A Comprehensive Study on Meshing Performances Compensation for Face-Hobbed Hypoid Gears: Coupled Analysis of Spatial Installation Errors and Manufactured Tooth Flank Characteristics

Abstract

In manufacturing face-hobbing hypoid gears, the coupling between tooth flank errors and installation errors has a significant impact on dynamic meshing behavior, yet quantitative models for their synergistic effects remain scarce. This study elucidates the combined effects of three-dimensional (3D) installation errors and real tooth flank deviations on transmission error. First, a geometric model of the real tooth flank, incorporating midpoint pitch deviation, is established based on theoretical flank equations and coordinate transformations. Then, a finite element model integrating 3D installation errors is developed. Finally, the combined effects of installation errors and real tooth flanks on meshing performance are analyzed. Results reveal a dual role of installation errors: when compensating for midpoint pitch deviation, the peak-to-peak transmission error (PPTE) decreases by 3.78%, while the contact pattern length and area increase. Under certain conditions, despite a 26.28% increase in PPTE, the contact pattern length grows by 2.29%, accompanied by a notable reduction in maximum contact stress on the tooth flanks.

1. Introduction

Face-hobbed hypoid gears (FH-HGs) are extensively applied in advanced engineering systems such as automotive differentials and aero-engines, due to their flexible spatial configuration, high load capacity, smooth transmission, and compact structure. In practical applications, tooth flanks often exhibit pitch deviations, and installation errors of gear pairs typically show multi-directional distribution characteristics. These factors significantly influence the meshing performance of gear pairs. However, the effects of manufacturing deviations and three-dimensional (3D) installation errors on meshing performance are highly coupled and uncertain, making their quantitative evaluation challenging. Therefore, elucidating the coupling mechanism between manufacturing and 3D installation errors in FH-HG is crucial to developing effective transmission error compensation methods and guiding precision assembly.

Enhancing the meshing performance of gear pairs has long been a central focus in gear transmission research. Numerous studies have sought to improve performance through innovative tooth flank design and modification strategies. Simon [1] performed tooth flank modification on FH-HG by optimizing tool geometry and machine settings. Litvin et al. [2] established a complete parametric mathematical model of the pitch cone based on the direct calculation method for nonlinear equation systems. Fan [3] established a comprehensive theoretical framework encompassing tool geometry, kinematics, and the complete process of tooth surface generation and contact analysis. Yuan et al. [4] proposed a tooth flank modification method that accounts for system flexibility, effectively improving meshing quality and reducing system vibration. Yin et al. [5] proposed a rack approximation-based tooth flank modification method, effectively increasing both contact area and contact ratio. Liu et al. [6] proposed an active asymmetric tooth profile modification approach that quantitatively regulates the point contact trajectory, thereby markedly enhancing meshing performance. Shih and Fong [7] proposes a flank modification approach based on ease-off topography, utilizing linear regression to optimize the tooth flank of hypoid gears, thereby mitigating noise and preventing edge contact. Huang et al. [8] achieved balanced wear distribution and vibration suppression by synergistically optimizing tip relief and helix angle modification. Ni et al. [9] developed a parabolic modification technique for bevel gears that reduces contact forces while significantly improving transmission error stability and stress distribution. Gu et al. [10] applied compound modification to optimize gear flanks, lowering contact stress and peak-to-peak transmission error while markedly mitigating meshing impacts and partial load conditions. Simon [11] conducted an investigation on micro-geometry modifications for hypoid gears, utilizing optimized tool parameters to enhance load distribution and reduce transmission error. Kolivand and Kahraman [12] develops a methodology for ease-off topography to quantify the effects of both global and local deviations on the load distribution and transmission error of hypoid gears, as analyzed under loaded tooth contact conditions.

Further, the formation mechanisms and effects of tooth flank geometric deviations and deformations have been extensively investigated and incorporated into tooth flank design processes to optimize and compensate for manufacturing errors. Duan et al. [13] developed a comprehensive gear meshing model that accounts for tooth flank modification, pitch errors, and geometric eccentricity. Liang et al. [14] proposed a novel tooth flank error modeling approach based on the relationship between discrete tooth flank points and measured error points. Fan [15] introduced modified machine tool setting correction techniques to compensate for tooth flank deviations by adjusting theoretical machine tool configurations. Zhou et al. [16] devised a matching point search algorithm that resolves the correspondence problem between the measured and theoretical tooth flanks. Li et al. [17] extracted 3D model surfaces from manufacturer designs to construct error computation models, thereby minimizing tooth flank deviations through reconstruction and optimal matching techniques. Liang et al. [18] established a tooth flank model incorporating errors by considering 40 machine tool error sources, and subsequently derived the corresponding tooth flank equation with error terms. Wei et al. [19] investigated the influence of manufacturing uncertainties on hypoid gear contact performance optimization and proposed a contact model that integrates Gaussian-distributed random tooth flank errors together with an associated optimization methodology. Li et al. [20] presents a numerical contact and sensitivity analysis method using CMM data to predict contact patterns and enhance performance in hypoid gear error-sensitivity evaluation. Gonzalez P [21] established a direct relationship between the meshing performance of hypoid gears and machine tool settings, analyzing the sensitivity of tooth flank errors to these parameters for error correction. Xue et al. [22] developed an analytical prediction formula for the relationship between transmission error and pitch error in face-hobbing hypoid gears and validated the method through experiments. Vivet et al. [23] predicted misaligned contact performance with high robustness by determining the instantaneous pose of the error tooth flank using a penetration-based gear contact model. Chen et al. [24] proposed a dynamic compensation method for geometric errors in gear milling machines, which reduces contact stress by establishing an error tooth flank model and an inverse kinematics relationship. Liu et al. [25] proposed an iterative search-based method for compensating tilt error in tooth flank measurement, providing a more accurate basis for machine tool parameter adjustment during manufacturing.

Furthermore, installation errors are another critical determinant of gear-pair performance and have been investigated extensively. Deng et al. [26] analyzed a coupled multi-parameter tooth contact analysis (TCA) model and clarified how installation errors shape the contact pattern. Ding et al. [27,28] systematically adjusted installation settings to chart the resulting changes in contact patterns. Tang et al. [29] proposed a profile-deviation evaluation method that accounts for installation errors, thereby reducing their influence on profile measurement. Wang et al. [30] investigated the contact characteristics associated with four types of misalignments and developed a sensitivity model for hypoid gears. An enhanced multi-population genetic algorithm was then employed to optimize the gear curvatures, effectively reducing their sensitivity to misalignments. Zhang et al. [31] developed a continuous-indexing grinding methodology that effectively mitigates the impact of installation errors on transmission performance. Huang et al. [32] examined the effect of cutting vibrations on tooth-flank accuracy, offering theoretical support for controlling installation errors at their source. Li and Xie [33] investigated how profile deviations, meshing errors, and lead modifications interact to affect tooth-flank contact-stress distributions. Liu et al. [34] demonstrated that eccentric error has a more pronounced effect on the geometrical dynamics of gears, contact patterns, and peak-to-peak transmission error than radial error. Han et al. [35] established a finite element model to analyze the evolution of root cracks under the influence of assembly errors. Song et al. [36] investigated the effects of assembly errors on meshing characteristics and proposed a method to eliminate such errors through mutual compensation of axial gear displacements. Wang et al. [37] based on an improved loaded tooth contact analysis method, revealed the coupled effects of assembly errors on time-varying mesh stiffness, transmission error, and contact stress. Xia et al. [38] investigated the influence of installation errors on gear meshing behavior under varying loads. Despite these advances, most studies treat installation errors and tooth-flank deviations as independent, and the coupled effects on meshing performance—particularly the combined influence of tooth-pitch deviations and 3D installation errors—remain insufficiently addressed. Therefore, it is essential to investigate the influence and coupling mechanisms of tooth pitch errors and three-dimensional installation errors on the meshing characteristics of the FH-HG.

Figure 1 illustrates the technical workflow of this study, which investigates the coupling mechanism between tooth pitch errors and three-dimensional (3D) installation errors in FH-HG. A tooth flank reconstruction model is first established based on the theoretical tooth flank geometry and measured tooth pitch error data. Subsequently, an analysis model considering 3D installation errors is developed using the orthogonal experimental design method. The coupling mechanism between tooth pitch errors and 3D installation errors is then systematically analyzed. The main research objectives of this work are as follows:

(1)

Establishing a coupled model of 3D installation errors and the tooth flank errors.

(2)

Optimizing 3D installation error compensation parameters based on the orthogonal experimental design method.

(3)

Determining the optimal 3D installation error compensation position, orientation, and parameter combination, using key indicators such as peak-to-peak transmission error (PPTE), contact pattern length, and contact pattern area as the basis.

2. Mathematical and Analysis Model of FH-HG

2.1. Mathematical Modelling of FH-HG

The mutually coupled motions and positional relationships among the cutter head, machine, and workpiece during the forming process of FH-HG are illustrated in Figure 2a. The schematic representations of the major cutting edge are provided in Figure 2b,c.

The governing equations for the blade and cutting edge are formulated as follows:

$$\left\{\begin{array}{l}{r}_q(u)=\left[\begin{array}{c}-2R_{BH}\mathit{sin}\left({\displaystyle \frac{u}{2R_{BH}}}\right)\mathit{sin}\left({\alpha }_{0c}-{\displaystyle \frac{u}{2R_{BH}}}\right)\\ 0\\ 2R_{BH}\mathit{sin}\left({\displaystyle \frac{u}{2R_{BH}}}\right)\mathit{cos}\left({\alpha }_{0c}-{\displaystyle \frac{u}{2R_{BH}}}\right)\\ 1\end{array}\right]\\ r_q({\theta }_f)={\mathit{M}}_{\mathit{D}}{\mathit{M}}_{\mathit{R}}\left[\begin{array}{c}{r}_{bh}cos{\theta }_f\\ 0\\ r_{bh}sin{\theta }_f\\ 1\end{array}\right]\end{array}\right.$$

(1)

where $u$ and ${\theta }_f$ are the variables of the major cutting edge and fittle, respectively. $R_{BH}$ is the radius of the main cutting edge, $r_{bh}$ is the radius of fillet, ${\mathbf{M}}_{\mathbf{D}}$ and ${\mathbf{M}}_{\mathbf{R}}$ are the transformation matrices from the $S_l$ to $S_j$ coordinate system, and ${\alpha }_0$ is the tool pressure angle.

The tool head center coordinate system $S_t$ is converted to the machine coordinate system $S_m$, which is then converted to the pinion coordinate system $S_k$ by the auxiliary coordinate system $S_w$. In the coordinate system $S_k$, the expression for the tool cutting trajectory can be expressed as:

$$\left\{\begin{array}{l}{\mathit{r}}_{\mathit{s}}\left(u\right)={\mathit{M}}_{\mathit{s}\mathit{q}}\left(\epsilon ,{\gamma }_a\right){\mathit{r}}_{\mathit{q}}\left(u\right)\\ {\mathit{r}}_{\mathit{t}}(\beta ,u)={\mathit{M}}_{\mathit{t}\mathit{s}}\left(\beta ,{\beta }_{\nu },R_c,{\delta }_s\right){\mathit{r}}_{\mathit{s}}\left(u\right)\\ {\mathit{r}}_{\mathit{k}}\left({\phi }_1,{\phi }_c,\beta ,u\right)={\mathit{M}}_{\mathit{k}\mathit{m}}\left({\phi }_1,A,{\gamma }_m,B,E\right){\mathit{M}}_{\mathit{m}\mathit{t}}\left({\phi }_c,q_0,S_R,j,i\right){\mathit{r}}_{\mathit{t}}(\beta ,u)\end{array}\right.$$

(2)

The symbols $\epsilon$, ${\gamma }_a$, ${\beta }_v$, $R_c$, ${\delta }_s$ are the structural parameters of the cutter plate. The $\beta$ is the rotation angle of the cutter head. The parameters $i$, $j$, $S_R$, $q_0$, $E$, ${\gamma }_m$, $B$ and $A$ are machine-tool settings. The ${\phi }_1$ is the blank rotation angle, and ${\phi }_c$ is the swing table rotation angle. ${\mathbf{M}}_{\mathbf{s}\mathbf{q}},$ ${\mathbf{M}}_{\mathbf{t}\mathbf{s}}$, ${\mathbf{M}}_{\mathbf{m}\mathbf{t}}$ and ${\mathbf{M}}_{\mathbf{k}\mathbf{m}}$ are transformation matrices from $S_q$ to $S_k$.

Based on the meshing principle of gears, the meshing equation is used to determine the tooth flank point set, expressed as:

$$f={\mathit{n}}_{\mathit{k}}\left(u,\beta ,{\phi }_c\right)\times {\mathit{v}}_{\mathit{k}}\left(u,\beta ,{\phi }_c\right)$$

(3)

where the normal vector to the tool cutting is ${\mathit{n}}_{\mathit{k}}$, and the relative velocity vector is ${\mathit{v}}_{\mathit{k}}$.

Simultaneous solution of the above equations yields a unique solution for the tooth flank equation. For the fillet segment, the solution procedure is similar to that for the major cutting edge: by replacing the variable in Equation (2) with ${\theta }_f$ and substituting the tooth flank discretization equation with the projection of the fillet segment.

Based on the theoretical tooth flank model generated via simulated gear machining, and according to the physical meaning of the midpoint tooth pitch error, a coordinate transformation matrix between the error tooth flank and the theoretical tooth flank is established to obtain the error tooth flank. Midpoint tooth pitch errors of the whole tooth are reconstructed through coordinate system transformation, as shown in Figure 3.

$${\mathit{r}}_{\mathit{i}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathit{cos}∆{\theta }_k& \mathit{sin}∆{\theta }_k& 0\\ 0& -\mathit{sin}∆{\theta }_k& \mathit{cos}∆{\theta }_k& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \left(k-1\right)cos{\displaystyle \frac{2\pi }{z_i}}& -\left(k-1\right)sin{\displaystyle \frac{2\pi }{z_i}}& 0\\ 0& \left(k-1\right)sin{\displaystyle \frac{2\pi }{z_i}}& \left(k-1\right)cos{\displaystyle \frac{2\pi }{z_i}}& 0\\ 0& 0& 0& 1\end{array}\right]{\mathit{r}}_{\mathit{k}}$$

(4)

The tooth flank errors of the pinion and gear are denoted as ${\mathbf{r}}_{\mathbf{i}}$ (i = p, g), where p and g refer to the pinion and gear, respectively. ${\theta }_k$ is the angular positioning error of the k-th flank, and ${\mathbf{r}}_{\mathbf{k}}$ is the theoretical tooth flank.

Analyzing the positional errors between the theoretical discrete coordinates and measured data points on the tooth flank surface, the angular deviations at corresponding points can be expressed as:

$$∆\theta ={\displaystyle \frac{d_i}{R_i}}=d_i/\sqrt{\left(y_i^2+z_i^2\right)}$$

(5)

where $∆\theta$ is the angular compensation at the contact point, $d_i$ is the normal directional deviation between the measured and theoretical tooth flank, and $R_i$ is the equivalent curvature radius at the projection point.

The deviation between the theoretical tooth flank and the error tooth flank is modeled as a sine function:

$$d_p=A_p\mathit{sin}\left({\displaystyle \frac{2\pi }{z_p}}\times k_p\right)\dots k_p=1~11$$

(6)

$$d_g=A_g\mathit{sin}\left({\displaystyle \frac{2\pi }{z_g}}\times k_g\right)\dots k_g=1~47$$

(7)

where $A_p$ and $A_g$ are the fluctuation amplitudes of midpoint tooth pitch error, the values which are defined as 20 µm and 50 µm, and $z_p$ and $z_g$ are the tooth numbers of the pinion and gear.

The modeling procedure of hypoid gears is illustrated in Figure 4. First, the theoretical tooth flank point set was generated in MATLAB 2021b. Subsequently, the cutting tooth profile was constructed by importing the pinion and gear point sets into CREO to establish a single-tooth blank. Finally, the complete assembly model was obtained according to the defined assembly relationships. The fundamental design parameters of the hypoid gears are listed in

Table 1. Basic parameters of hypoid gears.

Parameters		Pinion		Gear		Units
		Convex	Concave	Convex	Concave
Hypoid offset	E	38.1				mm
Shaft angle	$\Sigma$	90				deg
Mean normal module	$m_n$	3.4885				mm
Blade group	$z_0$	17				-
Teeth number	$z_i$	11		47		-
Pressure angle	${\alpha }_0$	22.3059	17.6933	19.0931	23.5060	deg
Nominal cutter radius	$R_{FI}$	75.9340	76.0660	75.9193	76.0807	mm
Reference point angle	${\beta }_V$	−21.1765	−10.5882	10.5882	21.1765	deg
Spheric radius	$R_I$	1093.65	210.0213	1046.9999	1113.1838	mm

.

2.2. Analysis Models Considering 3D Installation Errors

The analysis models considering 3D installation errors of hypoid gears are shown in Figure 5. The $S_i$ is the installation coordinate system of hypoid gears. The origin of coordinate $o_i$ is established at the axle’s cross-point of pinion. The theoretical axis of pinion and gear are consistent with $x_i$-axis and $y_i$-axis, respectively. The offset direction is coincident with $z_i$-axis. The $S_{pt}$ and $S_{gt}$ are the theoretical assembly coordinate system of pinion and gear in the coordinate system $S_i$. The parameter $XP$ and $XW$ are defined as the distance from the cross point ${cp}_1$ and ${cp}_2$ to theoretical installation point $o_{pt}^{\prime }$ and $o_{gt}^{\prime }$, respectively. The distance between ${cp}_1$ and ${cp}_2$ is the offset distance E. However, the assembly location of pinion and gear does not overlap with theoretical installation point. There are five kinds of errors between the theoretical position and the actual installation position, namely spatial rotation angle error $∆\theta$, shaft angle error $∆\Sigma$ and pinion axial distance error $∆XP$, offset distance error $∆E$ and gear axial distance error $∆XW$.

Among them, the spatial rotation angle error and shaft angle error represent that there are errors on the axes of the pinion and gear in two degrees of freedom around the X-axis and Y-axis, which is closer to the actual installation situation. In the modeling, the actual installation positions of pinion are considered as spatial rotation angle error $∆\theta$, shaft angle error $∆\Sigma$ and pinion axial distance error $∆XP$ and gear is offset distance error $∆E$ and gear axial distance error $∆XW$. The mathematical model of pinion and gear by considering the manufactured tooth flank error and 3D installation errors in the coordinate system $S_i$ can be expressed as:

$${\mathit{r}}_{\mathit{p}\mathit{a}}={\mathit{M}}_{\mathit{p}\mathit{a}\mathit{p}\mathit{t}}{\mathit{r}}_{\mathit{p}\mathit{f}}$$

(8)

$${\mathit{r}}_{\mathit{g}\mathit{a}}={\mathit{M}}_{\mathit{g}\mathit{a}\mathit{g}\mathit{t}}{\mathit{r}}_{\mathit{P}\mathit{f}}$$

(9)

where ${\mathbf{r}}_{\mathbf{p}\mathbf{f}}$ and ${\mathbf{r}}_{\mathbf{g}\mathbf{f}}$ are the total tooth flank form deviation of pinion and gear, respectively. ${\mathbf{r}}_{\mathbf{p}\mathbf{a}}$ and ${\mathbf{r}}_{\mathbf{g}\mathbf{a}}$ represent the mathematical models of the pinion and gear, respectively, which incorporate both manufactured tooth flank errors and 3D installation errors. Within the coordinate system $S_i$, ${\mathbf{M}}_{\mathbf{p}\mathbf{a}\mathbf{p}\mathbf{t}}$ denotes the transformation matrix from $S_{pt}$ to $S_p$, and ${\mathbf{M}}_{\mathbf{g}\mathbf{a}\mathbf{g}\mathbf{t}}$ denotes the transformation matrix from $S_{gt}$ to $S_g$. The expression of ${\mathbf{M}}_{\mathbf{p}\mathbf{a}\mathbf{p}\mathbf{t}}$ and ${\mathbf{M}}_{\mathbf{g}\mathbf{a}\mathbf{g}\mathbf{t}}$ can be calculated by

$${\mathit{M}}_{\mathit{p}\mathit{a}\mathit{p}\mathit{t}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& sin∆\Sigma & cos∆\Sigma & 0\\ 0& -cos∆\Sigma & sin∆\theta & ∆XP\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}-sin∆\theta & 0& cos∆\theta & 0\\ 0& 1& 0& 0\\ cos∆\theta & 0& sin∆\theta & XP\\ 0& 0& 0& 1\end{array}\right]$$

(10)

$${\mathit{M}}_{\mathit{g}\mathit{a}\mathit{g}\mathit{t}}=\left[\begin{array}{cccc}1& 0& 0& -E+∆E\\ 0& 1& 0& XW+∆XW\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(11)

The 3D installation error direction is specified: for $∆XW$ and $∆XP$, pinion and gear close to each other are positive, negative when moving away; for $∆E$, positive with axes separation, negative with axes convergence; for $∆E$, $S_g$ is positive when it is farther from $S_{gt}$ and negative when it is closer; for $∆\Sigma$ positive for decreased shaft angle, negative for increased shaft angle; for $∆\theta$ positive for counterclockwise rotation of the pinion axis, negative otherwise.

The installation error parameters in this paper are defined as follows: gear axial distance $∆XW$ (A), pinion axial distance $∆XP$ (B), offset distance $∆E$ (C), shaft angle $∆\Sigma$ (D), spatial rotation angle $∆\theta$ (E), and blank control group (F). There are 5 sets of test numbers (TN). The parameter ranges are set as shown in

Table 2. Orthogonal test factor level.

TN	A (mm)	B (mm)	C (mm)	D (deg)	E (deg)	F
1	−0.02	−0.02	−0.02	−0.02	−0.02	1
2	−0.01	−0.01	−0.01	−0.01	−0.01	2
3	0	0	0	0	0	3
4	0.01	0.01	0.01	0.01	0.01	4
5	0.02	0.02	0.02	0.02	0.02	5

.

To investigate the synergistic compensation mechanism between 3D installation errors and real tooth flank geometry on transmission error, a six-factor five-level orthogonal experimental design is implemented based on single-factor analysis, as shown in

Table 3. Installation error orthogonal design.

TN	A	B	C	D	E	F
TN1	−0.02	−0.02	−0.02	−0.02	−0.02	1
TN 2	−0.02	−0.01	0	0.01	0.02	2
TN 3	−0.02	0	0.02	−0.01	0.01	3
TN 4	−0.02	0.01	−0.01	0.02	0	4
TN 5	−0.02	0.02	0.01	0	−0.01	5
TN 6	−0.01	−0.02	0.02	0.01	0	5
TN 7	−0.01	−0.01	−0.01	−0.01	−0.01	1
TN 8	−0.01	0	0.01	0.02	−0.02	2
TN 9	−0.01	0.01	−0.02	0	0.02	3
TN 10	−0.01	0.02	0	−0.02	0.01	4
TN 11	0	−0.02	0.01	−0.01	0.02	4
TN 12	0	−0.01	−0.02	0.02	0.01	5
TN 13	0	0	0	0	0	1
TN 14	0	0.01	0.02	−0.02	−0.01	2
TN 15	0	0.02	−0.01	0.01	−0.02	3
TN 16	0.01	−0.02	0	0.02	−0.01	3
TN 17	0.01	−0.01	0.02	0	−0.02	4
TN 18	0.01	0	−0.01	−0.02	0.02	5
TN 19	0.01	0.01	0.01	0.01	0.01	1
TN 20	0.01	0.02	−0.02	−0.01	0	2
TN 21	0.02	−0.02	−0.01	0	0.01	2
TN 22	0.02	−0.01	0.01	−0.02	0	3
TN 23	0.02	0	−0.02	0.01	−0.01	4
TN 24	0.02	0.01	0	−0.01	−0.02	5
TN 25	0.02	0.02	0.02	0.02	0.02	1

. TN13 is a theoretical installation.

The pinion and gear were meshed in finite element analysis software as shown in Figure 6. The Poisson’s ratio and Young’s modulus were set to 0.3 and 210 GPa, respectively, and the torque load was 450 N·m. A global seed size of 5 was used for the pinion, resulting in approximately 121,982 simulation elements. For the gear, a global seed size of 6.5 was applied, also yielding about 121,982 elements. Additionally, local mesh refinement was performed on the contacting tooth surfaces of both gears, ensuring simulation accuracy while maintaining high computational efficiency.

3. Analysis of Meshing Performance Under Standard Installation

The transmission error and contact force fluctuations under standard installation are illustrated in Figure 7. The PPTE is measured to be 1308.57 μrad. Due to pitch deviations in both the pinion and the gear, the transmission error exhibits pronounced oscillatory behavior with synchronized fluctuations in the contact force. This dynamic interaction alters both the length and area of the contact pattern during meshing.

Figure 8 presents the contact pattern of TN13. The designation “13−6” in the upper left corner denotes the sixth tooth flank of the gear in TN13, and the other labels follow the same convention. The contact pattern is centrally distributed on the tooth flank without edge contact, with an average length of 23.14 mm. The maximum contact stress is within the range of 870–960 MPa.

4. Analysis of Meshing Performance Under Installation Error Compensation

4.1. Multi-Working Conditions Meshing Performance Analysis

A comparison of the transmission performance of TN1–TN5 is presented in Figure 9. Among them, TN1 demonstrates the highest transmission accuracy, with a PPTE of 1168.55 μrad. TN4 exhibited the lowest transmission accuracy, with a PPTE of 1496.03 µrad. From the analysis of the meshing performance of the 25th tooth flank, the contact pattern length of TN5 is measured to be 24.3mm, and the contact pattern length of TN2 is measured to be 22.7 mm. TN1 exhibited the largest contact area ratio at 32.89%, while TN2 showed the smallest at 30.43%. Additionally, TN1 demonstrated the lowest contact stress, measuring 814.33 MPa.

Figure 10 illustrates the dynamic response of the contact force for TN1–TN5. TN1 exhibits the largest fluctuation, with its peak-to-peak contact force (PPCF) reaching 135.48 N. In contrast, TN4 shows a PPCF of 96.09 N, which is 29.07% lower than that of TN1.

A comparison of the transmission performance of TN6–TN10 is presented in Figure 11. TN10 demonstrates the highest transmission accuracy, achieving a PPTE of 1130.66 μrad. Analysis of the meshing performance of the 25th tooth flank reveals that TN8 has a contact pattern length of 25.2 mm and a contact pattern area ratio of 35.36%, while TN7 exhibits the lowest contact stress at 816.76 MPa.

Figure 12 illustrates the dynamic response of the contact force for TN6–TN10. The maximum fluctuation is observed in TN10, with its PPCF reaching 130.96 N. In comparison, TN9 exhibits a markedly lower PPCF of 108.79 N, corresponding to a 16.93% reduction relative to TN10.

Figure 13 illustrates the transmission performance of TN11–TN15. Among the tested models, TN14 provides superior transmission accuracy, with a PPTE value of 1132.65 μrad. Meshing performance evaluation on the 25th tooth flank demonstrates that TN14 exhibits a contact length of 24.5 mm, a contact area ratio of 34.46%, and a minimum contact stress of 824.79 MPa.

Figure 14 illustrates the dynamic contact force response of TN11–TN15. Among them, TN14 exhibits the largest contact force fluctuation, reaching 140.20 N. In contrast, TN12 shows a significantly lower PPCF of 93.96 N, representing a 32.98% reduction compared with TN14.

Figure 15 presents a comparison of the transmission performance of TN16–TN20. Among these, TN20 achieves the highest transmission accuracy, with a PPTE of 1242.49 μrad. Examination of the meshing performance at the 25th tooth flank shows that TN17 exhibits a contact pattern length of 25.1 mm with an area ratio of 35.86%, whereas TN18 records the minimum contact stress of 819.82 MPa.

Figure 16 shows the dynamic response of contact force for TN16–TN20. TN16 exhibits the highest contact force fluctuation with a PPCF of 133.50 N. Comparatively, TN19 achieves a significantly reduced PPCF of 109.26 N, representing an 18.16% decrease relative to TN16.

Figure 17 presents a comparison of the transmission performance for TN21–TN25. Among these, TN23 attains the highest transmission accuracy, with a PPTE of 1152.07 μrad. Examination of the meshing performance of the 25th tooth flank indicates that TN24 exhibits a contact pattern length of 25.3 mm, a contact pattern area ratio of 35.12%, and a contact stress of 792.57 MPa.

Figure 18 illustrates the dynamic response of the contact force for TN21–TN25. TN22 exhibits the most pronounced contact force fluctuation, with a PPCF of 138.62 N. In contrast, TN25 demonstrates improved dynamic stability, achieving a reduced PPCF of 108.16 N, which corresponds to a 21.97% decrease.

Figure 19 presents a comparative analysis of PPTE under multiple working conditions. The analytical results are obtained by calculating the deviation of each working condition relative to the TN13 PPTE. Among them, TN25 exhibits the maximum PPTE of 1653.52 μrad, whereas TN10 shows the minimum PPTE of 1130.66 μrad. Compared with TN25, TN10 demonstrates a 31.62% reduction in PPTE.

Figure 20 illustrates the comparative analysis of PPCF under multiple working conditions. The analysis results are obtained by determining the differences between each working condition and the TN13 PPCF. TN14 shows the highest PPCF fluctuation at 140.20 N, whereas TN12 displays the lowest fluctuation at 93.96 N. In comparison to TN14, TN12 achieves a 32.98% reduction in PPCF.

Table 4. Range analysis results.

Parameters	Mean Values of PPTE (μrad)	Range (μrad)
$\theta$	2016.34–2549.36	533.02
$∆\Sigma$	2062.91–2509.62	268.03
$E$	2174.86–2287.79	112.93
$∆G$	2211.21–2296.92	85.71
$∆P$	2201.46–2267.61	66.15

presents the range analysis results of the orthogonal experimental design data, quantitatively evaluating the sensitivity between PPTE and installation parameters.

The range analysis revealed that the sensitivity ranking of PPTE to installation error parameters follows the descending order: $\theta$ > $∆\Sigma$ > $E$ > $∆G$ > $∆P$. To minimize PPTE, optimization efforts should prioritize $\theta$ and $∆\Sigma$ adjustments.

4.2. Analysis of Meshing Performance Under the Optimal Compensation Parameter

Figure 21 presents the contact patterns for TN10, which exhibit a tendency for top edge contact between the 6th and 39th tooth flanks. The maximum contact stress ranges from 811.15 MPa to 895.85 MPa, with a difference of 84.7 MPa. The average stress is 849.18 MPa. The maximum stress occurs at the 13th tooth, reaching 895.85 MPa, while the minimum stress of 811.15 MPa is observed at the 44th. The contact pattern area ratio varies between 29.27% and 35.86%, with a range of 6.59%, and the average area is 32.93%. The maximum contact pattern area ratio, 35.86%, is found at the 6th tooth, while the minimum value of 29.27% is at the 44th. The contact pattern length ranges from 22.6 mm to 25.7 mm, yielding a range of 3.1 mm. The 6th and 39th teeth exhibit the longest contact patterns, at 25.7 mm, while the 44th tooth shows the shortest length, at 22.6 mm. The average length of the contact patterns is 24.01 mm, which represents a 3.76% improvement compared to TN13.

A comparative analysis of the contact pattern areas ratio between TN10 and TN13 is shown in Figure 22a. The maximum relative deviation, 3.61%, occurs at the 39th tooth flank. At the 44th tooth flank, the contact pattern area ratio of TN13 is 1.08% larger than that of TN10. Similarly, a comparative analysis of the maximum contact stress between TN10 and TN13 is presented in Figure 22b. The largest relative deviation, 148 MPa, is observed at the 44th tooth flank. At the 13th tooth flank, the contact stress of TN10 is 3.15 MPa higher than that of TN13.

Comprehensive analysis shows that the average length and area ratio of the contact patterns increased, while the maximum tooth contact stress and PPTE decreased under TN10 installation compared to TN13. The TN10 installation parameters improve FH-GH transmission errors and meshing performance.

4.3. Analysis of Meshing Performance Under the Worst Compensation Parameter

Figure 23 shows the TN25 contact patterns, which are located in the middle of the tooth flank, with no edge contact observed. The maximum contact stress ranges from 803.06 MPa to 875.36 MPa, with a span of 72.3 MPa. The mean stress is 844.66 MPa. The maximum stress reaches 875.36 MPa at the 13th, while the minimum stress of 803.06 MPa occurs at the 44th. The contact pattern area ratio varies between 31.33% and 34.05%, with a range of 2.72%. The average contact pattern area ratio is 32.42%. The maximum contact pattern area ratio of 34.05% occurs at the 13th position, contrasting with the minimum value of 31.33% at the 39th. The contact pattern length ranges from 22.9 mm to 25 mm, with a span of 2.1 mm. The 31st exhibits the longest patterns at 25 mm, while the 39th shows the shortest length at 22.9 mm. The average length of the contact patterns is 23.67 mm, representing a 2.29% improvement over TN13.

A comparative analysis of the contact pattern area ratio between TN25 and TN13 is presented in Figure 24a. The maximum relative deviation of 2.63% is observed at the 21st tooth flank. At the 6th, 25th, and 39th tooth flanks, the contact pattern area ratio of TN13 increases by 0.83%, 0.57%, and 0.5%, respectively, compared to TN25. A comparative analysis of the maximum contact stress between TN25 and TN13 is presented in Figure 24b. The maximum relative deviation of 156.16 MPa occurs at the 44th tooth flank. The maximum contact stresses extracted for TN25 are all smaller than those for TN13.

Comprehensive analysis indicates that, although TN25 exhibits a 26.28% increase in PPTE compared to TN13, it demonstrates significant advantages in contact characteristics. Specifically, the average length of the contact patterns improves by 2.29%, partial tooth flanks show increased contact pattern area ratio, and the maximum contact stress is significantly reduced.

5. Conclusions

In this paper, an analytical model considering installation error and tooth pitch error is established. The coupling relationship between the real tooth flank, 3D installation error, and meshing performance is investigated. The results can be summarized as follows:

(1)

By introducing $\theta$, the construction of a 3D installation error model is enabled for the quantitative characterization of installation parameters. A coupling characteristics analysis between installation errors and tooth pitch deviations is conducted.

(2)

The influence of installation errors on meshing performance exhibits dual effects. Installation errors can compensate for midpoint tooth pitch errors, effectively reducing PPTE and thereby enhancing meshing performance. Installation errors may also increase PPTE, while reducing the maximum contact stress and increasing the contact pattern area ratio.

(3)

The sensitivity of PPTE to various installation error parameters, in descending order, is as follows: $\theta$ > $\mathrm{∆}\Sigma$ > $E$ > $\mathrm{∆}G$ > $\mathrm{∆}P$. To minimize PPTE, the adjustment of $\theta$ and $∆\Sigma$ should be prioritized.

(4)

Under TN10, the PPTE is reduced by 3.78% compared to TN13, thereby collectively enhancing meshing performance. Under TN25, the PPTE increases by 26.28% compared to TN13. However, the contact pattern average length improves by 2.29%, and the maximum contact stress is significantly reduced.