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DEVELOPMENT OF A SIX DEGREE OF FREEDOM MOTION SIMULATION MODEL FOR USE IN SUBMARINE DESIGN ANALYSIS. John Thomas Hammond « Vt ®?9
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DEVELOPMENT OF A SIX DEGREE OF
FREEDOM MOTION SIMULATIONMODEL FOR USE IN
SUBMARINE DESIGN ANALYSIS.
John Thomas Hammond
«Vt®?9
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DEVELOPMENT OF A SIX DEGREE OF FREEDOMMOTION SIMULATION MODEL FOR USE IN SUBMARINEDESIGN ANALYSIS
7. AuTHORfa;
HAMMOND, JOHN T.
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MOTION SIMJLATIQN; SUBMARINE DESIGN; SIX DEGREE OF FREEDOM
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L ASSI F I C A TION OF T^lS PtSgE (Vnon Dale Knlarad)
DEVELOPMENT OF A SIX DEGREE OF FREEDOM
MOTION SIMULATION MODEL FOR USE IN
SUBMARINE DESIGN ANALYSIS
by
John Thomas Hammond
Submitted to the Department of Ocean Engineering on
May 12, 1978 in partial fulfillment of the requirements for
the Degrees of Ocean Engineer and Master of Science in Naval
Architecture and Marine Engineering.
ABSTRACT
A computer mathematical model that will simulate the six degree
of freedom motion of a submarine has been developed. Hydrodynamic
coefficients obtained from the testing of physical models enable the
user to accurately simulate specific submarine designs. The simulation
model can be used as a design tool to study and predict submarine
dynamic response. Complete three dimensional motion is allowed for
the submarine while constraints in powering, rudder deflection, dive
plane angle, and response time can be selected by the user. Outputs
include a chronological history of all linear and angular velocities
and accelerations, the instantaneous angle of deflection of each
control surface, and the trajectory of the submarine center of
gravity.
Thesis Supervisor: Martin A. ADkowitz
Title: Professor of Ocean Engineering
Ipproved for publio release"?
distribution unlimited.
DEVELOPMENT OF A SIX DEGREE OF FREEDOM
MOTION SIMULATION MODEL FOR USE IN
SUBMARINE DESIGN ANALYSIS -
by
John Thomas Hammond
Lieutenant, United States Navy
B.S.M.E., University of Washington(1971)
Submitted in partial fulfillmentof the requirements for the
degrees of
Ocean Engineerand
Master of Sciencein Naval Architectureand Marine Engineering
at the
Massachusetts Institute of Technology
May, 1978
© John Thomas Hammond 1978
Hi
DEVELOPMENT OF A SIX DEGREE OF FREEDOM
MOTION SIMULATION MODEL FOR USE IN
SUBMARINE DESIGN ANALYSIS
by
John Thomas Hammond
Submitted to the Department of Ocean Engineering on
May 12, 1978 in partial fulfillment of the requirements for
the Degrees of Ocean Engineer and Master of Science in Naval
Architecture and Marine Engineering.
ABSTRACT
A computer mathematical model that will simulate the six degree
of freedom motion of a submarine has been developed. Hydrodynamic
coefficients obtained from the testing of physical models enable the
user to accurately simulate specific submarine designs. The simulation
model can be used as a design tool to study and predict submarine
dynamic response. Complete three dimensional motion is allowed for
the submarine while constraints in powering, rudder deflection, dive
plane angle, and response time can be selected by the user. Outputs
include a chronological history of all linear and angular velocities
and accelerations, the instantaneous angle of deflection of each
control surface, and the trajectory of the submarine center of
gravity.
Thesis Supervisor: Martin A. Abkowitz
Title: Professor of Ocean Engineering
ACKNOWLEDGEMENTS
This thesis was prepared under the auspices of the Department
of Ocean Engineering and Professor Martin A. Abkowitz. My
sincerest thanks to Professor Abkowitz for his expert guidance in
developing the simulation model used in this thesis. I would also
like to express special thanks to my wife Robin for her loving
support during the months of thesis preparation and for her
endless hours of expert typing.
J.T.H.
MAY 1978CAMBRIDGE, MASSACHUSETTS
TABLE OF CONTENTS
TITLE -1
ABSTRACT 2
ACKNOWLEDGEMENTS 3
LIST OF FIGURES 6
CHAPTER I - INTRODUCTION
1.1 BACKGROUND 7
1.2 MODEL DEVELOPMENT 8
CHAPTER II - THE SIMULATION MODEL
11.
1
GENERAL CAPABILITIES 12
11.
2
COORDINATE SYSTEMS 13
II. 3. EQUATIONS OF MOTION 13
II. 4 CONTROL SURFACES '.17
CHAPTER III - PROGRAM USER'S GUIDE
111.
1
GENERAL FEATURES 19
111.
2
SUBROUTINE FUNC 19
111.
3
SUBROUTINE RUDDER 20
111.
4
SUBROUTINE DEPTH 20
111.
5
SUBROUTINE STERN 21
111.
6
RANGE VARIABLES 22
111.
7
PROGRAM PARAMETERS 23
CHAPTER IV - TEST RESULTS AND CONCLUSIONS
IV. 1 FUNDAMENTAL MOTION TEST 25
IV. 2 HORIZONTAL MOTION TEST 26
IV. 3 VERTICAL MOTION TEST 28
5
IV.
4
COMPLETE MODEL TEST 31
IV.
5
USE AS A DESIGN TOOL 31
IV.
6
CONCLUSIONS AND RECOMMENDATIONS . . . 32
REFERENCES 34
APPENDIX A
A.l NOTATION 35
A.
2
EQUATIONS OF MOTION 45
A.
3
AXIS TRANSFORMATIONS 51
APPENDIX B
B.l LIST OF VARIABLES 53
B.2 COMPUTER PROGRAM LISTING 59
B.3 LEQT1F 87
LIST OF FIGURES
Figure 1
Local Submarine Coordinate System 10
Figure 2
Relationship of Axes, Angles, Velocities,
Forces, and Moments 14
Figure 3
Simulation Model Trajectory with Constant Dive
Plane Angle .- 27
Figure 4
Rudder Deflection and Angle of Turn as a Function
of Time 29
Figure 5
Depth as a Function of Time for a Simple Dive 30
CHAPTER I - INTRODUCTION
1.1 BACKGROUND
During the several years spent in designing a submarine, a
great deal of effort is expended trying to improve the design so that
the best possible vessel is eventually constructed. During the de-
sign phases the designer will frequently examine problems that were
found in older submarines and try to determine the cause of each
problem so that they can be eliminated in the new design. At several
stages during the design, computer models are used to assist in such
things as structural design, equilibrium calculations, and internal
arrangement. As the design progresses, physical models of the
proposed vessel are built and tested in a towing tank to measure the
hydrodynamic coefficients. Finally, when it is possible to deal with
the vessel as a whole, a dynamic analysis is conducted to gauge how
the submarine will perform in an underwater environment when all six
degrees of freedom are available.
While it is possible to gather dynamic information, such as the
hydrodynamic coefficients, from physical model tests, the models are
necessarily too restricted in their motion to permit a complete
8
analysis. It is necessary to construct a mathematical model of the
submarine to simulate the submarine's motion and thereby obtain
enough data for a complete analysis. The simulation model will use
the hydrodynamic coefficients obtained from the physical model as a
basis for the simulation. A properly constructed model will permit
the designer to simulate any conceivable maneuver and gauge the
submarine's response. The simulation model then becomes a tool to
assist in improving the design and achieving the best possible
vessel. After the submarine is designed, the model can continue to
be useful by assisting in the evaluation of operating procedures.
The model can be placed in any maneuvering situation without hazard
to crew or vessel. This is especially useful in evaluating casualty
situations. The model can also be used to estimate the effect of
proposed design changes to the submarine. For instance, the model
can show the effect of changing the maximum deflection of a control
surface or its rate of operation.
All of the few six degree of freedom submarine simulation models
in existence are too expensive for daily use in the design office.
Generally, the models are complicated to use and sometimes difficult
to obtain. These problems have created the need for a program that
is both inexpensive enough to permit daily use in the design office
and simple enough so that anyone can use it. This need has formed
the motivation for this thesis.
1.2 MODEL DEVELOPMENT
A mathematical model must use Newton's Law of Motion and the
9
differential equations resulting from a dynamic analysis of the
submarine. A submarine has movable appendages, so these must also
be accounted for in the model. Newton's Law of Motion can be
expressed as:
(1) Force = -nr (momentum)
(2) Moment = -nr (angular momentum)
If a submarine with a coordinate system such as shown in Figure 1 is
used, then equation (1) can be applied along each of the three axes.
Similarly, equation (2) can be applied around each axis. This gives
a total of six equations to represent the six degrees of freedom of
the submarine. These six equations of motion are well-known [1] so
they are not developed in detail here. They are, however, included
for reference in Appendix A.
While the equations of motion for the submarine form the heart
of the model, they are insufficient by themselves. On an actual
submarine, the officer of the deck orders the rudder or dive planes
moved in order to maneuver the ship. The routines will normally
sense where the ship currently is and where it is supposed to be and
then move the appendages in the appropriate direction just as the
deck officer would do. The six equations of motion together with
the appendage control subroutines constitute the vital components of
the simulation model. A main program is necessary to coordinate the
input, output, and to control the action of the subroutines.
The components of the model are discussed in detail in Chapter
II of this thesis. The model is written as a computer program whose
features are discussed in Chapter III. The final chapter will
10
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11
discuss the model tests and their results. A listing of the
program will be provided in Appendix B.
12
CHAPTER II - THE SIMULATION MODEL
II. 1 GENERAL CAPABILITIES
The simulation model developed in this thesis will provide
trajectory information for routine submarine maneuvers such as
turning or changing depth. The trajectory, which is referenced to
a fixed coordinate system, .is computed as a function of time. As
the trajectory of the model is developed, the velocities and
accelerations are calculated and stored for output to the user.
Both angular and linear velocities and accelerations are provided.
The angle of deflection of each control surface appendage is computed
for each step of the entire trajectory or maneuver.
The rate of deflection of each control surface and its angle
of maximum deflection are specified by the user. Any of the control
surfaces can be "jammed" by specifying a particular angle of
deflection. The model can be initially placed on any course and
depth and then ordered to come to any new course and depth. The
user can select the initial speed and the appropriate thrust
coefficients to cause a change in speed.
13
11.
2
COORDINATE SYSTEMS
The coordinate system plan used in the simulation model is
based on [1]. The model uses two orthogonal coordinate systems; one
remaining fixed at the water surface while the other travels with the
submarine to act as a local reference system. The fixed system,
designated x , y , z , is related to the moving system, designated
x, y, z, through the angles <£, 9 , <A as illustrated in Figure 2.
Quantities measured in the moving system can be referenced to the
fixed system by using the transformation [2] given in Appendix A.
The origin of the moving system is at the center of gravity of
the submarine. This has the advantage that only the principal
moments of inertia, I , I , I , are non-zero (i.e.: I »I «I = 0).
It has the additional advantage that the equations of motion in [1]
also use this reference point for the x, y, z system. It would have
been possible to use the centerline of the submarine for the origin.
This would have the advantage of making better use of the symmetry
of the vessel, however it would have the disadvantage of having to
correct both the equations of motion and some of the hydrodynamic
coefficients for the new origin.
Appendage movements are measured with respect to the moving
coordinate system. All velocities and accelerations are measured
along the axes of the moving system. This is also shown in Figure 2.
11.
3
EQUATIONS OF MOTION
The general nature of the six degree- of freedom model requires
the use of the six equations in their non-linear form. A large
14
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->r->
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CDt/f i_(D DCD O)C • —< Ll
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CD
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15
number of hydrodynamic coefficients are necessary for this model.
It would be desirable to derive each coefficient from existing hydro-
dynamic theory. Some coefficients have been obtained for bodies of
revolution or other mathematically amenable shapes;' however, when
appendages such as control surfaces, fairwaters, and propellers are
taken into account, the accuracy of the theoretical values is brought
into question. For this reason it is standard practice to obtain the
numerical value of the coefficients by experimental means. This
usually entails the towing and measurement of physical models.
Once the hydrodynamic coefficients are known, the equations
can be solved for the accelerations. The six equations of motion
must be written such that the highest order derivatives, namely
u, v, w, p, q, r and their coefficients, appear on the left hand
side of the equation. This gives each equation a form like:
• ••*••a,- ^ u + a. .., v + a. • , 9 w + a. •
. , p + a. ... q + a. .. c rl ,J l »J+1 l fJ+2 l ,j+3 K
l ,j+4 ^ l ,j+b
fj (u, Y, w,"p, q, r,<f>, 9, r> 8f
» 8b>
8$
, 1,bi, . . . )
The left hand side is placed in a matrix format. The six equations
are then represented as:
M• • • • —
u•
fi
V• hw•
= f3
p• Uq• hr h
16
The method of solution is an iterative technique in which initial
values are used in the right hand side functions and the six acceler-
ations are solved for on the left side. The accelerations are then
used to update the right side functions. A small time increment is
made and the accelerations are again solved by using the latest
value for the right side functions.
With each iteration the accelerations are used in a Taylor
series expansion to calculate the velocities. The calculations
have the form:
W>(t+At)=^
w\ (t) +
P( IP
q
r
q
r
'(At)
The pitch, roll, and yaw angles are calculated in a similar manner
using the angular velocity and acceleration.
(t) +
In order to plot the trajectory of the submarine, a coordinate
§•'*•' f]-®
system transformation is performed to obtain the linear velocities• • •
x , y , z in the fixed coordinate system. The velocities are used
in a Taylor series expansion to obtain the position of the center of
gravity of the submarine in the x , y , z system.
17
y t(t + At) = Jy/ (t) + Jy.L (At)
When the position of the submarine is calculated the time is advanced
one increment and another iteration begins.
The iteration loop will continue to operate until some test
criteria are met. For instance, if the model is performing a dive,
then the model tests the value of zn to see if the model is at theo
appropriate depth. A test is also made of the pitch angle to see if
it is within some specified range of zero. Lastly a check is made to
ensure that all of the dive planes are at zero deflection.
II. 4 CONTROL SURFACES
The motion of the model is controlled by the movement of the
control surfaces. The control surfaces include the rudder, the
stern planes, and the sail planes. For analysis of submarine designs
the movement of the control surfaces can be governed by an automatic
control system. In the model each control surface is provided with
its own automatic control system. The principle of operation for
all of the automatic control systems is the same. In each case the
deflection of the control surface is made proportional to an error
signal and the rate of change of the error. The sail planes, for
instance, control the depth of the model. The calculated deflection
of the planes is proportional to the depth error and the rate of
change of depth.
18S = K, (present depth - ordered depth)
+ IC, (rate of change of depth)
A large error will initially cause a large deflection, but as the
rate of change increases, the deflection will decrease until some
moderate rate is achieved. Since movement of the sailplanes does
not have any significant effect on any motion other than depth,
its control system uses only the variables associated with depth.
The stern planes control pitch angle as well as depth. The
control system accounts for this by using the pitch angle, 9* and the
rate of change of 9.
8 = K3
(present depth - ordered depth)
+ K. (rate of change of depth)
+ K5
(pitch angle)
+ K6
(rate of change of pitch angle)
The K's in the control systems are proportionality constants
known as gain. The value of the gain determines the sensitivity and
response of the control system. Since the error signal continually
varies during a maneuver, it is best to keep the gain low. It is
desirable to have just enough gain on the error signal to get the
control system moving and keep it moving in the right direction.
The gain on the rate signals should be much stronger to dampen out
the oscillations caused by response to the error signal.
The control surfaces move at a rate specified by the
user. Whenever the calculated deflection differs from the
present deflection, the control surface will move toward
the calculated value at the specified rate.
19
CHAPTER THREE - PROGRAM USER'S GUIDE
111.
1
GENERAL FEATURES
The simulation model consists of a main program and four
subroutines. The four subroutines are named FUNC, RUDDER, DEPTH,
and STERN. The main program reads the input data, changes units,
initializes values, and prints the output. The main program performs
the Taylor series expansions, the coordinate system transforms, and
provides the logical statements for calling the subroutines. On the
first iteration, when time equals zero, no subroutines are called;
the output from this first iteration is then a statement of the
initial conditions of the problem. Each succeeding iteration will
call subroutine FUNC to calculate the new position of the model.
Calls to the control surface subroutines are made on the basis of
need, with no subroutine being called more than once in the iteration.
The decision on whether or not to call RUDDER, DEPTH, or STERN is
contained in a series of logical IF statements in the main program.
111.
2
SUBROUTINE FUNC
Subroutine FUNC contains the six equations of motion. The
20
input parameters to FUNC consist of the hydrodynamic coefficients,
the model velocities, the orientation in space, all of the ship's
characteristics such as length and moments of inertia, the deflection
of each control surface, and the propulsive coefficients. The sub-
routine returns the value of the six accelerations: UDT, VDT, WDT,
PDT, QDT, RDT. The input and output for FUNC are all contained in
COMMON /FOUR/ and COMMON /FIVE/. The solution to the matrix equation
in subroutine FUNC is made possible by a call to the library function
LEQT1F [3]. LEQT1F performs a Gaussian reduction for the matrix
equation. Any similar Gaussian reduction could be substituted if
LEQT1F is not available. An explanation of the parameters used in
the call to LEQT1F is found in Appendix B.
111.
3
SUBROUTINE RUDDER
Subroutine RUDDER controls all horizontal motion for the
model. The input parameters include: K9, K10, T5, T6, Til, T12,
T17, T18, TLAGR, RRATE, RUDAMT, COURSE, DELR, DELT, R, PSI. The sub-
routine returns a new value for DELR, the rudder deflection, based on
a calculation using its present and ordered headings. The inputs are
all contained in COMMON /THREE/ and COMMON /FIVE/. Subroutine RUDDER
is called whenever the model is not on the desired course or when the
rudder is deflected.
111.
4
SUBROUTINE DEPTH
Subroutine DEPTH controls the forward set of dive planes. It
can control either bow planes or sail planes depending on what the
21
submarine is fitted with. This subroutine is sensitive to the depth
error and the rate of change of depth. Subroutine DEPTH will move
the forward planes in the direction necessary to bring the depth
error to zero. As input parameters, the subroutine uses: Kl , K2
Tl, T2, T13, T14, T15, T16, TLAGB, DELT, DIFF, ADIFF, ZDT, ATHETA,
MAXANG, DCRIT. The subroutine returns a new value for DELB, the
bow plane deflection, after each call.
If MAXANG, the maximum dive/ascent angle, has been exceeded,
then DEPTH will return a diagnostic write statement. The user may
specity MAXANG, the maximum pitch angle for the model. Whenever
MAXANG is exceeded the diving planes are moved to reduce the pitch
angle and a diagnostic signal is generated from within subroutine
DEPTH. The diagnostic will say "Maximum dive/ascent angle exceeded
at time . Standard fixup taken." Since the dive planes only
begin to react when MAXANG is exceeded, the submarine will overshoot
the angle before a reduction in the angle actually occurs. The
diving planes will continue to operate until the pitch angle is
less than MAXANG.
DEPTH makes only a minimal attempt to control the pitch angle
of the model; it only warns the user when the angle is exceeded, and
then moves the bow planes so that the pitch angle does not grow
larger. The principal purpose of DEPTH is to provide depth control;
it does this in conjunction with subroutine STERN.
III. 5 SUBROUTINE STERN
Subroutine STERN controls the movement of the stern planes to
22
achieve the desired depth and pitch angle. The movement of the
planes is sensitive to the depth error, the rate of change of depth,
the pitch angle, and the rate of change of pitch. The stern planes
will move to bring the depth error to zero and the pitch angle to
zero. As input parameters the subroutine uses: K5, K6, K7, K8, T3,
T7, T8, T10, TLAGS, STERAT, STERMX, THETA, Q, DELS, DIFF, ADIFF,
ZDT, ATHETA, MAXANG, NOPICH, DCRIT. The subroutine returns a new
value for the stern plane deflection, DELS. The input and output
parameters are all contained in COMMON /ONE/, COMMON /FIVE/, and
COMMON /SIX/. STERN is called to achieve a depth change or to correct
the pitch angle.
III. 6 RANGE VARIABLES
Since it is not usually possible to bring a computer simulation
model to a precise depth or angle, it is necessary to define ranges
around the desired depth or angle which will be acceptable to the
user. For example, if the submarine were to make a depth change of
500 feet, the dive may be considered complete if the model settled
within ten feet of the desired depth. The range is specified by the
user to enable him to achieve whatever precision is desired. Within
this range the main program will not make a call to the control sur-
face subroutine; since no call is made, the control surfaces cannot
be moved. In the program, the variable name for the depth range is
NODIFF; it is usually set at + 5 or + 10 feet. Subroutine DEPTH will
be called whenever the model is off of its ordered depth by more than
NODIFF. The range about zero pitch angle is given the name NOPICH;
23
it is usually set at + 1 degree. STERN is called if the pitch angle
is greater than NOPICH. The variable name for achieving the proper
course is ONCRS; it is usually set at + 1 degree. RUDDER will be
called whenever the model is off course by more than ONCRS.
An additional range variable is used during diving or surfacing.
When the model moves within some critical range of the desired depth,
it is time to begin moving the dive planes to zero angle and let the
vessel glide into the desired depth. This maneuver prevents
oscillation of the dive planes as the error signal and the rate
signal both become small. This also ensures that the planes are at
or near zero angle by the time the desired depth is reached. The
name of this variable is DCRIT; it is usually set at 50, 75 or 100
feet depending on the speed of the submarine. STERN will be called
whenever the model is off its ordered depth by more than DCRIT.
III. 7 PROGRAM PARAMETERS
The iterative nature of the program relies on a time increment
being made after each step. The size of the time increment is
optional; the smaller the increment the more accurate the calculations
In selecting the size of the time increment, one must bear in mind
the length of time required to complete the maneuver and the storage
capacity of the program. Due to the quantity of information that is
calculated by the program, it is practical to print only half of the
data at one time. The other half of the data is stored in the
array STOW. This data is then printed when the maneuver is complete.
STOW has the capacity to hold the data resulting from 600 iterations.
24
A time check is incorporated in the main program to enable the
user to limit the number of iterations. Achieving a number of
iterations equal to the variable ICNT will cause the program to stop
the present maneuver, print out all data, and see if another maneuver
is desired. ICNT should not be made larger than 600 so that the
available storage space is not exceeded.
Each control surface subroutine has a time lag scheme which
senses the initial command to the control surface and prevents
immediate action. A time lag occurs each time the direction of
movement is changed. A different time lag may be specified for
each control surface.
The variable INDEX is used to allow the user to perform more
•than one maneuver with a given submarine and then shift submarines
to perform more maneuvers. If INDEX is less than or equal to zero
then the same submarine coefficients will be used for each set of
initial conditions. If INDEX is greater than zero the program will
read new coefficients as well as a new set of initial conditions.
25
CHAPTER FOUR - TEST RESULTS AND CONCLUSIONS
IV. 1 FUNDAMENTAL MOTION TEST
The validation of the simulation model is a matter of impor-
tance. The method used was an independent test of each component of
the model, and then a series of tests with the components of the
model working together. The test maneuvers were selected either
because the correct dynamic response was known or because the maneuver
was simple enough so that the general nature of the response could
be predicted.
The first portion of the model to be tested was the subroutine
FUNC. FUNC obtains the solution to the six equations of motion. It
was important to establish that tnese equations were properly
installed in the program. A test of subroutine FUNC necessitated a
simultaneous test of the MAIN program to read in the hydrodynamic
coefficients, set up the "A" matrix for FUNC, perform the Taylor
series expansions, and write out the results. The test for FUNC
was to reduce the GM of the vessel to zero and then to assume a
constant angle on the dive planes. If the. model was working cor-
rectly it should traverse a perfect circle in the vertical plane.
26
It was known that the rudder should not move and the model should
not roll or yaw. The model should assume a circular trajectory with
a constant angular velocity and a constant vertical component of
velocity, w. Figure 3 shows the results of the maneuver. A circular
trajectory was quickly achieved. The angular velocity was constant,
w was constant, and the position of the model in the fixed coordinate
system confirmed the circular path. The model did not roll or yaw
and the rudder did not move.
IV. 2 HORIZONTAL MOTION TEST
With the knowledge that the equations of motion, the Taylor
series expansions, and the coordinate system transforms are operating
properly, the subroutine RUDDER was the next component to test. It
was decided that a simple left turn of 40 degrees would be an
appropriate test. A left turn was chosen because that presents the
greatest opportunity for error. The original course would be 000
degrees true and the new course would be 320 degrees true. Since the
yaw angle is positive when turning right, the model must cope with a
negative yaw angle as well as a proper method for dealing with course
headings given in true bearing. The subroutines DEPTH and STERN were
rendered inoperative to give the opportunity of seeing RUDDER operate
without interference. The model would be checked to ensure that the
proper rudder rate was used and that the rudder angle did not exceed
the maximum angle ordered. The model would be expected to roll and
squat in the turn. A change in depth was expected since the dive
planes could not act. As the model approached the new heading the
27
-1000
direction ofNi travel
•1500>X,
Simulation Model Trajectory With ConstantDive Plane Angle
Figure 3
28
rudder should be put amidships and the vessel should steady out.
In Figure 4 the relationship between the rudder and the heading
is shown as a function of time. The model reacted as predicted
except possibly at the end of the turn. The rudder' was amidships and
the new heading was achieved but the program did not run long enough
to ensure that the model was steady on the course. The model did roll,
squat, and change depth as predicted. The test for RUDDER was
considered to be accurate and sufficiently complete to warrant moving
to the next test.
IV. 3 VERTICAL MOTION TEST
Since DEPTH and STERN operate together to regulate the depth
and pitch of the model, they were tested together. The test consisted
of a simple dive with a depth change of 700 feet. The course was not
changed so the rudder should not move. The model should not roll or
yaw. The dive planes should move in the proper direction and at the
ordered rate. They should not deflect more than the ordered angle.
The model should pitch downward and if the maximum ordered pitch
angle is exceeded then the dive planes should move to reduce the
pitch angle. As it approaches the desired depth, the model should
slow its rate of descent and settle within ten feet of the depth,
with -1 degree of pitch angle.
The trajectory for this test is shown in Figure 5. The model
achieved steady state only six feet beyond the desired depth; the
pitch angle was -.03 degrees. The dive planes were all at zero
deflection. The model stayed within ten feet of the ordered depth
29
O
CO
<D
CD._ L.
co cdQ_ <D
33oCO a o o
co
<c
o(D CD
*
CDC E< i—
\T"O M—
<D c o CD
E < i_c ZJ
_ c o CD
o -t-> Llu
u cif
»+—
<D
QL.
<DT3"CDcr
co CO
L. +-» • CDCD u cd"D CD L.T3 m— CDD CD (I)
cr Q "D
30
E
o o o o o o o Oo o o o o o OT
—
C\J
Depth
3
Change
^r lO
(D
H
—
CD IS
CD
>QQJ
Q.
ECO
<L.
oLl
E
Ocg-t->
uc
<CO
<
Q.
Q
lO
i_
Ll
31
and within one degree of zero pitch angle for twenty seconds. This
was done to ensure that a steady state had been achieved.
IV. 4 COMPLETE MODEL TEST
As a final test the entire simulation model must work together.
This test repeated the forty degree left hand turn but this time the
dive planes were allowed to react to try to maintain the depth. The
program was kept running until the pitch angle was within one degree
of being zero, the model was within one degree of the proper course
and the depth was within ten feet of the ordered depth. The model
was initialized on course 000 degrees true at an initial speed of
twenty knots.
The maneuver was successfully completed. The model remained
steady at 320 -1 degrees true and the final depth was within one
foot of the ordered depth. Both the pitch and roll angles were near
one degree and were decreasing in magnitude. All of the control
surfaces were at zero angle of deflection. It should be noted that
the new course had been achieved during the first sixty seconds and
that the course was maintained by the model while the proper depth
and attitude were being obtained.
IV. 5 USE AS A DESIGN TOOL
There are numerous design tasks that could be used to demonstrate
the simulation model; a simple example is sufficient for this demon-
stration. Suppose a designer wanted to know the dynamic effects of
increasing the rudder rate in a turn. The designer would elect to
32
use a simulation model. Since he would not want the dive planes to
interfere with the analysis, he would set NODIFF, DCRIT, and NOPICH
at large values so that the dive planes would not move. Then he
would set RRATE, the rudder rate, at the value he desired to test
and order the model to come to a new course. For this run, let's
assume that the designer performed the left hand turn from 000
degrees true to 320 degrees true. He used a rudder rate of two
degrees/second. He then caused the model to perform the same
manuever again with a new rudder rate of four degrees/second. With
the output from these two maneuvers he could easily find the time
required for the turn, the advance and transfer of the submarine,
and the roll and pitch angles as a function of time. The designer
could use the information for whatever analysis he had in mind. The
model could be run againat new rudder rates or the dive planes could
be brought into play or virtually any other maneuver could be
simulated. This model can also do snap roll analysis [4].
The cost of running the simulation model is so low that it can
be used on a daily basis if desired. For the test turns mentioned
above, the cost was less than five dollars, which included reading
the cards, compilation, execution, and 900 lines of output. A
designer who used the model regularly could have the model as an on
line dataset which would greatly reduce the cost of using the model.
IV. 6 CONCLUSIONS AND RECOMMENDATIONS
This simulation model does accurately simulate the six degree
of freedom motion of a submarine for moderate maneuvers. The cost of
33
operating the model is very low, which will allow frequent usage.
The model is designed to permit a great deal of flexibility in the
application of the model. For those designers who use the simulation
model, it should be a useful design tool.
If work were continued on this simulation model, it is recom-
mended that some time be spent in improving the appendage control
subroutines. Further application of control theory would be helpful
with the appendage subroutines. The data input could be organized
more efficiently to remove some of the opportunity for error. It
could be very useful to have a plotting routine in the model to
visually display the information.
34
REFERENCES
(1) Gertler, M. and Hagen, G. R. , "Standard Equations of Motion for
Submarine Simulation," NSRDC Report No. 2510, Washington, D.C.,
June 1967.
(2) Abkowitz, M.A., Stability and Motion Control of Ocean Vehicles ,
Cambridge, Ma.: The M.I.T. Press, May 1969.
(3) (FORTRAN IV) IBM System/370-360, IMSL Library 1, Edition 6,
1977, Xerox Sigma s/6-7-9-11-560.
(4) Liberatore, D.J., "An Investigation Of Snap Roll in Submarines"
M.I.T. Thesis, Cambridge, Ma., May 1977.
(5) , IBM System/360 and System/370 FORTRAN IV Language,
GC28-6515-10, May 1974.
(6) Comstock, John P., ed. Principles of Naval Architecture , New
York: The Society of Naval Architects and Marine Engineers,
1967.
A.l NOTATION
Symbol Dimensionlesa Form
35
APPENDIX A
Definition
B B' =B
5P-t2UTTTl
Buoyancy force, positive upward
CB
CG
Center or" buoyancy of submarine
Center of mass of submarine
I
xy
y 'kp*>
s
i =-£»2 its
Ixvxy jp^»
Moment of inertia of submarine about x axis
Moment of inertia of submarine about y axis
Moment of inertia of submarine about z axis
Product of inertia about xy axis
y*i =
lYz Product of inertia about yz axes
zx
K
K.
zx ^K' =
K
K*
Product of inertia about zx axes
Hydrodynamic moment component about xaxis (rolling moment)
Rolling moment when body angle [ry, 8) andcontrol surface angles are zero
K*V
K
K.
KPlPl
Kpq
T ?p-t3 u*
K
K.
P £p-t 4 U
KK
IpI
K
pipI $pe
l££_pq spt
6
Coefficient used in representing K$ as a
function of (77- 1)
First order coefficient used in representingK as a function of p
Coefficient used in representing K as a functionof p
Second order coefficient used in reprssentingK as a fvinction of p
Coefficient used in representing K as a functionof the product pq
36
K
K-
K
K.v
Kvq
K
K
K6r
m
M
M*
MPP
M
M.qn
M.q
KK,
K
qr £ p*5
K
K
£p4*U
Kf
ip^"
v £p43 U
, _ Kv|v!K
K
K
KWP £p-t*
K • - Kv/r
v| V1 h^
vqKVq
•^vw
vw 5P43
1Kwp
6r ~*p^u 2
V - i
m 1 * mip*
3
M' =M
*p*3 u2
M l
M*M*
*p*3 u 2
MPP
, .MPP
?P*6
M '
qa
Mq
'
£pt*U
Mqn *pZ«U
M.'q 5P*
6
Coefficient used in representing K as afunction of the product qr
First order coefficient used in representingK as a function of r
Coefficient used in representing K as afunction of f
First order coefficient used in representingK as a function of v
Coefficient used in representing K as afunction of v
Second order coefficient used in representingK as a function of v
Coefficient used in representing K as a functionof the product vq
Coefficient used in representing K as a functionof the product vw
Coefficient used in representing K as a functionof the product wp
Coefficient used in representing K as a functionof the product wr
First order coefficient used in representingK as a function of 6
Overall length of submarine
Mass of submarine, including water in free-flooding spaces
Hydrodynamic moment component about y axis(pitching moment)
Pitching moment when body angles (or, 8) andcontrol surface angles are zero
Second order coefficient used in representingM as a function of p. First order coefficient is
zero.
First order coefficient used in representingM as a function of q
First order coefficient used in representingM as a function of (tj-1)
Coefficient used i-n representing M as a
function of q
37
M
l
|q|6s
MrP
M
Mvp
M
q|q| q|q| sp-l
M ,.. MH 6s
|ql 6s fp^u
M ' = T-rp 5pi&
rr ^ois
MVDvp
$p44
MM,
vr kpt*
MvvM M ' = —rrvv vv 5P-C
Second order coefficient used in representingM as a function of q
Coefficient used in representing M, as afunction q
Coefficient used ir. representing M as a
function of the product rp
Second order coefficient used in representingM as a function of r. First order coefficient
Coefficient used in representing M as afunction of the product vp
Coefficient used in representing M as afunction of the product vr
Second order coefficient used in representingM as a function of v
M Mw £p.£,3 U
First order coefficient used in representingM as a function of w
M M .-Atw* jpr u
M. M. 1 -w w jp*.*
MlM
i iM
i i' =;—s-^
M |w |q
l
w |q tot*
First order coefficient used i'n representingMw as a function of (n-l) „
Coefficient used in representing M as a functionof w
First order coefficient used in representing Mas a function of w; equal to zero for symmetricalfunction
Coefficient used in representing M. as a functionof w
Mw| w| Mw i w '
H» |wI
iP-t3
Second order coefficient used in representingMas a function of w
w|w|rj w|w \r)= ,3 ' '
M M ' =-^WW WW fp-t
M 5bM.. M., ' -6b 5b jpi3 U 2
First order coefficient used in representingM . . as a function of (tj-1)w w
|
'
Second order coefficient used in representingM as a function of w; equal to zero for sym-metrical function
First order coefficient used in representingM as .1 function of 6.
M5 s
6s 6s 3pl3 U zFirst order coefficient used in representingM as a function of 6 =
6srj
M 6 srj
6stj " s$l~ U'First order Coefficient used in representing2vlj s <ta a function of (tj-1)
38
N
N-
N
N.P
Npq
Nqr
N
Nrr?
N.r
N
N,
N
r!6r
NVTJ
N.v
N
N' - Nipl* u-
N*'
N ' *PP |pt4 U
N.'P
=NP
Npq
, =NPS
ip-t5
Nqr
,.Nqr
N '-Nr
r £p<.*U
N.'r *P*S
"r|, . -N rlr|
V , _ N|r!6rI6r |p£*U
M l
NvV ~ £p*
3 u
NVTJ
, _Nvr,
£p43 U
N '
NvV ipi*
N . _ Nvqvq vq yi*
Hydrodynamic moment component about z
axis (yawing moment)
Yawing moment when body Angles (or, j3) andcontrol surface angles arc zero
First order coefficient used in representing Nas a function of p
Coefficient used in representing N as a functionof p
Coefficient used in representing N as a functionof the product pq
Coefficient used in representing N as a function
of the product qr
First order coefficient used in representing Nas a function of r
First order coefficient used in representingNr as a function of (rj-1)
Coefficient used in representing N as a functionof f
Second order coefficient used in representingN as a function of r
Coefficient used in representing N-. as afunction of r
First order coefficient used in representing Nas a function of v
First order coefficient used in representing Nvas a function of (n- 1)
Coefficient used in representing N as afunction of v
Coefficient used in representing N as a functionof the product vq
N|vlr
N.N|vl
tpl*Coefficient used in representing Nr as afunction of v
Nv ' v l £p-t
3 Second order coefficient used in representingN as a function of v
N|v|tj
N ,_ NvMtl
tefFir:»t order coefficient used in representingN
. - as a mnction of (rj-1)
39
N
N
N,
wp
Nwr
N6r
N6rrj
VW J.„»3
N„N ' = Twp ^ p4+
NN6rTT7I6r ipt3 U
p u
Coefficient used in representing N as a functionof the product v\v
Coefficient used in representing N as a function
of the product wp
Coefficient used in representing N as a functionof the product wr
First order coefficient used in representing Nas a function of 5 r
First order coefficient used in representingN, as a function of (tj-1)
Angular velocity component about y^axisrelative to flu : d (roll)
P'
q'u
q'
r'r*
" U
f"
Angular acceleration component about x axisrelative to fluid
Angular velocity component about y axis relativeto fluid (pitch)
Angular acceleration component about y axisrelative to fluid
Angular velocity component about z axisrelative to fluid (yaw)
Angular acceleration component about z axisrelative to fluid
Uu
Linear velocity of origin of body axes relativeto fluid
" U
<!•at
*c u
Component of U in direction of the x axis
Time rate of change of u in direction of the
x axis
Command speed: steady value of ahead speedcomponent u for a jjivr-n propeller rpm whenbody angles (a, 8) and control surface anglesare zero. Sign changes with propeller reversal
Component of U in direction of the y axis
fr,_ vl
U 2Time rate of change of v in direction of the
y axis
40
w'U
Component of U in direction of the z axis
• ^4w ' = —
-
Time rate of change of v. in direction of thez axis
W W =W
spW Weight, including water in free flooding spaces
CB>*5l
Longitudinal body axis; also the coordinate cf apoint relative to the origin of body axes
The x coordinate of CB
*G x^-^°- The x coordinate of CG
A coordinate of the displacement of CG relativeto the origin of a set of fixed axes
X' =
rp
X.u
5P<.2U 2
X.-qqqq Ip-t
4
Xrrrp £p£*
X„rr £>£*
Xuwu "ipTT
Hydrodynamic force component along x axis(longitudinal, or axial, force)
Second order coefficient used in representingX as a function of q. First order coefficientis zero -
Coefficient used in representing X as a functionof the product rp
Second order coefficient used in representingX as a function of r. First order coefficient is
zero
Coefficient used in representing X as a functionof u
Second orr'er coefficient used in representingX as a function of u in the non-propelled case.First order coefficient is zero
X '
-vr -£
p<3Coefficient used in rcf-esenting X as a functionof the product vr
*pVSecond order coeffient used in representing Xas a function of v. First order coefficic.it is zero
vvrjx . _ AyvT
7 First order coefficient used in representing Xvvas a function of (77- 1)
Coefficient used in representing X as a function
of the product wq
41
Xww
WW17
'6b5b
'5r6r
k
6r6rrj
k
6s6s
'6s6s7j
X ' =WW y^
XWW?)
^wwr;
?P^
3C '
6b6b
x6b6b"
5 Pi2" 2
x . _A6rSr
6r6r '£p.£,
2U 2
"
Van?'6r6r^ " fplTu 2
, _ X6s6s17727726s6s sp-tAU
x , _ x6s6s77
6s6st] ip-t 2U 2
Second order coefficient used in representingX as a function of w. 'First order coefficient is
zero
First order coefficient used in representing Xwas a function of (77- 1)
Second ord^r coefficient used in representing Xas a function of 6^,. First order coefficientis zero
Second order coefficient used in representingX as a function of 6 r . First order coefficient iszero
First order coefficient used in representingX. . as a it. nction of {n- 1)6r6r '
Second order coefficient used in representing Xas a j
zeroas a function of 6S . First order coefficient is
First order coefficient used in representingX. . as a function of (77- 1)6s6s '
^B
^G
Y.P
Pip'
y- =
ys_ ^B
reYG
Yo_ Yo
y» ;
Y
*P*2U Z
Y-4-'Y
£p<,2U 2
Y '
P
YP
£pi3 U
Y.'P
_YP
he***
Y .
, _ypIp!
PIP 1 £pV
Lateral body axis; also the coordinate of apoint relative to the origin of body axes
The y coordinate of CB
The y coordinate of CG
A coordinate of th3 displacement of CG relative
to the origin of a set of fixed axes
Hydrodynamic force component along y axis(lateral force}
Lateral force when body angles (a, fi) and controlsurface angles are zero
First order coefficient used in representingY as a function of p
Coefficient used in representing Y as a function
of p
Second order coefficient used in representingY as a function of p
42
pqY • = 23-pq fpt*
Coefficient used in representing Y <is a functionof the product pq
qr q* ipZ*
rip4
Tu
Coefficient used in representing Y as a functionof the product qr
First order coefficient used in representing Y -
as a function of r
«)Y ' - YrT
>
rT7 " fpFUFirst order coefficient used in representingY as a function of (77- 1)
Yf
rl6r
Y.< =—
i
Ylr|6r "ipTTTT
Coefficient used in representing Y as a functionof f
Coefficient used in representing Y. as afunction of r
First order coefficient used in representingY as a function of v
VT?
Yv
l
vTT
Y-Y.' = -* ip*
a
First order coefficient used in representingYy as a function of (tj- 1)
Coefficient used in representing Y as afunction of v -"
vq
'•rl
V|V|TJ
wp
wr
' • - vq
vq " ?p-63
, ,_ Yvlr!vir| -j&r
. -Yvlvl
vivi -j^r
,
Y vlv|T7rv 1 v 1 tj
" TpT1
Y
«" ipt*
YY ' - pWP ~ ?pt3
YY ' - r
»r jpt3
Coefficient used in representing Y as a functionof the product vq
Coefficient used in representing Yv as ;i functionof r
Second order coefficient used in representingY as a function of v
First order coefficient used in representingYv | v i as a function of (77- 1
)
Coefficient used in representing Y as afunction of the product vw
Coefficient used in representing Y as a
function of the product wp
Coefficient used in representing Y as afunction of the product wr
6rY, ' =
6r
&* jpi'lHFirst order coefficient used in representingY as a function of 6r
6rrj
_6rij_
6r7J ^U* First order coefficient used in representing
Y t as a function of (T7-1)or
43
Normal body axis; also the coordinate of a
point 'relative to the origin of body axes
2G 6G 4
The z coordinate of CB
The z coordinate of CG
A coordinate of the displacement of CGrelative to the origin of a set of fixed axes
7.
PP
Z' =
z*
*P*2U 2
z.JH2.
PP tpt*
z„Z ' =1 £p<,
a U
Hydrodyn.imic force component along z
axis (normal force)
Normal force when body angles (a, /3) andcontrol surface angles are zero
Second order coefficient used in representingZ as a function of p. First order coefficient
is zero
First or.ler coefficient used in representingZ as a functicn of q
<tn
z.q
|q!6s
rp
rr
Z ' =-32_q*j £p-c
3.u
z^
q " jp^4
!q| 6 s ip<.3 U
1
rp *P<-4
t
Z rrrr ±pi<
First order coefficient used in representing
Z_ as a function of (tj-1)
Coefficient used in representing Z as a
function of q
Coefficient used in representing Z as a, 6s
function of q
Coefficient used in representing Z as afunction of the product rp
Second order coefficient used in representing
Z as a function of r. First order coefficientis zero
i _ v?
^7ip**UFirst ord'2r coefficient used in representingZ as a function of w
wT7z • = T-^n First order coefficient used in representing
Zw as a function of (77- 1)
Z .w Z •' =T-
7
w3- Coefficient used in representing Z as a
function of w
|wl
IpFuFirst order coefficient used in representingZ as a function of w; equal to zero for sym-metrical function
w|q|'w !hI '
ip*LaJ Coefficient uiicd in representing Z as a
function of q
44
w I w|Z
i i
Z • •' = -»—j-2 Second order coefficient used in representingW|W| W|W| § p<, _ function of u,
z.Z „•_ Z . I ' = —;
—
•—;—-t- First order coefficient used in representingWW 77 WW 77 *fli* 1 t r I ,\
b11'
' sP^ Z , ,as a function of (tj-1)w
jw
I
' '
WwZ ,
Z ' = -^——r- Second order coefficient used in representingWW WV/ i O/,- H Z as a function of w; equal to zero for sym-
metrical function
Z 6b
5P^ U as a function of 6^
Z.. Z., ' = ;—T- First order coefficient used in representing Z6b 6b io/Zn*
ZfiaZ Z, ' = - First order coefficient used in representing6s 5s
$p-t2U2 Z as a function of 6
3
Z. Z, ' = ^?—
,
First order coefficient used in representing6sr» ^TJ iP^U 2
Z6a
as a function of (7,-1)
a Angle of attack
P Angle of drift
6. Deflection of bowplane or sailplane
6 Deflection of rudderr
6 Deflection of sternplane
uc7j The ratio —
—
9 Angle of pitch
# Angle of yaw
Angle of roll
a., b., c. Sets of constants used in the representation ofiiipropeller thrust in the axial equation
A. 2 EQUATIONS OF MOTION
AXIAL FORCE
m [u - vr + wq - xG (q3 + r 2 ) + yG (pq - r) + Zq (pr +
q)J=
+ — X4|"x ' q
3 + X ' r2 + X • rp 1
7 L qq n rr rp r J
+ JL x3|"x.« i
2 L u+ X ' vr+X ' wq
vr wq J
?- i2 Tx ' u2 + X '"v
2 + X ' w2l
7 L UU VV WW J
+ — i2 u 2
[~X. ' ' 6r2 +X. . ' '6s2 +X.. * 6b2 ]2 L 6r 6r 6s 6s 6b 6b J
+ I p l? I a. u2 + b. uu + c. u 2I2 K Li icicJ
- (W - B) sin 9
45
|^ <•* [x „' v3
+ X J w2+ X. . _« 6 2 u3
VV7? ww?7 6r6r77 r
+ X, . „ 62 u2
Os6s77 s(T?-l)
46
LATERAL FORCE
i[_v - wp + ur - yG (ra
+ p2
) + zQ (qr - p) + xG (qp + r)j=
+-2- i4 [y- 1 r+Y.'p+Y ,'PlPl+Y ' pq + Y qr]
2 Lr pr
PlPl pq qr H J
+~ i3
T Y-' v + Y vq + Y ' wp + Y wr2 L v . vq ^ wp r wr J
+f i3[Y
r'Ur+ V UP + Y
|r|6r,a
lr
l6r+Tv|rr,T,I<va+w">*ll r l]
+t /2 Lv *a + v uv + Y
v|vi'v |(v2 +w3 )'J]
+4- *>
2Ty ' vw + Y, ' u3 6rl
2 L vw 6r J
+ (W - B) cos 9 sin<fr
+f *3-Y
rT?'ur(i/-l)
+T ** [YvV
uv + Yv]v|V
Vl <V* + w2 ^l + Y
6r7?« *, a
3
] (7,-1)
47
NORMAL FORCE
m[w - uq + vp - zQ (p2
+ q2
) + xQ(rp - q) + YG (^ + P> j =
+JL X3 [z^' w + Z
yr' vr + Z
vp' vp]
+ i. x» [Z • uq + Z a| q |6s + Zw 'w
|
(v» + w»,i| lq |]
+1 4-[z;'««-+zw'«w + zw1w1'w|(t» +w»)*l]
+f i"[Z |w|,a
lw
l+
-iww' Iw<v"+w">4 l].
*fIs [z— v* + Z
6g' u2 6s + Z
6b' a
a -8b]<
+ (W - B) cos 9 cos<J>
+f t3V uq<T," I)
48
ROLLING MOMENT
I p+ (I - I ) qr - (r + pq) I + (r2
- q2
) I + (pr - q) Ix f v z y n r^ xz n yz xy
+ m I y-. (w - uq + vp ) - z_ (v - wp + ur) I=
+-£- X5
i K- » p + K- ' r + K ' qr + K ' pq + K *p|p|l2 L P ^ r qr n
Pq P4p [p |
P 1P' J
+ ^ i4|~K ' up + K ' ur + K-' vl
2 L p r r v J
+-5- 4* j~K ' vq + K ' wp + K » wr2
L vq ^ wp.
r wr J
.+-£ X
3'[v u2 +K
v» uv+K
y|v |« v|(v»-+w»)*|]
+JL i3 [k 'vw + K £
' u2 6p]2 L vw 6r J
+ (y>, W - y_ B) cos cos 4> - (z^W - z_B) cos 8 sin«t>
Ci Jo \j Jd J
+f^ K*V
ufl CM)
49
PITCHING MOMENT
I q + (I - I ) rp - (p + qr) I + (p2
- r2
) I + (qp - r) Iy -i x z'Mr \r n / Xy \f ' zx nt^
yz
r »
~|
+ m I zG (u - vr + wq) - x_ (w - uq + vp) j=
+-f iS |M- ' q + M ' p
2+ M ' r
2+ M 'rp+M i I'qlq
2 L. q nPP rr rp qjq| HI4I
P 4 r 1+-=- i jM-'w + M 'vr+M 'vp
2 L w vr vp r J
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q'uq + M
lql6s'u|q|6s + M
|w|q'|(v 2 + w2
)-|q]
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|w |' w f(v
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i,"l
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1 u|w| + M ' |w (v2
+ w 2)
2|
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+ £ l*M ' uq (T]-l)
2 H7?
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s J
50
YAWING MOMENT
I r + (I - I ) pq - (q + rp) I + (q2
- p2
) I + (rq - p) Iz y x rn VM Vl yz *
n r xy tr ' zx
r "1
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r|r |
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+ _£_ x4
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)
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2 Lvt? v|v|77 lv '' 6r7? r _T '
51
A. 3 AXIS TRANSFORMATIONS [2]
1) A transformation of an axes system takes a quantity described
in one frame of reference and transforms it into another frame of
reference such that if we measured the same quantity in the second
frame of reference the transformed quantity and the measured quantity
would be identical
.
2) Transforms between frames are needed in the study of the motions
of ocean vehicles because the equations of motion for such a vehicle
are most easily derived in the inertial frame attached to the earth
(x , y , z ) frame, while the forces acting on the vehicle are most
easily evaluated in the frame attached to the vehicle (x, y, z).
Hence, we ultimately desire to transform the equations of motion
from the inertial frame into the non- inertial frame fixed in the
vehicle.
> >3) If V is some vector measure in the x , y , z frame and V
some vector measured in the x, y, z frame which is only changed
in orientation then:
V = T ( ^ , $ , 4> ) V where T ( <A , 8 , <f> ) = the transform.
Where
T (*, 9,f) =
COS0 cos<A
-sin^ cos<j5> + sin<£ sin<9 cos<A
sin<£ sin^ + cos<t> cos<A sin#
cosfl sin<A
cos<£ cds^ + sin^ sin# sin^
sin<?!> cos«A + cos^ sin# sin^
-sin#
sin<£ cos#
COS0 COS0
52
4) If V and V are the same vectors as in (3) above, then
Where
VQ
= T"1
( *, $,*) V
T"1
( + B 0,*) =
cose cos^
cose sin«A
sine
-sin«A cos</> + sin<£ sine cos<A
cos^ cos<A + sin<£ sine sin<A
sin</> cose
sin</> sin^ + cos<£ cos«A sine
•sin<£ cos^ + cos</> sine sin*
cos<£ cose
B.l LIST OF VARIABLES
53
APPENDIX B
VARIABLE MEANING
Six by six matrix containing coefficients
of UDT, VDT, WDT, PDT, QDT, RDT.
AA Six by six matrix set equal to matrix A
before each call to FUNC. The values of
AA are lost in the matrix reduction performed
by LEQT1F.
A1,A2 Limits used for selecting the proper
propeller thrust.
AI.BI.CI Set of constants representing the propeller
thrust in the X-equation.
B
BORATE
BOWMAX
COURSE
DECRIT
Ship's buoyancy.
Average bowplane rate.
Maximum ordered bowplane deflection.
New course for the ship.
Value of depth error when dive planes are
returned to zero deflection.
DELI
DEL3
Calculated deflection of the bowplane.
Calculated deflection of the stern plane.
54
DEL4
DELB
DELBO
DELGM
DELR
DELRO
DELS
DELSO
DELT
DIFF
ICNT
Calculated deflection of the rudder.
Actual bowplane deflection at time t.
Actual bowplane deflection (units changed
for output).
Amount the original GM is to be changed.
Actual rudder deflection at time t.
Actual rudder deflection (units changed
for output).
Actual stem plane deflection at time t.
Actual stern plane deflection (units
changed for output).
Time increment used in iteration.
Difference between present depth and
ordered depth.
Six by one matrix containing solution to
right hand side of equations of motion.
Counter used to count the number of
iterations
ID
INDEX
Forty- character alphanumeric heading.
If greater than zero, read new submarine
coefficients. If less than or equal to
zero, run same submarine for new initial
55
conditions.
IX,IY,IZ,IXY,IXZ,IYZ Moments of inertia.
KCOEFF
L
LEQT1F
M
MAXANG
MCOEFF
NCOEFF
NODIFF
NOPICH
ODEPTH
ONCRS
K - equation coefficients'.
Ship's overall length.
Matrix reduction subroutine from IMSLIB [3]
Ship's mass.
Maximum ordered dive/ascent angle.
M - equation coefficients.
N - equation coefficients.
Acceptable error range around ordered depth
Acceptable error range around zero pitch
angle.
Ordered depth.
Acceptable error range around ordered
course.
P.Q.R
PDT,QDT,RDT
Angular velocity about the x, y, and z
axes respectively.
Angular acceleration about the x, y, and z
axes respectively.
PDT0,QDT0,RDT0 Angular acceleration about the x, y, and z
PHI
PHIO
PO,QO,RO
PSI
PSIO
RHO
RRATE
RUDAMT
STERAT
STERHX
STOW
T
Tl . . . T18
THETA
56
axes, respectively (with units changed for
output).
Angle of roll
.
Angle of roll (with units changed for
output).
Angular velocity about the x, y, and z
axes respectively (with units changed for
output).
Angle of yaw.
Angle of yaw (with units changed for
output).
Sea water density.
Average rudder rate.
Maximum ordered rudder deflection.
Average stern plane rate.
Maximum ordered stern plane deflection.
Storage array for half of output data.
Present time.
Time lag signals.
Angle of pitch.
THETAO
57
Angle of pitch (with units changed for
output).
TLAGB
TLAGR
TLAGS
U,V,W
Time lag for bowplane control system.
Time lag for the rudder control system.
Time lag for stern plane control system.
Forward, lateral, and vertical velocities
respectively.
UDT,VDT,WDT Forward, lateral, and vertical accelerations
respectively (with units changed for output)
UO
U00,V0,W0
Initial forward velocity.
Forward, lateral, and vertical velocities
respectively (with units changed for output)
WT
X,Y,Z
XB,YB,ZB
Ship's weight.
Coordinate labels of the fixed (x , y ,
z ) coordinate system.
The x, y, z position of the center of
buoyancy.
XCOEFF
XDT,YDT,ZDT
X - equation coefficients
Velocities in the fixed coordinate system
along the x , y , z axes respectively.
XG,YG,ZG The x, y, z position of the center of
58
gravity.
YCOEFF Y - equation coefficients
ZCOEFF Z - equation coefficients
59
B.2 COMPUTER PROGRAM LISTING
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B.3 LEQT1F
A - input matrix of dimension N by N containing the coefficient
matrix of the equation AX = B.
On output, A is replaced by the LU decomposition of a
rowwise permutation of A.
M - number of right-hand sides, (input)
N - order of A and number of rows in B. (input)
IA - number of rows in the dimension statement for A and B in
the calling program, (input)
B - input matrix of dimension N by M containing right-hand
sides of the equation AX = B.
On output, the N by M solution X replaces B.
ID6T - input option.
If ID6T is greater than the elements of A and B are
assumed to be correct to ID6T Decimal digits and the
routine performs an accuracy test.
If IDGT equals zero, the accuracy test is bypassed.
WKAREA - work area of dimension greater than or equal to N.
IER - error parameter
terminal error = 128 + N
N = 1 indicates that A is algorithmically singular.
Warning error = 32 + N.
N = 2 indicates that the accuracy test failed. The
computed solution may be in error by more than can be
accounted for by the uncertainty of the data.
CALL LEQT1F (A, M, N, IA, B, IDGT, WKAREA, IER)
ThesisHI 753c.l
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degree of freedom mo-tion simulation model
for use in submarinedesign analysis.
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Pment of a SIX de9 ree of freedom m
3 2768 002 07594 7DUDLEY KNOX LIBRARY
