6-10 July 2008, Daejeon, Korea LASER DOPPLER VIBROMETER...

LASER DOPPLER VIBROMETER AND IMPULSE SIGNAL PHASE DEMODULATION IN ROTATION UNIFORMITY MEASUREMENTS

Jiri TUMA VSB Technical University of OstravaCzech Republic

6-10 July 2008, Daejeon, Korea

ICSV 15, 6-10 July 2008, Korea 2

Czech Republic


Outline

Terminology noteMotivationTransducers & Signal Processing Methods Principle of phase demodulation using Hilbert transformAccuracy of Incremental Rotary EncodersComparison of Measurements Done with Laser and IRCConclusion


Terminology Note: Nominal vs. Actual Angle of Rotation

Uniform rotation Non-uniform (irregular) rotation

360°360° 0°0°

360° 360°

0°0°

360°0°

Nominal angle of rotation, deg Nominal angle of rotation, deg

Actual angle of rotation, deg Actual angle of rotation, deg

Angle variation

Nominal angle of rotation, deg

It is assumed rotation at the steady-state rotational speed. In fact it is a time axis.

ϕ

ϕΔ

ϕΔ

ϕ


Motivation

Angular vibration of a gear versus linear vibration on the gearbox housing (geared axis systems)Transmission error measurement of a gear trainMeasuring and analyzing car engine crankshaft rotational irregularities, caused by dynamic changes in torque

Angular vibration as one of the important machine vibration and noise source.


Gear Angular VibrationTime : Time (Enhanced Time(Encoder))

-0,0016

0,0000

0,0016

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0Nominal Revolution

deg

Time : Time (Time (Enhanced Time(Encoder))) - 0 to 100 ord

-60000-30000

03000060000


deg/

s2̂

Time : Order Analyzer : Enhanced Time(Vibrace H)

-10-505

10


m/s

^2

Linear acceleration onthe gearbox housing

Angular acceleration

Difference between actual and nominal rotation angle

Double differentiation


Test Stand for Transmission Error (TE) Measurements

211

22 r

nnTE ⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ−Θ=

-4-2024

0,0 0,2 0,4 0,6 0,8 1,0Nominal tooth pitch rotation

TE in

mic

rons

Back-to-back test rig

Simple gear train, tooth numbers n1, n2

2 incremental rotary encoders

Pinion

Wheel

2Θ1Θ

r2

Measurement range: ± 10 μm (microns)

( ) 211

22 r

nnmTE ⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ−Θ=


Rotational irregularities of a car engine evaluated using an impulse signal for ECU

-200

-150

-100

-50

0

50

100

150

200

250

300

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

Revolution

rad/s2

-

+

Two complete revolutions

Compression strokes

Power strokes

Crankshaft angular acceleration

Engine idle speed: 800 RPM


Transducers & Signal Processing Methods for Angular Vibration Measurements

Tangentially mounted accelerometersTorsiografLaser Torsional Vibration Meter (Doppler effect)Incremental rotary encoders(several hundreds of pulses per revolution)

Time interval length measurementsSample number & InterpolationHigh frequency oscillator (10 GHz) & Impulse counter (Rotec)

Quadrature mixingPhase demodulation

Transducers How to process impulse signals?

ICSV 15, 6-10 July 2008, Korea 10

Transducers

Dual-beam laser transducer based on the Doppler effectLaser: Ga-Al-As diode, 780 nm lightInstantaneous changes in angular velocityMeasurement ranges: 10, 100, 1000, 10000°/s Frequency range: 0.3 to 1000 Hz Accuracy: ±1% of full scale

Laser Torsional Vibration Meter, Brüel & Kjær Type 2523

Heidenhain Incremental Rotary Encoders

ERN 460-500 type, 500 impulses per revolution

ERN 460-1024 type, 1024 impulses per revolution

The maximum directional deviation is within ± 1/20 grating period.

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Signal Analyzers for impulse signals

PULSE, the BK Signal Analyzers

Frequency range FR:

25.6 kHz (65536 kHz samp freq)

102.4 kHz (256 kHz sampl freq)

Phase demodulation, including possible signal resampling for order analysis, requires 5 or 6 samples per impulse period at least.

ICSV 15, 6-10 July 2008, Korea 12

Principle of Phase Demodulation

-1,5-1,0-0,50,00,51,01,5

0,0 0,2 0,4 0,6 0,8 1,0Time, s

yi

How to evaluate the instantaneous phase or frequency of a phase modulated harmonic signal?

( )iii Eyarcsin=ϕ

Instantaneous phase may be evaluated for given sample yi and envelope Ei using a formula

Solution: The creation of an analytic signal to evaluate phase and envelope.

Problems: estimation of instantaneous envelope, measurement noise.

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Analytic Signal

fP π=ω 2

Real harmonic signal Complex analytic signal

(vanishing XN and multiplication XP by 2 )

Analytic signalHilbert transform =+ jTime signal

( ) ( ) ( )22 tytxtE +=Envelope

( ) ( ) ( )( )txtyt atan=ϕWrapped Phase

π+

π−

t+

Unwrapped Phase

Wrapped Phase( )tϕ

ICSV 15, 6-10 July 2008, Korea 14

Phase demodulation based on the Hilbert transform

Using FFT and Inverse FFT Using digital filters as the Hilbert transformer

2π

2π Hilbert

Transformer

x(t)x(t)

y(t)

Real part

Imaginary part

( ) ( ){ }kxFFTjX =ω

( ) ( ){ }ω= jYIFFTky

Impulse Response

-1,0-0,50,00,51,0

-16 -12 -8 -4 0 4 8 12 16Index n

Delay

ICSV 15, 6-10 July 2008, Korea 15

Accuracy of Incremental Rotary Encoders

Two incremental rotary encoders

PinionWheel

2Θ1Θ

r2

Gear train transmission error measurement

Required error is less than

10-4 deg (0.0001 deg) for a mesh cycle

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Comparison of two encoders of the Heidenhain ERN 460-500 type

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,0000 0,0002 0,0004 0,0006 0,0008

Time, s

[V]

-100

-80

-60

-40

-20

0

0 10000 20000Frequency, Hz

RM

S dB

/ref 1

VE1E2

E1E2

500 impulses per revolution

Fs = 65536 Hz

104010441048105210561060

0,0 0,2 0,4 0,6 0,8Time, s

RPM

Both encoders rotate at the same rotational speed

Global uniformity of rotation

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Order of Averaging Operation and Phase Demodulation

Impulse time signals

Averaging in time domain

Phase demodulation

Frequency spectrum

Phase difference

Impulse time signals

Averaging in time domain

Phase demodulation

Frequency spectrum

Phase difference

Phase difference

Averaging in freq domain

Frequency spectrum

Inst frequency spectrum

BA

C

D E F

(Signal Enhancement in BK Signal Analyzers)

ICSV 15, 6-10 July 2008, Korea 18

Phase difference frequency spectra

-0,03-0,02-0,010,000,010,020,03

0,0 0,2 0,4 0,6 0,8 1,0

Nominal revolution

deg

-0,03-0,02-0,010,000,010,020,03

0,0 0,2 0,4 0,6 0,8 1,0

Nominal revolution

deg

AveragedPhaseDifferencesAveraged TimeSignals

Phase difference

1E-61E-51E-41E-31E-21E-1

1 10 100 1000

Order

RM

S d

eg

Averaged spectrumAveraged time signal

Phase difference

1E-61E-51E-41E-31E-21E-1

1 10 100 1000

Order

RMS

deg

Spectrum of the averaged impulse signals (D)

Averaged spectrum of phase differences (F) spectrum of averaged phase differences (E)

16 phase differences

Averaging of impulse signals (A) and phase differences (B)

C

A

B

FE

D

ICSV 15, 6-10 July 2008, Korea 19

Phase Difference between Two Encoders of the ERN 460-500 Type

E1E2

Phase difference

0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

1 10 100 1000Order

RM

S de

g 1040 RPM

Phase difference

-0,04

-0,03

-0,02

-0,01

0,00

0,01

0,02

0,03

0,04

0,0 0,2 0,4 0,6 0,8 1,0

Nominal Revolution

deg

Maximum directional deviation

360° 36° 3.6° 0.36°

Tooth-by-tooth rotation teethn

0360=

Wave length

Teeth

Averaged spectrum of phase differences

ICSV 15, 6-10 July 2008, Korea 20

Comparison of Measurements Done with Laser Torsional Vibration Meter and Incremental Rotary Encoder

Laser Torsional Vibration Meter Brüel & Kjær Type 2523 Incremental Rotary EncoderHeidenhain ERN 460-1024 typePULSE, the Brüel & KjærSignal AnalyzerSampling frequency 65536 Hz

Laser

Encoder Instrumentation

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Signal Processing Procedure

Band pass filtration with the centre frequency equal to the mean frequency of the impulse signal Computation of the Hilbert transform of the phase modulated harmonic signal using the Hilbert TransformerComputation of the wrapped phase using ATAN function for real and imaginary coordinates in the complex plane Unwrapping phaseComputation of the first derivative of the phase with respect to time to obtain a signal proportional to the instantaneous angular frequency

Laser Torsional Vibration MeterSensor output signal is proportional to the instantaneous angular velocity

Incremental rotary encoder

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Band Pass Filtration

-4

-2

0

2

4

0,000 0,001 0,001 0,002 0,002Time, s

V

-10123456

0,000 0,001 0,001 0,002 0,002Time, s

V

-100

-80

-60

-40

-20

0

20

0 5000 10000 15000 20000 25000Frequency, Hz

RM

S dB

/ref 1

V

Impulse signal Phase modulated harmonic signal

-100

-80

-60

-40

-20

0

20

0 5000 10000 15000 20000 25000Frequency, Hz

RM

S dB

/ref 1

V

ICSV 15, 6-10 July 2008, Korea 23

Differentiation with Respect to Time

Impulse Response : Ideal Diff FIR Filter

-150-100-50

050

100150

0,0000 0,0005 0,0010 0,0015 0,0020Time, s

gi

Phase demodulation

Differentiation with respect to time

Phase (Angle)

Angular velocity

Frequency Response : Ideal Diff FIR Filter

20304050607080

10 100 1000 10000Frequency, Hz

Mag

nitu

de in

dB

Impulse signal

Band pass filtration

Harmonic signal

Impulse response

ICSV 15, 6-10 July 2008, Korea 24

Averaged Time Signal

-200

-100

0

100

200

300

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9Nominal revolution

deg/

s

LaserEncoder

-200

-100

0

100

200

300

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10Nominal revolution

deg/

s

LaserEncoder

390 RPM6.5 Hz40.8 rad/s

2340 deg/s

Mean rotational speed

15.4 ms

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Averaged Frequency Spectrum

-6-3036

0 20 40 60 80 100 120 140 160Order

RM

S de

g/s

Difference

0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160Order

RM

S de

g/s

EncoderLaser

63 ord

1 kHz

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Conclusion

The paper describes two methods for angular vibration measurement during rotation The first one is based on the phase demodulation of impulse signals and the second one employs a two-beam Doppler laser vibrometer. The measurement method, which is based on using the phase demodulation, was demonstrated on the incremental rotary encoderaccuracy testing and measurement of the hand drill rotation uniformity. The second presented measurement demonstrates employing the laser vibrometer. Both angular vibration measurements result in almost the same time history and frequency spectrum.

ICSV 15, 6-10 July 2008, Korea 27

Thank you for your attention

6-10 July 2008, Daejeon, Korea LASER DOPPLER VIBROMETER...

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