Mechanical and Fluids Engineering Rotordynamics Tutorial: Theory ...

Mechanical and Fluids Engineering Rotordynamics Tutorial: Theory, Practical Applications and Case Studies Dr. J. Jeffrey Moore Southwest Research Institute

Gas Turbine Technology Center Southwest 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: ƒ Lateral ƒ Torsional ƒ Structural/Foundation

Rotordynamic Theory Single Degree of Freedom Theory

F (t ) = Meω cos ωt 2

K M

ωn =

Natural Frequency

X (t ) = Meω 2 A(ω ) cos(ω t + φ ) A(ω ) =

C

(

1

M ω −ω 2

K

2 n

)

2 2

− (Cω ) 2

⎡ ⎤ Cω φ (ω ) = tan ⎢ 2 2 ⎥ m ( ω ω )⎦ − n ⎣ −1

M

Jeffcott Rotor X(t)

F(t)

M X&& + C X& + K X = F (t )

Rotordynamic Theory Bode Plot - Amplitude

Light Damping

More Damping

Imbalance ω=ωn

Rotordynamic Theory Bode Plot - Phase 180

More Damping

90

Light Damping 0

ω=ωn

Rotordynamic Theory Solving Resonance Problems ™ Move natural frequency away from excitation frequency ƒ Increasing or decreasing stiffness ƒ Increasing or decreasing mass

™ Reduce the excitation magnitude ƒ Balancing

™ Add damping to the system ƒ Improved bearing design ƒ Squeeze film dampers

™ Change the excitation frequency ƒ Change rotation speed

Rotordynamic Theory Gyroscopic Effects Simple Overhung Disk Rotor

™ Important with overhung disks

0.3

0.2 Shaft1 12

ƒ Eg. Single-stage overhung

0.1

compressor

Shaft1 1

10

-0.1

™ Gyroscopic forces:

-0.2

ƒ Cxθ = Ip ω

Bearings

03

Rotordynamic Damped Natural Frequency Map Overhung Disk Example

5

Natural Frequency, Hz

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

5

0

4 3

Forward Backward

2 1 0 0.

2000.

4000.

6000.

8000.

Rotor Speed, rpm

10000.

12000.

Rotordynamic Theory Modeling Turbomachinery ƒ Continuous system modeled by a system of springs and

masses formulated using either finite element or transfer matrix methods

ƒ Results in following system of equations:

[M ] X&& + [C ] X& + [K ] X

= F (t )

ƒ Similar form as the single degree of freedom ƒ Use 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 Analysis ™ Damping is a Stabilizing Influence ™ Destabilizing Forces Arise from Cross-Coupling Effects that Generate Forces in the Direction of Whirl

Fx=-Kxy Y

™ Cross-Coupled Stiffness Yields a force in the Ydirection for a displacement in the X ™ Sources include: fixed arc bearings, floating ring oil seals, labyrinth seals, impeller/turbine stages

Fy=Kyx X

Y X

Rotordynamic Theory ™ Stability Calculated by Solving the Eigenvalue Problem:

[M ] X&& + [C ] X& + [K ] X = {0} ™ 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:

δ=

2πζ 1− ζ 2

™ 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.

Rotordynamic Theory Evaluation Using Log Dec(rement) Linear Vibration Neutrally Stable

XN-1

Rotor Vibration

XN

⎡X

⎤

δ = Ln ⎢ n-1 ⎥ = 0 ⎣ Xn ⎦ Unstable

Undesirable

δ 0

Rotordynamic Modeling 2nd Section

Rotordynamic Modeling ƒ Break the series of smaller

Division Wall Seal

segments at diameter steps

ƒ Components like impellers,

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

ƒ Stations added at bearings

Second Section Gas Balance Seal

Gas Flow Path

centerlines

1st Section

Typical High Pressure Centrifugal Compressor

Sample 10-Stage Compressor Model 40 15 haft1 1

5

10

20

25

30

35

45

50

55

60

65

75 70

Shaft1 79

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

Rotordynamic Modeling Rotordynamics Shaft FE Model 10-Stage Centrifugal Compressor

Coupling

SWRI Model - Nom Brngs

0.6

Shaft Radius, meters

0.2

40 15

Shaft1 1

5

20

25

Thrust Disk

Balance Drum

Impellers DGS

0.4

Red = Structural Green = Added Mass

30

45

35

50

55

60

10

65

75 70

Shaft1 79

0

-0.2

Bearings

-0.4

-0.6 0

0.4

0.8

1.2

1.6

Axial Location, meters

2

2.4

Rotordynamic Modeling Journal Bearing Cross-Coupling ™ Oil wedge causes a horizontal movement from a vertical load (cross-coupling)

Non-symmetric Pressure Profile

Rotordynamic Modeling Journal Bearing Modeling ™ Solution to the Reynolds’ equation provides the pressure profile on the pad

∂ ⎛ 3 ∂p ⎞ ∂ ⎛ 3 ∂p ⎞ ∂h ⎫ ⎧∂ ⎜h ⎟ + ⎜h ⎟ = 6μ ⎨ [hU ] + 2 ⎬ ∂t ⎭ ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂z ⎠ ⎩ ∂x •Assuming small perturbation results in 1st order equations that yield rotordynamic coefficients (Kxx, Kxy, etc.)

Rotordynamic Modeling Common Bearing Types Load

Load

Journal Radius R Bearing

Bearing

Groove 15° 15°

Clearance (C)

C = Clearance m = Preload

Plain Cylindrical Bearing

Elliptical Bearing Load Groove Bearing

Clearance (C)

Bearing Housing

Most Stable Bearing Pivot

4-Axial Groove Bearing

Tilting Pad Bearing Load Between Pads

Tilting Shoe Clearance C

Rotordynamic Modeling Journal Bearing Modeling ™ Plain journal bearings are the least stable ™ Elliptic and Axial Groove bearings introduce “preload” that improves the stability ™ Tilt-Pad bearings possess essentially no crosscoupling since the pads can pivot ƒ Most commonly used bearing in high speed turbomachinery ƒ More expensive than fixed pad designs ƒ Necessary when operating at speeds well above (> 3X) first

critical speed

ƒ Many parameters can be adjusted to achieve desired

stiffness and damping properties

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

Rotordynamic Modeling Undamped Critical Speed Map ƒ First six natural frequencies calculated for varying bearing

support stiffness

Undamped Critical Speed Map 10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

Critical Speed, cpm

100000

2nd Critical Speed

10000

1000 1.0E+06

MCOS

1st Critical Speed 1.0E+07

1.0E+08

1.0E+09

1.0E+10

1.0E+11

1.0E+12

Bearing Stiffness, N/m

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

Rotordynamic Modeling 1st Critical Speed Mode Shape

™ Cylindrical mode with flexibility Undamped C.S. Mode Shape Plot

10000

1000 1.0E+06

10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

100000

Critical Speed, cpm

™ Intersection between bearing stiffness curve and critical speed curve represents critical speed

Undamped Critical Speed Map

2nd Critical Speed 1st Critical Speed 1.0E+07

1.0E+08

1.0E+09

1.0E+10

Bearing Stiffness, N/m

forward backward f=3837.1 cpm K=200000000 N/m

1.0E+11

1.0E+12

Rotordynamic Modeling 2nd

Critical Speed Mode Shape

Undamped C.S. Mode Shape Plot

10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

100000

Critical Speed, cpm

™ Conical Mode with Flexibility

Undamped Critical Speed Map

10000

1000 1.0E+06

10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

2nd Critical Speed 1st Critical Speed 1.0E+07

1.0E+08

1.0E+09

1.0E+10

Bearing Stiffness, N/m

forward backward f=12631.6 cpm K=300000000 N/m

1.0E+11

1.0E+12

Rotordynamic Modeling API Requirements ™ Critical speeds separated from operating speed range ™ Separation margin function of amplification factor

1 ⎞ ⎛ SM 2 = 10 + 17⎜1 − ⎟ AF − 1 . 5 ⎠ ⎝ ™ =Unbalance Amount:

UB =

4W N

™ Unbalance Configuration 1st Mode

2nd Mode Reference: API 617, 7th Edition, Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services, American Petroleum Institute, July, 2002.

Rotordynamic Modeling Rotordynamic Response Plot Response, microns pk-pk

50

1st Critical Speed

40

NC1=4060 rpm AF1=5.84

Operating Speed

30

20

10

0 0

2000

4000

6000

8000

10000

12000

Rotor Speed, rpm

14000

16000

18000

20000

Rotordynamic Response Plot 50

Response, microns pk-pk

Unbalance Response Example ™ First critical speed excited by mid-span unbalance ™ Second critical speed excited by quarter-span unbalance ƒ Damping increased 2nd critical speed from 12600 to 15000 rpm ™ Separation margins meet API requirements for 1st critical speed ™ No separation margin required for 2nd critical speed since AF < 2.5

2nd Critical Speed

45 40 35

Operating Speed

NC2=15000 rpm AF2=2.05

30 25 20 15 10 5 0 0

5000

10000

Rotor Speed, rpm

15000

20000

Close Clearance Components

Journal Bearing

Honeycomb Seal Labyrinth Seal

Impeller Oil Seal

Rotordynamic Modeling Honeycomb Seal Damping Test Data vs. Predictions • Damper seals like honeycomb seals provide substantial damping • Damping increases with increasing pressure differential Ceff - Y-Direction

10000 5000 0 Re (H) (N/m)

-5000

0

100

200

300

400

-10000 -15000 -20000 -25000 -30000 -35000 Frequency (Hz)

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.

Rotordynamic Modeling Aero Cross-Coupling ™ Arises from Impellers of Centrifugal Compressors ™ Most Common Method version of Wachel Equation

( K XY )i

Mole Weight = 63, 000* 10

( N S )i

∑ j =1

( Horsepower )i , j

⎛ ρD ⎞ ⎜ ⎟ RPM * Di * hi ⎝ ρ S ⎠ j

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

Rotordynamic Modeling Stability Analysis ™ First 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 Plot 10-Stage Centrifugal Compressor SWRI Model - Nom Brngs

forward backward f=4016.3 cpm d=.1494 logd N=12000 rpm

Rotordynamic Modeling ™ 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

With Seals No Seals API Kxy

Log Dec

1

0.5

0 0.E+00

2.E+07

4.E+07

6.E+07

8.E+07

-0.5

-1

-1.5

-2

Mid-span Kxy (N/m)

1.E+08

1.E+08

Rotordynamic Modeling Measured Log Decrement in Centrifugal Compressor • Shows damper seal effectiveness • Log Dec increases as discharge pressure increases • A smooth seal was tested to simulate a “plugged-up” seal Smooth Seal - Test

3

Smooth Seal - Test

Smooth Seal - Prediction

Smooth Seal - Prediction

Hole Pattern - Test

Hole Pattern - Prediction

Log Dec

2

1

Hole Pattern - Prediction

Increasing

Division Wall Seal Leakage

Hole Pattern - Test

0 0

500

1000

1500

2000

2500

Discharge Pressure (psia)

3000

3500 0

500

1000

1500

2000

2500

3000

Discharge Pressure (psia)

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

3500

Rotordynamic Modeling Foundation Support Effects ™ Industrial 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 ƒ

Lowers critical speeds

ƒ

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 Types ™ Proximity Probe ƒ

Measures Relative Shaft Displacement (static and dynamic)

ƒ

Most Common

ƒ

Most Applicable to Fluid Film Bearings

ƒ

Subject to Electromechanical Runout (false vibration)

™ Velocity Transducer ƒ

Measures Absolute Casing Motion

ƒ

Types: magnetic coil or integrating accelerometer

ƒ

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

ƒ

Vibration severity independent of frequency

ƒ

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

Review of Transducers Transducer Types Cont. ™ Accelerometers ƒ Typically used in higher frequency measurement ƒ Not usually used on compressors due to low

motion of massive casing ƒ Severity a function of frequency ƒ Typically used with rolling element bearing (eg.

Aeroderivative gas turbines) and on gearboxes

Types of Vibration Instrumentation ™ Overall Level / Vibration Monitor ƒ Provides machinery protection ƒ Overall vibration level only ƒ No detailed information

™ Waveform/Orbit – Oscilloscope ƒ Good for viewing vibration data in real

time ƒ Orbit shape shows symmetry in system • Round=symmetric ƒ Shows transient data (impacts, bursts,

etc.)

Types of Vibration Instrumentation ™ Fast Fourier Transform (FFT’s) – Spectrum Analyzer ƒ Breaks down complex waveform into frequency components ƒ Characterize vibration: • Subsynchronous - < running speed • Synchronous = running speed • Supersynchronous > running speed ƒ Can display multiple spectra in time to make waterfall plot • Shows how vibration changes in time or during transient events

Types of Vibration Instrumentation

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.

Types of Vibration Instrumentation ™ Tracking Filter ƒ Provides amplitude and phase at running speed and

multiples of running speed

ƒ 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 Data ƒ Shows shaft position (for proximity probes) ƒ Used to characterize external loads on bearings ƒ Can indicate misalignment issues

Example Vibration Phenomena Faulty Instrumentation ™ Can result in random vibration (amplitude and frequency) ™ Check for: ƒ Loose connections ƒ Mis-wired leads ƒ Damaged probes ƒ Loose transducer mounting ƒ Probe or probe housing resonance

™ Incorrect transducer or signal conditioning ƒ Accelerometer resonant frequency (use low pass filter) ƒ Wrong proximity probe cable length

™ Calibrate instrumentation if suspect

Example Vibration Phenomena Unbalance ™ High synchronous vibration (1X) ™ Vibration increases with speed squared ƒ More rapid near critical speeds

™ Phase angle constant at constant speed and steadystate conditions ™ Can be balanced out if suitable balance planes exist

Example Vibration Phenomena Critical Speed in the Operating Speed Range ™ High sensitivity to unbalance ™ Can be caused by: worn bearings, loose foundation, poor initial design 30 Operating Speed

Amplitude

25 20 15 10 5 0 0

1000

2000

3000

4000

Speed (rpm)

5000

6000

7000

Example Vibration Phenomena Rotordynamic Instability Frequency < Running speed (subsynchronous) Usually does not track with speed Frequency at a natural frequency (usually first mode) Close to but not equal to the first critical speed Amplitude can grow suddenly with small changes in operating condition ™ Can be destructive (wiped seals, bearing, etc.) ™ Results when destabilizing forces exceed stabilizing ones

™ ™ ™ ™ ™

ƒ Cross-coupled forces > Damping forces

™ Analytically shown when log dec < 0 ™ Requires loaded operation to occur ƒ Often not discovered until field commissioning

™ Cannot be balanced!!

Example Vibration Phenomena Rotordynamic Instability Cont. ™ Typical Sources of Destabilizing Forces ƒ Annular Seals (labyrinth) ƒ Bearings (fixed pad types) ƒ Impeller excitation ƒ Secondary internal leakage paths ƒ Internal rotor friction ƒ Floating ring oil seals

™ Methods to Improve Stability ƒ Tilt-pad bearings ƒ Damper seals (honeycomb, hole pattern) ƒ Squeeze film damper bearings ƒ Swirl Brakes/Shunt Injection ƒ Thicker shafts / Shorter bearing span

Example Vibration Phenomena Instability Example: High Pressure Centrifugal Compressor Instability

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.

Example Vibration Phenomena Oil Whirl ™ Frequency Tracks at 1/2X Running Speed ™ Inner Loop Indicates Forward Subsynchronous Whirl

Example Vibration Phenomena Surge ™ Lower frequency and near first natural frequency ™ Surge control system ƒ Should prevent operation in surge at steady-state conditions ƒ May not keep compressor out of surge during upsets,

especially ESD’s

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

Example Vibration Phenomena Surge Detection Using Vibration and Process Variables During Rapid Shut-Down (ESD) Bearing Vibration (mils)

Flow Orifice Delta-P (in H20) Flow Drops Rapidly

Surge

Surge Valve Position (%Closed) Closed

Open

Surge Valve Opening Delayed by 2 Seconds

Speed (RPM)

Example Vibration Phenomena Blue = Decreasing Flow Red = Increasing Flow

Rotating Stall ƒ Diffuser Stall

Head

• 5-30% of running speed • Occurs while operating near surge • Tracks speed • Point of inception exhibits hysteresis with flow

Flow

Hysteresis

• Associated droop in headflow curve shape

1X Stall Reference: 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.

Example Vibration Phenomena Unsteady Aerodynamic Excitation ™ Caused by turbulence in the flow field at high load

Example Vibration Phenomena Wiped Journal Bearing ™ Example Spectrum ƒ Low frequency response

Example Vibration Phenomena Damaged Bearing Pads on Tilt-Pad Bearing ™ Produces Asymmetry Causing Backward Whirl

Rotation

Whirl

Example Vibration Phenomena Loose Component on the Shaft ™ Amplitude/Phase shows Hysteresis ƒ Does not track same path during run-up/shut-down ƒ Caused by dry-gas seal in this example Polar Plot Shut-Down

Run-Up

Example Vibration Phenomena Mis-alignment ™ Polar Plot Shows Phase Rolling the Wrong Way When Approaching the Critical Speed Decreasing Phase Angle

Example Vibration Phenomena Mis-alignment cont. ™ Shaft Position on Drive-End does not Drop in Bearing ƒ Actually rises in bearing during shutdown Drive End

Non-Drive End

Shaft Drops In Bearing During Shutdown Shaft Rises In Bearing During Shutdown

Example Vibration Phenomena 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

Example Vibration Phenomena Torsional Vibration ™ Steady-State – Avoid resonance of 1X running speed ™ Transient – Start-up or Short Circuit of Motors ™ Strain Gages or Torsiographs typically used for measurement Torsional Crack in Shaft

Measured Stress in Coupling During Synchronous Motor Start

Example Vibration Phenomena Torsional Vibration Cont. ™ Measured Coupling Stress of Gas Turbine Driven Compressor Package with Gear ƒ 1X Tracking ƒ 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 the Last 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 ƒ

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: ƒ

Choose the right type of instrumentation for the machine and vibration type

ƒ

Correct installation and wiring to prevent noise and false signals important

ƒ

Use the appropriate data acquisition equipment

ƒ

Correlate vibration with key process parameters

ƒ

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

™ Do Not be slow to ask for help ƒ

Down time and loss production can far out weigh cost of consultants fees

Questions??? www.swri.org Dr. J. Jeffrey Moore Southwest Research Institute (210) 522-5812 [email protected]