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Performance Analysis of a Conical Hydrodynamic Journal Bearing

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

Many turbo-machines such as turbines and compressors generate axial and radial load during their operation. Conical hydrodynamic journal bearing is proposed to sustain radial load and adjust the effect of axial load on rotating elements. The present study has proposed the performance analysis for conical hydrodynamic journal bearing of semi-cone angles (γ = 5°, 10°, 20°, 30°) for a wide range of radial load (\(\bar{W}_{\text{r}} = 0.1 - 0.9\)) on rotating journal. Finite element method has been used to solve the modified Reynolds equation for investigating the flow of lubricant in the clearance space between journal and bearing. Performance characteristics such as load-carrying capacity, fluid film thickness, stiffness coefficients, damping coefficients and threshold speed for various configurations of conical hydrodynamic journal bearing have been presented and discussed. Results show that threshold speed (\(\bar{\omega }_{\text{th}}\)) of conical journal bearing is considerably reduced as the semi-cone angle increases to γ = 30° as compared to the base bearing of semi-cone angle γ = 5°.

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Abbreviations

\(C_{ij}\) :

Damping coefficient (i, j = 1,2) (Ns/m)

\(c_{\text{r}}\) :

Radial clearance (µm)

\(c_{\text{r}} /R_{j}\) :

Clearance ratio

D :

Mean bearing diameter (mm)

d :

Mean journal diameter (mm)

e :

Journal eccentricity (mm) \((e^{2} = x^{2} + z^{2} )\)

\(e_{o} ;\varphi_{o}\) :

Journal eccentricity and attitude angle at initial equilibrium position

F :

Fluid film reaction \(\frac{\partial h}{\partial t} \ne 0\) (N)

F o :

Fluid film reaction \(\frac{\partial h}{\partial t} = 0\) (N)

F x, F z :

Fluid film reaction component in X and Z directions, \(\frac{\partial h}{\partial t} \ne 0\) (N)

F a, F r :

Fluid film reaction component in axial and radial directions, \(\frac{\partial h}{\partial t} \ne 0\) (N)

h :

Fluid film thickness (mm)

\(h_{\hbox{min} }\) :

Minimum fluid film thickness (mm)

L :

Bearing length (mm)

\(o_{1} , o_{2}\) :

Bearing and journal center

p :

Fluid film pressure (MPa)

\(p_{\hbox{max} }\) :

Maximum fluid film pressure (MPa)

\(p_{\text{s}}\) :

Supply pressure (MPa)

\(R_{\text{b}}\) :

Mean bearing radius (mm)

\(R_{j}\) :

Mean journal radius (mm)

\(R_{1} , R_{2}\) :

Minimum and maximum radius of conical journal (mm)

\(S_{ij}\) :

Fluid film stiffness coefficients (i, j = 1, 2) (N/m)

t :

Time (s)

\(t_{\text{s}}\) :

Supply temperature (°C)

W :

External load (N)

W a :

Axial load (N)

\(W_{\text{r}}\) :

Radial load (N)

X, Y, Z :

Cartesian coordinates (mm)

x, z :

Horizontal, vertical journal eccentricity (mm)

\(X_{j } ,Z_{j } ; \dot{X}_{j } ,\dot{Z}_{j }\) :

Journal center displacement and velocity components

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

Coordinates of steady-state equilibrium journal center from geometric center of bearing (mm)

α :

Circumferential coordinate (ϕ) (rad)

β :

Axial coordinate (\(r\sin \gamma /R_{j}\))

γ :

Semi-cone angles

ε :

Journal eccentricity ratio (e/\(c_{\text{r}}\))

r, \(\theta ,\phi\) :

Spherical coordinates (mm, rad) (\(\theta = \gamma )\)

\(\varphi\) :

Attitude angle (rad)

λ :

Aspect ratio (L/D)

μ :

Lubricating fluid absolute or dynamic viscosity (Ns/m2)

ρ :

Mass density of lubricating fluid (kg/m3)

\(\mu_{\text{r}}\) :

Dynamic viscosity at reference inlet temperature and atmospheric pressure (Ns/m2)

ν :

Journal relative velocity (m/s)

\(\omega\) :

Journal angular velocity (rad/s)

Ω :

Speed parameter \(\omega \left( {\mu R_{j}^{2} /c_{\text{r}}^{2} p_{\text{s}} } \right)\)

\(\bar{C}_{11}\), \(\bar{C}_{22}\) :

Direct damping coefficients

\(\bar{C}_{12}\), \(\bar{C}_{21}\) :

Cross-coupled damping coefficients

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

Damping coefficient

\(\bar{h}\) :

Fluid film thickness (\(h/c_{\text{r}}\))

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

Minimum fluid film thickness \((h_{\hbox{min} } /c_{\text{r}} )\)

\(\bar{M}_{\text{c}}\) :

Critical mass parameter \(M_{\text{c}} \left( {c_{\text{r}}^{5} p_{\text{s}} /\mu^{2} R_{j}^{6} } \right)\)

\(N_{j}\) :

Shape function

ξ, η :

Local coordinates system for shape function \(N_{j}\)

\(\bar{p}\) :

Pressure ratio (\(p/p_{\text{s}}\))

\(\bar{p}_{j}\) :

Pressure ratio at jth nodal point \(\left( {p_{j} /p_{\text{s}} } \right)\)

\(\bar{p}_{\hbox{max} }\) :

Maximum pressure ratio \(\left( {p_{\hbox{max} } /p_{\text{s}} } \right)\)

\(\bar{S}_{11}\), \(\bar{S}_{22}\) :

Direct stiffness coefficients

\(\bar{S}_{12}\), \(\bar{S}_{21}\) :

Cross-coupled stiffness coefficients

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

Stiffness coefficient

\(\bar{W}_{\text{a}}\) :

Axial load \(\left( { \bar{W}_{\text{a}} = \frac{{W_{\text{a}} }}{{p_{\text{s}} R_{j}^{2} }} } \right)\)

\(\bar{W}_{\text{r}}\) :

Radial load (\(\bar{W}_{\text{r}} = \frac{{W_{\text{r}} }}{{p_{\text{s}} R_{j}^{2} }}\))

\(\bar{\mu }\) :

Dynamic viscosity (\(\mu /\mu_{\text{r}} )\)

\(\bar{\omega }_{\text{th}}\) :

Threshold speed margin

\(\left[ {\bar{F}} \right]\) :

Assembled fluidity matrix

\(\left\{ {\bar{p}} \right\}\) :

Nodal pressure vector

\(\left\{ {\bar{Q}} \right\}\) :

Nodal flow vector

\(\left\{ {\bar{R}_{\text{H}} } \right\}\) :

Column vector due to hydrodynamic terms

\(\left\{ {\bar{R}_{{x_{j} }} } \right\},\left\{ {\bar{R}_{{z_{j} }} } \right\}\) :

Global vectors due to journal center linear velocities

\(\bar{t}\) :

Time t\(\left( {c_{\text{r}}^{2} p_{\text{s}} /\mu R_{j}^{2} } \right)\)

\(\Delta \bar{X}_{J}\), \(\Delta \bar{Z}_{J}\) :

Small amount of journal movement from steady-state equilibrium journal center

\(\bar{X}_{J} = X_{J} /C_{\text{r}}\), \(\bar{Z}_{J} = Z_{J} /C_{\text{r}}\) :

Coordinates of steady-state equilibrium journal center

\(\Delta \bar{\dot{X}}_{J}\), \(\Delta \bar{\dot{Z}}_{J}\) :

Velocity components of journal center

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Acknowledgements

The financial support from the All India Council for Technical Education (AICTE), New Delhi, India, under Research Promotion Scheme (RPS) Ref. No. 8-221/RIFD/RPS/Policy-1/2014-15 is gratefully acknowledged. The author is grateful to the Director, Veermata Jijabai Technological Institute (V.J.T.I), Mumbai, and Principal, K. J. Somaiya College of Engineering (KJSCE), Mumbai, for all support provided for this study which is respectfully acknowledged.

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Authors and Affiliations

  1. Machine Dynamics and Vibration Laboratory, Department of Mechanical Engineering, Veermata Jijabai Technological Institute (VJTI), Mumbai, 400019, India

    Ajay Kumar Gangrade, Vikas M. Phalle & S. S. Mantha

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  1. Ajay Kumar Gangrade
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  2. Vikas M. Phalle
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  3. S. S. Mantha
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Correspondence to Ajay Kumar Gangrade.

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Gangrade, A.K., Phalle, V.M. & Mantha, S.S. Performance Analysis of a Conical Hydrodynamic Journal Bearing. Iran J Sci Technol Trans Mech Eng 43, 559–573 (2019). https://doi.org/10.1007/s40997-018-0217-2

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