Gear transmission systems are widely applied across numerous industrial sectors, including railway locomotives, automotive vehicles, aerospace systems, wind turbines, and marine vessels. During actual operation, gear pairs are susceptible to various operational factors such as load variations, rotational speed fluctuations, and temperature gradients. The interactions among these factors can frequently induce compound faults. Once compound faults develop in gears, the stability, economic efficiency, and safety of the entire mechanical system become significantly compromised. However, the majority of existing research has predominantly focused on the influence of single gear faults on the Time-Varying Meshing Stiffness (TVMS) and dynamic responses. There remains a notable gap in the analytical calculation and dynamic modeling of TVMS for spur gears subjected to compound faults. Consequently, this thesis integrates theoretical derivations, simulation analyses, and experimental validations to systematically investigate the influence of compound faults, including cracks and spalling, on the TVMS and the associated dynamic characteristics of spur gears. The findings of this research are expected to provide a scientific foundation for the design, manufacturing, and maintenance of gear systems.
1. Research Background and Significance
Over extended operation periods, gear teeth frequently suffer from damage such as tooth root cracks, tooth surface spalling, and tooth breakage. These failures directly lead to gear transmission system malfunctions, severely reducing equipment safety and reliability, and in some cases resulting in catastrophic accidents. Gear transmission systems for spur gears often experience multiple interacting factors, making it increasingly complex to identify and analyze the specific modes of failure. For instance, material defects, poor lubrication, and manufacturing errors could combine with load cycles to cause cracks and pitting simultaneously. Understanding these compound faults is of great importance for early fault diagnosis and preventive maintenance.
Several methodologies exist for calculating TVMS, including the analytical energy method, the finite element method, and experimental approaches. Among these, the energy method has been widely adopted due to its accurate representation of gear tooth deformation and its computational efficiency. This thesis adopts an energy-based approach to compute TVMS for healthy spur gears and extends it to account for various fault types, specifically focusing on compound faults.
To enhance the accuracy of the TVMS calculations, a structural coupling effect has been incorporated. Traditional models often neglect the mutual influence of multiple teeth meshing simultaneously, potentially overestimating the mesh stiffness. This study employs Muskhelishvili’s elastic ring theory to model the gear body and quantify the foundation stiffness more accurately, thereby providing a robust baseline for fault analysis.
2. Healthy Spur Gears: TVMS Calculation and Dynamic Modeling
The TVMS of a healthy spur gear pair was first established using an improved energy method. In this model, the gear tooth is idealized as a non-uniform cantilever beam fixed on the root circle. The total meshing stiffness of a gear pair is expressed as:
$$k = \sum_{i=1}^{N} \frac{1}{1/k_{h,i} + 1/k_{t,i}^{p}+ 1/k_{t,i}^{g} + 1/k_{f,i}^{p} + 1/k_{f,i}^{g}}$$
where \( k \) is the total meshing stiffness, \( k_h \) is the Hertzian contact stiffness, \( k_t \) denotes the tooth stiffness, \( k_f \) represents the gear body foundation stiffness, and \( N \) is the number of simultaneously meshing tooth pairs (\( N=1 \) for single-tooth engagement and \( N=2 \) for double-tooth engagement in standard spur gears). The superscripts \( p \) and \( g \) refers to the driving gear and the driven gear, respectively.
The tooth stiffness \( k_t \) incorporates the bending stiffness \( k_b \), shear stiffness \( k_s \), and axial compressive stiffness \( k_a \) as:
$$1/k_t = 1/k_b + 1/k_s + 1/k_a$$
Detailed expressions for these stiffness components were derived by integrating over the tooth profile. The coordinate system and cantilever beam model of the gear tooth are essential for this calculation. The expressions involve the area \( A_x \) and the area moment of inertia \( I_x \) of the tooth cross-section, along with the geometric parameters of the involute and trochoid curves.
To verify the analytical model, a finite element model of the gear pair was constructed, as shown in Figure 2.5. The finite element method used a five-tooth model to ensure computational accuracy while reducing time. The angular displacement of the driving gear was obtained under an applied torque, and the TVMS was calculated using:
$$k = T / (r_{bp}^2 \Delta \theta)$$
where \( T \) is the input torque, \( r_{bp} \) is the base circle radius of the driving gear, and \( \Delta \theta \) is the angular displacement. The gear parameters for the simulation are detailed in Table 1.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of teeth | 40 | 40 |
| Module (mm) | 3 | 3 |
| Face width (mm) | 20 | 20 |
| Pressure angle (°) | 20 | 20 |
| Mass (kg) | 1.226 | 1.226 |
| Moment of inertia (kg·m²) | 2.577×10⁻³ | 2.577×10⁻³ |
| Young’s modulus (GPa) | 206 | 206 |
| Hub bore radius (mm) | 20 | 20 |
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of teeth | 26 | 31 |
| Module (mm) | 3 | 3 |
| Face width (mm) | 25 | 25 |
| Pressure angle (°) | 20 | 20 |
| Mass (kg) | 0.988 | 1.02 |
| Moment of inertia (kg·m²) | 1.81×10⁻³ | 1.87×10⁻³ |
| Young’s modulus (GPa) | 206 | 206 |
| Hub bore radius (mm) | 20 | 20 |
For the dynamic analysis, a six-degree-of-freedom (6-DOF) lumped parameter model was developed, capturing both translational and torsional vibrations. The differential equations of motion for the spur gear system are given by:
$$m_p \ddot{x}_p + c_{px} \dot{x}_p + k_{px} x_p = -k \delta \cos\alpha_{12} – c \dot{\delta} \cos\alpha_{12}$$
$$m_p \ddot{y}_p + c_{py} \dot{y}_p + k_{py} y_p = -k \delta \sin\alpha_{12} – c \dot{\delta} \sin\alpha_{12}$$
$$I_p \ddot{\theta}_p + c r_{bp} \dot{\delta} + k r_{bp} \delta = T_p$$
$$m_g \ddot{x}_g + c_{gx} \dot{x}_g + k_{gx} x_g = k \delta \cos\alpha_{12} + c \dot{\delta} \cos\alpha_{12}$$
$$m_g \ddot{y}_g + c_{gy} \dot{y}_g + k_{gy} y_g = k \delta \sin\alpha_{12} + c \dot{\delta} \sin\alpha_{12}$$
$$I_g \ddot{\theta}_g + c r_{bg} \dot{\delta} + k r_{bg} \delta = -T_g$$
In these equations, \( m_p \) and \( m_g \) are the masses of the driving and driven gears, \( I_p \) and \( I_g \) are their moments of inertia, \( k_{px}, k_{py}, k_{gx}, k_{gy} \) denote the bearing stiffness in the \( x \) and \( y \) directions, and \( c \) represents the meshing damping. The relative displacement \( \delta \) along the line of action is expressed as:
$$\delta = (x_p – x_g)\cos\alpha_{12} + (y_p – y_g)\sin\alpha_{12} + r_{bp}\theta_p – r_{bg}\theta_g – e(t)$$
where \( e(t) \) is the composite transmission error, modeled as a harmonic function of the meshing frequency. The experimental setup consisted of a controller, a three-phase asynchronous motor, a test gearbox, and a magnetic powder brake. The vibration acceleration was measured using a data acquisition system equipped with an accelerometer, operating at a sampling frequency of 20 kHz. This experimental configuration served as a crucial validation tool for the subsequent fault gear studies.
3. TVMS Calculation and Dynamic Analysis of Spur Gears with Multiple Cracks
In many practical scenarios, multiple cracks can be present on the spur gears, either on adjacent teeth, non-adjacent teeth, or even on the same tooth. To accurately evaluate the TVMS of such cracked spur gears, a sliicing method was employed. The gear tooth is divided into \( n \) thin slices along its face width, and the stiffness of each slice is computed separately. The total stiffness is then assembled from these slices:
$$k = \sum_{s=1}^{n} k_{slice,s}$$
When a crack is introduced, it alters the effective cross-sectional area and the area moment of inertia of the tooth at the cracked location. The crack geometry is parameterized by its depth \( q(z) \), angle \( \alpha_c(z) \), and position \( d_{cs}(z) \). The effective area \( A_x \) and inertia \( I_x \) are subsequently adjusted to account for the crack. For a through-crack reaching the gear body, the foundation stiffness calculation is also modified using adjusted geometric parameters.
To validate the proposed model, adjacent cracked teeth on the driving gear were analyzed using both the analytical method and the finite element method. The comparison, shown in Figure 3.4, indicates that both methods produce nearly identical results for various crack depths. The TVMS reduction was found to be most significant when both adjacent cracked teeth are simultaneously engaged, such as during the second meshing cycle in the studied configuration.
Furthermore, the influence of crack position on adjacent teeth was investigated. The results revealed that the TVMS varies not only with the crack depth but also with the crack location \( d_{cs} \). If a crack enters the gear body, the foundation stiffness reduction can be substantial, leading to a sharp decrease in TVMS. Conversely, cracks located farther from the tooth root have a relatively smaller influence.
When two non-adjacent teeth on the same gear contain cracks, their effects on TVMS were observed to be independent. Since the two cracked teeth do not engage simultaneously in a gear pair with a contact ratio between 1 and 2, the total stiffness reduction is simply the sum of the individual crack effects. This observation confirms the localized and independent nature of cracks on non-adjacent teeth.
The study further investigated the scenario of multiple cracks on a single tooth. Because cracks can occur at various positions, including the tooth root and the tooth flank, their failure regions may not overlap. In such cases, the tooth stiffness is calculated using the cracked inertia and area values corresponding to each crack. If the failure regions partially or fully overlap, the combined effective geometry must be considered. A parametric study involving six different crack configurations, as listed in Table 3.1, demonstrated that the TVMS is sensitive to the relative positions of multiple cracks on the same tooth. Failure region determination is crucial for accurate TVMS calculation.
The analysis was further extended to consider cracks on both the driving and driven spur gears, leading to eight distinct meshing cases. These cases, including scenarios such as a cracked gear meshing with a healthy gear on either the driving or driven side, were fully enumerated. The corresponding TVMS components were calculated and analyzed. These eight cases are: Hg-Hp-Hg-Hp, Cg-Cp-Hg-Hp, Hg-Hp-Hg-Cp, Hg-Hp-Cg-Hp, Cg-Hp-Hg-Cp, Hg-Cp-Cg-Hp, Cg-Hp-Hg-Hp-Hg-Cp, and Hg-Cp-Hg-Hp-Cg-Hp. The notation H denotes a healthy tooth, C denotes a cracked tooth, and the subscripts p and g refer to the driving and driven gears, respectively. The results highlight that the specific combination of crack locations on both gears dictates the overall TVMS periodic variation.
The dynamic responses of spur gears with multiple cracks were then computed. For adjacent tooth cracks, the time-domain acceleration signals displayed distinct periodic impulses caused by stiffness drops at each cracked tooth. The time interval between these impulses corresponds to the rotational period of the gear, which is approximately 0.06 seconds in the experimental setup. Both simulated and experimental FFT spectra exhibited prominent peaks at the mesh frequency and its harmonics, surrounded by dense sidebands attributed to the rotational frequency. The sidebands became broader and more numerous with an increase in the number of cracks. Moreover, the time-domain response differences between a gear with multiple cracked teeth and a gear with multiple cracks on a single tooth were highlighted. Generally, a multi-tooth crack results in multiple shock pulses, while a single-tooth multi-crack manifests as one large shock pulse where several cracks coalesce to cause significant local stiffness reduction.
4. TVMS Calculation and Dynamic Analysis of Spur Gears with Multiple Spalls
Spalling is another common gear surface failure that causes material loss from the tooth flank. In this study, a shape-independent model for spalls was adopted to account for the irregular geometry often observed in practice. The effects of spalling on the TVMS of spur gears were investigated comprehensively. The spall parameters include the length \( L_s \), depth \( h_s \), and the center offset \( p \) relative to the center plane of the gear face. A critical feature considered in this analysis is the torsion induced by non-symmetric spalls. Because the spall area is not centered, the stiffness of the contact region becomes non-uniform, generating an additional torsional stiffness \( k_{\tau} \). The effective tooth stiffness for a spalled gear was thus expressed as:
$$1 / k_t = 1 / k_a + 1 / k_b + 1 / k_s + 1 / k_{\tau}$$
The detailed expressions for these stiffness components were derived using angular displacement as the integration variable. The defect ratio was introduced to adapt the stiffness calculation to arbitrary spall geometries, such as elliptical or irregular shapes. These ratios, including the defect length ratio \( C_{Lsx} \), the defect depth ratio \( C_{hsx} \), and the defect offset ratio \( C_{psx} \), were used to modify the area and inertia calculation of the gear tooth. The modified formulas for the cross-sectional area and moment of inertia are:
$$A_{x1} = 2 y_1 L, \quad A_{x2} = 2 y_2 L, \quad A_{x3} = 2 y_2 L (1 – C_{hsx} C_{Lsx}), \quad A_{x4} = 2 y_2 L$$
$$I_{x1} = \frac{2}{3} y_1^3 L, \quad I_{x2} = \frac{2}{3} y_2^3 L, \quad I_{x3} = \frac{2}{3} (y_2 – C_{hsx} C_{Lsx})^3 L K_c, \quad I_{x4} = I_{x2}$$
For validation, adjacent spalled teeth on the driving gear were analyzed. The results from the analytical model and finite element model showed excellent agreement, confirming the accuracy of the proposed method. The TVMS showed a clear drop when the spall entered the meshing zone. Moreover, the effect of a spall on TVMS is typically less pronounced than that of a crack, as spall primarily affects the contact stiffness and local tooth geometry without necessarily compromising the tooth root strength in the same manner as a crack.
Using the model, the TVMS for non-adjacent spalled teeth on the same gear was computed. Similar to the crack analysis, each spall independently reduces the TVMS during its respective meshing cycle, and the stiffness returns to the healthy value once the spall leaves the contact zone.
The investigation was extended to cases with multiple spalls on a single tooth. The location of spalls relative to each other—whether on the same side or opposite sides of the tooth flank—significantly influenced the resultant TVMS reduction. Notably, when spalls were positioned on the same side, the torsion effect was amplified, leading to a larger reduction in TVMS compared to spalls on opposite sides, where the induced torsion tended to counteract. For example, when a spall of length 5mm and offset 3mm was placed alongside another spall of length 4mm and offset 2mm on the same flank, the stiffness reduction was markedly higher than when the spalls were placed on opposite flanks.
Moreover, for spur gears where both the driving and driven gears are affected by spalling, eight distinct meshing cases were considered, analogous to the multiple crack analysis. These cases encompass various combinations of healthy and spalled teeth on either gear and were systematically analyzed to understand their impact on the total TVMS.
Dynamic simulations were performed for adjacent tooth spalling. The time-domain responses exhibited three distinct impulse trains within one rotational period, corresponding to the three spalled teeth. The amplitude of each impulse was directly correlated with the spall dimensions and its location. Larger spalls with greater depths and more central positions on the tooth flank caused more significant vibration impulses. The FFT analysis displayed peaks at the mesh frequency and its harmonics. Importantly, additional peaks corresponding to the rotational frequency of the faulty gear were observed, indicating amplitude modulation of the meshing frequency. The dynamic response of spur gears with spalls distributed across multiple teeth was also examined. When multiple spalls are present on a single tooth, the vibration signature is dominated by a single major pulse, whereas multiple spalled teeth produce multiple pulses, each associated with a particular spall. This pattern is similar to that observed for multiple cracks and highlights the importance of understanding the spatial distribution of faults.
5. TVMS Calculation and Dynamic Analysis of Spur Gears with Crack-Spalling Compound Faults
In realistic gearbox operations, cracks and spalls often coexist on the same gear or even on the same tooth. The interaction between these two failure modes creates a more complex fault scenario that poses a substantial challenge to diagnostic techniques. This thesis devoted a dedicated chapter to the calculation of TVMS for spur gears with crack-spalling compound faults.
The compound fault model was established by combining the previously derived crack and spall models. The total stiffness of a tooth slice was calculated by sequentially applying the reductions in cross-sectional area, moment of inertia, and the additional torsional stiffness induced by the spall. The model was first validated against the finite element method for the case of adjacent teeth, one with a crack and the other with a spall. Both the analytical and finite element results showed a good match, as indicated in Figure 5.2. It was found that the order of engagement significantly affects the TVMS. When the cracked tooth meshes first, the stiffness drop is larger than when the spalled tooth meshes first.
The analysis was further refined for the case where a crack and a spall are present on the same tooth. The relative positions of the crack and the spall—whether the crack is located before, within, or after the spall region—were found to be critical. If the crack initiates before the spalled region, the stiffness reduction is maximal. In contrast, if the crack is located after the spall, the two failures act independently, and the total stiffness loss is roughly additive. These findings illustrate the strong dependence of compound fault effects on the spatial arrangement of the failures.
In addition, the case of a crack on one gear and a spall on the other gear was explored, resulting in eight distinct meshing cases, including Cg-Sp-Hg-Hp, Hg-Hp-Hg-Sp, Hg-Hp-Cg-Hp, Cg-Hp-Hg-Sp, Hg-Sp-Cg-Hp, Cg-Hp-Hg-Hp-Hg-Sp, and Hg-Sp-Hg-Hp-Cg-Hp. Each case dictated a unique pattern of stiffness variation over the meshing cycles. The TVMS for these cases, as shown in Figure 5.6, provided valuable insights into the combined effects of cracks and spalls on the transmission characteristics of spur gears.
The dynamic response of spur gears with crack-spalling compound faults was then analyzed. For the case of a single tooth with both a crack and a spall, the time-domain response showed a pronounced periodic impulse occurring once per rotation. Due to the combined effects of the two faults, a distinct impulse peak was observed. The FFT spectrum displayed a high peak at the mesh frequency with a dense sideband structure. The sidebands were observed to be denser than those observed for a single-crack defect. This increased complexity in the frequency domain arises from the interaction of the two failure modes, making compound faults more dectectable.
For adjacent teeth experiencing a crack-spalling compound fault, the time-domain response showed two consecutive impulses with a very short time interval between them. This interval corresponds to the tooth-to-tooth meshing period, which is approximately 0.0023 seconds in the experimental setup. The corresponding FFT analysis revealed multiple peaks at the mesh frequency and its harmonics, with noticeably more complex sidebands compared to single-fault cases. The experimental results confirmed these observations, and a very good agreement was obtained between the simulated and measured signals. This consistency validates the effectiveness of the developed analytical tool for modeling compound faults in spur gears.
6. Conclusion and Future Perspectives
This thesis presented a comprehensive investigation into the TVMS calculation and dynamic analysis of spur gears affected by compound faults, including multiple cracks, multiple spalls, and combined crack-spalling failures. The key contributions and findings of this work are summarized as follows:
1) The proposed analytical method, based on an improved energy method incorporating structural coupling, accurately calculates the TVMS of spur gears under various compound fault conditions. The validity of the method was verified through extensive comparisons with finite element analyses.
2) Compound faults significantly reduce the TVMS of spur gears. The presence of multiple cracks or spalls can lead to a more substantial stiffness reduction than a single fault, especially when the faulty zones are simultaneously engaged. Non-adjacent faulty teeth influence the stiffness independently. However, when faults are located on the same tooth, their interactions greatly influence the overall stiffness degradation pattern. Correct determination of the effective failure region is essential for accurate TVMS calculation.
3) The dynamic behaviors of spur gears with compound faults reveal complex characteristics that differ from those with single faults. Multi-tooth faults cause repeated vibration impulses, while single-tooth faults with multiple defects produce a single dominant impulse. The FFT spectra of compound fault gears exhibit more prominent peaks and broader, denser sidebands, providing an identifiable signature for fault diagnosis.
4) The research further emphasizes the importance of considering the spatial relationship between cracks and spalls. The relative positions and engagement sequence of these faults significantly impact TVMS and the resulting vibration characteristics. Understanding these complex interactions is crucial for precise health monitoring and maintenance of gear transmission systems.
Future research should focus on incorporating the effects of friction and variable friction coefficients along the tooth profile. Moreover, investigating the coupling effects of multiple faults in more complex gear systems, such as planetary gear trains, is an important direction for further work. Additionally, the integration of advanced signal processing methods with the developed analytical models could significantly improve the accuracy and efficiency of compound fault diagnosis in spur gears.