Gerando um PSD - VRU

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Gerando um PSD
VOLTAR PARA: Teste Aleatório
O software de Pesquisa de Vibração usa o método de Welch para estimativa de
PSD. O processo começa com dados de entrada no domínio do tempo
distribuídos por Gauss (um arquivo de histórico de tempo).
Aqui está um resumo das etapas para gerar um PSD:
Alguns pontos adicionais - janelas e sobreposição especificamente - também
influenciam o resultado geral. Vejamos cada etapa com mais detalhes.
Como gerar um PSD
A Figura 1.1 mostra cinco segundos de dados de vibração aleatória distribuída
gaussiana em um arquivo de histórico de tempo (Aceleração vs. Tempo). É difícil
extrair qualquer informação significativa do gráfico que não seja a aceleração de
pico, que parece estar em torno de -30G.
Os dados são particionados em quadros de igual duração no tempo; cada
quadro é transformado no domínio da frequência usando a Fast Fourier
Transform (FFT)
Os dados do domínio da frequência são convertidos em energia, tomando a
magnitude ao quadrado de cada ponto de frequência; as magnitudes ao
quadrado (valores de potência) de cada quadro são calculadas
A média é dividida pela taxa de amostragem para normalizar para um único
hertz
https://vru.vibrationresearch.com/course/random-testing/
Figura 1.1. Um gráfico do histórico de cinco segundos.
To extract useful information from the vibration data, it must be viewed in the
frequency domain. Moving to the frequency domain requires two calculations:
the FFT and the PSD.
FAST FOURIER TRANSFORM (FFT)
The FFT transforms the data into an Acceleration vs. Frequency graph. No
windowing, averaging, or normalizing functions are used to create an FFT graph.
An FFT graph is often used to monitor the frequency spectrum. It focuses on
changes in the frequency spectrum while viewing live data or playing through a
time history file. However, to view energy distribution across the frequency
spectrum, the PSD must also be calculated.
POWER SPECTRAL DENSITY (PSD)
The steps to calculating the power spectral density are as follows:
1. Divide the time history file into frames of equal time length
First, the time history file should be divided into frames of equal time length. To
do so, use lines of resolution and the sample rate to determine the width of each
frame. There are two samples per analysis line.
In Figure 1.2, the recorded time history was sampled at 8,192Hz and 4,096 lines of
resolution resulting in a frame width of one second. If the number of lines was
changed to 1,024, the frame width would be 0.25 seconds. This is an important
step to remember when initially selecting a sample rate. Using a sample rate
that is an exponential of 2 (2n) usually results in a PSD with lines spaced at a
convenient interval.
Figure 1.2. The time history graph divided into frames.
2. Calculate the FTT for each frame after applying a window function
APPLYING A WINDOW FUNCTION TO THE DATA
An FFT assumes data is an infinite series; therefore, the starting and ending
points of each frame are interpreted as though they were next to each other.
Normally, random data is not an infinite series, so a window function needs to be
applied before calculating the FTT.
Without a window function, the starting and ending points may be different,
resulting in a transient spike between the two points. The transient spike would
show up in the FFT as high-frequency energy. Think of a terminal peak shock
pulse: the sharp transition from the peak amplitude to zero acceleration
generates a larger amount of high-frequency energy than a smooth pulse like
the half-sine.
For the FFT, the sharp transition between starting and ending points is a
discontinuity. This discontinuity is called a spectral leakage and is reflected in an
FFT calculation. Applying a window function removes the emphasis on the
discontinuities and reduces the spectral leakage. Ideally, the real data would
have identical starting and ending points in every frame of data. As this is not
the case, we must minimize any discontinuity by using a window function
(Figure 1.3).
Figure 1.3. Frames of data after windowing.
There is more than one window function, so we must determine which function
is most suitable for the current application. A window function is evaluated by
two key components: the side lobe and the main lobe. The Hanning window
function is used in Figure 1.3 (the grey waveform is the original waveform and
the orange waveform is the windowed data.) The Hanning function has a very
high and wide main lobe and low side lobes at practically zero. There is little to
no discontinuity between the starting and ending points, which results in
accurate frequency measurement.
The most commonly used window functions in vibration testing are the
Hanning or Blackman. Both functions have good frequency resolution, minimal
discontinuity, and, therefore, minimal leakage. Other window functions may be
appropriate for other applications. For a comprehensive list of commonly (and
not-so-commonly) used window functions, see the VRU course Window
Functions for Signal Processing.
CALCULATING THE FFT FOR EACH FRAME
The windowed data is then used to calculate the FFT for each frame,
transforming the signal from the time domain into the frequency domain (see
Figure 1.4.) This linear transformation gives us the ability to observe the
frequency content of the time history waveform.
https://vru.vibrationresearch.com/course/windows-functions-for-signal-processing/
Figure 1.4. Calculating the FFT for each frame.
FFT is used widely in signal analysis and processing. Most importantly, it shows
the frequencies present in a waveform and in what proportions. This information
can be used to determine many things: what frequencies are being excited
during a section of time, the peak acceleration of each frequency inside of a
windowed frame of data, the distribution of peaks, harmonic content, and more.
Certain weighting factors can also be applied to an FFT. If our goal was to simply
generate an FFT, we might apply a weighting factor to ensure 1Gpk of the time
domain data is equal to 1Gpk in the FFT. When the goal is to generate a PSD, the
window function is normalized to preserve the input power.
3. Square the individual FFTs for each frame and find an average
Next, square the individual FFTs for each frame and then find the average
squared-amplitude (Figure 1.5).
Figure 1.5. Squaring and averaging FFTs.
The PSD shows the average energy at a single frequency over a period of time.
Initially, it will have a variance or “hashiness.” As more frames of similar data are
included in the average, the overall variance will decrease, the accuracy will
increase, and the PSD will appear much smoother.
The total amount of time included in a PSD is related to the averaging
parameter degrees of freedom (DOF). The more frames of data averaged
together, the higher the DOF.  Simply put, more FFTs will result in better PSD.
However, large numbers of FFTs require more time to collect and calculate.
4. Normalize the calculation to a single hertz
Finally, take the average squared FFT and divide it by the sample rate. By doing
so, the data is normalized to a single hertz and a power spectral density is
determined (Figure 1.6). For acceleration, the resulting unit is G2/Hz.
Figure 1.6. The power spectral density (PSD).
Using the PSD, the response of the product under test is clear. As more frames
of data are added to the PSD, the variance will continue to decrease and the
PSD will become smoother.
For a more in-depth look at variance and methods of PSD smoothing, watch the
webinar on Instant Degrees of Freedom, a patented feature from Vibration
Research that quickly and effectively reduces the variance and creates a smooth
PSD. This is the only mathematically-justif ied method for displaying a smooth
PSD trace in a short period of time.
https://vibrationresearch.com/webinar/idof-version-2015/
Additional Parameters to Consider
OVERLAPPINGThere is one additional technique that is often used, though not required, during
PSD creation: overlapping. Overlapping is used to include more original data in
the PSD and to generate more DOF for a period of time.
With a 0% overlap, each frame of data is completely separated. When frames are
overlapped, some data in each frame is not accounted for due to the applied
window functions. Additionally, for each frame of data included in the PSD, two
DOFs are calculated for the total average.
For example, if we create a PSD with 0% overlap, 120 DOF, an 8,192Hz sample
rate, 4,096 lines of resolution, and a Hanning window function (resulting in 1-
second frames), we will need to average 60 seconds worth of data to achieve 120
degrees of freedom.
With a 50% overlap, there would be 0.5 seconds between the start of each frame
(each frame would still be 1 second in length.) Frames do not result in 2 DOF per
FFT when they are overlapped. A 50% overlap would result in around 1.85 DOF
per FFT; a 75% overlap would result in around 1.2 DOF per FFT. Therefore, for a
50% overlap, 120 DOF, 8,192Hz, 4,096 lines of resolution, Hanning Window PSD,
we can achieve the desired PSD in 64.8 frames in 32.4 seconds.
Figure 1.7. Overlapping windowed frames.
In the original example with 0% overlap, there were five frames of data, resulting
in 5 FFTs. With a 50% overlap, as shown in Figure 1.7, the same section of data
results in 9 FFTs.
LINES OF RESOLUTION
The last parameter to consider in the PSD calculation is the lines of resolution.
Lines of resolution, along with the sample rate, determine how far apart each
analysis point is spaced on the PSD. A higher number of lines will result in a
more accurate PSD but requires a larger number of samples to properly
calculate.
Many test standards require a certain number of lines to be included inside a
resonance to properly display the peak. If too few lines are used in a PSD, the
result is similar to under-sampling a waveform: the distance between analysis
points will be too great and the gap between not appropriately accounted for. A
minimum requirement of three or more lines of resolution is needed to properly
resolve a resonance.
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What is the PSD?
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