Download - Case Study, 38th Turbomachinery Symposium
Case Study, 38th Turbomachinery Symposium
Title:
Torsional – Lateral Coupled Vibration of Centrifugal Compressor System at
Interharmonic Frequencies Related to Control Loop Frequencies in Voltage Source
PWM Inverter
Abstract:
Centrifugal compressor train in a refinery experienced high vibration problem due to
torsional resonance. Sidebands in the VFD output current based on VFD control loop
frequencies were identified as the root cause. In this VFD, stator current was used for
torque and speed control, hence control loop frequencies had a potential to generate
such sidebands. Frequencies of this type of sidebands widely vary with the rotation
speed (proportional to harmonics of the fundamental frequency), hence it is difficult to
avoid resonance at the train torsional natural frequency. In addition, even if a
compressor system is proven to have sufficient safety margin against high cycle fatigue
failure due to the torque pulsation by this mechanism, such minute torque pulsation may
have a potential to excite high lateral vibration at speed adjusting gear. If unpredicted or
overlooked during design stage, such high vibration may disturb plant operation. This
case study therefore proposes guidelines to predict such vibration levels by a simplified
torsional-lateral coupled vibration analysis.
Authors:
Akira Adachi, Toyo Engineering Corporation
Kenji Tanaka, Hitachi Plant Technologies, Ltd.
Naohiko Takahashi, Hitachi Plant Technologies, Ltd.
Yasuo Fukushima, Hitachi Plant Technologies, Ltd.
14-17 September 2009, 38th Turbomachinery Symposium
Kenji Tanaka
Hitachi Plant Technologies, Ltd.Akira Adachi Toyo Engineering CorporationNaohiko
Takahashi
Hitachi Plant Technologies, Ltd.
Yasuo
Fukushima
Hitachi Plant Technologies, Ltd.
Torsional-Lateral Coupled Vibration of Centrifugal Compressor System at
Interharmonic
Frequencies Related to Control Loop Frequencies in Voltage Source Inverter
114-17 September 2009, 38th Turbomachinery Symposium
High vibration was sometimes observed at gear LS shaft
Introduction
Figure 1. Compressor Train
Speed Increaser
Gear
Induction Motor
Centrifugal Compressor
VFD
LS Coupling
HS Coupling
Train Data
VFD Induction Motor + Gear + Centrifugal Compressor
4-Pole1200kW
8588- 12882rpm
1000-1500rpm
214-17 September 2009, 38th Turbomachinery Symposium
Onset of Problem
Lateral Vibration on Low Speed Gear Shaft
50μm (p-p) Vibration Detected on LS Gear Shaft around 1040-1140rpm (17.3-19.0rps), 570kW
Dominant Frequency Component: ca. 16Hz
Close to Calculated 1st Torsional
Natural Freq. fn1
(15.71Hz)No Significant Vibration on Compressor Shaft or Motor Shaft
Figure 2. Cascade Plot of LS Gear Shaft Vibration
fn1
=16Hz
0Hz 100Hz50Hz
Frequency [Hz]
Vibr
atio
n A
mpl
itude
(μm
p-p)
1145
1100
1040
Spe
ed(r
pm)
20μm
1×N 2×N 3×N 4×N 5×N
314-17 September 2009, 38th Turbomachinery Symposium
Onset of Problem
Lateral Vibration on High Speed Gear (Pinion) Shaft
Max. 10μm (p-p) Vibration with 16Hz Detected on HS Gear Shaft
Stiffer Pinion Bearings Than Bull Gear Bearings at Part-Load Due to Uploading Bull Gear and Downloading Pinion
Figure 3. Cascade Plot of HS Gear Shaft Vibration
fn1
=16Hz
0Hz 100Hz50Hz
Frequency [Hz]
Vibr
atio
n A
mpl
itude
(μm
p-p)
1145
1100
1040
Spe
ed(r
pm)
20μm
Pinion Lighter Than Bull Gear
Lower Vibration on HS Gear Shaft
+Concern: What is the Excitation Source?
414-17 September 2009, 38th Turbomachinery Symposium
Torsional Natural Frequencies
Analysis for Torsional Natural Frequencies (FEM)
0
1000
2000
3000
4000
5000
0 1000 2000 3000 4000 5000
Rotor speed [rpm]
Tors
iona
l nat
ural
freq
uenc
y [c
pm]
First natural frequency: 942.6cpm (15.71Hz)
Second natural frequency: 2752cpm (45.87Hz)
Max
. ope
ratin
g sp
eed
(LS
sha
ft): 1
500r
pm
1 x rotor speed
2 x rotor speed
Min
. ope
ratin
g sp
eed
(LS
sha
ft): 1
000r
pm
Figure 4. Campbell Diagramfor Torsional Vibration
Figure 5. Torsional Vibration Mode Shapes
Second Natural Frequency: 45.87 Hz (2752cpm)
First Natural Frequency: 15.71Hz (943cpm)
Third Natural Frequency: 422.1Hz (25324cpm)
CompressorMotor Gear
514-17 September 2009, 38th Turbomachinery Symposium
Critical Speeds Predicted by Rotor Analyses
Lateral & Torsional
Vibration Calculation
First Second Operating Speed
Lateral
Compressor 5220 17700 8588-12882
Pinion(HS Shaft)
21200-25600Dependent on
Load
- 8588-12882
Bull Gear(LS Shaft)
3380-9450Dependent on
Load
- 1000-1500
Motor 2078 - 1000-1500
Torsional 943 2752
Critical Speeds [rpm]
Shaft Synchronous Resonance is Excluded. Only Possibility = VFD?
614-17 September 2009, 38th Turbomachinery Symposium
Estimation of Shaft Torque and Motor Torque
Estimation of Shaft Torque Causing 50μm Shaft Vibration
Fr
Ft
LS GearHS Gear
Fnθ
Figure 6. Force Applied on Gear Teeth
(Double Helical)
Transverse Pressure Angle θθ= tan-1(tanα/cosβ)α= Normal Pressure Angleβ= Helix Angle
Force Applied on Teeth Causing 50μm (p-p) Shaft Vibration
Shaft Torque Deduced by One-Way Lateral-Torsional
Analysis
Excitation
(1/2)Ft
(1/2)Fr
(1/2)Facancelled
(1/2)Fn
θ
αβ(1/2)Ft
(1/2)Fr
(1/2)Facancelled
(1/2)Fn
θ αβ
Tt
= (PCD/2) ×
Ft= (PCD/2) ×
Fn ×
cos
θ
714-17 September 2009, 38th Turbomachinery Symposium
Estimation of Shaft Torque and Motor Torque
Lateral Vibration Analysis of LS Gear Shaft
Unit Excitation on Tooth Surface Fn_p.u.
=9.8・sin(2πfn1
t) [N]
Tt
= (PCD/2) ×
Fn
×cos
θ= (0.806/2) ×
4851 ×
cos
21.9°
= 1815 [N・m]
0.213 Times Rated Torque
Figure 7. Frequency Response to Unit Excitation Force of Gear Shaft
LS Gear Shaft Vibration Amplitude at Probe @ 16Hz=0.101μm (p-p) (Calculated)
(Bearing Stiffness and DampingCalculated @ 1140rpm, 570kW)
Fn
=9.8×(50/0.101)=4851 [N]0.00E+0
2.00E-2
4.00E-2
6.00E-2
8.00E-2
1.00E-1
1.20E-1
0 5 10 15 20 25
Excitation Frequency [Hz]
Am
plitude
[μ
mp-
p]
X
Y
X-Probe Angle=15 degree right of vertical
Y-Probe Angle=15 degree above horizontal
0.101μm
16Hz
814-17 September 2009, 38th Turbomachinery Symposium
0
5
10
15
20
0 10 20 30 40 50 60
Excitation frequency [Hz]
Torq
ue a
t bu
l ge
ar s
haf
t /
air
gap
torq
ue a
t m
oto
r sh
aft
[p.
u.]
Estimation of Shaft Torque and Motor Torque
Estimation of Excitation Torque at Motor Air Gap
Figure 8. Frequency Response at LS Gear Shaft
Assuming ζ=0.02
TAG
=1815 / 16.9=107 [N・m]
fn1
=15.7 Hz
16.9 p.u.
1.3%Rated Torque at Motor Air Gap Suspected.
fn1
Amplification Factor for Torsional
System
Close to Maximum Measured Data among Interharmonic
Frequencies
during Factory Test(1.5%Rated Torque)
Merely 1.3% of air gap torque fluctuation can cause 50 μm p-p
lateral vibration!
914-17 September 2009, 38th Turbomachinery Symposium
Cause and Effect
Sequence of Lateral-Torsional
Analysis
LS GearShaft
Vibration50μm16Hz
Effect
Effective Force on Bull Gear
Teeth 4854N16Hz
Torque at Bull Gear
Teeth 1815Nm
16Hz
Air Gap Torque on
Motor Shaft
107Nm (1.3%)16Hz
Torsional Frequency
Response
(ζ=0.02, Geometry)
AF16.9
Geometry(θ, PCD)
AF0.101/9.8
Cause
LateralFrequency Response
Bearing Stiffness,Damping
Actual Cause and Effect
1014-17 September 2009, 38th Turbomachinery Symposium
Strength Evaluation
High Cycle Fatigue Evaluation
Figure 9. Modified Goodman Diagram
0
200
400
600
800
1000
1200
0 200 400 600 800 1000 1200 1400
Mean stress σm [N/mm2]
Altern
atin
g st
ress
σa [
N/m
m2]
Fatigue Factor of Safety 1.25 for 107
Cycle Life
107
Cycle Life
Stress at the Condition
Mechanical Strength Verified
No Modification Made on Machinery or VFD
1114-17 September 2009, 38th Turbomachinery Symposium
1000
1050
1100
1150
1200
1250
1300
0 10 20 30 40 50 60
Investigation of Source of Excitation Torque
Detailed Measurement (LS Gear Shaft Vibration)
Figure 10. Cascade Plot of LS Gear Shaft Vibration
fn1
=16Hz
(N/60) (2N/60)
(3N/60)
fc
=256Hz, n=7, slope: -6
fc
=1024Hz, n=25, slope: -24
fc
=1024Hz, n=29, slope: +30fc
=1024Hz, n=31, slope: -30
fc
=1024Hz, n=23, slope: +24
Rot
or S
peed
(LS
Sha
ft) [
rpm
]
Frequency [Hz]
Am
plitu
de
(50
μmp-
p/di
v)
50μm
1005
1015
1025
1035
1045
10 15 20 50μm
Rot
or S
peed
[rpm
]
Frequency [Hz]
fc
=1024Hz, n=29, slope: +30
fc
=1024Hz, n=31, slope: -30
1214-17 September 2009, 38th Turbomachinery Symposium
Investigation of Source of Excitation Torque
Figure 11. Cascade Plot of VFD Output Current
Detailed Measurement (VFD Output Current)
1000
1050
1100
1150
1200
1250
1300
0 10 20 30 40 50 60
Rot
or S
peed
(LS
Sha
ft) [
rpm
]
Frequency [Hz]
VFD
Out
put
Cur
rent
(5
0dB
/div
, min
.0.1
% R
ated
Cur
ent)
50dB=32%
Fundamental Frequencyf
fc
=256Hz, n=7, slope: +7
fc
=1024Hz, n=25, slope: +25
fc
=1024Hz, n=29, slope: -29 fc
=1024Hz, n=31, slope: +31
fc
=1024Hz, n=23, slope: -23 f -
fn1 f + fn1fn1
=16Hz fn1
=16Hz
1314-17 September 2009, 38th Turbomachinery Symposium
Pattern of Interharmonic
Frequency Component
Inclined Streaks of Interharmonic
Frequencies in Shaft Vibration Frequencies and VFD Output Frequencies
Firm Correlation Between Shaft Vibration and VFD Output Current Suspected
LS Shaft Vibration Frequency ContentDifference of Harmonics of Multiples of 6 and Sampling Frequencies of VFD (1024Hz, 256Hz)
VFD Output Current Frequency ContentDifference of Harmonics of Odd Numbers Other Than Multiples of 3 and Sampling Frequencies of VFDInclination Opposite to That of Shaft Vibration
1414-17 September 2009, 38th Turbomachinery Symposium
Pattern of Interharmonic
Frequency Component
Relation of Between Shaft Vibration Frequencies and VFD Output Frequencies
Frequencies of Fluctuating Torque Generated by 3-Phase IM
Te
= pM’Is_a’Ir’・(9/4)・sin(2π(fa
-f)t+γ)If fa
= fbi
= |fc
-nf|Te
=pM’Is_a’Ir’・(9/4)・sin(2π(|fc
-(n±1)f|)t +γ)
Shaft Vibration Freq. [Hz]: fbt
=|fc
-(n±1)f|
VFD Output Freq. [Hz]: fbi
= |fc
-nf| f:VFD Fundamental Freq. [Hz] fc
:Arbitrarily Existing Constant Freq. [Hz]
n:Positive Odd Integer Other Than 3
Shaft Vibration Caused by Excitation of Motor Torque
Sideband Frequencies
fa
:Arbitrarily Existing Current Freq. [Hz]
1514-17 September 2009, 38th Turbomachinery Symposium
Sampling in VFD
VFD Control Loop
Figure 12. Block Diagram of VFD Control System
Frequency Compensat
ion
SpeedRef.
Automatic Torque Boost
Starting Torque Boost
Voltage 2-phase / 3-phase
Transform
ation
V/fPattern
+
-
++
Current 3-phase / 2-phaseTransfor
mation
Voltage Compens
ation
PWMControl
IM
InverterCell-
+
++
CompressorSystem
CT
Several Sampling Frequencies Used in VFD Control Loop
FluxRef.
ωre*
×ω1
∫ω1
dt
Λγω1
Λγ
1614-17 September 2009, 38th Turbomachinery Symposium
Assumed Cause of Sideband
Assumed Cause of Sideband in VFD Output Current
Harmonics of Odd Numbers Produced by Pulse (Rectangular Wave) of Fundamental Frequency
Coarse Pulse of Fundamental Frequency Remained in Current (e.g. Improper Dead Time Compensation)
Harmonics of Multiple of 3 Eliminated in Balanced Three-Wire System
Sideband Frequencies Raised by Modulation between Harmonic Frequencies and Sampling Frequencies Occurred in VFD Control Loop Harmonics enhance
sidebands.
1714-17 September 2009, 38th Turbomachinery Symposium
Triangular Carrier Wave fc_PWM
Fundamental Wave fET
- ET
0
θ1 θ2
Ed
/2
- Ed
/2
0
Inverter Output
π 2π 3π 4π
Other Possible Cause of Sideband
Sideband Frequencies Due to PWM
Sum and Difference of Frequencies of Harmonics of Triangular Carrier Wave (4.8kHz) and Harmonics of Signal Wave (Fundamental Frequency)
Sideband Frequencies Due to PWM Not Observed in This VFD Output Current Figure 13. Mechanism
of PWM
1814-17 September 2009, 38th Turbomachinery Symposium
Concluding Remarks
Measurement of torque & current in factory test is important in case VFD characteristics are unknown.
Strength evaluation by lateral-torsional analysis is essential to determine mechanical soundness.
Information of any possible frequencies used in VFD control
and the resulting amplitude of torque pulsation should be disclosed in advance by VFD vendor.
Reduction of amplitude of fundamental frequency harmonics would decrease amplitude of sideband frequencies.