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Abstract
A linear stability analysis of hydrodynamic journal bearings is presented, including the effects of elastic distortion of the liner and micropolar lubrication. Hydrodynamic equations of the lubricant and equations of motion of the journal are solved simultaneously with the deformation equations for the bearing surface to predict the fluid film pressure distributions theoretically. The components of stiffness and damping coefficients, critical mass parameter, and whirl ratio, which reflect the dynamic characteristic of the journal bearing, are calculated for varying eccentricity ratio taking into account the flexibility of the liner and the micropolar properties of the lubricant. The results presented show that stability decreases with an increase in the value of the elasticity parameter of the bearing liner and micropolar fluids exhibit better stability in comparison to Newtonian fluids.
Nomenclature
| a, b | = | inside and outside radius of bearing liner (m) |
| C | = | radial clearance (m) |
| C′ | = | 2 + λ/μ |
| D | = | journal diameter (m) |
| Dij, | = | damping coefficient of micropolar fluid for i = R, φ and j = R, φ (Ns/m), |
| dm,n | = | distortion of m, n harmonic |
| E | = | Young's modulus (N/m2) |
| e, ϵ | = | steady state eccentricity at mid-plane of bearing (z = 0) (m), ϵ = e/C |
| F | = | deformation factor, |
| Fi | = | force component along the R and φ directions for i = R and φ (N) |
| H = b-a | = | thickness of bearing liner (m) |
| = | local film thickness (m), | |
| Λ | = | characteristic length of the micropolar fluid (m) |
| ν | = | Poisson's ratio |
| lm | = | non-dimensional characteristic length of micropolar fluid (m), lm = C/Λ |
| L | = | bearing length (m) |
| m, n | = | axial and circumferential harmonic |
| M, | = | mass parameter (kg), |
| = | critical value of non-dimensional mass parameter | |
| N | = | coupling number |
| = | local film pressure (Pa), | |
| R | = | journal radius (m) |
| Sij, | = | stiffness coefficient of micropolar fluid for i = R, φ and j = R, φ (N/m), |
| t, | = | time (s), |
| U | = | velocity of the journal (m/s), U = ωR |
| Ω | = | whirl ratio |
| = | steady state load (N), | |
| α | = | constant describing the fluid-wall interaction |
| x, y, z | = | circumferential, radial, axial coordinates (m) |
| θ, | = | dimensionless coordinates, θ = x/R, |
| u′, v′, w′ | = | radial, circumferential and axial displacements |
| η | = | viscosity of Newtonian fluid (Pa s) |
| γ, χ | = | viscosity coefficient for the micropolar fluid (Pa s) |
| τij | = | stress in ith plane in jth direction (N/m2) |
| δ, | = | radial deformation at bearing liner surface (m), |
| ω | = | angular velocity of the journal (rad/s) |
| ωp | = | angular velocity of orbital motion of the centre (rad/s) |
| λ, μ | = | Lame's constant |
| φ | = | attitude angle (rad) |
| θ | = | angular coordinate of the bearing (rad) |
| θ1, θ2 | = | angles of end and start of hydrodynamic film at each axial plane of bearing (rad) |
| — | = | above a variable represents the non-dimensional value of parameter |
| S | = | Sommerfeld number |
| 0 | = | subscript represents the steady state value |
| 1 and 2 | = | subscript represents the first order perturbation along ϵ0 and φ directions |