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Ship motion models

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Title:
Ship motion models
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Stamile, Jennifer Patricia
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xiv, 121 leaves : ; 28 cm

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Subjects / Keywords:
Ships -- Hydrodynamics -- Mathematical models ( lcsh )
Motion -- Mathematical models ( lcsh )
Stability of ships -- Mathematical models ( lcsh )
Motion -- Mathematical models ( fast )
Ships -- Hydrodynamics -- Mathematical models ( fast )
Stability of ships -- Mathematical models ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 120-121).
Statement of Responsibility:
by Jennifer Patricia Stamile.

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University of Colorado Denver
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Auraria Library
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ocm47108271
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Full Text
SHIP MOTION MODELS
by
Jennifer Patricia Stamile
BA, University of Colorado at Denver, 1996
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Basic Science
2000
[alJ


This thesis for the Master of Basic Science
degree by
Jennifer Patricia Stamile
has been approved
'A
by
Alberto Sadun
(>
Date


Stamile, Jennifer Patricia (MBS)
Ship Motion Models
Thesis directed by Associate Professor Randall P. Tagg
ABSTRACT
Ship motion is a dynamic system with six degrees of freedom. The principle
movements studied are roll and heave. Two approaches help in forming a
comprehensive view of ship motion. One approach examines actual ship motion data
with numerical tools, primarily the tools in Chaos Data Analyzer: Professional
Version. The most significant result of the analysis is finding the correlation time.
The second approach is to construct a model of ship motion from basic naval-
architecture principles. With a model of two coupled differential equations, one can
learn about the important components of the data. Mathieu equations are similar in
structure to the modeled equations, have been used to model ship motion, and help
model a coupled non-autonomous four-dimensional system. This thesis helps form a
basis for future projects.
This abstract accurately represents the content of the candidate'^thesis. I recommend
its publication.
Signed
Randall P. Tagg
in


ACKNOWLEDGEMENT
My thanks to the Office of Naval Research Grants for the grants and Zenas
Hartvigson for the scholarship that helped fund this project.


CONTENTS
Figures.................................................................viii
Tables...................................................................xiv
Chapter
1. Introduction............................................................1
2. Analyzing Actual Ship Motion Data.......................................4
2.1 Each Window...........................................................5
2.1.1 Time Series..........................................................5
2.1.2 Power Spectrum.......................................................6
2.1.3 Correlation Function.................................................9
2.2 The Entire Day.......................................................11
2.2.1 Mean................................................................11
2.2.2 Variance............................................................12
2.2.3 Skewness............................................................13
2.2.4 Kurtosis............................................................14
2.2.5 Dominant Frequencies................................................15
2.2.6 Correlation Time....................................................16
2.2 Hurst Exponent.......................................................18
2.3 Summary..............................................................19
v


3. Modeling Moored Ships..................................................21
3.1 Rigid Body Motion......................................................21
3.2 Simplified Class of Hull Shapes......................................25
3.3 Parameters Derived from Ship Hull Shape and Mass Distribution........28
3.3.1 Ship Mass............................................................28
3.3.2 Volume...............................................................29
3.3.3 Center of Gravity....................................................32
3.3.4 Moment of Inertia....................................................44
3.4 Parameters Derived from Ship Position Relative to Sea Surface.......47
3.4.1 Equilibrium Waterline Depth..........................................47
3.4.2 Center of Buoyancy...................................................51
3.5 Modeling Sea Surface Variations........................................59
3.6 Equations of Motion...................................................60
3.7 Summary...............................................................63
4. Coupled Mathieu Equations..............................................64
4.1 Undamped Linear Mathieu Equation Two Degrees of Freedom.............64
4.2 Damped Linear Mathieu Equation Two Degrees of Freedom..............67
4.3 Nonlinear Damped Mathieu Equation Two Degrees of Freedom...........69
4.4 Nonlinear Damped Coupled Mathieu Equations -
Two Degrees of Freedom Each............................................70
4.5 Mathieu Equation Conclusion...........................................73
5. Conclusion.............................................................74
vi


Appendix
A. Plots of Different Days and Plots of Heave.............................76
B. Matlab Programs..........................................................108
C. Glossary of Naval Architecture...........................................118
References...................................................................120


FIGURES
Figure
2.1 Degrees of Freedom.......................................................4
2.2 Time Series of Roll 38 of July 19.........................................5
2.3 Time Series of Roll 47 of July 19.........................................6
2.4 Power Spectrum of Roll 38 of July 19......................................8
2.5 Power Spectrum of Roll 47 of July 19......................................8
2.6 Correlation Function of Roll 38 of July 19...............................10
2.7 Correlation Function of Roll 47 of July 19...............................10
2.8 Time Series of Roll of July 19...........................................11
2.9 Mean of Roll of July 19...................................................12
2.10 Variance of Roll of July 19.............................................13
2.11 Skewness of Roll of July 19.............................................14
2.12 Kurtosis of Roll of July 19.............................................15
2.13 Dominant Frequency of Raw Roll Data of July 19..........................16
2.14 Dominant Frequency of Smoothed Roll Data of July 19.....................16
2.15 Correlation Time of Raw Roll Data of July 19............................17
2.16 Correlation Time of Smoothed Roll Data of July 19.......................17
2.17 Correlation Time of Roll Data of July 19................................18
viii


2.18 Hurst Exponent of Raw Roll Data of July 19..............................19
2.19 Hurst Exponent of Smoothed Roll Data of July 19.........................19
3.1 Rigid Body Motion Diagram...............................................22
3.2 Views of the Trapezoidal Vessel.........................................26
3.3 Rectangular Transverse Hull..............................................27
3.4 Triangular Transverse Hull...............................................27
3.5 Volume of Trapezoidal Hull...............................................30
3.6 Center of Gravity for Right Triangle....................................32
3.7 Triangular Part of a Tilted Trapezoid...................................34
3.8 Center of Gravity of Trapezoidal Hull of Uniformly Distributed Mass......39
3.9 Center of Gravity of Trapezoidal Hull of Non-Uniformly Distributed Mass.42
3.10 Moment of Inertia of Trapezoidal Hull ..................................44
3.11 Equilibrium Waterline Depth.............................................47
3.12 The Actual Area of the Water...........................................52
3.13 The Area of the Vessel's Waterline.....................................52
3.14 The Areas of the Two Triangles Formed by the Waterline and Tilted Vessel... 53
3.15 Important Parameters of Buoyancy........................................54
3.16 Ship Position Relative to the Ocean Floor..............................54
3.17 Construction to Find Buoyancy Moment...................................54
3.18 Closer Look at Figure 3.17.............................................54
4.1 Ince-Strutt Stability Chart...............................................65
ix


4.2 Solutions of Mathieu Linear Equation with a = 1 and q = 0.005...........66
4.3 Solutions of Mathieu Linear Equation with a = 4 and q = 0.005...........66
4.4 Solutions of Mathieu Linear Equation with a = 2 and q = 0.005...........67
4.5 Solutions of Damped Linear Mathieu Equation with a = 1 and q = 0.0025 ..69
4.6 Solutions of Damped Nonlinear Mathieu Equation with a=l .469,
q=0.1102, and (3=0.42..................................................70
4.7 Weakly Coupled Mathieu Equations........................................71
4.8 Moderately Coupled Mathieu Equations....................................72
4.9 Highly Coupled Mathieu Equations........................................73
A.l Time Series of Roll of July 18..........................................76
A. 2 Mean of Roll of July 18................................................77
A.3 Variance of Roll of July 18.............................................77
A.4 Skewness of Roll of July 18.............................................78
A.5 Kurtosis of Roll of July 18.............................................78
A.6 Dominant Frequency of Roll of July 18...................................79
A. 7 Correlation Time of Roll of July 18....................................79
A. 8 Hurst Exponent of Roll of July 18......................................80
A.9 Time Series of Roll of July 17..........................................80
A. 10 Mean of Roll of July 17...............................................81
A.ll Variance of Roll of July 17............................................81
A. 12 Skewness of Roll of July 17...........................................82
x


A. 13 Kurtosis of Roll of July 17...........................................82
A. 14 Dominant Frequency of Roll of July 17.................................83
A. 15 Correlation Time of Roll of July 17...................................83
A. 16 Hurst Exponent of Roll of July 17.....................................84
A. 17 Time Series of Roll of July 15........................................84
A.18 Mean of Roll of July 15................................................85
A. 19 Variance of Roll of July 15...........................................85
A. 20 Skewness of Roll of July 15...........................................86
A. 21 Kurtosis of Roll of July 15...........................................86
A.22 Dominant Frequency of Roll of July 15..................................87
A. 23 Correlation Time of Roll of July 15...................................87
A. 24 Hurst Exponent of Roll of July 15.....................................88
A.25 Time Series of Heave of July 19........................................88
A. 26 Mean of Heave of July 19..............................................89
A.27 Variance of Heave of July 19...........................................89
A.28 Skewness of Heave of July 19...........................................90
A. 29 Kurtosis of Heave of July 19..........................................90
A.30 Dominant Frequency of Heave of July 19.................................91
A.31 Correlation Time of Heave of July 19...................................91
A. 3 2 Hurst Exponent of Heave of July 19...................................92
A.33 Time Series of July 18.................................................92
xi


A. 34 Mean of Heave of July 18............................................93
A. 3 5 Variance of Heave of July 18.......................................94
A. 3 6 Skewness of Heave of July 18.......................................94
A.37 Kurtosis of Heave of July 18.........................................95
A.38 Dominant Frequency of Heave of July 18...............................95
A.39 Correlation Time of Heave of July 18.................................96
A.40 Hurst Exponent of Heave of July 18...................................96
A. 41 Time Series of Heave of July 17.....................................97
A.42 Mean of Heave of July 17.............................................97
A.43 Variance of Heave of July 17.........................................98
A.44 Skewness of Heave of July 17.........................................98
A.45 Kurtosis of Heave of July 17.........................................99
A. 46 Dominant Frequency of Heave of July 17..............................99
A. 47 Correlation Time of Heave of July 17................................100
A. 48 Hurst Exponent of Heave of July 17..................................100
A. 49 Time Series of Heave of July 15.....................................101
A. 50 Mean of Heave of July 15............................................101
A. 51 Variance of Heave of July 15........................................102
A. 52 Skewness of Heave of July 15........................................102
A. 53 Kurtosis of Heave of July 15........................................103
A. 54 Dominant Frequency of Heave of July 15.............................103


A. 55 Correlation Time of Heave of July 15...................................104
A. 56 Hurst Exponent of Heave of July 15.....................................104
A.57 CDA File with Six Point Gaussian Smoothing of Roll of July 19..........105
A. 5 8 Skewed Derivative of Raw Roll Data of July 19..........................106
A. 59 Skewed Derivative of Smoothed Roll Data of July 199....................106
A.60 Correlation Dimension of Raw Roll Data of July 19......................107
A.61 Correlation Dimension of Smoothed Roll Data of July 19.................107
xiii


TABLES
Table
3.1 Volumes for Different Transverse Hull Shapes..............................31
3.2 Centers of Gravity (Distribute Mass) for Different Transverse Hull Shapes.41
3.3 Moments of Inertia of Uniformly Massed Hull Shapes........................47
3.4 Equilibrium Waterline Depths for Different Hull Shapes....................51
xiv


1. Introduction
Naval crane ships need to transport cargo from one ship to another ship
without injury to people or property. Problems arise from different sea states or
conditions causing oscillation in the crane's cable, thus creating a giant pendulum.
From the work of John Starrett, chaotic behavior can occur from pendulum
movements. People have approached this problem through mechanical engineering
solutions, which help most of the time, but do not solve the underlying problem. By
understanding the motion of the ship and predicting its movements, one is more likely
to transport cargo safely over a greater range of oceanic conditions.
One method for understanding ship motion is to study actual ship data to tease
out the important characteristics that allow for predicting the ship's movements.
However, to obtain ship motion data some noise is introduced into the system, which
impedes analysis. Noise can be somewhat stochastic and originate from human
activity and instrumentation. Humans may move cargo, which changes the
distribution of mass on the ship. Some noise may be intrinsic to the system such as
waves, currents, and fish swimming by the ship. To grasp the fundamental
characteristics of the system, a ship model needs to be created from fundamental
principles of ship behavior and shape. With these mathematical models, noise is
eliminated; thus, the basic principles of the system can be studied through numerical
integration, and the creation of sample time series for analysis.
A number of researchers have done work on ship motion. Some experts chose
to work strictly numerically, without experimentation. Other researchers chose to
justify their models with ship model data.
Chen, Shaw, and Troesch (1999) created a system of equations that involve
roll, sway, and heave (see Figure 2.1). They examined the nonlinear, large-amplitude
motion of the ship in response to beam seas, where waves strike the vessel broadside.
The authors found that heave was on a fast manifold, while roll and sway were on a
slower manifold.
Ohtsu (1990) created an optimum autopilot system with a single-input/single-
output, control-type autoregressive model. The autopilot system was fixed-gain and
noise-adaptive. This system controlled the rudder and thus, roll. Ohtsu created ship
components for a model ship and recorded actual ship motion data. He found that the
yaw (see Figure 2.1) of a small rudder induced roll motion.
1


Sanchez and Nayfeh (1997) used numerical means to analyze ship motion.
They examined parametric excitation (time varying), and external excitation
(inhomogeneity). The authors identified instabilities that appeared when one of the
excitations is slowly varied. They fixed the level of parametric excitation for a model
boat. They studied the stability and bifurcation of an equation with some heave and
roll coupling.
Falzarano, Esparza, and Taz U1 Mulk (1995) studied roll motion in isolation.
They produced an analogy between pendulums and ship motion. They performed
steady-state bifurcation analysis through their numerical studies. They observed the
changes to the restoring moment of roll by changing the height of the center of
gravity and damping. The authors altered damping by examining the presence and
size of bilge keels. They found that bilge keels add nonlinear damping and are
influential in resistance to capsizing.
Iseki (1990) focused on the cross spectrums of heave, pitch, and roll (see
Figure 2.1). The basic idea is that the ship is a giant wave probe and by
understanding the motion of the ship, one can estimate the directional wave spectra.
The directional wave spectrum describes the wave energy in terms of frequency and
direction. He created a model ship that was rigidly fixed to restrict surge motion and
used springs to loosely restrict sway and yaw motion. The ship was excited by long
crested irregular waves. Iseki found that the ship's pitch frequency was influenced by
the waves, while roll frequency was nearly independent of the waves.
Allievi and Soudack (1990) modeled roll motion with damping using basic
naval architectural principles. The authors used a Mathieu-like equation for their
analysis as they calculated the stable and unstable regions based on their parameters.
They examined undamped, linearly damped, and nonlinearly damped Mathieu
systems and the phase portraits of those systems.
Despite these hard-working researchers, a conclusive understanding of ship
motion has not been found. The thesis chapters are written as follows:
Chapter two examines the work of the Digital Sealegs Group. The Digital
Sealegs Group was a group of students and faculty of the University of Colorado at
Denver involved in the study of cranes and crane ships. This chapter focuses on the
numerical tools used to study the rolling of an actual crane ship. The most significant
result is the evidence of measurable correlation times for several windows of 1,024
seconds of data for each window.
2


Chapter three investigates the formation of a ship model for the coupled heave
and roll of a moored ship. Three different ship geometries are examined with
different parameters (e.g. "metacentric height"). An argument is made to restrict the
model to a coupling of roll (rotation about the ship's axis from bow to stern) and
heave (vertical translation), with forcing only in the heave direction.
Chapter four presents coupled Mathieu equations, describing the parametric
forcing of two pendula whose pivots are moved sinusoidally up and down. These
equations have been used to model theories and, when coupled, produced a non-
autonomous four-dimensional system. This system helps in visualizing the behavior
of such larger-dimensional systems for dynamics that, in the uncoupled case, is
understood and rich in behavior (i.e., periodic motion, period doubling, and chaos).
Chapter five presents the conclusion. In addition, this chapter assesses the
model as a tool for evaluating approaches to predicting real data. Further work is
suggested, including adding noise to the model-generated data and constructing an
experiment with a physical ship model.
Included in the appendix are MatLab programs, a glossary, and additional
plots of data. Some of the MatLab programs used to generate the plots in chapters
two, four, and five. Also, in the appendix are some additional graphs for the roll and
heave of the other data files.
3


2. Analyzing Actual Ship Motion Data
The Digital Sealegs Group analyzed the ship motion data for the NOAA ship,
Discoverer, and a naval crane ship. Preliminary investigations by Margo Martinez
and John Slavich involved finding coupling between degrees of freedom spectral
analysis. John Slavich analyzed the data from the Discoverer ship. Margo Martinez
analyzed crane ship data provided through the Carderock Division of the Naval
Surface Warfare Center. There were data from July 15, 17, 18, and 19, 1993. Most
of the analysis was performed on the July 19 roll data, which was a long contiguous
set of recorded data during sea state 3.
The ship data recorded from both ships had six degrees of freedom (Figure
2.1) (Gillmer & Johnson, 1982). There are three translational motions: surge, sway,
and heave. There are three rotational motions: roll, pitch, and yaw. Of the six
degrees of freedom, pitch, heave, and roll are predominately used in ship motion.
This section focuses on roll motion.
The software, Chaos Data Analyzer: The Professional Version (CDA), was
used to extract the Hurst exponent, the power spectrum, and the correlation function.
A MatLab program calculated the mean, kurtosis, variance, and skewness for
windows of 1,024 seconds (2,048 samples) throughout the entire data file. Two time
scales are examined: a single window of approximately 17 minutes of data, and the
entire day of approximately 19 hours of data. The purpose of these tests is to identify
the characteristic features of the data, such as dominant frequency, and to investigate
the data's degree of "stationarity," i.e., how the features and statistics behave over
longer time scales.
4


2.1 Each Window
Each file or window was sampled for 17 minutes and 4 seconds. Each
window had its time series, power spectrum, and correlation function analyzed as
plots. The sampling rate was two samples per second, well within observed
frequencies of motion.
2.1.1 Time Series
Events such as ship motion change over time creating a progression of data
points known as a time series (Williams, 1997). The time series plots of July 19,
1993 involved some files that seemed too erratic to have structure (Figure 2.2). Other
time series appeared organized and had a low frequency envelope (Figure 2.3).
Under a smaller period, the signal looks smooth and thus there is not a lot of
instrumental noise in the data.
2.5 F
O
Q)
O)
<1>
T3
O
a:
1 .5
0.5
200 400 600 800 1000 1200 1400 1600 1800 2000
Tim e (0.5 seconds)
Figure 2.2 Time Series of Roll 38 of July 19
5


I I-------------------------1------------------------1-------------------------1-------------------------1------------------------1-------------------------1-------------------------1-------------------------r
200 400 600 800 1000 1200 1400 1600 1800 2000
Time (0.5 seconds)
Figure 2.3 Time Series of Roll 47 of July 19
2.1.2 Power Spectrum
Data may have some periodic components. Differing amplitudes and phases
of sines and cosines form the periodic components of the data. Fourier analysis
allows for the extraction of the different waves. The power spectrum involves taking
a fast Fourier transform of the time series. The power is the mean square amplitude
and is plotted against the frequency. A broad spectrum often suggests random and/or
chaotic data. A spectrum of a few dominant peaks usually means periodic and quasi-
periodic data. Fourier analysis examines superimposed simultaneous multiple waves
with various heights (Williams, 1997). Within the signal is a standard or reference
wave, which is often the longest wave available, or the length of the record. This
wave is called the fundamental wave. The primary characteristics of the fundamental
wave are its wavelength (fundamental wavelength) and its frequency (fundamental
frequency). Fourier analysis uses waves whose frequencies are integer multiples of
the fundamental frequency. All waves are based on the wavelength of the composite
wave. Fourier analysis indicates which frequencies are in the signal and their relative
importance. The equation for the Fourier analysis is:
N_
y = 'Z(ah cosM + Z?, sinM) (2.1)
h= 0
6


The discrete Fourier coefficients are:
n
(2.2)
and
(2.3)
The variables in these equations are:
N= number of observations
h harmonic number (1 for first harmonic)
t time
y = data value at time t
Least-squares estimates involve minimizing the average squared difference
between the value of the composite wave and the sum of the components.
CDA uses 128 frequency intervals. A larger number of intervals would
improve resolution, but "exacerbates the spurious responses" (Sprott & Rowlands,
1995). Since CDA truncates the data to the largest power of two, the window chosen
for the data was a power of two. CDA uses non-overlapping segments. The
maximum frequency used by CDA is the Nyquist critical frequency, which is the
reciprocal of twice the interval between data points.
For most of the CDA windows, a dominant peak was found (Figure 2.4 and
2.5). Random and chaotic data often have broad spectrum, because of all the
contributions from different frequencies. Since in most windows in this system have
a pronounced peak, this system is to some degree periodic.
Ndata points (N = 2048) with sample time {At = 0.5 seconds) has a maximum
N 1
number frequency intervals with frequency interval Af =----. The maximum
2 NAt
(Nyquist) frequency is:
(2.4)
7


1
1
=1 Hz
(2.5)
, N_ Af = =________
Jnyq 2 X / ltd 2x0.55
If m intervals are averaged to end up with 128 averaged frequency intervals,
then:
m =
f N\
J
1 1024
128 ~~ 128
= 8
Therefore, the frequency interval of the averaged spectrum is:
8 1
AC = mtf
m
NAt 2048x0.55 12.8
Hz
(2.6)
(2.7)
The peaks appear to lie at approximately corresponding to a period of 12.8
seconds. This is probably the ship's natural roll period. The bottom axis for each plot
needs to be multiplied by 1/128.
0.12 1 1 1 1 1 1
0.1 -
0.08 -
0.06 - -
0.04 - -
0.02 \ -
20 40 60 80 100 120
Figure 2.4 Power Spectrum of Roll 38 of July 19
8


0.03
0.025
0.02
0 .0 1 5
0.01
0.005
40
60
_j____________i____________I
80 1 00 1 20
Figure 2.5 Power Spectrum of Roll 47 of July 19
2.1.3 Correlation Function
In a time series, there may be repeated data points separated by some time lag
(Williams, 1997). Autocorrelation shows to what extent two time segments with
certain time lags differ from each other. For a zero value, the two time segments are
not correlated or the sum of the products is close to zero. The correlation time states
the time it takes until two segments match each other in value. The CDA correlation
function was obtained by multiplying each x(t) with x(t-tau) and summing the result
over all of the data points (equation 2.8) (Fenny & Moon, 1989). The correlation
function is the sum plotted as a function of x or n (Sprott & Rowlands, 1995). CDA
calculates the correlation time as the tau when the correlation function first falls to 1/e
in value. N is the sample size minus one.
(2-8)
'v k=\
n 0,1,2,..., A (2.9)
There were correlation functions that seemed reasonable (Figure 2.6) and
other correlation functions that did not make sense (Figure 2.7). The reasonable
functions appeared to have slower oscillating envelopes. Roll 47 has the erratic time
9


series (Figure 2.2) as well as the erratic correlation function (Figure 2.7). Roll 38 has
a more structured time series (Figure 2.3) and also has a more structured correlation
function (Figure 2.6).
Figure 2.6 Correlation Function of Roll 38 of July 19
Figure 2.7 Correlation Function of Roll 47 of July 19
10


A problem with how CDA calculates the correlation time is that the time is
taken without consideration of dampened oscillating systems. The system may not
return to the same value, but the system may return to close in value. This semi-
memory in values is not considered in CDA's code and may be a more accurate
correlation time for this system.
2.2 The Entire Day
For July 19, the data were collected for 19 hours, 20 minutes, and 32 seconds.
The time series is shown in Figure 2.8.
a>
O)

o
o
cn
3.5 i--------1--------1--------1 i--------1 i
Tim (0.5 kilosec)
Figure 2.8 Time Series of Roll of July 19
For mean, variance, skewness, and kurtosis, time is based on the middle point
of the time for each window. After the statistics are discussed, dominant frequencies,
correlation times, and Hurst exponents will be discussed.
2.2.1 Mean
The sampling rate and the total time of observation limit a time series
(Williams, 1997). Increasing sampling rates allows for greater resolution; while
averaging over nearby sampling rates helps decrease noise, and aids in illustrating
11


general trends. In addition, some help in finding the general trend in the data is the
use of the arithmetic mean:
x
(2.10)
The mean of the roll data was approximately one until the 25th file at which
point the mean became very erratic (Figure 2.9). This change in the mean can be
observed through the time series (Figure 2.8) as the general trend is fairly constant for
the first 20 kiloseconds and then loses some stationarity. Figure 2.9 shows the mean
for each individual window.
Figure 2.9 Mean of Roll of July 19
2.2.2 Variance
It is useful to know how much the data deviates from the mean (Williams,
1997). Means and variances are tests of stationarity. The variance gives the
magnitude of the average deviation from the mean:
s
2
(2.11)
12


The standard deviation is the square root of the variance. For a small total
number of data points, N-l should be used as a divisor so as not to underestimate the
values. The variances of the roll windows are spaced fairly close together until the
20th window, then the variance of the windows are farther apart (Figure 2.10).
These changes in variance can be observed in the time series as the relative
"thickness" of the data. Between 30 kiloseconds and 40 kiloseconds, the elevated
data peaks in the time series that correlate elevated peaks on the variance plot (Figure
2.10). The variance for each window is calculated. The peaks in the plot correspond
to places of large variation in the time series.
Figure 2.10 Variance of Roll of July 19
2.2.3 Skewness
Skewness is defined by (Sprott & Rowlands, 1995):
-*)3
(2.12)
13


The skewness of the roll data deviated only slightly from zero (Figure 2.11). There
are no apparent trends. Some windows are skewed in one direction and other
windows are skewed in the other direction. Skewness describes the lack of symmetry
about the mean (James & James, 1992). When comparing the skewness to the time
series, it is not easy to verify the skewness plot since the data is so concentrated
around the mean.
Figure 2.11 Skewness of Roll of July 19
2.2.4 Kurtosis
Kurtosis is defined by (Sprott & Rowlands, 1995):
£(*, -*)4
M 4
(2.13)
Figure 2.12 shows the kurtosis for July 19 each point represents one window
of data. Kurtosis describes the concentration about the mean (James & James, 1992)
There are no apparent patterns in the plot. The 63rd window has the highest kurtosis
value and is the most skewed in the positive direction.
14


Figure 2.12 Kurtosis of Roll of July 19
2.2.5 Dominant Frequencies
In the dominant frequencies, correlation times, and the Hurst exponent, some
smoothing was performed. The process involved a nine-point low pass filter was
constructed by convolving the data with nine points of 0.1111 in value and
interpolating the result. Smoothing was performed on the data in order to reduce the
amount of noise in the signal. Noise will have more erratic peaks and valleys, which
with smoothing will be decreased in steepness to allow the underlying structure of the
data to be seen and analyzed.
The average dominant frequency for the raw roll data of July 19 was 0.10
0.04 Hz (Figure 2.13). With smoothing, the average dominant frequency for the roll
data of July 19 was 0.10 0.04 Hz (Figure 2.14).
15


Time (sec)
Figure 2.13 Dominant Frequency of Raw Roll Data of July 19
0.12
0.11
I 01
| 0.08
§ 0.07
| 0.06
q 0.05
0.01
0.03
4
- 0.09

+-
illlill
' I !
4-(-4.
-* r
4-t-
4' ,# ,4* 4 4' # ,4" xv ,4 u4" ,4 aT
> t.to Tims (sec)
Figure 2.14 Dominant Frequency of Smoothed Roll Data of July 19
2.2.6 Correlation Time
The correlation time is the amount of time it takes for a time segment to repeat
itself. The correlation time was 1.810.7 seconds (Figure 2.15) and with smoothing it
was 1.9 1 0.7 seconds (Figure 2.16). One approach to correlation time was to take
correlation functions from the CDA and construct an exponential fit to the sine wave
16


Tao (0.5 seconds) Tao (0.5 seconds)
with a declining amplitude. The reciprocal of the exponential power is the correlation
time (Figure 2.17).
Figure 2.15 Correlation Time of Raw Roll Data of July 19
Figure 2.16 Correlation Time of Smoothed Roll Data of July 19
17


8
0 t-----------1-----------1-----------1-----------1-----------1-----------1-----------
0 10000 20000 30000 40000 50000 60000 70000
time (sec)
Figure 2.17 Correlation Time of Roll Data of July 19
2.2.7 Hurst Exponent
The Hurst exponent represents how random or uncorrelated the data points are
to each other. White or uncorrelated noise has a Hurst exponent of 0.5. The Hurst
exponent is from "the slope of the root-mean-square displacement of various initial
conditions" (i.e., each point) versus time (Sprott & Rowlands, 1995). For data, the
line starts at the first data point and ends at the square root of the total duration of the
data record. Hurst exponents greater than 0.5 show that trends will continue into the
future. Hurst exponents less than 0.5 show that trends will reverse in the future.
Hurst exponents show how values move away from the initial value using each point
in time series as an initial condition.
The average Hurst exponent of raw roll data of July 19 was 0.32 0.07
(Figure 2.18). With smoothing, the average Hurst exponent was 0.34 0.07 (Figure
2.19). Smoothing hardly affected the Hurst exponent. The roll data had Hurst
exponents less than 0.5, so the data is correlated and trends are to be reversed in the
future.
18


0 10000 20000 30000 40000 50000 60000 70000
Time (sec)
Figure 2.18 Hurst Exponent of Raw Roll Data of July 19
Figure 2.19 Hurst Exponent of Smoothed Roll Data of July 19
2.3 Summary
Margo Martinez and John Slavich were able to obtain ship motion data and
perform numerical tests on the data. Further tests (i.e., Hurst exponent, power
spectrum, autocorrelation) were performed on a set of roll data of July 19. There was
19


no definitive answer to knowing the underlying ship dynamics, perhaps due to too
much noise in the system. In the short term there is a dynamical system behavior
"quasi-stationarity." For the entire day, there is non-stationairy behavior, which can
be explained by tidal changes.
20


3. Modeling Moored Ships
In order to gain insight into the dynamics represented by the July 19 roll data,
a model was developed. By simplifying the system to a mathematical expression, one
can obtain the fundamental attributes of the system without having to deal with noise.
With the construction of a mathematical model, there is always a question of how
complex to make the model. A complex model may be more realistic, but it often
results in large amounts of computing time, more debugging, and may impede the
conceptualizing of the basic components of the system. Therefore, a compromise
must exist between a simple and complex model.
The ship moves with six degrees of freedom (see Figure 2.1). All of the
literature describes models with roll, but there is some discrepancy as to what the next
important degree of freedom is after roll. To load and unload cargo on a ship, the
crane ship and the transport vessel must be parallel. Thus, the movement of the
transverse cross section is most important. Therefore, heave is an important
component of ship motion.
This chapter describes the basic knowledge needed to understand the
movement of a rigid body and then states a coupled system of equations describing
roll and heave. Three basic hull shapes will be examined. For these shapes, the
parameters of the coupled equations that can be derived from ship geometry include
mass, volume, center of gravity, and moment of inertia. Additional parameters
require analysis of the ship's position and orientation relative to the sea surface:
equilibrium waterline, center of buoyancy, and the metacenter.
3.1 Rigid Body Motion
The following equations form the basic mathematical theory behind rigid
body motion, which is the fundamental basis for the coupled equations. Harrison and
Nettleton (1997) and Ginsberg (1995) describe the dynamics of rigid body motion,
which are demonstrated below.
21


Figure 3.1 Rigid Body Motion Diagram
Imagine a solid body divided into a set of small masses, /,- The total mass, m,
is then:
m = Ydmi (3.1)
I
The center of mass, rCm, is defined as:
2>,r.
rcm =------------------------------------ (3.2)
m
Let r be the position of the /th mass relative to the center of mass:
r'i=ri-rCm (3.3)
Note that:
-rcm)= 0 (3.4)
I
Substituting equation 3.3 into equation 3.4:
2]m,r'i=0 (3.5)
Let the force be distributed over the body with force, Ft, acting at each small
mass. The total force is:
22


III -M (3.6)
From Newton's 2nd Law:
F = 2>,r, i (3.7)
= ^mfocm +r'i) i (3.8)
d2 ( V = Xm,rcm +-7rY.mir'i= Zmi Ycm + ZmiY' i at j \ i J i (3.9)
= mfcm + ^jmjr'i (3.10)
If the distribution of masses is fixed in time:
r. d2 ^
mr cm +- > mj ,
dtlLT
(3.11)
Then since
!>/. =o
(3 12)
The second term of equation 3.11 vanishes and the equation for the center of
mass motion is:
m r cm = F
(3.13)
Now examine the torques on the solid body about the center of mass. Let
T,=r'ixFi (314)
The total torque about the center of mass is:
23


= 2><=I '.xF,
(3 15)
The second term vanishes because of the property that for axa = 0 any
vector a (again assuming mass distribution remains fixed).
= ^rix mi (r'cm + r'i)
(3.16)
( _ "N _ _
^/w, r'i x rcm + ^mir'i x r'i
i ) i
(3.17)
The first term vanishes (equation 3.5) and the second term can be rearranges
to give:
r = ^/n,
d ( r< > f (it* i r i x f
dt d, J V
r i x r
dt dt j

dt ,
f n ^
f dr i
r i x
v
dt
y
Again assuming the mass distribution remains fixed.
For rigid body motion:
dr' - ~
L = 6) xr
dt
for some angular velocity vector co .
Substituting 3.20 into equation 3.19:
r = V mr'i x fc x r')
dt1? ^
(3.18)
(3.19)
(3.20)
(3.21)
24


(3.22)
= ^Zw.Mr'' r'^-r^i a>)
The result is:
- d /r\
T = UO)]
dr
where the moment of inertia tensor is:
1 =
2>,U2 +2,2) -YJmlx,y, -'ZrriiX'Z,
-Zm Zmiixi +z?) -Zm>y>z-
i i ' / s
- Z mtxiz> Z m>y>z. Z ^ lx-2 + ^)
V <
and



(3.23)
(3.24)
(3.25)
3.2 Simplified Class of Hull Shapes
A trapezoidal hull (Figure 3.2) is a good approximation to an actual ship. The
top left picture in Figure 3.2 shows the side view. The top right in Figure 3.2 shows
the transverse section of the ship. The bottom right picture shows the top view.
Through limits on the bottom of the ship, rectangular (Figure 3.3) and
triangular hulls (Figure 3.4) can be created from the trapezoidal hull. The width of
the deck for all the vessels is d. The height of the vessel is h. The length of the
vessel is /. The width of the bottom of the vessel is k.
25


Figure 3.2 Views of the Trapezoidal Vessel
The rectangles indicate a simplification of the ship geometry that ignores the
shape of the bow and stem. The equilibrium waterline depth coo is the value of the
neutral waterline depth for 0=0 (zero roll). Later, all dimensions will be scaled by
the width of the deck.
d (3.26)
| III (3.27)
d (3.28)
rJ *0 d (3.29)
26


Figure 3.3 Rectangular Transverse Hull
For the rectangular hull (Figure 3.3),
t = \ (3.30)
or
k = \ (3.31)
Figure 3.4 Triangular Transverse Hull
For the triangular hull (Figure 3.4),
*=0
d
(3.32)
27


or
k' = 0
(3.33)
3.3 Parameters Derived from Ship Hull
Shape and Mass Distribution
There are certain characteristics of the vessel that are independent of the ship's
position relative to the sea, such as ship mass, volume, center of gravity, and moment
of inertia. These factors will be described in order. To discuss these factors, certain
variables and scaled values must be defined.
Some important variables are:
c = center of gravity
po = density of water
V= volume
M = mass
I = moment of inertia about the x-axis
For a vessel symmetric about the midplane fore to aft, the center of gravity
lies a (scalar) distance c above the keel. Some scaled values are:
c' = -
d
I'
Md2
v-A
d3
(3.34)
(3.35)
(3.36)
3.3.1 Ship Mass
Let the vessel have a total mass, M.
M=Mi+Me (3.37)
28


Mi is the mass of the interior contents of the ship, which will be assumed to be
uniformly distributed with density pt. Me is the mass of the exterior surface of the
ship and is described:
Me Md +Mk + 2Mh (3.38)
Md is the mass of the deck, A4 is the mass of the keel, and Mh is the mass of
each side of the hull. Let po be the density of the water in which the vessel floats.
Masses will be scaled by the mass of water whose volume equals that of the total
volume of the vessel.
Thus, the scaled mass is:
M'
M
PoV
The scaled interior mass is:
M
PoV
py = a
p The scaled exterior mass is:
M
t
e
K_
PoV
(3.39)
(3.40)
(3.41)
In order for a vessel to float, M' < 1. A neutrally buoyant vessel (e g., a
submarine stationary at constant depth) hasM' =1. IfM'> 1, the vessel sinks.
3.3.2 Volume
The volume is the cross-sectional area of the ship multiplied by the length of
the ship. The trapezoidal hull transverse area can be viewed in Figure 3.5.
29


d-k
d-k
Figure 3.5 Volume of Trapezoidal Hull
The volume of the trapezoidal vessel is:
v = (a,+a2+a,)-i
(., 1 d-k ,
kh-\---------h +
V 2 2
1d-k,V
----h l
2 2)
f
V
, d-k)
k +---
2 J
2
hi
(3.42)
(3.43)
(3.44)
(3.45)
The resulting equation for the trapezoidal hull volume is:
V = Ihd
2
(
1+-
V.
(3.46)
By substituting equation 3.46 into equation 3.36, the scaled volume of a
trapezoidal hull is:
If, k\h l
^ d ) i
dd
(3.47)
30


(3.48)
-(l + k]hl'
By letting = 0, the volume of the triangular hull is:
d
or, in scaled form:
f = -Idh
,n 2
V' = -Vh'
m 2
By letting = 1, the rectangular hull's volume is:
d
or, in scaled form:
V=ldh
v' = hr
Table 3.1 summarizes the results of the volume equations.
Table 3.1 Volumes for Different Transverse Hull Shapes
Hull Shape V V'
T rapezoidal 1 ( k\ 1+- dhl A dJ
Rectangular dhl hr
Triangular -dhl 2 -hr 2
(3.49)
(3.50)
(3.51)
(3.52)
31


3.3.3 Center of Gravity
With an understanding of the volumes, the centers of gravity of these vessels
can be calculated. The centers of gravity will remain constant for the vessels as long
as their distributions of mass remain the same (i.e., no shifting of their masses).
In this section, there is a progression of calculations for different shapes.
First, the center of gravity for a right triangle is calculated. Then the center of gravity
of a more complex triangle is calculated. Finally, the center of gravity for the
different hull shapes is calculated for distributed and non-distributed masses.
To calculate the center of gravity, integrate over the volume of the vessel:
To analyze shapes, it is easier to have the centroid of a right triangle (Figure
The center of gravity, c, that has a uniform mass per unit area, cr, for a right
triangle is:
PoSv
(3.53)
3.6).
Figure 3.6 Center of Gravity for a Right Triangle
(3.54)
32


The mass per unit area divides out, because it is assumed constant.
1
ffdydz
triangle
vo o y
Integrating equation 3.55:
( P f \
1 1-^ dy
T* o l P)
UJ r i -qp \W r 2 dy
L Vo 2 K Pj y
_2_
qp
Integrating equation 3.57:

fy \
\.Z8 J
_2_
1
1 P
3V
y
v2 3 p
f 2 i 3 V
2 P 1 P
P- +
V
V
P 3 p*
yy
_2_
f 1 2
6V
1 2
C6qP
\
y
The resulting center of gravity for a right triangle is:
(3.55)
(3.56)
(3.57)
(3.58)
(3.59)
33


(3.60)
M (2 \
2
\8 y1 q
Kl J
This result will be useful below and in section 3.3.3.1 for the center of gravity
of the trapezoidal hull with distributed mass.
Many triangles involve the side of the trapezoid and require special attention
to find the center of gravity (Figure 3.7).
Figure 3.7 Triangular Part of Tilted Trapezoid
From trigonometry:
q 5 sin
W
From the Pythagorean Theorem:
(tftan(^)+/?)2 +q2 = s2
Substituting equation 3.61 into equation 3.62:
q2 tan2(^) +2# /?tan(^) + p2 +q2 = -r^~
sin
w
(3.61)
(3.62)
(3.63)
| A
1r-^v + tan2($0 +2<7-/?tan(6>)+/?2 =0
. sin \e) J
(3.64)
34


As a side note:
1_ 1 = sin2(^)~l
sin2 (d) sin2 (9)
cos
(e)
sm
(e)
tan2(0)
Substituting the result of equation 3.67 into equation 3.64:
tan2(^)-

+ 2p qtan()+ p2 0
Using the quadratic formula:
- 2ptan( y ^ tan
- p\ax\(rf>)
p2 tan2{$)-p2 tan2(^)+ p2
tan2(&)
mM-(J(e)
-p,anMpdm
tan 2(ff)-
tan2{6)
(3.65)
(3.66)
(3.67)
(3.68)
(3.69)
(3.70)
(3.71)
35


-tan(^)-
tan(^) +
tan(#)
-tan(^)-
tan(^) +
tan(6>),
tan(#)
tan(^)-
1
tan($)
tan
tan(6>)y
tan(^)y
-1
= <
,an<^)+anV)
-1
tan(^)-
tan(#)
Exclude the top root to get positive q when 0
tan(#)
? =
1 - tan(#)tan(^)
(3.72)
(3.73)
(3.74)
The area Aj of the triangle 7) is the area (Al + A2) of the larger triangle
(jj + T2) minus the area A2 of the smaller triangle.
A ~ ~^( (3.75)
Substituting equation 3.76 into equation 3.74:
A = P2 tan (0)
1 2 1 tan(6?)tan(^)
(3.76)
(3.77)
36


Let c\ be the center of gravity of triangle 7/ with respect to origin 9'. Let
9"9' be the vector from origin 9" to 9' origin. Let c"2 be the center of gravity of
triangle T2 with respect to origin 9. Let cn be the center of gravity of the 7) +T2
combined triangle with respect to origin 9". Then:
A^'i +&&')+ A2cl = {A, + A2)c2 (3.78)
c'l=-9"9' +
{A + A )u
A2c 2
(3.79)
-9n9'Ax +(A, +A2yn -A2c2
A
The vector between origins is:
9"9'
rq tan^)'
(3.80)
(3.81)
Using the result 3.60 for right triangles, the center of gravity for the combined
triangle Tj+ T2 relative to 9" is:
c 12 -
^(?tan ()+p)
1
v 3 y
The center of gravity for the triangle T2 relative to 9" is:
c 2 =
f 1 >
1
3 7
(3.82)
(3.83)
Substituting equations 3.76, 3.81, 3.82, and 3.83 into equation 3.80:
37


1 f --qp
v 0 j
+
-3P + -?2 tan(^)
l
^ N
Aq\M(t>) + p)
--^2tan{<£) 3 1
(\ ^
qtan{ V 3 y
1
-qp
i
-^VtanW+(^p + ^2 tan(^fetanW+p)-^3 tan2W
( qp
(3.84)
(3.85)
c i
q2p tan(^)+j q2p tan(^)+j#>2 + j?2/> tan(^)
1 2
-q P
\_
qp
y
(3.86)
c i =
~^q2ptan() + ^qp2
1
;q2p
J_
qp

C 1
p-qim(4>)
(3.87)
(3.88)
Substituting equation 3.74 into equation 3.88:
tan($)
c i
^ 1 tan(^)tan(#)
tan(9)
1 tan(^)tan(#)
tan ()p
P
c i
1 tan(^)tan($) tan((9)tan(^)
tan(^)
3 1 tan(^)tan(#)
(3.89)
(3.90)
38


C 1 =
1- 2tan(^)tan($)
tan($) J 3 1- tan(^)tan( (3.91)
This result will be useful in finding the center of buoyancy in section 3.4.2.
3.3.3.1 Center of Gravity with Distributed Mass
Figure 3.8 provides the parameters used to calculate the center of gravity for a
uniformly massed vessel.
d -----------
c
C,
O
k ---N
Figure 3.8 Center of Gravity of Trapezoidal Hull of Uniformly Distributed Mass
Figure 3.9 provides the parameters used to calculate the center of gravity of a
non-uniformly distributed mass of a trapezoidal hull.
A Aj + A2 + A3 (3.92)
= kh + -(-(d-kf\ + ((d-k) 2\2 j 2^2 ) (3.93)
(3.94)
c (-4,ci +A2c2 + A3ci) A (3.95)
39


c=-
~(d+k)h
f0>
kh 1 ,
-h
\2 )
1
+-
2
(c/-ArVi
2
r k Y\
k V-{d-k)h
2 3
v2 y
3
1
+-
2
yi

k 1
4
2 3
\ld-k)h
V ,
h
3
(3.96)
h(d + A:)
fl 0 V
kh U + (d -k)h h
\hJ 13 )_
d + k
k+-(d-k)
3 V
(3.97)
(3.98)
where the result from equation 3.60 has been used for finding c2 and c3.
The height c of the center of gravity above the keel in the trapezoid case is:
1 h
c =
3 d + k
hf 2d + k'\
(2d + k)
3 ,d + k j
( k\
h 2 H
d
3 k
l d)
(3.99)
(3.100)
(3.101)
In scaled form.
3
(2 + k'\
1 + k'
(3.102)
For the triangle, use = 0 :
40


or, in scaled form:
c*=\h' (3 104)
k
For the rectangle, = 1:
(3 105)
or, in scaled form:
(3.106)
Table 3.2 summarizes the center of gravity as a length measured from the
keel. Both scaled and unsealed values are presented in the table.
Table 3.2 Centers of Gravity (Distribute Mass) for Different Transverse Hull
Shapes
Hull Shape c c'
Rectangle I* 2 h' 2
Trapezoid 2 + ~ d 1+* l d) h 3 f2 + k'^h' U + yt'J 3
Triangle 3 3


3.3.3.2 Center of Gravity with Non-Distributed Mass
Some important variables in this section are:
m = total mass of vessel
rrtd = mass of deck
mk = mass of the bottom
mi = mass of left hull
mr- mass of right hull
c = center of gravity of the vessel
Cd center of gravity of deck
Ck = center of gravity of bottom
ci = center of gravity of left side of hull
cr center of gravity of right side of hull
/Lid mass per unit length of deck
Hh = mass per unit length of sides
//* = mass per unit length of bottom
md=Hdd
Figure 3.9 Center of Gravity of Trapezoidal Hull of Nonuniformly Distributed
Mass
mdCd +mkCk +mtci +mrcr
m
(3.107)
42


1
C -
Mkk + Mdd + 2juh^2 +^{d-k)2
X fjji
ro^
vy
+/^
fo)
\hj

'-*-W
2 4V 7
h
V 2 )
vhlh2 +-M-kf
-+-(d-k)
2 4V
h
V 2 J
1

Mkk + Mdd + 2 fuJh2 +^{d-k)2
0
*d + VHJb2 + ^(d ~kf
c-h
Vd +Mk-, + 2[ih
d
For the rectangle, let = 1:
Crec = h
Md +^m/I ~l
+ 0
Vd + Mk +2Mh
fh^
+ 0
\d J J
C rec h
Vd +Mh
0 ^
\UJ

Md +Mk+ 2 jut
\d J /
(3.108)
(3.109)
(3.110)
(3.111)
(3.112)
43


For the triangle, let = 0 :
Cm = h
2 1
Me +Mh^ UJ + 4
Md+Mk +2fih^

\u J
1
+ -
4
y
(3.113)
3.3.4 Moment of Inertia
The moment of inertia of a uniform distribution of mass for all three hull
shapes is given below. The moment usually restores the ship to an upright position
(Gillmer & Johnson, 1982). Figure 3.10 presents the important parameters of the
moment of inertia of a trapezoidal hull.
z
h
The ship is assumed to be symmetric part to starboard (a good assumption)
and fore and aft about its midplane (ignores change in cross-section at bow and stern).
With this degree of symmetry of this ship model, roll motions only deal with the
upper left element of the matrix in equation 3.24 (i.e., ^/w ,(y,2 + z,2)).
44


(3.114)
k d-k z
2 2/h
I ~ pl\dz ^[y2+z2)dy
k d-k 2
2 + ~2~h
Integrating equation 3.114:
= plj
o L
y 2
z + z v
3
k d-k z
2 + ^2~h
k d-k z
2 ~2~~h
= pl]
2fk d-kzV (k d-kz
- +----z + 2 +---
3(2 2 h) \2
2 h
dz
Pi\
2 ( k d-k z Y k d k z ^
3 l 2 + 2 h

A
2 h
k d-k z3
z + 2 z +---
2 2 h
dz
Using a mathematical software program such as MathCad results in:
I = plh(k3+d3 +k2d + kd2 +\2h2d + 4kh2)
48 v
= M
24
f k3 +d3 +k2d + kd2 +\2h2d + 4kh2^
\
k + d
where
M = pih
d + k
The result for the moment of inertia for the trapezoid is:
I = M
24
d2+k2+ 4
(
1 + 2-
V
k + d j
h2
(3.115)
(3.116)
(3.117)
(3.118)
(3.119)
(3.120)
(3.121)
45


/ = M
24
1 +
k_
\dj
+ 4
f \
, 2 2
1 + r
1+* UJ
l d)
(3.122)
The scaled moment of inertia is:
r =
24
f d >
1 + A'2 +4 1 + 2 h'2
^ k+d)
(3.123)
For the triangle, let = 0 :
I tn = Md2
24
1 + 12
h
\dj
(3.124)
For the scaled triangle:
I' =fl + 12 h'2}
m 24L J
(3.125)
For the rectangle, let = 1:
Irec = Aid1
rec 24
2 + 8
h_
\d j
(3.126)
For the scaled rectangle:
C = [2 + 8/7,2l
24
(3.127)
Table 3.5 summarizes the moments of inertia for uniformly massed hull
shapes.
46


Table 3.5 Moments of Inertia of Uniformly Massed Hull Shapes
Hull Shape Moment of Inertia Scaled Moment of Inertia
Rectangular Md2 24 2 + 8 \d y 2 [2 + 8/j'21 24L J
Triangular Md2 24 1 + 12 f/A UJ 2" [l + 12/?'2l 241 1
Trapezoidal M 24 fk\ 1+ UJ +4 1+ \ 1+- l d, \2 ') 1 r ( d \ 1 l+r2+4 1+2 h'2 24 k ~\~d J
3.4 Parameters Derived from Ship Position
Relative to Sea Surface
In this section, equilibrium waterline and center of buoyancy are described.
3.4.1 Equilibrium Waterline Depth
Figure 3.11 shows how the equilibrium water line can be determined.
Figure 3.11 Equilibrium Waterline Depth
47


The volume of the water is needed. Using the result from equation 3.46,
replacing h with a>0 and replacing d with k + (d k):
h
Vo =
k + ~~(d-k)
+ k
-6)0l
(3.128)
Archimedes' principle requires that the mass of displaced water equal the mass
of vessel.
p0V0=M
(3.129)
Substituting equation 3.128 into equation 3.129:
PqI(<.
k + -2.
(on d-k
h 2
M
(3.130)
Dividing both sides of the equation by Ipo'.
6)r
k + ^
(Or, d-k
h 2
M
Ipo
(3.131)
d-k 2 . M
------(Or, + k(Or,---= 0
2 h lp0
(3.132)
Dividing each term of equation 3.132 by (d-k):
2 2 hk 2Mh
0)0 + d-k** p0l{d-k)~
(3.133)
Using the quadratic formula:
2 hk
(On
d-k y
' hk y
.d-k,
+ 4
2 Mh
p0l(d-k)
(3.134)
48


hk \( hk ^2
d-k
\d~kj
+
2Mh
p0l{d-k)
(3.135)
Factoring (d k) from the terms of equation 3.135, gives the final result for the
trapezoidal equilibrium water level:
-hk +
a>n
V
(hkf+-(d-k)
Pol
d-k
(3.136)
Dividing both sides by d to scale the equations:
-hk +
co
V
(hk)2+~(d-k)
Pol
d d(d-k)
Scaling the denominator:
(3.137)
con
-hk + (hkf +2Mhd ( k\ 1--
V IPo l d)
f k^
1--
V aj
(3.138)
Scaling the other variables in the equation:
h k \fh_k}2
d d llyJ dj
+
2M- f
d r
=
pJ-/1
d
k\
1--
d)
( kA
1--
l dj
(3.139)
-h'k' +
V
{h'k'Y +2-^^{l-k')
Pod3 /'
1 -k'
(3.140)
49


As a side note:
. M _2MVp0
Pod Pod
M'pQ-(d + k)hl
= 2
d3Po
= M(\ + k)h'l'
Substituting equation 3.143 into equation 3.140:
co
, -h'k' + J(hk)2 +M'{\ + k')h'2(]-k')
For the scaled triangular hull, let k' = 0 :
o = h'yfM7
The unsealed expression for the triangular hull is obtained by letting k
equation 3.138:
\2Mh
~pjd
For the rectangular hull, let k = d in equation 3.133 and solve for cu0
o =
M
pjd
For the scaled rectangle, use M = M'pjdh in equation 3.147:
, M'pjdh
a>0 =
Pjd
(3.141)
(3.142)
(3.143)
(3.144)
(3.145)
= 0 in
(3.146)
(3.147)
(3.148)
50


Dividing through by d.
co'0 = Mh' (3.149)
Table 3.4 summarizes the equilibrium waterline depth for a variety of hull
shapes.
Table 3.4 Equilibrium Waterline Depths for Different Hull Shapes
Hull Shape o 6>0
Rectangle uM PoV hM
Trapezoid h 1-* d k + ~d+^U )2 A/f ft)2) + 1 P Triangle h M PoV hyfhT
3.4.2 Center of Buoyancy
Consider now a vessel that has rolled to angle # relative to the sea surface.
Figure 3.12 shows the situation in the ship's frame of reference with the origin 9 at
the keel. The shaded region A is the submerged portion. The center of buoyancy b
is identified as the centroid of this region. The location of b in ship coordinates is
found by constructing zl out of a trapezoid A] (Figure 3.13) plus a triangle A2 and
minus a triangle As (Figure 3.14).
51


3
Figure 3.12 The Actual Area of the Water
52


Figure 3.14 The Areas of the Two Triangles Formed by the Waterline and Tilted
Vessel
Figure 3.15 presents all relevant parameters used in computing the center of
buoyancy of the vessel from this construction. The vertical bisector of the ship cross
section is 9D and WxWr is the waterline, which intersects 9D at point Wc. Line $i$2
is the horizontal (in ship coordinates) through Wc. Note that represents the angle of
the sides of the hull away from the vertical and is a constant of the ship geometry.
53


Figure 3.15 Important Parameters of Ship Motion
The important parameters of Figure 3.15 are as follows.
The center of gravity of the ship is:
c = (o,c0) (3.150)
The center of buoyancy (the center of gravity of the displaced volume of
water) is:
MmJ (3151>
lim b=b0 (3.152)
0->O
The centroid of Ai is b\, &2 is the centroid of A 2, and £3 is the centroid of^j.
54


According to the construction in Figure 3.15:
P2=P3=P (3.153)
and p = +o) tan(^) (3.154)
where tan(^)= h (3.155)
The center of buoyancy in ship's coordinates relative to the keel involves:
1 Axb\ + A2b2 -A3b3 (3.156)
A
where A Ax + A2 A3 (3.157)
The submerged area A is found be evaluating the areas A;,A2, and A j replacing d with 2p and h with to in equation 3.45 for the area of trapezoid A using equation 3.77 with angle for A2 and -^for Aj. by and by
( 1 Ax = p + k CO V 2 J (3.158)
j tan(6>) p2 (3.159)
2 1 tan(#)tan(^) 2
j tan(0) p2 (3.160)
3 1 + tan(^)tan(^?) 2
Substitute equations 3.158, 3.159, and 3.160 into equation 3.157:
55


(3.161)
A =
1 ,
p +k
2
CO +
1 tan(0)p2 1 tan (0)p2
2 1 tan(#)tan(^) 2 1 + tan(#)tan
Substitute/? from equation 3.157 into equation 3.161 results in:
1 4cok + 4tv2 \an( A = -
4 (-1 + tan(0)tan(^))(l + tan(#)tan(^))
(3.162)
The centroid of^4/ is the centroid of a trapezoid (use equation 3.101 replacing
d with 2p and h with co):
b =
0
1 3 2 p + k
The vertex ft of triangle A2 (Figure 3.15):
r-p'
V J
The centroid of A2 relative to 32'.
1 2tan(i9)tan( 3(l tan(#)tan(^))l^ tan(t?)
The centroid of A2 relative to the keel thus:
\ a J
3(l tan(0)tan(^))l^ tan(#)
1- 2tan(#)tan(p)'
The vertex ft of triangle A3 (Figure 3.15) is:
V
(3.163)
(3.164)
(3.165)
(3.166)
(3.167)
The centroid of triangle A3 relative to ft:
56


(3.168)
P
3(l + tan(#)tan(p))
^1 + 2 tan(#)tan(p)N
, tan(/9) ,
The centroid of As relative to the keel is thus:
p (\ + 2 tan(0)tan(p)'N
K0)J 3(l tan(^)tan(^))l^ tan(#) ,
(3.169)
Substituting the centroids of the various areas (equations 3.158, 3.159, and
3.160) and their corresponding areas (equations 3.161, 3.163, 3.166, and 3.169) into
equation 3.156 enables one to find the centroid of the entire underwater region:
3 ^ \-l + tan(0)tan(^))2(l + tan(0)tan(0))2
-tan(6)2p2 3(0tan^ ~ 3(0tan^2 tan^3 +P + Ptan^2 tan(^)2
^ 3 (-1 + tan(#) tan(^))2 (l + tan(0) tan(^))2
(3.170)
Substitute for p from equation 3.154 into equation 3.170:
1 tan(#)(£ + 2 12 (-1 + tan(i9) tan(^))2 (l + tan(#) tan(^))2
1 tan(ff)2 (k + 2co tan(ff))2(totan(^) 4 v24 (-1 + tan(<9) tan(^))2 (l + tan(6>) tan(^))2 y
(3.171)
In the above expression, recall that is a constant given by the ship geometry:
tan(^) = ~~j~
(3.158). The angle 0is, however, one of the dynamical variables, i.e.,
it is a function of time 9(t).
The other dynamical variable is z the position of the center of gravity of ship
relative to earth coordinates. In this model, the sea surface lies a distance s above the
ocean floor (Figure 3.16).
57


Figure 3.16 Ship Position Relative to the Ocean Floor
From the construction in Figure 3.16 and recalling that the center of gravity of
the ship lies a distance c above the keel.
5 = z + (a> c)cos(0) (3.172)
Solving for co gives:
co- c
+
s-z
cos(#)
(3.173)
Thus the dynamical variable z enters into the model by substituting equation
3.174 for co in the expression for submerged area A (equation 3.162) and for center of
buoyancy b (equation 3.171).
58


3.5 Modeling Sea Surface Variations
The ship will be forced into motion by sea surface variations. Normally,
waves are thought of in this context, so that the ship does not sit in a level sea as
assumed in Figure 3.16. However, the waves that are significant in driving ship
motion will have periods comparable to the ship roll period. In Chapter 2, the roll
period of the crane ship was calculated to be about 12 seconds. The period T of deep-
water ocean waves is related to the wavelength X by (Aubrecht II, 1996):
<3174>
where g is the acceleration due to gravity.
Thus,
1 =
(3.175)
Substituting 9.80 m/s2 for g and 12 sec for Tgives X = 225 meters
The ships of interest have deck dimensions (approximating the "beam of the
ship"): d=30 meters, which is about 13% of the wavelength. Thus, to a first
approximation, the sea surface is a level surface that is periodically rising up and
down.
5 = s0 + 5, sin
T
\ wave J
(3.176)
where so is a sufficiently large average depth and Sj is a variation small relative to the
wavelength:
S] < 2 meters
(3.177)
or
59


(3.178)
< 0.07
d
Note that the maximum variation in surface height will be on the order of:
A (3.179)
. 2 i d (3.180)
225
* 6% of d (3.181)
This variation is ignored in the dynamical equations discussed below.
3.6 Equations of Motion
In this model, ship motion is governed by Newton's Laws as they describe:
(1) the acceleration z in the true vertical z direction as a result of imbalance
between the ship's weight Mg and the buoyancy force Fb .
(2) the angular acceleration 6 due to the moment of buoyancy around the
center of gravity.
The magnitude of the buoyancy force is given by:
FB=p,A\z (3.182)
where A is the submerged cross-sectional area given in equation
_ 1 4cok + 4 4 (-1 + tan(#)tan(^)Xl + tan(#)tan(^))
(3.162)
where
60


tan(^) =
v ' 2 h
and
s-z
cos(#)
The dynamics is then given by:
mz =p0A\g-Mg
(3.158)
(3.173)
(3.183)
The moment arm n of the buoyancy force Fb about the center of gravity is
found using Figures 3.17 and 3.18.
Figure 3.17 Construction to Find Buoyancy Moment
61


Figure 3.18 Closer Look at Figure 3.17
n = by cos(#)-(c-6z)sin(#) (3.184)
It is useful to identify the intersection of the true vertical through the center of
buoyancy with the ship's vertical through the center of gravity as a point called the
"metacenter."
If the distance from the metacenter to the center of gravity is m then:
n = msm(0) (3.185)
Thus, using equation 3.184:
m =
fc-O
(3.186)
This is called the metacentric height and must be positive for the buoyancy
force to exert a righting moment rather than cause the ship to capsize. That is, m
determines the static stability of the ship.
The roll dynamics involves:
10 = FBn (3.187)
62


where I is the moment of inertia of the ship.
Subsituting for Fb from equation 3.182 and n from equation 3.184:
^ = FBlbycos(0)-(co -&f)sin(0)] (3.188)
Introducing phenomenological damping terms y and /?, our model then arrives
at the coupled system of two second-order equations.
z =£7T-A(zfi)-g- (3.189)
M
0 = ^Y~A{z,e\by(z,e)cOs(0)-{co-bz{z,e))sm(e^[-y{z,e)0 (3.190)
Further elaboration of the model will require specifying dependence of the
damping coefficients on z and 9. Additionally, the effect of inertia of the water
surrounding the ship (the added-mass effect) would need to be included.
3.7 Summary
Modeling of ship motion involves ship geometries and the interaction between
the water and the ship. To simplify the calculations a simple trapezoidal transverse
hull is used with a rectangular side view. By taking limits of the bottom of the vessel,
rectangular and triangular hulls can be examined. Using the geometries, the volumes
of the ships, the centers of gravity, and the moments of inertia can be calculated.
With the addition of water, the centers of buoyancy can be calculated. Using all of
these parameters, one can understand the coupled dynamical equations for the roll
and heave motion of the ship.
63


4. Coupled Mathieu Equations
The Mathieu equation has been used for a variety of applications from springs
and pendulums, to ships. A coupled spring and pendulum system is similar to ship
motion where the pendulum represents the rolling motion of the ship and the spring
represents the heave motion of the ship.
Allievi and Soudack (1990) actually modeled ship roll motion using the
Mathieu equation. They analyzed naval architectural principles, such as the
metacentric height, to create a differential equation. They also examined the presence
of damping and examined Poincare sections to understand the stability and instability
regions based on changing parameters.
To understand a non-autonomous four-dimensional coupled system, a
progressively more complex series of Mathieu equations was used. For each step of
complexity, the solutions to the differential equations were plotted often with
Poincare sections (the x's or points on the graphs).
First, the equation was undamped with two degrees of freedom. By being
undamped, the parametric instability regions or "tongues" could easily be identified.
Second, the equation was damped and had two degrees of freedom. The introduction
of damping added realism to the system and aided in the understanding of the
influence of damping on instability. Third, nonlinearity was added to the Mathieu
equation, with two degrees of freedom, to explore full nonlinear behavior such as
limit cycles and chaos. Fourth, two nonlinear equations each with two degrees of
freedom were coupled in order to see what happens to the geometrical description of
dynamics as the coupling between these two equations was increased. MatLab
program B.2 was used to create all of the graphs used in this section.
4.1 Undamped Linear Mathieu Equation -
Two Degrees of Freedom
First, a linear equation with no damping was examined (equation 4.1)
(Anicin, Davidovic, and Babovic, 1993).
d2x
- + [a + 16<7cos(2r)]x = 0 (4.1)
dz
64


Values for a were chosen using the Ince-Strutt stability chart (Anicin,
Davidovic, and Babovic, 1993). Figure 4.1 shows a parameter space, which shows
the stable (outside the marked regions) and unstable (the p regions) regions.
Figure 4.1 Ince-Strutt Stability Chart
There is a series of alternating unstable and stable regions. The unstable
regions alternate between those regions with resonances drive frequencies that are
multiples of twice the natural frequencies (subharmonic response) (Figure 4.2) and
those regions with resonances at the natural frequency of the system (Figure 4.3).
The instability is noticed by the fact that the trajectory spirals outward from the initial
conditions of x(t) = 0.05 and x'(t) = 0.
65


0.3
0.2 -
0.1 -
0 -
x1
-0.1 -
-0.2 -
-0.3 -
_0 4 I____________I__________1 ___________l____________l___________I
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
x
Figure 4.2 Solutions of Mathieu Linear Equation with a = 1 and q = 0.005
X
Figure 4.3 Solutions of Mathieu Linear Equation with a = 4 and q = 0.05
66


The stable region is a circle (Figure 4.4), thus showing quasi-periodicity in the
system.
X
Figure 4.4 Solutions of Mathieu Linear Equation with a = 2 and q = 0.005
4.2 Damped Linear Mathieu Equation -
Two Degrees of Freedom
Second, damping was added (equation 4.2) to the linear equation (equation
4.1). The damping coefficient /? was chosen to be 0.5 for the different a values.
^-^ + [a + 16^cos(2r)]x + y9= 0 (4.2)
dx dx
A damped Mathieu equation can be converted to an equivalent undamped
Mathieu equation:
p
x = xe 2
or
-E
x-xe 2
(4.3)
(4.4)
67


£
2
(4.5)
dx
~dt
cK
e
dt
d2x
dt2
d2x ~
-e 3 -2e
E
2
(4.6)
So,
^ J3e pt +@xe ^ + (a + 16 dt2 dt 4 v v ^ H dt 2
(4.7)
Dividing through by e ^
c/2x
+
02
a - +16q cos(2r)
(4.8)
The damping raised the tapered ends of the unstable regions from the a axis of
Figure 4.1. Therefore, it takes a higher q value to remain in the unstable region for
the same a value. The stability can be seen by the fact that the trajectory is spiraling
inward from the initial conditions of x(t) = 0.05 and x'(t) = 0 (Figure 4.4 ).
68


0.02
Figure 4.5 Solutions of Damped Linear Mathieu Equation
with a = 1 and q = 0.0025
4.3 Nonlinear Damped Mathieu Equation -
Two Degrees of Freedom
Third, nonlinearity was added (equation 4.9) to the damped equation (equation
4.2).
d2x dx
- + \a + 16<7cos(2r)]sin(x) + /? = 0 (4.9)
dx dx
With certain parameters, a strange attractor can be observed (Figure 4.6). The
existence of a strange attractor implies chaotic activity in the system.
69


Figure 4.6 Solutions of Damped Nonlinear Mathieu Equation with a=1.469,
<7=0.1102, and ^=0.42
4.4 Nonlinear Damped Coupled Mathieu Equations -
Two Degrees of Freedom Each
Last, two coupled equations were used (equation 4.10 and equation 4.11).
The coupling coefficient is x The initial value for x is 0.05, while the initial value for
y is 0.5.
+ \a +16q cos(2r)]sin (x)+/? + x{x ~ >0 = 0 (4.10)
dr dr
py + [a +16<7 cos(2r)]sin(y)+/?^- + - x) = 0 (4.11)
dr dr
For figures 4.7 and 4.8, a = 1.469, 7 = 0.1102, and /?=0.42. For figure 4.8,
a = 1.5, q = 0.11, and /?= 0.42. Figure 4.7 shows low coupling of0.05. Thetwo
graphs look slightly different. Figure 4.8 shows moderate coupling 0.07, where there
appears to be some higher dimensional activity based on the fuzziness of the strange
attractor. The s-attractor can no longer be seen. High coupling Figure 4.9 has twice
as many points as Figures 4.8 and 4.7 to show the detail. With more points, another
s-curve can be seen.
70


Figure 4.7 Weakly Coupled Mathieu Equations
71


72


x y
Figure 4.9 Highly Coupled Mathieu Equations
4.5 Mathieu Equation Conclusion
Mathieu equations are versatile. They model various dynamic systems
including ship motion. These equations enable people to handle non-autonomous
four-dimensional systems in a relatively simple fashion. As a linear undamped
system, unstable spirals and circles are possible. By adding damping, regions of
instability are altered allowing for stable spirals. Adding a nonlinear term into the
equation allows for chaotic activity, such as the s-curve attractor. The coupling of
Mathieu equations shows that for a set of a, q, and /?, there is a coupling coefficient
that shows some higher dimensional activity. Future research will involve placing the
ship equations into this program and seeing how changing parameters demonstrates
the dynamics of the system.
73


5. Conclusion
This thesis strives to help understand the motion of ships and how to predict
ship motion. Analysis of the Carderock ship roll data, recorded on July 19th 1993,
began the whole progression of thought. The most important finding is the extraction
of a correlation time from the data. With a lack of insight into the ship motion
system, a different approach was taken. This new approach began with using the
basic principles of naval architecture to form equations. Next, I examined coupled
equations, from the dynamic system of Mathieu. By analyzing the well-known
Mathieu equation and coupling it with itself, one can understand the dynamics that
the ship model equations might exhibit. Despite all of this work, there are still some
problems.
Trying to tease apart the essence of a system is very difficult. Actual data is
limited by the technology used to collect it, as well as the researchers and the
experiment design. The placement of the sensors on the ships is important so that
only the degree of freedom is recorded. The designers need to ensure that the
instruments work properly and the data is taken at regular intervals. The experiment
design needs to ensure that the proper instruments are used and that the instruments
are used in such a way to get the proper data. To understand the results of the output,
one needs to look critically at how one analyzed the data.
As with any scientific endeavor, assumptions were made which directly
and/or indirectly tainted the data. CDA Professional Version was heavily used
without a strict validation of its assumptions. CDA's many tools, along with other
nonlinear tools, were not used (such as Liaponov exponent). The data was biased by
calculations based on strict 2,048 data point files in a contiguous fashion. Although
discussed, the idea of a moving window and/or using a different window size was not
implemented. CDA is limited to 32,000 or fewer data points; MatLab: Student
Version 5.0 does not accept arrays with more than 16,000 elements. Biases come
from both the analysis of actual ship data and the formation of models.
Models need validation through experimentation and must mimic real-life
behavior. Without experimental evidence validating the model, modeled equations
are theoretical or just mathematical exercises. A suggested experiment is as follows:
Construct a physical ship model with a scale of roughly 1:24. Place the ship in a
tank. Raise and lower the fluid in the tank to a frequency equal to the model's natural
roll frequency. This tests the idea that heave and roll are coupled as in the
74


demonstrated Mathieu's equations. Experimentation and noise help to test
mathematical equations.
Mathematical models simplify the complex interactions observed by the
instruments used in experiments. The instruments are influenced by human actions,
such as sailors moving cargo around, or uncontrollable events, such as a change in the
climate. The circuitry in the instruments often adds noise to the data collected. These
different components come together as the time series, full of noise that complicates
the analysis. The mathematical equations have the principle elements of the system
without the perturbations of instrumentation and events. For future study, it would be
helpful to add noise to the system and run through the same tests as in chapter two
(i.e., mean, variance, kurtosis, power spectrum, etc. . .).
Another future study would be to understand the ship model equations with
the same MatLab program that was used for the Mathieu equations in chapter four.
Another useful project from chapter four is to investigate the "fuzzy" Poincare section
results from the particular range of parameters for the coupled, nonlinear Mathieu
equations. There is some higher dimensional activity illustrated by the plots that
should be analyzed. Different parameters will need to be used to understand the
system of ship motions. Overall, this research is a significant beginning to
understand the complex nature of ship motion.
75


Appendix A
Plots of Different Days and of Heave
Here are some plots of time series, mean, variance, skewness, kurtosis,
dominant frequency, correlation time, and Hurst exponent of roll from the July 18,
July 17, and July 15 respectively. For July 18, the sampling was taken over 11 hours,
22 minutes, and 40 seconds. For July 17, the sampling was taken over 3 hours, 24
minutes, and 48 seconds. For July 15, the sampling was taken over 7 hours, 57
minutes, and 52 seconds. Next are the plots of time series, mean, variance, skewness,
kurtosis, dominant frequency, correlation time, and Hurst exponent of heave from
July 19, July 18, July 17, and July 15 respectively. Dominant frequencies and
correlation times were left out if CDA gave out a value of zero or a value that was
extremely high.
-1 -
.2-----------------------------------------------------
0 20 40 60 80 100
Time (0.5 kilosec)
Figure A.l Time Series of Roll of July 18
76


Variance (degreed) Mean (degrees)
77


Kurtosis Skewness
Figure A.5 Kurtosis of Roll of July 18
78


Time (sec)
Figure A.6 Dominant Frequency of Roll of July 18
The dominant frequency of roll of July 18 is 0.078 0.003 Hz.
Figure A.7 Correlation Time of Roll of July 18
The correlation time of roll of July 18 is 2.55 0.075 seconds.
79


Hurst Exponent
The Hurst exponent of roll of July 18 is 0.38 0.03.
80


Figure A.10 Mean of Roll of July 17
81


Kurtosis
Time (kilosec)
Figure A.12 Skewness of Roll of July 17
Figure A.13 Kurtosis of Roll of July 17
82


Figure A.14 Dominant Frequency of Roll of July 17
The dominant frequency of roll of July 17 is 0.078 0.000 Hz.
Figure A.15 Correlation Time of Roll of July 17
The correlation time of roll of July 17 is 2.1 . 0.9 seconds.
83


0 43
0.41
0 39
0.37
| 033
X
0.31
0.29
0.27
0.25
0 2000 4000 6000 8000 10000 12000
Time (sec)
| raw data fl smooth data [
Figure A.16 Hurst Exponent of Roll of July 17
For roll of July 17, the Hurst exponent value for the raw data was 0.35 0.05
and 0.37 + 0.05 for the smoothed data.
Time (0.5 kilosec)
Figure A.17 Time Series of Roll of July 15
84


Figure A.18 Mean of Roll of July 15
Figure A.19 Variance of Roll of July 15
85


Kurtosis Skewness
Figure A.21 Kurtosis of Roll of July 15
86


Full Text

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SHIP MOTION MODELS by Jennifer Patricia Stamile BA, University of Colorado at Denver 1996 A thesis submitted to the of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Basic Science 2000

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This thesis for the Master of Basic Science degree by Jennifer Patricia Stamile has been approved Alberto Sadun Date

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Stamile, Jennifer Patricia (MBS) Ship Motion Models Thesis directed by Associate Professor Randall P Tagg ABSTRACT Ship motion is a dynamic system with six degrees of freedom The principle movements studied are roll and heave Two approaches help in forming a comprehensive view of ship motion One approach examines actual ship motion data with numerical tools, primarily the tools in Chaos Data Analyzer : Professional Version The most significant result ofthe analysis is finding the correlation time. The second approach is to construct a model of ship motion from basic naval architecture principles With a model oftwo coupled differential equations, one can learn about the important components of the data Mathieu equations are similar in structure to the modeled equations, have been used to model ship motion, and help model a coupled non-autonomous four-dimensional system This thesis helps form a basis for future projects This abstract accurately represents the content ofthe candidate' its publication Randall P Tagg Ill

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ACKNOWLEDGEMENT My thanks to the Office ofNaval Research Grants for the grants and Zenas Hartvigson for the scholarship that helped fund this project.

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CONTENTS Figures ....... ...... .... .............. . ............ .... .... ..... . ...... ......... .... ...... ............................ viii Tables ........... ...... ........ ........ .......... ............ ... ........ ...... ...... ................ ...... ......... ...... xiv Chapter 1 Introduction .... ......... ...... . ................ .......... ....... ............ .............. .... .... ..... .... ....... 1 2 Analyzing Actual Ship Motion Data ........ ....... .... ...................... ............ ......... ...... 4 2 1 Each Window ........ ..... .... ..... . ................. .... ..... ............. ................... ..... .............. 5 2 1 1 Time Series .............. ..... .......... ..... ........ . .... ....... ....... . ..... .... ....... ......... ..... ....... 5 2 1 2 Power Spectrum ........ ....................... . ..... .......... .......... ......... ...... .............. ...... 6 2 1 3 Correlation Function ....... ............................... ................................ ............... 9 2 2 The Entire Day ......... ......... ........ ........ . ..................................... . ......... ....... ..... 11 2 2 1 Mean ...... ..... .......... ................................. ..................................... ............. .... 11 2 2 2 Variance .. ............................ .......... ...... ...... ........ ....... . ..... .............. ........... ..... 12 2 2 3 Skewness ............... ..... .... .......... . .................... .................. ................ ..... ...... 13 2.2.4 Kurtosis ........................................ .............. ...... ................................... ...... 14 2.2 5 Dominant Frequencies ........... ............................. ........ .......... ................ .... .... 15 2 2 6 Correlation Time ......... ....... ...... .... . ................... ....................... ....... .......... .... 16 2.2 Hurst Exponent ........ .... ...... .... ...... .................. .................. ............................ 18 2 3 Summary ................................................... .... .............. ..... ...... .................... ..... 19 v

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3 Modeling Moored Ships ............ ......... ................................. ........... ..... .... ... .... 21 3 1 Rigid Body Motion ..... .... . ........... . ............ . ........ ..... .................. .... ....... ... .... 21 3 2 Simplified Class of Hull Shapes . ........ ............. .......... .......... ...... . ................... 25 3 3 Parameters Derived from Ship Hull Shape and Mass Distribution ............ .... .... 28 3 3 1 ShipMass ....... .......... . ................. ......... ............... .... ............................... ..... 28 3 3 2 Volume .... ......................................................................................... ..... ...... 29 3 3 3 Center of Gravity ............ ................................................................ ....... ....... 32 3 3 4 Moment of Inertia ........... .............. ..... ....... ..... .............. .... .......... ................. 44 3 4 Parameters Derived from Ship Position Relative to Sea Surface ........................ 4 7 3.4 1 Equilibrium Waterline Depth ..................................... ........ ....... .................... 47 3.4 2 Center of Buoyancy ...................... ............. .... ...... ..... ..................................... 51 3 5 Modeling Sea Surface Variations . ........ .... ............ ... ................. ...................... 59 3 6 Equations of Motion ... ..... ............................................ ....... .................... .... ... 60 3 7 Summary ................................................................... ...................................... 63 4 Coupled Mathieu Equations ................ .... ........ .... .... . . .... .............. .... .............. .. 64 4 1 Undamped Linear Mathieu EquationTwo Degrees of Freedom ...................... 64 4.2 Damped Linear Mathieu EquationTwo Degrees of Freedom .......... ................ 67 4 3 Nonlinear Damped Mathieu EquationTwo Degrees ofFreedom ..................... 69 4.4 Nonlinear Damped Coupled Mathieu EquationsTwo Degrees of Freedom Each .................................................. ...... ................. 70 4 5 Mathieu Equation Conclusion ............... ..................... ............ ......................... 73 5 Conclusion ........................................ .................... ...................................... ...... 74 VI

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Appendix A. Plots ofDifferent Days and Plots ofHeave ... .... . ........................................... .... 76 B. Matlab Programs ................ . . ..... ........... .............. ................ ........................ 108 C. Glossary of Naval Architecture .................................................. ......... ...... ...... 118 References ............ .................. .......... ........ ................ .......................... .......... .... 120 Vll

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FIGURES Figure 2 1 Degrees ofFreedom ....... ....... ........................................ .................................... 4 2 2 Time Series ofRoll38 of July 19 ........................................................................ 5 2 3 Time Series ofRoll47 of July 19 ........................................................................ 6 2.4 Power Spectrum ofRoll38 ofJuly 19 ................................................................. 8 2 5 Power Spectrum of Roll 47 of July 19 ............................. .............................. ...... 8 2 6 Correlation Function ofRoll38 of July 19 ........................................................ 10 2 7 Correlation Function ofRoll47 ofJuly 19 ........................................................ 10 2 8 Time Series ofRoll of July 19 .... ..... ..................................................... ............. 11 2 9 Mean of Roll of Jul y 19 ...... ............................................................................. 12 2 .10 Variance ofRoll of July 19 ............................................................................. 13 2 .11 Skewness ofRoll ofJuly 19 ............................................................................ 14 2 .12 Kurtosis ofRoll of July 19 ................. .......................... ................................. 15 2 .13 Dominant Frequency ofRaw Roll Data of July 19 .......................................... 16 2 .14 Dominant Frequency of Smoothed Roll Data of July 19 ............................ ...... 16 2 .15 Correlation Time of Raw Roll Data of July 1 9 .... ........................................... 17 2 .16 Correlation Time of Smoothed Roll Data of July 19 ........................................ 17 2 .17 Correlation Time of Roll Data of July 19 ........................................................ 18 Vlll

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2 .18 Hurst Exponent of Raw Roll Data of July 19 .................... ....... ....................... 19 2 .19 Hurst Exponent of Smoothed Roll Data of July 19 . ........................... .... ......... 19 3 1 Rigid Body Motion Diagram . .......... .................... ............................... ...... ..... 22 3 2 Views of the Trapezoidal Vessel ....... ... ............ .................. ........................... 26 3 3 Rectangular Transverse Hull ..... ............. .... ..... .... . ............. .............................. 27 3.4 Triangular Transverse Hull ... .......... . ...... ............................... ......... ... . ........ . 27 3 5 Volume of Trapezoidal Hull ............... ..... ..................... ................. ......... ...... ... 30 3 6 Center of Gravity for Right Triangle ......... ............... ........................................ 3 2 3 7 Triangular Part of a Tilted Trapezoid ................................................................ 34 3 8 Center of Gravity ofTrapezoidal Hull ofUniformly Distributed Mass ........... . 39 3 9 Center of Gravity ofTrapezoidal Hull ofNon-Uniformly Distributed Mass . ... .42 3 1 0 Moment of Inertia of Trapezoidal Hull .................. ........ ....... ... . .................... 44 3 .11 Equilibrium Waterline Depth ................................... ...................................... .47 3 12 The Actual Area of the Water ........ .......... .............. ...... .......... ........... ..... ......... 52 3 .13 The Area of the Vessel's Waterline . ....... ............................................... .......... 52 3 .14 The Areas of the Two Triangles Formed by the Waterline and Tilted Vessel.. 53 3 .15 Important Parameters ofBuoyancy .... .... ............ .............................. ... .... .... . 54 3 .16 Ship Position Relative to the Ocean Floor ................ .................. .... ........ . ..... 54 3 .17 Construction to Find Buoyancy Moment .... .......... ........... . ............. ........... .... 54 3 .18 Closer Look at Figure 3 .17 .... ....... .............. ..................... ............. ................. 54 4 1 Ince-Strutt Stability Chart .... .............. ........... ...... ... .... ................. .... .............. 65 IX

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4 2 Solutions of Mathieu Linear Equation with a = 1 and q = 0 005 .................. ...... 66 4 3 Solutions of Mathieu Linear Equation with a = 4 and q = 0 005 ........ ............... 66 4.4 Solutions of Mathieu Linear Equation with a = 2 and q = 0 005 ...... ................. 67 4 5 Solutions of Damped Linear Mathieu Equation with a= 1 and q = 0 0025 ... .... 69 4 6 Solutions ofDamped Nonlinear Mathieu Equation with a=l.469 q = O .ll02, and ........... ................................ .................. ..... ........ .... .... 70 4 7 Weakly Coupled Mathieu Equations .......... ........ .... ..... ......... . . ... ................ ... 71 4 8 Moderately Coupled Mathieu Equations ............................ ....... ..... .......... ........ 72 4 9 Highly Coupled Mathieu Equations .................. .... . ...................... .... ..... ... ....... 73 A. 1 Time Series of Roll of July 18 .......... ........................................................ ....... 7 6 A.2 Mean of Roll of July 18 ....... . ..... ......... ......... .... .... ...... ....... .......... ......... ... ... 77 A.3 Variance ofRoll of July 18 ......... .... .................... . ........ ..... ..... .... ... ...... ..... ..... 77 A.4 Skewness ofRoll of July 18 . ........ ............ ............. ...... ....... ..................... ...... 78 A. 5 Kurtosis of Roll of July 18 .................... ....... ....... .................... ....... ..... .......... 78 A. 6 Dominant Frequency of Roll of July 18 ....................... ............... ...... . ....... ... .. 79 A. 7 Correlation Time of Roll of July 18 ... ................................. ... . .... ................... 79 A.8 Hurst Exponent of Roll of July 18 .... . . ........... ..... ....... ..... ..... ... ....................... 80 A.9 Time Series of Roll of July 17 ... ............. . ...... .... .... .... .......... ........... ........ ........ 80 A.10 Mean ofRoll of July 17 . .... ........ ............ ................. ..................................... 81 All Variance of Roll of July 17 ..... .......... ....... .... ...... . .... .... ................................ ... 81 A.l2 Skewness ofRoll of July 17 .......... .... ................ ..... .......................... ....... ... 82 X

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A.13 Kurtosis of Roll of July 17 .............................................. .... ............ ............... 82 A.14 Dominant Frequency of Roll of July 17 ............ ..... ...... ....................... ........... 83 A.15 Correlation Time of Roll of July 17 ........ ................... .............. .......... .......... 83 A.16 Hurst Exponent of Roll ofJuly 17 ........... ..... ................................ ............... . 84 A.17 Time Series of Roll of July 15 ...................................... . ..... .... ....................... 84 A.18 Mean ofRoll of July 15 ................................... .... ............ ....... . ..... ............ . . 85 A.19 Variance ofRoll ofJuly 15 .......... .... ........... .... ...... ............ . ...... ..................... 85 A.20 Skewness ofRoll of July 15 ........... .... . ...... ........ ........ ......................... ......... 86 A.21 Kurtosis ofRoll of July 15 .............. .... ................... .... .......... ....... ....... .... ........ 86 A.22 Dominant Frequency of Roll of July 15 .... ........................... ........ ............... 87 A.23 Correlation Time of Roll of July 15 ....... ................................ . ....................... 87 A.24 Hurst E x ponent of Roll of July 15 ........ ...................... ......................... ........ 88 A.25 Time Series ofHeave of July 19 ....... .... ........................... . ...................... .... 88 A.26 Mean of Heave of July 19 .............................................................................. 89 A.27 Variance ofHeave ofJuly 19 ............................... .............. ... .... ..................... 89 A.28 Skewness of Heave of July 19 ...................... ...... .............. ............ .................. 90 A.29 Kurtosis ofHeave of July 19 .......................................................................... 90 A.30 Dominant Frequency of Heave ofJuly 19 ................ .... ........ .............. ............. 91 A.31 Correlation Time of Heave of July 19 ............................................................. 91 A.32 Hurst Exponent of Heave of July 19 .. ................................................... .......... 92 A.33 Time Series of July 18 .............................................................. .. ........... .... ..... 92 Xl

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A.34 Mean ofHeave of July 18 ............................... . ..... .... . ....... ............. ......... ... 93 A.35 Variance ofHeave of July 18 ................................ ..... . .................... ......... ..... 94 A.36 Skewness of Heave of July 18 .... . . .... . ......... . .............. ................. .... ..... ..... 94 A.37 Kurtosis ofHeave of July 18 ........ ........ ............. ... ...... ...... ............................. 95 A.38 Dominant Frequency of Heave of July 18 ..... ..... ............. ...................... . ..... 95 A.39 Correlation Time ofHeave ofJuly 18 ...... ..... .... .............. .................. . . ......... 96 A.40 Hurst Exponent of Heave of July 18 ....... ...... ........................................ ....... 96 A. 41 Time Series of Heave of July 17 . ..... .... ..................... ..... ..... .................... ..... 97 A.42 Mean of Heave of July 17 .................. .................... ....................................... 97 A.43 Variance of Heave of July 17 .... .... .......... ............. ....... ................................ 98 A.44 Skewness of Heave of July 17 ...... ....... .................... ... .......... . ............. ....... 98 A.45 Kurtosis ofHeave ofJuly 17 ............. .... ................ ..... ....... ............................ 99 A.46 Dominant Frequency of Heave of July 17 ..... . .... ...... ..... ......... ....................... 99 A.47 Correlation Time ofHeave of July 17 ..... ..................................................... 100 A.48 Hurst Exponent of Heave of July 17 ............................ ........ .................... .... 100 A.49 Time Series ofHeave ofJuly 15 ... ....... ...... . .......................... ..... .......... ....... 101 A.50 Mean ofHeave ofJuly 15 . ..... . ........... ................................. ............ . ..... 101 A.51 Variance ofHeave of July 15 ........ ..... ....................................................... . 102 A. 52 Skewness of Heave of Jul y 15 ...... .................. ......... ................... ............ .... 102 A.53 Kurtosis ofHeave ofJuly 15 ........ .... ....... ........................ ..... ..... . ................ 103 A.54 Dominant Frequency ofHeave ofJuly 15 ............. ................ ...... ............ ... 103 Xll

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A.55 Correlation Time ofHeave ofJuly 15 .................. .............. . ....... .... ..... ........ 104 A. 56 Hurst Exponent of Heave of July 15 .... ...... ......... . .................... . ................. 104 A. 57 CDA File with Six Point Gaussian Smoothing of Roll of July 19 .... ............ 105 A.58 Skewed Derivative ofRaw Roll Data of July 19 ......... .... ............ ............ ...... 106 A. 59 Skewed Derivative of Smoothed Roll Data of July 199 ................... ............. 106 A.60 Correlation Dimension of Raw Roll Data of July 19 .................. ........... . ..... 107 A.61 Correlation Dimension of Smoothed Roll Data of July 19 .... .............. .......... 107 Xlll

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TABLES Table 3 1 Volumes for Different Transverse Hull Shapes .......................................... ....... 31 3 2 Centers of Gravity (Distribute Mass) for Different Transverse Hull Shapes ...... .41 3 3 Moments oflnertia ofUniformly Massed Hull Shapes ...... ............................ .... 47 3.4 Equilibrium Waterline Depths for Different Hull Shapes ................................... 51 XIV

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1 Introduction Naval crane ships need to transport cargo from one ship to another ship without injury to people or property Problems arise from different sea states or conditions causing oscillation in the crane's cable thus creating a giant pendulum From the work of John Starrett chaotic behavior can occur from pendulum movements People have approached this problem through mechanical engineering solutions which help most of the time but do not solve the underlying problem By understanding the motion of the ship and predicting its movements one is more likely to transport cargo safely over a greater range of oceanic conditions One method for understanding ship motion is to study actual ship data to tease out the important characteristics that allow for predicting the ship's movements However to obtain ship motion data some noise is introduced into the system which impedes analysis Noise can be somewhat stochastic and originate from human activity and instrumentation Humans may move cargo, which changes the distribution of mass on the ship Some noise may be intrinsic to the system such as waves currents and fish swimming by the ship To grasp the fundamental characteristics of the system a ship model needs to be created from fundamental principles of ship behavior and shape With these mathematical models noise is eliminated ; thus, the basic principles ofthe system can be studied through numerical integration, and the creation of sample time series for analysis A number of researchers have done work on ship motion Some experts chose to work strictly numerically without experimentation Other researchers chose to justify their models with ship model data. Chen Shaw and Troesch (1999) created a system of equations that involve roll, sway and heave (see Figure 2 .1). They examined the nonlinear, large-amplitude motion ofthe ship in response to beam seas where waves strike the vessel broadside The authors found that heave was on a fast manifold while roll and sway were on a slower manifold Ohtsu (1990) created an optimum autopilot system with a single-input/single output control-type autoregressive model. The autopilot system was fixed-gain and noise-adaptive This system controlled the rudder and thus roll. Ohtsu created ship components for a model ship and recorded actual ship motion data He found that the yaw (see Figure 2 .1) of a small rudder induced roll motion

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Sanchez and Nayfeh (1997) used numerical means to analyze ship motion They examined parametric excitation (time varying), and external excitation (inhomogeneity) The authors identified instabilities that appeared when one of the excitations is slowly varied They fixed the level of parametric excitation for a model boat. They studied the stability and bifurcation of an equation with some heave and roll coupling Falzarano, Esparza, and Taz Ul Mulk (1995) studied roll motion in isolation. They produced an analogy between pendulums and ship motion They performed steady-state bifurcation analysis through their numerical studies They observed the changes to the restoring moment of roll by changing the height of the center of gravity and damping The authors altered damping by examining the presence and size ofbilge keels They found that bilge keels add nonlinear damping and are influential in resistance to capsizing Iseki (1990) focused on the cross spectrums of heave, pitch, and roll (see Figure 2 1 ) The basic idea is that the ship is a giant wave probe and by understanding the motion of the ship, one can estimate the directional wave spectra The directional wave spectrum describes the wave energy in terms of frequency and direction He created a model ship that was rigidly fixed to restrict surge motion and used springs to loosely restrict sway and yaw motion The ship was excited by long crested irregular waves Iseki found that the ship's pitch frequency was influenced by the waves, while roll frequency was nearly independent of the waves Allievi and Soudack (1990) modeled roll motion with damping using basic naval architectural principles The authors used a Mathieu-like equation for their analysis as they calculated the stable and unstable regions based on their parameters They examined undamped, linearly damped, and nonlinearly damped Mathieu systems and the phase portraits of those systems Despite these hard-working researchers, a conclusive understanding of ship motion has not been found The thesis chapters are written as follows: Chapter two examines the work of the Digital Sealegs Group The Digital Sealegs Group was a group of students and faculty ofthe University of Colorado at Denver involved in the study of cranes and crane ships This chapter focuses on the numerical tools used to study the rolling of an actual crane ship The most significant result is the evidence of measurable correlation times for several windows of 1 024 seconds of data for each window 2

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Chapter three investigates the formation of a ship model for the coupled heave and roll of a moored ship Three different ship geometries are examined with different parameters (e g "metacentric height") An argument is made to restrict the model to a coupling of roll (rotation about the ship's axis from bow to stem) and heave (vertical translation) with forcing only in the heave direction Chapter four presents coupled Mathieu equations describing the parametric forcing oftwo pendula whose pivots are moved sinusoidally up and down These equations have been used to model theories and, when coupled produced a non autonomous four-dimensional system. This system helps in visualizing the behavior of such larger-dimensional systems for dynamics that, in the uncoupled case is understood and rich in behavior (i e periodic motion period doubling and chaos). Chapter five presents the conclusion In addition this chapter assesses the model as a tool for evaluating approaches to predicting real data Further work is suggested including adding noise to the model-generated data and constructing an experiment with a physical ship model. Included in the appendix are MatLab programs, a glossary, and additional plots of data Some of the MatLab programs used to generate the plots in chapters two, four and five. Also in the appendix are some additional graphs for the roll and heave of the other data files 3

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2 Analyzing Actual Ship Motion Data The Digital Sealegs Group analyzed the ship motion data for the NOAA ship Discoverer and a naval crane ship Preliminary investigations by Margo Martinez and John Slavich involved finding coupling between degrees of freedom spectral analysis John Slavich analyzed the data from the Discoverer ship Margo Martinez analyzed crane ship data provided through the Carderock Division of the Naval Surface Warfare Center. There were data from July 15, 17, 18, and 19, 1993 Most of the analysis was performed on the July 19 roll data, which was a long contiguous set of recorded data during sea state 3 The ship data recorded from both ships had six degrees of freedom (Figure 2 .1) (Gillmer & Johnson 1982). There are three translational motions : surge sway and heave There are three rotational motions : roll pitch and yaw Of the six degrees of freedom pitch heave and roll are predominately used in ship motion This section focuses on roll motion y Figure 2.1 Degrees of Freedom The software, Chaos Data Analyzer : The Professional Version (CDA) was used to extract the Hurst exponent the power spectrum and the correlation function A MatLab program calculated the mean kurtosis variance and skewness for windows of 1 024 seconds (2 048 samples) throughout the entire data file Two time scales are examined : a single window of approximately 17 minutes of data and the entire day of approximately 19 hours of data The purpose of these tests is to identify the characteristic features of the data such as dominant frequency and to investigate the data's degree of "stationarity i. e., how the features and statistics behave over longer time scales 4

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2 1 Each Window Each file or window was sampled for 17 minutes and 4 seconds Each window had its time series power spectrum and correlation function analyzed as plots The sampling rate was two samples per second well within observed frequencies of motion 2 1 1 Time Series Events such as ship motion change over time creating a progression of data points known as a time series (Williams 1997) The time series plots of July 19, 1993 involved some files that seemed too erratic to have structure (Figure 2 2) Other time series appeared organized and had a low frequency envelope (Figure 2.3) Under a smaller per i od the signal looks smooth and thus there is not a lot of instrumental noise in the data. 2 1 5 0 0:: 0 5 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (0.5 seconds) Figure 2.2 Time Series of Roll 38 of July 19 5

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1 5 1/) Q) :E C) Q) 0 5 0 0::: 0 0 5 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (0 5 seconds) Figure 2.3 Time Series of Roll 47 of July 19 2 1 2 Power Spectrum Data may have some periodic components Differing amplitudes and phases of sines and cosines form the periodic components of the data Fourier analysis allows for the extraction of the different waves The power spectrum involves taking a fast Fourier transform of the time series The power is the mean square amplitude and is plotted against the frequency A broad spectrum often suggests random and/or chaotic data A spectrum of a few dominant peaks usually means periodic and quasi periodic data Fourier analysis examines superimposed simultaneous multiple waves with various heights (Williams, 1997) Within the signal is a standard or reference wave, which is often the longest wave available or the length of the record This wave is called the fundamental wave The primary characteristics ofthe fundamental wave are its wavelength (fundamental wavelength) and its frequency (fundamental frequency). Fourier analysis uses waves whose frequencies are integer multiples of the fundamental frequency All waves are based on the wavelength of the composite wave Fourier analysis indicates which frequencies are in the signal and their relative importance The equation for the Fourier analysis is: N 2 y = L:(ah cos[hB]+ {Jh sin[heD h= O 6 (2 .1)

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The discrete Fourier coefficients are : 2 N I 2Jrht a h = -LYn cos --n N t .=O N and The variables in these equations are : N = number of observations h = harmonic number (1 for first harmonic) tn =time Yn = data value at time tn (2 2) (2 3) Least-squares estimates involve minimizing the average squared difference between the value of the composite wave and the sum of the components + p ; )/2 (2.4) CDA uses 128 frequency intervals A larger number of intervals would i mprove resolution but "exacerbates the spurious responses" (Sprott & Rowlands 1995) Since CDA truncates the data to the largest power of two the window chosen for the data was a power of two CDA uses non-o v erlapping segments The maximum frequency used by CDA is the Nyquist critical frequency which is the reciprocal of twice the interval between data points For most of the CDA windows a dominant peak was found (Figure 2.4 and 2 5) Random and chaotic data often have broad spectrum because of all the contributions from different frequenc i es. Since in most windows in this system have a pronounced peak this system is to some degree periodic N data points ( N = 2048) with sample time (L1t = 0 5 seconds) has a ma x imum number N frequency intervals with frequency interval !J.f = 1-. The maximum 2 N!J.t (Nyquist) frequency is : 7

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N 1 1 fny q =2x l1j= 2111 = 2 x 0 5s =1 Hz (2 5) If m intervals are averaged to end up with 128 averaged frequency intervals then : m = = = 8 (2 6) Therefore, the frequency interval of the averaged spectrum is: m 8 1 11/. =m11f=-= =-Hz av NM 2048 X 0 .5s 12. 8 (2 7) The peaks appear to lie at approximately corresponding to a period of 12 8 seconds. This is probably the ship's natural roll period The bottom axis for each plot needs to be multiplied by 1/128 0.12 0.1 0 .08 0.06 0 .04 0 .02 \ 20 40 60 80 100 120 Figure 2.4 Power Spectrum of Roll 38 of July 19 8

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0 0 3 0 .025 0 .02 0 .015 0 .01 0 0 0 5 20 40 60 80 Figure 2.5 Power Spectrum of Roll 47 of July 19 2 1 3 Correlation Function In a time series there may be repeated data points separated by some time lag (Williams, 1997) Autocorrelation shows to what extent two time segments with certain time lags differ from each other For a zero value the two time segments are not correlated or the sum of the products is close to zero The correlation time states the time it takes until two segments match each other in value The CDA correlation function was obtained by multiplying each x(t) with x(t-tau) and summing the result over all ofthe data points (equation 2 8) (Fenny & Moon 1989) The correlation function is the sum plotted as a function of -r or n (Sprott & Rowlands 1995) CDA calculates the correlation time as the tau when the correlation function first falls to 1 / e i n value. N is the sample size minus one (2 .8) n = 0 ,1 2 .. N (2 9) There were correlation functions that seemed reasonable (Figure 2 6) and other correlation functions that did not make sense (Figure 2. 7) The reasonable functions appeared to have slower oscillating en v elopes Roll 4 7 has the erratic time 9

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series (Figure 2 2) as well as the erratic correlation function (Figure 2 7) Roll38 has a more structured time series (Figure 2 3) and also has a more structured correlation function (Figure 2 6) 4 3 2 0 1 2 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 2.6 Correlation Function of Roll 38 of July 19 200 400 600 800 1000 1200 1400 1600 1800 2000 Figure 2.7 Correlation Function ofRoll47 of July 19 10

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A problem with how CDA calculates the correlation time is that the time is taken without consideration of dampened oscillating systems The system may not return to the same value, but the system may return to close in value. This semi memory in values is not considered in CDA's code and may be a more accurate correlation time for this system 2 2 The Entire Day For July 19, the data were collected for 19 hours 20 minutes and 3 2 seconds The time series is shown in Figure 2 8 3 5 3 2 5 2 u; 1 5 Q) Q) 0:, 1 Q) 0 05 0:: 0 -0. 5 -1 1 5 0 20 40 60 80 100 120 140 Time (0. 5 kilosec) Figure 2.8 Time Series of Roll of July 19 For mean variance, skewness and kurtosis time is based on the middle point of the time for each window After the statistics are discussed dominant frequencies correlation times and Hurst exponents will be discussed 2 2 1 Mean The sampling rate and the total time of observation limit a time series (Williams 1997) Increasing sampling rates allows for greater resolution ; while averaging over nearby sampling rates helps decrease noise and aids in illustrating 11

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general trends In addition some help in finding the general trend in the data is the use of the arithmetic mean : 1 N x=-l:x; N i = t (2 1 0) The mean of the roll data was approximately one until the 25th file at which point the mean became very erratic (Figure 2 9) This change in the mean can be observed through the time series (Figure 2 8) as the general trend is fairly constant for the first 20 kiloseconds and then loses some stationarity. Figure 2 9 shows the mean for each individual window 1.6 .------,-------.----.-----,.-----r---.------, 1.4 ............................... ............................... ; .............................. ; ............ 1 2 ............................... + ............................. ; .............................. .......... i ; 0 8 1........................... + .............................. ; .............................. ; ............................... ; .................... Q) 0 6 1............................. f ............................... .............................. .............................. .................. ...... : ...... !""""'"''W-0.4 1........................... + ......................... .. .. .............................. : .............................. : .. ............................. : ....................... .. .. \ 0 10 20 30 40 50 60 70 Time (kilosec) Figure 2.9 Mean of Roll of July 19 2 2 2 Variance It is useful to know how much the data deviates from the mean (Williams, 1997) Means and variances are tests of stationarity The variance gives the magnitude of the average deviation from the mean : 2 -) 2 s =L..J X; -x N i = t (2.11) 12

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The standard deviation is the square root of the variance For a small total number of data points, N-1 should be used as a divisor so as not to underestimate the values The variances of the roll windows are spaced fairly close together until the 20th window then the variance ofthe windows are farther apart (Figure 2 10) These changes in variance can be observed in the time series as the relative "thickness" ofthe data Between 30 kiloseconds and 40 kiloseconds, the elevated data peaks in the time series that correlate elevated peaks on the variance plot (Figure 2 1 0) The variance for each window is calculated The peaks in the plot correspond to places of large variation in the time series 0 35 rt -: rr: 0 3 -r-r......._ N :3 0 25 0) a.> 0 2 a.> (.) c: 0 15 > 0 1 0 05 . + + + + .. ? .. j ......... . ............ ............... + .............................. .......................... ............ ... ... ....... : : : : .... .............................................. 0 10 2 0 30 40 50 60 70 Time (kilosec) Figure 2.10 Variance of Roll of July 19 2 2 3 Skewness Skewness is defined by (Sprott & Rowlands 1995) : (2 12) 13

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The skewness ofthe roll data deviated only slightly from zero (Figure 2 11) There are no apparent trends Some windows are skewed in one direction and other windows are skewed in the other direction Skewness describes the lack of symmetry about the mean (James & James 1992) When comparing the skewness to the time series it is not easy to verify the skewness plot since the data is so concentrated around the mean 1 5 l ---t -r+--t-......... : : : . j 0 : + __ 1 : J + ......... 0 5 : : ; l ............ 1,,,,,,, ........... .... .......... 1, 1 : 1 1 0 10 20 30 40 50 60 70 Time (kilosec) Figure 2.11 Skewness of Roll of July 19 2 2.4 Kurtosis Kurtosis is defined by (Sprott & Rowlands 1995) : (2 13) Figure 2 .12 shows the kurtosis for July 19 each point represents one window of data Kurtosis describes the concentration about the mean (James & James 1992) There are no apparent patterns in the plot. The 63rd window has the highest kurtosis v alue and is the most skewed in the positive direction 14

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.!!! IJ) 55 ........................ .1 ............ ............... ........................... .;. .......................... ........................... ; ........................... ; ............ . i i i . . : :[: ::: : t :::: j :: : :::::: : .g 4 + .. !! ...... .... I I : 3 5 + + + . r --r ............. +.......... 2 5 ............ .i ........................... i ........................ 0 10 20 30 40 50 60 70 Time (kilosec) Figure 2.12 Kurtosis of Roll of July 19 2 2 5 Dominant Frequencies In the dominant frequencies correlation times and the Hurst exponent, some smoothing was performed The process involved a nine-point low pass filter was constructed by convolving the data with nine points of 0 .1111 in value and interpolating the result. Smoothing was performed on the data in order to reduce the amount of noise in the signal. Noise will have more erratic peaks and valleys, which with smoothing will be decreased in steepness to allow the underlying structure of the data to be seen and analyzed The average dominant frequency for the raw roll data of July 19 was 0 .10 0.04 Hz (Figure 2 13) With smoothing the average dominant frequency for the roll data of July 19 was 0 .10 0.04 Hz (Figure 2.14) 15

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0.12 0.11 g 0.1 Q) g. O W Q) O.(Jl O.CXJ 8 0.05 0.04 0.03 0 .11 TT"TTT"T....,..,.TT"TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT""TTT""TTT"TT"1 0 1 +H+Ht-t++ttt+lt-t+IH+t+H+H+H+H+H+H+H+H+H+H+H+H+H++t++t++t 0 .09 g 0 .08 +H+HI++I-lll-+lt+HH+++It-IH+H+H+++++++++++++++-IH+-IHI+++++H+H++t-IH .., :::> g' 0 .07 +H+Ht-t+l-llt+lt+HH+t+lt-IH+H+H+++++++++++++++-IH+-IHI++H+H++t++t-IH It 0 06 +H+HI++a-Jil.-I-*+HI++++I+..-H+H++++++++++++++++++111++-1Hf++++++++H+H..-H .5 E 0 .05 +H+HI++I-lll-+lt-t+IH+t+t+-1+++++++++++++++++ ++++++-IH+-IHI+++++H+H++t-IH 8 0 04 0.03 Tune (sec) Figure 2.13 Dominant Frequency of Raw Roll Data of July 19 I n I n Iii un I 1 n I I I I : I Ture(sec) Figure 2.14 Dominant Frequency of Smoothed Roll Data of July 19 2.2 6 Correlation Time The correlation time is the amount of time it takes for a time segment to repeat itself The correlation time was 1.8 0 7 seconds (Figure 2 15) and with smoothing it was 1.9 0 7 seconds (Figure 2 16) One approach to correlation time was to take correlation functions from the CDA and construct an exponential fit to the sine wave 16

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I with a declining amplitude The reciprocal of the exponential power is the correlation time (Figure 2 17) 7 2 ,....., "' -g 6 2 8 5 2 on e 4 2 0 3 2 2 2 1.2 9.2 8 2 ,....., 7.2 "' "0 = 6 2 @ 0 4 2 3 2 2 2 1.2 0 0 10000 20000 30000 40000 50000 60000 70000 Time (sec) Figure 2.15 Correlation Time of Raw Roll Data of July 19 10000 20000 30000 40000 50000 60000 70000 Time (sec) Figure 2.16 Correlation Time of Smoothed Roll Data of July 19 17

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8.-------------------------------------------------------, 7 6 \ 0 10000 20000 30000 40000 50000 60000 70000 time (sec) Figure 2.17 Correlation Time of Roll Data of July 19 2 2 7 Hurst Exponent The Hurst e x ponent represents how random or uncorrelated the data points are to each other White or uncorrelated noise has a Hurst exponent of0. 5 The Hurst exponent is from "the slope ofthe root-mean-square displacement ofvarious initial conditions" (i. e., each point) versus time (Sprott & Rowlands 1995) For data the line starts at the first data point and ends at the square root of the total duration of the data record Hurst exponents greater than 0 5 show that trends will continue into the future Hurst exponents less than 0 5 show that trends will reverse in the future Hurst e x ponents show how values move away from the initial value using each point in time series as an initial condition The average Hurst e x ponent ofraw roll data ofJuly 19 was 0 32 0 07 (Figure 2 18) With smoothing the average Hurst exponent was 0 34 0 07 (Figure 2 19) Smoothing hardly affected the Hurst e x ponent. The roll data had Hurst e x ponents less than 0 5 so the data is correlated and trends are to be re v ersed in the future 18

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I I 0 5 0.45 ..... 0.4 = Cl) = 0 0 .35 JJ.l ..... "' 0.3 0.25 0 2 0 10000 20000 30000 40000 50000 60000 70000 Tune (sec) Figure 2.18 Hurst Exponent of Raw Roll Data of July 19 0.2_ Ern 0 10000 20000 30000 40000 50000 60000 70000 Time (sec) Figure 2.19 Hurst Exponent of Smoothed Roll Data of July 19 2 3 Summary Margo Martinez and John Slavich were able to obtain ship motion data and perform numerical tests on the data Further tests (i. e., Hurst exponent, power spectrum autocorrelation) were performed on a set of roll data of July 19. There was 19

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no definitive answer to knowing the underlying ship dynamics perhaps due to too much noise in the system In the short term there is a dynamical system behavior "quasi-stationarity For the entire day there is non-stationairy behavior which can be explained by tidal changes 20

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3 Modeling Moored Ships In order to gain insight into the dynamics represented by the July 19 roll data, a model was developed By simplifying the system to a mathematical expression one can obtain the fundamental attributes of the system without having to deal with noise With the construction of a mathematical model there is always a question of how complex to make the model. A complex model may be more realistic, but it often results in large amounts of computing time more debugging and may impede the conceptualizing ofthe basic components ofthe system. Therefore a compromise must exist between a simple and complex model. The ship moves with six degrees of freedom (see Figure 2 .1). All of the literature describes models with roll but there is some discrepancy as to what the next important degree of freedom is after roll. To load and unload cargo on a ship the crane ship and the transport vessel must be parallel. Thus, the movement of the transverse cross section is most important. Therefore heave is an important component of ship motion This chapter describes the basic knowledge needed to understand the movement of a rigid body and then states a coupled system of equations describing roll and heave Three basic hull shapes will be examined For these shapes the parameters ofthe coupled equations that can be derived from ship geometry include mass volume, center of gravity and moment of inertia Additional parameters require analysis of the ship's position and orientation relative to the sea surface : equilibrium waterline center of buoyancy and the metacenter 3 1 Rigid Body Motion The following equations form the basic mathematical theory behind rigid body motion which is the fundamental basis for the coupled equations Harrison and Nettleton (1997) and Ginsberg (1995) describe the dynamics of rigid body motion which are demonstrated below 21

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0 Figure 3.1 Rigid Body Motion Diagram Imagine a solid body divided into a set of small masses, m;. The total mass m is then : (3.1) The center of mass r e m is defined as : 'Lm; ; i r e m=--' ; __ (3.2) m Let be the position of the ith mass relative to the center of mass : r i = r;r e m (3. 3) Note that: (3.4) Substituting equation 3 3 into equation 3.4 : (3. 5) Let the force be distributed over the body with force F ;, acting at each small mass The total force is : 22

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(3. 6) From Newton's 2nd Law: (3. 7) .. .., -) = L:mi r e m + r i (3. 8) (3. 9) (3 1 0) Ifthe distribution of masses is fixed in time : d 2 -.. ""' I = mrcm +-2 L.Jm; r ; dt i (3 11) Then since (3. 12) The second term of equation 3 .11 vanishes and the equation for the center of mass motion is : mfc m = F (3. 13) Now examine the torques on the solid body about the center of mass Let (3. 14) The total torque about the center of mass is : 23

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(3. 15) The second term vanishes because of the property that for a x a= 0 any vector a (again assuming mass distribution remains fixed) 00 -) = Lr i X m ; r em+ r i (3. 16) .. .. ( -J = X T c m + X r ; (3 17) The first term vanishes (equation 3 5) and the second term can be rearranges to give : d (dr'J r = -Lm. r'; x --1 dt i I dt Again assuming the mass distribution remains fixed For rigid body motion : dr' -_; = m x r' dt I for some angular velocity vector m Substituting 3 .20 into equation 3 19: d -(--) r = dt Lmr'; x xr;' I 24 (3. 18) (3. 19) (3. 20) (3. 21)

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The result is: d (T-) r=-vw dt where the moment of inertia tensor is: L:mi(yi2 +zn Lm;X;Y; i 1= Lm;X;Y; L:m; (x; + z;2 ) i L:m;X;Z; Lm;y;z; and 3 2 Simplified Class of Hull Shapes (3. 22) (3 .23) L:m; X ;Z; i Lm;y;z; (3.24) i Lm;(x;2 + Y;2) (3. 25) A trapezoidal hull (Figure 3 2) is a good approximation to an actual ship The top left picture in Figure 3 2 shows the side view The top right in Figure 3 2 shows the transverse section of the ship The bottom right picture shows the top view. Through limits on the bottom ofthe ship rectangular (Figure 3 3) and triangular hulls (Figure 3 .4) can be created from the trapezoidal hull. The width of the deck for all the vessels is d. The height of the vessel is h. The length of the vessel is I. The width of the bottom of the vessel is k. 25

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\ . / . \ ... ........ . ....... .... -.. ... . ........ , ... Figure 3.2 Views of the Trapezoidal Vessel k The rectangles indicate a simplification of the ship geometry that ignores the shape of the bow and stem The equilibrium waterline depth U-tJ is the value of the neutral waterline depth for (} = 0 (zero roll) Later all dimensions will be scaled by the width of the deck. h' = !!_ d d 26 (3. 26) (3. 27) (3. 28) (3. 29)

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r h i ffio i Fig ure 3.3 Rectangular Transverse Hull For the rectangular hull (Figure 3 3) d or k' = 1 !-+--d ---+1 Figure 3.4 Triangular Transverse Hull For the triangular hull (Figure 3.4) !:...= 0 d 27 (3. 30) (3. 31) (3. 32)

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or k'= 0 3 3 Parameters Derived from Ship Hull Shape and Mass Distribution (3.33) There are certain characteristics of the vessel that are independent of the ship's position relative to the sea such as ship mass, volume center of gravity and moment of inertia These factors will be described in order. To discuss these factors certain variables and scaled values must be defined Some important variables are : c = center of gravity P o = density of water V= volume M= mass I = moment of inertia about the x-axis For a vessel symmetric about the midplane fore to aft the center of gravity lies a (scalar) distance c above the keel. Some scaled values are : 3 3 1 Ship Mass c c=d I'=-IMd2 Let the vessel have a total mass M : M =M;+Me 28 (3.34) (3. 35) (3. 36) (3. 37)

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M i is the mass of the interior contents of the ship which will be assumed to be uniformly distributed with density Pi M e is the mass of the exterior surface of the ship and is described : (3. 38) M d is the mass of the deck M k is the mass of the keel and Mh is the mass of each side of the hull. Let p0 be the density of the water in which the vessel floats Masses will be scaled by the mass of water whose volume equals that of the total volume of the vessel. Thus the scaled mass is: The scaled interior mass is: The scaled exterior mass is: M'= M e e P o V In order for a vessel to float M' < 1 A neutrally buoyant vessel (e g a submarine stationary at constant depth) has M' = 1 If M' > 1 the vessel sinks. 3 3 2 Volume (3. 39) (3 .40) (3.41) The volume is the cross-sectional area of the ship multiplied by the length of the ship The trapezoidal hull transverse area can be viewed in Figure 3 5 29

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d-k d-k k Figure 3.5 Volume of Trapezoidal Hull The volume of the trapezoidal vessel is : = d +k hi 2 The resulting equation for the trapezoidal hull volume is : By substituting equation 3.46 into equation 3 36 the scaled volume of a trapezoidal hull is : = _!_( 1 + 2 d dd 30 (3 .42) (3.43) (3.44) (3.45) (3. 46) (3.47)

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= _!_(1 + k'Yz'l' 2 By letting !!._ = 0 the volume of the triangular hull is : d or, in scaled form : 1 = -ldh 2 V' = _!_l'h' In 2 By letting !!._ = 1 the rectangular hull's is: d vrec = ldh or, in scaled form : V' = h'l' rec Table 3 1 summarizes the results ofthe volume equations Table 3.1 Volumes for Different Transverse Hull Shapes Hull ShaJ>e v V' Trapezo idal + ; ) dhl _!_ (1 + k'Yz'l' 2 Rectangular dhl h'l' Triangular _!_dhl _!_h'l' 2 2 31 (3.48) (3.49) (3. 50) (3. 51) (3. 52)

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3 3 3 Center of Gravity With an understanding of the volumes the centers of gravity of these vessels can be calculated The centers of gravity will remain constant for the vessels as long as their distributions of mass remain the same (i e., no shifting oftheir masses) In this section there is a progression of calculations for different shapes. First, the center of gravity for a right triangle is calculated Then the center of gravity of a more complex triangle is calculated Finally the center of gravity for the different hull shapes is calculated for distributed and non-distributed masses To calculate the center of gravity integrate over the volume of the vessel: JJJ P o { Jdxdyck P ogV (3. 53) To analyze shapes it is easier to have the centroid of a right triangle (Figure 3 6) Figure 3.6 Center of Gravity for a Right Triangle The center of gravity, c, that has a uniform mass per unit area CT, for a right triangle is : (3 54) 32

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The mass per unit area divides out, because it is assumed constant. pq( l ; ) I I ydydz 0 0 p q ( l ; ) 1 = ----,-,,.--II dydz (3. 55) tria n gle I I zdydz 0 0 Integrating equation 3 .55: (3. 56) p( y 2 J qi y-dy =21 /( 2p 2J qp -q2 I 1-___l + y 2 dy 2 0 p p (3. 57) Integrating equation 3 57 : (3. 58) (3. 59) The resulting center of gravity for a right triangle is: 33

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(3.60) This result will be useful below and in section 3 3 3 1 for the center of gravity of the trapezoidal hull with distributed mass Many triangles involve the side of the trapezoid and require special attention to find the center of gravity (Figure 3 7) Figure 3. 7 Triangular Part of Tilted Trapezoid From trigonometry : q = ssin(B) (3. 61) From the Pythagorean Theorem : (3. 62) Substituting equation 3 .61 into equation 3 62 : (3. 63) (3. 64) 34

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As a side note : 1 1 sin 2 (B) 1 sin 2 (B) sin 2 (B) 1 Substituting the result of equation 3 67 into equation 3 64 : Using the quadratic formula : -2ptan() ptan() ptan()p_!__( ) tan B tan 2 () f ) tan \B 35 (3. 65) (3. 66) (3. 67) (3. 68) (3. 69) (3. 70) (3 71)

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-tan()+-1 -tan(B) q = (tan()+ }an()-ta:(ll r -tan()--1 -tan(B) ( tan()+ +)X tan() + )J p tan,e tan,e = 1 1 p tan() + -() tan e 1 1 p tan() tan(e) Exclude the top root to get positive q when 0 tan1e) q= \; p 1 -tan(B )tan() The areaA1 of the triangle T1 is the area (A1 + A2)ofthe larger triangle (I; + T2 ) minus the area A 2 of the smaller triangle 1 1 A1 = q(qtan() + p ) --q2 tan() 2 2 1 =-qp 2 Substituting equation 3 .76 into equation 3 74 : A _. tan(e) 1 -2 1 -tan(B )tan() 36 (3. 72) (3. 73) (3. 74) (3. 75) (3. 76) (3. 77)

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Let c; be the center of gravity of triangle T1 with respect to origin 9'. Let 9" 9' be the vector from origin 9" to 9' origin Let c" 2 be the center of gravity of triangle T 2 with respect to origin 9" Let c" I 2 be the center of gravity of the T1 + T 2 combined triangle with respect to origin 9" Then : (3.78) (3. 79) +(AI AI (3. 80) The vector between origins is : (3. 81) Using the result 3 60 for right triangles, the center of gravity for the combined triangle T1 + T 2 relative to 9" is : --; p )] c I 2 -1 -q 3 The center of gravity for the triangle T2 relative to 9" is: [.!_ q tan()] c" -3 2 -1 q 3 Substituting equations 3.76, 3.81, 3 82, and 3 .83 into equation 3 80 : 37 (3. 82) (3. 83)

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1 w{ qtan() ] ( 1 1 2 ( P)] 1 2 ( -2 0 + 2qp+2q tan 1 -2q tan 1 q q 3 3 -qp 2 Substituting equation 3 7 4 into equation 3 88: tan(B) ( ) p-1 tan( )tan(B) tan p 1 tan(B) 3 1 tan()tan(B)p (3. 84) (3. 85) (3. 86) (3. 87) (3. 88) (3. 89) ;;\ ( 1 tan()tan(B)-tan(B)tan()J p 1 ( 3 .90) -tan(B) 3 1 tan()tan(B) 38

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(12 tan()tan(B)J p 1 c 1 = tan(B) 31-tan()tan(B) (3. 91) This result will be useful in finding the center ofbuoyancy in section 3.4 2 3 3 3 1 Center of Gravity with Distributed Mass Figure 3 8 provides the parameters used to calculate the center of gravity for a uniformly massed vessel. Figure 3.8 Center of Gravity of Trapezoidal Hull of Uniformly Distributed Mass Figure 3 9 provides the parameters used to calculate the center of gravity of a non-uniformly distributed mass of a trapezoidal hull. h = -(d +k) 2 39 (3. 92) (3. 93) (3. 94) (3. 95)

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{ o.] ( 1 k Ih +I I(d -k)h 2 3 2 2 3 2 ') (3. 96) I(d+k)h 2 2 2 2 2 2 3 3 (3. 97) (3. 98) where the result from equation 3 60 has been used for finding c 2 and c3 The height c of the center of g ra v ity above the keel in the trapezoid case is: In scaled form : 1 h c=--(2d+k) 3d + k = h (2d + k ) 3 d + k c' = h'( 2 + k') 3 1 +k' For the triang l e use !!._ = 0 : d 40 (3. 99) (3. 100) (3. 101) (3. 102)

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or, in scaled form: k For the rectangle -= 1 : d or, in scaled form : 2 ccn = -h 3 (3. 103) 3.h' ccn -3 (3. 104) 1 Cree = 2h (3. 105) (3. 106) Table 3 2 summarizes the center of gravity as a length measured from the keel. Both scaled and unsealed values are presented in the table. Table 3.2 Centers of Gravity (Distribute Mass) for Different Transverse Hull Shapes Hull Shape c c' Rectangle _!_h _!_h' 2 2 Trapezoid [2<]h 3 1+k' 3 d Triangle 3.h 3.h' 3 3 41

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3 3 3 2 Center of Gravity with Non-Distributed Mass Some important variables in this section are : m = total mass of vessel m d = mass of deck m k = mass of the bottom m t = mass of left hull m r = mass of right hull c = center of gravity of the vessel c d = center of gravity of deck -ck = center of gravity of bottom c 1 = center of gravit y of left side of hull c = center of gravity of right side of hull fJd = mass per unit length of deck J.lh = mass per unit length of sides J.lk = mass per unit length of bottom =(O,h) c, = -+--(d-k) (k I (I ) h ) 2 2 2 2 m k = f.lk k Figure 3.9 Center of Gravity of Trapezoidal Hull of Non uniformly Distributed Mass m d c d + m kck + m 1 c 1 + m,cr c = ___::__ ___::. __ ---"-----'-m 42 (3. 107)

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k For the rectangle let -= 1 : d 0 43 (3. 110) (3 .111) (3.112)

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For the triangle let !!..._ = 0 : d 3 3.4 Moment of Inertia (3. 113) The moment of inertia of a uniform distribution of mass for all three hull shapes is given below The moment usually restores the ship to an upright position (Gillmer & Johnson 1982) Figure 3 .10 presents the important parameters ofthe moment of inertia of a trapezoidal hull. 0 k d-k z y=-+---2 2 h Figure 3.10 Moment of Inertia of Trapezoidal Hull The ship is assumed to be symmetric part to starboard (a good assumption) and fore and aft about its midplane (ignores change in cross-section at bow and stem) With this degree of symmetry of this ship model roll motions only deal with the upper left element of the matrix in equation 3 24 (i.e., :Lm;(y;2 + z;2 )). 44

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Integrating equation 3.114 : ( k d k z ) h[ 3 ] -+--/ = pi f L z + z 2 y 2 2 h d.z 0 3 d k !..) 2 2 h 2 k d -k z k d -k z 2 h [ ( )3 ( ) ] =pi I 3 2+-2-h z+2 2+-2-h z dz fh [ 2 ( k d k z )( k d k z) k 2 d k z 3 ] = p/0 3 2+-2-h 2+-2-h z+22z +-2-h d.z Using a mathematical software program such as MathCad results in : where =-1 M[k3 +d3 +k2d+kd2 +12h2d+4kh2J 24 k+d M = plhd+k 2 The result for the moment of inertia for the trapezoid is : I= -1 M[d2 +k2 24 k+d 45 (3. 114) (3. 115) (3. 116) (3. 117) (3.118) (3. 119) (3. 120) (3.121)

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The scaled moment of inertia is: 1'=-1 [ 1 + k '2 24 k + d For the triangle let !!.._ = 0 : d For the scaled triangle : I'. = -1 [1 + 12h'2 ] rn 24 k For the rectan g le let -= 1 : d For the scaled rectangle : I' = -1 [2 + 8h'2 ] r ec 24 (3. 122) (3. 123) (3. 124) (3. 125) (3. 126) (3. 127) Table 3 5 summarizes the moments of inertia for uniformly massed hull shape s 46

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Table 3.5 Moments of Inertia of Uniformly Massed Hull Shapes Hull Shape Moment of Inertia Rectangular Md' [ ) ] Triangular Md'[l + )'] Trapezoidal +{+ 3.4 Parameters Derived from Ship Position Relative to Sea Surface Scaled Moment of Inertia 1 [2 +8h'2 ] 24 1 [1 + 12h'2 ] 24 _I [1 +k' +{I+ 24 k+d In this section, equilibrium wate rline and center of buoyancy are described 3.4 1 Equilibrium Waterline Depth Figure 3 .11 shows how the equilibrium water line can be determined k + m o (d-k) h k Figure 3.11 Equilibrium Waterline Depth 47

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The volume ofthe water is needed Using the result from equation 3.46, replacing h with roo and replacing d with k + CVo (d-k): h (3. 128) Archimedes' principle requires that the mass of displaced water equal the mass of vessel. Substituting equation 3 128 into equation 3 129 : ( cv d-kJ P olcvo k+---;;--2=M Dividing both sides ofthe equation by lp0 : (k CV0 d-k J M CVo +-----h 2 lp0 d k 2 M --cv +kcv --= 0 2h 0 0 I P o Dividing each term of equation 3 132 by (d-k): Using the quadratic formula : 2hk ---+ d k 4( hk J2 4 2Mh d -k + p0l(d k) 2 48 (3. 129) (3. 130) (3. 131) (3. 132) (3. 133) (3. 134)

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( hk )2 2Mh d k + p0l(d k) hk =---+ d k (3. 135) Factoring (d-k) from the terms of equation 3 .135, gives the final result for the trapezoidal equilibrium water level : Dividing both sides by d to scale the equations : (hk y + 2Mh (d k) W o P o l d(d k) hk + d Scaling the denominator : -hk+ d Scaling the other variables in the equation : = hk ---+ dd -h'k'+ (h'k'Y 1 -k' 49 (3. 136) (3. 137) (3. 138) (3. 139) (3. 140)

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As a side note : 2_!:!_ = 2 M'Vpo P o d 3 P o d 3 (3. 141) 1 M'p0 (d +k)hl = 2 2 d 3 P o (3.142) = M'(1 + k''y-J'l' (3. 143) Substituting equation 3 143 into equation 3 140 : -h'k' + + M'(1 + k')h'2 (1-k') w 0 1-k' (3. 144) For the scaled triangular hull let k' = 0 : (3. 145) The unsealed expression for the triangular hull is obtained by letting k = 0 in equation 3 138 : For the rectangular hull, let k =din equation 3 133 and solve for w0 : M w --0 P old For the scaled rectangle use M = M'p0ldh in equation 3 147 : 50 (3. 146) (3. 147) (3. 148)

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Dividing through by d: = M'h' (3. 149) Table 3.4 summarizes the equilibrium waterline depth for a variety of hull shapes Table 3.4 Equilibrium Waterline Depths for Different Hull Shapes Hull Shape {1) 0 {1) 0 Rectangle hM h'M' P o V l [-k' +M'(l-k'2 )] Trapezoid d \ d p0V d d Triangle P o V h'JM' 3.4.2 Center of Buoyancy Consider now a vessel that has rolled to angle B relative to the sea surface Figure 3 .12 shows the situation in the ship's frame of reference with the origin 9 at the keel. The shaded region A is the submerged portion The center of buoyancy b is identified as the centroid ofthis region The location of b in ship coordinates is found by constructing A out of a trapezoid A1 (Figure 3 13) plus a triangle A 2 and minus a triangle A3 (Figure 3 14) 51

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9 Figure 3.12 The Actual Area of the Water Figure 3.13 The Area of the Vessel's Waterline 52

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Figure 3.14 The Areas of the Two Triangles Formed by the Waterline and Tilted Vessel Figure 3 15 presents all relevant parameters used in computing the center of buoyancy of the vessel from this construction The vertical bisector of the ship cross section is .9D and W,Wr is the waterline which intersects .9D at point We. Line is the horizontal (in ship coordinates) through We. Note that> represents the angle of the sides of the hull away from the vertical and is a constant of the ship geometry 53

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Figure 3.15 Important Parameters of Ship Motion The important parameters of Figure 3 .15 are as follows The center of gravity of the ship is: (3. 150) The center of buoyancy (the center of gravity of the displaced volume of water) is : (3. 151) (3. 152) -The centroid of A1 is b1, b z is the centroid of A2 and b 3 is the centroid of A3 54

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According to the construction in Figure 3 .15: and where 1 p = -k +wtan() 2 ( ) d-k tan, =--h (3. 153) (3. 154) (3. 155) The center of buoyancy in ship's coordinates relative to the keel involves: --b = A1bt + A2 b2-A3 b 3 A (3. 156) where (3. 157) The submerged area A is found be evaluating the areas A1 A2, and A3 by replacing d with 2p and h with ro in equation 3.45 for the area of trapezoid A1 and by using equation 3 77 with angle for A2 and -for A3 (3. 158) = tan(B) p2 1 tan(B)tan(cp) 2 (3. 159) = tan(B) p2 1 + tan(B )tan(cp) 2 (3. 160) Substitute equations 3 158 3 159 and 3 160 into equation 3 157 : 55

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A-(p+!_k)m+!_ tan(B)p2 _!_ tan(B)p2 2 2 1-tan(B )tan() 2 1 + tan(B )tan() Substitute p from equation 3 .15 7 into equation 3 .161 results in : A __ !_ 4mk +4m2 tan()+ k2 tan(BY tan() -4 ( -1 + tan(B )tan( )XI+ tan(B )tan()) (3. 161) (3.162) The centroid of A 1 is the centroid of a trapezoid (use equation 3 1 01 replacing d with 2p and h with m): (3.163) The vertex 8 2 of triangle A 2 (Figure 3 15) : (3. 164) The centroid of A 2 relative to 8 2 : p (1-2 tan(B )tan(lp )J 3(1-tan(B )tan(lp )) tan(B) (3. 165) The centroid of A2 relative to the keel thus : ( -pJ p (1-2 tan(B )tan(lp ) J m + 3(1-tan(B )tan(lp )) tan(B) (3.166) The vertex -93 of triangle A 3 (Figure 3 .15) is: (3. 167) The centroid of triangle A3 relative to -93: 56

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p (1 + 2 tan(B)tan(


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m cos( B) s(t) :z(t) Figure 3.16 Ship Position Relative to the Ocean Floor From the construction in Figure 3.16 and recalling that the center of gravity of the ship lies a distance c above the keel. Solving for m gives : s = :Z +(m -c)cos(B) s-z (I)=C+-....,.....,-COS(B) (3. 172) (3. 173) Thus the dynamical variable z enters into the model by substituting equation 3.17 4 for min the expression for submerged area A (equation 3 162) and for center of buoyancy b (equation 3 171) 58

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3 5 Modeling Sea Surface Variations The ship will be forced into motion by sea surface variations Normally, waves are thought of in this context so that the ship does not sit in a level sea as assumed in Figure 3 .16. However, the waves that are significant in driving ship motion will have periods comparable to the ship roll period In Chapter 2, the roll period of the crane ship was calculated to be about 12 seconds The period T of deep water ocean waves is related to the wavelength A by (Aubrecht II 1996) : T = wave g where g is the acceleration due to gravity Thus Substituting 9 80 rn/s2 for g and 12 sec forT gives A = 225 meters (3. 174) (3.175) The ships of interest have deck dimensions (approximating the "beam of the ship") : d:::JO meters which is about 13% of the wavelength Thus to a first approximation the sea surface is a level surface that is periodically rising up and down (3. 176) where s0 is a sufficiently large average depth and s1 is a variation small relative to the wavelength : s 1 :::;< 2 meters (3. 177) or 59

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(3. 178) Note that the maximum variation in surface height will be on the order of: 2 ::::::2;rx-d 225 ::::::6% ofd This variation is ignored in the dynamical equations discussed below 3 6 Equations of Motion (3. 179) (3. 180) (3. 181) In this model, ship motion is governed by Newton's Laws as they describe: ( 1) the acceleration z in the true vertical z direction as a result of imbalance between the ship's weight Mg and the buoyancy force F s (2) the angular acceleration B due to the moment of buoyancy around the center of gravity The magnitude of the buoyancy force is given by: where A is the submerged cross-sectional area given in equation where A=_ 1 4mk +4m2 tan()+ k2 tan( BY tan() 4 (-1 + tan(B)tan()X1 + tan(B)tan()) 60 (3. 182) (3. 162)

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() d k tan, =-2h and s z m=c+ f) cos,e The dynamics is then given by : mz = p0A lg -Mg (3. 158) (3. 173) (3.183) The moment arm n of the buoyancy force FB about the center of gravity is found using Figures 3 1 7 and 3 .18. Figure 3.17 Construction to Find Buoyancy Moment 61

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[b)\ -(cb.}tan(B )]cos(B) c-b z b Y -(cb.}tan(B) Figure 3.18 Closer Look at Figure 3.17 n =bY cos(B)-{cb.}sin(B) (3. 184) It is useful to identify the intersection of the true vertical through the center of buoyancy with the ship's vertical through the center of gravity as a point called the metacenter If the distance from the metacenter to the center of gravity is m then : n = msin(B) (3.185) Thus using equation 3 184 : (3. 186) This is called the metacentric height and must be positive for the buoyancy force to exert a righting moment rather than cause the ship to capsize That is m determines the static stability of the ship The roll dynamics involves : (3. 187) 62

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where I is the moment of inertia ofthe ship Subsituting for FB from equation 3 182 and n from equation 3 184 : (3.188) Introducing phenomenological damping terms rand p, our model then arrives at the coupled system of two second-order equations i = p/vtgl A(:z,o)gp(:z,o):i i} = Pogl A(z,Blb1(z,B)cos(B)-(c0 -bz(z,B))sin(B)]-r(z,B)9 I (3.189) (3. 190) Further elaboration of the model will require specifying dependence of the damping coefficients on z and B. Additionally, the effect of inertia of the water surrounding the ship (the added-mass effect) would need to be included 3.7 Summary Modeling of ship motion involves ship geometries and the interaction between the water and the ship To simplify the calculations a simple trapezoidal transverse hull is used with a rectangular side view By taking limits of the bottom of the vessel rectangular and triangular hulls can be examined. Using the geometries, the volumes of the ships the centers of gravity, and the moments of inertia can be calculated With the addition of water, the centers of buoyancy can be calculated. Using all of these parameters, one can understand the coupled dynamical equations for the roll and heave motion ofthe ship 63

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4 Coupled Mathieu Equations The Mathieu equation has been used for a variety of applications from springs and pendulums, to ships A coupled spring and pendulum system is similar to ship motion where the pendulum represents the rolling motion of the ship and the spring represents the heave motion ofthe ship Allievi and Soudack (1990) actually modeled ship roll motion using the Mathieu equation They analyzed naval architectural principles such as the metacentric height to create a differential equation. They also examined the presence of damping and examined Poincare sections to understand the stability and instability regions based on changing parameters To understand a non-autonomous four-dimensional coupled system a progressively more complex series ofMathieu equations was used For each step of complexity the solutions to the differential equations were plotted often with Poincare sections (the x's or points on the graphs) First the equation was undamped with two degrees of freedom By being undamped the parametric instability regions or "tongues" could easily be identified Second the equation was damped and had two degrees of freedom The introduction of damping added realism to the system and aided in the understanding of the influence of damping on instability Third, nonlinearity was added to the Mathieu equation, with two degrees of freedom to explore full nonlinear behavior such as limit cycles and chaos Fourth, two nonlinear equations each with two degrees of freedom were coupled in order to see what happens to the geometrical description of dynamics as the coupling between these two equations was increased MatLab program B.2 was used to create all of the graphs used in this section 4 1 Undamped Linear Mathieu EquationTwo Degrees of Freedom First a linear equation with no damping was examined (equation 4.1) (Anicin Davidovic and Babovic 1993) d2x 2 +[a+ 16qcos(2r)] x = 0 dr 64 (4 .1)

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Values for a were chosen using the Ince-Strutt stability chart (Aniein Davidovic and Babovic, 1993) Figure 4 1 shows a parameter space which shows the stable (outside the marked regions) and unstable (the ll regions) regions . I i . i I 0 2.0 a Figure 4.1 Ince-Strutt Stability Chart There is a series of alternating unstable and stable regions The unstable regions alternate between those regions with resonances drive frequencies that are multiples of twice the natural frequencies (subharmonic response) (Figure 4 2) and those regions with resonances at the natural frequency of the system (Figure 4 3) The instability is noticed by the fact that the trajectory spirals outward from the initial conditions ofx(t) = 0 .05 and x'(t) = 0 65

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0 3 0 2 0.1 0 X: -0 1 -0 2 -0 3 -0.4 -0 4 -0 3 -0 2 -0 1 0 0.1 0 2 0 3 X Figure 4.2 Solutions of Mathieu Linear Equation with a = 1 and q = 0.005 1.5 r-------.-----r----,------,-----,-----.------. 0 5 x' 0 -0.5 -1 -1.5 '--------'-----'----.L..------'------'------'-------' -0.8 -0. 6 -0.4 -0.2 0 0.2 0 4 0.6 X Figure 4.3 Solutions of Mathieu Linear Equation with a = 4 and q = 0.05 66

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The stable region is a circle (Figure 4.4) thus showing quasi-periodicity in the system 0 08 0 06 0 .04 0 02 ,: 0 -0 .02 -0 04 0 06 0 .08 -0 06 -0 04 -0 02 X Figure 4.4 Solutions of Mathieu Linear Equation with a = 2 and q = 0.005 4 2 Damped Linear Mathieu EquationTwo Degrees of Freedom Second damping was added (equation 4 2) to the linear equation (equation 4 .1). The damping coefficient fJwas chosen to be 0 5 for the different a values d2x dx 2 +[a+ 16qcos(2 r)]x+ fJ-= 0 dr dr A damped Mathieu equation can be converted to an equivalent undamped Mathieu equation : /!!_ :X= xe2 or _/!!_ X = xe 2 67 (4.2) (4 3) (4.4)

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dx dX _!!!_ p _!!_ -=-e 2 --xe 2 dt dt 2 (4.5) (4 6) So Dividing through by e -fk: ----#+ a--+ 16qcos(2r) x = 0 d2( /32 J dt 4 (4 8) The damping raised the tapered ends of the unstable regions from the a axis of Figure 4.1 Therefore it takes a higher q value to remain in the unstable region for the same a value The stability can be seen by the fact that the trajectory is spiraling inward from the initial conditions ofx(t) = 0 .05 and x'(t) = 0 (Figure 4.4 ) 68

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0 .01 0 x' 0 01 0 .02 -0.03 -0 0 4 '----------'--------'-------'--------'------_J_----_J_-----'-------' 0 .03 -0.02 0 01 0 0 .01 X 0.02 0 .03 0 .04 Figure 4.5 Solutions of Damped Linear Mathieu Equation with a = 1 and q = 0.0025 4 3 Nonlinear Damped Mathieu EquationTwo Degrees of Freedom 0 .05 Third nonlinearity was added (equation 4 9) to the damped equation (equation 4 2) d2x dx 2 +[a+ 16qcos(2r)]sin(x) + /3= 0 dr dr (4 9) With certain parameters a strange attractor can be observed (Figure 4 6) The existence of a strange attractor implies chaotic activity in the system 69

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0 5 x' 0 -0 5 -1 -1 5 -2 -2 5 -3 -2 -1 0 2 3 X Figure 4.6 Solutions of Damped Nonlinear Mathieu Equation with a=1.469, q=0.1102, and fJ=0.42 4.4 Nonlinear Damped Coupled Mathieu EquationsTwo Degrees of Freedom Each Last two coupled equations were used (equation 4 .10 and equation 4 11) The coupling coefficient is x The initial value for xis 0 05, while the initial value for y is 0.5. d2x dx 2 +[a+ 16qcos(2r)]sin{x )+ p+ x(x-y) = o dr dr (4 10) d2 d ---?+[a+ 16qcos(2r )]sin(y ) + p J[ + z(yx)= 0 dr dr (4 11) For figures 4 7 and 4 .8, a= 1.469, q = 0 1102, and p = 0.42 For figure 4 .8, a = 1.5 q = 0 .11, and P= 0.42 Figure 4 7 shows low coupling of0.05. The two graphs look slightly different. Figure 4 8 shows moderate coupling 0 07 where there appears to be some higher dimensional activity based on the fuzziness of the strange attractor The s-attractor can no longer be seen. High coupling Figure 4 9 has twice as many points as Figures 4 8 and 4 7 to show the detail. With more points another s-curve can be seen 70

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2 5 2 t \ 2 1 5 t 1 5 \ t t t ., t 0 5 \ t ' 0 5 \ \ \ 0 l t t x 0 \ -,., \ : \ \ -0. 5 t -0. 5 t \ t l l -1 -1 \ $ -1. 5 \ 1 5 \ -2 -2 -2. 5 -2. 5 -4 -2 0 2 -4 -2 0 2 4 X y Figure 4.7 Weakly Coupled Mathieu Equations .. 71

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2 5 .-------,----,----,-----, 2 1.5 0 5 x 0 -0. 5 -1 -1.5 -2 'f \ J .. .:: .,l ; .. \. # 't<\ .1 t' .. ;. 2 5 '------'----'----'------' -4 2 0 X 2 4 2 ">-0 1 -2 \ + ... : ;. +t \ ..... ....... + . \'f -4 -2 0 y 2 4 Figure 4.8 Moderately Coupled Mathieu Equations 72

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2 5 .----.--.---.----, + I 2 x 0 .. I . + .... . : ' I t ; \ s \ .. l t 1.5 0 5 -0. 5 1 \ -1.5 -2 -2. 5 L._ ___J_ __ __L_ __ L_ __J -4 -2 0 2 4 X 2 5 2 1 5 0 5 ":., 0 -0.5 -1 -1.5 -2 -2. 5 -4 ... I ] . i .. \ . .. :; l I i t J \ + + -2 + + 0 y Figure 4.9 H ighl y Co upl e d Mat h ie u Equati o ns 4 5 Math ieu Equati o n Conclusi on I \ 2 4 Mathieu equations are versatile They model various dynamic systems including ship motion These equations enable people to handle non-autonomous four-dimensional systems in a relatively simple fashion As a linear undamped system unstable spirals and circles are possible. By adding damping, regions of instability are altered allowing for stable spirals Adding a nonlinear term into the equation allows for chaotic activity such as the s-curve attractor The coupling of Mathieu equations shows that for a set of a, q and /3, there is a coupling coefficient that shows some higher dimensional activity Future research will involve placing the ship equations into this program and seeing how changing parameters demonstrates the dynamics of the system 73

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5 Conclusion This thesis strives to help understand the motion of ships and how to predict ship motion Analysis ofthe Carderock ship roll data recorded on July 19th 1993 began the whole progression of thought. The most important finding is the extraction of a correlation time from the data With a lack of insight into the ship motion system a different approach was taken This new approach began with using the basic principles of naval architecture to form equations. Next I examined coupled equations from the dynamic system ofMathieu By analyzing the well-known Mathieu equation and coupling it with itself, one can understand the dynamics that the ship model equations might exhibit. Despite all of this work there are still some problems Trying to tease apart the essence of a system is very difficult Actual data is limited by the technology used to collect it as well as the researchers and the e x periment design The placement of the sensors on the ships is important so that only the degree of freedom is recorded The designers need to ensure that the instruments work properly and the data is taken at regular intervals The e x periment design needs to ensure that the proper instruments are used and that the instruments are used in such a way to get the proper data To understand the results of the output one needs to look critically at how one analyzed the data As with any scientific endeavor assumptions were made which directly and/or indirectly tainted the data CDA Professional Version was heavily used without a stri ct validation of its assumptions CD A's many tools along with other nonlinear tools were not used (such as Liaponov exponent) The data was biased by calculations based on strict 2 048 data point files in a contiguous fashion Although discussed the idea of a moving window and/or using a different window size was not implemented CDA is limited to 32 000 or fewer data points ; MatLab : Student Version 5 0 does not accept arrays with more than 16, 000 elements Biases come from both the analysis of actual ship data and the formation of models Models need validation through experimentation and must mimic real-life behavior. Without experimental evidence validating the model, modeled equations are theoretical or just mathematical exercises A suggested experiment is as follows : Construct a physical ship model with a scale of roughly 1:24 Place the ship in a tank. Raise and lower the fluid in the tank to a frequency equal to the model's natural roll frequency. This tests the idea that heave and roll are coupled as in the 74

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demonstrated Mathieu's equations Experimentation and noise help to test mathematical equations Mathematical models simplify the complex interactions observed by the instruments used in experiments The instruments are influenced by human actions such as sailors moving cargo around, or uncontrollable events such as a change in the climate The circuitry in the instruments often adds noise to the data collected These different components come together as the time series, full of noise that complicates the analysis The mathematical equations have the principle elements of the system without the perturbations of instrumentation and events For future study it would be helpful to add noise to the system and run through the same tests as in chapter two (i e., mean variance kurtosis power spectrum etc .... ) Another future study would be to understand the ship model equations with the same MatLab program that was used for the Mathieu equations in chapter four. Another useful project from chapter four is to investigate the "fuzzy" Poincare section results from the particular range of parameters for the coupled nonlinear Mathieu equations There is some higher dimensional activity illustrated by the plots that should be analyzed Different parameters will need to be used to understand the system of ship motions. Overall this research is a significant beginning to understand the complex nature of ship motion 75

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Appendix A Plots of Different Days and of Heave Here are some plots oftime series mean, variance skewness kurtosis dominant frequency correlation time and Hurst exponent of roll from the July 18, July 17, and July 15 respectively For July 18, the sampling was taken over 11 hours 22 minutes and 40 seconds. For July 17, the sampling was taken over 3 hours 24 minutes and 48 seconds For July 15, the sampling was taken over 7 hours 57 minutes and 52 seconds Next are the plots of time series, mean variance skewness, kurtosis dominant frequency correlation time and Hurst exponent of heave from July 19, July 18, July 17, and July 15 respectively Dominant frequencies and correlation times were left out if CD A gave out a value of zero or a value that was extremely high 5 4 3 u;-cl) 2 cl) 0! cl) -.::::> C) 1 0::: 0 -1 -2 0 20 4 0 60 80 100 T i m e (0. 5 kilosec) Figure A.l Time Series of Roll of July 18 76

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3 2 5 t/) 2 ..... Cl -----r--r r -----r ---;----r r---................... ............... '0 c co 1 5 0 5 0 5 1 0 15 20 25 30 35 40 45 Time (kilosec) Figure A.2 Mean of Roll of July 18 0 9 0 .8 Q) Q) 0, 0 6 Q) Q) (.) 0 4 ;:: !I] > 0 3 0 2 0.1 ................ ;. .................. i ................... ................... l. ................... j l ..... ..... ..... f .... . ... J. ________ ................. f-------1 -.. r:::r:r -. ................. r ................ T ................ r ..... ............ r ................ T ............ 1 ................. T .............. .... I" ............. .. .. ........ ....... r ............... r ............. T ............... ']"........ .. T ........... T ............... T ................ T .............. .. r r r -. t_-. r... -.-.:. --.-.: . .. -.-.:.. . : .. -.. r :r : -.-. : : ; ................. i .............. """ '"'""""'['"" ........ ' """"""(""'"'"'"'''' ''"' """"' l""""'""""'f'""'"""'" .... ''l"'""'"""""t''' '"""""''!'"""""" + ................... ; ................... .. .. .............. 0 5 1 0 15 20 25 30 35 40 45 Time (kilosec) Figure A.3 Variance of Roll of July 18 77

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0 5 0 !Z 2! 3:: 0 5 (/) -1 -1. 5 : 0 5 10 15 20 25 30 35 40 45 Tim e (kilosec) Figure A.4 Skewness of Roll of July 18 0 5 10 15 20 25 30 35 40 45 Time (kilosec) Figure A.S Kurtosis of Roll of July 18 78

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0.05 O.o:tS :g O.C ;... g o.m ., ::I g" 0.02.5 ... '""' c 0.02 8 0.01 0 .005 0 Ture(sec) Figure A.6 Dominant Frequency of Roll of July 18 The dominant frequency of r oll ofJuly 18 is 0 078 0 00 3 Hz. "' ., .,.. e 0 oS ..... 10 9 8 7 6 5 4 3 2 0 Tune (sec) Figure A. 7 Correlation Time of Roll of July 18 T h e c orrela t ion time ofroll o f Jul y 18 is 2 .55 0 07 5 se cond s 79

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0 5 0.45 1 \ r'l. 'C 0.4 I'll ""' 0 11 "' !:! 0.35 :I: I & 0 3 1 .. 0.25 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 T ime (sec) Figure A .8 Hurst Ex ponent of Roll of Jul y 18 T h e Hurst ex p onent of roll of July 1 8 is 0 38 0 .03. 4 3 2 c.n Q) 1 0) Q) -o 0 0 0::: -1 -2 -3 0 5 10 15 20 25 Time (0 5 kilosec) F igure A 9 Time Serie s of Roll 17 80

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1 9 . rTrT: 1 s ......... ................. l ............................. L ........... . .............. l ..... ....................... L ........................... t ..................... 1 7 .......................... L ................... .... .I.. ....................... ..l. ........................... l ........................... .l. ..................... I 1 6 ................... j .......... L ] .. ........ CD 1 5 t''''''''' "''' ''''''''""''t''"'"'''''''''' ''""''''1"'''''''''''''''''''' ''''' oooooooooooooooooooooooo ooo(ooooooo oooo o o ooooooooo c: : : : ::l 1 4 :: J: :: r.. :r : :.1:::: ,-:: 1 0 2 4 6 8 10 12 Time (kilosec) Figure A.10 Mean of Roll of July 17 0 6 (1) '0 -(1) g 0.3 ca c ca > 0.2 0 1 r -. ........................... i ................ ... .... . ................. ..;. ... .. ... ... ................ .;. ............................... ... .. -------1-0 ---; --1---1------0 2 4 6 8 10 12 Time (kilosec) Figure A.ll Variance of Roll of July 17 81

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1/) 1/) !!! 3: Q) .> (/) 1/) u; 0 t:: :::J 0 5 0 -0.5 -1 -1.5 -2 -2. 5 -3 r --r : l r ............ r:r r r r rr -rt TT r 0 2 4 6 8 10 12 Time ( kilose c) Figure A.12 Skewness of Roll of July 17 18 16 14 12 :: 1 :J::: 1::::: ::: 10 8 ......................... l .................. .... .................... .. r r t oooo .. ooooo oooooooo o oooooooooooooooo 6 4 ----t r l l -+ t l r r 2 r t -r' 0 0 2 4 6 8 10 12 Time (ki losec ) Figure A.13 Kurtosis of Roll of July 17 82

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0.04 < 0 035 O.oJ W: i 0.025 x ;; ... 0.02 i :. 0 015 8 @:; 0 .01 :; .:?.!. m I Y. "'-' 'x I ;.;::.. % 0 }. :-:=::: " 0 005 "' ::!1 :-: .. ;:;:,. =-=-. lli 512 1536 2560 3584 4608 5632 6656 768 0 8 704 9728 10752 11776 Time (sec) Figure A.14 Dominant Frequency of Roll of July 17 The dominant frequency of roll of July 17 is 0 078 0 000 Hz. 512 1536 2560 3584 4608 5632 6656 7680 8704 9728 10752 11776 Tnne (sec) Figure A.15 Correlation Time of Roll of July 17 The correlation time of roll of July 17 is 2 1 0 9 seconds 83

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0 .43 0 .41 0 39 0 37 l 0 35 0 33 ::r: 0 .31 0 29 0 .27 0 .25 0 2000 4000 6000 Time (sec) 8000 1-mw data smooth data 10000 Figure A.16 Hurst Exponent ofRoll of July 17 12000 For roll of July 17, the Hurst exponent value for the raw data was 0 35 0 .05 and 0.37 0 .05 for the smoothed data 2 1 5 1 u:;-
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0 8 .------,.-------r----.----.-----,--------, 0 75 ............................................. """"""""'""'l""""""""""""""'""'t""" '""""" ""'"'""""+""'"""""""""' Vl 0 1 -Q) g 0 08 ro c 0 06 0.04 0 02 + .......................... ............................... . ; ................................. ( ............................................ t---l-----1-; .................................................... r ................................................................ r .......................................................... .. -_ : ; j :: --; : : :::L : f::: : : ............................. ............................ ... .; ................................. ................................. 1 ............... ! '................................. j ..................... (.. ..... . . '; i ; l i r 0 5 10 15 20 25 30 Time (kilosec) Figure A.19 Variance of Roll of July 15 8 5

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0 1 0 .05 0 !ll -0 05 (/) -0 1 -0 .15 -0 2 3 5 3.4 3 3 3.2 (/) 3 1 .(i) 0 t :I 3 2 9 2.8 2.7 2 6 0 5 10 15 20 25 30 Time (kilosec) Figure A.20 Skewness of Roll of July 15 5 10 15 20 25 30 Time (kilosec) Figure A.21 Kurtosis of Roll of July 15 86

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0 1 0.00 ;.., g j <()"" (sex::) Figure A.22 Dominant Frequency of Roll of July 15 The dominant frequency of roll of July 15 was 0 1 0 2 Hz for both raw and smooth data 4.5 4 3 5 "' 3 "" c 0 2.5 "' e 2 0 1.5 I 0.5 0 ,_,{" "''"' t-.'"'"' ,:t,o,b p,'>J"' ,.,:;fo v Tone (sec) I !ill raw data s mooth d ata I Figure A.23 Correlation Time for Roll of July 15 The correlation time for roll of July 15 was 1 05 0 .15 seconds for the raw data and 1.9 0 1 seconds for the smoothed data 87

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0 .42 "' '-il' j, 0.37 t:
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0 37 0 365 0 36 0 .355 (/) :s 0 35 c: !0 Q) 0 345 0 .34 0 335 0.33 0 l l ; l : .. .... -_i l : : ; : : ................................................. rr ...................... ; ......................... 1-i : I I ; 1 lt .. .. t .. .............. ........................... : ........................ 10 20 30 40 50 60 70 Time (kilosec) Figure A.26 Mean of Heave of July 19 0 .04 r----,----,...----.---..,..------,----,----, 0 035 0 .03 0 025 N $E. Q) 0 .02 0 c: (1J 0 015 > 0 .01 0.005 : : : : :! r ii ft .. .. :: .................. 10 20 30 40 50 60 70 Time (kilosec) Figure A.27 Variance of Heave of July 19 89

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0 15 r------,----.---------r-----.----..------.-----, 0 05 0 1 ......................... f ( . ......................... i............... ; ........................... ; ............. ............. ( ........................ :, _ _ I I .... rr ... ........ ... : i r ... (/) gj c: Q) -0 05 0 -0 1 0 15 0 2 .__ __ __._ ___ ......._ __ ___. ___ _,_ ___ ,__ __ __._ __ ___, 0 10 20 30 40 50 60 70 Time (kilosec) Figure A.28 Skewness of Heave of July 19 5 5 5 4 5 .!!1 (/) 0 4 t:: :J ::.::: 3 5 . : : ; :.: ::1.1 '! :::: 1.1 -.................... :t r ................ :. r ........... ......................... ......... T : .. r -:-10 20 30 40 50 60 70 Time (k i losec) Figure A.29 Kurtosis of Heave of July 19 90

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?Jo,rv rv"rv 1\.,rv ':b,rv :-,rv \::)1\.rv Time (sec) Figure A.30 Dominant Frequency of Heave of July 19 The dominant frequency ofheave of July 19 is 0 .15 0 .05 Hz. ,-..._ "' -e 1::: 0 u Q) "' "' 0 '-' 0 C1:l r-3 9 3.4 2 9 2.4 1.9 0 10000 20000 30000 40000 50000 60000 70000 Time (sec) Figure A.31 Correlation Time of Heave of July 19 The correlation time of heave ofJuly 19 is 1.2 0 2 seconds 91

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c .., t:: 0 0. ><: = :r: 0.33 0.31 0.29 0.27 0 .25 0 .23 0.21 0.19 0 IO
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0 355 0 35 ................ ) ...................... t .................. .. cq; -0 345 .... : .... ................. 1. .................... 1 .................. c: 0 34 ................ .................. + 0 335 .................... .................... .............. .. 0 33 0 5 10 15 20 25 30 35 40 45 Time (kilosec) Figure A.34 Mean of Heave of July 18 9 3

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0 025 0 02 ..._ 0 015 N Q) 0 c: 0.01 C'll > ............... f ...................... ........... f ...................... j ..... .............. .................. .L. ................... ; ............................................. J ................... ; ..................... r + --' ___ J__ -+ -+ ---,----0 005 ............. .. ..... ............ ...... 5 10 15 20 25 30 35 40 45 Time (kilosec) Figure A.35 Variance of Heave of July 18 0 j -1 -2 rn rn -3 Q) r:: Q) -4 .:.!. en -5 -6 -7 l ............... T ............... ; r r .. :r rii .. r ir r 8 0 5 10 15 20 25 30 35 40 45 Time (kilosec) Figure A.36 Skewness of Heave of July 18 94

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(/) iii 0 t :J ::.::: 90 so ...................................... f ................... r ................. ................... l ..................... j .......................................... f"' .............. 70 ...... """!"""""""''; ......... ......... 1... ............... .L ................. L. .................. i ..................... l ................... l.. ............... : ;:: l_l: :... ::! :::: lr:_ ::: : 60 50 I I .................................... T ............. T .... .. ,,,,_ ,,,_ ............... t ................... r ................. ; ::r :-<: ; : : : 40 30 20 ; : : : : ; 10 .................. .; .. ........................ .. """""1'""'""'""""'''" ................ ; ................... ; .................... : ................ .. 0 0 0 1 N -;: 0.08 g g. 0 .06 s 0.04 E 8 0 .02 0 5 10 15 20 25 30 35 40 Time (kilosec ) Figure A .37 Kurtosis of Heave of July 18 Tune (sec ) Figure A.38 Dominant Frequency of Heave of July 18 45 The dominant frequency ofheave of July 18 is 0 011 0 .03 Hz 95

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3.5 3 2 5 2 0 1.5 0.5 0 Tune (sec) Figure A.39 Correlation Time of Heave of July 18 The correlation time of heave ofJuly 18 is 1.4 0 2 seconds 0.35 0 3 11\ IJII' 'C !! 0.25 I 1 .... 0 !t I t 'I "" 0.2 0.15 0.1 0 5000 10000 15000 20000 25000 30000 35000 40000 Time (sec) Figure A.40 Hurst Exponent of Heave of July 18 The Hurst exponent of heave ofJuly 18 is 0 .25 0 .05. 96 45000

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1 4 1.2 1 0 8 ;:::::::;< 0 6 u Q) (f) b 0.4 Q) {U 0 2 Q) ::r::: 0 -0.2 -0.4 -0. 6 0 5 1 0 1 5 2 0 25 Time ( 0 5 kilosec) Figure A.41 Time Series of Heave of July 17 0 375 : 1 ___ ,_ --r -r ------+-----r0 .37 .,.,.,.,.,. "roooooooooooooooooooooooo.ooooToooooooooooooooooooooooooooooo -E-0.365 c ro Q) ::!: i .. ; ...... ........ ........... ............................... l ................................. l .............................. : : 0 36 0 355 ........................... ... .. ; .................. . ........... : ........ . ..... ................. : ... . . .... ... 0 2 4 6 8 10 12 Time (kilosec) Figure A.42 Mean of Heave of July 17 97

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X 163 7.-------.-------.--------.-------.-------.-------. 6.5 ............................... 1 ................................ 6 .r (/) 5.5 ........ Q) (.) c: ro c ro > (/) (/) Q) c: Q) .:,:. (/) 5 4 5 4 l l j : --r---j ---i -0 2 4 6 8 10 12 Time (kilosec) Figure A.43 Variance of Heave of July 17 0.4 0 2 0 -0 2 -0 4 -0.6 -0 8 1 1 2 1 4 0 2 4 6 8 10 12 Time (k i losec ) Figure A.44 Skewness of Heave of July 17 98

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35 30 25 20 (/) B ::; 15 10 5 0 0 0 1 0 08 g g" 0 06 s 0 0 4 0 02 0
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3 2 3 2.8 r--r--f:::;; = = I I I I 8 ; 2.6 e 2.4 1., I 22 2 =a l= ::a 512 1536 2560 3584 4608 5632 665 6 7680 8704 972 8 10752 11TI6 T ime ( sec) I raw data 0 s mooth da!B I Figure A.47 Correlation Time of Heave of July 17 The correlation time for heave of July 17 was 1 3 0 2 seconds for the raw data and 1.4 0 2 seconds for the smoothed data 0 35 0 .33 0 3 1 0.29 c 0 .27 :I: 0 .25 0 .23 0 .21 0 .19 0 2000 4000 6000 8000 10000 12000 Time (se c ) 1-+raw data-e-smooth data j Figure A.48 Hurst Exponent of Heave of July 17 For the Hurst exponent of heave of July 17, the value was for the raw data 0 27 0 .03 and 0 .31 0 .03 for the smooth data. 100

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0 8 0 6 N' 0.4 < (_) 0 33 ...... .............. !---J-r --j-;0.325 .......... .. 5 10 15 20 25 30 Time (kilosec) F igure A.50 M ean of Heave of Jul y 15 101

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0.018 !!! .......................... --r : : 0.016 :i ............................ r ................................ i ""!" 0 014 : : : ---:i : : :: ; : ---r: ::: : " l 0 006 0 004 ...................... .......... i .................................. i .................................. i .................................................................. r ........................... .. ............................ T ............................ : ................................ ,...... .. ......... :-............................ T ............................ .. 0 002 ............................... r ........................... .. OL_ ______ L_ ______ L_ ______ L_ ______ 0 5 10 15 20 25 Time (kilosec) Figure A.51 Variance of Heave of July 15 0 15 . . 0 1 .. + "''"l'' ' '''''''''''"''''"''''''''1'"""""'"''''''''''""' 0.05 ...... .. . ........ ............. .. .................... 'l' .. .... ........ ....... .... lZ 0 Q) c Q) en -a.o5 -0 1 ................................. -0 15 .............................. ; ............................... ;........................... ;............................... ; ............................... ;............................ -0.2 0 5 10 15 Time (kilosec) 20 Figure A.52 Skewness of Heave of July 15 102 25 30 30

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3 5 3 0 1 N 0 .08 ii g. s 0.04 E 80.Q2 0 5 10 15 Time (kilosec) 20 Figure A.53 Kurtosis of Heave of July 15 25 .,,rv )< !T>
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';;' -g 3 25 8 2 l! "' e u 0 .. E-0.5 0 ..,< """"'+> ..,-..,b 'tJ oJ''tJ ,_,g,b o,t}' f'f'"' A .._,"Jb "' bl 'd'" '+> ....... '-" .._,q; ,..., '),' Time (sec) Figure A.55 Correlation Time of Heave of July 15 The correlation time ofheave of July 15 is 1.44 0 06 seconds 0 3 3 0 .31 0.29 il 0 27 "'" ] 0 2 5 0 23 0 2 1 0 1 9 0 5000 10000 1 5000 Time (sec) -...... 20000 r-.. 2 5000 Figure A.56 Hurst Exponent of Heave of July 15 The Hurst exponent of heave of July 15 is 0 .23 0 04 104 30000

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0.0006 +++++-+1-H-H+t-H-+++-+1-H-H-H.t-++++++-+1-H-H+t-++++++1-H-H-H++-H-+++++-+1-Hf+-1 0 0005 +++++-+1-HI-H+t-H-++++1-HI-H--bllll-++++++-+1-H-H+t-++++++1-HH-1-H++-H-+++++-+1-Hf-t--1 0 0004 +++++-+1-H+t++-H-++++f-H-H--Jijjll-+++++1-HH-1-H+t-++++++1-HH-1++++-H-+++++-+1-Hf-t--1 m N M File Figure A.57 CDA File with Six Point Gaussian Smoothing 105

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0.1 0 .05 0 -0.05 (/) -0.1 (/) (1) -0.1 5 (1) ,.;..: C/J -0.2 -0.25 -0.3 -0.35 -0.4 Tirre (sec) Figure A.58 Skew Derivative for Raw Roll of July 19 0 .15 0.1 0.05 ... (/) (/) I"' !"""' (1) 0 (1) ,.;..: C/J .."- .-l lit UI'JI 1/\!1 I T w I'P" -0.05 ... IAII !!!t u -0. 1 II! -0.15 Tirre (sec) Figure A.59 Smoothed Skew Derivative for Roll of July 19 106

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5.3 .----------------------, 4 8 c:: 4 3 c:: <1) 3 8 Cl 3.3 2 8 0 10000 20000 30000 40000 50000 60000 70000 time (sec) Figure A.60 Correlation Dimension of Raw Roll Data of July 19 4 8 .-----------------------, 4.6 4.4 c:: 4.2 0 c;; 4 5 E 3.8 a 3.6 3.4 3.2 o 1 aro 2(ll) 3(ID) 4(XXX) 5(ID) ff..1XX'J 7aro tirre (sec) Figure A.61 Correlation Dimension of Smoothed Roll Data of July 19 107

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Appendix B MatLab Programs In this section, MatLab programs used in chapters two, four, and five are discussed Appendix B.1 Chapter 2 % This program calculates the mean skewness variance, and kurtosis for the user % chosen time series. The time series provided by the naval crane ship % have different numbers of data points dependin g on the day So, this % change is taken into account in both the graphs and the calculations clear all % Prompt the user for the appropriate time series file = menu('Choose a time series :', 'July 19 : roll' 'July 19 : heave', 'July 18 : roll' July 18 : heave', 'July 17 : roll' 'July 17 : heave' 'July 15 : roll' 'July 15 : heave') ; %Change the end point ofthe calculation and the titles ofthe plots % for the appropriate da y % July 19 time series if file == 1 I file == 2 num_end = 68; file num = 19; % July 18 time series elseif file = 3 I file = 4 num end = 40 ; file num = 18; % July 1 7 time series elseif file = 5 I file = 6 num end = 12; file num = 17; % July 15 time series 108

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else num_end = 28 ; file num = 15; end % Change the plot titles and the time series loading to the appropriate type oftime senes %(i.e., roll or heave) if mod(file,2) = 0 title_nam ='Heave'; file nam = ['hv' int2str(file num) _']; mean_unit = '(ft I s 2)' ; var unit = '(ft 2 I s 4)' ; else title nam ='Roll' ; file nam =['r' int2str(file num) _']; mean_unit = '(degrees)' ; var_unit = '(degrees"2)'; end % Initialize the matrices used for the entire program matrix_ kurt = zeros(num end, 1 ) ; matrix skew = zeros( num end 1 ) ; matrix_ vari = zeros( num end 1); matrix_ mean = zeros(num end, 1 ); % Increment for all of the data files fori= 1 : num end; %Initialize the variables that are used each loop kurtsum = 0 ; skewsum = 0 ; varsum = 0 ; %Load each file load([ file nam int2str(i) '. dat']) ; % Initialize a variable with the data information eval(['cda file = file nam int2str(i) ; ']) ; %The variable for the mean of the file 109

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xbar = mean(cda file) ; % Increment for each data point for x = 1 : 2048 % Calculate the kurtosis kurt = ( cda file( x ) xbar)/\4 ; kurtsum = kurtsum + kurt ; % Calculate the skewness skew = (cda_file( x)xbarY'3 ; skewsum = skewsum + skew ; % Calculate the variance var = (cda_file( x)xbarY'2 ; varsum = varsum + var ; end % Calculate the standard deviation of each file deviation = std(cda file) ; % Record the current file's kurtosis information matrix kurt(i 1) = (kurtsum I 2048) I deviation/\4 ; % Record the current file's mean information matrix mean(i 1) = x bar ; % Record the current file's skewness information matrix skew(i 1) = (skewsum I 2048) I deviation/\3 ; % Record the current file's v ariance information matrix vari(i ,1) = (varsum) I (2047) ; % Clear the data variable clear cda file ; end % Index for plotting time time_end = 512 : 1024 : num end*1024 ; time end = time end/10/\3 ; 110

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% The graph for kurtosis figure (1) plot (time end, matrix_kurt, '+-b') ; grid on title (['Graph of Kurtosis ofJuly file_ num ' title nam ]) ; xlabel ('Time (kilosec)') ; ylabel ('Kurtosis') ; % The graph for skewness figure (2) plot (time_end matrix skew '+-b') ; grid on title (['Graph of Skewness ofJuly' file num '' title_nam]) ; x label ('Time (kilosec)') ; ylabel ('Skewness') ; % The graph for variance figure (3) plot (time end matrix vari, '+-b'); grid on title (['Graph of Variance of July file num ' title nam ]) ; xlabel ('Time (kilosec)') ; ylabel (['Variance' var unit]) ; % The graph of the mean figure (4) plot (time end matrix mean + -b') ; title (['Graph ofMean of July' file_num' 'title_ nam]) ; grid on xlabel ('Time (kilosec)') ; ylabel (['Mean' mean unit]) ; Appendix B .2 Chapter 4 The functions used in the program follow the code of the program %This program plots a user inputed amount ofMatheiu equation graphs The user chooses the a value for the whole set of %equations. For each individual graph the user inputs a q value and the type of graph The user may choose a linear 111

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% with no damping graph a damped linear graph, a nonlinear damped graph and a coupled, nonlinear, damped graph %For the differential equations the initial values were 0 05 for non-prime variables and 0 prime variabes % ode45 was the solver used for the equations The user may choose the damping coefficients for the damped equations % and the coupling coeffiecient for the coupled equations clear all quit = 'n' ; fintime= 1 0000 ; strtim = 0 ; endtim = fintime ; yd1 = 0 05 ; y d2 = 0 ; tm = O : pi : fintime ; timestp=[O fintime ] ; while quit == 'n' clc % initializing parameters a = input('Enter the value for a : ') ; % Prompt for the number of plots To allow for long titles on the plots, % users are not allowed to have more than one column of graphs numans = input('Enter the number of plots including coupled : '); % initialize plot counter numbr = 1 ; % create plots until the total number of g raphs is reached while numbr <= numans clear ym clear y clear t % prompt for type of graph 112

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mansw = menu('Choose an equation','Linear' 'Damped' 'Damped Nonlinear' 'Coupled') ; % clear the screen clc %Prompt for parameter value q = input('Please enter the q value for the plot : ') ; %User chose the linear Mathieu case switch mansw case 1 % Calulate the linear case with no damping [t ,y] = ode45('Am' timestp [0.05 0], [], a q); i = 1 ; while t(i) < fintime if (y(i 1) < 0) y(i 1)= rem(y(i 1), -pi) ; else y(i 1 ) = rem(y(i 1 ) pi) ; end i =i+1; end ym = interp 1 (t y tm) ; % Plot the linear case with no damping and label the graph subplot(numans 1 numbr) plot(y( : 1), y(:,2) 'm' ym(:, 1), ym( : 2) 'b*') ; set (gca 'FontName' 'Times New Roman' 'FontSize' 10) ; title (['Free Section with a = num2str(a) and q = nurn2str(q)]) ; xlabel ('x' 'FontName' 'Times New Roman' 'FontSize' 10) ; Y label ('x'" 'FontName' 'Times New Roman' 'FontSize' 10) ' ' %User chose the damped Mathieu case case 2 % Prompt for the damping coefficient beta = input('Please enter damping coefficient : '); 113

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% Calculate the differential equation for giv en parameters [t ,y] = ode45('Am1', timestp [0.05 0] [], a q beta) ; i = 1 ; while t( i ) < fintime if (y(i 1) < 0) y(i 1 )=rem(y(i 1 ) -pi) ; else y(i 1 ) = rem(y(i 1 ) pi) ; end i =i+1; end ym = interp1(t y tm) ; % Plot the damped linear case and label the graph subplot(numans 1 numbr) plot(y(:, 1), y(:,2) 'm' ym(:, 1), ym(:,2), 'bx ) ; title (['Damped Section with a=', nurn2str(a) ', q = nurn2str(q) and \beta = num2str(beta)]) ; x label ( 'x'); ylabel (' x "') ; %User chose Nonlinear Mathieu case with dampening case 3 % Prompt for the damping coefficient beta = input('Please enter a damping coefficient: '); % Calculate the nonlinear differential equation with the selected parameters [t ,y] = ode45('Am2' timestp, [yd 1 yd2] [], a q beta) ; i = l ; while t(i) < fintime if (y(i 1) < 0) y(i 1 ) = rem(y(i 1 ) -pi) ; else y(i 1 ) = rem(y(i 1 ) pi) ; end 114

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i = i+ 1 ; end ym=interp 1(t y tm) ; % Plot the Nonlinear equation and label the graph subplot(numans 1 numbr), plot(ym( : 1), ym(:,2) 'b.'); title (['Damped and Nonlinear with a =', num2str(a) q = ', num2str(q) and \beta =', num2str(beta)]) ; xlabel ( 'x ') ; ylabel (' x "') ; %User chose the coupled equations otherwise % Prompt for the damping coefficient beta = input('Please enter a damping coefficient : ') ; % Prompt for the coupling coefficient gamma = input('Please enter a coupling coefficient: '); % Calculate the coupled differential equations with selected parameters [t ,y] = ode45('Am3' timestp [0.05 0 0 .05 0], [], a q beta gamma) ; i =1; while t(i) < fintime if(y(i, 1) < 0) y(i 1 ) = rem(y(i 1 ) -pi) ; else y(i 1 ) = rem(y(i 1 ) pi) ; end i = i+ 1 ; end i = 1 ; w hile t(i ) < fintime if (y(i 3) < 0) y(i 3 ) = rem(y(i 1 ) -pi) ; else y(i 3) = rem(y(i 3) pi) ; 115

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end i=i+ 1 ; end ym = interp1(t,y tm) ; % Plot the first coupled plot and label the graph subplot(numans, 1, numbr) plot(ym( : 1), ym( : 2) 'bx'); title (['Coupled System with a=', nurn2str(a) ', q = ', nurn2str(q), and \beta=', num2str(beta)]); xlabel ('x') ; ylabel ('x"') ; % Increment for the second coupled plot numbr = numbr + 1 ; % Plot the second coupled plot and label the graph subplot(numans, 1 numbr), plot (ym(:,3), ym( : 4), 'bx') ; title (['Dampened and Nonlinear and coupling = num2str(gamma)]); xlabel ('y') ; ylabel ('y"') ; end % end of switch statement % Increment the number of plots numbr = numbr + 1; end % end of while statement % present the graphs figure(1) % let the figure window fill the whole screen and clear the screen set(gcf,'Position' [-2 2 644, 442]) ; clc quit = input ('Do you want to quit ((y)es or (n)o)? ', 's') ; end Here are the functions used in the program above : 116

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%Am. m function ode = Fm(t y unused a q) % Original Mathieu's equation ode = [y(2) ;-y(l)*(a+ 16*q cos(2*t))]; % Aml.m function ode = Fm(t y unused a, q beta) % adding damping beta with Am ode = [y(2) ; -y(l )*(a+ 16*q*cos(2*t))-beta*y(2)] ; % Am2 m function ode = Fm(t y unused a q beta) % adding damping and nonlinearity beta with Am ode = [y(2) ; -sin(y(l ))*(a + 16*q*cos(2*t))-beta*y(2)] ; % Am3. m function ode = Fm(t y unused a q beta gamma) % adding damping ofbeta, nonlinearity and coupling gamma with Aml ode = [y(2) ; -sin(y(l ))*(a + 16*q*cos(2*t))-beta*y(2)-gamma*(y(l )-y(3)) ; y( 4) ; sin(y(3))*(a + 16*q *cos(2 t))-beta *y( 4)g amma *(y(3)-y( 1 ))] ; 117

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Appendix C Glossary of Naval Architecture All terms are from Tupper (1996), except for keel. Amidships -This is the vertical transverse section of the ship that has the largest section of area of the ship This is located in the midpoint of the ship. Beam-A term for the ship's greatest width in a transverse horizontal direction, often quoted at amidships Center of Buoyancy -Where the centroid of the displaced volume of water is located through which all buoyancy forces act. Center of Flotation-Where the centroid of the waterplane is located Center of Gravity-Where the centroid of the ship is located, through which all gravitational forces act. Displacement-The weight of water displaced by a floating ship, which is equal to the ship's weight. Draught -This term is the distance between the keel and the waterline Half Breadth Plan -This view of the ship shows how the waterlines are groupe d This view is a series ofz-x slices ofthe ship Heave -This degree of freedom is the vertical translation. Heave is also the movement along the z-axis Keel -This part of the ship is the lowest and most central part ofthe bottom of the vessel that runs from the fore to the aft ofthe vessel (Watson & Watson, 1991) Metacenter -The point where the direction of buoyancy from the center ofbuoyancy intersects with the original noninclined vertical line of the vessel. 118

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Metacentric Height-This is the distance from the center of gravity to the metacenter For small inclinations for a given center of gravity, the metacenter can be considered fixed in position Metacentric height is a good indicator of the stability of the ship Midships -See Amidships. Pitch -This degree of freedom is the rotation about a transverse axis Pitch is also the rotation about the y-axis Profile Plan -This view of the ship is a side elevation of a ship's form Righting Lever -This term is the distance between the lines of action of the gravity and buoyancy forces This is numerically equal to m*sin(8), where m is the metacentric height and 8 is the angle of roll. Roll -This degree of freedom is the rotation about a fore and aft axis Roll is also the rotation about the x-axis Sheer Plan -This is a vertical longitudinal center line section of a vessel which involves the intersections of vertical fore and aft planes Surge -This degree of freedom is the fore and aft translation. Surge can also be thought of as the movement along the x-axis. Sway-This degree of freedom is the transverse translation Sway is also the movement along the y-axis Transverse section-This term d escribes a z-y slice of the ship Trim -The difference between the draught of the fore and aft of the vessel. Waterplanes-These are the horizontal planes parallel to the surface of the water that mark the depth of the water along the vessel. Waterlines-These are the lines of intersection ofthe body ofwater and the hull Yaw-This degree of freedom is the rotation about a vertical axis Yaw is also the rotation about the z-axis 119

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References Allievi, A. & A. Soudack A. (1990) International Journal of Control Ship Stability via the Mathieu Equation 51(1) : 139-167 Anicin B A., Davidovic D M., & Babovic, V. M (1993). On the Linear Theory of the Elastic Pendulum European Journal of Physics. 14(3) : 132-135 Aubrecht II G. J. (1996). Wave Motion Macmillan Encyclopedia of Physics J. S. Rigden, ed. New York: Macmillan Chen, S Shaw S W & Troesch, A. W (1999) A Systematic Approach to Modeling Nonlinear Multi-DOF Ship Motions in Regular Seas Journal of Ship Research. 43(1) : 25-37 Falzarano, J. M Esparza L. & Taz U1 Mulk, M. (1995) A Combined Steady-State and Transient Approach to Study Large Amplitude Ship Rolling Motion and Capsizing Journal of Ship Research. 39( 3) : 213-224 Feeny, B. F and Moon, F. C. (1989) Autocorrelation on Symbol Dynamics for a Chaotic Dry Friction Oscillator Physics Letters A 141(8, 9) : 397-400 Gillmer, T. C. & Johnson, B. (1982) Introduction to Naval Architecture Anapolis Maryland : Naval Institute Press Ginsberg, J. H. ( 1995). Advanced Engineering Dynamics Cambridge : Cambridge University Press Harrison, H R., & Nettleton T. (1997) Advanced Engineering Dynamics London : Arnold. Iseki, T. (1999) Estimation of Directional Wave Spectra Using Ship Motion Data. The Practice of Time Series Analysis. Akaike, H & Kiragawa, G ed New York: SpringerVerlag James G., & James, R. C. (1992). Mathematics Dictionary New York: Van Nostran d Reinhold 120

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Ohtsu, K. (1999) The Statistical Optimum Control of Ship Motion and a Marine Main Engine The Practice of Time Series Analy s is Akaike H & Kiragawa G ed New York : Springer-Verlag Sanchez N E & Nayfeh A. H (1997) Global Behavior of a Biased Nonlinear Oscillator Under External and Parametric Excitations Journal of Sound and V ibration. 207(2) : 13 7-149 Sprott, J. C & Rowlands G (1995) C haos Data Analyzer : The Professional Version. New York : American Institute ofPhysics Tupper, E. (1998) Introduction to Naval Architecture Oxford England : Butterworth-Heinemann Watson, B. W. & Watson S M ed (1991) The Unit e d States Navy: A D ic tionary New York : Garland Publishing Williams G P (1997) C hao s Theory Tamed. Washington DC : Joseph Henry Press 121