Meshing performance investigations on the curve constructed worm drive

Abstract

To reduce the error sensitivity of cylindrical worm drive, the curve constructed worm drive is proposed. Depend on the theory of constructing tooth surfaces by the conjugate curve, the meshing theory, and mathematical model of the curve constructed worm drive are built. The tooth contact analysis of curve constructed worm drive is proposed, and it is simulated by the numerical method and finite element method. The meshing characteristic of two-point contact is verified, and the sensitivity is examined in terms of center distance, shaft angle, and axial errors, as well as the finite element analysis is utilized to examine the tooth contact in the presence of errors. The analyzed results show that this novel worm drive has well adaptation to the errors. This study is intended to provide a new way forward in the design of point contact worm drive.

Introduction

The cylindrical worm transmission is a type of spatially staggered shaft transmission device that offers numerous advantages, including a large transmission ratio, compact structure, and smooth operation. As such, it finds widespread use across various industries. ¹ When designing a cylindrical worm drive, it is typically expected to have a linear contact between the cylindrical worm and the worm gear. However, achieving linear contact in the meshing process is challenging due to manufacturing and mounting errors. Because the line contact in conventional worm gear pairs is a theoretically fragile state of perfection, it is highly sensitive to both manufacturing and alignment errors. The presence of such errors causes the theoretically infinite contact line to degenerate into one or several isolated contact points and edge contacts. This shift in load-bearing behavior leads to an exponential increase in localized contact stress, which can easily trigger early-stage pitting, scuffing, and wear. In contrast, the point contact area, after running-in, forms a small elliptical patch in practical engineering applications. When errors such as center distance deviation, shaft angle misalignment, or manufacturing inaccuracies occur, this contact area can freely shift and adjust its shape on the tooth surface, thereby avoiding edge contact and stress concentration. Thus, the conventional worm gear pair evolves from a theoretically ideal line contact—representing a brittle optimum—into a point contact in engineering practice, exemplifying a robust sub-optimum. Hence, the substitution of a linear contact with a point contact can effectively reduce the sensitivity to errors in the worm drive.^2–5

Regarding point-contact tooth meshing, some studies have been finished by researchers. Seol^2,3 investigated the effects of contact shift and transmission errors on line-contact worm gears. The transfer point for the linear contact has been determined and computerized simulations have been developed. Litvin and Donno ⁴ and Litvin et al. ⁵ proposed a worm gear transmission that features localized contact for load bearing. Moreover, the influence of misalignment was investigated by TCA computer program. Litvin et al. ⁶ also proposed design of involute crossed helical gears. Litvin et al. ⁷ and Gonzalez-Perez et al. ⁸ proposed a novel geometry for worm gearing that effectively reduces the sensitivity of the gearing to error. Application of TCA confirms the reduced sensitivity to deviation and improved meshing conditions of this worm gearing. Shi et al. ⁹ investigated the effect of errors on the contact area and improved the engagement performance by localizing the contact zone to reduce the sensitivity to the loads and the errors. Simon^10,11 suggested the utilization of an oversized hob for machining the worm wheel. Aiming to diminish the sensitivity to errors and the mismatch contact of the worm transmission. And computer aided contact analysis of load-bearing tooth flanks completed. Dudás ¹² has proposed a novel type of worm gearing characterized by continuous pointwise meshing. Its main advantage is the lack of transmission errors and its ability to tolerate misalignments well. Meng et al.^13,14 investigated the mismatched meshing of conical surface enveloping worm gear and the mismatched meshing of ZC1 worm gear. The two worm drive pairs not only have good meshing performance, but also are insensitive to assembly errors. Zhao et al ¹⁵ investigated the meshing theory of nonspecific mismatched transmission on the example of a type of mismatch transmission. Chi et al. ¹⁶ analyzed an hourglass worm transmission constituted by a hourglass worm and a modified worm gear. The study rescues the surface that this worm gearing is capable of achieving ideal point contact and is insensitive to positioning errors. Chen et al. ¹⁷ proposed a design for a multitooth point contact hourglass worm transmission. This hourglass worm gear transmission can grind precisely. Ye et al. ¹⁸ studied a novel point contact double lead worm gear. The ability of the hourglass worm to realize backlash-adjustable is confirmed by TCA and finite element simulation. Zheng et al. ¹⁹ offered a novel hourglass worm drive with point contact. The contact performance and backlash-adjustable capability of this hourglass worm gearing are verified by computer simulation and prototype experiments. Li et al. ²⁰ proposed a novel point-contact hourglass worm gear based on modified involutes by modifying the involute gears. The analysis of meshing performance and error adaptation is completed. Ye et al. ²¹ studied a mismatched TI worm gearing. The contact characteristics of the worm gear can be affected by adjusting the parameters of the intermediate gear. This mismatched TI worm gearing performs well in terms of load carrying capacity and error adaptation has been verified. Ren^22,23 presented a point contact worm transmission constructed from two generating surfaces. Additionally, A method is further proposed for the generation of point-contact worm drive using two conjugate involute helical surfaces. The aforementioned point contact transmissions primarily consist of mismatched engagements resulting in localized contacts, or alternatively. Generally, the origins of these transmission can be attributed to the conjugate surfaces. In recent years, a novel approach for point contact drives has been proposed through the theory of constructing tooth surfaces according to conjugate curves.^24–26

On the basic of the curve conjugate theory, the curve constructed worm drive is proposed, the meshing theory and mathematical model are investigated. The TCA of the curve constructed worm drive is presented, and the characteristic of the two-point contact of the curve constructed worm drive are verified. Meanwhile, the effects of different meshing parameters on the meshing performance are investigated. An accurate errors analysis model is established in three dimensions, and the low error sensitivity of the curve constructed worm drive is examined through the finite element analysis.

Mathematical model

Coordinate systems

To elucidate the curve constructed worm drive, the coordinate system depicted in Figure 1 is established. The coordinate system S_m (O_m, x_m, y_m, z_m) and S_n (O_n, x_n, y_n, z_n) rigidly connected to the absolute space. The rotatable coordinate system S₁ (O₁, x₁, y₁, z₁) and S₂ (O₂, x₂, y₂, z₂) are rigidly connected to the worm and worm wheel, respectively. Furthermore, rotate around the axes z₁ and z₂ with the angular velocity vectors ω₁ and ω₂. The curve constructed worm drive makes a continuous meshing motion, at a given time, the worm angle around the z₁ axis is φ₁ and the worm wheel angle around the z₂ axis is φ₂. The parameter denoting the distance between the center points of the curve constructed worm drive is a.

Figure 1.

Coordinate systems of the curve constructed worm drive.

The conversion relationship from S₁ to S₂ is in the following:

$${\mathit{r}}_2={\mathit{M}}_{21}{\mathit{r}}_1={\mathit{M}}_{2\mathrm{m}}{\mathit{M}}_{\mathrm{mn}}{\mathit{M}}_{\mathrm{n}1}{\mathit{r}}_1$$

(1)

Where, r ₁ is the tooth surface coordinate vector of worm, r ₂ is the tooth surface coordinate vector of worm gear, and M ₂₁, M _2m, M _mn, and M _n1 are the coordinate transformation matrix.

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{M}}_{\mathrm{n}1}=\left[\begin{array}{cccc}\cos {\phi }_1\hfill & \sin {\phi }_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin {\phi }_1}& \cos {\phi }_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right],}\\ \multicolumn{1}{c}{{\mathit{M}}_{\mathrm{mn}}=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & -a\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right],}\\ \multicolumn{1}{c}{{\mathit{M}}_{2\mathrm{m}}=\left[\begin{array}{cccc}\cos {\phi }_2\hfill & \sin {\phi }_2\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin {\phi }_2}& \cos {\phi }_2\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]}\end{array}$$

Conjugate curve

In the curve constructed worm drive, the trajectory of the contact points of the worm is a cylindrical helix and the equation of the cylindrical helix Г₁ can be expressed in the following:

$${\Gamma }_1:\{\begin{array}{c}{\mathit{r}}_1=x_1{\mathit{i}}_1+y_1{\mathit{j}}_1+{\mathit{z}}_1{k}_1\hfill \\ \multicolumn{1}{c}{x_1=R\cos \theta }\\ \multicolumn{1}{c}{y_1=R\sin \theta }\\ \multicolumn{1}{c}{z_1=p\theta }\end{array}$$

(2)

Where, R is the radius of the cylindrical helix, θ is the angle of helix, p is the pitch.

At any given point along the cylindrical helix Г₁, it is possible to determine three vectors that are mutually orthogonal to each other, as illustrated in Figure 2. The vector α represents the tangential direction, β represents the principal normal direction, and γ represents the binormal direction. At the point P on the curve Г₁, the normal vector n₁ can be expressed by the principal normal vector and the binormal vector in any given direction. Hence, the normal vector n₁ can be mathematically represented in any given direction as follows ²⁷ :

$${\mathit{n}}_1=u\mathbf{\beta }+v\mathbf{\gamma }$$

(3)

Where, u and v are coefficients determined by the direction of the curve at point P.

Figure 2.

Normal vector of cylinder helix and triangulation of curve.

Furthermore, according to the force relationship of the curve constructed worm drive at the contact points P, the solution to the relationship between the normal pressure angle and the principal normal and binormal vectors can be determined. This establishes that n _n can be represented as the following equation:

$${\mathit{n}}_n=\sin {\alpha }_n\mathbf{\beta }+\cos {\alpha }_n\mathbf{\gamma }$$

(4)

Where, α _n is the normal pressure angle.

In the rotating coordinate system S₁, according to equations (2) and (3) above, the nor-mal vector n _n in any direction of the cylindrical helix Г₁ can be represented by the following equation:

$$\{\begin{array}{c}{\mathit{n}}_{\mathrm{n}}=n_{x1}{\mathit{i}}_1+n_{y1}{\mathit{j}}_1+n_{z1}{\mathit{k}}_1\hfill \\ \multicolumn{1}{c}{n_{x1}=u{\mathbf{\beta }}_{\mathbf{x}1}+v{\gamma }_{x1}}\\ \multicolumn{1}{c}{n_{y1}=u{\mathbf{\beta }}_{\mathbf{y}1}+v{\gamma }_{y1}}\\ \multicolumn{1}{c}{n_{z1}=u{\mathbf{\beta }}_{\mathbf{z}1}+v{\gamma }_{z1}}\end{array}$$

(5)

Where, u = sin α_n, v = cos α _n , [β_x1, β_y1, β_z1]^T, and [γ_x1, γ_y1, γ_z1]^T can be formulated as:

$$\{\begin{array}{c}{\left[\begin{array}{ccc}{\mathbf{\beta }}_{\mathbf{x}1}\hfill & {\beta }_{y1}\hfill & {\beta }_{z1}\hfill \end{array}\right]}^T={\left[\begin{array}{ccc}-\cos \theta \hfill & -\sin \theta \hfill & 0\hfill \end{array}\right]}^T\hfill \\ \multicolumn{1}{c}{{\left[\begin{array}{ccc}{\mathbf{\gamma }}_{\mathbf{x}1}\hfill & {\gamma }_{y1}\hfill & {\gamma }_{z1}\hfill \end{array}\right]}^T={\left[\begin{array}{ccc}\frac{p\sin \theta }{{(R^2+p^2)}^{1/2}}\hfill & -\frac{p\cos \theta }{{(R^2+p^2)}^{1/2}}\hfill & \frac{R}{{(R^2+p^2)}^{1/2}}\hfill \end{array}\right]}^T}\end{array}$$

To simplify calculations, assume ω₁ to be 1 rad/s. Consequently, the angular velocity, ω₂, determined as i₂₁ rad/s. In consequence, the relative velocity ${\mathit{v}}_1^{(12)}$ within S₁ is able to be expressed according to the principle of relative motion as follows:

$$\begin{array}{c}{\mathit{v}}_1^{\left(12\right)}=(-y_1+i_{21}{z}_1\cos {\phi }_1){\mathit{i}}_1+(x_1-i_{21}{z}_1\sin {\phi }_1){\mathit{j}}_1\hfill \\ +(-i_{21}{x}_1\cos {\phi }_1+i_{21}{y}_1\sin {\phi }_1+i_{21}a){\mathit{k}}_1\hfill \end{array}$$

(6)

The trajectory of contact for the curve constructed worm wheel is a conjugate curve, that is, produced by the cylindrical helix Г₁. On the basic of the meshing principle, curves Г₁ and Г₂ should be tangential during meshing and satisfy the following properties:

$$\Phi ={\mathit{n}}_{\mathrm{n}}·{\mathit{v}}_1^{(12)}=0$$

(7)

Substituting equations (5) and (6) into equation (7), the engagement equation of the curve constructed worm drive is obtained as:

$$\begin{array}{c}\multicolumn{1}{c}{\Phi =(u{\beta }_{x1}+v{\gamma }_{x1})(-y_1+i_{21}{z}_1\cos {\phi }_1)}\\ \multicolumn{1}{c}{+(u{\beta }_{y1}+v{\gamma }_{y1})(x_1-i_{21}{z}_1\sin {\phi }_1)}\\ +(u{\beta }_{z1}+v{\gamma }_{z1})(-i_{21}{x}_1\cos {\phi }_1+i_{21}{y}_1\sin {\phi }_1+i_{21}a)\hfill \\ \multicolumn{1}{c}{=0}\end{array}$$

(8)

According to the coordinate change and meshing principle, the trajectory of contact points of the curve constructed worm wheel can be dealt with as:

$$\{\begin{array}{c}{\mathit{r}}_2={\mathit{M}}_{21}{\mathit{r}}_1={\mathit{M}}_{2\mathrm{m}}{\mathit{M}}_{\mathrm{mn}}{\mathit{M}}_{\mathrm{n}1}{\mathit{r}}_1\hfill \\ \multicolumn{1}{c}{\Phi ({\phi }_1,\theta )=0}\end{array}$$

(9)

Tooth surface

By equidistant the cylindrical spiral along the direction normal to the given pressure angle, as shown in Figure 3, with an equidistance of h₁, the equation for the equidistance curve Г_h1 can be obtained as follows:

$${\Gamma }_{\mathrm{h}1}:\{\begin{array}{c}{\mathit{r}}_{\mathrm{h}1}(\theta )=\mathit{r}(\theta )+h_1{\mathit{n}}_{\mathrm{n}}\hfill \\ \multicolumn{1}{c}{x_{\mathrm{h}1}(\theta )=x_1(\theta )+h_1·n_{x1}}\\ \multicolumn{1}{c}{y_{\mathrm{h}1}(\theta )=y_1(\theta )+h_1·n_{y1}}\\ \multicolumn{1}{c}{z_{\mathrm{h}1}(\theta )=z_1(\theta )+h_1·n_{z1}}\end{array}$$

(10)

Where h₁ is the equidistant distance of equidistant curve Г_h1 to worm contact trajectory.

Figure 3.

Contact trajectory curves and the equidistant curves.

A circle can be created by taking the equidistant distance h₁ as the radius and any point on the equidistant curve Г_h1 as the center of the circle. A circular coordinate system S_u (O_u, x_u, y_u, z_u) is established, as shown in Figure 4. Any point Q on the circle can be determined by the parameters ϕ, together with the parametric equation of the circle can be obtained as follows:

$$\{\begin{array}{c}{\mathit{r}}_{\mathrm{u}}(\varphi )=x_{\mathrm{u}}(\varphi ){\mathit{i}}_{\mathrm{u}}+y_{\mathrm{u}}(\varphi ){\mathit{j}}_{\mathrm{u}}+z_{\mathrm{u}}(\varphi ){\mathit{k}}_{\mathrm{u}}\hfill \\ \multicolumn{1}{c}{x_{\mathrm{u}}(\varphi )=h_1\cos \varphi }\\ \multicolumn{1}{c}{y_{\mathrm{u}}(\varphi )=h_1\cos \varphi }\\ \multicolumn{1}{c}{z_{\mathrm{u}}(\varphi )=0}\end{array}$$

(11)

Where ϕ is the spiral parameter.

Figure 4.

Coordinate system of the circle.

The surface Σ₁ is generated by the continuous translation of the circle along the equidistant curve Г_h1, as depicted in Figure 5. By employing a transformation relation from S_u to S₁, the surface Σ₁ can be represented:

$$\{\begin{array}{c}{\mathit{r}}^{\mathrm{I}}=x^{\mathrm{I}}{\mathit{i}}^{\mathrm{I}}+y^{\mathrm{I}}{\mathit{j}}^{\mathrm{I}}+z^{\mathrm{I}}{\mathit{k}}^{\mathrm{I}}\hfill \\ \multicolumn{1}{c}{{\mathit{r}}^{\mathrm{I}}={\mathit{M}}_{1\mathrm{u}}·{\mathit{r}}_{\mathrm{u}}}\end{array}$$

(12)

Where, M _1u represents the transformation matrix which from S_u to S₁.

Figure 5.

Circular swept round tube.

Through calculation, the tooth surface Σ₁ has been determined as follows:

$$\{\begin{array}{c}{\mathit{r}}^{\mathrm{I}}=x^{\mathrm{I}}{\mathit{i}}^{\mathrm{I}}+y^{\mathrm{I}}{\mathit{j}}^{\mathrm{I}}+z^{\mathrm{I}}{\mathit{k}}^{\mathrm{I}}\hfill \\ \multicolumn{1}{c}{x^{\mathrm{I}}=x_{\mathrm{h}1}+h\cos \theta \cos (a_n+\varphi )+\mathrm{ph}\sin \theta \sin (a_n+\varphi )/(p^2+R^2{)}^{1/2}}\\ \multicolumn{1}{c}{y^{\mathrm{I}}=y_{\mathrm{h}1}+h\sin \theta \cos (a_n+\varphi )-\mathrm{ph}\cos \theta \sin (a_n+\varphi )/(p^2+R^2{)}^{1/2}}\\ \multicolumn{1}{c}{z^{\mathrm{I}}=z_{\mathrm{h}1}+\mathrm{Rh}\sin (a_n+\varphi )/(p^2+R^2)}\end{array}$$

(13)

Where, R is the equidistant distance of constructed surface.

Similarly, the tooth surface Σ₂ has been determined to be as follows:

$$\{\begin{array}{c}{\mathit{r}}^{\mathrm{II}}=x^{\mathrm{II}}{\mathit{i}}^{\mathrm{II}}+y^{\mathrm{II}}{\mathit{j}}^{\mathrm{II}}+z^{\mathrm{II}}{\mathit{k}}^{\mathrm{II}}\hfill \\ \multicolumn{1}{c}{x^{\mathrm{II}}=x_{\mathrm{h}2}+h_2\cos {\phi }_2\sin ({\phi }_1+\theta )\cos (a_n+\varphi )}\\ \multicolumn{1}{c}{+(h_{2}R\sin {\phi }_2+h_{2}p\cos {\phi }_2\sin ({\phi }_1+\theta ))\sin (a_n+\varphi )/(p^2+R^2{)}^{1/2}}\\ \multicolumn{1}{c}{y^{\mathrm{II}}=y_{\mathrm{h}2}+h_2\sin {\phi }_2\cos ({\phi }_1+\theta )\cos (a_n+\varphi )}\\ \multicolumn{1}{c}{+(h_{2}R\cos {\phi }_2+h_{2}p\sin {\phi }_2\sin ({\phi }_1+\theta ))\sin (a_n+\varphi )/(p^2+R^2{)}^{1/2}}\\ \multicolumn{1}{c}{z^{\mathrm{II}}=z_{\mathrm{h}2}+h_2\sin ({\phi }_1+\theta )\cos (a_n+\varphi )}\\ \multicolumn{1}{c}{-h_{2}p\sin ({\phi }_1+\theta )\sin (a_n+\varphi )/(p^2+R^2{)}^{1/2}}\end{array}$$

(14)

As illustrated in Figure 6, the tooth surface on one side of the curve constructed worm drive, is respectively, generated by intercepting the above-mentioned surfaces Σ₁ and Σ₂ using the root and the tip cylindrical surface of the curve constructed worm and worm wheel, etc.

Figure 6.

Intercepted tooth surface of the curve constructed worm drive.

Contact types

By adjusting the equidistant direction and equidistant distance along the normal equidistant curve, this point contact worm gearing can obtain three types of contact as shown in Figure 7.

1. Convex to convex contact: Considering curves Г_h1 and Г_h2 equidistant in opposite directions. By taking the positive and negative values of h₁ and h₂ in equations (13) and (14), the convex-convex contact mode is obtained.

2. Convex to plane contact: When the equidistance distance h₂ between Г₂ and Г_h2 approaches infinity, the tooth profile curve of the worm wheel in the normal plane becomes a straight line, resulting in the attainment of the convex-plane contact mode.

3. Convex to concave contact: When the curves Г_h1 and Г_h2 have the same equidistant direction, that is, the parameters h₁ and h₂ take the same positive and negative, with h₂ taking a larger value compared to h₁, the convex concave contact is obtained.

Figure 7.

Three types of contact for the curve constructed worm drive.

Constructed worm drive

By the method of constructing the tooth surfaces of the curve constructed worm drive described earlier. Selecting any of the contact types and then taking the appropriate parameters, the curve constructed worm drive can be constructed. As an example, a convex-plane contact mode of the curve constructed worm drive is constructed as depicted in Figure 8.

Figure 8.

Convex-plane contact mode of the curve constructed worm drive.

Tooth contact analysis

Two-point contact characteristic

The above description indicates that the curve constructed worm drive is point contact. The surface of the worm tooth exhibits a cylindrical helix as the trajectory curve of contact. Obviously, the trajectory curve of contact on the worm wheel is part of the conjugate curve to the cylindrical helix on the worm. As shown in Figure 9, the green surface represents the worm gear tooth surface, and the red surface denotes the worm tooth surface. The intersecting area between the two is the meshing contact zone of the transmission pair. It can be observed that Figure 9(a) illustrates the meshing-in region of the curve constructed worm drive, presenting a long elliptical contact in the middle of the worm gear tooth surface. Figure 9(b) shows the mid-meshing region of the curve constructed worm drive, where the contact part splits into two short ellipses. Figure 9(c) depicts the meshing-out region of the curve constructed worm drive, with the two contact regions transforming into even shorter ellipses. During the course of the meshing process, these two contact points progressively move away along the contact trajectory. During the whole meshing process, the contact trajectory is V-shaped.

Figure 9.

Two-point contact for the curve constructed worm drive: (a) begin of meshing, (b) middle of meshing, and (c) end of meshing.

Principal directions and curvatures

The process how the worm tooth surface is generated may be described by the movement of a tooth profile curve along the cylindrical helix Г_h1. During meshing, at any contact point, the tangential vector of the tooth curve is always perpendicular to the tangential vector of the cylindrical helix. Therefore, $F={\mathit{r}}_{\theta }^{\mathrm{I}}·{\mathit{r}}_{\varphi }^{\mathrm{I}}=0$ and $M={\mathit{r}}_{\theta \varphi }^{\mathrm{I}}·{\mathit{n}}_{n1}=0$ are founded. In addition, it is worth mentioning that the two principal directions of the worm tooth surface are same as the tangential vector of the cylindrical helix and the profile tooth curve, respectively. Therefore, the principal curvature $K_i^{\mathrm{I}}(i=1,2)$ and principal direction $e_i^{\mathrm{I}}(i=1,2)$ of the worm tooth surface can be expressed as:

$$\{\begin{array}{c}{K}_1^{\mathrm{I}}=N^{\mathrm{I}}/G^{\mathrm{I}},e_1^{\mathrm{I}}=r_{\theta }/|r_{\theta }|,d_{\theta }\ne 0\hfill \\ \multicolumn{1}{c}{K_2^{\mathrm{I}}=L^{\mathrm{I}}/E^{\mathrm{I}},e_2^{\mathrm{I}}=r_{\varphi }/|r_{\varphi }|,d_{\varphi }\ne 0}\end{array}$$

(15)

Where, $N^{\mathrm{I}}={\mathit{r}}_{\theta \theta }^{\mathrm{I}}·{\mathit{n}}_{n1}$ ; $G^{\mathrm{I}}={\mathit{r}}_{\theta }^{\mathrm{I}}·{\mathit{r}}_{\theta }^{\mathrm{I}}$ ; $L^{\mathrm{I}}={\mathit{r}}_{\varphi \varphi }^{\mathrm{I}}·{\mathit{n}}_{n1}$ ; $E^{\mathrm{I}}={\mathit{r}}_{\varphi }^{\mathrm{I}}·{\mathit{r}}_{\varphi }^{\mathrm{I}}$ .

Similarly, the formation of the worm wheel tooth surface is easily described by the movement of a tooth profile curve along the conjugate curve Г_h2. During meshing, at any contact point, the tangential vectors of the tooth curve and the tangential vectors of the generating line are perpendicular to each other, and the $F={\mathit{r}}_{\theta }^{\mathrm{II}}·{\mathit{r}}_{\varphi }^{\mathrm{II}}=0$ and $M={\mathit{r}}_{\theta \varphi }^{\mathrm{II}}·{\mathit{n}}_{n1}=0$ still available.

Therefore, the principal curvature $K_i^{\mathrm{II}}(i=1,2)$ and principal direction $e_i^{\mathrm{II}}(i=1,2)$ of the worm wheel tooth surface is able to be obtained as:

$$\{\begin{array}{c}{K}_1^{\mathrm{II}}=N^{\mathrm{II}}/G^{\mathrm{II}},e_1^{\mathrm{II}}=r_{\theta }/|r_{\theta }|,d_{\theta }\ne 0\hfill \\ \multicolumn{1}{c}{K_2^{\mathrm{II}}=L^{\mathrm{II}}/E^{\mathrm{II}},e_2^{\mathrm{II}}=r_{\varphi }/|r_{\varphi }|,d_{\varphi }\ne 0}\end{array}$$

(16)

Where, $N^{\mathrm{II}}={\mathit{r}}_{\theta \theta }^{\mathrm{II}}·{\mathit{n}}_{n2}$ , $G^{\mathrm{II}}={\mathit{r}}_{\theta }^{\mathrm{II}}·{\mathit{r}}_{\theta }^{\mathrm{II}}$ , $L^{\mathrm{II}}={\mathit{r}}_{\varphi \varphi }^{\mathrm{II}}·{\mathit{n}}_{n2}$ , $E^{\mathrm{II}}={\mathit{r}}_{\varphi }^{\mathrm{II}}·{\mathit{r}}_{\varphi }^{\mathrm{II}}$ .

Based on the engagement relationship between the tooth surfaces of the curve constructed worm drive, at the contact point, $e_1^{\mathrm{I}}$ and $e_1^{\mathrm{II}}$ are oriented in the same direction. On the other hand, $e_2^{\mathrm{I}}$ and $e_2^{\mathrm{II}}$ may be oriented in the same or opposite directions, depending on the type of contact.

Further analyzed, the relationship at the point of contact between the curvature of the cylindrical helix and the principal curvature of the tooth surface in the direction $e_1^{\mathrm{I}}$ is shown in Figure 10.

Figure 10.

Curvature relations for curve and curved surface.

δ represents the angle between the osculating plane and the normal plane of the tooth surface. On the basic of Meusnier theorem, ²⁷ the mathematical equation that represents the curvature connecting the curve Г_h1 and the worm tooth surface Σ₁ in the direction $e_1^{\mathrm{I}}$ will be determined as follows:

$$|K_1^{\mathrm{I}}|=K_1|\cos \delta |$$

(17)

Combine with Figure 2, there exists the relation $\delta =90-{\alpha }_{\mathrm{n}}$ . Hence, the normal curvature $K_1^{\mathrm{I}}$ is obtained as:

$$K_1^{\mathrm{I}}=K_1\sin {\alpha }_{\mathrm{n}}$$

(18)

Solve the curvature of Г₁ and simplify. Substitute into equation (18) and the normal curvature $K_1^{\mathrm{I}}$ can be expressed as:

$$K_1^{\mathrm{I}}=R\sin {\alpha }_{\mathrm{n}}/(R^2+p^2)$$

(19)

Similarly, the normal curvature $K_1^{\mathrm{II}}$ can be obtained:

$$K_1^{\mathrm{II}}=R\sin {\alpha }_{\mathrm{n}}/(R^2+p^2)$$

(20)

In the direction $e_2^{\mathrm{I}}$ and $e_2^{\mathrm{II}}$ , at the point of contact, the curvature of the tooth surface is tantamount to the curvature of the tooth profile curve in the normal plane. Therefore, the curvature $K_2^{\mathrm{I}}$ and $K_2^{\mathrm{II}}$ can be expressed as:

$$\{\begin{array}{c}{K}_1^{\mathrm{I}}=1/h_1\hfill \\ \multicolumn{1}{c}{K_2^{\mathrm{II}}=1/h_2}\end{array}$$

(21)

Contact ellipse

The contact points of the two tooth surfaces experience pressure when the curve constructed worm drive meshes, resulting in elastic deformation. This deformation causes the formation of a transient contact ellipse as depicted in Figure 11. ξ and η are used for representing the directions of the semi-major axis and the semi-minor axis, respectively.

Figure 11.

Contact ellipse of the curve constructed worm drive.

Depending on Hertzian point contact theory ²⁸ and gear geometry theory, ²⁷ the mathematical description of the contact ellipse can be determined as:

$$A{\xi }^2+B{\eta }^2=\pm \mathrm{\Delta }$$

(22)

Where, Δ represents the amount of elastic deformation. A and B are represented as:

$$\begin{array}{c}A=\frac{1}{4}((K_1^{\mathrm{I}}+K_2^{\mathrm{I}})-(K_1^{\mathrm{II}}+K_2^{\mathrm{II}})-\sqrt{{(K_1^{\mathrm{I}}-K_2^{\mathrm{I}})}^2-2(K_1^{\mathrm{I}}-K_2^{\mathrm{I}})(K_1^{\mathrm{II}}-K_2^{\mathrm{II}})\cos \sigma +{(K_1^{\mathrm{II}}-K_2^{\mathrm{II}})}^2})\hfill \\ B=\frac{1}{4}((K_1^{\mathrm{I}}+K_2^{\mathrm{I}})-(K_1^{\mathrm{II}}+K_2^{\mathrm{II}})+\sqrt{{(K_1^{\mathrm{I}}-K_2^{\mathrm{I}})}^2-2(K_1^{\mathrm{I}}-K_2^{\mathrm{I}})(K_1^{\mathrm{II}}-K_2^{\mathrm{II}})\cos \sigma +{(K_1^{\mathrm{II}}-K_2^{\mathrm{II}})}^2})\hfill \end{array}$$

(23)

Where, $K_i^{\mathrm{I}}$ and $K_i^{\mathrm{II}}$ (i = 1, 2) obtained from equations (15) to (16), respectively; σ represents the angle between $e_1^{\mathrm{I}}$ and $e_1^{\mathrm{II}}$ at the point of contact.

Induced normal curvature

The induced curvature quantifies how close two surfaces are along a particular tangential direction. This is also the important indicator for evaluating the strength of the contact between two surfaces. It represents the relative principal curvature of both tooth surfaces in worm gear transmission. Theoretically, a reduction in the induced normal curvature enables worm transmission to have better meshing characteristics.

According to the Euler equation the normal curvatures $k_{\mathrm{n}}^{\mathrm{I}}$ and $k_{\mathrm{n}}^{\mathrm{II}}$ of the both tooth surfaces of the curve constructed worm drive along the same direction at the point of contact can be expressed, respectively as:

$$\begin{array}{c}{k}_{\mathrm{n}}^{\mathrm{I}}=k_1^{\mathrm{I}}{\cos }^2{q}_1+k_2^{\mathrm{I}}{\sin }^2{q}_1\hfill \\ k_{\mathrm{n}}^{\mathrm{II}}=k_1^{\mathrm{II}}{\cos }^2{q}_2+k_2^{\mathrm{II}}{\sin }^2{q}_2\hfill \end{array}$$

(24)

Where, q₁ is the angle between e and $e_1^{\mathrm{I}}$ , q₂ is the angle between e and $e_1^{\mathrm{II}}$ , e enable to be oriented at any angle relative to $e_1^{\mathrm{I}}$ .

Further, $k_{\mathrm{n}}^{\mathrm{I}-\mathrm{II}}$ can be expressed as:

$$k_{\mathrm{n}}^{\mathrm{I}-\mathrm{II}}=k_{\mathrm{n}}^{\mathrm{I}}-k_{\mathrm{n}}^{\mathrm{II}}$$

(25)

Since the angle σ between the principal directions $e_1^{\mathrm{I}}$ and $e_1^{\mathrm{II}}$ on tooth surface Σ₁ and Σ₂ is 0. Therefore, the two principal directions of the induced curvature are the same as $e_1^{\mathrm{I}}$ and $e_2^{\mathrm{I}}$ , respectively. These two principal directions of induced curvature will now be named $e_{\mathrm{n}}^1$ and $e_{\mathrm{n}}^2$ .

The expression for the curvature of both tooth surfaces in the $e_{\mathrm{n}}^1$ direction is the same at the contact point. Therefore, in this direction, the induced curvature $k_{\mathrm{n}}^1$ is 0. In the $e_{\mathrm{n}}^2$ direction, the induced curvature $k_{\mathrm{n}}^2$ can be obtained as follows:

$$k_{\mathrm{n}}^2=1/h_1-1/h_2$$

(26)

Numerical examples

The design parameters are given in Table 1 to further analyze the curve constructed worm drive.

Table 1.

Design parameters of the curve constructed worm drive.

Parameters	Value
Thread direction	Right
Center distance, a (mm)	41
Axial module of the worm, mx1 (mm)	2
Numbers of the worm threads, Z1	2
Numbers of the worm wheel teeth, Z2	33
reference lead angle of worm, γx1 (°)	14.04
Width of the curve constructed worm wheel, B (mm)	14
Equidistant distance of worm tooth contact curve, h1 (mm)	4
Equidistant distance of worm wheel tooth contact curve, h2 (mm)	∞

Contact trajectory

Contact points distribution is a significant index used to assess the meshing performance of worm transmission. Investigating the effect of key design parameters on the contact trajectory can contribute a valuable reference to the design of the curve constructed worm transmission.

While keeping parameters such as center distance unchanged, this analysis focuses on separately evaluating the effects of lead angle γ and normal pressure angle α_n on the contact trajectory. Based on equation (2), the connection between γ and R can be presented as:

$$\tan \gamma =\frac{m{z}_1}{2R}$$

(27)

That is, when exploring the effect of the lead angle on the contact trajectory, it is possible to analyze it by considering the helix radius of the cylindrical helix, the modulus, and the numbers of worm threads. Figure 12(a) to (c) show the change in contact trajectory when one of the parameters is changed, respectively. Besides, the effect of normal pressure angle α_n on the trajectory of contact is analyzed, as shown in Figure 12(d).

Figure 12.

Contact trajectory of the curve constructed worm wheel surface: (a) helix radius R, (b) worm threads Z₁, (c) axial module of the worm m _xl , and (d) normal pressure angle a _n .

From Figure 12, the contact points are projected onto the curve constructed worm wheel tooth surface whenever the worm is turned through an equal angle. The regularity of the contact point distribution throughout the meshing process is obtained.

Typically, the contact points exhibit a sparse distribution in proximity to the central region of the tooth surface, whereas they are densely distributed in close proximity to both sides of the tooth surface. In Figure 12(a) and (c), when the helical radius R of the cylindrical helix and the module m _{x 1} of the worm axial direction increase, the initial emergence location of the contact point moves towards the positive semi-axis of the y₂-axis. From Figure 12(b), as the number of worm heads increases, the more dispersed the contact points will be. From Figure 12(d), it can be seen that α_n exerts a marginal influence on the contact trajectory within the negative semi-axis of the y₂-axis. Nevertheless, within the positive semi-axis of the y₂-axis, the smaller the α_n, the shorter the contact trajectory and the denser the distribution.

Contact ellipse

Since the value of elastic deformation does not influence the shape of the contact ellipse, given the elastic deformation Δ = 0.00635 mm, the laws of different parameters on the contact ellipse shape are discussed. According to equations (22) and (23), the size of the contact ellipse is related to $K_i^{\mathrm{I}}(i=1,2)$ and $K_i^{\mathrm{II}}(i=1,2)$ . From equations (19) to (21) for the calculation of the principal curvatures, the effects of changes in $K_1^j(j=\mathrm{I},\mathrm{II})$ and $K_2^{\mathrm{II}}$ on the contact ellipse can be discussed in terms of α_n and h₂, respectively. The influence of analyzing h₂ and αn on the contact ellipse shape are analyzed is presented in Figure 13.

Figure 13.

Contact ellipse of the curve constructed worm wheel surface: (a) normal pressure angle α_n and (b) equidistant distance h₂.

As shown in Figure 13(a), a modification of α_n will lead to a change in the semi-major axis of the contact ellipse. As α_n increases, there is a corresponding decrease in the value of the semi-major axis of the contact ellipse. From Figure 13(b), combined with equation (21), the semi-minor axis of the contact ellipse gradually converges to the red ellipse as h₂ gradually converges to infinity.

Induced normal curvature

Based on equation (26), keeping h₁ constant and changing the value of h₂, the change of induced normal curvature $k_{\mathrm{n}}^2$ will be obtained as shown in Figure 14(a). When the value of h₂ tends to infinity, $k_{\mathrm{n}}^2$ converges to 1/h₁. In this particular case, the tooth contact mode exhibited by the curve constructed worm drive can be interpreted as convex to plane contact. For the convex-plane contact, the change of induced normal curvature is obtained by changing the value of h₁ as shown in Figure 14(b).

Figure 14.

Induced normal curvature of the curve constructed worm drive: (a) induced normal curvature of three contact types and (b) induced normal curvature of convex-plane contact.

Finite element analysis of tooth contact

An accurate 3D model of the curve constructed worm drive is built according to the previously derived equations and considered the parameters given in Table 1. As presented in Figure 15, the 3D model is convex-plane contact.

Figure 15.

Precision 3D modeling of the curve constructed worm drive.

To perform finite element static structural analysis on the 3D model, it firstly be simplified. Afterwards, the boundary conditions for the static structural analysis are set, as shown in Figure 16. Assuming the worm is stationary with respect to surface I which is fully constrained rigid. Besides, the worm wheel can only rotate around z₁ as a result of applying cylindrical constraints on the rigid surface II. Define the material of the worm as structural steel, while the material of the worm gear is defined as PA66-GF50. Hexahedral mesh with 0.1 mm is used, and there are 823,566 elements with 1,332,573 nodes in the finite element model. Additionally, applying a torque T₂ with 33 Nm on the rigid surface II.

Figure 16.

Boundary conditions in finite element analysis.

Upon completing the finite simulation, Figure 17 illustrates the contact state of the curve constructed worm drive. Evidently, there are two points of contact between the worm surface and worm wheel surface. Therefore, the practical contact points are consistent with the above theory.

Figure 17.

Contact points on the curve constructed worm wheel: (a) left tooth surface and (b) right tooth surface.

Now contact state is analyzed of the curve constructed worm drive at zero error. As presented in Figure 18, the contact state of the curve constructed worm drive is determined using finite element simulation. From Figure 18, the curve constructed worm drive exhibits the characteristic of two-point contact under the action of torque.

Figure 18.

Tooth contact state of the curve constructed worm drive under zero error.

Error sensitivity analysis

The curve constructed worm drive is mainly designed to diminish the sensitivity of cylindrical worm drive to errors. It serves to reduce the machining accuracy requirement, reduce the break-in time and improve the industrial efficiency.

As shown in Figure 19, the error model of the curve constructed worm drive is created. It includes center distance error Δa, shaft angle error Δλ, worm axial error ΔL₁, and worm wheel axial error ΔL₂.

Figure 19.

Error model schematic of the curve constructed worm drive.

Profile errors of tooth surface

Finite element analysis is carried out for the curve constructed worm drive, considering center distance errors of Δa = +0.1 mm and Δa = −0.1 mm, respectively. The tooth contact state is illustrated in Figure 20, taking center distance error into account.

Figure 20.

Tooth contact state of the curve constructed worm drive under center distance error: (a) tooth surface of worm gear with Δa = + 0.1 mm, (b) tooth surface of worm with Δa = + 0.1 mm, (c) tooth surface of worm gear with Δa = − 0.1 mm, and (d) tooth surface of worm with Δa = − 0.1 mm.

From Figure 20, with Δa = +0.1 mm and Δa = −0.1 mm, the contact state of the curve constructed worm drive is still two-point contact. When Δa = +0.1 mm, the tooth surface contact stress increases by 10% compared to the zero error case. When Δa = −0.1 mm, the tooth surface contact stress is almost unchanged compared to the zero errors case.

Shaft angle error

The analysis of the tooth contact state of the curve constructed worm drive is conducted in the case of shaft angle error. As illustrated in Figure 21, finite element simulation of the tooth contact is performed under Δλ = +0.5° and Δλ = −0.5°.

Figure 21.

Tooth contact state of the curve constructed worm drive under shaft angle error: (a) tooth surface of worm gear with Δλ = + 0.5°, (b) tooth surface of worm with Δλ = + 0.5°, (c) tooth surface of worm gear with Δλ = − 0.5°, and (d) tooth surface of worm with Δλ = − 0.5°.

From Figure 21, the contact points of the curve constructed worm drive are diminished under shaft angle error. In case of axis intersection angle error Δλ = +0.5°, the tooth contact stress increases by 26% compared to the zero error case. In case of axis intersection angle error Δλ = −0.5°, the tooth contact stress increases by 60% as com-pared to the zero errors case.

Axial error

As illustrated in Figure 22, the tooth contact state of the curve constructed worm drive under axial error is analyzed. The tooth contact state with axial error is analyzed using finite element simulation for the worm drive with variations of ΔL₁ = +0.1 mm, ΔL₁ = −0.1 mm, ΔL₂ = +0.1 mm, and ΔL₂ = −0.1 mm.

Figure 22.

Tooth contact state of the curve constructed worm drive under axial error: (a) tooth surface of worm gear with ΔL₁ = + 0.1 mm, (b) tooth surface of worm with ΔL₁ = + 0.1 mm, (c) tooth surface of worm gear with ΔL₁ = − 0.1 mm, (d) tooth surface of worm with ΔL₁ = − 0.1 mm, (e) tooth surface of worm gear with ΔL₂ = + 0.1 mm, (f) tooth surface of worm with ΔL₂ = + 0.1 mm, (g) tooth surface of worm gear with ΔL₂ = − 0.1 mm, and (h) tooth surface of worm with ΔL₂ = −0.1 mm.

From Figure 22, the tooth contact state of the curve constructed worm drive remains practically unchanged for the worm axial errors ΔL₁ = +0.1 mm and ΔL₁ = −0.1 mm. The stress on the tooth surface is even decreased compared to the zero error case. For the worm wheel axial error ΔL₂ = +0.1 mm, the tooth surface stress increase by 30%. However, when ΔL₂ = −0.1 mm, the tooth surface stress is basically unchanged. There-fore, positive the worm wheel axial errors should be avoided as much as possible of the curve constructed worm drive.

Conclusions

In the research performed in the paper, the following conclusions are obtained:

1. The curve constructed worm drive with curve conjugation is proposed, presenting a mathematical model for this novel transmission, and this transmission exhibits three types of tooth contact modes.

2. The TCA of the curve constructed worm drive is presented. The verification of the characteristic of two-point contact of the curve constructed worm drive is conducted through finite element static structure analysis.

3. The model to analyze the error sensitivity of the curve constructed worm drive has been developed. This study employs finite element analysis to investigate the error sensitivity in terms of center distance, shaft angle, and axial errors. The curve constructed worm drive exhibits excellent adaptability to center distance and axial errors. However, it is relatively poorly adapted to shaft angle errors.