Static load sharing performance of non-orthogonal and offset helical face gear dual-branch drivetrain

Abstract

To research the static load sharing performance of non-orthogonal offset helical (NOH) face gear dual-branch drivetrain, the non-orthogonal offset layout forms of the system are explored, and the meshing phase differences of split-stage and converge-stage are analyzed. Based on the torque balance condition, deformation coordination condition, and static balance condition of the system, a static torque distribution model of the system was generated, and the static load sharing coefficients (SLSCs) were obtained results. The influence of assembly errors, support stiffness, torsional stiffness, meshing stiffness, and input load on the static load sharing performance was analyzed. The results indicate that assembly errors have a strong influence on the SLSCs, with the shaft angle errors exerting the strongest influence on the SLSCs; When there are assembly errors, the impact of input load and torsional stiffness on the static load sharing performance is greater than without assembly error. To increase the load sharing performance of the system, the meshing phase differences should be eliminated, and the offset layout form should be adopted in the dual-branch transmission system. In addition, reducing the support stiffness of the input helical gear and the fluctuation amplitude of meshing stiffness can also improve the load sharing performance.

Introduction

Helical face gear drives ¹ are an advanced new type of angular gear transmission. It is composed of a helical gear and a face gear ² that can mesh with each other. Face gear drives have unique advantages, such as their large tooth surface overlap, strong load-bearing capacity, high single-stage transmission ratio, and compact structure.^3–5 Therefore, it is more suitable for branch drivetrains, especially for aviation drivetrains that are lightweight and work in a narrow space. ⁶ Helical face gear has larger tooth surface overlap, stronger load-bearing capacity, and more stable transmission performance when compared to spur face gear. ⁷ In addition, the structure of offset non-orthogonal layout forms is more compact, and the branch drives of face gear pairs have excellent load sharing performance in helicopters. ⁸ Therefore, research on the application of NOH face gear branch drivetrains in helicopters has attracted much attention.

Since Litvin first proposed a spur face gear branch drivetrain, and the finite element model was created. Litvin et al. ⁹ also calculated the torque distribution using the finite element method, many scholars or institutions have proposed a series of different types of face gear branch drivetrains and conducted in-depth research on their load sharing characteristic. Pias and Turro ¹⁰ designed a face gear branch drivetrain with multiple input shaft, which has an independent load sharing mechanism. The torque is first input by a pinion and divided into two face gears installed concentrically and facing each other. Then, the output shaft, which is connected with the bottom face gear, receives the converged torque through the idler gears. Gmirya and Kish ¹¹ designed a branch drive of face gear pairs similar to the one designed by Pias, and it can achieve multi-path torque branch transmission. The torque is first split through the cylindrical gear pairs, then transmitted to the face gear pairs composed of four concentric installed face gears and two pinions, achieving secondary split, and finally converged through the idler gears. Afterwards, Gmirya ¹² has also designed a branch transmission device that provides multi-path and three-stage power gear transmission. In this system, torque is delivered from the high-speed input shaft to the first stage face gear pairs, then divided by the second stage spur gear pairs, and finally converged by the third stage herringbone gear pairs to the low-speed output shaft, and achieves even load through the floating pinion.

In recent years, Kawasaki et al. ¹³ proposed a method for designing helical pinions by defining a reference point between helical pinions and face gears. The influence of helix angle on surface contact was analyzed, providing a design method for the application of helical face gear transmission instead of bevel gear and hypoid gear transmission. Zschippang et al. ¹⁴ proposed a method for calculating load sharing, transmission performance, and contact pressure of face gear transmission. This method combines the analysis of the contact path for rigid gear meshing with finite element analysis of load related compliance, greatly improving computational efficiency. In recent years, Zhao et al. ¹⁵ have studied the branch drivetrain of concentric face gears, established its static load sharing calculation equations, and revealed the reasons for the uneven load of the system. Dong et al.^16–18 researched the branch drivetrain of concentric face gears and the four-branch drivetrain of face gears, and analyzed the effects of differences, stiffness, and other factors on the static load sharing performance of two systems. He also conducted research on the twin rotors concentric face gear power-split drivetrain ¹⁹ and proposed optimization design methods for its load sharing and lightweight design. Gong et al. ²⁰ considered the elastic support conditions of the concentric face gear branch drivetrain and established its quasi-static analysis model to study its uniform load performance and contact characteristics. Mo et al. ²¹ proposed a double input face gear split-parallel transmission system and explored its dynamic vibration characteristics. In summary, the structural types of face gear branch drivetrains are diverse and varied. A series of related research achievements have greatly promoted the innovate of face gear branch transmission in the aviation field, especially in the field of helicopter power transmission. However, the aforementioned research mostly focusses on spur face gears, and research on helical face gears is very rare, with less consideration given to non-orthogonal and offset situations. The dual-branch transmission of face gears is the basic configuration of face gear branch transmission. Therefore, studying the load sharing performance of NOH face gear dual-branch drivetrain can provide theoretical reference for its application in other types of face gear branch drivetrains.

The load sharing performance of NOH face gear dual-branch drivetrain is studied in this article. Firstly, the dual-branch layout forms of NOH face gears, and meshing phase differences are analyzed. Secondly, considering factors, such as assembly errors, loads, support stiffness, torsional stiffness, and meshing phase stiffness, this study establishes a static torque distribution equations of NOH face gear dual-branch drivetrain. Finally, the influences of these factors on the static load sharing performance of this drivetrain are analyzed.

The structure of the drivetrain and the non-orthogonal offset layout form

The structure of the drivetrain

The NOH face gear dual-branch drivetrain mainly includes the NOH face gear pairs in the split-stage and the helical gear pairs in the converge-stage. And its structural diagram is shown as Figure 1. The torque enters the system through the input shaft, and is divided into two branches by the helical gear 1 to transfer to the NOH face gears 2 and 3. The elastic shaft 24 connects the NOH face gear 2 and the helical gear 4, the elastic shaft 35 connects the NOH face gear 3 and the helical gear 5, and the helical gear 4 and 5 converge and transfer the torque to the helical gear 6, which connects the output shaft.

Figure 1.

The structural diagram of NOH face gear dual-branch drivetrain.

The non-orthogonal offset layout form

The layout form of NOH face gear pairs is shown in Figure 1, which means that the offset direction is opposite and the offset angle is equal, and the shaft angle is equal and acute. Among them, the definition of the offset angle $\eta$ is as follows:

$$\eta =\arctan (E/L_0)$$

(1)

Where $E$ is the offset distance, and $L_0$ is the pitch diameter of the NOH face gear tooth width.

Due to the identical tooth surfaces of NOH face gears 2 and 3, the offset angles between NOH face gear pairs 12 and 13 are equal. The tooth surfaces geometry of the helical gear is asymmetric. So, the opposite offset direction are necessary to ensure that the meshing tooth surfaces of two NOH face gears are the same and have the same meshing characteristics, which is conducive to achieving equal load distribution between two branches. When the shaft angle is greater than 90°, the space of the input helical gear axis is easily limited. However, when the shaft angle is less than 90°, there is a large variation space in the input helical gear axis, which is conducive to adjusting input shaft direction flexibly. By adopting a non-orthogonal and offset layout form, not only can both NOH face gears have double span support conditions, improving support stiffness and system stability, but also can form almost any input and output angle and any configuration of a branch drivetrain. This can provide rich configuration options for aviation drivetrains that are compact and work in a narrow space, especially for the drivetrains in helicopters.

The force on NOH face gear

The NOH face gears have more complex forces than that of non-orthogonal and offset spur face gears, as shown in Figure 2. The angle $\alpha$ is the end face pressure angle of helical gear 1, the angle $\gamma$ is the shaft misalignment angle, and the angle $\beta$ is the base circle helix angle of helical gear 1.

Figure 2.

The force on NOH face gear.

In the coordinate system $O_r-x_r{y}_r{z}_r$ , the $x_r$ axis follows the direction of the NOH face gear tooth width, the $y_r$ axis follows the direction of the NOH face gear tooth thickness, and the $z_r$ axis follows the direction of the NOH face gear axis; the coordinate system $O_g-x_g{y}_g{z}_g$ is an auxiliary coordinate system. The force $F_n$ is the normal meshing force, and it is defined as the forward direction from the active helical gear to the passive face gear.

Firstly, decompose it into force $F_a$ parallel to the cone surface, force $F_c$ perpendicular to the cone surface, and force $F_b$ tangential to the cone surface, as follows:

$$\{\begin{array}{c}{F}_a=-F_n\cos \alpha \sin \eta \hfill \\ \multicolumn{1}{c}{F_b=F_n\sin \alpha }\\ \multicolumn{1}{c}{F_c=F_n\cos \alpha \cos \eta }\end{array}$$

(2)

Secondly, decomposing $F_a$ , $F_b$ , and $F_c$ in the coordinate system $O_r-x_r{y}_r{z}_r$ can be expressed as:

$$\{\begin{array}{c}{F}_{\mathrm{xr}}=F_a\sin \gamma -F_b\cos \gamma \hfill \\ \multicolumn{1}{c}{F_{\mathrm{yr}}=F_c}\\ \multicolumn{1}{c}{F_{\mathrm{zr}}=F_a\cos \gamma +F_b\sin \gamma }\end{array}$$

(3)

Finally, these forces are transformed into the coordinate system $O_g-x_g{y}_g{z}_g$ , as follows:

$$\left[\begin{array}{c}{F}_{\mathit{xg}}\hfill \\ \multicolumn{1}{c}{F_{\mathit{yg}}}\\ \multicolumn{1}{c}{F_{\mathit{zg}}}\end{array}\right]=\left[\begin{array}{ccc}\cos \beta \hfill & \sin \beta \hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \beta }& \cos \beta \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{c}{F}_{\mathit{xr}}\hfill \\ \multicolumn{1}{c}{F_{\mathit{yr}}}\\ \multicolumn{1}{c}{F_{\mathit{zr}}}\end{array}\right]$$

(4)

Static model and equations

Static model

Simplify the static modeling of the system as follows: the support of the gear and the teeth are treated as springs, and the wheel itself is treated as a rigid body; Not considering phenomena such as gear disengagement, reverse impact, torsion, and vibration between gears; Neglecting frictional force, inertial force, gravity, and thermal expansion force. The static model of NOH face gear dual-branch drivetrain is shown as Figure 3.

Figure 3.

Static model of NOH face gear dual-branch drivetrain.

In the coordinate system $O_p-x_p{y}_p{z}_p$ , the $x_p$ axis follows the axis direction of helical gear 1, the $y_p$ axis follows the tooth thickness direction of helical gear 1, and the $z_p$ axis follows the tooth height direction of helical gear 1.

$T_1$ and $T_6$ are the input and output torques of the system, respectively; $K_{x1}$ , $K_{y1}$ , and $K_{z1}$ are the equivalent support stiffness of helical gear 1 at the support points in the $x_p$ , $y_p$ , and $z_p$ directions, respectively; $K_{x2}$ , $K_{y2}$ , and $K_{z2}$ are the equivalent support stiffness of NOH face gear 2 at the support points in the $x_g$ , $y_g$ , and $z_g$ directions, respectively; $K_{x3}$ , $K_{y3}$ , and $K_{z3}$ are the equivalent support stiffness of NOH face gear 3 at the support points in the $x_g$ , $y_g$ , and $z_g$ directions, respectively. Similarly, $K_{x4}$ and $K_{y4}$ , $K_{x5}$ and $K_{y5}$ , $K_{x6}$ and $K_{y6}$ are the equivalent support stiffness of helical gear 4, helical gear 5, and helical gear 6 at the support points in the $x_g$ and $y_g$ directions, respectively; $K_{12}$ , $K_{13}$ , $K_{46}$ , and $K_{56}$ are the meshing stiffness of the NOH face gear pair 12, 13, and the helical gear pair 46, 56, respectively; $K_{24}$ and $K_{35}$ are the torsional stiffness of intermediate shafts 24 and 35.

Meshing phase differences

It is necessary to determine the meshing phase differences of each gear pair. The arc length between the meshing lines of two face gear pairs is measured on the helical gear 1. This arc length corresponds to a center angle which is defined as the meshing phase difference in the Kahraman, ²² as shown in Figure 4.

Figure 4.

The angular position relationship between two NOH face gear pairs under elastic support.

Let the meshing phase differences of helical gear 1 be ${\lambda }_1$ . Let the axial direction of helical gear 1 as the $x_p$ axis, the tooth thickness direction as the $y_p$ axis, and the tooth height direction as the $z_p$ axis in the coordinate system $S_p$ .

When the meshing point between helical gear 1 and NOH face gear 2 is determined, the relative meshing point between helical gear 1 and NOH face gear 3 depends on the magnitude of the meshing phase differences ${\lambda }_1$ . If ${\lambda }_1$ is an integer multiple of the meshing cycle, the relative meshing points of helical gear 1 and NOH face gear 3 remains unchanged. Conversely, if ${\lambda }_1$ is not an integer multiple of the meshing cycle, the relative meshing points of helical gear 1 and NOH face gear 3 changes on the meshing line. Due to one meshing cycle corresponds to one tooth pitch angle of a gear, the relationship between the meshing phase differences and the meshing cycle can be expressed as

$$A_1+a_1=\frac{{\lambda }_1}{2\pi /z_1}$$

(5)

Where $A_1$ is an integer corresponding to the multiple of the meshing cycles of ${\lambda }_1$ ; $a_1$ is the decimal of the multiple of the meshing cycles corresponding to ${\lambda }_1$ ; $z_1$ is the number of teeth of helical gear 1.

Let the meshing phase differences of helical gear 6 be ${\lambda }_2$ , as shown in Figure 5. Let the NOH face gear tooth width direction as the $x_g$ axis, the tooth thickness direction as the $y_g$ axis, and the axis direction as the $z_g$ axis in the coordinate system $S_g$ .

Figure 5.

The angular position relationship between two helical gear pairs under elastic support.

Similarly, the relationship between the meshing phase differences and the meshing cycles can be expressed as

$$A_2+a_2=\frac{{\lambda }_2}{2\pi /z_6}$$

(6)

Where $A_2$ is the integer of ${\lambda }_2$ for the multiple of meshing cycles; $a_2$ is the decimal of the multiple of the meshing cycles of ${\lambda }_2$ ; $z_6$ is the number of teeth on the gear 6.

Assembly errors and elastic deformation transform along the meshing line direction

The relationship between the angle and position of two NOH face gear pairs under elastic support is shown in Figure 3. ${\zeta }_{12}$ and ${\zeta }_{13}$ are the angles between the meshing lines of two NOH face gear pairs and the positive direction of the $z_p$ axis, respectively. The expression is:

$$\{\begin{array}{c}{\zeta }_{12}=1.5\pi -{\alpha }_{t1}\hfill \\ \multicolumn{1}{c}{{\zeta }_{13}=0.5\pi -{\alpha }_{t1}}\end{array}$$

(7)

Where ${\alpha }_{t1}$ is the end face pressure angle of helical gear 1.

Let $x_1$ , $y_1$ , and $z_1$ represent the elastic deformation of helical gear 1 supported in the $x_p$ , $y_p$ , and $z_p$ directions, respectively. $x_2$ and $x_3$ , $y_2$ and $y_3$ , $z_2$ and $z_3$ represent the elastic deformation of NOH face gear 2 and NOH face gear 3 supported in the $x_g$ , $y_g$ , and $z_g$ directions, respectively. $\mathrm{\Delta }{E}_m$ , $\mathrm{\Delta }{q}_m$ , and $\mathrm{\Delta }{\gamma }_m$ represent the offset errors of the $m$ -th ( $m=1,2,3$ ) gear in the $y_p$ direction, the axial displacement errors in the $z_g$ direction, and the shaft angle errors relative to the axis direction in two NOH face gear pairs. Translate the elastic deformation and assembly errors of two NOH face gear pairs into relative displacements $\mathrm{\Delta }{L}_{12}$ and $\mathrm{\Delta }{L}_{13}$ along their meshing lines, specifically:

$$\{\begin{array}{c}\mathrm{\Delta }{L}_{12}=-x_1\sin \beta \sin {\zeta }_{12}+[x_2(\sin \gamma \sin \eta \sin {\zeta }_{12}+\cos \gamma \cos {\zeta }_{12})\hfill \\ +(y_1-y_2\cos \eta +\mathrm{\Delta }{E}_1-\mathrm{\Delta }{E}_2)\sin {\zeta }_{12}+(z_1+L_{m2}\mathrm{\Delta }{\gamma }_1-L_{m2}\mathrm{\Delta }{\gamma }_2)\hfill \\ \cos {\zeta }_{12}-(z_2+\mathrm{\Delta }{q}_1-\mathrm{\Delta }{q}_2)(\sin \gamma \cos {\zeta }_{12}-\cos \gamma \sin \eta \sin {\zeta }_{12})]\cos \beta \hfill \\ \mathrm{\Delta }{L}_{13}=-x_1\sin \beta \sin {\zeta }_{13}+[x_3(\sin \gamma \sin \eta \sin {\zeta }_{13}+\cos \gamma \cos {\zeta }_{13})\hfill \\ +(y_1-y_3\cos \eta +\mathrm{\Delta }{E}_1-\mathrm{\Delta }{E}_3)\sin {\zeta }_{13}+(z_1+L_{m3}\mathrm{\Delta }{\gamma }_1-L_{m3}\mathrm{\Delta }{\gamma }_3)\hfill \\ \cos {\zeta }_{13}-(z_3+\mathrm{\Delta }{q}_1-\mathrm{\Delta }{q}_3)(\sin \gamma \cos {\zeta }_{13}-\cos \gamma \sin \eta \sin {\zeta }_{13})]\cos \beta \hfill \end{array}$$

(8)

Where $L_2$ and $L_3$ are the middle radius of two NOH face gears.

The angle position relationship between two pairs of helical gear pairs under elastic support is shown in Figure 4. ${\zeta }_{46}$ and ${\zeta }_{56}$ are the angles between the meshing lines of two helical gear pairs and the positive direction of the $x_g$ axis, respectively. The expression is:

$$\{\begin{array}{c}{\zeta }_{46}=\pi +0.5\Omega -{\alpha }_{t2}\hfill \\ \multicolumn{1}{c}{{\zeta }_{56}=\pi -0.5\Omega -{\alpha }_{t2}}\end{array}$$

(9)

Where $\Omega$ is the angle between the centerline connecting $O_4{O}_6$ and $O_5{O}_6$ during the initial assembly of the system, and ${\alpha }_{t2}$ is the end face pressure angle of the helical gear 6.

Let $x_n$ and $y_n$ ( $n=4,5,6$ ) represent the elastic deformation of each helical gear support in the $x_g$ and $y_g$ directions. Similarly, $\mathrm{\Delta }{x}_n$ and $\mathrm{\Delta }{y}_n$ represent the assembly errors of the $n$ -th helical gear in the $x_g$ and $y_g$ directions. Translate the elastic deformation and assembly errors of two helical gear pairs into relative displacements $\mathrm{\Delta }{L}_{46}$ and $\mathrm{\Delta }{L}_{56}$ along their meshing lines, specifically:

$$\{\begin{array}{c}\mathrm{\Delta }{L}_{46}=[(x_4-x_6+\mathrm{\Delta }{x}_4-\mathrm{\Delta }{x}_6)\cos {\zeta }_{46}\hfill \\ +(y_4-y_6+\mathrm{\Delta }{y}_4-\mathrm{\Delta }{y}_6)\sin {\zeta }_{46}]\cos {\beta }_b\hfill \\ \mathrm{\Delta }{L}_{56}=[(x_5-x_6+\mathrm{\Delta }{x}_5-\mathrm{\Delta }{x}_6)\cos {\zeta }_{56}\hfill \\ +(y_5-y_6+\mathrm{\Delta }{y}_5-\mathrm{\Delta }{y}_6)\sin {\zeta }_{56}]\cos {\beta }_b\hfill \end{array}$$

(10)

Where ${\beta }_b$ is the base helix angle of helical gear 6.

Torque distribution under elastic support

The torque transmission relationship of the system and the torque angle relationship of each gear pair are shown in Figure 6. ${\phi }_i$ is the meshing angle of gear $i$ . ( $i=1,2,\cdots ,6$ ), $\mathrm{\Delta }{\phi }_{12}$ , $\mathrm{\Delta }{\phi }_{13}$ , $\mathrm{\Delta }{\phi }_{46}$ , and $\mathrm{\Delta }{\phi }_{56}$ are the torque angle deformation of gear pairs 12, 13, 46, and 56, respectively. $\mathrm{\Delta }{\phi }_{24}$ and $\mathrm{\Delta }{\phi }_{35}$ are the torque angle deformation of shafts 24 and 35, respectively.

Figure 6.

System torque transmission relationship and torsion angle relationship.

The input torque of helical gear 1 is $T_1$ , which is split into $T_{12}$ and $T_{13}$ by NOH face gears 2 and 3. The torque acting on NOH face gears 2 and 3 is $T_{21}$ and $T_{31}$ ; The torques acting on both ends of shafts 24 and 35 are $T_{24}$ , $T_{42}$ , $T_{35}$ , $T_{53}$ , respectively; the torques acting on helical gears 4 and 5 are $T_{46}$ and $T_{56}$ , and the torques transmitted to helical gears 6 are $T_{64}$ and $T_{65}$ . The output torque of helical gears 6 is $T_6$ .

According to the torque transmission relationship of the system, the torque balance condition can be obtained as

$$\{\begin{array}{c}{T}_1-T_{12}-T_{13}=0\hfill \\ T_{12}-\frac{r_{b1}}{r_{b2}}{T}_{46}=0\hfill \\ T_{13}-\frac{r_{b1}}{r_{b3}}{T}_{56}=0\hfill \end{array}$$

(11)

Where $r_{b1}$ is the base circle radius of helical gear 1; $r_{b2}$ and $r_{b3}$ are the equivalent base circle radius of NOH face gears 2 and 3, respectively.

According to the system torsion angle relationship, the torsion angle deformation coordination condition is shown as follows:

$$\begin{array}{c}\frac{T_{12}}{K_{12}{r}_{p1}{r}_{b1}\cos {\alpha }_{t1}{\cos }^2\beta }-\frac{\mathrm{\Delta }{L}_{12}}{r_{b1}\cos \beta }+\frac{I_{12}{T}_{46}}{K_{24}}\hfill \\ +\frac{I_{12}{T}_{46}}{K_{46}{r}_{p4}{r}_{b4}\cos {\alpha }_{t2}{\cos }^2{\beta }_b}-\frac{I_{12}\mathrm{\Delta }{L}_{46}}{r_{b4}\cos {\beta }_b}\hfill \\ =\frac{T_{13}}{K_{13}{r}_{p1}{r}_{b1}\cos {\alpha }_{t1}{\cos }^2\beta }-\frac{\mathrm{\Delta }{L}_{13}}{r_{b1}\cos \beta }+\frac{I_{13}{T}_{56}}{K_{35}}\hfill \\ +\frac{I_{13}{T}_{56}}{K_{56}{r}_{p5}{r}_{b5}\cos {\alpha }_{t2}{\cos }^2{\beta }_b}-\frac{I_{13}\mathrm{\Delta }{L}_{56}}{r_{b5}\cos {\beta }_b}\hfill \end{array}$$

(12)

Where $r_{p1}$ is the pitch radius of helical gear 1; $r_{b4}$ and $r_{p4}$ are the base circle and pitch radius of helical gear 4; $r_{b5}$ and $r_{p5}$ are the base circle and pitch radius of helical gear 5; $I_{12}$ and $I_{13}$ are the transmission ratios of two NOH face gear pairs.

The static balance condition of the system under elastic support is shown as:

$$\{\begin{array}{c}\frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\tan \beta +\frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\tan \beta =K_{x1}{x}_1\hfill \\ -\frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{12}-\frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{13}=K_{y1}{y}_1\hfill \\ -\frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\cos {\zeta }_{12}-\frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\cos {\zeta }_{13}=K_{z1}{z}_1\hfill \\ -\frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{12}\sin \eta \sin \gamma -\frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\cos {\zeta }_{12}\cos \gamma =K_{x2}{x}_2\hfill \\ \frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{12}\cos \eta =K_{y2}{y}_2\hfill \\ \frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\cos {\zeta }_{12}\sin \gamma -\frac{T_{12}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{12}\sin \eta \cos \gamma =K_{z2}{z}_2\hfill \\ -\frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{13}\sin \eta \sin \gamma -\frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\cos {\zeta }_{13}\cos \gamma =K_{x3}{x}_3\hfill \\ \frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{13}\cos \eta =K_{y3}{y}_3\hfill \\ \frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\cos {\zeta }_{13}\sin \gamma -\frac{T_{13}}{r_{p1}\cos {\alpha }_{t1}}\sin {\zeta }_{13}\sin \eta \cos \gamma =K_{z3}{z}_3\hfill \\ -\frac{T_{46}}{r_{p4}\cos {\alpha }_{t2}}\cos {\zeta }_{46}=K_{x4}{x}_4\hfill \\ -\frac{T_{46}}{r_{p4}\cos {\alpha }_{t2}}\sin {\zeta }_{46}=K_{y4}{y}_4\hfill \\ -\frac{T_{56}}{r_{p5}\cos {\alpha }_{t2}}\cos {\zeta }_{56}=K_{x5}{x}_5\hfill \\ -\frac{T_{56}}{r_{p5}\cos {\alpha }_{t2}}\sin {\zeta }_{56}=K_{y5}{y}_5\hfill \\ \frac{T_{46}}{r_{p4}\cos {\alpha }_{t2}}\cos {\zeta }_{46}+\frac{T_{56}}{r_{p5}\cos {\alpha }_{t2}}\cos {\zeta }_{56}=K_{x6}{x}_6\hfill \\ \frac{T_{46}}{r_{p4}\cos {\alpha }_{t2}}\sin {\zeta }_{46}+\frac{T_{56}}{r_{p5}\cos {\alpha }_{t2}}\sin {\zeta }_{56}=K_{y6}{y}_6\hfill \end{array}$$

(13)

The torque balance condition, torsion angle deformation coordination condition, and static balance condition of the joint system are used to obtain the mathematical model of torque distribution of the system under elastic support.

Calculation method of the SLSC

At present, there is no unified standard definition for the average load coefficient of multi-stage gear branch drivetrains, and some research papers on face gear branch transmission^23,24 also have different definitions of the average load coefficient. According to Fu et al., ²⁵ the average load coefficient of a branch drivetrain is defined as the ratio of the actual distributed torque of each gear pair to the theoretical distributed torque of that branch.

Based on the torque distribution mathematical model of the system under elastic support, the torque distribution of each gear pair can be solved. The SLSCs $j{z}_{\mathrm{jk}}$ ( $j=1,k=2,3$ and $j=4,5,k=6$ ) of the gear pair vary with time $t$ , and their expressions are shown as:

$$\{\begin{array}{c}j{z}_{12}(t)=\frac{2T_{12}(t)}{T_1}\hfill \\ j{z}_{13}(t)=\frac{2T_{13}(t)}{T_1}\hfill \\ j{z}_{46}(t)=\frac{2T_{46}(t)r_{b1}}{T_1{r}_{b2}}\hfill \\ j{z}_{56}(t)=\frac{2T_{56}(t)r_{b1}}{T_1{r}_{b3}}\hfill \end{array}$$

(14)

Within one meshing cycle, the maximum SLSC between two NOH face gear pairs is defined as the SLSC in the split-stage. The maximum SLSC between two pairs of helical gear pairs is defined as the SLSC in the converge-stage, that is,

$$\{\begin{array}{c}{j}_1=max(j_{12}(t),j_{13}(t))\hfill \\ j_2=max(j_{46}(t),j_{56}(t))\hfill \end{array}$$

(15)

Where $j_1$ is the SLSC of the split-stage; $j_2$ is the SLSC of the converge-stage.

Since face gear 2 and face gear 3 are identical, $r_{b2}=r_{b3}$ , as can be seen from equations (11) to (14):

$$\{\begin{array}{c}j{z}_{12}(t)=j{z}_{46}(t)\hfill \\ \multicolumn{1}{c}{j{z}_{13}(t)=j{z}_{56}(t)}\end{array}$$

(16)

From equations (15) to (16), it can be seen that $j_1=j_2$ , and let $j_1=j_2=j$ , $j$ is collectively referred to as the SLSC.

The SLSC can characterize the non-uniformity of load distribution in each branch of the system. The larger the SLSC, the more uneven the load distribution in each branch of the system. The SLSC more accurately reflects the causes of uneven load in the branch drivetrain, such as assembly errors, support stiffness, torsional stiffness, meshing stiffness, and other factors affecting the load sharing performance.

Static load sharing performance of the system

Table 1 displays the main simulation parameters and operating conditions of the system. From equation (16), it can be seen that the instantaneous load sharing coefficients of split-stage and converge-stage are the same. Therefore, in the example, the SLSC of the split-stage is taken as an example to analyze the load sharing performance of the system.

Table 1.

Main simulation parameters and operating conditions of the system.

Parameters	Symbol (unit)	Numerical value	Parameters	Symbol (unit)	Numerical value
Module of the NOH face gear pairs	$m_1\left(\mathrm{mm}\right)$	4	Radial support stiffness of face gear 3	$K_{x3}/K_{y3}(\mathrm{N}·{\mathrm{m}}^{-1})$	1.62 × 109
Pressure angle of the NOH face gear pairs	${\alpha }_{n1}(°)$	25	Module of the helical gear pairs	$m_2\left(\mathrm{mm}\right)$	4
Helix angle of the NOH face gear pairs	$\beta (°)$	10	Pressure angle of the helical gear pairs	${\alpha }_{n2}(°)$	20
Shaft angle of the NOH face gear pairs	$\gamma (°)$	80	Helix angle of the helical gear pairs	${\beta }_b(°)$	12
Offset angle of the NOH face gear pairs	$\eta (°)$	3.1364	Teeth number of helical gears 4 and 5	$z_4/z_5$	65
Teeth number of helical gear 1	$z_1$	27	Teeth number of helical gear 6	$z_6$	27
Teeth number of face gears 2 and 3	$z_2/z_3$	86	Tooth width of the helical gear pairs	$B_2\left(\mathrm{mm}\right)$	25
Tooth width of helical gear 1	$B\left(\mathrm{mm}\right)$	25	Radial support stiffness of helical gear 4	$K_{x4}/K_{y4}(\mathrm{N}·{\mathrm{m}}^{-1})$	9.05 × 108
Middle radius of face gears 2 and 3	$L_{m2}/L_{m3}\left(\mathrm{mm}\right)$	182.5	Radial support stiffness of helical gear 5	$K_{x5}/K_{y5}(\mathrm{N}·{\mathrm{m}}^{-1})$	9.05 × 108
Axial support stiffness of helical gear 1	$K_{x1}(\mathrm{N}·{\mathrm{m}}^{-1})$	2.5 × 107	Radial support stiffness of helical gear 6	$K_{x6}/K_{y6}(\mathrm{N}·{\mathrm{m}}^{-1})$	8.12 × 108
Radial support stiffness of helical gear 1	$K_{y1}/K_{z1}(\mathrm{N}·{\mathrm{m}}^{-1})$	6.72 × 106	Torsional stiffness of elastic shafts 24 and 35	$K_{24}/K_{35}(\mathrm{N}·\mathrm{m}·\mathrm{ra}{\mathrm{d}}^{-1})$	2.55 × 105
Axial support stiffness of face gear 2	$K_{z2}(\mathrm{N}·{\mathrm{m}}^{-1})$	4.34 × 108	Input load	$T_1(\mathrm{N}·\mathrm{m})$	150
Radial support stiffness of face gear 2	$K_{x2}/K_{y2}(\mathrm{N}·{\mathrm{m}}^{-1})$	1.62 × 109	Input speed	$n_1(\mathrm{r}·\mathrm{mi}{\mathrm{n}}^{-1})$	900
Axial support stiffness of face gear 3	$K_{z3}(\mathrm{N}·{\mathrm{m}}^{-1})$	4.34 × 108

In the example, the number of teeth of helical gear 1 and helical gear 6 is odd. When ${\lambda }_1={\lambda }_2=\pi$ , it can be seen from equations (5) to (6) that $a_1=a_2=0.5$ . The relative meshing points of two NOH face gear pairs differ by half a meshing cycle, and the helical gear pairs are also the same.

When there is no error, the instantaneous static load distribution curve of the system is shown in Figure 7. The SLSC curves of the split-stage fluctuates around amplitude 1, and the waveform has a clear meshing periodicity of the split-stage.

Figure 7.

Instantaneous SLSC curves of the system without error.

The shape of the instantaneous average SLSC curves for the converge-stage and the split-stage are the same, but due to the unequal meshing cycles of two stages, the instantaneous average SLSC curves for each cycle are not exactly the same.

The influence of meshing phase differences on the SLSCs

When there is no error, the average SLSC of the system is 1.021. The uneven distribution of two branch loads is caused by the meshing phase differences, but the meshing phase differences have little effect on the average SLSCs. When the phase difference is 0.5, it has the greatest impact on the average SLSC, as shown in Figure 8.

Figure 8.

The influence of meshing phase differences $a_1$ and $a_2$ on the SLSCs.

Therefore, to ensure the equal load of two branch diversion system, the first step is to eliminate the influence of the meshing phase differences of each gear pair. For example, when ${\lambda }_1={\lambda }_2=\pi$ , $a_1=a_2=0$ . At this time, the number of teeth of the helical gear 1 and the helical gear 6 should be even.

The impact of assembly errors on the SLSCs

Due to the symmetrical structural parameters of the double branch, the impact of assembly errors on the load sharing performance of face gears 2 and 3 is consistent. Therefore, only one of them can be studied. The impact of assembly errors in the split-stage on the system’s load sharing is shown in Figures 9 and 10.

Figure 9.

The impact of assembly errors $\mathrm{\Delta }{E}_m$ and $\mathrm{\Delta }{q}_m$ in the split-stage on the SLSCs.

Figure 10.

The impact of assembly errors $\mathrm{\Delta }{\gamma }_1$ and $\mathrm{\Delta }{\gamma }_2$ in the split-stage on the SLSCs.

As the assembly errors increase, the SLSCs approximately increase linearly; Under the same assembly errors conditions, the assembly errors of helical gear 1 have a greater impact on the SLSCs than that of face gear 2; In the assembly errors of the split-stage, the shaft angle errors ( $\mathrm{\Delta }{\gamma }_m$ ) have the greatest impact on the SLSCs, followed by the offset errors ( $\mathrm{\Delta }{E}_m$ ), and the axial displacement errors ( $\mathrm{\Delta }{q}_m$ ) is the smallest.

The assembly errors of helical gears 4 and 5 can also be studied only one of them. The impact of assembly errors in the converge-stage on the SLSCs is shown in Figure 11.

Figure 11.

The impact of assembly errors $\mathrm{\Delta }{x}_n$ and $\mathrm{\Delta }{y}_n$ in the converge-stage on the SLSCs.

As the assembly errors increase, the SLSCs also increase approximately linearly; Under the same assembly errors conditions, the assembly errors of helical gear 6 have a greater impact on the SLSCs than helical gear 4; In the assembly errors of the converge-stage, $\mathrm{\Delta }{y}_n$ have a greater impact than $\mathrm{\Delta }{x}_n$ on the SLSCs.

The impact of input load on the SLSCs

Figure 12 shows the variation curves of the SLSC with the input load. When there is no assembly error, the average load factor of the system hardly changes with the increase of input load. However, when there are assembly errors in the system, the SLSCs decrease with the increase of input load. For example, when $\mathrm{\Delta }{E}_1=0.05\mathrm{mm}$ , the SLSC becomes smaller and smaller with the increase of input load. This is because the system’s elastic deformation can compensate for assembly errors, and the larger the input load, the greater the elastic deformation, and the more obvious the compensation effect. Therefore, when the input load is small, the SLSCs are greatly affected by assembly errors, and the SLSCs are large.

Figure 12.

The impact of input load on the SLSCs.

The influence of multiple of support stiffness on the SLSCs

Figure 13 shows the influence curves of the support stiffness multiple ( $n_z$ ) of helical gear 1 and face gear 2 on the SLSCs under the condition of no assembly error. As the multiple of the support stiffness of helical gear 1 increases, the SLSC also increases. Floating assembly ²⁶ of helical gear 1 can be considered to reduce its support stiffness and decrease the SLSCs; As the support stiffness multiple of face gear 2 increases, the SLSC first decreases and then slightly increases, with the minimum SLSC of the system near 1 multiple the support stiffness; The closer the support stiffness multiple is to 1 multiple, the more symmetrical two branch structures become. This indicates that the offset arrangement shown in Figure 1 has a symmetrical branch structure, which can ensure that face gear 2 and face gear 3 have the same support stiffness, thereby improving load sharing performance of the system.

Figure 13.

The influence of multiple of support stiffness on the SLSCs.

The influence of multiple of torsional stiffness on the SLSCs

Figure 14 shows the variation curves of the SLSC with the multiple of two shafts torsional stiffness ( $n_t$ ).

Figure 14.

The influence of multiple of torsional stiffness on the SLSCs.

As the torsional stiffness increases, the SLSC also increases, especially when there are assembly errors, the increase in the SLSC is more significant. By weakening the torsional stiffness of two shafts, the load sharing performance of the system can be effectively improved.

The influence of amplitude multiples of mesh stiffness fluctuations on the SLSCs

Figure 15 shows the variation curves of the SLSC with the amplitude multiple ( $n_k$ ) of the fluctuation of the meshing stiffness of the split-stage and converge-stage. As the amplitude of mesh stiffness fluctuation increases, the SLSC also increases, and the mesh stiffness fluctuation amplitude of the split-stage has a greater impact on the SLSC than that of the converge-stage. By weakening the fluctuation amplitude of the meshing stiffness of the split-stage and converge-stage and lowering the SLSCs, it is possible to consider weakening the fluctuation amplitude of the meshing stiffness through tooth surface modification. ²⁷

Figure 15.

The influence of amplitude multiples of mesh stiffness fluctuations on the SLSCs.

Conclusions

• (1) The uneven load of the system without error is mainly caused by the meshing phase differences of the split-stage, but the meshing phase differences have little effect on the SLSCs.

• (2) Assembly errors have a significant impact on the load sharing performance of the system, especially the shaft angle errors of the split-stage face gear pairs, which have the greatest impact on the SLSCs; The assembly errors of helical gear 1 have a greater impact on the SLSCs than that of face gears 2 and 3, while the assembly errors of helical gear 6 have a greater impact on the SLSCs than that of helical gears 4 and 5.

• (3) When there are assembly errors, the input load and torsional stiffness have a significant impact on the SLSCs.

• (4) The closer the support stiffness multiple is to 1 multiple, the more symmetrical two branch structures become. Using a symmetrical offset layout of branch structures can ensure that face gear 2 and face gear 3 have the same support stiffness; Consider floating assembly of helical gear 1 to reduce its support stiffness and decrease the SLSCs.

• (5) It can be considered to reduce the fluctuation amplitude of the meshing stiffness of the split-stage and converge-stage and lower the system load sharing performance by modifying the tooth surface.