Vibrationdata
Unit 11
Power Spectral Density Function
PSD
1
PSD Introduction
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• A Fourier transform by itself is a poor format for representing random
vibration because the Fourier magnitude depends on the number of
spectral lines, as shown in previous units
• The power spectral density function, which can be calculated from a
Fourier transform, overcomes this limitation
• Note that the power spectral density function represents the magnitude,
but it discards the phase angle
• The magnitude is typically represented as G2/Hz
• The G is actually GRMS
2
Sample PSD Test Specification
0.1
Vibrationdata
Overall Level = 6.0 grms
2
0.04 g / Hz
-3 dB / octave
2
PSD ( g / Hz )
+3 dB / octave
0.01
0.001
20
80
350
2000
FREQUENCY (Hz)
3
Calculate Final Breakpoint G^2/Hz
0.1
Overall Level = 6.0 grms
2
0.04 g / Hz
-3 dB / octave
+3 dB / octave
Number of octaves between
two frequencies
2
PSD ( g / Hz )
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n
0.01
ln (f 2 / f1 )
ln (2)
Number of octaves from
350 to 2000 Hz = 2.51
0.001
20
80
350
2000
FREQUENCY (Hz)
The level change from 350 to 2000 Hz = -3 dB/oct x 2.51 oct = -7.53 dB
For G^2/Hz calculations: dB  10 log A / B
The final breakpoint is: (2000 Hz, 0.007 G^2/Hz)
4
Overall Level Calculation
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Note that the PSD specification is in log-log format.
Divide the PSD into segments.
The equation for each segment is
 
y1  n

y (f ) 
f
f n 
 1 
The starting coordinate is (f1, y1)
5
Vibrationdata
Overall Level Calculation (cont)
The exponent n is a real number which represents the slope.
The slope between two coordinates
y 
log  2 
 y1 
n
 f2 
log  
 f1 
The area a1 under segment 1 is


f
y
a1  2  1  f n df
f1  n 
 f1 

6
Overall Level Calculation (cont)
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There are two cases depending on the exponent n.




a1  




 
 y1   1  f n 1  f n 1  , for n  1
1

 f n   n  1  2
 1 
  f 2 
y1f1 ln   , for n  1
  f1 
7
Overall Level Calculation (cont)
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Finally, substitute the individual area values in the summation
formula.
The overall level L is the “square-root-of-the-sum-of-the-squares.”
L
m
 ai
i 1
where m is the total number of segments
8
Vibrationdata
dB Formulas
dB difference between two levels
If A & B are in units of G2/Hz,
A
dB  10 log  
B
If C & D are in units of G or GRMS,
C
dB  20 log  
D
9
dB Formula Examples

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Add 6 dB to a PSD
The overall GRMS level doubles
The G^2/Hz values quadruple

Subtract 6 dB from a PSD
The overall GRMS level decreases by one-half
The G^2/Hz values decrease by one-fourth
10
PSD Calculation Methods
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Power spectral density functions may be calculated via three methods:
1. Measuring the RMS value of the amplitude in successive frequency
bands, where the signal in each band has been bandpass filtered
2. Taking the Fourier transform of the autocorrelation function. This is the
Wierner-Khintchine approach.
3. Taking the limit of the Fourier transform X(f) times its complex conjugate
divided by its period T as the period approaches infinity.
11
PSD Calculation Method 3
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The textbook double-sided power spectral density function XPSD(f) is
lim X(f)X * (f)
XPSD (f) 
T
T 
The Fourier transform X(f)

has a dimension of [amplitude-time]

is double-sided
12
PSD Calculation Method 3, Alternate
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Let be the one-sided power spectral density function.
lim G(f) G * (f)
ˆ

X
(f)
PSD
Δf
Δf  0
The Fourier transform G(f)

has a dimension of [amplitude]

is one-sided
( must also convert from peak to rms by dividing by 2 )
13
Recall Sampling Formula
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The total period of the signal is
T = Nt
where
N is number of samples in the time function and in the Fourier transform
T is the record length of the time function
t is the time sample separation
14
More Sampling Formulas
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Consider a sine wave with a frequency such that one period is equal to the
record length.
This frequency is thus the smallest sine wave frequency which can be
resolved. This frequency f is the inverse of the record length.
f = 1/T
This frequency is also the frequency increment for the Fourier transform.
The f value is fixed for Fourier transform calculations.
A wider f may be used for PSD calculations, however, by dividing the data
into shorter segments
15
Statistical Degrees of Freedom
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• The f value is linked to the number of degrees of freedom
• The reliability of the power spectral density data is proportional to the
degrees of freedom
• The greater the f, the greater the reliability
16
Statistical Degrees of Freedom (Continued)
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The statistical degree of freedom parameter is defined follows:
dof = 2BT
where
dof is the number of statistical degrees of freedom
B is the bandwidth of an ideal rectangular filter
Note that the bandwidth B equals f, assuming an ideal
rectangular filter
The BT product is unity, which is equal to 2 statistical degrees of
freedom from the definition in equation
17
Trade-offs
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• Again, a given time history has 2 statistical degrees of freedom
• The breakthrough is that a given time history record can be subdivided
into small records, each yielding 2 degrees of freedom
• The total degrees of freedom value is then equal to twice the number of
individual records
• The penalty, however, is that the frequency resolution widens as the
record is subdivided
• Narrow peaks could thus become smeared as the resolution is widened
18
Example: 4096 samples taken over 16 seconds, rectangular filter.
Vibrationdata
Period of
Each
Record Ti
(sec)
Frequency
Resolution
Bi=1/Ti
(Hz)
dof per
Record
=2Bi TI
Total dof
1
Number
of Time
Samples
per
Record
4096
16
0.0625
2
2
2
2048
8
0.125
2
4
4
1024
4
0.25
2
8
8
512
2
0.5
2
16
16
256
1
1
2
32
32
128
0.5
2
2
64
64
64
0.25
4
2
128
Number of
Records
NR
19
Summary
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• Break time history into individual segment to increase degrees-of-freedom
• Apply Hanning Window to individual time segments to prevent leakage
error
• But Hanning Window has trade-off of reducing degrees-of-freedom because
it removes data
• Thus, overlap segments
• Nearly 90% of the degrees-of-freedom are recovered with a 50% overlap
20
Original Sequence
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Segments,
Hanning Window,
Non-overlapped
21
Original Sequence
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Segments,
Hanning Window,
50% Overlap
22