Spur gears are fundamental components in mechanical transmission systems, widely used in industries such as aerospace, automotive, and manufacturing. However, gear failures like cracks and pitting can lead to significant performance degradation, increased noise, and even catastrophic system failures. Traditional experimental methods for simulating different failure types and degrees in spur gears are often costly and complex. To address this, we develop a dynamic model using the lumped parameter method, incorporate time-varying mesh stiffness calculations via the energy method, and modify the model parameters based on experimental data. This approach enables accurate simulation of various fault conditions in spur gears, providing a foundation for fault diagnosis databases. The model’s validity is verified through experimental vibration signal analysis, demonstrating its capability to identify multiple gear failure types.
In this study, we focus on the dynamic behavior of spur gears under normal and faulty conditions. The primary excitation in gear systems stems from time-varying mesh stiffness, which is influenced by factors like tooth deformation, base deformation, and contact deformation. By accurately modeling these aspects, we can predict the dynamic response of spur gears and identify fault characteristics. The use of a bending-torsion coupled vibration model allows for comprehensive analysis of the system’s behavior. Furthermore, model modification techniques ensure that the simulated responses align closely with experimental data, enhancing the reliability of our approach.
Dynamic Modeling of Spur Gear Systems
We employ the lumped parameter method to establish a six-degree-of-freedom bending-torsion coupled vibration model for spur gears. This model accounts for translational and rotational motions, as well as the elasticity of shafts and support bearings. The generalized displacement vector is defined as:
$$ \{\delta\} = \{x_p, y_p, \theta_p, x_g, y_g, \theta_g\}^T $$
where \(x_p\) and \(x_g\) are the translational displacements along the x-axis for the pinion and gear, \(y_p\) and \(y_g\) are along the y-axis, and \(\theta_p\) and \(\theta_g\) are the rotational displacements. The relative displacement along the mesh line is given by:
$$ y = y_p + R_p \theta_p – y_g + R_g \theta_g $$
Here, \(R_p\) and \(R_g\) are the base circle radii of the pinion and gear, respectively. The dynamic mesh forces \(F_p\) and \(F_g\) acting on the pinion and gear are expressed as:
$$ F_p = c_m \dot{y} + k_m y $$
$$ F_g = -F_p = -c_m \dot{y} – k_m y $$
where \(c_m\) is the mesh damping and \(k_m\) is the time-varying mesh stiffness. The friction force \(F_f\) on the tooth surface is approximated as \(F_f = f F_p\), with \(f\) being the equivalent friction coefficient. The equations of motion are derived as follows:
$$ m_p \ddot{x}_p + c_{px} \dot{x}_p + k_{px} x_p = F_f $$
$$ m_p \ddot{y}_p + c_{py} \dot{y}_p + k_{py} y_p = F_g $$
$$ I_p \ddot{\theta}_p = T_p – F_p R_p + F_f (R_p \tan \beta – H) $$
$$ m_g \ddot{x}_g + c_{gx} \dot{x}_g + k_{gx} x_g = -F_f $$
$$ m_g \ddot{y}_g + c_{gy} \dot{y}_g + k_{gy} y_g = -F_g $$
$$ I_g \ddot{\theta}_g = -T_g – F_g R_g + F_f (R_g \tan \beta – H) $$
In these equations, \(m_p\) and \(m_g\) represent the masses, \(I_p\) and \(I_g\) the moments of inertia, \(k_{px}, k_{py}, k_{gx}, k_{gy}\) the support stiffnesses, and \(c_{px}, c_{py}, c_{gx}, c_{gy}\) the support dampings for the pinion and gear. \(T_p\) and \(T_g\) are the external torques, and \(H\) is the distance from the mesh point to the pitch point. The system can be written in matrix form as:
$$ M \ddot{\delta} + C \dot{\delta} + K \delta = F $$
where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrices, respectively, and \(F\) is the force vector. The matrices are defined as follows:
$$ M = \begin{bmatrix}
m_p & 0 & 0 & 0 & 0 & 0 \\
0 & m_p & 0 & 0 & 0 & 0 \\
0 & 0 & I_p & 0 & 0 & 0 \\
0 & 0 & 0 & m_g & 0 & 0 \\
0 & 0 & 0 & 0 & m_g & 0 \\
0 & 0 & 0 & 0 & 0 & I_g
\end{bmatrix} $$
The damping matrix \(C\) and stiffness matrix \(K\) incorporate terms related to mesh damping, support damping, mesh stiffness, and support stiffness. For example, the stiffness matrix includes components like \(k_{px} – f k_m\) for the x-direction of the pinion. Solving these equations using numerical methods like the Runge-Kutta method yields the dynamic response of the spur gear system.
Time-Varying Mesh Stiffness Calculation
The time-varying mesh stiffness is a critical parameter in the dynamic analysis of spur gears. We use the energy method to calculate the mesh stiffness, considering potential energies due to bending, shear, axial compression, Hertzian contact, and fillet foundation deformation. The total potential energy \(U\) stored in a gear tooth is given by:
$$ U = U_b + U_s + U_a + U_h + U_f $$
where \(U_b = \frac{F^2}{2K_b}\) is the bending potential energy, \(U_s = \frac{F^2}{2K_s}\) is the shear potential energy, \(U_a = \frac{F^2}{2K_a}\) is the axial compression energy, \(U_h = \frac{F^2}{2K_h}\) is the Hertzian contact energy, and \(U_f = \frac{F^2}{2K_f}\) is the fillet foundation deformation energy. The corresponding stiffnesses are calculated as follows:
$$ \frac{1}{K_b} = \int_0^l \frac{[(l – x) \cos \alpha_p – h \sin \alpha_p]^2}{E I_x} dx $$
$$ \frac{1}{K_s} = \int_0^l \frac{1.2 \cos^2 \alpha_p}{G A_x} dx $$
$$ \frac{1}{K_a} = \int_0^l \frac{\sin^2 \alpha_p}{E A_x} dx $$
$$ \frac{1}{K_h} = \frac{4(1 – \nu^2)}{\pi E W} $$
$$ \frac{1}{K_f} = \frac{\cos^2 \alpha_p}{W E} \left\{ L^* \left( \frac{u_f}{s_f} \right)^2 + M^* \left( \frac{u_f}{s_f} \right) + P^* (1 + Q^* \tan^2 \alpha_p) \right\} $$
In these equations, \(E\) is the elastic modulus, \(G\) is the shear modulus, \(I_x\) is the area moment of inertia at distance \(x\) from the root, \(A_x\) is the cross-sectional area, \(\nu\) is Poisson’s ratio, and \(W\) is the tooth width. The parameters \(L^*, M^*, P^*, Q^*\) are derived from empirical formulas for fillet foundation deformation. The total mesh stiffness for a gear pair is the sum of the stiffnesses of all contacting tooth pairs:
$$ k_m(t) = \sum_{i=1}^N k_i^j $$
where \(N\) is the number of simultaneously engaged tooth pairs, and \(k_i^j\) is the stiffness of the \(i\)-th tooth pair at contact point \(j\).
Mesh Stiffness for Faulty Spur Gears
For spur gears with cracks, the crack is modeled as a straight line starting from the root, with parameters such as crack length \(q_0\), crack angle \(\alpha_c\), and distance from the root \(h_c\). The effective area moment of inertia \(I_x\) and cross-sectional area \(A_x\) are modified as:
$$ I_x = \begin{cases}
\frac{1}{12} (h_x + h_x)^3 W & \text{if } h_x \leq h_q \\
\frac{1}{12} (h_x + h_q)^3 W & \text{if } h_x > h_q
\end{cases} $$
$$ A_x = \begin{cases}
(h_x + h_x) W & \text{if } h_x \leq h_q \\
(h_x + h_q) W & \text{if } h_x > h_q
\end{cases} $$
where \(h_q = h_c – q_0 \sin \alpha_c\). These modified values are substituted into the stiffness equations to compute the mesh stiffness for cracked spur gears.
For spur gears with pitting, the pit is modeled as a rectangular凹坑 with length \(a_s\), width \(w_s\), and depth \(h_s\). The effective tooth width \(\Delta W_x\), cross-sectional area \(\Delta A_x\), and area moment of inertia \(\Delta I_x\) are altered as:
$$ \Delta W_x = \begin{cases}
w_s & \text{if } x \in [\mu – \frac{a_s}{2}, \mu + \frac{a_s}{2}] \\
0 & \text{otherwise}
\end{cases} $$
$$ \Delta A_x = \begin{cases}
\Delta W_x h & \text{if } x \in [\mu – \frac{a_s}{2}, \mu + \frac{a_s}{2}] \\
0 & \text{otherwise}
\end{cases} $$
$$ \Delta I_x = \begin{cases}
\frac{1}{12} \Delta W_x h^3 + \frac{A_x \Delta A_x (h_x – \frac{h}{2})^2}{A_x – \Delta A_x} & \text{if } x \in [\mu – \frac{a_s}{2}, \mu + \frac{a_s}{2}] \\
0 & \text{otherwise}
\end{cases} $$
where \(\mu\) is the distance from the root to the pit center. The updated parameters \(I_x’ = I_x – \Delta I_x\), \(A_x’ = A_x – \Delta A_x\), and \(W_x’ = W – \Delta W_x\) are used in the stiffness calculations. This approach allows us to simulate the effects of pitting on the mesh stiffness of spur gears.
Simulation Analysis
We conduct simulations for spur gears with parameters as listed in Table 1. The pinion is the driving gear with a speed of 1800 rpm. We analyze normal spur gears, as well as those with cracks and pitting faults.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 23 | 84 |
| Module (mm) | 2 | 2 |
| Pressure Angle (°) | 20 | 20 |
| Tooth Width (mm) | 20 | 20 |
| Mass (kg) | 0.22 | 1.9 |
| Moment of Inertia (kg·m²) | 4.86 × 10⁻⁵ | 3.51 × 10⁻³ |
For normal spur gears, the time-varying mesh stiffness exhibits periodic variations due to single and double tooth contact regions, as shown in Figure 1. The stiffness remains consistent across cycles in the absence of faults.
For cracked spur gears, we simulate cracks with angles of 45° and depths of 1 mm, 1.5 mm, and 2 mm. The mesh stiffness decreases with increasing crack depth, as illustrated in Figure 2. Similarly, for pitted spur gears, we consider pits with depths of 1 mm and varying lengths and widths. The mesh stiffness reduction is more pronounced for larger pits, as depicted in Figures 3 and 4.
The dynamic responses, including displacement and acceleration, are computed using the Runge-Kutta method. Normal spur gears show stable amplitude responses, while faulty spur gears exhibit increased amplitudes and periodic impulses. The frequency domain analysis reveals characteristic features such as mesh frequency harmonics and sidebands due to fault-induced modulations.
System Model Modification
To enhance the accuracy of our dynamic model for spur gears, we modify the support stiffness and damping parameters based on experimental data. The gear shaft support is modeled as a two-degree-of-freedom system in the x and y directions. The equations of motion are:
$$ m \ddot{x} + 2(c_{xx} \dot{x} + c_{xy} \dot{y} + k_{xx} x + k_{xy} y) = f_x(t) $$
$$ m \ddot{y} + 2(c_{yx} \dot{x} + c_{yy} \dot{y} + k_{yx} x + k_{yy} y) = f_y(t) $$
Assuming harmonic excitation and response, the acceleration response can be expressed as:
$$ \begin{bmatrix} X \\ Y \end{bmatrix} = \frac{1}{H} \begin{bmatrix}
m – \frac{2j c_{yy}}{\omega} – \frac{2 k_{yy}}{\omega^2} & \frac{2 k_{xy}}{\omega^2} + \frac{2j c_{xy}}{\omega} \\
\frac{2 k_{yx}}{\omega^2} + \frac{2j c_{yx}}{\omega} & m – \frac{2j c_{xx}}{\omega} – \frac{2 k_{xx}}{\omega^2}
\end{bmatrix} \begin{bmatrix} F_x \\ F_y \end{bmatrix} $$
where \(H\) is a determinant derived from the system matrices. The frequency response functions (FRFs) \(R_{ij}(\omega)\) relate input forces to output responses. We define an objective function to minimize the difference between simulated and experimental FRFs:
$$ \epsilon = \min \sum_{i=x,y} \sum_{j=x,y} \sum_{\omega} | R_{ij}(\omega)_s – R_{ij}(\omega)_e | $$
By optimizing the parameters \(k_{ij}\) and \(c_{ij}\), we obtain modified support stiffness and damping values. For example, the equivalent support stiffness for the pinion in the x-direction is \(k_{px} = k_{xx} + k_{xy}\). This modification ensures that the dynamic model of spur gears closely matches the experimental setup.
Experimental Verification
We validate our model using a gearbox test rig with spur gears exhibiting cracks and pitting faults. Vibration acceleration signals are acquired in the radial direction at a sampling frequency of 51,200 Hz over 30 seconds. The signals are processed using time synchronous averaging (TSA) to reduce noise.
For normal spur gears, the simulation and experimental signals show stable amplitudes in the time domain and dominant mesh frequency components in the frequency domain, with no significant sidebands. For cracked spur gears, both simulation and experiment reveal periodic impulses in the time domain, with intervals corresponding to the faulted gear’s rotational period. In the frequency domain, sidebands around the mesh frequencies indicate modulation due to the crack. For pitted spur gears, similar characteristics are observed, though with less pronounced modulations due to the smaller fault size.
We further simulate varying degrees of cracks and pitting in spur gears. The results demonstrate that the dynamic responses differ significantly with fault severity, providing a basis for fault quantification. Tables 2 and 3 summarize the key features observed in the simulations for different fault types and degrees.
| Crack Depth (mm) | Time-Domain Feature | Frequency-Domain Feature |
|---|---|---|
| 1.0 | Moderate periodic impulses | Weak sidebands around mesh frequencies |
| 1.5 | Strong periodic impulses | Pronounced sidebands |
| 2.0 | Severe periodic impulses | Wide sideband distribution |
| Pit Size (mm²) | Time-Domain Feature | Frequency-Domain Feature |
|---|---|---|
| 0.5 × 4 | Slight amplitude modulation | Minor sidebands |
| 0.7 × 4 | Noticeable amplitude modulation | Moderate sidebands |
| 0.9 × 4 | Significant amplitude modulation | Distinct sidebands |
The close agreement between simulation and experimental results confirms the accuracy of our dynamic model for spur gears with faults. This model can be used to generate extensive datasets for fault diagnosis, supporting the development of statistical indicators for gear health monitoring.
Conclusion
In this work, we develop a comprehensive dynamic model for spur gears that incorporates time-varying mesh stiffness calculations for normal and faulty conditions. The model is modified using experimental data to improve its accuracy. Simulations and experiments demonstrate that the model effectively captures the dynamic responses of spur gears with cracks and pitting faults. The results show distinct time-domain and frequency-domain features for different fault types and severities, enabling reliable fault identification. This approach provides a valuable tool for generating fault diagnosis databases and enhancing the reliability of spur gear systems in practical applications.