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An Enveloping Function for Signal
Processing of Mechanical Vibration
- Frequency Analysis
沈毓泰
南台科技大學 機械系 副教授
2004/06/5~6
南台科技大學
1
1. Introduction

High Frequency Resonance Technique

The traditional methods



The filter-based Hilbert transform
The FFT-based Hilbert transform
The wavelet methods


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The wavelet transform
The wavelet-based Hilbert transform
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Modulation

Defect impacts  system resonance
1.5
1
Amplitude
0.5
0
-0.5
-1
-1.5
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0
0.005
0.01
Time (Sec)
0.015
南台科技大學
0.02
3
Demodulation

Band-pass filter

Envelope detection
1
Amplitude
0.75
0.5
0.25
0
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0
0.005
0.01
0.015
Time (Sec)
0.02
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0.025
0.03
4
2. A enveloping function for
signal processing

ha , fC , fW (t ) 



1
j t
e
t
- ( )2
a
(e
- j 2 ( f C 
fW
)t
2
e
- j 2 ( f C 
fW
)t
2
)
a : the scaling factor
fC : the center frequency
fW : the passband width
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4000
4000
2000
2000
Amplitude
Amplitude
a=0.1
0
- 2000
- 4000
- 0.002 - 0.001
HL
0
Time sec
0.001
0
- 2000
- 4000
- 0.002 - 0.001
0.002
Amplitude
Amplitude
0.75
0.5
0.75
0.5
3000
4000
0
1000
5000
HL
2000
90
Phas e degree
HL
Phas e degree
0.002
0.25
0.25
45
0
- 45
- 90
1000
0.001
1
1
0
1000
HL
0
Time sec
2000
HL
3000
4000
Frequency Hz
5000
2000
3000
4000
5000
90
45
0
- 45
- 90
1000
2000
HL
3000
4000
Frequency Hz
5000
4000
4000
2000
2000
Amplitude
Amplitude
a=0.001
0
- 2000
- 4000
- 0.002 - 0.001
HL
0
Time sec
0.001
0
- 2000
- 4000
- 0.002 - 0.001
0.002
Amplitude
Amplitude
0.75
0.5
4000
0.75
0.5
3000
4000
0
1000
5000
HL
2000
90
Phas e degree
HL
Phas e degree
3000
0.002
0.25
0.25
45
0
- 45
- 90
1000
0.001
1
1
0
1000
HL
0
Time sec
2000
HL
3000
4000
Frequency Hz
5000
2000
5000
90
45
0
- 45
- 90
1000
2000
HL
3000
4000
Frequency Hz
5000
Parameter characteristics



the slope attenuation is increased with
the dilation a
the range 0.007 ≤ a ≤ 0.015
the waveform interval is chosen to be
0.01 second
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High frequency resonance technique

For a vibration signal x(t), the demodulated
signal could then be described by the
convolution integral :

ya , fC , fW (t )   x( ) ha , fC , fW (t   ) d


The frequency-domain analysis


Ea , fC , fW ( f )   ea , fC , fW (t )
3. Simulation
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The effect of waveform interval
with additive noise
3
Frequency
2
1
0
-1
-2
-3
0
0.02
0.04
0.06
Time(sec)
0.08
0.1
3D envelope spectrum
4. Experimental study
Normal bearing
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Defect bearing
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5. Conclusion

Three parameters of the enveloping
function are capable of designating the
passband and adjusting the passband
quality , which are center frequency,
passband width and dilation scale.
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
The enveloping function provides an easy
way to sweep filtering passband from a low
frequency band to a high frequency band,
and to derive a 3D envelope spectrum. It
would be helpful in diagnosing the defect
of a mechanical system.
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敬 請 指 正
謝 謝
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