This material is supported by the TAMU Turbomachinery Research Consortium. Parts of the investigation were conducted under NASA NRA on Subsonic Rotary.

This material is supported by the TAMU Turbomachinery Research Consortium. Parts of the investigation were conducted under NASA NRA on Subsonic Rotary Wing, SSRW2-1.3 Oil-Free Engine Technology (Foil Gas Bearing Modeling). Grant Cooperative Agreement NNX07P98A.

Luis San AndrésLuis San AndrésMast-Childs Professor

Fellow ASMETexas A&M University

Keun RyuKeun RyuSr. Development Engineer

BorgWarner Turbo Systems

ASME Turbo Expo 2011: Power for Land, Sea and Air June 6-10, 2011, Vancouver, BC

GT2011-45763

On the Nonlinear Dynamics of Rotor-Foil Bearing Systems:

Effects of Shaft Acceleration, Mass Imbalance and Bearing Mechanical Energy

Dissipation

Presentation available at http://rotorlab.tamu.edu

Series of corrugated foil structures (bumps) assembled within a bearing sleeve.

Integrate a hydrodynamic gas film in series with one or more structural layers.

Applications: ACMs, micro gas turbines, turbo expanders, blowers, etc

Reliable with adequate load capacity and high temperature capability

Tolerant to misalignment and debris Need coatings to reduce friction at start-up

& shutdown Damping from dry-friction and operation

with limit cycles

Gas Foil Bearings – Bump type

OIL-FREE Systems! reduce overall system weight, complexity, and maintenance cost increase system efficiency due to low power losses extend maintenance intervals.

Gas Foil Bearings Issues

Endurance: performance at start up & shut down

Little test data for rotordynamic force coefficients

Thermal management for high temperature applications (gas turbines, turbochargers)

Prone to subsynchronous whirl and limit cycle operation – Forced nonlinearity!

NOT rotordynamic instability (San Andrés, 2007)

AIAA2007-5094

San Andrés, L. and Kim, T. H., 2008, “Forced Nonlinear Response of Gas Foil

Bearing Supported Rotors,” Tribol. Int., 41(8)

TAMU research on foil bearingsyear Topic

2008-11 Metal Mesh Foil Bearings: construction, verification of lift off performance and load capacity, identification of structural stiffness and damping coefficients, identification of rotordynamic force coefficients

2008-10 Extend nonlinear rotordynamic analysisPerformance at high temperatures, temperature and rotordynamic measurements

2007-09 Thermoelastohydrodynamic model for prediction of GFB static and dynamic forced performance at high temperatures

2005-07 Integration of Finite Element structure model for prediction of GFB static and dynamic forced performance

Effect of feed pressure and preload (shims) on stability of FBS. Measurements of rotordynamic response.

2005-07 Rotordynamic measurements: instability vs. forced nonlinearity?

2005-06 Model for ultimate load capacity, Isothermal model for prediction of GFB static and dynamic forced performance

2004-09 Measurement of static load capacity, Identification of structural stiffness and damping coefficients. Ambient and high temperatures

Overview – Subsynchronous motions

Heshmat (2000): Operation of a flexible rotor-GFB system at super critical bending mode rotor speeds. Large amplitude subsynchronous motions suddenly appear while crossing system bending critical speed.

Heshmat (1994): GFB operates at max speed of 132 krpm, i.e. 4.61 ×106 DN, showing stable limit cycle operation with large amplitude subsynchronous motions at frequency = rigid body mode natural frequency .

Lee, et al. (2004, 2003): GFBs with viscoelastic layer eliminate large subsynchronous whirl motions appearing in flexible rotor-GFB system (2004) and a two stage centrifugal compressor (2003).

San Andrés et al. (2006): Small imbalances lead to mainly synchronous rotor motions. Large mass imbalances cause sub harmonic motions at rotor speeds > 2 x system natural frequency (whirl frequency ratio ~ 50%) => nonlinear forced rotor responses

San Andrés et al. (2007): Introduce simple GFB model as a nonlinear structure. rotor-GFB performs as a Duffing oscillator with multiple frequency response. Agreement between predictions and test data. 1/2 and 1/3 WFRs due to nonlinearity. (First paper predicting NL forced response of rotor-GFB systems with validations to reliable test data)

Example 1 – Subsynchronous motions

Heshmat (1994)- Maximum speed 132 krpm, i.e. 4.61 ×106 DN.- Stable limit cycle operation with large amplitude sub harmonic motions at whirl frequency = rigid body mode natural frequency .

Subsynchronous amplitude at 350 Hz

Synchronous, 2,200 Hz (132 krpm)

Supersynchronous amplitude at 3,300 Hz (bending mode)

Subsynchronous amplitude recorded during rotor speed

coastdown from 132 krpm (2,200 Hz)

Whirl amplitude remains ~ constant as subsynchronous frequency drops from 350 Hz to 180 Hz

Heshmat (1994)- Maximum speed 132 krpm, i.e. 4.61 ×106 DN.- Stable limit cycle operation but with large amplitude subsynchronous motions. Whirl frequency tracks rotor speed


Heshmat (2000) Flexible rotor- GFB system operation to 85 krpm (1.4 kHz): 1st bending critical speed:34 krpm (560 Hz)

Waterfall plot recorded during rotor speed coastdown test from 45 krpm (750 Hz)

Rotor orbit shape at 45k rpm

Large amplitude limit cycle motions above bending critical speed, whirl frequency = natural frequency (rigid body)


Lee, et al. (2003, 2004)Flexible rotor supported on GFBs with viscoelastic layer

Viscoelastic layer eliminates large motions at natural frequency & appearing above 1st bending critical speed.

50 kRPM (833 Hz)

Bump type GFBViscoelastic GFB

Synchronous vibration

1st bending mode

Rigid body mode

Bum

p ty

pe G

FB

Vis

coel

astic

GF

BSynchronous

vibration


0 200 400 600 800 1000 1200 1400 1600 1800 20000

5

10

15

20

25

30

35

40Waterfall -Horizontal

Frequency [Hz]

Am

plitu

de

.

.Frequency [Hz]

Dis

pla

cem

ent

[um

]

1 X

Whirl and bifurcation at high rotor speeds

Ro

tor

coas

tin

g d

ow

n

Max. Rotor speed = 69 krpm

San Andrés, L., et al., 2011, “Identification of Rotordynamic Force Coefficients of a Metal Mesh Foil Bearing Using Impact Load Excitations,” ASME J. Eng. Gas Turbines Power, Vol. 133


Metal Mesh Foil Bearing

RudDloff, L., Arghir, M., et al., 2011, “Experimental Analysis of a First generation foil Bearing. Start-Up Torque and Dynamic

Coefficients,” ASME GT2010-22966


Unloaded FB: “Self-Excited” whirl motions at speed 30 krpm (500 Hz) with whirl frequency=165 Hz (WFR=0.33)

Kim, D., Shetty P., Lee. D., 2011, “Imbalance Response of a Rotor Supported

by Hybrid Air Foil Bearings,” ASME GT2011-45576

Example 6a – Subsynchronous motions

Loaded hybrid FB (vertical): 2.67 bar gauge supply pressure Sub sync whirl motions start at 20 krpm with (nat) freq 5900 rpm (WFR=0.30). Too large amplitudes at 30 krpm, test stopped

Kim, D., Shetty P., Lee. D., 2011, “Imbalance Response of a Rotor Supported

by Hybrid Air Foil Bearings,” ASME GT2011-45576

Example 6b – Subsynchronous motions

Loaded hybrid FB (vertical): 4 bar gauge supply pressure Large amplitude whirl motions start at 34 krpm (567 Hz) with whirl frequency~natural frequency 7200 rpm (WFR=0.21)

Amplitudes of subsynchronous

motions INCREASE as

imbalance increases (forced

nonlinearity!)

last two indices are multipliers for X & Y axis offset

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

Frequency [Hz]

Am

plit

ud

e [

mic

ron

s]

last two indices are multipliers for X & Y axis offset

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

Frequency [Hz]

Am

plit

ud

e [

mic

ron

s]D

ispl

acem

ent

Am

plitu

de

(μm

) D

ispl

acem

ent

Am

plitu

de

(μm

)

Frequency (Hz)

25.7 krpm

2.6 krpm u = 7.4 μm

1X 0.5X

2X

1X 0.5X

2X

8.5 krpm

25.7 krpm

2.6 krpm

12.5 krpm

20.5 krpm

u = 10.5 μm

Frequency (Hz)

Rotor speed +

Imbalance +

San Andrés, L., Rubio, D., and Kim, T.H, 2007, “Rotordynamic Performance of a Rotor

Supported on Bump Type Foil Gas Bearings: Experiments and Predictions,” ASME J. Eng.

Gas Turbines Power, 129


Gen II foil bearings

2 krpm

25 krpm

50 krpm-400 -200 0 200 400 600 800 1000

Frequency [Hz] 1X

100

80

60

40

20

0

Am

pli

tud

e [µ

m]

Large amplitudes locked at natural frequency (50

krpm to 27 krpm) ……

but stable limit cycle!

Kim, T.H., and San Andrés, L., 2009, “Effects of a Mechanical Preload on the Dynamic Force Response of

Gas Foil Bearings - Measurements and Model Predictions,” STLE Tribol. Trans., 52

Rotor speeddecrea

ses



Analysis vs. test data

0 100 200 300 400 500 6000

20

40

60

80

PredictionExperiment

Frequency [Hz]

Am

plitu

de [

um]

0 5 10 15 20 25 300

40

80

120

160

200


Rotor speed [krpm]

Fre

quen

cy [

Hz]

(a) Subsynchronous Amplitude vs frequency (b) Subsynchronous frequency vs speed

Am

plitu

de [

μm]

Subsynchronous whirl frequencies concentrate in a narrow band enclosing natural frequency (132 Hz) of test system

0 100 200 300 400 500 6000

20

40

60

80


Frequency [Hz]

Am

plitu

de [

um]

0 5 10 15 20 25 300

40

80

120

160

200


Rotor speed [krpm]F

requ

ency

[H

z]

(a) Subsynchronous Amplitude vs frequency (b) Subsynchronous frequency vs speed

Amplitude vs. frequency Frequency vs. rotor speed

Test data

Predictions

Test data

Predictions

Rotor speed (krpm)Frequency (Hz)

San Andrés, L. and Kim, T. H., 2008, “Forced Nonlinear Response of Gas Foil Bearing Supported Rotors,” Tribol. Int., 41(8)

AIR SUPPLY

Cooling flow/feed pressure on FB motions

Ps

Rotating journal

Pa

Bump spring

Top foil Bearing housing

Circumferential velocity

ΩRJ Axial

velocity Ω RJ

Outer gap

Inner gas film

X

Y

z x

Typically foil bearings DO not require pressurization.

Cooling flow is for thermal management: to remove

heat from drag or to reduce thermal gradients in hot/cold

engine sections

Side effect: Axial flow retards evolution of circumferential flow velocity

San Andres et al, ASME JGT, 209, v31

Effect of side flow on rotordynamics

(a) 0.35 bar

(b) 1.4 bar

(c) 2.8 bar

Whirl frequency locks at RBS

natural frequency ( not

affected by level of feed pressure

For Ps ≥ 2.8 bar rotor subsync. whirl motions

disappear;(stable rotor

response)

ωsub= 132 Hz

ωsub= 147 Hz

ωsub= 127 Hz

Subsynchronous ωsyn= 508 Hz

Synchronous

FFT of shaft motions at 30 krpm


Onset of subsynchronous whirl motions

(a) 0.35 bar

(b) 1.4 bar

(c) 2.8 bar

SynchronousSubsynchronous

NOS: 25 krpm

NOS: 30.5 krpm

NOS: 27 krpm

Delay of large

amplitude subsynchro

nous rotor motions

with increase in

axial cooling flow

(feed pressure)

Effect of side flow on rotordynamics


Objectives

To extend earlier analysis to predict the forced response of a rigid rotor supported on FBs modeled as nonlinear structure with material damping.To determine the effects of rotor acceleration, imbalance mass, and the FB structural loss factor on the dynamic forcedresponse of simple RBS.

Most GFB analyses are complex; coupling top foil & under spring models with gas film flow model.

butGFB forced performance depends mainly on the material properties of the support elastic structure

Dynamic Stiffness & Damping Mechanism for Foil Bearing

Dynamic Stiffness & Damping Mechanism for Foil Bearing

GT2011-45763

Fast accelerations are typical in MTM due to small rotor mass moment of inertia. This work provides design and operation considerations for the appropriate selection and use of GFBs to avoid the build up of excessive nonlinear RBS response.

FB load–deflection structural test

Nonlinear bearing forced deflection. Hysteresis loop shows energy dissipation

Loading

Loading

Unloading

Unloading

Stiffness hardening is

likely to induce internal

resonances at rotor speeds

greater than the RBS natural

frequency

Kim and San Andrés (2007): Eight cyclic load - unload

structural tests on Gen II foil bearing

Load–Deflection Structural tests

Nonlinear bearing forced deflection: test data, polynomial fit & model prediction

F = r (0.0675 -0.002 r + 0.0001 r2 )

Test data

Prediction

2 31 2 3sFBF K r K r K r

Kim and San Andrés (2007): Eight cyclic load - unload

structural tests on Gen II foil bearing

r

FFB

FB load–deflection structural test

FB Structural Loss Factor

0 20 40 60 80 100

0

0.05

0.1

0.15

0.2

FB deflection [um]

Str

uctu

ral l

oss

fact

or [

-]

*

FB deflection [μm]

Loss factor (γ) represents structural damping and is obtained from load-deflection hysteresis loop

TYP, loss factor is large at small displacements BUT decreases for large displacements. Typical of structural system with dry-friction

2

1FBs

S

F drK r

where

sBS

FK r

local stiffness coefficient

r

21 2 32 2 31

2

E E

n

K K x K xf

M

Natural frequency for small amplitude motions about SEP:

= 130 Hz (7.8 krpm)

Rotor-GFB systemRotor mass, M =1.02 kg

4 9 141 2 32 3

6.75 10 , 2 10 , 10N N N

K K Km m m



FB structure(static):

Static equilibrium

21 2 3 ; 0

2E E E E

M gx K K x K x y

XE=37.5 m

X

Y

Replicates laboratory set-up

Shaft length (L) = 209.5 mmShaft diameter (D) = 38.1 mm

( 25 bumps) LB=38.1 mm

Hollow rotor

L

D

Equations of MotionEOMs: rigid rotor & in-phase imbalance

2 2; ; with s s

x x

FB FBFB FB

F FF x x F y y r x y

r r

Assumption: minute gas film with infinite film stiffness


2

2

2 cos( ) sin( )

2 sin( ) cos( )

x

y

FB

FB

M x F M u Mg

M y F M u

FB dynamic reaction force

Varying rotor speed (t)

Rotor speed21

20 0

;t t

i idt t dt t t 21

20 0

;t t

i idt t dt t t Rotor angle

Use Runge-Kutta scheme Rotor speeds: 2 - 36 krpm (600 Hz)

Numerical solution

• Rotor speed ramp rate: α=±35 Hz/s, ±71 Hz/s, and ±283 Hz/s (2 krpm ↔ 36 krpm)

• Sampling rate: 12 k/s, 24 k/s, and 96 k/s • Time step: 0.0833 ms, 0.04167 ms, and 0.01042 ms

• # integration points = 192,000

Simple MATHLAB or MATHCAD code: numerical integration and post-processing of results in frequency domain

The fast rotor acceleration requires of a smaller time step (faster acquisition rate) since the speed changes quickly.

Typical rotor response

0 5 10 15 20 25 30 35 40

100

0

100

Xspeed (kRPM)

Res

pon

se (

um

)

Vertical response (x)

0 10 20 30 40 Rotor speed [krpm]

Am

pli

tud

e [µ

m]

100

-100

0

1 × fn 2 × fn 3 × fn

0

30

60

90

120

150

180

210

0 200 400 600 800

Frequency [Hz]

Am

plit

ud

e [µ

m] 1X

2 krpm

35 krpm

1 × fn

2 × fn

3 × fn

Vertical response (x )

0 5 10 15 20 25 30 35 40

100

0

100

Xspeed (kRPM)

Resp

on

se (

um

)

0 10 20 30 40 Rotor speed [krpm]

100

-100

0

Am

pli

tud

e [µ

m] 1 × fn 2 × fn 3 × fn


0

30

60

90

120

150

180

210

0 200 400 600 800

Frequency [Hz]

Am

pli

tud

e [

µm

]

1X35 krpm

2 krpm

Vertical response (x )

3 × fn

2 × fn

1 × fn

Rotor acceleration +283 Hz/s

Rotor deceleration -283 Hz/s

u=8 µm; FB γ=0.14

Solutions obtained in a few seconds. Post-processing filters responses and finds synchronous and subsynchronous components of motion

Effect of rotor acceleration/deceleration on RBS forced response

29

Effect of rotor acceleration (+)

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

sp

eed

[k

rpm

]

2

13

2

4

35

1X

1 × fn

2 × fn

3 × fn


½WFR½ WFR3½ WFR3

0

20

40

60

80

0 10 20 30 40

Rotor speed [krpm]

Am

plit

ud

e [µ

m]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

2

1

3

24

3

5

WFM_YWFM_X


1X½WFR½ WFR3½ WFR3

1 × fn

2 × fn

3 × fn

0

20

40

60

80

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

1 × fn

3 × fn

2 × f n 3 × f n1 × f n

α= +283 Hz/s (FAST) α= +35 Hz/s (SLOW)

u=8 µm. FB γ=0.14

WFR=ω/Ω

Subsynchronous motion amplitudes more severe for the slowest rotor acceleration: More elapsed time for the whirl motions to build up!

Ramp rate

30

Effect of rotor deceleration u=8 µm. FB γ=0.14

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

36

2

4

1

3

2

1X

3 × fn

2 × fn

1 × fn


½WFR½ WFR3½ WFR3

α= -283 Hz/s (FAST DECEL) α= -35 Hz/s (SLOW DECEL)

0

20

40

60

80

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

0

20

40

60

80

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

36

2

4

1

3

2

1X

3 × fn

2 × fn

1 × fn


½WFR½ WFR3½ WFR3

Notable differences in the onset speed and persistence of whirl motions show the RBS has a marked mechanical hysteresis

Ramp rate

31

Whirl frequencies: +α

α= +35 Hz/s (SLOW acceleration)

u=8 µm. FB γ=0.14

WFR=ω/Ω

Subsynchronous whirl motions from 11 to 20 krpm with WFR=½ at first, and later from 20 to 36 krpm jump to WFR=⅓. Above 28 krpm, more complex WFRs ranging from 0.31 to 0.37, slightly above and below ⅓.

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Rotor speed [krpm]

WF

R

2 × f n 3 × f n1 × f n

Once a subsynchronous frequency motion appears, its

amplitude rapidly increases with rotor speed.

Significant motion amplitudes

with WFR=½ and WFR=⅓ appear at ~twice and ~three times the

system natural frequency.

0

20

40

60

80

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

1 × fn

3 × fn

2 × f n 3 × f n1 × f n

32

Whirl frequencies: -α

α= -35 Hz/s (SLOW deceleration)

u=8 µm. FB γ=0.14

WFR=ω/Ω

For rotor speeds > ~20 krpm (333 Hz), motions with WFRs ranging from 0.27 to 0.41, i.e., a chaotic regime, are apparent.

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Rotor speed [krpm]

WF

R

2 × f n 3 × f n1 × f n

0

20

40

60

80

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

The motions with a 50% WFR are not as severe in

amplitude as when the rotor accelerates,

occurring over a shorter rotor speed span.

33

Effect of rotor acceleration u=8 µm. FB γ=0.14

0

10

20

30

40

50

60

0 10 20 30 40

Rotor speed [krpm]

Am

plitu

de [

µm

]

Acceleration

Deceleration

Linear response

Acceleration

Linear response

Deceleration

Natural frequency (f n )

The peak amplitude during rotor deceleration is ~10 µm smaller than the one predicted during rotor acceleration

xL =u/γ = 8 µm/0.14 = 57.1 µm

Synchronous response ± 71 Hz/s

2

22 2

( ')

1 ( ') ( ')L

rx u

r r

'n

fr

f

Linearized model

=7800 rpm

Effect of mass imbalance on RBS forced response

35

Effect of rotor mass imbalance α= +283 Hz/s, FB γ=0.14

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

sp

ee

d [

krp

m]

2

13

2

4

35

1X

1 × fn

2 × fn

3 × fn


½WFR½ WFR3½ WFR3

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

2

1

3

24

3

5

1X

1 × fn

2 × fn

3 × fn


½WFR½ WFR3½ WFR3

0

20

40

60

80

100

120

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

0

20

40

60

80

100

120

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

u=4 μm u=20 μm

Imbalance

Mass imbalance exacerbates the bearings’ nonlinearity and showcases a distinctive jump phenomenon

36

Effect of rotor mass imbalance u=20 µm. FB γ=0.14

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

2

1

3

24

3

5

1X

1 × fn

2 × fn

3 × fn


½WFR½ WFR3½ WFR3

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

AccelerationDecelerationLinear response

Linear response Acceleration

Deceleration


Jump

WFM_YWFM_X

3 × fn

2 × fn

1 × fn

1X


Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

36

2

4

1

3

2

½WFR½ WFR3½ WFR3

α= -283 Hz/s

α= +283 Hz/s

Synchronous amplitude shows stiffness hardening effect with jump phenomenon while rotor accelerates

Synchronous response

37

Whirl frequencies: Imbalance ↑

u= 20 µm α= +283 Hz/s, FB γ=0.14

WFR=ω/Ω

As imbalance mass increases, and for rotor speeds above (3×fn), the rotor motion has a broader whirl motions at ⅓ WFR.

½ frequency whirl disappears for u>20 µm!

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Rotor speed [krpm]

WF

R

2 × f n 3 × f n1 × f n

0

20

40

60

80

100

120

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

Effect of FB loss factor (material damping) on RBS forced response

39

Effect of FB loss factor

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

spee

d [

krp

m]

2

1

3

24

35 1X½WRF½ WFR3½ WFR3

1 × fn

2 × fn

3 × fn

Vertical response (x)WFM_Y

WFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

sp

ee

d [

krp

m]

2

13

2

4

35

1X

1 × fn

2 × fn

3 × fn

½WRF½ WFR3½ WFR3


0

20

40

60

80

100

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR ~1/3WFR

2 × f n 3 × f n1 × f n

0

20

40

60

80

100

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR

2 × f n 3 × f n1 × f n

γ=0.07 γ=0.28

α= +283 Hz/s. FB u=8 µm

Bearing loss factor affects the onset and persistence of rotor sub harmonic motions

FB loss factor

Whirl frequency: FB loss factor ↑

WFR=ω/Ω

½WFR motions are apparent from 8 ~ 15 krpm (1×fn ~ 2× fn)

For γ>0.2, ⅓WFR frequency

components disappear!

γ=0.28 α= +283 Hz/s. u=8 µm

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Rotor speed [krpm]

WF

R

2 × f n 3 × f n1 × f n

0

20

40

60

80

100

0 10 20 30 40

Rotor speed [krpm]

Am

pli

tud

e [

µm

]

1X ~1/2WFR

2 × f n 3 × f n1 × f n

41

Effect of FB loss factor

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35 40

Rotor speed [krpm]

Am

plit

ud

e [µ

m]

AccelerationDecelerationLinear response

Linear response

Acceleration

Deceleration


Jump

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

sp

eed

[k

rpm

]3

6

2

4

13

2

3 × fn

2 × fn

1 × fn

1X


½WFR½ WFR3½ WFR3

WFM_YWFM_X

Frequency [Hz]

0 200 400 600 800

Ro

tor

sp

ee

d [

krp

m]

2

13

2

4

35

1X½WRF½ WFR3½ WFR3

1 × fn

2 × fn

3 × fn


u=8 µm. γ=0.07

α= -283 Hz/s

α= +283 Hz/s

As rotor accelerates, the FB hardening stiffness with little damping ( low) affects more the response (multi-frequency). On deceleration, the rotor synchronous response appears free of nonlinearities

Synchronous response

42

Other NL response: small u & γ

u=1 µm. γ=0.015

For speeds above RBS natural frequency (fn =130 Hz), whirl motions of

large amplitudes with whirl frequency locked at the natural frequency!

α=+283 Hz/s

0 200 400 600 8000

8

16

24

32

40

Frequency [Hz]

Am

plitu

de [

um]

Am

plit

ud

e [

μm

]

Frequency [Hz]2 krpm

35 krpm1X

System natural frequency (fn =130 Hz)

Bearing with little damping! (poor mechanical energy

dissipation)

2 krpm

25 krpm

50 krpm-400 -200 0 200 400 600 800 1000

Frequency [Hz] 1X

100

80

60

40

20

0

Am

pli

tud

e [µ

m]

Example TEST DATA(not same FBs)

ConclusionsFor operation above the RBS system critical speed and as the rotor accelerates, large amplitude whirl motions appear with a main subsynchronous frequency tracking rotor speed, first at 50% speed and later bifurcating into at 33% whirl frequency.

Large rotor imbalances produce both jump phenomenon and a stronger hysteresis during slow acceleration and deceleration cases.

Slow rotor accelerations result in responses with more abundant subsynchronous whirl patterns, increased amplitudes of whirl, and accompanied by a pronounced mechanical hysteresis when the rotor decelerates.

Material damping (dry friction) in the FB aids to reduce and delay the nonlinear response, eventually eliminating the multiple frequency behavior.

GT2011-45763

Recommendation GT2011-45763

Fast rotor start up and coast down procedures Reductions in rotor inertia or larger drive torque

To ameliorate subsynchronous rotor motions resulting from nonlinear effect of hardening support structure

Minimize rotor imbalance (a forcing function) Well-balanced rotor is a must!

Large FB material damping Improvements in GFB design and materials

AcknowledgmentsThanks support of• TAMU TRC (2004-2008)• NASA GRC (2007-09) & Dr. Samuel Howard • NSF (2003-06),• Foster-Miller (FBs)

Learn more http://rotorlab.tamu.edu

Questions (?)

© 2011 Luis San Andres

This material is supported by the TAMU Turbomachinery Research Consortium. Parts of the investigation were conducted under NASA NRA on Subsonic Rotary.

Documents

nonlinear response of

foil bearingsyeartopic2008

prediction of gfb static

rotordynamic measurements2007

forced nonlinearity

metal mesh foil bearings

flexible rotorgfb system

high temperatures2005