# Lubricated Loaded Tooth Contact Analysis and Non-Newtonian Thermoelastohydrodynamics of High-Performance Spur Gear Transmission Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Tooth Contact Analysis (TCA) and Lubricated Loaded Tooth Contact Analysis (LLTCA)

#### 2.2. Elastohydrodynamic Lubrication (EHL)

^{−5}and 1.9609 × 10

^{8}, respectively and the piezoviscosity index, Z, can be found from:

#### 2.3. Tractive Analysis

#### 2.4. Thermal Network Model

## 3. Method of Solution

- Inputs from LLTCA at the start of the meshing cycle are used.
- An initial guess is made for the film thickness at the center of the contact.
- The computational domain is set with an inlet length of $12.42b$ and contact exit position of $4.42b$. The number of elements used in the direction of lubricant entrainment is 2051.
- Iterative pressure residuals are found using the under-relaxed Effective Influence Newton (EIN) method, including local surface deflection calculated through the use of Equation (9), which is based on Equation (38), where n denotes the iteration step and $\mathrm{\Omega}$ is the under-relaxation factor, typically ${10}^{-2}$–${10}^{-1}$:$${p}^{n}={p}^{n-1}+\mathrm{\Omega}\mathsf{\Delta}{p}^{n}.$$
- The iterative procedure evaluates the contact pressure and continues until the pressure convergence criterion is satisfied:$${\sum}_{i}{\sum}_{j}\left|\frac{{p}_{i,j}^{n}-{p}_{i,j}^{n-1}}{{p}_{i,j}^{n}}\right|\le {10}^{-3}.$$
- When pressure convergence is satisfied, the contact load-carrying capacity is calculated through the integration of pressure distribution over the computational domain as:$${W}_{p}=\int pdx.$$
- The following equilibrium condition should be satisfied in order to achieve a load balance condition where $W$ is the applied load:$$\left|\frac{W-{W}_{p}}{W}\right|\le {10}^{-3}.$$
- If Equation (41) is not satisfied, the film thickness is updated through modification of the undeformed gap, using Equation (42):$${h}_{0}={h}_{0}{\left(\frac{{W}_{t}}{W}\right)}^{\varsigma}.$$
- Once the film thickness is determined, the thermal network model (highlighted in Section 2.4) is used to find the temperature of the lubricant as well as those of the adjacent meshing surfaces.
- Steps 2 to 9 are repeated for each point on the meshing cycle until the entire meshing cycle is completed.

## 4. Shear Stress and Friction

## 5. Sub-Surface Stress Field

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${A}_{EHL}$ | Area of contact |

${a}_{v}$ | Vogel viscosity constant |

$b$ | Hertzian semi-half-width of the contact |

${b}_{v}$, ${c}_{v}$ | Vogel viscosity constants |

${c}_{p}$ | Specific heat capacity of the lubricant |

${c}_{s}$ | Specific heat capacity of the solids body |

$D$ | Deborah number |

${E}^{\prime}$ | Equivalent/reduced Young’s modulus of elasticity: $\left(2/\left(\frac{1-{\nu}_{1}^{2}}{{E}_{1}}+\frac{1-{\nu}_{2}^{2}}{{E}_{2}}\right)\right)$ |

$F$ | Havriliak and Negami non-Newtonian function |

${f}_{r}$ | Friction |

${G}_{e}$, ${G}_{v}$ | Parameters for the Greenwood chart |

${G}^{*}$ | Dimensionless materials’ parameter |

${H}_{c}$ | Dimensionless film thickness $\left({H}_{c}={h}_{c}/{R}^{\prime}\right)$ |

$h$ | Film thickness |

${h}_{0}$ | Minimum film thickness for the undeformed (rigid) profile |

${h}_{c}$ | Central contact film thickness |

$i,j$ | Nodal position identifier |

${k}_{l}$ | Thermal conductivity of the lubricant |

${k}_{s}$ | Thermal conductivity of the solids |

$l$ | Contact length |

$\dot{m}$ | Lubricant mass flow rate |

$n$ | Iteration counter |

$p$ | Pressure |

$\overline{p}$ | Average contact pressure $\left(W/4bl\right)$ |

${p}_{max}$ | Maximum Hertzian elastic line contact pressure $\left(\sqrt{W{E}^{\prime}/2\pi {R}^{\prime}}\right)$ |

$\dot{Q}$ | Rate of heat generation |

${\dot{Q}}_{1,2}$ | Heat conducted away through the bounding surfaces |

${\dot{Q}}_{cv}$ | Heat convected away by the lubricant |

$q$ | Shear field |

${R}^{\prime}$ | Radius of curvature of equivalent solid |

${R}_{i}$ | Conductive thermal resistivity of the lubricant |

${R}_{e}$ | Convective thermal resistivity of the lubricant |

${r}_{x}$ | Radius of curvature in the direction of entraining motion |

${r}_{y}$ | Radius of curvature in the side leakage direction |

$U$ | Rolling velocity in the direction of lubricant entrainment $\left({u}_{1}+{u}_{2}\right)$ |

$\overline{U}$ | Speed of entraining motion $\left(\left({u}_{1}+{u}_{2}\right)/2\right)$ |

$\mathsf{\Delta}U$ | Relative sliding velocity $\left({u}_{1}-{u}_{2}\right)$ |

${U}^{*}$ | Dimensionless (rolling) velocity parameter |

$V$ | Velocity in the slide leakage direction $\left({v}_{1}+{v}_{2}\right)$ |

$W$ | Applied load |

${W}_{p}$ | Load carrying capacity of the lubricant |

${W}^{*}$ | Dimensionless load parameter |

$s$ | Undeformed geometrical contact profile |

$x,y$ | Cartesian coordinate set |

$T$ | Bulk temperature |

${T}_{0}$ | Atmospheric reference temperature |

$t$ | Time |

## Greek Symbols

${\alpha}_{0}$ | Piezoviscosity coefficient at ambient temperature |

$\overline{\alpha}$ | Piezoviscosity coefficient at specified temperature |

${\alpha}_{HN}$, ${\beta}_{HN}$ | Havriliak and Negami parameters |

$\dot{\gamma}$ | Shear rate |

$\mathsf{\delta}$ | Localized elastic deflection |

$\zeta $ | Evans and Johnson’s friction parameter |

${\eta}_{0}$ | Viscosity at rest temperature and atmospheric pressure |

$\eta $ | Dynamic viscosity of the lubricant |

${\eta}_{c}$ | Dynamic viscosity at the center of the contact |

${\eta}_{p}$ | Piezoviscosity of the lubricant (solely dependent on pressure) |

${\theta}_{0}$ | Bulk flow temperature |

$\mathsf{\Delta}\theta $ | Temperature rise |

$\lambda $ | Relaxation time |

$\mu $ | Coefficient of friction |

$\rho $ | Density of the lubricant |

${\rho}_{c}$ | Density at the center of the contact |

${\rho}^{\prime}$ | Density of the solid bodies |

$\varsigma $ | Load relaxation parameter |

${\sigma}_{e}$ | Equivalent stress |

$\tau $ | Shear stress |

${\tau}_{l}$ | Limiting shear stress |

${\tau}_{0}$ | Characteristic shear stress |

$\mathrm{\Omega}$ | Pressure relaxation parameter |

${\chi}_{i}$ | Thermal partitioning coefficient |

## Abbreviations

1D | One-Dimensional |

CMM | Coordinate Measuring Machine |

EHL | Elastohydrodynamic Lubrication |

EIN | Effective Influence Newton–Raphson |

FEA | Finite Element Analysis |

HN | Havriliak–Negami |

LLTCA | Lubricated Loaded Tooth Contact Analysis |

NVH | Noise, Vibration, and Harshness |

TCA | Tooth Contact Analysis |

## Appendix A

**Table A1.**Gear pair specification [19].

Gear Type | Spur |
---|---|

Pinion No. Teeth | 13 |

Wheel No. Teeth | 35 |

Gear Module | 3.8 |

Center Distance | 90 mm |

Gear Width | 13.3 mm |

**Figure A1.**Gear geometry assembly [19].

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**Figure 1.**Overview of (

**a**) Tooth Contact Analysis (TCA) and (

**b**) Lubricated Loaded Tooth Contact Analysis (LLTCA).

**Figure 2.**LLTCA procedure flow chart [19].

**Figure 4.**Entrainment speed through the meshing cycle [19].

**Figure 5.**Meshing cycle on the Greenwood Chart: I.R.—Isoviscous Rigid, V.R—Piezoviscous Rigid, I.E.—Isoviscous Elastic, V.E.—Piezoviscous Elastic [38].

**Figure 6.**Variation of Deborah number during a meshing cycle with input data predicted through LLTCA and TCA.

**Figure 7.**The thermal network model [39].

**Figure 8.**Central contact thickness across the meshing cycle: N—Newtonian, HN—Havriliak, and Negami shear thinning.

**Figure 9.**Pressure and film thickness comparison between Newtonian and Havriliak and Negami Non-Newtonian behavior (LLTCA input).

**Figure 10.**Pressure and film thickness across the contact for mesh point (

**a**) 0.056, (

**b**) 0.25, (

**c**) 0.472, (

**d**) 0.666, (

**e**) 0.917, and (

**f**) 0.972 for the Havriliak and Negami cases.

**Figure 12.**Surface flash temperature of meshing bodies for Newtonian and non-Newtonian conditions for a complete meshing cycle.

**Figure 14.**Sub-surface orthogonal shear stress ${\tau}_{xz}$ contours for meshing point (

**a**) 1/18

^{th}, and (

**b**) 35/36

^{th}position in the meshing cycle.

**Table 1.**Lubricant base properties [22].

Parameter | Symbol | Value | Units |
---|---|---|---|

Young’s modulus (both pinion and wheel) | ${E}_{1}$, ${E}_{2}$ | 206 | GPa |

Poisson ratio (both pinion and wheel) | ${\nu}_{1}$, ${\nu}_{2}$ | 0.3 | - |

Dynamic viscosity (at 40 °C) | ${\eta}_{0}$ | 0.0304 | Pa·s |

Lubricant density (at 40 °C) | ${\rho}_{0}$ | 851 | kg/m^{3} |

Pressure coefficient of viscosity (at 40 °C) | ${\alpha}_{0}$ | 1.69 × 10^{−8} | 1/Pa |

Density of the solid | ${\rho}^{\prime}$ | 7800 | kg/m^{3} |

Limiting shear stress | ${\tau}_{L}$ | 2 | MPa |

Specific heat capacity of lubricant | ${c}_{p}$ | 1670 | J/kg·K |

Specific heat capacity of solid | ${c}_{s}$ | 470 | J/kg·K |

Thermal conductivity of lubricant | ${k}_{l}$ | 0.137 | W/m·K |

Thermal conductivity of the solid | ${k}_{s}$ | 46.7 | W/m·K |

**Table 2.**Non-Newtonian lubricant properties [50].

Parameter | Value | Units |
---|---|---|

${\lambda}_{f}$ | 7.9 × 10^{−8} | S |

${\alpha}_{HN}$ | 0.7 | - |

${\beta}_{HN}$ | 1 | - |

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**MDPI and ACS Style**

Sivayogan, G.; Rahmani, R.; Rahnejat, H.
Lubricated Loaded Tooth Contact Analysis and Non-Newtonian Thermoelastohydrodynamics of High-Performance Spur Gear Transmission Systems. *Lubricants* **2020**, *8*, 20.
https://doi.org/10.3390/lubricants8020020

**AMA Style**

Sivayogan G, Rahmani R, Rahnejat H.
Lubricated Loaded Tooth Contact Analysis and Non-Newtonian Thermoelastohydrodynamics of High-Performance Spur Gear Transmission Systems. *Lubricants*. 2020; 8(2):20.
https://doi.org/10.3390/lubricants8020020

**Chicago/Turabian Style**

Sivayogan, Gajarajan, Ramin Rahmani, and Homer Rahnejat.
2020. "Lubricated Loaded Tooth Contact Analysis and Non-Newtonian Thermoelastohydrodynamics of High-Performance Spur Gear Transmission Systems" *Lubricants* 8, no. 2: 20.
https://doi.org/10.3390/lubricants8020020