The analysis of nonlinear dynamic behavior in gear transmission systems is paramount for ensuring reliability, efficiency, and quiet operation in high-performance applications. Among various gear types, the herringbone gear stands out for its exceptional load-carrying capacity and inherent ability to cancel out axial forces, making it a cornerstone of power transmission in marine propulsion and other heavy-duty, high-speed machinery. However, the very nature of gear meshing introduces complex nonlinearities. Factors such as time-varying mesh stiffness, gear backlash, and notably, tooth surface friction, can induce significant vibrations and noise. These phenomena not only compromise cabin comfort on vessels but also pose potential threats to the structural integrity and safety of the entire transmission system. Therefore, developing a refined dynamic model that accurately incorporates these nonlinearities, especially tooth surface friction, is of critical engineering importance for the design and optimization of herringbone gear systems.
This article presents a comprehensive investigation into the nonlinear amplitude-frequency characteristics of a marine herringbone gear transmission system, with a focused analysis on the influence of tooth surface friction. We establish a high-fidelity, coupled bending-torsional-axial-pendular nonlinear dynamic model and employ numerical methods to unravel the complex interactions within the system.
1. Development of a 24-DOF Coupled Nonlinear Dynamic Model for Herringbone Gears
To capture the complete dynamic response, a lumped mass parameter approach is adopted. The herringbone gear pair is conceptually decomposed into two simultaneously meshing helical gear halves. The model accounts for six degrees of freedom (DOF) for each of the four main inertial components: the left and right halves of the driving pinion and the driven wheel. This results in a total of 24 degrees of freedom, enabling a fully coupled analysis of translational vibrations in the x, y, and z directions, rotational vibrations (tilting) about the x and y axes, and torsional vibration about the z-axis for each component.
The model integrates several key nonlinear and time-varying factors:
- Time-varying Mesh Stiffness (k_m(t)): Represented as a Fourier series expansion of the average mesh stiffness.
- Gear Backlash (2b): Modeled as a piecewise linear function, creating a dead-zone nonlinearity essential for predicting impacts and separations.
- Composite Transmission Error (e(t)): Treated as a kinematic excitation source, also expressed as a Fourier series.
- Tooth Surface Friction: Incorporated through friction forces and moments calculated based on elastohydrodynamic lubrication (EHL) theory, acting along the tooth profile tangent direction.
- Elastic Supports and Intermediate Shaft Sections: The stiffness and damping of bearings (supporting translational and tilting motions) and the flexibility of the shaft connecting the two halves of each herringbone gear are included.
The generalized coordinate vector q for the system is defined as:
$$ \mathbf{q} = \{ x_{pL}, y_{pL}, z_{pL}, \theta_{xpL}, \theta_{ypL}, \theta_{pL}, x_{gL}, y_{gL}, z_{gL}, \theta_{xgL}, \theta_{ygL}, \theta_{gL}, x_{pR}, y_{pR}, z_{pR}, \theta_{xpR}, \theta_{ypR}, \theta_{pR}, x_{gR}, y_{gR}, z_{gR}, \theta_{xgR}, \theta_{ygR}, \theta_{gR} \}^T $$
The equations of motion are derived using Newton’s second law. For conciseness, the governing differential equation for the left-half driving pinion in the x-direction is presented below. Similar equations are formulated for all 24 DOFs.
$$ m_{pL}\ddot{x}_{pL} + c_{pLx}\dot{x}_{pL} + k_{pLx}x_{pL} + c_{ptx}(\dot{x}_{pL} – \dot{x}_{pR}) + k_{ptx}(x_{pL} – x_{pR}) = – (k_{mL}\delta_L + c_{mL}\dot{\delta}_L)\cos\beta_L \sin\psi – F_{fpLx} $$
Where $F_{fpLx}$ is the x-component of the friction force on the left pinion. The relative displacement $\delta_L$ along the left gear pair’s line of action is the fundamental coupling term:
$$ \delta_L = [(x_{pL} – x_{gL})\sin\psi + (y_{pL} – y_{gL})\cos\psi + r_{bpL}\theta_{pL} – r_{bgL}\theta_{gL}]\cos\beta_L + [(r_{pL}\theta_{xpL} – r_{gL}\theta_{xgL})\sin\psi + (r_{pL}\theta_{ypL} – r_{gL}\theta_{ygL})\cos\psi + (z_{pL} – z_{gL})]\sin\beta_L – e_L(t) $$
The backlash nonlinearity is applied to this relative displacement:
$$ f(\delta_j) = \begin{cases}
\delta_j – b, & \delta_j > b \\
0, & -b \le \delta_j \le b \\
\delta_j + b, & \delta_j < -b
\end{cases} \quad (j = L, R) $$
The system of equations can be compactly written in matrix form:
$$ \mathbf{M}\ddot{\mathbf{q}} + \mathbf{C}\dot{\mathbf{q}} + \mathbf{K}\mathbf{q} = \mathbf{F}(t, \mathbf{q}, \dot{\mathbf{q}}) $$
where M, C, and K are the mass, damping, and stiffness matrices, respectively, and F is the force vector containing excitations from transmission error, external torque, and nonlinear friction forces.
2. Analysis of Meshing Characteristics and Friction Force Modeling
The calculation of time-varying mesh forces and friction effects requires a detailed understanding of the herringbone gear contact geometry. The contact line between mating teeth sweeps across the plane of action. Its length varies periodically depending on the transverse contact ratio ($\varepsilon_\alpha$) and the overlap ratio ($\varepsilon_\beta$).
The length of the contact line for the i-th tooth pair, $l_i$, is a piecewise function of its position in the mesh cycle. For the common case where $\varepsilon_\alpha \ge \varepsilon_\beta$:
$$ l_i(s) = \begin{cases}
\frac{s}{\sin\beta_b}, & 0 \le s < \varepsilon_\beta P_{bt} \\
\frac{b}{\cos\beta_b}, & \varepsilon_\beta P_{bt} \le s < \varepsilon_\alpha P_{bt} \\
\frac{-s + b\tan\beta_b + l_a}{\sin\beta_b}, & \varepsilon_\alpha P_{bt} \le s < (\varepsilon_\alpha+\varepsilon_\beta)P_{bt} \\
0, & (\varepsilon_\alpha+\varepsilon_\beta)P_{bt} \le s < NP_{bt}
\end{cases} $$
where $s$ is the distance traveled along the line of action, $\beta_b$ is the base helix angle, $P_{bt}$ is the base pitch, $b$ is the face width, and $l_a$ is the length of the path of contact.
The total normal mesh force $F_n$ is obtained by integrating the product of the linear mesh stiffness $k_v$ and the deflection $\delta$ along the total contact line length $L$ for all simultaneously engaged tooth pairs $N$:
$$ F_n = \sum_{i=1}^{N} \int_{a_i}^{b_i} k_v \, \delta \, dl $$
The limits of integration $a_i$ and $b_i$ are determined by the piecewise function for $l_i(s)$.
2.1 Elastohydrodynamic Lubrication (EHL) Friction Model
Accurately modeling tooth surface friction is critical. The friction coefficient $\mu$ is not constant but varies with operating conditions. We employ an EHL-based empirical model that accounts for sliding-rolling conditions, lubricant properties, and surface roughness. The model is expressed as:
$$ \mu = e^{f(S_R, P_h, \nu_0, S)} P_h^{b_2} |S_R|^{b_3} V_e^{b_6} \nu_0^{b_7} R^{b_8} $$
where the function $f$ is given by:
$$ f(S_R, P_h, \nu_0, S) = b_1 + b_4|S_R| P_h \log_{10}(\nu_0) + b_5 e^{-|S_R| P_h \log_{10}(\nu_0)} + b_9 S $$
The parameters in the model ($b_1$ through $b_9$) are empirically derived constants. The key input variables are:
- $S_R$: Slide-to-roll ratio ($2v_s/v_r$).
- $P_h$: Maximum Hertzian contact pressure.
- $\nu_0$: Dynamic viscosity of the lubricant at atmospheric pressure.
- $V_e$: Entrainment or rolling velocity ($v_r/2$).
- $R$: Equivalent radius of curvature at the contact point.
- $S$: Composite root-mean-square (RMS) surface roughness.
The friction force $f_i$ for the i-th tooth pair is then calculated by integrating the product of the local friction coefficient $\mu_{0i}$, linear stiffness, and deflection along its contact line:
$$ f_i = \int_{a_i}^{b_i} \mu_{0i} \, k_v \, \delta \, dl $$
The total friction force $F_f$ is the sum over all contacting pairs. This friction force acts tangentially to the tooth profile. Furthermore, it generates a friction moment $M_f$ about the gear center because the line of action of the friction force does not pass through the gear’s axis of rotation. This moment is calculated similarly by including the moment arm $r_{tk}$ (the transverse radius to the point of contact) in the integral.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $b_1$ | -8.9164 | $b_6$ | -0.1006 |
| $b_2$ | 1.0330 | $b_7$ | 0.7527 |
| $b_3$ | – | $b_8$ | – |
| $b_4$ | -0.3540 | $b_9$ | 0.6203 |
| $b_5$ | 2.8120 |
3. Non-Dimensionalization of the Dynamic Equations
To generalize the analysis and improve numerical stability, the equations of motion are non-dimensionalized. We define a characteristic frequency $\omega_n = \sqrt{k_{me}/m_c}$, where $k_{me}$ is the average mesh stiffness and $m_c$ is the equivalent mass. The dimensionless time and excitation frequency are defined as $\bar{t} = \omega_n t$ and $\Omega = \omega / \omega_n$, respectively. The displacement scale is chosen as the half-backlash $b$.
The dimensionless parameters are:
$$ \bar{x}_{ij} = \frac{x_{ij}}{b}, \quad \dot{\bar{x}}_{ij} = \frac{\dot{x}_{ij}}{b \omega_n}, \quad \ddot{\bar{x}}_{ij} = \frac{\ddot{x}_{ij}}{b \omega_n^2}, \quad \Delta_j = \frac{\delta_j}{b} $$
Applying this transformation to all equations yields a dimensionless system. The dimensionless mesh stiffness and error excitation become:
$$ \bar{k}_{mj}(\bar{t}) = 1 – \sum_{l=1}^{L} \varepsilon_l \cos(l\Omega\bar{t}) $$
$$ \bar{e}”_j(\bar{t}) = -\sum_{l=1}^{L} (l\Omega)^2 e_l \sin(l\Omega\bar{t}) $$
Importantly, the backlash function simplifies to a clearance of $\pm1$ in the dimensionless relative displacement $\Delta_j$:
$$ f(\Delta_j) = \begin{cases}
\Delta_j – 1, & \Delta_j > 1 \\
0, & -1 \le \Delta_j \le 1 \\
\Delta_j + 1, & \Delta_j < -1
\end{cases} $$
4. Numerical Solution and Analysis of Amplitude-Frequency Characteristics
The resulting set of strongly nonlinear, coupled ordinary differential equations is solved numerically using the fourth-order Runge-Kutta method. A representative set of system parameters for a marine herringbone gear transmission is used for the analysis, as summarized below.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 25 | 75 |
| Helix Angle (deg) | 24.43 | 24.43 |
| Normal Module (mm) | 6 | 6 |
| Face Width per Half (mm) | 55 | 55 |
| Mass (kg) | 7.11 | 64.75 |
| Moment of Inertia (kg·m²) | 0.012 | 0.7279 |
The input torque is set to 1400 Nm, and the dimensionless damping ratio $\xi$ is 0.145. The dynamic response is analyzed in terms of the dimensionless displacement $\Delta$ along the line of action over a range of dimensionless excitation frequencies $\Omega$.
4.1 Time-Domain Response and Phase Portraits
The system’s response is examined with and without the inclusion of tooth surface friction. Phase portraits and time history plots of $\Delta$ at key frequencies reveal the nature of the vibration.
- At $\Omega = 1$ (Primary Resonance Region): Both models (with and without friction) exhibit a periodic, near-harmonic response. The phase trajectory is a limit cycle, approximately elliptical. The gear pair is in a persistent single-sided contact state (impact), never crossing into the backlash region or separating. The inclusion of friction visibly reduces the amplitude of vibration, demonstrating its damping effect.
- At Subharmonic Frequencies ($\Omega = 1/2, 1/3$): The system exhibits superharmonic resonance. The phase portraits form complex, closed non-elliptical curves, indicating a periodic but non-sinusoidal response. The gear pair operates in a no-impact state within the backlash zone for portions of the cycle. Again, friction acts to suppress the vibration amplitude, smoothing the response.
4.2 Amplitude-Frequency Response and the Role of Friction
The peak-to-peak amplitude of the dimensionless dynamic transmission error $\Delta$ is plotted against the dimensionless excitation frequency $\Omega$ to construct the nonlinear frequency response curve. Comparing the curves with and without friction reveals profound insights.
| Frequency Band ($\Omega$) | System Behavior | Effect of Tooth Surface Friction |
|---|---|---|
| Low-Frequency (0 – 0.89) | Stable, low-amplitude response. Gear pair primarily in no-impact or light single-sided impact state. | Significant damping effect. Friction consistently reduces vibration amplitude across this band. |
| Non-Stationary Resonance Zone (0.95 – 1.23) | Region containing the primary resonance peak. Exhibits a classical “jump” phenomenon due to nonlinear stiffness/backlash. The amplitude is highly sensitive and shows hysteresis. | Damping effect is present but less pronounced relative to the large resonant amplitudes. The primary influence of friction here is to shift the jump-up frequency slightly higher (from ~0.89 to ~0.94), effectively increasing the apparent damping in the system. |
| High-Frequency (1.42 – 2.0) | Post-resonance, stable response. Amplitudes are lower than at resonance but may exhibit complex multi-periodic or chaotic behavior at specific frequencies. | Significant damping effect. Friction effectively attenuates vibrations, similar to the low-frequency band. |
A critical observation is that within the studied parameter range, the inclusion of friction does not qualitatively change the fundamental meshing state of the herringbone gear pair. The transitions between no-impact, single-sided impact, and double-sided impact (or separation) states are governed primarily by the combined effects of backlash, time-varying stiffness, and excitation force. Friction acts principally as a dissipative (damping) mechanism, extracting energy from the system and reducing the magnitude of oscillation without altering the state sequence.
5. Conclusion
This analysis provides a detailed theoretical framework for assessing the nonlinear dynamics of herringbone gear transmission systems, with a specific emphasis on the role of tooth surface friction. The 24-DOF coupled bending-torsional-axial-pendular model offers a high-degree-of-fidelity representation suitable for precision analysis. The use of an EHL-based friction model introduces a realistic, physically-grounded representation of the energy dissipation occurring at the gear mesh interface.
The key findings are:
- Tooth surface friction exhibits a consistent damping effect on the vibration amplitude of the herringbone gear pair along the line of action. This damping is most significant in the low-frequency ($\Omega < 0.89$) and high-frequency ($\Omega > 1.42$) regimes, where it can be a primary factor in controlling vibration levels.
- Within the intense non-stationary resonance zone ($0.95 < \Omega < 1.23$), while friction still provides damping, its effect is less dominant relative to the large amplitudes driven by parametric and nonlinear resonance. Its primary role in this region is to modestly increase the effective system damping, shifting resonance features like the jump frequency.
- For the operational conditions analyzed, tooth surface friction does not induce a change in the fundamental meshing impact state (e.g., from single-sided to double-sided impact) of the herringbone gear. Its influence is dissipative rather than state-altering.
These insights underscore the importance of incorporating detailed friction models in the dynamic simulation of high-performance herringbone gear systems. Accurate prediction of friction-induced damping is essential for reliable vibration and noise prognosis, which directly informs gear design, lubrication selection, and system operational guidelines for marine and other critical applications. Future work could explore the coupling between friction, thermal effects, and wear progression in a transient dynamic framework for herringbone gear systems.