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


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


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


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?


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


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?


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



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


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 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!


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


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


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


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


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


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]


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


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.


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


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.

case study, 38th turbomachinery symposium

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