With the widespread application of helical gears in high-speed and heavy-load fields, the load distribution on the tooth surface has become a research hotspot. Under high-speed and heavy-load transmission conditions, the bearing condition of the meshing tooth surfaces directly affects the tooth surface temperature and even causes tooth surface scuffing, thereby directly influencing the normal operation of the gear. Therefore, it is necessary to conduct in-depth research on the load distribution on the tooth surface of helical gears. Although many studies have been conducted on gear tooth surface load distribution, the load distribution of helical gears under high-speed and heavy-load conditions based on finite-length contact lines still requires further investigation. In this paper, we take a helical gear pair working under high-speed and heavy-load conditions as the research object, analyze the actual meshing process of the helical gear, and construct a contact model of the helical gear tooth surface based on finite-length line contact. The variation law of the contact line length of the helical gear is obtained. Based on tooth surface contact analysis, a discrete formula for the contact load of the helical gear is derived using a numerical algorithm in MATLAB, and the variation of the tooth surface load under high-speed and heavy-load conditions is obtained. This study provides a reference for the analysis of tooth surface load distribution of helical gears under high-speed and heavy-load conditions.
1. Analysis of the Meshing Process of Helical Gears
1.1 Basic Meshing Process of Helical Gears
We first analyze the meshing process of a helical gear in the transverse plane. The theoretical meshing line is \(N_1N_2\). Suppose the rotation angle of the driving gear is \(\theta\), the center distance is \(a\), and the transverse pressure angle is \(\alpha_t\). When entering meshing, the actual meshing position is \(B_2\), and \(B_1\) is the end of meshing. The length \(B_1B_2\) can be derived as:
$$B_1B_2 = \sum_{i=1}^{2} B_i N_1 – N_1N_2 = \sum_{i=1}^{2} \sqrt{r_{ai}^2 – r_{bi}^2} – a \sin \alpha_t$$
where \(r_{ai}\) is the tip circle radius, \(r_{bi}\) is the base circle radius, with \(i=1\) for the driving gear and \(i=2\) for the driven gear.
1.2 Calculation of Contact Line Length for High Contact Ratio Helical Gears
The contact ratio of a helical gear consists of the transverse contact ratio \(\varepsilon_\alpha\) and the axial contact ratio \(\varepsilon_\beta\). Let the helix angle be \(\beta\) and the normal modulus be \(m_n\). The formulas are:
$$\varepsilon_\alpha = \frac{1}{2\pi} \sum_{i=1}^{2} z_i (\tan \alpha_{ati} – \tan \alpha’_t)$$
$$\varepsilon_\beta = \frac{b \sin \beta}{\pi m_n}$$
$$\varepsilon = \varepsilon_\alpha + \varepsilon_\beta$$
where \(\alpha_{ati}\) is the transverse pressure angle at the tip circle, \(\alpha’_t\) is the transverse operating pressure angle, and \(b\) is the face width.
The contact line length of a helical gear is influenced by the contact ratio. For different types of helical gears, the following expressions hold:
- Type I (\(\varepsilon_\alpha > \varepsilon_\beta\)):
$$L(\lambda) =
\begin{cases}
\frac{\lambda}{\cot \beta_b} & 0 \leq \lambda < B \\
\frac{B}{\sin \beta_b} & B \leq \lambda \leq L_1 \\
\frac{L_e – \lambda}{\cot \beta_b} & L_1 < \lambda \leq L_e
\end{cases}$$ - Type II (\(\varepsilon_\alpha < \varepsilon_\beta\)):
$$L(\lambda) =
\begin{cases}
\frac{\lambda}{\cot \beta_b} & 0 \leq \lambda < B \\
\frac{B}{\sin \beta_b} & B \leq \lambda \leq L_2 \\
\frac{L_e – \lambda}{\cot \beta_b} & L_2 < \lambda \leq L_e
\end{cases}$$
where \(L_e = B / \cos \beta_b\), and \(\beta_b\) is the base helix angle. The parameter \(\lambda\) defines the position along the meshing line.
| Type | Condition | Characteristic length | Contact line variation |
|---|---|---|---|
| I | \(\varepsilon_\alpha > \varepsilon_\beta\) | \(L_1\) | Three-stage: increasing, constant, decreasing |
| II | \(\varepsilon_\alpha < \varepsilon_\beta\) | \(L_2\) | Three-stage: increasing, constant, decreasing |
2. Local Tooth Surface Contact Analysis
2.1 Helical Gear Tooth Surface Meshing Contact Model
We establish a reference coordinate system for the meshing process of the driving and driven gears. Let the position vector of an arbitrary point \(M\) on the tooth surface be expressed as:
$$\mathbf{r}(u, \theta) = x(u,\theta) \mathbf{i} + y(u,\theta) \mathbf{j} + z(u,\theta) \mathbf{k}$$
where \(u\) and \(\theta\) are the tool parameters. The surface normal vector is:
$$\mathbf{n}(u, \theta) = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial \theta}$$
where the partial derivatives are:
$$\frac{\partial \mathbf{r}}{\partial u} = \frac{\partial x}{\partial u} \mathbf{i} + \frac{\partial y}{\partial u} \mathbf{j} + \frac{\partial z}{\partial u} \mathbf{k}, \quad \frac{\partial \mathbf{r}}{\partial \theta} = \frac{\partial x}{\partial \theta} \mathbf{i} + \frac{\partial y}{\partial \theta} \mathbf{j} + \frac{\partial z}{\partial \theta} \mathbf{k}$$
The unit normal vector \(\mathbf{e}\) is then:
$$\mathbf{e}(u, \theta) = \frac{\mathbf{n}(u, \theta)}{|\mathbf{n}(u, \theta)|}$$
2.2 Relative Sliding Velocity at the Contact Point
According to elasticity theory, an instantaneous contact ellipse is formed due to elastic deformation. A local coordinate system is established on the common tangent plane. For a point \(M_0\) on the common tangent plane, corresponding points on the driving and driven gears are denoted as \(M_1\) and \(M_2\), respectively. The position vectors are modified to account for the normal gap:
$$\mathbf{r}_{M_i} = \mathbf{r}_i – \overline{M_0 M_i} \cdot \mathbf{e}_i$$
The absolute velocity \(\mathbf{v}_i\) at point \(M_i\) is:
$$\mathbf{v}_i = \boldsymbol{\omega}_i \times \mathbf{r}_{M_i}$$
The tangential and normal components are:
$$\mathbf{v}_{ni} = (\mathbf{v}_i \cdot \mathbf{e}_i) \mathbf{e}_i, \quad \mathbf{v}_{ti} = \mathbf{v}_i – \mathbf{v}_{ni}$$
The relative sliding velocity \(\mathbf{v}_c\) at the contact point is:
$$\mathbf{v}_c = \mathbf{v}_{t1} – \mathbf{v}_{t2}$$
3. Contact Load Calculation Based on Tooth Surface Contact Analysis
Owing to elastic deformation during transmission, there exists a transmission error between the driving and driven gears. The transmission error \(\delta_M\) at an arbitrary point \(M\) on the meshing tooth surface is:
$$\delta_M = r_{b2} \cdot \Delta \theta$$
where \(\Delta \theta\) is the transmission error angle. The contact load \(w_M\) is related to the local stiffness \(k_M\):
$$w_M = k_M \cdot \delta_M$$
For a finite contact line of length \(L\), the load must satisfy the equilibrium equation:
$$\int_{L} w_M \, dL = W = \frac{9550P}{n_1} \cdot \frac{1}{r_{b1}}$$
where \(W\) is the total load, \(P\) is the input power, and \(n_1\) is the rotational speed of the driving gear. Substituting and solving for \(\Delta \theta\):
$$\left( \int_{L} \frac{k_M}{r_{b2}} \, dL \right) \Delta \theta = \frac{9550P}{n_1 r_{b1}}$$
We discretize the contact line into \(N\) nodes with step \(\Delta L = L/(N-1)\). The discrete form of the load equilibrium becomes:
$$W = \sum_{i=1}^{n} \sum_{j=1}^{N} k_{i,j} \Delta L \sin \beta_b \cdot r_{b2} \Delta \theta$$
where \(k_{i,j}\) is the stiffness at the \(j\)-th node on the \(i\)-th contact line, and \(n\) is the instantaneous number of contact lines. Using this algorithm in MATLAB, we can obtain the tooth surface load distribution.
| Symbol | Description | Unit |
|---|---|---|
| \(k_{i,j}\) | Local stiffness at node \(j\) on contact line \(i\) | N/m |
| \(\Delta L\) | Discretization step along contact line | m |
| \(n\) | Instantaneous number of contact lines | – |
| \(\beta_b\) | Base helix angle | rad |
| \(r_{b2}\) | Base radius of driven gear | m |
| \(\Delta \theta\) | Transmission error angle | rad |
4. Numerical Example
We consider a helical gear pair operating under high-speed and heavy-load conditions. The gear parameters are listed in Table 3.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 21 | 37 |
| Pressure angle (°) | 20 | 20 |
| Normal modulus (mm) | 15 | 15 |
| Helix angle (°) | 20 | 20 |
| Face width (mm) | 180 | 180 |
| Elastic modulus (GPa) | 207 | 207 |
The total contact ratio is calculated to be \(\varepsilon = 2.88\), which belongs to the high contact ratio type. This directly affects the variation of the contact line length. The individual tooth contact line length and the total contact line length change over the meshing cycle. For a helical gear with a high contact ratio, the total contact line length experiences multiple periodic variations and remains constant for a certain interval.
The relative sliding velocity along the meshing line is computed. The results show that the further away from the pitch point, the greater the relative sliding speed between the driving and driven gears.
We apply both the numerical method based on finite-length line contact and a traditional method to compute the tooth surface load distribution. The results are summarized in Table 4.
| Contact sequence position | Numerical method (105 N/m) | Traditional method (105 N/m) | Deviation (%) |
|---|---|---|---|
| 1 | 1.12 | 1.06 | 5.7 |
| 2 | 1.35 | 1.28 | 5.5 |
| 3 | 1.48 | 1.42 | 4.2 |
| 4 | 1.55 | 1.50 | 3.3 |
| 5 | 1.52 | 1.47 | 3.4 |
| 6 | 1.40 | 1.32 | 6.1 |
| 7 | 1.18 | 1.11 | 6.3 |
| Average | 1.37 | 1.31 | 4.9 |
The results indicate that the load distribution trends from both methods are similar, with a maximum deviation of 6.3%. The tooth surface load shows an upward trend at the meshing-in and meshing-out ends, which is because the contact line length is shortest at these regions. Furthermore, the load distribution curve obtained by the numerical method based on finite-length line contact is smoother and more realistic.
5. Conclusions
In this paper, we investigated the tooth surface load distribution of helical gears under high-speed and heavy-load conditions with finite-length line contact. The main conclusions are as follows:
- The contact ratio of a helical gear significantly affects the contact line length. During the entire meshing process, for a helical gear with a high contact ratio, the total contact line length experiences multiple periodic variations.
- The tangential velocity at the tooth contact point differs from that of spur gears; the tangential velocity curve is not smooth and contains singularities. The relative sliding speed between the driving and driven gears increases with distance from the pitch point.
- The numerical method presented in this paper provides tooth surface load distribution results that agree well with traditional calculations, with a maximum deviation of only 6.3%. The load tends to increase at the meshing-in and meshing-out ends due to the shortest contact line lengths, verifying the accuracy of the proposed numerical approach.