Case Study, 38th Turbomachinery Symposium Title: Torsional – Lateral Lateral Coupled Vibration of Centrifugal Compressor System at Interharmonic Frequencies Related to Control Loop Frequencies in Voltage Source PWM Inverter
Abstract: Centrifugal compressor 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 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 Frequencies of this type of sidebands sidebands widely vary with the t he rotation speed (proportional to harmonics of the fundamental frequency), hence it is difficult to avoid resonance at the train torsional t orsional 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
Torsional-Lateral Coupled Vibration of Centrifugal Compressor System at Interharmonic Frequencies Related to Control Loop Frequencies in Voltage Source Inverter Kenji Tanaka Akira Adachi Naohiko Takahashi Yasuo Fukushima
Shaft Synchronous Resonance is Excluded. Only Possibility = VFD?
Estimation of Shaft Torque and Motor Torque Estimation of Shaft Torque Causing 50μm Shaft Vibration Shaft Torque Deduced by One-Way Lateral-Torsional Analysis Force Applied on Teeth Causing 50μm (p-p) Shaft Vibration (1/2)Ft
α θ
Ft Fn θ
(1/2)Fa cancelled
(1/2)Fn
β(1/2)Ft (1/2)Fr
θ α
(1/2)Fn
β (1/2)Fr
Fr
(1/2)Fa cancelled
Figure 6. Force Applied on Gear Teeth (Double Helical)
Tt = (PCD/2) × Ft = (PCD/2) × Fn × cos θ
Estimation of Shaft Torque and Motor Torque Lateral Vibration Analysis of LS Gear Shaft X-Probe Angle=15 degree right of vertical Y-Probe Angle=15 degree above horizontal
1.20E-1
0.101μm
] 1.00E-1 p p m8.00E-2 μ [ e6.00E-2 d u t i l 4.00E-2 p m A2.00E-2
X Y
16Hz
0.00E+0 0
5
10
15
20
25
Excitation Frequency [Hz]
Figure 7. Frequency Response to Unit Excitation Force of Gear Shaft (Bearing Stiffness and Damping Calculated @ 1140rpm, 570kW)
Unit Excitation on Tooth Surface Fn_p.u.=9.8・sin(2πfn1t) [N] LS Gear Shaft Vibration Amplitude at Probe @ 16Hz =0.101μm (p-p) (Calculated) Fn=9.8×(50/0.101)=4851 [N] Tt = (PCD/2) × Fn×cos θ = (0.806/2) × 4851 × cos 21.9° = 1815 [N・m] 0.213 Times Rated Torque
Estimation of Shaft Torque and Motor Torque Estimation of Excitation Torque at Motor Air Gap Amplification Factor for Torsional System TAG=1815 / 16.9 =107 [N・m]
20
] . u . p [ t f a15 / h s t f r a o h t s o r m a t10 e a g l e u u b q t r a o t 5 e p u a q r g r o i T a
1.3%Rated Torque at Motor Air Gap Suspected.
0 0
10
f n1
20
30
40
50
60
Close to Maximum Measured Data among Figure 8. Frequency Interharmonic Frequencies Response at LS Gear Shaft during Factory Test Assuming ζ=0.02 (1.5%Rated Torque) Merely 1.3% of air gap torque fluctuation Excitation frequency [Hz]
Cause and Effect Actual Cause and Effect Sequence of Lateral-Torsional Analysis Effect LS Gear Shaft Vibration 50μm 16Hz AF 0.101/9.8 Lateral Frequency Response
Cause Effective Force on Bull Gear Teeth 4854N 16Hz Bearing Stiffness, Damping
Torque at Bull Gear Teeth 1815Nm 16Hz
Geometry (θ, PCD)
Air Gap Torque on Motor Shaft 107Nm (1.3%) AF 16Hz 16.9 (ζ=0.02, Geometry) Torsional Frequency Response
Strength Evaluation High Cycle Fatigue Evaluation
Fatigue Factor of Safety 1.25 for 107 Cycle Life 107 Cycle Life
Stress at the Condition
Figure 9. Modified Goodman Diagram
Mechanical Strength Verified No Modification Made on Machinery or VFD
Investigation of Source of Excitation Torque Detailed Measurement (LS Gear Shaft Vibration) ]1045 m p r [ d1035 e e p S r o t1025 o R
1300
] m 1250 p r [ ) t f a 1200 h S S L ( 1150 d e e p S r1100 o t o R
1015
1005 10
) v i e d d / u p t i l p p m m μ A 0 5 (
15
Frequency [Hz]
1050 50μm
1000 0
10
20
30
Frequency [Hz]
40
50
60
Figure 10. Cascade Plot of LS Gear Shaft Vibration
20
50μm
Investigation of Source of Excitation Torque Detailed Measurement (VFD Output Current)
1300
1050
) t n e r u C t d n e t e r r a R u C t % 1 u . p t 0 . u n i O m D , F i v V d / B d 0 5 (
1000
50dB =32%
] m1250 p r [ ) t f a h 1200 S S L ( d 1150 e e p S r o t 1100 o R
0
10
20
30
40
50
60
Frequency [Hz]
Figure 11. Cascade Plot of VFD Output Current
Pattern of Interharmonic Frequency Component Inclined Streaks of Interharmonic Frequencies in Shaft Vibration Frequencies and VFD Output Frequencies
LS Shaft Vibration Frequency Content Difference of Harmonics of Multiples of 6 and Sampling Frequencies of VFD (1024Hz, 256Hz)
VFD Output Current Frequency Content Difference of Harmonics of Odd Numbers Other Than Multiples of 3 and Sampling Frequencies of VFD Inclination Opposite to That of Shaft Vibration Firm Correlation Between Shaft Vibration and VFD Output Current Suspected
Pattern of Interharmonic Frequency Component Relation of Between Shaft Vibration Frequencies and VFD Output Frequencies VFD Output Freq. [Hz]: f:VFD Fundamental Freq. [Hz] fbi = |fc -nf| Sideband fc:Arbitrarily Existing Frequencies Constant Freq. [Hz] Shaft Vibration Freq. [Hz]: n:Positive Odd Integer fbt =|fc -(n±1)f| Other Than 3 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|
fa:Arbitrarily Existing Current Freq. [Hz]
Te=pM’Is_a’Ir’・(9/4)・sin(2π(|fc -(n±1)f|)t +γ) Shaft Vibration Caused by Excitation of Motor
Sampling in VFD VFD Control Loop
Automatic Torque Boost Speed Ref.
ω
* re
ω1
+
Flux Ref.
+
Λγ
ω1Λγ + × +
-
+ V/f Pattern
+ -
Voltage 2-phase / 3-phase Transform ation
PWM Control
Inverter Cell
Starting Torque Boost
∫ω1dt Frequency Compensat ion
Voltage Compens ation
Several Sampling Frequencies Used in VFD Control Loop
Current 3-phase / 2-phase Transfor mation
CT IM
Compressor System
Figure 12. Block Diagram of VFD Control System
Assumed Cause of Sideband Assumed Cause of Sideband in VFD Output Current Coarse Pulse of Fundamental Frequency Remained in Current (e.g. Improper Dead Time Compensation) Harmonics of Odd Numbers Produced by Pulse (Rectangular Wave) of Fundamental Frequency Harmonics of Multiple of 3 Eliminated in Balanced Three-Wire System Sideband Frequencies Raised by Modulation between Harmonic Frequencies and Sampling Frequencies Harmonics enhance Occurred in VFD Control Loop sidebands.
Other Possible Cause of Sideband Sideband Frequencies Due to PWM Sum and Difference of Frequencies of Harmonics of Triangular Carrier
ET
Wave (4.8kHz) and Harmonics of
0
Signal Wave (Fundamental Frequency) Sideband Frequencies Due to PWM Not Observed in This VFD Output Current
Triangular Carrier Wave f c_PWM Fundamental Wave f
π
- ET θ1
2
π
3
π
4
π
θ2
Ed/2 0 - Ed/2
Inverter Output
Figure 13. Mechanism of PWM
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.