# Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions could ideally be reduced by a factor of 4.5 [3]. However, this only applies if the electricity used is generated completely from renewable sources. Realistically, this might be possible only in a few countries, and even there, only within the next decades. As climate change is a global challenge, more efficient internal combustion engines (ICE) will be needed in the coming years [4,5]. Since a complete switch to electromobility is not feasible for resource reasons alone, alternative fuels, such as hydrogen, could become relevant in the combustion sector [6].Considering these aspects, the development of more efficient internal combustion engines will remain an important research focus in the upcoming years.

## 2. Materials and Methods

#### 2.1. Load Spectrum of the Cam–Tappet Contact

#### 2.2. Fluid Properites

#### 2.3. Numerical Modelling

#### 2.3.1. Dimensionless Scaling of the Contact Area

#### 2.3.2. Hydrodynamics

#### 2.3.3. Contact Mechanics

#### 2.3.4. Equilibrium of Forces

#### 2.3.5. Film Thickness Equation

#### 2.3.6. Cavitation

#### 2.3.7. Mixed Lubrication

#### 2.3.8. Numerical Implementation

#### 2.4. Target Values of the Simulation

## 3. Results

#### 3.1. Pressure and Lubriant Gap Distribution

#### 3.2. Friction in the Cam–Tappet Contact

#### 3.3. Influence of Surface Routhness on the Tribological Behaviour

## 4. Discussion

## 5. Conclusions

- Effects at the edges of the line contact appear to have an important influence on the tribological behavior. The narrower lubrication gap and the increased pressure at these areas suggest that the edge areas might contribute decisively to increased wear;
- The cam–tappet contact is in the mixed friction region, with the solid contact clearly dominating in the total friction force. The friction forces determined in the simulation agree well with those from experimental bench tests;
- The surface properties of cams and tappets have a considerable effect on the lubricant film structure and thus on the friction and wear behavior of the tribological system. The influence of roughness outweighs many other influencing factors, and thus deserves special attention;
- The selection of the simulation approach and the influencing variables should always be adapted to the aspect of the contact to be considered in order to find an optimal balance between accuracy and computational efficiency. The presented model is particularly suitable for the investigation of geometry adaptations and time-dependent friction force curves over the cycle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{c}$ | Carreau parameter |

${b}_{Hertz}$ | Herzian contact half-wide |

$C$ | Elasticity tensor |

${C}_{u}$ | Speed correction factor |

${C}_{F}$ | Force correction factor |

${C}_{R}$ | Radius correction factor |

${E}_{g}$ | Function of the cam edge geometry |

${F}_{n}$ | Contact normal force |

${F}_{R}$ | Friction force |

${F}_{R,solid}$ | Solid friction force |

${F}_{R,fluid}$ | Fluid friction force |

${G}_{c}$ | Critical shear stress |

$h$ | Lubricant gap height |

${h}_{\mathrm{liq}}$ | Gap height with fluid |

${h}_{min}$ | Minimum lubricant gap height |

$H$ | Dimensionless lubricant gap height |

${H}_{0}$ | Dimensionless distance between undeformed bodies |

$L$ | Dimensionless contact length in y-direction |

${n}_{c}$ | Carreau parameter |

$p$ | Hydrodynamic pressure |

${p}_{solid}$ | Solid contact pressure |

${p}_{cav}$ | Cavitation pressure |

${p}_{\mathrm{max}}$ | Maximum pressure |

$P$ | Dimensionless hydrodynamic pressure |

${P}_{solid}$ | Dimensionless solid contact pressure |

${P}_{tot}$ | Dimensionless total contact pressure |

${p}_{Hertz}$ | Hertzian contact pressure |

$R$ | Cam radius |

$S$ | y-axis scaling factor |

$t$ | Time |

$T$ | Dimensionless time |

${u}_{m}$ | Mean entrainment velocity |

$x,y,z$ | Coordinates |

$X,Y,Z$ | Dimensionless coordinates |

${\alpha}_{\eta}$ | Pressure viscosity coefficient |

$\gamma $ | Penalty function |

$\gamma $ | Ratio of the x and y correlation lengths |

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

$\overline{\delta}$ | Dimensionless elastic deformation |

$\epsilon $ | Strain tensor |

$\eta $ | Viscosity |

${\eta}_{0}$ | Base viscosity |

$\overline{\eta}$ | Dimensionless viscosity |

${\eta}_{\infty}$ | Second plateau viscosity |

${\eta}_{liq}$ | Viscosity of the liquid phase |

$\mathsf{\theta}$ | Fractional film content |

$\lambda $ | Lubricant gap height ratio |

$\mu $ | Coefficient of friction |

$\xi $ | Penalty factor |

$\rho $ | Density |

${\rho}_{0}$ | Base density |

${\rho}_{liq}$ | Density of the liquid phase |

$\overline{\rho}$ | Dimensionless density |

$\sigma $ | Stress tensor |

$\tau $ | Shear stress |

$\phi $ | Cam angle |

$\psi $ | Term of the Reynolds equation |

$\mathsf{\Omega}$ | Calculation area |

${\mathsf{\Omega}}_{\mathrm{c}}$ | Central calculation area |

$\nabla $ | Nabla operator |

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**Figure 1.**Simplified structure of the cam–tappet contact (

**a**) and the corresponding sections of the cam (

**b**).

**Figure 2.**Load collective of the cam–tappet contact at different camshaft speeds. Mean entrainment speed ${u}_{m}$ (

**a**) and contact normal force ${F}_{n}$ (

**b**) are shown over cam cycle from pre-cam (approx. ±90°…±55°) via rising flank (approx. ±55°…±40°) to cam tip (approx. ±40°…0°).

**Figure 3.**Scaled FE calculation area $\mathsf{\Omega}$ and representation of the mesh in the central contact area ${\mathsf{\Omega}}_{\mathrm{c}}$.

**Figure 4.**Solid-state contact pressure curve (

**a**) and flux factors (

**b**) depending on the height ratio $\lambda =\frac{h}{\sigma}$.

**Figure 5.**Lubricant gap $h$ (

**a**,

**b**) and total pressure p (

**c**,

**d**) for 500 rpm (

**a**,

**c**) and 2000 rpm (

**b**,

**d**) camshaft speed at pre-cam, rising flank and cam tip. In the center of the contact area, uniform distributions typical for line contacts were formed; at the edges, these deviated due to the edge geometry. The color scales of the enlargements are adjusted to show the edge effects. It should be noted that the length of the simulation area was extended to $10{b}_{Hertz}$ for the 2000 rpm load case.

**Figure 6.**Maximum pressure ${p}_{max}$ (

**a**) and minimum lubricant gap ${h}_{min}$ (

**b**) over one cam cycle for different load cases. The pressure curves were similar for all load cases, and the minimum lubricant film height increased considerably with increasing speed.

**Figure 7.**Fluid (

**a**) and solid (

**b**) friction forces over one cam cycle for different load cases using a combined surface roughness of $\sigma =0.1\mathsf{\mu}\mathrm{m}$. Solid-state friction was dominant in all load cases and occurred primarily in cam tip contact. It decreased with increasing speed. Fluid friction occurred mainly in the area of the rising flank. The solid friction was additionally evaluated for a lower coefficient of friction. Note different axis scaling.

**Figure 8.**Fluid (

**a**) and solid (

**b**) friction forces dependent on the surface roughness at 2000 rpm. With increasing roughness, the solid friction increased considerably, whereas the proportion of fluid friction decreased. Note different axis scaling.

Base density ${\rho}_{0}$ | $805\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}}$ |

Base viscosity ${\eta}_{0}$ | $0.03\mathrm{P}\mathrm{a}\cdot \mathrm{s}$ |

Pressure viscosity coefficient ${\alpha}_{\eta}$ | $1.31\cdot {10}^{-8}\mathrm{P}{\mathrm{a}}^{-1}$ |

Critical shear stress ${G}_{c}$ | $6\mathrm{M}\mathrm{P}\mathrm{a}$ |

Second plateau viscosity ${\eta}_{\infty}$ | $0.2{\eta}_{0}$ |

Carreau parameter ${a}_{c}$ | 2.2 |

Carreau parameter ${n}_{c}$ | 0.8 |

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**MDPI and ACS Style**

Orgeldinger, C.; Tremmel, S. Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects. *Lubricants* **2021**, *9*, 106.
https://doi.org/10.3390/lubricants9110106

**AMA Style**

Orgeldinger C, Tremmel S. Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects. *Lubricants*. 2021; 9(11):106.
https://doi.org/10.3390/lubricants9110106

**Chicago/Turabian Style**

Orgeldinger, Christian, and Stephan Tremmel. 2021. "Understanding Friction in Cam–Tappet Contacts—An Application-Oriented Time-Dependent Simulation Approach Considering Surface Asperities and Edge Effects" *Lubricants* 9, no. 11: 106.
https://doi.org/10.3390/lubricants9110106