Circular Spline Tooth Longitudinal Modification Design and Contact Analysis for Harmonic Drives with Short Flexspline

Abstract

Harmonic drives (HDs) with short flexspline (FS) always suffer from small meshing areas and severe stress concentration caused by large cone angles when a short FS is assembled and loaded. To address this issue, a tooth longitudinal modification method for the circular spline (CS) with a double circular arc common-tangent tooth profile (DCTP) is proposed. Using neutral layer and envelope conjugation theories, a mathematical model of the conventional straight CS tooth was developed. A shaping cutter for this tooth profile was then designed through coordinate transformation and meshing principles. The proposed longitudinal modification for the CS was achieved by adjusting the cutter’s trajectory. A precise finite element model of the HD was developed, revealing that tooth longitudinal modification can reduce the maximum contact pressure by 69.6% and significantly increase the contact area for HDs with short FS. This work provides valuable technical references for improving the contact state of HDs with short FS.

1. Introduction

HDs are favored in the applications of the aerospace industry, robots, and machine tools [1,2,3], for their compactness, precision, and large transmission ratios. As industrial demands evolve towards lighter and smaller solutions, HDs with short FS are emerging as the leading trend over traditional counterparts. A short FS yields a compact design with a ratio of FS length to diameter lower than 0.5 [4]. However, this reduction intensifies FS coning deformation, increasing the cone angle from φ₁ to φ₂, as shown in Figure 1a. This leads to challenges like reduced gear tooth meshing area (black area as shown in Figure 1b) and heightened pressure concentration. These factors can lead to contact fatigue. Hence, it is necessary to study how to overcome these issues to enhance the contact performance and longevity of HDs with short FS.

Contact fatigue emerges as the predominant failure mode in HD systems, attributed to the transmission of motion via the cyclical elastic deformation of the FS [5,6]. Numerical methods stand out as the primary approach for analyzing contact in harmonic reducers. Li et al. [7] and Yang et al. [8] employed finite element models to investigate the influence of tooth width, chamfering, and FS length on the uneven load distribution resulting from FS deformation. Their findings indicate that increasing FS length, tooth width, and chamfering effectively mitigate partial axial loads. To address cone deformation of the flexible gear shell, Chen et al. [9] and Song et al. [10] introduced a method for calculating the conjugate tooth profile based on multi-section analysis, thereby circumventing inaccuracies inherent in traditional designs focused solely on the middle section’s meshing characteristics. Zhang et al. [11,12] developed separate finite element models and a modified energy method to comprehensively explore various aspects of disposable harmonic gears under full load conditions, encompassing meshing stiffness, load distribution, and contact pressure. Ma et al. [13] adopted a rigid–flexible coupling method to investigate the causes of time-varying stiffness in HD gears under different torque conditions, validating their findings through experimental research. Yao et al. [14] proposed a piecewise method for accurately calculating FS deformation when assembled with a cam wave generator (WG), thereby enhancing HD contact analysis. Elasticity theory and experimental techniques emerge as vital tools for analyzing contact interactions in HD systems. Wang et al. [15] and Qiu et al. [16] devised a rapid pressure calculation method based on mechanics analysis to address pressure and deformation in short FS of varying lengths. Additionally, Ma et al. [17] suggested a non-contact method for measuring FS deformation, revealing a nonlinear distribution along the rotating axis, where radial displacement near the closed end tends to decrease, approaching zero, as driving speed increases.

Tooth shape plays a crucial role in the contact performance of HDs, encompassing both the tooth’s longitudinal shape and tooth profile shape [18]. The longitudinal shape of a tooth typically follows a linear contour along its width. However, advancements in gear design have led to proposals for tooth longitudinal multi-section conjugation, especially when considering cone deformation in flexible gear shells [9,10]. However, they did not provide a design method for the hob tooth surface or its trajectory during modification. Additionally, the tooth profile shape defines the contour of the tooth in cross-section, with common shapes including involute and circular arc profiles. The involute tooth profile has emerged as a prevalent choice due to its ease of implementation in manufacturing and matured production techniques. In conventional HDs with involute teeth, Maiti et al. [19,20] analyzed variations in tooth pitches and pressures using finite element methods, revealing secondary contacts. To mitigate meshing interference, Yang et al. [21] and Routh et al. [22] proposed design methods for short tooth and asymmetrical involute profiles separately. Further analysis by Yang et al. [23] compared involute, cycloid common tangent, and DCTP profiles, showing that the latter two offer larger conjugate existent domains and contact ratios. To address issues such as tooth interference and improper meshing, circular arc tooth profiles have been proposed to replace involute ones. Ishikawa [24,25] introduced an ‘S-tooth’ profile with two circular arc segments, ensuring continuous contact angles without deformation, thereby enhancing meshing performance and load-bearing capacity. However, its application is limited due to the abstraction of gear teeth as racks with identical profiles during design. Building upon curve mapping and bidirectional conjugation, Song et al. [26] presented a design method for DCTP in harmonic gears, analyzing the impacts of design parameters on meshing characteristics. Their findings demonstrated the feasibility of achieving multi-tooth meshing in HDs. Drawing upon gearing theory, Cheng [27] and Chen et al. [28] derived tooth profiles for both the FS and CS of an HD with DTCP. They further simulated meshing characteristics using finite element analysis.

The above mentioned references have made great contributions to improving the contact performance of HDs by solely enhancing the tooth profile shape of the FS. However, until now, some issues have still been scarcely studied. Firstly, edge contact and stress concentration caused by FS coning deformation are often overlooked, leading to discrepancies between theory and reality. Research shows that modifying the tooth direction can effectively address these issues. Therefore, designing optimal tooth surface geometry to account for FS coning deformation is crucial for improving the contact performance of HDs with short FS. Secondly, few scholars [10,29] have proposed longitudinal tooth modification for FS to mitigate edge contact and stress concentration caused by conical deformation. Given the thinness of the FS body, adding radial feed motion of the hob can further thin the end, impacting fatigue life. Thus, exploring processing methods and cutter design technology for CS longitudinal tooth modification is also essential. Lastly, finite element models used in contact analysis often oversimplify or ignore key components like the WG and thin wall bearing, neglecting true FS deformation. Consequently, establishing a parametric high precision finite element model is crucial for obtaining accurate contact analysis results in HDs.

To accomplish the research objectives outlined above, this paper proposes a novel method for tooth longitudinal modification in CS with DCTP and a shaping cutter. Utilizing the neutral layer concept and gear meshing theory, mathematical models are developed for both the longitudinal modification of CS teeth and the corresponding shaping cutter. Subsequently, to accurately simulate the operational conditions of the HDs, a parameterized finite element model encompassing all components is established. The grid nodes and 8-node elements of this model are meticulously generated using MATLAB. Finally, the paper delves into an analysis and discussion of the impacts of the FS length-to-diameter ratio, WG profile, modification radius, and torque on contact characteristics.

2. Tooth Longitudinal Modification for CS with DCTP

The generating processes for tooth longitudinal modification in CS with DTCP and the shaping cutter are depicted in Figure 2. This main process can be segmented into three distinct parts: straight tooth CS modeling, shaping cutter tooth profile modeling, and longitudinal modification of CS modeling. Firstly, by considering the conjugate relationship between the CS and the FS with DTCP, a mathematical model for the straight CS tooth is established based on the neutral layer and envelope conjugation theories. Subsequently, a shaping cutter capable of generating the straight CS tooth is derived through coordinate transformation and meshing principles. Lastly, longitudinal modification of the CS tooth is accomplished by adjusting the center of the shaping cutter relative to the CS center for arc motion.

2.1. Straight Tooth CS

As shown in Figure 3, the DTCP is illustrated within the FS local coordinate system. The origin O_r is defined as the intersection point of the neutral layer of the FS and the symmetry axis of the gear teeth. A local coordinate system S_r(O_r, X_r, Y_r) for the FS is then established, rigidly connected to the FS gear teeth. Segment AB represents a convex arc tooth profile, segment BC represents a straight segment tooth profile, and segment CD represents a concave arc tooth profile. The parameter definitions of the DTCP are shown in

Table 1. Double arc tooth profile parameter table.

Parameter	Parameter Definition	Parameter	Parameter Definition
ha	Addendum	c1	Center displacement of convex arc
hf	Dedendum	c2	Center displacement of concave arc
h1	Straight tooth profile height	δ	Common tangent inclination angle
e1	Offset of convex arc center	ds	Wall thickness of flexspline cylinder
e2	Offset of concave arc center	r1	Convex arc radius
α	Tooth tip pressure angle	r2	Concave arc radius
kt	Tooth thickness coefficient	m	Modulus

.

From Figure 3, it can be seen that s denotes the arc length from any point on the tooth profile to point A. The piecewise functions of the convex tooth profile, straight tooth profile, and concave tooth profile of the DCTP are as follows:

The position vector for segment AB of the convex arc tooth profile in coordinate system S_r is determined with the following equation:

$$R_{r1}(s)=\left[\begin{array}{l}{x}_{r1}\\ y_{r1}\\ 1\end{array}\right]=\left[\begin{array}{l}{r}_1\cos (\alpha -{\displaystyle \frac{s}{r_1}})-c_1\\ r_1\sin (\alpha -{\displaystyle \frac{s}{r_1}})+h_f+d_s-e_1\\ 1\end{array}\right],s\in (0,l_1)$$

(1)

where r₁ is the convex arc radius, α is the tooth tip pressure angle, c₁ and e₁ are the center displacement of the convex arc in the X_r and Y_r directions, respectively, h_f is the dedendum, and d_s is half of the wall thickness of the FS.

The position vector for segment BC of the linear tooth profile in coordinate system S_r is determined with the following equation:

$$R_{r2}(s)=\left[\begin{array}{l}{x}_{r2}\\ y_{r2}\\ 1\end{array}\right]=\left[\begin{array}{l}{r}_1\cos \delta -c_1+(s-l_1)\sin \delta \\ r_1\sin \delta +h_f+d_s-e_1-(s-l_1)\cos \delta \\ 1\end{array}\right],s\in (l_1,l_2)$$

(2)

where l₁ is the length of the convex tooth profile, l₂ is the length of the straight tooth profile, and δ is the common tangent inclination angle.

The position vector for segment CD of the concave arc tooth profile in coordinate system S_r is determined with the following equation:

$$R_{r3}(s)=\left[\begin{array}{l}{x}_{r3}\\ y_{r3}\\ 1\end{array}\right]=\left[\begin{array}{l}-r_1\cos (\delta +{\displaystyle \frac{s-l_2}{r_2}})+{\displaystyle \frac{p}{2}}+c_2\\ -r_2\sin (\delta +{\displaystyle \frac{s-l_2}{r_2}})+h_f+d_s+e_2\\ 1\end{array}\right],s\in (l_2,l_3)$$

(3)

where l₃ is the length of the concave tooth profile, p is the tooth pitch, and r₂ is the concave arc radius.

Coordinate systems applied for the derivation of the CS tooth surface mathematical model are shown in Figure 4. The local coordinate systems S(O, X, Y) and the global S_g(O_g, X_g, Y_g) are fixed on the WG and CS, respectively. The origin O of the coordinate system S is positioned at the rotation center of the HD. The Y-axis and the X-axis point to the long-axis and short-axis direction of the WG, respectively, and rotate with the WG. S_g is a global coordinate system, and its origin, O_g, is also positioned at the rotation center of the HD, with Y_g located on the angle bisector of the CS teeth. Assuming that point A₀ is a point on the undeformed neutral layer, A₀ will move to point A when the FS rotates through an angle φ₁ without considering deformation. However, due to the WG deforming the FS, the position of point A changes to point O_r. Additionally, ρ is the length between the point O_r and the point O_g, r_m is the radius of the neutral circle of the FS, μ is the normal angle formed by the deformation of the FS, φ₂ is the rotation angle of the WG, φ is the angle between the undeformed end of the FS and Y_g, and β is the angle between the axes Y_g and Y_r.

Based on the coordinate transformation depicted in Figure 4, the position vector of the CS tooth surface can be expressed in the coordinate system S_g and can be determined by the following equations:

$$R_g({\phi }_1,s)=M_{gr}{R}_r(s)$$

(4)

$$f({\phi }_1,s)=({\displaystyle \frac{\partial R_g}{\partial s}}\times \left[\begin{array}{l}0\\ 0\\ 1\end{array}\right])·{\displaystyle \frac{\partial R_g}{\partial {\phi }_1}}=0$$

(5)

where Equation (5) is the meshing equation between the CS and FS, and

$${\mathit{M}}_{gr}=\left[\begin{array}{ccc}\cos \beta & \sin \beta & \rho \sin {\phi }_1\\ -\sin \beta & \cos \beta & \rho \cos {\phi }_1\\ 0& 0& 1\end{array}\right].$$

Parameter β can be solved using the following approach based on the neutral layer concept.

The cosine cam acts as a WG, and its outer contour is in the shape of an elliptical line. After assembling it with the FS, the neutral layer of the FS will deform similarly to the outer contour of the WG. Therefore, the deformed neutral layer equation is as follows:

$$\rho =r_m+w_0\cos (2{\phi }_3)$$

(6)

where φ₃ is the relative rotation angle between WG and FS, and φ₃ = φ₁ + φ₂.

From Figure 3, it can be observed that:

$$\beta ={\phi }_1+\mu$$

(7)

The transmission ratio of the harmonic reducer can be expressed as:

$${\phi }_2={\displaystyle \frac{Z_r}{Z_{\mathrm{g}}-Z_r}}\phi$$

(8)

According to Ref. [12], normal rotation angle μ, radial deformation ω, circumferential deformation v, and rotational angle φ1 can be derived by the neutral layer concept as follows:

$$\mu =\arctan ({\displaystyle \frac{{\rho }^{\prime }}{\rho }})$$

(9)

$$\omega =\rho -r_m$$

(10)

$$\nu =-{\displaystyle \underset{0}{\overset{\phi +{\phi }_2}{\int }}\omega }\mathrm{d}{\phi }_3$$

(11)

$${\phi }_1=\phi +{\displaystyle \frac{\nu }{r_m}}$$

(12)

Combining Equations (7)–(10), parameter β can be solved.

As Equation (4) involves two unknown variables, the tooth profile variable s and the rotation angle φ₁, a stepwise iterative method is employed for its solution. The calculation process is illustrated in Figure 5. Initially, compute the discrete s value based on the input parameters and solve the meshing equation Equation (5) to derive the corresponding φ₁ value. Subsequently, substitute the determined s value and φ₁ value into the CS coordinate equation to obtain the corresponding position vector R_g.

Curve fitting is performed on the discrete tooth profile points to derive the analytical equation of the straight tooth profile. Figure 6 illustrates the local coordinate system of a straight tooth in the CS. Based on the arc radius and circle center obtained from the arc fitting of the straight tooth CS profile, curve fitting is conducted to obtain the analytical equation for the straight tooth CS profile. The parameter definitions of the CS profile are shown in

Table 2. Table of tooth profile parameters for CS.

Parameter	Parameter Definition	Parameter	Parameter Definition
hag	Addendum	c1g	Center displacement of convex arc
hfg	Dedendum	c2g	Center displacement of concave arc
h1g	Linear tooth profile height	δg	Common tangent inclination angle
c1g	Offset of convex arc center	αg	Tooth tip pressure angle
c2g	Offset of concave arc center	rg	Reference radius
r1g	Convex arc radius	m	Modulus
r2g	Concave arc radius

.

The position vector R_g1, after curve fitting of the concave tooth profile of the CS, is expressed as follows:

$$R_{g1}=\left[\begin{array}{l}{X}_{g1}\\ Y_{g1}\\ 1\end{array}\right]=\left[\begin{array}{l}{r}_{g1}\cos ({\alpha }_g-{\displaystyle \frac{s}{r_{g1}}})-c_{g1}\\ -r_{g1}\sin ({\alpha }_g-{\displaystyle \frac{s}{r_{g1}}})+e_{g1}+r_g\\ 1\end{array}\right],s\in (0,l_{g1})$$

(13)

where r_g is the CS indexing circle radius, r_g1 is the convex arc radius of the CS, l_g1 is the length of the convex tooth profile of the CS, α_g is the tooth tip pressure angle of the CS, c_g1 and e_g1 are the center displacement of the convex arc in the X_g and Y_g directions, respectively, and s serves as a variable representing the profile function of the CS.

The position vector R_g2, after curve fitting of the straight tooth profile of the CS, is expressed as follows:

$$R_{g2}=\left[\begin{array}{l}{X}_{g2}\\ Y_{g2}\\ 1\end{array}\right]=\left[\begin{array}{l}{r}_{g1}\cos {\delta }_g-c_{g1}+(s-l_{g1})\sin {\delta }_g\\ r_{g1}\sin ({\delta }_g)-e_{g1}-(s-l_{g1})\cos {\delta }_g+r_g\\ 1\end{array}\right],s\in (l_{g1},l_{g2})$$

(14)

where l_g2 is the length of the straight tooth profile, and δ_g is the common tangent inclination angle of the CS.

The position vector R_g3, after curve fitting of the convex tooth profile of the CS, is expressed as follows:

$$R_{g3}=\left[\begin{array}{l}{X}_{g3}\\ Y_{g3}\\ 1\end{array}\right]=\left[\begin{array}{l}-r_{g2}\cos ({\delta }_g+{\displaystyle \frac{s-l_{g2}}{r_{g2}}})+{\displaystyle \frac{p_g}{2}}+c_{g2}\\ r_{g2}\sin ({\delta }_g+{\displaystyle \frac{s-l_{g2}}{r_{g2}}})-e_{g2}+r_g\\ 1\end{array}\right],s\in (l_{g2},l_{g3})$$

(15)

where l_g3 is the length of the concave tooth profile, p_g is the tooth pitch of CS, r_g2 is the concave arc radius, and c_g2 and e_g2 are the center displacement of the concave arc in the X_g and Y_g directions, respectively, of the CS.

2.2. Shaping Cutter Tooth Profile

As shown in Figure 7, the processing coordinate system of the shaping cutter for machining the CS is illustrated. The local coordinate systems S_c1(O_c1, X_c1, Y_c1) and S_g1(O_g1, X_g1, Y_g1) are attached to the shaping cutter and the CS, respectively. Additionally, Y_g1 is located on the angle bisector of CS teeth, and Y_c1 is located on the angle bisector of shaping cutter teeth. O_c represents the center of the shaping cutter, while O_g denotes the center of the CS. In addition, φ₁ and φ₂ are the rotation angles of the CS and the shaping cutter, respectively, and E_cg is the center distance between the shaping cutter and the CS. Δφ is the angle between Y_g1 and Y_c1, and Δφ is half of the central angle between two adjacent teeth of the CS.

Based on the processing coordinate transformation depicted in Figure 7, the position vector of the shaping cutter tooth surface can be expressed in the coordinate system S_g1 can be determined by the following equations:

$${\mathit{R}}_c(s,{\phi }_1)={\mathit{M}}_{cc1}{\mathit{M}}_{c1g1}{\mathit{M}}_{g1g}{\mathit{R}}_g(s)$$

(16)

$$f({\phi }_1,s)=({\displaystyle \frac{\partial R_c}{\partial s}}\times \left[\begin{array}{l}0\\ 0\\ 1\end{array}\right])·{\displaystyle \frac{\partial R_c}{\partial {\phi }_1}}=0$$

(17)

where ${\mathit{M}}_{g1g}=\left[\begin{array}{ccc}\cos {\phi }_1& \sin {\phi }_1& 0\\ -\sin {\phi }_1& \cos {\phi }_1& 0\\ 0& 0& 1\end{array}\right]$, ${\mathit{M}}_{c1g1}=\left[\begin{array}{ccc}\cos △\phi & -\sin △\phi & 0\\ \sin △\phi & \cos △\phi & -E_{cg}\\ 0& 0& 1\end{array}\right]$, ${\mathit{M}}_{cc1}=\left[\begin{array}{ccc}\cos {\phi }_2& -\sin {\phi }_2& 0\\ \sin {\phi }_2& \cos {\phi }_2& 0\\ 0& 0& 1\end{array}\right]$, φ₁ is the rotation angle of the CS, and φ₂ is the rotation angle of the shaping cutter.

The relationship between angles φ₁ and φ₂ satisfies the following equation:

$${\displaystyle \frac{{\phi }_1}{{\phi }_2}}={\displaystyle \frac{Z_c}{Z_g}}$$

(18)

where Z_c and Z_g represent the tooth number of the shaping cutter and the CS, respectively.

Curve fitting is conducted on the discrete tooth profile points to derive the analytical equation of the shaping cutter tooth profile. Figure 8 illustrates the curve fitting of discrete tooth profile points to obtain the shaping cutter tooth profile curve, followed by fitting the shaping cutter tooth profile curve into the tooth profile analytical equation.

The position vector R_c1, after curve fitting of the convex tooth profile of the shaping cutter, is as follows:

$${\mathit{R}}_{c1}=\left[\begin{array}{l}{x}_{c1}\\ y_{c1}\\ 1\end{array}\right]=\left[\begin{array}{l}{r}_{c1}\cos ({\alpha }_c-{\displaystyle \frac{s}{r_{c1}}})-c_{c1}\\ r_{c1}\sin ({\alpha }_c-{\displaystyle \frac{s}{r_{c1}}})-e_{c1}+r_c\\ 1\end{array}\right],s\in (0,l_{c1})$$

(19)

where r_c is the shaping cutter indexing circle radius, l_c1 is the length of the convex tooth profile of the shaping cutter, r_c1 is the convex arc radius of the shaping cutter, α_c is the tooth tip pressure angle of the shaping cutter, and c_c1 and e_c1 are the center displacement of the convex arc in the X_c and Y_c directions, respectively, of the shaping cutter.

The position vector R_c2, after curve fitting of the straight tooth profile of the shaping cutter, is as follows:

$${\mathit{R}}_{c2}=\left[\begin{array}{l}{x}_{c2}\\ y_{c2}\\ 1\end{array}\right]=\left[\begin{array}{l}{r}_{c1}\cos {\delta }_c-c_{c1}+(s-l_{c1})\sin {\delta }_c\\ r_{c1}\sin ({\delta }_c)-e_{c1}-(s-l_{c1})\cos {\delta }_c+r_c\\ 1\end{array}\right],s\in (l_{c1},l_{c2})$$

(20)

where l_c2 is the length of the straight tooth profile of the shaping cutter, and δ_c is the common tangent inclination angle.

The position vector R_c2, after curve fitting of the concave tooth profile of the shaping cutter, is as follows:

$${\mathit{R}}_{c3}=\left[\begin{array}{l}{x}_{c3}\\ y_{c3}\\ 1\end{array}\right]=\left[\begin{array}{l}-r_{c1}\cos (\delta +{\displaystyle \frac{s-l_{c2}}{r_{c2}}})+{\displaystyle \frac{p_c}{2}}+c_{c2}\\ -r_{c2}\sin (\delta +{\displaystyle \frac{s-l_{c2}}{r_{c2}}})+e_{c2}+r_c\\ 1\end{array}\right],s\in (l_{c2},l_{c3})$$

(21)

where l_c3 is the length of the concave tooth profile of the shaping cutter, p_c is the tooth pitch of the shaping cutter, r_c2 is the concave arc radius of the shaping cutter, and c_c2 and e_c2 are the center displacement of the concave arc in X_c and Y_c directions, respectively, of the shaping cutter.

2.3. Longitudinal Modification of CS

The longitudinal modification of the CS tooth can be achieved by adjusting the center of the shaping cutter relative to the CS center for arc motion, as shown in Figure 9. From Figure 9, it can be observed that the arc motion can be accomplished by changing the center distance E_cg between the CS and the shaping cutter. In addition, point O is the center of the arc motion trajectory, R is the longitudinal modification radius, b is the distance from the shaping cutter cutting point to the centerline of the CS tooth width, and d_b is the pitch diameter of the CS.

The center distance E_cg between the CS and the shaping cutter can be represented as follows:

$$E_{\mathit{cg}}={\displaystyle \frac{m(Z_g-Z_c)}{2}}-\sqrt{R^2-b^2}+R$$

(22)

According to the processing coordinate system of the shaping cutter for machining the CS shown in Figure 7, any tooth profile equation along the tooth width cross-section can be represented as:

$${\mathit{R}}_g(s,{\phi }_1)={\mathit{M}}_{g{g}_1}{\mathit{M}}_{g_1{c}_1}{\mathit{M}}_{c_{1}c}{\mathit{R}}_c(s)$$

(23)

$$f({\phi }_1,s)=({\displaystyle \frac{\partial R_g}{\partial s}}\times \left[\begin{array}{l}0\\ 0\\ 1\end{array}\right])·{\displaystyle \frac{\partial R_g}{\partial {\phi }_1}}=0$$

(24)

where

$${\mathit{M}}_{gg1}=\left[\begin{array}{ccc}\cos {\phi }_1& -\sin {\phi }_1& 0\\ \sin {\phi }_1& \cos {\phi }_1& 0\\ 0& 0& 1\end{array}\right], {\mathit{M}}_{g_{1}c1}=\left[\begin{array}{ccc}\cos △\phi & \sin △\phi & E_{cg}\sin △\phi \\ -\sin △\phi & \cos △\phi & E_{cg}\cos △\phi \\ 0& 0& 1\end{array}\right], {\mathit{M}}_{cc}=\left[\begin{array}{ccc}\cos {\phi }_2& \sin {\phi }_2& 0\\ -\sin {\phi }_2& \cos {\phi }_2& 0\\ 0& 0& 1\end{array}\right].$$

Providing a value for b, Equation (22) can be used to determine the center distance E_cg. Then, by substituting E_cg into Equations (23) and (24), the tooth profile can be established with the distance from the shaping cutter cutting point to the centerline of the CS tooth width set as b. Continuously adjusting the value of b allows for the creation of a three-dimensional tooth surface model of the CS with longitudinal modification.

3. Parametric Finite Element Model of HDs

An HD always consists of four main components: FS, CS, flexible bearing (FB), and WG. The accuracy of finite element mesh modeling and assembly of the aforementioned main components determines the precision of contact analysis. Hence, utilizing the discrete nodes generated by the parametric model, a precise finite element model is constructed for the HD with the longitudinally modified tooth CS. Parametric modeling of the HD is conducted using MATLAB 2016a. Because the geometric mathematical models of FS and CS tooth surfaces are very complex, it is difficult to directly establish a solid model in the form of equations and divide the mesh in 3D modeling analysis software such as ABAQUS 2022. Moreover, the accuracy of shape modeling has a significant impact on contact solutions, and approximate modeling methods cannot be used. Only discrete methods can be used to form point clouds and establish high precision tooth surface models. The advantage of MATLAB is that it can achieve a high precision point cloud solution of tooth surfaces, establish high precision tooth surface models, and use spatial coordinate changes to achieve high precision assembly between various components. It can also be parameterized. Initially, the entire model is discretized into spatial points, which are then interconnected point by point to create the hexahedral finite element model. Subsequently, the material properties, element attributes, assembly relationships, and boundary conditions of each part are documented in the INP file, which is then imported into ABAQUS for further calculation.

The structural diagram, single tooth nodes, and finite element model of the FS are shown in Figure 10a–c, separately. From Figure 10a, it can be observed that L is the FS length, B is the tooth width, r_f is the inner diameter of FS, d_st is the wall thickness, r_k is the Inner diameter of FS bottom, and L_d is the thickness of the bottom wall of the FS.

The structural diagram, single tooth nodes, and finite element model of the CS are shown in Figure 11a–c, respectively. From Figure 11a, it can be observed that R is the modification radius, and d_w is the rim diameter of the CS. In Figure 11c, the small image on the right shows end view diagrams of the CS longitudinal modification profile and straight profile.

The structural diagram, ball and ring nodes, and finite element model of the FB are shown in Figure 12a–c, respectively. From Figure 12a, it can be observed that di and do are the inner and outer diameters, respectively.

The structural diagram, WG nodes, and finite element model of the WG are shown in Figure 13a–c, respectively. From Figure 13a, it can be observed that a and b are the short axis length and the long axis length, respectively. The distance from a point on the WG to the center of the WG can be represented as:

$${\rho }_{\mathrm{wg}}={\displaystyle \frac{d_i}{2}}+\epsilon \cos (2\phi )$$

(25)

where ε is the WG profile coefficient.

During the discretization of the computational model, 8-node 3D solid elements (C3D8R) were used, as they provide a good balance between computational efficiency and accuracy. Furthermore, to ensure precise contact simulation, the Penalty contact algorithm in Abaqus was employed, which enables more accurate contact simulation without increasing model stiffness. Additionally, a friction coefficient was applied to the contact surfaces, set according to the relative motion between the FS and CS to closely mimic real working conditions. For all other contact pairs, a surface-to-surface contact was adopted, with a friction coefficient of 0.08 between the rollers and outer race, and 0.15 for the remaining contact pairs.

The accuracy of the simulation analysis still heavily relies on the boundary conditions imposed on the model. The simulation process is divided into three distinct load steps, each building upon the previous one. Initially, constraints are imposed on the FS displacement along its long axis, while contact is established between the FB and WG, as well as between the FB and FS. This initial step aims to eliminate interference between the FB and WG, illustrated in Figure 14a. The subsequent step focuses on establishing tooth contact between the CS and FS without applying any loads. This is accomplished by fully constraining the CS, WG, and FS through coupling points, as depicted in Figure 14b,c. In the final step, the rotational freedom of the WG and FS along the axis direction is released. Additionally, torque is applied to the coupling point of the FS, as shown in Figure 14d.

4. Results and Discussion

To validate the effectiveness of the CS tooth longitudinal modification method proposed in this paper, a comparison of contact characteristics between modified and unmodified CS tooth profiles is conducted. Additionally, to assess its contact characteristics for HDs with a short FS, the impacts of FS length-to-diameter ratio, WG profile, modification radius, and torque on contact pressure and load sharing among teeth are also investigated. The basic parameters of FS and CS for analysis are listed in

Table 3. Basic parameters of FS and CS for analysis.

Parameter	Value	Parameter	Value
ha (mm)	0.3	hag (mm)	0.35
hf (mm)	0.4	hfg (mm)	0.35
h1 (mm)	0.1508	h1g (mm)	0.1463
c1 (mm)	0.3912	c1g (mm)	0.3416
c2 (mm)	0.2874	c2g (mm)	0.3327
e1 (mm)	0.0732	e1g (mm)	0.0746
e2 (mm)	0.15	e2g (mm)	0.1485
r1 (mm)	0.67	r1g (mm)	0.6724
r2 (mm)	0.8	r2g (mm)	0.7964
α (°)	33.79	αg (°)	38.53
δ (°)	14.74	δg (°)	14.56
ds (mm)	0.4	rg (mm)	25.5
m (mm)	0.5	Zr	100
Zg	102	Q	0.4
ω0	0.5	R (mm)	200
T (N·m)	30	ε	0.5

, and the material mechanics properties of parts of HD components are listed in

Table 4. Material mechanics properties of parts of HD components.

Parameter	Material	Young’s Modulus E (GPa)	Poisson’s Ratio v
FS	30CrMnNi	204	0.3
CS	QT400	207	0.3
WG	GCr15	217	0.3

.

The FS teeth are numbered in the clockwise direction, which starts from the upper tooth, as shown in Figure 15a. From Table 1, it is observed that the FS and CS have 100 and 102 teeth, respectively. Since the number of teeth is even, the two contacting surfaces with a relative angular spacing of 180° between FS teeth exhibit identical contact states, as depicted in Figure 15b. Moreover, the FS teeth’s contact area is situated near the long axis. Therefore, due to the large number of teeth, it is impractical to display the contact status of every tooth’s working surface in the contact distribution diagram. Instead, this paper solely illustrates the contact status of FS teeth near the lower end of the long axis.

4.1. Comparison Between CS Teeth with and Without Longitudinal Modification

In order to verify the effectiveness of the proposed method, two sets of examples, with and without modification radius R = 200 mm, are selected for analysis. The analysis results are shown in Figure 16 and Figure 17.

Figure 16 illustrates a comparison of the contact pressure between CS teeth with and without modification. When the CS teeth are unmodified longitudinally, as depicted in Figure 16, it is evident that contact areas are predominantly concentrated near the tooth end, with the maximum contact pressure peaking at 2649 MPa, indicating severe stress concentration. It can be observed from Figure 16 that when the cylinder length is relatively short and without modification, there is indeed stress concentration in the meshing of the gear teeth, consistent with the findings in references [10,30], further validating the rationality of the model. Conversely, with a modification radius of R = 200 mm, the contact area expands and shifts from the tooth end toward the middle along the width of the tooth. Consequently, the maximum contact pressure notably decreases to 804 MPa, representing a significant reduction of 69.6%. Therefore, longitudinal modification of CS teeth effectively increases the contact area and avoids stress concentration. By comparing the contact stress analysis results between CS teeth with and without modification in this study, it is evident that the longitudinal modification of CS teeth effectively addresses the issues of edge contact and stress concentration in HDs with short FS. This verifies the superiority of the proposed method over traditional approaches.

Figure 17 depicts the load sharing among teeth for CS teeth with and without modification. From Figure 17, it can be seen that the number of engaged teeth is 13, and the tooth carrying the heaviest load is #41 near the lower end of the long axis, regardless of modification. Therefore, the appropriate amount of CS tooth longitudinal modification hardly alters the load sharing among teeth.

4.2. Effect of FS Length-to-Diameter Ratio on Contact Pressure and Load Sharing

A compact design is achieved with a short FS, characterized by an FS length-to-diameter ratio (Q) of less than 0.5. To investigate the impact of FS length on tooth surface contact pressure and load sharing among teeth, five sets of different Q values (Q = 0.2, 0.4, 0.6, 0.8, and 8) are selected for contact analysis under two conditions: CS teeth with and without longitudinal modification.

(1)

CS teeth without longitudinal modification

The results of the analysis are presented in Figure 18 and Figure 19, illustrating the variations in contact pressure and load sharing across teeth corresponding to different Q values when the CS teeth are without longitudinal modification.

As shown in Figure 18, the maximum contact pressure follows a sequence of 4496 MPa, 2649 MPa, 2261 MPa, 1417 MPa, and 568 MPa as Q gradually increases from 0.2 to 8. And, it is evident that the contact area shifts from the tooth end to the middle of the tooth width, resulting in a gradual decrease in contact pressure and an increase in contact area. When Q reaches 8.0, the contact area spans the entire tooth surface along the tooth width. This phenomenon arises due to nearly axial uniform deformation of the FS when the length significantly exceeds the diameter.

From Figure 19, it can be observed that the number of engaged teeth near the lower end of the long axis gradually changes as Q increases from 0.2 to 8, with values of 11, 13, 13, 13, and 8, respectively. Initially, as Q increases, the number of engaged teeth increases, reaching a peak before gradually decreasing. When Q reaches 8.0, the teeth carrying the load are from #40 to #47. However, when Q is 0.2, the teeth carrying the load are from #36 to #41 and from #48 to #53. Hence, with an increase in Q, load sharing becomes more continuous.

(2)

CS teeth with longitudinal modification

When the CS teeth are with longitudinal modification, the results of the analysis are presented in Figure 20 and Figure 21, illustrating the variations in contact pressure and load sharing across teeth corresponding to different Q values.

After modification, as illustrated in Figure 20, the maximum contact pressure follows a sequence of 1964 MPa, 804 MPa, 774 MPa, 752 MPa, and 806 MPa as Q gradually increases from 0.2 to 8. Compared with the maximum contact pressure in the non-modified case, as shown in Figure 18, modification results in a reduction of 56%, 70%, 66%, 47%, and 42% in maximum contact pressure, respectively. When Q is large, such as 8.0, the modification does not decrease the maximum contact pressure; instead, it increases. This occurs when Q is sufficiently large, causing the modification to change the line contact to point contact, resulting in a smaller contact area and higher contact pressure.

After modification, as shown in Figure 21, the number of engaged teeth near the lower end of the long axis gradually changes as Q increases from 0.2 to 8, with values of 11, 13, 13, 13, and 9, respectively. Comparing this with the non-modified case shown in Figure 21, it is evident that modification hardly affects the number of engaged teeth, and the tooth carrying the heaviest load remains #41 or near #41 regardless of Q. Additionally, modification results in the maximum load becoming closer as Q increases.

4.3. Effects of WG Profile on Contact Pressure and Load Sharing

The length of the long axis of WG directly determines the maximum height at which FS teeth are lifted, which significantly affects the meshing between the FS and CS. To analyze the influence of different profiles of WG on contact pressure and load sharing, four sets of WG profile coefficients (ε = 0.4, 0.45, 0.5, and 0.55) are selected for analysis. The analysis results are shown in Figure 22 and Figure 23.

Figure 22 illustrates the impact of ε on contact pressure. It is evident from Figure 22 that as the ε value gradually increases, the maximum contact pressure follows a sequence of 2187 MPa, 1283 MPa, 804 MPa, and 1001 MPa, respectively. Initially, with an increase in ε, the maximum contact pressure decreases before rising again. When ε deviates from ω₀ (0.5), contact occurs at the FS tooth tip, leading to stress concentration. This is because the design and actual values of radial deformation are consistent when ε is 0.5.

Figure 23 illustrates the impact of ε on load sharing. It is evident from Figure 23 that as the ε value increases from 0.4 to 0.55, the number of engaged teeth near the lower end of the long axis gradually changes with values of 10, 14, 13, and 4, respectively. The tooth carrying the heaviest load follows a sequence of #37, #38, #41, and #50. This indicates that the variation in ε significantly alters the clearance distribution between adjacent gear teeth working surfaces, changes the positions of teeth carrying the heaviest load and the number of engaged teeth, and thus affects the load sharing among teeth.

4.4. Effects of Modification Radius on Contact Pressure and Load Sharing

In order to gain deeper insights into the influence of modification radius on contact pressure and load sharing among teeth, the analysis encompasses four different modification radius (R = 50 mm, 200 mm, 350 mm, and 500 mm). The results of this analysis are presented in Figure 24 and Figure 25, illustrating the variations in contact pressure and load sharing among teeth corresponding to different modification radii.

From Figure 24, it is observed that as R gradually increases from 50 mm to 500 mm, the maximum contact pressure follows a sequence of 1238 MPa, 804 MPa, 1412 MPa, and 1991 MPa, respectively. The maximum contact pressure exhibits a trend of initially decreasing and then increasing. This phenomenon is primarily due to longitudinal modification shifting the contact position toward the middle of the tooth width, thereby avoiding edge contact and reducing contact stress. However, as the modification radius decreases, resulting in an increased modification amount, it also leads to a reduction in contact area, consequently increasing contact pressure.

In Figure 25, it is observed that as R increases incrementally from 50 mm to 500 mm, the number of engaged teeth near the lower end of the long axis follows a sequence of 12, 13, 13, and 13, respectively. Furthermore, regardless of the modification radius, tooth #41 consistently bears the heaviest load. Consequently, the modification radius has minimal influence on both the number of engaged teeth and the load sharing trend among teeth as the gear teeth’s position changes.

4.5. Effects of Torque on Contact Pressure and Load Sharing

In order to analyze the influence of torque on contact pressure and load sharing among teeth, four sets of different T values (T = 15 N·m, 30 N·m, 45 N·m, and 60 N·m) are taken for contact analysis. The results of this analysis are presented in Figure 26 and Figure 27, illustrating the variations in contact pressure and load sharing among teeth corresponding to different torque.

From Figure 26, it is observed that as torque T gradually increases from 15 N·m to 60 N·m, the maximum contact pressure follows a sequence of 608 MPa, 804 MPa, 947 MPa, and 1071 MPa, respectively. With the increase in torque, the maximum contact pressure also gradually increases. However, the rate of growth slows down. This phenomenon is primarily attributed to the increase in torque, leading to greater deformation of the teeth, reduced clearance between adjacent teeth working surfaces, and more gear teeth entering mesh. Additionally, it is notable that the increase in torque does not alter the position of the contact area of the teeth; rather, it only modifies the size of the contact area.

In Figure 27, it is observed that as torque T increases incrementally from 15 N·m to 60 N·m, the number of engaged teeth near the lower end of the long axis follows a sequence of 10, 13, 15, and 16, respectively. Therefore, as torque T increases, the number of engaged teeth also increases. Furthermore, regardless of the torque T, tooth #41 consistently bears the heaviest load. Hence, the torque T has minimal influence on the load sharing trend among teeth as the position of the gear teeth changes.

5. Conclusions

Shortening the FS increases deformation, reduces gear teeth meshing area, and heightens pressure concentration due to the increased cone angle. Hence, a longitudinal modification method for CS with DCTP teeth was proposed, along with an easy-to-manufacture shaping cutter for processing CS teeth. Additionally, an HD precision and parameterized finite element model was developed. The influences of several important HD parameters on contact pressure and load sharing have been studied. The main conclusions are as follows:

(1)

The longitudinal modification of CS teeth can significantly decrease contact pressure, enhance contact area, and prevent stress concentration at the tooth end for short FS HDs.

(2)

As the FS length-to-diameter ratio increases, the contact area spans the entire tooth surface along the tooth width. Conversely, with a shorter ratio, contact pressure rises sharply, and the contact area diminishes.

(3)

With a fixed modification radius, contact pressure initially decreases and then increases as the FS length-to-diameter ratio rises. Modification improves tooth contact for short FS HDs, but becomes unnecessary or even harmful to achieve optimal contact performance when the ratio exceeds a certain threshold.

(4)

When the WG profile coefficient deviates from the intended radial deformation for the tooth, it leads to tooth contact at the tip of the FS, causing stress concentration. Additionally, this ratio has the most significant influence on load sharing among teeth compared to other parameters.

(5)

As the modification radius increases, the contact pressure exhibits a trend of decreasing first and then increasing. For short FS HDs, there exists an optimal modification radius that can achieve the best contact state.

(6)

With the increase in torque, the maximum contact pressure gradually rises, although at a slower rate. It is notable that the increase in torque does not change the position of the teeth’s contact area but rather adjusts its size.