Hydrodynamic Bearing Theory

Hydrodynamic Bearings Theory Lecture 25 Engineering 473 Machine Design

Lubrication Zones Boundary Lubrication

Contact between journal and bearing

Mixed-film Lubrication Intermittent contact

Coefficient of Friction

Boundary Lubrication

Hydrodynamic Lubrication Journal rides on a fluid film. Film is created by the motion of the journal.

Mixed-film Lubrication

Hydrodynamic Lubrication

Bearing Parameter

µn Bearing Parameter ≡ p µ ≡ dynamic viscosity, lb - sec/in 2 n ≡ rotational speed, rev/sec p ≡ pressure (force/projected area), psi

Stable/Unstable Lubrication Coefficient of Friction

Boundary Lubrication

Mixed-film Lubrication

Bearing Parameter

Hydrodynamic Lubrication

µn p

Hydrodynamic Lubrication is often referred to as stable lubrication. If the lubrication temperature increases, the viscosity drops. This results in a lower coefficient of friction, that causes the lubrication temperature to drop. => Self Correcting.

Mixed-film lubrication is unstable – an increase in lubrication temperature causes further increases in lubrication temperature.

Newtonian Fluid A Newtonian fluid is any fluid whose shear stress and transverse rate of deformation are related through the equation.

du τ=µ dy

Dynamic Viscosity du µ=τ dy Units ips lbf in 2

lbf − sec = = reyn 2 in in sec in

SI N m2

m sec m

=

N − sec m2

Other common units are discussed in the text.

Pumping Action When dry, friction will cause the journal to try to climb bearing inner wall.

When lubricant is introduced, the “climbing action” and the viscosity of the fluid will cause lubricant to be drawn around the journal creating a film between the journal and bearing. The lubricant pressure will push the journal to the side.

Journal Bearing Nomenclature β is equal to 2π for a full bearing If β is less than 2π, it is known as a partial bearing. We will only be considering the full bearing case.

Analysis Assumptions 1. Lubricant is a Newtonian fluid 2. Inertia forces of the lubricant are negligible 3. Incompressible 4. Constant viscosity 5. Zero pressure gradient along the length of the bearing 6. The radius of the journal is large compared to the film thickness

Analysis Geometry

Actual Geometry

Unrolled Geometry

From boundary layer theory, the pressure gradient in the y direction is constant.

X-Momentum Equation

æ dp ö ∂τ ö æ å Fx = 0 = çè p + dx dx ÷ødydz + τdxdz − çç τ + ∂y dy ÷÷dxdz − pdydz è ø

dp ∂τ = dx ∂y

∂u τ=µ ∂y

dp ∂ 2u =µ 2 dx ∂y

X-Momentum Equation (Continued) X-Momentum Eq.

dp ∂ 2u =µ 2 dx ∂y General Solution

∂ 2 u 1 dp = 2 ∂y µ dx ∂u 1 dp = y + C1 (x ) ∂y µ dx 1 dp 2 y + C1 (x )y + C2 (x ) u= 2µ dx

Boundary Conditions

y = 0, u = 0 y = h(x), u = -U

X-Momentum Equation (Continued)

1 dp 2 u= y + C1 (x )y + C2 (x ) 2µ dx

y = 0, u = 0

C2 (x ) = 0 U h (x ) dp − C1 (x ) = − h (x ) 2µ dx

y = h(x), u = -U

(

)

1 dp 2 U u= y − h (x )y − y 2µ dx h (x )

Note that h(x) and dp/dx are not known at this point.

Mass Flow Rate

h (x )

 = ρ ò udy m 0

 =ρ m

h (x )

ò 0

(

)

æ 1 dp 2 U ö çç y − h (x )y − y ÷÷dy h (x ) ø è 2µ dx

é h (x )3 dp Uh (x )ù  = ρ ê− m − ú 2 û ë 12µ dx

Conservation of Mass é h (x )3 dp Uh (x ) ù  = ρ ê− m − ú 2 û ë 12µ dx Conservation of Mass Requires

 dm =0 dx

3 æ d h (x ) dp ö U dh ÷− − çç =0 ÷ dx è 12µ dx ø 2 dx 3 æ d h (x ) dp ö dh ç ÷ = −6 U ç dx è µ dx ÷ø dx

Reynold’s Equation

h(x) Relationship cr = radial clearance

θ

e ε= cr

h (θ ) = c r (1 + ε ⋅ cos θ )

e

h min = c r (1 − ε )

h max = c r (1 + ε ) æ æ 2x ö ö h (x ) = c r çç1 + ε ⋅ cos ç ÷ ÷÷ è D øø è

Sommerfeld Solution 3 æ d h (x ) dp ö dh ç ÷ = −6 U ç dx è µ dx ÷ø dx

A. Sommerfeld solved these equations in 1904 to find the pressure distribution around the bearing.

æ æ 2x ö ö h (x ) = c r çç1 + ε ⋅ cos ç ÷ ÷÷ è D øø è

It is known as a “long bearing” solution because there is no flow in the axial direction.

µUr é 6ε ⋅ sin θ ⋅ (2 + ε cos θ )ù p= 2 ê + po 0 ≤ θ ≤ π 2 ú 2 c r ë 2 + ε (1 + ε cos θ ) û

(

)

r is the journal radius, ε is a chosen design parameter.

Ocvirk Short-Bearing Solution A “short bearing” allows lubricant flow in the longitudinal direction, z, as well as in the circumferential direction, x. 3 3 ∂ æ h (x ) dp ö ∂ æ h (x ) dp ö ∂h ç ÷− ç ÷ = −6U ∂x çè µ dx ÷ø ∂z çè µ dz ÷ø ∂x

Governing Equation

The Ocvirk solution (1955) neglects the first term as being small compared to the axial flow.

µU æ l2 3ε ⋅ sinθ 2ö p = 2 çç − z ÷÷ 3 rcr è 4 ( ) 1 + ε ⋅ cosθ ø

0≤θ≤π

Short-Bearing Pressure Distributions

Norton Fig. 10-8 & 10-9

Short & Long Bearing Comparisons

%

Assignment Use Matlab to plot the pressure distribution predicted by the Sommerfeld equation for a journal bearing having a clearance ratio of 0.0017, journal radius of 0.75 in, ε of 0.6, µ=2.2µreyn, shaft rotational speed=20 rev/sec, and po=o. First, generate the plot only for the range θ equals 0 to π. Second, generate the plot for the range θ equals 0 to 2π. What happens to the pressure distribution from π to 2π. Is this physically possible? Discuss what would happen to the lubricant if this pressure distribution occurred.