Spacecraft Vibration Analysis

Jorge Juarez (jjuarez7@gatech.edu), Yuwei Lin (ylin387@gatech.edu)

Abstract—Aerospace vehicles are subject to high dynamics

which may degrade performance and structural integrity over
time. One method of monitoring the rigors of flight is through
the use of an accelerometer. This sensor may also be used to
monitor engine performance, rocket motor combustion, or any
other vibration-producing mechanism. The application of
Fourier analysis on a vibration signal is explored in this paper,
along with the tradeoffs of Fast Fourier Transform (FFT)
windowing, and the application of Power Spectral Density (PSD)
analysis.

I. INTRODUCTION
Many kinds of accelerometers exist, some use a proof
mass to measure force, others use displacement to measure
capacitance, but the result is the same: the sensor measures
uniaxial force directly and produces an output voltage
proportional to the experienced acceleration. Fig 1. Fast Fourier Transform of Cosine Signal Swept
From t = 0 at 100 Hz to t = 2 at 400 Hz
We shall focus on a special type of accelerometer sensor
which is used to measure vibration. These vibration- a rocket launch is not analyzed in depth in real time, rather it
monitoring accelerometers differ from other sensors in that the is post-processed after flight. Further, we shall assume that all
data input is AC coupled. In other words, a vibration data has been sampled at a fast enough rate as to satisfy the
accelerometer (henceforth “vibrometer”) will not produce a Shannon-Nyquist sampling theorem. Lastly, we will assume
signal output if subjected to a constant acceleration, say 7 g’s. that all data has been appropriately low-pass sampled and that
For the vibrometer to produce a signal output, it must all calibrations have been appropriately implemented. For
experience oscillating accelerations. Then the sensor illustration purposes, we will show plots from a swept cosine
amplitude and frequency output will be proportional to the function from 100-400 [Hz] over a period of two seconds.
amplitude and frequency of the sensed vibration. At this point the dataset is ready for Fourier analysis.
This makes the vibrometer amenable to spectral analysis Consider the result in Fig.1 when we perform a single FFT on
since one can directly apply the Fourier Transform to the entire set of vibration data, where the FFT is a
determine the frequency and magnitude of vibration on the computationally efficient version of eq.(1), the Discrete
instrumented vehicle. This data can then be compared to Fourier Transform (DFT).
structural design limitations of air frames to determine if the 𝑁−1 −𝑖2𝜋𝜏𝑣
1
vehicle has been overstressed. Another application would be 𝐹(𝑣) = ∑ 𝑓(𝜏) 𝑒 𝑁 [1, eq. (1)]
𝑁 𝜏=0
in the analysis of solid rocket motor combustion, the nominal
profiles of which would be very well known to the Clearly this will produce a plot of frequency vs. amplitude
manufacturer. For example, some of the Solid Rocket but note that we have lost all information about the time when
Boosters (SRB’s) manufactured by Magellan Aerospace used a certain frequency was prominent. Note that we have selected
by NASA’s Sounding Rocket Program (NSRP) are a constant-power frequency sweep such that the amplitude of
instrumented with vibrometers to monitor for combustion the signal decreases as the frequency increases to keep the
instability, a phenomena where large chunks of propellant are amount of power constant across all frequencies.
shot out of the motor instead of igniting and producing thrust.
The SRB vibration profile is shared with Magellan and with
White Sands Missile Range (the launch site) authorities to
verify that the motors are operating in a safe and effective
manner. Undoubtedly, one of the primary tools used by the
engineers in those organizations to interpret the vibration data
is the Fourier Transform. Since the nominal vibration profile
of each motor is known to some degree of accuracy; if the
measured profile exceeds or falls below the expected
operation then suitable actions may be taken to correct the
anomaly in the future; such as refurbishment or a change in
manufacturing or shipping process.

II. FOURIER TRANSFORM

In this analysis we will assume that the entire set of
vibration data is given in advance; in other words, we will not
assume that data is continuously streaming in real time for Fig. 2. STFT Using Hamming Window with Window Length
analysis. This is a fair assumption since most of the data from L = 100 and 15 Samples Overlapped
 To get around the issue of losing knowledge of time we
must use the Short Time Fourier Transform (STFT) on small
chunks of the dataset. In other words, instead of performing a
single FFT on the entire dataset, we can define a window sizes
and evaluate several FFT’s on small chunks of the dataset.
Then we combine the data together into a 3-D plot called a
spectrogram which plots Time vs. Frequency vs. Amplitude.
In this way, we introduce a dimension of time that we did not
have before, and we would be able to identify which
frequencies were prominent in each window of the dataset as
shown in Fig. 2.
Notice that now that we have knowledge of Fig.2, we can
clearly see that the vibration dataset is exponentially
increasing in frequency as time increases, and that near the end
of the recording, there is no longer any low frequency signals
present.
Now that we have started windowing the dataset, we have Fig. 4. STFT Using Kaiser Window with Window Length L
control over two parameters which will alter the results in the = 100, β = 5, 15 Samples Overlapped
spectrogram: window size and window shape.
As we decrease the number of samples in the window, we III. WINDOW FUNCTIONS
will achieve increased temporal resolution. This means we A window function is a mathematical function that is zero-
have a better idea of exactly when in time a certain frequency valued outside of some chosen interval. The window functions
occurred. The tradeoff here is that a smaller window size process in the time domain first to help improve results of later
necessarily means that longer wavelength (low frequency) processing in the frequency domain.
data will be much harder, or impossible to detect [2] as
illustrated in Fig. 3. Fig. 2 and Fig. 3 are identical except for We will introduce two window functions and compare
the window size, notice how Fig. 3 is much smoother and does their characteristics.
not exhibit sharp discontinuities as Fig. 2 does. However, the
smaller window size has made it more difficult to locate low A. Kaiser Window
frequencies, as shown in the wider band of Fig. 3. The Kaiser window is a discrete sequence window that
Although this tradeoff between temporal resolution and maximizes the energy concentration in the main lobe. The
low-frequency resolution seems like a conundrum, we believe function of the Kaiser window is:
this issue is easily managed. In every dataset there is going to
be a minimum frequency that the users are interested in, and
this minimum frequency is known ahead of time. Then, the
user may simply select a window size large enough to detect
this minimum frequency of interest. If this selected window [3, eq. (2)]
size does not provide the required temporal resolution, the
where I0 is the zeroth-order modified Bessel function and the
user may simply make another Spectrogram with smaller
window length is L = N+1. Increasing β widens the main lobe
window sizes and use both plots.
and decreases the amplitude of the side lobe [4], [5]. This
effect can be observed by comparing Fig. 4 and Fig. 5. Using
the Kaiser window often reveal signals close to the
background noise that other windows may obscure.

Fig. 3. STFT Using Hamming Window with Window Fig. 5. STFT Using Kaiser Window with Window Length L
Length L = 20 and 15 Samples Overlapped = 100, β = 15, and 15 Samples Overlapped
B. Hamming Window
The Hamming window is the member of the generalized
Hamming window family. The generalized hamming
window is constructed by multiplying a rectangular window
by one period of cosine. The cosine term lowers the side lobes
but with a tradeoff of doubling the width of the main lobe [6],
[7]. The function of the Hamming window is:

[8, eq. (3)]

where the window length is L = N + 1. The Hamming

window does a better job canceling the first side lobe but with
a tradeoff of slow side lobe roll off with discontinuity in
amplitudes at its endpoints. The effect of slow side lobe roll-
off can be seen by comparing the background color of Fig. 2
and Fig. 4. Fig. 7. Comparison of PSD with Sine Tones on Noise [9]
C. Choosing the Window Function For Our Application The PSD is then the right tool to compare vibration profiles
By comparing Fig. 2 and Fig. 4, we obtained the against each other because the amplitudes are normalized to
conclusion that the Kaiser window does a better job at the time duration of each recording [10]. The PSD is defined
revealing signals close to the background. With a higher β as the Fourier Transform of the autocorrelation function. In
value, (compare Fig. 5 to Fig. 4), we can obtain a better other words, the PSD multiplies the amplitude of the FFT by
visualization and analysis for the interested signal at a given its complex conjugate and normalizes it to the frequency bin
frequency. We believe that using a Kaiser window with high width.
β value will be a better fit to analyze the vibration data as the 𝐹(𝑣)𝐹 ∗ (𝑣)
recorded dataset is likely to be rife with random vibration. 𝑋𝑃𝑆𝐷 (𝑓) = lim [10, eq. (4)]
𝛥𝑓→0 𝛥𝑓

Fig. 7 uses the same data as Fig. 6 except that the PSD is
IV. ANALYSIS & PSD plotted instead of the FFT. Note how to amplitudes of each
Using the FFT for vibration analysis is fine for analyzing PSD do not vary with the size of each dataset.
one dataset in a vacuum, but if one wants to compare a A clear downside regarding the PSD is that we have once
vibration profile to another, the FFT is not the right tool for again lost information on the time dependence of the signal.
the job. The reason for this is that the amplitude of the FTT As it turns out, no single plot is likely to give a user all the
for a given dataset depends on the duration of the dataset [9]. information they are looking for. It is likely that a combination
For example, Fig. 6 compares the same vibration profile over of STFT plots of different window sizes and a PSD plot will
different durations. Note that the longer the duration of the be needed to be used together to get a full picture of the
recording, the lower the amplitude. When comparing different vibration envelope.
vibration profiles against each other, it is impractical to
assume all the data will be the same time duration.
V. CONCLUSION
We have explored how Fourier techniques can be used to
analyze vibration data for aerospace vehicles, but this analysis
can be extended to any system, like a car engine or train. The
STFT was used to reinsert a dimension of time to our analysis,
and the effect of window size and shape was examined. The
window size should be selected such that the window is large
enough to detect the lowest frequency of interest. The window
size should be no larger than necessary in order to keep as
much time-domain resolution as possible. The sample size
dependence on signal amplitude of the STFT was identified as
a shortcoming and the PSD was proposed as a solution to this
problem. Since the PSD will normalize the dataset to the size
of the recording, this shortcoming was eliminated, but the
dimension of time was once again lost. Using a STFT in
conjunction with a PSD plot now allows separate vibration
recordings to easily be compared to each other, which is
needed in practice because rockets are flown many times
throughout their lifecycle. For example, the NSRP has flown
over 2900 rockets as of 2015 [11]. Clearly this space program
and many others would have an interest in comparing the
Fig. 6. Comparison of FFT with Sine Tones on Noise [9] vibration profiles of each flight against each other to monitor
engine performance and airframe stress.
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