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Columbia UniversityMechanical Engineering
Control Lab Signal Processing Lab1
Signal Processing Experiment
• Two parts– Investigate the Fourier spectrum of
several periodic nonlinear signals– Investigate the artifacts of sampling (data
acquisition)
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Control Lab Signal Processing Lab2
Agilent Arbitrary Waveform Generator
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Control Lab Signal Processing Lab3
Arbitrary Waveform Generator
Control Panel
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Control Lab Signal Processing Lab4
Arbitrary Waveform GeneratorRear Panel
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Control Lab Signal Processing Lab5
Programming• Use LabVIEW program to take analog signals from the
Arbitrary Waveform Generator• Display analog data vs. time (compare to oscilloscope)• Display analog data vs. frequency (program spectrum
analyzer)
• Open Intuilink software to communicate with the Arbitrary Waveform Generator
• Program generator waveform• Start Generator (continuous waveform)• Start data acquisition
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Control Lab Signal Processing Lab6
Fourier SpectrumInvestigate spectrum of non-linear waveforms using
Fast Fourier Transform (FFT) in LabVIEW1. Investigate spectrum of pure sine wave2. Investigate popular nonlinear waveforms – square
wave, pulse train, triangle wave (compare results to theory)
3. Investigate typical modulated waveforms – amplitude modulation (AM) & frequency modulation (FM) (compare results to theory)
4. Investigate the effect of adding bias (offset) to signals5. Investigate spectrum of arbitrary nonlinear signal
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Control Lab Signal Processing Lab7
Different Waveforms
• Look up Fourier Spectrum for these signals– Single frequency sine wave– Square wave– Triangle wave– Pulse Waveform (non symmetric square wave)
• Fourier Spectrum of arbitrary waveform• Effects of adding offset
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Control Lab Signal Processing Lab8
Single frequency sine wave
Square wave
Triangle wave
Pulse Waveform
Effects of adding offset
0
0
0
0
0
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Control Lab Signal Processing Lab9
Arbitrary Waveform
• Free-hand draw some arbitrary waveform
• Provide the waveform (vs. time) and the spectrum (vs. frequency) in report
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Control Lab Signal Processing Lab10
Programming theArbitrary Waveform Generator
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Control Lab Signal Processing Lab11
Programming theArbitrary Waveform Generator
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Control Lab Signal Processing Lab13
Use Waveform Editor to Draw Arbitrary Waveform
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Control Lab Signal Processing Lab14
Use Waveform Math to Draw Arbitrary Waveform
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Control Lab Signal Processing Lab15
Sending Waveform to Generator
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Control Lab Signal Processing Lab16
Use Equation Calculator to Draw Standard Waveforms (Sine, Square, Triangle) & Modulated Waveforms
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Control Lab Signal Processing Lab17
Amplitude Modulated (AM) Waveform
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Control Lab Signal Processing Lab18
Sine Waves at 2 Frequencies
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Control Lab Signal Processing Lab19
Frequency Modulated (FM) Waveform
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Control Lab Signal Processing Lab20
Modulated Waveforms
• Single frequency tone modulation in both cases– Amplitude modulated signal (AM wave)– Frequency modulated signal (FM wave)
• Vary tone frequency keeping modulation constant
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Control Lab Signal Processing Lab25
Part 2 – Sampling Artifacts
• Signal-to-Noise Ratio– Signal Detection– Effects of Quantization
• Discrimination & Dynamic Range (sample size)
• Aliasing & Bandwidth (sampling frequency)
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Control Lab Signal Processing Lab26
Why Does Spectrum of Pure Sine Wave Appear to Have Noise Associated with It?
frequency
amplitude
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Control Lab Signal Processing Lab27
Sampling in Time
100 200150500Time (s)
0
1.25
5.00
2.50
3.75
6.25
7.50
8.75
10.00
Amplitude(volts)
40kHz Sampled System(5kHz Sine Wave)
| ||||
Analog SignalSampled Signal
As sampling frequency increases, the digital signal better approximates the analog signal.
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Control Lab Signal Processing Lab28
Quantization in Amplitude
100 200150500Time (s)
0
1.25
5.00
2.50
3.75
6.25
7.50
8.75
10.00
Amplitude(volts)
16 Bit Versus 3 Bit Resolution(5kHz Sine Wave)
16-bit
3-bit
000
001
010
011
100
101
110
111
| ||||
As resolution increases, the digital signal better approximates the analog signal.
The maximum quantization error is ½ the increment size.
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Control Lab Signal Processing Lab29
Quantization Noise
4 bit (16 level) A/D
Difference looks like noise on signal
If you assume that the noise is uniformly distributed & uncorrelated with the signal then the rms (root mean square) value of the noise is
(the standard deviation of the uniform distribution) LSB
12 1
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Control Lab Signal Processing Lab30
Quantization Noise 16 bit Converter
Green – analog signalRed – D/A output
Blue - noise
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Control Lab Signal Processing Lab31
Sampling Effects
• Single frequency sine wave with white noise (Signal-to-Noise (S/N) ratio)– Add White Noise to single frequency until noise
obscures the signal– Change sampling frequency & note effect
• Frequency discrimination versus sample size– Produce Sine waves at 2 very closely spaced
frequencies– Change sample size until both frequencies appear in
spectrum• Aliasing
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Control Lab Signal Processing Lab32
Aliasing• Nyquist Theorem – sample at greater than twice the rate of the
maximum frequency component in the input signal
• Nyquist frequency – half the sampling frequency
• Only possible to recover frequencies at or below Nyquist (so what is the bandwidth of the data acquisition system?)
• Frequencies above Nyquist will “alias” between DC and Nyquist
• Alias frequency = Absolute Value (closest integer multiple of sampling frequency - input frequency)
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Control Lab Signal Processing Lab33
Aliasing Example
Sampling Frequency
ƒs=100 Sampling Frequency
ƒs/2=50 Nyquist Frequency
5000
F3 160Hz
F4 510Hz
F1 25Hz
F2 70Hz
ƒs=100 ƒs/2=50 Nyquist Frequency
5000
Arrow – Actual Frequency Arrows – Alias
F3 160Hz
F4 510Hz
F3 alias 40Hz
F1 25Hz
F4 alias 10Hz
F2 alias 30Hz
F2 70Hz
Alias F2 = |100 - 70| = 30 HzAlias F3 = |(2)100 - 160| = 40 HzAlias F4 = |(5)100 - 510| = 10 HzRed
Black
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Control Lab Signal Processing Lab34
Preventing Aliasing• Oversampling
• Use filters designed to eliminate frequencies over Nyquistfrequency
• Use combination of both because real-world filters exhibit some roll-off
• Use special antialiasing filters that have sharp roll-off
Ideal Filter Real-World Filter
FrequencyNyquist0.0
1.0
V outV in
0.0
1.0
V outV in
FrequencyCutoff(3-dB down)
Passband
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Control Lab Signal Processing Lab35
Why is Signal Processing Important for Mechanical Engineering?
Comment on this in report.
Using Spectrum Analysis & Signal Analysis Techniques to Determine Faults in Motors & Motor Driven Systems
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Control Lab Signal Processing Lab36
Vibration Signature Analysis• What is Vibration Signature Analysis?
– Measure the vibration characteristic of a system as a function of time.– Take the frequency response of the signal.– Correlate a specific frequency component or set of frequency components
with a failure of the system to predict the failure before it occurs.
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Control Lab Signal Processing Lab37
What is Vibration?
• Vibration : Any motion that repeats itself after an interval of time is called a vibration or oscillation.
• The swinging of a pendulum and the motion of a plucked string are typical examples of vibration.
• For a motor driven system, the vibration characteristic can give an indication of impending system failure.
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Control Lab Signal Processing Lab38
Now Consider the Situation Where the Force on a Housing is Caused by the Rotating Unbalance of a Motor in the Housing
e = eccentricitymo = mass unbalancer = rotation frequency
Machine of total mass m i.e. m0 is included in m
m0
kc
e
rt
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Control Lab Signal Processing Lab39
What force is imparted on the structure? Note it rotateswith x component:
From dynamics,Rx m0ax moer
2 sin moer2 sinrt
Ry m0ay moer2 cos moer
2 cosrt
e
m0
Rx
Ry
r
texa
tex
rrrx
rr
sin
sin2
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Control Lab Signal Processing Lab40
The problem is now just like the situation of a Single Degree of Freedom (SDOF) system with a harmonic excitation
Note the influences on the forcing function
(we are assuming that the mass m is held in place in the y direction as indicated in the figure)
m0er2sin(rt)
kc
x(t)
mxx
xxm c kx moe r2 sin rt
or 2n n2x mo
me r
2 sin rt
..
..
.
.
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Control Lab Signal Processing Lab41
• Just another SDOF oscillator with a harmonic forcing function
• Expressed in terms of frequency ratio r
x (t) Xsin(r t )
X moem
r2
(1 r2 )2 2r 2
tan1 2r1 r2
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Control Lab Signal Processing Lab42
Displacement magnitude vs frequency caused by rotating unbalance
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Control Lab Signal Processing Lab43
Transducers for Vibration Signature Analysis
Using Displacement Transducers to measure more than one degree of
freedom
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Control Lab Signal Processing Lab44
• Displacement Transducers– Measures the displacement of the housing– Non-contact type are typically capacitive or inductive (eddy
current) essentially measuring the change in air gap of a capacitor or magnetic circuit
– Contact type can be resistive or LVDTs (Linear Variable Differential Transformer)
– Capacitive, inductive and LVDT require electronic processing of the signals
– Good for measuring low frequency vibrations– Very limited output for low amplitude high frequency
vibrations
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Control Lab Signal Processing Lab45
• Accelerometers– Measures the acceleration of the housing– Piezoelectric type measures the force produced by an
accelerating mass (seismic mass) – cannot measure DC acceleration
– MEMS type often measure the displacement of the seismic mass mounted on a cantilever beam using a non-contact (capacitive or inductive) displacement transducer – can be used to measure DC
– Requires electronic processing of the signals– Good for measuring high frequency vibrations– Very limited output for low frequency vibrations (unless high
amplitude)
Columbia UniversityMechanical Engineering
Control Lab Signal Processing Lab46
• Velocity Transducers– Measures the velocity of the housing– Essentially moving coil or moving magnet linear tachometers– May not require any additional electronic processing of the
signals– Good for measuring both low & high frequency vibrations
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Control Lab Signal Processing Lab47
Velocity & Acceleration are Related to Displacement
) t ( sin A ) t ( sin A- td
y(t)d (t)y
) 2
t ( sin A ) t ( cos A td
y(t)d (t)y
) 2
t ( cos A ) t ( sin A y(t)
222
2
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Control Lab Signal Processing Lab48
Displacement, Velocity & Acceleration forSimple Harmonic Motion
) 2
t 2 ( sin 1.0 y(t)
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Control Lab Signal Processing Lab49
Transducer Usefulness Vs. Frequency
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Control Lab Signal Processing Lab50
Vibration Signature Analysis Process
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Control Lab Signal Processing Lab51
• Air Gap Eccentricity
In 3 degrees of Freedom
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Control Lab Signal Processing Lab52
Rotor Static Unbalance