319.818.0Preset (″)568.610.9Optimized transmission error curve 1 (″)4.95.88.510.4Optimized transmission error curve 2 (″)7.67.912.914.1


### 5.2. Loaded Tooth Contact Analysis

Because this study was based on the optimization of unloaded transmission errors and only the deviation of the contact point of the tooth surface is corrected, it was necessary to use loaded tooth contact analysis for the initial and optimized hypoid gear pairs to obtain the initial and optimized loaded contact pattern and loaded transmission errors. First, the tooth surfaces of the gear and pinion were solved according to the cutting mathematical model of the hypoid gear pair. Second, the TCA was used to obtain the installation position of the hypoid gear pair, to establish the assembly model of the hypoid gear pair through Pro/E software, and single teeth were cut on the pinion and the gear in the assembly model. Third, the single tooth assembly model was imported into Hypermesh software for 3D meshing. Finally, the single tooth assembly model with 3D mesh is imported into Abaqus software, and the single tooth model array is a five-tooth model; thus, the five-tooth model was subjected to loading tooth contact analysis based on the finite element method. For finite element analysis in Abaqus software, the density, elastic modulus, and Poisson’s ratio were set to 7.85 × 103 kg/m3, 1.88 × 105 MP, and 0.3, respectively. In addition, the rotation speed of the pinion and the load torque of the gear were set to 1000 r/min and 200 N∗m, respectively. Figure 9 shows the finite element model of loaded tooth contact analysis of a hypoid gear pair with five pairs of teeth.Figure 9
Finite element model of loaded tooth contact analysis of hypoid gear.The loaded contact pattern was derived from a third tooth pair in the finite element model. Figures10 and 11 show the loaded contact patterns for the concave and convex surface of the initial and optimized pinions, respectively. Figure 12 shows the initial and optimized loaded transmission error curves for the concave and convex surfaces of the pinion. The solid and dashed lines in Figure 12 represent the initial and optimized loaded transmission error curves, respectively.Figure 10
Loaded contact pattern for the concave surface of the initial and optimized pinion.
(a)(b)Figure 11
Loaded contact pattern for the convex surface of the initial and optimized pinion.
(a)(b)Figure 12
Loaded transmission error for the initial and optimized pinion.
(a)(b)Figures10 and 11 show that the contact areas on the concave and convex surfaces of the initial and optimized hypoid gear pairs are the same. The maximum contact pressure for the concave surface of pinion is reduced from 333 MP initially to 329 MP after optimization. The maximum contact pressure for the convex surface of the pinion increased from 331 MP initially to 344 MP after optimized. As can be seen from Figures 7, 8, and 12, compared with the unloaded transmission error, the amplitudes of the initial and optimized loaded transmission error curves at the meshing transition point are reduced. For the concave surface, the amplitudes of the loaded transmission error curves of the initial and optimized pinions at the meshing transition point are 4 μm and 8.9 μm, respectively. For the convex surface, the amplitudes of the loaded transmission error curves of the initial and optimized pinions at the meshing transition point are 4.6 μm and 8.1 μm, respectively.


### 5.3. Contact Area Rolling Test of the Optimized Gear Pair

The H650 C spiral bevel gear milling machine produced by Haarang Kaishuai Precision Machinery Co., Ltd. was used to cut the gear pairs optimized by the Levenberg–Marquard algorithm with trust region strategy, and the contact area rolling test was carried out for the gear pairs after cutting. Figure13 shows the cutting machine of the gear pair; Figure 14 shows the optimized hypoid gear pair that have been machined; and Figure 15 shows the optimized contact areas of the concave and convex surfaces of the pinion.Figure 13
Cutting machine of the gear pair.Figure 14
Optimized hypoid gear pair that has been machined.Figure 15
Optimized contact areas of the concave and convex surfaces of the pinion. (a) Concave. (b) Convex.
(a)(b)Figure15 shows that the rolling contact areas of the concave and convex surfaces of the pinion are good and are basically the same as the optimized loading contact areas in Figures 10 and 11. The contact area rolling test further shows that, after correcting the deviation of the tooth surface contact point to achieve high-precision optimization of the transmission error, the optimized tooth surface contact area can also meet the tooth surface design requirements. Therefore, for a hypoid gear machined by the duplex helical method, only the deviation of the tooth surface contact point needs to be corrected to obtain the corresponding machine tool parameters and, thereby, to obtain the required transmission error curve of the gear pair. Thus, it is not necessary to obtain the transmission error curve by cutting the tooth surface multiple times. The design and manufacturing efficiency of gear pair are improved. In addition, the proposed method can also be applied to the spiral bevel gear processed by the five-cut method.


## 5.1. Nonlinear Optimization of Transmission Error

In this study, we used a pair of hypoid gear pairs machined by the forming method for the gear and the duplex helical method for the pinion as an example. Table2 lists the main geometric parameters of the hypoid gear pair; Table 3 lists the parameters of the preset second-order parabolic transmission error curve; and Table 4 is the initial head-cutter parameters and machine tool parameters of the hypoid gear pair.Table 2
Main geometric parameters of the hypoid gear pairs.
GearPinionNumber of teeth438Module6.861Face width (mm)4144.691Mean cone distance (mm)130.762128.323Pinion offset (mm)25.4Mean spiral angle (°)33.845Hand of spiralRightLeftHand of spiral (°)90Pitch angle (°)77.21512.549Face angle of blank (°)78.12416.154Root angle (°)73.54711.655Table 3
Parameters of the preset second-order parabolic transmission error curve.
ConcaveConvexAmplitude of meshing transition point (″)56Table 4
Initial head-cutter and machine tool parameters of the hypoid gear pair.
GearPinionBlade angleap (°)ConcaveConvexConcaveConvex20252025Point radiusrc1 (mm)115.824112.776114.966112.206Tilt anglei (°)015.572Swivel anglej (°)0−35.332Radial distancesr1 (mm)116.573117.385Center roll positionq1 (°)60.46565.626Work offsete1 (mm)028.568Machine center to backxg1 (mm)3.1416−0.147Sliding basexb1 (mm)016.637Ratio of rollmb105.326Machine root angleγm1 (°)72.2366−3.631Velocity coefficient of helical motionhl (mm/rad)00.766The target pinion can be obtained using the two constraints of the preset transmission error curve and the coincidence of the contact line with the initial pinion. According to the flowchart shown in Figure4, the Levenberg–Marquard algorithm with trust region strategy is used to optimize the transmission error and converges after 35 iterations. The deviation of the concave surface is reduced from a maximum of −13 um to a maximum of −1.3 um; and the deviation of the convex surface is reduced from a maximum of 12.9 μm to a maximum of 1.5 μm. In addition, the sequential quadratic programming algorithm [17, 18] was also used to optimize the transmission error; and the results were compared with those obtained using the Levenberg–Marquardt algorithm with the trust region strategy. After optimization using the sequential quadratic programming algorithm, the deviation of the concave surface is reduced from a maximum of −13 μm to a maximum of −5.6 μm; and the deviation of the convex surface is reduced from a maximum of 12.9 um to a maximum of 4.5 um. The comparison between the Levenberg–Marquard algorithm with trust region strategy and the sequential quadratic programming algorithm shows that the Levenberg–Marquard algorithm with trust region strategy has a better optimization effect. The optimized machine tool parameters of the pinion are listed in Table 5. TCA was applied to the hypoid gear pair that corresponding to the initial and optimized machine tool parameters. The optimized transmission error curve obtained by the Levenberg–Marquardt algorithm with trust region strategy was defined as optimized transmission error curve 1, whereas that obtained by the sequential quadratic programming algorithm was defined as optimized transmission error curve 2. The transmission error curve obtained from the initial machine tool parameters was defined as the initial transmission error curve. The transmission error curve obtained from the initial machine tool parameters is was defined as the initial transmission error curve. Figure 7 shows the initial and preset curves, as well as optimized transmission error curves 1 and 2, for the concave surfaces of the pinion. Figure 8 shows the initial and preset curves, as well as optimized transmission error curves 1 and 2, for the convex surfaces of the pinion. Lines ○, ●, □, and ∆ in Figures 7 and 8 correspond to the initial curve, preset curve, optimized transmission error curve 1, and optimized transmission error curve 2, respectively.Table 5
Optimized machine tool parameters of the pinion.
Levenberg–Marquard algorithm with trust region strategySequential quadratic programming algorithmTilt anglei (°)14.64115.049Swivel anglej (°)−36.249−36.018Radial distancesr1 (mm)117.968117.721Center roll positionq1 (°)65.62665.626Work offsete1 (mm)29.10028.880Machine center to backxg1 (mm)−0.138−0.117Sliding basexb1 (mm)15.93816.303Ratio of rollmb15.3765.358Machine root angleγm1 (°)−2.671−3.077Velocity coefficient of helical motionhl (mm/rad)0.5160.595Figure 7
Initial curve, preset curve, optimized transmission error curve 1, and optimized transmission error curve 2 for the concave surfaces of the pinion.Figure 8
Initial curve, preset curve, optimized transmission error curve 1, and optimized transmission error curve 2 for the convex surfaces of the pinion.Furthermore, Figure7, Figure 8, and Table 3 show that the Levenberg–Marquard algorithm with trust region strategy is better than sequential quadratic programming algorithm in optimizing transmission error. Therefore, the following content only discusses the optimization effect of the Levenberg–Marquard algorithm on the transmission error. Figures 7 and 8 show that the optimized transmission error curve 1 is consistent with the preset transmission error curve. Further analysis of the transmission error curve was conducted, and the results are summarized in Table 6. When comparing with the preset values, it was found that the amplitudes of the meshing transition points of the concave and convex surfaces of the optimized pinion differ by 0.1″ and 0.2″, i.e., 2% and 3.3%, respectively. The amplitudes of the maximum transmission error of the concave and convex surfaces of the optimized pinion differ by 0.1″ and 0.5″, i.e., 1.2% and 4.6%, respectively. When comparing with the initial values, the maximum amplitude and the amplitude of the meshing transformation point of optimized transmission error curve 1 are reduced. When performing further analysis of the transmission errors in Table 6, they were compared with the initial transmission error curve. The amplitudes of the meshing transition points of the concave and convex surfaces of the optimized pinion are reduced by 7.2″and 4.5″, i.e., 59.5% and 43.7%, respectively. The amplitudes of the maximum transmission error for the concave and convex surfaces of the optimized pinion are reduced by 11.3″ and 7.6″, i.e., 57.1% and 42.2%, respectively. In summary, optimized transmission error curve 1 is consistent with the preset transmission error curve. Furthermore, the amplitudes of the meshing transition point and maximum transmission error are significantly reduced in comparison to the initial transmission error curve. These characteristics verify the effectiveness of the proposed method, which involves the precise, nonlinear optimization of the hypoid gear transmission error driven by the duplex helical method.Table 6
Maximum amplitude and amplitude of the meshing transition point for the transmission error.
Amplitude of meshing transition pointAmplitude of maximum transmission errorConcaveConvexConcaveConvexInitial (″)12.110.319.818.0Preset (″)568.610.9Optimized transmission error curve 1 (″)4.95.88.510.4Optimized transmission error curve 2 (″)7.67.912.914.1


## 5.2. Loaded Tooth Contact Analysis

Because this study was based on the optimization of unloaded transmission errors and only the deviation of the contact point of the tooth surface is corrected, it was necessary to use loaded tooth contact analysis for the initial and optimized hypoid gear pairs to obtain the initial and optimized loaded contact pattern and loaded transmission errors. First, the tooth surfaces of the gear and pinion were solved according to the cutting mathematical model of the hypoid gear pair. Second, the TCA was used to obtain the installation position of the hypoid gear pair, to establish the assembly model of the hypoid gear pair through Pro/E software, and single teeth were cut on the pinion and the gear in the assembly model. Third, the single tooth assembly model was imported into Hypermesh software for 3D meshing. Finally, the single tooth assembly model with 3D mesh is imported into Abaqus software, and the single tooth model array is a five-tooth model; thus, the five-tooth model was subjected to loading tooth contact analysis based on the finite element method. For finite element analysis in Abaqus software, the density, elastic modulus, and Poisson’s ratio were set to 7.85 × 103 kg/m3, 1.88 × 105 MP, and 0.3, respectively. In addition, the rotation speed of the pinion and the load torque of the gear were set to 1000 r/min and 200 N∗m, respectively. Figure 9 shows the finite element model of loaded tooth contact analysis of a hypoid gear pair with five pairs of teeth.Figure 9
Finite element model of loaded tooth contact analysis of hypoid gear.The loaded contact pattern was derived from a third tooth pair in the finite element model. Figures10 and 11 show the loaded contact patterns for the concave and convex surface of the initial and optimized pinions, respectively. Figure 12 shows the initial and optimized loaded transmission error curves for the concave and convex surfaces of the pinion. The solid and dashed lines in Figure 12 represent the initial and optimized loaded transmission error curves, respectively.Figure 10
Loaded contact pattern for the concave surface of the initial and optimized pinion.
(a)(b)Figure 11
Loaded contact pattern for the convex surface of the initial and optimized pinion.
(a)(b)Figure 12
Loaded transmission error for the initial and optimized pinion.
(a)(b)Figures10 and 11 show that the contact areas on the concave and convex surfaces of the initial and optimized hypoid gear pairs are the same. The maximum contact pressure for the concave surface of pinion is reduced from 333 MP initially to 329 MP after optimization. The maximum contact pressure for the convex surface of the pinion increased from 331 MP initially to 344 MP after optimized. As can be seen from Figures 7, 8, and 12, compared with the unloaded transmission error, the amplitudes of the initial and optimized loaded transmission error curves at the meshing transition point are reduced. For the concave surface, the amplitudes of the loaded transmission error curves of the initial and optimized pinions at the meshing transition point are 4 μm and 8.9 μm, respectively. For the convex surface, the amplitudes of the loaded transmission error curves of the initial and optimized pinions at the meshing transition point are 4.6 μm and 8.1 μm, respectively.


## 5.3. Contact Area Rolling Test of the Optimized Gear Pair

The H650 C spiral bevel gear milling machine produced by Haarang Kaishuai Precision Machinery Co., Ltd. was used to cut the gear pairs optimized by the Levenberg–Marquard algorithm with trust region strategy, and the contact area rolling test was carried out for the gear pairs after cutting. Figure13 shows the cutting machine of the gear pair; Figure 14 shows the optimized hypoid gear pair that have been machined; and Figure 15 shows the optimized contact areas of the concave and convex surfaces of the pinion.Figure 13
Cutting machine of the gear pair.Figure 14
Optimized hypoid gear pair that has been machined.Figure 15
Optimized contact areas of the concave and convex surfaces of the pinion. (a) Concave. (b) Convex.
(a)(b)Figure15 shows that the rolling contact areas of the concave and convex surfaces of the pinion are good and are basically the same as the optimized loading contact areas in Figures 10 and 11. The contact area rolling test further shows that, after correcting the deviation of the tooth surface contact point to achieve high-precision optimization of the transmission error, the optimized tooth surface contact area can also meet the tooth surface design requirements. Therefore, for a hypoid gear machined by the duplex helical method, only the deviation of the tooth surface contact point needs to be corrected to obtain the corresponding machine tool parameters and, thereby, to obtain the required transmission error curve of the gear pair. Thus, it is not necessary to obtain the transmission error curve by cutting the tooth surface multiple times. The design and manufacturing efficiency of gear pair are improved. In addition, the proposed method can also be applied to the spiral bevel gear processed by the five-cut method.


## 6. Conclusions

This report proposed a transmission error optimization method for hypoid gears based on the duplex helical method, that is, only by correcting the deviations of the contact points between the target and initial pinion tooth surfaces, the transmission error of the concave and convex surfaces are optimized synchronously. The optimization algorithm used is the Levenberg–Marquard algorithm with trust region strategy, which was compared with the sequential quadratic programming algorithm. In terms of the consistency between the optimized and the preset transmission error curves, the Levenberg–Marquard algorithm with trust region strategy was better than the sequential quadratic programming algorithm. In addition, the transmission error curve optimized by the Levenberg-Marquard algorithm with trust region strategy was compared with the initial and preset transmission error curves, and the following conclusions were obtained.(1)
When comparing with the preset values, the amplitudes of the meshing transition points of the concave and convex surfaces of the optimized pinion differ by 0.1″ and 0.2″, i.e., 2% and 3.3%, respectively. The amplitudes of the maximum transmission error of the concave and convex surfaces of the optimized pinion differ by 0.1″ and 0.5″, i.e., 1.2% and 4.6%, respectively. When comparing with the initial values, the amplitudes of the meshing transition points of the concave and convex surfaces of the optimized pinion are reduced by 7.2″and 4.5″, i.e., 59.5% and 43.7%, respectively. The maximum transmission error amplitudes for the concave and convex surfaces of the optimized pinion are reduced by 11.3″ and 7.6″, i.e., 57.1% and 42.2%, respectively. In other words, the optimized transmission error curve not only maintains the consistency with the preset transmission error curve but also exhibited reduced amplitudes of the mesh conversion point and maximum transmission error compared with those of the initial transmission error curve.(2)
The loaded contact areas on the concave and convex surfaces of the initial and optimized hypoid gear pairs were basically the same. The amplitudes of loaded transmission error curves at the meshing transition points of the concave and convex surfaces of the initial and optimized pinion were reduced. The effectiveness of the nonlinear optimization method for the transmission error of the hypoid gear was further verified by loaded tooth surface contact analysis. However, the maximum contact pressure of the concave surfaces of the pinions increased from 331 MP initially to 344 MP after optimization. Therefore, factors such as contact pressure and transmission efficiency need to be considered in research.(3)
The tooth surface cutting test revealed that the rolling contact areas of the concave and convex surfaces of the optimized pinion were basically the same as the loading contact areas obtained by the finite element method. That is, the optimized tooth surface contact area was good, and there is no edge contact problem, which further supports the effectiveness of the proposed transmission error curve optimization method.



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*Source: 9626089-2020-11-29.xml*