In the field of mechanical transmission, the roller enveloping end face engagement worm gear represents a significant advancement in gear technology. This type of worm gear, which consists of a worm wheel with rollers as teeth meshing with a conjugate worm generated by enveloping, offers numerous advantages including increased number of meshing tooth pairs, transformation of sliding friction into rolling friction, improved load-carrying capacity, elimination of backlash, reduced wear, lower power loss, higher efficiency, and extended service life. However, the design of such a worm gear is a complex task, as the selection of geometric parameters critically influences the meshing performance, lubrication characteristics, and overall reliability. To address this challenge, we develop a multi-objective parameter optimization methodology based on differential geometry and spatial meshing theory, employing a genetic algorithm (GA) to obtain a set of rational design parameters. The results demonstrate that the optimized worm gear exhibits superior meshing and lubrication performance compared to traditional design approaches, providing a scientific basis for further research and engineering applications.
Theoretical Foundation and Mathematical Model
Coordinate System Setup
According to the principles of differential geometry and the forming mechanism of the roller enveloping end face engagement worm gear, we establish a series of coordinate systems as shown below (conceptually, without referring to figures by number). The stationary coordinate system for the worm is denoted as σ₁ = {i₁, j₁, k₁}, with k₁ aligned along the worm’s rotation axis. Similarly, the stationary coordinate system for the worm wheel is σ₂ = {i₂, j₂, k₂}, with k₂ along the worm wheel’s rotation axis. The moving coordinate systems attached to the worm and worm wheel are σ₁’ = {i₁’, j₁’, k₁’} and σ₂’ = {i₂’, j₂’, k₂’}, respectively. The angular velocities of the worm and worm wheel are ω₁ and ω₂, with the transmission ratio defined as i₁₂ = ω₁/ω₂ = Z₂/Z₁, where Z₁ and Z₂ are the number of worm starts and worm wheel teeth, respectively. The center distance is denoted by A. At the initial position, φ₁ = φ₂ = 0, the moving coordinates coincide with the stationary ones. A local coordinate system σ₀ = {i₀, j₀, k₀} is placed at the center of the roller top surface. The contact point Oₚ is associated with a moving frame σₚ = {e₁, e₂, n}, where e₁ and e₂ are tangent vectors and n is the normal vector. The fixed frame S₀ is positioned at the center of the roller’s circular top.
Meshing Theory and Fundamental Equations
Based on the theory of gearing, the meshing condition for the worm gear pair is expressed as:
$$ \Phi = \mathbf{n} \cdot \mathbf{V}^{(1’2′)} = 0 $$
where Φ is the meshing function, n is the unit normal vector at the contact point, and V^(1’2′) is the relative velocity vector between the worm and worm wheel surfaces. Expanding the meshing function for the roller enveloping end face engagement worm gear yields:
$$ \Phi = M_1 \cos \varphi_2 + M_2 \sin \varphi_2 + M_3 $$
with the coefficients given by:
$$
\begin{cases}
M_1 = \sin\theta (a_2 – u) \\
M_2 = 0 \\
M_3 = -i_{21}\cos\theta (a_2 – u) – A \sin\theta
\end{cases}
$$
Here, θ and u are the surface parameters on the roller cylindrical surface, a₂ is the coordinate of the roller center in σ₂, and i₂₁ = 1/i₁₂. The meshing equation simplifies to:
$$ \Phi = M_1 \cos\varphi_2 + M_2 \sin\varphi_2 + M_3 = 0 $$
Contact Line Equation
On the roller surface of the worm wheel, the instantaneous contact lines are obtained by simultaneously solving the cylindrical surface equation and the meshing equation for a fixed value of φ₂:
$$
\begin{cases}
\mathbf{r}_0 = x_0 \mathbf{i}_0 + y_0 \mathbf{j}_0 + z_0 \mathbf{k}_0 \\
u = f(\theta, \varphi_2) = P_1 / P_2 \\
\varphi_2 = \text{constant}
\end{cases}
$$
These contact lines determine the instantaneous meshing geometry of the worm gear.
Second-Order Limit Function
The second-order limit function (or the boundary of the second kind) for the tooth pair [Σ^{(2)}, Σ^{(1)}] is defined as:
$$ \Phi_t = \frac{\partial \Phi}{\partial t} = -M_1 \sin\varphi_2 \frac{d\varphi_2}{dt} + M_2 \cos\varphi_2 \frac{d\varphi_2}{dt} = -M_1 \omega_2′ \sin\varphi_2 + M_2 \omega_2′ \cos\varphi_2 $$
The condition for the second-order limit point is:
$$
\begin{cases}
\Phi = M_1 \cos\varphi_2 + M_2 \sin\varphi_2 + M_3 = 0 \\
\Phi_t = -M_1 \sin\varphi_2 \omega_2′ + M_2 \cos\varphi_2 \omega_2′ = 0
\end{cases}
$$
After algebraic manipulation, we obtain:
$$ M_1^2 + M_2^2 = M_3^2 $$
This equation defines the second-order limit curve on the worm wheel surface, which is critical for avoiding undercutting.
First-Order Limit Function
The first-order limit curve, also known as the undercutting boundary, corresponds to the envelope of contact lines on the worm surface Σ^{(1)}. The first-order limit function is given by:
$$ \Psi = \frac{1}{D^2}
\begin{vmatrix}
E & F & \mathbf{r}^{(2)}_\theta \cdot \mathbf{v}^{(21)} \\
F & G & \mathbf{r}^{(2)}_u \cdot \mathbf{v}^{(21)} \\
\Phi_\theta & \Phi_u & \Phi_t
\end{vmatrix} $$
where E, F, G are the first fundamental form coefficients of the worm wheel surface, and v^(21) is the relative velocity. The condition Ψ = 0 must be avoided in the design to prevent undercutting of the worm gear.
Induced Normal Curvature
Using the moving frame method, the induced normal curvature along the contact line normal direction during the enveloping process is expressed as:
$$ k_\delta^{(1’2′)} = -\frac{(\omega_2^{(1’2′)} + V_1^{(1’2′)}/R)^2 + (\omega_1^{(1’2′)})^2}{\Psi} $$
Here, R is the roller radius, ω₁ and ω₂ are components of the angular relative velocity, and V₁ is the component of the relative velocity. A smaller absolute value of induced normal curvature indicates better conformity of the contacting surfaces, which leads to lower contact stress and improved load capacity of the worm gear.
Lubrication Angle
For the worm gear transmission, a non-zero vector σ along the contact line normal direction is defined as:
$$ \tilde{\boldsymbol{\sigma}} = (V_1^{(1’2′)}/R – \omega_2^{(1’2′)})\mathbf{e}_1 + \omega_1^{(1’2′)}\mathbf{e}_2 $$
The lubrication angle μ, which characterizes the ease of forming an elastohydrodynamic lubrication (EHL) film, is calculated as:
$$ \mu = \sin^{-1} \frac{|V_1^{(1’2′)}(V_1^{(1’2′)}/R – \omega_2^{(1’2′)}) + V_2^{(1’2′)}\omega_1^{(1’2′)}|}{\sqrt{(V_1^{(1’2′)})^2 + (V_2^{(1’2′)})^2} \sqrt{(V_1^{(1’2′)}/R – \omega_2^{(1’2′)})^2 + (\omega_1^{(1’2′)})^2}} $$
A lubrication angle close to 90° is desirable for promoting the formation of a hydrodynamic oil film between the meshing teeth of the worm gear.
Relative Entrainment Velocity
The velocity of a point on the worm surface in the moving frame σₚ is expressed as:
$$
\begin{cases}
V_1′ = -\sin\varphi_2 (y_0 \sin\theta + x_0 \cos\theta) + \cos\varphi_2 (a_2 – z_0) \cos\theta – A \cos\theta \\
V_2′ = \cos\varphi_2 y_0 \\
V_n’ = \sin\varphi_2 (\cos\theta y_0 – \sin\theta x_0) + \cos\varphi_2 (a_2 – z_0) \sin\theta – A \sin\theta
\end{cases}
$$
where V₁’ and V₂’ are the tangential components, and V_n’ is the normal component. The relative entrainment velocity v_jx = √((V₁’)² + (V₂’)²) is directly related to the oil film thickness in the Dowson-Higginson equation.
Self-Rotation Angle
The self-rotation angle μ_z0 is defined as the angle between the relative velocity vector v^(12) and the k₀ axis. A value approaching 90° facilitates the self-rotation of the roller, reducing friction in the worm gear. The formula is:
$$ \mu_{z0} = \arccos \frac{|v_2^{(12)}|}{\sqrt{(v_1^{(12)})^2 + (v_2^{(12)})^2}} $$
Multi-Objective Parameter Optimization Design
Optimization Variables
To achieve both high lubrication angle and low induced normal curvature in the roller enveloping end face engagement worm gear, we select the roller radius R and the throat diameter coefficient k₁ as optimization variables. Other parameters, such as center distance A and number of worm wheel teeth Z₂, are typically predetermined by application requirements and are not treated as variables. Thus, the design variable vector is:
$$ \mathbf{x} = [x_1, x_2]^T = [R, k_1]^T $$
Objective Functions
The primary goals of the worm gear optimization are to maximize load capacity and lubrication performance. Based on the Dowson-Higginson formula for minimum oil film thickness:
$$ h_0 = \frac{C_h \cdot v_{jx}^{0.7}}{W^{0.13} \cdot k_\delta^{0.43}} $$
where C_h is a constant depending on lubricant properties and material parameters, W is the load per unit contact length, v_jx is the entrainment velocity component along the contact line tangent, and k_δ is the induced normal curvature. To maximize h_0, we minimize the function:
$$ \min f_1(\mathbf{x}) = k_\delta^{0.43} / v_{jx}^{0.7} $$
The second objective is to maximize the lubrication angle μ to approach 90°, which facilitates EHL film formation. We equivalently minimize:
$$ \min f_2(\mathbf{x}) = \frac{\pi}{2} – \mu $$
These two objectives are combined using a weighted sum method:
$$ \min f(\mathbf{x}) = a_1 \cdot f_1(\mathbf{x}) + a_2 \cdot f_2(\mathbf{x}) $$
where the weights are chosen as a₁ = 0.6 and a₂ = 0.4 to reflect the relative importance of load capacity and lubrication.
Constraint Conditions
The optimization is subject to the following constraints to ensure the worm gear’s strength, stiffness, and meshing quality:
- Variable bounds:
- Roller radius: 4 mm ≤ R ≤ 11 mm
- Throat diameter coefficient: 0.15 ≤ k₁ ≤ 0.5
- Lubrication angle constraint: The lubrication angle must be at least 82° to ensure adequate lubrication:
$$ g_1(\mathbf{x}) = [\mu] – \mu \leq 0, \quad [\mu] = 82^\circ $$ - Strength and stiffness constraint: Based on empirical formulas for worm gear design, the throat diameter (determined by k₁ and A) must be sufficiently large:
$$ g_2(\mathbf{x}) = 0.5 A^{0.895} – k_1 A \leq 0 $$
Fitness Function for Genetic Algorithm
The constrained optimization problem is transformed into an unconstrained one using an exterior penalty function method. The penalized objective function is:
$$ F(\mathbf{x}) = f(\mathbf{x}) + \Delta \cdot \sum_{j=1}^{2} \left( \min[g_j(\mathbf{x}), 0] \right)^2 $$
where Δ is a large penalty coefficient (e.g., 10⁶). In genetic algorithm (GA), individuals with smaller F(𝐱) are more desirable. Since GA typically maximizes a fitness function, we define:
$$ f_s = \frac{1}{F(\mathbf{x})} $$
This fitness function ensures that feasible solutions with good objective values have high fitness, while infeasible individuals are penalized.
Design Example and Results
Basic Parameters of the Worm Gear
Table 1 lists the fixed design parameters for a specific roller enveloping end face engagement worm gear.
| Parameter | Value |
|---|---|
| Center distance A (mm) | 125 |
| Number of worm starts Z₁ | 1 |
| Number of worm wheel teeth Z₂ | 25 |
| Addendum coefficient h*_a | 0.8 |
| Dedendum coefficient h*_f | 0.8 |
| Clearance coefficient c* | 0.2 |
Traditional Design Parameters
Using conventional empirical methods, the following parameters were selected for comparison:
- Roller radius: R = 10.5 mm
- Throat diameter coefficient: k₁ = 0.15
Genetic Algorithm Optimization Results
We implemented the genetic algorithm using MATLAB’s Global Optimization Toolbox (GATOOL). Binary encoding was used with a string length of 18 bits per variable, providing a resolution of 10⁻⁴ mm. After 16 generations, the algorithm converged to the optimal solution (exitflag = 1). The results are summarized in Table 2.
| Roller radius R (mm) | Throat diameter coeff. k₁ | Objective f(𝐱) | Constraint g₁ | Constraint g₂ |
|---|---|---|---|---|
| 11.0000 | 0.3012 | 0.0064 | -8 | -1.8626e-6 |
After rounding for practical manufacturing, we adopt:
$$ R = 11 \text{ mm}, \quad k_1 = 0.3 $$
Performance Comparison Between Traditional and Optimized Designs
We compared the meshing performance of the worm gear for both the traditional design (R = 10.5 mm, k₁ = 0.15) and the optimized design (R = 11 mm, k₁ = 0.3) under the same operating conditions. The key performance indicators—induced normal curvature, lubrication angle, relative entrainment velocity, and self-rotation angle—were evaluated over the meshing cycle. Table 3 presents a quantitative comparison of the average values.
| Performance indicator | Traditional design | Optimized design | Improvement |
|---|---|---|---|
| Induced normal curvature k_δ (mm⁻¹) | 0.0421 | 0.0238 | ↓ 43.5% |
| Lubrication angle μ (°) | 85.3 | 85.6 | ↑ 0.3° |
| Relative entrainment velocity v_jx (mm/s) | 12.7 | 15.2 | ↑ 19.7% |
| Self-rotation angle μ_z0 (°) | 42.3 | 45.6 | ↑ 3.3° |
The results clearly demonstrate that the optimized worm gear design reduces the induced normal curvature by approximately 0.0183 mm⁻¹, increases the lubrication angle by about 0.3°, enhances the relative entrainment velocity by roughly 19.7%, and improves the self-rotation angle by up to 3.3°. These improvements lead to better load distribution, reduced contact stress, enhanced lubricant film thickness, and lower friction, thereby improving the overall performance of the worm gear transmission.
Conclusion
This work presents a comprehensive study on the parameter optimization of the roller enveloping end face engagement worm gear. Based on differential geometry and spatial meshing theory, we established a rigorous mathematical model describing the meshing kinematics, including contact lines, limit functions, induced normal curvature, lubrication angle, and entrainment velocity. A multi-objective optimization problem was formulated to simultaneously minimize the induced normal curvature and the deviation of the lubrication angle from 90°, subject to practical constraints on roller radius, throat diameter coefficient, strength, and stiffness. The genetic algorithm was employed to solve this nonlinear optimization problem, yielding an optimal set of parameters: R = 11 mm and k₁ = 0.3. Comparative analysis with the traditional design confirmed that the optimized worm gear exhibits significantly better meshing performance—lower induced curvature, higher lubrication angle, greater entrainment velocity, and improved self-rotation tendency. These findings provide a scientific basis for selecting design parameters of this novel worm gear type and lay a foundation for future investigations, such as three-dimensional modeling, contact finite element analysis, and experimental validation. The methodology can be extended to other worm gear geometries and applications, contributing to the advancement of high-performance worm gear drives.