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Moore-RD-Tutorial.pdf
```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:
 Lateral
 Torsional
 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
 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 Effects
Simple Overhung Disk Rotor
 Important with overhung
disks
0.3
0.2
Shaft1
12
 Eg. Single-stage overhung
0.1
compressor
Shaft1
1
10
-0.1
 Gyroscopic forces:
-0.2
 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
 Continuous system modeled by a system of springs and
masses formulated using either finite element or transfer
matrix methods
 Results in following system of equations:
[M ] X&& + [C ] X& + [K ] X
= F (t )
 Similar form as the single degree of freedom
 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
Stable
Desirable
δ >0
Rotordynamic Modeling
2nd Section
Rotordynamic Modeling
 Break the series of smaller
Division
Wall Seal
segments at diameter steps
 Components like impellers,
couplings, thrust disks do not
add shaft stiffness are modeled
as added mass
 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
 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
Rotordynamic Modeling
Undamped Critical Speed Map
 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
 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
 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
 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

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 Types
 Proximity Probe

Measures Relative Shaft Displacement (static and
dynamic)

Most Common

Most Applicable to Fluid Film Bearings

Subject to Electromechanical Runout (false
vibration)
 Velocity Transducer

Measures 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 Monitor
 Provides machinery protection
 Overall vibration level only
 No detailed information
 Waveform/Orbit – Oscilloscope
 Good for viewing vibration data in real
time
 Orbit shape shows symmetry in system
• Round=symmetric
 Shows transient data (impacts, bursts,
etc.)
Types of Vibration Instrumentation
 Fast Fourier Transform (FFT’s) – Spectrum Analyzer
 Breaks 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
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
 Provides 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 Data
 Shows shaft position (for proximity probes)
 Used to characterize external loads on bearings
 Can indicate misalignment issues
Example Vibration Phenomena
Faulty Instrumentation
 Can 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
Example Vibration Phenomena
Unbalance
 High synchronous vibration (1X)
 Vibration increases with speed squared
 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





 Cross-coupled forces > Damping forces
 Analytically shown when log dec < 0
 Requires loaded operation to occur
 Often not discovered until field commissioning
 Cannot be balanced!!
Example Vibration Phenomena
Rotordynamic Instability Cont.
 Typical Sources of Destabilizing Forces
 Annular Seals (labyrinth)
 Bearings (fixed pad types)
 Impeller excitation
 Secondary internal leakage paths
 Internal rotor friction
 Floating ring oil seals
 Methods to Improve Stability
 Tilt-pad bearings
 Damper seals (honeycomb, hole pattern)
 Squeeze film damper bearings
 Swirl Brakes/Shunt Injection
 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
 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
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
 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
 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
 Does not track same path during run-up/shut-down
 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
 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
 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 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

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 help

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]
```
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