The study of gear system dynamics is crucial not only for analyzing and predicting vibration and noise but also for informing new requirements in gear design and manufacturing. Among the various gear types, hyperboloidal gears represent the most complex form of transmission. Characterized by smooth operation, large transmission ratios, high power capacity, and compact structure, they are widely used in diverse machinery and equipment. Therefore, establishing the fundamental theory for the nonlinear dynamics of hyperboloidal gear systems holds significant importance for both theoretical research and practical application.
First, during the meshing process of gear pairs, the magnitude and direction of the tooth surface friction force undergo periodic changes. This constitutes an internal excitation source, which, when coupled with nonlinear factors such as backlash and time-varying mesh stiffness, leads to complex nonlinear dynamic behavior in the gear transmission system. Second, in practical operation, perfectly smooth gear tooth surfaces do not exist. Almost all gear transmissions operate under mixed elastohydrodynamic lubrication (mixed-EHL) conditions, meaning the tooth contact involves both lubricant film and asperity contact. The presence of surface asperities undoubtedly has a significant impact on the friction force and the sub-surface stress distribution in the contact zone. Building upon the analysis of friction effects in spur and helical gears, this investigation extends to the more complex case of hyperboloidal gears.
To accurately reflect the realistic lubrication situation in gear transmissions, a mixed-EHL friction model for hyperboloidal gears tooth contact is proposed. This model accounts for the coexistence of full-film and boundary contact. Under these mixed lubrication conditions, the effects of a time-varying friction coefficient on gear dynamics are analyzed in depth. A comprehensive 14-degree-of-freedom (14-DOF) nonlinear dynamic model for a hyperboloidal gear pair system is developed, incorporating this time-varying friction model. The simulation results across a wide range of operating conditions indicate that the influence of the time-varying friction coefficient on the dynamic response of hyperboloidal gears is marginal compared to using a constant average value.
Tooth Contact Analysis and Relative Sliding Velocity
The study of gear meshing analysis began with spur gears and progressed to helical gears. Hyperboloidal gears exhibit a gradual engagement and disengagement process, where the contact point location and the line of action of the mesh force are both time-varying. Dynamically, hyperboloidal gears generally exhibit lower vibration and better performance than spur gears. However, due to the existence of the spiral angle, the line of contact in hyperboloidal gears is inclined relative to the gear axes. Consequently, the mesh force is a spatial force vector, making their dynamic characteristics considerably more complex than those of spur gears and necessitating a three-dimensional spatial dynamic model.
The actual contact area between two tooth surfaces is typically an ellipse. For analysis, this elliptical area is discretized into numerous small cells. The position vector of each point (cell) in the coordinate system $S_l$ ($l=1,2$ for pinion and gear respectively) is expressed as $\mathbf{r}_i^{(l)} = \{x_i^{(l)}, y_i^{(l)}, z_i^{(l)}\}^T$, and its unit normal vector is $\mathbf{n}_i^{(l)} = \{n_{ix}^{(l)}, n_{iy}^{(l)}, n_{iz}^{(l)}\}^T$. The spatial meshing parameter $\lambda_{iu}^{(l)}$ ($u=x,y,z$) represents the moment arm in the direction of the $u$-axis. The relative sliding velocity of the two contact points at cell $i$ with respect to the fixed coordinate system $S_0$ is given by:
$$
\mathbf{v}_i^{(12)} = \mathbf{v}_i^{(1)} – \mathbf{v}_i^{(2)} = \boldsymbol{\omega}^{(1)} \times \mathbf{r}_i^{(1)} – \boldsymbol{\omega}^{(2)} \times \mathbf{r}_i^{(2)} = (\boldsymbol{\omega}^{(1)} – \boldsymbol{\omega}^{(2)}) \times \mathbf{r}_i^{(2)} + \boldsymbol{\omega}^{(1)} \times \mathbf{r}^{(12)}
$$
where $\boldsymbol{\omega}^{(1)}$ and $\boldsymbol{\omega}^{(2)}$ are the angular velocity vectors of the pinion and gear, respectively. Defining them specifically, along with the position vectors, we can derive the components:
$$
\begin{aligned}
\boldsymbol{\omega}^{(1)} &= \{0, \omega_1, 0\}^T, \quad \boldsymbol{\omega}^{(2)} = \{0, 0, -\omega_2\}^T, \\
\mathbf{r}_i^{(2)} &= \{x_i, y_i, z_i\}^T, \quad \mathbf{r}^{(12)} = \{E, 0, 0\}^T
\end{aligned}
$$
where $E$ is the offset. The sliding velocity components become:
$$
\mathbf{v}_i^{(12)} = \begin{Bmatrix}
v_{ix} \\ v_{iy} \\ v_{iz}
\end{Bmatrix} = \omega_1 \begin{Bmatrix}
z_i – \frac{\omega_2}{\omega_1} y_i \\ \frac{\omega_2}{\omega_1} x_i \\ -x_i – E
\end{Bmatrix} = \omega_1 \begin{Bmatrix}
z_i – \frac{N_p}{N_g} y_i \\ \frac{N_p}{N_g} x_i \\ -x_i – E
\end{Bmatrix}
$$
where $N_p$ and $N_g$ are the number of teeth on the pinion and gear, respectively. This relative sliding velocity is a key parameter for calculating the friction force on the tooth surface of hyperboloidal gears.
The Mixed Elastohydrodynamic Lubrication (Mixed-EHL) Model for Gears
In gear transmissions, the tooth surface friction coefficient changes significantly with variations in rotational speed, load distribution, and tooth flank topography. It has been observed that due to these influencing factors, the lubrication state of the gear teeth constantly fluctuates between fluid friction and boundary friction. In fact, mixed elastohydrodynamic lubrication is the prevalent contact state in actual gear transmissions, representing a combination of fluid lubrication, boundary lubrication, and thin-film lubrication.
Load Sharing Coefficient
Under mixed-EHL conditions, the total normal contact load ($P_n$) is shared between the fluid pressure in the full-film lubrication region ($P_n^{EHL}$) and the asperity contact pressure in the boundary lubrication region ($P_n^{BDR}$).
$$
P_n = P_n^{EHL} + P_n^{BDR}
$$
Defining $f_\lambda$ ($0 \le f_\lambda \le 1$) as the percentage of the load carried by the full-film lubricant, the asperity contact load ratio is $(1 – f_\lambda)$. Thus, the pressure distribution under mixed-EHL conditions can be expressed as:
$$
P_n^{EHL} = P_n \cdot f_\lambda, \quad P_n^{BDR} = P_n \cdot (1 – f_\lambda)
$$
For point contacts, such as those in hyperboloidal gears, a load-sharing coefficient has been proposed that relates directly to the film thickness ratio $\lambda$:
$$
f_\lambda = \frac{1.21 \lambda^{0.64}}{1 + 0.37 \lambda^{1.26}}
$$
The film thickness ratio $\lambda$ is a crucial parameter for distinguishing lubrication states and is defined as the ratio of the minimum lubricant film thickness $H_{min}$ to the composite surface roughness $\sigma$:
$$
\lambda = \frac{H_{min}}{\sigma}, \quad \sigma = \sqrt{\sigma_p^2 + \sigma_g^2}
$$
where $\sigma_p$ and $\sigma_g$ are the root-mean-square roughness of the pinion and gear surfaces, respectively. Typically, a regime of $0.2 < \lambda < 2$ is considered mixed lubrication, while $\lambda < 0.2$ and $\lambda > 2$ correspond to boundary and full-film lubrication, respectively. The minimum film thickness for isothermal elliptical contacts in the full-film regime can be estimated by the well-known Hamrock and Dowson formula:
$$
H_{min} = 3.63 \bar{U}^{0.68} G^{0.49} \bar{W}^{-0.073} (1 – e^{-0.68k})
$$
where $\bar{U}$ is the dimensionless speed parameter, $G$ is the dimensionless material parameter, and $\bar{W}$ is the dimensionless load parameter. The load-sharing coefficient $f_\lambda$, and hence the lubrication state for hyperboloidal gears, is thus influenced by operating conditions (speed, load) and surface characteristics.
Friction Coefficient under Mixed-EHL Conditions
The friction coefficient between meshing tooth surfaces is time-varying and strongly nonlinear, dependent on material, roughness, geometry, load, temperature, and lubricant properties. For hyperboloidal gears with point contact, the instantaneous rolling velocity, slide-to-roll ratio, normal load, and effective radius of curvature at each contact point further influence the friction coefficient.
Under mixed lubrication, the total normal force $F_n$ is the sum of the forces carried by the EHL film and the asperities. Similarly, the total tangential friction force $F_t$ comprises contributions from both regimes:
$$
F_t = F_t^{EHL} + F_t^{BDR}
$$
The tangential forces are related to their respective normal loads through average friction coefficients for each regime:
$$
F_t^{BDR} = \mu_i^{BDR*} \cdot F_n^{BDR} = \mu_i^{BDR*} \cdot F_n (1 – f_\lambda)
$$
$$
F_t^{EHL} = \mu_i^{EHL*} \cdot F_n^{EHL} = \mu_i^{EHL*} \cdot F_n f_\lambda
$$
Here, $\mu_i^{EHL*}$ and $\mu_i^{BDR*}$ are the average friction coefficients under full-film and boundary lubrication for the given contact conditions at point $i$. Therefore, the overall mixed-EHL friction coefficient is:
$$
\mu_i^{MIX} = \frac{F_t}{F_n} = \frac{F_t^{EHL} + F_t^{BDR}}{F_n} = \mu_i^{EHL*} f_\lambda + \mu_i^{BDR*} (1 – f_\lambda)
$$
In boundary lubrication, for run-in ground or hobbed surfaces, the asperity contacts remain elastic with an intact boundary film, and the friction coefficient can be assumed constant. Experimental data suggests a typical range of 0.1 to 0.15. We assume $\mu_i^{BDR*} = \mu^{BDR} = 0.15$.
For the full-film (EHL) regime, a reference model based on extensive experimental data for gear contacts can be used to predict $\mu_i^{EHL}$. This model is a function of slide-to-roll ratio ($SR$), maximum Hertzian pressure ($P_h$), inlet dynamic viscosity ($\eta_0$), and RMS surface roughness ($S$):
$$
\begin{aligned}
\mu_i^{EHL} &= e^{f(SR, P_h, \eta_0, S)} P_h^{b_2} SR^{b_3} V_e^{b_6} \eta_0^{b_7} R^{b_8} \\
f(SR, P_h, \eta_0, S) &= b_1 + b_4 SR P_h \log_{10}(\eta_0) + b_5 e^{-SR P_h \log_{10}(\eta_0)} + b_9 e^{S}
\end{aligned}
$$
where $V_e$ is the entrainment velocity, $R$ is the effective radius, and $b_1$ to $b_9$ are empirical constants. However, this model calculates the friction for the full nominal load. Under mixed-EHL, the EHL film carries only a fraction $f_\lambda$ of the load. Studies suggest that the EHL friction coefficient scales with load approximately as $\mu^{EHL} \propto (P_n)^{0.2}$. Therefore, the effective EHL friction coefficient under mixed conditions can be scaled as:
$$
\frac{\mu_i^{EHL*}}{\mu_i^{EHL}} \approx (f_\lambda)^{0.2} \quad \Rightarrow \quad \mu_i^{EHL*} = \mu_i^{EHL} (f_\lambda)^{0.2}
$$
Substituting this into the mixed friction coefficient equation yields:
$$
\mu_i^{MIX} = \mu_i^{EHL} (f_\lambda)^{1.2} + \mu^{BDR} (1 – f_\lambda)
$$
The total friction force at the tooth contact of hyperboloidal gears is then:
$$
F_f = F_L + F_C = \mu_i^{MIX} F_n
$$
where $F_L$ and $F_C$ represent the lubricant film and asperity contact contributions, respectively.
Dynamic Model Incorporating Time-Varying Friction
Based on previous work on spur and helical gears, and considering the specific meshing characteristics of hyperboloidal gears, a 14-DOF nonlinear dynamic model is established using a lumped-parameter approach. The degrees of freedom include three translational and three rotational motions for both the pinion and gear, plus the rotations of the input (engine) and output (load) inertias.
The dynamic transmission error (DTE), a key excitation in gear dynamics, is defined as the relative displacement between the pinion and gear along the line of action at the contact point:
$$
\delta = \{\mathbf{h}_p\}^T \{\mathbf{q}_p\} – \{\mathbf{h}_g\}^T \{\mathbf{q}_g\}
$$
where $\{\mathbf{h}_l\} = \{n_{lx}, n_{ly}, n_{lz}, \lambda_{lx}, \lambda_{ly}, \lambda_{lz}\}$ is the directional rotation radius vector, and $\{\mathbf{q}_l\} = \{x_l, y_l, z_l, \theta_{lx}, \theta_{ly}, \theta_{lz}\}^T$ is the displacement vector for body $l$ ($l=p,g$).
The dynamic mesh force $F_m$ includes both elastic and damping components, and accounts for backlash nonlinearity $f(\delta – e)$:
$$
F_m = k_m f(\delta – e) + c_m g(\dot{\delta} – \dot{e})
$$
The tooth surface friction force is:
$$
F_f = \mu \cdot F_m
$$
The total friction force components in the global coordinate system are obtained by summing contributions from all discretized contact cells $i$ across the contact ellipse:
$$
\begin{aligned}
\mathbf{n}_{fi} &= \frac{(\mathbf{n}_i \times \mathbf{v}_i^{(12)}) \times \mathbf{n}_i}{\|(\mathbf{n}_i \times \mathbf{v}_i^{(12)}) \times \mathbf{n}_i\|} \\
f_{fi} &= \text{sign}(\omega_1) \cdot f_{ni} \cdot \mu_i \\
F_{fx} &= \sum_{i=1}^{N} n_{fix} f_{fi}, \quad F_{fy} = \sum_{i=1}^{N} n_{fiy} f_{fi}, \quad F_{fz} = \sum_{i=1}^{N} n_{fiz} f_{fi} \\
F_f &= \sqrt{F_{fx}^2 + F_{fy}^2 + F_{fz}^2} \\
\mu_f &= F_f / F_n
\end{aligned}
$$
The system of 14 second-order nonlinear differential equations can be summarized in matrix form, incorporating stiffness, damping, time-varying mesh parameters, and the friction force vector derived above. For brevity, the full expanded set is represented conceptually. The equations for the pinion and gear translations and rotations, as well as the input/output shaft dynamics, all include terms from the mesh stiffness $k_m$, damping $c_m$, static transmission error excitation $e(t)$, and the friction force components $F_f$ with their respective directional cosines $n_{f}$ and moment arms $\lambda_f$.
| Component | Generalized Equation of Motion |
|---|---|
| Input Shaft | $I_E \ddot{\theta}_E – k_{py}^r(\theta_{py} – \theta_E) – c_{py}^r(\dot{\theta}_{py} – \dot{\theta}_E) = T_E$ |
| Pinion Translations (x,y,z) | $m_p \ddot{x}_p + k_{px}^t x_p + c_{px}^t \dot{x}_p + [k_m f(\cdot) n_{px} + c_m g(\cdot) n_{px}] – F_f n_{fpx} = 0$ |
| Pinion Rotations ($\theta_x, \theta_y, \theta_z$) | $I_{px} \ddot{\theta}_{px} + k_{px}^r \theta_{px} + c_{px}^r \dot{\theta}_{px} + [k_m f(\cdot) \lambda_{px} + c_m g(\cdot) \lambda_{px}] – F_f \lambda_{fpx} = 0$ |
| Gear Translations (x,y,z) | $m_g \ddot{x}_g + k_{gx}^t x_g + c_{gx}^t \dot{x}_g – [k_m f(\cdot) n_{gx} + c_m g(\cdot) n_{gx}] + F_f n_{fgx} = 0$ |
| Gear Rotations ($\theta_x, \theta_y, \theta_z$) | $I_{gx} \ddot{\theta}_{gx} + k_{gx}^r \theta_{gx} + c_{gx}^r \dot{\theta}_{gx} – [k_m f(\cdot) \lambda_{gx} + c_m g(\cdot) \lambda_{gx}] + F_f \lambda_{fgx} = 0$ |
| Output Shaft | $I_L \ddot{\theta}_L – k_{gy}^r(\theta_{gy} – \theta_L) – c_{gy}^r(\dot{\theta}_{gy} – \dot{\theta}_L) = -T_L$ |
Note: $f(\cdot)=f(\delta-e)$, $g(\cdot)=g(\dot{\delta}-\dot{e})$. Similar equations exist for y and z translations and rotations.
Computational Results and Discussion
The analysis is performed on a hyperboloidal gear pair with design parameters as listed in the table below. The pinion has 10 teeth, making one mesh cycle correspond to 36 degrees of pinion rotation.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 10 | 43 |
| Face Width (mm) | 52.2 | 47.7 |
| Offset (mm) | 31.7 | |
| Mass (kg) | 11.4 | 49.5 |
| Mass Moment of Inertia (Input/Output) (kg·m²) | $I_E=0.006$ | $I_L=0.1$ |
Key simulation parameters include: Input torque $T_{in}$ = 50 N·m (light), 500 N·m (medium), 2000 N·m (heavy); Pinion speed $N_p$ = 1000, 3000, 5000 rpm; Maximum Hertzian pressure $P_h$ = 2 GPa; Lubricant inlet viscosity $\eta_0$ = 10 cP; Kinematic viscosity $\nu_k$ = 13 cSt; Composite RMS roughness $S$ = 0.07 µm. The mesh cycle is divided into 20 steps for numerical analysis.
First, the instantaneous friction coefficient is calculated throughout the mesh cycle using the mixed-EHL model. Figure X(a) shows the variation of $\mu_i^{MIX}$ over one mesh cycle at different speeds. A critical observation is that for hyperboloidal gears, the friction coefficient remains nearly constant within a single mesh cycle at a given speed. This is a distinct characteristic compared to spur gears, where the friction coefficient reverses sign at the pitch point. In hyperboloidal gears, the direction of the friction force only oscillates within a small range without a complete reversal.
Figure X(b) illustrates that the average friction coefficient decreases significantly with increasing rotational speed. This is because higher speeds increase the entrainment velocity and the film thickness ratio $\lambda$, leading to a larger load-sharing coefficient $f_\lambda$ and a greater proportion of load carried by the lower-friction fluid film. Figure X(c) shows that varying the input torque has a comparatively minor effect on the predicted friction coefficient. Therefore, speed variation exerts a more pronounced influence on the friction coefficient in hyperboloidal gears than load variation does.
These computed instantaneous friction coefficients are incorporated into the 14-DOF dynamic model. The system of differential equations is solved using a 5th-order variable-step Runge-Kutta numerical integration method. The dynamic mesh forces in the X, Y, and Z global directions and the dynamic transmission error are primary response metrics.
The results for dynamic mesh forces under a medium load of 500 N·m indicate that the influence of the time-varying friction coefficient is relatively small. The force responses calculated using a constant average friction coefficient (e.g., $\mu=0.1$) are very similar to those obtained with the detailed mixed-EHL time-varying model. The effect is slightly more noticeable in the X-direction compared to the Y and Z directions. Most importantly, the impact on the Dynamic Transmission Error (DTE) is negligible. The DTE response with the time-varying coefficient is virtually indistinguishable from that computed with a constant friction coefficient.
Conclusion
Extending the dynamic analysis of friction effects from spur and helical gears, this study focuses on the complex case of hyperboloidal gears. A mixed elastohydrodynamic lubrication (mixed-EHL) model is established to predict the time-varying friction coefficient, reflecting the realistic lubrication state in gear transmissions. This model is integrated into a comprehensive 14-degree-of-freedom nonlinear dynamic model for a hyperboloidal gear pair.
The simulation results lead to two main conclusions. First, for hyperboloidal gears, the friction coefficient predicted under mixed-EHL conditions remains essentially constant throughout a single meshing cycle at a fixed speed, in contrast to the sign reversal seen in spur gears. Its value decreases with increasing rotational speed, while variations in load have a less significant effect.
Second, comparing dynamic responses obtained with the detailed time-varying friction coefficient and a constant average value reveals that the influence on key metrics like dynamic mesh forces and particularly dynamic transmission error is marginal. Therefore, for the dynamic analysis of hyperboloidal gears, where solution efficiency and model simplification are often priorities, using a representative constant friction coefficient is a justifiable approach that yields sufficiently accurate results for predicting overall dynamic behavior.