Spur Gear Lubrication Dynamics

The oil film stiffness is a critical parameter that connects the lubrication model with the gear dynamic model. I define the oil film stiffness based on the change in normal load with respect to the change in film thickness:

$$
k_{o}=\frac{\Delta F}{\Delta h}
$$

For the discretized contact region, the oil film stiffness at a single node is:

$$
k_{oj}=L\cdot\frac{\Delta p_{j}}{\Delta h_{j}}\cdot\Delta x
$$

where $L$ is the contact width and $\Delta x$ is the mesh spacing. The total oil film stiffness of the entire contact region is obtained by summing the contributions from all nodes:

$$
k_{o}=\sum_{j=44}^{100}k_{oj}=L\sum_{j=44}^{100}\frac{\Delta p_{j}}{\Delta h_{j}}\Delta x
$$

In dimensionless form:

$$
k_{o}=\sum_{j}\frac{P_{H}\cdot\Delta\overline{p}_{j}}{b\cdot\Delta\overline{h}_{j}}\cdot L\cdot\Delta\overline{x}
$$

The calculation procedure involves solving the EHL equations at two slightly different load conditions to obtain the corresponding film thickness distributions. The ratio of load difference to film thickness difference gives the oil film stiffness. I applied this calculation over an entire meshing cycle using the time-varying load obtained from the gear dynamic model. The oil film stiffness shows a periodic variation with meshing position. As the rotational speed increases, the oil film stiffness decreases rapidly at first, then levels off. As the load increases, the oil film stiffness increases nonlinearly.

I compared my computed oil film stiffness trends with published results. The comparison confirms that the oil film stiffness decreases with increasing speed and increases with increasing load. These trends are consistent with the theoretical expectation that higher speeds produce thicker films, which are more compliant, while higher loads compress the film and thus increase its stiffness.

5. Dynamic Model of the Two-Stage Spur Gear System

Building an accurate dynamic model of the spur gear transmission requires the integration of multiple components. I used the finite element method to discretize the continuous system into elements, then assembled the global mass, damping, and stiffness matrices.

5.1 Shaft Element

The shafts are modeled using Timoshenko beam theory, which accounts for both bending and torsional deformation. Each shaft element has two nodes, with three degrees of freedom per node: one translational displacement in the radial direction, one translational displacement in the orthogonal radial direction, and one rotational displacement around the axial direction. The stiffness matrix of a shaft element is:

$$
\mathbf{K}_{e}^{s}=\begin{bmatrix}
\frac{GA}{Kl} & 0 & 0 & -\frac{GA}{Kl} & 0 & 0\\
0 & \frac{GA}{Kl} & 0 & 0 & -\frac{GA}{Kl} & 0\\
0 & 0 & \frac{GJ}{l} & 0 & 0 & -\frac{GJ}{l}\\
-\frac{GA}{Kl} & 0 & 0 & \frac{GA}{Kl} & 0 & 0\\
0 & -\frac{GA}{Kl} & 0 & 0 & \frac{GA}{Kl} & 0\\
0 & 0 & -\frac{GJ}{l} & 0 & 0 & \frac{GJ}{l}
\end{bmatrix}
$$

where $G$ is the shear modulus, $A$ is the cross-sectional area, $J$ is the polar moment of inertia, $l$ is the element length, and $K$ is the shear correction factor. The mass matrix is derived using the consistent mass formulation.

I used Rayleigh damping to represent the energy dissipation in the shaft elements:

$$
\mathbf{C}_{s}=\alpha\mathbf{M}_{s}+\beta\mathbf{K}_{s}
$$

with coefficients $\alpha=3$ and $\beta=8\times10^{-7}$ for steel shafts.

5.2 Bearing Element

The deep groove ball bearings are modeled as spring-damper elements. The bearing stiffness depends on the load distribution among the rolling elements. I calculated both the static stiffness and the time-varying stiffness due to the varying number of balls in the loaded zone. The bearing stiffness has the following form:

$$
k_{b}(t)=k_{bs}+k_{a}\sin(2\pi f_{b}t+\beta_{b})
$$

where $k_{bs}$ is the average static stiffness, $k_{a}$ is the fluctuation amplitude, $f_{b}$ is the ball passage frequency, and $\beta_{b}$ is the phase angle. For the bearings in my system, the stiffness in the odd-pressure case is $8.21\times10^{8}\,\mathrm{N/m}$ and in the even-pressure case is $8.95\times10^{8}\,\mathrm{N/m}$.

5.3 Lubricated Gear Mesh Element

The lubricated gear meshing element is modeled as an equivalent spring-damper system with the combined stiffness of the gear tooth and the oil film. For the engagement of a spur gear pair, the relative displacement along the line of action considering the torsion and lateral vibrations is:

$$
\delta_{i}=x_{p i}-x_{g i}-e_{i}(t)
$$

where $x_{p i}$ and $x_{g i}$ represent the equivalent displacements of the pinion and gear, and $e_{i}(t)$ is the loaded static transmission error. The equivalent displacements include the influence of the transverse vibrations at the bearing locations and the torsional vibrations of the gears.

The combined stiffness is the series connection of the tooth contact stiffness and the oil film stiffness. However, when the film thickness is extremely thin, the lubricating film may rupture, leading to direct metal-to-metal contact. Therefore, I adopt the following piecewise expression:

$$
k_{m}(t)=\begin{cases}
k_{z}(t), & \delta(t)<h_{c}(t)\\[2mm] $$="" $k_{z}(t)$="" &=""

5.4 System Assembly

The overall system is divided into 95 nodes with a total of 192 degrees of freedom. The input shaft has 13 nodes, the intermediate shaft has 9 nodes, and the output shaft has 10 nodes, with additional nodes at bearing locations. The gear pairs connect the shafts at specific nodes: gear pair 1 connects node 7 on the input shaft with node 16 on the intermediate shaft, while gear pair 2 connects node 20 on the intermediate shaft with node 29 on the output shaft.

The global equation of motion is:

$$
\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K}(t)\mathbf{x}(t)=\mathbf{F}(t)
$$

where $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}(t)$ are the global mass, damping, and time-varying stiffness matrices, respectively, and $\mathbf{F}(t)$ is the external force vector. I solve this equation using the Newmark-$\beta$ integration method with a constant average acceleration assumption, which ensures unconditional stability for linear systems.

6. Natural Characteristics Analysis

I first examined the undamped free vibration of the system by neglecting damping and external forces:

$$
\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{K}(t)\mathbf{x}(t)=\mathbf{0}
$$

Assuming harmonic motion $\mathbf{x}(t)=\boldsymbol{\phi}e^{j\omega t}$, the eigenvalue problem becomes:

$$
\left[\mathbf{K}-\omega_{n}^{2}\mathbf{M}\right]\boldsymbol{\phi}=\mathbf{0}
$$

Solving this eigenvalue problem yields the natural frequencies and mode shapes of the spur gear transmission system. Table 1 shows the first 12 natural frequencies for both the dry contact condition and the condition with lubrication.

Table 1: Comparison of system natural frequencies (Hz)

| Mode | Without oil film | With oil film | Change (%) |
|——|—————-|————–|————|
| 1 | 127 | 121 | -4.7 |
| 2 | 203 | 198 | -2.5 |
| 3 | 1054 | 1044 | -0.9 |
| 4 | 1129 | 1128 | -0.1 |
| 5 | 1130 | 1122 | -0.7 |
| 6 | 1230 | 1226 | -0.3 |
| 7 | 2118 | 2100 | -0.8 |
| 8 | 2798 | 2786 | -0.4 |
| 9 | 2969 | 2967 | -0.1 |
| 10 | 5120 | 5109 | -0.2 |
| 11 | 5837 | 5713 | -2.1 |
| 12 | 6691 | 6688 | -0.04 |

The most significant reduction occurs at the 11th natural frequency, where the percentage reduction is approximately 2.1 percent. The influence of the oil film on the natural frequencies is relatively small for most modes because the oil film stiffness is much larger than the mesh stiffness contribution, so the series combination only slightly reduces the overall stiffness.

When the rotational speed varies, the natural frequencies also vary because the operating condition affects the oil film stiffness. The 11th natural frequency as a function of the input speed is shown for both conditions. In the low-speed region, the natural frequency difference is minimal. In the medium- and high-speed regions, the natural frequency decreases as the speed increases due to the reduction in oil film stiffness. At certain resonance speeds, the natural frequency approaches that of the dry contact condition. This is because resonance amplifies the dynamic load, which compresses the oil film and increases its stiffness, making the overall stiffness closer to the dry condition.

The mode shapes at the 11th order show that in the dry condition, the system exhibits bending vibration modes across all three shafts. In the lubricated condition, the bending order of the shafts increases, and the vibration amplitudes at the characteristic points (gear locations) decrease by varying amounts. For example, the vibration amplitude at the first pinion location decreases slightly, while at the second gear location, the amplitude decreases by about 33 percent. This demonstrates that the oil film damps the lateral vibrations to some extent, contributing to smoother power transmission.

7. Rough Tooth Surface Analysis

Real gear tooth surfaces are not perfectly smooth. The roughness amplitude is typically in the order of 0.1 to 1.6 micrometers, which is comparable to the oil film thickness in elastohydrodynamic lubrication. The roughness peaks penetrate the thin film and cause localized pressure spikes. To study this effect, I extended the model to incorporate a cosine roughness profile on both tooth surfaces:

$$
S_{p}(x,t)=A_{p}\cos\left[\frac{2\pi}{l_{p}}\left(x-\int_{0}^{t}u_{p}\,dt\right)\right]
$$

$$
S_{g}(x,t)=A_{g}\cos\left[\frac{2\pi}{l_{g}}\left(x-\int_{0}^{t}u_{g}\,dt\right)\right]
$$

where $A_{p}$, $A_{g}$ are the roughness amplitudes and $l_{p}$, $l_{g}$ are the wavelengths for the pinion and gear, respectively. For my analysis, the dimensionless amplitudes are 0.1 and 0.05, and the dimensionless wavelengths are 0.4 and 0.2.

The film thickness equation is modified to:

$$
h(x,t)=h_{0}(t)+\frac{x^{2}}{2R(t)}+\delta(x,t)-S_{p}(x,t)-S_{g}(x,t)
$$

The elastic deformation term $\delta(x,t)$ is calculated using the Boussinesq integral:

$$
\delta(x,t)=-\frac{2}{\pi E’}\int_{x_{in}}^{x_{out}}p(x’,t)\ln|x-x’|\,dx’
$$

Considering the shear heating in the lubricant film, I incorporated the energy equation. For the Ree-Eyring non-Newtonian fluid model, the effective viscosity depends on the shear stress through:

$$
\mu^{*}=\mu\frac{\tau_{0}/\tau}{\sinh(\tau_{0}/\tau)}
$$

The governing Reynolds equation for the non-Newtonian case becomes:

$$
\frac{\partial}{\partial x}\left(\varepsilon\frac{\partial p}{\partial x}\right)=\frac{\partial(\rho_{e}h)}{\partial x}+\frac{\partial(\rho_{e}h)}{\partial t}
$$

where the effective coefficients account for the non-Newtonian behavior.

Tables 2 through 4 summarize the lubricating oil parameters and the operating conditions used in my simulations.

Table 2: Spur gear and lubricant parameters

| Parameter | Value |
|———–|——-|
| Elastic modulus of gear material (Pa) | $2.06\times10^{11}$ |
| Poisson’s ratio of gear material | 0.3 |
| Lubricant density (kg/m³) | 780 |
| Lubricant ambient viscosity (Pa·s) | 0.075 |
| Viscosity-pressure coefficient (m²/N) | $2.2\times10^{-8}$ |
| Lubricant specific heat (J/(kg·K)) | 2000 |
| Lubricant thermal conductivity (W/(m·K)) | 0.14 |
| Gear material specific heat (J/(kg·K)) | 470 |
| Gear material thermal conductivity (W/(m·K)) | 46 |
| Lubricant initial temperature (K) | 313 |

Table 3: Roughness parameters

| Parameter | Pinion | Gear |
|———–|——–|——|
| Roughness amplitude (dimensionless) | 0.1 | 0.05 |
| Roughness wavelength (dimensionless) | 0.4 | 0.2 |

Table 4: Gear parameters for the two-stage spur gear system

| Parameter | Gear pair 1 | Gear pair 2 |
|———–|————-|————-|
| Tooth numbers (pinion/wheel) | 36/90 | 29/100 |
| Module (mm) | 1 | 0.8 |
| Pressure angle (°) | 20 | 20 |
| Face width (mm) | 12 | 12 |
| Mass of pinion (kg) | 0.16 | 0.09 |
| Mass of wheel (kg) | 1.3 | 1.6 |
| Moment of inertia of pinion (kg·m²) | $2\times10^{-4}$ | $1\times10^{-4}$ |
| Moment of inertia of wheel (kg·m²) | $30.4\times10^{-4}$ | $87.1\times10^{-4}$ |

I employed a multigrid method to solve the coupled system of equations for the rough-surface line contact EHL problem. The computational domain extends from $x_{in}/b=-4.134$ to $x_{out}/b=1.5$, with a fine grid of 257 nodes in the $x$-direction and 11 nodes across the film thickness. The meshing cycle is divided into 140 time instants, and for each instant, the time-varying coefficients for curvature radius, entrainment speed, and load are updated based on the meshing position.

The comparison between smooth and rough tooth surfaces reveals that the roughness creates localized fluctuations in the oil film pressure. The pressure profile exhibits multiple peaks corresponding to the roughness asperities. The film thickness follows an inverse pattern, with troughs at the pressure peaks. On average, the film thickness with rough surfaces is slightly larger than that with smooth surfaces, which can be attributed to the pumping effect of the transverse roughness that draws additional lubricant into the contact zone. However, the peak pressures in the rough-surface case are significantly higher, increasing the risk of asperity contact and surface fatigue.

In the first spur gear pair, the maximum film pressure with rough surfaces is approximately 25 percent higher than with smooth surfaces at the same operating condition. For the second spur gear pair, the mean film pressure is higher but the peak pressure is lower compared with the first pair, because the second pair operates at a lower speed and a higher torque, resulting in a thicker hydrodynamic film.

8. Dynamic Load Response with Rough Surfaces

I introduced the oil film stiffness computed from the rough-surface model into the gear dynamics program. The time-varying combined mesh stiffness for each gear pair is obtained by the series combination of the tooth contact stiffness and the oil film stiffness. Figure 11 shows the time-varying combined stiffness for gear pair 1 over several meshing cycles. In the dry contact condition, the stiffness exhibits the characteristic variation with two deep valleys corresponding to the transitions between single- and double-tooth meshing. When the oil film is included, the stiffness values in these valleys increase slightly, making the overall variation smoother. This demonstrates that the oil film effectively reduces the severity of mesh stiffness excitation.

For gear pair 2, a similar trend is observed. The oil film fills the stiffness drop at the transition points and reduces the difference between the maximum and minimum stiffness values. The results imply that the lubricating film contributes to the smooth operation of the spur gear transmission by reducing the magnitude of parametric excitation.

Using the Newmark-$\beta$ integration method, I solved the system equations of motion for a duration of 3 seconds. The time step was set to $1\times10^{-6}$ seconds. The dynamic load for each gear pair was extracted during the last 0.5 seconds of the simulation to ensure steady-state behavior.

Table 5: Dynamic load variation with and without oil film

| Parameter | Gear pair 1 without oil | Gear pair 1 with oil | Gear pair 2 without oil | Gear pair 2 with oil |
|———–|————————|———————-|————————|———————-|
| Mean dynamic load (N) | 500 | 500 | 1000 | 1000 |
| Peak dynamic load (N) | 1250 | 1000 | 1600 | 1400 |
| Peak reduction (%) | — | 20.0 | — | 12.5 |

The time-domain history of the dynamic load for gear pair 1 shows that the beat phenomenon, which is prominent in the dry contact condition, is greatly weakened when the oil film is considered. The peak amplitude decreases from 1250 N to 1000 N, representing a reduction of 20 percent.

For gear pair 2, the peak load is reduced from 1600 N to 1400 N, a reduction of 12.5 percent. The trough value also decreases from 700 N to 600 N. These reductions indicate that the lubrication condition is more favorable for the first gear pair, which operates at higher speed and lower load, allowing the formation of a thicker and more resilient oil film.

The frequency spectrum of the dynamic load provides further insight. I performed the fast Fourier transform on the time-domain signal over a duration of 0.02 seconds. The spectrum for gear pair 1 in the dry contact condition shows multiple discrete frequency components, including the second-stage meshing frequency ($f_{m2}$), its second harmonic ($2f_{m2}$), the first-stage meshing frequency ($f_{m1}$), the combined frequency ($f_{m1}+2f_{m2}$), and the second harmonic of $f_{m1}$. When the oil film is included, the amplitude at the second harmonic of the second-stage frequency drops dramatically from 171.0 N to 51.6 N, a reduction of about 70 percent.

Table 6: Frequency components and amplitudes for gear pair 1 (N)

| Frequency component | Without oil film | With oil film |
|——————–|—————–|————–|
| $f_{m2}$ (1063 Hz) | 24.3 | 40.4 |
| $2f_{m2}$ (2126 Hz) | 171.0 | 51.6 |
| $f_{m1}$ (3300 Hz) | 46.0 | 121.1 |
| $f_{m1}+2f_{m2}$ | 117.2 | — |
| $2f_{m1}$ | 34.0 | — |
| $4/3 f_{m1}$ | — | 108.1 |

The spectrum for gear pair 2 in the dry condition contains the second-stage meshing frequency and its multiples (up to the seventh harmonic). With the oil film, the number of significant harmonics is reduced, and the amplitudes at the fundamental and second harmonic frequencies are reduced from 198.2 N to 130.0 N and from 234.6 N to 182.5 N, respectively.

Table 7: Frequency components and amplitudes for gear pair 2 (N)

| Frequency component | Without oil film | With oil film |
|——————–|—————–|————–|
| $f_{m2}$ (1063 Hz) | 198.2 | 130.0 |
| $2f_{m2}$ (2126 Hz) | 234.6 | 182.5 |
| $3f_{m2}$ | 35.9 | — |
| $4f_{m2}$ | 33.9 | — |
| $5f_{m2}$ | — | 36.2 |
| $7f_{m2}$ | 34.8 | — |

The second-stage meshing frequency’s second harmonic falls close to the seventh natural frequency of the system (around 2118 Hz), which explains the prominent amplitude at that frequency. The oil film reduces this resonance response by approximately 22 percent, demonstrating its vibration-damping capability.

From the spectrum analysis, I also observe that in the first-stage spur gear system, the second-stage meshing frequency is clearly present, indicating that the second-stage spur gear pair transmits its vibration to the first stage through the intermediate shaft. However, in the second-stage system, the first-stage meshing frequency does not appear significantly. This one-way coupling indicates that the second gear pair has a greater influence on the first gear pair than vice versa. When the oil film is included, the influence of the second-stage vibration on the first stage is weakened, as evidenced by the reduced amplitude at the corresponding frequency components.

9. Conclusion

My research has systematically investigated the dynamic behavior of a two-stage spur gear transmission system under lubricated conditions. The main findings can be summarized as follows:

First, the line-contact elastohydrodynamic lubrication model effectively captures the film pressure and thickness distributions in the spur gear meshing zone. The Newton-Raphson method converges efficiently for the smooth-surface problem, and the multigrid method is suitable for the rough-surface problem with non-Newtonian fluids. The oil film thickness is strongly influenced by the entrainment speed and weakly influenced by the initial viscosity.

Second, the oil film stiffness in the spur gear system decreases with increasing rotational speed and increases with increasing load. These trends have been verified against published simulations. The oil film stiffness is incorporated into the gear mesh stiffness as a series connection, and the resulting time-varying combined stiffness provides a more realistic representation of the gear meshing condition.

Third, including the oil film stiffness in the dynamic model causes a slight reduction in the natural frequencies of the spur gear transmission system. The 11th natural frequency is the most affected, with a reduction of 2.1 percent. The mode shapes show increased bending complexity of the shafts when the oil film is considered. At resonance speeds, the difference in natural frequencies between lubricated and dry conditions diminishes due to the enhanced dynamic load compressing the film.

Fourth, the analysis of rough tooth surfaces reveals that the transverse roughness induces a pumping effect that slightly increases the mean film thickness. However, the roughness also produces sharp pressure spikes, which can lead to surface fatigue. The dynamic load analysis shows that the oil film reduces the peak dynamic load by 20 percent for the first gear pair and 12.5 percent for the second gear pair. The frequency spectrum indicates that the oil film attenuates the vibration energy, particularly at the resonance-related frequencies, and weakens the vibration transmission from the second-stage to the first-stage spur gear pair.