This paper presents a comprehensive three-dimensional finite element modeling and modal analysis of a star-planetary herringbone gear transmission system. The primary objective is to identify the root cause of resonance observed in experimental tests, particularly those induced by rotational frequency and meshing frequency excitations. The study establishes a novel 3D finite element model that incorporates gear meshing stiffness matrices to accurately represent the coupling effects between herringbone gears. The model is validated against experimental data, and the modal characteristics are analyzed in detail. The results show that the simulated rotational frequency resonance modes and meshing frequency resonance modes have a unique one-to-one correspondence with experimental measurements. The meshing frequency resonance is predominantly characterized by a five-nodal diameter vibration mode of the internal ring gear coupled with torsional vibration modes of the gear pairs. This work provides a valuable reference method for resonance prediction and root cause analysis in herringbone gear transmission systems used in star-planetary configurations.
Herringbone gears are widely employed in high-power-density transmissions due to their high load capacity, smooth operation, and elimination of axial thrust forces. In advanced aircraft engine architectures, such as geared turbofan engines, a star-planetary gear system with herringbone gears is used to decouple the fan rotor from the low-pressure turbine, allowing each to operate at its optimal speed. However, the complex dynamic behavior of such systems, particularly the interaction between multiple meshing pairs and flexible structural components, can lead to significant vibration issues. Understanding the modal properties of the entire system is crucial for avoiding resonance and ensuring reliable operation.
Traditional analytical approaches, such as lumped parameter models, have been extensively used to study the natural frequencies and vibration modes of planetary gear systems. While these models provide valuable insights, they often oversimplify the system by ignoring three-dimensional effects, gear body flexibility, and the precise coupling between herringbone gear pairs. Finite element methods offer a more realistic representation, but the direct use of contact algorithms for modal analysis is computationally expensive and often impractical due to convergence issues, especially for systems with multiple gear pairs. In this work, we propose a hybrid approach where the gear meshing action is represented by stiffness matrices derived from detailed quasi-static contact simulations. These matrices are then integrated into a 3D finite element model using specialized spring elements (MATRIX27 in ANSYS), enabling efficient and accurate modal analysis of the entire herringbone gear planetary system.
We begin by describing the experimental setup and key observations from vibration tests. Then, we detail the 3D finite element modeling procedure, including the simplification of gear bodies, the calculation and implementation of herringbone gear meshing stiffness matrices, and the modeling of bearings and other connections. Finally, we present and discuss the modal analysis results, comparing them with experimental data and identifying the critical vibration modes responsible for the observed resonances.
Experimental Observations
The star-planetary herringbone gear system under investigation was tested on a power-circulating test rig using a differential planetary loading method. The test article consists of a sun gear, five planetary gears, and a ring gear, all with herringbone tooth profiles. The gears are made of high-strength steel and have the parameters listed in Table 1. Accelerometers were mounted on the support structure (torque bracket) to measure radial and axial vibrations. Sweep tests were conducted over a range of input speeds, and the root-mean-square (RMS) velocity amplitudes at the rotational frequency and the first meshing frequency were extracted.
| Parameter | Sun Gear | Planet Gear | Ring Gear |
|---|---|---|---|
| Number of teeth | 43 | 41 | 127 |
| Module (mm) | 3.5 | 3.5 | 3.5 |
| Pressure angle (°) | 22.5 | 22.5 | 22.5 |
| Helix angle (°) | 28.019 | 28.019 | 28.019 |
| Total face width (mm) | 60 (with 30 mm gap) | 59 (with 31 mm gap) | 57 (with 33 mm gap) |
Three repeated sweep tests showed consistent trends, confirming the repeatability of the resonance phenomena. Figure 1 (see link below) shows the typical test setup. The key resonance speeds and corresponding frequencies are summarized in Table 2. It is evident that both rotational frequency and meshing frequency excitations produce significant vibration peaks. The rotational frequency resonances appear at speeds around 3420, 4500, 5245, and 6120 rpm, corresponding to frequencies of 57, 75, 87.4, and 102 Hz, respectively. The meshing frequency resonances occur at 4300, 5300, 6100, and 6400 rpm, corresponding to frequencies of 3079.3, 3814.2, 4424.7, and 5097.1 Hz. These experimental data serve as the benchmark for validating our finite element model.
The radial and axial vibration amplitudes at the rotational frequency were both high, indicating that the system experiences complex three-dimensional motion. The meshing frequency vibration amplitudes were significantly larger than those at the rotational frequency, suggesting that the meshing excitation is the dominant source of vibration in the star-planetary herringbone gear system.
| Resonance Type | Resonance Speed (rpm) | Corresponding Frequency (Hz) |
|---|---|---|
| Rotational frequency (radial) | 4500 | 75 |
| Rotational frequency (axial) | 3420 | 57 |
| Rotational frequency (axial) | 6120 | 102 |
| Rotational frequency (axial) | 5245 | 87.4 |
| Meshing frequency (radial) | 4300 | 3079.3 |
| Meshing frequency (radial) | 5300 | 3814.2 |
| Meshing frequency (radial) | 6100 | 4424.7 |
| Meshing frequency (radial) | 6400 | 5097.1 |
Three-Dimensional Finite Element Modeling
Overall Model Strategy
Due to the complexity of the test rig, we modeled only the test article itself, excluding the auxiliary gearboxes. This decision is justified because the auxiliary gearboxes have the same transmission ratio and meshing frequency as the test article; their influence is primarily on the excitation amplitude, not the excitation frequency. For the modal analysis, only the natural frequencies and mode shapes of the test article are of interest. The finite element model was built using ANSYS Workbench. The system consists of five planetary gears equally spaced around the sun gear, with herringbone tooth profiles. The input shaft is connected through a spline, a diaphragm coupling, and a splined coupling, all modeled using bushing elements with appropriate stiffness values (Table 3). Two support bearings connect the input shaft to the housing, which is fixed. The output shaft is connected to the ring gear via a rigid bond, and the planet gears are mounted on planet pins through double-row cylindrical roller bearings.
| Component | Radial Stiffness (N/m) | Axial Stiffness (N/m) | Tilting Stiffness (N·m/rad) | Torsional Stiffness (N·m/rad) |
|---|---|---|---|---|
| Spline (sun gear) | 10⁹ | 0 | 0 | 1.54×10⁸ |
| Diaphragm coupling | 10⁸ | 8×10⁶ | 51,182 | 1.05×10⁷ |
| Splined coupling | 10⁷ | 0 | 100 | 10⁶ |
| Inner bearing | 10⁶ | 10⁶ | 10³ | – |
| Input shaft bearing 1 | 1.5×10⁸ | 8.22×10⁶ | 4×10⁴ | – |
| Input shaft bearing 2 | 1.5×10⁶ | 8.22×10⁶ | 4×10⁴ | – |
| Output shaft ball bearing | 1.715×10⁸ | 0.85×10⁸ | 10⁶ | – |
| Output shaft roller bearing | 7.2×10⁸ | – | 10⁶ | – |
| Planet gear bearing (per row) | 1.61×10⁹ | – | – | – |
Simplification of Gear Bodies
To reduce computational cost, the gear bodies were simplified by removing the tooth profiles and a portion of the rim. The removed mass was compensated by adding point masses at the gear centers. The validity of this simplification was verified by comparing the free-free modal frequencies of the original and simplified models. Table 4 shows the free-free modal frequencies of the sun gear before and after simplification. The maximum error is only 1.16%, and the mode shapes are identical. Similar validations were performed for the planet gears (maximum error 2.22%) and the ring gear (maximum error 6.59%). This simplification allows us to retain the correct inertia and flexibility of the gear bodies while enabling the use of spring elements for meshing connections.
| Mode | Original (Hz) | Simplified (Hz) | Error (%) |
|---|---|---|---|
| 1 | 1946.7 | 1969.1 | 1.15 |
| 2 | 1946.8 | 1969.3 | 1.16 |
| 3 | 2617.8 | 2621.1 | 0.13 |
| 4 | 2620.0 | 2621.7 | 0.06 |
| 5 | 5328.2 | 5353.5 | 0.47 |
| 6 | 5334.8 | 5354.6 | 0.37 |
Modeling of Herringbone Gear Meshing
The herringbone gear is treated as two separate helical gear halves with opposite helix directions. For each half, we created remote points on the base circles of the sun gear, planet gears, and ring gear. The sun gear and each planet gear were connected by a spring element (COMBIN14 replaced by MATRIX27) representing the meshing stiffness. Similarly, each planet gear was connected to a central remote point on the ring gear’s base circle via springs. The stiffness matrix for each meshing pair is a 12×12 matrix derived from a quasi-static contact analysis using Abaqus. The meshing force along the line of action is given by the relative displacement and stiffness.
The relative displacement at the mesh for an external helical gear pair (sun-planet) in the mesh plane is:
$$
\delta_m = \left[ (x_p – x_g)\sin\alpha_n + (y_p – y_g)\cos\alpha_n + (r_p\theta_{pz} + r_g\theta_{gz}) \right]\cos\beta + \left[ (r_p\theta_{py} – r_g\theta_{gy})\cos\alpha_n + (r_p\theta_{px} – r_g\theta_{gx})\sin\alpha_n + (z_g – z_p) \right]\sin\beta – E_{ste}(t)
$$
where $x_p, y_p, z_p, \theta_{pj}$ are the translations and rotations of the pinion remote point, $x_g, y_g, z_g, \theta_{gj}$ are those of the gear, $\alpha_n$ is the pressure angle, $\beta$ is the helix angle, $r_p, r_g$ are the base circle radii, and $E_{ste}$ is the static transmission error.
The time-varying meshing stiffness is expressed as:
$$
K(t) = K_m + k_0 \sin(\Omega_{mi} t)
$$
where $K_m$ is the mean stiffness and $k_0$ is the fluctuation amplitude. For the herringbone gears under study, the values obtained from Abaqus contact simulations are given in Table 5.
| Parameter | Value (N/m) |
|---|---|
| $K_m$ | 1.65×10⁹ |
| $k_0$ | 0.05×10⁹ |
The mesh force vector is related to the displacement vector by:
$$
F_m = K(t)\boldsymbol{\Gamma} \mathbf{q} + c_m \boldsymbol{\Gamma} \dot{\mathbf{q}}
$$
where $\boldsymbol{\Gamma}$ is the transformation vector from gear degrees of freedom to the mesh line of action, and $\mathbf{q}$ is the 12-degree-of-freedom vector for the gear pair. The stiffness matrix in the global coordinate system is then assembled as:
$$
\mathbf{K}^m = K_m \begin{bmatrix} \mathbf{K}_{11}^m & \mathbf{K}_{12}^m \\ \mathbf{K}_{21}^m & \mathbf{K}_{22}^m \end{bmatrix}
$$
where each submatrix $\mathbf{K}_{ij}^m$ is a 6×6 matrix whose components depend on the pressure angle, helix angle, and base radius. For the sun-planet external mesh, the components of $\mathbf{K}_{11}^m$ (for example) are:
$$
\mathbf{K}_{11}^m = \begin{bmatrix}
c^2bs^2 & cc^2bs & -cbssb & cb r_p s^2 sb & ccb r_p s sb & c^2b r_p s \\
cc^2bs & c^2c^2b & -ccbsb & ccb r_p s sb & c^2cb r_p s b & cc^2b r_p \\
-cbssb & -ccbsb & s^2b & -r_p s s^2b & -c r_p s^2b & -cb r_p s b \\
cb r_p s^2 sb & ccb r_p s sb & -r_p s s^2b & r_p^2 s^2 s^2b & c r_p^2 s s^2b & cb r_p^2 s sb \\
ccb r_p s sb & c^2cb r_p s b & -c r_p s^2b & c r_p^2 s s^2b & c^2 r_p^2 s^2b & ccb r_p^2 s b \\
c^2b r_p s & cc^2b r_p & -cb r_p s b & cb r_p^2 s sb & ccb r_p^2 s b & c^2b r_p^2
\end{bmatrix}
$$
where $c = \cos\alpha_n$, $s = \sin\alpha_n$, $cb = \cos\beta$, $sb = \sin\beta$, and $\alpha_n$ is modified by the planet angular position. For the internal meshing (planet-ring), the matrix has a similar form but with sign changes in the off-diagonal blocks due to the internal contact geometry. The complete set of ten external and ten internal meshing stiffness matrices were implemented in ANSYS using MATRIX27 elements via APDL commands, replacing the initial COMBIN14 elements.
Modal Analysis Results and Discussion
Before performing modal analysis, a static structural analysis was conducted to verify the torque transmission capability. An input torque of 25,590 N·m was applied, and the output shaft was constrained in the circumferential direction. The reaction torque at the output was 75,438 N·m, which is within 0.19% of the theoretical value of 75,580 N·m, confirming the model’s load path accuracy. The modal analysis was then performed to extract all modes up to 5500 Hz, covering the first two meshing harmonics. A total of over 200 modes were identified, but only those with significant participation in the experimental resonance frequencies are discussed here.
Figure 2 shows the distribution of natural frequencies in the low-frequency range (below 125 Hz). Three modes were found that correspond to rotational frequency excitations. Their mode shapes are predominantly axial or radial at the input couplings. Table 6 compares these simulated frequencies with the experimental values from Table 2. There is an excellent match for three of the four rotational frequency resonances. The mode at 56.14 Hz (simulated) corresponds to the experimental 57 Hz axial resonance; the mode at 84.5 Hz (simulated) corresponds to the experimental 87.4 Hz axial resonance; and the mode at 104 Hz (simulated) corresponds to the experimental 102 Hz axial resonance. The experimental 75 Hz radial resonance was not found in the simulation, likely due to influences from the test rig’s auxiliary components that were not modeled. The direction of vibration (axial or radial) in the simulation matches the experimental observations, confirming the reliability of the model for rotational frequency excitations.
| Simulated Frequency (Hz) | Experimental Frequency (Hz) | Error (%) | Mode Description |
|---|---|---|---|
| 56.14 | 57 | 1.53 | Axial at splined coupling |
| – | 75 | – | Not captured (likely external) |
| 84.5 | 87.4 | 3.43 | Axial at diaphragm and splined coupling |
| 104 | 102 | -1.92 | Radial at splined coupling |
For the meshing frequency resonances, we focused on modes that involve significant gear pair deformation, particularly those of the ring gear and the gear bodies. After screening hundreds of modes, we identified a family of modes characterized by a five-nodal diameter (5D) vibration pattern on the ring gear, coupled with torsional vibrations of the sun and planet gears. Table 7 lists the simulated frequencies that match the experimental meshing frequency resonances. The maximum error is 6.79%, which is acceptable given the simplifications and uncertainties in damping and boundary conditions. All these modes exhibit a clear 5D ring gear pattern. For example, the mode at 3078.9 Hz corresponds to the experimental 3079.3 Hz; the mode at 3822.1 Hz corresponds to 3814.2 Hz; the mode at 4351.2 Hz corresponds to 4424.7 Hz; and the mode at 5468.5 Hz corresponds to 5097.1 Hz. The relatively larger error at the highest frequency may be due to increased sensitivity to model parameters at higher-order modes.
| Simulated Frequency (Hz) | Experimental Frequency (Hz) | Error (%) | Mode Description |
|---|---|---|---|
| 3078.9 | 3079.3 | 0.01 | 5D ring + torsional gears |
| 3822.1 | 3814.2 | -0.21 | 5D ring + torsional gears |
| 4351.2 | 4424.7 | 1.69 | 5D ring + torsional gears |
| 5468.5 | 5097.1 | -6.79 | 5D ring + torsional gears |
Furthermore, we found three additional modes with the same characteristic (5D ring gear coupled with torsional gear modes) at frequencies of 3839.5 Hz, 4055.9 Hz, and 4221.4 Hz. These correspond to resonance speeds of 5357, 5659, and 5890 rpm, respectively. Although these peaks were not as prominent in the experiment as the four listed in Table 2, they were indeed present as smaller resonance peaks in the sweep data (see Figure 5 of the original paper). This confirms that the 5D ring gear mode combined with torsional gear vibration is the fundamental mechanism behind the meshing frequency resonances in this star-planetary herringbone gear system.
The physical explanation is as follows: The ring gear, being a thin-shell structure, has natural modes at specific nodal diameters. The five-nodal diameter mode is particularly susceptible to excitation by the meshing forces because the five planetary gears (with their equally spaced 72° intervals) create a periodic forcing pattern that can strongly couple with a 5D mode shape. The torsional vibration of the sun and planet gears adds an additional degree of freedom that modifies the overall modal effective mass. Therefore, in the design of star-planetary herringbone gear systems, it is essential to avoid the coincidence of the meshing frequency (and its harmonics) with any of the ring gear’s 5D modes to prevent severe vibration.
Conclusion
A comprehensive three-dimensional finite element model for the modal analysis of a star-planetary herringbone gear transmission system has been developed and experimentally validated. The key findings are:
- The proposed modeling method, which replaces gear teeth with stiffness matrices derived from quasi-static contact simulations and integrates them into a 3D FE model via spring elements, accurately captures the dynamic coupling of herringbone gear pairs without the computational burden of contact algorithms.
- The simulated rotational frequency resonance modes (below 125 Hz) show excellent agreement with experimental data, with errors within 3.4%. The mode shapes correctly predict the direction (axial/radial) of vibration.
- The meshing frequency resonance modes are dominated by a five-nodal diameter vibration pattern of the ring gear, coupled with torsional vibrations of the sun and planet gears. Up to seven such modes were identified within the frequency range of interest, and their frequencies match the experimental resonance peaks with errors under 7%.
- This work demonstrates that the 5D ring gear mode is a critical design concern for star-planetary herringbone gear systems. The developed 3D FE model provides a reliable tool for predicting and avoiding such resonances during the design phase.
Future work will include parametric studies to investigate the influence of ring gear thickness, planet mesh phasing, and bearing stiffness on the critical modal frequencies. Additionally, the model will be extended to include damping effects to enable forced response predictions and quantitative vibration level assessments.