In our research on high-power marine transmission systems, planetary herringbone gears have been increasingly adopted due to their superior load capacity, compact dimensions, and high reliability. However, the inherent complexity of planetary herringbone gear systems—with multiple meshing interfaces, centrifugal forces on the planet gears, and sensitivity to manufacturing errors—often leads to significant vibration and noise issues. Meeting the stringent noise requirements of modern naval vessels demands effective vibration reduction strategies. Through our systematic investigation, we have developed a phase tuning based design methodology that can suppress specific vibration modes in planetary herringbone gear transmissions by appropriately selecting gear tooth numbers and the count of planet gears. This approach relies on the analytical relationship between the dynamic meshing forces and the natural vibration patterns of the system, and it has been validated through numerical simulations.
Our work focuses on a typical planetary herringbone gear set used in a ship power system, where the sun gear acts as the input, the carrier as the output, and the ring gear is fixed. Each herringbone gear is modeled as two helical gears with opposite helix angles connected by an intermediate shaft section. This representation captures the coupled bending, axial, and torsional vibrations. The dynamic model we constructed includes all essential components: sun gear, planet gears, carrier, and ring gear, with each helical gear segment treated as a lumped mass node having four degrees of freedom: two translational radial displacements (x, y), an axial displacement (z), and a rotational angle (θ). Left and right sides of the herringbone gear are denoted by subscripts L and R. With N equally spaced planet gears, the total degrees of freedom for the system are 12 + 8N.
To establish the governing equations, we adopted a generalized finite element approach. The system is discretized into several basic sub-units: external meshing pairs (sun-planet), internal meshing pairs (planet-ring), carrier-planet support couplings, coupling between the two helical gear halves of a herringbone gear, and bearing support elements. For each sub-unit, we derived the mass, damping, and stiffness matrices, and then assembled them to form the global equation of motion:
This figure illustrates the lumped parameter model we employed, where each herringbone gear is split into left and right helical segments, and meshing interfaces are represented by springs and dampers along the line of action. The model fully accounts for the coupling between bending, axial, and torsional motions, which is crucial for accurately predicting the vibrational behavior of planetary herringbone gears.
$$ M \ddot{q}(t) + C \dot{q}(t) + K q(t) = F(t) $$
Here, \(M\), \(C\), and \(K\) are the global mass, damping, and stiffness matrices, respectively, while \(F(t)\) is the forcing vector arising from time-varying meshing stiffness and other excitations. The displacement vector \(q(t)\) contains all degrees of freedom for the sun, planet, and carrier components. The detailed expressions for these matrices follow the procedures outlined in our prior work.
Phase Tuning Theory for Vibration Suppression
The essence of phase tuning in planetary herringbone gear systems lies in the fact that the dynamic meshing forces on the sun gear (or ring gear) can be expressed as a Fourier series with harmonic orders \(l\) corresponding to the mesh frequency \(\omega_m\). By aligning or canceling these harmonic components through proper selection of tooth numbers and planet count, we can eliminate certain excitations and thus suppress the associated vibration modes. We begin by defining a local coordinate system for each planet gear. Let the radial direction from the carrier center to the n-th planet be the en1 axis, with en2 perpendicular to it. The dynamic meshing force between the sun gear and the n-th planet gear can be written in these local coordinates:
$$ F_n = F_{n1} e_{n1} + F_{n2} e_{n2} $$
The components \(F_{n1}\) and \(F_{n2}\) are expanded in Fourier series:
\[
\begin{aligned}
F_{n1} &= \sum_{l=0}^\infty \left[ a_l^n \sin(l(\omega_m t + \phi_n)) + b_l^n \cos(l(\omega_m t + \phi_n)) \right] \\
F_{n2} &= \sum_{l=0}^\infty \left[ c_l^n \sin(l(\omega_m t + \phi_n)) + d_l^n \cos(l(\omega_m t + \phi_n)) \right]
\end{aligned}
\]
where \(\phi_n\) is the meshing phase of the n-th planet gear. For equally spaced planets, the angular position of planet n is \(\psi_n = 2\pi (n-1)/N\). A transformation between the local \(\{e_{n1}, e_{n2}\}\) frame and the fixed \(\{i, j\}\) frame yields:
\[
\begin{bmatrix} F_{nx} \\ F_{ny} \end{bmatrix}
= \begin{bmatrix} \cos\psi_n & \sin\psi_n \\ -\sin\psi_n & \cos\psi_n \end{bmatrix}
\begin{bmatrix} F_{n1} \\ F_{n2} \end{bmatrix}
\]
The net force on the sun gear from all N planets is the sum over n. After substituting the Fourier expansions and using trigonometric identities, we obtain expressions for the l-th harmonic components of \(F_x\) and \(F_y\). The key step is the introduction of a phase tuning factor:
$$ k = \text{mod}\left( \frac{l Z_s}{N} \right) $$
where \(Z_s\) is the number of teeth on the sun gear, and \(N\) is the number of planet gears. This factor dictates whether the harmonic contributions from different planets cancel or reinforce each other. Using the well-known identities for sums of cosines and sines over equally spaced angles:
\[
\sum_{n=1}^N \cos\left( \frac{2\pi (n-1) m}{N} \right) =
\begin{cases}
0, & \text{if } m/N \text{ is not integer} \\
N, & \text{if } m/N \text{ is integer}
\end{cases}
\]
\[
\sum_{n=1}^N \sin\left( \frac{2\pi (n-1) m}{N} \right) = 0, \quad \text{for all } m
\]
We can then systematically evaluate the net force and moment on the sun gear. The results are summarized in Table 1, which lists the resulting vibration patterns for different values of the phase tuning factor \(k\) (which can be 0, 1, 2, …, N-1).
Table 1: Phase tuning laws for planetary herringbone gear systems with N planet gears (N=5 shown as example)
| \( \text{mod}(l Z_s / N) \) | \( l=0 \) | \( l=1, N-1 \) | \( l=2, N-2 \) | \( l=3, N-3 \) | \( l=4, N-4 \) |
|---|---|---|---|---|---|
| 0 | Torsional vibration | Torsional vibration | Torsional vibration | Torsional vibration | Torsional vibration |
| 1 | Translational vibration | Planet gear vibration | Planet gear vibration | Translational vibration | Torsional vibration |
| 2 | Planet gear vibration | Translational vibration | Translational vibration | Planet gear vibration | Torsional vibration |
| 3 | Planet gear vibration | Translational vibration | Translational vibration | Planet gear vibration | Torsional vibration |
| 4 | Translational vibration | Planet gear vibration | Planet gear vibration | Translational vibration | Torsional vibration |
The interpretation is as follows:
- When \(k = 0\), the net force components (\(F_x^l, F_y^l\)) both vanish, but the net moment \(T^l\) is nonzero. This means the sun gear experiences only torsional vibration (and axial motion), while its radial translational vibration is suppressed.
- When \(k = 1\) or \(k = N-1\), the net force components are nonzero, but the net moment is zero. In this case, translational vibration of the sun gear (and ring gear) is excited, while torsional vibration is suppressed.
- For other values of \(k\) (e.g., 2, 3, … when \(N \ge 4\)), both net force and net moment vanish, leading to complete suppression of both translational and torsional vibrations of the sun gear. However, the planet gears themselves may still vibrate in certain patterns (planet gear modes).
These relationships hold for any harmonic order \(l\), and they reveal that by simply changing the sun gear tooth number (thus altering the remainder \(Z_s \mod N\)), we can switch between different excitation patterns. This forms the basis of our phase tuning vibration reduction design.
Numerical Validation Using Two Design Cases
To verify the theoretical predictions, we designed two contrasting parameter sets for a planetary herringbone gear transmission, as detailed in Table 2. Both cases use the same helix angle and pressure angle, but differ in the tooth numbers of the sun gear, planet gears, and ring gear, as well as the input speed, while maintaining the same fundamental mesh frequency (1196 Hz) and its second harmonic (2392 Hz).
Table 2: Two design schemes for the planetary herringbone gear system
| Parameter | Scheme 1 | Scheme 2 |
|---|---|---|
| Sun gear teeth \(Z_s\) | 41 | 35 |
| Planet gear teeth \(Z_p\) | 31 | 32 |
| Ring gear teeth \(Z_r\) | 103 | 99 |
| Pressure angle (°) | 22.5 | 22.5 |
| Helix angle (°) | 30 | 30 |
| Number of planets \(N\) | 5 | 5 |
| Sun gear speed (r/min) | 1750 | 2050 |
| 1st mesh frequency (Hz) | 1196 | 1196 |
| 2nd mesh frequency (Hz) | 2392 | 2392 |
For Scheme 1, \(\text{mod}(Z_s / N) = 41 \mod 5 = 1\). According to Table 1, the first harmonic (l=1) and fourth harmonic (l=4) of the mesh frequency will excite translational vibration of the sun gear; the second and third harmonics (l=2,3) will excite planet gear vibration; while the fifth harmonic (l=5) will produce torsional vibration. For Scheme 2, \(\text{mod}(Z_s / N) = 35 \mod 5 = 0\). In this case, all harmonics will excite only torsional vibration of the sun gear. Therefore, by comparing the dynamic responses of the two schemes, we can directly validate the phase tuning predictions.
Natural Vibration Modes
Before analyzing the forced response, we first computed the natural frequencies and mode shapes of the planetary herringbone gear system for both schemes. Solving the eigenvalue problem associated with the undamped free vibration equation gave three distinct categories of modes: (1) central component axial-torsional modes (involving sun gear and ring gear undergoing axial and torsional motions), (2) planet gear modes (where the planet gears vibrate while the central components remain nearly stationary), and (3) central component translational modes (where the sun and ring translate in the radial plane). These mode categories are fundamental to understanding the response to harmonic excitation.
Forced Response Comparison
We applied the time-varying meshing stiffness excitation as the primary internal forcing. The waveform and harmonic content of the meshing stiffness are illustrated in the figure below (the actual numerical values were computed from the gear geometry).
The dynamic equations were solved in the frequency domain, and the steady-state displacement response of the sun gear was extracted. Since acceleration is proportional to displacement scaled by \(-\omega^2\), the displacement spectrum directly reflects the vibration intensity at each harmonic.
Results for Scheme 1:
The time-domain displacement of the sun gear shows a complex pattern. In the frequency domain, we observed that the first and fourth harmonics of the mesh frequency produce pronounced peaks in the radial (translational) displacement components, while the second and third harmonics exhibit negligible radial motion. Instead, the second and third harmonics produce strong planet gear vibrations (not shown here, but confirmed by examining planet gear displacements). The fifth harmonic excites mainly torsional (and axial) motion of the sun gear. These observations are in excellent agreement with the predictions of Table 1 for the case \(k=1\).
Results for Scheme 2:
The sun gear response shows that all harmonics (first through fifth) excite almost exclusively torsional vibration. The radial translational vibrations are suppressed across all harmonics. This matches the expectation for \(k=0\). The small residual radial motions that appear are due to unavoidable manufacturing errors and slight asymmetry in the numerical model, but they are orders of magnitude smaller than the torsional response.
The comparisons clearly demonstrate that by modifying only the sun gear tooth count (and correspondingly the planet and ring gear counts to maintain meshing compatibility), we can fundamentally alter the vibration pattern of the planetary herringbone gear system. This ability to selectively suppress torsional or translational resonances is extremely valuable in naval applications where low noise and precise positioning are critical.
Practical Implementation of Phase Tuning Design
Based on our analytical and numerical findings, we can outline a practical design procedure for planetary herringbone gear transmissions:
- Select the number of planet gears \(N\) (typically 3 to 6 for manufacturing and load-sharing reasons).
- Determine which vibration mode is most critical for the specific application. For example, if the system has a natural frequency close to the mesh frequency that would cause a torsional resonance of the sun gear, we need to suppress torsional excitation at that harmonic.
- Calculate the required remainder \(R = Z_s \mod N\) that yields the desired suppression. From Table 1, if we want to suppress torsional vibration at harmonic \(l\), we need \(k = \text{mod}(l Z_s / N) = 1\) or \(N-1\) (which excites translation instead). Conversely, if translational vibration is problematic, choose \(k=0\) to suppress it while leaving torsional excitation.
- Adjust the sun gear tooth count \(Z_s\) (and correspondingly planet gear and ring gear counts) to achieve the required remainder, while maintaining the desired gear ratio, center distance, and strength requirements.
- Verify through dynamic simulation as we have done here.
Note that when the remainder equals 2, 3, etc. (for N ≥ 4), both translational and torsional vibrations are suppressed for that harmonic, leaving only planet gear modes. This is often the most desirable outcome as it minimizes the vibration transmitted to the housing through the central components. Therefore, in many practical designs, we recommend selecting \(Z_s\) such that its remainder modulo N is not 0 or 1 (or N-1), i.e., the vibration pattern belongs to the planet gear mode family.
Conclusion
Through a rigorous analytical derivation and numerical verification, we have established a comprehensive phase tuning methodology tailored for planetary herringbone gear transmissions. The key conclusions are:
- When the sun gear tooth count is a multiple of the number of planet gears (remainder 0), the system experiences only torsional vibration under mesh frequency excitation, while translational vibrations are suppressed.
- When the remainder is 1 or N-1, translational vibrations are excited and torsional vibrations are suppressed at the corresponding harmonics. For other remainders, both translational and torsional vibrations of the sun gear are suppressed, leaving only planet gear vibrations.
- By strategically choosing the sun gear tooth number, we can eliminate the most dangerous resonance (either torsional or translational) and redirect the vibration energy into less critical planet gear modes.
- Our method is particularly effective for marine planetary herringbone gear systems, where noise and vibration control are paramount. The ability to suppress central component vibrations without adding mass or damping is a significant advantage.
Future work will extend this theory to consider the effects of manufacturing errors, tooth surface modifications, and more realistic bearing supports. Additionally, experimental validation on a test rig is planned to further confirm the practical applicability of the phase tuning approach for herringbone gears.