Mismatched ZN-type worm pair meshing theory

Abstract

Cylindrical worms are sensitive to errors; mismatch can solve this problem. However, applying the finite element method to calculate the meshing characteristics of mismatched worm pairs has several limitations. Therefore, the meshing theory of mismatched ZN-type worm pairs is established using a numerical method. The contact equation of the mismatched ZN-type worm pair is solved numerically, and the error curves of the worm gear angle, transmission ratio, and worm gear angular velocity are obtained. The influence of different mismatched parameters on meshing conditions is studied, and the research results can guide how to configure different mismatched parameters. The results show that the mismatched ZN-type worm pair has a small transmission error, slight angular acceleration, stable operation, and good meshing characteristics. The numerical method has good reusability, which is more conducive to observing the influence of related factors on mismatch.

Keywords

ZN-type worm pair contact ellipse half axis mismatched worm pair meshing characteristic nonlinear equation

Introduction

Characterized by their simple structure and broad industrial applicability, ZN-type worm pairs are commonly employed to transmit torque between spatially offset shafts.¹ The worm features a straight-sided normal tooth profile, manufactured using a straight-edged lathe tool to ensure geometric precision.²

As a type of line-contact cylindrical worm drive, ZN-type worm pairs exhibit inherent sensitivity to various manufacturing and assembly errors.³ In practical applications, however, unavoidable deviations from ideal geometry mean that theoretical line contact rarely fully materializes.⁴ Converting nominal line contact to point contact has been identified as an effective solution to this challenge,⁵ as it mitigates the detrimental effects of real-world imperfections on meshing performance.

Point contact is known to mitigate error sensitivity in worm gear drives.⁶ During the meshing of a mismatched worm gear pair, the instantaneous contact point evolves into a contact ellipse under external load, with successive elliptical contact regions forming a continuous contact path. The geometric characteristics of this cumulative contact area ensure adequate load-carrying capacity by effectively distributing contact stresses, thereby balancing error tolerance and transmission performance.

Currently, the finite element method (FEM) is commonly employed for numerical calculations of tooth surface contact paths, yet it has inherent limitations. This approach is constrained by the accuracy of tooth surface modeling and is susceptible to external factors, leading to poor reusability; even under identical conditions, results may vary significantly. Moreover, its substantial computational workload hinders systematic investigations into the influences of relevant parameters. To address these issues, numerical calculations are instead conducted by establishing analytical expressions for tooth surface contact, offering a more controlled and efficient framework for such analyses.

To determine the contact area of mismatched ZN-type worm pair tooth surfaces, the worm tooth surface equation must first be established, followed by deriving the worm gear tooth surface equation using the meshing principle. Once the worm gear tooth surface equation is obtained, the system must satisfy the point-contact meshing condition, which serves as a constraint for formulating the governing system of equations. Typically, such systems are highly complex, incorporating numerous trigonometric functions that complicate the solution process. Direct computation often struggles to determine suitable initial values for the system, making it challenging to obtain convergent solutions.⁷

Litvin et al.⁸ proposed an approach to estimate meshing initial values by iteratively optimizing the closest points between gear surfaces and minimizing the deviation of their normal vectors relative to the contact surface. However, this method has practical limitations. The geometric construction elimination method⁹ can reduce the number of parameters in the equations to one or a few, enabling more precise selection of initial values for iterative solution of the system. These initial values are then substituted back into the original equations to determine the solutions for all parameters. Once the parameters are solved, a series of checks are performed: verifying whether the datum point lies near the mid-region of the tooth surface, detecting any curvature interference on the tooth surfaces, confirming that the worm gear’s angular acceleration curve forms a downward-opening parabola with its vertex at zero, and assessing the overall rationality of the parameters. The mismatch parameters are then iteratively adjusted until all the above conditions are satisfied, ensuring the solution meets both geometric and kinematic requirements.

Litvin and Fuentes-Aznar⁴ introduced the local synthesis approach, systematically analyzing various worm gear types and verifying that point contact exhibits lower error sensitivity compared to line contact in worm drives. Subsequently, Mu et al.¹⁰ proposed further development of mismatching modification techniques in grid theory through physical information neural networks. Meng et al.¹¹ delved into the meshing theory of mismatched conical surface-enveloping conical worm pairs, deriving their tooth surface and meshing equations while establishing a systematic framework for point-conjugate gear transmission kinematics, from which kinematic error curves were obtained. Chi et al.¹² proposed a novel mismatched meshing worm drive mechanism and elucidated its meshing theory, expanding the practical applications of point-contact worm systems. These studies collectively advance the theoretical foundation and engineering applications of point-contact worm drives, highlighting their error-tolerant advantages and design flexibility.

Simon¹³ proposed a new method for tooth contact analysis of mismatched spiral bevel gears, assuming that point contact under load extends into surface contact along the “potential” contact line. The influence of machine tool settings on tooth contact was also studied by minimizing the spacing function. Simon¹⁴ studied the influence of cylindrical worm gear tooth errors and shaft misalignment on load tooth contact, developed a load tooth contact analysis method and computer program, explored contact lines, load distribution, transmission errors, and analyzed the effects of various factors.

When simulating tooth surface contact, understanding the curvatures of the two interacting surfaces and the dimensions of the contact ellipse is critical. This is primarily because tooth surface mismatch transforms idealized line contact into point contact, and the contact region near this point evolves into an ellipse under applied load.¹⁵ Li et al.¹⁶ proved that modifying parameters can affect the meshing performance of the transmission pair, thereby affecting the transmission performance. Liu et al.¹⁷ analyzed point contact, line contact along tooth width, and line contact along tooth height, and the results showed that the contact form is the main factor affecting vibration amplitude and fluctuation.

Building on these foundations, this paper presents a computational analysis of the meshing theory for mismatched ZN-type worm pairs, employing numerical methods to conduct tooth surface contact analysis. The study systematically derives the computational formula for the relative principal curvature at the tooth surface contact points of such worm pairs. Through numerical simulations, key characteristics including the contact path, contact ellipse geometry, and kinematic error curve of mismatched ZN-type worm pairs are obtained, offering quantitative insights into their meshing behavior and contact mechanics.

The surface equation of the ZN-type worm pair and the hob

This section calculates the tooth surface equation when the worm gear and worm point are conjugate mesh. As shown in Figure 1, the worm tooth surface $S_{1}$ and the worm gear hob tooth surface $S_{d}$ are the track surfaces obtained by the spiral motion of the straight line around the cylinder. The worm gear hob grinds the tooth surface of the worm gear. The contact type of the conjugate tooth surface couple $[S_{d}, S_{2}]$ is line contact. When $S_{d} = S_{1}$ , the contact type of the conjugate tooth surface couple $[S_{1}, S_{2}]$ is a line contact. When $S_{d} ≅ S_{1}$ , the contact type of the conjugate tooth surface pair $[S_{1}, S_{2}]$ is point contact, which is called mismatch.

Figure 1.

Meshing of mismatched ZN-type worm pairs.

Worm tooth surface equation

The tooth surface formation process of the ZN-type worm is shown in Figure 2. As stated in Zhao et al.,³ the tooth surface equation and unit normal vector of the ZN-type worm can be obtained as follows:

{({\vec{r}}_{1})}_{1} = x_{1} {\vec{i}}_{1} + y_{1} {\vec{j}}_{1} + z_{1} {\vec{k}}_{1}

(1)

where $x_{1} = u_{1} \cos α_{1} \cos θ_{1} - r_{01} \sin θ_{1}$ , $y_{1} = u_{1} \cos α_{1} \sin θ_{1} + r_{01} \cos θ_{1}$ , $z_{1} = u_{1} \sin α_{1} + p θ_{1}$ .

Figure 2.

Schematic diagram of worm helical surface formation.

In the coordinate system $σ_{1}$ , the normal vector of the spiral surface of the ZN-type worm is expressed as

{({\vec{n}}_{1})}_{1} = \frac{{({\vec{r}}_{u})}_{1} \times {({\vec{r}}_{θ})}_{1}}{| {({\vec{r}}_{u})}_{1} \times {({\vec{r}}_{θ})}_{1} |} = n_{x}^{(1)} {\vec{i}}_{1} + n_{y}^{(1)} {\vec{j}}_{1} + n_{z}^{(1)} {\vec{k}}_{1}

(2)

where $n_{x}^{(1)} = \frac{1}{D_{1}} [- u_{1} \sin α_{1} \cos θ_{1} + (p + r_{01} \tan α_{1}) \sin θ_{1}]$ , $n_{y}^{(1)} = \frac{1}{D_{1}} [- u_{1} \sin α_{1} \sin θ_{1} - (p + r_{01} \tan α_{1}) \cos θ_{1}]$ ,

\begin{matrix} n_{z}^{(1)} = \frac{u_{1} \cos α_{1}}{D_{1}}, D_{1} = \frac{\sqrt{E_{1} G_{1} - F_{1}^{2}}}{\cos α_{1}} \\ = \sqrt{u_{1}^{2} + {(p + r_{01} \tan α_{1})}^{2}} . \end{matrix}

The second kind of fundamental quantity³ of the surface $S_{1}$ are

\begin{matrix} L_{1} = 0, M_{1} = - \frac{1}{D_{1}} (p \cos α_{1} + r_{01} \sin α_{1}), \\ N_{1} = \frac{1}{D_{1}} [u_{1}^{2} \sin α_{1} \cos α_{1} + r_{01} (p + r_{01} \tan α_{1})] \end{matrix}

(3)

Establishing a moving unit orthogonal frame ${{\vec{α}}_{ξ}^{(1)}, {\vec{α}}_{η}^{(1)}, {\vec{n}}_{1}}$ on $S_{1}$ yields the following

{({\vec{α}}_{ξ}^{(1)})}_{1} = \frac{{({\vec{r}}_{u})}_{1}}{| {({\vec{r}}_{u})}_{1} |} = \cos α_{1} \vec{e} (θ_{1}) + \sin α_{1} {\vec{k}}_{1}

(4)

From equations (2) and (4) ${\vec{α}}_{η}^{(1)}$ in the coordinate system $σ_{1}$ can be expressed as

\begin{matrix} {({\vec{α}}_{η}^{(1)})}_{1} = {({\vec{n}}_{1})}_{1} \times {({\vec{α}}_{ξ}^{(1)})}_{1} = - \frac{1}{D_{1}} (p + r_{01} \tan α_{1}) \sin α_{1} \vec{e} (θ_{1}) \\ + \frac{u_{1}}{D_{1}} \vec{g} (θ_{1}) + \frac{1}{D_{1}} (p + r_{01} \tan α_{1}) \cos α_{1} {\vec{k}}_{1} \end{matrix}

(5)

Since ${\vec{α}}_{ξ}^{(1)}$ is the tangent vector along the $u_{1}$ direction, the normal curvature along this direction is zero, that is, $k_{ξ}^{(1)} = 0$ .

According to differential geometry,¹⁸ the mean curvature of $S_{1}$ is

\begin{matrix} H_{1} = \frac{N_{1} - 2 F_{1} M_{1}}{2 \cos^{2} α_{1} D_{1}^{2}} = \frac{\tan α_{1}}{2 D_{1}^{3}} \\ [u_{1}^{2} + 2 (p + r_{01} \tan α_{1}) (p - r_{01} \cot 2 α_{1})] \end{matrix}

(6)

According to $k_{ξ}^{(1)} = 0$ and equation (6), it can be concluded that¹⁹

\begin{matrix} k_{η}^{(1)} = 2 H_{1} - k_{ξ}^{(1)} = \frac{\tan α_{1}}{D_{1}^{3}} \\ [u_{1}^{2} + 2 (p + r_{01} \tan α_{1}) (p - r_{01} \cot 2 α_{1})], \\ τ_{ξ}^{(1)} = - \frac{1}{D_{1}^{2}} (p + r_{01} \tan α_{1}) \end{matrix}

(7)

where $k_{η}^{(1)}$ is the normal curvature along the ${({\vec{α}}_{η}^{(1)})}_{1}$ direction and $τ_{ξ}^{(1)}$ is the geodesic torsion along the ${({\vec{α}}_{ξ}^{(1)})}_{1}$ direction.

Worm gear and worm gear hob tooth surface equation

Due to the similarity between the worm tooth surface and the worm gear hob tooth surface, the parameters of the worm gear hob can be obtained according to the worm calculation process. The tooth surface equation of the worm gear hob is

{({\vec{r}}_{d})}_{d} = x_{d} {\vec{i}}_{d} + y_{d} {\vec{j}}_{d} + z_{d} {\vec{k}}_{d}

(8)

where $x_{d} = u_{2} \cos α_{2} \cos θ_{2} - r_{02} \sin θ_{2}$ , $y_{d} = u_{2} \cos α_{2} \sin θ_{2} + r_{02} \cos θ_{2}$ , $z_{d} = u_{2} \sin α_{2} + p θ_{2}$ , $u_{2}$ and $θ_{2}$ are the parameters that define the surface of the worm gear hob. Meanwhile, $r_{02}$ and $α_{2}$ have the same meanings in the worm gear hob face as $r_{01}$ and $α_{1}$ in the worm gear tooth face.

In the coordinate system $S_{d}$ , the orthogonal frame ${{\vec{α}}_{ξ}^{(d)}, {\vec{α}}_{ξ}^{(d)}, {\vec{n}}_{d}}$ of the worm gear hob surface can be expressed as

\begin{matrix} {({\vec{n}}_{d})}_{d} = \frac{- u_{2} \sin α_{2}}{D_{2}} \vec{e} (θ_{2}) + \frac{1}{D_{2}} (p + r_{02} \tan α_{2}) \vec{g} (θ_{2}) \\ + \frac{u_{2} \cos α_{2}}{D_{2}} {\vec{k}}_{d}, \end{matrix}

{({\vec{α}}_{ξ}^{(d)})}_{d} = \cos α_{2} \vec{e} (θ_{2}) + \sin α_{2} {\vec{k}}_{d},

\begin{matrix} {({\vec{α}}_{η}^{(d)})}_{d} = - \frac{1}{D_{2}} [(p + r_{02} \tan α_{2}) \sin α_{2}] \vec{e} (θ_{2}) \\ + \frac{u_{2}}{D_{2}} \vec{g} (θ_{2}) + \frac{1}{D_{2}} (p \cos α_{2} + r_{02} \sin α_{2}) {\vec{k}}_{d} \end{matrix}

(9)

where $D_{2} = \sqrt{u_{2}^{2} + {(p + r_{02} \tan α_{2})}^{2}}$ .

Similarly, the normal curvature and geodesic torsion can be derived as follows

\begin{matrix} k_{ξ}^{(d)} = 0, k_{η}^{(d)} = \frac{\tan α_{2}}{D_{2}^{3}} \\ [u_{2}^{2} + 2 (p + r_{02} \tan α_{2}) (p - r_{02} \cot 2 α_{2})], \\ τ_{ξ}^{(d)} = - \frac{p + r_{02} \tan α_{2}}{D_{2}^{2}} \end{matrix}

(10)

The meshing coordinate system of the worm gear hob and the worm gear is illustrated in Figure 3. The angular velocity vector of the worm gear hob coincides with that of the worm gear, denoted as ${\vec{ω}}_{2}$ . When the worm gear hob rotates by an angle $φ_{d}$ , the worm gear rotates by an angle $φ_{d} / i_{d 2}$ . The center distance between the worm gear hob and the worm gear is $a_{d}$ , and $i_{d 2}$ represents the process transmission ratio.

Figure 3.

Coordinate system of the worm gear grinding process with a hob.

According to equation (8), the dynamic coordinate system $σ_{d} {O_{d}; {\vec{i}}_{d}, {\vec{j}}_{d}, {\vec{k}}_{d}}$ can be converted to the fixed coordinate system $σ_{od} {O_{d}; {\vec{i}}_{od}, {\vec{j}}_{od}, {\vec{k}}_{od}}$ . It can be obtained that

{({\vec{r}}_{d})}_{od} = R [{\vec{k}}_{od}, φ_{d}] {({\vec{r}}_{d})}_{d} = x_{od} {\vec{i}}_{od} + y_{od} {\vec{j}}_{od} + z_{d} {\vec{k}}_{od}

(11)

where $R [{\vec{k}}_{od}, φ_{d}] = [\begin{matrix} \cos φ_{d} & - \sin φ_{d} & 0 \\ \sin φ_{d} & \cos φ_{d} & 0 \\ 0 & 0 & 1 \end{matrix}]$ is the transformation coordinate system matrix,¹⁹ which means that the clockwise rotation angle of the coordinate system around axis ${\vec{k}}_{od}$ is $φ_{d}$ , and $x_{od} = u_{2} \cos α_{2} \cos (θ_{2} + φ_{d}) - r_{02} \sin (θ_{2} + φ_{d})$ , $y_{od} = u_{2} \cos α_{2} \sin (θ_{2} + φ_{d}) + r_{02} \cos (θ_{2} + φ_{d})$ .

Similarly, the equation of the unit normal vector ${\vec{n}}_{d}$ of the surface $S_{d}$ in the coordinate system $σ_{od}$ is

{({\vec{n}}_{d})}_{od} = R [{\vec{k}}_{od}, φ_{d}] {({\vec{n}}_{d})}_{d} = n_{x}^{(od)} {\vec{i}}_{od} + n_{y}^{(od)} {\vec{j}}_{od} + n_{z}^{(od)} {\vec{k}}_{od}

(12)

where $n_{x}^{(od)} = \frac{1}{D_{2}} [- u_{2} \sin α_{2} \cos (θ_{2} + φ_{d}) + (p + r_{02} \tan α_{2}) \sin (θ_{2} + φ_{d})]$ ,

\begin{matrix} n_{y}^{(od)} = \frac{1}{D_{2}} [- u_{2} \sin α_{2} \sin (θ_{2} + φ_{d}) - (p + r_{02} \tan α_{2}) \cos (θ_{2} + φ_{d})], \\ n_{z}^{(od)} = \frac{u_{2} \cos α_{2}}{D_{2}} . \end{matrix}

The relative speeds of the worm gear hob and worm gear are calculated according to the gear meshing principle²⁰ $Φ_{d 2} = {\vec{n}}_{d} \cdot {\vec{V}}_{d 2} = 0$ . Let the rotational speed of the worm gear hob be $ω_{d} = 1 (rad / s)$ ; then, $ω_{2} = 1 / i_{d 2} (rad / s)$ . From Figure 3, it is possible to have

(\vec{O_{2} O_{d}})_{od} = - a_{d} {\vec{j}}_{od}, {({\vec{ω}}_{d})}_{od} = {\vec{k}}_{od}, {({\vec{ω}}_{2})}_{od} = \frac{1}{i_{d 2}} {\vec{i}}_{od}, {({\vec{ω}}_{d 2})}_{od} = {({\vec{ω}}_{d})}_{od} - {({\vec{ω}}_{2})}_{od} = - \frac{1}{i_{d 2}} {\vec{i}}_{od} + {\underset{⇀}{k}}_{od}

(13)

From equations (11) and (13), the relative velocity of the worm pair in the coordinate system $σ_{od}$ can be expressed as

{({\vec{V}}_{d 2})}_{od} = {({\vec{ω}}_{d 2})}_{od} \times {({\vec{r}}_{d})}_{od} - {({\vec{ω}}_{2})}_{od} \times {(\vec{O_{2} O_{d}})}_{od} = - y_{od} {\vec{i}}_{od} + (x_{od} + \frac{z_{d}}{i_{d 2}}) {\vec{j}}_{od} + \frac{a_{d} - y_{od}}{i_{d 2}} {\vec{k}}_{od}

(14)

Using equations (12) and (14), the meshing equation is

Φ_{d 2} = - y_{od} n_{x}^{(od)} + n_{y}^{(od)} (x_{od} + \frac{z_{d}}{i_{d 2}}) + \frac{(a_{d} - y_{od}) n_{z}^{(od)}}{i_{d 2}} = \frac{A \sin (θ_{2} + φ_{d}) + B \cos (θ_{2} + φ_{d}) + C}{i_{d 2} D_{2}}

(15)

where $A = - u_{2} z_{d} \sin α_{2} - u_{2}^{2} \cos^{2} α_{2}$ , $B = - z_{d} (p + r_{02} \tan α_{2}) - r_{02} u_{2} \cos α_{2}$ , and $C = u_{2} \cos α_{2} (a_{d} - i_{d 2} p)$ .

After obtaining the meshing equation, it is necessary to convert the worm gear coordinate system to the coordinate system $σ_{o 2} {O_{2}; {\vec{i}}_{o 2}, {\vec{j}}_{o 2}, {\vec{k}}_{o 2}}$ . From equation (11), the following can be obtained

{({\vec{r}}_{d})}_{o 2} = R [{\vec{j}}_{o 2}, - \frac{π}{2}] {({\vec{r}}_{d})}_{od} = - z_{d} {\vec{i}}_{o 2} + y_{od} {\vec{j}}_{o 2} + x_{od} {\vec{k}}_{o 2}

(16)

By changing the coordinate of equation (16) and combining it with $Φ_{d 2} = 0$ obtained from equation (15), the worm gear tooth surface equation in the coordinate system $σ_{2} {O_{2}; {\vec{i}}_{2}, {\vec{j}}_{2}, {\vec{k}}_{2}}$ can be obtained as

{({\vec{r}}_{2})}_{2} = R [{\vec{k}}_{o 2}, - \frac{φ_{d}}{i_{d 2}}] [{({\vec{r}}_{d})}_{o 2} + {(\vec{O_{2} O_{d}})}_{o 2}], Φ_{d 2} = 0

(17)

This section calculates the worm and worm gear tooth surface equation and some fundamental quantities. The accurate calculation of the tooth surface equation of the worm gear and worm is the basis of subsequent calculations.

Mismatched ZN-type worm pair meshing equation solution

Transforming line contact into point contact is referred to as mismatch. This involves modifying the tooth surface of the original line contact gear pair so that it no longer satisfies the criteria for line contact but instead adheres to the local conjugate conditions for point contact. Consequently, the original line contact gear pair deviates from its initial meshing pattern, which is precisely what the term “mismatch” denotes.

To calculate the meshing principle of the mismatched ZN-type worm pair, several steps are required. First, the tooth surface equations of the worm and the worm gear must be expressed within the same coordinate system. Subsequently, the common-point condition, common-normal-line condition, and meshing equations of the worm gear and the worm gear hob are employed to formulate a system of equations. Finally, the elimination method is utilized to solve these equations.

Coordinate transformation of the worm gear and worm

The worm gear and worm meshing coordinate system is similar to the meshing of the worm gear and worm gear hob, as shown in Figure 4, where $φ_{1}$ is the rotation angle of the worm, $φ_{2}$ is the rotation angle of the worm gear, and the center distance $| \vec{O_{1} O_{2}} | = a$ .

Figure 4.

Worm gear and worm meshing coordinate system.

Transforming the worm equation into the coordinate system $σ_{o 2}$ yields the following

\begin{matrix} {({\vec{r}}_{2})}_{o 2} = R [{\vec{k}}_{o 2}, φ_{2}] {({\vec{r}}_{2})}_{2} = R [{\vec{k}}_{o 2}, φ_{2} - \frac{φ_{d}}{i_{d 2}}] {({\vec{r}}_{d})}_{o 2} = [- z_{d} \cos (φ_{2} - \frac{φ_{d}}{i_{d 2}}) - (y_{od} - a_{d}) \sin (φ_{2} - \frac{φ_{d}}{i_{d 2}})] {\vec{i}}_{o 2} \\ + [- z_{d} \sin (φ_{2} - \frac{φ_{d}}{i_{d 2}}) + (y_{od} - a_{d}) \cos (φ_{2} - \frac{φ_{d}}{i_{d 2}})] {\vec{j}}_{o 2} + x_{od} {\vec{k}}_{o 2} \end{matrix}

(18)

The equation of the unit normal vector of the tooth surface of the worm in the coordinate system $σ_{o 2}$ is

\begin{matrix} {({\vec{n}}_{2})}_{o 2} = R [{\vec{k}}_{o 2}, φ_{2} - \frac{φ_{d}}{i_{d 2}}] R [{\vec{j}}_{od}, - \frac{π}{2}] {({\vec{n}}_{d})}_{od} = [- n_{z}^{(od)} \cos (φ_{2} - \frac{φ_{d}}{i_{d 2}}) - n_{y}^{(od)} \sin (φ_{2} - \frac{φ_{d}}{i_{d 2}})] {\vec{i}}_{o 2} \\ + [- n_{z}^{(od)} \sin (φ_{2} - \frac{φ_{d}}{i_{d 2}}) + n_{y}^{(od)} \cos (φ_{2} - \frac{φ_{d}}{i_{d 2}})] {\vec{j}}_{o 2} + n_{x}^{(od)} {\vec{k}}_{o 2} \end{matrix}

(19)

In the coordinate system $σ_{o 1} {O_{1}; {\vec{i}}_{o 1}, {\vec{j}}_{o 1}, {\vec{k}}_{o 1}}$ , the equation of the tooth surface of the worm is

{({\vec{r}}_{1})}_{o 1} = R [{\vec{k}}_{o 1}, φ_{1}] {({\vec{r}}_{1})}_{1} = x_{o 1} {\vec{i}}_{o 1} + y_{o 1} {\vec{j}}_{o 1} + z_{1} {\vec{k}}_{o 1}

(20)

where $x_{o 1} = u_{1} \cos α_{1} \cos (θ_{1} + φ_{1}) - r_{01} \sin (θ_{1} + φ_{1})$ , $y_{o 1} = u_{1} \cos α_{1} \sin (θ_{1} + φ_{1}) + r_{01} \cos (θ_{1} + φ_{1})$ .

{({\vec{r}}_{1})}_{o 2} = R [{\vec{j}}_{o 2}, - \frac{π}{2}] {({\vec{r}}_{1})}_{o 1} = - z_{1} {\vec{i}}_{o 2} + y_{o 1} {\vec{j}}_{o 2} + x_{o 1} {\vec{k}}_{o 2}

(21)

Similarly, the equation for the unit normal vector of the tooth surface of the worm in the coordinate system $σ_{o 2}$ is as follows:

{({\vec{n}}_{1})}_{o 2} = R [{\vec{j}}_{o 2}, - \frac{π}{2}] R [{\vec{k}}_{o 1}, φ_{1}] {({\vec{n}}_{1})}_{1} = - n_{z}^{(o 1)} {\vec{i}}_{o 2} + n_{y}^{(o 1)} {\vec{j}}_{o 2} + n_{x}^{(o 1)} {\vec{k}}_{o 2},

(22)

where $n_{x}^{(o 1)} = \frac{- u_{1} \sin α_{1} \cos (θ_{1} + φ_{1}) + (p + r_{01} \tan α_{1}) \sin (θ_{1} + φ_{1})}{D_{1}}$ ,

n_{y}^{(o 1)} = \frac{- u_{1} \sin α_{1} \sin (θ_{1} + φ_{1}) - (p + r_{01} \tan α_{1}) \cos (θ_{1} + φ_{1})}{D_{1}}, n_{z}^{(o 1)} = \frac{u_{1} \cos α_{1}}{D_{1}} .

Solution of the nonlinear equation

The instantaneous contact point shall meet the common point condition ${({\vec{r}}_{2})}_{o 2} = {({\vec{r}}_{1})}_{o 2} + {(\vec{O_{2} O_{1}})}_{o 2}$ , common normal line ${({\vec{n}}_{2})}_{o 2} = {({\vec{n}}_{1})}_{o 2}$ , and the meshing conditions of the worm gear hob and worm gear $Φ_{d 2} = 0$ to ensure correct engagement in the case of mismatch.²¹ The above nonlinear equation contains seven unknowns and six equations. Therefore, one variable of the system of nonlinear underdetermined equations shall be assigned, or a nonlinear equation shall be supplemented.

Let $ϕ_{1} = θ_{1} + φ_{1}$ , $ϕ_{d} = θ_{2} + φ_{d}$ , and $ϕ_{2} = φ_{2} - \frac{φ_{d}}{i_{d 2}}$ ; according to the common normal condition, it can be obtained that

\frac{1}{D_{1}} [u_{1} \sin α_{1} \cos ϕ_{1} - (p + r_{01} \tan α_{1}) \sin ϕ_{1}] + \frac{1}{D_{2}} [(p + r_{02} \tan α_{2}) \sin ϕ_{d} - u_{2} \sin α_{2} \cos ϕ_{d}] = 0

(23)

\frac{u_{1} \cos α_{1}}{D_{1}} + \frac{\sin ϕ_{2}}{D_{2}} [u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}] - \frac{u_{2} \cos α_{2} \cos ϕ_{2}}{D_{2}} = 0

(24)

\frac{1}{D_{1}} [u_{1} \sin α_{1} \sin ϕ_{1} + (p + r_{01} \tan α_{1}) \cos ϕ_{1}] - \frac{1}{D_{2}} [u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}] \cos ϕ_{2} - \frac{1}{D_{2}} u_{2} \cos α_{2} \sin ϕ_{2} = 0

(25)

According to the conditions of common points, it can be obtained that

u_{2} \cos α_{2} \cos ϕ_{d} - r_{02} \sin ϕ_{d} - u_{1} \cos α_{1} \cos ϕ_{1} + r_{01} \sin ϕ_{1} = 0

(26)

(u_{2} \cos α_{2} \sin ϕ_{d} + r_{02} \cos ϕ_{d} - a_{d}) \cos ϕ_{2} - z_{d} \sin ϕ_{2} - (r_{01} \cos ϕ_{1} + u_{1} \cos α_{1} \sin ϕ_{1}) + a = 0

(27)

z_{1} + (a_{d} - u_{2} \cos α_{2} \sin ϕ_{d} - r_{02} \cos ϕ_{d}) \sin ϕ_{2} - z_{d} \cos ϕ_{2} = 0

(28)

Using the meshing function $Φ_{d 2} = 0$ , the following can be obtained

\begin{matrix} (u_{2} z_{d} \sin α_{2} + u_{2}^{2} \cos^{2} α_{2}) \sin ϕ_{d} + [z_{d} (p + r_{02} \tan α_{2}) + r_{02} u_{2} \cos α_{2}] \cos ϕ_{d} \\ - u_{2} \cos α_{2} (a_{d} - i_{d 2} p) = 0 \end{matrix}

(29)

Equations (23)–(29) contain six independent equations and seven unknown quantities, which can be solved by the elimination method. Since the normal vectors of both surfaces are unit vectors, Equations (26)–(28) contains only two independent equations. The seven unknowns are $u_{1}$ , $u_{2}$ , $ϕ_{1}$ , $ϕ_{d}$ , $ϕ_{2}$ , $z_{1}$ , and $z_{d}$ .

That is to say, the system of equations contains six independent equations and seven unknowns, which are eliminated until it becomes one equation containing two unknowns. Among them, only equation (28) contains the unknown variable $z_{1}$ , which means that the solution of this unknown variable can only be calculated through equation (28), so equation (28) does not need to be considered when eliminating it.

Step 1: Eliminate the unknown $z_{d}$ .

After removing equation (28), the unknown $z_{d}$ appears solely in equations (27) and (29). From equation (29), $z_{d}$ can be expressed as follows

z_{d} = \frac{u_{2} [\cos α_{2} (a_{d} - i_{d 2} p) - r_{02} \cos α_{2} \cos ϕ_{d} - u_{2} \cos^{2} α_{2} \sin ϕ_{d}]}{u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}}

(30)

By introducing equation (30) into equation (27), in z_dequation (27) can be eliminated to obtain the following equation

\begin{matrix} - \frac{u_{2} [\cos α_{2} (a_{d} - i_{d 2} p) - r_{02} \cos α_{2} \cos ϕ_{d} - u_{2} \cos^{2} α_{2} \sin ϕ_{d}]}{u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}} \sin ϕ_{2} \\ + (u_{2} \cos α_{2} \sin ϕ_{d} + r_{02} \cos ϕ_{d} - a_{d}) \cos ϕ_{2} - r_{01} \cos ϕ_{1} - u_{1} \cos α_{1} \sin ϕ_{1} + a = 0 \end{matrix}

(31)

At this time, only equations (24) and (31) are left, including four independent equations and five unknowns $u_{1}$ , $u_{2}$ , $ϕ_{1}$ , $ϕ_{d}$ , and $ϕ_{2}$ .

Step 2: Eliminate the unknown $ϕ_{1}$ .

Using equations (23) and (26), the following can be obtained

\sin ϕ_{1} = \frac{| \begin{matrix} - \frac{1}{D_{1}} u_{1} \sin α_{1} & \frac{1}{D_{2}} [(p + r_{02} \tan α_{2}) \sin ϕ_{d} - u_{2} \sin α_{2} \cos ϕ_{d}] \\ u_{1} \cos α_{1} & u_{2} \cos α_{2} \cos ϕ_{d} - r_{02} \sin ϕ_{d} \end{matrix} |}{| \begin{matrix} - \frac{1}{D_{1}} u_{1} \sin α_{1} & \frac{1}{D_{1}} (p + r_{01} \tan α_{1}) \\ u_{1} \cos α_{1} & - r_{01} \end{matrix} |} = S_{S} \sin ϕ_{d} + S_{C} \cos ϕ_{d}

(32)

where $S_{S} = \frac{| \begin{matrix} \sin α_{1} & \frac{D_{1}}{D_{2}} (p + r_{02} \tan α_{2}) \\ - \cos α_{1} & - r_{02} \end{matrix} |}{p \cos α_{1}}$ , $S_{C} = \frac{| \begin{matrix} \sin α_{1} & - \frac{D_{1}}{D_{2}} u_{2} \sin α_{2} \\ - \cos α_{1} & u_{2} \cos α_{2} \end{matrix} |}{p \cos α_{1}}$ .

Similarly, according to equations (23) and (26), the following can be obtained

\cos ϕ_{1} = \frac{| \begin{matrix} \frac{1}{D_{2}} [(p + r_{02} \tan α_{2}) \sin ϕ_{d} - u_{2} \sin α_{2} \cos ϕ_{d}] & \frac{1}{D_{1}} (p + r_{01} \tan α_{1}) \\ u_{2} \cos α_{2} \cos ϕ_{d} - r_{02} \sin ϕ_{d} & - r_{01} \end{matrix} |}{| \begin{matrix} - \frac{1}{D_{1}} u_{1} \sin α_{1} & \frac{1}{D_{1}} (p + r_{01} \tan α_{1}) \\ u_{1} \cos α_{1} & - r_{01} \end{matrix} |} = C_{S} \sin ϕ_{d} + C_{C} \cos ϕ_{d}

(33)

where $C_{S} = \frac{| \begin{matrix} \frac{D_{1}}{D_{2}} (p + r_{02} \tan α_{2}) & p + r_{01} \tan α_{1} \\ r_{02} & r_{01} \end{matrix} |}{p u_{1} \cos α_{1}}$ , $C_{C} = \frac{| \begin{matrix} \frac{D_{1}}{D_{2}} u_{2} \sin α_{2} & p + r_{01} \tan α_{1} \\ - u_{2} \cos α_{2} & - r_{01} \end{matrix} |}{p u_{1} \cos α_{1}}$ .

By using equations (32) and (33) to eliminate $ϕ_{1}$ in equations (25) and (31), it can be obtained that

u_{2} \cos α_{2} \sin ϕ_{2} + [u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}] \cos ϕ_{2} - \frac{D_{2}}{D_{1}} (S_{2}^{(d)} \sin ϕ_{d} + C_{2}^{(d)} \cos ϕ_{d}) = 0

(34)

\begin{matrix} \frac{u_{2} \cos α_{2} (r_{02} \cos ϕ_{d} + u_{2} \cos α_{2} \sin ϕ_{d} - a_{d} + i_{d 2} p)}{u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}} \sin ϕ_{2} + (u_{2} \cos α_{2} \sin ϕ_{d} + r_{02} \cos ϕ_{d} - a_{d}) \cos ϕ_{2} \\ - S_{1}^{(d)} \sin ϕ_{d} - C_{1}^{(d)} \cos ϕ_{d} + a = 0, \end{matrix}

(35)

where $S_{1}^{(d)} = u_{1} \cos α_{1} S_{S} + r_{01} C_{S}$ , $C_{1}^{(d)} = u_{1} \cos α_{1} S_{C} + r_{01} C_{C}$ , $S_{2}^{(d)} = u_{1} \sin α_{1} S_{S} + (p + r_{01} \tan α_{1}) C_{S}$ , and $C_{2}^{(d)} = u_{1} \sin α_{1} S_{C} + (p + r_{01} \tan α_{1}) C_{C}$ .

In this case, equations (24), (34), and (35) contain four unknowns $u_{1}$ , $u_{2}$ , $ϕ_{d}$ , and $ϕ_{2}$ .

Step 3: Eliminate the unknown $ϕ_{2}$ .

From equation (34), $\cos ϕ_{2}$ can be expressed as follows

\cos ϕ_{2} = \frac{\frac{D_{2}}{D_{1}} [S_{2}^{(d)} \sin ϕ_{d} + C_{2}^{(d)} \cos ϕ_{d}] - u_{2} \cos α_{2} \sin ϕ_{2}}{u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}} .

(36)

By using equation (36) to eliminate $\cos ϕ_{2}$ in equation (35), the following can be obtained

\begin{matrix} \frac{D_{2}}{D_{1}} (u_{2} \cos α_{2} \sin ϕ_{d} + r_{02} \cos ϕ_{d} - a_{d}) (S_{2}^{(d)} \sin ϕ_{d} + C_{2}^{(d)} \cos ϕ_{d}) + i_{d 2} p u_{2} \cos α_{2} \sin ϕ_{2} \\ + (a - S_{1}^{(d)} \sin ϕ_{d} - C_{1}^{(d)} \cos ϕ_{d}) [u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d}] = 0 \end{matrix}

(37)

From equations (24) and (34), it can be calculated as follows

\sin ϕ_{2} = \frac{D_{2}}{D_{1}} \frac{| \begin{matrix} - u_{1} \cos α_{1} & - u_{2} \cos α_{2} \\ S_{2}^{(d)} \sin ϕ_{d} + C_{2}^{(d)} \cos ϕ_{d} & u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d} \end{matrix} |}{| \begin{matrix} u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d} & - u_{2} \cos α_{2} \\ u_{2} \cos α_{2} & u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d} \end{matrix} |}

(38)

Using the nature of the determinant, the calculation yields

\sin ϕ_{2} = \frac{D_{2}}{D_{1}} \frac{| \begin{matrix} - u_{1} \cos α_{1} & - u_{2} \cos α_{2} \\ S_{2}^{(d)} \sin ϕ_{d} & u_{2} \sin α_{2} \sin ϕ_{d} \end{matrix} | + | \begin{matrix} - u_{1} \cos α_{1} & - u_{2} \cos α_{2} \\ C_{2}^{(d)} \cos ϕ_{d} & (p + r_{02} \tan α_{2}) \cos ϕ_{d} \end{matrix} |}{{(u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d})}^{2} + u_{2}^{2} \cos^{2} α_{2}}

= \frac{{S'}_{S} \sin ϕ_{d} + {S'}_{C} \cos ϕ_{d}}{D_{S} \sin^{2} ϕ_{d} + D_{C} \cos ϕ_{d} \sin ϕ_{d} + u_{2}^{2} - D_{S}}

(39)

where $S'_{S} = \frac{D_{2}}{D_{1}} | \begin{matrix} - u_{1} \cos α_{1} & - u_{2} \cos α_{2} \\ S_{2}^{(d)} & u_{2} \sin α_{2} \end{matrix} |$ , $S'_{C} = \frac{D_{2}}{D_{1}} | \begin{matrix} - u_{1} \cos α_{1} & - u_{2} \cos α_{2} \\ C_{2}^{(d)} & (p + r_{02} \tan α_{2}) \end{matrix} |$ ,

D_{S} = u_{2}^{2} \sin^{2} α_{2} - {(p + r_{02} \tan α_{2})}^{2}, D_{C} = 2 u_{2} \sin α_{2} (p + r_{02} \tan α_{2}) .

By substituting equation (39) into equation (37), the calculation yields

A_{1} \sin^{4} ϕ_{d} + A_{2} \sin^{3} ϕ_{d} + A_{3} \sin^{2} ϕ_{d} + A_{4} \sin ϕ_{d} + A_{5} + \cos ϕ_{d} (B_{1} \sin^{3} ϕ_{d} + B_{2} \sin^{2} ϕ_{d} + B_{3} \sin ϕ_{d} + B_{4}) = 0

(40)

where $A_{1} = | \begin{matrix} \frac{D_{2}}{D_{1}} u_{2} \cos α_{2} & D_{S} u_{2} \sin α_{2} - D_{C} (p + r_{02} \tan α_{2}) \\ S_{1}^{(d)} & D_{S} S_{2}^{(d)} - D_{C} C_{2}^{(d)} \end{matrix} | - | \begin{matrix} \frac{D_{2}}{D_{1}} r_{02} & D_{C} u_{2} \sin α_{2} + D_{S} (p + r_{02} \tan α_{2}) \\ C_{1}^{(d)} & D_{C} S_{2}^{(d)} + D_{S} C_{2}^{(d)} \end{matrix} |$ ,

A_{2} = | \begin{matrix} a & a_{d} \frac{D_{2}}{D_{1}} \\ D_{S} S_{2}^{(d)} - D_{C} C_{2}^{(d)} & D_{S} u_{2} \sin α_{2} - D_{C} (p + r_{02} \tan α_{2}) \end{matrix} |,

A_{3} = | \begin{matrix} \frac{D_{2}}{D_{1}} u_{2} \cos α_{2} & (u_{2}^{2} - D_{S}) u_{2} \sin α_{2} + D_{C} (p + r_{02} \tan α_{2}) \\ S_{1}^{(d)} & (u_{2}^{2} - D_{S}) S_{2}^{(d)} + D_{C} C_{2}^{(d)} \end{matrix} |

+ | \begin{matrix} \frac{D_{2}}{D_{1}} r_{02} & D_{C} u_{2} \sin α_{2} - (u_{2}^{2} - D_{S}) (p + r_{02} \tan α_{2}) + D_{S} (p + r_{02} \tan α_{2}) \\ C_{1}^{(d)} & D_{C} S_{2}^{(d)} - (u_{2}^{2} - D_{S}) C_{2}^{(d)} + D_{S} C_{2}^{(d)} \end{matrix} |,

A_{4} = | \begin{matrix} a & a_{d} \frac{D_{2}}{D_{1}} \\ (u_{2}^{2} - D_{S}) S_{2}^{(d)} + D_{C} C_{2}^{(d)} & (u_{2}^{2} - D_{S}) u_{2} \sin α_{2} + D_{C} (p + r_{02} \tan α_{2}) \end{matrix} | + i_{d 2} p u_{2} \cos α_{2} S'_{S},

A_{5} = (u_{2}^{2} - D_{S}) | \begin{matrix} \frac{D_{2}}{D_{1}} r_{02} & (p + r_{02} \tan α_{2}) \\ C_{1}^{(d)} & C_{2}^{(d)} \end{matrix} |

B_{1} = | \begin{matrix} \frac{D_{2}}{D_{1}} u_{2} \cos α_{2} & D_{C} u_{2} \sin α_{2} + D_{S} (p + r_{02} \tan α_{2}) \\ S_{1}^{(d)} & D_{C} S_{2}^{(d)} + D_{S} C_{2}^{(d)} \end{matrix} | + | \begin{matrix} \frac{D_{2}}{D_{1}} r_{02} & D_{S} u_{2} \sin α_{2} - D_{C} (p + r_{02} \tan α_{2}) \\ C_{1}^{(d)} & D_{S} S_{2}^{(d)} - D_{C} C_{2}^{(d)} \end{matrix} |,

B_{2} = | \begin{matrix} a & a_{d} \frac{D_{2}}{D_{1}} \\ D_{C} S_{2}^{(d)} + D_{S} C_{2}^{(d)} & D_{C} u_{2} \sin α_{2} + D_{S} (p + r_{02} \tan α_{2}) \end{matrix} |,

B_{3} = (u_{2}^{2} - D_{S}) | \begin{matrix} \frac{D_{2}}{D_{1}} u_{2} \cos α_{2} & (p + r_{02} \tan α_{2}) \\ S_{1}^{(d)} & C_{2}^{(d)} \end{matrix} | + | \begin{matrix} \frac{D_{2}}{D_{1}} r_{02} & (u_{2}^{2} - D_{S}) u_{2} \sin α_{2} + D_{C} (p + r_{02} \tan α_{2}) \\ C_{1}^{(d)} & (u_{2}^{2} - D_{S}) S_{2}^{(d)} + D_{C} C_{2}^{(d)} \end{matrix} |,

B_{4} = (u_{2}^{2} - D_{S}) | \begin{matrix} a & a_{d} \frac{D_{2}}{D_{1}} \\ C_{2}^{(d)} & (p + r_{02} \tan α_{2}) \end{matrix} | + i_{d 2} p u_{2} \cos α_{2} S'_{C} .

At this time, the equations (32) and (42) remain, which contain three unknowns, $u_{1}$ , $u_{2}$ , and $ϕ_{d}$ .

Step 4: Eliminate the unknown $ϕ_{d}$ .

From equations (32) and (33), the following results can be obtained

2 \sin ϕ_{d} \cos ϕ_{d} (S_{S} S_{C} + C_{S} C_{C}) = 1 - S_{C}^{2} - C_{C}^{2} - (S_{S}^{2} + C_{S}^{2} - S_{C}^{2} - C_{C}^{2}) \sin^{2} ϕ_{d}

(41)

The square of the two sides of the equal sign of equation (41) can eliminate the unknown $\cos ϕ_{d}$ , and we thus derive the following

a_{1} \sin^{4} ϕ_{d} + a_{2} \sin^{2} ϕ_{d} + a_{3} = 0

(42)

where $a_{1} = 4 {(S_{S} S_{C} + C_{S} C_{C})}^{2} + {(S_{S}^{2} + C_{S}^{2} - S_{C}^{2} - C_{C}^{2})}^{2}$ , $a_{3} = {(S_{C}^{2} + C_{C}^{2} - 1)}^{2}$ , and $a_{2} = 2 (S_{S}^{2} + C_{S}^{2} - S_{C}^{2} - C_{C}^{2}) (S_{C}^{2} + C_{C}^{2} - 1) - 4 {(S_{S} S_{C} + C_{S} C_{C})}^{2}$ .

Square the two sides of equation (40) to obtain

\sum_{i = 1}^{9} b_{i} \sin^{9 - i} ϕ_{d} = 0

(43)

where $b_{1} = A_{1}^{2} + B_{1}^{2}$ , $b_{2} = 2 (A_{1} A_{2} + B_{1} B_{2})$ , $b_{3} = A_{2}^{2} + 2 A_{1} A_{3} + B_{2}^{2} + 2 B_{1} B_{3} - B_{1}^{2}$ , $b_{4} = 2 (A_{1} A_{4} + A_{2} A_{3} + B_{1} B_{4} + B_{2} B_{3} - B_{1} B_{2}), b_{5} = A_{3}^{2} + 2 A_{1} A_{5} + 2 A_{2} A_{4} + B_{3}^{2} + 2 B_{2} B_{4} - B_{2}^{2} - 2 B_{1} B_{3},$ $b_{6} = 2 (A_{2} A_{5} + A_{3} A_{4} + B_{3} B_{4} - B_{1} B_{4} - B_{2} B_{3}), b_{7} = A_{4}^{2} + 2 A_{3} A_{5} + B_{4}^{2} - B_{3}^{2} - 2 B_{2} B_{4}$ , $b_{8} = 2 (A_{4} A_{5} - B_{3} B_{4}), b_{9} = A_{5}^{2} - B_{4}^{2} .$

By applying the resultant elimination method to eliminate the unknown $ϕ_{d}$ , the following determinant of the resultant matrix can be obtained

M_{0} = | \begin{matrix} a_{1} & 0 & a_{2} & 0 & a_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{1} & 0 & a_{2} & 0 & a_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{1} & 0 & a_{2} & 0 & a_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a_{1} & 0 & a_{2} & 0 & a_{3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{1} & 0 & a_{2} & 0 & a_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{1} & 0 & a_{2} & 0 & a_{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & a_{1} & 0 & a_{2} & 0 & a_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{1} & 0 & a_{2} & 0 & a_{3} \\ b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} & b_{7} & b_{8} & b_{9} & 0 & 0 & 0 \\ 0 & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} & b_{7} & b_{8} & b_{9} & 0 & 0 \\ 0 & 0 & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} & b_{7} & b_{8} & b_{9} & 0 \\ 0 & 0 & 0 & b_{1} & b_{2} & b_{3} & b_{4} & b_{5} & b_{6} & b_{7} & b_{8} & b_{9} \end{matrix} | = 0

(44)

When and only when $M_{0} = 0$ , equations (42) and (43) have common solutions, that is, when $M_{0} = 0$ , the values of $u_{1}$ and $u_{2}$ are the solutions at the contact region.

Datum point and intersection point of the contact path and worm pair boundary

When the worm gear and worm are engaged, the value of $u_{2}$ gradually decreases from the root of the worm gear to the top of the worm gear. The value of $u_{2}$ on the tooth surface of the worm gear ranges from $u_{min}$ to $u_{max}$ , where $u_{\min} = \frac{\sqrt{r_{f 2}^{2} - r_{02}^{2}}}{\cos α_{2}}$ , $u_{max} = \frac{\sqrt{r_{a 2}^{2} - r_{02}^{2}}}{\cos α_{2}}$ , $r_{f 2}$ and $r_{a 2}$ are the radius of the root circle and the top circle of the worm gear, respectively. To make the datum point at the tooth surface’s center, the solution for $u_{2}$ should also be near $\frac{u_{\min} + u_{\max}}{2}$ . The mismatched parameters are adjusted if the datum point is too far from the worm wheel center.

Usually, a nonlinear equation of the point where the instantaneous transmission ratio error is zero is supplemented as the datum point of the conjugate tooth surface pair $[S_{1}, S_{2}]$ . The datum point of the gear surface is the point where the instantaneous transmission ratio is the same as the nominal transmission ratio,²² and the instantaneous transmission ratio $i_{12}^{*}$ is

i_{12}^{*} = \frac{({({\vec{k}}_{o 2})}_{o 2}, {({\vec{r}}_{2})}_{o 2}, {({\vec{n}}_{2})}_{o 2})}{({({\vec{k}}_{o 1})}_{o 1}, {({\vec{r}}_{1})}_{o 1}, {({\vec{n}}_{1})}_{o 1})}

(45)

From equations (21) to (26), it can be derived that:

i_{12}^{*} = \frac{n_{z}^{(od)} (y_{od} - a_{d}) - z_{d} n_{y}^{(od)}}{x_{o 1} n_{y}^{(o 1)} - y_{o 1} n_{x}^{(o 1)}}

(46)

When the instantaneous transmission ratio equals the nominal transmission ratio, the equation should be satisfied at the datum point is $i_{12} = i_{12}^{*}$ .

From $Φ_{d 2} = 0$ , it can be derived that:

n_{z}^{(od)} (y_{od} - a_{d}) - z_{d} n_{y}^{(od)} = (x_{od} n_{y}^{(od)} - y_{od} n_{x}^{(od)}) i_{d 2}

(47)

By substituting equations (21)–(25) into equation (47), it can be derived that:

i_{12} = \frac{(x_{od} n_{y}^{(od)} - y_{od} n_{x}^{(od)}) i_{d 2}}{x_{o 1} n_{y}^{(o 1)} - y_{o 1} n_{x}^{(o 1)}} = - \frac{i_{d 2} u_{2} D_{1} \cos α_{2}}{u_{1} D_{2} \cos α_{1}}

(48)

The relationship between the unknowns $u_{1}$ and $u_{2}$ at the datum point is as follows

u_{1} = \frac{i_{d 2} u_{2} \cos α_{2} (p + r_{01} \tan α_{1})}{\sqrt{i_{12}^{2} D_{2}^{2} \cos^{2} α_{1} - i_{d 2}^{2} u_{2}^{2} \cos^{2} α_{2}}}

(49)

Equation (49) is taken as the supplementary equation of the contact equations at the datum point.

When the datum point is determined, the contact path intersects with the worm top circle and the worm gear gorge circle. At the top circle of the worm $u_{1} = u_{max}$ , taking $u_{1} = u_{max}$ into the resultant $M_{0}$ , the value of $u_{2}$ at the intersection of the contact path and the worm top circle can be solved. The other parameters are similar to those at the datum point. In this way, the value of all unknowns at the point is obtained, and the specific position of the point is determined.

The third point is the contact path intersection and the worm gear’s gorge circle. The intersection of the contact path and the gorge circle must meet the following equation³

z_{d}^{2} + {(a_{d} - \sqrt{x_{od}^{2} + {(y_{od} - a_{d})}^{2}})}^{2} = r_{g 2}^{2},

that is

\begin{matrix} M_{1} = 2 a_{d} (u_{2} \cos α_{2} \sin ϕ_{d} + r_{02} \cos ϕ_{d}) + 2 a_{d} \sqrt{u_{2}^{2} \cos^{2} α_{2} + r_{02}^{2} + a_{d}^{2} - 2 a_{d} (u_{2} \cos α_{2} \sin ϕ_{d} + r_{02} \cos ϕ_{d})} \\ - \frac{u_{2}^{2} {[\cos α_{2} (a_{d} - i_{d 2} p) - r_{02} \cos α_{2} \cos ϕ_{d} - u_{2} \cos^{2} α_{2} \sin ϕ_{d}]}^{2}}{{(u_{2} \sin α_{2} \sin ϕ_{d} + (p + r_{02} \tan α_{2}) \cos ϕ_{d})}^{2}} - 2 a_{d}^{2} - u_{2}^{2} \cos^{2} α_{2} - r_{02}^{2} + r_{g 2}^{2} = 0 \end{matrix}

(50)

where $r_{g 2}$ is the radius of the worm gear gorge circle.

Equation (29) is used to obtain

\sin ϕ_{d} = \pm \sqrt{\frac{- a_{2} \pm \sqrt{a_{2}^{2} - 4 a_{1} a_{3}}}{2 a_{1}}}, \cos ϕ_{d} = \pm \sqrt{\frac{2 a_{1} + a_{2} \mp \sqrt{a_{2}^{2} - 4 a_{1} a_{3}}}{2 a_{1}}}

(51)

By substituting equation (50) into equation (51) to obtain the relation between the unknowns $u_{1}$ and $u_{2}$ , the resultant $M_{0}$ is also a function of the unknown $u_{1}$ and $u_{2}$ . The equations $M_{0}$ and $M_{1}$ are solved for the intersection of the contact path of the worm gear and the throat circle.

Previously, the datum and intersection points between the contact path and the root circle of the worm gear were determined. By constructing a line connecting these two points, the approximate location of the intersection between the contact path and the worm gear’s gorge circle can be preliminarily estimated. Using the datum point as the initial value in an iterative algorithm, the coordinates of the intersection point between the contact path and the gorge circle are solved. Once these three characteristic points are fully defined, all contact points along the path are derived through interpolation, forming a complete contact trajectory.

Kinematic error

Because the instantaneous transmission ratio is usually not equal to the nominal transmission ratio, the corner error and angular acceleration error of the driven wheel will increase. Except at the datum point, there is a transmission ratio error at every point, which is $Δ i_{12}^{*} = i_{12}^{*} - i_{12}$ .

Since the transmission ratio is not constant, there is also an error in the angle of the worm gear, and the error in the angle of the worm gear is $Δ φ_{2}^{*} = φ_{2} - φ_{2 J} - \frac{1}{i_{12}} (φ_{1} - φ_{1 J})$ , where $φ_{1 J}$ and $φ_{2 J}$ are the values of $φ_{1}$ and $φ_{2}$ at the datum point, respectively.

The worm gear angle error curve must be a parabola with a downwards opening to ensure good contact and reduce impact. Because the transmission ratio is not constant, there will also be errors in the angular velocity of the worm gear. The angular velocity error of the worm gear is $Δ ω_{2}^{*} = \frac{1}{i_{12}^{*}} - \frac{1}{i_{12}}$ .

Calculation method for the long and short semiaxes of the contact ellipse

Relative principal curvature and relative principal direction of points on the contact path

Once the coordinates of each point on the contact path have been calculated, it must be ensured that all contact points are free from curvature interference.

After coordinate transformation, the following can be obtained:

\begin{matrix} {({\vec{α}}_{ξ}^{(d)})}_{od} = R [{\vec{k}}_{od}, φ_{d}] {({\vec{α}}_{ξ}^{(d)})}_{d} = \cos α_{2} \cos ϕ_{d} {\vec{i}}_{od} + \cos α_{2} \sin ϕ_{d} {\vec{j}}_{od} + \sin α_{2} {\vec{k}}_{od}, \\ {({\vec{α}}_{η}^{(d)})}_{od} = R [{\vec{k}}_{od}, φ_{d}] {({\vec{α}}_{η}^{(d)})}_{d} = α_{η x}^{(d)} {\vec{i}}_{od} + α_{η y}^{(d)} {\vec{j}}_{od} + α_{η z}^{(d)} {\vec{k}}_{od} \end{matrix}

(52)

where $α_{η x}^{(d)} = \frac{- (p + r_{02} \tan α_{2}) \sin α_{2} \cos ϕ_{d} - u_{2} \sin ϕ_{d}}{D_{2}}$ , $α_{η y}^{(d)} = \frac{- (p + r_{02} \tan α_{2}) \sin α_{2} \sin ϕ_{d} + u_{2} \cos ϕ_{d}}{D_{2}}$ , and $α_{η z}^{(d)} = \frac{(p + r_{02} \tan α_{2}) \cos α_{2}}{D_{2}}$ .

According to the meshing principle, the normal vector of the contact path between the worm gear and the worm gear hob is

{({\vec{N}}_{d})}_{od} = N_{ξ}^{(d)} {({\vec{α}}_{ξ}^{(d)})}_{od} + N_{η}^{(d)} {({\vec{α}}_{η}^{(d)})}_{od}

(53)

where $N_{ξ}^{(d)} = τ_{ξ}^{(d)} {({\vec{V}}_{d 2})}_{od} \cdot {({\vec{α}}_{η}^{(d)})}_{od} + {({\vec{ω}}_{d 2})}_{od} \cdot {({\vec{α}}_{η}^{(d)})}_{od}$ ,

N_{η}^{(d)} = τ_{ξ}^{(d)} {({\vec{V}}_{d 2})}_{od} \cdot {({\vec{α}}_{ξ}^{(d)})}_{od} + k_{η}^{(d)} {({\vec{V}}_{d 2})}_{od} \cdot {({\vec{α}}_{η}^{(d)})}_{od} - {({\vec{ω}}_{d 2})}_{od} \cdot {({\vec{α}}_{ξ}^{(d)})}_{od} .

The curvature interference limit line function of the hob and worm gear is

Ψ_{d} = {({\vec{N}}_{d})}_{od} \cdot {({\vec{V}}_{d 2})}_{od} + \frac{A \cos (θ_{2} + φ_{d}) - B \sin (θ_{2} + φ_{d})}{i_{d 2} D_{2}}

(54)

The normal curvatures of the worm gear tooth surface along the ${({\vec{α}}_{ξ}^{(d)})}_{od}$ and ${({\vec{α}}_{η}^{(d)})}_{od}$ directions and the geodesic torsion along the ${({\vec{α}}_{ξ}^{(d)})}_{od}$ direction are

k_{ξ}^{(2)} = - \frac{{(N_{ξ}^{(d)})}^{2}}{Ψ_{d}}, k_{η}^{(2)} = k_{η}^{(d)} - \frac{{(N_{η}^{(d)})}^{2}}{Ψ_{d}}, τ_{ξ}^{(2)} = τ_{ξ}^{(d)} - \frac{N_{ξ}^{(d)} N_{η}^{(d)}}{Ψ_{d}}

(55)

For the convenience of calculation, the unit orthogonal vectors on the worm gear and worm tooth surface are transformed into the same fixed coordinate system $σ_{o 2}$ . After coordinate transformation, it can be derived that:

\begin{matrix} {({\vec{α}}_{ξ}^{(1)})}_{o 2} = R [{\vec{j}}_{o 2}, - \frac{π}{2}] R [{\vec{k}}_{o 1}, φ_{1}] {({\vec{α}}_{ξ}^{(1)})}_{1}, {({\vec{α}}_{η}^{(1)})}_{o 2} R [{\vec{j}}_{o 2}, - \frac{π}{2}] R [{\vec{k}}_{o 1}, φ_{1}] {({\vec{α}}_{η}^{(1)})}_{1}, \\ {({\vec{α}}_{ξ}^{(d)})}_{o 2} = R [{\vec{k}}_{o 2}, φ_{2} - \frac{φ_{d}}{i_{d 2}}] R [{\vec{j}}_{od}, - \frac{π}{2}] {({\vec{α}}_{ξ}^{(d)})}_{od} . \end{matrix}

(56)

As shown in Figure 5, the four-unit vectors at the contact point are in the same plane, and the angle $ζ$ of ${({\vec{α}}_{ξ}^{(1)})}_{o 2}$ and ${({\vec{α}}_{ξ}^{(d)})}_{o 2}$ can be expressed as

\cos ζ = {({\vec{α}}_{ξ}^{(1)})}_{o 2} \cdot {({\vec{α}}_{ξ}^{(d)})}_{o 2}, \sin ζ = {({\vec{α}}_{η}^{(1)})}_{o 2} \cdot {({\vec{α}}_{ξ}^{(d)})}_{o 2}

(57)

Figure 5.

Contact ellipse at the instantaneous contact point.

The normal curvatures of the surface $S_{1}$ in the ${({\vec{α}}_{ξ}^{(d)})}_{o 2}$ direction and ${({\vec{α}}_{η}^{(d)})}_{o 2}$ direction are

k_{1 ξ} = k_{η}^{(1)} \sin^{2} ζ + 2 τ_{ξ}^{(1)} \sin ζ \cos ζ, k_{1 η} = k_{η}^{(1)} \cos^{2} ζ - 2 τ_{ξ}^{(1)} \sin ζ \cos ζ

(58)

In the directions ${({\vec{α}}_{ξ}^{(d)})}_{o 2}$ , the geodesic torsion of $S_{1}$ is

τ_{1 ξ} = k_{η}^{(1)} \sin ζ \cos ζ + τ_{ξ}^{(1)} (\cos^{2} ζ - \sin^{2} ζ)

(59)

The induced normal curvature and induced geodesic torsion of tooth surfaces $S_{1}$ and $S_{2}$ are

k_{ξ}^{(12)} = k_{1 ξ} - k_{ξ}^{(2)}, k_{η}^{(12)} = k_{1 η} - k_{η}^{(2)}, τ_{ξ}^{(12)} = τ_{1 ξ} - τ_{ξ}^{(2)}

(60)

According to equation (57), the relative mean curvature and the relative Gauss curvature at the contact point can be obtained

H^{(12)} = \frac{k_{ξ}^{(12)} + k_{η}^{(12)}}{2}, K^{(12)} = k_{ξ}^{(12)} k_{η}^{(12)} - {(τ_{ξ}^{(12)})}^{2}

(61)

Then, the relative principal curvature can be expressed as

\begin{matrix} k_{1}^{(12)} = H^{(12)} - \sqrt{{(H^{(12)})}^{2} - K^{(12)}}, k_{2}^{(12)} = H^{(12)} \\ + \sqrt{{(H^{(12)})}^{2} - K^{(12)}} \end{matrix}

(62)

The normal vector of the worm is pointed from the gear surface to the inside, and if there is no curvature interference, the relative principal curvature $k_{1, 2}^{(12)}$ needs to be greater than zero, then $H^{(12)} > 0$ , and $k_{2}^{(12)} > k_{1}^{(12)} > 0$ .

From the Euler formula and Bertrand formula,

γ_{1} = \arctan (\frac{τ_{ξ}^{(12)}}{k_{ξ}^{(12)} - k_{2}^{(12)}})

(63)

where $γ_{1}$ is the included angle from the first principal direction ${({\vec{α}}_{ξ}^{(d)})}_{o 2}$ to ${({\vec{g}}_{1}^{(12)})}_{o 2}$ .

Then the two main directions are

\begin{matrix} {({\vec{g}}_{1}^{(12)})}_{o 2} = \cos γ_{1} {({\vec{α}}_{ξ}^{(d)})}_{o 2} + \sin γ_{1} {({\vec{α}}_{η}^{(d)})}_{o 2}, {({\vec{g}}_{2}^{(12)})}_{o 2} \\ = - \sin γ_{1} {({\vec{α}}_{ξ}^{(d)})}_{o 2} + \cos γ_{1} {({\vec{α}}_{η}^{(d)})}_{o 2} \end{matrix}

(64)

Half-axis calculation method for the contact ellipse

The worm gear and worm contact can be regarded as the contact between two isotropic uniform objects with different elastic constants. In the case of no curvature interference, after the two tooth surfaces contact at point $P$ and bear the load, an elliptical contact path is formed at the contact point. The points with a normal gap less than $Δ η$ in the neighborhood of point $P$ will contact each other to form an instantaneous contact elliptical surface. The long and short semiaxes of the instantaneous contact ellipse are²¹

a_{e} = \sqrt{\frac{2 Δ η}{| k_{1}^{(12)} |}}, b_{e} = \sqrt{\frac{2 Δ η}{| k_{2}^{(12)} |}}

(65)

where $a_{e}$ is the long semiaxis of the contact ellipse, $b_{e}$ is the short semiaxis of the contact ellipse, and $Δ η$ is the diameter of the red lead powder particles used in the tooth surface contact path for color inspection, $Δ η = 0.0064 ~ 0.011 (mm)$ .

Results and discussion

Determination of mismatched parameters and contact path

Point contact analysis necessitates parameter adjustments based on line contact results, thus requiring parameter settings that ensure consistency and avoid mismatch. The primary geometric parameters of the worm mechanism are defined as follows: transmission ratio $i_{12} = 12.5$ , number of worm threads $Z_{1} = 4$ , and center distance $a = 315 (mm)$ . All other relevant parameters can be derived from these three foundational values. Specific parameter details are listed in Table 1.

Table 1.

The calculation results of the worm pair parameters.

Description	Symbol	Value
Transmission ratio	$i_{12}$	$12.5$
Center distance	$a$	$315 (mm)$
Reference diameter of the worm	$d_{1} \approx 0.681 a^{0.875}$	$110 (mm)$
Number of threads	$Z_{1}$	$4$
Tooth number of worm gear	$Z_{2} = i_{12} Z_{1}$	$50$
Module	$m \approx (1.4 ~ 1.7) a / Z_{2}$	$10$
Normal profile angle of the worm	$α_{n 1}$	$20 (°)$
Lead angle	$γ = \arctan (m Z_{1} / d_{1})$	$19.9831 (°)$
Spiral parameters	$p = m Z_{1} / 2$	$20$
Worm addendum	$h_{a 1} = m$	$10 (mm)$
Worm dedendum	$h_{f 1} = 1.2 m$	$12 (mm)$
Addendum circle radius of the worm	$r_{a 1} = d_{1} / 2 + h_{a 1}$	$65 (mm)$
Dedendum circle radius of worm	$r_{f 1} = d_{1} / 2 - h_{f 1}$	$43 (mm)$
Modification coefficient	$X_{2}$	$0.99$
Reference radius of worm gear	$r_{2} = m Z_{2} / 2$	$500 (mm)$
Worm gear addendum	$h_{a 2} = (1 + X_{2}) m$	$19.9 (mm)$
Worm gear dedendum	$h_{f 2} = (1.2 - X_{2}) m$	$2.1 (mm)$
Addendum circle radius of worm gear	$r_{a 2} = r_{2} + h_{a 2}$	$269.9 (mm)$
Dedendum circle radius of worm gear	$r_{f 2} = r_{2} - h_{f 2}$	$247.9 (mm)$
The radius of the gorge circle	$r_{g 2} = a_{d} - r_{a 2}$	$45 (mm)$

The adjustable mismatch parameters primarily consist of the geometric design parameters for both the worm and the worm gear hob. According to the forming principle of the worm gear and worm gear hob, a straight line tangent to a cylinder with the radius of the guide cylinder spirally inclines around a fixed axis. Adjustments are made to specific parameters: namely, the geometric parameters $r_{01}$ and $α_{1}$ defining the worm tooth surface $S_{1}$ , and the parameters $r_{02}$ and $α_{2}$ characterizing the worm gear hob tooth surface $S_{d}$ . The key process parameters for grinding the worm using the worm gear hob are the center distance $a_{d}$ and the process transmission ratio $i_{d 2}$ .

When setting the mismatched parameter, the datum point should be in the center of the tooth surface, and the resultant formula $M_{0} = 0$ , and the value of $u_{2}$ obtained should be near $\frac{u_{min} + u_{max}}{2}$ . From the data in Table 1, $u_{min} = 45.0285 (mm)$ and $u_{max} = 68.4355 (mm)$ can be obtained. The preliminarily designed mismatch parameters are shown in Table 2, and an image of $M_{0} = 0$ is shown in Figure 6. There are five zero points in the value range of $u_{min} ~ u_{max}$ , which are $u_{2} = 48.5743 (mm)$ , $u_{2} = 49.6004 (mm)$ , $u_{2} = 54.4459 (mm)$ , $u_{2} = 58.1572 (mm)$ , and $u_{2} = 64.7889 (mm)$ . The results must meet the common point conditions, the common normal line conditions, and the meshing function between the worm gear hob and the worm gear. After considering the above results, it is found that only $u_{2} = 58.1572 (mm)$ can meet all the equations, indicating that $u_{2} = 58.1572 (mm)$ is the solution at the datum point. At this time, $\sin ϕ_{d} = 0.9895$ . See Table 3 for the values of the other unknowns.

Table 2.

Mismatch parameter selection.

Item	Symbol	Value
Worm guide cylinder radius	$r_{01}$	$3.9547 (mm)$
Radius of the worm gear guide cylinder	$r_{02}$	$5.9547 (mm)$
Worm production line inclination angle	$α_{1}$	$18 . 7493^{°}$
Worm gear hob production line inclination angle	$α_{2}$	$18 . 9493^{°}$
Process transmission ratio	$i_{d 2}$	$12.58$
Process center distance	$a_{d}$	$314.9 (mm)$

Figure 6.

Image of $M_{0}$ at the datum point.

Table 3.

The calculation results of each point on the contact path and their relative principal curvature.

Item	Point $A$	Point $B$	Point $B_{1}$	Point $B_{2}$	Point $B_{3}$	Point $B_{4}$	Point $C$
$u_{1} (mm)$	58.7711	68.4355	64.8697	61.3231	57.8027	54.3196	50.8917
$θ_{1} (°)$	15.8205	−65.0646	−34.0722	−4.5418	23.3411	49.4158	73.5886
$θ_{2} (°)$	68.0103	155.9687	122.4814	90.3379	59.7246	30.8066	3.6878
$φ_{d} (°)$	58.1572	68.2806	64.5643	60.8479	57.1315	53.4151	49.6987
$θ_{2} (°)$	3.6191	−74.7264	−44.4551	−15.8699	10.7541	35.1164	56.8937
$φ_{d} (°)$	78.0801	163.8593	130.9851	99.6528	70.1289	42.7040	17.6883
$φ_{2} (°)$	5.2649	12.2935	9.6192	7.0505	4.6020	2.2863	0.1112
$Δ i_{12}^{*}$	0	0.0253	0.0180	0.0087	−0.0038	−0.0209	−0.0451
$k_{1}^{(12)} (μ m^{- 1})$	0.3997	0.2822	0.3203	0.3645	0.4135	0.4602	0.4794
$k_{2}^{(12)} (m m^{- 1})$	0.0076	0.0119	0.0100	0.0085	0.0072	0.0060	0.0049

The calculation in section “Datum point and intersection point of the contact path and worm pair boundary” shows the intersection of the contact path and the worm gear top circle in Figure 7(a). When $u_{1} = u_{max}$ , the zero point of $M_{1}$ is at $u_{2} = 68.2806 (mm)$ . The intersection between the contact path and the worm gear gorge circle is shown in Figure 7(b). With the dichotomy method, when $u_{2} = 49.6987 (mm)$ , $M_{1} = 0$ , and then $u_{1} = 50.8917 (mm)$ . Other parameters for these two points are shown in Table 3.

Figure 7.

The function image at the intersection of the contact path and worm gear boundary: (a) intersection of the contact path and the worm top circle and (b) the intersection of the contact path and worm gear gorge circle.

The complete contact path can be derived by assigning different values to the parameter $u_{2}$ using an interpolation method, with the calculated coordinates of the resulting points tabulated in Table 3. The relative principal curvatures computed from equation (62) all exhibit positive values, confirming the absence of curvature interference under these mismatched parameters.

As depicted in Figure 8, the contact path is centrally positioned on the worm gear and extends across its tooth surface. The major axis of the contact ellipse is nearly perpendicular to the contact line, and the geometries of these ellipses vary systematically with different applied loads.

Figure 8.

Contact path and contact ellipse of worm gear surface.

Kinematic error curves of the mismatched ZN-type worm pair

The computed angular error curves of the worm gear over three meshing cycles are illustrated in Figure 9. Each curve manifests a downward-opening parabolic profile, tangential to the horizontal axis at the datum reference point. Specifically, negative angular errors are observed at all non-datum contact points along the curves. A critical observation is the intersection of these curves across the three cycles, which quantitatively demonstrates that the contact ratio of the mismatched ZN-type worm gear pair exceeds unity. This geometric-mechanical characteristic enables continuous meshing transmission, ensuring overlapping tooth contact throughout the engagement period.

Figure 9.

Worm gear angle error curve.

The transmission ratio error and worm gear angular velocity error profiles are presented in Figure 10. The transmission ratio error curve demonstrates a monotonic increasing trend, with its null point—where the theoretical and actual transmission ratios coincide—positioned symmetrically at the curve’s midpoint. Notably, the maximum value of this error satisfies $| Δ i_{12}^{*} | < 0.05$ (corresponding to a peak error below $0.4 %$ in dimensionless terms), indicating minimal fluctuations in the instantaneous transmission ratio. Such low-amplitude variations are critical for maintaining consistent power transfer and reducing kinematic inconsistencies in precision drive systems.

Figure 10.

Kinematic error curve of the worm pair: (a) worm pair transmission ratio error curve and (b) worm gear angular velocity error curve.

Concurrently, the angular velocity error curve of the worm gear exhibits a low-magnitude characteristic, manifesting as a monotonically decreasing function throughout the meshing cycle. This behavior suggests that angular velocity deviations diminish as the contact point progresses from the entry to the exit of the meshing zone, likely due to the gradual alignment of tooth profiles and the distribution of contact stresses. The small order of magnitude, typically an order lower than the transmission ratio error, signifies that rotational speed fluctuations are effectively mitigated, which is essential for minimizing dynamic loads, torsional vibrations, and associated noise generation. Together, these error characteristics validate the design’s compliance with high-precision transmission requirements, where both kinematic accuracy and dynamic stability are paramount.

For ZN-type worm gear pairs with different degrees of diameter mismatch, the conclusions drawn are similar. When adjusting the mismatch parameters, it is necessary to ensure that the guide cylinder radius of the worm gear hob is slightly larger than that of the worm, that is, $r_{02} > r_{01}$ . Appropriately increasing the generating line inclination angle $α_{2}$ of the worm gear hob and the processing transmission ratio $i_{d 2}$ , while ensuring that the processing center distance $a_{d}$ between the worm gear hob and the worm gear is smaller than the theoretical center distance $a$ .

Conclusion

This paper conducts a computational investigation into the meshing theory of geometrically mismatched ZN-type worm gear pairs. The analytical tooth surface equations for the worm and worm gear are systematically derived, followed by the establishment of contact equations specific to mismatched ZN-type configurations. The seven embedded unknowns within the contact equations are sequentially eliminated by employing the resultant elimination method, thereby reducing the system to a resultant matrix containing a single remaining variable. Numerical solutions are obtained for critical geometric features: the datum point, the intersection of the contact path with the worm gear’s gorge circle, and its intersection with the root circle. This numerical framework enables comprehensive parametric analyses with minimized computational effort during subsequent parameter adjustments, offering robust reusability for iterative design optimizations.

The mathematical formulation describing the relative principal curvature at the contact point is also rigorously deduced. Numerical simulations were conducted on the mismatching contact path and motion error curve of the ZN-type worm gear pair. The results show that the mismatched ZN-type worm gear pair exhibits excellent contact performance, a substantial contact area, and a well-defined motion error curve. Through minor adjustments to a few worm gear parameters, the mismatched worm and worm gear can achieve smooth meshing.

Footnotes

Handling Editor: Chenhui Liang

ORCID iD

Yaping Zhao

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (52075083), the Fundamental Research Funds for the Central Universities (N25ZLL045), and the Open Fund of the Key Laboratory for Metallurgical Equipment and Control of Education Ministry in Wuhan University of Science and Technology (MECOF2022B04 and MECOF2023B01).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Liou

Rakuff

The development of worm drives. Power Transm Eng 2018; 38–43.

AGMA 6022-C93. Design manual for cylindrical wormgearing. AGMA, 2014.

Zhao

Novel method to establish equation of ZN helicoid and meshing limit line of related worm drive. Proc IMechE, Part C: J Mechanical Engineering Science 2024; 238: 479–490.

Litvin

Fuentes-Aznar

Gear geometry and applied theory. 2nd ed. Cambridge University Press, 2004.

Ren

, et al. A point-contact conjugate hourglass worm drive based on the meshing theory of conjugate tooth surfaces generated by two generating surfaces. Mech Mach Theory 2022; 174: 104877.

Chen

Luo

, et al. Study on meshing performance of a novel point-contact hourglass worm drive based on modified involute. Mech Mach Theory 2023; 185: 105339.

Litvin

Gonzalez-Perez

Yukishima

, et al. Design, simulation of meshing, and contact stresses for an improved worm gear drive. Mech Mach Theory 2007; 42(8): 940–959.

Litvin

Vecchiato

Fuentes

, et al. Automatic determination of guess values for simulation of meshing of gear drives. Comput Methods Appl Mech Eng 2004; 193(33–35): 3745–3758.

Zhao

Kong

Meshing principle of conical surface enveloping spiroid drive. Mech Mach Theory 2018; 123: 1–26.

10.

Sun

Dong

, et al. Meshing theory of point-contact conical-envelope cylindrical worm-face worm gear drive. Mech Mach Theory 2025; 205: 105870.

11.

Meng

Zhao

Cui

, et al. Meshing theory of offsetting Archimedes cylindrical worm drive. Mech Based Des Struct Mach 2024; 52(1): 152–168.

12.

Chi

Zhao

Zhu

, et al. Mismatched gearing composed of hourglass worm and spur gear: meshing theory, tooth contact simulation, comprehensive design. Mech Mach Theory 2022; 174: 104883.

13.

Simon

Computer simulation of tooth contact analysis of mismatched spiral bevel gears. Mech Mach Theory 2007; 42(3): 365–381.

14.

Simon

Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears. Mech Mach Theory 2006; 41(6): 707–724.

15.

Guo

An improved approach for tooth contact analysis of mismatched involute spiroid gears. Mech Mach Theory 2023; 180: 105157.

16.

Chen

Wang

, et al. Local conjugate meshing performance analysis of an envelope hourglass worm drive based on medium gear. J Mech Des 2025; 147(3): 033501.

17.

Liu

Chen

, et al. Study on the vibration characteristics of worm drive with different contact forms. Nonlinear Dyn 2025; 113(11): 12915–12935.

18.

Differential geometry and gear meshing principle. Petroleum University Press, 1999.

19.

Zhao

Engineering differential geometry of curves and surfaces. Science Press, 2023.

20.

Dong

Foundation of gear engagement theory. China Machine Press, 1989.

21.

Zhao

Huai

Zhang

Compound modification of globoidal worm drive with variable parameters. Appl Math Modell 2017; 50: 17–38.

22.

Dong

Design and modification of hourglass worm drives. China Machine Press, 2004.