Gear surfaces three-dimensional slope deviation evaluation method

Abstract

Addressing the limitation of traditional gear slope deviation evaluation methods confined to two-dimensional (2D) characteristic line analysis, this paper proposes a new three-dimensional (3D) slope deviation evaluation index based on the normal vectors of discrete points on the tooth flank. A 3D point cloud average deviation plane is established based on the least squares method. The normal vector of this plane is extracted and decomposed into the profile and helix directions to construct the evaluation index. The relationship between the 3D slope deviation evaluation index and the 2D slope deviation indicators is analytically derived, thereby achieving a quantitative characterization of the degree of inclination of the entire tooth flank. The effectiveness of the proposed 3D tooth flank slope deviation evaluation method is verified through simulation and experiment. Under complex deviation conditions such as twist, the new method can more comprehensively reflect the 3D topographic characteristics of the tooth flank. This research provides a theoretical foundation for the 3D gear accuracy evaluation system and significantly enhances the completeness of tooth flank quality control.

Introduction

The performance of gear transmission systems highly depends on the manufacturing accuracy of the tooth flanks,^1–3 among which slope deviation is a core evaluation indicator that directly determines the meshing characteristics and service life of gear pairs.^4,5 The current mainstream method, referenced from standard ISO 1328-1:2013, ⁶ involves extracting profile characteristic lines and helix characteristic lines to calculate the profile slope deviation and helix slope deviation, respectively, thus constructing a 2D deviation evaluation system for tooth flanks.^7–9 Gear measurement technology has achieved leapfrog development from contact probes to optical 3D scanning.^10,11 Literature^12,13 established a unified mathematical model as a theoretical benchmark and developed an advanced point cloud inversion algorithm capable of comprehensively and accurately comparing actually measured gear data with the theoretical model to assess various geometric errors in one go. Although many current methods can acquire high-density point cloud data for the entire tooth flank, the evaluation methods still focus on 2D characteristic line analysis.^8,14 This evaluation mode, based on local cross-sectional lines, essentially reduces the 3D tooth flank topography to discrete 2D projections. Consequently, complex spatial forms such as tooth flank twist and coupled multi-type deviations cannot be effectively characterized, which has become a bottleneck problem restricting high-precision gear manufacturing.

The limitations of existing methods stem from their lack of spatial evaluation dimension. On the one hand, characteristic line analysis can only reflect linear deviations in specific sections and cannot capture the overall tilt trend of the tooth flank under the synergistic effect of the normal, profile, and face width directions. ¹⁵ On the other hand, when the tooth flank exhibits nonlinear distortions like mid-concavity/convexity^16,17 or compound deviations like superimposed tilt and twist,^18,19 traditional methods ignore the spatial correlation between points on the tooth flank. This not only leads to distorted evaluation results but also makes it difficult to guide the tracing of machining errors. ²⁰ Especially with the continuously increasing accuracy requirements for gears in high-end equipment such as wind turbine gearboxes^21–23 and high-speed train gearboxes,^24,25 the contradiction between this localized evaluation and the need for full-tooth-flank quality control is becoming increasingly acute.^26,27 Therefore, there is an urgent need to establish a new full-tooth-flank evaluation theory that matches 3D measurement technology.

To break through the above limitations, this study proposes a new slope deviation evaluation method based on 3D tooth flank topography. It extracts the normal vector components characterizing the overall tilt degree of the tooth flank from a 3D spatial perspective, then constructs an evaluation index incorporating the evaluation range, establishing a mathematical mapping model between the 3D evaluation results and the traditional 2D slope deviations. This achieves theoretical compatibility between the new method and the ISO standard. The specific research route is shown in Figure 1. This study overcomes the limitations inherent in traditional two-dimensional feature line analysis, achieving a significant advancement to 3D full-tooth flank topography evaluation. In contrast to conventional methods that only capture local cross-sectional deviations, the new approach utilizes full-tooth flank point cloud fitting to comprehensively identify complex spatial deviations such as distortion. This effectively addresses the challenges associated with 2D evaluations when characterizing compound deviations and offers a more holistic solution for high-precision gear quality control. Under complex conditions such as twist deviation, the method significantly enhances evaluation completeness by leveraging its capability for quantitative analysis of spatial correlations on the tooth flank, providing a theoretical basis for 3D gear accuracy control.

Figure 1.

Shows the logic diagram of this study.

Tooth flank normal deviation modeling

Mapping the normal deviation obtained in the gear coordinate system to the tooth flank coordinate system is the foundation for evaluating and optimizing tooth flank performance. ²⁸ To assess the deviation of the tooth flank, the tooth flank coordinate system and the parametric equation of the tooth flank should first be constructed, based on which the normal deviation of the profile is established.

Tooth flank coordinate system

The tooth flank coordinate system (o_n–x_n, y_n, z_n) is shown in Figure 2. Here, y_n is along the direction of the involute tangent, pointing from the gear root to the tip; z_n is along the helix tangent direction, pointing from z_n = 0 to z_n = b, where b is the face width; x_n is the tooth flank normal direction, and δ is the tooth flank normal deviation, coaxial with x_n.

Figure 2.

Tooth flank coordinate system.

Mathematical model of the involute helicoid

The involute helicoid is a spatial surface formed by a straight line rolling along the base cylinder. The line is tangent to the base cylinder and maintains a fixed angle with the axis, as shown in Figure 3.

Figure 3.

Involute helicoid.

In Figure 3, (o–x, y, z) is the coordinate system of the involute helicoid, r_b is the base circle radius, μ is the sum of the involute pressure angle and the roll angle, θ is the rotation angle of the section involute relative to the initial section involute, β_b is the base helix angle of the involute helicoid, P₀ is the start point of the transverse involute, P is the start point of the involute in any section of the helicoid, points A, M, C are points on the involute, points F and E are the tangent points of points M and C with the base circle in the section, respectively, and the straight line AC is the generating line of the involute helicoid and also the tooth contact line. For any point M on the tooth flank, its position can be expressed according to spatial geometric relationships:

$$\{\begin{array}{c}\mathit{r}=\mathrm{OE}+\mathrm{EF}+\mathrm{FM}=x\mathit{i}+y\mathit{j}+z\mathit{k}\hfill \\ x(\mu ,z)=\mathrm{OE}\cos (\mu +\theta )+\mathrm{FM}\sin (\mu +\theta )=r_{\mathrm{b}}\cos (\mu +\theta )+\theta r_{\mathrm{b}}\sin (\mu +\theta )\hfill \\ y(\mu ,z)=\mathrm{OE}\sin (\mu +\theta )-\mathrm{FM}\cos (\mu +\theta )=r_{\mathrm{b}}\sin (\mu +\theta )-\theta r_{\mathrm{b}}\cos (\mu +\theta )\hfill \\ z(\mu ,z)=\mathrm{EF}=z\hfill \end{array}$$

(1)

Therefore, based on the geometric relationships, the involute helicoid can be represented by the following equations:

$$\{\begin{array}{c}x(\mu ,z)=r_{\mathrm{b}}[\cos (\mu (z)+\theta )+\theta \sin (\mu (z)+\theta )]\hfill \\ y(\mu ,z)=r_{\mathrm{b}}[\sin (\mu (z)+\theta )-\theta \cos (\mu (z)+\theta )]\hfill \\ z(\mu ,z)=r_{\mathrm{b}}\cot {\beta }_{\mathrm{b}}(\theta (z))\hfill \end{array}$$

(2)

Profile normal deviation

The profile normal deviation is a core element in gear evaluation, representing the perpendicular distance from a measured actual discrete point to the theoretical involute. It can be measured or calculated directly in the transverse involute direction. ⁴

Figure 4 shows a schematic of the profile normal deviation, where P_d is the actual measured discrete point, d is the perpendicular distance from P_d to the involute tooth profile, that is, the profile normal deviation, and P₀ is the foot of the perpendicular. To calculate the profile normal deviation, first determine an actual discrete point P_d, then calculate the minimum distance from this point to the involute tooth profile. The tangent t−t to the involute at point P₀ is determined by the derivative at that point.

Figure 4.

Profile normal deviation.

Let the coordinates of the foot of the perpendicular P_o be (x₀, y₀) and the coordinates of the discrete point P_d be (x_d, y_d), satisfying equation (3).

$$\begin{array}{c}(x_0({\mu }_0,z_0)-x_{\mathrm{d}}({\mu }_{\mathrm{d}},z_0))·\frac{\partial x({\mu }_0,z_0)}{\partial \mu }+(y_0({\mu }_0,z_0)\hfill \\ -y_{\mathrm{d}}({\mu }_{\mathrm{d}},z_0))·\frac{\partial y({\mu }_0,z_0)}{\partial \mu }=0\hfill \end{array}$$

(3)

Thus, the profile normal deviation is:

$$d=\sqrt{{(x_0({\mu }_0,z_0)-x_{\mathrm{d}}({\mu }_{\mathrm{d}},z_0))}^2+{(y_0({\mu }_0,z_0)-y_{\mathrm{d}}({\mu }_{\mathrm{d}},z_0))}^2}$$

(4)

Based on the tangent vector $t=\left[\partial x\left({\mu }_0,z_0\right)/\partial \mu ,\partial y\left({\mu }_0,z_0\right)/\partial \mu \right]$ of the involute at point P_o, the relative position of the discrete point to the involute can be determined by the sign of the result calculated from equation (5).

$$(x_{\mathrm{d}}-x_0)·(-\partial y({\mu }_0,z_0))+(y_{\mathrm{d}}-y_0)·\partial x({\mu }_0,z_0)=0$$

(5)

3D tooth flank slope deviation evaluation

Evaluation method

To quantitatively characterize the overall inclination of the tooth flank, the normal error point cloud within the tooth flank coordinate system is fitted to an average deviation plane. To ensure that this method effectively captures the global trend of surface deviations while minimizing the influence of local random fluctuations, the least squares method is employed for plane fitting in this study. The specific method is as follows: after calculating the normal deviation of the 3D tooth flank point cloud, the point cloud data is transformed into the tooth flank coordinate system, obtaining a point cloud of normal errors at various points on the tooth flank. This normal error point cloud data in the tooth flank coordinate system is then fitted to a 3D average deviation plane, analogous to the 2D average profile trace and average helix trace. As shown in Figure 5, the components of the normal vector ρ of the average deviation plane in the x_n, y_n, and z_n directions are δ_h, y_hβ, and z_hα, respectively. ε_α and ε_β are the deviation angles of the normal vector ρ. Based on the average deviation plane equation, the normal vector ρ (δ_h, y_hβ, z_hα) of the plane is calculated. This vector, combined with the tooth flank evaluation range (L_α, L_β), serves as the new evaluation index for 3D tooth flank slope deviation.

Figure 5.

3D slope deviation.

Evaluation index

The plane normal vector can indicate the degree of tilt of the plane. The tooth flank coordinate system has three directions: the profile direction, the face width direction, and the normal deviation direction. The normal vector of the tooth flank error point cloud can be decomposed into two important directions: the profile direction and the helix direction. This decomposition helps describe variations of the tooth flank in different directions, thereby providing a more precise basis for gear design and optimization.

The fitted plane normal vector ρ can be expressed as:

$$\rho =\left[\begin{array}{c}{\rho }_{\mathrm{x}}\hfill \\ \multicolumn{1}{c}{{\rho }_{\mathrm{y}}}\\ \multicolumn{1}{c}{{\rho }_{\mathrm{z}}}\end{array}\right]=\left[\begin{array}{c}{\delta }_{\mathrm{h}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{h}\mathrm{\beta }}}\\ \multicolumn{1}{c}{z_{\mathrm{h}\mathrm{\alpha }}}\end{array}\right]$$

(6)

Where ρ_x, ρ_y, and ρ_z are respectively the components of the normal vector ρ in the direction of the tooth profile, the direction of the tooth width, and the direction of the normal deviation. Normalize the normal vector:

$$e_{\rho }=\left[\begin{array}{c}{e}_{\delta }\hfill \\ \multicolumn{1}{c}{e_y}\\ \multicolumn{1}{c}{e_z}\end{array}\right]=\left[\begin{array}{c}\frac{{\delta }_{\mathrm{h}}}{\sqrt{{{\delta }_{\mathrm{h}}}^2+{y_{\mathrm{h}\mathrm{\beta }}}^2+{z_{\mathrm{h}\mathrm{\alpha }}}^2}}\hfill \\ \multicolumn{1}{c}{\frac{y_{\mathrm{h}\mathrm{\beta }}}{\sqrt{{{\delta }_{\mathrm{h}}}^2+{y_{\mathrm{h}\mathrm{\beta }}}^2+{z_{\mathrm{h}\mathrm{\alpha }}}^2}}}\\ \multicolumn{1}{c}{\frac{z_{\mathrm{h}\mathrm{\alpha }}}{\sqrt{{{\delta }_{\mathrm{h}}}^2+{y_{\mathrm{h}\mathrm{\beta }}}^2+{z_{\mathrm{h}\mathrm{\alpha }}}^2}}}\end{array}\right]$$

(7)

Where e _δ is the component of the unit normal vector in the deviation value direction; e _y is the component of the unit normal vector in the profile direction; e _z is the component of the unit normal vector in the face width direction. The fitted plane’s normal vector e _ρ (e _δ , e _y , e _z ), combined with the evaluation range (L_α, L_β), collectively forms the new evaluation index F _H for 3D tooth flank slope deviation, denoted as F _H (e _δ , e _y , e _z ; L_α, L_β).

Mapping relationship between 2D and 3D slope deviation evaluation indices

By projecting the normal vector of the average deviation plane of the tooth flank error point cloud onto the profile direction and the helix direction respectively, the normals of the average profile line and the average helix line in the traditional 2D slope deviation definition can be obtained. Using the projected normal vectors and the evaluation range allows for the conversion of the 3D slope deviation to the 2D slope deviation. Figure 6 shows a comparison between the projection of the 3D tooth flank slope deviation and the traditional 2D slope deviation. Here, F_Hα and F_Hβ are the profile slope deviation and helix slope deviation, respectively; L_α and L_β are the profile and helix evaluation ranges; g_α is the length of path of contact, C_f is the profile control point, N_f is the effective root point, F_a is the tip form point, and a is the tip point.

Figure 6.

3D tooth flank slope deviation versus 2D slope deviation.

The projection of the normal vector determines the angle between the average deviation plane and the profile and helix directions. The magnitude of its angle with the coordinate axes can intuitively reflect the degree of tilt of the plane in the profile and helix directions. These two components, the inclination angle along the profile direction ε_α and the inclination angle along the helix direction ε_β, can be obtained by decomposing the normal vector:

$${\epsilon }_{\alpha }=\arctan \left(\frac{e_y}{e_{\delta }}\right)$$

(8)

$${\epsilon }_{\beta }=\arctan \left(\frac{e_x}{e_{\delta }}\right)$$

(9)

According to the ISO 1328-1:2013 standard for evaluating 2D profile slope deviation, the slope deviations resulting from projecting the 3D tooth flank slope deviation onto the x_n–y_n plane and the x_n–z_n plane are given by equations (10) and (11), respectively.

$$f_{\mathrm{H}\mathrm{\beta }}=\tan ({\epsilon }_{\mathrm{\beta }})·L_{\mathrm{\beta }}$$

(10)

$$f_{\mathrm{H}\mathrm{\alpha }}=\tan ({\epsilon }_{\mathrm{\alpha }})·L_{\mathrm{\alpha }}$$

(11)

Tooth flank slope deviation experiment and analysis

Simulation analysis

To validate the effectiveness of the proposed 3D gear slope deviation evaluation method, a simulation verification was conducted. The gear parameters used for the simulation are shown in Table 1, and common types of tooth flank deviations were adopted as the research objects. To more intuitively and systematically analyze the influence of deviation magnitude on tooth flank error, an adjustable coefficient was introduced. By varying this coefficient, the degree of deviation can be conveniently controlled, and corresponding numerical simulation analysis can be performed to study the influence of different slope deviation magnitudes on tooth flank error. The slope deviation adjustment coefficient τ is used to adjust the simulated tooth flank tilt angle; the mid-concavity/convexity deviation adjustment coefficient ω is used to adjust the amplitude of the simulated tooth flank; the sinusoidal deviation adjustment coefficient ξ is used to adjust the amplitude of the simulated tooth flank. The deviation types are listed in Table 2.

Table 1.

Parameters of gear 1.

Parameter	Value
Number of teeth z	23
Module m (mm)	1
Pressure angle α (°)	20
Addendum modification coefficient x	0
Helix angle β (°)	0
Tooth width b (mm)	6

Table 2.

Characteristic deviation types.

No.	Deviation type	Adjustment coefficient	Value	No.	Deviation type	Adjustment coefficient	Value
1	Profile slope	${\tau }_{\alpha 1}$	0.01°	9	Profile slope and helix mid-concavity	${\tau }_{\alpha 1}$	0.01°
		0	0			${\omega }_{\beta 1}$	0.01 (mm)
2	Helix slope	0	0	10	Profile slope and profile mid-concavity	${\tau }_{\alpha 1}$	0.01°
		${\tau }_{\beta 1}$	0.01°			${\omega }_{\alpha 1}$	0.01 (mm)
3	Profile mid-concavity	${\omega }_{\alpha 1}$	0.01 (mm)	11	Helix slope and helix mid-concavity	${\tau }_{\beta 1}$	0.01°
		0	0			${\omega }_{\beta 1}$	0.01 (mm)
4	Helix mid-concavity	0	0	12	Helix slope and profile mid-concavity	${\tau }_{\beta 1}$	0.01°
		${\omega }_{\beta 1}$	0.01 (mm)			${\omega }_{\alpha 1}$	0.01 (mm)
5	Profile sinusoidal	${\xi }_{\alpha 1}$	0.01 (mm)	13	Profile slope and helix sinusoidal	${\tau }_{\alpha 1}$	0.01°
		0	0			${\xi }_{\beta 1}$	0.01 (mm)
6	Helix sinusoidal	0	0	14	Profile slope and profile sinusoidal	${\tau }_{\alpha 1}$	0.01°
		${\xi }_{\beta 1}$	0.01 (mm)			${\xi }_{\alpha 1}$	0.01 (mm)
7	Profile slope and helix slope	${\tau }_{\beta 1}$	0.01°	15	Helix slope and helix sinusoidal	${\tau }_{\beta 1}$	0.01°
		${\tau }_{\alpha 1}$	0.01°			${\xi }_{\beta 1}$	0.01 (mm)
8	Tooth flank twist	${\upsilon }_{\alpha }$	0.01°	16	Helix slope and profile sinusoidal	${\tau }_{\beta 1}$	0.01°
		${\upsilon }_{\beta }$	0.01°			${\xi }_{\alpha 1}$	0.01 (mm)

In the tooth flank coordinate system, point cloud data of 1000 × 1000 points is generated along the roll length and face width directions to construct the 3D point cloud diagram of tooth flank deviations, as shown in Figure 7.

Figure 7.

Characteristic error point clouds: (a) profile slope, (b) profile sinusoidal, (c) helix slope, (d) profile mid-concavity, (e) helix sinusoidal, (f) helix mid-concavity, (g) profile slope and profile mid-concavity, (h) helix slope and profile mid-concavity, (i) profile slope and profile sinusoidal, (j) helix slope and profile sinusoidal, (k) helix slope and helix sinusoidal, (l) profile slope and profile sinusoidal, (m) profile slope and helix mid-concavity, (n) helix slope and helix mid-concavity, (o) tooth flank twist, and (p) profile slope and helix slope.

A comparison between the simulation results of the 3D slope deviation evaluation method and the traditional profile slope deviation and helix slope deviation is shown in Table 3. When the tooth flank exhibits single deviations such as tilt, mid-concavity, or sinusoidal deviation individually, the results of the proposed 3D tooth flank slope deviation evaluation method show good agreement with those from ISO 1328-1:2013. However, under complex deviation conditions such as tooth flank twist, the difference between the 3D method and ISO 1328-1:2013 increases slightly compared to cases with single or pairwise superimposed deviations. This is because ISO 1328-1:2013 uses characteristic lines for evaluation, which can only reflect 2D local information and cannot represent the 3D tooth flank shape. The proposed 3D slope deviation index can better reflect the 3D shape of the tooth flank, providing an effective means for subsequent research on 3D tooth flank shape discrimination methods.

Table 3.

Comparison between 3D slope deviation evaluation method and ISO 1328-1:2013 results.

No.	Profile slope deviation (μm)			Helix slope deviation (μm)
	3D slope deviation along the tooth profile	Profile slope deviation	Relative error	3D slope deviation along the helical line	Helix slope deviation	Relative error
1	−1.0620	−1.0620	0	0	0	0
2	0	0	0	0.0226	0.0226	0
3	0	0	0	0	0	0
4	0	0	0	0	0	0
5	−18.5300	−18.5300	0	0	0	0
6	0	0	0	18.5300	18.5300	0
7	−1.0620	−1.0620	0	0.0226	0.0226	0
8	0.0377	0.0366	0.0011	0.0176	−0.0010	0.0175
9	−1.0620	−1.0620	0	0	0	0
10	−1.0620	−1.0620	0	0	0	0
11	0	0	0	0.0232	0.0232	0
12	0	0	0	0.0238	0.0232	0
13	−1.1060	−1.1060	0	37.0500	37.0500	0
14	35.9400	35.9400	0	0	0	0
15	0	0	0	37.0800	37.0800	0
16	37.0500	37.0500	0	0.0226	0.0238	0

Experimental verification

The proposed evaluation method was used to assess gear measurement data obtained from a rapid gear measurement platform, and the correctness of the method was verified using measurement results from a Klingelnberg P26 ²⁹ gear measuring center. The gears selected for the experiment contained characteristic deviations: Characteristic Gear 1 contained tooth flank slope deviation, and Characteristic Gear 2 contained profile mid-concavity deviation. The measuring equipment is shown in Figure 8. Their parameters are listed in Table 4, and the measurement was carried out in a constant-temperature laboratory, ignoring the influence of environmental temperature fluctuations on the data. The testing process was repeated 10 times, and the average value of the samples was calculated. The measured 3D point cloud data is shown in Figure 9.

Figure 8.

P26 and the gear under measurement.

Table 4.

Characteristic gear parameters.

Parameter	Characteristic gear 1	Characteristic gear 2
Number of teeth z	56	56
Module m (mm)	2.25	2.25
Pressure angle α (°)	20	20
Addendum modification coefficient x	0	0
Helix angle β (°)	0	0
Tooth width b (mm)	28	28

Figure 9.

Characteristic gear deviation point clouds: (a) characteristic gear1 and (b) characteristic gear2.

The calculation results for Characteristic Gears 1 and 2 were compared with the evaluation results from the Klingelnberg P26 gear measuring center, as shown in Table 5. For Gear 1, the projected profile slope deviation is consistent with the P26 profile slope deviation result, and the relative error between the projected helix slope deviation and the P26 helix slope deviation is 0.12 μm. For Characteristic Gear 2, the relative error between the projected profile slope deviation and the P26 profile slope deviation is 0.03 μm, and the relative error between the projected helix slope deviation and the P26 helix slope deviation is 0.03 μm.

Table 5.

Characteristic gear slopes deviation calculation results.

Deviation type	Profile slope deviation (μm)			Helix slope deviation (μm)
Measurement item	Projected profile Slope deviation	P26 profile slope deviation	Relative error	Projected helix slope deviation	P26 helix slope deviation	Relative error
Gear 1	7.22	7.22	0	6.94	6.82	0.12
Gear 2	−30.47	−30.50	0.03	1.30	1.27	0.03

Analysis of the data in Table 5 shows that the 3D tooth flank slope deviation is closely related to the traditional 2D slope deviations and is consistent with the results from the P26 gear measuring center, verifying the theoretical correctness.

Conclusion

Aiming at the limitations of traditional gear slope deviation evaluation methods in characterizing 3D tooth flank topography, this study proposed a new 3D slope deviation evaluation method based on a normal average deviation plane. Simulations and experiments were conducted, leading to the following conclusions:

• (1) A 3D tooth flank slope deviation evaluation index was proposed. By fitting the measured tooth flank point cloud to a normal average deviation plane and extracting the normal vector components combined with the evaluation range to construct a comprehensive evaluation index, a 3D quantitative characterization of the overall tilt degree of the tooth flank was achieved. This compensates for the deficiency of traditional 2D characteristic line analysis in global topography evaluation.

• (2) By analyzing the projection components of the normal vector in the profile and helix directions, the mathematical conversion relationship between the 3D slope deviation and the profile slope deviation and helix slope deviation in the traditional ISO 1328-1:2013 standard was derived, verifying the compatibility of the new method with existing standards.

• (3) Simulations based on 16 types of deviation patterns and experiments using a gear measuring center show that the results are consistent with traditional evaluation results under single deviation conditions. Under complex deviation conditions such as twist, the evaluation accuracy is significantly improved due to the capability for spatial correlation analysis, solving the characterization challenge of complex tooth flank spatial forms.

• (4) The evaluation method of this study is applicable to all types of gears, but for different gears, new evaluation indicators and systems need to be constructed based on high-precision theoretical tooth flank mathematical models.

• (5) The next step of this research will combine tooth flank contact analysis with transmission performance simulation to explore the mapping law between 3D slope deviation and service performance such as gear vibration noise and fatigue life, and construct a closed-loop evaluation system of “manufacturing deviation – morphological features – performance output.”