Keywords: spur gears; elastohydrodynamic lubrication; time-varying oil film stiffness; tooth surface roughness; dynamic characteristics.
Spur gears are widely used in wind turbine gearboxes, aerospace transmissions, marine propulsion systems, and many industrial machines. Although spur gears have a simple tooth geometry compared with helical or bevel gears, their dynamic behaviour is still difficult to predict with high accuracy because of the combined effects of time-varying mesh stiffness, friction, sliding, lubrication, and support flexibility. In practice, gearboxes are usually lubricated by oil, and the lubricating film formed between meshing teeth changes the contact stiffness, damping, and load transmission. This thesis focuses on the dynamic characteristics of a two-stage fixed-axis spur gear transmission while explicitly considering the lubricating film formed between the meshing teeth. The influence of the oil film is represented by a time-varying oil film stiffness, which is combined with the conventional mesh stiffness of dry spur gears. By doing so, I build a parametric dynamic model that connects elastohydrodynamic lubrication with the vibration response of spur gears.
The main research contents include the line-contact elastohydrodynamic lubrication model of spur gear pairs, the time-varying oil film stiffness calculation, the finite element dynamic modelling of the two-stage spur gear transmission, and the rough-surface lubrication analysis. I derive a series of governing equations for pressure, film thickness, viscosity, density, and load balance. The Newton–Raphson method is used to solve the steady state elastohydrodynamic lubrication equations for smooth surfaces, while a multigrid method is used for the rough-surface transient thermal elastohydrodynamic lubrication problem. The time-varying meshing parameters, including curvature radii, entrainment velocity, sliding velocity, and load distribution, are obtained from gear meshing theory. Then the oil film stiffness is calculated from the numerical pressure and film-thickness distributions. Finally, the oil film stiffness is superimposed with the gear mesh stiffness in the system stiffness matrix, and the dynamic response of the two-stage spur gear transmission is solved with the Newmark integration method. The results reveal the important role of gear tooth lubrication in modifying the natural frequencies, mode shapes, and dynamic loads.
The work is organized as follows. First, the elastohydrodynamic lubrication equations for a spur gear pair are introduced, and the effects of initial viscosity, entrainment speed, and load are discussed. Second, the time-varying meshing parameters of two-stage spur gears are derived. Third, a finite element model of the two-stage spur gear system is established, in which the lubricated tooth mesh element is represented by the combination of the dry mesh stiffness and the oil film stiffness. Fourth, the model is extended to rough tooth surfaces by using a non-Newtonian Ree–Eyring fluid model and a cosine roughness function. Finally, the dynamic loads in the time and frequency domains are compared for dry contact, smooth-surface lubrication, and rough-surface lubrication.
1. Introduction and Research Significance
Gear transmission systems play an irreplaceable role in aerospace equipment, heavy machinery, wind turbines, ships, and vehicles. With the trend toward higher power density, lower noise, and longer service life, the lubrication design of spur gears has become a critical issue. More than half of gearbox failures are related to gear tooth damage. Tooth breakage is often caused by excessive load and stress concentration, while many other failure modes, such as pitting, scuffing, and adhesive wear, are directly related to the lubrication condition. In a lubricated gear pair, the oil film between the meshing teeth can separate the surfaces, reduce friction, and mitigate impact. The stiffness of the oil film, together with the dry contact stiffness of the gear teeth, forms one of the main excitations in the gear transmission system. Therefore, it is necessary to explore the influence of the oil film stiffness on the vibration and dynamic load of spur gears.
Previous studies on gear dynamics often neglected the lubrication effect and used the dry contact stiffness to approximate the meshing stiffness. However, the oil film has a finite stiffness because it can resist deformation under load. The oil film stiffness can be of the same order as the dry contact stiffness, especially under heavily loaded conditions. If the oil film stiffness is omitted, the predicted natural frequencies, mode shapes, and dynamic loads may deviate significantly from the actual values. This thesis therefore aims to establish a bridge between elastohydrodynamic lubrication and gear dynamics by introducing a time-varying oil film stiffness into the dynamic model of a two-stage spur gear transmission. The results can provide useful guidance for gearbox optimization, lubrication design, and vibration reduction.
There are many challenges in the coupled lubrication–dynamics analysis of spur gears. First, the contact geometry changes continuously along the line of action, so the curvature radius, entrainment velocity, and load at the contact point are time-varying. Second, the lubrication regime can be full film, mixed, or boundary depending on the operating conditions and surface roughness. Third, the dynamic loads from the gear system change the elastic deformation and consequently affect the film thickness. A complete description therefore requires an iterative coupling between the lubrication calculation and the dynamic analysis. In this thesis, I use a quasi-steady approach for smooth surfaces and a transient thermal elastohydrodynamic lubrication approach for rough surfaces. The time-varying oil film stiffness is updated according to the instantaneous load and motion state, and then inserted into the system stiffness matrix.
2. Line-Contact Elastohydrodynamic Lubrication Model for Spur Gear Pairs
A pair of meshing spur gear teeth can be simplified as two parallel cylinders in line contact. This is a standard assumption because the tooth surfaces are conjugate surfaces and the contact zone is small compared with the tooth dimensions. The two cylinders have instantaneous radii equal to the radii of curvature of the tooth profiles at the contact point. For the first gear pair of the studied two-stage spur gear transmission, the instantaneous radii are denoted by \(R_{p1i}\) for the pinion and \(R_{g1i}\) for the gear. The equivalent radius of curvature is defined as:
$$R_i = \frac{R_{p1i} \, R_{g1i}}{R_{p1i} + R_{g1i}}$$
where the subscript \(i\) indicates the instantaneous meshing position. The motion is treated as the rolling-sliding contact of an equivalent cylinder against a smooth plane. The oil is dragged into the contact by the entrainment velocity \(u_i\), which is the average of the two surface velocities.
The steady-state line-contact elastohydrodynamic lubrication equation is based on the Reynolds equation:
$$\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{\mu}\frac{\partial p}{\partial x}\right) = 12 u \frac{\partial (\rho h)}{\partial x}$$
where \(p\) is the oil film pressure, \(h\) is the local film thickness, \(\rho\) is the density of the lubricant, \(\mu\) is the dynamic viscosity, and \(u\) is the entrainment velocity. The equation describes the balance between the Poiseuille flow caused by the pressure gradient and the Couette flow caused by the moving surfaces.
The film thickness equation includes the rigid gap, the elastic deformation of the surfaces, and, in the case of rough surfaces, the roughness profile. For smooth surfaces, the film thickness at position \(x\) is:
$$h(x) = h_c + \frac{x^{2}}{2R} + v(x)$$
where \(h_c\) is the central film thickness, \(R\) is the equivalent radius of curvature, and \(v(x)\) is the elastic deformation. The elastic deformation can be calculated by the superposition integral of the pressure distribution.
The lubricant density is pressure-dependent. In this work, the following relation is used:
$$\rho = \rho_0 \left(1 + \frac{0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p}\right)$$
where \(\rho_0\) is the density at atmospheric pressure. The viscosity-pressure relation is described by the Roelands equation:
$$\mu = \mu_0 \exp\left\{(\ln \mu_0 + 9.67)\left[\left(1 + 5.1 \times 10^{-9} p\right)^{z} – 1\right]\right\}$$
where \(\mu_0\) is the ambient viscosity and \(z\) is a dimensionless pressure-viscosity index. The Roelands equation is suitable for spur gear contacts because it remains accurate over the high-pressure range encountered in gear meshing. The dimensionless parameters used in the numerical solution are:
$$G = \alpha E’, \quad U = \frac{\mu_0 u}{E’ R}, \quad W = \frac{w}{E’ R}$$
where \(\alpha\) is the pressure-viscosity coefficient, \(E’\) is the equivalent elastic modulus, and \(w\) is the load per unit width. The load balance equation is:
$$w = \int_{x_{in}}^{x_{out}} p(x) \, dx$$
The Reynolds, film thickness, density, viscosity, and load balance equations form a nonlinear system. I solve this system with the Newton–Raphson method. The contact region is discretized into a finite number of nodes, and the pressure and film thickness are updated iteratively until convergence is achieved.
Using the numerical solution, I investigated the influence of initial viscosity, entrainment speed, and load on the pressure and film thickness distributions. Table 1 summarizes the qualitative trends.
| Parameter | Film thickness | Film pressure | Secondary pressure peak |
|---|---|---|---|
| Initial viscosity | Increases with diminishing increment | Increases with diminishing increment | Peak increases; location unchanged |
| Entrainment speed | Increases noticeably | Increases slightly in central region | Peak increases; location moves toward the inlet |
| Load | Increases first, then decreases when the film is strongly compressed | Increases and may fluctuate under heavy load | Peak fluctuates; location moves toward the outlet |
The numerical results show that the initial viscosity has a relatively weak influence on the film pressure and film thickness compared with the entrainment speed and load. The oil film has a compression limit; when the load exceeds a certain value, the film thickness cannot decrease significantly because the lubricant becomes extremely viscous at high pressure. This phenomenon is important for the design of heavily loaded spur gears.
3. Gear Meshing Parameters and Time-Varying Oil Film Stiffness
The two-stage spur gear transmission considered in this thesis consists of a first gear pair and a second gear pair. The first gear pair has a pinion with 36 teeth and a gear with 90 teeth. The second gear pair has a pinion with 29 teeth and a gear with 100 teeth. The module, pressure angle, face width, material properties, and lubricant properties are listed in Table 2.
| Parameter | Value |
|---|---|
| Teeth numbers of first pinion/gear | 36 / 90 |
| Teeth numbers of second pinion/gear | 29 / 100 |
| Pressure angle | 20° |
| Face width | 12 mm |
| Gear material elastic modulus | 2.06 × 1011 Pa |
| Gear material Poisson ratio | 0.3 |
| Lubricant density | 780 kg/m3 |
| Lubricant ambient viscosity | 0.075 Pa·s |
| Pressure-viscosity coefficient | 2.2 × 10−8 |
To calculate the time-varying oil film stiffness, I first obtain the time-varying geometric and kinematic parameters of the spur gear pair. A static coordinate system is fixed at the pitch point, and a moving coordinate system translates along the line of action with the contact point. The distance between the instantaneous contact point and the pitch point is denoted as \(s\). The boundaries of the contact path are determined by the addendum circles of the two gears. For the first gear pair, the instantaneous radii of curvature are:
$$R_{p1i} = r_{bp1} \tan\alpha + s$$
$$R_{g1i} = r_{bg1} \tan\alpha – s$$
where \(r_{bp1}\) and \(r_{bg1}\) are the base circle radii of the pinion and gear, and \(\alpha\) is the pressure angle. The equivalent radius of curvature at the contact point is then given by the previous expression. The pinion and gear surface velocities are:
$$u_{p1i} = \omega_{p1} R_{p1i}, \quad u_{g1i} = \omega_{g1} R_{g1i}$$
where \(\omega_{p1}\) and \(\omega_{g1}\) are the angular velocities. The entrainment velocity is:
$$u_{1i} = \frac{u_{p1i} + u_{g1i}}{2}$$
The sliding velocity is \(u_{s1i}=u_{p1i}-u_{g1i}\). The sliding coefficients of the pinion and gear are important for the lubrication performance:
$$\eta_1 = \frac{u_{s1i}}{u_{g1i}}, \quad \eta_2 = \frac{u_{s1i}}{u_{p1i}}$$
The load distribution along the line of action is not constant because the number of tooth pairs in contact changes. In the double-tooth contact region, the load is shared by two pairs of teeth, while in the single-tooth contact region the load is carried by one pair. The instantaneous load per unit width can be expressed as:
$$
w_i(s)=
\begin{cases}
w_1\left[\frac{1}{3}+\frac{2}{3}\frac{s-s_{\min}}{s_{d1}}\right], & s_{\min} \le s < s_{\min}+s_{d1},\\[4pt]
w_1, & s_{\min}+s_{d1} \le s \le s_{\max}-s_{d2},\\[4pt]
w_1\left[\frac{1}{3}+\frac{2}{3}\frac{s_{\max}-s}{s_{d2}}\right], & s_{\max}-s_{d2} < s \le s_{\max},
\end{cases}
$$
where \(w_1\) is the nominal load per unit width in the single-tooth contact region, and \(s_{d1}\), \(s_{d2}\) are the distances corresponding to the double-tooth contact transitions. This piecewise load distribution causes a sudden change in the contact force at the transitions, which is one of the main excitations in spur gear dynamics.
The oil film stiffness is defined as the derivative of the normal load with respect to the normal displacement of the contacting surfaces. Since the pressure and film thickness are obtained at discrete nodes in the elastohydrodynamic lubrication solution, the oil film stiffness can be calculated by:
$$k_{oj} = L \, \Delta x \, \frac{\Delta p_j}{\Delta h_j} \frac{R \, p_H}{b}$$
where \(L\) is the face width, \(\Delta x\) is the dimensionless grid spacing, \(p_H\) is the maximum Hertzian pressure, \(b\) is the Hertzian contact half-width, \(\Delta p_j\) and \(\Delta h_j\) are the changes in dimensionless pressure and film thickness at node \(j\). The total oil film stiffness at a given meshing position is:
$$k_o = \sum_{j} k_{oj}$$
Figure 1 shows a typical calculation model of the oil film stiffness in the contact region. The pressure distribution and film thickness distribution are obtained by solving the elastohydrodynamic lubrication equations. The film behaves like a set of small springs connected in parallel.
Table 3 presents the influence of speed and load on the oil film stiffness.
| Operating condition | Oil film stiffness trend |
|---|---|
| Increasing input speed | Stiffness decreases sharply at first, then becomes nearly constant at high speed |
| Increasing load | Stiffness increases nonlinearly |
4. Dynamic Model of the Two-Stage Spur Gear Transmission
The dynamic model is constructed using the finite element method. The complete transmission system includes three shafts, six deep-groove ball bearings, two spur gear pairs, and the gearbox housing. The input shaft is divided into twelve elements, the intermediate shaft into eight elements, and the output shaft into nine elements. Each shaft node has three degrees of freedom: one translational displacement in the direction normal to the shaft, one translational displacement in the perpendicular radial direction, and one torsional rotation. Figure 2 shows the finite element arrangement of the shafts, bearings, and gear mesh elements.
For each shaft segment, I use a Timoshenko beam formulation, which accounts for both bending deformation and shear deformation. The shaft element mass and stiffness matrices are obtained from the kinetic energy and potential energy of the beam. Rayleigh damping is used for the shaft elements:
$$[C_s] = \alpha_s [M_s] + \beta_s [K_s]$$
where \(\alpha_s\) and \(\beta_s\) are the mass-proportional and stiffness-proportional coefficients. The deep-groove ball bearing is modelled as a spring–damper element with time-varying stiffness. The bearing stiffness can be written as:
$$k_b(t) = k_{bs} + k_a \sin(2\pi f_b t + \beta_b)$$
where \(k_{bs}\) is the static bearing stiffness, \(k_a\) is the fluctuation amplitude, \(f_b\) is the ball passage frequency, and \(\beta_b\) is the phase angle.
The gear mesh element is the most important part of the dynamic model because it includes the effect of the lubricating film. The gear mesh stiffness in a dry contact condition is \(k_z(t)\), which is obtained from finite element contact analysis under the assumption of ideal smooth rigid surfaces. The oil film stiffness \(k_o(t)\) is calculated from the elastohydrodynamic lubrication solution. The total time-varying mesh stiffness \(k_m(t)\) is then determined as:
$$
k_m(t)=
\begin{cases}
k_z(t), & \delta(t) \ge h_c(t),\\[4pt]
\dfrac{k_z(t) \, k_o(t)}{k_z(t) + k_o(t)}, & \delta(t) < h_c(t),
\end{cases}
$$
where \(\delta(t)\) is the tooth deformation and \(h_c(t)\) is the central oil film thickness. The first case represents the condition when the oil film breaks down and direct metal-to-metal contact occurs. The second case represents full-film lubrication, in which the dry mesh stiffness and the oil film stiffness act like two springs in series. The series combination is reasonable because the total deformation is the sum of the tooth deformation and the film deformation.
After assembling all element matrices, the global dynamic equation of the two-stage spur gear system is:
$$[M]\{\ddot{x}(t)\} + [C]\{\dot{x}(t)\} + [K(t)]\{x(t)\} = \{F(t)\}$$
where \([M]\), \([C]\), and \([K(t)]\) are the global mass, damping, and stiffness matrices, \(\{x(t)\}\) is the displacement vector, and \(\{F(t)\}\) is the external excitation vector. The time-varying stiffness matrix includes the time-varying mesh stiffness and the time-varying bearing stiffness. I solve the equation with the Newmark-β integration method, which is unconditionally stable for the parameters chosen in this thesis.
The natural frequencies of the system are obtained by solving the eigenvalue problem corresponding to the undamped free vibration equation:
$$\left([K] – \omega_n^2 [M]\right) \{\phi_n\} = \{0\}$$
where \(\omega_n\) is the natural frequency and \(\{\phi_n\}\) is the mode shape. Table 4 lists the first twelve natural frequencies of the system with and without the oil film effect. The operating condition is an input speed of 3500 r/min and a load torque of 70 N·m.
| Mode | Without oil film (Hz) | With oil film (Hz) |
|---|---|---|
| 1 | 127 | 121 |
| 2 | 203 | 198 |
| 3 | 1054 | 1044 |
| 4 | 1129 | 1128 |
| 5 | 1130 | 1122 |
| 6 | 1230 | 1226 |
| 7 | 2118 | 2100 |
| 8 | 2798 | 2786 |
| 9 | 2969 | 2967 |
| 10 | 5120 | 5109 |
| 11 | 5837 | 5713 |
| 12 | 6691 | 6688 |
It can be seen that the oil film reduces the natural frequencies to different degrees. The largest reduction occurs at the eleventh mode, where the frequency decreases from 5837 Hz to 5713 Hz, a reduction of about 2.1%. The reduction is caused by the lower combined stiffness when the oil film is placed between the meshing teeth. The mode shapes show that the first few modes are dominated by the torsional motion of the gear pairs, while the higher modes are dominated by the bending vibration of the shafts. When the oil film is considered, the bending order of the shafts increases in some modes, which means that the lubricated spur gear system is more sensitive to high-frequency excitations. This phenomenon is important for condition monitoring of gearboxes: the lubricating film changes the natural frequencies and may shift the resonance frequencies away from or toward the operating frequency.
5. Dynamic Characteristics with Rough Tooth Surfaces
Real spur gear tooth surfaces are not perfectly smooth. The roughness height is of the same order as the oil film thickness in elastohydrodynamic lubrication. Therefore, the effects of tooth surface roughness must be included in the analysis. In this chapter, I use a non-Newtonian Ree–Eyring fluid model and a cosine roughness function to represent the transverse roughness pattern on the tooth surfaces. The Ree–Eyring model is appropriate for mineral oils under moderate and high shear rates:
$$\frac{\partial u}{\partial y} = \frac{\tau_0}{\mu} \sinh\left(\frac{\tau}{\tau_0}\right)$$
where \(\tau\) is the shear stress and \(\tau_0\) is the characteristic shear stress of the lubricant. When \(\tau_0\) tends to infinity, the model reduces to the Newtonian fluid model. The surface roughness functions of the pinion and gear are:
$$S_p(x,t)=A_p \cos\left(\frac{2\pi}{l_p}\left[x – \int_0^t u_p \, dt\right] + \varphi_p\right)$$
$$S_g(x,t)=A_g \cos\left(\frac{2\pi}{l_g}\left[x – \int_0^t u_g \, dt\right] + \varphi_g\right)$$
where \(A_p\), \(A_g\) are the roughness amplitudes, \(l_p\), \(l_g\) are the roughness wavelengths, \(u_p\), \(u_g\) are the surface velocities, and \(\varphi_p\), \(\varphi_g\) are the initial phase angles. The film thickness equation for rough surfaces is:
$$h(x,t) = h_0(t) + \frac{x^2}{2R(t)} + v(x,t) – S_p(x,t) – S_g(x,t)$$
The Reynolds equation for the rough-surface transient thermal elastohydrodynamic lubrication problem is written in a generalized form:
$$\frac{\partial}{\partial x}\left(\varepsilon \frac{\partial p}{\partial x}\right) = \frac{\partial(\bar{\rho} h)}{\partial t} + \frac{\partial(\bar{\rho} u_e h)}{\partial x}$$
where \(\varepsilon\) is the flow coefficient, \(\bar{\rho}\) is the average density, and \(u_e\) is the effective entrainment velocity. The energy equation and the heat conduction equations for the gear bodies are solved to obtain the temperature distribution in the oil film and in the tooth surfaces. The viscosity–temperature relation is included in the Roelands equation, while the density–temperature relation is included in the Dowson–Higginson density equation.
The multigrid method is used to solve the rough-surface elastohydrodynamic lubrication equations because the roughness introduces high-frequency pressure oscillations that are difficult to handle with direct iteration methods. The solution domain is discretized with a fine mesh in the contact region, and the pressure, film thickness, temperature, and shear stress are updated recursively on different grid levels.
Table 5 compares the lubrication behaviour of smooth and rough tooth surfaces for the first and second spur gear pairs. The input speed is 5500 r/min and the load torque is 100 N·m.
| Surface type | Film thickness | Film pressure |
|---|---|---|
| Smooth | Uniform plateau in the central region; obvious necking at the outlet | Smooth distribution; distinct secondary pressure peak |
| Rough | Wavy distribution; mean thickness is larger because of the pump effect of transverse roughness | High pressure spikes at roughness peaks; secondary peak is less obvious |
For the first gear pair, the oil film pressure is higher in the rough-surface case than in the smooth-surface case at the roughness peaks. The film thickness oscillates around a larger mean value because the transverse roughness acts like small pumping elements that enhance the oil entrainment. However, the pressure spikes are very sharp, and these local pressure peaks can cause fatigue damage on the tooth surfaces. The second gear pair has a lower entrainment speed and a larger load than the first gear pair; therefore, the average pressure is higher and the film thickness is smaller. This means that the second gear pair is more prone to oil film rupture and direct surface contact.
The time-varying oil film stiffness obtained from the rough-surface lubrication analysis is combined with the dry mesh stiffness. Figure 3 shows the time-varying oil film stiffness of the first and second gear pairs, and Figure 4 compares the comprehensive mesh stiffness with and without the oil film. The oil film stiffness has a periodic distribution because the meshing position changes periodically. When the oil film is considered, the total mesh stiffness decreases noticeably. In the single-tooth contact region, the original deep depression of the dry mesh stiffness is partially filled by the oil film stiffness, so the variation of the comprehensive mesh stiffness is smaller. This can reduce the vibration excitation caused by the sudden change of mesh stiffness at the single-tooth/double-tooth transitions.
Table 6 lists the dynamic load reduction due to the oil film for both gear pairs.
| Item | Gear pair 1 (dry) | Gear pair 1 (lubricated) | Gear pair 2 (dry) | Gear pair 2 (lubricated) |
|---|---|---|---|---|
| Peak dynamic load | 1250 N | 1000 N | 1600 N | 1400 N |
| Load reduction percentage | — | 20% | — | 12.5% |
| Dominant frequency component | 2×\(f_{m2}\) | \(f_{m1}\) | 2×\(f_{m2}\) | 2×\(f_{m2}\) |
| Amplitude at dominant component | 171.0 N | 121.1 N | 234.6 N | 182.5 N |
The time-domain dynamic load of the first gear pair shows a beating phenomenon in the dry-contact case. The beating phenomenon is weakened when the oil film is considered. The peak dynamic load is reduced from about 1250 N to 1000 N, which indicates that the oil film can effectively cushion the impact between the meshing teeth. For the second gear pair, the peak dynamic load is reduced from 1600 N to 1400 N, and the valley load is reduced from 700 N to 600 N. The reduction effect is smaller than that of the first gear pair because the second gear pair operates at a lower speed and a higher load, which makes the oil film thinner and less effective.
The frequency spectra of the dynamic loads reveal important information about the vibration transmission between the two stages. The meshing frequencies of the first and second gear pairs are:
$$f_{m1} = \frac{n_{p1}}{60} Z_{p1}, \quad f_{m2} = \frac{n_{p1}}{60} \frac{Z_{p1}}{Z_{g1}} Z_{p2}$$
For the operating condition with an input speed of 5500 r/min, the first-stage meshing frequency is 3300 Hz and the second-stage meshing frequency is about 1063 Hz. In the dry-contact case, the dynamic load spectrum of the first gear pair contains not only the first-stage meshing frequency but also the second-stage meshing frequency and its harmonics. This indicates that the vibration of the second gear pair is transmitted to the first gear pair through the intermediate shaft. When the oil film is considered, the amplitude at \(2 \times f_{m2}\) is reduced by about 69.8%, and the dominant frequency component changes to the first-stage meshing frequency. The oil film therefore weakens the vibration transmission from the second stage to the first stage.
For the second gear pair, the dry-contact spectrum contains several harmonics of the second-stage meshing frequency. The highest amplitude occurs at \(2 \times f_{m2}\), which is close to the seventh natural frequency of the system. This is a resonance condition, and the oil film cannot eliminate the resonance. However, the oil film can reduce the resonant amplitude significantly. The amplitude at \(f_{m2}\) is reduced by 34.4%, and the amplitude at \(2 \times f_{m2}\) is reduced by 22.2%. The oil film also filters out some high-frequency components, so the frequency content of the dynamic load becomes simpler in the lubricated case.
6. Conclusions
In this thesis, I have developed a coupled elastohydrodynamic lubrication and dynamics model for a two-stage spur gear transmission. The following conclusions are drawn from the numerical results.
First, the line-contact elastohydrodynamic lubrication analysis shows that the oil film pressure and thickness are strongly affected by the entrainment velocity and load, while the initial viscosity has a relatively weak influence. The oil film has a compression limit, so the film thickness cannot be reduced indefinitely under heavy load. The secondary pressure peak is a characteristic feature of elastohydrodynamic lubrication; its magnitude and location depend on the operating conditions.
Second, the time-varying meshing parameters of spur gears, including the curvature radius, entrainment velocity, sliding velocity, and load distribution, change significantly along the line of action. The oil film stiffness decreases with increasing speed and increases with increasing load. The trend of the oil film stiffness is useful for the lubrication design of spur gears.
Third, the oil film stiffness changes the natural frequencies and mode shapes of the two-stage spur gear system. The natural frequencies are reduced when the oil film is considered, especially at the eleventh mode. The mode shapes show that the oil film increases the bending order of the shafts, which makes the spur gear system more sensitive to dynamic excitation. This finding is important for the design of gearbox structures and for the selection of operating speed ranges.
Fourth, the transverse roughness on the tooth surfaces produces a pump effect that increases the mean film thickness. However, the roughness peaks cause pressure spikes, which may lead to surface fatigue and pitting. The oil film reduces the dynamic load amplitude and weakens the vibration transmission from the second gear pair to the first gear pair. The frequency spectrum analysis shows that the oil film can significantly reduce the resonant amplitude at the second-stage meshing frequency harmonics.
Finally, the proposed model provides a systematic and parametric way to analyse the lubrication and dynamic characteristics of spur gears. The model can be extended to helical gears, planetary gears, and other types of gear transmissions. Future work should include the effect of the oil film damping, the bearing oil film stiffness, and real measured roughness profiles. In addition, experimental validation of the time-varying oil film stiffness and its effect on the gear vibration response should be carried out to further improve the modelling accuracy.