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Understanding spectra and spectral densities
Introduction This document demonstrates some properties of the
frequency spectra and spectral densities and
how they related to their corresponding signals in the time
domain.
Spectra This section begins by explaining how periodic sine
waves can be summed and shown in the
frequency domain. It then continues to describe how impulse
response functions and random
vibrations appear in the frequency domain and how they compare
to their time domain
counterparts.
Properties of sine wave A sine wave can be defined by:
( ) ( ) Equation 1 Where is the amplitude of the sine wave, is
time and is the angular frequency which is related
to frequency in Hz by:
Equation 2 Where the period of the signal in seconds is defined
as:
Equation 3 This example will consider the resulting spectra of
two sine waves with frequencies of 2.5 Hz and 4.0
Hz. The sine wave parameters are shown in Table 1, and are
plotted in Figure 1 between 0 and 10
seconds. It is important to note that the total time period used
is an integer multiple of the sine
wave period. As such, if it was desired to extend the signal, it
can simply be appended onto the end.
(Hz) (s)
1 2.5 0.4
2 4.0 0.25 Table 1 Sine wave parameters.
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Figure 1 - Sine waves using parameters from Table 1.
Some properties can be calculated for the sine wave. The mean
can be defined as:
Equation 4
For a sign wave that oscillates around the zero position, the
mean will always be zero so long as the
signal length is an integer multiple of the sine wave period. A
more appropriate measure is the root
mean square (RMS) which is defined as:
(
)
Equation 5
Another common parameter is the power of a signal which is
defined as:
(
)
Equation 6
It can be seen that RMS is the square root of the power. As the
signals are harmonic, i.e. repeat
exactly after a certain time period, these properties are
constant, irrespective of the signal length.
If two sine waves are summed:
( ) ( ) ( ) Equation 7 The period of the combined signal is now
the minimum common multiple of the individual sine wave
periods. In this example this is 2 seconds, i.e. 4 oscillations
of and 8 oscillations of . In this
case, the power is simply the sum of the individual waves.
The properties for the signals in this example are shown in
Table 2.
0 1 2 3 4 5 6 7 8 9 10-2
-1
0
1
2
Time (s)
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2.5 Hz Sine Wave
0 1 2 3 4 5 6 7 8 9 10-2
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0
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Time (s)
4.0 Hz Sine Wave
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(s)
0.4 0 0.5
0.25 0 2
2 0 .5 2.5 Table 2 Sine wave properties.
Steady state sine wave spectra The sum of the sine waves is
shown in the top plot of Figure 2. In this case it is difficult to
distinguish
the properties of each sine wave. To aid in this the signal can
be shown in the frequency domain
using a type of Discrete Fourier Transform (DFT), know as the
Fast Fourier Transform (FFT). This is a
complex valued transform that contains information about each
individual sine waves frequency,
amplitude and phase. The transform essentially decomposes the
signal into individual sine waves at
discrete frequencies. Each sine wave can be summed to recreate
the original signal. The bottom plot
of Figure 2 shows the magnitude FFT of the summed sine waves
with a time step of 0.01 s and a
maximum time of 10 seconds. It is clear that there are two sine
waves at 2.5 Hz and 4 Hz, with
amplitudes of 1 and 2 respectively.
Figure 2 Summed sine waves using parameters from Table 1 (top)
and their corresponding frequency domain representation (bottom).
dt=0.01s, tmax=10s.
As the FFT can be used to recreate the original time history, it
should be obvious that some of the
properties of the time history signal effect the FFT result;
namely the time step,
, and the maximum time, . These are related to the frequency
step and maximum frequency
with the following.
Equation 8
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
2
2.5
Frequency (Hz)
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Frequency Domain
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Equation 9
For example, the time signal is now sampled with and and is
shown in Figure 3.
The time domain plot is now much more jagged and the maximum
frequency has reduced from 50
to 5. However, the frequency step is still small and it is clear
that there are two sine wave
components at 2.5 Hz and 4 Hz with amplitudes of 1 and 2
respectively.
Figure 4 shows the time history now sampled with and . In this
case the time
history looks nice and smooth. The middle plot shows the full
frequency spectrum, which goes up to
50 Hz. The bottom plot shows the spectrum between 0 and 5 Hz
clearly showing the lower
frequency resolution.
Figure 3 - Summed sine waves using parameters from Table 1 (top)
and their corresponding frequency domain representation (bottom).
dt=0.1s, tmax=10s.
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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Time Domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
Frequency (Hz)
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Frequency Domain
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Figure 4 - Summed sine waves using parameters from Table 1 (top)
and their corresponding frequency domain representation (middle and
bottom). The bottom plot is a cropped region between 0 and 5 Hz.
dt=0.01s, tmax=2s.
Impulse response spectra Up to this point, every frequency
spectrum, although have different and , the amplitude has
been constant for each frequency component. This is due to the
periodicity of the signal and is true
for any steady state harmonic oscillation.
The same two sine waves are used for the basis of the next
comparison, but viscous damping is also
applied to give an impulse response. In this example and have
damping of 3% and 2%
respectively and the response is described by:
( ) ( ) ( )
Equation 10
Where is the damping ratio. As damping is small it is assumed
that the damped natural frequency
equals the undamped natural frequency.
Figure 5 shows the time history response with and corresponding
frequency
spectrum with and . Both plots have the characteristic shape of
a 2 degree of
freedom oscillation with viscous damping.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-2
0
2
4
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
1
2
Frequency (Hz)
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Frequency Domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
Frequency (Hz)
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Frequency Domain
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Figure 5 Impulse response of summed sine waves using parameters
from Table 1 (top) and their corresponding frequency domain
representation (bottom). dt=0.01s, tmax=10s.
Figure 6 shows the time history response with and corresponding
frequency
spectrum with and . Although the time history looks more jagged,
the frequency
domain is identical to Figure 5 from 0 5 Hz.
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
Frequency (Hz)
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Figure 6 Impulse response of summed sine waves using parameters
from Table 1 (top) and their corresponding frequency domain
representation (bottom). dt=0.1s, tmax=10s.
Figure 7 shows the time history response with and corresponding
frequency
spectrum with and . The bottom plot shows a zoomed plot of the
frequency
spectrum between 0 and 5 Hz. In this case, the time domain
signal is smooth and the frequency
spectrum is course. Although more coarse, the shape of the
frequency spectrum is the same is in
Figure 6, but the amplitude is much larger. This shows that the
amplitude of a non-periodic signal is
a function of the length of the time window. In this case is may
be expected as the response
decreases with time, reducing the average power. However, it
will be shown that the same occurs
for deterministic, stationary random signals.
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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Time Domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
Frequency (Hz)
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Frequency Domain
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Figure 7 Impulse response of summed sine waves using parameters
from Table 1 (top) and their corresponding frequency domain
representation (middle and bottom). The bottom plot is a cropped
region between 0 and 5 Hz.
dt=0.01s, tmax=2s.
Random response spectra A pseudo random signal was developed
that has equal power at all frequencies. As power is the
square of the RMS, the RMS in the time domain should be
constant. Table 3Table 3 Pseudo
random properties. defines the properties of the pseudo random
signal.
(s)
2 0 0.5 0.25
10 0 0.5 0.25 Table 3 Pseudo random properties.
Figure 8 shows the pseudo random signal for and and Figure 9
shows the
pseudo random signal for and ; the black line represents the RMS
of the whole
signal. In both cases the RMS and power of the signal is the
same, however the amplitude of the
frequency spectra are very different. The amplitude of the
frequency spectra are related by
amplitude given by:
Equation 11
Which can be rewritten as:
Equation 12
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-2
0
2
4
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
Frequency (Hz)
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Frequency Domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
Frequency (Hz)
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Frequency Domain
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In this case,
.
Figure 8 Random response using properties from Table 3 (top) and
its corresponding frequency domain representation (bottom).
dt=0.01s, tmax=10s.
0 1 2 3 4 5 6 7 8 9 10-2
-1
0
1
2
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
Frequency (Hz)
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Frequency Domain
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Figure 9 Random response using properties from Table 3 (top) and
its corresponding frequency domain representation (bottom).
dt=0.01s, tmax=2s.
Spectral densities This section begins by explaining the
characteristics of the spectral densities of sine waves,
impulse
response signals and random signals and how they compare to
their time domain counterparts.
This section specifically deals with power spectral densities
(PSDs) which are used to represent
random signals and are used to calculate the RMS of the signal.
A one-sided PSD can be defined as
defined as:
( )
| ( )|
Equation 13
Where ( ) is the single sided DFT of ( ). Using the power
spectrum, the total power can be
defined as the numeric integral of the PSD:
( )
Equation 14
And the RMS canbe defined as:
Equation 15
Steady state sine wave spectra The top plot in Figure 10 shows
the time history of the same summed sine waves as in Figure 2,
however this time the bottom plot shows the PSD. It should be
noticed that the amplitude is now
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
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2
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
Frequency (Hz)
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Frequency Domain
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units squared per Hz and the amplitude is no longer a discrete
frequency point as it is a density.
Strictly speaking, the data points should be represented as a
histogram, but it is commonly shown as
a line graph. As with the FFT, changing changes , but does not
change the amplitude of the
PSD. This is shown in Figure 11.
As with the FFT there are clear peaks at the sine wave
frequencies and the amplitudes are lager for
sine waves with larger magnitudes; however, the amplitudes do
not match the sine wave amplitude.
Using Figure 11 as an example, this is easy to calculate. For
the sine wave at 4 Hz with an amplitude
of 2 and a 10 s time window, using Equation 8:
and the amplitude of the spectrum is
define by Equation 13: ( ) | |
. In addition to this, the power due to the 4 Hz sine wave
is
the area under the graph: ( )
( )
, and the RMS in which is the
same as the value reported in Table 2. So although the amplitude
of the PSD may be high, sensible
values of power and RMS are achieved.
Figure 12 shows the time history and resulting PSD for the two
summed sine waves for a 2 s time
window. In this case the amplitude of the peaks has
significantly reduced. This characteristic is not
the same as a FFT where the amplitude of the spectrum stays
constant for periodic signals. For a
periodic signal, so long as the time window is a multiple of the
period, the power of the signal is
constant, regardless of the length of the time window. This fact
must be replicated in the PSD. As
the time window reduced, so did the frequency resolution. The
power in the signal, and therefore
the area under the graph, must stay constant, therefore the
amplitude must decrease.
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Figure 10 Summed sine waves using parameters from Table 1 (top)
and their corresponding PSD (bottom). dt=0.01s,
tmax=10s.
Figure 11 Summed sine waves using parameters from Table 1 (top)
and their corresponding PSD (bottom). dt=0.1s,
tmax=10s.
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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2/H
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
Frequency (Hz)
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2/H
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PSD
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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Time Domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
5
10
15
20
Frequency (Hz)
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2/H
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PSD
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Figure 12 Summed sine waves using parameters from Table 1 (top)
and their corresponding PSD (middle and bottom). The bottom plot is
a cropped region between 0 and 5 Hz. dt=0.01s, tmax=2s.
Impulse response spectra As it has been shown that only changes
the maximum frequency, 0.01 s will be used in the rest of
the PSD examples.
Figure 13 and Figure 14 show the impulse response for a 10 s and
2 s window respectively. In this
case the spectral densities are different, and because the
signal is a transient (i.e. non-periodic), the
power and RMS of the two signals is not the same. As a result,
the area under the PSD is not the
same, which is different to the previous summed sine wave
example. In this case, the signal with the
2 s window has a larger power and RMS value.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
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0
2
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Time (s)
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Time Domain
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2
4
Frequency (Hz)
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2/H
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PSD
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
5
10
Frequency (Hz)
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2/H
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PSD
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Figure 13 Impulse response of summed sine waves using parameters
from Table 1 (top) and their corresponding PSD (bottom). dt=0.01s,
tmax=10s.
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4
Time (s)
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Time Domain
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
2
Frequency (Hz)
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2/H
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PSD
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Figure 14 Impulse response of summed sine waves using parameters
from Table 1 (top) and their corresponding PSD (bottom). dt=0.01s,
tmax=2s.
Random response spectra As with the FFT example, a pseudo random
signal is created with equal power at all frequencies. As
power is the square of the RMS, the RMS in the time domain
should is constant with an amplitude of
0.5.
The pseudo random signal, and corresponding PSDs, for a 10 s and
a 2 s time window are shown in
Figure 15 and Figure 16 respectively. In this case, the PSDs for
both signals have the same
amplitude, which is the opposite to the FFT spectra in Figure 8
and Figure 9. This is because
although the signals are not periodic, the properties are time
invariant. As such, a PSD will have the
same amplitude, regardless of the length of the time window. In
this case it is easy to calculate the
power and RMS of the signal. It is simply and respectively,
which
matches Table 3.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-2
0
2
4
Time (s)
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Time Domain
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1
2
Frequency (Hz)
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2/H
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PSD
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
Frequency (Hz)
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2/H
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PSD
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Figure 15 Random response using properties from Table 3 (top)
and its corresponding PSD (bottom). dt=0.01s, tmax=10s.
0 1 2 3 4 5 6 7 8 9 10-2
-1
0
1
2
Time (s)
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0 5 10 15 20 25 30 35 40 45 500
0.005
0.01
Frequency (Hz)
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PSD
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Figure 16 Random response using properties from Table 3 (top)
and its corresponding PSD (bottom). dt=0.01s,
tmax=2s.
Conclusions From the examples shown here it can be seen
that:
The DFT is a good method to display the frequency content of
periodic signals, so long as the
time window as a whole number multiplier of the period. The
amplitude of the DFT is
constant, regardless of the time window size and time step.
A spectral density is not a good method to display the frequency
content of a periodic signal.
The amplitude of the spectral density is a function of the
length of the time window.
Neither the DFT or spectral density give a good estimation of an
impulse response function
as the amplitude of both methods are dependent on the length of
the time window.
Although not shown here, these responses can be normalised by
calculating a transfer
function between the impulse response, and the impulse. This is
the same for any non-
deterministic, time varying excitation/response.
The DFT is not a good method of displaying the frequency content
of a random signal. The
amplitude of the DFT is a function of the length of the time
window.
A spectral density is a good method of displaying the frequency
content of a random signal,
so long as the properties of the random response are
deterministic and time invariant. In
this case, the amplitude of the spectral density is constant,
regardless of the time window
length.
PSDs can be used to simply calculate the total power of the
signal and RMS by integration.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
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Time (s)
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0.005
0.01
Frequency (Hz)
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PSD
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Real example This example shows two simulated stress PSDs from
two locations on a flare stack.
The top plot in Figure 17 shows the stress PSD at a location of
relatively low stress, but many
vibrations modes contribute to the response, which is shown by
the numerous peaks in the PSD.
The middle plot shows the cumulative integration of the PSD,
which shows the increase in power as
the frequency increases. The bottom plot shows the cumulative
RMS of the PSD. These plots allow
you to see how much the peaks in the PSD spectrum contribute to
the response. There are clear
steps as the peaks are passed. The total RMS is approximately
0.28 MPa.
Figure 17 Stress PSD (top), cumulative stress power (middle) and
cumulative RMS stress (bottom) for low stress location.
Figure 18 shows the response at another node. This node reported
the highest stress and minimum
fatigue life in the model. In this case the peak in the PSD
spectrum has a large amplitude, which may
initially give concern. However if the cumulative RMS plot is
considered, it is clear that the response
is due to the peak between 2 and 3 Hz, and due to the peaks
being narrow, the total RMS response
is only approximately 1.6 MPa.
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0.2
0.4
Frequency (Hz)
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PSD
0 1 2 3 4 5 6 7 8 9 100
0.05
0.1
Frequency (Hz)
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Cumulative power spectrum
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
Frequency (Hz)
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a
Cumulative RMS spectrum
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Figure 18 Stress PSD (top), cumulative stress power (middle) and
cumulative RMS stress (bottom) for high stress location.
0 1 2 3 4 5 6 7 8 9 100
10
20
Frequency (Hz)
MP
a2/H
z
PSD
0 1 2 3 4 5 6 7 8 9 100
2
4
Frequency (Hz)
MP
a2
Cumulative power spectrum
0 1 2 3 4 5 6 7 8 9 100
1
2
Frequency (Hz)
MP
a
Cumulative RMS spectrum
