In modern mechanical transmission systems, helical gears play a critical role due to their ability to transmit power smoothly and efficiently between non-parallel shafts. Non-orthogonal helical gears, which operate with intersecting axes at angles other than 90 degrees, are particularly valuable in applications such as automotive differentials, helicopter reducers, robotics, and conveyor drives. These gears often employ point contact meshing, which can enhance load distribution and reduce noise. However, designing such gears requires precise mathematical modeling to avoid issues like undercutting, high sliding rates, and excessive contact stress. In this study, we develop a comprehensive mathematical model for non-orthogonal helical gear systems based on conjugate curve theory, focusing on contact characteristics to improve load capacity and transmission performance. We derive equations for tooth profiles, analyze root cutting conditions, sliding rates, and contact stresses, and validate our model through simulations and experiments. The results demonstrate that non-orthogonal helical gears exhibit superior contact properties compared to traditional involute gears, making them suitable for high-performance applications.
To establish a foundation for our analysis, we begin by defining the coordinate systems and geometric relationships for non-orthogonal helical gears. We consider fixed coordinate systems \( S(O – x, y, z) \) and \( S_p(O_p – x_p, y_p, z_p) \), along with moving coordinate systems \( S_1(O_1 – x_1, y_1, z_1) \) and \( S_2(O_2 – x_2, y_2, z_2) \) attached to the pinion and gear, respectively. The angular velocities are denoted by \( \omega^{(1)} \) and \( \omega^{(2)} \), with rotation angles \( \phi_1 \) and \( \phi_2 \). The shaft angle is \( \Sigma \), the center distance is \( a \), and the contact point is \( P \). The relative velocity at point \( P \) is crucial for understanding the meshing behavior and is given by:
$$v^{(12)}_1 = [-y_1 (1 + i_{21} \cos\Sigma) – z_1 i_{21} \cos\phi_1 \sin\Sigma – a i_{21} \sin\phi_1 \cos\Sigma] i_1 + [x_1 (1 + i_{21} \cos\Sigma) + z_1 i_{21} \sin\phi_1 \sin\Sigma – a i_{21} \cos\phi_1 \cos\Sigma] j_1 + i_{21} \sin\Sigma (x_1 \cos\phi_1 – y_1 \sin\phi_1 – a) k_1$$
where \( i_{21} = \phi_2 / \phi_1 \) is the transmission ratio, and \( i_1, j_1, k_1 \) are unit vectors along the coordinate axes. This equation accounts for the spatial kinematics of helical gears and forms the basis for deriving conjugate curves. The tooth profiles are generated using the concept of spatial curve meshing, where a space curve \( \Gamma_1 \) on the pinion conjugates with a corresponding curve \( \Gamma_2 \) on the gear. The coordinates of \( \Gamma_2 \) can be expressed as:
$$x_2 = x_1 (\cos\phi_1 \cos\phi_2 – \sin\phi_1 \sin\phi_2 \cos\Sigma) + y_1 (-\sin\phi_1 \cos\phi_2 – \cos\phi_1 \sin\phi_2 \cos\Sigma) – z_1 \sin\phi_2 \sin\Sigma – a \cos\phi_2$$
$$y_2 = x_1 (\cos\phi_1 \sin\phi_2 + \sin\phi_1 \cos\phi_2 \cos\Sigma) + y_1 (-\sin\phi_1 \sin\phi_2 + \cos\phi_1 \cos\phi_2 \cos\Sigma) + z_1 \cos\phi_2 \sin\Sigma – a \sin\phi_2$$
$$z_2 = (1 + i_{21} \cos\Sigma) (n_{nx1} y_1 – n_{ny1} x_1) + n_{nz1} a i_{21} \sin\Sigma$$
Here, \( n_{nx1}, n_{ny1}, n_{nz1} \) are components of the unit normal vector. The tooth surface is then developed using an equidistant envelope method, which involves offsetting the conjugate curves by a distance equal to the tooth profile radius. The general equation for the tooth profile is:
$$x_{\Gamma_i} = r_{xi} \pm \rho_i n^0_{nxi} + \rho_i \cos\phi_i \cos\alpha_i$$
$$y_{\Gamma_i} = r_{yi} \pm \rho_i n^0_{nyi} + \rho_i \cos\phi_i \sin\alpha_i$$
$$z_{\Gamma_i} = r_{zi} \pm \rho_i n^0_{nzi} + \rho_i \sin\phi_i$$
$$\Phi(t, \phi, \alpha) = (r_{it}, r_{i\phi}, r_{i\alpha}) = 0$$
where \( \rho_i \) is the profile radius, \( \phi_i \) is the rotation angle, \( \alpha_i \) is the pressure angle, and \( \Phi(t, \phi, \alpha) \) represents the generalized curve equation. This approach allows for the creation of convex-concave tooth profiles that optimize contact conditions. For helical gears, the initial curve is often a cylindrical helix on the pinion, described by:
$$x_1 = R \cos\theta$$
$$y_1 = R \sin\theta$$
$$z_1 = p \theta$$
where \( R \) is the pitch radius, \( \theta \) is the curve parameter, and \( p \) is the helix parameter. The tooth profile equations for the pinion and gear are derived as follows:
$$x_{\Sigma_1} = R \cos\theta + h_1 n^{0′}_{nx1} + h_1 \cos\phi_1 \cos\alpha_1$$
$$y_{\Sigma_1} = R \sin\theta + h_1 n^{0′}_{ny1} + h_1 \cos\phi_1 \sin\alpha_1$$
$$z_{\Sigma_1} = 0$$
and
$$x_{\Sigma_2} = R \cos\phi_2 \cos(\theta + \phi_1) + R \sin\phi_2 \cos\Sigma \sin(\theta + \phi_1) + p \theta \sin\phi_2 \sin\Sigma – a \cos\phi_2 – h_2 n^{0′}_{nx2} + h_2 \cos\phi_2 \cos\alpha_2$$
$$y_{\Sigma_2} = R \sin\phi_2 \cos(\theta + \phi_1) + R \cos\phi_2 \cos\Sigma \sin(\theta + \phi_1) + p \theta \cos\phi_2 \sin\Sigma – a \sin\phi_2 – h_2 n^{0′}_{ny2} + h_2 \cos\phi_2 \sin\alpha_2$$
$$z_{\Sigma_2} = 0$$
where \( h_1 \) and \( h_2 \) are the convex and concave profile radii, respectively. These equations define the geometry of non-orthogonal helical gears and are used for subsequent analysis and simulation.
To illustrate the application of our model, we consider a case study with a shaft angle of \( 15^\circ \). The key design parameters for the non-orthogonal helical gear pair are summarized in the table below:
| Parameter | Value |
|---|---|
| Shaft Angle \( \Sigma \) (degrees) | 15 |
| Pinion Pitch Radius \( R_1 \) (mm) | 36 |
| Center Distance \( a \) (mm) | 136 |
| Normal Module \( m_n \) (mm) | 5 |
| Pressure Angle \( \alpha \) (degrees) | 30 |
| Transmission Ratio \( i_{21} \) | 31/11 |
| Pinion Tooth Number \( Z_1 \) | 11 |
| Gear Tooth Number \( Z_2 \) | 31 |
| Helix Parameter \( p \) | 39.3 |
| Convex Profile Radius \( h_1 \) (mm) | 5 |
| Concave Profile Radius \( h_2 \) (mm) | 6 |
| Face Width \( B \) (mm) | 32 |
| Curve Parameter \( \theta \) (rad) | [0, 0.75] |
| Coefficient \( u \) | -0.58 |
| Coefficient \( v \) | -0.85 |
Using these parameters, we simulate the meshing process via computational tools. The results show continuous motion with a constant transmission ratio, and point contact is maintained along the tooth profiles without interference. This confirms the validity of our mathematical model for non-orthogonal helical gears.
One critical aspect in gear design is avoiding undercutting, which occurs when the generating process creates singularities on the tooth surface. For non-orthogonal helical gears, the root cutting condition is derived from the singularity condition of the generated surface. The equation of the contact region is given by:
$$x = \rho_a \sin\alpha_a – e_a$$
$$y = -(\rho_a \cos\alpha_a – l_a) \cos\beta + u_a \sin\beta$$
$$z = (\rho_a \cos\alpha_a – l_a) \sin\beta + u_a \cos\beta$$
where \( \rho_a \) is the convex profile radius, \( e_a \) and \( l_a \) are center offsets, \( \alpha_a \) is the pressure angle, \( \beta \) is the helix angle, and \( u_a \) is the axial displacement. The singularity condition is expressed as \( v^{(2)}_r = 0 \), which leads to the system:
$$\frac{d}{ds} [\Phi(t, \phi, \alpha)] = 0$$
$$\frac{\partial r_1}{\partial t} \frac{dt}{ds} + \frac{\partial r_1}{\partial \phi} \frac{d\phi}{ds} = -v^{(12)}_1$$
$$\frac{\partial \Phi}{\partial t} \frac{dt}{ds} + \frac{\partial \Phi}{\partial \phi} \frac{d\phi}{ds} + \frac{\partial \Phi}{\partial \alpha} \frac{d\alpha}{ds} = 0$$
By solving this system, we obtain the root cutting condition as \( \Delta_1 = 0 \), \( \Delta_2 = 0 \), \( \Delta_3 = 0 \), where:
$$\Delta_1 = \begin{vmatrix}
\frac{\partial x_1}{\partial t} & \frac{\partial x_1}{\partial \phi} & -v^{(12)}_{x1} \\
\frac{\partial y_1}{\partial t} & \frac{\partial y_1}{\partial \phi} & -v^{(12)}_{y1} \\
\frac{\partial \Phi}{\partial t} & \frac{\partial \Phi}{\partial \phi} & -\frac{\partial \Phi}{\partial \alpha} \frac{d\alpha}{ds}
\end{vmatrix}$$
$$\Delta_2 = \begin{vmatrix}
\frac{\partial x_1}{\partial t} & \frac{\partial x_1}{\partial \phi} & -v^{(12)}_{x1} \\
\frac{\partial z_1}{\partial t} & \frac{\partial z_1}{\partial \phi} & -v^{(12)}_{z1} \\
\frac{\partial \Phi}{\partial t} & \frac{\partial \Phi}{\partial \phi} & -\frac{\partial \Phi}{\partial \alpha} \frac{d\alpha}{ds}
\end{vmatrix}$$
$$\Delta_3 = \begin{vmatrix}
\frac{\partial y_1}{\partial t} & \frac{\partial y_1}{\partial \phi} & -v^{(12)}_{y1} \\
\frac{\partial z_1}{\partial t} & \frac{\partial z_1}{\partial \phi} & -v^{(12)}_{z1} \\
\frac{\partial \Phi}{\partial t} & \frac{\partial \Phi}{\partial \phi} & -\frac{\partial \Phi}{\partial \alpha} \frac{d\alpha}{ds}
\end{vmatrix}$$
This condition ensures that the tooth profiles of helical gears are free from undercutting, thereby maintaining structural integrity.
Another important characteristic is the sliding rate, which quantifies the relative motion between meshing teeth and affects efficiency and wear. For non-orthogonal helical gears, the sliding rates \( U_1 \) and \( U_2 \) for the pinion and gear are defined as:
$$U_1 = \lim_{\Delta S_1 \to 0} \frac{\Delta S_1 – \Delta S_2}{\Delta S_1}$$
$$U_2 = \lim_{\Delta S_2 \to 0} \frac{\Delta S_2 – \Delta S_1}{\Delta S_2}$$
where \( \Delta S_1 \) and \( \Delta S_2 \) are the arc lengths on the pinion and gear curves, respectively. The arc lengths are computed as:
$$\Delta S_1 = \sqrt{R^2 + p^2}$$
$$\Delta S_2 = \sqrt{ [R \cos\phi_2 \cos(\theta + \phi_1) + R \sin\phi_2 \cos\Sigma \sin(\theta + \phi_1) + p \theta \sin\phi_2 \sin\Sigma – a \cos\phi_2]’^2 + [R \sin\phi_2 \cos(\theta + \phi_1) + R \cos\phi_2 \cos\Sigma \sin(\theta + \phi_1) + p \theta \cos\phi_2 \sin\Sigma – a \sin\phi_2]’^2 + [-R \cos\theta \sin\phi_1 \sin\Sigma – R \sin\theta \cos\phi_1 \sin\Sigma + p \theta \cos\Sigma]’^2 }$$
Based on our parameters, the sliding rates for non-orthogonal helical gears are calculated and compared to those of involute gears. The results, presented graphically, show that the sliding rates for non-orthogonal helical gears are lower, with the maximum absolute value occurring at the tooth root. This reduction in sliding contributes to improved transmission efficiency and reduced wear in helical gears.
Contact stress analysis is essential for assessing the load-bearing capacity of helical gears. We employ the Hertzian contact model and finite element analysis (FEA) to evaluate stress distributions. The gear pair is discretized using Solid185 hexahedral elements, with contact and target surfaces defined by CONTA173 and TARGE170 elements, respectively. The extended Lagrangian algorithm is used for solution. The material is 20CrMnTi steel with a Poisson’s ratio of 0.25 and Young’s modulus of 205 GPa. A torque of 200 N·m is applied to the pinion. The FEA results for non-orthogonal helical gears indicate a maximum contact stress of 1266.5 MPa at the mid-point of the tooth profile, with stresses distributed in an elliptical pattern along the face width. The maximum Von Mises stress on the pinion is 793.69 MPa. In contrast, involute gears under the same conditions exhibit a higher maximum contact stress of 1639 MPa and a Von Mises stress of 1233.8 MPa. This demonstrates that non-orthogonal helical gears have superior contact characteristics, reducing the risk of failure.
To validate our model, we conduct machining experiments using a five-axis CNC center. The tooth profiles are fabricated via milling based on the derived equations. The experimental setup includes torque sensors and variable-speed motors to measure input and output parameters. The transmission efficiency \( \eta \) is calculated as \( \eta = n_o T_o / (n_i T_i) \), where \( n_i \) and \( T_i \) are input speed and torque, and \( n_o \) and \( T_o \) are output speed and torque. Tests are performed at speeds of 200, 400, 600, 800, and 1000 rpm with loads of 200, 300, 400, 500, and 600 N·m. The results, summarized in the table below, show that efficiency increases with both speed and load, reaching a peak of 95.9% at 600 N·m. The overall efficiency ranges from 91.2% to 95.9%, confirming the practical viability of our non-orthogonal helical gear design.
| Load (N·m) | Efficiency at 200 rpm (%) | Efficiency at 400 rpm (%) | Efficiency at 600 rpm (%) | Efficiency at 800 rpm (%) | Efficiency at 1000 rpm (%) |
|---|---|---|---|---|---|
| 200 | 91.2 | 92.5 | 93.1 | 93.8 | 94.2 |
| 300 | 92.0 | 93.0 | 93.7 | 94.3 | 94.8 |
| 400 | 92.8 | 93.6 | 94.2 | 94.9 | 95.3 |
| 500 | 93.5 | 94.2 | 94.8 | 95.4 | 95.7 |
| 600 | 94.1 | 94.7 | 95.2 | 95.7 | 95.9 |
In conclusion, we have developed a robust mathematical model for non-orthogonal helical gear systems based on conjugate curve theory. Our analysis covers tooth profile generation, root cutting conditions, sliding rates, and contact stresses, demonstrating that non-orthogonal helical gears offer enhanced performance over traditional involute gears. The lower sliding rates and contact stresses contribute to higher efficiency and durability, making these gears ideal for demanding applications. Experimental validation confirms the model’s accuracy, with transmission efficiencies exceeding 95% under optimal conditions. This work provides a foundation for further optimization of helical gears in advanced mechanical systems.