In the field of mechanical transmission, worm gears are widely used due to their high reduction ratio, compact structure, and smooth operation. However, conventional worm gears suffer from relatively low transmission efficiency due to sliding friction. To address this issue, I have investigated a novel type of worm gears — the single-roller enveloping end face worm gears — which incorporate rolling elements to reduce friction losses. This paper presents a comprehensive mathematical model for the transmission efficiency of such worm gears, analyzes the influence of key parameters, and performs optimization using MATLAB. The goal is to provide design guidelines for achieving higher efficiency in worm gears.
Worm gears with movable teeth, such as the single-roller enveloping end face type, convert sliding friction into rolling friction by allowing the rollers on the worm gear to rotate about their own axes. This innovation significantly reduces friction losses compared to conventional worm gears. In this work, I focus on the transmission efficiency of these worm gears, deriving instantaneous and average efficiency formulas, and examining how geometrical and frictional parameters affect performance.
1. Mathematical Modeling of Single-Roller Enveloping End Face Worm Gears
The three-dimensional model of the single-roller enveloping end face worm gears consists of an end face worm and a roller-type worm gear. The rollers are evenly distributed on the end face of the worm gear disc and can rotate freely around their central axes. This structure increases the contact area and improves stability. The meshing theory forms the basis for force analysis and efficiency calculation.
At a given meshing point \( p \), the forces acting on the worm and the gear tooth are analyzed. Let \( F_{n1} \) and \( F_{n2} \) be the normal forces, \( F_{t1} \) and \( F_{t2} \) the tangential forces, and \( F_{a1} \) and \( F_{a2} \) the axial forces. According to meshing theory, the relative velocity vector \( \mathbf{v}_{12} \) has zero normal component at the meshing point, implying that friction forces lie in the common tangent plane. The roller rotates under the action of the tangential force, converting sliding into rolling. Let \( f \) be the sliding friction coefficient and \( f_g \) the ratio of rolling friction coefficient to sliding friction coefficient. The resultant force affecting efficiency on the gear tooth side is:
$$ \mathbf{F}_2 = \mathbf{F}_{n2} + \mathbf{F}_{a2} + f_g \mathbf{F}_{t2} $$
Similarly for the worm side:
$$ \mathbf{F}_1 = \mathbf{F}_{n1} + \mathbf{F}_{a1} + f_g \mathbf{F}_{t1} $$
By equilibrium, \( \mathbf{F}_2 = -\mathbf{F}_1 \). The friction forces are:
$$ F_{f1} = F_{n1} f \quad , \quad F_{f2} = F_{n2} f $$
Projecting onto tangential and axial directions:
$$ \begin{cases} F_{t1} = F_{f1} \cos \alpha_1 \\ F_{a1} = F_{f1} \sin \alpha_1 \end{cases} \quad , \quad \begin{cases} F_{t2} = F_{f2} \cos \alpha_2 \\ F_{a2} = F_{f2} \sin \alpha_2 \end{cases} $$
where \( \alpha_1 = \alpha_2 = \arctan(V_2 / V_1) \). The velocity components \( V_1 \) and \( V_2 \) are functions of the center distance \( A \), roller radius \( R \), worm gear tooth number \( z_2 \), transmission ratio \( i_{21} \), worm gear rotation angle \( \varphi_2 \), and tooth parameters \( \theta, u \). The intermediate variable \( a_2 \) is given by:
$$ a_2 = \frac{A (2 – k)(8 – 5 z_2)}{10 z_2} $$
where \( k \) is the throat neck coefficient. Detailed expressions for \( V_1 \) and \( V_2 \) are omitted for brevity but are derived from the geometry of the worm gears.
2. Transmission Efficiency Model
In the moving coordinate system, the instantaneous power of the worm \( P_w \) and the worm gear \( P_g \) are:
$$ P_w = \mathbf{F}_1 \cdot \mathbf{v}_1 \quad , \quad P_g = \mathbf{F}_2 \cdot \mathbf{v}_2 $$
The instantaneous transmission efficiency \( \eta \) at meshing point \( p \) is:
$$ \eta = \frac{P_g}{P_w} = \frac{ ( – \mathbf{e}_n \pm f \sin\alpha_2 \mathbf{e}_2 \pm f_g f \cos\alpha_2 \mathbf{e}_1 ) \cdot \mathbf{v}_2 }{ ( \mathbf{e}_n \mp f \sin\alpha_1 \mathbf{e}_2 \mp f_g f \cos\alpha_1 \mathbf{e}_1 ) \cdot \mathbf{v}_1 } $$
where \( \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_n \) are the base vectors of the local coordinate system on the roller.
The single-tooth instantaneous efficiency \( \eta_u \) is obtained by integrating \( \eta \) along the instantaneous contact line height \( h \):
$$ \eta_u = \frac{1}{h} \int_{0}^{h} \eta \, du $$
The tooth height \( h \) is the sum of addendum \( h_a \) and dedendum \( h_f \), with \( h_f = h_{fc} m \) (usually \( h_{fc} = 1 \) or 0.8). The single-tooth average efficiency \( \eta_p \) is the average of \( \eta_u \) over the meshing period from worm gear angle \( \varphi_{20} \) to \( \varphi_{2e} \):
$$ \eta_p = \frac{1}{\varphi_{2e} – \varphi_{20}} \int_{\varphi_{20}}^{\varphi_{2e}} \eta_u \, d\varphi_2 $$
For multiple teeth meshing simultaneously, let \( n \) be the number of teeth in contact. The instantaneous efficiency \( \eta_{un} \) and average efficiency \( \eta_{pn} \) are:
$$ \eta_{un} = \frac{ \sum_{i=1}^{n} \int_{0}^{h} \eta_i \, du }{ n h } $$
$$ \eta_{pn} = \frac{1}{n} \sum_{i=1}^{n} \frac{ \int_{\varphi_{20i}}^{\varphi_{2ei}} \eta_{ui} \, d\varphi_2 }{ \varphi_{2ei} – \varphi_{20i} } $$
where \( \varphi_{20i} = \varphi_{201} + (i-1)\gamma \), \( \varphi_{2ei} = \varphi_{201} + i\gamma \), \( \gamma = 360^\circ / z_2 \) is the angular pitch, and \( \varphi_{201} = -90^\circ – n\gamma/2 \).
3. Influence of Main Parameters on Average Efficiency
To analyze the effects, I used a baseline set of parameters for the single-roller enveloping end face worm gears, listed in Table 1. The worm is single-start. A single-variable method was adopted: varying one parameter while keeping others fixed at baseline values.
| Parameter | Value |
|---|---|
| Center distance \( A \) (mm) | 160 |
| Number of worm starts \( z_1 \) | 1 |
| Number of worm gear teeth \( z_2 \) | 24 |
| Number of simultaneously meshing teeth \( n \) | 4 |
| Throat neck coefficient \( k \) | 0.4 |
| Roller radius \( R \) (mm) | 10 |
| Worm rotational speed \( \omega_1 \) (rad/s) | 1 |
| Sliding friction coefficient \( f \) | 0.15 |
| Ratio of rolling to sliding friction \( f_g \) | 0.2 |
3.1 Worm Gear Rotation Angle
Figure (not shown) illustrates the effect of worm gear angle \( \varphi_2 \) on instantaneous efficiency. The efficiency increases almost linearly from meshing entry to exit. For \( f = 0.05 \), the efficiency increase from entry to exit is 4.65%; for \( f = 0.2 \), the increase is 11.27%. Larger \( f \) leads to a stronger dependence on \( \varphi_2 \). At a fixed angle, higher \( f \) reduces instantaneous efficiency.
3.2 Sliding Friction Coefficient
Table 2 summarizes the average efficiency \( \eta_{pn} \) for different \( f \) values. As \( f \) increases, efficiency drops significantly. For example, when \( f \) rises from 0.05 to 0.2, average efficiency decreases from 75.29% to 43.59% — a reduction of 31.7 percentage points.
| \( f \) | 0.05 | 0.10 | 0.15 | 0.20 |
|---|---|---|---|---|
| \( \eta_{pn} \) (%) | 75.29 | 60.53 | 50.67 | 43.59 |
3.3 Throat Neck Coefficient
Figure (not shown) shows that efficiency decreases with increasing \( k \). The reduction magnitude depends on \( f \). For \( f = 0.05 \), efficiency drops by 3.09% when \( k \) goes from 0.3 to 0.5; for \( f = 0.2 \), the drop is 7.07%.
3.4 Roller Radius
Increasing roller radius \( R \) slightly reduces efficiency. When \( R \) increases from 5 mm to 10 mm, the efficiency reduction ranges from 0.71% (at \( f = 0.05 \)) to 1.66% (at \( f = 0.20 \)). Higher \( f \) amplifies this sensitivity.
3.5 Number of Worm Gear Teeth
Efficiency decreases as \( z_2 \) increases. On average, each additional tooth reduces efficiency by about 1.00% to 2.33% depending on \( f \). Table 3 illustrates the trend for \( f = 0.15 \).
| \( z_2 \) | 20 | 22 | 24 | 26 | 28 | 30 |
|---|---|---|---|---|---|---|
| \( \eta_{pn} \) (%) | 52.17 | 50.13 | 48.09 | 46.05 | 44.01 | 41.97 |
3.6 Center Distance
Efficiency improves slightly with larger center distance. For \( A \) increasing from 160 mm to 200 mm, the gain is 0.28% to 0.67% depending on \( f \). The effect is relatively small.
Overall, for these worm gears, efficiency increases with worm gear rotation angle and center distance, and decreases with sliding friction coefficient, throat neck coefficient, roller radius, and number of worm gear teeth.
4. Optimization of Transmission Efficiency
To maximize the average efficiency \( \eta_{pn} \) of the single-roller enveloping end face worm gears, I formulated an optimization problem with design variables \( X = [k, R, z_2, A, h_{fa}, h_{fc}, n] \). The objective is to minimize the reciprocal of efficiency: \( \min f(X) = 1 / g(X) \), subject to geometric and structural constraints:
$$ \begin{cases} s_2 > R \\ s_1 = \frac{(2 – k)A}{2} – (h_{fc} + c_c)m \\ s_2 = s_1 \tan\beta \\ \beta = \gamma / 2 \end{cases} $$
The constraints on each variable are:
$$ \begin{aligned} f &\in [0.05,\ 0.2] \\ k &\in [0.3,\ 0.5] \\ R &\in [5,\ 10]\ \text{mm} \\ z_2 &\in [20,\ 30] \\ A &\in [160,\ 200]\ \text{mm} \\ h_{fa} &\in [0.8,\ 1.0] \\ h_{fc} &\in [0.8,\ 1.0] \\ n &\in [3,\ 5] \end{aligned} $$
The clearance coefficient \( c_c \) is taken as 0.2. The initial baseline efficiency with Table 1 parameters is 83.2%, which is already higher than conventional worm gears. Using MATLAB’s optimization toolbox (both the fmincon function and a genetic algorithm), I found optimal solutions for different numbers of simultaneously meshing teeth \( n \). Table 4 presents the optimized results.
| Parameter | fmincon | Genetic Algorithm | ||||
|---|---|---|---|---|---|---|
| \( n=3 \) | \( n=4 \) | \( n=5 \) | \( n=3 \) | \( n=4 \) | \( n=5 \) | |
| \( k \) | 0.3635 | 0.3615 | 0.3623 | 0.3613 | 0.3618 | 0.3604 |
| \( R \) (mm) | 8.1917 | 8.1919 | 8.1922 | 8.1914 | 8.1921 | 8.1919 |
| \( z_2 \) | 23.361 | 23.534 | 23.602 | 23.354 | 23.421 | 23.524 |
| \( A \) (mm) | 179.63 | 179.70 | 179.75 | 179.61 | 179.68 | 179.72 |
| \( h_{fa} \) | 0.9180 | 0.9184 | 0.9190 | 0.9178 | 0.9187 | 0.9184 |
| \( h_{fc} \) | 0.9068 | 0.9073 | 0.9067 | 0.9065 | 0.9068 | 0.9073 |
| \( \eta_{pn} \) (%) | 84.81 | 86.44 | 85.45 | 84.78 | 86.42 | 85.26 |
The highest efficiency of 86.44% is achieved with \( n = 4 \) and the following approximate values: \( k = 0.36 \), \( R = 8.19 \) mm, \( A = 179.7 \) mm, \( h_{fa} = 0.92 \), \( h_{fc} = 0.91 \). This represents an improvement of 3.2% over the baseline design. Both optimization methods yield similar results, confirming the robustness of the optimum.
5. Conclusion
In this study, I developed a comprehensive mathematical model for the transmission efficiency of single-roller enveloping end face worm gears. The efficiency model accounts for sliding friction, rolling friction, and geometric parameters. The main findings are:
- The average transmission efficiency of these worm gears is negatively correlated with the sliding friction coefficient, throat neck coefficient, roller radius, and number of worm gear teeth, and positively correlated with the center distance.
- Among all parameters, the sliding friction coefficient has the most significant impact on efficiency, followed by the number of worm gear teeth; the center distance has the least influence.
- Optimization using both gradient-based and genetic algorithms shows that the maximum efficiency can reach 86.4% with a carefully chosen set of parameters, a 3.2% improvement over a baseline design.
- To achieve high efficiency in single-roller enveloping end face worm gears, in addition to high-precision manufacturing, designers should avoid excessively large throat neck coefficients, roller radii, and numbers of worm gear teeth for a given center distance.
This work provides a theoretical basis for the design and application of high-efficiency worm gears, particularly in scenarios where energy loss must be minimized.