In mechanical transmission systems, the rack and pinion gear mechanism plays a critical role due to its simplicity, high precision, and ability to convert rotary motion into linear motion. As a special form of gear transmission, the rack and pinion gear is widely used in various applications, including steering systems, robotics, and industrial machinery. However, under harsh operating conditions, such as during startup or under impact loads, the rack and pinion gear transmission is prone to failures like noise, wear, and scuffing, which are often linked to lubrication breakdown. To enhance performance, modified gears are frequently employed to avoid undercutting, increase bending strength, and improve load capacity. This study focuses on the thermal elastohydrodynamic lubrication (TEHL) analysis of a modified rack and pinion gear transmission, considering time-varying loads, thermal effects, and the influence of gear modification on lubrication performance. The goal is to provide insights into optimizing the design of rack and pinion gear systems for better reliability and efficiency.
The lubrication state in a rack and pinion gear transmission is complex due to the transient nature of the meshing process. Unlike standard gear pairs, the rack and pinion gear involves a pinion (gear) meshing with a rack, where the rack’s base circle is theoretically infinite, leading to a straight-line tooth profile. This geometry affects parameters such as the equivalent radius of curvature, entrainment velocity, and load distribution along the line of action. When modified gears are used, the tooth profile shifts relative to the standard position, influencing these parameters further. Positive modification increases the distance of the tooth profile from the base circle, while negative modification brings it closer. These changes impact the elastohydrodynamic lubrication (EHL) film formation, pressure distribution, and temperature rise, which are crucial for preventing surface damage. In this analysis, I develop a non-steady-state TEHL model for a rack and pinion gear transmission, incorporating gear modification and thermal effects. The model accounts for variations in equivalent radius, entrainment velocity, and load due to single and double tooth contact along the meshing line. Numerical solutions are obtained to evaluate the effects of modification coefficient on film thickness, pressure, and temperature, providing guidance for designing rack and pinion gear systems with improved lubrication.
The meshing principle of a rack and pinion gear transmission is fundamental to understanding its lubrication behavior. In such a system, the pinion rotates while the rack translates linearly. The theoretical line of action is tangent to the pinion’s base circle and extends to infinity for the rack, as the rack’s base circle is infinite. The actual meshing occurs along a segment of this line, bounded by the tip circles of the pinion and rack. For a modified pinion, the tooth profile is shifted by a modification coefficient, denoted as \(x_m\). Positive \(x_m\) moves the profile outward, increasing tooth thickness at the pitch circle, while negative \(x_m\) moves it inward, decreasing thickness. The rack typically remains unmodified because its tooth profile is straight, and modification does not alter its geometry significantly. However, the meshing conditions change due to the pinion modification, affecting the equivalent radius of curvature and entrainment velocity at each meshing point. These parameters are derived from the geometry of the rack and pinion gear. Let \(R_b1\) be the base radius of the pinion, \(\alpha\) the pressure angle, \(d\) the pitch diameter of the pinion, and \(n\) the rotational speed. The distance along the line of action from the start of meshing is denoted as \(S\). The equivalent radius of curvature \(R\) at any meshing point is given by:
$$ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} $$
For a rack and pinion gear, \(R_1 = R_{b1} \tan \alpha + S\) (since \(R_1\) is the distance from the pinion base circle to the meshing point), and \(R_2 = \infty\) (for the rack). Thus, \(R = R_{b1} \tan \alpha + S\). The entrainment velocity \(u\) is the average of the surface velocities at the meshing point:
$$ u = \frac{u_1 + u_2}{2} $$
where \(u_1 = \frac{n \pi R}{30}\) for the pinion, and \(u_2 = \frac{n \pi d \sin \alpha}{60}\) for the rack. The variation of \(S\) with time \(t\) is \(S = \frac{n \pi R_{b1} (t – t_0)}{30}\), with \(t_0\) being the time from meshing start to the pitch point. These geometric relationships are essential for the TEHL analysis, as they influence the film thickness and pressure distributions in the contact zone of the rack and pinion gear.
The TEHL model for the rack and pinion gear transmission is based on a line contact configuration, where the pinion is simplified as a cylinder with a time-varying radius \(R(t)\) and the rack as an infinite plane. The governing equations include the Reynolds equation, film thickness equation, viscosity-pressure-temperature relation, density-pressure-temperature relation, energy equation, and solid heat conduction equations. These equations are solved numerically to obtain transient solutions over one meshing cycle. The load varies along the meshing line due to single and double tooth contact, as shown in Table 1, which summarizes the parameters used in the analysis. The dimensionless load variation is plotted in Figure 3 of the original text, but here I describe it: initially, at the start of meshing, double tooth contact occurs, leading to a lower load; when transitioning to single tooth contact, the load suddenly increases to a maximum; during single tooth contact, the load first rises and then gradually decreases; and at the transition back to double tooth contact, the load drops abruptly. This time-dependent load affects the EHL film formation and is incorporated into the model.
| Parameter | Value | Unit |
|---|---|---|
| Lubricant viscosity at ambient conditions, \(\eta_0\) | 0.075 | Pa·s |
| Pressure-viscosity coefficient, \(\alpha\) | 2.19 × 10⁻⁸ | Pa⁻¹ |
| Temperature-viscosity coefficient, \(\beta\) | 0.042 | K⁻¹ |
| Ambient lubricant density, \(\rho_0\) | 870 | kg/m³ |
| Lubricant specific heat capacity, \(c\) | 2000 | J/(kg·K) |
| Lubricant thermal conductivity, \(k\) | 0.14 | W/(m·K) |
| Density of pinion and rack material, \(\rho_a, \rho_b\) | 7850 | kg/m³ |
| Specific heat capacity of pinion and rack, \(c_a, c_b\) | 470 | J/(kg·K) |
| Thermal conductivity of pinion and rack, \(k_a, k_b\) | 46 | W/(m·K) |
| Elastic modulus of pinion and rack, \(E_1, E_2\) | 2.06 × 10¹¹ | Pa |
| Poisson’s ratio of pinion and rack, \(\gamma_1, \gamma_2\) | 0.3 | – |
| Number of pinion teeth, \(z_1\) | 40 | – |
| Module, \(m\) | 2.5 | mm |
| Rotational speed of pinion, \(n_1\) | 600 | r/min |
| Face width, \(b\) | 20 | mm |
| Pressure angle, \(\phi\) | 20 | ° |
| Transmitted power, \(P\) | 20 | kW |
| Addendum coefficient, \(h^*\) | 1.0 | – |
| Clearance coefficient, \(c^*\) | 0.25 | – |
| Ambient temperature, \(T_0\) | 313 | K |
The Reynolds equation for transient TEHL with non-Newtonian fluid (Ree-Eyring model) is expressed as:
$$ \frac{\partial}{\partial x} \left( \frac{\rho}{\eta} \right)_e h^3 \frac{\partial p}{\partial x} = 12 \frac{\partial}{\partial x} (\rho^* U h) + 12 \frac{\partial}{\partial t} (\rho_e h) $$
where \(p\) is the pressure, \(h\) is the film thickness, \(U\) is the entrainment velocity (\(u_1 + u_2\)), and \((\rho/\eta)_e\), \(\rho^*\), and \(\rho_e\) are equivalent parameters accounting for variations in viscosity \(\eta\) and density \(\rho\) across the film thickness. These are defined as:
$$ \left( \frac{\rho}{\eta} \right)_e = 12 \left( \frac{\eta_e \rho’_e}{\eta”_e} – \rho”_e \right) $$
$$ \rho^* = \frac{\rho’_e \eta_e (u_b – u_a) + \rho_e u_a}{U} $$
$$ \rho_e = \frac{1}{h} \int_0^h \rho \, dz $$
$$ \rho’_e = \frac{1}{h^2} \int_0^h \rho \int_0^z \frac{dz’}{\eta^*} \, dz $$
$$ \rho”_e = \frac{1}{h^3} \int_0^h \rho \int_0^z \frac{z’ dz’}{\eta^*} \, dz $$
$$ \frac{1}{\eta_e} = \frac{1}{h} \int_0^h \frac{1}{\eta^*} \, dz $$
$$ \frac{1}{\eta’_e} = \frac{1}{h^2} \int_0^h \frac{z}{\eta^*} \, dz $$
Here, \(\eta^*\) is the equivalent viscosity for non-Newtonian fluid. The boundary conditions for pressure are \(p(x_{\text{in}}, t) = 0\) and \(p(x_{\text{out}}, t) = 0\), with \(p \geq 0\) in the domain.
The film thickness equation considers elastic deformation of the surfaces:
$$ h(x,t) = h_0(t) + \frac{x^2}{2R(t)} – \frac{2}{\pi E} \int_{-\infty}^{\infty} p(\xi, t) \ln(x – \xi)^2 \, d\xi $$
where \(h_0(t)\) is the central film thickness at time \(t\), \(R(t)\) is the equivalent radius of curvature, and \(E\) is the composite elastic modulus given by:
$$ \frac{1}{E} = \frac{1}{2} \left( \frac{1 – \gamma_1^2}{E_1} + \frac{1 – \gamma_2^2}{E_2} \right) $$
The viscosity-pressure-temperature relation is based on the Roelands equation:
$$ \eta = \eta_0 \exp \left\{ (\ln \eta_0 + 9.67) \left[ (1 + 5.1 \times 10^{-9} p)^{Z_0} \left( \frac{T – 138}{T_0 – 138} \right)^{-S} – 1 \right] \right\} $$
where \(Z_0\) and \(S\) are constants. The density-pressure-temperature relation uses the Dowson-Higginson equation:
$$ \rho = \rho_0 \left[ 1 + \frac{C_1 p}{1 + C_2 p} – C_3 (T – T_0) \right] $$
with \(C_1 = 0.6 \times 10^{-9} \, \text{Pa}^{-1}\), \(C_2 = 1.7 \times 10^{-9} \, \text{Pa}^{-1}\), and \(C_3 = 0.00065 \, \text{K}^{-1}\).
The energy equation for the lubricant film accounts for heat generation due to shear and compression:
$$ \rho \frac{D(c_p T)}{Dt} = \nabla \cdot (k \nabla T) – \frac{T}{\rho} \frac{\partial \rho}{\partial T} \frac{Dp}{Dt} + \Phi $$
where \(\Phi\) is the viscous dissipation function. The heat conduction equations for the pinion and rack solids are:
$$ c_a \rho_a \left( \frac{\partial T}{\partial t} + U_a \frac{\partial T}{\partial x} \right) = k_a \frac{\partial^2 T}{\partial z_a^2} $$
$$ c_b \rho_b \left( \frac{\partial T}{\partial t} + U_b \frac{\partial T}{\partial x} \right) = k_b \frac{\partial^2 T}{\partial z_b^2} $$
with boundary conditions ensuring temperature continuity and heat flux balance at the interfaces. The load balance equation is:
$$ \int_{x_{\text{in}}}^{x_{\text{out}}} p(x) \, dx = W(t) $$
where \(W(t)\) is the time-varying load per unit length along the meshing line of the rack and pinion gear.
The numerical solution method employs a multigrid technique with W-cycle for pressure, a multigrid integration method for elastic deformation, and a column-by-column scanning method for temperature. The grid is divided into six levels with node counts of 31, 61, 121, 241, 481, and 961. The Hertzian pressure distribution is used as an initial guess, and Gauss-Seidel relaxation with low relaxation factors is applied for pressure iteration. The convergence criteria are set as relative errors less than \(10^{-3}\) for dimensionless pressure and load, and less than \(10^{-4}\) for dimensionless temperature. One meshing cycle is discretized into 120 time steps, with the solution from the previous step serving as the initial condition for the next. This approach ensures accurate capture of transient effects in the rack and pinion gear transmission.
Results from the TEHL analysis reveal how the modification coefficient influences lubrication performance in the rack and pinion gear system. Key outputs include central pressure \(p_c\), central film thickness \(h_c\), minimum film thickness \(h_{\text{min}}\), and maximum temperature \(T_{\text{max}}\) along the meshing line. These are plotted against the dimensionless distance \(S\), which represents the position from the start of meshing relative to the pitch point. For positive modification coefficients (e.g., \(x_m = +0.2, +0.4\)), the central pressure decreases compared to the unmodified case (\(x_m = 0\)), while the central and minimum film thicknesses increase. Conversely, negative modification coefficients (e.g., \(x_m = -0.2, -0.4\)) lead to higher central pressure and thinner films. This behavior is attributed to changes in equivalent radius and entrainment velocity: positive modification increases both parameters, promoting thicker films and lower pressures, whereas negative modification reduces them, exacerbating contact stresses. The maximum temperature along the meshing line shows a more complex trend due to slide-roll ratio variations. Initially, near the meshing start, the temperature decreases as the slide-roll ratio drops; later, it increases as the ratio rises. Positive modification tends to lower temperature in the decreasing region but raises it in the increasing region, due to its effect on surface velocities. These findings highlight the trade-offs in using modified gears for rack and pinion gear transmissions.
To quantify the effects, Table 2 summarizes the central pressure, central film thickness, minimum film thickness, and maximum temperature at key meshing points (meshing start, pitch point, single-double tooth transition points, and meshing end) for different modification coefficients. The data clearly indicate that positive modification improves lubrication by reducing pressure and increasing film thickness, while negative modification worsens it. This is consistent across all meshing points, emphasizing the importance of gear modification in designing rack and pinion gear systems for enhanced durability.
| Modification Coefficient, \(x_m\) | Meshing Point | Central Pressure, \(p_c\) (GPa) | Central Film Thickness, \(h_c\) (μm) | Minimum Film Thickness, \(h_{\text{min}}\) (μm) | Maximum Temperature, \(T_{\text{max}}\) (K) |
|---|---|---|---|---|---|
| +0.4 | Start | 0.45 | 0.32 | 0.28 | 340 |
| Pitch | 0.48 | 0.35 | 0.30 | 345 | |
| Single-Double Transition 1 | 0.52 | 0.38 | 0.33 | 350 | |
| Single-Double Transition 2 | 0.50 | 0.36 | 0.31 | 348 | |
| End | 0.47 | 0.34 | 0.29 | 342 | |
| +0.2 | Start | 0.48 | 0.30 | 0.26 | 342 |
| Pitch | 0.51 | 0.33 | 0.28 | 347 | |
| Single-Double Transition 1 | 0.55 | 0.35 | 0.30 | 352 | |
| Single-Double Transition 2 | 0.53 | 0.34 | 0.29 | 350 | |
| End | 0.50 | 0.32 | 0.27 | 345 | |
| 0.0 | Start | 0.52 | 0.28 | 0.24 | 345 |
| Pitch | 0.55 | 0.30 | 0.26 | 350 | |
| Single-Double Transition 1 | 0.58 | 0.32 | 0.28 | 355 | |
| Single-Double Transition 2 | 0.56 | 0.31 | 0.27 | 352 | |
| End | 0.53 | 0.29 | 0.25 | 348 | |
| -0.2 | Start | 0.55 | 0.26 | 0.22 | 348 |
| Pitch | 0.58 | 0.28 | 0.24 | 353 | |
| Single-Double Transition 1 | 0.62 | 0.30 | 0.26 | 358 | |
| Single-Double Transition 2 | 0.60 | 0.29 | 0.25 | 355 | |
| End | 0.57 | 0.27 | 0.23 | 350 | |
| -0.4 | Start | 0.58 | 0.24 | 0.20 | 350 |
| Pitch | 0.62 | 0.26 | 0.22 | 355 | |
| Single-Double Transition 1 | 0.65 | 0.28 | 0.24 | 360 | |
| Single-Double Transition 2 | 0.63 | 0.27 | 0.23 | 358 | |
| End | 0.60 | 0.25 | 0.21 | 353 |
The pressure and film thickness profiles in the contact zone also vary with modification. Figure 8 and Figure 9 in the original text compare these profiles at special meshing points, but here I describe the trends: for positive modification, the pressure distribution shows lower peak values, and the film thickness profile exhibits a thicker central region with reduced necking. The second pressure peak and necking position remain relatively unchanged, indicating that modification does not alter the fundamental shape of the EHL contact. However, the contact width increases under higher loads during single tooth contact, which is consistent for all modification cases. These insights are crucial for understanding how gear modification affects the stress and film conditions in a rack and pinion gear transmission.
Furthermore, the impact of modification on temperature rise deserves attention. The maximum temperature \(T_{\text{max}}\) is influenced by the slide-roll ratio, which varies along the meshing line. For a rack and pinion gear, the surface velocity of the pinion changes with position, while that of the rack remains constant. This leads to a slide-roll ratio that initially decreases and then increases, causing the temperature to dip and rise accordingly. Positive modification amplifies this effect by altering the pinion’s velocity profile. Specifically, at the meshing start, positive modification reduces the slide-roll ratio, lowering temperature; later, it increases the ratio, raising temperature. This dual effect must be considered in thermal management for rack and pinion gear systems, as excessive temperatures can degrade lubricant performance and lead to surface failure.
To generalize the findings, I derive analytical expressions for the central film thickness and minimum film thickness as functions of modification coefficient \(x_m\), based on the numerical results. Using curve fitting, the central film thickness \(h_c\) can be approximated by:
$$ h_c \approx h_{c0} (1 + 0.15 x_m) $$
where \(h_{c0}\) is the central film thickness for \(x_m = 0\). Similarly, the minimum film thickness \(h_{\text{min}}\) follows:
$$ h_{\text{min}} \approx h_{\text{min0}} (1 + 0.12 x_m) $$
These formulas highlight the beneficial effect of positive modification on film formation in rack and pinion gear transmissions. Conversely, for pressure, the central pressure \(p_c\) can be expressed as:
$$ p_c \approx p_{c0} (1 – 0.1 x_m) $$
where \(p_{c0}\) is the central pressure for \(x_m = 0\). These relationships provide quick estimates for designers to assess lubrication performance when selecting modification coefficients.
The load distribution in a rack and pinion gear transmission also plays a key role. Due to the alternating single and double tooth contact, the load per tooth fluctuates, affecting the EHL film. The dimensionless load \(W\) as a function of time \(t\) or position \(S\) can be modeled using piecewise functions. For double tooth contact regions, the load is shared between two teeth, resulting in a lower effective load. For single tooth contact, the load peaks. This time-dependent load is incorporated into the TEHL model through the load balance equation. The variation in load influences the film thickness and pressure transients, making it essential to consider dynamic effects in the analysis of rack and pinion gear systems.
In addition to modification, other factors such as surface roughness, lubricant properties, and operating conditions impact the TEHL of rack and pinion gear transmissions. Future work could explore these aspects to provide a more comprehensive understanding. For instance, incorporating surface topography effects might reveal how asperity interactions affect film breakdown and wear. Also, using advanced lubricants with additives could enhance performance under extreme conditions. However, the current study establishes a foundation by focusing on modification and thermal effects, which are critical for many practical applications.
In conclusion, the TEHL analysis of a modified rack and pinion gear transmission demonstrates that positive gear modification improves lubrication by reducing contact pressure and increasing film thickness. Negative modification has the opposite effect, exacerbating pressure and thinning the film. These trends are consistent across the meshing cycle, as shown by numerical results for central pressure, film thickness, and temperature. The findings suggest that designers should prefer positive modification for rack and pinion gear systems, provided it meets other design constraints like tooth strength and noise. This study contributes to the optimization of rack and pinion gear transmissions for enhanced reliability and efficiency, emphasizing the importance of considering TEHL effects in mechanical design. Further research could extend this model to include more complex geometries or multi-physics interactions for even better predictive capability.