Basics of Spectrum Analysis/Measurements and the FFT Analyzer

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Basics of Spectrum Analysis/Measurements and the FFT Analyzer
Mechanical Engineering - 22.403 ME Lab II
ME 22.403
Mechanical Lab II
Basics of
Spectrum Analysis/Measurements
and
the FFT Analyzer
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 1 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Transformation of Time to Frequency
Many times a transformation is performed to provide a better or
clearer understanding of a phenomena. The time representation
of a sine wave may be difficult to interpret. By using a Fourier
series representation, the original time signal can be easily
transformed and much better understood.
Transformations are also performed
to respresent the same data with
significantly less information.
Notice that the original time signal
was defined by many discrete time
points (ie, 1024, 2048, 4096 …)
whereas the equivalent Fourier
representation only requires 4
amplitudes and 4 frequencies.
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 2 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
The Anatomy of the FFT Analyzer
The FFT Analyzer can be broken down
into several pieces which involve the
digitization, filtering, transformation
and processing of a signal.
ANALOG
SIGNAL
ANALOG
FILTER
ADC
DISPLAY
DIGITAL
FILTER
FFT
Several items are important here:
Digitization and Sampling
Quantization of Signal
Aliasing Effects
Leakage Distortion
Windows Weighting Functions
The Fourier Transform
Measurement Formulation
DISCRETE
DATA
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 3 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
The Anatomy of the FFT Process
ANALOG SIGNALS
OUTPUT
INPUT
ANTIALIASING FILTERS
Actual time signals
Analog anti-alias filter
AUTORANGE ANALYZER
ADC DIGITIZES SIGNALS
OUTPUT
INPUT
APPLY WINDOWS
INPUT
OUTPUT
COMPUTE FFT
LINEAR SPECTRA
LINEAR
OUTPUT
SPECTRUM
LINEAR
INPUT
SPECTRUM
Digitized time signals
Windowed time signals
Compute FFT of signal
AVERAGING OF SAMPLES
COMPUTATION OF AVERAGED
INPUT/OUTPUT/CROSS POWER SPECTRA
INPUT
POWER
SPECTRUM
OUTPUT
POWER
SPECTRUM
CROSS
POWER
SPECTRUM
Average auto/cross spectra
COMPUTATION OF FRF AND COHERENCE
Compute FRF and Coherence
FREQUENCY RESPONSE FUNCTION
Dr. Peter Avitabile
COHERENCE FUNCTION
University of Massachusetts Lowell
Spectrum Analysis 082702 - 4 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Aliasing (Wrap-Around Error)
WRAP-AROUND
ACTUAL SIGNAL
OBSERVED
ACTUAL
ALIASED SIGNAL
f max
Aliasing results when the sampling does not occur fast enough.
Sampling must occur faster than twice the highest frequency to be
measured in the data - sampling of 10 to 20 times the signal is
sufficient for most time representations of varying signals
However, in order to accurately represent a signal in the
frequency domain, sampling need only occur at greater than twice
the frequency of interest
Anti-aliasing filters are used to prevent aliasing
These are typically Low Pass Analog Filters
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 5 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Anti-Aliasing Filters
Anti-aliasing filters are typically specified with a cut-off
frequency. The roll-off of the filter will determine how quickly
the signal will be attenuated and is specified in dB/octave
FILTER ROLLOFF
G dB = 20 log10 G = 20 log10
Vout
Vin
fc
The cut-off frequency is usually specified at the 3 dB down point
(which is where the filter attenuates 3 dB of signal).
Butterworth, Chebyshev, elliptic, Bessel are common filters
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 6 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Digitization of a Signal
Sampling rate of the ADC is specified as a maximum that is
possible. Basically, the digitizer is taking a series of “snapshots”
at a very fast rate as time progresses
Digital
Representation
Analog Signal
ADC
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 7 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Sampling
Each sample is spaced delta t seconds apart. Sufficient sampling
is needed in order to assure that the entire event is captured.
The maximum observable frequency is inversely proportional to the
delta time step used
Digital Sample
Rayleigh Criteria
fmax = 1 / 2 ∆t
∆t spacing
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 8 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Sampling Theory
In order to extract valid frequency information, digitization of the
analog signal must occur at a certain rate.
Shannon's Sampling Theorem states
fs > 2 fmax
That is, the sampling rate must be at least twice the desired
frequency to be measured.
For a time record of T seconds, the lowest frequency component
measurable is
∆f = 1 / T
With these two properties above, the sampling parameters can be
summarized as
fmax = 1 / 2 ∆t
∆t = 1 / 2 fmax
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 9 Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Sampling Parameters
Due to the Rayleigh Criteria and Shannon’s Sampling Theorum, the
following sampling parameters must be observed.
T=N ∆ t
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 10
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Sampling Parameters
Due to the Rayleigh Criteria and Shannon’s Sampling Theorum, the
following sampling parameters must be observed.
PICK
THEN
AND
∆t
fmax = 1 / (2 ∆t)
T = N ∆t
fmax
∆t = 1 / (2 fmax )
∆f = 1/(N ∆t)
∆f
T = 1 / ∆f
∆t = T / N
T
∆f =1 / T
fmax = N ∆f / 2
If we choose ∆f = 5 Hz and N = 1024
Then
T = 1 / ∆f = 1 / 5 Hz = 0.2 sec
fs = N ∆f = (1024) (5 Hz) = 5120 Hz
fmax = fs = (5120 Hz) / 2 = 2560 Hz
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 11
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Sampling Relationship
An inverse relationship between time and frequency exists
T
BW
Given delta t = .0019531 and N = 1024 time points,
then T = 2 sec and BW= 256 Hz and delta f = 0.5 Hz
TIME DOMAIN
FREQUENCY DOMAIN
T
BW
Given delta t = .000976563 and N = 1024 time points,
then T = 1sec sec and BW = 512 Hz and delta f = 1 Hz
TIME DOMAIN
T
FREQUENCY DOMAIN
BW
Given delta t = .0019531 and N = 512 time points,
then T = 1 sec and BW = 256 Hz and delta f = 1 Hz
TIME DOMAIN
Dr. Peter Avitabile
FREQUENCY DOMAIN
University of Massachusetts Lowell
Spectrum Analysis 082702 - 12
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Quantization Error
Sampling refers to the rate at which the signal is collected.
Quantization refers to the amplitude description of the signal.
A 4 bit ADC has 24 or 16 possible values
A 6 bit ADC has 26 or 64 possible values
A 12 bit ADC has 212 or 4096 possible values
ADC BIT STEPS
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 13
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Quantization Error
Quantization errors refer to the accuracy of the amplitude
measured. The 6 bit ADC represents the signal shown much
better than a 4 bit ADC
Dr. Peter Avitabile
A
D
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University of Massachusetts Lowell
Spectrum Analysis 082702 - 14
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Quantization Error
Underloading of the ADC causes amplitude errors in the signal
10 volt
range
on
ADC
All of the available
dynamic range of the
analog to digital
converter is not used
effectively
0.5 volt signal
This causes amplitude
and phase distortion
of the measured
signal in both the time
and frequency domains
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 15
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
AC Coupling
A large DC bias can cause amplitude errors in the alternating part
of the signal. AC coupling uses a high pass filter to remove the
DC component from the signal
All of the available
dynamic range of the
analog to digital
converter is dominated
by the DC signal
10 volt
range
on
ADC
The alternating part of
the signal suffers from
quantization error
This causes amplitude
and phase distortion of
the measured signal
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 16
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Clipping and Overloading
Overloading of the ADC causes severe errors also
The ADC range is set
too low for the signal
to be measured and
causes clipping of the
signal
1 volt
range
on
ADC
A
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Dr. Peter Avitabile
1.5 volt signal
University of Massachusetts Lowell
This causes amplitude
and phase distortion
of the measured
signal in both the time
and frequency domains
Spectrum Analysis 082702 - 17
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
The Fourier Transform
Forward Fourier Transform
+∞
Sx (f )= ∫ x ( t )e − j2 πft dt
−∞
and Inverse Fourier Transform
+∞
x ( t )= ∫ Sx (f )e j2 πft df
−∞
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 18
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Discrete Fourier Transform
Even though the actual time signal is continuous, the signal is
discretized and the transformation at discrete points is
+∞
Sx (m∆f )= ∫ x ( t )e − j2πm∆f t dt
−∞
This integral is evaluated as
Sx (m∆f )≈∆t
+∞
∑ x(n∆t )e− j2πm∆f n∆t
n = −∞
However, if only a finite sample is available (which is generally the
case), then the transformation becomes
N −1
Sx (m∆f )≈∆t ∑ x(n∆t )e − j2πm∆f n∆t
n =0
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 19
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Fourier Transform and FFT
Actual Time
Signal
ACTUAL
Captured Time
Signal
CAPTURED
DATA
DATA
T
Reconstructed
Time Signal
RECONTRUCTED
DATA
Frequency
Spectrum
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 20
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Fourier Transform and FFT
Actual Time
Signal
ACTUAL
Captured Time
Signal
CAPTURED
DATA
DATA
T
Reconstructed
Time Signal
RECONTRUCTED
DATA
Frequency
Spectrum
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 21
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Leakage
F
R
E
Q
ACTUAL
DATA
CAPTURED
DATA
T
I
M
E
Periodic Signal
T
RECONTRUCTED
DATA
Non-Periodic Signal
U
E
N
C
Y
ACTUAL
DATA
CAPTURED
DATA
T
RECONTRUCTED
DATA
T
Leakage due to
signal distortion
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 22
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Leakage
When the measured signal is not periodic in the sample interval,
incorrect estimates of the amplitude and frequency occur. This
error is referred to as leakage.
Basically, the actual energy distribution is smeared across the
frequency spectrum and energy leaks from a particular ∆f into
adjacent ∆f s.
Leakage is probably the most common and most serious digital
signal processing error. Unlike aliasing, the effects of leakage
can not be eliminated.
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 23
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Minimize Leakage
In order to better satisfy the periodicity requirement of the FFT
process, time weighting functions, called windows, are used.
Essentially, these weighting functions attempt to heavily weight the
beginning and end of the sample record to zero - the middle of the
sample is heavily weighted towards unity
ACTUAL
DATA
CAPTURED
DATA
T
I
M
E
Periodic Signal
T
RECONTRUCTED
DATA
Non-Periodic Signal
ACTUAL
DATA
CAPTURED
DATA
T
RECONTRUCTED
DATA
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 24
F
R
E
Q
U
E
N
C
Y
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Rectangular/Hanning/Flattop
In order to better satisfy the periodicity requirement of the FFT
process, time weighting functions, called windows, are used.
Essentially, these weighting functions attempt to heavily weight the
beginning and end of the sample record to zero - the middle of the
sample is heavilty weighted towards unity
Rectangular - Unity gain applied to entire sample interval; this
window can have up to 36% amplitude error if the signal is not
periodic in the sample interval; good for signals that inherently
satisfy the periodicity requirement of the FFT process
Hanning - Cosine bell shaped weighting which heavily weights the
beginning and end of the sample interval to zero; this window can
have up to 16% amplitude error; the main frequency will show
some adjacent side band frequencies but then quickly attenuates;
good for general purpose signal applications
Flat Top - Multi-sine weighting function; this window has excellent
amplitude characteristics (0.1% error) but very poor frequency
resolution; very good for calibration purposes with discrete sine
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 25
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Rectangular/Hanning/Flattop
Time weighting functions
are applied to minimize
the effects of leakage
AMPLITUDE
ROLLOFF
Rectangular
WIDTH
General window
frequency characteristics
Hanning
Flat Top
and many others
Windows DO NOT eliminate leakage !!!
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 26
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Rectangular
The rectangular window function is shown below. The main lobe is narrow, but the side lobes are very large
and roll off quite slowly. The main lobe is quite rounded and can introduce large measurement errors. The
rectangular window can have amplitude errors as large as 36%.
0
-10
-20
Amplitude
-30
-40
-50
-60
dB
-70
- 80
- 90
-100
-16
-14
-12
Dr. Peter Avitabile
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
15.9375
University of Massachusetts Lowell
-3
-2.5
-2
-1.5
-1
-0.5
0
Spectrum Analysis 082702 - 27
0.5
1
1.5
2
2.5
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Hanning
The hanning window function is shown below. The first few side lobes are rather large, but a 60 dB/octave
roll-off rate is helpful. This window is most useful for searching operations where good frequency
resolution is needed, but amplitude accuracy is not important; the hanning window will have amplitude errors
of as much as 16%.
0
-10
-20
Amplitude
-30
-40
-50
-60
dB
-70
- 80
- 90
-100
-16
-14
-12
Dr. Peter Avitabile
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
15.9375
University of Massachusetts Lowell
-3
-2.5
-2
-1.5
-1
-0.5
0
Spectrum Analysis 082702 - 28
0.5
1
1.5
2
2.5
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Flat Top
The flat top window function is shown below. The main lobe is very flat and spreads over several frequency
bins. While this window suffers from frequency resolution, the amplitude can be measured very accurately
to 0.1%.
0
-10
-20
Amplitude
-30
-40
-50
-60
dB
-70
- 80
- 90
-100
-16
-14
-12
Dr. Peter Avitabile
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
15.9375
University of Massachusetts Lowell
-3
-2.5
-2
-1.5
-1
-0.5
0
Spectrum Analysis 082702 - 29
0.5
1
1.5
2
2.5
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows
Rectangular
Hanning
Flat Top
0
0
0
-10
-10
-10
-20
-20
-20
-30
-30
-30
-40
-40
-40
-50
-50
-50
-60
-60
-60
dB
dB
dB
-70
-70
-70
- 80
- 80
- 80
- 90
- 90
- 90
-100
-16
-14
-12
-10
-8
-6
Dr. Peter Avitabile
-4
-2
0
2
4
6
8
10
12
14
15.9375
-100
-16
-14
-12
-10
-8
University of Massachusetts Lowell
-6
-4
-2
0
2
4
6
8
10
12
14 15.9375
-100
-16
-14
-12
-10
Spectrum Analysis 082702 - 30
-8
-6
-4
-2
0
2
4
6
8
10
12
14 15.9375
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Force/Exponential for Impact Testing
Special windows are used for impact testing
Force
window
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 31
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Windows - Force/Exponential for Impact Testing
Special windows are used for impact testing
Exponential
window
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 32
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Hammer Tips for Impact Testing
METAL TIP
HARD PLASTIC TIP
Real
Real
-976.5625us
TIME PULSE
123.9624ms
dB Mag
-976.5625us
123.9624ms
dB Mag
0Hz
6.4kHz
FREQUENCY SPECTRUM
0Hz
FREQUENCY SPECTRUM
6.4kHz
RUBBER TIP
SOFT PLASTIC TIP
Real
Real
-976.5625us
TIME PULSE
-976.5625us
123.9624ms
TIME PULSE
123.9624ms
dB Mag
dB Mag
0Hz
Dr. Peter Avitabile
TIME PULSE
FREQUENCY SPECTRUM
University of Massachusetts Lowell
6.4kHz
0Hz
FREQUENCY SPECTRUM
Spectrum Analysis 082702 - 33
6.4kHz
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Pretrigger Delay and Double Impacts
Pretrigger delay used to
reduce the amount of
frequency spectrum distortion
DOUBLE IMPACT
Real
t=0
NO PRETRIGGER
USED
-976.5625us
998.53516ms
TIME PULSE
dB Mag
DOUBLE IMPACT
t=0
0Hz
FREQUENCYRealSPECTRUM
800Hz
PRETRIGGER
SPECIFIED
-976.5625us
Double impacts should be avoided due to
the distortion of the frequency spectrum
and force dropout that can occur
Dr. Peter Avitabile
University of Massachusetts Lowell
TIME PULSE
998.53516ms
dB Mag
0Hz
Spectrum Analysis 082702 - 34
FREQUENCY SPECTRUM
800Hz
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Exponential Window
If the signal does not
naturally decay within the
sample interval, then an
exponentially decaying window
may be necessary.
However, many times changing
the signal processing
parameters such as bandwidth
and number of spectral lines
may produce a signal which
requires less window weighting
Dr. Peter Avitabile
University of Massachusetts Lowell
T = N∆ t
T = N∆ t
Spectrum Analysis 082702 - 35
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Measurement - Linear Spectra
x(t)
INPUT
Sx(f)
Dr. Peter Avitabile
h(t)
y(t)
TIME
SYSTEM
OUTPUT
FFT & IFT
H(f)
Sy(f)
FREQUENCY
x(t)
- time domain input to the system
y(t)
- time domain output to the system
Sx(f)
- linear Fourier spectrum of x(t)
Sy(f)
- linear Fourier spectrum of y(t)
H(f)
- system transfer function
h(t)
- system impulse response
University of Massachusetts Lowell
Spectrum Analysis 082702 - 36
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Measurement - Linear Spectra
+∞
x ( t )= ∫ Sx (f )e j2 πft df
−∞
+∞
y( t )= ∫ S y (f )e
j2 πft
h ( t )= ∫ H (f )e
Sx (f )= ∫ x ( t )e − j2 πft dt
−∞
+∞
df
−∞
+∞
+∞
S y (f )= ∫ y( t )e − j2 πft dt
−∞
+∞
j2 πft
df
−∞
H(f )= ∫ h ( t )e − j2 πft dt
−∞
Note: Sx and Sy are complex valued functions
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 37
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Measurement - Power Spectra
Rxx(t)
INPUT
Gxx(f)
Ryx(t)
Ryy(t)
TIME
SYSTEM
OUTPUT
FFT & IFT
Gxy(f)
Gyy(f)
FREQUENCY
Rxx(t) - autocorrelation of the input signal x(t)
Ryy(t) - autocorrelation of the output signal y(t)
Ryx(t) - cross correlation of y(t) and x(t)
Gxx(f) - autopower spectrum of x(t)
G xx ( f ) = S x ( f ) • S*x ( f )
Gyy(f) - autopower spectrum of y(t)
G yy ( f ) = S y ( f ) • S*y ( f )
Gyx(f) - cross power spectrum of y(t) and x(t)
G yx ( f ) = S y ( f ) • S*x ( f )
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 38
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Measurement - Power Spectra
R xx (τ)=E[ x ( t ), x ( t + τ)]=
lim 1
x ( t )x ( t + τ)dt
∫
T → ∞TT
+∞
G xx (f )= ∫ R xx (τ)e − j2 πft dτ=Sx (f )•S*x (f )
−∞
R yy (τ)=E[ y( t ), y( t + τ)]=
lim 1
y( t )y( t + τ)dt
∫
T → ∞T T
+∞
G yy (f )= ∫ R yy (τ)e − j2 πft dτ=S y (f )•S*y (f )
−∞
R yx (τ)=E[ y( t ), x ( t + τ)]=
lim 1
y( t )x ( t + τ)dt
∫
T → ∞TT
+∞
G yx (f )= ∫ R yx (τ)e − j2 πft dτ=S y (f )•S*x (f )
−∞
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 39
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
The Frequency Response Function and Coherence
S y =HSx
H1 formulation
- susceptible to noise on the input
- underestimates the actual H of the system
S y •S*x =HSx •S*x
Other
formulations
for H exist
Sy •S*x G yx
H=
=
*
Sx •Sx G xx
COHERENCE
γ
Dr. Peter Avitabile
2
xy
(S y •S*x )(Sx •S*y )
G yx / G xx H1
=
=
=
*
*
(Sx •Sx )(S y •S y ) G yy / G xy H 2
University of Massachusetts Lowell
Spectrum Analysis 082702 - 40
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Typical Measurements
Measurements - Auto Power Spectrum
Measurements - Cross Power Spectrum
x(t)
y(t)
AVERAGED INPUT
AVERAGED OUTPUT
POWER SPECTRUM
POWER SPECTRUM
Gyy (f)
G xx (f)
OUTPUT RESPONSE
INPUT FORCE
G xx(f)
G yy (f)
AVERAGED INPUT
AVERAGED OUTPUT
POWER SPECTRUM
POWER SPECTRUM
AVERAGED CROSS
POWER SPECTRUM
G yx (f)
Measurement Definitions
12
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Frequency Response Function
Measurement Definitions
13
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - FRF & Coherence
Coherence
1
Real
AVERAGED INPUT
AVERAGED CROSS
AVERAGED OUTPUT
POWER SPECTRUM
POWER SPECTRUM
POWER SPECTRUM
Gyx(f)
G yy(f)
G xx(f)
0
0Hz
AVG: 5
200Hz
COHERENCE
Freq Resp
40
dB Mag
-60
0Hz
AVG: 5
200Hz
FREQUENCY RESPONSE FUNCTION
FREQUENCY RESPONSE FUNCTION
H(f)
Measurement Definitions
Dr. Peter Avitabile
14
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
University of Massachusetts Lowell
Measurement Definitions
15
Spectrum Analysis 082702 - 41
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Hammers and Tips
40
COHERENCE
dB Mag
FRF
INPUT POWER SPECTRUM
-60
0Hz
800Hz
COHERENCE
40
FRF
dB Mag
INPUT POWER SPECTRUM
-60
0Hz
Dr. Peter Avitabile
University of Massachusetts Lowell
200Hz
Spectrum Analysis 082702 - 42
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Leakage and Windows for Impact Testing
ACTUAL TIME SIGNAL
SAMPLED SIGNAL
WINDOW WEIGHTING
WINDOWED TIME SIGNAL
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 43
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Simple time-frequency response relationship
RESPONSE
increasing rate of oscillation
WOW !!!
FORCE
time
frequency
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 44
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Sine Dwell to Obtain Mode Shape Characteristics
MODE 1
MODE 2
Dr. Peter Avitabile
University of Massachusetts Lowell
MODE3
MODE 4
Spectrum Analysis 082702 - 45
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Mode Shape Characteristics for a Simple Beam
MODE # 1
MODE # 2
MODE # 3
DOF # 1
DOF #2
DOF # 3
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 46
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Mode Shape Characteristics for a Simple Plate
MODE 2
2
1
4
3
6
MODE 1
5
2
4
1
3
6
5
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 47
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
Why and How Do Structures Vibrate?
OUCH !!!
Motor or disk
unbalance
can cause unwanted
vibrations or worse
OOOPS !!!
INPUT TIME FORCE
OUTPUT TIME RESPONSE
f(t)
y(t)
FFT
Dr. Peter Avitabile
IFT
INPUT SPECTRUM
FREQUENCY RESPONSE FUNCTION
OUTPUT SPECTRUM
f(jω )
h(j ω)
y(j ω)
University of Massachusetts Lowell
Spectrum Analysis 082702 - 48
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
HP 35660 FFT Dual Channel Analyzer
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 49
Copyright © 2001
Mechanical Engineering - 22.403 ME Lab II
HP 35660 FFT Dual Channel Analyzer
Dr. Peter Avitabile
University of Massachusetts Lowell
Spectrum Analysis 082702 - 50
Copyright © 2001

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