Gear vibration and noise have long been critical concerns in transmission design. Extensive literature identifies the transmission error (TE), arising from the combined effects of time-varying mesh stiffness and manufacturing inaccuracies, as the primary excitation source. While profile modifications are commonly employed to minimize TE, the dynamic excitation from tooth surface friction often receives less attention. Frictional forces not only induce translational vibration perpendicular to the line of action but also generate frictional moments that constrain torsional vibration. Particularly under poor lubrication or when microscopic surface damage occurs, friction becomes a significant source of vibration. Therefore, a thorough analysis of the dynamic excitation mechanisms due to tooth friction, especially the impact of microscopic surface changes, is essential.
This investigation focuses on the influence of tooth surface friction on the dynamic behavior of spur gear systems. We develop a nonlinear dynamic model incorporating the effects of friction and analyze how variations in surface roughness alter the system’s vibration signatures. This work aims to enhance the understanding of friction-induced vibrations and provide insights useful for the fault diagnosis of micro-damaged gear surfaces.
1. Development of the Dynamic Model
1.1 Gear Meshing Process Analysis
Consider a spur gear pair with a contact ratio greater than 1 but less than 2. The pinion rotates clockwise under an input torque \(T_p\), while the gear carries a load torque \(T_g\). The analysis focuses on the forces during the meshing cycle. Two pairs of teeth are in contact simultaneously during the double-pair contact zones. For the pinion, Tooth Pair 1 is subject to a normal force \(N_{p1}\) and a corresponding friction force \(F_{p1}\), while Tooth Pair 2 experiences \(N_{p2}\) and \(F_{p2}\). The direction of the friction force on a given tooth reverses as the contact point passes the pitch point C. The moment arms of these friction forces relative to the pinion center, \(H_{p1}\) and \(H_{p2}\), vary with the angular position.
1.2 Six-Degree-of-Freedom Nonlinear Model
A lumped-parameter model with six degrees of freedom (6-DOF) is established to comprehensively capture the effects of both normal meshing forces and friction forces. The model accounts for translational motions (\(x, y\)) and rotational motions (\(\theta\)) for both the pinion and the gear.
The dynamic relative displacement along the line of action, known as the Dynamic Transmission Error (DTE), is defined as:
$$ \delta(t) = R_p \theta_p(t) – R_g \theta_g(t) + y_p(t) – y_g(t) + e(t) $$
Here, \(R\) denotes the base circle radius, and \(e(t)\) represents the composite static transmission error, which includes manufacturing errors and surface topography effects:
$$ e(t) = (e_r + e_{S_{avg}}) \cos(Z_p \Omega_p t + \phi) $$
where \(e_r\) is the error from manufacturing precision, \(e_{S_{avg}}\) is the error related to surface roughness, and \(\phi\) is an initial phase.
The normal meshing force for a tooth pair \(i\) (\(i=1,2\)) is modeled as:
$$ N_{pi}(t) = -K_{mi}(t) \delta(t) – c_{mi}(t) \dot{\delta}(t), \quad N_{gi}(t) = -N_{pi}(t) $$
The friction force is proportional to the normal force via a time-varying friction coefficient \(f_i(t)\) and a direction coefficient \(\lambda(t)\):
$$ F_{pi}(t) = \lambda(t) f_i(t) N_{pi}(t), \quad F_{gi}(t) = -F_{pi}(t) $$
The direction coefficient \(\lambda(t)\) changes sign at the pitch point. It can be defined using the roll angle \(\alpha\):
$$ \lambda(t) = \text{sign}\left[ (\alpha_C – \alpha_A) – \text{mod}(\Omega_p t, \alpha_D – \alpha_A) \right] $$
The governing equations of motion for the 6-DOF system are derived from Newton’s second law:
$$
\begin{aligned}
m_p \ddot{x}_p + c_{px} \dot{x}_p + K_{px} x_p &= \sum_{i=1}^{2} F_{pi}(t) \\
m_p \ddot{y}_p + c_{py} \dot{y}_p + K_{py} y_p &= \sum_{i=1}^{2} N_{pi}(t) \\
I_p \ddot{\theta}_p &= T_p – \sum_{i=1}^{2} R_p N_{pi}(t) – \sum_{i=1}^{2} H_{pi}(t) F_{pi}(t) \\
m_g \ddot{x}_g + c_{gx} \dot{x}_g + K_{gx} x_g &= \sum_{i=1}^{2} F_{gi}(t) \\
m_g \ddot{y}_g + c_{gy} \dot{y}_g + K_{gy} y_g &= \sum_{i=1}^{2} N_{gi}(t) \\
I_g \ddot{\theta}_g &= -T_g + \sum_{i=1}^{2} R_g N_{gi}(t) + \sum_{i=1}^{2} H_{gi}(t) F_{gi}(t)
\end{aligned}
$$
The dynamic bearing forces, which are key excitations for the gearbox housing, are calculated as:
$$
\begin{aligned}
F_{bpx}(t) &= -K_{px}x_p(t) – c_{px}\dot{x}_p(t) \\
F_{bpy}(t) &= -K_{py}y_p(t) – c_{py}\dot{y}_p(t) \\
F_{bgx}(t) &= -K_{gx}x_g(t) – c_{gx}\dot{x}_g(t) \\
F_{bgy}(t) &= -K_{gy}y_g(t) – c_{gy}\dot{y}_g(t)
\end{aligned}
$$
2. Case Study and Parameter Determination
2.1 Gear Pair Parameters
We analyze an FZG Type A test spur gear pair, designed with profile shifts to achieve high sliding velocities, making friction effects pronounced. The main parameters are summarized below.
| Parameter / Unit | Pinion | Gear |
|---|---|---|
| Number of Teeth, \(Z\) | 16 | 24 |
| Module, \(m_n\) (mm) | 4.5 | |
| Pressure Angle, \(\alpha_0\) (deg) | 20 | |
| Profile Shift Coefficient, \(x\) | 0.8635 | -0.5103 |
| Base Circle Radius, \(R\) (mm) | 33.829 | 50.744 |
| Contact Ratio, \(\varepsilon_\alpha\) | 1.327 | |
| Mass, \(m\) (kg) | 0.676 | 1.084 |
| Moment of Inertia, \(I\) (kg·mm²) | 407 | 1168 |
| Bearing Stiffness, \(K_b\) (N/m) | 3.5e9 | |
| Operating Speed (rpm) | 900 | 600 |
| Torque, \(T\) (Nm) | 90 | 135 |
The corresponding roll angles for key meshing points (A: start of contact, B: start of single-pair contact, C: pitch point, D: end of single-pair contact, E: end of contact) are calculated based on the actual meshing geometry after profile shift.
| Roll Angle (rad) | Value |
|---|---|
| \(\alpha_A\) | 0.324 |
| \(\alpha_B\) | 0.453 |
| \(\alpha_C\) | 0.413 |
| \(\alpha_D\) | 0.717 |
| \(\alpha_E\) | 0.845 |
| Operating Pressure Angle, \(\alpha’\) (rad) | 0.392 |
2.2 Time-Varying Mesh Stiffness
The time-varying mesh stiffness \(K_m(t)\) is a primary internal excitation for a spur gear system. Accurate calculation is crucial. Using dedicated gear analysis software, the stiffness for a single tooth pair over its path of contact is obtained. For a spur gear pair with a contact ratio between 1 and 2, the total mesh stiffness at any time is the sum of the stiffnesses of the two simultaneously engaged tooth pairs, which are phased according to the base pitch. The stiffness exhibits significant variations at the transitions between single and double tooth contact zones (points B, D, E).
2.3 Time-Varying Friction Coefficient
The friction coefficient on gear tooth surfaces is not constant. It depends on the lubrication regime, relative sliding velocity, and surface roughness. For the mixed lubrication conditions typical in such gears, the Benedict & Kelly model is employed:
$$ f_i(t) = \frac{0.0137 \times 1.13}{1.13 – S_{avg}} \log_{10} \left[ \frac{29700 T_p}{\eta \xi_{si}(t) \xi_{ei}^2(t) Z_p R_p \cos\alpha_0 \cos\alpha’} \right] $$
where \(S_{avg}\) is the composite surface roughness (\(\mu m\)), \(\eta\) is the dynamic viscosity of the lubricant, \(\xi_s(t)\) is the sliding velocity, and \(\xi_e(t)\) is the entraining velocity at the contact point for tooth pair \(i\). The sliding and entraining velocities are functions of the friction force arms \(H_{pi}(t)\) and \(H_{gi}(t)\) and the rotational speeds \(\Omega_p\) and \(\Omega_g\). This model explicitly links the friction coefficient to the surface roughness \(S_{avg}\). Calculations for three different roughness values show that the coefficient increases with roughness and reaches a maximum near the pitch point where sliding velocity is zero but the \(\log()\) argument is minimized.
3. Numerical Simulation Results and Discussion
The nonlinear time-varying system of equations is solved using a variable-step 4th-5th order Runge-Kutta numerical integration method. The dynamic responses for spur gear systems with three different tooth surface roughness values (\(S_{avg} = 0.325 \mu m, 0.53 \mu m, 0.991 \mu m\)) are compared to investigate the effect of surface micro-geometry.
3.1 Dynamic Meshing and Friction Forces
The dynamic normal force between the gears shows periodic impulses at the transition points (B, D, E) due to changes in the number of load-sharing teeth. Interestingly, as surface roughness increases, the amplitude of these normal force fluctuations shows a decreasing trend. For instance, the peak normal force reduces from approximately 3.6 kN for the smoothest surface to about 3.0 kN for the roughest surface. This suggests that increased friction may have a damping effect on vibrations along the line of action.
In contrast, the dynamic friction force is significantly affected. The total friction force exhibits a sharp impulse when a tooth pair passes the pitch point (point C), as the friction direction reverses. The magnitude of this friction force increases substantially with increased surface roughness, directly due to the higher friction coefficient predicted by the lubrication model.
3.2 Dynamic Bearing Forces
The bearing forces are the primary excitation transmitted to the gearbox housing. The bearing force in the line-of-action (Y-direction) shows clear impulses at the meshing frequency and its harmonics, corresponding to the normal force shocks. The frequency spectrum reveals that increasing surface roughness amplifies low-frequency components while slightly reducing some high-frequency content.
The bearing force in the friction direction (X-direction, off-line-of-action) displays a markedly different character. A dominant impulse occurs at the pitch point passage frequency (which equals the meshing frequency for a spur gear), caused by the reversal of friction force direction. The time-domain amplitude of this force increases noticeably with increasing roughness. Its frequency spectrum is also dominated by the meshing frequency and its lower-order harmonics, with amplitudes growing as roughness increases.
3.3 Dynamic Transmission Error (DTE)
The Dynamic Transmission Error is a fundamental metric for gear vibration. Analysis of the DTE signal reveals a counter-intuitive result: as tooth surface roughness increases, the amplitude of the DTE fluctuations decreases. Specifically, the lower-order harmonic components of the DTE spectrum show significant reduction, while higher-order components are less affected. This phenomenon can be explained by the role of friction: the friction force generates a moment that opposes the driving torque over a significant portion of the meshing cycle. This opposition acts to suppress the relative torsional vibration between the pinion and gear, thereby reducing the primary vibration along the line of action, i.e., the DTE. This indicates that signals purely in the line-of-action direction may not be sensitive indicators of increasing tooth surface roughness or incipient micro-pitting.
4. Experimental Validation
4.1 Test Setup
Experiments were conducted on a standard FZG test rig, which is a back-to-back (power recirculating) gear test setup. The test specimens were the FZG Type A spur gear pairs. Vibration acceleration was measured using piezoelectric accelerometers mounted on the bearing housings of both the pinion and gear, in both the line-of-action (Y) and off-line-of-action (X) directions. An additional accelerometer was placed on the central surface of the gearbox housing. Data was acquired and processed using a commercial signal analyzer. Tests were performed under identical load and speed conditions (900 rpm pinion speed, 90 Nm input torque, oil bath lubrication) for two gear states: one with a new, undamaged tooth surface and one with a deliberately induced scuffed (damaged) surface exhibiting significantly higher roughness.
4.2 Comparison of Results and Analysis
The time-domain acceleration signals from the experiment for the damaged surface condition show good qualitative agreement with the simulation predictions. On the pinion bearing housing, the line-of-action (Y) acceleration shows clear periodic shocks matching the theoretical meshing period. The off-line-of-action (X) acceleration shows a sustained, slightly oscillatory decay following each major impulse, corresponding to the friction force reversal event, which also aligns with simulation results.
A comparative frequency-domain analysis of the experimental data for the undamaged vs. damaged spur gear surfaces provides strong validation:
Off-Line-of-Action (X-direction): The spectral structure (dominant meshing harmonics) remains largely unchanged between the two states. However, the amplitude at specific high harmonics (e.g., the 22nd multiple of meshing frequency) increased by approximately 2.6 m/s² for the damaged (rougher) surface. This confirms the simulation finding that increased roughness exacerbates vibration in the friction direction.
Line-of-Action (Y-direction): The change in vibration amplitude for the damaged surface was minimal, with a slight decrease (0.3 m/s²) observed at a dominant meshing harmonic. This aligns with the simulation prediction that DTE and line-of-action vibration are somewhat suppressed by increased friction, or alternatively, that increased roughness may promote better damping from the lubricant film.
Gearbox Housing: The housing vibration represents a superposition of forces from all directions and includes structural resonances. The spectrum for the damaged surface condition showed the same dominant meshing harmonics as the undamaged condition, but with a clear overall increase in amplitude (up to 1.5 m/s²). This demonstrates that while the effect on individual directional bearings forces might be complex, the overall vibration severity detectable on the gearbox increases with tooth surface deterioration.
5. Conclusions
This study presented a comprehensive nonlinear dynamic analysis of a spur gear system, explicitly incorporating the effects of time-varying tooth surface friction. A 6-DOF model was developed, and the friction coefficient was linked to surface roughness via a mixed lubrication model. Numerical simulations and experimental tests lead to the following key conclusions:
- Tooth surface friction has a dual role: While it acts as an excitation source causing vibration in the direction perpendicular to the line of action (OLOA), the friction moment can simultaneously suppress the relative torsional vibration, leading to a reduction in the Dynamic Transmission Error (DTE) and vibration along the line of action (LOA).
- Surface roughness significantly influences dynamics: An increase in tooth surface roughness leads to a higher time-varying friction coefficient. This results in amplified dynamic bearing forces and vibration acceleration in the off-line-of-action direction. Conversely, the vibration along the line of action may show little increase or even a slight decrease.
- Diagnostic implication: Monitoring vibration in a single direction, particularly the line-of-action, may be insufficient for detecting early-stage tooth surface degradation (e.g., micro-pitting, increased roughness). A more robust fault diagnosis strategy for spur gear surface conditions should involve the comparative analysis of vibration signals from both the line-of-action and the off-line-of-action (friction) directions. An increase in OLOA vibration concurrent with stable or slightly reduced LOA vibration can be a characteristic signature of worsening tooth surface micro-geometry.
The findings underscore the importance of including friction models in high-fidelity spur gear dynamics simulations, especially for applications where surface conditions are critical, such as in fault diagnosis and prognostics for geared transmission systems.