Calibration of Sensors for Angular Vibration...

Calibration of Sensors for Angular Vibration MeasurementsVSB – Technical University of Ostrava Faculty of Mechanical Engineering Ostrava, Czech Republic

Lubomir Smutny, Jiri Tuma, Radim Farana

IMEKO XVIII Word Congress, Rio de Janeiro, RJ, BrazilSeptember 16 – 22, 2006

© VSB-TU Ostrava, 2006 2

Outline

MotivationCalibration problemPrinciple of phase demodulation using Hilbert transformCalibration procedureSoftware tools for phase demodulationConclusion


Motivation

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

Emitted gearbox noise level is proportional to the transmission error (TE) level resulting from the gear angular vibration.

Low noise gear design …


Transducers & Signal Processing Methods for Angular Vibration Measurements

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

Time interval length measurements

Sample number & InterpolationHigh frequency oscillator (10 GHz) & Impulse counter (Rotec)

Phase demodulation

Transducers How to process impulse signals


Test Stand for Transmission Error (TE) Measurements

Back-to-back test rig

Simple gear train, tooth numbers n1, n2

Two incremental rotary encoders

PinionWheel

211

22 r

nnTE ⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ−Θ=

2Θ1Θ

r2

Measurement range: ± 10 µm (microns)


Calibration problem

Rotational speed up to 6000 RPMSampling frequency less than 65 kHz (or 256 kHz)Low price encoders (IRC)Easy to use, measurements in situ

Requirements for Measurements

Phase demodulation of impulse signals is reliable to useBoth the encoders for TE measurements are enough accurate

It was needed to prove that


Heidenhain Incremental Rotary Encoders

500 impulses per revolution 1024 impulses per revolution

Price less than 300 €

ERN 460-500 type ERN 460-1024 type

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


Laser Torsional Vibration MeterDual-beam laser transducer based on the Doppler effectLaser: Ga-Al-As diode producing 780 nm lightOutput: instantaneous changes in angular velocityMeasurement ranges:10, 100, 1000, 100000/s Frequency range: 0.3 to 1000 Hz Accuracy: ±1% of full scale Brüel & Kjær Type 2523


Signal Analyzers for measurements

9/2 channel PULSE65 kHz sampling

frequency

9/2 and 2/1-channel PULSE Analyzers

65 kHz and 256 kHz


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 of the harmonic signal?

Instantaneous envelope Ei = ?( )iii Eyarcsin=ϕ


Phase Modulation

Real phase modulated signal

Pω

-2,0-1,00,01,02,0

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

Nominal Revolution

x(t) = A cos(ωPt+∆φM(t))

PhaseModulation signal Sideband

components

Carrying component

Analytic signal


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


( )πϕ ≤∆⇒≤ samplff2

-4-2024

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

Nominal Revolution

rad

Phase Unwrapping and Linear Trend Removing

-0,15-0,10-0,050,000,050,100,15

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

Nominal Revolution

rad

02468

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

Nominal Revolution

rad

Discontinuities removing

π−

π+

π2

ϕ→π−ϕ⇒π+>ϕ∆ϕ→π+ϕ⇒π−<ϕ∆ 2,2


Calibration Procedure

Verifying the method of the phase demodulation by comparing simultaneous encoder and laser angular vibration measurementsComparing of two identical encoders mounted on the common shaft


Comparison of Laser and Encoder Angular Velocity Measurements

Laser

Encoder

1188 RPM

-20246

0,0000 0,0005 0,0010Time [s]

V-80-60-40-20

020

15000 17500 20000 22500 25000Frequency [Hz]

RM

S dB

/ref 1

V

Impulse signal 65536 Smps/s

Frequency spectrum


Differentiation with Respect to TimeImpulse 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

Impulse signal

Phase (Angle)

Angular velocity

Frequency Response : Ideal Diff FIR Filter

20304050607080

10 100 1000 10000Frequency [Hz]

Mag

nitu

de in

dB


Averaged Time Signal and Frequency Spectrum

-20

-10

0

10

20

30

40

50

60

0 50 100 150

Order [-]R

MS

dB/re

f 1 d

eg/s

Encoder Laser

-200-100

0100200

0,0 0,2 0,4 0,6 0,8 1,0Nominal Revolution [-]

Ang

ular

vel

ocity

[deg

/s]

-200-100

0100200


Ang

ular

vel

ocity

[deg

/s]

Laser Encoder


Comparison of two encoders of the ERN 460-500 type

-2024

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

Volt E1

E2

E1E2 500 impulses per revolution

Autospectrum

-100-80-60-40-20

0

300 400 500 600 700

Order [-]

RM

S dB

/ref 1

E1E2


Accuracy of Position Measurement

ε1 ε2 ε3 ε4 ε5 ε6 εN

Ideally accurate impulse position

Actual impulse position

Errors in degs …..

One complete revolution, N impulses

η1

η2

ActualIdeal

Rotation by the 1/order - part of a circle

Mean value { } 0=εE

Dispersion { } 22εσ=εE

Mean value { } 0=ηE

Dispersion { } order22 NE εσ=η

∑=

ε=ηorder

1

N

ii

Nii ,...,1, =ε

Cumulative errors

……..

Std deviation orderNεη σ=σ

Grating period

Mean value

Cumulative errors

If , then{ } ( )kiE ki <>=εε 0


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 634 RPM1040 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

Uncorrelated errors


Indoor Software Tools for Angular Vibration Measurements


Automation Program for PULSE, the BK Signal Analyser


Signal Analyser


ConclusionThe paper is focused on the calibration of the angular vibration sensorsThere are many approaches to angular vibration measurements, e.g. laser vibrometers, encodersPhase demodulation based on the theory of analytical signals gives reliable result of measurementsThe low impulse number encoders seem to be more accurate at the 1/20-part of a circle than the maximum directional error stated in the Heidenhain brochureThe theory is illustrated by the verification and comparison of the encoders.


Thank you for your attention


Measurement Examples


Effect of Phase Modulation on Impulse Signal Frequency Spectrum

Pinion 21 T Wheel 44 TEnhanced Spectrum , 21-Tooth Gear

-90-80-70-60-50-40-30-20-10

010

395 416 437 458 479 500 521 542 563 584 605

Order [-]

RM

S dB

/ref 1

V

Enhanced Spectrum , 44-Tooth Gear

-90-80-70-60-50-40-30-20-10

010

324 368 412 456 500 544 588 632 676

Order [-]R

MS

dB/re

f 1 V

Scaled in multiples of the rotational frequency

Carrying component

Sidebands


Angular Vibration of the 21and 44-Tooth Gear in Degs (after Comb Filtration)

Toothmeshing frequency harmonics with some sideband components

Time History : Pinion 21T : Enhanced Time(Impulsy500)

-0,002-0,0010,0000,0010,002

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

deg

Time History : Wheel 44T : Enhanced Time(Impulsy500)

-0,006-0,0030,0000,0030,006


deg


An Alternative Procedure

( ) ( )( ) ( ) ( ) ( )

( ) ( )tytxdt

tdytxtydt

tdx

dttdt 22 +

−=

ϕ=ω

( ) ( )( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=ϕ

txtyt arctan

( ) ( ) ( )tytxtE 22 +=

Phase ……………....

Angular frequency ...

Envelope ..………….

( ) ( ) τ∫ τω=ϕ dtt

0

Phase ……………….

Calibration of Sensors for Angular Vibration...

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