Helical gears are widely used in various industrial applications such as vehicles, machine tools, ships, and power generation due to their compact structure, high load-bearing capacity, and efficient transmission. The meshing characteristics of helical gears are critical indicators of transmission system performance, and internal excitations significantly impact vibration, noise, and service life. This study investigates the effect of center distance error on the dynamic behavior of helical gears, focusing on time-varying meshing stiffness, dynamic meshing force, and vibration characteristics. A pair of subway helical gears is analyzed to understand how deviations in center distance alter meshing conditions and dynamic responses.
In ideal conditions, helical gears are installed at the theoretical center distance, where the pitch circles are tangent and the meshing angle equals the pressure angle. However, practical installations often involve center distance errors, leading to changes in the axial distance between gear centers. This error affects the meshing state, causing the operating pitch circle radius to differ from the theoretical pitch circle radius and altering the transverse meshing angle. The transverse meshing angle for helical gears with center distance error is given by:
$$ \cos(\alpha_t’) = \frac{a \cos(\alpha_t)}{a’} $$
where \( a’ = a \pm \Delta a \) is the actual center distance, \( a \) is the ideal center distance, and \( \alpha_t \) is the ideal transverse pressure angle. The transverse contact ratio under center distance error is calculated as:
$$ \varepsilon_{\alpha} = \frac{1}{2\pi} \left[ z_1 (\tan(\alpha_{at1}’) – \tan(\alpha_t’)) + z_2 (\tan(\alpha_{at2}’) – \tan(\alpha_t’)) \right] $$
where \( z_1 \) and \( z_2 \) are the numbers of teeth on the driving and driven helical gears, respectively, and \( \alpha_{at1}’ \) and \( \alpha_{at2}’ \) are the operating transverse pressure angles at the addendum circles. The length of the contact line for helical gears varies with the meshing phase and is expressed as:
$$ L'(t) = \begin{cases}
\frac{L_{\max} t}{\beta_{\varepsilon} t_z} & t \in [0, \beta_{\varepsilon} t_z] \\
L_{\max} & t \in [\beta_{\varepsilon} t_z, \alpha_{\varepsilon} t_z] \\
L_{\max} \frac{t – \gamma_{\varepsilon} t_z}{t_z (\beta_{\varepsilon} – \gamma_{\varepsilon})} & t \in [\alpha_{\varepsilon} t_z, \gamma_{\varepsilon} t_z]
\end{cases} $$
Here, \( \beta_{\varepsilon} \) is the axial contact ratio, \( \alpha_{\varepsilon} \) is the transverse contact ratio, \( \gamma_{\varepsilon} = \alpha_{\varepsilon} + \beta_{\varepsilon} \) is the total contact ratio, and \( t_z \) is the meshing period. The geometric parameters of the subway helical gears used in this study are summarized in Table 1.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth, \( Z \) | 16 | 101 |
| Module, \( m \) (mm) | 6 | 6 |
| Pressure Angle, \( \alpha \) (°) | 25 | 25 |
| Face Width, \( b \) (mm) | 97 | 90 |
| Helix Angle, \( \beta \) (°) | 12.5 | 12.5 |
The time-varying meshing stiffness of helical gears is a key internal excitation that influences dynamic behavior. To compute this, the slicing method is employed, where the helical gear is divided into multiple slices along the face width, each treated as a spur gear. The potential energy method is then applied to determine the meshing stiffness of each slice, and the total stiffness is obtained by integration. The stiffness components include bending stiffness \( k_b \), shear stiffness \( k_s \), axial compressive stiffness \( k_a \), and Hertzian contact stiffness \( k_h \), derived from the respective potential energies:
$$ U_b = \int_0^d \frac{[F_a (d – x) – F_b h]^2}{2EI_x} dx = \frac{F^2}{2k_b} $$
$$ U_s = \int_0^d \frac{1.2 F_b^2}{2GA_x} dx = \frac{F^2}{2k_s} $$
$$ U_a = \int_0^d \frac{F_a^2}{2EA_x} dx = \frac{F^2}{2k_a} $$
$$ U_h = \frac{F^2}{2k_h} $$
where \( E \) is the elastic modulus, \( G \) is the shear modulus, \( I_x \) is the area moment of inertia, \( A_x \) is the cross-sectional area, and \( F \) is the meshing force. The stiffness values are calculated as:
$$ \frac{1}{k_b} = \sum_{i=1}^N \int_{\alpha_1}^{\alpha_2} \frac{3\{1 + \cos(\alpha_1i)[(\alpha_2 – \alpha)\sin(\alpha) – \cos(\alpha)]\}^2}{2EL[\sin(\alpha) + (\alpha_2 – \alpha)\cos(\alpha)]^3} \cos(\alpha) d\alpha $$
$$ \frac{1}{k_s} = \sum_{i=1}^N \int_{\alpha_1}^{\alpha_2} \frac{1.2(1+\nu)(\alpha_2 – \alpha)\cos(\alpha)\cos(\alpha_1i)}{EL[\sin(\alpha) + (\alpha_2 – \alpha)\cos(\alpha)]} d\alpha $$
$$ \frac{1}{k_a} = \sum_{i=1}^N \int_{\alpha_1}^{\alpha_2} \frac{(\alpha_2 – \alpha)\cos(\alpha)\sin(\alpha_1i)}{2EL[\sin(\alpha) + (\alpha_2 – \alpha)\cos(\alpha)]} d\alpha $$
$$ k_h = \frac{\pi E L}{4(1-\nu^2)} $$
The fillet foundation stiffness \( k_f \) is given by:
$$ \frac{1}{k_f} = \frac{\cos^2(\alpha)}{EL} \left\{ L^* \left( \frac{u_f}{S_f} \right)^2 + M^* \left( \frac{u_f}{S_f} \right) + P^* (1 + Q^* \tan^2(\alpha)) \right\} $$
The single-tooth meshing stiffness \( k_1 \) is then:
$$ \frac{1}{k_1} = \frac{1}{k_h} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_{f2}} $$
and the total time-varying meshing stiffness \( k_m \) is the sum of the stiffnesses of all engaged tooth pairs:
$$ k_m = \sum_{n=1}^{M} k_n $$
Under quasi-static conditions, the dynamic transmission error \( e \) is related to the applied torque and meshing stiffness:
$$ e = \frac{T_1}{r_{b1} k_m} = \frac{T_2}{r_{b2} k_m} $$
To analyze the dynamic behavior, a lumped mass model with eight degrees of freedom (8-DOF) is established, considering bending, torsion, and axial vibrations. The generalized displacement vector is:
$$ \{q\} = [x_1, y_1, z_1, \theta_1, x_2, y_2, z_2, \theta_2]^T $$
The dynamic meshing force and its components in the X, Y, and Z directions are:
$$ F_n = k_m(t) X + c_m \dot{X} $$
$$ F_x = F_n (-\sin(\alpha_n) \cos(\phi) – \cos(\alpha_n) \cos(\beta_n) \sin(\phi)) $$
$$ F_y = F_n (-\sin(\alpha_n) \sin(\phi) + \cos(\alpha_n) \cos(\beta_n) \cos(\phi)) $$
$$ F_z = F_n (-\cos(\alpha_n) \sin(\beta_n)) $$
where \( c_m \) is the meshing damping, calculated as:
$$ c_m = 2\xi \sqrt{\frac{k_m r_1^2 r_2^2 I_1 I_2}{r_1^2 I_1 + r_2^2 I_2}} $$
The nonlinear vibration differential equations for the helical gear system are:
$$ m_1 \ddot{x}_1 + c_{1x} \dot{x}_1 + k_{1x} x_1 = F_{x1} $$
$$ m_1 \ddot{y}_1 + c_{1y} \dot{y}_1 + k_{1y} y_1 = F_{y1} $$
$$ m_1 \ddot{z}_1 + c_{1z} \dot{z}_1 + k_{1z} z_1 = F_{z1} $$
$$ I_1 \ddot{\theta}_1 = T_1 – F_n r_{b1} $$
$$ m_2 \ddot{x}_2 + c_{2x} \dot{x}_2 + k_{2x} x_2 = -F_{x2} $$
$$ m_2 \ddot{y}_2 + c_{2y} \dot{y}_2 + k_{2y} y_2 = -F_{y2} $$
$$ m_2 \ddot{z}_2 + c_{2z} \dot{z}_2 + k_{2z} z_2 = -F_{z2} $$
$$ I_2 \ddot{\theta}_2 = -T_2 + F_n r_{b2} $$
For dynamic analysis, the subway helical gears operate at a speed of 1800 rpm with an input power of 138 kW. Material properties include an elastic modulus of \( 2.06 \times 10^{11} \) Pa, Poisson’s ratio of 0.3, and density of 7850 kg/m³. The influence of center distance errors (±0.4 mm, ±0.2 mm, and 0 mm) on internal excitations and dynamic responses is examined. The time-varying meshing stiffness decreases with positive center distance errors due to reduced contact lines and engagement regions, as shown in Table 2.
| Center Distance Error (mm) | Time-Varying Meshing Stiffness (N/m) | Dynamic Transmission Error (mm) |
|---|---|---|
| -0.4 | 1.29 × 109 | 1.25 × 10-5 |
| -0.2 | 1.27 × 109 | 1.28 × 10-5 |
| 0 | 1.25 × 109 | 1.30 × 10-5 |
| +0.2 | 1.23 × 109 | 1.33 × 10-5 |
| +0.4 | 1.21 × 109 | 1.35 × 10-5 |
The dynamic meshing force and vibration acceleration in the Y-direction are evaluated for different center distance errors. The mean and standard deviation of these parameters are summarized in Table 3. Negative center distance errors result in larger fluctuations in dynamic meshing force and higher vibration acceleration, indicating reduced meshing stability. Positive errors initially increase fluctuations but then stabilize, with minimal impact on the mean dynamic meshing force.
| Center Distance Error (mm) | Mean Dynamic Meshing Force (N) | Standard Deviation of Dynamic Meshing Force (N) | Mean Y-direction Vibration Acceleration (μm/s²) | Standard Deviation of Y-direction Vibration Acceleration (μm/s²) |
|---|---|---|---|---|
| -0.4 | 20410.57 | 537.09 | 3633.74 | 91.52 |
| -0.2 | 20410.57 | 536.95 | 3622.85 | 91.49 |
| 0 | 20410.57 | 545.56 | 3610.44 | 90.19 |
| +0.2 | 20410.57 | 543.33 | 3582.56 | 92.58 |
| +0.4 | 20410.57 | 545.56 | 3555.82 | 92.96 |
The analysis reveals that center distance errors significantly alter the meshing characteristics of helical gears. As the error increases, the time-varying meshing stiffness decreases, leading to higher dynamic transmission errors. The dynamic meshing force remains relatively stable in mean value but exhibits varying fluctuations, with negative errors causing more pronounced vibrations. The Y-direction vibration acceleration decreases with positive errors, suggesting improved stability under certain conditions. These findings highlight the importance of controlling center distance errors in helical gear systems to minimize vibrations and noise, thereby enhancing reliability and performance in practical applications such as subway transmissions. Further research could explore the combined effects of other manufacturing errors and operating conditions on the dynamic behavior of helical gears.