Lecture on Angular Vibration Measurements Based on Phase Demodulation

Transcription

1 Lecture on Angular Vibration Measurements Based on Phase Demodulation JiříTůma VSB Technical University of Ostrava Czech Republic

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 Jiri Tuma,

3 Motivation Angular vibration as the source of the machine vibration and noise

4 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, S F T Forces F T and F S result in torque Force F S excites gearcase vibration

5 Gear Angular Vibration deg deg/s^2 0,0016 0,0008 0,0000-0,0008-0, 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^ 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,

6 Source of car shaking while running at idle speed Crankshaft angular vibration Engine linear vibration Car body linear vibration Jiri Tuma,

7 Variation of the Angular Acceleration Variation in 3D Surface Plot Jiri Tuma, 2002 Jiri Tuma,

8 Transducers and signal processing methods

9 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,

10 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,

11 Principle of the Hilbert transform

12 Analytic Signal Property ω = 2π P f Real harmonic signal (vanishing X N ) Complex analytic signal Jiri Tuma,

13 Analytic Signal in a Helix Shape ω = 2π P f Jiri Tuma,

14 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,

15 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,

16 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,

17 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, Index n 0,0 0,0 0,2 0,4 0,6 0,8 1,0 Normalised Frequency [-] Jiri Tuma,

18 Principle of phase demodulation

19 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,

20 Phase Unwrapping and Linear Trend Removing 2π + π π 4 2 Unit ,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 rad ,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,

21 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 () t = x () t + y () t () t + y () t dy dt ( t) Jiri Tuma,

22 Gear Angular Vibration Measurements Solving the gearbox noise problem at the very source

23 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

24 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 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,

25 Instrumentation 9/2 channels PULSE Order Analysis Heidehain encoders of the ERN type (less than 300 ) Jiri Tuma,

26 Encoder Accuracy E2 E1 1, Phase difference Circle part RMS deg 0, , , , RPM 1040 RPM 1 order 0, , Order [-] Heidehain encoders of the ERN type (500 pulses per revolution) Jiri Tuma,

27 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 Heidehain encoders of the ERN type Axle Jiri Tuma,

28 Using the Fourier to evaluate the Hilbert transform

29 Effect of Phase Modulation on Pulse Frequency Spectrum RMS db/ref 1 V Enhanced Spectrum, 21-Tooth Gear RMS db/ref 1 V Enhanced Spectrum, 44-Tooth Gear Order [-] Orde r [-] Pinion 21 T Wheel 44 T Jiri Tuma,

30 Pinion Angular Vibration deg Time history : Pinion 21T : Enhanced Time(Impulsy500) 0,0 0,2 0,4 0,6 0,8 1,0 Revolution [-] Unwrapped phase deg 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,

31 Phase Modulation Frequency Spectrum -20 Autospectrum : Pinion 21T : Enhanced Time(Impulsy500) -20 Autospectrum : Wheel 44T : Enhanced Time(Impulsy500) RMS db/ref RMS db/ref 1 deg Order [-] Order [-] Pinion 21 T Wheel 44 T Jiri Tuma,

32 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 f 0 toothmeshing frequency 5 f f 0 Jiri Tuma,

33 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,

34 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,

35 Comb Filter 2 - Frequency Response H 1 ( j f ) f 0 Pass Band 0 Only harmonics of the toothmeshing frequency without sidebands f 0 toothmeshing frequency 5 f f 0 Jiri Tuma,

36 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,

37 Transmission Error (average per a tooth pitch rotation) RPM, +40 Nm 500 RPM, +80 Nm TE [micron) Tooth pitch rotation Jiri Tuma,

38 Truck Gearbox Jiri Tuma,

39 Transmission Error 2R 2N micron Nm 867 Nm 1300 Nm micron Nm 697 Nm 1045 Nm ,2 0,4 0,6 0, ,2 0,4 0,6 0,8 1 Tooth pitch rotation Tooth pitch rotation Jiri Tuma,

40 Using the FIR filter to evaluate the Hilbert transform

41 Measured Impulse Signals Impulse signals V 6 Time 3 : Time Capture Analyzer : Expanded Time(Encoder1) ; Expanded Time(Encoder2) ,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 Time [s] Frequency spectra RMS db/ref 1E Autospectrum : Time Capture Analyzer : Expanded Time(Encoder1) ; Expanded Time(Encoder2) Frequency [Hz] Jiri Tuma,

42 Filtered Impulse Signals Filtered impulse signals RMS db/ref 1 V 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] Frequency spectra of filtered signals Autospectrum 1 : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1)) ; Time 2: Real (Expanded Time(Encoder2)) Frequency [Hz] Jiri Tuma,

43 Phase Difference Unwrapped phase of impulse signals FIR Filters : Time Capture Analyzer : Time: Real (Expanded Time(Encoder1));Time: Real (Expanded Time(Encoder2)) deg 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,

44 Phase Spectrum RMS db/ref 1 deg 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)))) Frequency [Hz] Jiri Tuma,

45 Time Domain Signal IFFT of phase spectrum micron 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,

46 Transmission Error Time History Pinion micron 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)))))) Revolution [-] Wheel micron 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)))))) Revolution [-] Jiri Tuma,

47 Averaged Transmission Error 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 [-] 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,

48 Results of the gear design improvements Effect of the design improvements on the gearbox noise

49 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,

50 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,

51 Effect of Tooth Surface Modification Hluk v db S Gear train S RPM Torque Nm Hluk v db Gear train T RPM Torque Nm T1 T2 Jiri Tuma,

52 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

53 Engine rotation uniformity at idle speed Average RPM during 250 consecutive double revolutions RPM Index 800 RPM = 13.3 Hz Hz Jiri Tuma,

54 Measurements of a Car Engine Rotational Speed and Acceleration Impulse signals crankshaft 4/2 channels PULSE Order Analysis tacho & (Divider) camshaft Jiri Tuma,

55 Source of an Impulse Signal Jiri Tuma,

56 Impulse Signal = 58 impulses per revolution Impulse signal for engine control unit 4 2 V ,5 1 1,5 2 Addition of missing impulses Revolution V ,9 0,92 0,94 0,96 0,98 1 Revolution Jiri Tuma,

57 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,

58 Engine rotation uniformity at idle speed Instantaneous RPM during the 2-revolution engine cycle 800 RPM = 13.3 Hz Hz RPM ,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Revolution Jiri Tuma,

59 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 Orders RPM Filtered out Orders rad/s2 Filtered out Orders Jiri Tuma,

60 Engine Crankshaft Angular Velocity and Acceleration Angular velocity ord limit RPM ,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Revolution Angular acceleration 300 Angular acceleration ra d /s ,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,

61 Angular acceleration variation during two engine revolutions 4-cylinder // 4-stroke engine combustion cycle rad/s ,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Revolution compression cycle Jiri Tuma,

62 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 ,5 1 1,5 2 full half zero 1.5 ord = 6.6 Hz Hz Jiri Tuma,

63 Crankshaft angular acceleration frequency spectrum Index 50 rad/s ,5 1 1,5 2 2,5 3 3,5 4 Order 6.6 Hz 13.3 Hz 26.6 Hz rad/s Order Jiri Tuma,

64 Linear acceleration frequency spectra Absorber effect Engine Car body Hz 6.6 Hz Human body extra sensitive Jiri Tuma,

65 Ride comfort RMS of Acceleration 4 8 Frequency [Hz] Jiri Tuma,

66 Results Original absorber Improved absorber 0,14 0, ord = 6.6 Hz 0,1 0,08 m/s2 0,06 0,04 0, 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, Index Jiri Tuma,

67 Software Tools for Transmission Error Evaluation

68 Automation Program for PULSE, the BK Signal Analyser Jiri Tuma,

69 Signal Analyser Jiri Tuma,

70 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,

71 Thank you for your attention Jiri Tuma,