1. Introduction
Face gear drives have wide application prospects under the working conditions of high speed and heavy loads, such as the transmission systems in helicopters and machine tools, owing to their small size, light weight, low noise, large capacity, and high reliability. The transmission performances of face gears are closely related to many factors, such as design, manufacture, and installation. However, edge contact, contact area distribution, and maximum contact stress can restrict the service life of face gears. Unfortunately, current gear design methods, including finite element analysis (FEA) and numerical tooth contact analysis (TCA), do not automatically identify edge contact and minimize the maximum contact stress, despite the complexity of the process and the high technical experience requirements of the personnel. Namely, the automatic comparison and optimization of gear design parameters cannot be realized at present. Therefore, under the condition without repeated modeling, determining how to automatically avoid edge contact, quickly calculate the optimal maximum contact stress, output the optimal design parameters, and show the qualified contact path is of great significance.
In terms of the research on face gear geometry design, many publications can be referred to. Litvin et al. made outstanding contributions to the theory and application of face gears. They deduced the surface equations of the standard face gear [
1] and the face gear with offset [
2]; introduced the FEA method of stress analysis [
3,
4,
5,
6]; presented the theory of spur face gears in a handbook [
7]; and summarized the technologies of gears, including face gears, in a book [
8]. Their work provides significant references for researchers in the field of gears. In addition, Zhou et al. presented a new closed-form model for the geometry establishment of face gears without solving nonlinear equations [
9], and the model was applied to the modeling and milling of face gears [
10,
11,
12,
13,
14,
15]. In addition, they put forward a time-saving CAD/CAE integration method [
16,
17] for optimizing design parameters with FEA. Liu et al. [
18] proposed a new type of face gear drive that can achieve non-uniform transmission ratios. This face gear drive, which consists of an undulating face gear and a planar noncircular gear, is applied to transmit the angular velocity of change. Zschippang et al. [
19] elaborated a general method for the generation of face gears with shaft angle, helical angle, and axial offset. In addition, they described the procedure for determining geometry quality. Tan [
20,
21,
22] studied the face gear that meshes with a conical involute pinion and discussed the generation and geometry modeling methods with the conditions of gear integrity obtained. In order to investigate the strength variation of face gears, Li et al. [
23] constructed an equivalent face gear based on ISO 6336 standard. Lin et al. [
24] researched a non-circular face gear pair and established multiple analytical models and equations for the classification of transmission patterns, transmission ratios, and relative motions. Moreover, Lin [
25] also provided a discrete algorithm for the curve-face gear pair and analyzed the limiting points in the determination rule. Zhang et al. [
26] investigated the tooth geometry and contact characteristics of offset-axis face gear drives in detail by simulating the conjugate motion. Guo et al. [
27] simulated the computerized generation of face gear drives enveloped by circular cutters.
For the purpose of improving the contact performances of face gear drives, the methods of profile modification are applied in geometry design. Litvin et al. [
28] investigated two kinds of face gear drives generated by modified shapers, including the design, generation, and stress analysis. They also compared the contact stress of the two kinds of modified face gears. Meanwhile, they proposed the modification geometry of an asymmetric face gear and applied a tooth contact analysis (TCA) algorithm to calculate the contact path, which was verified by the FEA method [
29]. Furthermore, they analyzed a helical face gear that was enveloped by two mismatched parabolic racks corresponding to the pinion and the shaper from the aspects of design, generation, and stress analysis. Wu et al. [
30] established a model of a parabolically modified face gear to perform TCA and researched the factors affecting the meshing path. Moreover, they obtained the ideal position of the contact area. For the aim of investigating the behavior of a transmission composed of a face gear and a modified pinion, Barone et al. [
31] established geometric models based on enveloping theory and applied FEA models for simulation. Peng et al. [
32] provided the ease-off surface modification to the manufacturing process of face gears to control the unloaded meshing performance. To reduce the sensitivity of the modified face gear drive to misalignments, Zanzi et al. [
33] worked on an enhanced approach of longitudinal plunging to generate a double-crowned face gear drive.
Transmission systems composed of face gears are also investigated by scholars. Dong et al. explored the characteristics of concentric face gear split-torque transmission systems, for instance, the assembly conditions, power flow directions, load sharing performances [
34], and the method of mesh stiffness calculation [
35]. Mo et al. [
36] conducted an investigation on the load sharing of a power-split system that consists of face gears by using an analytical method, and the investigation method was applied to study a herringbone planetary gear system that contains a floating sun gear and flexible support [
37].
However, the above studies on TCA of face gears are based on the existing commercial software for FEA [
38], or the traditional analytical TCA model is used to solve the coordinates of contact points. The FEA method not only is time-consuming and may result in unstable convergence may, but also often requires repeated manual modeling work. As the traditional TCA model is used to calculate the contact points of the tooth surface analytically, the solution of multiple equations is too complex and the stability is poor. Moreover, neither of the two methods can automatically identify and avoid the edge contact, achieve the parametric optimization of the maximum contact stress of the tooth surface, or automatically return the final optimized design parameters. The research on helical face gears, especially the modified helical ones, is not involved numerically.
In this paper, a simplified algorithm of the tooth contact analysis with errors (ETCA) of modified helical face gears is provided [
14], based on which an automatic optimization algorithm of the loaded tooth contact analysis with errors (ELTCA) that integrates the functions of contact stress optimization and edge contact avoidance is proposed. Firstly, an ETCA algorithm, derived from the traditional algorithm by simplifying the contact equations and solving the rotation angles of gears separately, is introduced. Secondly, the coordinates of contact points are obtained with the ETCA algorithm. The geometric parameters, such as the contact ellipses and curvatures of the contact points, are calculated based on the coordinates, and the ELTCA algorithm is presented according to the Hertz theory to calculate the contact stress. The effectiveness of the ELTCA algorithm is verified through comparison with the finite element simulation. Furthermore, the optimization model considering the edge contact avoidance and the minimization of maximum contact stress is established, and it is verified based on the simplified ELTCA algorithm. Finally, the influence of the geometric parameters and assembly errors on the contact path and the maximum contact stress is explored by applying the optimization model.
4. A Comprehensive Optimization Model of the Contact Stress Based on the Simplified ELTCA
4.1. The Optimization Model of the Contact Stress
In the Hertz theory, the maximum contact stress is formulated as Equation (9). However, this formula is not suitable for the case of edge contact, nor can it be applied to optimize the contact calculation, for example by filtering tooth surface parameters and reducing the maximum contact stress.
The optimization model is proposed based on the following ideas:
- (i)
The edge contact should be avoided.
- (ii)
The maximum contact stress should be minimized by optimizing the modification parameters.
As shown in
Figure 9a, point
PIJ (
I∈[1, M],
J∈ [1, N]) is sampled from the cross-section of the face gear blank, and the sampling points correspond to the M × N points on the tooth surface of the modified helical face gear.
If edge contact occurs (as shown in
Figure 9b), at least one point will fall on the boundary line of tooth surfaces. Accordingly, the value of
I is equal to
M, or the value of
J is equal to
N. For purpose of avoiding edge contact, the following restrictions should be met.
The objective function of contact stress optimization is to minimize the value of contact stress. Based on the constraints in Equation (10), the optimal load contact pattern (LCP) can be simulated automatically in the program through cyclic comparison so that the corresponding optimization parameters
arM and
u0M can be calculated. The optimization model is formulated as
Finally, according to the optimal modification coefficients
arM and
u0M, the contact stress can be calculated as
To realize the programming of the optimization algorithm, the global optimization algorithm that integrates the grid searching with the downhill simplex method is employed in the solving process. The solving procedure can be summarized in the following steps as demonstrated in
Figure 10.
Step 1 is to initialize the search range and the step size in grid searching. The search range of the parabolic offset distance is set as u0∈[−1, 2], and the step size is 0.1. The search range of the parabolic coefficient is set as ar∈[−0.01, 0.01], and the step size is 0.001.
Step 2 is to calculate the objective function of modification parameters in each group as well as store the value of the function and corresponding parameters that meet the constraints of Equation (10). A group of modification parameters that minimize the value of the objective function is selected as the initial value of the optimization algorithm of the downhill simplex.
Step 3 is to initialize the position and the size of simplex vertices. The initial size of the simplex is set as 0.1, and the number of iterations is set as 40.
Step 4 is to calculate the objective function value of each simplex vertex. Through four methods of reflection, reflection and expansion, contraction, and multidimensional contraction, new vertices are obtained to form a new simplex by replacing the worst point in the original simplex.
Step 5 is to repeat Step 4 until the residual requirements are met.
4.2. Contact Path and Contact Stress Calculation Based on the Optimization Model
The basic design parameters of the helical face gear drive in this section are the same as those in
Section 3.2. As described in
Section 4.1, the two coefficients
u0 and
ar that determine the geometry of the parabola of the modified shaper are the main optimization parameters. Within the given intervals of the modification coefficients
u0∈[−1, 2] and
ar∈[−0.01, 0.01], the calculated optimal parameters as well as the minimum value of the maximum contact stress are presented in
Table 3.
The minimum value of the maximum contact stress is 758.36 Mpa and is shown as a black point in
Figure 11, the corresponding parameter
u0 which determines the position of the parabola apex is 1.6406 mm, and the quadratic coefficient
ar is −0.001125. In other words, the maximum contact stress without edge contact is not less than 758.36 MPa as each group of modification parameters is in the intervals of
u0∈[−1, 2] and
ar∈[−0.01, 0.01].
Based on the optimization model, the contact path on the tooth surface of the face gear corresponding to the parameters in
Table 1 can be obtained, as shown in
Figure 11.
The edge contact has been avoided, and the contact area and contact path have also changed. Compared with the result before optimization (as shown in
Figure 8), the optimized contact area and contact path are more inclined, and the contact path is longer, which is conducive to improving the contact ratio and transmission stability. Moreover, the contact area is further away from the inner portion of the face gear, reducing the risk of edge contact.
4.3. Verification of the Optimization Model
In order to further verify the effect of the proposed optimization model, some numerical calculations are carried out. The optimization effect to be verified includes three aspects: avoiding edge contact, minimizing the maximum contact stress, and efficiency.
As shown in
Figure 11, the effect of the proposed model in avoiding edge contact is obvious. To verify the effect of the proposed model in minimizing the maximum contact stress, the contact stress values of different parameters in the intervals of u
0∈[−1, 2] and a
r∈[−0.01, 0.01] are calculated by permutation and combination and are shown in
Table 4 and
Table 5.
All values of the maximum contact stresses in
Table 4 and
Table 5, except for the cases of edge contact (which will not be displayed in the optimization results), are greater than 758.36 MPa, which means that the optimization algorithm minimizes the maximum contact stress. According to the data in
Table 4 and
Table 5, when the value of parameter u
0 becomes relatively small, or the absolute value of coefficient a
r becomes relatively large, edge contact is easily prone to emerge.
The parameter ranges are u
0∈[−1, 2] and a
r∈[−0.01, 0.01]. The step size of parameter u
0 is 0.1, and the step size of parameter a
r is 0.001. Thus, there are 31 × 21 groups of modification parameters as a selection. For each group of parameters, the maximum value of the contact stress is selected from the values of contact stress at all contact points. The minimum is selected from these maximum values as the result of optimization that demotes the optimal values. In
Table 4 and
Table 5, 21 sets of data are presented which are the most representative of the two dimensions. Hence, these sets of data supporting the conclusion are considered credible. According to
Table 6, the proposed optimization model improves the efficiency by 10%, and the simplified TCA equation makes the algorithm more stable and less sensitive to the initial values of the numerical calculation.