Three-Dimensional Conjugate Tooth Surface Design and Contact Analysis of Harmonic Drive with Double-Circular-Arc Tooth Profile

The Harmonic Drive (HD) consists mainly of three components: the flexspline (FS), circular spline (CS), and wave generator (WG). It is widely utilized in aerospace, industrial robotics, and other fields due to its compact structure, high transmission ratio, and precise transmission capabilities. However, the FS exhibits spatial features, particularly in the case of short-cup FS, due to its elastic coning deformation. To achieve optimal contact characteristics, it is essential to design the tooth surfaces of both the FS and CS as spatially conjugate surfaces. Therefore, establishing an HD kinematics model based on the coning deformation of the FS and designing a three-dimensional (3D) conjugate tooth surface for the FS hold critical significance.

Many researchers have carried out a lot of work in the field of HD structure design [1, 2], meshing theory [3‐5], tooth profile design [5‐9], backlash and meshing force distribution [10]. In terms of the design of HD conjugate tooth profile, Yang et al. [11] proposed an exact solution for two-dimensional (2D) conjugate profiles of zero-backlash HDs with elliptical cam WG. Dong et al. [1, 4] proposed an HD kinematics model considering the deformation of FS. They emphasized that in order to avoid unnecessary interference and deformation, one of the tooth surfaces of FS and CS must be a spatial tooth surface. Wu et al. [12] proposed that the tooth surface of CS adopts the spatial tooth surface, and that of the FS adopts the spur gear tooth surface. However, it is difficult to achieve mass production of CS spatial tooth surfaces. Considering the coning deformation of FS, Liu et al. [13] converted the design of the 3D tooth surface of CS into a 2D tooth profile design in multiple cross-sections. Wang et al. [14] discretized the FS tooth surface into many cross-sections normal to the rotating axis, and designed the FS tooth surface by reasonably adjusting the position of the FS tooth profile along the radius. Chen et al. [15] analyzed the spatial coning deformation features of the FS based on the straight generatrix assumption. In view of the problem that there is a certain deviation between the actual deformation and the assumed linear deformation, Li et al. [16] uses the finite element method to explore the influence of the structural parameters on the radial deformation of FS. The tooth surface of the FS is modified based on the simulation results of the FS deformation. Papers cited above treat the tooth surface of the harmonic gear as a series of discrete cross-sectional tooth profiles for designing the conjugate tooth surface. Some researchers have taken into account the impact of the FS coning deformation on the design of the tooth surface of harmonic gears. However, they have not thoroughly analyzed the coning deformation of the FS, resulting in qualitative conclusions instead of quantitative results. Hence, it is crucial to consider the manufacturing technology of both the FS and the CS, conduct a detailed analysis of the coning deformation of the FS, and develop a 3D tooth surface design method that aligns more closely with practical engineering requirements.

This paper presents a novel method for designing a 3D conjugate tooth surface with a double-circular-arc tooth profile in HD. The approach incorporates the radial deformation function of the FS obtained through FEA into the kinematics model of HD. By considering the hobbing process of FS, the position of the main section tooth profile along the radial direction is adjusted to achieve the 3D conjugate tooth surface of FS. Additionally, accounting for the coning deformation of FS, a multi-section assembly and meshing motion simulation of the 3D conjugate tooth surface are performed. Finally, the loaded contact characteristics of the FS are analyzed and compared using the finite element method, both with the spur gear tooth surface and the 3D conjugate tooth surface.

If the combined effect of complex factors such as the actual geometry of the flexible bearing and the transition fillet of the gear ring is considered, the kinematics of HD will be very complicated. Therefore, to simplify the problem, the following assumptions are made.

As shown in Figure 1, the coning deformation of the FS is considered to establish an HD coordinate system. The CS is fixed, the WG is the input, and the FS is the output. The coordinate system S(O-XYZ) is fixedly connected to the WG, the Z-axis coincides with the rotating axis, and the X-axis is the major axis of the WG. The coordinate system S₁(O₁-X₁Y₁Z₁) is fixedly connected to the FS tooth, the X₁ axis is the symmetrical line of the FS tooth profile, and the origin O₁ is the intersected point of the X₁ axis and the FS neutral curve. The coordinate system S₂(O₂-X₂Y₂Z₂) is connected to the CS, and the Z-axis also coincides with the rotating axis. At the initial position, the coordinate system S and S₂ coincide, the origin O₁ is located at the vertex of the WG major axis, and the 3 axes X₁, X₂ and X are collinear. The WG rotates counterclockwise by angle φ₂ relative to the X₂ axis, the designated point O₁ at the open end of the FS rotates clockwise by angle γ, and the closed-end rotates clockwise by angle α. The WG forces the neutral curve of the FS to undergo elliptical deformation, so that the position of O₁ point moves from O₁' to O₁ in Figure 1, and the normal direction changes from OO₁' to X₁ axis. After the WG is installed in the FS cylinder, the FS neutral curve in different sections normal to the rotating axis has different maximum radial displacements. a(z) is the semi-length of the major axis of the neutral curve in each section of the FS, which can be expressed as a function of z.where \({r}_{m}\) is the radius of the FS neutral surface before deformation, \({d}_{a}(z)\) is the function of the maximum radial displacement of the FS, which will be obtained by finite element analysis of the FS assembly deformation in Section 3.1.

The semi-length of the minor axis b(z) can be determined according to the assumption (4).

To facilitate elliptic integration, all kinematic parameters are expressed as a function of φ₁. φ₁ is the angle that designates the clockwise rotation of point O₁, located at the open end of the FS, relative to the X axis. According to the parametric equation of the standard ellipse, the polar radius of the neutral curve in each section can be expressed in polar coordinates aswhere \(r\) is the polar radius of OO₁.

According to Ref. [11], the equations of the kinematic parameters and their derivatives in Figure 1 can be derived as follows.where \(\mu\) is the angle between the polar radius OO₁ and the X₁ axis. \(\varepsilon\) is the second eccentricity of the ellipse, \(\varepsilon =\sqrt{{a\left(z\right)}^{2}-{b\left(z\right)}^{2}}/b\left(z\right)\).

The coordinate system S₂ shown in Figure 1 can be converted to S₁, the coordinate transfer matrix M₁₂ and the base vector transfer matrix W₁₂ can be expressed as follows.

According to the conjugate theory, the normal vector of the tooth surface and the relative velocity at the meshing contact point are perpendicular to each other. That is to say, the conjugation condition can be expressed as follows [11].where n_i is the normal vector of the contact point of two conjugate surfaces, and the subscript i represents the coordinate system S_i. v_i⁽¹²⁾ is the relative velocity vector of the contact point.

In the coordinate system S₁, substituting \({{\varvec{n}}}_{\mathbf{1}}={{\varvec{W}}}_{\mathbf{12}}{{\varvec{n}}}_{2}\) and \({{\varvec{v}}}_{\mathbf{1}}^{\left(\mathbf{12}\right)}=\frac{\mathrm{d}{{\varvec{r}}}_{\mathbf{1}}}{\mathrm{d}t}=\frac{\mathrm{d}{{\varvec{M}}}_{\mathbf{12}}}{\mathrm{d}t}{{\varvec{r}}}_{\mathbf{2}}\) into Eq. (12), it can be transformed intowhere,

Substituting \(\frac{\mathrm{d}{{\varvec{M}}}_{\mathbf{12}}}{\mathrm{d}{\varphi }_{1}}=\left[\begin{array}{ccc}-\dot{\beta }\mathrm{sin}\beta & -\dot{\beta }\mathrm{cos}\beta & r\dot{\mu }\mathrm{cos}\mu +\dot{r}\mathrm{sin}\mu \\ \dot{\beta }\mathrm{cos}\beta & -\dot{\beta }\mathrm{sin}\beta & r\dot{\mu }\mathrm{sin}\mu -\dot{r}\mathrm{cos}\mu \\ 0& 0& 0\end{array}\right]\) and \({{\varvec{W}}}_{\mathbf{12}}^{{\mathbf{T}}}=\left[\begin{array}{ccc}\mathrm{cos}\beta & \mathrm{sin}\beta & 0\\ -\mathrm{sin}\beta & \mathrm{cos}\beta & 0\\ 0& 0& 1\end{array}\right]\) into Eq. (15), then the following matrix can be obtained

In the cross-section where the Z coordinate value is z_i, the kinematic parameters and their derivatives are obtained according to Eqs. (1)–(9) and the maximum radial displacement function \({d}_{a}(z)\). The CS tooth profile curve \(\widetilde{C}\) is expressed as a function of arc length s. For each point on the tooth profile curve \(\widetilde{C}\), there is a parameter value s_j(j = 1, 2,…, s). The vectors r₂ and n₂ of each point are substituted into Eq. (13) to obtain the angle φ₁ when this point performs conjugate motion, which is recorded as φ_1j(j = 1, 2, …, s). Substituting φ_1j into Eq. (17), the theoretical tooth profile curve \(\widetilde{F}\) of the FS conjugated with the CS tooth profile curve \(\widetilde{C}\) can be obtained.

To thoroughly analyze the detailed elastic coning deformation of the FS, the finite element method is employed to study how the FS deforms when subjected to the influence of the WG. The design parameters of HD are shown in Table 1. Figure 2 shows the double-circular-arc tooth profile with common-tangent, and Table 2 shows the specific tooth profile parameters.

To assess the rationality and superiority of the designed 3D conjugate tooth surface, a multi-section assembly and meshing motion simulation of the machinable tooth surface is conducted. Figure 26 shows the assembly simulation results in 4 typical cross-sections of the machinable tooth surface. At the major axis of the WG, teeth 1 to 3 are in full engagement, and teeth 25 to 26 at the minor axis are completely disengaged. Affected by the coning deformation of the FS, in the fully engaged state, the tooth profile in the section near the closed end (z = 12 mm) is at the innermost side, and the tooth profile in the open end section (z = 0 mm) is at the outermost side. In the disengaged state, the position of the tooth profile in the two sections is just exchanged.

Figure 27 shows the meshing motion simulation results in 4 typical cross-sections of the machinable tooth surface. The red curve represents the movement trail of the origin O₁ fixedly connected with the FS tooth. When the WG rotates 360°, the FS rotates two more teeth relative to the CS. In the process of gear tooth meshing, there is no meshing interference between CS and FS gear teeth. Part of the tooth profile in each section is always in the meshing contact state. From the open end to the closed end, the movement trail of the FS teeth gradually becomes smoother. In addition, the radial displacement is gradually reduced, which is consistent with the actual situation. In the section near the closed end, the tooth pair of the FS and the CS only engage at the top of the tooth profile. In the remaining sections, the tooth pairs of the FS and the CS maintain a continuous meshing with a significant arc length. Figure 28 shows the backlash distribution in 4 typical cross-sections of the machinable tooth surface. The abscissa represents the number of teeth. The FS tooth at the major axis of the WG is marked as 1, and the other teeth are marked as 2, 3, 4, ..., n in the clockwise direction. The ordinate represents the backlash, and if there is interference between the tooth profiles, the backlash is negative. The backlash distribution in 4 typical cross-sections is drawn with different line types. Take the backlash value of less than 0.003 mm as the possible condition for meshing contact [20], and mark the backlash value involved in the meshing with a red circle. The results indicate that there is meshing engagement between certain gear teeth in each section of the FS. The number of gear teeth participating in the meshing from the section z = 0 mm to the section z = 12 mm are 3 (1st–3rd), 5 (1st–5th), 11 (3rd–13th), and 5 (9th–14th) respectively, and the gear teeth involved in meshing gradually move away from the major axis. Although the backlash distribution varies across each section, the overall backlash distribution of the FS, comprising all sections, is notably more uniform with minimal fluctuations.

The finite element model of the loaded harmonic gear contact analysis is established by adding the CS component to the finite element model of the FS coning deformation in Section 3.1. All parts are divided into hexahedral element shapes, as shown in Figure 29. The material properties are shown in Table 3. Three reference points RP-Wave, RP-Flex and RP-Rigid are established, which are coupled to the inner wall of the WG cam, the end face of the FS flange and the outer wall of the CS respectively. The analysis steps are set as follows:

Figure 30 shows the contact pressure distribution in the meshing area of the FS with spur gear tooth surface and the 3D machined tooth surface, respectively. In the FS with spur gear tooth surfaces, there are 7 teeth on one side and 14 on both, which accounts for 14% of the teeth. The contact patterns are mainly concentrated at the edge of the opening end, and the maximum contact pressure appears on the 6th tooth, with a value of 1256.5 MPa. The FS with the 3D machinable tooth surface has 12 teeth on one side (24 on both sides) to engage in meshing, accounting for 24%. More than 4/5 of tooth surfaces in the tooth width direction are engaged in meshing, and the meshing condition is favorable. The maximum contact pressure appears on the 8th tooth.

Figure 31 shows the maximum contact pressure distribution of the contacting gear teeth of the FS with a spur gear tooth surface and a 3D machinable tooth surface. The tooth surface of the 3D machinable gear engages with more teeth compared to the tooth surface of a spur gear, resulting in a more uniform load distribution among the teeth with reduced magnitude.

Figure 32 displays the contact pattern of FS with both a spur gear tooth surface and a 3D machinable tooth surface. In the case of FS with a spur gear tooth surface, contact occurs at the edge of the opening end. The contact pattern accounts for 16.46% of the full tooth surface, and the maximum contact pressure is 1210.14 MPa. The full tooth width direction of the FS with the 3D machinable tooth surface basically participates in meshing. The contact pattern accounts for 37.40% of the full tooth surface, which is 2.272 times that of the spur gear tooth surface. The main contact area is in the middle of the tooth width, and the maximum contact pressure (492.16 MPa) is only 40.67% of the spur gear tooth surface. Figure 33 shows the transmission error of a harmonic gear with a spur gear tooth surface and a 3D machinable tooth surface during the 180° rotation of the WG. Figure 34 shows the peak-to-peak value of transmission error (TE).

The results show that the transmission errors of the two tooth surfaces have no long-period TE. The peak-to-peak value and average value of TE of spur gear tooth surface are 0.0984 and 1.781 μrad respectively, and that of 3D machinable tooth surface are 0.0701 and 1.393 μrad respectively. The results confirm that the 3D machinable tooth surface designed in this paper exhibits superior transmission performance compared to the tooth surface of spur gears.

This paper presents a novel method for designing a 3D conjugate tooth surface of HD with a double-circular-arc tooth profile. Additionally, an HD kinematics model is established, which considers the FS coning deformation. The designed 3D machinable tooth surface is verified through motion simulation and finite element analysis, ensuring accurate backlash distribution and loaded tooth contact characteristics. The following are the main conclusions drawn from this study: