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Analysis of a double layer porous hybrid journal bearing considering the combined influence of wear and non-Newtonian behaviour of lubricant

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Abstract

Use of porous materials in fluid film is well established so as to make more uniform distribution of pressure along the journal surface. The work presented in this paper deals with theoretical examination into the effect of worn bearing surface on the behaviour of a hybrid double layer porous journal bearing system (DLPJBS) operating with power-law lubricants. The governing equation for the flow of non-Newtonian lubricant in the bearing porous clearance space is solved by using the FEM. The effects of non-Newtonian lubricant on the bearing characteristics of a worn bearing have been studied. Findings of this study indicates that the hybrid DLPJBS operating under non-Newtonian lubricant offers enhanced values of \({\bar{h}}_{min}\), \({\bar{\omega}}_{th}\) and rotor dynamic coefficients (stiffness and damping coefficients).

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Abbreviations

\( {\text{c}} \) :

Radial clearance (mm)

\( C_{ij} \) :

Fluid film damping coefficients (\( i,j = x,z \)) (N s mm−1)

\( D \) :

Diameter of journal (mm)

\( e \) :

Journal eccentricity (mm)

\( F_{xo} , F_{zo} \) :

Components of fluid film reaction (N)

\( F_{o} \) :

Fluid film reaction (N)

\( g \) :

Acceleration due to gravity (mm s−2)

\( h \) :

Nominal fluid film thickness (mm)

\( \Delta h \) :

Change in bearing geometry due to wear (mm)

\( H \) :

Wall thickness of porous bearing, (mm), (Fig. 1b)

\( H_{1} \) :

Thickness of inner layer of porous bearing (mm), (Fig. 1b)

\( L \) :

Length of bearing (mm)

\( k_{1} \) :

Permeability of inner layer of porous material (mm2)

\( k_{2} \) :

Permeability of outer layer porous material (mm2)

\( M_{j} ,M_{c} \) :

Journal mass, critical mass (kg)

\( n \) :

Power-law index

\( Q \) :

Lubricant flow of oil (mm3 s−1)

\( p \) :

Pressure (N mm −2)

\( p_{1} \) :

Pressure in the inner layer of porous matrix (N mm −2)

\( p_{2} \) :

Pressure in the outer layer of porous matrix (N mm −2)

\( p_{s} \) :

Supply pressure (N mm −2)

\( R_{j} \) :

Radius of journal (mm)

\( S_{ij} \) :

Fluid film stiffness coefficients (\( i,j = x,z \)) (N mm −1)

\( T_{f} \) :

Frictional torque, (N mm)

\( t \) :

Time (s)

\( U \) :

Velocity of Journal (mm s −1)

\( W_{o} \) :

External load (N)

\( X,Y,Z \) :

Cartesian coordinates

\( X_{j} , Z_{j} \) :

Journal centre coordinates

\( \alpha \) :

\( \frac{x}{{R_{j} }} \), Circumferential coordinate

\( \beta \) :

\( \frac{y}{{R_{j} }} \), Axial coordinate

\( \mu \) :

Apparent viscosity of lubricant (N s m −2)

\( \mu_{r} \) :

Reference viscosity of lubricant (N s m −2)

\( \omega_{j} \) :

Journal rotational speed (rad s −1)

\( \delta_{w} \) :

Wear depth (mm)

\( \tau \) :

Shear stress (N mm −2)

\( \dot{\gamma } \) :

Shear strain rate (s −1)

\( \emptyset \) :

Attitude angle, (rad)

\( \omega_{I} \) :

\( \left( {\frac{g}{c}} \right)^{1/2} \) (rad s−1)

\( \omega_{th} \) :

Threshold speed (rad s−1)

\( \varepsilon \) :

\( \left( {\frac{e}{c}} \right) \)

\( \bar{h} \) :

\( \left( {\frac{h}{c}} \right) \)

\( \Delta \bar{h} \) :

\( \left( {\frac{\Delta h}{c}} \right) \)

\( \bar{h}_{min} \) :

\( \left( {\frac{{h_{min} }}{c}} \right) \)

\( \overline{H}_{1} \) :

\( \left( {\frac{{H_{1} }}{L}} \right) \)

\( \overline{H} \) :

\( \left( {\frac{H}{L}} \right) \)

\( \overline{F}_{xo} ,\overline{F}_{zo} \) :

\( \left( {\frac{{F_{xo} ,F_{zo} }}{{p_{s} R_{j}^{2} }}} \right) \)

\( \overline{C}_{ij} \) :

\( C_{ij} \left( {\frac{{c^{3} }}{{\mu R_{j}^{4} }}} \right) \)

\( \overline{Q} \) :

\( Q\left( {\frac{\mu }{{c^{3} p_{s} }}} \right) \)

\( \overline{M}_{j} ,\overline{M}_{c} \) :

\( M_{j} ,M_{c} \left( {\frac{{\mu R_{j}^{4} }}{{\omega_{j} c^{3} }}} \right) \)

\( \overline{p} \) :

\( \frac{p}{{p_{s} }} \)

\( \overline{p}_{1} \) :

\( \frac{{p_{1} }}{{p_{s} }} \)

\( \overline{p}_{2} \) :

\( \frac{{p_{2} }}{{p_{s} }} \)

\( \overline{p}_{max} \) :

\( \frac{{p_{max} }}{{p_{s} }} \)

\( \overline{\delta }_{w} \) :

\( \left( {\frac{{\delta_{w} }}{c}} \right) \)

\( \overline{S}_{ij} \) :

\( S_{ij} \left( {\frac{{c^{3} }}{{p_{s} R_{j}^{2} }}} \right) \)

\( S_{O} \) :

\( \frac{{\mu \omega_{j} LD}}{{W_{o} }}\left( {\frac{{R_{j} }}{c}} \right)^{2} \), Sommerfeld number

\( \overline{t} \) :

\( t\left( {\frac{{c^{2} p_{s} }}{{\mu_{r} R_{j}^{2} }}} \right) \)

\( \overline{T}_{f} \) :

\( T_{f} \left( {\frac{1}{{p_{s} cR_{j}^{2} }}} \right) \)

\( \overline{\mu } \) :

\( \frac{\mu }{{\mu_{r} }} \)

\( \overline{W}_{o} ,\overline{F}_{o} \) :

\( \left( {\frac{{W_{o} ,F_{o} }}{{p_{s} R_{j}^{2} }}} \right) \)

\( \overline{X}_{j} ,\overline{Z}_{j} \) :

\( \left( {\frac{{X_{j} , Z_{j} }}{c}} \right) \)

\( \lambda \) :

\( \frac{L}{D} \), Aspect ratio

\( \overline{\tau } \) :

\( \tau \left( {\frac{{R_{j} }}{{p_{s} c}}} \right) \)

\( \overline{z} \) :

\( \left( {\frac{z}{h}} \right) \), Coordinates across fluid film thickness

\( \overline{{\dot{\gamma }}} \) :

\( \dot{\gamma }\left( {\frac{{\mu_{r} R_{j} }}{{p_{s} c}}} \right) \)

\( {{\Omega }} \) :

\( \omega_{j} \left( {\frac{{\mu_{r} R_{j}^{2} }}{{c^{2} P_{s} }}} \right) \)

\( \overline{\omega }_{th} \) :

\( \frac{{\omega_{th} }}{{\omega_{I} }} \)

\( {{\Psi }} \) :

\( \frac{{k_{1} H}}{{c^{3} }} \), Permeability parameter

\( \left[ {\overline{F}_{ij} } \right] \) :

Fluidity matrix

\( N_{i} ,N_{j} \) :

Shape functions

\( \left[ {\overline{P}_{j} } \right] \) :

Nodal pressure vector

\( \left[ {\overline{Q}_{j} } \right] \) :

Nodal flow vector

\( \left[ {\overline{R}_{Hj} } \right] \) :

RHS vector due to hydrodynamic terms

\( \left[ {\overline{R}_{Xj} } \right] \) :

RHS vector due to squeeze velocity \( \left( {\overline{{\dot{X}}} } \right) \)

\( \left[ {\overline{R}_{Zj} } \right] \) :

RHS vector due to squeeze velocity \( \left( {\overline{{\dot{Z}}} } \right) \)

\( e \) :

eth Element

j :

Journal

\( o \) :

Steady-state condition

\( - \) :

Nondimensional parameter

\( min/max \) :

Minimum/Maximum value

\( r \) :

Reference value

/:

First/second derivative w.r.t, time

s :

Supply

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  1. Department of Mechanical and Industrial Engineering, Tribology Laboratory, Indian Institute of Technology, Roorkee, Roorkee, 247667, India

    Anil Singh & Satish C. Sharma

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  1. Anil Singh
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  2. Satish C. Sharma
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Correspondence to Anil Singh.

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Singh, A., Sharma, S.C. Analysis of a double layer porous hybrid journal bearing considering the combined influence of wear and non-Newtonian behaviour of lubricant. Meccanica 56, 73–98 (2021). https://doi.org/10.1007/s11012-020-01259-2

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  • DOI: https://doi.org/10.1007/s11012-020-01259-2

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