In the field of gear design, non-circular helical gears offer significant advantages, including smooth transmission, high overlap ratios, and enhanced load-bearing capacity. However, the curvature variations along the pitch curve of non-circular gears pose challenges in achieving uniform tooth alignment when using traditional methods. This article presents a novel approach to designing high-order non-circular helical gears by integrating the conversion method of tooth profile with an improved spiral curve derived from the intersection of an involute helicoid and the pitch curve cylinder. We address the issue of uneven tooth deviation by replacing the conventional helix with a spatially derived intersection curve, ensuring better alignment and performance. The methodology is validated through kinematic analysis and interference checks, demonstrating the feasibility of this design for practical applications. Throughout this work, the focus remains on optimizing helical gears for complex motion transmission, with repeated emphasis on the role of helical gears in non-circular systems.
The design of non-circular gears typically employs methods such as the analytical approach, conversion method of tooth profile, and envelope method. The conversion method, which approximates each tooth position’s pitch curve as a standard spur gear’s reference circle, is computationally efficient and widely applicable. For helical gears, this method involves determining the spiral line at each tooth based on the base circle radius and helix angle. However, due to the varying curvature of the pitch curve, the projected spiral lines do not fully coincide with the pitch curve in the plane, leading to biased tooth orientation. This issue is particularly pronounced in high-order non-circular helical gears, where the pitch curve’s complexity increases. To overcome this, we propose using the intersection curve between the involute helicoid and the pitch curve cylinder, which we refer to as the “intersection curve,” to replace the traditional helix. This curve ensures that its projection onto the pitch curve plane aligns perfectly with the pitch curve, eliminating deviation problems.
The fundamental theory of non-circular gears begins with the pitch curve design. For a pair of non-circular gears, the pitch curves must remain tangent during operation. The polar equations for the driving and driven gears are given by:
$$ \rho_1(\theta_1) = \frac{p_1}{1 – k_1 \cos n_1 \theta_1} $$
$$ p_1 = A_1 (1 – k_1^2) $$
and
$$ \rho_2(\theta_2) = \frac{p_2}{1 + k_2 \cos n_2 \theta_2} $$
$$ p_2 = \frac{n_2 p_1}{n_2 – k_1^2 (n_2 – 1)} $$
$$ k_2 = \frac{k_1}{n_2 – k_1^2 (n_2 – 1)} $$
where \( n_1 \) and \( n_2 \) are the orders of the driving and driven gears, respectively, \( k_1 \) and \( k_2 \) are their eccentricities, and \( A_1 \) and \( A_2 \) are the semi-major axis lengths. The center distance \( a \) is calculated as:
$$ a = A_1 [1 + n_2 – k_1^2 (n_2 – 1)] $$
The number of teeth \( z_i \) for each gear is determined based on the pitch curve circumference, ensuring uniform distribution. Key validations include pressure angle checks, undercutting checks, and convexity checks. For instance, the pressure angle \( \alpha_{12L} \) for the driving gear must satisfy \( \alpha_{12L} \leq 65^\circ \), and the minimum curvature radius \( r_{\text{min}} \) must prevent undercutting:
$$ r_{\text{min}} = \min \left\{ \frac{ \left[ \rho^2(\theta) + \left( \frac{d\rho}{d\theta} \right)^2 \right]^{3/2} }{ \rho^2(\theta) + 2 \left( \frac{d\rho}{d\theta} \right)^2 – \frac{d^2\rho}{d\theta^2} } \right\} $$
$$ h_a^* m \leq r_{\text{min}} \sin^2 \alpha $$
Convexity is ensured when \( k_i \leq \frac{1}{n_i^2 – 1} \) for \( n_i > 1 \). These checks are critical for designing functional non-circular helical gears.
In traditional helical gear design, the spiral line is derived from the intersection of the tooth flank surface and the reference cylinder. However, for non-circular helical gears, the varying curvature causes the projected spiral lines to deviate inward or outward relative to the pitch curve. To address this, we utilize the involute helicoid, which is generated by a straight line inclined at the base helix angle \( \beta_b \) rolling on the base cylinder. The parametric equations of the involute helicoid in a local coordinate system are:
$$ x_1 = r_b \cos \phi + u \cos \beta_b \sin \phi $$
$$ y_1 = r_b \sin \phi – u \cos \beta_b \cos \phi $$
$$ z_1 = p \phi – u \sin \beta_b $$
where \( p = r_b \tan \beta_b \), \( r_b \) is the base radius, \( \phi \) is the roll angle, and \( u \) is the parameter along the generating line. The intersection curve is obtained by solving the equations of the involute helicoid and the pitch curve cylinder. For each tooth position \( j \), the curvature center \( c_j \) in the global coordinate system is calculated as:
$$ c_j = \begin{pmatrix} x_j(\theta_j) \\ y_j(\theta_j) \\ 0 \end{pmatrix} = \begin{pmatrix} \rho_j(\theta_j) \cos \theta_j – r_j(\theta_j) \cos n_j(\theta_j) \\ \rho_j(\theta_j) \sin \theta_j – r_j(\theta_j) \sin n_j(\theta_j) \\ 0 \end{pmatrix} $$
where \( n_j(\theta_j) \) is the inclination angle of the normal line. The intersection curve \( G_j \) for tooth \( j \) is derived by combining the involute helicoid equation and the pitch cylinder equation:
$$ F_j(\theta_j, \phi_j, u_j) = 0 $$
$$ F(\theta, \rho(\theta), C) = 0 $$
This curve replaces the helix and ensures proper alignment. The modeling process involves calculating parameters such as base radius \( r_{bj} \), rotation angles for the end-face involute \( \text{ROT1}_j \), and twist angles for sweeping \( \text{ROT2}_j \):
$$ r_{bj} = r_j \cos \alpha_t $$
$$ \text{ROT1}_j = (-1)^i \left( \frac{\pi m_t}{4 r_j} + \tan \alpha_t – \alpha_t \right) \frac{180^\circ}{\pi} $$
$$ \text{ROT2}_j = \frac{(-1)^{i+1} 180^\circ \cdot B \tan \beta}{\pi r_j} $$
where \( i = 1, 2 \) for driving and driven gears, \( m_t \) is the transverse module, \( \alpha_t \) is the transverse pressure angle, and \( B \) is the face width. The lead \( S_j \) of the base cylinder helix is:
$$ S_j = \frac{2 \pi r_j}{\tan \beta} $$
To illustrate the parameter setup, consider a driving gear with the following specifications:
| Parameter | Symbol | Value |
|---|---|---|
| Normal module | \( m_n \) | 3 mm |
| Number of teeth | \( z_1 \) | 21 |
| Face width | \( B \) | 50 mm |
| Helix angle | \( \beta \) | 15° (right-hand) |
| Normal pressure angle | \( \alpha_n \) | 20° |
| Normal addendum coefficient | \( h_{an}^* \) | 1 |
| Normal dedendum coefficient | \( c_n^* \) | 0.25 |
| Eccentricity | \( k_1 \) | 0.15 |
| Order | \( n_1 \) | 1 |
The transverse parameters are derived as \( m_t = m_n / \cos \beta \), \( \alpha_t = \arctan(\tan \alpha_n / \cos \beta) \), and \( \beta_b = \arctan(\tan \beta \cdot \cos \alpha_t) \). For the driven gear, the order is set to \( n_2 = 3 \) to avoid concavity, with eccentricity \( k_2 = 0.050508 \) and center distance \( a = 130.197 \) mm. The number of teeth for the driven gear is \( z_2 = n_2 z_1 = 63 \).
The modeling process begins with drawing the pitch curve using parametric equations. For example, the pitch curve equation in CAD software is input as:
$$ r = \frac{p}{1 – k \cos(n \cdot \theta)} $$
$$ \theta = 360 \cdot t $$
$$ z = 0 $$
for \( 0 \leq t \leq 1 \). The addendum and dedendum curves are offset from the pitch curve by \( h_a \) and \( h_f \), respectively. Points for curvature centers and tooth centers are plotted based on calculated coordinates. Next, the base cylinder helix is drawn for each tooth, followed by the construction of tangent planes and generating lines. The involute helicoid is created by sweeping the generating line along the base helix. The intersection curve is obtained by intersecting the involute helicoid with the pitch cylinder within the face width. The end-face tooth profile, an involute curve, is generated by rotating and mirroring the base involute. Finally, the helical tooth is formed by sweeping the tooth profile along the intersection curve with a twist angle, and the gear body is completed through Boolean operations and array patterning.
Kinematic simulation is performed to validate the design. The driving gear is rotated at a constant angular velocity \( \omega_1 = 60^\circ / \text{s} \), while the driven gear follows a variable angular velocity based on the transmission ratio function:
$$ \omega_2 = \frac{\omega_1}{i_{12}} = \frac{60}{3.0612 – 0.6093 \cos(60t)} \quad \text{for} \quad 0 \leq t \leq 30 $$
Interference checks using global collision detection reveal minor interferences, which are resolved by adjusting the center distance from \( a = 130.197 \) mm to \( a’ = 130.658 \) mm, an increase of less than 0.36%. This adjustment ensures smooth operation without interference, confirming the robustness of the design for helical gears in non-circular applications.
In conclusion, the integration of the conversion method with the intersection curve approach effectively addresses the tooth deviation problem in high-order non-circular helical gears. The use of CAD and Excel for parameter management streamlines the modeling process, while kinematic analysis validates the design’s functionality. This method enhances the performance of helical gears in non-circular systems, paving the way for further research on meshing stiffness and dynamic behavior. The repeated focus on helical gears underscores their importance in achieving precise and efficient motion transmission.