# Skidding Analysis of Exhaust Cam-Roller Unit in the Steady/Startup Operation of Internal Combustion Engine

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model and Solver

_{cr}R

_{r1}w and f

_{rp}R

_{r2}w are, respectively, the torque of the cam and the pin on the roller, and I

_{r}ώ

_{r}is the inertia torque of the roller.

#### 2.1. Cam-Roller Contact Pair

_{in}, y) = p(x

_{out}, y) = p(x, y

_{in}) = p(x, y

_{out})

_{in}≤ x ≤ x

_{out}, y

_{in}≤ y ≤ y

_{out})

_{Δ}is a sign function, f

_{Δ}= 1 if $\left|y\right|>{L}_{1}/2$, and f

_{Δ}= 0 if $\left|y\right|\le {L}_{1}/2$.

_{1}= lnη

_{0}+ 9.67, A

_{2}= 5.1 × 10

^{−9}Pa

^{−1}, A

_{3}= 1/(T

_{0}− 138), A

_{4}= 138/(T

_{0}− 138), Z

_{0}= α/(A

_{1}A

_{2}), and S

_{0}= β

_{0}/(A

_{1}A

_{3}).

_{1}= 0.6 × 10

^{−9}Pa

^{−1}, C

_{2}= 1.7 × 10

^{−9}Pa

^{−1}, and C

_{3}= 0.00065 K

^{−1}.

_{1}are the measured and modified load values, respectively.

#### 2.2. Roller-Pin Contact Pair

#### 2.3. Numerical Calculation Process

## 3. Comparison between Theoretical Simulation and the Experiment

_{r}/u

_{c}under relatively low viscosity conditions. Therefore, the coupling simulation model of cam-roller and roller-pin can reflect the skidding situation between the cam and the roller.

## 4. Results and Discussion

_{c}= u

_{r1}). Otherwise, there is a relative motion between the cam and the roller, which is described by the slide-roll ratio (SRR = 2(u

_{c}− u

_{r1})/(u

_{c}+ u

_{r1})).

#### 4.1. Skidding at Steady Running Process

_{c}≠ u

_{r1}) and pure rolling (u

_{c}= u

_{r1}), as shown in Figure 5, where the lubricant viscosity (η

_{0}) is equal to 0.152 Pa · s and the modified load (w

_{1}) is null. On the one hand, it can be deduced from the overall perspective trend of the minimum film thickness, maximum pressure, maximum temperature increase, and friction coefficient that the lubrication state of the cam-roller unit is stable at the rotation angle of the cam base circle part, while the lubrication parameters change significantly at the other rotation angles. This corresponds to the changes in the operating condition parameters such as cam curvature, cam surface velocity, and applied load in Figure 4. The lubrication analysis results can reflect the lubrication state of the cam-roller unit under different working conditions in real-time. On the other hand, the existence of skidding between the cam and the roller has almost no effect on the lubrication state of the roller-pin pair. In fact, even after skidding occurs, it will not greatly change the roller speed, which can be verified by the slide-roll ratio shown in Figure 6. For the same reason, the influence of skidding on the minimum film thickness and the maximum pressure is also very small in the cam-roller pair, while the maximum temperature rise and the friction coefficient will be greatly affected. Even a small skid on the surfaces between the cam and the roller can make the friction coefficient no longer equal to zero, which changes the temperature of the contact zone. Therefore, the skidding phenomenon between the cam and the roller mainly affects the temperature rise and the friction coefficient of the lubrication state in the cam-roller unit.

#### 4.1.1. Effect of Oil Viscosity

_{1}= 0 N. It can be clearly seen that the skidding condition is affected by the oil viscosity, and this effect is only reflected in the numerical value of the slide-roll ratio and not its regularity in a cam rotation cycle. The change in the slide-roll ratio is larger in the rotation angle of the cam base circle part. The oil viscosity can change the friction coefficient of the roller-pin contact (Figure 8), which in turn changes the driving torque required for the roller motion. However, at the rotation angle of the cam base circle, the contact load is relatively small, and the friction coefficient of the cam-roller contact should significantly change, resulting in an obvious change in the slide-roll ratio. The contact load is relatively large in the rotation angle of the cam opening and closing ramp, and thus the change in the slide-roll ratio is small. In addition, the oil viscosity also affects the minimum film thickness and the maximum temperature increase in the cam-roller unit, which is consistent with the classical lubrication theory. Therefore, the oil viscosity can change the skidding between the cam and the roller in the cam-roller unit.

#### 4.1.2. Effect of the Modified Load

#### 4.2. Skidding at the Startup Running Process

_{c}) at 180°. Figure 15 shows the variation of the cam surface velocity, where the accelerated rotation stage is 0–180° and the constant rotation stage is 180–360° during the startup running process.

_{0}= 0.152 Pa · s and w

_{1}= 0 N (Figure 16) to understand the characteristics of skidding during the startup process. It can be clearly seen that the skidding situation during the acceleration stage of the startup running process is different from that in the steady running process, but similar to the situation in the constant speed stage. This is related to the lubrication state of the cam-roller unit (Figure 15). At the beginning of the startup stage (0–18°), the difference in slide-roll ratio is greater than 50% because the cam needs to drive the stationary roller to start moving. Although a large slide-roll ratio produces a large roller driving torque (that is, a large friction coefficient f

_{cr}in Figure 17), it cannot realize the synchronous acceleration of the cam and the roller. However, it can gradually reduce the slide-roll ratio. In addition, a lubrication film is formed in two contact zones, and its thickness gradually increases. Within the cam rotation angle of 34–54°, the slide-roll ratio during startup is slightly lower than that in steady process (the difference is less than 5%). This may be due to the cam and the roller reaching a relatively stable state before entering the cam opening ramp. During the acceleration stage, the roller has a relatively lower rotational speed than that in the steady running condition, which indicates that the impedance torque (see from f

_{rp}in Figure 17) and the moment of inertia exerted on the roller are small. Therefore, the driving torque generated by the relatively small slide-roll ratio can achieve the desired acceleration. The change in working conditions then breaks the dynamic balance of the cam-roller unit, making the slide-roll ratio higher than that in the steady running process again. As the cam speed gradually approaches the rated speed, the gap between the slide-roll ratio and the lubrication state in the two conditions also gradually decreases. When the cam rotation angle exceeds 180°, the conditions for steady and startup processes are the same. That is, the slide-roll ratio and the lubrication state in the cam-roller unit are no longer different. In general, skidding is more obvious during the startup running process, especially at the beginning of the acceleration stage, where skidding is very severe and the lubricating oil film thickness is small.

#### 4.2.1. Effect of Oil Viscosity

_{rp}of the roller-pin contact (Figure 19). High viscosity can increase the impedance torque of the roller. It is worth mentioning that during the steady running process, this is consistent with the previously discussed influence of oil viscosity, and the effect of oil viscosity is higher than that in the startup process.

#### 4.2.2. Effect of the Modified Load

## 5. Conclusions

- Skidding mainly affects the temperature rise and the friction coefficient of the cam-roller contact, especially the friction coefficient;
- In a complete cam rotation cycle, the skidding situation is not constant, and even negative values of the slide-roll ratio may appear. In general, skidding is more serious at the rotation angle of the cam base part (light load) than at the other rotation angles (heavy load);
- Compared with the steady running process, skidding is more serious during the startup running process, especially at the beginning of the acceleration stage;
- Appropriately reducing the oil viscosity or increasing the initial load can effectively reduce skidding. In addition, the effect is more pronounced during the steady running process than the startup running process.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

b | Hertzian contact half width, m |

c | radial clearance, mm |

c_{f} | specific heat capacity of the lubricant, J/kg·K |

c_{s} | specific heat capacity of the solid, J/kg·K |

d | thickness of the thermal boundary layer (d = 3.15b), m |

e | eccentricity, mm |

E’ | reduced elastic modulus, Pa |

h | film thickness, μm |

h_{0} | rigid gap between two bounding bodies at the contact center, μm |

h_{min} | minimum film thickness, μm |

h_{td} | height of the roller convexity, mm |

I_{r} | moment of inertia of the roller, kg·m^{2} |

k_{f} | thermal conductivity of the lubricant, W/m·K |

k_{s} | thermal conductivity of the solid, W/m·K |

k_{r} | convective heat transfer coefficient, W/m^{2}·K |

L | total length of the roller, mm |

L_{1} | middle length of the roller, mm |

l | length of the rotation angle (l = R_{r2}ψ), m |

n | outside normal direction |

n_{c} | rated rotation speed of the cam, rpm |

p | pressure, Pa |

p_{max} | maximum pressure, Pa |

w, w_{1} | measured and later modified load value, N |

R_{x} | comprehensive curvature radius of the cam and the roller, mm |

R_{d} | end profile radius of the roller, mm |

R_{r1,}R_{r2} | outer and inner radius of the roller, mm |

SRR | slide to roll ratio |

t | time, s |

T | temperature, K |

T_{0} | ambient temperature, K |

T_{max} | maximum temperature increase, K |

u, v | velocity of the fluid along the x and y directions, m/s |

u_{c,}u_{r,} | surface velocity of the cam and the roller, m/s |

u_{r1,}u_{r2} | outer and inner surface velocity of the roller, m/s |

x, y, z | coordinates, m |

x_{in}, x_{out}y _{in}, y_{out} | boundary coordinates of the computation domain, m |

z_{c}, z_{r} | coordinates in the cam and the roller, m |

α | pressure-viscosity coefficient, Pa^{−1} |

β_{0} | temperature viscosity coefficient of the lubricant, K^{−1} |

η | viscosity, Pa·s |

η_{0} | ambient viscosity, Pa·s |

θ | cam rotation angle, deg |

λ | axial coordinate |

μ | friction coefficient |

ρ | density, kg/m^{3} |

ρ_{0} | ambient density, kg/m^{3} |

ρ_{s} | density of the solid, kg/m^{3} |

φ | attitude angle |

ψ | circumferential coordinate |

ω | rotation velocity of the cam, rpm |

ώ_{r} | acceleration of the roller, rad/s^{2} |

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**Figure 1.**Cam-roller unit in the valve mechanism. (

**a**) Physical photograph; (

**b**) contact simplified model.

**Figure 4.**Changes in the comprehensive curvature radius, cam surface velocity, and load spectrum in a complete cam rotation cycle.

**Figure 5.**Lubrication state of the cam-roller unit under the condition of skidding or rolling between the cam and the roller in a complete cam rotation cycle. (

**a**) Cam-roller contact; (

**b**) Roller-pin contact.

**Figure 6.**Change of the slide-roll ratio between the cam and the roller in a complete cam rotation cycle.

**Figure 7.**Impact of the viscosity on the slide-roll ratio between the cam and the roller in a complete cam rotation cycle.

**Figure 8.**Impact of the viscosity on the lubrication state of the cam-roller unit. (

**a**) Cam-roller contact; (

**b**) Roller-pin contact.

**Figure 10.**Variation of the minimum film thickness (

**a**), maximum temperature (

**b**), and fiction coefficient (

**c**) with the viscosity in the cam-roller unit.

**Figure 12.**Impact of the modified load on the lubrication state of the cam-roller unit. (

**a**) Cam-roller contact; (

**b**) roller-pin contact.

**Figure 14.**Variation of the minimum film thickness (

**a**), maximum pressure (

**b**), maximum temperature (

**c**), and friction coefficient (

**d**) with the modified load in the cam-roller unit.

**Figure 17.**Lubrication states of the cam-roller unit in the steady/startup running process. (

**a**) Cam-roller contact; (

**b**) roller-pin contact.

**Figure 19.**Impact of the viscosity on the lubrication state of the cam-roller unit in the startup running process. (

**a**) Cam-roller contact; (

**b**) roller-pin contact.

**Figure 21.**Impact of the modified load on the lubrication state in the startup running process. (a) Cam-roller contact; (b) Cam-roller contact.

Parameters | Numerical Values |
---|---|

Reduced elastic modulus of the contact solids E’/Pa | 2.1 × 10^{11} |

Density of the lubricant ρ_{0}/(kg/m^{3}) | 870 |

Density of the solid ρ_{s}/(kg/m^{3}) | 7850 |

Specific heat of the lubricant c_{f}/(J/kg·K) | 1700 |

Specific heat of the solid c_{s}/(J/kg·K) | 470 |

Thermal conductivity of the lubricant k_{f}/(W/m·K) | 0.14 |

Thermal conductivity of the solid k_{s}/(W/m·K) | 46 |

Convective heat transfer coefficient k_{r}/(W/m^{2}·K) | 80 |

Viscosity-pressure coefficient α/Pa^{−1} | 2.2 × 10^{−8} |

Viscosity-temperature β_{0}/K^{−1} | 0.042 |

Ambient temperature T_{0}/K | 313 |

Outer radius of the roller R_{r1}/mm | 25.5 |

Inner radius of the roller R_{r2}/mm | 16 |

Total length of the roller L/mm | 14 |

Middle length of the roller L_{1}/mm | 12 |

Surface convexity of the roller h_{td}/mm | 4.5 |

End profile radius of the roller R_{d}/mm | 12.5 |

Radial clearance between the roller and the pin c/μm | 10 |

Related rotation speed of the cam n_{c}/rpm | 950 |

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**MDPI and ACS Style**

Li, S.; Guo, F.; Wong, P.L.; Liang, P.; Huang, J.
Skidding Analysis of Exhaust Cam-Roller Unit in the Steady/Startup Operation of Internal Combustion Engine. *Lubricants* **2023**, *11*, 361.
https://doi.org/10.3390/lubricants11090361

**AMA Style**

Li S, Guo F, Wong PL, Liang P, Huang J.
Skidding Analysis of Exhaust Cam-Roller Unit in the Steady/Startup Operation of Internal Combustion Engine. *Lubricants*. 2023; 11(9):361.
https://doi.org/10.3390/lubricants11090361

**Chicago/Turabian Style**

Li, Shuyi, Feng Guo, Pat Lam Wong, Peng Liang, and Jisong Huang.
2023. "Skidding Analysis of Exhaust Cam-Roller Unit in the Steady/Startup Operation of Internal Combustion Engine" *Lubricants* 11, no. 9: 361.
https://doi.org/10.3390/lubricants11090361