1. Planing Hull Acceleration Time Series Processing and Analysis
Independent Study Report, Winter Semester, 2011
Prepared For:
Armin Troesch
Professor, Department of Naval Architecture and Marine Engineering
The University of Michigan
Prepared By:
Daniel Place
Graduate Student, Department of Naval Architecture and Marine Engineering
The University of Michigan

2. 2
Table of Contents
List of Tables .................................................................................................................................. 3
List of Figures................................................................................................................................. 3
1.0 – Introduction............................................................................................................................ 4
2.0 – Background............................................................................................................................ 4
3.0 – Vertical Acceleration Time Series Filtration ......................................................................... 5
3.1 – Filter Selection................................................................................................................... 5
3.2 – Filter Recreation and Testing............................................................................................. 7
3.3 – Acceleration Time Series Filtration................................................................................... 8
3.4 – Attenuation of Slamming Events Due to Filtration ......................................................... 10
4.0 – Statistical Analysis of Vertical Acceleration Time Series................................................... 13
4.1 – Peak Acceleration Values from Time Series ................................................................... 13
4.2 – Random Forcing Time Series .......................................................................................... 14
4.3 – Fitting the Vibro-Impact Model to Experimental Data.................................................... 17
4.4 – Extending Vibro-Impact Simulations .............................................................................. 17
5.0 – Conclusions.......................................................................................................................... 18
Acknowledgements....................................................................................................................... 19
References..................................................................................................................................... 20

3. 3
List of Tables
Table 1 – Testing Conditions for unspecified 36’ planing vessel................................................... 4
Table 2 – Vessel parameters used for simulated planing vessel impact....................................... 11
Table 3 – Raw and filtered peak accelerations from Heller-Jasper/Wagner simulation............... 12
List of Figures
Figure 1 – Wave slamming event with substantial noise content................................................... 5
Figure 2 – Fast Fourier transform of vertical acceleration time series ........................................... 6
Figure 3 – Comparison of filter frequency output from Michigan and Butterworth filters............ 7
Figure 4 – Michigan filter gain and phase plot, 12 Hz cutoff, 50 point lag window...................... 8
Figure 5 – FFT plots of the sample acceleration time series before and after filtration................. 9
Figure 6 – Original and filtered acceleration time series.............................................................. 10
Figure 7 – Histogram of maximum accelerations from time series (filtered, 10 Hz cutoff freq.) 13
Figure 8 – Effect of filter cutoff frequency on CDF of acceleration peaks in Weibull space ...... 14
Figure 9 – Impact oscillator schematic (Rose, 2010) ................................................................... 15
Figure 10 – Simulated random forcing example time series ........................................................ 15
Figure 11 – Comparison of forcing maxima CDF to a Rayleigh distribution .............................. 16
Figure 12 – Comparison of simulated impact local peak CDF to experimental CDFs ................ 16
Figure 13 – Ensemble of forcing time series using different initial random numbers ................. 18

4. 4
1.0 – Introduction
Data obtained experimentally in a harsh working environment is prone to contamination from
sources other than what is being measured. The vertical acceleration of a high speed planing
vessels operating in a moderate sea state and experiencing non-linear slamming events can excite
secondary vibrations. These will be seen as high frequency signals in the acceleration data being
collected. The high frequency signals prevent the rigid body maximum accelerations from being
accurately observed. These signals must be removed through digital post-processing. Digital
low-pass filtration can attenuate the important information contained in a signal, thus the
properties of the filter must be known in order to interpret and understand the filtered results.
The maximum accelerations experienced by a vessel over its lifetime dictate its capability and
the health of those operating the vessel. Using a simulation to model the slamming accelerations
experienced by a high speed planing vessel is useful because long time series can be generated
that would be impractical to capture in the real world. The statistics and properties of a non-
linear vibro-impact oscillator have been studied as a model to simulate vessel slamming (Rose,
2010). This model can be used to simulate long acceleration time series which can provide
insight into the probability of extreme slamming events.
The goal of the work presented was to understand the development and implementation of a
digital filtering algorithm and its effects on measured data. Also studied were extreme value
statistics and their role in predicting extreme values.
2.0 – Background
The research performed throughout the last semester has concentrated on analyzing acceleration
time series obtained by the Naval Surface Warefare Center Carderock, Combatant Craft Division
(CCD), Test and Evaluation Branch at Little Creek. Their primary mission is high speed vessel
research, development and testing for the Navy. CCD tested an unspecified 36’ planing patrol
vessel in two different sea states at two different speeds and captured 100 second acceleration
time series each time (Table 1).
Testing Date
Sig. Wave
Height [m]
Average
Period [s]
Dominant
Period [s]
Average Craft
Speed [kts]
27 Apr 2005 3.5 5.2 7.2
32 (head seas),
40 (stern seas)
05 May 2005 4.4 3.7 4.6
28 (head seas),
31 (stern seas)
Table 1 – Testing conditions for unspecified 36’ planing vessel
The vertical acceleration data that was captured over the 18 minutes was found to have large
spikes that were believed to be erroneous or due to vibrations other than the boat responding to
wave forcing. Investigating the maximum accelerations that are experienced on a planing vessel
is very important for the health of the people using the vessel and the integrity of the vessel
itself. However, if the actual peak accelerations in time series are diluted with erroneously large
accelerations from other sources, it is difficult to determine the actual peak accelerations being
experienced on board. This concern prompted investigation into the best way to retrieve useful

5. 5
information from the acceleration time series. Figure 1 contains an excerpt of the raw
acceleration data showing the substantial amount high frequency content it contains.
0 2 4 6 8 10
-5
0
5
10
Original Signal
Amplitude[g's]
Time [s]
Figure 1 – Wave slamming event with substantial noise content
The work done with the acceleration time series at the University of Michigan concentrated on
using different digital filters to post-process the data and remove high frequency content while
being able to recover the actual peak accelerations experienced by the vessel. Once a good
estimate of the true acceleration time series was obtained, the statistics of the time series were
found to provide insight into predicting the frequency of extreme acceleration values. The work
done for predicting extreme values was not completed. However, it is clear that being able to
predict the maximum accelerations that a vessel and its passengers could potentially experience
on a high speed planing vessel is valuable.
3.0 – Vertical Acceleration Time Series Filtration
3.1 – Filter Selection
The first step taken in filtering the acceleration noise out of the time series was to choose an
appropriate digital filtering scheme. It was known that the rigid hull accelerations due to wave
slamming would occur in the 1-10 Hertz range whereas accelerations occurring at higher
frequencies (20-100 Hertz) were most likely vibrations from local flexure of the deck and plating
(Riley, et al., 2010). Spectral analysis of the acceleration time series verified this trend. The
FFT plot is shown in Figure 2. The most prevalent frequencies occurred in the 1-10 Hertz range
as expected and another cluster was observed around 25 Hz. The 40-160 Hz range also
contained vibration frequencies which may have been due to the engine running and subsequent
vibration of it and other parts of the vessel.

6. 6
0 20 40 60 80 100 120 140 160 180 200
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Single-Sided Amplitude Spectrum of Acceleration (original signal)
Frequency (Hz)
|Acceleration(g's)|
Figure 2 – Fast Fourier transform of vertical acceleration time series
A low-pass digital filter was the best choice for eliminating the unwanted frequencies because
the rigid body acceleration data was concentrated at lower frequencies whereas the noise in the
data existed above 15 Hertz. Many types of low-pass digital filtering schemes exist and have
varying degrees of complexity and effectiveness. The Butterworth filter is a common digital
low-pass filter which is popular because of its robustness and was initially considered as a good
filter to use. A second filter was considered which was designed by Ken Bongort circa 1979 at
the University of Michigan NA&ME department (this filter will henceforth be referred to as the
“Michigan” filter). It is a symmetric, non-recursive filter which means that its weights are
calculated using data on either side of the point being filtered. This means that there will be no
phase shift in the data once it is filtered. Because it uses information before and after the point
being filtered, the filter cannot be used as data is captured and can only be used in post-
processing. It is also a simple filter algorithm and is computationally cheap for the filtering
quality obtained. The Butterworth filter was not chosen because of the amount of frequencies
beyond the cutoff frequency that can “leak” into the data. Figure 3 compares the gain response
of a 5th
order Butterworth filter to the Michigan filter (10 Hertz cutoff frequency). It was found
that the Butterworth did have a broader range of frequencies leaking into the output signal above
the cutoff frequency as well as more signal attenuation below the cutoff frequency than the
Michigan filter. The Michigan filter’s steeper roll off was advantageous for filtering the
acceleration data because the frequencies above 10 Hertz needed to remain intact while ensuring
that the frequencies around 25 Hertz were eliminated.

7. 7
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Frequency [Hertz]
Gain
Magnitude frequency response
Michigan
Butterworth, 5th order
Figure 3 – Comparison of filter frequency output from Michigan and Butterworth filters
3.2 – Filter Recreation and Testing
The Michigan low-pass filter scheme was originally written in FORTRAN and was translated to
the MATLAB programming language for this research project. Before the filter could be used
on the random acceleration time series, it was important to understand how the different filter
parameters affected its performance. The following parameters are user defined and must be
understood to ensure accurate results:
o Cutoff frequency - Highest frequency desired to remain in the filtered signal, set at 12 Hz
o Lag window – Number of points that will be used on either side of a data point to
compute its weight, set at 50
o Sampling period - Rate at which the original data was sampled, 512 Hz
It was important to note that the number of points specified for the lag window is eliminated
from the beginning and end of the output signal. This needs to be considered because if
important information exists at either end of the signal, it may not be represented in the filter
output. This was not a concern with the acceleration time series because the important events
began occurring after 5000 data points.
The filter characteristics and behavior was analyzed by using it to filter known signals and
comparing the output to the input. This was done with a series of sine waves of the following
form:
( ) 1.0sin( )n nf x tω= [1]

8. 8
The signal frequency ωn was varied from 3 to 80 Hertz using a lag window of 50 points and a
cutoff frequency of 12 Hertz. Each signal was filtered and ratios of the output to input peak
height were found to determine the amount of signal attenuation at each frequency (Figure 4).
The phase shift of the output signal was also found and plotted.
0 10 20 30 40 50 60 70
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Signal Gain, fcutoff
= 12 hz, lag window = 50
Gain[FilteredAmp/OriginalAmp]
Frequency [Hz]
0 10 20 30 40 50 60 70
0
0.005
0.01
0.015
0.02
0.025
Signal Phase, fcutoff
= 12 hz, lag window = 50
Phase[rad]
Frequency [Hz]
Figure 4 – Michigan filter gain and phase plot, 12 Hz cutoff, 50 point lag window
The filter was effective at completely eliminating signals with frequencies above 20 Hertz. The
signal was attenuated to 50 percent of its original amplitude at the cutoff frequency and rolled off
to 1.7 percent of its original amplitude at 20 Hertz. The important rigid body accelerations of the
planing vessel occurred in the 1 to 5 Hertz range, thus it was important to verify that this range
was minimally attenuated by the Michigan filter. It was found that at 5 Hertz, the signal was
attenuated to 98 percent of its original amplitude. This provided confidence that using the
Michigan filter on the vertical acceleration time series would not significantly attenuate the peak
accelerations.
3.3 – Acceleration Time Series Filtration
The original vertical time series contained 18 minutes of data collected at 512 Hertz, equaling
more than 500,000 data points. It was not possible with the computers used to perform a fast
Fourier transform (FFT) on the full set of data, thus a 50,000 point sample was taken and used
for analysis purposes. The sample time series will also be used for plotting in this report because
plotting the full 18 minute time series renders it unreadable.
The acceleration time series sample was sent through the Michigan filter and the output’s
frequency content was found with an FFT and compared to the frequency content of the original
signal (Figure 5). The filter succeeded in removing virtually all of the high frequency content
above 15 Hertz. Figure 6 shows the raw and filtered acceleration time series and also an
expanded view of a major slamming event which occurred at 10.3 seconds. The shape of the

9. 9
slamming event appears to remain intact while the high frequency oscillations occurring after the
slam are removed.
0 20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
Single-Sided Amplitude Spectrum of Acceleration (original signal)
Frequency (Hz)
|Acceleration(g's)|
0 20 40 60 80 100 120 140 160 180 200
0
0.1
0.2
0.3
0.4
0.5
Single-Sided Amplitude Spectrum of Acceleration (filtered signal)
Frequency (Hz)
|Acceleration(g's)|
Figure 5 – FFT plots of the sample acceleration time series before and after filtration

10. 10
0 10 20 30 40 50 60 70 80 90
-5
0
5
10
Original Signal
Acceleration[g's]
Time [s]
0 10 20 30 40 50 60 70 80 90
-5
0
5
10
Filtered Signal, fcutoff
= 12 hz, lag window = 50
Acceleration[g's]
Time [s]
10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11
-5
0
5
10
Slamming Event, Original Signal
Acceleration[g's]
Time [s]
10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11
-5
0
5
10
Slamming Event, Filtered Signal
Acceleration[g's]
Time [s]
Figure 6 – Original and filtered acceleration time series
3.4 – Attenuation of Slamming Event Accelerations Due to Filtration
Although the slamming events in the filtered acceleration time series did not appear to have been
significantly attenuated, a method for quantifying the filter’s effect on the peak rigid body
accelerations was sought. It was first necessary to generate a simulated planing boat wave
impact event in order to measure its characteristics before and after the Michigan filter scheme
was applied to it. University of Michigan NA&ME student Don Norman produced a MATLAB
code that uses a solution by Wagner which approximates the vertical force on a planing vessel as
linearly increasing with time until the chine becomes wet. The following equations describe the
force on the body as described by Wagner.
2.39917
3 49546.12443fC β −
= ⋅ [2]
tan( )
w
B
t
V
β
π
⋅
=
⋅
[3]

11. 11
2
33 VtCF wf ⋅⋅⋅= ρ [4]
3
vessel
F
A
m
= [5]
Where: Cf3 is the interpolated F3 coefficient by Faltinsen
tw is the time to chines wet
β is the deadrise angle
B is the beam of the vessel
V is the forward velocity
F3 is the vertical force on the body
ρ is water density
A is the vertical acceleration of the vessel
mvessel is the mass of the planing craft
The time when the chine becomes wet is the time of maximum vertical acceleration. After, the
acceleration is suggested by Heller and Jasper to decay exponentially with time. The following
equations describe the Heller-Jasper solution.
( )
3 0 1
wt t
wt t
F F e α
α
− −
− 
= ⋅ − 
 
[4]
3
vessel
F
A
m
= [5]
Where: F0 is the vertical force on the body at time of chines wet according to Wagner
α is the time after impact between the maximum force and subsequent zero
down crossing
The impact simulation code output one impact event as an acceleration time series. This was
filtered by the Michigan low-pass filter to determine attenuation effects. Because the exact
dimensions and type of vessel the original vertical acceleration time series were captured on
were not known, many of the parameters necessary to run the simulation were estimated based
on experience and existing vessels (Table 2).
Length Beam Deadrise Displacement
30.4 m 8.4 m 25 deg. 8.16 t
Table 2 – Vessel parameters used for simulated planing vessel impact
The peak accelerations from the impact simulation are shown in Table 3 using α values of 0.1
and 0.2 based on two references: Price, et al. SAVIAC 2003, and Riley, et al. NSWC-CD 2010.
Also shown are the results from filtering the data at different cutoff frequencies with the
Michigan filter which reveals attenuation of the peak acceleration. Adjusting the cutoff

12. 12
frequency does, however, change the filter properties. It was found by NA&ME student
Xiaofeng Jiang that a constant magnitude frequency response could be maintained if the
relationship that lag window times the cutoff frequency (CF) equals 2000 held. Thus, each
cutoff frequency tested required a different lag window to be used during filtration.
α = 0.1 α = 0.2
Vessel
Speed
Raw CF = 10Hz 20 Hz 50 Hz 100 Hz Raw CF = 10Hz 20 Hz 50 Hz 100 Hz
40 kts 5.24 4.59 5.09 5.19 5.22 5.24 4.88 5.09 5.19 5.22
45 kts 6.63 6.14 6.40 6.55 6.59 6.63 6.13 6.42 6.56 6.60
50 kts 8.21 7.52 7.88 8.08 8.15 8.21 7.52 7.91 8.10 8.16
55 kts 9.87 8.99 9.48 9.75 9.83 9.87 9.02 9.53 9.76 9.84
60 kts 11.74 10.59 11.24 11.58 11.69 11.74 10.67 11.31 11.61 11.70
65 kts 13.76 12.31 13.13 13.72 13.70 13.76 12.45 13.23 13.60 13.71
70 kts 15.98 14.18 15.17 15.72 15.89 15.98 14.37 15.32 15.77 15.91
75 kts 18.46 16.22 17.39 18.07 18.26 18.46 16.47 17.61 18.15 18.33
Table 3 – Raw and filtered peak accelerations from Heller-Jasper/Wagner simulation
The results show that filtration does have an attenuation effect on the peak impact accelerations.
This indicates that the peak accelerations from the filtered acceleration time series results are
probably not the true peak values the rigid vessel was experiencing. However, the attenuation is
relatively small, with the worst case being the 50 kts condition using α=0.1 and filtered with a
cutoff frequency of 10 Hertz. The simulated peak acceleration at this condition is attenuated to
87.6 percent of its original size by the Michigan filter. This is a significant finding and indicates
that the impact peaks of the filtered vertical acceleration time series are probably smaller than
what was actually experienced on the planing vessel.

13. 13
4.0 – Statistical Analysis of Vertical Acceleration Time Series
Having obtained a method for extracting and determining the maximum slamming acceleration
values in the vertical acceleration time series, statistical methods were used to examine the
likelihood of the planing vessel experiencing extreme slamming events.
4.1 – Peak Acceleration Values from Time Series
The local maximum peaks in the vertical acceleration data were found using an algorithm which
looks for a maximum value between zero crossings, after a zero up-crossing. Figure 7 shows the
distribution of maximum acceleration values found in the time series. The largest acceleration
values, while not occurring very often, are arguably the most important values to consider. The
vessel and those riding in it must be prepared for the less common and rare impacts where
“extreme” accelerations are experienced.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
15
20
25
30
35
40
45
Accelerations [g's]
NumberofOccurences
Figure 7 – Histogram of maximum accelerations from time series (filtered, 10 Hz cutoff freq.)
The cumulative density function (CDF) of maximum values for this type of non-linear process
has been described as approaching the CDF of a Rayleigh distribution. The Rayleigh
distribution CDF can be plotted in Weibull space on a quantile-quantile (Q-Q) plot as a line with
a slope of two, where each quantile is described by the following equations.
( )ξ10log=xQ [6]
( )( )( )ξPQy −−= 1loglog 1010 [7]
Where: P(ξ) is the CDF, which is the probability that an event does not exceed the
value of ξ

14. 14
Different cutoff frequencies were used on the raw acceleration time series to observe the
Michigan filter’s effect on the peak distribution in Weibull space (Figure 8). It was found that
increasing the cutoff frequency tended to distance the acceleration peak distribution away from
the Rayleigh distribution, especially for the smaller acceleration peaks. This indicates that
leaving the higher frequency signals in the acceleration time series forces the algorithm that finds
the local peaks to select the high frequency, low amplitude peaks and dilute the rigid body
acceleration low amplitude peaks. Therefore, for a higher cutoff frequency, there is a smaller
probability of experiencing a larger acceleration event.
-1 -0.5 0 0.5 1 1.5 2
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Qx
Qy
Cutoff-10 hz
Cutoff-20 hz
Cutoff-30 hz
Cutoff-40 hz
Cutoff-50 hz
Rayleigh Distribution
Figure 8 – Effect of filter cutoff frequency on CDF of acceleration peaks in Weibull space
4.2 – Random Forcing Time Series
The vertical acceleration data taken from the planing vessel was an 18 minute time series. This
is not enough time to appropriately sample for extreme slamming events which may only occur a
few times over the life of the vessel. In order to estimate and predict these events, a simulation
that can run for an arbitrary amount of time is a useful tool. A model was developed (Rose,
2010) to produce nonlinear acceleration response of a vessel due to slamming and has been used
for prediction of extreme values and their probability. The model used to produce the nonlinear
acceleration was a vibro-impact oscillator two degree of freedom system shown in Figure 9. The
non-linear slamming event occurs when the oscillator hits the wall, modeling a ship hull
impacting with the water.

15. 15
Figure 9 – Impact oscillator schematic (Rose, 2010)
The non-linear spring stiffness used for the model follows Equation 8.
[8]
Where: γ is a non-dimensional modifier of the linear spring constant k that changes the
spring stiffness as a function of displacement (Rose, 2010)
A random forcing time series was first generated which drove the impact oscillator and produced
the nonlinear accelerations. The forcing maxima follow a Gaussian distribution with a broadness
parameter which converges to a Rayleigh CDF at higher forcing values. Figure 10 shows a
sample randomly generated forcing time series and Figure 11 compares the CDF of forcing
maxima to the Rayleigh distribution.
0 5 10 15 20 25
-2
-1
0
1
2
Number of Natural Cycles
Forcing(F/kL)
Figure 10 – Simulated random forcing example time series

16. 16
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Qx
Qy
Forcing
Rayleigh Distribution
Figure 11 – Comparison of forcing maxima CDF to a Rayleigh distribution
The nonlinear equations of motion were numerically solved for the acceleration of the oscillator
using code previously developed (Rose, 2010). The resulting acceleration time series received
the same post processing as the filtered vertical acceleration time series in order to record the
local peak accelerations. The CDF of the peaks was found and plotted in Weibull space and
compared to the CDF of the actual acceleration time series local peaks (Figure 12).
Figure 12 – Comparison of simulated impact local peak CDF to experimental CDFs

17. 17
The simulated non-linear acceleration time series converges to the Rayleigh CDF until Qx is
equal to about 0.5 which corresponds to about 3.2 g’s of acceleration. At this point the
simulation CDF slope decreases from 2 to about 0.5, joining the trend which the actual
acceleration data follows. This indicates that the probability the rigid hull will experience
vertical accelerations larger than about 3.2 g’s is greater than what the Rayleigh distribution
predicts.
4.3 – Fitting the Vibro-Impact Model to Experimental Data
Although the vibro-impact maximum acceleration CDF had a similar trend to the experimental
data at larger acceleration values, it was desired to have the model represent the experimental
data at the smaller accelerations as well. Fitting the impact oscillator CDF to the experimental
data was a trial and error process that involved changing the system parameters governing the
random forcing generation and the vibro-impact model. The results from the best simulation are
shown in Figure 12. It was not possible to closely match the trend of the smaller peak
accelerations regardless of the combination of parameters used. The parameters that were
optimized and the best values selected for each were as follows.
ζ – Damping coefficient – 0.25
γ – Wall parameter: non-dimensional term that changes the wall stiffness. Increasing it
effectively shortens the distance between the vibro-impact oscillator and the wall – 50
Fσ – RMS force: non-dimensional root mean squared forcing term – 1.0
η0 – Initial frequency ratio: smallest frequency component defining the forcing – 1.25
ηm – Maximum frequency ratio: largest frequency component defining the forcing – 1.75
N – Number of Fourier components used to generate the forcing – 1000
τt – Number of natural cycles – defines the length of the forcing – 1000
4.4 – Extending Vibro-Impact Simulations
The vibro-impact acceleration simulation did fit the experimental data well in Weibull space
above 3.2 g’s as was seen in Figure 12. It was believed that the model could be used to simulate
a much longer exposure time than was performed experimentally to determine the likelihood of
even larger impacts occurring. This information could then be used for to plan for extreme
loadings on the real world planing vessel and its passengers.
Generating a sufficiently long forcing time series presented a problem because there were only a
certain number of Fourier components that could be used without the computation becoming
excessively expensive. This problem was handled by generating an ensemble of forcing time
series which all used a different starting random number. This ensured that, during each run, the
time series were different from each other. Figure 13 compares five randomly generated forcing
time series.

18. 18
Figure 13 – Ensemble of forcing time series using different initial random numbers
Generating an ensemble of random time series was only viable for a single MATLAB session
because, upon restarting MATLAB, it was observed that the same random numbers would be
used again, which would generate identical forcing time series. This was unacceptable because
the forcing was then not truly random which could result in a bias in the results. The best
solution to this problem was not solved by then end of the semester, but a few options were
investigated. One imperfect solution was to use the computer clock to initialize the random
number stream so as to always have a different set of random numbers when MATLAB started.
The issue was that if the computer and MATLAB are started around the same time each day, the
random number streams would not be sufficiently different from each other. Starting MATLAB
at different times each day relative to the computer boot time would be a temporary solution but
still presents issues.
The accelerations were never generated for the forcing time series ensembles due to the issues
with ensuring the forcing time series were truly random. If the forcing ensemble had been
solved for acceleration, it would have output an ensemble of acceleration time series, one for
each forcing series. Post-processing the ensemble would have been important to ensure the
results were representative of a single, long exposure time series. Discarding the beginning of
the acceleration time series from all but the first series in the ensemble would have been very
important because each would have contained transients. Removing the first five cycles was
suggested although this would have been verified in the post processing.
5.0 – Conclusions
The work documented here concentrated on the processing and interpretation of planing vessel
vertical accelerations. A low-pass filter was selected and tested to ensure it was an appropriate
choice for the vertical acceleration time series collected experimentally. A filter developed in
the University of Michigan NA&ME department was chosen because of its shorter transition
between its pass and stop bands compared to a Butterworth filter. The Michigan filter was used
to remove high frequency signals from secondary vibrations on the planing vessel which mainly
occured near 25 Hertz. A cutoff frequency of 12 Hertz was used which removed the high
frequency signals effectively. The filter was also tested to document its attenuation effects on
the important rigid body slamming accelerations experienced by the vessel. Based on the
filtering theoretical slamming events, the Michigan filter was found to attenuate the maximum

19. 19
acceleration peaks to about 87 percent of their original values using a cutoff frequency of 10
Hertz.
Statistical analysis of the maximum acceleration values obtained from the filtered time series
was performed to investigate the probability of the vessel experiencing extreme slamming
accelerations. The acceleration time series was filtered using different cutoff frequencies and the
CDF of the peaks was plotted in Weibull space and compared to the Rayleigh distribution. It
was found that a lower cutoff frequency shifts the CDF of the peaks closer to the Rayleigh
distribution, which indicates that the high frequency peaks removed by filtration are low
amplitude, thus the probability of exceeding them increases when they are removed by the filter.
A vibro-impact oscillator was used to simulate slamming accelerations experienced by a planing
vessel and the results were compared to the experimental acceleration CDF in Weibull space. It
was found that the impact oscillator results converged to the Rayleigh distribution until
accelerations exceed 3.2 g’s. After this point the simulation results began to follow the trend of
the experimental results. It was found that both the impact oscillator results and the experimental
results indicated that there was a higher probability of experiencing accelerations above 3.2 g’s
than the Rayleigh distribution suggested. This conclusion is important because it means the
vessel and its passengers must plan for accelerations higher than what a simple Rayleigh
distribution would have suggested. Work was also done to simulate much longer acceleration
time series, unfortunately problems with maintaining random forcing time series were
encountered and the research for the rest of the semester concentrated on random number
generation. Once these problems are solved, it would be very useful to simulate longer time
series to investigate the probability of even larger extreme slamming events being experienced.
Acknowledgements
The original MATLAB code used for analyzing the statistics of the vertical acceleration time
series was written by Christopher Rose for his Master’s thesis (Rose, 2010). Although parts
were modified for the research performed this semester, credit is given to Christopher for the
creation of the code necessary to perform this research.

20. 20
References
[Rose, 2010] Rose, C. (2010). Stochastic Vibro-Impact Model for Planing Craft Acceleration
Prediction, unpublished thesis (M.S.), The University of Michigan.
[Riley, et al., 2010] Riley, M., Coats, T., Haupt, K., Jacobson, D. (2010) ‘The Characterization
of Individual Wave Slam Acceleration Responses for High Speed Craft’, paper presented
at The 29th
American Towing Tank Conference, Annapolis Maryland, 12 August 2010.