We investigate the vibration characteristics of a straight spur gear pair under the combined influence of random transmission error and random tooth surface friction parameters. By integrating statistical methods with a lumped-mass approach, we develop a three-degree-of-freedom bending‑torsional coupled dynamic model for a straight spur gear system. The random nature of gear errors and friction parameters is mathematically characterized using Gaussian white noise, and the coupled effects are numerically solved via the fourth‑order Runge–Kutta method. Our results reveal that the randomness in both error and friction significantly amplifies the dynamic response, leading to richer frequency sidebands and more chaotic phase trajectories. Among the two sources, the randomness of the transmission error exerts a stronger destabilizing effect on the gear system. This work provides theoretical insights for the robust dynamic design of straight spur gear transmissions.
Introduction
Modern gear transmissions are increasingly required to operate at high speeds and high power densities, making vibration and noise reduction a critical challenge. Manufacturing imperfections, assembly variations, and fluctuating operating conditions introduce randomness in the transmission error and the micro‑topography of the tooth surface. These random factors lead to unpredictable dynamic behaviors of the straight spur gear system. Although many studies have separately addressed deterministic gear errors or friction effects, the combined randomness of both error and friction in a unified straight spur gear model remains insufficiently explored. In this work, we aim to fill this gap by establishing a comprehensive dynamic model that simultaneously accounts for random transmission error and random friction parameters, and by analyzing their distinct and coupled impacts on the gear vibration response.
Dynamic Model of a Straight Spur Gear Pair with Random Error and Friction
We consider a straight spur gear pair (driving pinion and driven gear) as a three‑degree‑of‑freedom (3‑DOF) lumped‑mass system, as shown schematically below. The degrees of freedom include translational displacements along the x‑ and y‑ axes and the rotational displacement about the gear axis for each gear. The displacement vector is:
$$ \mathbf{q} = [x_p,\ y_p,\ \theta_p,\ x_g,\ y_g,\ \theta_g]^T $$
The bending‑torsional coupled equations of motion, including time‑varying mesh stiffness, tooth surface friction, and random error, are expressed as:
$$
\begin{cases}
m_p \ddot{x}_p + k_{xp} x_p + \sin\alpha\, k_m(t)\,\delta(t) = F_f(t)\sin\alpha \\
m_p \ddot{y}_p + k_{yp} y_p + \cos\alpha\, k_m(t)\,\delta(t) = -F_f(t)\cos\alpha \\
I_p \ddot{\theta}_p + R_p(t)\,k_m(t)\,\delta(t) = T_p(t) \\
m_g \ddot{x}_g + k_{xg} x_g – \sin\alpha\, k_m(t)\,\delta(t) = -F_f(t)\sin\alpha \\
m_g \ddot{y}_g + k_{yg} y_g – \cos\alpha\, k_m(t)\,\delta(t) = F_f(t)\cos\alpha \\
I_g \ddot{\theta}_g – R_g(t)\,k_m(t)\,\delta(t) = -T_g(t)
\end{cases}
$$
where the relative mesh displacement $\delta(t)$ is:
$$ \delta(t) = \sin\alpha (x_p – x_g) + \cos\alpha (y_p – y_g) + R_p(t)\theta_p – R_g(t)\theta_g + e(t) $$
Here, $m_i$, $I_i$ are the mass and moment of inertia of gear $i$ ($i=p,g$); $k_{xi}, k_{yi}$ are the equivalent support stiffness; $k_m(t)$ is the time‑varying mesh stiffness; $T_i(t)$ is the input/output torque; $R_i(t)$ is the instantaneous curvature radius along the line of action; $\alpha$ is the pressure angle; $F_f(t)$ is the time‑varying friction force; and $e(t)$ is the composite transmission error including random components. The gear pair parameters used in this study are listed in Table 1.
Table 1. Straight spur gear pair parameters
| Parameter | Driving pinion | Driven gear |
|---|---|---|
| Number of teeth | 33 | 26 |
| Accuracy grade | 6GJ | 6GJ |
| Mass (kg) | 10.6 | 7.43 |
| Module (mm) | 7 | 7 |
| Moment of inertia (kg·mm²) | 147 670 | 61 426 |
| Pressure angle (°) | 20 | 20 |
| Face width (mm) | 69 | 69 |
| Input speed (r/min) | 2 000 | — |
| Input torque (N·m) | 2 340.7 | — |
Randomness in Error and Tooth Surface Friction of Straight Spur Gears
Random Transmission Error
The total transmission error $e(t)$ of a straight spur gear pair consists of a deterministic periodic part and a random part. According to gear manufacturing tolerances (GB/T 10095), the tooth profile error $\Delta f_f$ and base pitch error $\Delta f_{pb}$ are treated as independent Gaussian random variables. The deterministic part follows a sinusoidal function:
$$ e_i(t) = e_m + E_i \sin(\omega t + \varphi_i) $$
where $\omega$ is the mesh frequency. Superposing a Gaussian white noise $\xi(t)$ with zero mean and variance $\sigma^2 = 0.0005$, we obtain the random error:
$$ e(t) = e_i(t) + \xi(t) $$
Figure 4 in the original paper (not reproduced here) shows the time history and frequency spectrum of the random error; the mesh frequency peak (1 100 Hz) and its second harmonic (2 200 Hz) are clearly observed, along with broadband noise due to the random component.
Tooth Surface Friction with Randomness
The friction force acting on a straight spur gear tooth is given by:
$$ F_f(t) = \mu(t)\,k_m(t)\,\delta(t) $$
where $\mu(t)$ is the coefficient of friction. The random surface roughness is assumed to follow a Gaussian distribution, leading to a random friction coefficient:
$$ \mu(t) = \mu_0 + \sigma_\mu\,\xi(t) $$
with mean $\mu_0 = 0.109$ and standard deviation $\sigma_\mu = 0.05$. The instantaneous curvature radii $R_p(t)$ and $R_g(t)$ also become random because the distance $s(t)$ from the pitch point to the instantaneous contact point inherits the randomness from the surface roughness:
$$
\begin{cases}
s(t) = s_\mu + \xi(t) \\
R_p(t) = r_1 \sin\alpha_p + s(t) \\
R_g(t) = r_2 \sin\alpha_g + s(t)
\end{cases}
$$
Here $s_\mu$ is the mean distance determined by the involute geometry. The mean value $s_\mu$ is computed from the geometric model shown in Fig. 8 of the original paper (not replicated). The resulting time‑varying curvature radii (Fig. 9 in the original) exhibit clear random fluctuations that affect the friction torque and ultimately the dynamic response of the straight spur gear system.
Dynamic Response Characteristics
We solve the 3‑DOF system using the fourth‑order fixed‑step Runge–Kutta method with a step size of 0.00015 s. The vibration responses in the x‑, y‑, and rotational directions for both the driving and driven gears are presented in Fig. 10 of the original paper. The statistical characteristics (standard deviation) of the acceleration responses are summarized in Table 2. The driven gear shows larger bending vibrations than the driving gear, while the torsional vibrations are comparable. In all cases, the y‑direction (line‑of‑action direction) exhibits larger variability, indicating that this direction is most sensitive to the random excitations.
Table 2. Standard deviation of acceleration responses
| Gear | x‑direction (mm/s²) | y‑direction (mm/s²) | Torsional (rad/s²) |
|---|---|---|---|
| Driving pinion | 5.5127 | 15.1342 | 0.1254 |
| Driven gear | 7.8657 | 21.5914 | 0.2375 |
The power spectral density (PSD) of the torsional acceleration (Fig. 11 in the original) shows dominant peaks at the mesh frequency (1 100 Hz) and its second harmonic (2 200 Hz), surrounded by numerous sidebands. These sidebands arise from the convolution of the random error and the random friction parameters with the mesh stiffness harmonics.
Influence of Randomness Components
Effect of Random Error on Straight Spur Gear Vibration
We compare two cases: (A) without any random error (only deterministic error), and (B) with random error as described. The time‑domain acceleration responses for the driving pinion are shown in Fig. 12 of the original paper. The deterministic case yields smooth, nearly periodic waveforms, while the random error causes large, unpredictable fluctuations. Table 3 quantifies the increase in standard deviation: approximately 65% across all three directions. The frequency spectra (Fig. 13) further reveal that random error broadens the peaks and introduces continuous spectral content, especially around the mesh frequency, where symmetric sidebands at 860 Hz and 1 340 Hz appear. The phase portraits (Fig. 14) transition from closed periodic orbits to a chaotic, diffuse attractor under random error, confirming the loss of deterministic periodicity.
Table 3. Effect of random error on standard deviation of driving pinion acceleration
| Case | x‑direction (mm/s²) | y‑direction (mm/s²) | Torsional (rad/s²) |
|---|---|---|---|
| Without random error | 3.3406 | 9.1597 | 0.0759 |
| With random error | 5.5127 | 15.1342 | 0.1254 |
| Increase (%) | 65.02% | 65.23% | 65.21% |
Effect of Random Friction Parameters on Straight Spur Gear Vibration
We fix the random error and compare two scenarios: (i) friction coefficient and curvature radii are deterministic constants (mean values); (ii) they are random as per the model. The time‑domain responses (Fig. 15 in the original) show that the random friction parameters further amplify the vibration amplitudes. The standard deviations increase by approximately 27% in all directions (Table 4). The PSD comparison (Fig. 16) indicates that random friction elevates the peak amplitudes at the mesh frequency by about 40% and at the second harmonic by 53%. This demonstrates that even when the transmission error is already random, the additional randomness in friction noticeably worsens the dynamic behavior of the straight spur gear.
Table 4. Effect of random friction parameters on standard deviation of driving pinion acceleration
| Case | x‑direction (mm/s²) | y‑direction (mm/s²) | Torsional (rad/s²) |
|---|---|---|---|
| Friction parameters deterministic | 5.5127 | 15.1342 | 0.1254 |
| Friction parameters random | 6.9896 | 19.2008 | 0.1591 |
| Increase (%) | 26.79% | 26.87% | 26.84% |
Conclusions
We have developed a comprehensive dynamic model for a straight spur gear pair accounting for the randomness of both transmission error and tooth surface friction parameters. The probability‑based representation of error and friction, combined with the lumped‑mass bending‑torsional coupling, effectively captures the stochastic nature of the real gear system. Our numerical investigations lead to the following conclusions:
- The inclusion of random error significantly increases the vibration amplitudes (by ~65%) and introduces broadband frequency components and chaotic phase trajectories, indicating a strong destabilizing effect on the straight spur gear system.
- The randomness of friction parameters, even in the presence of random error, further exacerbates the vibration by ~27%, highlighting the importance of considering surface roughness variability in dynamic analyses.
- The combined effect of both random sources leads to more complex sideband structures in the power spectrum, with peak amplitude increases of 40–53% at the mesh frequency harmonics.
- Among the two sources, the randomness of transmission error dominates the deterioration of dynamic stability, making it a critical factor for the robust design of straight spur gear transmissions.
This work provides a theoretical framework for engineers to evaluate and mitigate vibration risks in high‑performance straight spur gear applications, and serves as a foundation for future reliability‑based optimization studies.