Helical gears are critical components in mechanical systems, widely used in aerospace, railway transportation, and marine equipment due to their superior meshing performance, high contact ratio, and strong load-bearing capacity. The modification design, such as profile shifting, is employed to avoid undercutting, enhance load capacity, adjust center distance, and reduce geometric dimensions. In this study, we investigate the dynamic characteristics of helical gears under different modification designs by analyzing time-varying meshing stiffness and its impact on system vibrations. The time-varying meshing stiffness serves as a key internal excitation in helical gear systems, significantly influencing vibration noise and fatigue failure. We establish a dynamic model based on Lagrange dynamics and employ an analytical method to compute the stiffness, examining how modification coefficients affect meshing behavior and dynamic responses.
The calculation of time-varying meshing stiffness for modified helical gears involves considering geometric changes due to profile shifts. For a modified helical gear pair, the tooth profile shifts alter the meshing line, base circle, and pressure angle, leading to variations in stiffness. The potential energy method, combined with slice theory, is applied to derive the Hertzian contact stiffness, bending stiffness, shear stiffness, axial compressive stiffness, and fillet foundation stiffness. The comprehensive meshing stiffness is then obtained by summing these components over the contact segments, accounting for the contact ratio. The formulas for these stiffness components are as follows:
$$k_h = \frac{\pi E L}{4(1 – \nu^2)}$$
where \(E\) is the elastic modulus, \(L\) is the face width, and \(\nu\) is Poisson’s ratio. The fillet foundation stiffness \(k_f\) is given by:
$$k_f = \frac{1}{\left\{ \frac{\cos^2 \alpha_m}{E L} \left[ L^* \left( \frac{\mu_f}{S_f} \right)^2 + M^* \left( \frac{\mu_f}{S_f} \right) + P^* \left(1 + Q^* \tan^2 \alpha_m \right) \right] \right\}}$$
Here, \(\alpha_m\) is the mesh angle, and other parameters are derived from gear geometry. The bending stiffness \(k_b\), shear stiffness \(k_s\), and axial compressive stiffness \(k_a\) for each tooth slice are computed as:
$$k_b = \sum_{i=1}^N \left( \Delta y / \int_{-\alpha’_1}^{\alpha_2} \frac{(\alpha_2 – \alpha) \cos \alpha}{2E [\sin \alpha + (\alpha_2 – \alpha) \cos \alpha]^3} \cdot 3 \left\{ 1 + \cos \alpha’_1 \cdot [(\alpha_2 – \alpha’_1) \sin \alpha – \cos \alpha] \right\}^2 d\alpha \right)$$
$$k_s = \sum_{i=1}^N \left( \Delta y / \left\{ \int_{-\alpha’_1}^{\alpha_2} \frac{\cos \alpha \cos^2 \alpha’_1}{E [\sin \alpha + (\alpha_2 – \alpha) \cos \alpha]} \cdot 1.2 (1 + \nu) (\alpha_2 – \alpha) d\alpha \right\} \right)$$
$$k_a = \sum_{i=1}^N \left( \Delta y / \left\{ \int_{-\alpha’_1}^{\alpha_2} \frac{(\alpha_2 – \alpha) \cos \alpha \cos^2 \alpha’_1}{2E [\sin \alpha + (\alpha_2 – \alpha) \cos \alpha]} d\alpha \right\} \right)$$
In these equations, \(\alpha_2\) is the approach angle, \(\alpha’_1\) is the recess angle, and \(\Delta y\) represents the slice thickness. The total meshing stiffness \(k\) for helical gears depends on the number of teeth in contact, determined by the contact ratio \(\epsilon\). For double-tooth contact, it is:
$$k = \sum_{i=1}^2 \left[ 1 / \left( \frac{1}{k_{h,i}} + \frac{1}{k_{b1,i}} + \frac{1}{k_{s1,i}} + \frac{1}{k_{a1,i}} + \frac{1}{k_{f1,i}} + \frac{1}{k_{b2,i}} + \frac{1}{k_{s2,i}} + \frac{1}{k_{a2,i}} + \frac{1}{k_{f2,i}} \right) \right]$$
and for triple-tooth contact, it extends to three pairs. This analytical approach allows efficient computation of stiffness variations under different modification designs for helical gears.
To analyze the dynamic behavior, we develop an 8-degree-of-freedom dynamic model for the helical gear system, considering three translational and one rotational direction for each gear. The generalized coordinates are \(\mathbf{q} = \{ x_p, y_p, z_p, \theta_p, x_g, y_g, z_g, \theta_g \}\), and the generalized velocities are \(\dot{\mathbf{q}}\). The Lagrangian function is constructed as:
$$L = E_k – E_p = \frac{1}{2} \dot{\mathbf{q}}^T \mathbf{M} \dot{\mathbf{q}} – \frac{1}{2} \mathbf{q}^T \mathbf{K} \mathbf{q}$$
where \(E_k\) is the kinetic energy, \(E_p\) is the potential energy, \(\mathbf{M}\) is the mass matrix, and \(\mathbf{K}\) is the stiffness matrix. Incorporating damping through the Rayleigh dissipation function:
$$D = \frac{1}{2} \dot{\mathbf{q}}^T \mathbf{C} \dot{\mathbf{q}}$$
where \(\mathbf{C}\) is the damping matrix, the equations of motion are derived using Lagrange’s equation:
$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \right) – \frac{\partial L}{\partial \mathbf{q}} + \frac{\partial D}{\partial \dot{\mathbf{q}}} = \mathbf{Q}$$
The generalized force vector \(\mathbf{Q}\) includes meshing forces and friction. The meshing force \(F_m\) is expressed as:
$$F_m = k_m e_{\text{DTE}} + c_m \dot{e}_{\text{DTE}}$$
where \(k_m\) is the time-varying meshing stiffness, \(c_m\) is the damping coefficient, and \(e_{\text{DTE}}\) is the dynamic transmission error. The friction force \(F_f\) is given by \(F_f = -\mu F_m\), with \(\mu\) as the friction coefficient. The dynamic transmission error in the transverse plane is:
$$e_{\text{DTEt}} = R_{bp} \beta_p – R_{bg} \beta_g + (x_p – x_g) \cos \alpha + (y_p – y_g) \sin \alpha – e(t)$$
where \(R_{bp}\) and \(R_{bg}\) are base circle radii, \(\alpha\) is the pressure angle, and \(e(t)\) is static transmission error. The equations of motion for the system are:
$$m_p \ddot{x}_p + c_{bx} \dot{x}_p + k_{bx} x_p = -F_{mx} – F_f \sin \alpha$$
$$m_p \ddot{y}_p + c_{by} \dot{y}_p + k_{by} y_p = -F_{my} – F_f \cos \alpha$$
$$m_p \ddot{z}_p + c_{bz} \dot{z}_p + k_{bz} z_p = -F_{mz}$$
$$I_p \ddot{\theta}_p = F_{my} R_{bp} – T_p$$
$$m_g \ddot{x}_g + c_{bx} \dot{x}_g + k_{bx} x_g = F_{mx} + F_f \sin \alpha$$
$$m_g \ddot{y}_g + c_{by} \dot{y}_g + k_{by} y_g = F_{my} + F_f \cos \alpha$$
$$m_g \ddot{z}_g + c_{bz} \dot{z}_g + k_{bz} z_g = F_{mz}$$
$$I_g \ddot{\theta}_g = -F_{my} R_{bg} + T_g$$
The components of the meshing force along the coordinate axes are:
$$F_{mx} = F_m \cos \beta_b \cos \alpha$$
$$F_{my} = F_m \cos \beta_b \sin \alpha$$
$$F_{mz} = F_m \sin \beta_b$$
where \(\beta_b\) is the base helix angle. To evaluate dynamic responses, we use statistical indicators like root mean square (RMS) and kurtosis value (KV), defined as:
$$X_{\text{RMS}} = \sqrt{\frac{1}{N} \sum_{i=1}^N x_i^2}$$
$$X_{\text{KV}} = \frac{1}{N} \sum_{i=1}^N \left( \frac{x_i – \bar{x}}{\sigma} \right)^4$$
where \(\bar{x}\) is the mean, and \(\sigma\) is the standard deviation. Normalized changes in these indicators are computed to compare different modification designs for helical gears.
We analyze the influence of modification design on time-varying meshing stiffness and dynamic characteristics. The parameters for the helical gear pair are summarized in the table below:
| Parameter | Value |
|---|---|
| Module (mm) | 5.5 |
| Number of teeth (pinion/gear) | 17/107 |
| Face width (mm) | 70 |
| Helix angle (°) | 17 |
| Pressure angle (°) | 20 |
Different modification cases are considered, as shown in the following table:
| Case | Pinion Modification Coefficient | Gear Modification Coefficient |
|---|---|---|
| 1 | 0.1 | 0.3 |
| 2 | -0.1 | 0.3 |
| 3 | 0 | 0 |
| 4 | 0.1 | -0.3 |
| 5 | -0.1 | -0.3 |
For a rotational speed of 1000 rpm and applied torques, the time-varying meshing stiffness is computed. Positive modification (Case 1 and 4) reduces the overall stiffness, while negative modification (Case 2 and 5) increases it. This is due to changes in the transverse contact ratio and pressure angle. The stiffness variations affect the dynamic meshing force; for instance, when stiffness decreases, the meshing force amplitude in the time domain reduces, lowering dynamic loads. Conversely, increased stiffness leads to higher force amplitudes. In the frequency domain, reduced stiffness results in lower mesh frequency amplitudes and fewer sidebands, whereas increased stiffness enhances these components.
The statistical indicators for meshing force under different modification designs are computed and normalized. The table below summarizes the percentage changes in RMS and KV relative to the standard helical gears:
| Case | RMS Change (%) | Kurtosis Change (%) |
|---|---|---|
| 1 | -5.2 | -3.8 |
| 2 | 4.1 | 2.9 |
| 3 | 0 | 0 |
| 4 | -6.7 | -4.5 |
| 5 | 5.3 | 3.6 |
These results indicate that positive modification generally decreases dynamic responses, while negative modification increases them. The changes are more pronounced with larger absolute modification coefficients. For helical gears, this implies that proper modification design can optimize dynamic performance by controlling stiffness and force variations.
In conclusion, our study demonstrates that modification design significantly influences the time-varying meshing stiffness and dynamic characteristics of helical gears. Positive modification reduces stiffness and meshing forces, leading to improved dynamic behavior with lower vibrations and fewer sidebands in the frequency domain. Negative modification has the opposite effect. The analytical model and dynamic analysis provide a foundation for optimizing helical gear systems in practical applications, ensuring enhanced performance and reliability. Future work could explore combined effects with other parameters like helix angle or load conditions for comprehensive design guidelines.