Vibration Analysis: FFT, PSD, and Spectrogram Basics
FFT, PSD and spectrograms don't need to be so complicated. Once you
understand the basics they can really help with your vibration analysis.
This topic contain the right mix of theory and practical information,
with examples, so that you can hopefully take your vibration analysis to the
next level!
In order to effectively do your job of vibration analysis, you may be
more interested in some practical information; but it is important to
understand a bit of the theory behind FFTs, PSDs and spectrograms. Let's go
through the important differences between an FFT, PSD, and spectrogram.
Simple Vibration Analysis in the Time Domain
When analyzing vibration data in the
time domain (acceleration/vibration amplitude plotted against time) we’re
limited to a few parameters in quantifying the strength of a vibration profile:
amplitude, peak-to-peak value, and RMS. A simple sine wave is shown
below with these parameters identified.
1. The peak
or amplitude is valuable for shock events but it doesn’t take into
account the time duration and thus the energy in the event.
2. The
same is true for peak-to-peak with the added benefit of
providing the maximum excursion of the wave, useful when looking at
displacement information, specifically clearances.
3. The RMS (root
mean square) value is generally the most useful because it is directly related
to the energy content of the vibration profile and thus the destructive
capability of the vibration. RMS also takes into account the time history of
the wave form.
Vibration is an oscillating motion about an equilibrium so most
vibration analysis looks to determine the rate of that oscillation, or the
frequency. The number of times a complete motion cycle occurs during
a period of one second is the vibration’s frequency and is measured in hertz
(Hz). For simple sine waves the vibration frequency could be
determined from looking at the waveform in the time domain; but as we add
different frequency components and noise, we need to perform spectrum analysis
to get a clearer picture of the vibration frequency.
Fast Fourier Transform (FFT)
FFT
Background
Any waveform is actually just the sum of a series of simple sinusoids of
different frequencies, amplitudes, and phases. A Fourier series is
that series of sine waves; and we use Fourier analysis or spectrum analysis to
deconstruct a signal into its individual sine wave components. The
result is acceleration/vibration amplitude as a function of frequency, which lets
us perform analysis in thefrequency domain (or spectrum) to gain a
deeper understanding of our vibration profile. Most vibration analysis will
typically be done in the frequency domain.
Fourier analysis works by “testing” for the presence of each frequency
component. A discrete Fourier transform (DFT) multiplies the raw
waveform by sine waves of discrete frequencies to determine if
they match and what their corresponding amplitude and phase are. A
fast Fourier transform (FFT) is just a DFT using a more efficient algorithm
that takes advantage of the symmetry in sine waves. The FFT requires a
signal length of some power of two for the transform and splits the process
into cascading groups of 2 to exploit these symmetries. This dramatically
improves processing speed; if N is the length of the signal, a DFT needs N2
operations while a FFT needs N*log2(N)
operations. So for example, let’s take a signal length of 1024
(210); the DFT needs 1,048,576 operations compared to the much more efficient
FFT and its 10,240 operations. Although the execution time of an FFT
will be fastest when the signal length is some power of 2, today's computing
power and modern FFT algorithms make this time difference insignificant.
The number of discrete frequencies that are tested as part of
a Fourier transform is directly proportional to the number of samples in the
original waveform. With N being the length of the signal, the number
of frequency lines or bins is equal to N/2. These frequency bins
occur at intervals (∆f) equal to the sample rate of the raw waveform
(Fs) divided by the number of samples (N), which is another way of saying that
the frequency resolution is equal to the inverse of the total acquisition time
(T). To improve the frequency resolution, you must extend the
recording time.
The lowest frequency tested is 0 Hz, the DC component; and the highest
frequency is the Nyquist frequency (Fs/2). Windowing can be used to
scale the frequency range of interest and manipulate the frequency bin
width. There’s a lot that goes into understanding how windowing
works and how to choose the right function for yourapplication;Let’s dive into some examples to understand how
spectrum analysis can be used for your vibration testing application!
Constructed
Sine Wave and FFT Example
To illustrate how an FFT can be used, let’s build a simple waveform with
and use an FFT for vibration analysis. This constructed waveform
will consist of three different frequency components: 22 Hz, 60 Hz, and 100
Hz. These frequencies will have an amplitude of 1g, 2g, and 1.5g
respectively. The following figure shows how this waveform looks a
little confusing in the time domain and also illustrates how the signal length
affects the frequency resolution of the FFT. Again, the MATLAB script which
generated this data is available to download.
If we sample this wave at a 500 Hz rate (500 samples per second) and
take an FFT of the first 50 samples we’re left with a pretty jagged FFT due to
our bin width being 10 Hz (Fs of 500 divided by N of 50). The
amplitude of these frequency components are also a bit low. But if
the range is extended to the first 250 samples as shown then the FFT is able to
accurately calculate both the frequency and amplitude of the individual sine
wave components.
Not that the “pure” waveform didn’t look confusing enough in the time
domain; but if broadband noise is added as shown in the bottom plots then the
waveform becomes even less distinguishable. This is the power of an
FFT, it is able to clearly identify the major frequencies that exist to help
theanalyzer determine the cause of any vibration
signal.
Real World FFT and Spectrogram Examples
Vibration
Analysis of a Car Engine
In real world applications there will typically be many different
frequency components of a vibration profile as well as mechanical and
electrical noise. Let’s look at some data taken on a passenger car engine
while it was idling and do some vibration analysis. This data was
generated with a Slam
Stick vibration data logger as part of a how-to
video series if you're interested in some more details
about the test setup.
We can use spectrum analysis of the vibration profile to indicate what
the engine’s crank shaft rotation speed was. This is a 4-cylinder
4-cycle engine. The engine operates with two pairs of pistons moving out of
phase with each other and two piston combustions per crank shaft rotation; so
the dominant frequency of the engine’s vibration will be twice the crank shaft
rotation speed (here’s a nice video on how a 4-stroke
engine works). In the FFT there is clearly a
dominate frequency at 30 Hz or 1,800 RPM which tells us that at idle the crank
shaft is rotating at 900 RPM (or 15 Hz) where there is also a peak in the FFT.
The use of an FFT in our vibration analysis gave clues on what was causing the
measured vibration.
In many applications the vibration frequency will change with time and
you can run into trouble if you only look at the FFT. Let's zoom out
of the area where the car engine is running at a relatively fixed rate, and
compute an FFT of the entire signal. In this test the engine sat off
for a period of time, idled, then the engine was revved before letting it idle
again and finally turning it off. The vibration frequency changed
pretty dramatically throughout the test; but the FFT doesn't capture
that. We know from the previous plot that when it was idling there
was a fairly significant dominate vibration frequency of 30 Hz; but this peak
gets muted when you try and look at the FFT of a changing vibration environment.
In this example, and others where the vibration frequency changes with
time, we need a spectrogram. A spectrogram works by breaking the
time domain data into a series of chunks and taking the FFT of these time
periods. These series of FFTs are then overlapped on one another to
visualize how both the amplitude and frequency of the vibration signal changes
with time. Turn this three dimensional surface plot of FFTs on its
side, add a color scale to represent the
amplitude (often works best when you look at the color/amplitude
on a logarithmic scale) and you're left with a spectrogram!
Back to that car engine example where the engine was revved for a
bit. The spectrogram shown below illustrates how the dominate
frequencies change with time in relation to when the car engine was idled and
revved. Using a spectrogram theanalyzer gains
a much deeper understanding of the vibration profile and how it changes with
time.
Vibration
Analysis on the Exterior of an Aircraft during Climb
A spectrogram doesn't have to be viewed in two dimensions. In
the below example I kept the surface plot in the three dimensional
view. This is from data taken by a Slam
Stick on the outside of an aircraft as it climbed from
23,000 ft to 40,000 feet. The
temperature also dropped from 14°C to -31°C (58°F to -24°F) during the test.
These three dimensional surface plots look pretty cool; but I think that
the 2D view better illustrates how the spectrum changes with time as shown
below. I wish I could say more about this data and the exact
application; but this vibration data was acquired testing our aerospace
customer’s proprietary system. The vibration data is really interesting though,
and illustrates how enjoyable vibration analysis can be!
Vibration
Analysis on a Semi-Trailer Truck during Transit
Now let’s quickly take a look at some data I gathered with our Slam
Stick vibration loggers as part of a vibration analysis and energy harvesting
study. We had a customer interested in harvesting energy on his
trailer bed during shipments to power an asset tracking device using ourpiezoelectric
energy harvesters. Below is an excerpt of the data
with a spectrogram, the raw data can be downloaded by clicking the link at the
end of this post.
The spectrogram helps us see that that there is definitely some
interesting vibration content; and there appears to be a fairly consistent
signal around 25 Hz (it’s a must to have a relatively consistent vibration
frequency for vibration energy harvesting). We can use an FFT to
take a closer look at the vibration profile during a time period where there
isn't a shock, shown below.
The FFT reveals a modest peak around 27 Hz; but the vibration in this
environment is still mostly dominated by “random” vibration levels which brings
us to how and why PSDs are used in vibration analysis.
Power Spectral Density (PSD)
What is a
PSD?
A lot of vibration in the real world, especially during transit, can be
called “random” vibration because it is motion at many frequencies at the same
time. FFTs are great at analyzing vibration
when there are a finite number of dominant frequency components; but power
spectral densities (PSD) are used to characterize random vibration
signals. A PSD is computed by multiplying each frequency bin in an FFT by
its complex conjugate which results in the real only spectrum of amplitude in
g2. The key aspect of a PSD which makes it more useful than a FFT
for random vibration analysis is that this amplitude value is then normalized to
the frequency bin width to get units of g2/Hz. By normalizing the
result we get rid of the dependency on bin width so that we can compare
vibration levels in signals of different lengths.
PSDs and
Vibration Test Standards
Because of how PSDs cancel out the effect of bandwidth of a frequency
spectrum, many test standards will describe their vibration environment with a
PSD. Let’s go through an example from MIL-STD-810G. Figure
514.6C-5 (page 312) describes the typical acceleration levels that jet aircraft
cargo are exposed to as shown below.
If you were developing something for the government that was going to be
transported with a jet aircraft, you would be required to do some testing on
your device/equipment to prove it can survive prolonged exposure to those
vibration levels. Most shaker control systems will have these
exposure profiles built in but they can also be constructed easily given some
known PSD levels and rise/decline rates. Let’s take a look at some
data captured by a Slam
Stick X when it was being excited with these vibration
levels; all this data is available to download.
Obviously the raw data in the time domain doesn’t tell us much although
I was admittedly surprised by the nearly 20g amplitude of the
vibration. Let’s do some vibration analysis and compute both FFTs
and PSDs of these signals to see how the signal length affects the amplitude
for the FFT but not the PSD.
The red lines in the PSD are the input error bounds that the shaker is
trying to keep the signal within. As you see, the PSD of different signal
lengths just fills in this area but the amplitude doesn’t change overall. The
FFT amplitude however shifts down as the bandwidth is increased. The PSD
amplitude does not shift because it is normalized to the frequency bin
width. This normalization that occurs in a PSD calculation makes it
so much more desirable to be used when analysing random vibration signals.
Vibration
Analysis of the Seat on a Commercial Airline
Now let’s put ourselves in the shoes of someone buying equipment to be
integrated into a larger system. We will want to make sure this
equipment can handle the vibration levels in this environment so we may require
a test organization to quantify that environment. I used a Slam
Stick recently on a commercial flight to do just this, but I
wanted to understand what type of vibration levels us humans were exposed to by
measuring the seat in front of me. Check out the data below along
with a PSD (again this is all available to download).
There is definitely a resonance of that seat around 250 Hz; but there is
surprisingly steady broadband vibration of 10-5 g2/Hz from 1 Hz to 1
kHz. Now if I was designing a system to operate on/in that airline
seat, I could use this PSD to program an exposure profile in our laboratory
shaker. Then I’d be able to do some in-house testing ahead of a
field test; and because we measured the actual data in the actual environment,
we’d have confidence that our system can survive. It’s incredibly
valuable to go out and actually measure the environment than to simply rely on
some test standard. These test standards will recommend using the
standard’s data as a guide; but they typically try and encourage the engineer
to go out and get the actual vibration
data. Nothing beats the real data!
Summary
FFT
The fast Fourier transform (FFT) is an efficient algorithm used to
compute a discrete Fourier transform (DFT). This Fourier transform
outputs vibration amplitude as a function of frequency so that the analyser can
understand what is causing the vibration. The frequency resolution
in an FFT is directly proportional to the signal length and sample
rate. To improve the resolution the time of the recording must be
extended; but be careful of a changing vibration environment.
Spectrogram
A spectrogram takes a series of FFTs and overlaps them to illustrate how
the spectrum (frequency domain) changes with time. If vibration
analysis is being done on a changing environment, a spectrogram can be a
powerful tool to illustrate exactly how that spectrum of the vibration changes.
PSD
A power spectral density (PSD) takes the amplitude of the FFT,
multiplies it by its complex conjugate and normalizes it to the frequency bin
width. This allows for accurate comparison of random vibration
signals that have different signal lengths. For this reason, PSDs
are typically used to describe random vibration environments like those
specified in military and commercial test standards.
Getting the
Vibration Data
Before you can get started with your vibration
analysis, you'll need to gather some vibration data! If you haven't gotten that
far, explore some vibration measurement options; and if you're going the route
of accelerometers, make sure you select the right accelerometer type.