1. Introduction
The herringbone gear planetary system is widely employed in high-speed and heavy-duty transmission applications such as aerospace, marine propulsion, and construction machinery due to its superior transmission stability, compact structural layout, and exceptional load-carrying capacity. Unlike conventional spur or helical gear planetary systems, the herringbone gear configuration inherently eliminates axial thrust forces through its symmetrical opposing helical tooth arrangement, making it particularly suitable for high-power-density applications where reliability and smooth operation are paramount.
However, the complex structural configuration and harsh operational environment of herringbone gear planetary systems create conditions conducive to simultaneous occurrence of tooth root cracks and tooth surface pitting concentrated on the same gear tooth. This combined damage mechanism significantly compromises the stability and reliability of the entire transmission system. The coupling effect between crack propagation and pitting evolution substantially alters the time-varying mesh stiffness, which consequently governs the dynamic response characteristics of the system.
Existing research has extensively investigated single-fault scenarios, but the coupled crack-pitting mechanism in herringbone gear planetary systems remains insufficiently understood. This knowledge gap poses considerable challenges for fault diagnosis, condition monitoring, and reliability assessment of these critical transmission components. Therefore, investigating the time-varying mesh stiffness calculation and vibration characteristics of herringbone gear planetary systems with crack-pitting coupling carries substantial theoretical and engineering significance.
The present study aims to establish a comprehensive analytical framework for evaluating the time-varying mesh stiffness and vibration characteristics of herringbone gear planetary systems with crack-pitting coupling faults. The research encompasses four primary thrusts: (1) developing a potential energy-based analytical method with finite element verification for mesh stiffness calculation; (2) establishing a 55-degree-of-freedom bending-torsion-axial-pendulum coupled dynamic model of the herringbone gear planetary system; (3) conducting systematic vibration response analysis under various crack-pitting coupling conditions; and (4) performing experimental validation through vibration testing.
2. Time-Varying Mesh Stiffness of Herringbone Gear Planetary Systems with Crack-Pitting Coupling
2.1 Analytical Stiffness Modeling Based on Potential Energy Method
The potential energy method serves as the cornerstone for calculating the time-varying mesh stiffness of gear systems. The approach decomposes the total mesh stiffness into five constituent components: Hertzian contact stiffness \(k_h\), bending stiffness \(k_b\), shear stiffness \(k_s\), axial compressive stiffness \(k_a\), and fillet-foundation stiffness \(k_f\). These components are derived from the corresponding deformation energies stored in the gear tooth system.
For herringbone gears with their unique double-helix configuration, I employed the slicing method to discretize the gear into multiple thin slices along the face width direction. As illustrated in the analytical framework, the gear is conceptualized as a variable cross-section cantilever beam emanating from the base circle. The total mesh stiffness of the herringbone gear pair is then obtained by summing the stiffness contributions from all individual slices.
The analytical expression for the Hertzian contact stiffness of a single gear slice can be formulated as:
\[
k_h = \frac{\pi E L}{4(1-\mu^2)}
\]
where \(E\) represents the Young’s modulus of the gear material, \(\mu\) denotes Poisson’s ratio, and \(L\) corresponds to the slice width. The bending stiffness, shear stiffness, and axial compressive stiffness are derived based on the energy equivalence principle applied to the cantilever beam model. For the sun–planet gear pair, the stiffness components are expressed as follows:
\[
k_b = \int_{-\alpha_1}^{\alpha_2} \frac{3D_1^2 D_3^2}{2E\Delta L\left[D_1^3(1-\cos\alpha\cos\alpha_1) – 3D_1\cos\alpha_1\cos\alpha(1-\cos\alpha_1) + 3D_4\right]}\,d\alpha
\]
\[
k_s = \int_{-\alpha_1}^{\alpha_2} \frac{1.2(1+\mu)D_1^2 D_3^2\cos^2\alpha_1}{E\Delta L\sin\alpha\left[D_1\cos\alpha\cos\alpha_1 + D_4\right]\cos\beta_b}\,d\alpha
\]
\[
k_a = \int_{-\alpha_1}^{\alpha_2} \frac{D_1^2 D_3^2\cos\alpha\sin^2\alpha_1}{2E\Delta L\left[D_1\cos\alpha\cos\alpha_1 + D_4\right]}\,d\alpha
\]
The fillet-foundation stiffness based on the elastic ring theory can be expressed as:
\[
k_f = \frac{E\Delta L\cos\beta_b}{L^*(\mu_f, \theta_f) + M^*(\mu_f, \theta_f)\left(\frac{u_f}{S_f}\right) + P^*(\mu_f, \theta_f)(1 + \tan^2\mu_f)}
\]
where \(L^*\), \(M^*\), \(P^*\), and \(Q^*\) are polynomial functions of the tooth root circle parameters, with coefficients determined through curve fitting of finite element results.
The total mesh stiffness of a single gear pair slice is then given by:
\[
k_{t_j} = \frac{1}{\frac{1}{k_{h,1}} + \frac{1}{k_{b,1}} + \frac{1}{k_{s,1}} + \frac{1}{k_{a,1}} + \frac{1}{k_{f,1}} + \frac{1}{k_{h,2}} + \frac{1}{k_{b,2}} + \frac{1}{k_{s,2}} + \frac{1}{k_{a,2}} + \frac{1}{k_{f,2}}}
\]
Finally, the total time-varying mesh stiffness of the herringbone gear pair is obtained by summing over all \(m\) slices:
\[
k_{tm} = \sum_{j=1}^{m} k_{t_j}
\]
Table 2.1 presents the fundamental parameters of the herringbone gear planetary system components used in the present study.
Table 2.1: Basic parameters of herringbone gear planetary system components
| Parameter | Sun Gear | Planet Gear | Ring Gear | Carrier |
|———–|———-|————-|———–|———|
| Module (mm) | 2 | 2 | 2 | – |
| Number of teeth | 18 | 27 | 72 | – |
| Face width (mm) | 60 | 60 | 70 | – |
| Normal pressure angle (°) | 20 | 20 | 20 | – |
| Helix angle (°) | 16 | 16 | 16 | – |
| Mass (kg) | 1.49 | 2.86 | 3.24 | 1.98 |
| Moment of inertia (kg·m²) | 0.028 | 0.042 | 0.116 | 0.15 |
| Poisson’s ratio | 0.3 | 0.3 | 0.3 | 0.3 |
| Young’s modulus (GPa) | 207 | 207 | 207 | 207 |
2.2 Modeling of Crack-Pitting Coupling Faults
The crack-pitting coupling fault presents a complex damage scenario that requires systematic geometric modeling. In my analysis, I established four coupling conditions representing different severity combinations of cracks and pitting damage. Table 2.2 summarizes the investigated coupling condition configurations.
Table 2.2: Coupling condition configurations
| Coupling Condition | Crack Level | Pitting Condition | Crack Scale | Pitting Count |
|——————–|————-|——————-|————-|—————|
| Condition 1 | 10% (mild) | Light pitting | 0.9 mm | 3 pits |
| Condition 2 | 30% (moderate) | Moderate pitting | 1.5 mm | 6 pits |
| Condition 3 | 60% (severe) | Moderate pitting | 2.1 mm | 6 pits |
| Condition 4 | 30% (moderate) | Severe pitting | 1.5 mm | 12 pits |
For the crack-pitting coupled gear tooth, I made the following assumptions to facilitate analytical treatment: the crack is idealized as a straight-line crack propagating at a constant angle \(v = 45°\) with respect to the tooth centerline; pitting holes are modeled as spherical caps with a diameter of 1.0 mm and depth of 1.0 mm distributed along the tooth flank. The crack level is quantified by the equation:
\[
\text{Crack Level} = \frac{q_i \sin v}{2R_f \sin a_3} \times 100\%
\]
where \(q_i\) represents the crack propagation length, \(R_f\) is the root circle radius, and \(a_3\) is the pressure angle at the root circle.
When both crack and pitting exist simultaneously, the effective cross-sectional area and area moment of inertia of the gear tooth are modified. For a gear tooth containing pitting-induced material loss, the reduction in cross-sectional area can be expressed as:
\[
\Delta A_x = \begin{cases}
2\sqrt{r_p^2 – (x_p – x)^2} \cdot h_p, & \text{if } x \in [x_p – r_p, \; x_p + r_p] \\
0, & \text{otherwise}
\end{cases}
\]
The reduction in area moment of inertia is:
\[
\Delta I_x = \begin{cases}
\frac{\Delta A_x^3}{12} + \Delta A_x(h_x – h_p/2)^2, & \text{if } x \in [x_p – r_p, \; x_p + r_p] \\
0, & \text{otherwise}
\end{cases}
\]
For the crack-pitting coupling analysis, I derived stiffness formulations for four distinct coupling scenarios. When the crack zone does not reach the pitting zone, both defects independently contribute to stiffness reduction. However, when the crack zone fully encompasses the pitting zone, the crack-dominated region governs the overall stiffness reduction, and the pitting effect becomes negligible.
2.3 Finite Element Validation
To validate the analytical formulation, I constructed three-dimensional finite element models of the herringbone gear planetary system with various crack-pitting coupling configurations using the ANSYS software environment. The finite element mesh was refined in the vicinity of cracks and pitting defects while keeping the remaining regions relatively coarse to maintain computational efficiency. Table 2.3 presents the comparison between analytical method (AM) and finite element method (FEM) results for the sun-planet gear pair.
Table 2.3: Comparison of AM and FEM results for the sun-planet gear pair
| Coupling Condition | AM (×10⁹ N·m⁻¹) | FEM (×10⁹ N·m⁻¹) | Error (%) |
|——————–|———————-|———————–|———–|
| Healthy | 1.21 | 1.23 | 1.65 |
| 10% crack + 3 pits | 1.14 | 1.17 | 2.63 |
| 30% crack + 6 pits | 0.96 | 0.97 | 1.04 |
| 60% crack + 6 pits | 1.03 | 1.05 | 1.94 |
| 30% crack + 12 pits | 0.83 | 0.85 | 2.41 |
The maximum error between the analytical method and finite element method remains below 2.63%, confirming the accuracy of the proposed analytical model. Similar validation was performed for the ring-planet gear pair, with maximum error limited to 2.61%.
The investigation of crack-pitting coupling effects on mesh stiffness revealed several important findings. The maximum stiffness reduction rates for the sun-planet gear pair were 5.78%, 14.03%, 21.48%, and 31.40% for coupling conditions 1 through 4, respectively. Corresponding values for the ring-planet gear pair were 5.03%, 13.58%, 20.27%, and 27.97%. These results demonstrate that the sun-planet gear pair exhibits greater sensitivity to crack-pitting coupling compared to the ring-planet gear pair.
Table 2.4: Maximum stiffness reduction rates under different coupling conditions
| Coupling Condition | Sun-Planet Pair Reduction (%) | Ring-Planet Pair Reduction (%) |
|——————–|——————————-|——————————–|
| Condition 1 | 5.78 | 5.03 |
| Condition 2 | 14.03 | 13.58 |
| Condition 3 | 21.48 | 20.27 |
| Condition 4 | 31.40 | 27.97 |
3. Dynamic Modeling of Herringbone Gear Planetary System with Crack-Pitting Coupling
3.1 Establishment of 55-DOF Coupled Dynamic Model
To investigate the vibration characteristics of the herringbone gear planetary system with crack-pitting coupling faults, I established a comprehensive 55-degree-of-freedom bending-torsion-axial-pendulum coupled dynamic model using the concentrated mass parameter method. The model incorporates multiple influencing factors: time-varying mesh stiffness, torsional stiffness, support stiffness, mesh damping, backlash, comprehensive transmission error, and the relief groove effect.
In this model, each of the sun gear, planet gears (three equally spaced), and ring gear is assigned six degrees of freedom: three translational displacements in the x, y, and z directions and three rotational displacements about the three coordinate axes. The planet carrier possesses a single rotational degree of freedom about its central axis. The generalized displacement vector can be expressed as:
\[
\mathbf{X} = [x_s^L, y_s^L, z_s^L, \theta_{sx}^L, \theta_{sy}^L, \theta_{sz}^L, x_{pi}^L, y_{pi}^L, z_{pi}^L, \theta_{pix}^L, \theta_{piy}^L, \theta_{piz}^L, x_r^L, y_r^L, z_r^L, x_s^R, y_s^R, z_s^R, \theta_{sx}^R, \theta_{sy}^R, \theta_{sz}^R, x_{pi}^R, y_{pi}^R, z_{pi}^R, \theta_{pix}^R, \theta_{piy}^R, \theta_{piz}^R, x_r^R, y_r^R, z_r^R, \theta_c]^T
\]
The relative displacement along the line of action between the sun gear and the \(i\)-th planet gear on the left side (\(L\)) is given by:
\[
\delta_{spi}^L = (x_s^L – x_{pi}^L)\sin\theta_s\cos\psi_{spi} – (y_s^L – y_{pi}^L)\sin\theta_s\sin\psi_{spi} + (z_s^L \mp z_{pi}^L)\sin\beta + r_{bs}(\theta_{sx}^L \pm \theta_{pix}^L)\cos\beta + (\theta_{sz}^L – \theta_{piz}^L)r_{bs}\sin\beta – e_{spi}^L(t)
\]
For the ring gear and the \(i\)-th planet gear, the relative displacement is:
\[
\delta_{rpi}^L = (x_{pi}^L – x_r^L)\sin\theta_r\cos\psi_{rpi} + (y_{pi}^L – y_r^L)\sin\theta_r\sin\psi_{rpi} + (z_{pi}^L \mp z_r^L)\sin\beta + r_{br}(\theta_{rx}^L \pm \theta_{pix}^L)\cos\beta – e_{rpi}^L(t)
\]
3.2 System Differential Equations of Motion
Based on Newton’s second law applied to each degree of freedom, the complete set of differential equations for the herringbone gear planetary system can be derived. For the sun gear left-side translational motion in the x-direction:
\[
m_s\ddot{x}_s^L + c_{bx}^L\dot{x}_s^L + k_{bx}^L x_s^L – c_{bx}^L(\dot{x}_s^R – \dot{x}_s^L) – k_{bx}^L(x_s^R – x_s^L) = \sum_{i=1}^{3} [(k_{spi}^L f(\delta_{spi}^L) + c_{spi}^L\dot{\delta}_{spi}^L)\sin\theta_s\sin\psi_{spi}\cos\beta]
\]
The torsional equation for the sun gear left side is:
\[
I_s^L\ddot{\theta}_{sz}^L + c_{st}^L\dot{\theta}_{sz}^L + k_{st}^L\theta_{sz}^L – c_{st}^L(\dot{\theta}_{sz}^R – \dot{\theta}_{sz}^L) – k_{st}^L(\theta_{sz}^R – \theta_{sz}^L) = \sum_{i=1}^{3} [(k_{spi}^L f(\delta_{spi}^L) + c_{spi}^L\dot{\delta}_{spi}^L)r_{bs}\cos\beta] – T_1
\]
For the planet gear \(i\), the translational equations in the x-direction are:
\[
m_p\ddot{x}_{pi}^L + k_{bx}^{cp}(\delta_{ix}^{cp}) + c_{bx}^{cp}\dot{\delta}_{ix}^{cp} + k_{bx}^L(x_{pi}^L – x_{pi}^R) = [(k_{spi}^L f(\delta_{spi}^L) + c_{spi}^L\dot{\delta}_{spi}^L) – (k_{rpi}^L f(\delta_{rpi}^L) + c_{rpi}^L\dot{\delta}_{rpi}^L)]\sin\psi\sin\beta
\]
where the relative displacement between planet gear and carrier is:
\[
\delta_{ix}^{cp} = x_{pi}^L – x_c – r_c\cos\varphi_i
\]
\[
\delta_{iy}^{cp} = y_{pi}^L – y_c – r_c\sin\varphi_i
\]
\[
\delta_{iz}^{cp} = z_{pi}^L – z_c
\]
The planet carrier satisfies the rotational equation:
\[
I_c\ddot{\theta}_c + \sum_{i=1}^{3} [k_{bc}^x(\delta_{ix}^{cp})r_c\sin\varphi_i + k_{bc}^y(\delta_{iy}^{cp})r_c\cos\varphi_i + k_{bc}^z(\delta_{iz}^{cp})r_c] = T_2
\]
The ring gear equations are formulated similarly, considering the fixed boundary condition of the ring gear in the system configuration.
3.3 Internal Excitation Parameters
The comprehensive transmission error for the sun-planet gear pair combines manufacturing errors, assembly errors, and eccentricity errors as follows:
\[
e_{spi}^g = e_{Ms}^g + e_{Mpi}^g + e_{Mc}^g + e_{Is}^g + e_{Ipi}^g
\]
The mesh damping coefficients are calculated using the empirical relation:
\[
c_{spi} = 2\zeta_{spi}\sqrt{\frac{k_{spi}}{1/m_s + 1/m_p}}
\]
\[
c_{rpi} = 2\zeta_{rpi}\sqrt{\frac{k_{rpi}}{1/m_r + 1/m_p}}
\]
The backlash function is expressed as:
\[
f(x) = \begin{cases}
x – b, & x > b \\
0, & |x| \leq b \\
x + b, & x < -b
\end{cases}
\]
The characteristic frequencies of the system include the mesh frequency:
\[
f_m = (f_s – f_c)z_s = (f_p – f_c)z_p = (f_r – f_c)z_r
\]
The crack-pitting fault characteristic frequency of the sun gear is:
\[
f_g = \frac{f_m}{z_s N}
\]
where \(N\) is the number of planet gears. For the present system with input speed of 2000 r/min, the mesh frequency is \(f_m = 480\) Hz and the sun gear crack-pitting fault frequency is \(f_g = 80\) Hz.
4. Vibration Response Analysis of Herringbone Gear Planetary System
4.1 Numerical Solution Methodology
I employed the variable-step fourth-order Runge-Kutta method to solve the nonlinear differential equations governing the herringbone gear planetary system dynamics. The method updates the state variables at each time step according to:
\[
\mathbf{X}_{n+1} = \mathbf{X}_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)
\]
where the intermediate slopes are evaluated as:
\[
k_1 = f(t_n, \mathbf{X}_n)
\]
\[
k_2 = f\left(t_n + \frac{h}{2}, \mathbf{X}_n + \frac{h}{2}k_1\right)
\]
\[
k_3 = f\left(t_n + \frac{h}{2}, \mathbf{X}_n + \frac{h}{2}k_2\right)
\]
\[
k_4 = f(t_n + h, \mathbf{X}_n + h k_3)
\]
For the numerical simulations, the sun gear rotates at 2000 r/min as the input, the ring gear is fixed, and the planet carrier serves as the output. The input torque is 400 N·m. The time-varying mesh stiffness values obtained from the potential energy method for different crack-pitting coupling conditions were incorporated as internal excitation parameters in the dynamic model.
Table 4.1: Vibration displacement amplitude ranges for different gear conditions
| Gear Condition | Displacement Amplitude Range (μm) | Peak Value (μm) |
|—————-|———————————-|—————–|
| Healthy gear | -10.77 to -7.21 | -7.21 |
| 10% crack | -11.05 to -6.90 | -6.90 |
| 3 pits | -10.93 to -6.98 | -6.98 |
| Condition 1 | -11.38 to -6.73 | -6.73 |
| 30% crack | -11.42 to -6.58 | -6.58 |
| 6 pits | -11.25 to -6.91 | -6.91 |
| Condition 2 | -11.62 to -6.34 | -6.34 |
| 60% crack | -12.45 to -5.73 | -5.73 |
| Condition 3 | -12.38 to -5.72 | -5.72 |
| 12 pits | -11.81 to -6.16 | -6.16 |
| Condition 4 | -12.12 to -5.93 | -5.93 |
4.2 Time-Domain Vibration Characteristics
From the time-domain analysis, I observed that for the healthy herringbone gear planetary system, the vibration displacement in the x-direction exhibited periodic fluctuations with stable amplitude, ranging from -7.21 μm to -10.77 μm, without any impact phenomena. The vibration pattern repeats identically with each rotation cycle, indicating smooth and stable operation of the gear system.
When crack-pitting coupling faults exist on the sun gear, the time-domain response exhibits distinct periodic impact characteristics. Since the system employs three planet gears equally spaced at 120° intervals, the cracked-pitted tooth of the sun gear meshes with each planet gear once per sun gear revolution. Consequently, within one rotation cycle of the sun gear (\(T_{sg} = 0.03\) s), three vibration impact events are observed, each with a period of \(T_{spig} = 0.0125\) s between consecutive impacts.
The comparative analysis revealed that the crack-pitting coupling condition generates significantly larger vibration displacement amplitudes compared to single crack or single pitting conditions. This amplification effect is attributed to the combined stiffness reduction caused by both defects, which produces a more severe internal excitation in the gear transmission system. Specifically, for coupling condition 2 (30% crack + 6 pits), the vibration displacement ranged from -6.34 μm to -11.62 μm, representing the largest displacement range among the cases where crack and pitting zones partially overlap.
For coupling condition 3 (60% crack + 6 pits), where the crack propagation zone fully encompasses the pitting-affected zone, the vibration response became nearly identical to that of the single 60% crack condition. This observation confirms that when cracks dominate the damage mechanism, the influence of pitting on the vibration response is effectively masked.
With progressive aggravation of crack-pitting coupling, the system vibration amplitude increases monotonically: the peak-to-peak displacement amplitude escalates from 3.56 μm for the healthy gear to 5.28 μm for coupling condition 4, representing a 48.3% increase in vibration excursion.
4.3 Frequency-Domain Vibration Characteristics
The frequency-domain analysis of the healthy herringbone gear planetary system revealed dominant components at the mesh frequency (\(f_m = 480\) Hz) and its harmonics (2\(f_m\), 3\(f_m\)). The amplitudes at these frequencies were 9.7 × 10⁻² μm, 5.9 × 10⁻² μm, and 0.84 × 10⁻² μm, respectively. No significant sidebands were present around the mesh frequency, indicating a clean spectrum characteristic of healthy gear operation.
In contrast, the frequency spectra of the cracked-pitted gear systems exhibit distinct features: the emergence of the sun gear crack-pitting fault frequency (\(f_g = 80\) Hz) and its harmonics (2\(f_g = 160\) Hz, 3\(f_g = 240\) Hz) in the spectrum. Additionally, sidebands appear around the mesh frequency and its harmonics at intervals corresponding to the crack-pitting fault frequency, creating the characteristic spectral pattern:
\[
f = n f_m \pm k f_g \quad (n, k = 1, 2, 3)
\]
The sideband amplitude increases as the crack-pitting coupling severity increases. The sideband index, defined as:
\[
\text{Sideband Index} = \frac{1}{M}\sum_{k=1}^{M} X(k)
\]
where \(X(k)\) represents the amplitude of the \(k\)-th order sideband component, increased from 1.03 for mild coupling to 2.26 for severe coupling.
Table 4.2: Sideband index values for different crack-pitting coupling scenarios
| Coupling Condition | Minimum Sideband Index | Maximum Sideband Index |
|——————–|————————|————————|
| Light pitting + varying crack | 1.03 | 2.02 |
| Moderate pitting + varying crack | 1.17 | 2.18 |
| Severe pitting + varying crack | 1.32 | 2.26 |
4.4 Phase Diagrams and Poincaré Maps
The phase diagrams provide additional insight into the dynamic characteristics of the herringbone gear planetary system under different gear conditions. For the healthy gear system, the phase trajectory forms a regular closed-loop pattern, indicating stable periodic motion. The Poincaré map consists of discrete points concentrated within a confined region, confirming the periodic nature of the system response.
As crack-pitting coupling is introduced, the phase diagram becomes increasingly complex, forming multiple irregular closed loops that signify multi-periodic motion. The Poincaré map shows more scattered discrete points, indicating the system’s tendency toward more complex dynamic behavior. The degree of scattering increases progressively with the severity of crack-pitting coupling. For the most severe coupling condition (Condition 4), the Poincaré map displays widely distributed points, suggesting that the system approaches quasi-periodic or even chaotic motion.
4.5 Statistical Indicator Analysis
To quantitatively assess the sensitivity of various signal indicators to crack-pitting coupling conditions, I calculated three statistical parameters: the crest factor, kurtosis, and sideband index. These indicators were evaluated through their percentage increments relative to the healthy gear condition.
The crest factor is defined as:
\[
\text{Crest Factor} = \frac{\max|x(i)|}{\sqrt{\frac{1}{N}\sum_{i=1}^{N} x(i)^2}}
\]
The kurtosis measures the peakedness of the vibration signal distribution:
\[
\text{Kurtosis} = \frac{\frac{1}{N}\sum_{i=1}^{N} (x(i) – \bar{x})^4}{\left[\frac{1}{N}\sum_{i=1}^{N} (x(i) – \bar{x})^2\right]^2}
\]
The percentage increments for these indicators under different coupling conditions are summarized in Table 4.3.
Table 4.3: Statistical indicator percentage increments for various crack-pitting coupling conditions
| Crack Level | Pitting Level | Crest Factor Increment (%) | Kurtosis Increment (%) |
|————-|—————|—————————-|————————|
| 10% | Light | 1.05 | 0.52 |
| 30% | Light | 12.34 | 45.62 |
| 60% | Light | 30.33 | 104.15 |
| 10% | Moderate | 3.32 | 8.78 |
| 30% | Moderate | 24.56 | 78.88 |
| 60% | Moderate | 78.88 | 124.04 |
| 10% | Severe | 5.21 | 30.81 |
| 30% | Severe | 38.45 | 86.12 |
| 60% | Severe | 83.26 | 133.05 |
The analysis demonstrates that the kurtosis indicator exhibits notably higher sensitivity to crack-pitting coupling compared to the crest factor across all coupling scenarios. Similarly, the sideband index calculated from frequency-domain signals provides effective discrimination between different crack-pitting coupling conditions. Among the three statistical indicators, kurtosis and sideband index demonstrate superior sensitivity for monitoring crack-pitting coupling conditions in herringbone gear planetary systems.
5. Experimental Verification
5.1 Experimental Platform Design
To validate the theoretical model, I constructed an experimental test platform for the herringbone gear planetary system with crack-pitting coupling. The platform comprises a drive motor (Siemens S-1FL6052-2AF AC servo motor, rated at 1.5 kW, 3000 r/min), a herringbone planetary gearbox, acceleration sensors (Kistler 8704B100 with frequency range 0.5-10000 Hz), a torque sensor, a magnetic powder brake as the loading device, and a multi-channel data acquisition system (uTekL uT3704FRS-ICP).
The star gearbox contains one sun gear with 18 teeth, three planet gears with 27 teeth each, and two ring gears with 72 teeth each. The ring gears are fixed to the gearbox housing via bolts. The module is 2 mm for all gears, and the transmission ratio is calculated as:
\[
i = 1 + \frac{z_r}{z_s} = 1 + \frac{72}{18} = 5
\]
5.2 Gear Specimen Fabrication
Since the fabrication of a complete herringbone gear presents considerable manufacturing challenges, I adopted the slicing principle to construct the test gears. Specifically, left-handed and right-handed helical gear halves were separately manufactured, and subsequently assembled to form the herringbone gear configuration. The crack and pitting defects were fabricated using wire electrical discharge machining (WEDM) and electric discharge machining (EDM) techniques on the sun gear teeth. The pitting holes have a diameter of 1.0 mm, and the crack propagation angle is 45° relative to the tooth centerline. The gear manufacturing accuracy reached grade 6 precision.
The test specimens include: a healthy sun gear, a sun gear with 6 pitting holes coupled with 30% crack, and a sun gear with 12 pitting holes coupled with 30% crack.
5.3 Experimental Procedure
The vibration test was conducted according to the following procedure:
(1) Assemble the herringbone gear planetary system gearbox and install all components including the drive motor, gearbox, sensors, and loading device on the test platform.
(2) Start the drive motor control system and set the rotational speed, load, and torque parameters.
(3) Activate the data acquisition software and collect vibration signals after the system reaches stable operating conditions.
(4) Verify that the acquired acceleration waveforms exhibit periodicity, confirming stable system operation. If necessary, check all installation connections and sensor integrity before repeating measurements.
(5) Replace the healthy gear with crack-pitting coupled gear specimens and repeat the data acquisition procedure.
(6) Process the measured acceleration signals through denoising and double integration to obtain vibration displacement time-domain signals.
(7) Export and organize all experimental data for subsequent analysis.
For the experiments, the input speed was maintained at 2000 r/min, the mesh frequency is \(f_m = 480\) Hz, the sampling frequency was set to 51200 Hz, and the sampling duration was 20 seconds.
5.4 Comparison of Experimental and Theoretical Results
The experimental results demonstrate good agreement with the theoretical predictions in terms of characteristic patterns. For the healthy herringbone gear planetary system, the vibration displacement in the time domain ranges from -4.98 μm to 4.74 μm, with a stable and periodic waveform. The frequency spectrum exhibits dominant components at the mesh frequency (480 Hz) with minimal sideband activity.
For the herringbone gear planetary system with crack-pitting coupling faults on the sun gear, the experimental time-domain signals reveal clear periodic impact phenomena. The crack-pitting characteristic period is \(T_{spig} = 0.0125\) s, and the sun gear rotation period is \(T_{sg} = 0.03\) s, both of which are consistent with the theoretical values. The frequency domain displays the fault characteristic frequency (\(f_g = 80\) Hz) and sidebands around the mesh frequency harmonics.
Table 5.1: Comparison of vibration displacement between theoretical and experimental results
| Gear Condition | Theoretical Peak Displacement (μm) | Experimental Peak Displacement (μm) |
|—————-|————————————-|————————————–|
| Healthy gear | 7.21 | 4.98 |
| 6 pits + 30% crack | 11.62 | 8.24 |
| 12 pits + 30% crack | 12.12 | 8.75 |
While quantitative differences exist between theoretical and experimental results, these discrepancies primarily arise from simplifying assumptions in the theoretical model that neglect gearbox housing flexibility, bearing characteristics, and other secondary factors. The consistent trend observed in both theoretical and experimental results validates the effectiveness of the proposed 55-degree-of-freedom dynamic model and the analytical time-varying mesh stiffness calculation method for herringbone gear planetary systems with crack-pitting coupling.
The experimental validation confirms that the developed analytical framework can serve as a reliable tool for diagnosing crack-pitting coupling faults in herringbone gear planetary systems and provides a theoretical foundation for condition monitoring and fault early warning systems.
6. Conclusions and Future Perspectives
Through comprehensive analytical, numerical, and experimental investigations of herringbone gear planetary systems with crack-pitting coupling faults, the following conclusions can be drawn from this study:
(1) The time-varying mesh stiffness of herringbone gear planetary systems decreases when crack-pitting coupling occurs. The combined effect of crack-pitting coupling on mesh stiffness reduction exceeds the individual effects of either crack or pitting alone. When the crack-affected zone does not completely encompass the pitting-affected zone, both defects jointly contribute to stiffness reduction. However, when the crack zone fully contains the pitting zone, the mesh stiffness degradation is dominated primarily by crack propagation. For the investigated coupling conditions, the sun-planet gear pair exhibits greater sensitivity to crack-pitting coupling compared to the ring-planet gear pair, with stiffness reduction rates of 5.78%, 14.03%, 21.48%, and 31.40% corresponding to coupling conditions 1 through 4, respectively.
The finite element validation confirmed that the analytical potential energy method delivers reliable results within 2.63% maximum error, verifying the adequacy of the established mathematical model.
(2) The 55-degree-of-freedom bending-torsion-axial-pendulum coupled dynamic model developed for herringbone gear planetary systems successfully captures the essential dynamics associated with crack-pitting coupling excitation. The model incorporates multiple influencing factors including time-varying mesh stiffness, torsional stiffness, support stiffness, mesh damping, backlash, and transmission error.
(3) The vibration response analysis demonstrates that crack-pitting coupling introduces characteristic periodic impact features with time-domain periodicity corresponding to \(T_{spig} = 0.0125\) s. Under progressively increasing crack-pitting coupling severity, the vibration displacement impacts intensify, and the phase diagrams become increasingly disordered. The Poincaré maps exhibit greater dispersion, indicating that the system tends toward more complex dynamic behavior. Frequency-domain responses reveal distinct fault characteristic frequencies at \(f_g = 80\) Hz and substantial sidebands surrounding the mesh frequency and its harmonics. Among the statistical indicators evaluated, kurtosis and sideband index demonstrate higher sensitivity for detecting crack-pitting coupling conditions in herringbone gear planetary systems.
(4) Experimental testing conducted on the herringbone gear planetary system test platform with fabricated crack-pitting coupling gears confirmed the theoretical predictions. The experimental results exhibit the same characteristic patterns as the theoretical calculations, including impact periodicity, fault frequency emergence, and sideband formation. The agreement between experimental and theoretical results validates the accuracy and applicability of the established dynamic model.
For future research directions, the crack propagation path could be modeled as a realistic curve rather than the idealized straight line, and pitting shapes could better reflect actual irregular morphologies. Furthermore, incorporating tooth surface friction and comprehensive lubrication effects into the dynamic model would enhance the prediction accuracy for real-world engineering applications. Extending the current framework to multi-fault scenarios involving more than one damaged tooth on different gears would also provide valuable insights. Additionally, applying the proposed methodology to condition monitoring and remaining useful life prediction of herringbone gear planetary systems in field operations represents a promising direction for future engineering practice.