Lecture on Angular Vibration Measurements Based on Phase Demodulation JiříTůma VSB Technical University of Ostrava Czech Republic
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 Jiri Tuma, 2005 2
Motivation Angular vibration as the source of the machine vibration and noise
Angular and Linear Vibration Excitation Line of action wheel F S Pressure angle Pitch point Centre line Support point Basic circle F T Pitch circle F T force acting to the wheel at the pitch point F S force acting at the wheel support bearing F = Jiri Tuma, 2005 4 S F T Forces F T and F S result in torque Force F S excites gearcase vibration
Gear Angular Vibration deg deg/s^2 0,0016 0,0008 0,0000-0,0008-0,0016 60000 30000 0-30000 -60000 Time : Time (Enhanced Time(Encoder)) 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Revolution [-] Time : Time (Time (Enhanced Time(Encoder))) - 0 to 100 ord 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Revolution [-] Angular vibration Double differentiation Angular acceleration m/s^2 10 5 0-5 -10 Time : Order Analyzer : Enhanced Time(Vibrace H) 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Revolution [-] Linear acceleration on the gearbox housing Jiri Tuma, 2005 5
Source of car shaking while running at idle speed Crankshaft angular vibration Engine linear vibration Car body linear vibration Jiri Tuma, 2005 6
Variation of the Angular Acceleration Variation in 3D Surface Plot Jiri Tuma, 2002 Jiri Tuma, 2005 7
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) Jiri Tuma, 2005 9
How to Process Impulse Signals Time interval length measurements Sample number & Interpolation High frequency oscillator (10 GHz) & Impulse counter (Signal analyzer Rotec) Phase demodulation Jiri Tuma, 2005 10
Principle of the Hilbert transform
Analytic Signal Property ω = 2π P f Real harmonic signal (vanishing X N ) Complex analytic signal Jiri Tuma, 2005 12
Analytic Signal in a Helix Shape ω = 2π P f Jiri Tuma, 2005 13
Evaluation of Analytic Signal X = X P + X N X N π 2 Y N = j X N j π 2 Y N π 2 = j X P Z = 2X P YP = j X P j X = X N N Evaluation of the Hilbert transform using Fast Fourier Transform (FFT) Digital filters Time signal + j Hilbert transform = Analytic signal Jiri Tuma, 2005 14
Evaluation of the Hilbert Transform using FFT ( jω) FFT{ x( k) } X = X ( jω) Y ( jω) ( k) IFFT{ Y ( jω) } y = π 2 Y N = j X N YP = j X P π 2 Jiri Tuma, 2005 15
Evaluation of Analytic Signal using Digital Filter x(t) y(t) Real part Hilbert Transformer z(t) Imaginary part Frequency response function G HT ( jω e ) = j, + π > ω > 0 j, π < ω < 0 Impulse response g HT 1 + π 2π π 0, = 2 πn, ( ) ( jω n = G e ) HT e n = 2k n = 2k jωn + 1 dω = Jiri Tuma, 2005 16
Hilbert Transformer 160-order FIR filter Impulse response n = -80,,80 Frequency response function 0,8 FIR Filter Coefficients : hy160 1,2 FIR Filter FRF : ; Coefficients : hy160 0,6 0,4 0,2 0,0-0,2-0,4 Magnitude 1,0 0,8 0,6 0,4 Hilbert Transformer -0,6 0,2-0,8-20 -16-12 -8-4 0 4 8 12 16 20 Index n 0,0 0,0 0,2 0,4 0,6 0,8 1,0 Normalised Frequency [-] Jiri Tuma, 2005 17
Principle of phase demodulation
Phase Modulation 1,5 1,0 0,5 0,0-0,5-1,0-1,5 Real phase modulated signal x(t) = A cos(ω P t+ φ M (t)) Modulation signal Phase 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Revolution ω P Analytic signal Carrying component Sideband components Jiri Tuma, 2005 19
Phase Unwrapping and Linear Trend Removing 2π + π π 4 2 Unit 0-2 -4 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Revolution Discontinuities removing ( 2 f f ϕ π ) sampl ϕ < π ϕ + 2π ϕ, ϕ > +π ϕ 2π ϕ 7 6 5 4 rad 3 2 1 0 0 0,2 0,4 0,6 0,8 1 0,15 0,1 0,05 rad 0-0,05-0,1-0,15 0 0,2 0,4 0,6 0,8 1 Revolution Revolution Jiri Tuma, 2005 20
An alternative procedure Phase... Angular frequency Phase Envelope.. ϕ ω ϕ e () t () y = arctan x = dϕ dt () t = ( t) () t ( t) dx dt x y () t x() t t 2 2 t () t = ω( τ) dτ 0 2 2 () t = x () t + y () t () t + y () t dy dt ( t) Jiri Tuma, 2005 21
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 decreasing the noise level by 7 db
Measurement Principle TE Transmission error TE TE n n ( ) 2 rad = Θ2 Θ1 n n ( ) 2 m = Θ2 Θ1 r2 n, n 1 2 Θ 1, Θ 2 r 2 1 1. Teeth number. Angle of rotation [rad]. Wheel radius E 1, E 2. Incremental rotary encoders Θ 1 Θ 2 n 1 E 1 E 2 n 2 pinion wheel Jiri Tuma, 2005 24
Instrumentation 9/2 channels PULSE Order Analysis Heidehain encoders of the ERN 460-500 type (less than 300 ) Jiri Tuma, 2005 25
Encoder Accuracy E2 E1 1,000000 Phase difference Circle part RMS deg 0,100000 0,010000 0,001000 0,000100 634 RPM 1040 RPM 1 order 0,000010 0,000001 1 10 100 1000 Order [-] Heidehain encoders of the ERN 460-500 type (500 pulses per revolution) Jiri Tuma, 2005 26
Measurement Arrangement Car gearbox 21 V I REV II III IV 21 Engine E 1 E 2 4/2 channels PULSE Order Analysis & Special software 44 44 Heidehain encoders of the ERN 460-500 type Axle Jiri Tuma, 2005 27
Using the Fourier to evaluate the Hilbert transform
Effect of Phase Modulation on Pulse Frequency Spectrum RMS db/ref 1 V 10 0-10 -20-30 -40-50 -60-70 -80-90 Enhanced Spectrum, 21-Tooth Gear 395 416 437 458 479 500 521 542 563 584 605 RMS db/ref 1 V 10 0-10 -20-30 -40-50 -60-70 -80-90 Enhanced Spectrum, 44-Tooth Gear 324 368 412 456 500 544 588 632 676 Order [-] Orde r [-] Pinion 21 T Wheel 44 T Jiri Tuma, 2005 29
Pinion Angular Vibration deg 200000 180000 160000 140000 120000 100000 80000 60000 40000 20000 0 Time history : Pinion 21T : Enhanced Time(Impulsy500) 0,0 0,2 0,4 0,6 0,8 1,0 Revolution [-] Unwrapped phase deg 12 10 8 6 4 2 0-2 -4-6 -8 Time fázová demodulace pastorek : Pinion 21T : Enhanced Time(Impulsy500) 0,0 0,2 0,4 0,6 0,8 1,0 Revolution [-] Phase variation Jiri Tuma, 2005 30
Phase Modulation Frequency Spectrum -20 Autospectrum : Pinion 21T : Enhanced Time(Impulsy500) -20 Autospectrum : Wheel 44T : Enhanced Time(Impulsy500) -30-30 -40-40 RMS db/ref 1-50 -60-70 -80-90 RMS db/ref 1 deg -50-60 -70-80 -90-100 -100-110 -110-120 0 21 42 63 84 105 126-120 0 44 88 132 176 220 Order [-] Order [-] Pinion 21 T Wheel 44 T Jiri Tuma, 2005 31
Comb Filter 1 - Frequency Response H 1 ( j f ) f 0 Pass Band 5 harmonics of the toothmeshing frequency with the limited number of sidebands 0 0 1 2 3 4 f 0 toothmeshing frequency 5 f f 0 Jiri Tuma, 2005 32
Angular Vibration of the 21-Tooth Gear in Deg (after Comb Filtration) Toothmeshing frequency harmonics with 3 sideband components deg Time History : Pinion 21T : Enhanced Time(Impulsy500) 0,0020 0,0015 0,0010 0,0005 0,0000-0,0005-0,0010-0,0015-0,0020 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Revolution [-] Jiri Tuma, 2005 33
Angular Vibration of the 44-Tooth Gear in Deg (after Comb Filtration) Toothmeshing frequency harmonics with 6 sideband components 0,006 Time History : Wheel 44T : Enhanced Time(Impulsy500) 0,004 0,002 deg 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 1,0 Revolution [-] Jiri Tuma, 2005 34
Comb Filter 2 - Frequency Response H 1 ( j f ) f 0 Pass Band 0 Only harmonics of the toothmeshing frequency without sidebands 0 0 1 2 3 4 f 0 toothmeshing frequency 5 f f 0 Jiri Tuma, 2005 35
Phase Delay Between Signals Original delay Zero delay 1,5 1,5 1,0 1,0 0,5 0,5 m/s^2 0,0-0,5-1,0 m/s^2 0,0-0,5-1,0-1,5-1,5-2,0 0,0 0,2 0,4 0,6 0,8 1,0-2,0 0,0 0,2 0,4 0,6 0,8 1,0 Tooth pitch rotation [-] Tooth pitch rotation [-] Vibration signal synchronized with pinion rotation Vibration Signal synchronized with pinion rotation Vibration signal synchronized with wheel rotation Vibration signal delayed by phase shift Jiri Tuma, 2005 36
Transmission Error (average per a tooth pitch rotation) 4 3 2 500 RPM, +40 Nm 500 RPM, +80 Nm TE [micron) 1 0-1 -2-3 -4 0 1 2 3 Tooth pitch rotation Jiri Tuma, 2005 37
Truck Gearbox Jiri Tuma, 2005 38
Transmission Error 2R 2N 6 6 4 4 micron 2 0-2 433 Nm 867 Nm 1300 Nm micron 2 0-2 348 Nm 697 Nm 1045 Nm -4-4 -6 0 0,2 0,4 0,6 0,8 1-6 0 0,2 0,4 0,6 0,8 1 Tooth pitch rotation Tooth pitch rotation Jiri Tuma, 2005 39
Using the FIR filter to evaluate the Hilbert transform
Measured Impulse Signals Impulse signals V 6 Time 3 : Time Capture Analyzer : Expanded Time(Encoder1) ; Expanded Time(Encoder2) 4 2 0-2 0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 Time [s] Frequency spectra RMS db/ref 1E-6 140 120 100 80 60 40 Autospectrum : Time Capture Analyzer : Expanded Time(Encoder1) ; Expanded Time(Encoder2) 0 5000 10000 15000 20000 25000 Frequency [Hz] Jiri Tuma, 2005 41
Filtered Impulse Signals Filtered impulse signals RMS db/ref 1 V 4 2 0-2 -4 20 Time : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1)) ; Time 2: Real (Expanded Time(Encoder2)) 0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 Time [s] 0-20 -40-60 -80 Frequency spectra of filtered signals Autospectrum 1 : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1)) ; Time 2: Real (Expanded Time(Encoder2)) 0 5000 10000 15000 20000 25000 Frequency [Hz] Jiri Tuma, 2005 42
Phase Difference Unwrapped phase of impulse signals 4000000 FIR Filters : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1));Time: Real (Expanded Time(Encoder2)) 3000000 deg 2000000 1000000 deg 0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Time [s] Phase difference (Signal1 Signal2 * 27/44) 0,10 0,05 0,00-0,05-0,10 Difference : Time Capture Analyzer : FIR Filters: Unwrapped Phase (Time: Real (Expanded Time(Encoder1))) 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Time [s] Jiri Tuma, 2005 43
Phase Spectrum RMS db/ref 1 deg 40 30 20 10 0-10 -20-30 -40 Phase 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)))) 0 1000 2000 3000 4000 5000 Frequency [Hz] Jiri Tuma, 2005 44
Time Domain Signal IFFT of phase spectrum micron 60 50 40 30 20 10 0-10 -20-30 -40-50 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))))) 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Time [s] Jiri Tuma, 2005 45
Transmission Error Time History Pinion micron 10 5 0-5 -10 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)))))) 0 2 4 6 8 10 12 14 Revolution [-] Wheel micron 10 5 0-5 -10 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)))))) 0 1 2 3 4 5 6 7 8 Revolution [-] Jiri Tuma, 2005 46
Averaged Transmission Error 4 3 2 1 0-1 -2-3 -4-5 Pinion 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 0,0 0,2 0,4 0,6 0,8 1,0 Revolution [-] 4 3 2 1 0-1 -2-3 -4-5 Wheel 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)))))) 0,0 0,2 0,4 0,6 0,8 1,0 Revolution [-] Jiri Tuma, 2005 47
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 Truck Gearbox ( ε 1.0) β ε γ ε α total contact ratio = profile contact ratio + face contact ratio ε β LCR HCR Jiri Tuma, 2005 49
Effect of Contact Ratio on the Noise Level in db db(a) 100,0 98,0 96,0 94,0 92,0 90,0 88,0 86,0 Truck gearbox noise level at the distance of 1m 3R 3N 4R 4N 5R 5N LCR 92,0 92,9 95,0 95,4 95,0 96,5 HCR 90,0 91,8 90,4 89,7 88,2 90,3 Speed Jiri Tuma, 2005 50
Effect of Tooth Surface Modification Hluk v db 88-92 84-88 80-84 76-80 72-76 68-72 64-68 60-64 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 S Gear train S RPM Torque Nm 805 1001 1245 1549 1968 2448 3080 22003831 Hluk v db 88-92 84-88 80-84 76-80 72-76 68-72 64-68 60-64 Gear train T2 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 RPM Torque Nm 1771 2203 2740 3408 4330 22005386 T1 T2 Jiri Tuma, 2005 51
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 idle speed Average RPM during 250 consecutive double revolutions 820 810 RPM 800 790 780 0 50 100 150 200 Index 800 RPM = 13.3 Hz Hz Jiri Tuma, 2005 53
Measurements of a Car Engine Rotational Speed and Acceleration Impulse signals crankshaft 4/2 channels PULSE Order Analysis tacho & (Divider) camshaft Jiri Tuma, 2005 54
Source of an Impulse Signal Jiri Tuma, 2005 55
Impulse Signal 6 60 2 = 58 impulses per revolution Impulse signal for engine control unit 4 2 V 0-2 -4-6 0 0,5 1 1,5 2 Addition of missing impulses Revolution 6 4 2 V 0-2 -4-6 0,9 0,92 0,94 0,96 0,98 1 Revolution Jiri Tuma, 2005 56
Angular Variation 1,5 1 0,5 deg 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 Jiri Tuma, 2005 57
Engine rotation uniformity at idle speed Instantaneous RPM during the 2-revolution engine cycle 800 RPM = 13.3 Hz Hz 830 820 810 RPM 800 790 780 770 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Revolution Jiri Tuma, 2005 58
Differentiation in the Frequency Domain Angle Velocity Acceleration ϕ t, Φ jω ω = d ϕ dt, Ω = jωφ ε = d ω dt, Ε = jωω 1,2 1 0,8 0,6 0,4 0,2 ( ) ( ) 0 deg 0 6 12 Orders 8 7 6 5 4 3 2 1 0 RPM Filtered out 0 6 12 Orders 120 100 80 60 40 20 0 rad/s2 Filtered out 0 6 12 Orders Jiri Tuma, 2005 59
Engine Crankshaft Angular Velocity and Acceleration Angular velocity 830 6 ord limit 820 810 RPM 800 790 780 770 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Revolution Angular acceleration 300 Angular acceleration 250 200 150 100 ra d /s 2 5 0 0-5 0-1 0 0-1 5 0-2 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 lu tio n Jiri Tuma, 2005 60
Angular acceleration variation during two engine revolutions 4-cylinder // 4-stroke engine + 300 250 200 150 100 combustion cycle rad/s2 50 0-50 -100 - -150-200 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Revolution compression cycle Jiri Tuma, 2005 61
Effect of sinusoidal signal distortion on its frequency spectrum 1 1 0,5 0-0,5 full half zero 0,8 0,6 0,4 0,2 0.5 ord 1 ord 1.5 ord 2 ord -1 0 0 0,5 1 1,5 2 full half zero 1.5 ord = 6.6 Hz Hz Jiri Tuma, 2005 62
Crankshaft angular acceleration frequency spectrum 90 80 70 240 60 Index 50 rad/s2 40 30 20 10 0 0 0,5 1 1,5 2 2,5 3 3,5 4 Order 6.6 Hz 13.3 Hz 26.6 Hz 200 160 120 80 40 rad/s2 70-80 60-70 50-60 40-50 30-40 20-30 10-20 0-10 0 0 1 2 3 4 Order Jiri Tuma, 2005 63
Linear acceleration frequency spectra Absorber effect Engine Car body 6.6-13.3-26.6 Hz 6.6 Hz Human body extra sensitive Jiri Tuma, 2005 64
Ride comfort RMS of Acceleration 4 8 Frequency [Hz] Jiri Tuma, 2005 65
Results Original absorber Improved absorber 0,14 0,12 0.5 ord = 6.6 Hz 0,1 0,08 m/s2 0,06 0,04 0,02 0 1 11 21 31 41 51 61 71 81 91 101 111 121 Index 0,09 0,08 0,07 1 ord = 13.3 Hz 0,06 m/s2 0,05 0,04 0,03 0,02 0,01 0 1 11 21 31 41 51 61 71 81 91 101 111 121 Index Jiri Tuma, 2005 66
Software Tools for Transmission Error Evaluation
Automation Program for PULSE, the BK Signal Analyser Jiri Tuma, 2005 68
Signal Analyser Jiri Tuma, 2005 69
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. Jiri Tuma, 2005 70
Thank you for your attention Jiri Tuma, 2005 71