In the field of gear transmission, the lubrication condition directly affects the efficiency, noise, and service life of the system. Under actual working conditions, gearboxes often experience non-steady elastohydrodynamic lubrication (EHL) states due to varying loads, speeds, and contact geometries. Impact loads, in particular, cause significant transient effects on oil film pressure and thickness, making the study of dynamic lubrication essential. Ferrofluids, as a novel intelligent lubricant, combine the magnetic properties of solid particles with the fluidity of liquid carriers, offering self-sealing and enhanced thermal conductivity. I focused on analyzing the EHL behavior of involute straight spur gears lubricated by ferrofluids with different carrier fluids under impact loads. By applying multigrid methods and multigrid integration techniques, I obtained the full numerical solution of the non-steady EHL problem and investigated the effects of carrier fluid type, transmission ratio, and transmitted power on pressure and film thickness.
My research established a comprehensive EHL model for a pair of involute straight spur gears. The geometric parameters and coordinate system are crucial for describing the time-varying contact conditions. I defined a static coordinate system \(XPY\) and a moving coordinate system \(xKy\) whose origin always coincides with the instantaneous meshing point. The meshing point moves along the line of action with a velocity \(v_1(t) = \omega_1 R_{b1}(t) = \omega_2 R_{b2}(t)\), where \(\omega_1\) and \(\omega_2\) are the angular velocities of the driving and driven gears, respectively. The instantaneous radii \(R_{b1}(t)\) and \(R_{b2}(t)\) change as the meshing progresses. The equivalent radius of curvature at the contact point is given by \(R(t) = \frac{R_1(t) R_2(t)}{R_1(t) + R_2(t)}\), with \(R_1(t) = R_{b1} \tan\varphi – s(t)\) and \(R_2(t) = R_{b2} \tan\varphi + s(t)\), where \(\varphi\) is the pressure angle and \(s(t)\) is the distance along the line of action. The entrainment velocity \(u(t)\) is the average of the tangential velocities of the two gear tooth surfaces: \(u(t) = \frac{u_1(t) + u_2(t)}{2}\), with \(u_1(t) = \omega_1 (R_{b1} \tan\varphi + s)\) and \(u_2(t) = \omega_2 (R_{b2} \tan\varphi – s)\).
The core of the lubrication analysis is the Reynolds equation. For an isothermal line contact under transient conditions, the pressure distribution \(p(x,t)\) satisfies:
$$ \frac{\partial}{\partial x} \left( \frac{\rho h^3}{\eta} \frac{\partial p}{\partial x} \right) = 12 \frac{\partial (\rho u h)}{\partial x} + 12 \frac{\partial (\rho h)}{\partial t} $$
where \(\rho\) is the density of the ferrofluid, \(\eta\) is its viscosity, \(h\) is the film thickness, and \(x\) is the coordinate along the rolling direction. The boundary conditions are \(p(x_{\text{in}}) = 0\), \(p(x_{\text{out}}) = 0\), and \(p(x) \ge 0\) within the computational domain \([x_{\text{in}}, x_{\text{out}}]\). The film thickness equation accounts for the geometry of the undeformed surfaces and the elastic deformation:
$$ h(x,t) = h_0(t) + \frac{x^2}{2R(t)} – \frac{2}{\pi E} \int_{x_{\text{in}}}^{x_{\text{out}}} p(\zeta,t) \ln (x-\zeta)^2 \, d\zeta $$
where \(h_0(t)\) is the rigid central film thickness, \(E\) is the equivalent elastic modulus of the two gear materials. The viscosity of the ferrofluid varies with pressure according to the Roelands equation:
$$ \eta = \eta_0 \exp\left\{ (\ln \eta_0 + 9.67) \left[ (1 + 5.1 \times 10^{-9} p)^{z_0} – 1 \right] \right\} $$
where \(\eta_0\) is the ambient viscosity and \(z_0\) is the pressure-viscosity index. The density variation follows the Dowson-Higginson relation:
$$ \rho = \rho_0 \frac{1 + 0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p} $$
The impact load applied to the gear tooth is modeled as a transient function:
$$ w_i(t) = w_0 e^{-0.2t} \sin\left( \frac{\pi t}{4} \right) $$
where \(w_0\) is the amplitude of the impact load. To facilitate numerical computation, I non-dimensionalized all equations. The dimensionless variables are defined as: \(X = x/b\), \(W_0 = w_0/(ER_0)\), \(U_0 = \eta_0 u_0/(ER_0)\), \(\bar{h} = h R_0 / b^2\), \(P = p/p_H\), \(\bar{\eta} = \eta/\eta_0\), \(\bar{\rho} = \rho/\rho_0\), \(C_{Rt} = R/R_0\), \(C_{ut} = u/u_0\), \(C_{wtt} = w/w_0\), \(t = t u_0/b\). Here, \(b\) is the half-width of the Hertzian contact, \(R_0\) is the reference radius of curvature, \(u_0\) is the reference entrainment velocity, and \(p_H\) is the maximum Hertzian pressure.
I employed the multigrid method to solve the discretized Reynolds equation and the multigrid integration technique to compute the elastic deformation efficiently. The pressure domain is discretized on six grid levels, with 961 nodes on the finest level. Gauss-Seidel iterations were performed on each grid, and a W-cycle was used for inter-grid transfer. The steady-state solution from the previous time step served as the initial guess for the next transient step. A full meshing cycle was divided into 120 time instants, and convergence was achieved when the relative error of pressure and load between successive iterations was less than \(10^{-3}\). The time-varying load on a single tooth pair was considered, as shown in the typical load-sharing diagram where points A, B, C, D, and E represent the start of meshing, transition from double to single tooth contact, the pitch point, transition back to double tooth contact, and the end of meshing, respectively.
The relevant lubrication parameters for the numerical calculation are summarized in the table below:
| Parameter | Value |
|---|---|
| Ambient viscosity of lubricant \(\eta_0\) (Pa·s) | 0.075 |
| Pressure-viscosity coefficient \(\alpha\) (Pa-1) | 2.19 × 10-8 |
| Temperature-viscosity coefficient \(\beta\) (K-1) | 0.042 |
| Ambient density \(\rho_0\) (kg·m-3) | 870 |
| Specific heat capacity of lubricant \(C\) (J·kg-1·K-1) | 2000 |
| Thermal conductivity of lubricant \(K\) (W·m-1·K-1) | 0.14 |
| Density of gear material \(\rho_{1,2}\) (kg·m-3) | 7850 |
| Specific heat of gear \(C_{1,2}\) (J·kg-1·K-1) | 470 |
| Thermal conductivity of gear \(K_{1,2}\) (W·m-1·K-1) | 46 |
| Elastic modulus of gear \(E_{1,2}\) (Pa) | 2.6 × 1011 |
| Poisson’s ratio \(\gamma_{1,2}\) | 0.3 |
| Number of teeth \(z_1, z_2\) | 35, 140 |
| Module \(m\) (mm) | 2.5 |
| Rotational speed \(n_1\) (r·min-1) | 600 |
| Face width \(B\) (mm) | 20 |
| Pressure angle \(\theta\) (°) | 20 |
| Power transmitted \(P\) (kW) | 20 |
| Addendum coefficient \(h^*\) | 1.0 |
| Clearance coefficient \(c^*\) | 0.25 |
| Ambient temperature \(T_0\) (K) | 313 |
The physical properties of the three different carrier fluid ferrofluids I investigated are listed below:
| Carrier fluid | Material code | Viscosity \(\eta\) (Pa·s) | Density \(\rho\) (kg·m-3) |
|---|---|---|---|
| Ester-based | H02 | 0.006 | 1250 |
| Hydrocarbon-based | E02 | 0.030 | 1300 |
| Diester-based | D01 | 0.075 | 1185 |
Results and Discussion
Effect of Different Carrier Fluid Ferrofluids under Impact Load
I first compared the pressure and film thickness distributions for the three ferrofluids under the same impact load condition at the pitch point. The results clearly show that the viscosity of the carrier fluid dominates the EHL behavior. The diester-based D01 ferrofluid, with the highest viscosity (0.075 Pa·s), produced the largest film thickness and the smallest pressure peak among the three. In contrast, the ester-based H02 ferrofluid (0.006 Pa·s) yielded the thinnest film and the highest pressure. The hydrocarbon-based E02 ferrofluid (0.030 Pa·s) showed intermediate performance. The second pressure peak in the outlet region shifted toward the inlet zone as the viscosity increased, indicating that higher viscosity impedes the outflow of lubricant, causing accumulation. The central pressure in the contact zone remained relatively stable across different fluids, but the pressure spike near the outlet was more pronounced for lower-viscosity fluids.
Effect of Transmission Ratio on Diester-based D01 Ferrofluid under Impact Load
For the diester-based D01 ferrofluid, I examined the influence of the transmission ratio \(i = z_2/z_1\) on the lubrication characteristics at the pitch point. With the driving gear teeth number fixed at 35, I varied the driven gear teeth number to achieve ratios of 2, 3, and 4. The results indicate that as the transmission ratio increases, the entrainment velocity and the equivalent radius of curvature both increase, which promotes the formation of a thicker oil film. Consequently, the film thickness grows and the contact pressure decreases. This trend suggests that, within allowable design limits, increasing the transmission ratio can improve the lubrication state of the straight spur gear and reduce friction. The pressure profiles also showed a slight shift of the second pressure peak toward the inlet with higher ratio, consistent with the viscosity effect.
Effect of Transmitted Power on Diester-based D01 Ferrofluid under Impact Load
I then studied the influence of the transmitted power (10 kW, 20 kW, and 30 kW) on the transient film thickness and pressure of the D01 ferrofluid at the pitch point under impact loading. The transmitted power directly determines the nominal load on the tooth. Higher power results in a larger normal load, which compresses the lubricant film more severely. The film thickness decreases while the pressure increases. At 30 kW, the pressure peak was approximately 15% higher than at 10 kW, and the central film thickness reduced by about 20%. The impact load superimposed on the steady load created additional transient spikes, but the overall trend remained consistent: power increase worsens the lubrication condition. The study also revealed that the impact load caused more severe pressure oscillations at higher powers, which could lead to fatigue and surface damage if not properly managed.
Conclusions
From my numerical analysis of non-steady EHL in involute straight spur gears lubricated by ferrofluids under impact loads, I draw the following key conclusions:
- Under impact loading, the diester-based D01 ferrofluid exhibits the largest film thickness and the smallest pressure among the tested carrier fluids, while the ester-based H02 ferrofluid shows the thinnest film and highest pressure. Viscosity is the primary factor governing the EHL performance.
- For the diester-based D01 ferrofluid, increasing the transmission ratio leads to a decrease in contact pressure and an increase in film thickness. Therefore, a higher transmission ratio is beneficial for enhancing lubrication and reducing friction in straight spur gears, provided other design constraints are satisfied.
- Elevating the transmitted power intensifies the load on the tooth, resulting in higher contact pressure and thinner oil film. The impact load further exacerbates these effects, potentially compromising the gear durability. Careful selection of lubricant viscosity and gear geometry is essential under high-power conditions.