rotordynamics tutorial

Mechanical and Fluids Engineering

Rotordynamics Tutorial: Theory, Rotordynamics Tutorial: Theory, Practical Applications and Practical Applications and

Case StudiesCase Studies

Gas TurbineGas TurbineTechnology CenterTechnology Center

Southwest Research InstituteSouthwest Research Institute

Dr. J. Jeffrey MooreDr. J. Jeffrey Moore

Southwest Research InstituteSouthwest Research Institute

Goal for this Tutorial

To Familiarize The Attendee with The Basic Concepts Of Rotordynamics, API Requirements, Analysis and

Design Techniques, and Vibration Behavior

Overview

Rotordynamics Theory

Rotordynamic Analysis of Turbomachinery

API Requirements

Transducers and Instrumentation

Types of Vibration Data

Example Vibration Phenomena

Rotordynamic Theory

Rotordynamics is the study of the dynamics of rotating equipment

Types of Dynamics:Lateral

Torsional

Structural/Foundation

Rotordynamic Theory

Single Degree of Freedom Theory

)cos()()( 2 φωωω += tAMetX

)(tFXKXCXM =++ &&&

M

K C

X(t)F(t)

( ) 22222 )(

1)(ωωω

ωCM

An −−

=

Jeffcott Rotor

tMetF ωω cos)( 2=

⎥⎦

⎤⎢⎣

⎡−

= −

)(tan)( 22

1

ωωωωφ

nmC

MK

n =ω Natural Frequency

Rotordynamic Theory

Bode Plot - Amplitude

Light Damping

More Damping

ω=ωn

Imbalance

Rotordynamic Theory

Bode Plot - Phase

0

90

180

Light Damping

More Damping

ω=ωn

Rotordynamic Theory

Solving Resonance ProblemsMove natural frequency away from excitation frequency

Increasing or decreasing stiffness

Increasing or decreasing mass

Reduce the excitation magnitude

Balancing

Add damping to the system

Improved bearing design

Squeeze film dampers

Change the excitation frequency

Change rotation speed

Rotordynamic Theory

Gyroscopic EffectsImportant with overhung disks

Eg. Single-stage overhung compressor

Gyroscopic forces:Cxθ = Ip ω

Creates radial damping force due to rotation velocityForward critical speeds increase with speed (gyroscopic stiffening effect)Backward critical speeds decrease with speedCauses rotors to whirl rather than translate

Simple Overhung Disk Rotor

Shaft112

105Shaft1

1

0 3

-0.2

-0.1

0

0.1

0.2

0.3

Bearings

Rotordynamic Damped Natural Frequency Map

0

1

2

3

4

5

0. 2000. 4000. 6000. 8000. 10000. 12000.

Rotor Speed, rpm

Nat

ural

Fre

quen

cy, H

z

ForwardBackward

Overhung Disk Example

Rotordynamic Theory

Modeling TurbomachineryContinuous system modeled by a system of springs and masses formulated using either finite element or transfer matrix methods

Results in following system of equations:

[ ] [ ] [ ] )(tFXKXCXM =++ &&&

Similar form as the single degree of freedomUse Matrix solution techniques to solve for natural

frequencies, unbalance response, and stability

Rotordynamic Theory

Stability Analysis

Unstable Stable

A Rotor System Is Unstable When The Destabilizing Forces Exceed Stabilizing (Damping) Forces

Rotordynamic Theory

Stability AnalysisDamping is a Stabilizing Influence

Destabilizing Forces Arise from Cross-Coupling Effects that Generate Forces in the Direction of Whirl

Cross-Coupled Stiffness Yields a force in the Y-direction for a displacement in the X

Sources include: fixed arc bearings, floating ring oil seals, labyrinth seals, impeller/turbine stages

Fy=Kyx X

Fx=-Kxy Y

Y

X

Rotordynamic Theory

Stability Calculated by Solving the Eigenvalue Problem:

Eigenvalues of the form: s = - ζ ωn + i ωd

Imaginary part gives the damped natural frequency

Real part gives the damping ratio (ζ), or stability

Logarithmic decrement (log dec) is related by:

Instability characterized by subsynchronous vibration near the first whirling frequency that rapidly grows to a large amplitude bounded only by rotor/stator rubbing

Can be brought on by small changes in load, pressure, or speed.

[ ] [ ] [ ] { }0=++ XKXCXM &&&

212

ζπζδ−

=

Rotordynamic Theory

0 X

X Ln

1-nn =⎥

⎦

⎤⎢⎣

⎡=δ

0<δ

0>δ

Linear Vibration

XN-1 XN

Rotor Vibration

Undesirable

Desirable

Evaluation Using Log Dec(rement)

Neutrally Stable

Unstable

Stable

Rotordynamic Modeling

Rotordynamic ModelingBreak the series of smaller segments at diameter steps

Components like impellers, couplings, thrust disks do not add shaft stiffness are modeled as added mass

Stations added at bearings centerlines

Second Section Gas Balance Seal

Division Wall Seal

2nd Section

1st Section

Gas Flow Path

Typical High Pressure Centrifugal CompressorSample 10-Stage Compressor Model

Shaft179

7570

656055504540

3530252015105

haft11

Reference: Moore, J.J., Soulas, T.S., 2003, “Damper Seal Comparison in a High-Pressure Re-Injection Centrifugal Compressor During Full-Load, Full-Pressure Factory Testing Using Direct Rotordynamic Stability Measurement,” Proceedings of the DETC ’03 ASME 2003 Design Engineering Technical Conference, Chicago, IL, Sept. 2-6, 2003


Shaft179

7570

656055504540

3530252015105

Shaft11

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.4 0.8 1.2 1.6 2 2.4

Axial Location, meters

Shaf

t Rad

ius,

met

ers

10-Stage Centrifugal Compressor

SWRI Model - Nom Brngs

Bearings

Impellers BalanceDrum

ThrustDisk

Rotordynamics Shaft FE ModelRed = StructuralGreen = Added MassCoupling

DGS


Journal Bearing Cross-Coupling

Oil wedge causes a horizontal movement from a vertical load (cross-coupling)

Non-symmetric Pressure Profile


[ ]⎭⎬⎫

⎩⎨⎧

∂∂

+∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

thhU

xzph

zxph

x2633 μ

Journal Bearing Modeling

Solution to the Reynolds’ equation provides the pressure profile on the pad

•Assuming small perturbation results in 1st order equations that yield rotordynamic coefficients (Kxx, Kxy, etc.)


4-Axial Groove Bearing

Load

Bearing

Clearance (C)

Plain Cylindrical Bearing

Clearance (C)

Bearing

Groove

Groove

LoadJournal Radius R

Bearing

C = Clearancem = Preload

Elliptical Bearing

15°15°

Load

Tilting ShoeClearance C

Pivot

Tilting Pad BearingLoad Between Pads

Bearing Housing

Common Bearing Types

MostStableBearing


Plain journal bearings are the least stable

Elliptic and Axial Groove bearings introduce “preload” that improves the stability

Tilt-Pad bearings possess essentially no cross-coupling since the pads can pivot

Most commonly used bearing in high speed turbomachinery

More expensive than fixed pad designs

Necessary when operating at speeds well above (> 3X) first critical speed

Many parameters can be adjusted to achieve desired stiffness and damping properties

• Preload, L/D, Clearance, Offset, Pad orientation

Journal Bearing Modeling


Undamped Critical Speed MapFirst six natural frequencies calculated for varying bearing support stiffness

Undamped Critical Speed Map

1000

10000

100000

1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 1.0E+12

Bearing Stiffness, N/m

Crit

ical

Spe

ed, c

pm

10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

1st Critical Speed

2nd Critical Speed

Intersection between bearing stiffness curve and mode curve is the undamped critical speed

MCOS

Undamped C.S. Mode Shape Plot10-Stage Centrifugal Compressor


f=3837.1 cpmK=200000000 N/m

forwardbackward


1000

10000

100000

1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 1.0E+12


Crit

ica

l Sp

ee

d, c

pm


1st Critical Speed

2nd Critical Speed


1st Critical Speed Mode ShapeIntersection between bearing stiffness curve and critical speed curve represents critical speed

Cylindrical mode with flexibility

Undamped C.S. Mode Shape Plot10-Stage Centrifugal Compressor


f=12631.6 cpmK=300000000 N/m

forwardbackward


1000

10000

100000

1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 1.0E+12


Crit

ica

l Sp

ee

d, c

pm


1st Critical Speed

2nd Critical Speed


2nd Critical Speed Mode ShapeConical Mode with Flexibility

API RequirementsCritical speeds separated from operating speed range

Separation margin function of amplification factor

=Unbalance Amount:

Unbalance Configuration

1st Mode 2nd Mode


NWUB 4

=

⎟⎠⎞

⎜⎝⎛

−−+=

5.11117102 AF

SM

Reference: API 617, 7th Edition, Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services, American Petroleum Institute, July, 2002.


Unbalance Response ExampleFirst critical speed excited by mid-span unbalanceSecond critical speed excited by quarter-span unbalance

Damping increased 2nd critical speed from 12600 to 15000 rpm

Separation margins meet API requirements for 1st critical speedNo separation margin required for 2nd critical speed since AF < 2.5

Rotordynamic Response Plot

0

5

10

15

20

25

30

35

40

45

50

0 5000 10000 15000 20000

Rotor Speed, rpm

Resp

onse

, mic

rons

pk-

pk OperatingSpeed

NC2=15000 rpmAF2=2.05

Rotordynamic Response Plot

0

10

20

30

40

50

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000Rotor Speed, rpm

Res

pons

e, m

icro

ns p

k-pk Operating

Speed

NC1=4060 rpmAF1=5.84

1st Critical Speed

2nd Critical Speed

Close Clearance Components

Honeycomb Seal

Oil Seal

Labyrinth Seal

Journal Bearing

Impeller


Ceff - Y-Direction

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

10000

0 100 200 300 400

Frequency (Hz)

Re

(H) (

N/m

)

Honeycomb Seal Damping Test Data vs. Predictions

Reference: Camatti, M., Vannini, G., Fulton, J.W., Hopenwasser, F., 2003, “Instability of a High Pressure Compressor Equipped with Honeycomb Seals,” Proc. of the Thirty-Second Turbomachinery Symposium, Turbomachinery Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station, Texas.

• Damper seals like honeycomb seals provide substantial damping• Damping increases with increasing pressure differential


Aero Cross-Coupling

Arises from Impellers of Centrifugal Compressors

Most Common Method version of Wachel Equation

CFD Methods Have Been Developed Show good correlation to experimental data for pump impellers

( )( ) ( ) ,

1

63,000* 10 * *

S iNi j D

XY ij i i S j

HorsepowerMole WeightKRPM D h

ρρ=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑


Stability AnalysisFirst Forward Whirling Mode at Maximum Continuous Speed

Log Decrement = 0.149 (no seal effects or cross-coupling)

No aero cross-coupling or seal effects included

Damped Eigenvalue Mode Shape Plot10-Stage Centrifugal Compressor


f=4016.3 cpmd=.1494 logdN=12000 rpm

forwardbackward


Stability map shows sensitivity to destabilizing cross-coupling at rotor mid-span

Rotor would be unstable without seal effects

Damper seal greatly improves stability

Stability Map

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.E+00 2.E+07 4.E+07 6.E+07 8.E+07 1.E+08 1.E+08

Mid-span Kxy (N/m)

Log

Dec

With SealsNo SealsAPI Kxy

0

1

2

3

0 500 1000 1500 2000 2500 3000 3500

Discharge Pressure (psia)

Log

Dec

Smooth Seal - Test

Smooth Seal - Prediction

Hole Pattern - Test

Hole Pattern - Prediction

Rotordynamic ModelingMeasured Log Decrement in Centrifugal Compressor

Reference: Moore, J.J., Soulas, T.S., 2003, “Damper Seal Comparison in a High-Pressure Re-Injection Centrifugal Compressor During Full-Load, Full-Pressure Factory Testing Using Direct Rotordynamic Stability Measurement,” Proceedings of the DETC ’03 ASME 2003 Design Engineering Technical Conference, Chicago, IL, Sept. 2-6, 2003

• Shows damper seal effectiveness• Log Dec increases as discharge pressure increases• A smooth seal was tested to simulate a “plugged-up” seal

0 500 1000 1500 2000 2500 3000 3500

Discharge Pressure (psia)

Divis

ion W

all S

eal L

eaka

ge

Smooth Seal - Test

Smooth Seal - Prediction

Hole Pattern - Test

Hole Pattern - Prediction

Incre

asing


Foundation Support EffectsIndustrial Gas Turbine Casing/Rotor Model

Finite element casing model coupled to rotor model

Casing and foundation flexibility had a great effect on location of critical speeds

Lowers critical speeds

Increases amplification factor

According to API 617, if the foundation flexibility is less than 3.5 times the bearing stiffness, then a foundation model should be included.

Review of Transducers

Transducer TypesProximity Probe

Measures Relative Shaft Displacement (static and dynamic)

Most Common

Most Applicable to Fluid Film Bearings

Subject to Electromechanical Runout (false vibration)

Velocity TransducerMeasures Absolute Casing Motion

Types: magnetic coil or integrating accelerometer

Indicates dynamic force transmitted to casing • Function of flexibility of casing

Vibration severity independent of frequency

Not usually used on compressors due to low motion of massive casing

Review of Transducers

Transducer Types Cont.Accelerometers

Typically used in higher frequency measurement

Not usually used on compressors due to low motion of massive casing

Severity a function of frequency

Typically used with rolling element bearing (eg. Aeroderivative gas turbines) and on gearboxes

Types of Vibration Instrumentation

Overall Level / Vibration MonitorProvides machinery protection

Overall vibration level only

No detailed information

Waveform/Orbit – OscilloscopeGood for viewing vibration data in real time

Orbit shape shows symmetry in system

• Round=symmetric

Shows transient data (impacts, bursts, etc.)


Fast Fourier Transform (FFT’s) – Spectrum AnalyzerBreaks down complex waveform into frequency components

Characterize vibration:

• Subsynchronous - < running speed

• Synchronous = running speed

• Supersynchronous > running speed

Can display multiple spectra in time to make waterfall plot

• Shows how vibration changes in time or during transient events


Waterfall Plot Courtesy of: Memmott, E.A., 1992, “Stability of Centrifugal Compressors by Application of Tilt Pad Seals, Damper Bearings, and Shunt Holes,” Proceedings of the Institute of Mechanical Engineers, IMechE 1992-6, 7-10 September, 1992.


Tracking FilterProvides amplitude and phase at running speed and multiples of running speed

Used to generate Bode plots

• Amplitude/Phase vs Speed (Bode and Polar plot formats)

• Shows Critical Speed Locations

• Used for balancing

• Used to indicate rubs and changes in system behavior

DC DataShows shaft position (for proximity probes)

Used to characterize external loads on bearings

Can indicate misalignment issues


Faulty InstrumentationCan result in random vibration (amplitude and frequency)

Check for:

Loose connections

Mis-wired leads

Damaged probes

Loose transducer mounting

Probe or probe housing resonance

Incorrect transducer or signal conditioning

Accelerometer resonant frequency (use low pass filter)

Wrong proximity probe cable length

Calibrate instrumentation if suspect


Unbalance

High synchronous vibration (1X)

Vibration increases with speed squaredMore rapid near critical speeds

Phase angle constant at constant speed and steady-state conditions

Can be balanced out if suitable balance planes exist


Critical Speed in the Operating Speed Range

High sensitivity to unbalance

Can be caused by: worn bearings, loose foundation, poor initial design

0

5

10

15

20

25

30

0 1000 2000 3000 4000 5000 6000 7000

Speed (rpm)

Am

plitu

de

OperatingSpeed


Rotordynamic InstabilityFrequency < Running speed (subsynchronous)Usually does not track with speedFrequency at a natural frequency (usually first mode)Close to but not equal to the first critical speedAmplitude can grow suddenly with small changes in operating conditionCan be destructive (wiped seals, bearing, etc.)Results when destabilizing forces exceed stabilizing ones

Cross-coupled forces > Damping forces

Analytically shown when log dec < 0Requires loaded operation to occur

Often not discovered until field commissioning

Cannot be balanced!!


Rotordynamic Instability Cont.Typical Sources of Destabilizing Forces

Annular Seals (labyrinth)Bearings (fixed pad types)Impeller excitationSecondary internal leakage pathsInternal rotor frictionFloating ring oil seals

Methods to Improve StabilityTilt-pad bearingsDamper seals (honeycomb, hole pattern)Squeeze film damper bearingsSwirl Brakes/Shunt InjectionThicker shafts / Shorter bearing span


Instability Example: High Pressure Centrifugal CompressorInstability

Reference: Memmott, E.A., 1992, “Stability of Centrifugal Compressors by Application of Tilt Pad Seals, Damper Bearings, and Shunt Holes,” Proceedings of the Institute of Mechanical Engineers,IMechE 1992-6, 7-10 September, 1992.


Oil Whirl

Frequency Tracks at 1/2X Running Speed

Inner Loop Indicates Forward Subsynchronous Whirl


Surge

Lower frequency and near first natural frequency

Surge control system Should prevent operation in surge at steady-state conditions

May not keep compressor out of surge during upsets, especially ESD’s

Record surge control valve command and position along with vibration to troubleshoot


Surge Detection Using Vibration and Process Variables During Rapid Shut-Down (ESD)

Bearing Vibration (mils)

Speed (RPM)Surge Valve Position (%Closed)

Flow Orifice Delta-P (in H20)

Surge Valve Opening Delayed by 2 Seconds

Flow Drops Rapidly

Closed

Open

Surge


Rotating StallDiffuser Stall

• 5-30% of running speed

• Occurs while operating near surge

• Tracks speed

• Point of inception exhibits hysteresis with flow

• Associated droop in head-flow curve shape

Blue = Decreasing FlowRed = Increasing Flow

Hysteresis Flow

Head

1X

StallReference: Sorokes, J.M., Kuzdzal, M.J., Sandberg, M.R., Colby, G.M., 1994, “Recent Experiences in Full Load Full Pressure Shop Testing of a High Pressure Gas Injection Centrifugal Compressor,” Proceedings of the 23rd Turbomachinery Symposium.


Unsteady Aerodynamic ExcitationCaused by turbulence in the flow field at high load


Wiped Journal BearingExample Spectrum

Low frequency response


Damaged Bearing Pads on Tilt-Pad BearingProduces Asymmetry Causing Backward Whirl

Whirl

Rotation


Loose Component on the ShaftAmplitude/Phase shows Hysteresis

Does not track same path during run-up/shut-down

Caused by dry-gas seal in this example

Run-Up

Shut-Down

Polar Plot


Mis-alignment

Polar Plot Shows Phase Rolling the Wrong Way When Approaching the Critical Speed

Decreasing PhaseAngle


Mis-alignment cont.

Shaft Position on Drive-End does not Drop in BearingActually rises in bearing during shutdown

Drive End Non-Drive End

Shaft RisesIn Bearing DuringShutdown

Shaft DropsIn BearingDuring Shutdown


Mis-alignment cont.

Orbit showing 2X vibration

Reference: Simmons, H.R., Smalley, A.J., 1989, “Effective Tools for Diagnosing Elusive Turbomachinery Dynamics Problems in the Field,” Presented at the Gas Turbine and Aeroengine Congress and Exposition, June 4-8, 1989, Toronto, Ontario, Canada


Torsional VibrationSteady-State – Avoid resonance of 1X running speed

Transient – Start-up or Short Circuit of Motors

Strain Gages or Torsiographs typically used for measurement

Measured Stress in Coupling During Synchronous Motor StartTorsional Crack in Shaft


Torsional Vibration Cont.Measured Coupling Stress of Gas Turbine Driven Compressor Package with Gear

1X Tracking

Shows Location of Torsional Natural Frequencies

Reference: Smalley, A.J., 1977, “Torsional System Damping,” Presented at the Vibration Institute Machinery Vibration Monitoring and Analysis Meeting, Houston, TX, April 19-21, 1983.

Summary

Our Understanding of Rotordynamics has Greatly Improved over theLast 50 years Including Complex Rotor/Fluid Interaction

Modern Analysis Tools Can Minimize the Risk of Encountering a Critical Speed or Stability Problem on New Equipment

Tools validated against test rig and full-scale testing results

Vibration Equipment in the Hands of the Right Expertise can Solve a Variety of Vibration Issues

Key Steps:Choose the right type of instrumentation for the machine and vibration type

Correct installation and wiring to prevent noise and false signals important

Use the appropriate data acquisition equipment

Correlate vibration with key process parameters

Troubleshooting often requires controlled changes of process parameters (eg. Speed, load, pressure, temperature, etc.)

Do Not be slow to ask for helpDown time and loss production can far out weigh cost of consultants fees

Questions???Questions???

www.swri.orgDr. J. Jeffrey MooreSouthwest Research Institute(210) [email protected]

rotordynamics tutorial

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