A Method for Analyzing the Meshing Contact Performance of Real Tooth Surfaces of Spiral Bevel Gears

Abstract

The meshing contact performance of spiral bevel gears is critical for transmission accuracy and service life but is inevitably influenced by manufacturing deviations. Existing tooth contact analysis (TCA) and lubrication-related studies for spiral bevel gears are mostly based on ideal theoretical tooth surfaces, failing to reflect the actual meshing state of as-machined gears with inherent machining deviations, and have poor robustness for complex deviated spatial surfaces. To accurately assess the actual meshing state, this paper proposes a novel contact performance analysis method based on a high-precision digital tooth surface reconstructed from one-dimensional probe measurement data. Unlike traditional TCA methods that rely on complex principal curvature calculations, this approach eliminates the mounting distance parameter by simplifying the meshing coordinate system, and employs a variable-radius cylindrical cutting method combined with a binary search algorithm to determine the instantaneous contact ellipse, effectively reducing computational complexity and improving solution robustness for deviated tooth surfaces. Experimental validation demonstrates that the digital tooth surface achieves a reconstruction accuracy of 2.6 × 10⁻⁵ mm. Furthermore, the method accurately predicts the contact pattern location and transmission error, with a discrepancy of only 4.7% compared to theoretical design values, which is highly consistent with the no-load rolling test results. This study confirms that the proposed method effectively reflects the actual meshing condition of machined gears, providing a practical theoretical foundation for the high-quality manufacturing and control of spiral bevel gears. Meanwhile, the high-fidelity contact characteristics of as-machined tooth surfaces output by this method can provide reliable input boundaries for thermoelastohydrodynamic lubrication (TEHL) simulation, friction loss prediction and anti-scuffing design of spiral bevel gears considering machining deviations.

1. Introduction

As a key component in mechanical transmission systems, the meshing contact performance of spiral bevel gears directly influences the accuracy and service life of the transmission system [1,2]. The rapid and precise assessment of the meshing condition of gear teeth, based on the actual tooth surface of spiral bevel gears, has long been a focal point of interest across various industries.

The geometric accuracy of the tooth surface serves as a critical indicator of the actual machining quality of spiral bevel gears. With the rapid advancement of gear inspection technology, gear measuring centers have gradually become indispensable for measuring and evaluating the geometric accuracy of tooth surfaces. Extensive and thorough research has been conducted by various researchers on the actual measurement of tooth surfaces in spiral bevel gears and the analysis of their meshing conditions. Shih et al. [3] integrated a three-dimensional probe and five-axis CNC machine tool to establish a mathematical measurement model for spiral bevel gears; Hua et al. [4] proposed a global error measurement and evaluation method for spiral bevel gear tooth surfaces based on a coordinate measuring machine; Hu et al. [5] examined the impact of geometric errors in coordinate measuring machines on gear deviation measurement;. Li et al. [6,7] analyzed the structure of gear measuring centers equipped with one-dimensional probes, established a coordinate system transformation model between measurement and workpiece systems, and derived theoretical tooth surface design equations based on gear meshing theory; Dai et al. [8] established a tooth surface deviation model for spiral bevel gears based on the tooth surface generation principle, which laid the theoretical foundation for the acquisition of actual deviated tooth surface data in this study. Liu et al. [9,10] proposed an installation error compensation method for spiral bevel gears on coordinate measuring machines, and achieved precise matching between actual and theoretical tooth surfaces through an iterative search algorithm; Zhou et al. [11] compensated for the alignment angle error in gear measuring centers with three-dimensional probes, and reconstructed the actual tooth surface using the NURBS algorithm. While most of the aforementioned studies rely on three-dimensional probes, ongoing research and advancements in one-dimensional probe measurement technology present novel approaches and a theoretical foundation for the measurement of spiral bevel gears and the construction of digital tooth surfaces.

The contact accuracy of the tooth surface directly reflects the actual meshing condition of spiral bevel gears. For tooth contact analysis (TCA), domestic and foreign scholars have carried out multi-dimensional improvements on traditional methods: Cao et al. [12] proposed a spiral bevel gear contact performance analysis method based on instantaneous conjugate contact curves, which realized the separation of transmission errors; Ding et al. [13] enhanced the traditional TCA model to realize the identification of tooth surface contact points under meshing uncertainty; Wang and Lu et al. [14,15] simplified the meshing equation parameters, solved the edge contact problem of traditional models, and improved the computational efficiency of TCA; He et al. [16] established a gear meshing coordinate system and contact model, and obtained the actual contact point sequence using a quadratic adaptive trust region algorithm; Lin et al. [17] derived the expression of the tooth contact path based on the meshing equation and coordinate transformations; Chen et al. [18] and Wang et al. [19] realized the meshing performance control of spiral bevel gear drives based on the local synthesis method and local conjugation theory.

Beyond geometric conformity, meshing contact performance is directly correlated with gear tribological behaviors. The local contact ellipse, sliding-rolling kinematic characteristics and load distribution derived from contact analysis are critical inputs for Thermoelastohydrodynamic Lubrication (TEHL) simulation, friction and efficiency assessment [20,21]. Precise contact characteristics from TCA and Loaded Tooth Contact Analysis (LTCA) are also indispensable foundational inputs for Elastohydrodynamic Lubrication (EHL) simulation and friction-loss calculation models [22]. Relevant studies have fully verified the application value of TCA/LTCA in gear performance optimization: Simon et al. [23] established a refined LTCA framework coupling tooth contact behavior, friction power loss and lubrication characteristics; Ding and Rong [24] developed an improved LTCA method for real tooth surfaces with manufacturing deviations, and realized multi-objective optimization of spiral bevel gears; Kahraman et al. [25,26,27] reported experimental benchmarks for gear efficiency and loss mechanisms, and established friction loss and mesh efficiency prediction models for hypoid gears; Mohammadpour et al. [28,29] conducted TEHL analysis of differential hypoid gears under high loads, and extended the framework to transient mixed non-Newtonian TEHL, which provides an important theoretical reference for the lubrication application of the method proposed in this study.

Although existing studies have improved traditional TCA methods, two key technical gaps for spiral bevel gears remain. First, most TCA and lubrication studies for spiral bevel gears are based on ideal theoretical tooth surfaces, ignoring inherent machining deviations and leading to inconsistency between analysis results and actual meshing conditions of as-machined gears. Second, traditional TCA methods rely on complex principal curvature calculation, resulting in high computational cost and poor robustness for deviated complex spatial surfaces, especially for spiral bevel gears.

To address the above limitations, this paper proposes a novel contact performance analysis method for spiral bevel gears that incorporates real tooth surface deviations. High-precision digital reconstruction of the actual machined tooth surface is achieved through measurement path planning for a one-dimensional probe and the construction of a tooth surface deviation model. By simplifying the meshing coordinate system to eliminate the mounting distance parameter, a solution strategy for the instantaneous contact ellipse combining a variable-radius cylindrical cutting method with a binary search algorithm is proposed, which eliminates the need for complex principal curvature calculation in traditional methods and improves the solution efficiency and robustness. This method enables accurate determination of the meshing condition directly based on the real tooth surface, providing a theoretical basis for the high-quality manufacturing and performance control of spiral bevel gears. Moreover, the contact ellipse geometry and meshing kinematics obtained on as-machined measured tooth surfaces can facilitate TEHL-based investigations of frictional losses, efficiency and scuffing propensity in spiral bevel gears, solving the inconsistency between ideal tooth surface-based lubrication simulation and the actual initial service performance of as-machined gears.

2. Construction of Digital Tooth Surface

During the machining process of spiral bevel gears, various error factors—such as inaccuracies in machining parameters, errors in the machine tool setup, and deviations in gear mounting—inevitably lead to discrepancies between the actual machined tooth surfaces and the theoretically designed ones [30]. Consequently, the actual meshing contact performance of the gear teeth becomes unpredictable. Therefore, constructing a high-precision digital tooth surface that accurately represents the actual machined tooth surface is a critical prerequisite for determining the real meshing contact behavior with precision.

2.1. Discrete Data Acquisition for the Error-Containing Tooth Surface Based on a One-Dimensional Probe

There exists a certain deviation between the actual machined tooth surface and the theoretically designed tooth surface, which is usually referred to as the deviated tooth surface. The theoretically designed tooth surface serves as the reference for acquiring discrete data of the deviated tooth surface and accurately evaluating the machining quality of the tooth surface; therefore, it is necessary to first construct the theoretically designed tooth surface. The theoretical tooth surface of a spiral bevel gear is an envelope surface formed by the generating motion between the cutting edge surface of the face milling cutter head and the gear blank, and its mathematical model is completely determined by the machine tool setting parameters. Based on the generating machining principle of Gleason spiral bevel gears, this paper derives the theoretically designed tooth surface and its normal vector model based on the machine tool setting parameters. Whether machining the gear or the pinion, the machine tool setting parameters typically include blank tilt angle γ_m, horizontal wheel position X_m, radial setting S_r, angular setting q, vertical wheel position E_m, axial wheel position correction value X_a, sliding base setting X_b, roll ratio R_a, total cutter tilt angle t_i, and basic cutter rotation angle s_j. During the machining process of the gear and pinion, different machining parameters can be selected according to different machining methods.

To obtain a theoretically designed tooth surface with ideal meshing performance, this paper conducts a systematic optimization of the aforementioned machine tool setting parameters with meshing contact performance as the core objective. First, based on the generating machining principle of the Gleason cutter tilt method, combined with the geometric parameters of the gear pair listed in

Table 1. Geometric parameters of spiral bevel gear pair.

Parameter	Gear	Pinion
Number of Teeth	27	18
Module/mm	7
Hand of Spiral	Right-hand	Left-hand
Mounting Distance/mm	116	125
Face Width/mm	36
Pressure Angle/°	20
Shaft Angle/°	90
Spiral Angle/°	35
Pitch Cone Angle/°	56.3099	33.6901
Face Cone Angle/°	59.1961	38.1236
Root Cone Angle/°	51.8764	30.8039
Addendum/mm	4.41	7.49
Whole Depth/mm	13.216
Outer Diameter/mm	287.346	233.867

, the initial values of the machine tool parameters that satisfy the basic conjugate meshing of the tooth surfaces are solved. Subsequently, the core optimization objectives are established as the minimization of deviation for key point positions on the contact path, a reasonable coverage ratio of the contact pattern, and a symmetric transmission error curve with a small parabolic amplitude. Meanwhile, the constraint conditions are set as no meshing interference, parameters matching the machining stroke of the machine tool, and guaranteed machinability of the tooth surface. The sensitivity law of key parameters including radial setting, ratio of roll, and blank tilt angle to contact performance is clarified through single-factor analysis. Combined with Tooth Contact Analysis (TCA) and the Newton–Raphson algorithm, iterative optimization is performed until the contact performance meets the design requirements. Finally, through pre-machining tests and rolling test verification, the optimal machine tool setting parameters shown in

Table 2. Machining parameters of spiral bevel gear pair.

Parameter	Gear Convex Side	Gear Concave Side	Pinion Convex Side	Pinion Concave Side
Cutter Radius/mm	114.3		114.8783	113.2378
Cutter Blade Angle/°	22	−18	−22	18
Radial Setting/mm	98.2356		97.1595	99.1355
Angular Setting/mm	72.3843		68.0297	75.7726
Ratio of Roll	1.19825		1.79981	1.79903
Vertical Wheel Position/mm	0		−1.8673	1.859
Axial Wheel Position/mm	0		1.2731	−1.376
Sliding Base Setting/mm	0		−0.652	0.705
Blank Tilt Angle/°	51.88		30.8039

are determined, and the construction of the theoretically designed tooth surface is completed on this basis.

Spiral bevel gears feature complex geometry, local conjugation of tooth surfaces, and excellent controllability, where the tooth surface structure is determined by machine tool setting parameters. The theoretically designed tooth surface is the benchmark for acquiring tooth surface data and evaluating machining accuracy. The tooth surface profile depends on cutter geometry, machining process, and gear cutting parameters. Taking the cutter tilt method as an example, a general mathematical model can be established as shown in Figure 1.

In Figure 1, the coordinate system S_m (x_m, y_m, z_m) is rigidly attached to the machine tool, with O_m as the machine center. The plane X_m O_m Y_m lies within the machine plane, and the orientation of each coordinate axis of the coordinate system S_m remains constant throughout the machining process. The coordinate systems S_t (x_t, y_t, z_t), S_c (x_c, y_c, z_c), and S_w (x_w, y_w, z_w) are rigidly attached to the cutter head, cradle, and machined gear, respectively. Here, O_c is the cradle center, and the coordinate system S_c coincides with S_m at the initial position, rotating together with the cradle around the Z_m axis of the coordinate system S_m during the generating process. The angle ϕ_c is the current rotation angle of the coordinate system S_c and the cradle, with an angular velocity denoted by ω^(c). O_t is the center of the milling cutter head, and the plane X_t O_t Y_t lies within the cutter tip plane (coinciding with the machine plane). S_r and angle q represent the radial setting and angular setting, respectively. O_w is located at the center of the gear coordinate system, coinciding with the design intersection point (or pitch cone vertex) of the machined gear. The coordinate system S_w coincides with the auxiliary coordinate system S_f at the initial position, rotating together with the gear blank around the X_f axis of the coordinate system S_f during the machining process. The angle ϕ_w is the current rotation angle of the coordinate system S_w and the gear blank, with an angular velocity denoted by ω^(w). The coordinate systems S_b (x_b, y_b, z_b) and S_a (x_a, y_a, z_a) are rigidly attached to the cutter tilt mechanism and cutter rotation mechanism, respectively. O_b is the adjustment center of the cutter rotation angle, and the coordinate system S_a coincides with the coordinate system S_b at the initial position, rotating by an angle s_j (cutter rotation adjustment angle) around the Z_b axis of the coordinate system S_b during the cutter rotation angle adjustment process. The coordinate systems S_d (x_d, y_d, z_d) and S_f (x_f, y_f, z_f) are auxiliary coordinate systems.

In the machining process of spiral bevel gears, straight-edge cutters are typically used for gear cutting. These cutters are divided into inner and outer blades, which form inner and outer conical surfaces, respectively, by rotating around the cutter head axis, corresponding to the convex and concave surfaces of the machined gear. The coordinate system model of the straight-edge milling cutter head is shown in Figure 2. In the figure, r_t is the cutter tip radius (measured in the X_t O_t Y_t plane); ϕ_t and S_F are the cutting point parameters; α_t is the cutter blade angle (for the gear, α_t takes a negative value for the concave surface and a positive value for the convex surface, ensuring that the normal vector of the gear tooth surface points from the space to the solid body; for the pinion, α_t takes a positive value for the concave surface and a negative value for the convex surface, ensuring that the normal vector of the pinion tooth surface points from the solid body to the outer space).

In the coordinate system,

$$H_t\left(S_F,{\varphi }_t\right)=\left[\begin{array}{c}\left(r_t+S_F\sin \alpha t\right)\cos {\varphi }_t\\ \left(r_t+S_F\sin \alpha t\right)\sin {\varphi }_t\\ -S_F\cos \alpha t\\ 1\end{array}\right]$$

(1)

$$n_t\left({\varphi }_t\right)=\left[\begin{array}{c}-\cos {\alpha }_t\cos {\varphi }_t\\ -\cos {\alpha }_t\sin {\varphi }_t\\ -\sin {\alpha }_t\end{array}\right]$$

(2)

Based on the gear meshing theory, the tooth surface equation and normal vector in the coordinate system S_w (x_w, y_w, z_w) can be derived as follows:

$$\begin{array}{cc}\hfill H_w\left(S_F,{\varphi }_t,{\varphi }_c,{\varphi }_w\right)& =M_{wf}\left({\varphi }_w\right)M_{fd}{M}_{dm}{M}_{mc}\left({\varphi }_c\right)M_{ca}{M}_{ab}{M}_{bt}\cdot H_t\left(S_F,{\varphi }_t\right)\hfill \\ & =M_{wt}\left({\varphi }_w,{\varphi }_c\right)\cdot H_t\left(S_F,{\varphi }_t\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill n_w\left({\varphi }_t,{\varphi }_c,{\varphi }_w\right)& =L_{wf}\left({\varphi }_w\right)L_{fd}{L}_{dm}{L}_{mc}\left({\varphi }_c\right)L_{ca}{L}_{ab}{L}_{bt}\cdot n_t\left({\varphi }_t\right)\hfill \\ & =L_{wt}\left({\varphi }_c,{\varphi }_w\right)\cdot n_t\left({\varphi }_t\right)\hfill \end{array}$$

(4)

Here,

$$\begin{array}{cc}\hfill M_{wt}& =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\varphi }_{\mathrm{w}}& \sin {\varphi }_{\mathrm{w}}& 0\\ 0& -\sin {\varphi }_{\mathrm{w}}& \cos {\varphi }_{\mathrm{w}}& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}\cos {\gamma }_m& 0& \sin {\gamma }_m& -X_a\\ 0& 1& 0& 0\\ -\sin {\gamma }_m& 0& \cos {\gamma }_m& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& E_m\\ 0& 0& 1& -X_b\\ 0& 0& 0& 1\end{array}\right]\cdot \hfill \\ & \left[\begin{array}{cccc}\cos {\varphi }_c& -\sin {\varphi }_c& 0& 0\\ \sin {\varphi }_c& \cos {\varphi }_c& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}\sin q& -\cos q& 0& S_r\cos q\\ \cos q& \sin q& 0& -S_r\sin q\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}\cos S_j& -\sin S_j& 0& 0\\ \sin S_j& \cos S_j& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \hfill \\ & \left[\begin{array}{cccc}\cos t_i& 0& \sin t_i& 0\\ 0& 1& 0& 0\\ -\sin t_i& 0& \cos t_i& 0\\ 0& 0& 0& 1\end{array}\right]\hfill \end{array}$$

The matrix L_wt is a 3 × 3 matrix obtained by removing the last row and last column of the matrix M_wt.

According to the gear meshing theory, when two spatial tooth surfaces mesh, they share a common tangent plane and normal line at the meshing point, and the relative velocity at the meshing point must be perpendicular to the common normal line to ensure continuous contact transmission of the two curved surfaces. Thus, the meshing equation between the cutter cutting surface and the gear tooth surface can be expressed in the fixed machine coordinate system S_m (x_m, y_m, z_m) as follows:

$${\mathit{n}}_m\cdot {{\mathit{v}}_m}^{\left(cw\right)}=0$$

(5)

where:

$$\left\{\begin{array}{c}{n}_m=L_{mc}\left({\varphi }_c\right)L_{ca}{L}_{ab}{L}_{bt}\cdot n_t\left({\varphi }_t\right)\\ {{\mathit{v}}_m}^{\left(cw\right)}=\left({{\omega }_m}^{\left(c\right)}-{{\omega }_m}^{\left(w\right)}\right)\times H_m-R_m\times {{\omega }_m}^{\left(w\right)}\\ H_m=M_{mc}\left({\varphi }_c\right)M_{ca}{M}_{ab}{M}_{bt}\cdot H_t\left(S_F,{\varphi }_t\right)\\ R_m={\left[\left(X_a\cos {\gamma }_m\right)\left(-E_m\right)\left(X_b+X_a\cos {\gamma }_m\right)\right]}^T\end{array}\right.\left\{\begin{array}{c}{H}_w=H_w\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)\\ n_w=n_w\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)\end{array}\right.$$

(6)

To simplify the calculation, we can take |ω_m^(w)| = 1, and then we obtain the following:

$$\left\{\begin{array}{c}{{\omega }_m}^{\left(c\right)}={\displaystyle \frac{1}{Ra}}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T\\ {{\omega }_m}^{\left(w\right)}={\left[\begin{array}{ccc}\cos {\gamma }_m& 0& \sin {\gamma }_m\end{array}\right]}^T\end{array}\right.$$

(7)

The roll ratio during the modified roll machining process is controlled by the second-order and third-order modified roll coefficients, and the following roll ratio relationship exists during machining:

$${\varphi }_w=R_a\left({\varphi }_c-C{\varphi }_c^2-D{\varphi }_c^3\right)$$

(8)

In the equation, C and D represent the second-order modified roll coefficient and the third-order modified roll coefficient, respectively.

When machining with the cutter tilt method, which differs from the modified roll method, a constant roll ratio can be ensured by setting C = 0 and D = 0 in Equation (8) [31].

During the gear machining process, the machine tool setting parameters r_t, α_t, γ_m, S_r, q, X_a, X_b, E_m, t_i, s_j, R_a, C, D, etc., are all known. By combining Equations (3)–(5) and (8), the specific expressions of the tooth surface H_w and the unit normal vector n_w in the workpiece coordinate system S_w can be determined.

$$\left\{\begin{array}{c}{H}_w=H_w\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)\\ n_w=n_w\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)\end{array}\right.$$

(9)

In the equation, Ω_j (where j = 1, 2, …, m, with m representing the number of machining parameters) denotes the set of machining parameters for the gear. The parametric coordinates of the tooth surface, (ϕ_t, ϕ_c), are determined by Ω_j.

The gear measuring center serves as a critical device for acquiring discrete data on the deviated tooth surface [32]. With the rapid advancement of gear inspection technology, gear measuring centers based on a one-dimensional probe have overcome significant challenges, such as measurement path planning and data compensation for complex tooth surfaces, including bevel gears, thereby enabling high-precision measurement of tooth surfaces. The mechanical structure of such a measuring center is illustrated in Figure 3. It consists of three linear axes (X, Y, and Z axes), a rotary table axis (C-axis), and a rotary axis (B-axis) that allows for limited motion of the one-dimensional probe. During the inspection process, coordinated movement of these axes facilitates the measurement of tooth surface deviations and the acquisition of discrete tooth surface data.

The unification of the spatial positions of the workpiece coordinate system and the measurement coordinate system is essential for accurately measuring tooth surface deviations. A transformation model that relates the workpiece coordinate system to the measuring coordinate system has been established, as illustrated in Figure 4. In this model, S_w (X_w, Y_w, Z_w) denotes the workpiece coordinate system, S_c (X_c, Y_c, Z_c) represents the measuring coordinate system, and S_d (X_d, Y_d, Z_d) is an auxiliary coordinate system. The origin O_c of the measuring coordinate system S_c is positioned at the center of the rotary table. The variable L denotes the axial distance between the origins of the two coordinate systems, while β represents the rotation angle of the gear.

The theoretical designed tooth surface of the spiral bevel gear, denoted as H_c, and its unit normal vector n_c, can be expressed in the measurement coordinate system S_c as follows:

$$\left\{\begin{array}{c}{\mathit{H}}_c\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)=M_{cw}\cdot {\mathit{H}}_w\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)\\ {\mathit{n}}_c\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)=M_{cw}\cdot {\mathit{n}}_w\left({\varphi }_t,{\varphi }_c;{\mathsf{\Omega }}_j\right)\end{array}\right.$$

(10)

Here, H_w and n_w represent the theoretical tooth surface and its unit normal vector, respectively, in the workpiece coordinate system S_w. The coordinates on the theoretical tooth surface are denoted by (ϕ_t, ϕ_c), and the transformation matrix is provided as follows:

$${\mathit{M}}_{cw}=\left[\begin{array}{cccc}0& \cos \beta & -\sin \beta & 0\\ 0& \sin \beta & \cos \beta & 0\\ -1& 0& 0& L\\ 0& 0& 0& 1\end{array}\right]$$

Rational planning of the measurement path for the one-dimensional probe is crucial for the accurate acquisition of discrete tooth surface data. As illustrated in Figure 5, a total of 45 grid points are selected, comprising 5 points along the tooth profile from the root to the tip and 9 points across the tooth width from the toe to the heel. This distribution ensures maximal coverage of the meshing region and enhances measurement efficiency. During the actual measurement, the midpoint of the surface grid nodes (point 1 in Figure 5) is defined as the measurement reference point, with its deviation value assumed to be zero. The one-dimensional probe then traverses along the tooth height direction from point 1 to point 45, ultimately retracting from the measurement endpoint. Consequently, the tooth surface deviation δ of the remaining measured points relative to the reference point can be accurately determined.

Clarifying the complex relationship among the theoretically designed tooth surface H_w, the tooth surface deviation δ, and the deviated tooth surface T is particularly important for acquiring discrete tooth surface data. The tooth surface deviation δ is typically evaluated along the normal direction n_w of the theoretically designed tooth surface H_w. Since each point on H_w corresponds to a specific deviation δ, δ can be defined as a function of the tooth surface coordinates (ϕ_t, ϕ_c). Therefore, the mathematical relationship among the theoretical designed tooth surface H_w, the tooth surface deviation δ, and the deviated tooth surface T can be expressed as follows:

$$\delta \left({\varphi }_{ti},{\varphi }_{ci};{\mathsf{\Omega }}_j\right)=\left(\mathit{T}\left(u_i,v_i\right)-{\mathit{T}}_t\left({\varphi }_{ti},{\varphi }_{ci};{\mathsf{\Omega }}_j\right)\right)\cdot \mathit{n}\left({\varphi }_{ti},{\varphi }_{ci};{\mathsf{\Omega }}_j\right)$$

(11)

where the subscript i (i = 1, 2, …, 45) denotes the index of the measurement grid point; T (u, v) represents the digital tooth surface, with u and v being the parameters of its construction function.

2.2. Reconstruction of the Digital Tooth Surface and Analysis of Its Accuracy

After measuring the deviated tooth surface using a gear measuring center, only discrete coordinate points of the tooth surface are obtained. These discrete points cannot be directly utilized for analyzing tooth contact performance. Therefore, it is essential to reconstruct a high-precision digital representation of the tooth surface to replace the deviated surface for subsequent analysis. The Non-Uniform Rational B-Spline (NURBS) method is capable of accurately and flexibly representing complex three-dimensional geometries, playing a significant role in fields such as product design and manufacturing. Consequently, the NURBS algorithm is employed to reconstruct the digital tooth surface.

During the reconstruction of the digital tooth surface, bicubic surfaces are selected in the u and v directions. The reconstruction of the deviated tooth surface is illustrated in Figure 6. For any given set of parameters (u, v), the corresponding coordinate point on the reconstructed deviated tooth surface can be obtained. The reconstructed digital tooth surface can be expressed as follows:

$$\mathit{T}\left(u,\nu \right)={\displaystyle \frac{{\displaystyle \sum _{i=0}^n}{\displaystyle \sum _{j=0}^m}{N}_{i,3}\left(u\right)N_{j,3}\left(\nu \right){\omega }_{i,j}{W}_{i,j}}{{\displaystyle \sum _{i=0}^n}{\displaystyle \sum _{j=0}^m}{N}_{i,3}\left(u\right)N_{j,3}\left(\nu \right){\omega }_{i,j}}}$$

(12)

In this context, N_i,3 and N_j,3 represent the B-spline basis functions in the u and v directions, respectively, with p = q = 3. Here, W_i,j denotes the control points that form the control polygon, while ω_i,j signifies the weight factor.

The normal vector of the NURBS surface is essential for determining whether two digital tooth surfaces are in proper mesh. However, the procedure for obtaining the normal vector of a digital surface differs from conventional methods. By exploiting the specific characteristics of the NURBS surface equations, the partial derivatives in the u and v directions are calculated via the derivatives of the basis functions. Subsequently, the surface normal vector is derived using the cross-product method.

The derivative formula for the basis function N_i,p is given by

$$N_{i,p}^{\left(k\right)}(u)={\displaystyle \frac{p}{u_{i+p}-u_i}}{N}_{i,p-1}^{(\mathrm{k}-1)}\left(u\right)+{\displaystyle \frac{p}{u_{i+p+1}-u_{i+1}}}{N}_{i+1,p-1}^{(\mathrm{k}-1)}\left(u\right),$$

(13)

where k denotes the order of the derivative and p is the degree of the basis function.

When solving for the partial derivative vectors of the digital tooth surface T (u, v), it follows from the above expression that the partial derivative vectors of the surface S (u, v) in the u and v directions are given by

$${\displaystyle \frac{{\partial }^{k+l}}{{\partial }^{k}u{\partial }^l\nu }}S\left(u,\nu \right)=\sum _{i=0}^n\sum _{j=0}^m{N}_{i,p}^{\left(k\right)}\left(u\right)N_{j,q}^{\left(l\right)}\left(\nu \right)W_{i,j},$$

(14)

where k and l denote the order of the partial derivative vectors in the u-and v-directions, respectively, and W represents the control points.

Taking an arbitrary point (u, v) on the digital tooth surface T, set k = 1, l = 0 and k = 0, l = 1, respectively. According to the above equation, the first-order tangent vectors S_u and S_v along the u-and v-directions of the surface can be solved. The plane formed by the tangent vectors S_u and S_v is the tangent plane at point (u, v). The normal vector of this tangent plane, i.e., the unit normal vector at this point, is given by

$$n={\displaystyle \frac{S_u\times S_v}{|S_u\times S_v|}}.$$

(15)

The reconstruction accuracy of the digital tooth surface directly determines the reliability of tooth contact performance analysis; therefore, verifying the reconstruction accuracy is of particular importance. The validation is based on the normal distance l between the theoretically designed tooth surface T_t and the digital tooth surface T. However, the reconstructed digital surface T is defined only within a limited parametric domain (0 ≤ u, v ≤ 1), with no explicit expression available outside this range. If the normal vectors are derived from the theoretical surface T_t, their extensions at the surface boundary may not intersect the digital surface T, which renders the calculation of the normal distance between the two surfaces infeasible. Hence, this study proposes a method that takes the digital surface T as the starting point: extending along its normal direction n until intersection with the theoretical surface T_t is achieved, thereby determining the normal distance l between the two surfaces.

In this paper, the digital tooth surface T is discretized into a grid comprising 25 rows and 45 columns. Based on Equation (15), the normal direction n at each grid point is determined, and the normal distance l between each grid point and the theoretical tooth surface T_t is calculated. The maximum normal distance l from each column is then extracted to construct a fitted error curve. As illustrated in Figure 7, taking a point Q (x_q, y_q, z_q) on the digital surface T as an example, the normal direction n (n_x, n_y, n_z) at this point is traced until it intersects the theoretically designed surface T_t at point Q* (x^*_q, y^*_q, z^*_q). The relationship among point Q, point Q*, and the normal distance l is established using the Newton–Raphson algorithm as follows:

$$\left\{\begin{array}{c}{x}_q^{\ast }=x_q+l\cdot n_x\\ y_q^{\ast }=y_q+l\cdot n_y\\ z_q^{\ast }=z_q+l\cdot n_z\end{array}\right.$$

(16)

3. Tooth Contact Model and Determination of Meshing Parameters

3.1. Construction of the Tooth Contact Model

In the measurement coordinate system, the origins of the workpiece coordinate systems for both the pinion and the gear coincide with the intersection point of their respective rotational axes during meshing. Consequently, the meshing coordinate system for spiral bevel gears is established as illustrated in Figure 8. Since the intersection point of the two gear axes serves as the origin of each workpiece coordinate system, the traditional mounting distance is eliminated, leading to a more streamlined meshing coordinate system. In the figure, θ₁ and θ₂ denote the rotational angles of the pinion and gear, respectively; Σ represents the shaft angle; S₁ (X₁,Y₁,Z₁) and S₂ (X₂,Y₂,Z₂) are the rotating coordinate systems associated with the pinion and gear, respectively; S_s (X_s,Y_s,Z_s) is the fixed coordinate system; and S_q (X_q,Y_q,Z_q) is an auxiliary coordinate system.

The digital tooth surfaces of the pinion and gear, denoted as T₁ (u₁,v₁) and T₂ (u₂,v₂), along with their corresponding normal vectors n₁ (u₁,v₁) and n₂ (u₂,v₂), are transformed from their respective workpiece coordinate systems S₁ and S₂ to the fixed coordinate system S_s.

$$\left\{\begin{array}{c}{\mathit{T}}_{si}\left(u_i,{\nu }_i,{\theta }_i\right)=M_{si}\left({\theta }_i\right)\cdot {\mathit{T}}_i\left(u_i,{\nu }_i\right)\\ {\mathit{n}}_{si}\left(u_i,{\nu }_i,{\theta }_i\right)=M_{si}\left({\theta }_i\right)\cdot {\mathit{n}}_i\left(u_i,{\nu }_i\right)\end{array}\right.\left(\begin{array}{c}i=1,2\end{array}\right)$$

(17)

Using appropriately defined transformation matrices:

$$M_{s1}\left({\theta }_1\right)=\left[\begin{array}{cccc}\cos \Sigma & 0& -\sin \Sigma & 0\\ 0& 1& 0& 0\\ \sin \Sigma & 0& \cos \Sigma & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_1& \sin {\theta }_1& 0\\ 0& -\sin {\theta }_1& \cos {\theta }_1& 0\\ 0& 0& 0& 1\end{array}\right], M_{s2}\left({\theta }_2\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_2& \sin {\theta }_2& 0\\ 0& -\sin {\theta }_2& \cos {\theta }_2& 0\\ 0& 0& 0& 1\end{array}\right]$$

During the meshing transmission of spiral bevel gears, it is essential that the position vectors and normal vectors of the two tooth surfaces coincide at any contact point. Based on this condition, the tooth contact equations in the fixed coordinate system S_s are established as follows:

$$\left\{\begin{array}{c}{\mathit{T}}_{s1}\left(u_1,{\nu }_1,{\theta }_1\right)={\mathit{T}}_{s2}\left(u_2,{\nu }_2,{\theta }_2\right)\\ {\mathit{n}}_{s1}\left(u_1,{\nu }_1,{\theta }_1\right)={\mathit{n}}_{s2}\left(u_2,{\nu }_2,{\theta }_2\right)\end{array}\right.$$

(18)

This system comprises five nonlinear equations with six unknowns (u₁, v₁, u₂, v₂, θ₁, θ₂). By specifying the pinion rotation angle θ₁, the Newton–Raphson algorithm is employed to solve for the remaining parameters (u₁, v₁, u₂, v₂, θ₂), thereby determining a single meshing point.

3.2. Acquisition of the Normal Distance Between Digital Tooth Surfaces

The normal distance between digital tooth surfaces serves not only as a criterion for determining whether the tooth surfaces are in mesh but also as a key factor in solving the sequence of contact points and the instantaneous contact ellipse. Given that the parametric equations of the digital surfaces are confined to a limited domain (0 ≤ u, v ≤ 1), traditional methods encounter difficulties in determining this normal distance. For instance, consider an arbitrary point M (x_M, y_M, z_M) on the pinion tooth surface T₁. The normal direction n_M at point M is traced until it intersects the gear tooth surface T₂ at point P (x_P, y_P, z_P), as illustrated in Figure 9.

Determining the coordinates (u_P, v_P) of the intersection point P on the gear tooth surface T₂ is crucial for acquiring the normal distance between the two digital tooth surfaces T₁ and T₂. As illustrated in Figure 10 (where the dashed lines represent the transformed configuration), the procedure is executed as follows: First, point M on the pinion tooth surface is translated to the coordinate origin O (see Figure 10a). Next, the normal vector n_M is rotated clockwise about the Z-axis by an angle α until it lies in the XOZ plane (see Figure 10b). Finally, n_M is rotated clockwise about the Y-axis by an angle γ until it aligns with the Z-axis (see Figure 10c). The gear tooth surface T₂ is then rigidly translated and rotated accordingly, resulting in the normal vector n_M intersecting the transformed gear surface at point P′ (x′_P, y′_P, z′_P), as shown in Figure 11.

The correspondence between point P and point P′ on the gear tooth surface T₂ before and after the transformation is given by the following:

$$\begin{array}{c}\left[\begin{array}{c}{x}_P^{\prime }\\ y_P^{\prime }\\ z_P^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cccc}\cos \gamma & 0& -\sin \gamma & 0\\ 0& 1& 0& 0\\ \sin \gamma & 0& \cos \gamma & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \alpha & \sin \alpha & 0& 0\\ -\sin \alpha & \cos \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -z_M\\ 0& 1& 0& -z_M\\ 0& 0& 1& -z_M\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_P\\ y_P\\ z_P\\ 1\end{array}\right]\end{array}$$

(19)

To determine the location (u_P, v_P) of the intersection point P, the gear tooth surface T₂ is projected onto the XOY plane, as illustrated in Figure 12. The intersection point P′′ is also projected onto this plane and is correspondingly denoted as point P′′. The procedure is as follows: First, a preliminary judgment is made to identify the grid cell in which point P′′ is located. Specifically, the four grid points of each of the 4 × 8 surface patches on T₂ are connected to point P′′, and the corresponding angles v₁, v₂, v₃, and v₄ are calculated. If v₁ + v₂ + v₃ + v₄ = 360°, it can be confirmed which specific grid cell contains point P′′; otherwise, point P′′ is not located on T₂. Next, the identified grid cell is refined, and the above method is repeated to determine which sub-patch contains point P″. The distances between point P′′ and the four grid points of the sub-patch are computed. This process is iterated until the distance is less than 10⁻⁵ mm. Finally, the coordinates (u_P, v_P) corresponding to the shortest grid-point distance are taken as the location of point P″. Substituting P″ into Equation (15) yields the three-dimensional coordinates of point P. Consequently, the normal distance l between the digital tooth surfaces of the pinion and gear, T₁ and T₂, is obtained as follows:

$$l=\sqrt{{\left(x_M-x_P\right)}^2+{\left(y_M-y_P\right)}^2+{\left(z_M-z_P\right)}^2}$$

(20)

3.3. Determination of Actual Meshing Point Sequence and Contact Path

The contact path, a crucial element of the contact pattern, is defined by the sequence of actual meshing points on the tooth surface. To accurately determine this sequence, it is essential to first establish the location of the initial meshing point. By designating the midpoint of the tooth surface as the initial iteration point D, the pinion is held stationary while the gear undergoes rotation. Substituting the rotation angles of both gears and the initial iteration point D into Equation (18) and employing the Newton–Raphson algorithm yields a set of converged precise solutions (u₁⁽¹⁾, v₁⁽¹⁾, u₂⁽¹⁾, v₂⁽¹⁾) along with the corresponding gear rotation increment Δθ₂. The point associated with this solution set is defined as the initial meshing point E, as depicted in Figure 13.

In the figure, the rotation angles of both the pinion and gear at the initial meshing point E are set to θ₁ = θ₂ = 0. A fixed step size is established, allowing the solution to progress from point E towards both the tooth tip and the tooth root directions. Each subsequent meshing point in the sequence is resolved iteratively based on the Newton–Raphson algorithm, using the previous meshing point as a reference, until the teeth disengage from the mesh. The resulting series of converged precise solutions, denoted as (u₁^(j), v₁^(j), u₂^(j), v₂^(j)) for j = 1, 2,…, k (where k represents the number of meshing points), are substituted into Equation (15) to ascertain the actual sequence of meshing points and their corresponding normal vectors. The line connecting these meshing points delineates the contact path. This method eliminates the need for meshing or refinement of the digital tooth surfaces, leading to faster computation; however, caution must be exercised during the solution process to prevent exceeding the surface boundaries.

The tooth surface of a spiral bevel gear is characterized by a complex spatial curved surface, which complicates the direct visualization of the meshing point sequence. To facilitate this visualization, the tooth surface can be rotated, allowing the meshing point sequence to be projected onto an axial cross-section. The coordinates (x*_j, y*_j) of the meshing point sequence on the projection plane can be expressed as follows:

$$\left\{\begin{array}{c}{x}_j^{\ast }=x_j\\ y_j^{\ast }=\sqrt{{\left(y_j\right)}^2+{\left(z_j\right)}^2}\end{array}\right.$$

(21)

where (x_j, y_j, z_j) are the coordinates of the meshing point before projection, j = 1, 2,…, k, and k is the number of meshing points.

3.4. Solution of Instantaneous Contact Ellipse Based on Binary Search Method

The shape, size, and location of the instantaneous contact ellipse significantly influence the geometric form of the contact pattern. Consequently, accurately determining the instantaneous contact ellipse is a critical step in obtaining the actual contact pattern. Traditional methods address this issue by searching based on the principal directions and principal curvatures of the pinion and gear tooth surfaces, which rely on the prior Hertz contact ellipse assumption for loaded conditions, as elaborated in detail in reference [33]. However, these methods are computationally intensive, sensitive to errors, and face challenges in calculating principal curvatures under the condition of quadratic surface enveloping. It should be noted that the instantaneous contact ellipse in this study is defined for the no-load meshing condition, which is consistent with the industry’s universal no-load dye penetration rolling test specification, rather than the prior elastic contact assumption.

This paper employs the method of truncating two meshing tooth surfaces T₁ and T₂, from a cylinder to obtain the instantaneous contact ellipse. As shown in Figure 14, point M is the meshing point of the two meshing tooth surfaces T₁ and T₂. The normal line at the meshing point M is denoted as n_M. Using n_M as the axis and r as the radius, a cylinder is cut to form the two meshing tooth surfaces T₁ and T₂.

To determine the directions of the major and minor axes, we utilize the binary search method. Given that the processes for solving the major and minor axes are analogous, we will use the solution for the major axis as an example. We establish a two-dimensional coordinate system with U as the vertical axis, V as the horizontal axis, and the meshing point M as the origin, as depicted in Figure 15. It is important to note that the directions of the major axes, ς₁ and ς₂, may not be symmetrical about the coordinate origin M; therefore, each direction is solved separately.

The normal distance at any point on the boundary of the instantaneous contact ellipse is 0.00635 mm. To determine the direction of the major axis, it is essential to first calculate the normal distance between the meshing tooth surfaces and subsequently identify the boundary endpoints of the instantaneous contact ellipse. The binary search method is employed for this purpose. Taking the positive V-direction as an example (refer to Figure 16), the endpoints are defined as the meshing point M and the boundary point M′ on the pinion tooth surface T₁. The midpoint M″ between M and M′ is then established. The corresponding normal distances at points M, M″, and M′ are denoted as l₁ (where the normal distance equals 0), l₂, and l₃, respectively. At this stage, two scenarios may arise:

(1)

If the normal distance l₃ ≤ 0.00635 mm, then the point M′ on the pinion tooth surface corresponds to the V-direction endpoint M*. Consequently, the endpoint N* on the gear tooth surface T₂ can also be determined using the same normal distance calculation method.

(2)

If the normal distance l₃ > 0.00635 mm holds, we determine the search direction according to the relationship between normal distance l₂ and 0.00635 mm, where two distinct scenarios exist. First scenario: If l₂ < 0.00635 mm, set l₁ = l₂, regard point M″ as point M, and take the midpoint M′ between M and M′. Then we assign l₂ as the normal distance corresponding to this midpoint, after which we then re-evaluate the relationship between l₂ and 0.00635 mm. Second scenario: If l₂ > 0.00635 mm, then set l₁ = l₂, regard point M′′ as point M′ and we select the midpoint M′′ between M and M′. We set its corresponding normal distance to l₂, and subsequently re-evaluate the relationship between l₂ and 0.00635 mm. In both sub-cases, the iteration continues until the normal distance reaches 0.00635 mm, at which point the corresponding boundary endpoints M* on the pinion tooth surface T₁ and N* on the gear tooth surface T₂ of the instantaneous contact ellipse are determined.

In practice, when searching for the boundary endpoints M* and N* of the instantaneous contact ellipse at any specified direction from the meshing point M on the pinion and gear tooth surfaces T₁ and T₂, we can identify these endpoints by following the procedures outlined above.

Since the methods for determining the directions of the semi-major axes ς₁ and ς₂ are identical, the solution for a1, as shown in Figure 15, is taken as an example and illustrated in Figure 17. Starting from the meshing point M, the positive V-direction, positive U-direction, and negative V-direction are followed sequentially to identify points (u, v) on the pinion tooth surface where the normal distance equals 0.00635 mm. These points are then connected to point M, resulting in segment lengths L₁, L₂, and L₃, with corresponding angles Ang₁ = 180°, Ang₂ = 90°, and Ang₃ = 0°. Points M₁, M₂, and M₃ in Figure 17 represent the projected points where the normal distance between the two tooth surfaces T₁ and T₂ equals 0.00635 mm in the negative V-direction, positive U-direction, and positive V-direction, respectively. Furthermore, any directional vector (du: dv) originating from the meshing point can be obtained from the tangent of an angle Ang, where Ang denotes the angle between that direction and the positive V-direction. When applying the binary search method to determine the direction of the semi-major axis a₁, the search termination condition is defined as the length variation satisfying |L₁−L₂| ≤ 0.000001. The following two cases are considered:

(1)

If L₁ ≤ L₃, set Ang₁ = Ang₂ and Ang₂ = (Ang₁+ Ang₃)/2; also set L₁ = L₂. To determine L₂, first follow the direction tanAng₂ to obtain the point on the pinion tooth surface where the normal distance equals 0.00635 mm. The length L₂ is then determined by connecting this point to the meshing point M, after which the search termination condition is checked.

(2)

If L₁ > L₃, set Ang₃ = Ang₂ and Ang₂ = (Ang₁+ Ang₃)/2; also set L₃ = L₂. The procedure for determining L₂ is the same as in case (1).

During the search for the semi-major axis direction, the appropriate search scheme is selected according to the above cases. The iteration continues until the termination condition is satisfied. The direction tanAng₂ corresponding to the final L₂ is taken as the semi-major axis direction ς₁, and the corresponding points (u₁, v₁) on the pinion and (u₂, v₂) on the gear serve as the endpoints of the semi-major axis in direction ς₁. By performing the binary search over the full 360° range, a complete instantaneous contact ellipse is obtained.

4. Digital Representation of Tooth Contact Performance

Transmission error and contact pattern serve as evaluation metrics for gear contact performance, enabling an accurate assessment of the precision and service life of spiral bevel gear transmission systems. Consequently, their precise characterization is particularly critical.

Utilizing Equation (18), the corresponding parameters (u₁^(j), v₁^(j), u₂^(j), v₂^(j)) for each meshing point along the contact path, along with the actual rotation angles θ₁ and θ₂ of the pinion and gear at each meshing point, can be determined. The transmission error at each meshing point is expressed as follows:

$$\mathsf{\Delta }\theta =\left(\begin{array}{c}{\theta }_2-{\theta }_{20}\end{array}\right)-{\displaystyle \frac{Z_1}{Z_2}}\left(\begin{array}{c}{\theta }_1-{\theta }_{10}\end{array}\right)$$

(22)

where θ₁₀ and θ₂₀ represent the initial meshing rotation angles of the pinion and gear, respectively, and Z₁ and Z₂ denote the number of teeth of the pinion and gear, respectively.

The digital representation of the transmission error is illustrated in Figure 18. Point E indicates the initial meshing point, while points A and A* represent the actual entry and exit points of contact, and points C and C* signify the potential entry and exit points of contact.

The tooth contact pattern offers an intuitive representation of the meshing state of spiral bevel gears. Its fundamental components encompass the contact path, the instantaneous contact ellipse, and the boundary curve. To visually depict the contact pattern, the sequence of meshing points is projected onto an axial cross-section, and the contact path is fitted from this sequence utilizing the least squares method. The instantaneous contact ellipse at each meshing point is determined, with its major axis represented as a straight line. Subsequently, the endpoints of these major axes are connected to form the boundary curve through an NURBS fitting algorithm. The resulting digital tooth contact pattern is illustrated in Figure 19.

5. Experimental Verification

To validate the effectiveness of the contact performance analysis method based on digital tooth surfaces, a spiral bevel gear pair designed for automotive applications is utilized as a case study. The specific geometric parameters and optimal machine settings are detailed in Table 1 and Table 2, respectively. The established testing platform comprises a JD45+ gear measuring center and a Y9550 rolling tester, as illustrated in Figure 20. The JD45+ gear measuring center is equipped with a high-precision one-dimensional probe, features a high level of automation, and accommodates a maximum measurable outer diameter of 500 mm, a module range of 1–20 mm, and a repeatability of 0.5 μm. The Y9550 rolling tester is outfitted with an advanced CNC system, can handle gears with a maximum outer diameter of 500 mm, and provides high accuracy and efficiency in rolling tests. Both the measuring and rolling test equipment fulfill the experimental requirements.

To validate the effectiveness of the contact performance analysis method, the theoretical contact performance results obtained from the proposed method in this study were compared with those derived from the conventional method. Using the working side as an example, a comparison of the contact performance analysis results simulated by both methods is illustrated in Figure 21, where the blue solid line represents the results of the proposed method and the red solid line denotes the results of the conventional method. It can be observed that the shape and trend of the contact pattern, as well as the transmission error curve, are generally consistent across both approaches.

The pinion and gear were mounted on the JD45+ gear measuring center. A probe with a diameter of 2 mm was selected, utilizing a straight probe for the pinion and a vertical probe for the gear. Based on the geometric parameters of the input gear and the theoretically optimal machining parameters, the measuring center generated the measurement path. To ensure that the path avoided the boundaries of the tooth tip, root, heel, and toe, the planned path was offset inward by specified clearances: 2 mm from the tooth tip, 2 mm from the tooth root, 3 mm from the heel, and 3 mm from the toe. Subsequently, the measuring system performed the tooth surface measurement according to this planned path, as illustrated in Figure 22 and Figure 23.

The results of tooth surface deviation for both the pinion and gear are illustrated in Figure 24 and Figure 25, respectively. The thick solid line represents the modified theoretical tooth surface with point contact generated by the optimized machine tool setting parameters in Table 2, while the thin solid line and the dashed line denote positive and negative deviations, respectively. The actual machined tooth surfaces of both the pinion and gear exhibit minor discrepancies when compared to their theoretical designs, demonstrating a high degree of geometric accuracy and indicating compliance with the design requirements.

To validate the accuracy of tooth surface deviations, the same tooth surface of both the pinion and the gear was measured using a Gleason 650GMS gear measuring center, as illustrated in Figure 26. The resulting tooth surface deviations for the pinion and gear are presented in Figure 27 and Figure 28, respectively, where the red and green curves represent the theoretical and actual tooth surfaces. A comparison of the deviations obtained from the two measuring devices is summarized in

Table 3. Comparison of tooth surface deviation (Unit: μm).

		JD45+		650GMS		Difference (%)		Deviation Position
		Convex	Concave	Convex	Concave	Convex	Concave	Convex	Concave
Pinion	MPD	3.9	3.5	3.8	3.6	2.6%	2.9%	Top of toe	Root of heel
	MND	−7.5	−7.3	−7.3	−7.4	2.7%	1.4%	Root of heel	Top of toe
Gear	MPD	2.3	9.1	2.2	9.2	4.5%	1.1%	Root of toe	Top of heel
	MND	−2.7	−8.9	−2.8	−8.8	3.7%	1.1%	Top of heel	Root of toe

, with MPD and MND denoting the maximum positive deviation and minimum negative deviation, respectively. The trends of the two measurement sets are consistent, with a maximum comparative error of 4.5%, which is below the 5% threshold. This minor discrepancy may be attributed to variations introduced during the reinstallation of the gears. Consequently, the tooth surface deviations obtained from the JD45+ gear measuring center meet the required accuracy standards.

Following the aforementioned reconstruction accuracy analysis method, the reconstruction error curve of the digital tooth surface was fitted, as illustrated in Figure 29. The maximum reconstruction error for both the pinion and the gear is merely 2.618 × 10⁻⁵ mm, which satisfies the accuracy requirements. Consequently, the digital tooth surface can be utilized in lieu of the actual machined tooth surface for subsequent analyses.

Taking the working side of the gear as an example, after accurately positioning the tooth surfaces of the pinion and gear within the correct meshing zone, the gear tooth surface was rotated using the aforementioned theoretical method to establish contact between the two surfaces. The corresponding initial meshing point on the gear was identified at (0.475, 0.523) (refer to the green meshing point in Figure 29). Utilizing this initial meshing point as the reference and adopting a step size of 0.01 rad, the gear was rotated toward both the pinion’s tooth tip and root directions to obtain the sequence of meshing points and their corresponding gear rotation angles. The resulting sequence of meshing points on the gear is illustrated in Figure 30. Subsequently, the meshing tooth surfaces were intersected by a variable-radius cylinder, and the instantaneous contact ellipse was determined using the binary search method. The instantaneous contact ellipse at the initial meshing point is depicted in Figure 31.

Substituting the rotation angles at each meshing point into Equation (22) yields the corresponding error values. The digitally characterized transmission error curve for the working side is illustrated in Figure 31. This transmission error curve demonstrates a periodic variation that closely resembles a parabolic form. The intersection value of the error curve is −26.573 arcseconds, which deviates from the theoretically designed transmission error by 4.7%. This discrepancy is below the 5% threshold, thus satisfying the requirements for mechanical transmission.

The instantaneous contact ellipse at each meshing point was obtained using the binary search method, with its major axis represented as a straight line. The digitally characterized contact patterns for the working sides of the pinion and gear are illustrated in Figure 32. The dashed lines delineate the boundaries of the complete tooth surface, while the solid lines indicate the boundaries of the measured tooth surface region. The contact patterns of both the pinion and gear are situated in the central area of the tooth surface, signifying high contact accuracy and favorable meshing conditions.

To validate the correctness of the simulated contact performance, the spiral bevel gear pair was mounted according to the standard on the Y9550 rolling tester shown in Figure 20. Subsequently, a rolling test experiment for tooth contact accuracy of the spiral bevel gears was conducted. The rolling test results for the working sides of the pinion and gear are presented in Figure 33a,b, respectively.

A comparison of Figure 33a,b reveals that both the actual contact pattern and the digital contact pattern exhibit a roughly parallelogram-like shape located in the central region of the tooth surface, demonstrating strong overall agreement. This finding indicates that the proposed contact performance analysis method for spiral bevel gears accurately reflects the actual meshing state of the gear teeth. However, slight discrepancies persist between the actual and digital contact patterns. These differences can primarily be attributed to various influencing factors during the experiment, such as lubrication conditions, vibration, and mounting errors of the rolling tester. Additionally, the projection of the meshing point sequence from the spatial tooth surface onto the axial cross-section also exerts subtle effects on the position and shape of the digital contact pattern.

6. Conclusions

This paper presents a method for analyzing the contact performance of spiral bevel gears that incorporates actual tooth surface deviations. By directly linking the digital tooth surface to the meshing behavior, the following conclusions are drawn:

• Digital Reconstruction Strategy: A measurement path planning method for one-dimensional probes has been established. By constructing a tooth surface deviation model, high-precision digital tooth surfaces are reconstructed, serving as a reliable substitute for actual machined surfaces in the analysis.
• Simplified Contact Model: The traditional meshing model has been optimized by eliminating the mounting distance parameter. A novel solution for the instantaneous contact ellipse is proposed using a variable-radius cylindrical cutting method and a binary search algorithm, effectively bypassing the complexity associated with calculating principal curvatures and directions.
• Validation of Accuracy: The proposed method has been validated through gear measuring center data and rolling tests. The reconstruction error of the digital surface is negligible (10⁻⁹ mm), and the simulated transmission error curve deviates from the theoretical design by only 4.7%, thereby satisfying mechanical transmission standards (<5%).
• Practical Application: The digital contact patterns demonstrate strong agreement with the rolling test results in terms of location and shape, verifying that this method can accurately predict the meshing state of actual manufactured gears.

For gear tribology and lubrication research, the proposed method has important application value. Most existing TEHL simulation and performance prediction methods for spiral bevel gears rely on ideal tooth surface contact results, which cannot reflect the influence of inherent machining deviations on contact characteristics. The accurate time-varying contact characteristics of as-machined tooth surfaces output by this method can be used as high-fidelity input for LTCA and TEHL simulation, effectively improving the prediction accuracy of transmission efficiency and scuffing risk of spiral bevel gears, and providing a reliable basis for lubrication performance optimization.