Hydrodynamic fluid film bearings and their effect on the stability - - PowerPoint PPT Presentation
Notes 5 Modern Lubrication Hydrodynamic fluid film bearings and their effect on the stability of rotating machinery Dr. Luis San Andres Mast-Childs Professor Lsanandres@tamu.edu http://rotorlab.tamu.edu/me626 1 September 2010
1
September 2010
http://rotorlab.tamu.edu/me626
Mast-Childs Professor Lsanandres@tamu.edu
2
Advantages
Do not require external source of pressure. Support heavy loads. The load support is a function of the lubricant viscosity, surface speed, surface area, film thickness and geometry of the bearing. Long life (infinite in theory) without wear of surfaces. Provide stiffness and damping coefficients of large magnitude.
Disadvantages
Thermal effects affect performance if film thickness is too small or available flow rate is too low. Potential to induce hydrodynamic
instability, i.e. loss of effective
damping for operation well above critical speed of rotor-bearing system
Radial and axial load support of rotating machinery – low friction and long life
Typically use MINERAL OIL as
replace with working fluid (water)
3
Fundamentals of Thin Film Lubrication
Geometry of flow region in a thin fluid film bearing (h << Lx, Lz)
DB=2 RB DJ=2 RJ
Cylindrical bearing
TYP (c/L*) = 0.001
μ ρ c U* Re =
SMALL Couette flow Reynolds #
( )
( )
( )
= ∂ ∂ + ∂ ∂ + ∂ ∂ z v y v x v
z y x 2 2 2 2
; y v z P y v x P
x x
∂ ∂ + ∂ ∂ − = ∂ ∂ + ∂ ∂ − = μ μ
Flow equations: continuity + momentum (x,y)
Quasi-static (pressure forces = viscous forces) Figures 1 & 2
x z
Lz Lx h(x,z,t)
U V
(U,V) surface velocities
V x Vy Vz
h << Lx,Lz
y x
Lx h(x,z,t)
U V
Vz Vy Vx
4
Importance of fluid inertia in thin film flows
Importance of fluid inertia effects on several fluid film bearing
9,296 930 0.163 13.30 R134 refrigerant 8,477 848 0.179 13.93 Liquid nitrogen 7,942 794 0.191 10.47 Liquid oxygen 7,052 705 0.216 1.075 Liquid hydrogen 1,588 159 1.00 64 Water 711 71 2.14 120 Light oil 51 5.1 30.0 1,682 Thick oil 99 9.9 15.4 1.23 Air
Re at 10,000 rpm Re at 1,000 rpm Kinematic viscosity (ν) centistoke Absolute viscosity (µ) lbm.ft.s x 10-5
fluid
Fluid inertia is important for operation at high speeds and with process fluids. These are prevalent conditions in HP turbomachinery Reynolds numbers
Table 1
5
Fluid inertia effects at inlet & edges
Fluid inertia (Bernoulli’s effect) causes sudden pressure drop (or raise) at sharp inlets (exits). Most important effect on annular pressure seals and hydrostatic bearings with process fluids Pressure drop & rise at sudden changes in film thickness
Figure 3
ΔP ~ ½ ρU2 P P U U ΔP ~ ½ ρU2 P P U U
6
Thin Film Lubrication: Reynolds Equation
Cylindrical journal bearing & coordinates
{ } { }
⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Θ ∂ ∂ Θ ∂ ∂ = Θ ∂ ∂ Ω + ∂ ∂ z P h z P h R h h t μ ρ μ ρ ρ ρ 12 12 1 2
3 3 2
Pressure = ambient on sides Pressure > Pcavitation
θ sin sin cos e e e c h
Y X
= Θ + Θ + =
Figure 4
X Y
Θ
journal
e
Bearing center
Ω
cos sin
X Y
h C e e θ θ = + +
θ
Elliptical PDE in film region Film thickness eX = e cos(φ ); eY = e sin(φ )
Kinematics of journal motion:
X Y
journal
Bearing center
φ
eY eX
7
e Vr OJ OB eY eX φ t
X Y
Vt r
Kinematics of journal motion
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ φ φ φ φ φ
e e e
Y X
cos sin sin cos
Θ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Ω − + Θ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Ω + = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Θ ∂ ∂ Θ ∂ ∂ sin 2 cos 2 12 12 1
3 3 2 X Y Y X
e e e e z P h z P h R
μ
Reynolds Eqn. in fixed coordinates (X,Y)
θ φ θ μ θ μ θ sin 2 cos 12 12 1
3 3 2
⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Ω − + = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ ∂
e z P h z P h R
Reynolds Eqn. in moving coordinates) Set: incompressible fluid (oil) For circular centered orbits:: radius (e) and
2 / Ω = φ
Loss of load capacity eX = e cos(φ ); eY = e sin(φ )
θ Θ
x=RΘ
Y r t OB OJ e
Ω
h
y Bearing Journal
φ A
Θ=θ+φ
8
Journal bearing reaction force
Fluid film force acting on journal surface
Dynamic forces = fn. of journal position and velocities, rotational speed (Ω), viscosity (μ) and geometry (L, D, c)
( )
dz d R t z P F F
L t r
θ θ θ θ
π
⋅ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡
sin cos , ,
2
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡
t r Y X
F F F F φ φ φ φ cos sin sin cos
( )
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Ω − = Ω = 2 , , , φ
α α α
e F e e F F
Y X
θ P.cosθ P.sinθ P
r θ Θ t X Y P
journal
Ft Fr
Force = integration of pressure field on journal surface
Figure 5
9
LONG journal bearing (limit geometry)
LONG BEARING MODEL
L/D >>> 1
Pressure does not vary axially. Not applicable for most practical cases, except sealed squeeze film dampers
{ } { }
⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ ∂ = Θ ∂ ∂ Ω + ∂ ∂ z P h z h h t μ 12 2
3
Figure 6
Ω
L D
journal Axial pressure field bearing
L/D >> 1 dP/dz → 0
10
SHORT journal bearing (limit geometry)
SHORT JOURNAL BEARING MODEL
L/D < 0.50
Applicable to actual rotating machinery
{ } { }
h h t P h R Θ ∂ ∂ Ω + ∂ ∂ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ ∂ 2 12 1
3 2
θ μ θ
⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω − + = −
2 2 3 3
2 sin 2 cos 6 ) , , ( L z H C e e P t z P
a
θ φ θ μ θ
proportional to viscosity (μ), speed (Ω), and most important to: 1/C
3
Control of tolerances in machined clearance is critical for reliable performance Figure 7
Ω journal L D Axial pressure field bearing
L/D << 1 dP/dθ → 0
11
STATIC LOAD PERFORMANCE
Force Balance for Static Load
Bearing reaction force = applied static load (% of rotor weight)
( ) ( )
2 / 3 2 2 3 2 2 2 3 3
1 4 ; 1 ε ε π μ ε ε μ − ⋅ Ω + = − Ω − = c L R F c L R F
t r
0.2 0.4 0.6 0.8 1 100 1 .103 1 .104 1 .105
Ft Static Forces for short length bearing
journal eccentricity (e/C) Radial and Tangential forces [N]
*
Radial and tangential forces for L/D=0.25 bearing. μ=0.019 Pa.s, L=0.05 m, c=0.1 mm, 3, 000 rpm,
Journal bearing can generate large reaction forces. Highly nonlinear functions of journal eccentricity Ftangential Fradial Figures 8 & 9
X Y W bearing Rotor (journal) fluid film Journal Rotation Ω e φ Static load X Y r t W
Ft φ
12
DESIGN PARAMETER: STATIC LOAD PERFORMANCE
Given S, iterative solution to find
and attitude angle (φ):
Sommerfeld number
N rotational speed (rev/s) W static load L, D=2R, c : clearance & μ viscosity Attitude angle
2
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = c R W D L N S μ
( )
( ) ( ) { }
2 2 2 2 2 2 2
1 16 1 4 ε π ε ε ε μ π σ − + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = = c L W R L D L S
( )
ε ε π φ 4 1 tang
2
− = − =
r t
F F
Locus of journal center for short length bearing
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ey/c ex/c Journal locus Clearance circle
load increases, low speed, low viscosity
e/c
attitude angle
speed increases, load loads, high viscosity
clearance circle W load spin direction
Low load, high speed, large viscosity
Low load, high speed, large viscosity High load, low speed, small viscosity
Figure 12
13
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.01 0.1 1 10 journal eccentricity (e/c) Sommerfeld number
*
DESIGN PARAMETER: STATIC LOAD PERFORMANCE
Sommerfeld number Sommerfeld # vs journal eccentricity
Low load, high speed, large viscosity High load, low speed, small viscosity
( )
2 2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = = c L W R L D L S μ π σ
Large e
Centered journal
Figure 10 N rotational speed (rev/s) W static load L, D=2R, c : clearance & μ viscosity
14
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50 60 70 80 90 journal eccentricity (e/c) Attitude angle
*
DESIGN PARAMETER: STATIC LOAD PERFORMANCE
Sommerfeld number Attitude angle # vs journal eccentricity
Low load, high speed, large viscosity High load, low speed, small viscosity
( )
2 2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = = c L W R L D L S μ π σ
Large e Centered journal
Figure 11 N rotational speed (rev/s) W static load L, D=2R, c : clearance & μ viscosity
15
DYNAMICS OF ROTOR-BEARING SYSTEM
Symmetric - rigid rotor supported on short length journal bearings
Rigid rotor supported on journal bearings. (u) imbalance, (e) journal eccentricity
Equations of motion:
2 2
Y
X Y 2Fo disk
Clearance circle
Ωt e Static load u Disk 2M journal bearing Rigid shaft
16
DYNAMICS OF ROTOR-BEARING SYSTEM
Consider small amplitude motions about static equilibrium position (SEP). SEP defined by applied static load. Small amplitude journal motions about an equilibrium position
O O Y X Y
e e e F F F
O O O O
φ ,
, , , ⇒ = − =
) ( ), ( t e e e t e e e
Y Y Y X X X
O O
Δ + = Δ + =
Let:
Y Y F X X F Y Y F X X F F F Y Y F X X F Y Y F X X F F F
Y Y Y Y Y Y X X X X X X
O O
∂ ∂ + Δ ∂ ∂ + Δ ∂ ∂ + Δ ∂ ∂ + = Δ ∂ ∂ + Δ ∂ ∂ + Δ ∂ ∂ + Δ ∂ ∂ + = Expansion of forces abut SEP Figure 14
W
φo ΔX ΔY eXo eY eo
X
clearance circle
Y
Static load
Journal center
Ω
W
φo ΔX ΔY eXo eY eo
X
clearance circle
Y
Static load
Journal center
Ω
X Y r t W
Ft φ
17
ROTORDYNAMIC FORCE COEFFICIENTS
Strictly valid for small amplitude motions. Derived from SEP The “physical representation” of stiffness and damping coefficients in lubricated bearings
;
j i ij
X F K ∂ ∂ − =
j i ij
X F C
∂ − =
Stiffness: Damping: Inertia:
;
j i ij
X F M
∂ − =
i,j = X,Y Figure 15
Kxx, Cxx
journal bearing X Y
Kxy, Cxy Kyx, Cyx Kyy Cyy
18
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Y X C C C C Y X K K K K F F t F t F
YY YX XY XX YY YX XY XX Y X Y X
O O
( ) (
ROTORDYNAMIC FORCE COEFFICIENTS
Stiffness Matrix: Damping Matrix: Static reaction force:
Inertia ~ 0 in journal bearings Strictly valid for small amplitude motions. Derived from SEP
Linearized Equations of motion
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Ω Ω Ω = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ t t u M Y X K K K K Y X C C C C Y X M O O M
YY YX XY XX YY YX XY XX
sin cos
2
19
0.01 0.1 1 10 0.1 1 10 Sommerfeld # Stiffness
y
0.2 0.4 0.6 0.8 1 0.1 1 10 journal eccentricity (e/c) Stiffness
Journal Bearing: STIFFNESS COEFFICIENTS
Care with non dimensional value interpretation
Eccentricity (e/c) Sommerfeld # (σ) High speed Low load Large viscosity High speed Low load Large viscosity Low speed Large load Low viscosity
kαβ = Kαβ (c/Fo)
kxx kxx kyy kyy kxy kxy
Figure 16 & 17
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
Bearing stiffnesses versus eccentricity and design number (σ)
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
20
0.01 0.1 1 10 1 10 100
Cxx Cyy Cxy Cyx
S# Damping
*
0.2 0.4 0.6 0.8 1 1 10 100 journal eccentricity (e/c) Damping
Journal Bearing: DAMPING COEFFICIENTS
Care with non dimensional value interpretation
Eccentricity (e/c) Sommerfeld # (σ) High speed Low load Large viscosity High speed Low load Large viscosity
cαβ = Cαβ (cΩ/Fο)
cxx
cxx
cyy
cyy cxy cxy =cyx =cyx
Figure 16 & 17
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
Bearing damping versus eccentricity and design number (σ)
Low speed Large load Low viscosity
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
21
Journal Bearing: OPERATION at CENTERED CONDITION
High speed Low load Large viscosity
eo→ 0, φo = 90 deg
Significance of cross-coupled effect in journal bearing
Pure cross-coupling effect
2 ; 2 4
3 3 3 3
π μ π μ c L R c C C c c L R k K K
YY XX YX XY
= = = Ω = Ω = = − =
no direct stiffness
Ω F
Non-rotating structure
F
Rotating structure
F F
22
STABILITY OF ROTOR-BEARING SYSTEM
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Y X K K K K Y X C C C C Y X M O O M
YY YX XY XX YY YX XY XX
threshold speed of rotation (Ωs) with rotor performing
(undamped) orbital motions at a whirl frequency (ωs)
1 ; ; − = = = = = j e B e B y e A e A x
j t j j t j
s s
τ ω ω τ ω ω
X Y 2Fo disk
Clearance circl
Ωt e Static load u Disk 2M journal bearing Rigid shaft
23
STABILITY OF ROTOR-BEARING SYSTEM
= whirl frequency (ωs)/threshold speed instability (Ωs)
YY XX YX XY XY YX XX YY YY XX eq s s
F M C c c k c k c c k c k k p
2 2 2
ω ω = + − − + = =
( )( )
2 2
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Ω = − ⋅ − − − =
s s YX XY YY XX YX XY YY eq XX eq s
c c c c k k k k k k ω ω
Equivalent support stiffness
Whirl frequency ratio The WFR is independent of the rotor characteristics (rotor mass and flexibility)
eq
s
K C F k M = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =
2
ω
n eq s
M K ω ω = =
whirl frequency equals the natural frequency of rigid rotor supported on journal bearings
X Y 2Fo disk
Clearance circl
Ωt e Static load u Disk 2M journal bearing Rigid shaft
24
0.01 0.1 1 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 S# whirl frequency ratio .5
* *
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 e/c whirl frequency ratio .5
*
WHIRL FREQUENCY RATIO
High speed Low load Large viscosity High speed Low load Large viscosity Eccentricity (e/c) Sommerfeld # (σ)
Rotor becomes unstable at speed = twice system natural frequency
as 50 .
→
= = ε Ω ω
XX XY s s
c k
Whirl frequency ratio
; ; ; = = − = = = =
YX XY YX XY YY XX YY XX
c c k k c c k k
( )
XX XY XY XX XX eq
c k c c k k + =
=0 At centered condition 0.50 Figure 18
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
Low speed Large load Low viscosity
25
0.01 0.1 1 10 1 2 3 4 5 6 7 8 9 10 S#
*
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 e/c
*
Threshold speed of instability
High speed Low load Large viscosity High speed Low load Large viscosity Eccentricity (e/c) Sommerfeld # (σ)
unstable stable unstable stable Figure 19
Fully stable for operation with ε > 0.75, all bearings (L/D). Threshold speed decreases as eccentricity (e/c) 0
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
Threshold speed of instability versus eccentricity and design number (σ)
Low speed Large load Low viscosity
Ps = M Ωs
2 c/Fo
26
0.01 0.1 1 10 1 2 3 4 5 6 7 8 9 10 S#
*
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 e/c
*
High speed Low load Large viscosity High speed Low load Large viscosity Eccentricity (e/c) Sommerfeld # (σ)
Critical mass equals maximum mass rotor is able to support stably if current operating speed = threshold speed of instability. Critical mass decreases for centered condition. Unlimited for large (e/c)
unstable unstable stable stable Figure 20
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
Critical mass versus eccentricity and design number (σ)
Low speed Large load Low viscosity
27
0.01 0.1 1 10 2 4 6
rigid T/c=0.1 T/c=1 T/c=10 Threshold speed (ps) for flexible rotor
Modified Sommerfeld number Threshold speed (ps)
EFFECTS OF ROTOR FLEXIBILITY
Static sag Sommerfeld # (σ) High speed Small load High viscosity Low speed Large load Low viscosity
Rotor flexibility decreases system natural frequency, thus lowering threshold speed of
unstable More flexibility stable ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = C T k p p
eq s sf
1
2 2
rot
F T =
Figure 21
2
4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω = c L W R L μ σ
bearing
2M
KRot
28
PHYSICS of WHIRL MOTION
At centered condition: No radial support, tangential force must be < 0 to oppose whirl motion
Figure 22
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ φ φ
e C C C C e e K K K K F F
tt tr tr rr tt tr rt rr d t r
2 ; 2
3 3 π
μ C L R C C C C K K K C C K K
rr tt tr rt tr rt tt rr
= = = Ω = − = = = = = =
Forces in rotating coordinate system Bearing force coefficients at (e/c)=0 Resultant forces
rt tt t r
d d
whirl
X Y
Ft= -(Cttω + Ktr) Δe
Rotor spin Ω
Fr= -(Crtω + Krr) Δe Δe
29
PHYSICS of WHIRL MOTION
Figure 22
Force diagram for circular centered whirl motions
Loss of damping for speeds above ωs
) 1 ( < = −
eq rt tt
C K C ω
whirl
X Y
Ft= -(Cttω + Ktr) Δe
Rotor spin Ω
Fr= -(Crtω + Krr) Δe Δe
30
PHYSICS of WHIRL MOTION
Figure 23 Forces driving and retarding rotor whirl motion
Cross-coupled force is a FOLLOWER force
) 1 ( < = −
eq rt tt
C K C ω
whirl
X Y Cross-coupled force = Krt Δe
Damping force =
Rotor spin, Ω
31
eq
rt tt
2
PHYSICS of WHIRL MOTION
Figure 24 Follower force from cross-coupled stiffnesses
Work from bearing forces. E<0 is dissipative; E>0 adds energy to whirl motion
FX=-KXY ΔY
X Y
whirl
FY=-KYX ΔX
KXY > 0, KYX < 0 ΔX<0, ΔY>0
32
PHYSICS of WHIRL MOTION
Figure 24 Influence of bearing asymmetry on whirl orbits
Bearing asymmetry creates strong stiffness asymmetry – a remedy to reduce potential for hydrodynamic instability
Energy from cross-coupled forces = Area (Kxy-Kyx)
X Y X Y
33
EXPERIMENTAL EVIDENCE of INSTABILITY
Figures 25 & 26
Amplitudes of rotor motion versus shaft
evidence of rotordynamic instability
Waterfall of recorded rotor motion demonstrating subsynchronous whirl
34
EXPERIMENTAL EVIDENCE of INSTABILITY
WFR ~ 0.47 X Transition from
whip (sub sync
system natural frequency)
35
EXPERIMENTAL EVIDENCE of INSTABILITY
Automotive Turbocharger
FRB FRB FRB FRB
WFR ~ 0.50 X
500 1000 1500 2000 2500 0.2 0.4 0.6 0.8
Frequency [Hz] Amplitude [-]
Compressor End - Y 6
1X
12.5 krpm 65 krpm
500 1000 1500 2000 2500 0.2 0.4 0.6 0.8
Frequency [Hz] Amplitude [-]
Compressor End - Y 6
1X
12.5 krpm 65 krpm
TC supported on floating ring bearings
36
EXPERIMENTAL EVIDENCE of INSTABILITY
Automotive Turbocharger Multiple sub- synchronous motions
1000 2000 3000 4000 5000 6000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
TEST-Vertical displacement
Frequency [Hz] Amplitude [-]
Ymax 0.306 =
29.76 krpm 243.8 krpm 127.7 krpm
1X
TC supported on semi-floating ring bearings
37
EXPERIMENTAL EVIDENCE of INSTABILITY
Metal Mesh Gas Foil Bearing
200 400 600 800 1000 1200 1400 1600 1800 2000 5 10 15 20 25 30 35 40
Waterfall -Horizontal
Frequency [Hz] Amplitude
.
Frequency [Hz] Displacement [um] 1 X Whirl and bifurcation at high rotor speeds Rotor coasting down
69 krpm
38
Cutting axial grooves in the bearing to supply oil flow into the lubricated surfaces generates some of these geometries. Other bearing types have various patterns of variable clearance (preload and
thus generating a direct stiffness for operation even at the journal centered position. In tilting pad bearings, each pad is able to pivot, enabling its own equilibrium position. This feature results in a strongly converging film region for each loaded pad and the near absence of cross-coupled stiffness coefficients.
Commercial rotating machinery implements bearing configurations aiming to reduce and even eliminate the potential of hydrodynamic instability (sub synchronous whirl)
39
OTHER BEARING GEOMETRIES
Used primarily on high speed turbochargers for PV and CV engines
whirl frequencies from inner and outer films (50% shaft speed, 50% [shaft + ring] speeds)
make
Floating Ring
Round bearings are nearly always “crushed” to make elliptical or multi- lobe
Axial Groove
Bearing used only
machines
resistance
contained
loss
Partial Arc
Round bearings are nearly always “crushed” to make elliptical bearings
Plain Journal
Comments Disadvantages Advantages Bearing Type
Table 2 Fixed Pad Non-Pre Loaded Journal Bearings
40
OTHER BEARING GEOMETRIES
Currently used by some manufacturers as a standard bearing design
properly
speeds
whirl
performance
Three and Four Lobe
High horizontal stiffness and low vertical stiffness - may become popular - used
at moderate speeds
known
suppression of whirl at high speeds
Offset Half (With Horizontal Split)
Probably most widely used bearing at low or moderate rotor speeds
high speeds
known
critical speeds
Elliptical
Comments Disadvantages Advantages Bearing Type
Fixed Pad Pre-Loaded Journal Bearings Table 2
41
OTHER BEARING GEOMETRIES
Fixed Pad Pre-Loaded & Hydrostatic Bearings Table 2
Generally high stiffness properties used for high precision rotors
critical speeds
design
pressure lubricant supply
suppression of oil whirl
design parameters
Hydrostatic
Used as standard design by some manufacturers
requiring detailed analysis
whirl due to non bearing causes
relatively easy to place in existing bearings
suppression of whirl
cost
performance
Multi-Dam Axial Groove or Multiple- Lobe
Very popular in the petrochemical
convert elliptical
dam
little warning
to wear or build up
be known
suppression of whirl
critical speeds
Pressure Dam (Single Dam)
Comments Disadvantages Advantages Bearing Type
42
OTHER BEARING GEOMETRIES
Tilting Pad Bearings & Foil Bearings Table 3
Used mainly for low load support
machinery (APU units).
not well known for heavily loaded machinery.
subsynchronous whirl 1.Tolerance to misalignment. 2.Oil-free
Foil bearing
Widely used bearing to stabilize machines with subsynchronous non-bearing related excitations
speeds
actual clearances
known
whirl (no cross coupling)
Tilting Pad journal bearing Flexure pivot, tilting pad bearing Comments Disadvantages Advantages Bearing Type
Bump foils Top foil Spot weld Journal Bearing sleeve