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Hydrodynamic lubrication in line contacts improved by the boundary slippage

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Abstract

The present paper proposes three types of the boundary slippage augmentations in hydrodynamic lubricated line contacts for improving the load-carrying capacity but reducing the friction coefficient. One augmentation is at the slower moving contact surface, the second augmentation is at the faster moving contact surface, and the third is at both of the contact surfaces. They are respectively suitable for different operating conditions. The analysis was respectively carried out for the load-carrying capacity and the friction coefficient of hydrodynamic lubricated line contacts augmented with these three types of boundary slippage. The obtained results were compared with those for conventional hydrodynamic lubricated line contacts (without artificial introduction of the boundary slippage) for the same operating conditions. It was shown that in certain operating conditions these three types of the boundary slippage augmentations can respectively significantly increase the load-carrying capacity but reduce the friction coefficient of the contact. The potential application values of the proposed boundary slippage augmentations are evident for reducing the energy loss and the temperature rise as well as for improving the anti-scuffing ability of the contact.

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Abbreviations

b :

half Hertzian contact width

E′:

equivalent Young’s elastic modulus of two contact surfaces [1]

f a ,f b :

friction coefficients at the faster and slower moving contact surfaces in the present contact respectively

f a,n ,f b,n :

friction coefficients at the faster and slower moving contact surfaces in the conventional hydrodynamic lubricated line contact respectively

F f,a ,F f,b :

friction forces per unit contact length at the faster and slower moving contact surfaces in the present contact respectively

F f,a,n ,F f,b,n :

friction forces per unit contact length at the faster and slower moving contact surfaces in the conventional hydrodynamic lubricated line contact respectively

G :

= αE′

h :

film thickness

h c :

film thickness at the location where dp/dx=0

H :

= h/R

H c :

= h c /R

H c,N :

central film thickness in a hydrodynamic lubricated line contact when the fluid is Newtonian (with no interfacial slippage at both the contact surfaces)

J 1, J 4, J 5 :

integrations

m w ,r it :

dimensionless variables dependent on G, U and W [2]

p :

film pressure

p s :

interfacial solidification pressure in the present designed contacts

p s,conv :

interfacial solidification pressure in a conventional contact

p max :

maximum pressure in the contact

P :

= αp

R :

radius of the cylinder or the equivalent radius of two contact surfaces

S :

slide-roll ratio, 2(u a −u b )/(u a +u b )

u a :

circumferential speed of the faster moving contact surface

u b :

circumferential speed of the slower moving contact surface

\(\bar{u}_{a}\) :

film velocity at the faster moving contact surface

\(\bar{u}_{b}\) :

film velocity at the slower moving contact surface

u :

rolling speed, (u a +u b )/2

U :

= uη a /(E′R)

U a :

= u a η a /(E′R)

U b :

= u b η a /(E′R)

w :

load per unit contact length of the contact

W :

= w/(E′R)

x :

coordinate

X :

= x/b

λ :

\({=}\, G\sqrt{8W/\pi}\)

η :

fluid viscosity

η a :

fluid viscosity at ambient pressure

α :

fluid viscosity-pressure index

α τ :

interfacial shear strength-pressure proportionality at high pressures in a conventional hydrodynamic lubricated line contact

β τ :

interfacial shear strength-pressure proportionality at high pressures in the present designed contacts

ρ :

fluid density

ρ c :

fluid density at the location where dp/dx=0

τ a :

shear stress at the faster moving contact surface

τ b :

shear stress at the slower moving contact surface

τ sa :

fluid-contact interfacial shear strength at the faster moving contact surface

τ sb :

fluid-contact interfacial shear strength at the slower moving contact surface

τ sb,1 :

fluid-contact interfacial shear strength at the stationary contact surface in the inlet zone in Fig. 1(c)

Fig. 1
figure 1

The studied contacts. A—The inlet zone; B—The Hertzian contact zone

Full size image
τ l0, \(\tau_{sa,0}'\), τ sa,0, τ s0 :

constants

Δu a :

interfacial slipping velocity at the faster moving contact surface

Δu b :

interfacial slipping velocity at the slower moving contact surface

\(\bar{\tau}_{c,1}\) :

a dimensionless critical shear stress, Eq. (40)

\(\bar{\tau}_{c,2}\) :

a dimensionless critical shear stress, Eq. (38)

\(\bar{\tau}_{l0}\) :

= τ l0/E′

\(\bar{\tau}_{sa}\) :

= τ sa /E′

\(\bar{\tau}_{sb}\) :

= τ sb /E′

\(\bar{\tau}_{s0}\) :

= τ s0/E′

\(\bar{\tau}_{sa,0}'\) :

\({ =}\, \tau_{sa,0}'/E'\)

\(\bar{\tau}_{sa,0}\) :

= τ sa,0/E′

\(\bar{\tau}_{sb,1}\) :

= τ sb,1/E′

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Acknowledgements

The author would like to express thanks to the Dean Project of the Natural Science Foundation of China (51145016), the project from Changzhou Science and Technology Bureau (CJ20120033) and the Qing Lan Project of Jiangsu Provincial Education Bureau.

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  1. College of Mechanical Engineering, Changzhou University, Changzhou, Jiangsu Province, China

    Yongbin Zhang

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Zhang, Y. Hydrodynamic lubrication in line contacts improved by the boundary slippage. Meccanica 49, 503–519 (2014). https://doi.org/10.1007/s11012-013-9808-6

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