In my research, I have focused on the static and dynamic vibration characteristics of herringbone gears during meshing. Herringbone gears are widely used in marine main transmissions due to their ability to cancel axial thrust forces, making them highly suitable for high-power applications. However, the complexity of their geometry and the coupling of torsional, lateral, and axial vibrations pose significant challenges for accurate analytical modeling. I have employed finite element analysis (FEA) using NX MasterFEM to perform static contact analysis and free modal analysis on a pair of herringbone gears from a ship gearbox. This approach allows me to investigate the contact deformation and natural vibration characteristics, providing insights into reducing gear system vibration damage.
1. Introduction
The herringbone gear, consisting of two helical gears with opposite helix angles on the same shaft, is a critical component in high-power marine propulsion systems. The traditional method for calculating contact strength of spur gears relies on the Hertzian model, which approximates the maximum contact stress on tooth surfaces. However, for herringbone gears, the analytical solution under contact conditions is extremely difficult due to the complex geometry and load distribution. Dynamic characteristics, such as natural frequencies and mode shapes, are equally challenging to obtain analytically. Classical dynamic modeling often simplifies the system by assuming linearized stiffness and damping, which may not capture the true behavior. In recent years, many researchers have studied gear contact and vibration, but most work has focused on spur or helical gears, with limited attention to herringbone gears. In this paper, I use the finite element method to systematically analyze the static contact and dynamic vibration characteristics of herringbone gear pairs, including the effects of bearing stiffness and shaft flexibility.
2. Gear Meshing Load Distribution and Stiffness Model
During gear meshing, the number of tooth pairs in contact changes due to the contact ratio. For a double-tooth contact model, the load distribution between the two tooth pairs is determined by the compatibility of displacements along the line of action. Let the total normal load be \(F_L\) and the displacement along the line of action be \(x\). Then the loads on tooth pair 1 and tooth pair 2, denoted \(F_{s1}\) and \(F_{s2}\), satisfy:
\[
\begin{aligned}
F_{s1} + F_{s2} &= F_L \\
F_{s1} &= K_{c1} x \\
F_{s2} &= K_{c2} x
\end{aligned}
\]
where \(K_{c1}\) and \(K_{c2}\) are the mesh stiffness of each tooth pair. The mesh stiffness varies with the position of the contact point along the tooth profile and the number of teeth in contact. The total deformation at the meshing point consists of three components: the Hertzian contact deformation \(\delta_H\), the bending and shear deformation of the tooth \(\delta_T\), and the deformation due to the elasticity of the gear body \(\delta_A\). The mesh stiffness for the \(j\)-th tooth pair is:
\[
K_{cj} = \frac{F_L}{\delta_H + \delta_{T1} + \delta_{T2} + \delta_{A1} + \delta_{A2}}, \quad j = 1, 2
\]
In NX MasterFEM, I can obtain these deformation quantities through finite element analysis by applying the meshing load and boundary conditions.
3. Dynamic Model of Gear Transmission
Vibration in gear meshing is primarily caused by the time-varying mesh stiffness and manufacturing errors, which produce a continuous excitation. Neglecting manufacturing errors for simplicity, the dynamic model of a single-stage gear pair can be represented as a two-degree-of-freedom system along the line of action. The equivalent masses \(m_1\) and \(m_2\) of the two gears at the meshing point are:
\[
m_i = \frac{I_i}{r_{bi}^2}, \quad i = 1, 2
\]
where \(I_i\) is the moment of inertia of gear \(i\), and \(r_{bi}\) is its base circle radius. Let \(x_1\) and \(x_2\) be the displacements of \(m_1\) and \(m_2\) along the line of action. The relative displacement is \(x = x_1 + x_2\). The equation of motion is:
\[
m \ddot{x} + K_c x = F_L
\]
where the equivalent mass \(m = \frac{m_1 m_2}{m_1 + m_2}\) and \(K_c\) is the instantaneous mesh stiffness. Since \(K_c\) is a function of the meshing position (or time), a full meshing cycle must be divided into small time intervals within which \(K_c\) is assumed constant. An iterative procedure is required to solve the equation. However, for complex systems with multi-mesh, multi-stage, and coupled vibrations, analytical modeling becomes intractable. The finite element method offers a powerful alternative.
4. Finite Element Model and Analysis Procedure
The analysis flow consists of the following steps: geometry modeling, finite element meshing, applying boundary conditions and loads, solving static contact and dynamic free modal problems, and post-processing results. I used I-DEAS NX for solid modeling and NX MasterFEM for meshing and analysis.
4.1 Physical Model
The gearbox under study is used for reducing the speed of a 25,000 kW gas turbine and transmitting torque to a propulsion device. The herringbone gear pair consists of a pinion (small gear, active) with tooth number \(Z_1\), input power 25,000 kW, and stable speed \(n_1 = 1000\) rpm, and a gear (large gear, passive) with tooth number \(Z_2\). The material properties and geometry are given in the design.
4.2 Mathematical Model Parameters
From the geometry: base circle radius of pinion \(r_{b1} = 0.20\) m, base circle radius of gear \(r_{b2} = 1.04\) m; moments of inertia \(I_1 = 21.3\) kg·m², \(I_2 = 4192.4\) kg·m². The tangential force at the meshing point is calculated as:
\[
F_L = \frac{M_1}{r_{b1}} = \frac{9.55 \times 10^3 P}{n_1 r_{b1}} = \frac{9.55 \times 10^3 \times 25000}{1000 \times 0.20} = 1.19375 \times 10^6 \text{ N}
\]
The equivalent mass is:
\[
m = \frac{m_1 m_2}{m_1 + m_2} = \frac{ \frac{21.3}{0.20^2} \cdot \frac{4192.4}{1.04^2} }{ \frac{21.3}{0.20^2} + \frac{4192.4}{1.04^2} } = 468.45 \text{ kg}
\]
4.3 Finite Element Meshing
I used hexahedral elements (brick elements) with a combination of manual partitioning and extrusion techniques. The contact region on the tooth flanks was locally refined to capture the high stress gradients. The resulting finite element model has good element quality, with a total of approximately 300,000 elements. The pinion and gear were meshed separately and then assembled in the contact definition.
4.4 Boundary Conditions for Static Contact Analysis
Two contact pairs were defined using surface-to-surface contact elements, which can handle large sliding and friction, as well as large deformations. The contact region on the pinion teeth (102 elements) was assigned as the contact surface, and the corresponding region on the gear teeth (106 elements) as the target surface. Local cylindrical coordinate systems were created at the center of each gear. For the pinion, the nodes on the inner bore surface were constrained in all degrees of freedom except the tangential (θ) direction. For the gear, all degrees of freedom on the inner bore surface were fixed. The torque was applied as a tangential force on the pinion bore nodes, equivalent to the driving torque.
4.5 Static Contact Results
The deformation contour plot shows that the maximum displacement occurs at the tooth tip region of the pinion and gear, with values on the order of 0.1 mm. The contact stress distribution is consistent with the Hertzian theory but shows local variations due to the actual tooth profile and meshing position. The mesh stiffness at different meshing positions can be extracted from the load–displacement relationship. This stiffness matrix is used as input for the subsequent dynamic analysis.
5. Dynamic Analysis – Free Vibration Characteristics
5.1 Finite Element Model for Dynamic Analysis
For the free modal analysis, the gear pair is not subjected to external loads, but the bearings and housing must be represented as elastic supports. I modeled each bearing as four linear springs connecting the shaft journal to ground: two horizontal and two vertical springs. The spring stiffness values were obtained from a separate theoretical analysis of the gearbox housing and bearing properties. The values are listed in Table 1.
| Stiffness | Pinion bearing horizontal | Pinion bearing vertical | Gear bearing horizontal | Gear bearing vertical |
|---|---|---|---|---|
| \(K\) (N/m) | \(8.45 \times 10^8\) | \(8.92 \times 10^8\) | \(9.35 \times 10^8\) | \(9.13 \times 10^8\) |
5.2 Modal Solution
I used the Lanczos method to extract the natural frequencies and mode shapes. The subspace iteration method (SVI) or Guyan reduction could also be used, but Lanczos is efficient for large models with many degrees of freedom. I extracted the first 50 modes; however, for vibration response analysis, typically the first 5–10 modes are sufficient. The first 16 natural frequencies and their mode shape descriptions are summarized in Table 2.
| Mode | Frequency (Hz) | Mode Shape Description |
|---|---|---|
| 1 | 32.86 | Radial expansion of large gear body |
| 2 | 64.32 | Horizontal bending of large gear body in opposite direction |
| 3 | 65.39 | Vertical bending of large gear body in opposite direction |
| 4 | 76.18 | Horizontal bending of large gear disc in opposite direction |
| 5 | 77.09 | Vertical bending of large gear disc in opposite direction |
| 6 | 126.50 | Axial stretching of large gear body |
| 7 | 195.05 | Horizontal bending of large gear shaft |
| 8 | 199.10 | Vertical bending of large gear shaft |
| 9 | 230.19 | Horizontal bending of small gear body in same direction |
| 10 | 231.77 | Vertical bending of small gear body in same direction |
| 11 | 296.43 | Radial expansion of small gear body |
| 12 | 318.35 | Partial radial expansion of large gear shaft |
| 13 | 343.64 | Torsion of large gear disc in alternating direction |
| 14 | 343.65 | Torsion of large gear disc in same direction |
| 15 | 422.60 | Horizontal bending of small gear body in opposite direction |
| 16 | 427.97 | Vertical bending of small gear body in opposite direction |
5.3 Discussion of Modal Results
The first mode appears at 32.86 Hz, corresponding to the large gear body radially expanding. This low frequency indicates that the large gear has a relatively low stiffness compared to its mass. The pinion’s first bending mode occurs at 230.19 Hz. The frequency range from 32.86 Hz to 427.97 Hz covers many potential resonant frequencies. If the gear meshing frequency (which is the product of shaft rotation speed and number of teeth) or its harmonics fall within this range, severe vibration could occur. For this gearbox, the meshing frequency at 1000 rpm is \(f_m = n_1 \times Z_1 / 60\). For the given tooth numbers (not explicitly stated but can be inferred), the meshing frequency is around 1000–2000 Hz, which is above the first 16 modes. However, subharmonics or sidebands may still excite lower modes. The mode shapes also reveal that the bearing supports have a significant influence: the large gear shaft bending modes (7 and 8) are near 195–199 Hz, suggesting that the bearing stiffness is a critical design parameter.
6. Comparison with Damped Support Systems
Although not directly investigated in this work, the use of damped spring hangers or dampers in steam piping systems has been shown to effectively reduce vibration and impact responses. In the context of herringbone gears, similar principles apply: adding damping at the bearing supports or using damped elastic elements can suppress resonant peaks and reduce fatigue damage. The optimal design of support location, stiffness, and damping requires further study, but the modal analysis provides a foundation for selecting appropriate damping parameters.
7. Conclusions
Through the three-dimensional finite element modeling and analysis of a herringbone gear pair, I have obtained the static contact deformation and the modal characteristics. The contact deformation plot reveals the regions of high stress and displacement, which helps in understanding load distribution and contact stiffness. The modal analysis provides the natural frequencies and mode shapes, which are essential for avoiding resonance and for designing damping treatments. The results show that the large gear has its first natural frequency as low as 32.86 Hz, indicating that it is susceptible to low-frequency excitation. The pinion’s first bending mode is at 230.19 Hz. To mitigate vibration damage, the gear system should be designed so that the operating meshing frequencies and their harmonics do not coincide with these natural frequencies. Increasing the bearing stiffness or adding damping can shift the natural frequencies upward or reduce the response amplitude. The finite element approach used here offers a quantitative and visual method for assessing the dynamic behavior of herringbone gears, which is far more accurate than traditional analytical models. Future work should incorporate the time-varying mesh stiffness and non-linear effects such as backlash and friction to fully simulate the transient response under varying load conditions.