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Balancing Wedge Action: A Contribution of Textured Surface to Hydrodynamic Pressure Generation

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Abstract

This paper suggests a new mechanism called ‘balancing wedge action’, which is based on the hydrodynamic lubrication theory for textured surfaces. While past studies have considered the local wedge film action produced by textured feature, this new mechanism is based on the promotion of a wedge film action between surfaces by the incorporation of a textured feature. The analytical model used in the current study is a one-dimensional centrally pivoted pad bearing having a single dimple on the pad, which considers the equilibrium of the moment applied to the surfaces. Analytical equations are derived for the pressure, shear stress, load, friction, and moment by integrating the Reynolds equation. A series of parametric simulations of the depth, width, and location of a dimple were conducted. The analytical results showed that the incorporation of a single dimple on the pad surface increases the convergence ratio between the surfaces, producing a load capacity and reducing the friction. No negative pressure can be found within the dimple, where a positive pressure with a greater positive gradient causes a higher shear stress than that outside the dimple. The trends for the load and friction in relation to the dimple depth and location are complex. The creation of the dimple closer to the centre results in a failure to obtain an equilibrium solution for the moment.

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Abbreviations

F :

Dimensionless friction, F = fh 0/(ηlu)

H :

Dimensionless film thickness, H = h/h 0

H 1 :

Dimensionless film thickness at inlet, H 1 = h 1/h 0

H 2 :

Dimensionless film thickness at left side of dimple, H 2 = h 2/h 0

H 2d :

Dimensionless film thickness at left side of dimple, H 2d = h 2d/h 0

H 3 :

Dimensionless film thickness at right side of dimple, H 3 = h 3/h 0

H 3d :

Dimensionless film thickness at right side of dimple, H 3d = h 3d/h 0

H d :

Dimensionless dimple depth, H d = h d/h 0

K :

Convergence ratio, K = (h 1 − h 0)/h 0

L 2 :

Dimensionless position at left side of dimple, L 2 = l 2/l

L 3 :

Dimensionless position at right side of dimple, L 3 = l 3/l

L pv :

Dimensionless position of pivot, L pv = l pv/l

M :

Dimensionless moment, M = h 20 m/(6ηl 3 u)

P :

Dimensionless pressure, P = h 20 p/(6ηlu)

P min :

Dimensionless minimum pressure

Q :

Dimensionless mass flow rate, Q = q/(h 0 u)

Q c :

Dimensionless Couette flow rate, Q c = q c/(h 0 u)

Q p :

Dimensionless Poiseuille flow rate, Q p = q p/(h 0 u)

S :

Dimensionless shear stress, S = −h 0 s/(ηu)

S c :

Dimensionless shear stress by Couette flow, S c = 1/H

S p :

Dimensionless shear stress by Poiseuille flow, S p = H/2(dP/dX)

X :

Dimensionless coordinate in direction of surface motion, X = x/l

W :

Dimensionless load, W = h 20 w/(6ηl 2 u)

f :

Friction (N/m)

h :

Film thickness (m)

h 1 :

Film thickness at inlet (m)

h 2 :

Film thickness at left side of dimple (m)

h 2d :

Film thickness at left side of dimple (m), h 2d = h 2 + h d

h 3 :

Film thickness at right side of dimple (m)

h 3d :

Film thickness at right side of dimple (m), h 3d = h 3 + h d

h 0 :

Minimum film thickness (m)

h d :

Dimple depth (m)

l :

Width of pad (m)

l 2 :

Position at left side of dimple (m)

l 3 :

Position at right side of dimple (m)

l pv :

Position of pivot (m)

n max :

Maximum number of series terms

p :

Pressure of fluid film (Pa)

q :

Mass flow rate, q = q c + q p (m3/(ms))

q c :

Couette mass flow rate (m3/(ms))

q p :

Poiseuille mass flow rate (m3/(ms))

s :

Shear stress (N/m)

u :

Sliding speed of moving surface (m)

x :

Coordinate in direction of surface motion (m)

w :

Load (N/m)

η :

Viscosity (Pas)

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Kyushu University, Fukuoka, Japan

    Kazuyuki Yagi & Joichi Sugimura

Authors
  1. Kazuyuki Yagi
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  2. Joichi Sugimura
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Corresponding author

Correspondence to Kazuyuki Yagi.

Appendix: Expansions of Load, Friction, and Moment for Accurate Calculation in Region of Low Convergence Ratios

Appendix: Expansions of Load, Friction, and Moment for Accurate Calculation in Region of Low Convergence Ratios

1.1 Load

Dimensionless load W is given by integrating the pressure distribution over the contact area.

$$ W = \int\limits_{0}^{{L_{2} }} {P{\text{d}}X} + \int\limits_{{L_{2} }}^{{L_{3} }} {P{\text{d}}X} + \int\limits_{{L_{3} }}^{1} {P{\text{d}}X} $$
(54)

Integrating the pressure distribution in the left land zone, dimple zone, and right land zone gives

$$ \begin{aligned} W & = \frac{1}{{K^{2} }}\left( { - \ln \left( {\frac{{H_{2} }}{{H_{1} }}} \right) - K\left( {\frac{{L_{2} }}{{H_{1}^{{}} }}} \right)} \right) - \frac{Q}{{K^{2} }}\left( { - K\left( {\frac{{L_{2} }}{{H_{1}^{2} }}} \right) + \left( {\frac{1}{{H_{2} }} - \frac{1}{{H_{1} }}} \right)} \right) \\ & \quad + \frac{1}{{K^{2} }}\left( { - \ln \left( {\frac{{H_{{3{\text{d}}}} }}{{H_{{2{\text{d}}}} }}} \right) - K\left( {\frac{{L_{3} - L_{2} }}{{H_{{2{\text{d}}}} }}} \right)} \right) - \frac{Q}{{K^{2} }}\left( { - K\left( {\frac{{L_{3} - L_{2} }}{{H_{{2{\text{d}}}}^{2} }}} \right) + \left( {\frac{1}{{H_{{3{\text{d}}}} }} - \frac{1}{{H_{{2{\text{d}}}} }}} \right)} \right) + P_{2} \left( {L_{3} - L_{2} } \right) \\ & \quad + \frac{1}{{K^{2} }}\left( { - \ln \left( {\frac{1}{{H{}_{3}}}} \right) - K\left( {1 - L_{3} } \right)} \right) - \frac{Q}{{K^{2} }}\left( { - K\left( {1 - L_{3} } \right) + \left( {1 - \frac{1}{{H_{3} }}} \right)} \right) \\ \end{aligned} $$
(55)

where P 2 is the dimensionless pressure at the left step point and Q is the dimensionless flow rate given by

$$ P_{2} = \frac{{L_{2} }}{{H_{1} H_{2} }}\left( {1 - \left( {\frac{1}{{H_{1} }} + \frac{1}{{H_{2} }}} \right)Q} \right) $$
(56)
$$ Q = \frac{{\left( {\frac{{L_{2} }}{{H_{1} H_{2} }} + \frac{{(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }} + \frac{{(1 - L_{3} )}}{{H_{3} }}} \right)}}{{\left( {\frac{{L_{2} (H_{1} + H_{2} )}}{{H_{1}^{2} H_{2}^{2} }} + \frac{{(L_{3} - L_{2} )(H_{{2{\text{d}}}} + H_{{3{\text{d}}}} )}}{{H_{{2{\text{d}}}}^{2} H_{{3{\text{d}}}}^{2} }} + \frac{{(1 - L_{3} )(1 + H_{3} )}}{{H_{3}^{2} }}} \right)}} $$
(57)

After further expansion, Eq. (55) can be modified to

$$ \begin{aligned} W & = \frac{1}{{K^{2} }}\left( { - \ln \left( {1 - \frac{{KL_{2} }}{{H_{1} }}} \right) - K\left( {\frac{{L_{2} }}{{H_{1} }}} \right)} \right) - \frac{Q}{{K^{2} }}\left( { - K\left( {\frac{{L_{2} }}{{H_{1}^{2} }}} \right) + \left( {\frac{{KL_{2} }}{{H_{1} H_{2} }}} \right)} \right) \\ & \quad + \frac{1}{{K^{2} }}\left( { - \ln \left( {1 - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }}} \right) - K\left( {\frac{{L_{3} - L_{2} }}{{H_{{2{\text{d}}}} }}} \right)} \right) - \frac{Q}{{K^{2} }}\left( { - K\left( {\frac{{L_{3} - L_{2} }}{{H_{{2{\text{d}}}}^{2} }}} \right) + \left( {\frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }}} \right)} \right) + P_{2} \left( {L_{3} - L_{2} } \right) \\ & \quad + \frac{1}{{K^{2} }}\left( { - \ln \left( {1 - \frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) - K\left( {1 - L_{3} } \right)} \right) - \frac{Q}{{K^{2} }}\left( { - K\left( {1 - L_{3} } \right) + \left( {\frac{{K(1 - L_{3} )}}{{H_{3} }}} \right)} \right) \\ \end{aligned} $$
(58)

Substituting the Maclaurin expansion of the log terms and the film thickness equations into Eq. (58), Eq. (58) is further modified to

$$ \begin{aligned} W & = \frac{1}{{K^{2} }}\left( {\frac{{KL_{2} }}{{H_{1} }} + \frac{{K^{2} L_{2}^{2} }}{{2H_{1}^{2} }} + \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} L_{2}^{n} }}{{nH_{1}^{n} }}} - K\left( {\frac{{L_{2} }}{{H_{1} }}} \right)} \right) - \frac{Q}{{K^{2} }}\left( {\frac{{KL_{2} (H_{1} - H_{2} )}}{{H_{{^{{_{1} }} }}^{2} H_{2} }}} \right) \\ & \quad + \frac{1}{{K^{2} }}\left( {\frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} + \frac{{K^{2} (L_{3} - L_{2} )^{2} }}{{2H_{{_{{2{\text{d}}}} }}^{2} }} + \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} (L_{3} - L_{2} )^{n} }}{{nH_{{_{{2{\text{d}}}} }}^{2} }}} - K\left( {\frac{{L_{3} - L_{2} }}{{H_{{2{\text{d}}}} }}} \right)} \right) - \frac{Q}{{K^{2} }}\left( {\frac{{K(L_{3} - L_{2} )(H_{{2{\text{d}}}} - H_{{3{\text{d}}}} )}}{{H_{{_{{2{\text{d}}}} }}^{2} H_{{3{\text{d}}}} }}} \right) + P_{2} \left( {L_{3} - L_{2} } \right) \\ & \quad + \frac{1}{{K^{2} }}\left( {\frac{{K(1 - L_{3} )}}{{H_{3} }} + \frac{{K^{2} (1 - L_{3} )^{2} }}{{2H_{{_{3} }}^{2} }} + \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} (1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} - K\left( {1 - L_{3} } \right)} \right) - \frac{Q}{{K^{2} }}\left( {\frac{{K(1 - L_{3} )(1 - H_{3} )}}{{H_{3} }}} \right) \\ & \quad - 1 < \frac{{KL_{2} }}{{H_{1} }} < 1, - 1 < \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} < 1, - 1 < \frac{{K(1 - L_{3} )}}{{H_{3} }} < 1 \\ \end{aligned} $$
(59)

The elimination of K in the denominators gives the following arranged expression

$$ \begin{aligned} W & = \frac{{L_{2}^{2} }}{{2H_{1}^{2} }} - Q\frac{{L_{2}^{2} }}{{H_{1}^{2} H_{2} }} + \sum\limits_{n = 3}^{\infty } {\frac{{L_{2}^{n} }}{{nH_{1}^{n} }}} K^{n - 2} \\ & \quad + \frac{{(L_{3} - L_{2} )^{2} }}{{2H_{{2{\text{d}}}}^{2} }} - Q\frac{{(L_{3} - L_{2} )^{2} }}{{H_{{2{\text{d}}}}^{2} H_{{3{\text{d}}}} }} + P_{2} (L_{3} - L_{2} ) + \sum\limits_{n = 3}^{\infty } {\frac{{(L_{3} - L_{2} )^{n} }}{{nH_{{2{\text{d}}}}^{n} }}} K^{n - 2} \\ & \quad + \frac{{(1 - L_{3} )^{2} }}{{2H_{3}^{2} }} + Q\frac{{(1 - L_{3} )^{2} }}{{H_{3} }} - \frac{{(1 - L_{3} )^{2} }}{{H_{3} }} + \sum\limits_{n = 3}^{\infty } {\frac{{(1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} K^{n - 2} \\ & \quad - 1 < \frac{{KL_{2} }}{{H_{1} }} < 1, - 1 < \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} < 1, - 1 < \frac{{K(1 - L_{3} )}}{{H_{3} }} < 1 \\ \end{aligned} $$
(60)

In the parallel case (K = 0), the dimensionless flow rate Q becomes

$$ Q = \frac{{\left( {L_{2} + \frac{{(L_{3} - L_{2} )}}{{(1 + H_{\text{d}} )^{2} }} + (1 - L_{3} )} \right)}}{{2\left( {L_{2} + \frac{{(L_{3} - L_{2} )}}{{(1 + H_{\text{d}} )^{3} }} + (1 - L_{3} )} \right)}} $$
(61)
$$ P_{2} = L_{2} \left( {1 - 2Q} \right) $$
(62)

Equation (59) becomes

$$ W = \frac{{L_{2}^{2} }}{2} - QL_{2}^{2} + \frac{{(L_{3} - L_{2} )^{2} }}{{2(1 + H_{\text{d}} )^{2} }} - Q\frac{{(L_{3} - L_{2} )^{2} }}{{(1 + H_{\text{d}} )^{3} }} + P_{2} (L_{3} - L_{2} ) - \frac{{(1 - L_{3} )^{2} }}{2} + Q(1 - L_{3} )^{2} $$
(63)

1.2 Friction

Dimensionless friction F is given by integrating the shear stress over the contact area

$$ F = \int\limits_{0}^{{L_{2} }} {S{\text{d}}X} + \int\limits_{{L_{2} }}^{{L_{3} }} {S{\text{d}}X} + \int\limits_{{L_{3} }}^{1} {S{\text{d}}X} $$
(64)

The integrated expression is as follows:

$$ \begin{aligned} F & = 4\frac{1}{K}\ln \left( {1 - \frac{{KL_{2} }}{{H_{1} }}} \right) + 6\frac{Q}{K}\left( {\frac{{KL_{2} }}{{H_{1} H_{2} }}} \right) \\ & \quad + 4\frac{1}{K}\ln \left( {1 - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }}} \right) + 6\frac{Q}{K}\left( {\frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }}} \right) \\ & \quad + 4\frac{1}{K}\ln \left( {1 - \frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) + 6\frac{Q}{K}\left( {\frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) \\ \end{aligned} $$
(65)

After further expansion in the same manner as the dimensionless load W, Eq. (65) can be modified to

$$ \begin{aligned} F & = 4\frac{1}{K}\left( { - \frac{{KL_{2} }}{{H_{1} }} - \sum\limits_{n = 2}^{\infty } {\frac{{K^{n} L_{2}^{n} }}{{nH_{1}^{n} }}} } \right) + 6\frac{Q}{K}\left( {\frac{{KL_{2} }}{{H_{1} H_{2} }}} \right) \\ & \quad + 4\frac{1}{K}\left( { - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} - \sum\limits_{n = 2}^{\infty } {\frac{{K^{n} (L_{3} - L_{2} )^{n} }}{{nH_{{_{{2{\text{d}}}} }}^{2} }}} } \right) + 6\frac{Q}{K}\left( {\frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }}} \right) \\ & \quad + 4\frac{1}{K}\left( { - \frac{{K(1 - L_{3} )}}{{H_{3} }} - \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} (1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} } \right) + 6\frac{Q}{K}\left( {\frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) \\ & \quad - 1 < \frac{{KL_{2} }}{{H_{1} }} < 1, - 1 < \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} < 1, - 1 < \frac{{K(1 - L_{3} )}}{{H_{3} }} < 1 \\ \end{aligned} $$
(66)

An arranged form is given by

$$ \begin{aligned} F & = - 4\left( {\frac{{L_{2} }}{{H_{1} }} + \frac{{(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} + \frac{{(1 - L_{3} )}}{{H_{3} }}} \right) + 6Q\left( {\frac{{L_{2} }}{{H_{1} H_{2} }} + \frac{{(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }} + \frac{{(1 - L_{3} )}}{{H_{3} }}} \right) \\ & \quad - 4\sum\limits_{n = 2}^{\infty } {\left( {\left( {\frac{{L_{2} }}{{H_{1} }}} \right)^{n} + \left( {\frac{{(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }}} \right)^{n} + \left( {\frac{{(1 - L_{3} )}}{{H_{3} }}} \right)^{n} } \right)} \frac{{K^{n - 1} }}{n} \\ & \quad - 1 < \frac{{KL_{2} }}{{H_{1} }} < 1, - 1 < \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} < 1, - 1 < \frac{{K(1 - L_{3} )}}{{H_{3} }} < 1 \\ \end{aligned} $$
(67)

In the parallel case (K = 0), Eq. (67) becomes

$$ F = - 4\left( {L_{2} + \frac{{(L_{3} - L_{2} )}}{{(1 + H_{\text{d}} )}} + (1 - L_{3} )} \right) + 6Q\left( {L_{2} + \frac{{(L_{3} - L_{2} )}}{{(1 + H_{\text{d}} )^{2} }} + (1 - L_{3} )} \right) $$
(68)

1.3 Moment

Dimensionless moment M is given by

$$ M = \int\limits_{0}^{{L_{2} }} {P(X - L_{\text{pv}} ){\text{d}}X} + \int\limits_{{L_{2} }}^{{L_{3} }} {P(X - L_{\text{pv}} ){\text{d}}X} + \int\limits_{{L_{3} }}^{1} {P(X - L_{\text{pv}} ){\text{d}}X} $$
(69)

The integrated expression is as follows.

$$ \begin{aligned} M & = - \frac{1}{{K^{2} }}\left( {\ln \left( {1 - \frac{{KL_{2} }}{{H_{1} }}} \right) + \frac{1}{K}\ln \left( {1 - \frac{{KL_{2} }}{{H_{1} }}} \right) + L_{2} + \frac{{KL_{2}^{2} }}{{2H_{1} }}} \right) \\ & \quad + \frac{Q}{{K^{2} }}\left( { - \left( {\frac{1}{{H_{2} }} - \frac{1}{{H_{1} }}} \right) - \frac{1}{K}\ln \left( {1 - \frac{{KL_{2} }}{{H_{1} }}} \right) - \frac{1}{K}\left( {\frac{1}{{H_{2} }} - \frac{1}{{H_{1} }}} \right) + \frac{{KL_{2}^{2} }}{{2H_{1}^{2} }}} \right) \\ & \quad - \frac{1}{{K^{2} }}\left( {\left( {1 + \frac{{(1 + H_{\text{d}} )}}{K}} \right)\ln \left( {1 - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }}} \right) + (L_{3} - L_{2} ) + \frac{{K(L_{3}^{2} - L_{2}^{2} )}}{{2H_{{2{\text{d}}}} }}} \right) \\ & \quad + \frac{Q}{{K^{2} }}\left( { - \left( {\frac{1}{{H_{{3{\text{d}}}} }} - \frac{1}{{H_{{2{\text{d}}}} }}} \right) - \frac{1}{K}\ln \left( {1 - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }}} \right) - \frac{{(1 + H_{\text{d}} )}}{K}\left( {\frac{1}{{H_{{3{\text{d}}}} }} - \frac{1}{{H_{{2{\text{d}}}} }}} \right) + \frac{{K(L_{3}^{2} - L_{2}^{2} )}}{{2H_{{2{\text{d}}}}^{2} }}} \right) + \frac{1}{2}P_{2} \left( {L_{3}^{2} - L_{2}^{2} } \right) \\ & \quad - \frac{1}{{K^{2} }}\left( {\left( {1 + \frac{1}{K}} \right)\ln \left( {1 - \frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) + (1 - L_{3} ) + \frac{{K(1 - L_{{_{3} }}^{2} )}}{2}} \right) \\ & \quad + \frac{Q}{{K^{2} }}\left( { - \left( {1 - \frac{1}{{H_{3} }}} \right) - \frac{1}{K}\ln \left( {1 - \frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) - \frac{1}{K}\left( {1 - \frac{1}{{H_{3} }}} \right) + \frac{{K(1 - L_{3}^{2} )}}{2}} \right) \\ & \quad - WL_{\text{pv}} = 0 \\ \end{aligned} $$
(70)

Substituting the Maclaurin expansion of the log terms into Eq. (70), M can be calculated as follows:

$$ \begin{aligned} M & = - \frac{1}{{K^{2} }}\left( { - \frac{{KL_{2} }}{{H_{1} }} - \frac{{K^{2} L_{2}^{2} }}{{2H_{1}^{2} }} - \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} L_{2}^{n} }}{{nH_{1}^{n} }}} - \frac{{L_{2} }}{{H_{1} }} - \frac{{KL_{2}^{2} }}{{2H_{{_{1} }}^{2} }} - \frac{{K^{2} L_{2}^{3} }}{{3H_{{_{1} }}^{3} }} - \sum\limits_{n = 4}^{\infty } {\frac{{K^{n - 1} L_{2}^{n} }}{{nH_{1}^{n} }}} + L_{2} + \frac{{KL_{2}^{2} }}{{2H_{1} }}} \right) \\ & \quad + \frac{Q}{{K^{2} }}\left( { - \left( {\frac{{KL_{2} }}{{H_{1} H_{2} }}} \right) + \frac{{L_{2} }}{{H_{1} }} + \frac{{KL_{2}^{2} }}{{2H_{{_{1} }}^{2} }} + \frac{{K^{2} L_{2}^{3} }}{{3H_{{_{1} }}^{3} }} + \sum\limits_{n = 4}^{\infty } {\frac{{K^{n - 1} L_{2}^{n} }}{{nH_{1}^{n} }}} - \frac{{L_{2} }}{{H_{1} H_{2} }} + \frac{{KL_{2}^{2} }}{{2H_{1}^{2} }}} \right) \\ & \quad - \frac{1}{{K^{2} }}\left( { - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} - \frac{{K^{2} (L_{3} - L_{2} )^{2} }}{{2H_{{_{{2{\text{d}}}} }}^{2} }} - \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} (L_{3} - L_{2} )^{n} }}{{nH_{{_{{2{\text{d}}}} }}^{2} }}} } \right) \\ & \quad - \frac{1}{{K^{2} }}\left( {\left( {\frac{{1 + H_{\text{d}} }}{K}} \right)\left( { - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} - \frac{{K^{2} (L_{3} - L_{2} )^{2} }}{{2H_{{_{{2{\text{d}}}} }}^{2} }} - \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} (L_{3} - L_{2} )^{n} }}{{nH_{{_{{2{\text{d}}}} }}^{2} }}} } \right) + (L_{3} - L_{2} ) + \frac{{K(L_{3}^{2} - L_{2}^{2} )}}{{2H_{{2{\text{d}}}} }}} \right) \\ & \quad + \frac{Q}{{K^{2} }}\left( { - \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }} + \frac{{(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} + \frac{{K(L_{3} - L_{2} )^{2} }}{{2H_{{_{{2{\text{d}}}} }}^{2} }} + \frac{{K^{2} (L_{3} - L_{2} )^{3} }}{{3H_{{_{{2{\text{d}}}} }}^{3} }} + \sum\limits_{n = 4}^{\infty } {\frac{{K^{n - 1} (L_{3} - L_{2} )^{n} }}{{nH_{{_{{2{\text{d}}}} }}^{2} }}} - \frac{{(1 + H_{\text{d}} )(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} H_{{3{\text{d}}}} }} + \frac{{K(L_{3}^{2} - L_{2}^{2} )}}{{2H_{{2{\text{d}}}}^{2} }}} \right) + \frac{1}{2}P_{2} \left( {L_{3}^{2} - L_{2}^{2} } \right) \\ & \quad - \frac{1}{{K^{2} }}\left( { - \frac{{K(1 - L_{3} )}}{{H_{3} }} - \frac{{K^{2} (1 - L_{3} )^{2} }}{{2H_{{_{3} }}^{2} }} - \sum\limits_{n = 3}^{\infty } {\frac{{K^{n} (1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} - \frac{{(1 - L_{3} )}}{{H_{3} }} - \frac{{K(1 - L_{3} )^{2} }}{{2H_{{_{3} }}^{2} }} - \frac{{K^{2} (1 - L_{3} )^{3} }}{{3H_{{_{3} }}^{3} }} - \sum\limits_{n = 4}^{\infty } {\frac{{K^{n - 1} (1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} + (1 - L_{3} ) + \frac{{K(1 - L_{{_{3} }}^{2} )}}{2}} \right) \\ & \quad + \frac{Q}{{K^{2} }}\left( { - \left( {\frac{{K(1 - L_{3} )}}{{H_{3} }}} \right) + \frac{{(1 - L_{3} )}}{{H_{3} }} + \frac{{K(1 - L_{3} )^{2} }}{{2H_{{_{3} }}^{2} }} + \frac{{K^{2} (1 - L_{3} )^{3} }}{{3H_{{_{3} }}^{3} }} + \sum\limits_{n = 4}^{\infty } {\frac{{K^{n - 1} (1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} - \frac{{(1 - L_{3} )}}{{H_{3} }} + \frac{{K(1 - L_{3}^{2} )}}{2}} \right) \\ & \quad - WL_{\text{pv}} = 0 \\ & \quad - 1 < \frac{{KL_{2} }}{{H_{1} }} < 1, - 1 < \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} < 1, - 1 < \frac{{K(1 - L_{3} )}}{{H_{3} }} < 1 \\ \end{aligned} $$
(71)

An arranged form is expressed as follows.

$$ \begin{aligned} M & = \frac{{L_{2}^{3} }}{{3H_{1}^{3} }} - Q\frac{{L_{2}^{3} }}{{H_{1}^{2} H_{2} }} + Q\frac{{L_{2}^{3} }}{{3H_{1}^{3} }} + \sum\limits_{n = 3}^{\infty } {\frac{{L_{2}^{n} }}{{nH_{1}^{n} }}} K^{n - 2} + (1 + Q)\sum\limits_{n = 4}^{\infty } {\frac{{L_{2}^{n} }}{{nH_{1}^{n} }}} K^{n - 3} \\ & \quad + \frac{{(L_{3} - L_{2} )^{2} L_{2} }}{{2H_{{2{\text{d}}}}^{2} }} + (1 + H_{\text{d}} )\frac{{(L_{3} - L_{2} )^{3} }}{{3H_{{2{\text{d}}}}^{3} }} - Q\frac{{(L_{3} - L_{2} )^{2} L_{3} }}{{H_{{2{\text{d}}}}^{2} H_{{3{\text{d}}}} }} + Q\frac{{(L_{3} - L_{2} )^{3} }}{{3H_{{2{\text{d}}}}^{3} }} \\ & \quad + \sum\limits_{n = 3}^{\infty } {\frac{{(L_{3} - L_{2} )^{n} }}{{nH_{{2{\text{d}}}}^{n} }}} K^{n - 2} + (1 + H_{\text{d}} + Q)\sum\limits_{n = 4}^{\infty } {\frac{{(L_{3} - L_{2} )^{n} }}{{nH_{{2{\text{d}}}}^{n} }}} K^{n - 3} + \frac{{P_{2} }}{2}(L_{3}^{2} - L_{2}^{2} ) \\ & \quad + \frac{{(H_{3} - 1)(1 - L_{3} )^{3} }}{{2H_{3}^{2} }} - \frac{{(1 - L_{3} )^{2} }}{{2H_{3}^{2} }} + \frac{{(1 - L_{3} )^{3} }}{{3H_{3}^{3} }} - Q\left( {\frac{{(H_{3} + 1)(1 - L_{3} )^{3} }}{{2H_{3}^{2} }} - \frac{{(1 - L_{3} )^{2} }}{{H_{3} }} - \frac{{(1 - L_{3} )^{3} }}{{3H_{3}^{3} }}} \right) \\ & \quad + \sum\limits_{n = 3}^{\infty } {\frac{{(1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} K^{n - 2} + (1 + Q)\sum\limits_{n = 4}^{\infty } {\frac{{(1 - L_{3} )^{n} }}{{nH_{3}^{n} }}} K^{n - 3} \\ & \quad - WL_{\text{pv}} = 0 \\ & \quad - 1 < \frac{{KL_{2} }}{{H_{1} }} < 1, - 1 < \frac{{K(L_{3} - L_{2} )}}{{H_{{2{\text{d}}}} }} < 1, - 1 < \frac{{K(1 - L_{3} )}}{{H_{3} }} < 1 \\ \end{aligned} $$
(72)

In the parallel case (K = 0), Eq. (72) becomes

$$ \begin{aligned} M &= \frac{{L_{2}^{3} }}{3} - Q\frac{{2L_{2}^{3} }}{3} \hfill \\ &\quad+ \frac{{(L_{3} - L_{2} )^{2} L_{2} }}{{2(1 + H_{\text{d}} )^{2} }} + \frac{{(L_{3} - L_{2} )^{3} }}{{3(1 + H_{\text{d}} )^{2} }} - Q\frac{{(L_{3} - L_{2} )^{2} L_{3} }}{{(1 + H_{\text{d}} )^{3} }} + Q\frac{{(L_{3} - L_{2} )^{3} }}{{3(1 + H_{\text{d}} )^{3} }} \hfill \\ &\quad +\frac{{P_{2} }}{2}(L_{3}^{2} - L_{2}^{2} ) - \frac{{(1 - L_{3} )^{2} }}{2} + \frac{{(1 - L_{3} )^{3} }}{3} - \frac{{2Q(1 - L_{3} )^{3} }}{3} + Q(1 - L_{3} )^{2} \hfill \\ &\quad- WL_{\text{pv}} = 0 \hfill \\ \end{aligned} $$
(73)

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Yagi, K., Sugimura, J. Balancing Wedge Action: A Contribution of Textured Surface to Hydrodynamic Pressure Generation. Tribol Lett 50, 349–364 (2013). https://doi.org/10.1007/s11249-013-0132-z

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