Helical gears are widely used in various mechanical equipment due to their advantages such as smooth transmission, high load capacity, strong reliability, and accurate transmission ratio. In this study, I focus on the dynamic behavior of helical gear pairs when crack faults occur. Gear systems often operate under high-speed and heavy-load conditions, making them prone to failures like pitting, cracks, wear, and tooth breakage. Among these, tooth crack is a common fault that alters the meshing state and internal dynamic excitation. By analyzing the dynamic characteristics of helical gear pairs with crack faults, I aim to reveal the influence of cracks on vibration response, providing a theoretical basis for early crack detection and fault diagnosis in mechanical gear transmissions.
Extensive research has been conducted on cracked gear analysis. For instance, Lin et al. proposed a correction algorithm for cracked gear meshing and established a dynamic model, finding that gear pair stiffness decreases with increasing crack length. Li et al. developed a deformation calculation formula for planetary gears. Ma et al. studied the dynamic mechanism of cracked gears, revealing different bifurcation patterns caused by transmission errors and cracks. Wan et al. proposed a meshing stiffness algorithm for cracked gears. Cui et al. established a dynamic model for gear-rotor systems and found that increased backlash enhances jump phenomena. Wang et al. emphasized that axial bending stiffness, axial torsional stiffness, and axial foundation stiffness should be considered for helical gear time-varying meshing stiffness. Fakher et al. derived a formula for spur gear meshing stiffness considering crack size. Zhang et al. improved the meshing stiffness calculation method for non-standard gears. Xiao et al. studied fault mechanisms in planetary gear systems from a dynamic perspective using an improved energy method. Gao et al. established a six-degree-of-freedom dynamic model for single-stage gear transmission systems, finding that transmission error, vibration impact, and severity vary with wear form and degree. Wang et al. explored crack fault mechanisms using dynamic methods.
Building on these studies, I simplify the gear tooth as a variable cross-section cantilever beam on the root circle and tooth surface circle. I investigate the effects of different crack parameters and dynamic parameters on the meshing stiffness response and dynamic contact characteristics of helical gear systems. The results provide a reference for crack fault prediction and identification in helical gears.
| Parameter | Value |
|---|---|
| Number of teeth (z1 / z2) | 19 / 48 |
| Elastic modulus E (GPa) | 206 |
| Poisson’s ratio ν | 0.3 |
| Normal module m (mm) | 3.175 |
| Normal pressure angle αn (°) | 20 |
| Center distance a (mm) | 106.3625 |
| Tooth width B (mm) | 16 |
| Dedendum radius rf (mm) | 27.12 |
| Base circle radius rb (mm) | 29.11 |
| Moment of inertia (kg·m²) | Ip=7.5, Ig=0.184 |
| Helix angle at pitch circle β (°) | 0.2443 |
| Equivalent mass Me (kg) | 0.188 |
| Meshing damping coefficient ζ | 0.10 |
| Bearing damping coefficient ξ | 0.05 |
| Radial modification coefficient cx | 0.25 |
1. Time-Varying Meshing Stiffness Calculation for Cracked Spur Gears
1.1 Root Crack in Spur Gears
When a root crack occurs on a spur gear tooth, the Hertz contact stiffness and axial compression stiffness remain unchanged, while the bending stiffness and shear stiffness are affected. I derive the new bending and shear stiffness for a tooth with a root crack. The cracked tooth is modeled as a variable cross-section cantilever beam. Assuming the crack extends uniformly along the tooth width with constant length q and angle V relative to the horizontal line from the base circle center. The effective area moment of inertia and effective cross-sectional area are given by:
$$ I_{xc} = \begin{cases} \frac{1}{12}(h + h_x)^3 \times L, & x \le d \\ \frac{2}{3} h_x^3 \times L, & x > d \end{cases} $$
$$ A_{xc} = \begin{cases} (h + h_x) \times L, & x \le d \\ 2 h_x \times L, & x > d \end{cases} $$
where:
$$ h = r_b \sin \alpha_2 – q \sin V $$
$$ h_x = \begin{cases} (r_f + r_0) \sin(\alpha_2 + \alpha_0) – \sqrt{r_0^2 – [r_0 \cos(\alpha_2 + \alpha_0) – x]^2}, & x \le x_D \\ r_b \theta \cos(\alpha_2 – \theta) – r_b \sin(\theta – \alpha_2), & x > x_D \end{cases} $$
The potential energies for bending and shear are:
$$ U_b = \int_0^d \frac{ [F_b (d – x) – M]^2 }{2E I_{xc}} dx $$
$$ U_s = \int_0^d \frac{1.2 F_b^2}{2G A_{xc}} dx $$
Thus, the bending stiffness and shear stiffness of the cracked tooth are:
$$ \frac{1}{K_{b,crack}} = U_b $$
$$ \frac{1}{K_{s,crack}} = U_s $$
The total effective meshing stiffness for the gear pair (pinion 1 and gear 2) becomes:
$$ K_{t,crack} = \frac{1}{ \frac{1}{K_h} + \frac{1}{K_{b1,crack}} + \frac{1}{K_{a1}} + \frac{1}{K_{b2}} + \frac{1}{K_{s2}} + \frac{1}{K_{a2}} } $$
1.2 Tooth Surface Crack in Spur Gears
For a tooth surface crack (e.g., leftward crack), the geometry differs. The expressions for h and hx are modified as:
$$ h = r_b \delta \cos(\delta – \alpha_2) – r_b \sin(\delta – \alpha_2) – q \sin V $$
$$ h_x = \begin{cases} (r_f + r_0) \sin(\alpha_2 + \alpha_0) – \sqrt{r_0^2 – [r_0 \cos(\alpha_2 + \alpha_0) – x]^2} – q \sin V, & x \le x_D \\ r_b \theta \cos(\alpha_2 – \theta) – r_b \sin(\theta – \alpha_2) – q \sin V, & x > x_D \end{cases} $$
The remaining formulas for stiffness are similar to those for root cracks.
1.3 Cracked Helical Gears: Slice Method
To analyze helical gears, I employ the slice method, which discretizes the gear face width into thin slices. Each slice is treated as a spur gear with a specific angular position, and the total meshing stiffness is obtained by summing the stiffness contributions of all slices. This approach captures the gradual engagement characteristic of helical gears. The figure below illustrates a healthy helical gear versus those with varying root crack lengths in the slice representation.
2. Influence of Crack Parameters on Time-Varying Meshing Stiffness of Helical Gears
2.1 Effect of Root Crack Parameters
Considering a root crack with length q and angle V in helical gears, I compute the bending and shear stiffness. Assuming V = 45°, the combined meshing stiffness Kt for different crack lengths is shown in the following table (qualitative trends). As q increases, the stiffness decreases significantly. Similarly, for a fixed q = 0.5 mm, varying V from 30° to 60° produces a minor stiffness reduction.
| Crack Length q (mm) | Stiffness Trend |
|---|---|
| 0 (Healthy) | Highest |
| 0.5 | Slight decrease |
| 1.0 | Noticeable decrease |
| 1.5 | Further decrease |
| 2.0 | Maximum reduction |
2.2 Effect of Tooth Surface Crack Parameters
For a leftward tooth surface crack with fixed q = 0.5 mm and varying distance s from the root, the stiffness reduction is less pronounced than root cracks. The results indicate that as s decreases (crack closer to root), stiffness reduces more, but overall impact is smaller.
| Distance s (mm) | Stiffness Trend |
|---|---|
| Healthy | Base level |
| 4.0 | Slight reduction |
| 3.5 | Moderate reduction |
| 3.0 | More reduction |
| 2.0 | Most reduction among surface cracks |
2.3 Periodic Effects of Crack Parameters on Helical Gear Stiffness
The helix angle introduces a gradual engagement, leading to periodic fluctuations in meshing stiffness. For a root-cracked helical gear, as the crack length increases, the minimum stiffness during the engagement cycle decreases substantially. For a tooth surface crack, the periodic dip is less severe. Figure A (not shown) demonstrates these periodic variations.
3. Torsional Vibration Dynamic Model of Helical Gear Pairs
I establish a single-stage helical gear pair torsional vibration model using the lumped-mass method. The model incorporates time-varying meshing stiffness, meshing damping, static transmission error, and friction excitation. The governing equations based on Newton’s second law are:
$$ I_p \ddot{\theta}_p + c_m r_{b1} [r_{b1} \dot{\theta}_p – r_{b2} \dot{\theta}_g – \dot{e}(t)] \cos\beta + k(t) r_{b1} [r_{b1} \theta_p – r_{b2} \theta_g – e(t)] \cos\beta – M_1 = T_p $$
$$ I_g \ddot{\theta}_g – c_m r_{b2} [r_{b1} \dot{\theta}_p – r_{b2} \dot{\theta}_g – \dot{e}(t)] \cos\beta – k(t) r_{b2} [r_{b1} \theta_p – r_{b2} \theta_g – e(t)] \cos\beta + M_2 = -T_g $$
where Ip, Ig are moments of inertia; Tp, Tg are external torques; M1, M2 are friction moments; θp, θg are torsional displacements; rb1, rb2 are base radii; β is helix angle; k(t) is time-varying meshing stiffness; cm is meshing damping; e(t) is composite transmission error. I solve the system using the variable-step Runge-Kutta method to obtain vibration responses.
4. Dynamic Characteristics of Cracked Helical Gear Torsional Vibration
4.1 Effect of Root Crack on Dynamic Response
4.1.1 Influence of Rotational Speed
For a root crack length q = 1 mm, I compare speeds of 3785 r/min and 5785 r/min. The dynamic transmission error (DTE) and vibration velocity exhibit periodic shock responses. At lower speed, the interval between shocks is 0.016 s with DTE peak 1.647×10⁻⁵ mm; at higher speed, interval is 0.011 s and peak 1.679×10⁻⁵ mm. Higher speed amplifies DTE fluctuations. Vibration velocity peak increases from 0.02215 mm/s to 0.02499 mm/s, indicating intensified meshing impact.
4.1.2 Influence of Torque
Comparing torques 500 N·m and 50,000 N·m (q=1 mm, 3785 r/min), DTE peak is larger under high torque. Vibration velocity peak rises from 1.406 mm/s to 2.735 mm/s. Higher torque exacerbates shock energy.
| Torque (N·m) | Vibration Velocity Peak (mm/s) |
|---|---|
| 500 | 1.406 |
| 50,000 | 2.735 |
4.1.3 Influence of Crack Length
At 500 N·m and 3785 r/min (rotational frequency 63 Hz, meshing frequency 1198 Hz), I compare healthy and root-cracked helical gears with q = 1.0, 1.5, 2.0 mm. Time-domain DTE and vibration velocity show periodic shocks at intervals equal to the gear rotational period. As crack length increases, shock amplitudes grow. In the frequency domain, sidebands appear around the meshing frequency. For DTE, sideband spacing is 68.7 Hz; for vibration velocity, 58.9 Hz. The sideband amplitudes increase with crack length, confirming that deeper cracks intensify modulation effects.
| Crack Length q (mm) | Vibration Velocity Peak (mm/s) |
|---|---|
| 0 (Healthy) | 0.02212 |
| 1.0 | 0.02215 |
| 1.5 | 0.02218 |
| 2.0 | 0.02222 |
4.2 Effect of Tooth Surface Crack on Dynamic Response
Tooth surface cracks have a milder effect on stiffness, hence their dynamic impact is less pronounced. At 50,000 N·m and 3785 r/min, for a surface crack with s=4 mm, q=1 mm, the frequency spectrum of DTE shows sidebands with spacing 58.9 Hz around 1198 Hz, but the sideband amplitudes are noticeably smaller than those from root cracks. Vibration velocity sidebands are almost negligible. This confirms that root cracks are more critical for fault detection in helical gears.
5. Conclusion
From this study, I draw the following conclusions:
- Crack faults reduce the meshing stiffness of helical gear pairs. Root cracks have a much more significant influence than tooth surface cracks for the same crack length.
- The slice method effectively captures the periodic stiffness variation in cracked helical gears. Both crack location and length affect the stiffness reduction pattern.
- Dynamic simulation reveals that cracked helical gears exhibit periodic shock responses in time-domain displacement and velocity. The shock interval corresponds to the gear rotation period. In the frequency domain, sidebands appear around the meshing frequency and its harmonics, with sideband spacing equal to the rotational frequency. Root cracks produce stronger sideband amplitudes compared to surface cracks.
- The findings provide a theoretical basis for identifying crack faults in helical gear transmission systems by monitoring vibration characteristics such as shock intervals and sideband energy.
This work advances the understanding of how crack faults affect the dynamic behavior of helical gears, aiding in early fault diagnosis and preventive maintenance.