Helical gears are critical components in mechanical transmission systems, operating under demanding conditions such as high speeds, elevated temperatures, and heavy loads. In such environments, tooth breakage faults can occur, leading to reduced reliability, premature failure, and potential catastrophic incidents. This study focuses on investigating the dynamic performance of helical gears under tooth breakage faults, with an emphasis on time-varying mesh stiffness and vibration responses. We employ a combination of analytical modeling, numerical simulations, and dynamic analysis to explore the effects of fault parameters on system behavior. By considering factors like Hertz contact, bending, shear, axial compression, and friction excitation, we aim to provide insights into fault detection and system health monitoring for helical gears.
To model the tooth breakage fault, we first develop a three-dimensional representation of a helical gear pair using parametric design software. The pinion gear is modeled with a single broken tooth to simulate the fault condition. This model allows us to analyze the geometric alterations and their impact on mesh characteristics. The helical gears are designed with specific parameters to ensure realistic simulation, including tooth numbers, module, pressure angle, and helix angle. The inclusion of a broken tooth introduces asymmetries in the gear pair, which significantly influences the time-varying mesh stiffness and dynamic responses.
The calculation of time-varying mesh stiffness is central to understanding the dynamic behavior of helical gears. We use the potential energy method, which accounts for various energy components: Hertz contact energy, bending energy, shear energy, and axial compression energy. For helical gears, the slice method is applied to handle the helical angle, where the gear tooth is divided into multiple slices along the face width. The effective mesh stiffness for each slice is computed and integrated to obtain the overall stiffness. The Hertz contact stiffness for helical gears is given by:
$$k_h = \frac{\pi E L}{4(1-\nu^2)}$$
where \(E\) is Young’s modulus, \(L\) is the contact length, and \(\nu\) is Poisson’s ratio. The bending stiffness \(k_b\), shear stiffness \(k_s\), and axial stiffness \(k_a\) are derived from the potential energy equations. For a healthy helical gear pair, the total mesh stiffness \(K_t\) is the sum of contributions from both gears:
$$K_t = \frac{1}{\frac{1}{k_{h1}} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{h2}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}}}$$
Under tooth breakage, the stiffness in the fault region decreases dramatically. For a broken tooth on the pinion, the effective mesh stiffness during the fault engagement becomes:
$$K_{t,\text{crack}} = \frac{1}{\frac{1}{k_{h1}} + \frac{1}{k_{b1,\text{crack}}} + \frac{1}{k_{s1,\text{crack}}} + \frac{1}{k_{a1,\text{crack}}} + \frac{1}{k_{h2}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}}}$$
where the subscript “crack” denotes the modified stiffness due to the broken tooth. This reduction in stiffness alters the dynamic forces and vibrations in the helical gear system.
The geometric and material parameters used in our analysis are summarized in the following table to provide a clear reference for the helical gears under study:
| Parameter | Value |
|---|---|
| Number of teeth, z1/z2 | 19/48 |
| Young’s modulus, E (Pa) | 2.06 × 1011 |
| Poisson’s ratio, ν | 0.3 |
| Normal module, m (mm) | 3.175 |
| Normal pressure angle, α (degrees) | 20 |
| Center distance, a (mm) | 106.3625 |
| Face width, B (mm) | 16 |
| Pinion root radius, r_f (mm) | 27.12 |
| Pinion base radius, r_b (mm) | 29.11 |
| Mass moment of inertia (kg·mm²) | I_p = 184, I_g = 7500 |
| Helix angle, β (rad) | 0.2443 |
| Equivalent mass, M_e (kg) | 0.188 |
| Mesh damping coefficient, ζ_m | 0.10 |
| Bearing damping coefficient, ζ_b | 0.05 |
| Profile shift coefficient, C_x | 0.25 |
To analyze the dynamic response, we establish a torsional vibration model for the helical gear pair. The model considers a single degree of freedom, incorporating time-varying mesh stiffness \(k(t)\), damping \(c_m\), static transmission error \(e(t)\), and friction-induced moments. The equations of motion are derived using Newton’s second law:
$$I_p \ddot{\theta}_p + c_m r_{b1} \cos \beta \left( r_{b1} \dot{\theta}_p – r_{b2} \dot{\theta}_g – \dot{e}(t) \right) + k(t) r_{b1} \cos \beta \left( r_{b1} \theta_p – r_{b2} \theta_g – e(t) \right) = T_p – M_1$$
$$I_g \ddot{\theta}_g – c_m r_{b2} \cos \beta \left( r_{b1} \dot{\theta}_p – r_{b2} \dot{\theta}_g – \dot{e}(t) \right) – k(t) r_{b2} \cos \beta \left( r_{b1} \theta_p – r_{b2} \theta_g – e(t) \right) = -T_g + M_2$$
where \(I_p\) and \(I_g\) are the mass moments of inertia, \(\theta_p\) and \(\theta_g\) are the angular displacements, \(r_{b1}\) and \(r_{b2}\) are the base circle radii, \(\beta\) is the helix angle, \(T_p\) and \(T_g\) are the input and output torques, and \(M_1\) and \(M_2\) are the friction moments. The static transmission error \(e(t)\) is modeled as a periodic function to account for manufacturing inaccuracies and assembly errors. The dynamic transmission error \(\delta\) is defined as:
$$\delta = r_{b1} \theta_p – r_{b2} \theta_g – e(t)$$
This model allows us to simulate the vibration responses, including velocity and acceleration, under various operating conditions.
We solve the dynamic equations using the Runge-Kutta numerical method, implemented in computational software. The time-varying mesh stiffness is input as a periodic function, and the solutions are obtained for both healthy and faulty helical gears. The results are analyzed in the time domain and frequency domain to identify characteristic patterns associated with tooth breakage. For instance, the mesh frequency \(f_m\) is calculated as:
$$f_m = \frac{z_1 n}{60}$$
where \(n\) is the rotational speed in rpm, and \(z_1\) is the number of teeth on the pinion. The presence of sidebands around the mesh frequency in the frequency spectrum is a key indicator of faults in helical gears.
Our analysis reveals that tooth breakage significantly reduces the time-varying mesh stiffness of helical gears. In the double-tooth engagement region, where the broken tooth is supposed to be in contact, the stiffness drops to a level equivalent to single-tooth engagement. This reduction causes periodic impacts in the dynamic responses. For example, at a rotational speed of 3785 rpm and a torque of 500 N·m, the dynamic transmission error shows impulsive peaks every 0.016 seconds, corresponding to the fault period. The vibration velocity and acceleration also exhibit similar periodic shocks, with magnitudes higher than those in healthy helical gears. The following table summarizes the maximum responses under different conditions for helical gears with tooth breakage:
| Operating Condition | Dynamic Transmission Error (mm) | Vibration Velocity (mm/s) | Vibration Acceleration (mm/s²) |
|---|---|---|---|
| n = 3785 rpm, T = 500 N·m | 0.010 | 1.406 | 6716 |
| n = 3785 rpm, T = 50000 N·m | 0.015 | 2.735 | 56500 |
| n = 5785 rpm, T = 50000 N·m | 0.012 | 0.2604 | 5017 |
The frequency domain analysis further confirms the fault presence. For helical gears with tooth breakage, the spectrum of dynamic transmission error, vibration velocity, and acceleration shows distinct sidebands around the mesh frequency. The sideband spacing equals the rotational frequency of the faulty gear, which is 63 Hz at 3785 rpm. This modulation effect is more pronounced in the acceleration response, indicating higher sensitivity to impacts. In contrast, healthy helical gears exhibit minimal sidebands and lower vibration levels. The mesh stiffness variation over one engagement cycle for faulty helical gears is plotted numerically, showing a dip in the stiffness during the broken tooth’s passage, which correlates with the observed dynamic responses.
The influence of operating parameters on the dynamic behavior of helical gears is also investigated. Higher rotational speeds lead to shorter impact intervals and increased vibration severity. For instance, at 5785 rpm, the impact interval reduces to 0.01 seconds, and the vibration acceleration peaks at 5017 mm/s², compared to 3785 rpm where it is 4009 mm/s². Similarly, higher torque levels amplify the dynamic transmission error and vibration responses due to increased load sharing and friction effects. At 50000 N·m torque, the vibration acceleration reaches 56500 mm/s², highlighting the critical role of load conditions in fault manifestation for helical gears.
In discussion, the periodic impacts in the time domain responses are attributed to the sudden loss of stiffness when the broken tooth enters the mesh. This excites the natural frequencies of the helical gear system, resulting in amplified vibrations. The sidebands in the frequency spectrum arise from the modulation of the mesh frequency by the fault-induced periodic impulses. These characteristics can be leveraged for condition monitoring and fault diagnosis in helical gear transmissions. Practical applications include predictive maintenance strategies, where vibration analysis can detect early signs of tooth breakage, preventing unexpected downtime and enhancing system reliability.
In conclusion, our study demonstrates that tooth breakage faults in helical gears cause significant reductions in time-varying mesh stiffness and induce periodic dynamic responses. The dynamic transmission error, vibration velocity, and acceleration exhibit impulsive behaviors, with frequency spectra showing sidebands around the mesh frequency. Operating parameters such as speed and torque intensify these effects. The findings provide a foundation for fault detection in helical gear systems, enabling improved design and maintenance practices. Future work could explore the effects of multiple faults, varying helix angles, and advanced signal processing techniques for real-time monitoring of helical gears in industrial applications.