VSB Technical University of Ostrava Czech Republic

Lecture on Angular Vibration

Measurements Based on Phase Demodulation

Jiří TůmaVSB Technical University of Ostrava

Czech Republic

© Jiri Tuma, 2005 2

Outline

• Motivation• Principle of phase demodulation using

Hilbert transform• Gear angular vibration measurements• Transmission error (TE) measurements• Measurements of the car engine rotational speed

uniformity• Software tools for phase demodulation

Motivation

Angular vibration as the source of the machine vibration and noise


Angular and Linear Vibration Excitation

Centre line Pitch

circle

Basic circle

Line of action

Pressure angle

Pitch point

Support point

FTFSTS FF =

FT force acting to the wheel at the pitch point

FS force acting at the wheel support bearing

Forces FT and FSresult in torquewheel Force FS excites gearcase vibration


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

-0,0016-0,00080,00000,00080,0016

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

Revolution [-]

deg

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

-60000

-30000

0

30000

60000

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

Revolution [-]

deg/

s^2

Time : Order Analyzer : Enhanced Time(Vibrace H)

-10

-5

0

5

10

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0Revolution [-]

m/s

^2

Linear acceleration on the gearbox housing

Angular acceleration

Angular vibration

Double differentiation


Source of car shaking while running at idle speed

Crankshaft angular vibration

Engine linear vibration

Car body linear vibration


Variation of the Angular Acceleration Variation in 3D Surface Plot

© Jiri Tuma, 2002

Transducers and signal processing methods


Transducers for Angular Vibration Measurements

• Tangentially mounted accelerometers• Laser Torsional Vibration Meter

(Doppler effect)• Incremental rotary encoders

(several hundreds of pulses per revolution)


How to Process Impulse Signals

• Time interval length measurements– Sample number & Interpolation– High frequency oscillator (10 GHz) & Impulse

counter (Signal analyzer Rotec)• Phase demodulation

Principle of the Hilbert transform


Analytic Signal Property

fP π=ω 2

Real harmonic signal

Complex analytic signal

(vanishing XN)


Analytic Signal in a Helix Shape

fP π=ω 2


Analytic signal

PXZ 2=

PP XjY −=NN XjY =

NNN XXjjYj −==

NP XXX +=

Evaluation of Analytic Signal

PXNX

2π

2π

−

2π

Hilbert transform =+ jTime signal

• Fast Fourier Transform (FFT)

• Digital filters

Evaluation of the Hilbert transform using …


Evaluation of the Hilbert Transform using FFT

2π

2π

( ) ( ) ωjYIFFTky =( ) ( )ωω jYjX →

( ) ( ) kxFFTjX =ω

NN XjY = PP XjY −=


Evaluation of Analytic Signal using Digital Filter

HilbertTransformer

x(t)y(t)

z(t)

Real part

Imaginary part

Frequency response function

( )⎩⎨⎧

<<−>>+−

=0,0,

ωπωπω

jj

eG jHT

( ) ( )

⎩⎨⎧

+==

=

=∫=+

−

12,22,0

21

knnkn

deeGng njjHTHT

π

ωπ

π

π

ωω

Impulse response


Hilbert Transformer

FIR Filter Coefficients : hy160

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

-20 -16 -12 -8 -4 0 4 8 12 16 20Index n

Frequency response functionImpulse response n = -80,…,80FIR Filter FRF : ; Coefficients : hy160

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0Normalised Frequency [-]

Mag

nitu

de

160-order FIR filter

Hilbert Transformer

Principle of phase demodulation


Phase ModulationReal phase modulated

signal

Pω

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

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

Revolution

x(t) = A cos(ωPt+∆φM(t))

PhaseModulation signal Sideband

components

Carrying component

Analytic signal


( )πϕ ≤∆⇒≤ samplff2

-4

-2

0

2

4

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

Revolution

Unit

Phase Unwrapping and Linear Trend Removing

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

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

Revolution

rad

01234567

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

Revolution

rad

Discontinuities removing

π−

π+

π2

ϕ→π−ϕ⇒π+>ϕ∆ϕ→π+ϕ⇒π−<ϕ∆ 2,2


An alternative procedure

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

( ) ( )tytxdt

tdytxtydt

tdx

dttdt 22 +

−=

ϕ=ω

( ) ( )( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=ϕ

txtyt arctan

( ) ( ) ( )tytxte 22 +=

Phase ……………....

Angular frequency …

Envelope ..…………

( ) ( ) τ∫ τω=ϕ dtt

0Phase ………………

Gear Angular Vibration Measurements

Solving the gearbox noise problem at the very source

Transmission error measurements

Emitted gearbox noise level is proportional to the transmission error level

decreasing TE by 10 dB results in decreasingthe noise level by 7 dB


Measurement Principle

( ) 11

22 Θ−Θ=

nnradTE

( ) 211

22 r

nnmTE ⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ−Θ=

21, nn

2r…. Angle of rotation [rad]21, ΘΘ

…. Teeth number

…. Wheel radius

TE – Transmission error

E1

E2

n1

n2

1Θ

2Θ

pinion

wheel

E1, E2…. Incremental rotary encoders


Instrumentation

9/2 channels PULSEOrder Analysis

Heidehain encoders of the ERN 460-500 type

(less than 300 €)


Encoder Accuracy

E1E2

order1

Circle part

Heidehain encoders of the ERN 460-500 type(500 pulses per revolution)

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


Measurement Arrangement


&Special software

Heidehain encoders of the ERN 460-500 type

I REV II III IVV

21

44

21

44

E1

E2

Axle

Engine

Car gearbox

Using the Fourier to evaluate the Hilbert transform


Effect of Phase Modulation on Pulse Frequency Spectrum

Pinion 21 T Wheel 44 T

Enhanced 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


Pinion Angular Vibration

Unwrapped phase Phase variation

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

0

20000

40000

60000

80000

100000120000

140000

160000

180000

200000

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

Revolution [-]

deg

Time fázová demodulace pastorek : Pinion 21T : Enhanced Time(Impulsy500)

-8

-6

-4

-2

0

2

4

6

8

10

12

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

Revolution [-]de

g


Phase Modulation Frequency Spectrum

Pinion 21 T Wheel 44 T

Autospectrum : Pinion 21T : Enhanced Time(Impulsy500)

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

0 21 42 63 84 105 126

Order [-]

RM

S d

B/re

f 1

Autospectrum : Wheel 44T : Enhanced Time(Impulsy500)

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

0 44 88 132 176 220

Order [-]R

MS

dB/

ref 1

deg


Comb Filter 1 - Frequency Response

0 1 2 3 40

1

5

( )0ffjH

0fff0 …toothmeshing frequency

Pass Band

5 harmonics of the toothmeshing frequency with

the limited number of sidebands


Angular Vibration of the 21-Tooth Gear in Deg (after Comb Filtration)

Toothmeshing frequency harmonics with 3 sideband components

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

-0,0020-0,0015-0,0010-0,00050,00000,00050,00100,00150,0020

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0Revolution [-]

deg


Angular Vibration of the 44-Tooth Gear in Deg (after Comb Filtration)Toothmeshing frequency harmonics with 6 sideband components

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

-0,006

-0,004

-0,002

0,000

0,002

0,004

0,006

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0Revolution [-]

deg


Comb Filter 2 - Frequency Response

0 1 2 3 40

1

5

( )0ffjH

0fff0 …toothmeshing frequency

Pass Band 0

Only harmonics of the toothmeshing frequency

without sidebands


Phase Delay Between Signals

Original delayOriginal delay Zero delayZero delay

-2,0-1,5-1,0-0,50,00,51,01,5

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

Tooth pitch rotation [-]

m/s

^2

V ibration s ignal synchronized w ith pinion rotation Vibration s ignal synchronized w ith w heel rotation

-2,0-1,5-1,0-0,50,00,51,01,5

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

Tooth pitch rotation [-]

m/s

^2V ibration Signal synchronized w ith pinion rotation Vibration s ignal de layed by phase shift


Transmission Error(average per a tooth pitch rotation)

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3Tooth pitch rotation

TE [m

icro

n)

500 RPM, +40 Nm 500 RPM, +80 Nm


Truck Gearbox


Transmission Error

2R

-6

-4

-2

0

2

4

6

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

Tooth pitch rotation

mic

ron 433 Nm

867 Nm1300 Nm

2N

-6

-4

-2

0

2

4

6

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

Tooth pitch rotation

mic

ron 348 Nm

697 Nm1045 Nm

Using the FIR filter to evaluate the Hilbert transform


Measured Impulse Signals

Time 3 : Time Capture Analyzer : Expanded Time(Encoder1) ; Expanded Time(Encoder2)

-20246

0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035

Time [s]

V

Impulse signals

Frequency spectraAutospectrum : Time Capture Analyzer : Expanded Time(Encoder1) ; Expanded Time(Encoder2)

40

60

80

100

120

140

0 5000 10000 15000 20000 25000

Frequency [Hz]

RM

S dB

/ref 1

E-6


Filtered Impulse Signals

Filtered impulse signals

Frequency spectra of filtered signals

Time : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1)) ; Time 2: Real (Expanded Time(Encoder2))

-4-2024

0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035Time [s]

Autospectrum 1 : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1)) ; Time 2: Real (Expanded Time(Encoder2))

-80

-60

-40-20

0

20

0 5000 10000 15000 20000 25000Frequency [Hz]

RM

S dB

/ref 1

V


Phase Difference

Unwrapped phase of impulse signals

Phase difference (Signal1 – Signal2 * 27/44)

FIR Filters : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1));Time: Real (Expanded Time(Encoder2))

0

1000000

2000000

3000000

4000000

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

Time [s]

deg

Difference : Time Capture Analyzer : FIR Filters: Unwrapped Phase (Time: Real (Expanded Time(Encoder1)))

-0,10-0,050,000,050,10

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9Time [s]

deg


Phase SpectrumPhase spectrum

Autospectrum 2 : Time Capture Analyzer : Difference (FIR Filters: Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) - FIR Filters: Unwrapped Phase (Time 2: Real (Expanded

Time(Encoder2))))

-40

-30

-20

-10

0

10

20

30

40

0 1000 2000 3000 4000 5000

Frequency [Hz]

RM

S dB

/ref 1

deg


Time Domain SignalIFFT of phase spectrum

Time History : Time Capture Analyzer : Time 1: Real (Difference (FIR Filters: Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) - FIR Filters: Unwrapped Phase (Time 2: Real (Expanded

Time(Encoder2)))))

-50-40-30-20-10

0102030405060

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

Time [s]

mic

ron


Transmission Error Time History

Time History : Time Capture Analyzer : Resampling (Time 1: Real (Difference (FIR Filters: Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) - FIR Filters: Unwrapped Phase (Time 2: Real (Expanded

Time(Encoder2))))))

-10-505

10

0 1 2 3 4 5 6 7 8Revolution [-]

mic

ron

Time History : Time Capture Analyzer : Resampling 1 (Time 1: Real (Difference (FIR Filters: Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) - FIR Filters: Unwrapped Phase (Time 2: Real (Expanded

Time(Encoder2))))))

-10-505

10

0 2 4 6 8 10 12 14Revolution [-]

mic

ron

Pinion

Wheel


Averaged Transmission Error

Time History : Time Capture Analyzer : Resampling 1: Averaged (Time 1: Real (Difference (FIR Filters:

Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) - FIR Filters: Unwrapped Phase

(Time 2: Real (Expanded Time(Encoder2))))))1

-5-4-3-2-101234

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

Revolution [-]

Time History : Time Capture Analyzer : Resampling: Averaged (Time 1: Real (Difference (FIR Filters:

Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) - FIR Filters: Unwrapped Phase

(Time 2: Real (Expanded Time(Encoder2))))))

-5-4-3-2-101234

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

Revolution [-]

Pinion Wheel

Results of the gear design improvements

Effect of the design improvements on the gearbox noise


Effect of Contact Ratio on the Average Toothmesh Acceleration Signal

total contact ratio = profile contact ratio + face contact ratioγε

( )0.1≈εβ

αε βε

Truck Gearbox

LCR HCR


Effect of Contact Ratio on the Noise Level in dB

Truck gearbox noise level at the distance of 1m

86,0

88,0

90,0

92,0

94,0

96,0

98,0

100,0

Speed

dB(A

)

LCR 92,0 92,9 95,0 95,4 95,0 96,5HCR 90,0 91,8 90,4 89,7 88,2 90,3

3R 3N 4R 4N 5R 5N


Effect of Tooth Surface Modification

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

5386

4330

3408

2740

2203

1771Hluk v

dB

RPM

88-9284-8880-8476-8072-7668-7264-6860-64

Torque Nm

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

3831

3080

2448

1968

1549

1245

1001

805Hluk v

dB

RPM

88-9284-8880-8476-8072-7668-7264-6860-64

Torque NmGear train S Gear train T2

S

T1T2

Measurements of a car engine rotational speed variation

Solving the problem of a car with random burst shaking while its engine is running in idle

Car body vibrations correlate with changes in engine rotational speed


Engine rotation uniformity at idlespeed

780

790

800

810

820

0 50 100 150 200

Index

RPM

Average RPM during 250 consecutive double revolutions

800 RPM = 13.3 Hz800 RPM = 13.3 Hz


Measurements of a Car EngineRotational Speed and Acceleration


&tacho

crankshaft

camshaft

Impulse signals

(Divider)


Source of an Impulse Signal


Impulse Signal

-6

-4

-2

0

2

4

6

0 0,5 1 1,5 2

Revolution

V

-6-4-20246

0,9 0,92 0,94 0,96 0,98 1

Revolution

V

Impulse signal for engine control unit

Addition ofmissing impulses

60 – 2 = 58 impulses per revolution


Angular Variation

-1,5

-1

-0,5

0

0,5

1

1,5

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

Revolution

deg


Engine rotation uniformity at idlespeed

770

780

790

800

810

820

830

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

Revolution

RPM

Instantaneous RPM during the 2-revolution engine cycle

800 RPM = 13.3 Hz800 RPM = 13.3 Hz


Differentiation in the FrequencyDomain

0

0,2

0,4

0,6

0,8

1

1,2

0 6 12Orders

deg

Angle Velocity Acceleration

0

1

2

3

4

5

6

7

8

0 6 12Orders

RPM

0

20

40

60

80

100

120

0 6 12Orders

rad/s2

Filteredout

Filteredout

( ) ( )ωΦϕ jt , Φω=Ωϕ=ω jdtd , Ωω=Εω=ε jdtd ,


Engine Crankshaft Angular Velocityand Acceleration

7 7 0

7 8 0

7 9 0

8 0 0

8 1 0

8 2 0

8 3 0

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

R e vo lu tio n

R P M

- 2 0 0

- 1 5 0

- 1 0 0

- 5 0

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

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

R e v o l u t i o n

r a d / s 2

Angular accelerationAngular acceleration

Angular velocityAngular velocity 6 ord limit6 ord limit


Angular acceleration variation during two engine revolutions

-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

4-cylinder / 4-stroke engine4-cylinder / 4-stroke engine

compression cyclecompression cycle

combustion cyclecombustion cycle

-

+


Effect of sinusoidal signal distortion on its frequency spectrum

-1

-0,5

0

0,5

1

0 0,5 1 1,5 2

full

half

zero

0

0,2

0,4

0,6

0,8

1

full half zero

0.5 ord

1 ord

1.5 ord

2 ord

1.5 ord = 6.6 Hz1.5 ord = 6.6 Hz


Crankshaft angular acceleration frequency spectrum

0

10

20

30

40

50

60

70

80

90

0 0,5 1 1,5 2 2,5 3 3,5 4

Order

rad/s2

0 1 2 3 40

40

80

120

160

200

240

rad/s 2

Order

Index

70-8060-70

50-6040-50

30-4020-3010-20

0-10

6.6 Hz 13.3 Hz 26.6 Hz


Linear acceleration frequency spectra

EngineEngine Car bodyCar body

6.6 Hz6.6 - 13.3 - 26.6 Hz

Absorber effect

Human body extra sensitive


Ride comfort

Frequency [Hz]

RMS of Acceleration

84


Results

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

1 11 21 31 41 51 61 71 81 91 101 111 121Index

m/s

2

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

1 11 21 31 41 51 61 71 81 91 101 111 121Index

m/s

2

0.5 ord = 6.6 Hz

1 ord = 13.3 Hz

Original absorber Improved absorber

Software Tools for Transmission Error

Evaluation


Automation Program for PULSE, the BK Signal Analyser


Signal Analyser


Conclusion

• The lecture is focused on the problem of the angular vibration measurements using phase demodulation

• The shaft angular vibration excite the housing linear vibration and consequently machine noise

• The theory is illustrated by experimental data.


Thank you for your attention

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VSB Technical University of Ostrava Czech Republic