# Methodological Approach in the Simulation of the Robustness Boundaries of Tribosystems under the Conditions of Boundary Lubrication

^{1}

^{2}

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## Abstract

**:**

_{f}and wear rate RR

_{I}, and must be used when designing new constructions of tribosystems. Theoretical calculations of such criteria and the dependence of their change on changing the predicted operating modes will allow for justifying rational operating modes within their stability.

## 1. Introduction

- Stable tribosystems, which, after being brought out of equilibrium by any external disturbance (change in load or sliding speed, or the short-term cessation of lubricant supply), after the removal of this disturbance, return to the original stable state, i.e., the established operating mode.
- Neutral tribosystems, which, after removing the disturbance, switch to a state of stable operation in a new mode, which is different from the original one.
- Unstable tribosystems, which, as they are brought out of equilibrium by any external disturbance, do not return to the original stable state after the removal of this disturbance, but switch to the mode of accelerated wear or to burr, i.e., cease operation.

## 2. Materials and Methods

_{1}, T

_{2}, T

_{3}are the time constants in the dimension of seconds (s);

_{1}, K

_{2}, K

_{3}are the gain coefficients, which are dimensionless quantities.

_{1}characterizes the change in the structure of the materials of the surface layers during run-in. It is defined by the following expression:

_{run}ia the tribosystem run-in time in the dimension of seconds.

_{2}has a physical sense of the time, during which the temperature is equalized by the volumes of triboelement materials when the load and sliding speed change. It is defined by the following expression:

_{run}is the stated volume of materials of the moving and stationary triboelements of the tribosystem, in the dimension of m

^{3}, and is determined by Expressions (2) and (3), which are given in [40], taking into account the total volume of the material of the friction node;

_{run}is the given coefficient of the thermal conductivity of the materials of the moving a

_{r}and stationary a

_{s}triboelements, in the dimension of m

^{2}/s, and is calculated according to Expression (6), which is given in [40];

_{acs}is the diameter of the actual contact spot, in the dimension of m, and it is calculated according to the expression given in [41];

_{acs}is the number of contact spots on the friction surface, and is calculated according to the expression given in [41].

_{3}characterizes the time until the friction and wear parameters stabilize in the new operating mode of the tribosystem. Time constant T

_{3}has different expressions to the model wear rates T

_{3(I)}and friction coefficient T

_{3(f)}:

_{def}is the given volume of deformable surface layers, measured in m

^{3}, and is determined by the expression given in [40], using Formulas (15) and (16);

^{−1}, which is calculated according to the formulas given in [30], using Formulas (8)–(10).

_{1}includes the degree of external influence on the tribosystem, considering the design features, which was proven in [40]:

_{0}and Q

_{max}are the initial value of the Q-factor of the tribosystem and the value of the Q-factor that was formed during the run-in. They are determined by the formulas given in [41].

_{2}characterizes the sensitivity of the tribosystem to changes in the external conditions:

_{TR}is the speed of dissipation in the tribosystem, measured in J/s, which is calculated according to the equations given in [1], using Formula (8);

_{f}is the shape coefficient of the tribosystem, in the dimension of m

^{−1}, and includes the friction areas and the volumes located under the friction areas of the moving and stationary triboelements. It is calculated according to the equation given in [41].

_{3}characterizes the ability of the tribosystem to change the structure of the surface layers of the triboelement materials during transient processes:

_{TS(max)}is the maximum value of the rheological properties of the combined materials in the tribosystem after running-in, measured in dimension m

^{−1}, and is calculated according to the equations given in [42].

_{f}. The shape factor takes into account the magnitude of the friction areas and the volumes of material that are placed under the working areas of friction. Values of 5500 and 170 were introduced to improve the accuracy of the simulation. When changing the structure of the friction unit, it is necessary to adjust the values of these coefficients.

_{i}of the characteristic Equation (9) are more than zero. This is a necessary condition for stability:

_{i}of the characteristics in Equation (9) are more than zero.

_{3}in the following form:

_{3}, measured in dimension s

_{3}, the more the margin of robustness for the tribosystem. When the value Δ

_{3}= 0, the tribosystem works on the verge of loss of stability, with negative values for Δ

_{3}, and there is burr or accelerated wear and the tribosystem has “lost stability”.

_{sl}is the sliding speed, m/s;

_{l}is the load change time, s.

_{f}and RR

_{I}for each load of the operating range of tribosystems, including the analysis of the obtained values. If the value of the criterion is more than one, then the tribosystem works in a stable range. The more the value of the robustness criterion, the more the margin of sustainable work.

_{o}, J/m

^{3}, [41,42].

_{b(i)}, W

_{b(aw)}are the values of external influence on the tribosystem, formula (21), at which there is a loss of stability (burr or accelerated wear), which is measured during the experiment and is averaged over the number of repetitions n.

_{b(exp)}, W

_{b(s)}is the value of the magnitude of the external influence on the tribosystem at which a loss of stability (burr or accelerated wear) occurs, which is measured during the experiment and according to the results of the simulation.

## 3. The Results of the Experimental Research

_{b(s)}, at which a loss of stability occurs (burr or accelerated wear), according to formulas (24) and (25), when RR

_{f}and RR

_{I}are equal to the unit, with the results of the experiment W

_{b(exp)}.

_{b(exp)}for various designs of tribosystems occurs, which are evaluated by the shape factor K

_{f}, m

^{−1}, is presented in the first block of Table 1. Experimental studies of the limits of the stable operation of tribosystems are performed for the tribosystem steel 5140 + bronze C61900, and lubricating medium E

_{y}= 3.6·10

^{14}J/m

^{3}, (motor oil SAE 40, API CC). The parameter E

_{y}takes into account the tribological properties of the lubricating medium. The technique for determining E

_{y}is described in [41]. The sliding speed is not changed and is equal υ

_{sl}= 0.5 m/s. During the experiment, the values of the shape factor of the tribosystem vary, K

_{f}= 6.25–22.6 m

^{−1}. This value is obtained by changing the friction area of the stationary triboelement. The load is increased from the initial value of 400 N, to the value when the tribosystem loses stability, i.e., burr or accelerated wear occur. The initial roughness of the friction surfaces for all experimental conditions was Ra = 0.2 μm.

_{f}= 6.25 m

^{−1}the loss of stability for the tribosystem occurs after the appearance of a burr. When increasing the form factor to K

_{f}= 22.6 m

^{−1}, the loss of stability of the tribosystem occurs after the appearance of accelerated wear.

_{W}= 8.3–15.0%, at the coefficient of variation v

_{W}= 16.6–20.0%. As it follows from the obtained results, an increase in the shape factor leads to an increase in the modelling error.

_{y}is presented in the second block of Table 1.

_{f}= 12.5 m

^{−1}. As a factor, the following changes have been selected: hydraulic oil HH, ISO-L-HL, (E

_{y}= 2.43·10

^{14}J/m

^{3}); motor oil SAE 40, API CC, (E

_{y}= 3.6·10

^{14}J/m

^{3}); transmission oil SAE 120, API GL-4, (E

_{y}= 4.18·10

^{14}J/m

^{3}). The sliding speed did not change and was equal υ

_{sl}= 0.5 m/s.

_{W}= 10.6–15.0%, at the coefficient of variation v

_{W}= 14.8–20.0%. A greater error is inherent in the use of lubricants with low values of tribological properties.

_{TS(max)}are presented in the third block of the Table 1.

_{f}= 12.5 m

^{−1}. Lubricating medium—motor oil SAE 40, API CC, (E

_{y}= 3.6·10

^{14}J/m

^{3}). As a factor that changes, have been selected: steel 5140 + steel 5140, (RS

_{TS(max)}= 326.7 m

^{−1}); steel 5140 + bronze C61900, (RS

_{TS(max)}= 436.0 m

^{−1}); steel 5140 + brass CW723R, (RS

_{TS(max)}= 460.9 m

^{−1}).

_{W}= 8.3–13.0%, at a coefficient of variation v

_{W}= 16.6–21.7%. A greater error is inherent in the use of composite materials with low values of rheological properties.

_{0}, are presented in the fourth block of Table 1. The determination of the Q-factor of the tribosystem is given in [41].

- Tribosystem №1: «steel 5140 + steel 5140», (RS
_{TS(max)}= 326.7 m^{−1}); K_{f}= 6.25 m^{−1}; lubricating medium E_{y}= 2.43·10^{14}J/m^{3}, (HH, ISO-L-HL). The value of the Q–factor of the tribosystem Q_{0}= 1.12·10^{10}J/m^{3}. - Tribosystem №2: «steel 5140 + bronze C61900», (RS
_{TS(max)}= 436.0 m^{−1}); K_{f}= 12.5 m^{−1}; lubricating medium E_{y}= 3.6·10^{14}J/m^{3}, (SAE 40, API CC). The value of the Q–factor of the tribosystem Q_{0}= 5.5·10^{10}J/m^{3}. - Tribosystem №3: «steel 5140 + brass CW723R», (RS
_{TS(max)}= 460.9 m^{−1}); K_{f}= 14.5 m^{−1}; lubricating medium E_{y}= 4.18·10^{14}J/m^{3}, (SAE 120, API GL-4). The value of the Q–factor of the tribosystem Q_{0}= 7.69·10^{10}J/m^{3}.

_{0}= 1.12·10

^{10}J/m

^{3}, (tribosystem №1), to values Q

_{0}= 7.69·10

^{10}J/m

^{3}, (tribosystem №3). An increase in the Q-factor contributes to an increase in the robustness of tribosystems.

_{W}= 14.8–18.7%, at the coefficient of variation v

_{W}= 12.9–18.7%. A bigger error is inherent in the application of tribosystems with low Q-values.

_{d(f)}= 3.42; k

_{d(I)}= 4.37, formulas (22) and (23). When the load time is reduced, to 1 s, the coefficients increase significantly, for example: k

_{d(f)}= 6.4; k

_{d(I)}= 7.38. The introduction of the coefficient for the change in the load speed (22) and (23) into the calculation expressions for determining the robustness of tribosystems (24) and (25) allows for reducing the modelling error.

## 4. Discussion

_{f}, formula (24), and RR

_{I}, formula (25), makes it possible to determine the rational modes of operation of tribosystems at the stage of design and engineering development. The operation of tribosystems in a certain range will increase their resource and reliability.

_{f}and RR

_{I}consider two options for the loss of stability.

## 5. Conclusions

_{f}and wear rate RR

_{I}, must be used when projecting new designs of tribosystems. Theoretical calculations of such criteria and the dependence of their change on changing the predicted operating modes will allow to justify rational operating modes within the limits of their stability.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

T_{1}, T_{2}, T_{3} | time constants, dimension s. |

K_{1}, K_{2}, K_{3} | amplification factors, dimensionless values. |

t_{run} | tribosystem run-in time, dimension s. |

V_{run} | the given volume of materials of the moving and stationary triboelements of the tribosystem, dimension m^{3} |

a_{run} | given coefficient of thermal conductivity of materials of movable a_{p} and fixed a_{n} triboelements, dimension m^{2}/s. |

d_{acs} | diameter of the actual spot of contact, dimension m. |

n_{acs} | the number of contact spots on the friction surface |

V_{def} | given volume of deformable surface layers, dimension m^{3}. |

${\dot{\epsilon}}_{run}$ | the magnitude of the rate of deformation of the surface layers of the materials of the movable and fixed triboelements, dimension s^{−1}. |

Q_{0} and Q_{max} | the initial Q-factor value of the tribosystem and the Q-factor value after run-in, dimension J/m^{3}. |

W_{TR} | speed of dissipation in the tribosystem, dimension J/s. |

K_{f} | tribosystem form factor, dimension m^{−1}. |

RS_{TC(max)} | the maximum value of the rheological properties of the combined materials in the tribosystem after the completion of running-in, dimension m^{−1}. |

N | load on the tribosystem, dimension N. |

v_{sl} | sliding speed, dimension m/s. |

t_{l} | load change time, dimension s. |

k_{d(f)} | coefficient of change of load speed according to the friction coefficient parameter, dimensionless value. |

k_{d(I)} | coefficient of change of the load speed according to the parameter of the wear rate, dimensionless value. |

RR_{f} | robustness of the tribosystem according to the friction coefficient, dimensionless value. |

RR_{I} | robustness of the tribosystem according to the rate of wear, dimensionless value. |

W_{b(i)}, W_{b(aw)} | the value of the magnitude of the external influence on the tribosystem at which occurs the loss of stability (burr or accelerated wear), according to the results of the experiment and according to the results of simulation, dimension J/s. |

E_{y} | tribological properties of the lubricating medium, dimension J/m^{3}. |

S_{Wb} | root mean square deviation of values of external influence during experimental studies, dimension J/s. |

v_{Wb} | coefficient of variation of measurements of external influence, at which the event of loss of stability of the tribosystem occurs, %. |

e_{Wb} | relative error of modeling the robustness of tribosystems, %. |

R_{a} | initial roughness of friction surfaces, dimension µm. |

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**Figure 1.**Dependencies of changes in the robustness criterion of various designs of tribosystems according to the coefficient of friction on the value of the input impact: 1—steel 5140 + steel 5140, K

_{f}= 6.25 m

^{−1}, hydraulic oil HH, ISO-L-HL, Q

_{0}= 1.12·10

^{10}J/m

^{3}; 2—steel 5140 + bronze C61900, K

_{f}= 12.5 m

^{−1}, motor oil SAE 40, APICC, Q

_{0}= 5.5·10

^{10}J/m

^{3}; 3—steel 5140 + brass CW723R, K

_{f}= 14.5 m

^{−1}; transmission oil SAE120, APIGL-4, Q

_{0}= 7.69·10

^{10}J/m

^{3.}

**Figure 2.**Block diagram of experimental equipment for recording and processing AE signals: 1—movable triboelement; 2—fixed triboelement; N—load.

**Figure 3.**Samples for testing: 1—movable sample steel 5140; 2—fixed samples: steel 5140, bronze C61900, brass CW723R with different friction areas on the end surfaces; 3—friction surfaces of the moving sample; 4—friction surfaces of a fixed sample.

**Table 1.**The results of testing the error of modelling the range of robustness of various designs of tribosystems.

Construction of the Tribosystem | Wb(s), N·m/s | Wb(exp), N·m/s | SWb, N·m/s | vWb, % | eWb, % |
---|---|---|---|---|---|

Steel 5140 + bronze C61900, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3}, K_{f} = 6.25 m^{−1} | 1300 (I) * | 1200 (I) * | 200 | 16.6 | 8.3 |

Steel 5140 + bronze C61900, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3}, K_{f} = 12.5 m^{−1} | 1900 (I, f) * | 1700 (I, f) * | 300 | 17.6 | 11.7 |

Steel 5140 + bronze C61900, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3}, K_{f} = 22.6 m^{−1} | 2300 (f) * | 2000 (f) * | 400 | 20.0 | 15.0 |

Steel 5140 + bronze C61900, K_{f} = 12.5 m^{−1}, hydraulic oil HH, ISO-L-HL, (E_{y} = 2.43·10^{14} J/m^{3}) | 850 (f) * | 1000 (f) * | 200 | 20.0 | 15.0 |

Steel 5140 + bronze C61900, K_{f} = 12.5 m^{−1}, motor oil SAE 40, APICC, (E_{y} = 3.6·10^{14} J/m^{3}) | 1900 (I, f) * | 1700 (I, f) * | 300 | 17.6 | 11.7 |

Steel 5140 + bronze C61900, K_{f} = 12.5 m^{−1}, transmission oil SAE 120, APIGL-4, (E_{y} = 4.18·10^{14} J/m^{3}) | 2600 (I) * | 2350 (I) * | 350 | 14.8 | 10.6 |

Steel 5140 + steel5140, (RS_{TS(max)} = 326.7 m^{−1}), K_{f} = 12.5 m^{−1}, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3} | 1300 (f) * | 1150 (f) * | 250 | 21.7 | 13.0 |

Steel 5140 + bronze C61900, (RS_{TS(max)} = 436.0 m^{−1}), K_{f} = 12.5 m^{−1}, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3} | 1900 (I, f) * | 1700 (I, f) * | 300 | 17.6 | 11.7 |

Steel 5140 +brass CW723R, (RS_{TS(max)} = 460.9 m^{−1}), K_{f} = 12.5 m^{−1}, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3} | 1950 (I, f) * | 1800 (I, f) * | 300 | 16.6 | 8.3 |

Tribosystem No. 1: Steel 5140 + Steel 5140, (RS_{TS(max)} = 326.7 m^{−1}), K_{f} = 6.25 m^{−1}, hydraulic oil HH, ISO-L-HL, (E_{y} = 2.43·10^{14} J/m^{3}), Q_{0} = 1.12·10^{10} J/m^{3} | 650 (f) * | 800 (f) * | 150 | 18.7 | 18.7 |

Tribosystem No. 2: Steel 5140 + bronze C61900, (RS_{TS(max)} = 436.0 m^{−1}), K_{f} = 12.5 m^{−1}, motor oil SAE 40, APICC, E_{y} = 3.6·10^{14} J/m^{3}, Q_{0} = 5.5·10^{10} J/m^{3} | 2000 (I, f) * | 1700 (I, f) * | 300 | 17.6 | 17.6 |

Tribosystem No. 3: Steel 5140 + brass CW723R, (RS_{TS(max)} = 460.9 m^{−1}), K_{f} = 14.5 m^{−1},transmission oil SAE 120, APIGL-4, E_{y} = 4.18·10^{14} J/m^{3}, Q_{0} = 7.69·10^{10} J/m^{3} | 3100 (I) * | 2700 (I) * | 350 | 12.9 | 14.8 |

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**MDPI and ACS Style**

Al-Quraan, T.M.A.; Alfaqs, F.; Alrefo, I.F.S.; Vojtov, V.; Voitov, A.; Kravtsov, A.; Miroshnyk, O.; Kondratiev, A.; Kučera, P.; Píštěk, V.
Methodological Approach in the Simulation of the Robustness Boundaries of Tribosystems under the Conditions of Boundary Lubrication. *Lubricants* **2023**, *11*, 17.
https://doi.org/10.3390/lubricants11010017

**AMA Style**

Al-Quraan TMA, Alfaqs F, Alrefo IFS, Vojtov V, Voitov A, Kravtsov A, Miroshnyk O, Kondratiev A, Kučera P, Píštěk V.
Methodological Approach in the Simulation of the Robustness Boundaries of Tribosystems under the Conditions of Boundary Lubrication. *Lubricants*. 2023; 11(1):17.
https://doi.org/10.3390/lubricants11010017

**Chicago/Turabian Style**

Al-Quraan, Tareq M. A., Fadi Alfaqs, Ibrahim F. S. Alrefo, Viktor Vojtov, Anton Voitov, Andrey Kravtsov, Oleksandr Miroshnyk, Andrii Kondratiev, Pavel Kučera, and Václav Píštěk.
2023. "Methodological Approach in the Simulation of the Robustness Boundaries of Tribosystems under the Conditions of Boundary Lubrication" *Lubricants* 11, no. 1: 17.
https://doi.org/10.3390/lubricants11010017