**4. Estimation of stability of the third body on the base of EHD theory of lubrication**

The most complete mathematical model of lubrication is the elastohydrodynamic (EHD) theory of lubrication [44]. The effectiveness of the EHD theory of lubrication is described by ratio λ or film parameter [45], which is the ratio of

At progressive damage of the third body, the friction torque increases and corresponds to positive friction. Our experimental researches have shown that in other equal conditions the variation of the friction coefficient mainly depends on degree of destruction of the third body. Therefore, preservation of the third body between interacting surfaces and avoidance the scuffing, has a crucial importance for decrease of the friction coefficient, wear rate, etc. This issue became burning especially for wheels and rails in the last 50 years and many works appeared that are devoted to enhancing stability of the wheel flanges against the operational impacts. In **Figure 14** are shown dependences of the friction factor and various damage types on relative sliding velocity (a) and of the wear rate (types) on slip (b) [32]. Three zones can be distinguished in **Figure 14a**. The low relative sliding velocity, full separation of the interacting surfaces and continuous third body provide high wear resistance of the interacting surfaces and relatively stable friction coefficient (zone 1, **Figure 14a**) that corresponds to "mild" [32] wear rate (**Figure 14b**). In such conditions, the main damage types are the fatigue and plastic deformations. Small increase of the sliding velocity leads to appearance of small damage sources in multiple places and emergence of small surges of the friction torque (zone 2, **Figure 14a**). The rise of the third body destruction, as well as the magnitude of the friction coefficient and its instability, are clearly reflected in the

*Tribology in Materials and Manufacturing - Wear, Friction and Lubrication*

*Dependences of the friction factor and various damage types on relative sliding velocity (a) and of the wear rate*

**Figure 13.**

**Figure 14.**

**142**

*(types) on slip (b).*

*Friction/creep relationship.*

film minimum thickness at the Hertzian contact zone to the r.m.s. of the rolling element surface finish:

$$
\lambda = \frac{h\_{\min}}{\sqrt{R\_{a\_1}^2 + R\_{a\_2}^2}} \tag{1}
$$

onset of the friction torque sharp increase is considered as beginning of the third

*A New Concept of the Mechanism of Variation of Tribological Properties of the Machine…*

experimental researches considering formula (1), criterion of the third body

On the base of system of equations of EHD, theory of lubrication and results of

*: Pn<sup>β</sup> R* � �0,6

As it follows from the formula (2), a criterion of the third body destruction depends on the mechanical and thermo-physical characteristics of interacting surfaces, geometric and kinematic parameters, thermo-physical and tribological parameters of the third body. The properties and stability of the boundary layers are revealed in values of coefficient K and exponent e. The researches have also shown special sensitivity of the third body stability to thermal loads and relative sliding velocities, which must be taken into account to improve working conditions.

The criterion of the third body destruction that is developed on the base of EHD theory of lubrication and results of experimental researches considering stability of

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *R*2 *<sup>a</sup>*<sup>1</sup> <sup>þ</sup> *<sup>R</sup>*<sup>2</sup> *a*2

� � q *<sup>f</sup>*

where VΣ<sup>k</sup> is a total rolling velocity; Vsl – sliding velocity; Pll – linear load; μ dynamic viscosity of the lubricant; R – reduced radius of curvature of the surfaces; Ra1 and Ra2- average standard deviation of the interacting surfaces; β – piezo coefficient of the lubricant viscosity; ζ– the lubricant thermal conductivity; α – thermal coefficient of the lubricant viscosity; a – thermal diffusivity; The exponents a, b, c, … , n and coefficient K are specified on the base of the experimental data obtained by T.I. Fowle, Y.N. Drozdov, Vellawer, G. Niemann, A.I. Petrusevich, I.I.

As it was already mentioned, one of the indicators of the third body destruction (scuffing) is appearance of signs of scuffing on the surfaces. According to criteria of destruction of the third body, its destruction is supposed when values of the corresponding criteria are less than 1. K. Schauerhammer experimentally ascertains the conditions of the third body destruction (scuffing) for the gear drive on the gear drive test bench TUME 11 [46]. To predict the destruction of the third body

**a b c d l fg h i j n**

�1 0.6 0.18 to 0.66

(�0.18) to (�0.66)

0.09 to 0.33

0.045 to 0.165

0.25 to 0.36

Sokolov, K. Shawerhammer, G. Tumanishvili and are given in the **Table 1**. As it is seen from the **Table 1**, destruction of the third body is especially sensitive to the degree b of sliding velocity. It follows from formulae (2) and (3) that with increase of the rolling velocity, radius of curvature, piezo-coefficient of viscosity, heat conductivity factor, thermal diffusivity and coefficient of elasticity, the stability of the third body increases and with increase of the sliding velocity, linear loading, roughness of surfaces and thermal coefficient of viscosity it

*: <sup>ζ</sup> αμVCK* 2 *Pe*1,2 2

� �*<sup>e</sup>*

• β<sup>g</sup> • ζ<sup>h</sup> • α<sup>i</sup> • a<sup>j</sup>

≤1 (3)

• En ≤ 1 (4)

body destruction.

destruction was developed that has a form:

*DOI: http://dx.doi.org/10.5772/intechopen.93825*

<sup>2</sup> <sup>þ</sup> *Ra*<sup>2</sup> 2

1

CA*: <sup>μ</sup>V<sup>Ξ</sup><sup>K</sup> Pn* � �0,7

<sup>C</sup> <sup>¼</sup> *<sup>K</sup> <sup>R</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Ra*<sup>1</sup>

the boundary layers has the form:

<sup>a</sup> • Vsl

<sup>b</sup> • Pll

<sup>c</sup> • μ<sup>0</sup>

<sup>d</sup> • Rl •

C ¼ K• VΣ<sup>k</sup>

decreases.

0.37 to 0.7

**Table 1.**

**145**

(�0.36) to (�1.32)

*The exponents of formula (3).*

(�0.15) to (�0.265)

0.04 to 0.52

q

0

B@

where Ra1 and Ra2 are the mean roughnesses of the surfaces.

Below are given the integro-differential equations of EHD theory of lubrication with the consideration of the thermal processes that take place in the lubricant film and on the boundaries of surfaces, and the corresponding boundary conditions:

$$\frac{dp}{dx} = \delta\mu (V\_1 + V\_2) \frac{h - h\_0}{h^3}, \text{when } x = -\infty, p = 0 \text{ and } x = \infty, p = \frac{dp}{dx} = 0;$$

$$h = h\_0 + \frac{\mathbf{x}^2 - \mathbf{x}\_0^2}{2R} + \frac{2}{\pi} \left(\frac{1 - \nu\_1^2}{E\_1} + \frac{1 - \nu\_2^2}{E\_2}\right) \int\_{-\infty}^{x\_0} p(\xi) \ln\left|\frac{\xi - x}{\xi - x\_0}\right| d\xi;$$

$$\rho c \mathbf{V} \frac{\partial t}{\partial \mathbf{x}} = \zeta \frac{\partial^2 t}{\partial \boldsymbol{\eta}^2} + \mu \left(\frac{\partial V}{\partial \boldsymbol{\eta}}\right)^2, \text{when } x = -\infty, t = t\_0;\tag{2}$$

$$t(\mathbf{x}, \mathbf{0}) = \left(\frac{1}{\pi \mu\_1 c\_1 \Lambda\_1 V\_1}\right)^{0.5} \int\_{-\infty}^{\infty} \zeta \frac{\partial t}{\partial \boldsymbol{\eta}} \bigg|\_{\boldsymbol{y} = \mathbf{0}} \frac{\partial e}{(\mathbf{x} - \boldsymbol{\varepsilon})^{0.5}} + t\_0;$$

$$t(\mathbf{x}, h) = \left(\frac{1}{\pi \mu\_2 c\_2 \Lambda\_2 V\_2}\right)^{0.5} \int\_{-\infty}^{\infty} -\zeta \frac{\partial t}{\partial \boldsymbol{\eta}} \bigg|\_{\boldsymbol{y} = \mathbf{k}} \frac{\partial e}{(\mathbf{x} - \boldsymbol{\varepsilon})^{0.5}} + t\_0;$$

$$\mu = \mu\_0 \exp\left(\beta \boldsymbol{p} - a \mathbf{d} t\right).$$

where *p* is pressure; *V*<sup>1</sup> and *V*<sup>2</sup> – peripheral speeds; *μ* – dynamic viscosity of lubricant oil in normal conditions; *h* – clearance; *h*<sup>0</sup> – minimum clearance; *R* – radius of curvature; *E*<sup>1</sup> and *E*<sup>2</sup> – modulus of elasticity; ν – Poisson's ratio of body materials; *t* – temperature; *ρ*, *c*, *ζ*, *ρ*1, *c*1, *ζ* 1, *ρ*2, *c*2, *ζ* <sup>2</sup> – correspondingly density, specific heat capacity and thermal conductivity of lubricant and interacting surfaces; *μ*<sup>0</sup> – dynamic viscosity of the lubricant; *β* – piezo coefficient of lubricant viscosity; *ζ* – lubricant thermal conductivity; *α* – thermal coefficient of lubricant viscosity; *ξ, ε* – complementary variables; *x*<sup>0</sup> – abscissa in the place of lubricant outlet from the gap.

Calculation of the oil film thickness, which separates the bodies, is the main problem of the EHD lubrication theory and there are numerous literature sources about it (Dowson, 1995; Ham rock and Dowson, 1981, etc.). There are various formulas for isothermal and anisothermal solutions for EHD problems describing the behavior of oil film thickness with various accuracies.

The modern friction modifiers contain tribochemically active products that have great influence on their operational properties. The various aspects of properties of these components are not sufficiently studied and they cannot be expressed mathematically. EHD theory of lubrication only considers the mechanical phenomena proceeding in the lubricant film of the contact zone, ignoring other layers.

The thickness of the rough surface boundary layers cannot be measured with the use of the modern methods of measurement of the oil layer thickness. Information about destruction of the boundary layers (and about onset of scuffing as well) can be obtained by sharp increase of the friction torque on the oscillogram. Therefore,

*A New Concept of the Mechanism of Variation of Tribological Properties of the Machine… DOI: http://dx.doi.org/10.5772/intechopen.93825*

onset of the friction torque sharp increase is considered as beginning of the third body destruction.

On the base of system of equations of EHD, theory of lubrication and results of experimental researches considering formula (1), criterion of the third body destruction was developed that has a form:

$$\mathbf{C} = K \left( \frac{R}{\sqrt{{R\_{d\_1}}^2 + {R\_{d\_2}}^2}} \right) \cdot \left( \frac{\mu V\_{\Xi K}}{P\_n} \right)^{0,7} \cdot \left( \frac{P\_n \theta}{R} \right)^{0,6} \cdot \left( \frac{\zeta}{a \mu V\_{\subset \mathbb{K}} 2^p P\_{\epsilon 1, \mathbb{Z}}} \right)^{\epsilon} \leq 1 \tag{3}$$

As it follows from the formula (2), a criterion of the third body destruction depends on the mechanical and thermo-physical characteristics of interacting surfaces, geometric and kinematic parameters, thermo-physical and tribological parameters of the third body. The properties and stability of the boundary layers are revealed in values of coefficient K and exponent e. The researches have also shown special sensitivity of the third body stability to thermal loads and relative sliding velocities, which must be taken into account to improve working conditions.

The criterion of the third body destruction that is developed on the base of EHD theory of lubrication and results of experimental researches considering stability of the boundary layers has the form:

$$\mathbf{C} = \mathbf{K} \bullet \mathbf{V}\_{\Sigma \mathbf{k}} \mathbf{^a} \bullet \mathbf{V}\_{\mathrm{sl}} \mathbf{^b} \bullet \mathbf{P}\_{\mathrm{ll}} \mathbf{^c} \bullet \boldsymbol{\mu}\_0 \mathbf{^d} \bullet \mathbf{R}^{\mathbf{l}} \bullet \left(\sqrt{R\_{a1}^2 + R\_{a2}^2}\right)^f \bullet \mathbf{j} \mathbf{^g} \bullet \boldsymbol{\zeta}^{\mathbf{h}} \bullet \mathbf{a^i} \bullet \mathbf{a^j} \bullet \mathbf{E}^n \le \mathbf{1} \tag{4}$$

where VΣ<sup>k</sup> is a total rolling velocity; Vsl – sliding velocity; Pll – linear load; μ dynamic viscosity of the lubricant; R – reduced radius of curvature of the surfaces; Ra1 and Ra2- average standard deviation of the interacting surfaces; β – piezo coefficient of the lubricant viscosity; ζ– the lubricant thermal conductivity; α – thermal coefficient of the lubricant viscosity; a – thermal diffusivity; The exponents a, b, c, … , n and coefficient K are specified on the base of the experimental data obtained by T.I. Fowle, Y.N. Drozdov, Vellawer, G. Niemann, A.I. Petrusevich, I.I. Sokolov, K. Shawerhammer, G. Tumanishvili and are given in the **Table 1**.

As it is seen from the **Table 1**, destruction of the third body is especially sensitive to the degree b of sliding velocity. It follows from formulae (2) and (3) that with increase of the rolling velocity, radius of curvature, piezo-coefficient of viscosity, heat conductivity factor, thermal diffusivity and coefficient of elasticity, the stability of the third body increases and with increase of the sliding velocity, linear loading, roughness of surfaces and thermal coefficient of viscosity it decreases.

As it was already mentioned, one of the indicators of the third body destruction (scuffing) is appearance of signs of scuffing on the surfaces. According to criteria of destruction of the third body, its destruction is supposed when values of the corresponding criteria are less than 1. K. Schauerhammer experimentally ascertains the conditions of the third body destruction (scuffing) for the gear drive on the gear drive test bench TUME 11 [46]. To predict the destruction of the third body


**Table 1.** *The exponents of formula (3).*

film minimum thickness at the Hertzian contact zone to the r.m.s. of the rolling

<sup>λ</sup> <sup>¼</sup> *hmin* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *R*2 *<sup>a</sup>*<sup>1</sup> <sup>þ</sup> *<sup>R</sup>*<sup>2</sup> *a*2

Below are given the integro-differential equations of EHD theory of lubrication with the consideration of the thermal processes that take place in the lubricant film and on the boundaries of surfaces, and the corresponding boundary conditions:

*<sup>h</sup>*<sup>3</sup> , when *<sup>x</sup>* ¼ �∞, *<sup>p</sup>* <sup>¼</sup> 0 and *<sup>x</sup>* <sup>¼</sup> *<sup>x</sup>*0, *<sup>p</sup>* <sup>¼</sup> *dp*

�∞

<sup>1</sup> � *<sup>ν</sup>*<sup>2</sup> 2 *E*2

> *ζ ∂t ∂y* � � � � � *y*¼0

� *<sup>ζ</sup> <sup>∂</sup><sup>t</sup> ∂y* � � � � � *y*¼*h*

� � *<sup>x</sup>*ð<sup>0</sup>

�∞

�∞

*μ* ¼ *μ*<sup>0</sup> exp ð Þ *βp* � *αΔt :*

where *p* is pressure; *V*<sup>1</sup> and *V*<sup>2</sup> – peripheral speeds; *μ* – dynamic viscosity of

*R* – radius of curvature; *E*<sup>1</sup> and *E*<sup>2</sup> – modulus of elasticity; ν – Poisson's ratio of body materials; *t* – temperature; *ρ*, *c*, *ζ*, *ρ*1, *c*1, *ζ* 1, *ρ*2, *c*2, *ζ* <sup>2</sup> – correspondingly density, specific heat capacity and thermal conductivity of lubricant and interacting surfaces; *μ*<sup>0</sup> – dynamic viscosity of the lubricant; *β* – piezo coefficient of lubricant viscosity; *ζ* – lubricant thermal conductivity; *α* – thermal coefficient of lubricant viscosity; *ξ, ε* – complementary variables; *x*<sup>0</sup> – abscissa in the place of lubricant

Calculation of the oil film thickness, which separates the bodies, is the main problem of the EHD lubrication theory and there are numerous literature sources about it (Dowson, 1995; Ham rock and Dowson, 1981, etc.). There are various formulas for isothermal and anisothermal solutions for EHD problems describing

The modern friction modifiers contain tribochemically active products that have great influence on their operational properties. The various aspects of properties of these components are not sufficiently studied and they cannot be expressed mathematically. EHD theory of lubrication only considers the mechanical phenomena proceeding in the lubricant film of the contact zone, ignoring other layers.

The thickness of the rough surface boundary layers cannot be measured with the use of the modern methods of measurement of the oil layer thickness. Information about destruction of the boundary layers (and about onset of scuffing as well) can be obtained by sharp increase of the friction torque on the oscillogram. Therefore,

lubricant oil in normal conditions; *h* – clearance; *h*<sup>0</sup> – minimum clearance;

where Ra1 and Ra2 are the mean roughnesses of the surfaces.

*Tribology in Materials and Manufacturing - Wear, Friction and Lubrication*

2 *π*

*t <sup>∂</sup>y*<sup>2</sup> <sup>þ</sup> *<sup>μ</sup>*

πρ1*c*1*λ*1*V*<sup>1</sup> � �0,5ð*<sup>x</sup>*

πρ2*c*2*λ*2*V*<sup>2</sup> � �0,5ð*<sup>x</sup>*

the behavior of oil film thickness with various accuracies.

<sup>1</sup> � *<sup>ν</sup>*<sup>2</sup> 1 *E*1 þ

> *∂V ∂y* � �<sup>2</sup>

*h* � *h*<sup>0</sup>

*<sup>x</sup>*<sup>2</sup> � *<sup>x</sup>*<sup>2</sup> 0 2*R* þ

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>ζ</sup> <sup>∂</sup>*<sup>2</sup>

*t x*ð Þ¼ , 0 <sup>1</sup>

*t x*ð Þ¼ , *<sup>h</sup>* <sup>1</sup>

*<sup>ρ</sup>*cV *<sup>∂</sup><sup>t</sup>*

<sup>q</sup> (1)

*<sup>p</sup>*ð Þ*<sup>ξ</sup>* ln *<sup>ξ</sup>* � *<sup>x</sup> ξ* � *x*<sup>0</sup>

� � � �

, when *x* ¼ �∞, *t* ¼ *t*0; (2)

*∂ε* ð Þ *<sup>x</sup>* � *<sup>ε</sup>* 0,5 <sup>þ</sup> *<sup>t</sup>*0;

*∂ε* ð Þ *<sup>x</sup>* � *<sup>ε</sup>* 0,5 <sup>þ</sup> *<sup>t</sup>*0; *dx* <sup>¼</sup> 0;

� � � � *dξ*;

element surface finish:

*dp*

outlet from the gap.

**144**

*dx* <sup>¼</sup> <sup>6</sup>*μ*ð Þ *<sup>V</sup>*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*<sup>2</sup>

*h* ¼ *h*<sup>0</sup> þ

discontinuous but restorable third body at the initial stage of destruction and progressively destructing third body have quite different properties. In the first case the said properties are stable and depend on the properties of the third body and in the second case, these properties are instable and worsened that are characterized by increasing friction coefficient, catastrophic wear and

conditions by estimation of the friction torque variation and with the use of the criterion of destruction of the third body, with ascertained beforehand values

• Prediction of destruction of the third body is possible in the laboratory

*A New Concept of the Mechanism of Variation of Tribological Properties of the Machine…*

• The friction coefficient (negative, neutral and positive), wear rate of the interacting surfaces (mild, severe and catastrophic), damage types (scuffing,

fatigue, plastic deformation, adhesive wear) and vibrations and noise generated in the contact zone depend on tribological properties of the third body, its degree of destruction and area of the factual contact zone seized

having due tribological properties at the initial stage of destruction.

Georgia (SRNSFG) under GENIE project CARYS-19-588.

George Tumanishvili\*, Tengiz Nadiradze and Giorgi Tumanishvili

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: ge.tumanishvili@gmail.com

The authors declare no conflict of interest.

Institute of Machine Mechanics, Tbilisi, Georgia

provided the original work is properly cited.

• For the improvement of tribological properties of the interacting surfaces, it is necessary to provide the contact zone with continuous or restorable third body

This work was supported by Shota Rustaveli National Science Foundation of

typical noise.

places;

**Acknowledgements**

**Conflict of interest**

**Author details**

**147**

of the experimental coefficients;

*DOI: http://dx.doi.org/10.5772/intechopen.93825*

#### **Figure 15.**

*Dependences of the fields of deviations of the values of λ parameter (1) and criterion C of destruction of the third body (2) developed by us, on the gear wheels circular velocity.*

#### **Figure 16.**

*Dependences of the fields of deviations of the temperature criterion (θ, 1) of H. block, criterion (SF, 2) of G. Niman and Saitzinger and offered criterion (C, 3) of destruction of the third body, on the gear wheels circular velocity.*

(scuffing), we used the well-known Dowson and Higginson formulas to determine the lubricating layer parameter (λ) [44, 45] and the criterion C developed by us at the values of the coefficient K = 2.7 and the exponent e = 0.336 in formula (3). Dependences of the fields of deviations of the values of these criteria on the gear wheels circular velocity are shown in **Figure 15**.

As it is seen from the graphs, deviations of the criterion C of destruction of the third body developed by us, are small and constant, while deviations of the parameter λ and its values increase with increase of the gear wheels velocity.

**Figure 16** shows the results of similar calculations using the C criterion with the values of the coefficient K = 1.55 and the exponent e = 0.29 in formula (3) and the formulas of H. Block [47] and Niemann G. and Saitzinger K. [48]. The studies were carried out on gear drive test bench FZG for transmissions A, L, N 141, 142, 143, 201, 202, and 203 with lubricant k1.

It is seen from the graphs that deviations of the offered criterion C of destruction of the third body little differ from the unit in the whole range of variation of the circular velocity, while deviations of other criteria significantly differ from the unit and they increase with increase of the circular velocity.

### **5. Conclusions**

• Tribological properties of the interacting surfaces mainly depend on tribological properties of the third body, degree of its destruction, disposition of the surfaces to seizure etc. The researches have shown that the continuous or *A New Concept of the Mechanism of Variation of Tribological Properties of the Machine… DOI: http://dx.doi.org/10.5772/intechopen.93825*

discontinuous but restorable third body at the initial stage of destruction and progressively destructing third body have quite different properties. In the first case the said properties are stable and depend on the properties of the third body and in the second case, these properties are instable and worsened that are characterized by increasing friction coefficient, catastrophic wear and typical noise.

