Influence of dimple geometry and micro-roughness orientation on performance of textured hybrid thrust pad bearing

Abstract

The present study deals with numerical simulation of textured hybrid thrust pad bearing. Influence of providing micro-dimples of different cross-sectional shapes on the bearing surface has been theoretically investigated on the performance of thrust pad bearing. Reynolds equation has been solved using mass-conserving algorithm based on Jakobsson–Floberg–Olsson cavitation boundary conditions. A parametric study is carried out to optimize dimple shapes from the viewpoint of load carrying capacity of bearings. The textured bearing surface is noticed to be beneficial in reducing the frictional power losses. Providing half-section dimples (second half in the direction of runner rotation) towards the leading edge of thrust pad, enhance the load carrying capacity and fluid film stiffness coefficient of bearings. Micro-roughness in a textured surface having transverse orientation is seen to improve the dynamic characteristics of hybrid thrust pad bearings.

Appendix

1.1 Appendix 1: Fluid film thickness expression for micro-dimples of different cross section shapes (Refer Fig. 1c)

Micro dimple shape	Expression for $h_{tex}$	Domain	Equations
Spherical: Sp	$\sqrt {\left( {\frac{{h_{d}^{2} + r_{d}^{2} }}{{2h_{d} }}} \right)^{2} - \left( {x_{l}^{2} + z_{l}^{2} } \right)} - \frac{{r_{d}^{2} }}{{2h_{d} }} + \frac{{h_{d} }}{2}$	$\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d}$—Sp	(14)
Spherical half: Sp(h)		$\left( {\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d} } \right)$ and $z_{l} > 0$—Sp(h)
Conical: Con	$\left[ {1 - \frac{{\sqrt {\left( {x_{l}^{2} + z_{l}^{2} } \right)} }}{{r_{d} }}} \right]h_{d}$	$\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d}$— Con	(15)
Conical half: Con(h)		$\left( {\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d} } \right)$ and $z_{l} > 0$—Con(h)
Conical frustum: Confr	$\left[ {1 - \frac{{\sqrt {\left( {x_{l}^{2} + z_{l}^{2} } \right)} }}{{r_{d} }}} \right]h_{d}$	$\frac{{r_{d} }}{2} < \left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d}$—Confr
	$h_{d}$	$\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} \le \frac{{r_{d} }}{2}$—(confr)	(16)
Conical frustum half: Confr(h)	$\left[ {1 - \frac{{\sqrt {\left( {x_{l}^{2} + z_{l}^{2} } \right)} }}{{r_{d} }}} \right]h_{d}$	$\left( {\frac{{r_{d} }}{2} < \left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d} } \right)$ and $z_{l} > 0$—Confr(h)	(17)
	$h_{d}$	$\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} \le \frac{{r_{d} }}{2}$ and $z_{l} > 0$—Confr(h)	(18)
Cylindrical: Cyl	$h_{d}$	$\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d}$—Cyl
Cylindrical half: Cyl(h)		$\left( {\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} < r_{d} } \right)$ & $z_{l} > 0$—Cyl(h)
Untextured zone	$0$	$\left( {\left( {x_{l}^{2} + z_{l}^{2} } \right)^{1/2} \geq r_{d} } \right)$	(19)

1.2 Appendix 2: Finite element formulation of modified Reynolds equation

The residue of Reynolds equation in FEM has been obtained as follows:

$$\begin{aligned} R_{es} & = \frac{\partial }{{\partial \bar{x}}}\left( {\bar{h}^{3} \emptyset_{x} g\frac{\partial }{{\partial \bar{x}}}\left\{ {\mathop \sum \limits_{j = 1}^{4} \left( {\bar{p}_{j} N_{j} } \right)} \right\}} \right) \\ & \quad + \frac{\partial }{{\partial \bar{z}}}\left( {\bar{h}^{3} \emptyset_{z} g\frac{\partial }{{\partial \bar{z}}}\left\{ {\mathop \sum \limits_{j = 1}^{4} \left( {\bar{p}_{j} N_{j} } \right)} \right\}} \right) \\ & \quad -\uplambda\bar{u}\frac{{\partial \bar{h}_{T} }}{{\partial \bar{x}}} -\uplambda\bar{w}\frac{{\partial \bar{h}_{T} }}{{\partial \bar{z}}} - \varUpsilon \frac{{\partial \bar{h}_{T} }}{{\partial \bar{t}}} \\ \end{aligned}$$

(20)

Weak formulation of modified Reynolds equation:

$$\iint\limits_{{\Omega ^{e} }} {}\left( {N_{j} *R_{es} } \right)d\bar{x}d\bar{z} = 0$$

(21)

Elemental system of equation:

$$\left[ {\bar{F}_{ij}^{e} } \right]\left\{ {\bar{p}} \right\} = \left\{ {\bar{Q}_{i}^{e} } \right\} + \left[ {\overline{RH}_{ij}^{e} } \right]_{{\bar{u},\bar{w}}} + \overline{{\dot{h}}}_{T} \left[ {\overline{RS}_{ij}^{e} } \right]$$

(22)

Matrix terms for eth element in the elemental system of equations can be expressed as:

Fluidity matrix:

$$\bar{F}_{ij}^{e} = \iint\limits_{{\Omega ^{e} }} {g\left[ {\bar{h}^{3} \emptyset_{x} \frac{{\partial N_{i} }}{{\partial \bar{x}}}\frac{{\partial N_{j} }}{{\partial \bar{x}}} + \bar{h}^{3} \emptyset_{z} \frac{{\partial N_{i} }}{{\partial \bar{z}}}\frac{{\partial N_{j} }}{{\partial \bar{z}}}} \right]d\bar{x}d\bar{z}}$$

(23)

Flow matrix:

$$\bar{Q}_{i}^{e} = \int\limits_{{\Gamma ^{e} }} {\left[ {\left( {\bar{h}^{3} \emptyset_{x} \frac{\partial p}{{\partial \bar{x}}}} \right)m_{1} + \left( {\bar{h}^{3} \emptyset_{z} \frac{\partial p}{{\partial \bar{z}}}} \right)m_{2} } \right]N_{i} {\text{d}}\Gamma ^{e} }$$

(24)

Hydrodynamic term matrix:

$$\overline{RH}_{ij}^{e} = \iint\limits_{{\Omega ^{e} }} {\lambda \left[ {\bar{u}\bar{h}_{T} \frac{{\partial N_{i} }}{{\partial \bar{x}}} + \bar{w}\bar{h}_{T} \frac{{\partial N_{j} }}{{\partial \bar{z}}}} \right]d\bar{x}d\bar{z}}$$

(25)

Squeeze matrix:

$$\overline{RS}_{ij}^{e} = \iint\limits_{{\Omega ^{e} }} {\varUpsilon N_{i} d\bar{x}d\bar{z}}$$

(26)

Derivative of nodal pressure with respect to squeeze velocity and film thickness is expressed as:

$$\frac{{\partial \bar{p}}}{{\partial \bar{h}_{T} }} = \left[ {\bar{F}} \right]^{ - 1} \left[ {\frac{{\partial \left\{ {\bar{Q}} \right\}}}{{\partial \bar{h}_{T} }} + \frac{{\partial \left[ {\overline{RH}_{ij} } \right]}}{{\partial \bar{h}_{T} }} + \overline{{\dot{h}}}_{T} \frac{{\partial \left[ {\overline{RS}_{ij} } \right]}}{{\partial \bar{h}_{T} }} - \left\{ p \right\}\frac{{\partial \left[ {\bar{F}} \right]}}{{\partial \bar{h}_{T} }}} \right];$$

(27)

$$\frac{{\partial \bar{p}}}{{\partial \overline{{\dot{h}}}_{T} }} = \left[ {\bar{F}} \right]^{ - 1} \left[ {\frac{{\partial \left\{ {\bar{Q}} \right\}}}{{\partial \overline{{\dot{h}}}_{T} }} + \frac{{\partial \left[ {\overline{RH}_{ij} } \right]}}{{\partial \overline{{\dot{h}}}_{T} }} + \left[ {\overline{RS}_{ij} } \right] + \overline{{\dot{h}}}_{T} \frac{{\partial \left[ {\overline{RS}_{ij} } \right]}}{{\partial \overline{{\dot{h}}}_{T} }} - \left\{ {\bar{p}} \right\}\frac{{\partial \left[ {\bar{F}} \right]}}{{\partial \overline{{\dot{h}}}_{T} }}} \right]$$

(28)