Research on the Collaborative Design of Spiral Bevel Gear Transmission Considering Uncertain Misalignment Errors

Abstract

To extend the time between the overhauls of helicopters, a novel collaborative methodology that takes into account uncertain misalignment errors by considering the shape and performance of the gear is built. Firstly, the digital characteristics of contact patterns, such as the reference point and direction angle, are extracted. Secondly, an optimization model calculates the equivalent misalignment by minimizing deviations in the reference point and direction angle between two contact patterns. This equivalent misalignment accounts for uncertainty misalignment errors introduced by complex gear support deformation. Thirdly, the ease-off is utilized to derive the pinion target surface that can sustain meshing performance under an equivalent misalignment, similar to the original gear in real conditions. This way it integrates with the optimization theory for flank reconstruction to redesign the pinion surface. Simulations reveal that the critical digital characteristics of the contact path on the original gear under the equivalent misalignment mirror those of the original gear in real conditions. Moreover, the surface parameters of the redesigned pinion result in an identical surface under a different equivalent misalignment, maintaining similar contact and dynamic performance. This proposed collaborative design approach, considering the shape and performance while accounting for uncertain misalignment errors through ease-off, greatly improves the gear transmission behavior.

1. Introduction

In a helicopter transmission system, the meshing performances of the input stage drive mechanism, known as the spiral bevel gear (SBG), greatly influences the helicopter transmission system. The uncontrollable vibration in the SBG caused by high-speed and heavy-load conditions will lead to a significant reduction in the time between overhauls (TBOs) of domestic reducers and affect the helicopter’s usage efficiency. The contact pattern on the gear surface is a crucial indicator for assessing its performance. Due to the challenging working conditions of helicopters, the bearing deformation of the gearbox body, bearing system, and shaft system in the helicopter reducer results in uncertain misalignment errors in the SBG. The uncertain misalignment errors cause the contact pattern to deviate from its ideal position, known as equivalent misalignment [1], which has a severe impact on gear performance.

Research on gear transmissions with misalignment has been conducted by numerous researchers in recent years to address the issues related to their performance. Several innovative approaches have been proposed by researchers such as Simon [2], Tang [3,4,5], Hu [6], and Guan [7] for analyzing the loaded performance of the misaligned gears. The effect of misalignment on gear performance, including transmission error (TE) and contact path (CP), has also been talked over by experts like Litvin [8], Simon [9], Liu [10], Vivet [11], Su [12], Wang [13], Fang [14], Ni [15], Kumar [16], Yang [17], Guan [18], and Song [19]. Given the increasing requirements for gear dynamic performance in practical engineering applications, extensive analysis has been carried out by experts like Yang [20], Lu [21], Wang [22], Hu [23], Zhao [24], and Xie [25] for exploring the influence of misalignment on the vibration issues through nonlinear dynamic modeling. Additionally, optimization design methods have been proposed by researchers like Simon [26,27] and Bai [28] for improving the loaded performance of gears with misalignment by lowering the impact of misalignment.

However, due to the complexity of the SBG deformation, it is difficult to accurately calculate and measure their equivalent misalignment. To solve this problem, Cai [29] and Mu [30] proposed a novel identification strategy of equivalent misalignment and set up a redesign mechanism for the SBG with misalignment according to the TE and contact pattern. Ease-off technology, which can control the TE and contact pattern well, may provide novel methods for the redesign of the misaligned SBG. Many researchers have conducted research on the design of a high-performance SBG with ease-off. Chen [31], Wang [32,33], Mu [34,35,36], and Li [37] proposed novel active topography modification approaches for the SBG according to the TE and contact pattern to improve performance, and they also verified the effectiveness of those approaches in improving loaded performance and dynamic performance. Despite numerous studies conducted by scholars on the ease-off for SBG, they did not take into account the impact of misalignment.

To guarantee the stable functioning of the SBG in extreme circumstances, the collaborative design of shape and performance for the SBG considering uncertain misalignment errors was put up. Firstly, the digital characteristics of the contact pattern were obtained through the parameterized recognition of the contact pattern. Secondly, the equivalent misalignment was obtained through the optimization model aiming at minimizing deviation in the reference point and direction angle of CP between two contact patterns, and the uncertain misalignment errors of the SBG caused by the complex support deformation were transformed into the corresponding equivalent misalignment. Thirdly, combined with optimization design theory, the ease-off technology was used for flank reconstruction to redesign the pinion surface, ensuring that its meshing performance remained consistent with the original SBG despite encountering equivalent misalignment. Simulations reveal that the percentage deviation in the maximum contact stress between the redesigned SBG and original SBG is less than 0.2%, the percentage deviation in the maximum tensile stresses on the gear and pinion between the redesigned SBG and original SBG is less than 0.4% and 0.7%, respectively, and the percentage deviation of the RMA based on the lumped mass method and the structural dynamics method between the redesigned and original SBG is less than 0.6% and 0.5%, respectively. Therefore, this research provides a legitimate approach for enhancing the meshing characteristics of the SBG in the most extreme circumstances.

2. Parametric Description of Contact Pattern

The pattern photos (Figure 1) serve as the actual physical evidence of the contact patterns on gear surfaces in actual working conditions.

These contact patterns are crucial for conducting in-depth studies on the high-performance design for the SBG. Additionally, the proposal and parameterized description of the numerical features of the contact patterns lay the foundation for the quantitative analysis of the relationship between the misalignment and contact pattern. Figure 2 illustrates the parameterized description of the contact pattern on the rotating projection of gear surface, which mainly consists of the reference point P_M (x_M, y_M) (the horizontal ordinate of the reference point is equal to the horizontal ordinate of the midpoint on the gear surface) and direction angle α.

The detailed analysis method for the parametric description of contact pattern is as follows [29,30].

Based on the machining parameters of the spiral bevel gear, the long axis direction of the contact ellipse can be calculated through the TCA. Within the contour of the tooth surface imprint, parallel lines are selected at certain intervals in the direction of the long axis of the contact ellipse. The intersection points of each parallel line and the imprint contour are calculated separately, and then a polygonal area composed of these points is used to replace the continuous contact imprint. The calculation of the imprint area and center point coordinates is converted into the calculation of the polygonal area and center of gravity.

With the flank equation, the contact ellipse’s major-axis direction can be calculated by incorporating the geometric parameters and machining parameters of the SBG. Within the contour of the contact pattern, the equidistant parallel lines along the major-axis direction are selected, and the intersections between the equidistant parallel lines and the boundary line of contact pattern are calculated; then, the polygonal area formed by these intersections represents the contact pattern.

Assuming the number of vertices of the polygonal area is N, the N-Polygon can be divided into n − 2 triangles. The formula for calculating the area of any triangle is as follows:

$$s_i=\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_{i+1}& y_{i+1}& 1\\ x_{i+2}& y_{i+2}& 1\end{array}\right|,\hspace{1em}(i=1,2,\dots ,n-2)$$

(1)

The calculation formula for the reference point P_M (x_M, y_M) is as follows:

$$\left\{\begin{array}{l}{x}_M={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n-2}(x_1+x_{i+1}+x_{i+2})s_i}}{3{\displaystyle \sum _{i=1}^{n-2}{s}_i}}}\\ y_M={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n-2}(y_1+y_{i+1}+y_{i+2})s_i}}{3{\displaystyle \sum _{i=1}^{n-2}{s}_i}}}\end{array}\right.$$

(2)

The instantaneous contact ellipse’s major-axis corresponds to the parallel line segment determined by the intersection points of the equidistant parallel lines and contact-pattern boundary line. The midpoint of the parallel line segment corresponds to the current meshing point, and the combination of these meshing points forms the CP on the gear surface. The CP can be mathematically described as a quadratic function, obtained by employing the least square method to fit the meshing points.

$$y=a_1{x}^2+a_{2}x+a_3$$

(3)

where a_i (i = 1, 2, 3) denotes the factors of the second-order CP equation.

The direction angle α should theoretically be the angle between the tangent line of the contact trace at the reference point and the pitch cone line. Here, the calculation of the direction angle α is simplified as the angle between the line connecting the entry meshing point (P_I (x_I, y_I)) and the exit meshing point (P_O (x_O, y_O)), and the pitch cone line, as follows:

$$\alpha =\arctan \sqrt{{\displaystyle \frac{y_O-y_I}{x_O-x_I}}}$$

(4)

3. Misalignment Identification

The harsh conditions experienced by helicopters cause the gearbox body, bearing system, and shaft system in the main reducers to deform when loaded, leading to the occurrence of uncertain misalignment errors in the SBG. These uncertain misalignment errors result in the actual meshing position deviating from its ideal position, which is reflected as equivalent misalignment (Figure 3). The coordinate system S_h is the meshing coordinate system of the spiral bevel gear transmission. The coordinate systems S₁ and S₂ are rigidly connected to the pinion and gear, respectively, and an auxiliary coordinate system S_c is added to describe the misalignment. φ₁ and φ₂ are the meshing angles of the pinion and gear. ΔΣ is the shaft angle error, ΔE is the center distance error, ΔP is the pinion axial error, and ΔG is the gear axial error. The misalignment of the SBG primarily includes the shaft angle error (ΔΣ), center distance error (ΔE), pinion axial error (ΔP), and gear axial error (ΔG) [29].

The CP is the collection of meshing points during entire meshing process of the SBG. For the CP of the SBG under any group of misalignments, if the reference point and direction angle remain the same, then the points on the gear surface when meshing occurs will be exactly the same. The relative curvature relationship between the gear and pinion surfaces will also remain consistent, resulting in other numerical characteristics of the contact pattern being identical. By controlling the reference point and CP direction angle, the precision identification of equivalent misalignment can be achieved.

3.1. Tooth Contact Analysis

The position and normal vectors are as follows.

$$\left\{\begin{array}{l}{\mathit{r}}_h^{(1)}={\mathit{M}}_{h1}({\phi }_1,\Delta P){\mathit{r}}_1^{(1)}({\theta }_p,{\phi }_p)\\ {\mathit{n}}_h^{(1)}={\mathit{M}}_{h1}({\phi }_1,\Delta P){\mathit{n}}_1^{(1)}({\theta }_p,{\phi }_p)\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}{\mathit{r}}_h^{(2)}={\mathit{M}}_{hc}(\Delta E,\Delta \Sigma ){\mathit{M}}_{c2}({\phi }_2,\Delta G){\mathit{r}}_2^{(2)}({\theta }_g,{\phi }_g)\\ {\mathit{n}}_h^{(2)}={\mathit{M}}_{hc}(\Delta E,\Delta \Sigma ){\mathit{M}}_{c2}({\phi }_2,\Delta G){\mathit{n}}_2^{(2)}({\theta }_g,{\phi }_g)\end{array}\right.$$

(6)

here, φ_j and θ_j (j = p, g) represent the flank parameters. φ_i (i =1, 2) is the meshing angle. M_h1, M_c2, and M_hc are the coordinate transformation matrices.

$${\mathit{M}}_{h1}=\left[\begin{array}{cccc}1& 0& 0& \Delta P\\ 0& \cos {\phi }_1& -\sin {\phi }_1& 0\\ 0& \sin {\phi }_1& \cos {\phi }_1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(7)

$${\mathit{M}}_{c2}=\left[\begin{array}{cccc}1& 0& 0& \Delta G\\ 0& -\cos {\phi }_2& -\sin {\phi }_2& 0\\ 0& \sin {\phi }_2& -\cos {\phi }_2& 0\\ 0& 0& 0& 1\end{array}\right]$$

(8)

$${\mathit{M}}_{hc}=\left[\begin{array}{cccc}\cos (\Sigma +\Delta \Sigma )& 0& \sin (\Sigma +\Delta \Sigma )& 0\\ 0& 1& 0& \Delta E\\ -\sin (\Sigma +\Delta \Sigma )& 0& \cos (\Sigma +\Delta \Sigma )& 0\\ 0& 0& 0& 1\end{array}\right]$$

(9)

Based on the meshing coordinate system shown in Figure 3, the TCA equations are obtained as follows [38]:

$$\left\{\begin{array}{c}{\mathit{r}}_h^{\left(1\right)}={\mathit{r}}_h^{\left(2\right)}\\ {\mathit{n}}_h^{\left(1\right)}={\mathit{n}}_h^{\left(2\right)}\end{array}\right.$$

(10)

In the case where a group of misalignments is known, the TCA equation consists of five independent equations and six unknown variables, and the six unknown variables are as follows: φ_j and θ_j (j = p, g) represent the flank parameters; φ_i (i =1, 2) is the meshing angle. By solving the TCA equation using the provided pinion meshing angle φ₁, the other flank parameters φ_p, θ_p, φ_g, and θ_g and the meshing angle of the gear named φ₂ are obtained. And the current meshing point is determined. The CP is composed of multiple meshing points, which are obtained by iteratively solving the TCA equation system with incremental changes in the pinion meshing angle. Thus, the reference point P′_M (x′_M, y′_M), the entry meshing point P′_I (x′_I, y′_I), and exit meshing point P′_O (x′_O, y′_O) on the CP are obtained.

3.2. Equivalent Misalignment Calculate

The positioning of the engaging on the gear surface is determined by the flank parameters, meshing angle, and misalignment. When identifying misalignment, it is sufficient to determine a set of equivalent misalignments based on the CP. This approach eliminates the need for precise calculations of the exact misalignment. Instead of searching for the “true misalignment,” the focus shifts to finding the “equivalent misalignment”. This alternative approach allows for a more accurate and efficient solution to the calculation and measurement of the misalignment in the SBG. In Figure 4, the parametric diagram of gear surface’s CP is depicted, illustrating its numerical properties such as reference point coordinates P′_M (x′_M, y′_M) and direction angle α′.

With the provided misalignment, the CP (Figure 4) marked as l_d is obtained with the TCA, and the reference point, entry meshing point, and exit meshing point on the CP l_d are

$${\mathit{P}}_j^{\prime }={\mathit{P}}_j^{\prime }\left(\mathit{D}\right)\begin{array}{ccc}& & j=I,M,O\end{array}$$

(11)

where D represents misalignment and P′_j represents a point on the CP.

Then the direction angle of the CP l_d can be obtained as follows:

$${\alpha }^{\prime }=\arctan \sqrt{{\displaystyle \frac{y_O^{\prime }-y_I^{\prime }}{x_O^{\prime }-x_I^{\prime }}}}$$

(12)

When the CP direction angle and reference point are identical, the contact pattern remains unchanged, resulting in the same performance for the SBG in both cases. Then, a mathematical model (Equation (11)) is introduced, where the objective is to minimize deviation in the CP direction angle and reference point between two CPs. The variables represent equivalent misalignment. The equivalent misalignment of the SBG under actual operating conditions is obtained by solving the optimization model using the simulated annealing-gravity search cooperative intelligent optimization algorithm.

$$\left\{\begin{array}{c}\min f_1(\mathit{D})=\sqrt{(x_M^{\prime }-x_M{)}^2+(y_M^{\prime }-y_M^{\prime }{)}^2}\\ \min f_1\left(\mathit{D}\right)=\left|{\alpha }^{\prime }-\alpha \right|\hfill \end{array}\right.$$

(13)

4. Flank Redesign Under Equivalent Misalignment

To improve the performance of the SBG under equivalent misalignment, the redesign of the pinion surface incorporating the identified misalignment is carried out. The pinion’s processing parameters are then adjusted using ease-off to achieve a SBG that exhibits a favorable meshing performance. The design of the pinion functional surface follows these basic principles: Firstly, the TE curve and CP curve, which should align with the original SBG, are planned. Secondly, the pinion’s auxiliary surface is created with the TE. Thirdly, the correction value ζ, along the instantaneous meshing line, is computed and added to pinion auxiliary surface. This process results in a pinion functional surface that exhibits a meshing performance consistent with the original SBG when positioned correctly. The position vector of the pinion functional surface is

$${\mathit{p}}_i^{*}={\mathit{p}}_i^0+{\zeta }_i{\mathit{n}}_i^0\begin{array}{cc}& (i=1,2,3,\dots ,k)\end{array}$$

(14)

where ${\mathit{p}}_i^0$ and ${\mathit{n}}_i^0$ denote the auxiliary surface’s position and unit normal vectors.

The following are the modified surface’s position and unit normal vectors:

$$\left\{\begin{array}{c}{\mathit{p}}_1={\mathit{p}}_1(s_{p,}{\theta }_p,{\phi }_p,{\phi }_1)\\ {\mathit{n}}_1={\mathit{n}}_1(s_{p,}{\theta }_p,{\phi }_p,{\phi }_1)\end{array}\right.$$

(15)

where s_p and θ_p stand for coordinate parameters, φ_p and φ₁ stand for the cradle and pinion meshing angle.

Hence, the normal deviations between the modified and functional surfaces are as follows:

$${\mathit{h}}_i\left(\mathit{d}\right)=({\mathit{p}}_i^{*}-{\mathit{p}}_i)\begin{array}{cc}{\mathit{n}}_i& (i=1,2,3,\dots ,k)\end{array}$$

(16)

$$\mathit{H}\left(\mathit{d}\right)=\left[{\mathit{h}}_1\right(\mathit{d}),{\mathit{h}}_2(\mathit{d}),{\mathit{h}}_3(\mathit{d}),\dots ,{\mathit{h}}_k(\mathit{d}){]}^T$$

(17)

where d stands for the machining parameters.

Finally, the optimization model (Equation (18)) is able to produce the redesigned pinion surface with the objective of lowering the normal deviation between the two surfaces.

$$\min f\left(\mathit{d}\right)=H\left(\mathit{d}{)}^{T}H\right(\mathit{d})$$

(18)

5. Dynamics Analysis

5.1. The Lumped Mass Method

The lumped mass method is used to model the SBG, which creates a dynamic model with multiple degrees of freedom (Figure 5a) [39]. To simplify the dynamics analysis, the dynamic model is simplified to a two-dof dynamic model (Figure 5b), which only considers the rotational degrees of freedom of the gear and pinion, and the two-dof can be simplified to the direction of the meshing line. The dynamic equation is then derived by incorporating the excitation from the meshing stiffness, loaded TE, and meshing impact (F_s) into the dynamic model as described below:

$$\left\{\begin{array}{l}{I}_1{\theta }_1=T_1-F_n{r}_1+F_s{r}_1\\ I_2{\theta }_2=-T_2+F_n{r}_2-F_s{r}_2\end{array}\right.$$

(19)

where F_n stands for the normal load, I_i (i = 1, 2) represents a moment of inertia, T_i (i = 1, 2) represent torque, and r_i (i = 1, 2) represents the equivalent base circle radius at the reference point.

However, the meshing stiffness excitation and loaded TE excitation are determined using LTCA, and the meshing impact excitation is determined using the impact theory [40,41]. Then the relative displacement along the normal direction of the meshing point between the tooth surfaces of the gear and pinion is calculated, and the vibration velocity and acceleration of the SBG are obtained by calculating the first and second derivatives of the relative displacement.

5.2. Structural Dynamics Method

The fundamental principle of structural dynamics for the SBG is depicted in Figure 6.

(1)

Establishment of meshing model for the spiral bevel gear

According to the machining principle of the spiral bevel gear, the coordinates of discrete points on the tooth surface are generated using machining parameters as variables and tooth surface equations as the basis.

The array of discrete points on the tooth surface is filled to form a point cloud. By programming according to the definition of elements in the finite element theory, an accurate mesh model of the spiral bevel gear can be obtained. Based on the meshing equation of the spiral bevel gear, the contact relationship and relative position between the gear and pinion, the coordinate transformation matrix, is obtained. The node coordinates of the gear and pinion can be transformed from the local coordinate system to the assembly coordinate system through a coordinate transformation matrix to achieve the assembly of the spiral bevel gear (Figure 7a). The mesh generation of the web plates, shafts, and support structures can draw on the generation process of gear models, generate discrete nodes through basic dimensional parameters, and then write mesh elements.

(2)

Establishment of structural dynamic model

A simplified calculation model of a 3D hexahedron element is built using the structural mechanics theory. By utilizing a reverse motion solving method, the calculation expression for the hexahedron element is obtained. The novel ball element bearing model is proposed based on the characteristics of helicopter ball bearings. The force on ball element is obtained based on the contact analysis between the inner and outer rings, ball element, and cage (Figure 7b). By leveraging established spatial contact algorithms and integrating the contact features of the high-speed SBG and bearings, the contact calculation model and spatial contact algorithm tailored for structural dynamics are developed. Drawing from the meshing damping model of the gear dynamics theory, the relative velocity between the pinion and gear is utilized for calculating the meshing damping force, which is then used to define damping and introduce a variable damping model. Furthermore, the dynamic control equations for improving the structural dynamics are obtained.

(3)

Establishment of structural dynamic model for spiral bevel gears

Combining the meshing model for the spiral bevel gear with the structural dynamic model, a structural dynamic model for the spiral bevel gear is built up. The vibration acceleration of the spiral bevel gear is determined with the structural dynamic model of the spiral bevel gear.

6. Numerical Example

To assess the success of the collaborative design of shape and performance for the SBG while accounting for uncertain misalignment errors, an analysis of a helicopter SBG is conducted as a case study. The geometric and processing parameters of the SBG can be found in

Table 1. Geometric parameters.

Parameter	Pinion	Gear
Tooth number	27	74
Gear ratio	2.74
Module (mm)	3.85
Face width (mm)	40
Shaft angle (°)	87
Pressure angle (°)	20
Machining method	Gear grinding
Hand of spiral	LH	RH
Mean cone distance(mm)	134.3815	134.3815
Working depth(mm)	6.5450	6.5450
Mean spiral angle (°)	30	30
Pitch angle (°)	19.6739	67.3261
Root angle (°)	18.8971	66.0053
Face angle (°)	20.9947	68.1029

and

Table 2. Processing parameters.

Parameter	Original		Redesign
	Gear	Pinion	Pinion_D1	Pinion_D2
Roll ratio	1.0834	2.8523	2.9429	2.9351
Initial cradle angle setting	−43.5898	−43.3032	−43.5669	−43.5210
Sliding base feed setting (mm)	−1.7409	0.3103	−0.7812	−0.7145
Increment of machine center to back (mm)	/	−2.8061	0.5638	0.3581
Vertical offset (mm)	/	−4.8480	−2.8485	−3.1281
Third-order coefficients	/	0.0135	−0.0064	−0.0044
Second-order coefficients	/	0.0438	0.0572	0.0566
Machine root angle	66.0382	18.9057	18.9057	18.9057

. The actual misalignment of the original SBG under real operating conditions was named as D_a, and the actual CP shown in Figure 8 and

Table 3. Numerical characteristics of CP.

Parameter	Da		D1		D2
	Gear	Pinion	Gear	Pinion	Gear	Pinion
The direction angle (°)	23.0272	153.0194	22.9722	153.0079	22.9903	153.0014
The horizontal coordinate of reference point (mm)	134.3815	134.1618	134.3815	134.1502	134.3815	134.1625
The vertical coordinate of reference point (mm)	0.8514	0.6049	0.8627	0.6061	0.8471	0.5909

was obtained based on the parametric description of the contact pattern.

The standard position of the original SBG was named D₀; two groups of equivalent misalignments of the original SBG in real working conditions were determined using the identification method for the equivalent misalignment (Equation (11)) and named D₁ and D₂ (

Table 4. Values of misalignment.

Parameter	D0	D1	D2
The pinion axial error ΔP (mm)	0	0.2351	0.2941
The gear axial error ΔG (mm)	0	0.0382	0.1390
The center distance error ΔE (mm)	0	−0.5511	−0.5062
The shaft angle error ΔΣ (°)	0	0.5707	0.5017

). The meshing quality of the original SBG in different groups of misalignments were analyzed using the TCA, and the corresponding CPs and TEs were obtained and listed in Figure 8 and Table 3. Compared with the actual CP, the deviations in the reference points on the CPs of the gear and pinion under the equivalent misalignment D₁ were 0.0113 mm and 0.0099 mm, respectively; the deviations in the CP direction angle of the gear and pinion under the equivalent misalignment D₁ was 0.0550° and 0.0115°, respectively. The deviations in the reference points on the CPs of the gear and pinion under the equivalent misalignment D₂ were 0.0043 mm and 0.0140 mm, respectively, the deviations in the CP direction angle of the gear and pinion under the equivalent misalignment D₂ were 0.0369° and 0.0180°, respectively. Therefore, the numerical characteristics of the CPs of the gear and pinion under the equivalent misalignment generated based on the identification method for the equivalent misalignment were very close to the actual CP of the SBG under the actual working conditions. For the CP of the SBG under any groups of misalignments, when their reference points and direction angles are the same, the meshing points on gear surface are exactly the same. The relative curvature relationship between the gear and pinion surfaces is also consistent, which will result in other numerical characteristics of the contact pattern being the same, and indicating that the equivalent misalignment has been successfully rectified with high precision.

The pinion surfaces whose CP and TE under the equivalent misalignments D₁ and D₂ were consistent with the original SBG without misalignment were created using ease-off. The pinions’ modified surfaces and their machining parameters (Table 2) were derived based on the pinion redesign method considering equivalent misalignment. Based on the gear surface equation, the flank deviation of the redesigned pinion is generated (Figure 9). It is shown that the maximum deviation of the redesigned pinion surface under two groups of equivalent misalignments is only 1 µm. As shown in Figure 10, the CPs and TEs of the redesigned SBG under the equivalent misalignments D₁ and D₂ are obtained with the TCA.

Through analysis, it was found that their TEs and CPs were basically consistent. For different groups of equivalent misalignments, the absolute positions of the gear and pinion in the meshing coordinate system are different, which results in the different machining parameters for the redesigned pinion surface, as shown in Table 2. But the relative position of the gear and pinion is the same, which determines that the micro geometry of the redesigned pinion surface is also the same, with the same gear, and the meshing performances of the SBG with the redesigned pinion are also basically the same. In theory, for completely equivalent different misalignment combinations, the machining parameters of the redesigned pinion surface using the same method will result in completely identical flank geometry, and the SBG composed of the redesigned pinion and same gear have exactly the same performance. Therefore, for the redesign of pinion under the actual working conditions, there is no need to accurately identify the uncertain misalignment errors caused by the deformation of the gear support system. There is only a need to calculate a group of equivalent misalignments and redesign the pinion surface under the group of equivalent misalignments using the collaborative design method.

The identification and equivalence analysis of the equivalent misalignment based on the TCA only reflect the no-load meshing performance of the SBG. To verify the equivalence of the equivalent misalignment obtained using the collaborative design method for the SBG considering the uncertain misalignment errors, it is necessary to analyze the contact pattern of the SBG under working conditions. And there is a strict correspondence between the contact pattern and tooth stress under the specific loads and boundary conditions. Here, when the load is 2000 Nm, the loaded performances of the original SBG under a standard position and the redesigned SBG under the groups of equivalent misalignments D₁ and D₂ are calculated with LTCA and illustrated in Figure 11 and

Table 5. LTCA results.

	D0	D1	D2
Maximum contact stress/(MPa)	1001.5444	1000.3946	1002.2740
Maximum tensile stress on gear/(MPa)	309.7505	310.9788	309.9731
Maximum tensile stress on pinion/(MPa)	202.5837	202.2701	201.2235

. In Figure 11a, the maximum contact stress of the original SBG under the standard position is 1001.5444 MPa, and the maximum contact stresses of the redesigned SBG under equivalent misalignments D₁ and D₂ are 1000.3946 MPa (0.1148%) and 1002.2740 MPa (0.0728%); therefore, the percentage deviation of the maximum contact stress between the redesigned SBG and original SBG is less than 0.2%, and the percentage deviation of the maximum contact stress of the redesigned SBG under the different groups of equivalent misalignment is only 0.1879%. Analysis of Figure 11b reveals that the maximum tensile stress on gear of the original SBG under the standard position is 309.7505 MPa, and the maximum tensile stresses on the gear of the redesigned SBG under the equivalent misalignments D₁ and D₂ are 310.9788 MPa (0.3965%) and 309.9731 MPa (0.0719%). Through analysis, it is found that the percentage deviation of the maximum tensile stresses on the gear between the redesigned SBG and original SBG is less than 0.4%, and the percentage deviation of the maximum tensile stress on the gear of the redesigned SBG under the different groups of equivalent misalignments is only 0.3234%. As shown in Figure 11c, it is found that the maximum tensile stress on the pinion of the original SBG under the standard position is 202.5837 MPa, and the maximum tensile stresses on the pinion of the redesigned SBG under the equivalent misalignments D₁ and D₂ are 202.2701 MPa (0.1548%) and 201.2235 MPa (0.6714%). Through data analysis, it is found that the percentage deviation of the maximum tensile stress on the pinion between the redesigned and original SBG is less than 0.7%, and the percentage deviation of the maximum tensile stress on the pinion of the redesigned SBG under the different groups of equivalent misalignments is only 0.5174%. In summary, the different groups of equivalent misalignments result in the different machining parameters of the redesigned pinion, but the maximum tooth stresses of the original SBG under the standard position and redesigned SBG under equivalent misalignment are basically consistent, and their tooth stress curves throughout the entire meshing process are also basically match, indicating that their loaded deformation and tooth force are basically the same, resulting in an identical contact pattern on the gear surface. So, the collaborative design method can obtain a high-performance SBG whose meshing performance considering the uncertainty misalignment errors is consistent with the original SBG in the perfect environment. That is to say, this method can eliminate the impact of the uncertain misalignment errors caused by loaded deformation of the gear support system.

To substantiate the effectiveness of the collaborative design method in reducing the impact of uncertainty errors caused by the loaded deformation of the gear support system on the meshing performance of the SBG, it is necessary to compare and analyze the dynamic performance of the SBG under working conditions. Here, when the load is 2000 Nm and the input speed is 10,000 rpm, the vibration acceleration of the original SBG under the standard position and the redesigned SBG under equivalent misalignment are calculated with the help of the lumped mass method and the structural dynamics theory, and presented in Figure 12 and

Table 6. Dynamic performance.

	Equivalent Misalignment	the RMA/(m/s2)
Lumped mass method	D0	260.6903
	D1	262.5036
	D2	259.8249
Structural dynamics theory	D0	250.3621
	D1	251.1382
	D2	249.8363

.

In Figure 12a, according to the lumped mass method, the root mean square of vibration acceleration (RMA) of the original SBG under the standard position is 260.6903 m/s², and the RMA of the redesigned SBG under the equivalent misalignments D₁ and D₂ are 262.5036 MPa (0.5805%) and 259.8249 MPa (0.3320%); therefore, the percentage deviation of the RMA between the redesigned and original SBG is less than 0.6%, and the percentage deviation of the RMA of the redesigned SBG under the different groups of equivalent misalignments is only 1.0216%. As seen from Figure 12b, based on the method of structural dynamics, the RMA of the original SBG under the standard position is 250.3621 MPa, and the RMA of the redesigned SBG under the equivalent misalignments D₁ and D₂ are 251.1382 MPa (0.3100%) and 249.8363 MPa (0.2100%). Through data analysis, it is found that the percentage deviation of the RMA between the redesigned SBG and original SBG is less than 0.5%, and the percentage deviation of the RMA of the redesigned SBG under the different groups of equivalent misalignments is only 0.5184%. In summary, the different groups of equivalent misalignments result in different machining parameters of the surface, but the RMA of the original SBG under the standard position and the redesigned SBG under equivalent misalignment are basically consistent. Although there is a slight deviation between the structural dynamics results and the lumped mass method results, the fluctuation rules of the vibration acceleration curves obtained by the two methods also basically match, indicating that their dynamic performances are basically the same. Therefore, the collaborative design method can obtain a high-performance SBG whose meshing performance considering the uncertainty errors is consistent with the original SBG under ideal working conditions. That is to say, this method can eliminate the impact of uncertainty errors caused by loaded deformation on the transmission performance of the SBG.

7. Conclusions

To ensure the stable operation of an aviation transmission system and enhance the meshing quality of the SBG in real-world conditions, the collaborative design of shape and performance for the SBG considering the uncertain misalignment errors caused by the loaded deformation of the gear support system was put up. The simulation analysis led to the following insightful conclusions.

(1)

With the digital feature extraction of the contact pattern of the SBG in real-world scenarios, the equivalent misalignment can be identified using the optimal model aimed at reducing the deviation in reference point and direction angle of the CP. The reference point coordinate and the direction angle of the CP named l_d are very similar to those of the CP named l_a. For the CP of the SBG under any groups of misalignments, when their reference point and direction angle are the same, the meshing points on gear surface are exactly the same. The relative curvature relationship between the gear and pinion surfaces is also consistent, which will result in the other numerical characteristics of the contact pattern being the same, and indicating that the equivalent misalignment has achieved a high level of solving accuracy.

(2)

According to ease-off, the pinion working surfaces with good performances are redesigned with different groups of equivalent misalignments. And the maximum deviation of the redesigned pinion surfaces with different groups of equivalent misalignments is only 1 µm. For different equivalent misalignments, the absolute positions of the gear pair in the meshing coordinate system are different, which results in different machining parameters for the redesigned pinion. But the relative position of the gear pair is basically the same, which determines that the redesigned pinion surface is also basically the same.

(3)

Using the TCA and LTCA, the meshing performances of the redesigned SBG under equivalent misalignment are compared with the original SBG in real-world scenarios. The TCA results showed that the CP and TE of the redesigned SBG under equivalent misalignment are consistent with the original SBG in real-world scenarios. LTCA results showed that the percentage deviation in the maximum contact stress between the redesigned SBG and original SBG is less than 0.2%, the percentage deviation of the maximum tensile stresses on gear between the redesigned SBG and original SBG is less than 0.4%, and the percentage deviation in the maximum tensile stress on the pinion between the redesigned and original SBG is less than 0.7%. In other words, the maximum contact stresses and maximum tensile stresses of the redesigned SBG under equivalent misalignment are consistent with the original SBG in real-world scenarios, and the tooth stress curves throughout the entire meshing process also basically match, indicating that the loaded deformation and tooth force of the SBG are basically the same. This indicates that the redesigned gear has a good meshing performance, and the equivalence of equivalent misalignment is also verified.

(4)

Based on the lumped mass method and structural dynamics theory, the dynamic performances of the redesigned SBG under the different groups of equivalent misalignments are analyzed and compared with the original SBG in real-world scenarios. The dynamic analysis results showed that the percentage deviation in the RMA with the lumped mass method between the redesigned and original SBG is less than 1%, and the percentage deviation in the RMA with the structural dynamics method between the redesigned SBG and original SBG is less than 0.5%. That is to say, the RMA of the original SBG in real-world scenarios and the redesigned SBG under the equivalent misalignment are basically consistent. Although there is a slight deviation between the structural dynamics results and the lumped mass method results, the fluctuation rules of the vibration acceleration curves obtained by the two methods also basically match, indicating that the dynamic performance of the SBG is basically the same. This indicates that the redesigned gear has a good meshing performance, and the equivalence of equivalent misalignment is also verified.