My research focuses on the vibration characteristics of a two-stage herringbone gear transmission system under the combined influence of tooth crack and pitting faults. Herringbone gears are widely used in high-speed and heavy-load applications such as marine propulsion, wind turbines, and aerospace transmissions because of their high load capacity, large overlap ratio, and balanced axial forces. However, under harsh operating conditions such as insufficient lubrication, alternating contact stresses, and manufacturing or assembly errors, the tooth surfaces and roots are prone to develop pitting and cracks. These faults rarely occur independently. In practice, crack-pitting coupling is common, and it significantly alters the time-varying mesh stiffness (TVMS), thereby affecting the dynamic response of the whole gear system. My work establishes a comprehensive analytical model for the TVMS of herringbone gears with crack-pitting coupling, constructs a 48-degree-of-freedom (DOF) dynamic model of the two-stage transmission, and validates the theoretical findings through finite element analysis and vibration experiments.
Introduction and Objectives
The herringbone gear system I investigated consists of a two-stage reduction gearbox. The first stage has a pinion with 25 teeth and a gear with 60 teeth; the second stage has a pinion with 25 teeth and a gear with 75 teeth. All gears have a module of 2 mm, a pressure angle of 20°, and a helix angle of 15°. The total face width of each herringbone gear is 43 mm, with a groove width of 3 mm. The gears operate at an input speed of 1800 r/min and a torque of 200 N·m. My objective is to quantify how different severities of crack-pitting coupling affect the TVMS and the resulting vibration responses in both the time and frequency domains.
Analytical Model for Time-Varying Mesh Stiffness of Herringbone Gears
I modeled each tooth of the herringbone gear as a cantilever beam with a variable cross-section. Because the herringbone gear has a helix angle, the contact line length varies with time and position along the face width. I used the slicing method, dividing the gear into thin slices along the axial direction. Each slice behaves approximately as a spur gear tooth. The total mesh stiffness is obtained by integrating the stiffness contributions of all slices. The stiffness components considered are the Hertzian contact stiffness, bending stiffness, shear stiffness, axial compressive stiffness, and gear foundation stiffness. In addition, I included the effect of the central groove (relief groove) on the foundation stiffness, which is often neglected in traditional models.
For a single slice, the mesh stiffness is given by the reciprocal sum of the individual stiffness components:
$$ \frac{1}{k_{m,i}} = \frac{1}{k_{h,i}} + \frac{1}{k_{b1,i}} + \frac{1}{k_{s1,i}} + \frac{1}{k_{a1,i}} + \frac{1}{k_{f1,i}} + \frac{1}{k_{b2,i}} + \frac{1}{k_{s2,i}} + \frac{1}{k_{a2,i}} + \frac{1}{k_{f2,i}} $$
where subscripts 1 and 2 denote the pinion and the gear, respectively. The total TVMS is the sum over all slices involved in meshing:
$$ k_m = \sum_{j} k_{m,j} $$
The Hertzian contact stiffness is calculated using the nonlinear contact model:
$$ k_h = \frac{1.275 E^{*0.9} F^{0.1}}{l^{0.8}} $$
with
$$ \frac{1}{E^{*}} = \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2} $$
where \(E_1, E_2\) are Young’s moduli, \(\nu_1, \nu_2\) are Poisson’s ratios, \(F\) is the normal contact force, and \(l\) is the instantaneous contact line length.
The bending stiffness \(k_b\), shear stiffness \(k_s\), and axial compressive stiffness \(k_a\) are derived from the potential energy method. For a cantilever beam model, the stored energies are expressed as integrals along the tooth height direction. After deriving the expressions, I obtained closed-form integrals that depend on the tooth geometry and the position of the contact point. For the case where the base circle radius \(R_b\) is larger than the root circle radius \(R_r\), the stiffness components are:
$$ k_b = \int_{\alpha_1}^{\alpha_2} \frac{3\left[1 – \cos\alpha_1\cos\alpha – \sin\alpha_1(\alpha_2-\alpha)\sin\alpha\right]^2(\alpha_2-\alpha)\cos\alpha}{2E L \left( \frac{\sin\alpha + (\alpha_2-\alpha)\cos\alpha}{\cos\alpha} \right)^3} \, d\alpha $$
and similarly for \(k_s\) and \(k_a\). The limits of integration are determined by the geometric angles \(\alpha_1\), \(\alpha_2\), and \(\alpha_3\). I derived two distinct formulations depending on whether \(R_b\) is greater or less than \(R_r\).
The foundation stiffness \(k_f\) consists of two parts: the radial foundation stiffness \(k_{tf}\) and the axial foundation stiffness \(k_{af}\) due to the central groove. The radial part is computed using the classical formula for gear body-induced compliance:
$$ k_{tf} = \frac{\cos^2\alpha_1}{E L \left[ L^{*}\left(\frac{u_f}{S_f}\right)^2 + M^{*}\left(\frac{u_f}{S_f}\right) + P^{*}(1 + Q^{*}\tan^2\alpha_1) \right]} $$
where \(L^{*}, M^{*}, P^{*}, Q^{*}\) are dimensionless coefficients, \(u_f\) is the distance from the tooth root to the bore, and \(S_f\) is the tooth root thickness.
The axial foundation stiffness \(k_{af}\) accounts for the bending deformation of the gear body due to the symmetric herringbone loads. I calculated the bending energy of the gear body as:
$$ U_{af} = \int_{R_0}^{R_r} \frac{\left( \sum_{i=1}^{N} F_i \sin\beta \cos\alpha \left( x – x_i \right) \right)^2}{2 E I_{af}(x)} \, dx $$
where \(I_{af}(x)\) is the area moment of inertia of the gear body at axial position \(x\), \(R_0\) is the shaft bore radius, and \(R_r\) is the root radius. The total foundation stiffness is then:
$$ k_f = \frac{k_{tf} k_{af}}{k_{tf} + k_{af}} $$
Thus, the TVMS of a healthy herringbone gear pair can be accurately computed by considering all these contributions.
Modeling of Pitting on Herringbone Gears
Pitting is a contact fatigue phenomenon that removes material from the tooth surface, typically near the pitch line. I represented each pit as a cylindrical depression with a circular base on the tooth surface. The parameters defining a pit are its radius \(R_{sp}\), depth \(d_{sp}\), and the axial position \(y_{sp}\) along the face width. Multiple pits with different sizes and positions create various pitting severity levels. I defined four pitting levels: slight (3 pits of radius 0.3 mm), mild (6 pits of radius 0.3 mm), moderate (6 pits of radius 0.4 mm), and severe (12 pits of radius 0.4 mm). The pitting affects the TVMS in two ways: it reduces the effective cross-sectional area and the second moment of area of the tooth, and it reduces the local contact line length.
For a slice containing a pit, the reduced cross-sectional area \(A’_x\) and second moment of area \(I’_x\) are computed by subtracting the pit geometry. The bending stiffness of a pitted tooth is then obtained by modifying the integrals. For example, the bending stiffness for a pitted tooth becomes:
$$ k_{db} = \int_{\alpha_1}^{\alpha_{spi}} \frac{3\left[1 – \cos\alpha_1\cos\alpha – \sin\alpha_1(\alpha_2-\alpha)\sin\alpha\right]^2(\alpha_2-\alpha)\cos\alpha}{2 E L \left( \frac{\sin\alpha + (\alpha_2-\alpha)\cos\alpha}{\cos\alpha} \right)^3} \, d\alpha + \int_{\alpha_{spi}}^{\alpha_{spo}} \frac{3\left[1 – \cos\alpha_1\cos\alpha – \sin\alpha_1(\alpha_2-\alpha)\sin\alpha\right]^2(\alpha_2-\alpha)\cos\alpha}{2 E L \left( \frac{\sin\alpha + (\alpha_2-\alpha)\cos\alpha}{\cos\alpha} – \frac{d_{sp}}{R_{sp}} \right)^3} \, d\alpha + \cdots $$
where \(\alpha_{spi}\) and \(\alpha_{spo}\) are the angular positions corresponding to the start and end of the pit along the profile direction. The reduced contact line length due to a pit at any time is given by:
$$ l_d(t) = l_0(t) – \Delta l_x $$
with \(\Delta l_x\) being the length of the pit crossing the contact line. For a circular pit of radius \(R_{sp}\), the chord length along the contact line at a distance \(u\) from the pit center is:
$$ \Delta l_x = 2 \sqrt{R_{sp}^2 – u^2} \quad \text{if } u \in [-R_{sp}, R_{sp}] $$
Hence the instantaneous contact line length for the pitted gear is reduced, which directly lowers the Hertzian stiffness and also affects the load sharing among multiple tooth pairs.
Modeling of Crack-Pitting Coupling
Cracks often initiate at the tooth root fillet due to stress concentration. I modeled a crack as a straight line propagating from the root at an angle \(\xi\) relative to the tooth centerline. The crack depth is \(q_i\). I defined four crack levels: slight (10% of the maximum crack depth), mild (30%), moderate (50%), and severe (70%). When both crack and pitting are present, the effective tooth cross-section is reduced by both defects. Depending on the relative positions of the crack tip and the pitting region, I identified four regimes:
- Regime 1: The crack has not reached the pitting region. The stiffness reduction is due to both crack and pitting acting independently.
- Regime 2: The crack tip lies within the pitting region. The crack and pitting overlap, leading to a combined reduction.
- Regime 3: The crack has passed the pitting region and dominates the cross-section reduction. The pitting only reduces the contact line length.
- Regime 4: The crack is so severe that it reduces the effective tooth thickness significantly, and the pitting has negligible additional effect on the cross-section.
I derived the effective cross-sectional area \(A_x”\) and second moment of area \(I_x”\) for each regime. For example, in Regime 2 where \(h_a \ge h_0\) and \(\alpha_{spi} < \alpha_l \le \alpha_{spo}\), the cross-section at a section located at angle \(\alpha\) is expressed as:
$$ A_x” = \begin{cases} (h_x + h_{x,l}) \, dy & \alpha \le \alpha_l \\ (h_x + d_{sp} – h_{x,l}) \, dy & \alpha_l < \alpha \le \alpha_{spo} \\ h_x \, dy & \alpha > \alpha_{spo} \end{cases} $$
Similarly, the second moment of area becomes:
$$ I_x” = \begin{cases} \frac{1}{12} (h_x + h_{x,l})^3 \, dy & \alpha \le \alpha_l \\ \frac{1}{12} (h_x + d_{sp} – h_{x,l})^3 \, dy & \alpha_l < \alpha \le \alpha_{spo} \\ \frac{1}{12} (h_x)^3 \, dy & \alpha > \alpha_{spo} \end{cases} $$
where \(h_{x,l}\) is the distance from the crack tip to the tooth centerline at the section. These expressions are substituted into the energy integrals to obtain the coupled bending and shear stiffness. The axial compressive stiffness and Hertzian stiffness depend only on the pitting because the crack does not change the tooth profile. The contact line length reduction remains as in the pitting case.
For a gear with \(N\) pits, the Hertzian stiffness is modified as:
$$ k_{h, coupled} = \frac{1.275 E^{*0.9} F^{0.1}}{(l – N \Delta l_x)^{0.8}} $$
By combining all the stiffness contributions, I obtained the total mesh stiffness for a herringbone gear pair with crack-pitting coupling at any instant of meshing.
Finite Element Validation
To verify the analytical TVMS model, I built finite element models of the herringbone gear pair using ANSYS. The gears were meshed with tetrahedral and hexahedral elements, and the contact regions were refined, especially around the pits and crack. The model constrained the inner bore of the gear while allowing rotation. A torque of 200 N·m was applied to the driven gear, and the pinion was given a constant rotational speed of 10 rad/s. The contact was set as face-to-face with a friction coefficient of 0.05.
I created four coupled fault models with the following severity combinations:
| Fault level | Crack depth ratio | Pitting configuration |
|---|---|---|
| Slight | 10% | 3 pits, radius 0.3 mm |
| Mild | 30% | 6 pits, radius 0.3 mm |
| Moderate | 50% | 6 pits, radius 0.4 mm |
| Severe | 70% | 12 pits, radius 0.4 mm |
The finite element simulation gave the nodal forces and deformations along the line of action. The mesh stiffness was extracted by dividing the total normal force by the total deformation. I compared the analytical results with the finite element results for both healthy and faulty gears. The comparison is shown in the following table for the mesh stiffness at a representative instant:
| Condition | Analytical TVMS (×10⁹ N/m) | FEM TVMS (×10⁹ N/m) | Error (%) |
|---|---|---|---|
| Healthy | 3.32 | 3.27 | 1.5 |
| Slight crack-pitting | 3.25 | 3.21 | 1.2 |
| Mild crack-pitting | 3.18 | 3.14 | 1.3 |
| Moderate crack-pitting | 3.02 | 2.98 | 1.3 |
| Severe crack-pitting | 2.78 | 2.71 | 2.5 |
The maximum error between the two methods was less than 2.5%, which confirms the validity of the analytical model for predicting the TVMS of herringbone gears with crack-pitting coupling.
Effects of Crack-Pitting Coupling on TVMS
Using the analytical model, I calculated the TVMS for a single-stage herringbone gear pair under different fault severities. The results showed that the groove width affects the foundation stiffness. Increasing the groove width from 0 to 15 mm increased the average mesh stiffness slightly, but the fluctuation amplitude remained almost unchanged.
For the crack-pitting coupling, the TVMS decreased as the severity increased. The reduction was not a simple superposition of the individual crack and pitting effects. The crack predominantly reduces the bending and shear stiffness by reducing the effective cross-section, while the pitting reduces the Hertzian stiffness and also reduces the bending and shear stiffness locally. The combined effect is often more severe than either fault alone.
I quantified the contribution of each fault in the coupling by calculating the percentage of stiffness reduction caused by the crack alone and by the pitting alone relative to the total reduction. The following table lists these percentages at the moment when the contact line passes the first pit row:
| Coupling condition | Pitting contribution (%) | Crack contribution (%) |
|---|---|---|
| Slight crack + slight pitting | 82.6 | 17.4 |
| Mild crack + slight pitting | 57.4 | 46.4 |
| Moderate crack + slight pitting | 35.6 | 65.5 |
| Severe crack + slight pitting | 21.0 | 85.4 |
| Slight crack + mild pitting | 91.1 | 8.9 |
| Mild crack + mild pitting | 74.3 | 27.6 |
| Moderate crack + mild pitting | 62.2 | 52.7 |
| Severe crack + mild pitting | 38.8 | 72.7 |
| Slight crack + moderate pitting | 93.6 | 6.4 |
| Mild crack + moderate pitting | 84.1 | 22.0 |
| Moderate crack + moderate pitting | 73.2 | 43.7 |
| Severe crack + moderate pitting | 49.1 | 65.0 |
| Slight crack + severe pitting | 83.9 | 16.1 |
| Mild crack + severe pitting | 56.4 | 49.3 |
| Moderate crack + severe pitting | 49.0 | 65.0 |
| Severe crack + severe pitting | 29.0 | 82.4 |
These results clearly show that at low crack levels, the pitting dominates the stiffness reduction, whereas at high crack levels, the crack becomes the dominant factor. However, the pitting still contributes by reducing the contact line length, which increases the compliance of the gear pair.
Dynamic Model of the Two-Stage Herringbone Gear System
I established a 48-DOF lumped parameter dynamic model of the two-stage herringbone gear transmission system. Each gear body has six degrees of freedom: three translational displacements along the x, y, and z axes, and three rotational displacements about these axes. The system includes four gears: the first-stage pinion (left and right hands), first-stage gear, second-stage pinion, and second-stage gear. Each hand of the herringbone gear is modeled as a separate body, and the two hands are connected by a torsionally flexible shaft element representing the central groove and the intermediate shaft.
The model incorporates the time-varying mesh stiffness of the gear pairs, the mesh damping, the static transmission error, the tooth friction, the bearing support stiffness and damping, the torsional stiffness of the shafts, and the gyroscopic effect. The mesh force is expressed as:
$$ F_{m,p}^j = k_{m,p}^j(t) \cdot \delta_{p}^j + c_{m,p}^j \cdot \dot{\delta}_{p}^j $$
where \(k_{m,p}^j(t)\) is the TVMS of the j-th gear pair (j = L or R for left and right hands), \(c_{m,p}^j\) is the mesh damping, and \(\delta_{p}^j\) is the relative displacement along the line of action. The relative displacement for the first-stage left hand is:
$$ \delta_L^1 = [x_{AL}^1 – x_{DL}^1]\cos\beta\cos\alpha + [y_{AL}^1 – y_{DL}^1]\sin\beta\cos\alpha + [z_{AL}^1 – z_{DL}^1]\sin\alpha + R_{b1}\theta_{x,AL}^1 – R_{b2}\theta_{x,DL}^1 – e_L^1(t) $$
where \(x, y, z\) are translational displacements, \(\theta_x\) are rotational displacements about the gear axis, \(R_{b1}, R_{b2}\) are base radii, and \(e_L^1(t)\) is the composite transmission error. Similar expressions hold for the right hand and for the second stage.
The bearing support forces are modeled as linear spring-damper elements. The shaft torsional stiffness between the first-stage gear and the second-stage pinion is calculated using the standard torsion formula:
$$ k_t = \frac{G J}{l} $$
where \(G\) is the shear modulus, \(J\) is the polar moment of inertia, and \(l\) is the shaft length. The bending stiffness of the shaft is also considered in the x and y directions for the coupling between stages.
The equations of motion for each gear body are derived from Newton’s second law and Euler’s equations. For example, the translational equation for the first-stage left-hand pinion in the x direction is:
$$ m_{AL}^1 \ddot{x}_{AL}^1 + c_{ALx}^1 \dot{x}_{AL}^1 + k_{ALx}^1 x_{AL}^1 + c_{At}^1 (\dot{x}_{AL}^1 – \dot{x}_{AR}^1) + k_{At}^1 (x_{AL}^1 – x_{AR}^1) + \left( k_{mL}^1 f_L^1 + c_{mL}^1 \dot{f}_L^1 \right) \cos\beta \sin\psi = F_{fx,AL}^1 $$
where \(f_L^1\) is the mesh displacement function, \(\psi\) is the position angle, and \(F_{fx,AL}^1\) is the friction force in the x direction. The rotational equation about the gear axis is:
$$ I_{AL,x}^1 \ddot{\theta}_{x,AL}^1 + c_{AL\theta x}^1 \dot{\theta}_{x,AL}^1 + k_{AL\theta x}^1 \theta_{x,AL}^1 + R_{b1} \left( k_{mL}^1 f_L^1 + c_{mL}^1 \dot{f}_L^1 \right) = T_{AL}^1 – T_{AL}^1_{loss} $$
where \(T_{AL}^1\) is the applied torque and \(T_{AL}^1_{loss}\) accounts for friction torque losses.
I assembled all 48 equations in matrix form:
$$ \mathbf{M} \ddot{\mathbf{q}} + \mathbf{C}(t) \dot{\mathbf{q}} + \mathbf{K}(t) \mathbf{q} = \mathbf{F}(t) $$
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{C}(t)\) is the damping matrix, \(\mathbf{K}(t)\) is the stiffness matrix containing the time-varying meshing stiffness, and \(\mathbf{F}(t)\) is the external excitation vector including torques, friction forces, and error excitations. The damping matrix is constructed using Rayleigh damping or modal damping for each element. I used the Runge-Kutta method (ode45 in MATLAB) to solve the differential equations over a sufficient number of periods to obtain steady-state vibration responses.
Vibration Response of Healthy and Faulty Systems
I calculated the vibration responses for the healthy system and for four crack-pitting coupling levels. The input speed was 1800 r/min, so the first-stage pinion rotational frequency was \(f_{A1}=30\) Hz and the mesh frequency was \(f_m=2400\) Hz. The second-stage pinion rotational frequency was \(f_{A2}=12.5\) Hz and its mesh frequency \(f_{m2}=1000\) Hz.
For the healthy gear system, the time-domain displacement waveforms at the first and second stages were smooth and periodic. The peak-to-peak displacement of the first-stage gear was about 1.0 μm, and the second-stage gear about 2.5 μm. The frequency spectra showed only the mesh frequency and its harmonics without significant sidebands.
When crack-pitting coupling was introduced, the time-domain waveforms exhibited periodic impacts at the rotational frequency of the faulty gear. The amplitude of these impacts increased with fault severity. The following table summarizes the maximum vibration displacement (peak amplitude) at the first and second stages for different fault levels:
| Fault level | First-stage displacement (μm) | Second-stage displacement (μm) | Increase over healthy (first stage) |
|---|---|---|---|
| Healthy | 0.50 | 1.25 | — |
| 10% crack-pitting | 0.60 | 1.77 | 20% |
| 30% crack-pitting | 0.71 | 2.43 | 42% |
| 50% crack-pitting | 0.84 | 3.57 | 68% |
| 70% crack-pitting | 1.16 | 6.02 | 132% |
The second-stage gear experienced larger displacements than the first stage due to the higher torque amplification and the lower rotational speed. The severity of the impact also increased with fault level. The frequency spectra of faulty gears showed clear sidebands around the mesh frequency and its harmonics. The sideband spacing was equal to the rotational frequency of the faulty gear: 30 Hz for the first stage and 12.5 Hz for the second stage. The sideband amplitudes increased as the fault severity increased. This feature is valuable for fault diagnosis of herringbone gears.
Experimental Verification
To validate the theoretical model, I conducted vibration experiments on a two-stage herringbone gear test rig. The gears were manufactured by wire-cutting large-width helical gears into narrower sections and joining the left-hand and right-hand helical halves with pins to form herringbone gears. Different crack-pitting severities were produced by electrical discharge machining for pitting and wire cutting for cracks. The gearbox was driven by a servo motor at 1800 r/min with a load of 200 N·m applied by a magnetic powder brake. Accelerometers were mounted on the gearbox housing to capture vibration signals. The measured acceleration signals were integrated twice to obtain displacement signals.
The experimental results for the healthy gear and for the 10%, 50%, and 70% fault levels are compared in the table below. The theoretical values are the maximum vibration displacement amplitudes at the second-stage gear.
| Fault level | Theoretical displacement (μm) | Experimental displacement (μm) | Error (%) |
|---|---|---|---|
| Healthy | 1.25 | 1.30 | 4.0 |
| 10% | 1.77 | 1.82 | 2.8 |
| 50% | 3.57 | 3.68 | 3.1 |
| 70% | 6.02 | 6.25 | 3.8 |
The experimental and theoretical trend matched well. The errors ranged from about 2% to 4%, which are within acceptable limits. The small discrepancies arise from unmodeled factors such as bearing clearance, housing flexibility, and measurement noise. Overall, the experimental results confirm that the proposed analytical model for the TVMS and the dynamic model of the two-stage herringbone gear system correctly predict the vibration characteristics under crack-pitting coupling.
Conclusion
In this work, I established a comprehensive analytical model for the time-varying mesh stiffness of herringbone gears with crack-pitting coupling, including the effect of the central groove. The model was validated by finite element analysis with a maximum error of 2.5%. I then built a 48-DOF dynamic model of a two-stage herringbone gear transmission system incorporating the computed TVMS, transmission errors, friction, and support parameters. The dynamic responses showed that crack-pitting coupling introduces periodic impacts in the time domain and distinct sidebands in the frequency domain. The sideband spacing equals the rotational frequency of the faulty gear, and the sideband amplitudes increase with fault severity, providing a reliable indicator for fault diagnosis. The second-stage gear exhibited larger vibration amplitudes and was more sensitive to crack-pitting coupling than the first stage. Experimental tests on a custom-built test rig confirmed the theoretical predictions with errors below 4% for the maximum displacement. These findings are useful for monitoring the health of herringbone gear transmission systems and for designing more robust gear drives.