Simulation of Submarine Manoeuvring

Sebastian ThuneA thesis presented for the degree of

Master of Science

Royal Institute of TechnologySweden

June 2015

Abstract

The best way to save money in a submarine design project is to, in an as early stage as possible, be able tofind errors in a design or construction. A powerful method to find such errors is to put the design throughrigorous testing in simulation software. This project has created a simulation program for a submarinescruising fully submerged. The software is using well-known and established equations of motion (EOM’s)that has been developed through the years. These equations describe the forces and moments acting on thesubmarine in six degrees of freedom and can be described with Newton’s second law of motion. The equationsaccelerations are integrated using the scipy.integrate.odeint module in Python to obtain the translational androtational velocities for a time series. By integrating the EOM’s with different initial values and conditionsa library of standard manoeuvring tests have been created together with graphically tailored representationfor each test result. The software has been verified by comparing a few chosen test results with simulationsoftware currently used by FOI. Through this comparison it is seen that the software correlates well withthe current one in every aspect except one. There is an uncorrelated phenomenon when stern-plane diving issimulated. The error is identified to the stern-plane drag term in the surge equation, Xδsδs . Evidence showsthat an error might exist in the current software and a dialogue with the manufacturer has been commenced.Until the error is solved, results from tests using stern-plane diving motions should be handled with care.

I

Sammanfattning

Simulering av ubatsmanovrering

Det basta sattet att spara pengar inom ett ubatdesignprojekt ar att, i ett sa tidigt skede som mojligt, hitta feli designen. En kraftfull metod att hitta sadana fel ar att fora designen genom rigorosa tester i ett simuleringprogram. Detta projekt har skapat ett sadant program for en ubat som ror sig i undervattenslage. Pro-grammet anvander sig av valkanda och etablerade rorelseekvationer som har utvecklats genom aren. Dessaekvationer beskriver krafter och moment som ar verksamma pa en ubat i sex frihetsgrader som kan beskrivasmed Newton’s andra rorelselag. Ekvationernas accelerationer integreras med hjalp av scipy.integrate.odeintmodulen i Python for att erhalla ubatens translation- och rotations-hastigheter for en tidsserie. Genom attintegrera rorelseekvationerna med olika begynnelsevarden och villkor sa har ett bibliotek av manovrering-stester blivit framtaget tillsammans med anpassad grafisk representation av testresultaten. Programmet harverifierats genom jamforelse av nagra valda test gentemot programmet som i dagslaget anvands av FOI.Denna jamforelse har visat att programmen korrelerar bra i alla fall utom ett. Vid dykning genom attanvanda aktra djuproder sa korrelerar inte programmen gentemot varandra. Felet har blivit identifierattill dragmotstandet som uppstar pa rodren i x-ekvationen, Xδsδs . Det ar inte otroligt att felet kan liggai programmet som anvands i dagslaget och en dialog med skaparna har initierats. Tills detta problem arutrett bor man vara medveten om att resultat fran simuleringar dar aktra djuproder anvands kan ha lagreprecision.

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Acknowledgement

The author would like to offer his deepest gratitude to FOI for the opportunity to perform this thesis attheir institution, and allowing me to test and improve my engineering skills.

A special thanks goes out to my supervisor and Senior Scientist Lennart Bosser for his endless support,discussions and guidance, which made my time at FOI very pleasant and fun. I would also like to thank himfor believing that I was capable to fulfil the task at hand. He has been a great mentor during my time at FOI.

A thanks goes out PhD Mats Nordin and to my friend and classmate MsD Johan Fridstrom for their inputsand support during the thesis. Johan should have praise for not going crazy while sharing an office with me.I would also like to thank him for suggesting me for the task to FOI.

Naturally I would like to thank my examiner PhD Ivan Stenius for the discussions, encouragement andenthusiastic support.

Finally I would like to thank the entire underwater department at FOI for all the intriguing small talks atcoffee and lunch breaks. The knowledge within the facility is truly amazing and it was sincerely interestingto hear all the discussions that took place.

Stockholm June 2015

Sebastian Thune

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Contents

Nomenclature 1

1 Introduction 4

2 Thesis structure and goals 7

3 Limitations 8

4 Equations of motion structure 94.1 Rigid body dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Hydromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Added mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Non-linear cross flow drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Horse-shoe vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 Control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Simulation method 295.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Non-linear cross flow drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Horse-shoe vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.6 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.8 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Validation and verification 376.1 Straight line test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 Bow-plane step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Acceleration test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4 Turning circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.5 Stern-plane deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Results 547.1 Meander test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.2 Control-system failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3 Hydroplane failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.4 Zig-zag tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.4.1 Horizontal zig-zag test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

IV

7.4.2 Vertical zig-zag test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.5 Step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.5.1 Horizontal step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.5.2 Vertical step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8 Conclusion 62

9 Further work 63

References 64

A Determination of hydrodynamic derivatives A1A.1 Straight-line test and rotating-arm technique . . . . . . . . . . . . . . . . . . . . . . . . . . . A1A.2 Planar motion mechanism (PMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3A.3 Strip theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5

B Adams’ Method B1

C Selection of vortex data points C1

D General arrangement D1

E Hydrodynamic derivatives E1

F Program structure F1

V

Nomenclature

vFW Lateral velocity due to lift on the bridge fairwater [m/s]

β Drift angle [rad]

δ1 First stern hydroplane deflection [rad]

δ2 Second stern hydroplane deflection [rad]

δ3 Third stern hydroplane deflection [rad]

δ4 Fourth stern hydroplane deflection [rad]

δb Deflection of bow-plane [rad]

δr Deflection of rudder [rad]

δs Deflection of stern-plane [rad]

η The ratio uc

u [ - ]

φ Roll angle [rad]

ψ Yaw angle [rad]

ρ Density of water [kg/m3]

τ(x) Time interval for vortex [s]

θ Pitch angle [rad]

ξ Bottom clearing bc [ - ]

B Buoyancy force [N]

b Beam of the vessel [m]

C Revolution scaling ratio [ - ]

c Clearing between baseline and bottom [m]

CD Drag coefficient [ - ]

CL Lift coefficient [ - ]

D Propeller diameter [m]

g Gravitational acceleration [kg/s2]

Ixx Moment of inertia about x-axis [kgm2]

Ixz Product of inertia with respect to x- and z-axis [kgm2]

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Iyy Moment of inertia about y-axis [kgm2]

Iyz Product of inertia with respect to y- and z-axis [kgm2]

Izz Moment of inertia about z-axis [kgm2]

J Advance ratio [ - ]

K Moments about the x-axis [Nm]

KQ Torque coefficient [ - ]

KT Thrust coefficient [ - ]

l Length of the vessel [m]

M Moments about the y-axis [Nm]

m The vessel mass [kg]

N Moments about the z-axis [Nm]

n Propeller revolution velocity [rpm]

p Angular velocity about x-axis [rad/s]

Q Propeller torque [Nm]

q Angular velocity about y-axis [rad/s]

r Angular velocity about z-axis [rad/s]

T Propeller thrust [N]

tdf Thrust deduction factor [ - ]

u Linear velocity in x-direction [m/s]

uc Commanded velocity [m/s]

v Linear velocity in y-direction [m/s]

v(x) Velocity component v at any x coordinate [m/s]

W Weight of vessel [N]

w Linear velocity in z-direction [m/s]

w(x) Velocity component w at any x coordinate [m/s]

wf Wake fraction [ - ]

X Force in x-direction [N]

x1 Vortex starting point [m]

x2 Vortex detachment point [m]

xB x coordinate for centre of buoyancy [m]

xG x coordinate for centre of gravity [m]

Y Force in y-direction [N]

yB y coordinate for centre of buoyancy [m]

2

yG y coordinate for centre of gravity [m]

Z Force in z-direction [N]

zB z coordinate for centre of buoyancy [m]

zG z coordinate for centre of gravity [m]

Abbreviations

CB Centre of Buoyancy

CG Centre of Gravity

DOF Degrees of Freedom

EOM Equations of Motion

FOI Totalforsvarets forskningsinstitut (Swedish defence research agency)

HS Hydrostatics

HD Hydrodynamics

RBD Rigid body dynamics

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Chapter 1

Introduction

A quick Google search of the sentence ”definition of submarine” gave the short but rather good answer:”A nautical vessel capable of operating submerged”. In other words, a submarine is a manned atmosphericmulti-purpose vessel that removed the constraints of just moving in the 2D horizontal plane of the vastocean to the 3D ocean space in a controlled way. The idea of submarines can be traced all the way backto the Renaissance, where sketches of an underwater vessel drawn by nobody else then the universal-geniusLeonardo Da Vinci has been found. The first practical submarine designed and manufactured was, however,done 1620 by the Dutch inventor Cornelis van Drebbel. This vessel was propelled by paddles. Today, thesubmarines are considered one of the most complex machines that exist, but seldom seen as they manoeuvrethe dark oceans beneath.

The Swedish history of submarine design begins with the development and building of Nordenfelt I in 1883.Based on an original idea from the English reverend George Garrett, the Swedish inventor and entrepreneurThorsten Nordenfelt built the Swedish first machine driven submarine at AB Palmcrantz & Co at Karlsvikon Kungsholmen, Stockholm. The steam engine came from J. & C. G. Bolinders Mekaniska Verkstads ABat lake Klara at Kungsholmen, Fleminggatan 4, in Stockholm where today the Tekniska namndhuset issituated. The submarine was finally assembled at Ekensbergs yard AB, Kungsholmen, Figure 1.1.

Figure 1.1: Nordenfelt I at Ekensbergs yard AB in Stockholm 1883. Picture: Sjohistoriska Museet.

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During 1884 the Nordenfelt I was further refined at Eriksbergs yard in Goteborg. The submarine was finallyshown for a collection of high-level individuals and royalties from Europe, just north of Landskrona in 1885.In spite of a successful demonstration interest among the nobility did not result in any further business. Thesubmarine was later sold to Greece after being inspected by the Royal Swedish Navy with the commentsthat the interior of the submarine was too hot.

The Swedish navy history of submarine design begins with HMUb Hajen, Figure 1.2, launched July 16,1904. The vessel was designed based on blue-prints that the Swedish marine engineer, Carl Richson had gothis hands on during his apprentice period at Mr. Hollands submarine yard in America 1900. After someredesign and royal acceptance by the Swedish King Oscar II, the submarine was built at the Naval yard inStockholm 1902-04. After a one year test period performed by the KMF (later FMV), and performed by theformer chief engineer of Nordenfelt I, HMUb Hajen became part of the Swedish navy.

Figure 1.2: HMUb Hajen on display in Karlskrona.

In 1945 the German submarine U3503 sank in Swedish territorial waters and was later salvaged by Swedenin 1946[13]. This was the starting point of a huge initiative for national development of submarines withinthe Swedish navy , Swedish industry and governmental agencies and technical university’s. A little bit morethan twenty years later, 1967, Sweden launched their new model, Sjoormen (project A11), again influencedby American submarine development, with a new hull-form giving the submarine superior manoeuvrabilitycompared to the older models. After this, Sweden continued to develop its own submarines and followedSjoormen with three more classes: Nacken (project A14), Vastergotland (project A17) and Gotland (projectA19). Three out of the five submarines that are in use today in the Swedish navy are of the Gotland class.The other two submarines are the modernisation of the Vastergotland class submarines that was updatedand re-launched as the Sodermanland class[8]. This class will be replaced by a new type, project A26,launching in 2019-2021.

Through the 18 classes of Swedish submarines, 68 submarines have been manufactured in Sweden. That is,68 submarines over a span in over 100 years of submarine history in Sweden. It is therefore understandablethat the amount of work invested in the design of the submarine is extremely high. The more quantifiableresults that can be obtained in early phases of the design process, the better and more precise informationcan form the basis for decision. FMV and FOI together with Kockums, SSPA and Chalmers university of

5

technology (CTH) has for more than a decade been developing methods and computer tools for this purpose,analysing and designing for a well-balanced and feasible requirements specification. They are taking vitalaspects of the submarine into consideration such as; hull strength, hydrostatics, hydrodynamics, energysupplies and handling of power, on-board environment, signature and general-arrangement. One of thesetools is manoeuvring simulation software. However, the current manoeuvring software relies on proprietaryand outdated software, which makes it hard to maintain and update. Therefore, FOI instigated this thesiswork at the Royal technical university (KTH) to investigate the implications and amount of work neededto develop a new simulation software. This thesis is the beginning of that simulation software where thecore functions of the new simulation software are implemented. The established and well-known equationsof motion (EOM’s) for a submarine is reformulated into a system of ordinary differential equations andimplemented into the programming language Python. The Python module scipy.intgrate.odeint is used tointegrate the time series solutions for couple of input signals creating some of the standard manoeuvringtests for a submarine. The program takes as input a file of measured or predicted hydrodynamics derivatives,coefficients and then, depending on what test is chosen to be simulated, user defined input signals to createthe time solutions and plot the solutions needed to make a decision of the submarines abilities and design.In the following report the EOM’s are broken down to clarify the theory of how to simulate a body movingthrough a fluid mathematically. The report also treats numerical integration methods for solving the problem,how the different terms in the EOM’s are interpreted and what tests the program is able to run and why.To verify that the program is working to satisfaction, the current program is used as a verification tool.

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Chapter 2

Thesis structure and goals

This thesis has two goals. The first is to create the core functions of a submarine manoeuvring simulationsoftware that can simulate some of the common submarine standard manoeuvring test. The second is toverify the program simulation results against the currently used software.

The thesis structure follows the thesis work. It starts with setting up the limitations of the program createdand follows with a chapter of literature study where the theory and different terms and structure of the EOM’sare presented. In Chapter 5 the different methods that have been used to solve the EOM’s for different timeintervals are presented together with the assumptions that have been been done for the model. Here, allthe state variables that will be solved during the simulation are presented together with their own dynamiclimits and controlling statements. In this chapter the post-processing model is described, presenting whatvariables besides the state variables that are computed and saved for presenting and analysing purposes.Chapter 6 consist of a comprehensive description of the verification process together with the results anddiscussion surrounding it. The tests that were not used for the verification process are later presented anddiscussed in the following chapter. The thesis is finalized with a conclusion of the thesis and a descriptionof further work that needs to be done to create a complete software in Chapter 8 and 9, respectively.

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Chapter 3

Limitations

The programs biggest limitation is that it is only applicable on submarines fully submerged. There is how-ever room to evolve the program into handling wave interactions, surface propulsion hydrodynamics andhydrostatics.

The program cannot handle the use and manipulation of the ballast tanks to use as trim water or compen-sation water or how the free liquid movements will affect the submarine.

Submarines today are dependent of a control-system. However, in the current version of this program, thereis no autopilot available. This is a limitation for some of the tests that should benefit from the option ofusing an autopilot.

The program can only handle forward motion of the submarine. To simulate the submarine moving in abackward motion, the backward EOM’s have to be implemented to the program in similar fashion as de-scribed in this thesis.

The program should preferably be used for conventional submarine designs since the hydrodynamic deriva-tives are estimated from regression analysis made from model test of the A17 and A24 projects. Estimationsof hydrodynamic derivatives for a vessel that has no correlation to these models will probably be far fromthe truth.

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Chapter 4

Equations of motion structure

Modelling a submarines motion involves a study of both statics and dynamics. Statics are considered whenthe body is in rest or moving at a constant velocity, the dynamics are concerned with the bodies accelerations.A submarine moving through the oceans has six DOF (degrees of freedom), giving six different components.These six components are conventionally defined as: surge, sway, heave, roll, pitch and yaw. The notationsused for the vessel in this report are based on Fossen (1994)[6] and are shown in Table 4.1.

Table 4.1: Notations for the six DOF forces/moments, linear/angular velocities,position and Euler angles.

DOF forces &moments

linear &angular vel.

position &Euler angles

1 motion in the x-direction (surge) X u x2 motion in the y-direction (sway) Y v y3 motion in the z-direction (heave) Z w z4 rotation about the x-axis (roll) K p φ5 rotation about the y-axis (pitch) M q θ6 rotation about the z-axis (yaw) N r ψ

The first three coordinates and their corresponding time derivatives in Table 4.1 are the position and trans-lation motions along x, y, and z-axis. The last three corresponds to the orientation angles and rotationalmotions, respectively. When analysing the motions it is convenient to define two coordinate systems: Thebody-fixed and the earth-fixed reference frames. The body-fixed frame is fixed to the moving vessel in itsorigin O, often chosen to coincide with the vessels center of gravity, CG. The body axes X0, Y0 and Z0 aredefined as the longitudinal axis, transverse axis and vertical axis, respectively. The axes and motions aredefined in Figure 4.1.

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Figure 4.1: The body- and earth-fixed reference frames.

Since marine vessels tend to move in low velocities, it is approximated that the acceleration of a point on theearth can be neglected. This results in the earth-fixed reference frame being inertial. Because of this, thepositions and orientations are described relative the inertial, earth-fixed, reference frame while the linear andangular velocities are described in the body fixed frame. The vehicles travelling path relative the earth-fixedcoordinate system is then given by velocity transformation, J , as seen in Equation 4.1[6].

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J =

c(ψ)c(θ) −s(ψ)c(φ) + c(ψ)s(θ)s(φ) s(ψ)s(φ) + c(ψ)c(φ)s(θ) 0 0 0s(ψ)c(θ) c(ψ)c(φ) + s(φ)s(θ)s(ψ) −c(ψ)s(φ) + s(θ)s(ψ)c(φ) 0 0 0−s(θ) c(θ)s(φ) c(θ)c(φ) 0 0 0

0 0 0 1 s(φ)t(θ) c(φ)t(θ)0 0 0 0 c(φ) −s(φ)0 0 0 0 s(φ)/c(θ) c(φ)/c(θ)

(4.1)

where s(·) = sin(·), c(·) = cos(·) and t(·) = tan(·). For the deriving of the transformation matrix in detailsee Fossen (1994)[6]. The earth-fixed velocities will then be transformed according to Equation 4.2

η = Jν, (4.2)

where η is the time derivative of the positions and Euler angles. ν is the translation and rotational motionsaccording to Equation 4.3 and 4.4

η =[x y z φ θ ψ

]T, (4.3)

ν =[u v w p q r

]T. (4.4)

This transformation is vital for extraction of the position and movements of a moving vessel as it is describedmathematically with the six EOM’s. The following sections describes the structure of the derived EOM’sand the theory behind their terms such as hydrostatics, hydrodynamics and rigid-body dynamics. Throughthis chapter the dynamic equation of motion expression is used repeatedly and is, for this report, written as

Mν + C(ν)ν +D(ν)ν + g(η) = τ , (4.5)

whereM is the inertia matrix including the added mass, C(ν) is the Coriolis and centripetal terms includingthe added mass, D(ν) is the damping matrix, g(η) is the vector of gravitational and buoyant forces andmoments, also know as hydrostatics, and τ is the vector of control inputs.

4.1 Rigid body dynamics

The rigid body dynamics are derived through application of Newton’s second law of motion that relates themass m, and the acceleration ν, to the force fC as

fC = νm. (4.6)

Euler’s first and second axioms, states that Newton’s second law of motion can be expressed in terms ofconservation of the linear and angular momentums, pC and hC respectively as

pC = mvC,

hC = ICω,(4.7)

where vC, IC and ω is the linear velocity, inertia tensor and angular velocity about the body’s CG, respec-tively. The time derivatives of Equation 4.7 yields the forces and moments at the body’s CG, fC and mC

respectively as

pC = fC,

hC = mC.(4.8)

The derived rigid-body dynamics work under the two assumptions. The first is that the vehicle is rigid,thus eliminating the forces acting between individual elements of mass. The second assumption is that theearth-fixed reference frame is inertial, eliminating forces due to the earth’s motion. With these assumptionsthe translation motions are described as

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f0 = m(v0 + ω × v0 + ω × rG + ω×(ω × rG))), (4.9)

where rG is the distance from the origin to the centre of gravity expressed in the body coordinates [xG, yG, zG].Therotational motions derived from the second axiom is

m0 = I0ω + ω×(I0ω) +mrG × v0 + ω × v0, (4.10)

where I0 is the body’s inertia tensor defined as:

I0 =

Ixx −Ixy −Ixz

−Iyx Iyy −Iyz

−Izx −Izy Izz

. (4.11)

For complete deduction of the expressions in Equations 4.9 and 4.10 see Fossen[6]. The other notations arewritten in component form as:

f0 = τ1 = [X,Y, Z]T External Forces

m0 = τ2 = [K,M,N ]T Moments from external forces

v0 = ν1 = [u, v, w]T Linear velocities

ω = ν2 = [p, q, r]T Angular velocities

Input of these components into 4.9 and 4.10 gives six equations that can be broken down into a vectorialrepresentation in form of Equation 4.5 as

MRBν + CRB(ν)ν = τRB, (4.12)

where τRB is a sum of all the external forces and moments acting on the submarine further explained insection 4.7. MRB is the rigid-body inertia matrix:

MRB =

m 0 0 0 mzG −myG

0 m 0 −mzG 0 mxG

0 0 m myG −mxG 00 −mzG myG Ixx −Ixy −Ixz

mzG 0 −mxG −Iyx Iyy −Iyz

−myG mxG 0 −Izx −Izy Izz

, (4.13)

and where the rigid-body Coriolis and centripetal matrix CRB(ν) is

CRB(ν) =

0 0 0 m(yGq+zGr) −m(xGq−w) −m(xGr+v)0 0 0 −m(yGp+w) m(zGr+xGp) −m(yGr−u)0 0 0 −m(zGp−v) −m(zGq+u) m(xGp+yGq)

m(yGq+zGr) −m(xGq−w) −m(xGr+v) 0 −Iyxq−Ixzp+Izzr Iyzr+Ixyp−Iyyq−m(yGp+w) m(zGr+xGp) −m(yGr−u) −Iyzq+Ixzp−Izzr 0 −Ixzr−Ixyq+Ixxp−m(zGp−v) −m(zGq+u) m(xGp+yGq) −Iyzr−Ixyp+Iyyq Ixzr+Ixyq−Ixxp 0

. (4.14)

4.2 Hydromechanics

The hydromechanics can, as for the rigid-body, be described with Equation 4.5 where the sum of thehydromechanic forces and moments can be written as

τH = −MAν − CA(ν)ν −D(ν)ν − g(η), (4.15)

where MA and CA(ν) are the hydrodynamic added inertia and Coriolis and centripetal terms, respectively.g(η) is the hydrostatic term presented in section 4.2.3. D(ν) is the combined damping matrix consistingof the potential damping DP(ν), the skin friction DS(ν), the wave drift damping DW(ν) and the vortexshedding damping DM(ν) written as

D(ν) = DP(ν) +DS(ν) +DW(ν) +DM(ν). (4.16)

12

4.2.1 Added mass

Added mass should be understood as the the pressure-induced forces and moments due to a forced harmonicmotion of the body, which are proportional to the acceleration of the body. In other terms, as the bodyis performing a motion in the fluid it is pushing a fluid mass as it is moving through it, generating forcesand moments. If the vehicle is completely submerged, the added mass coefficients will be constant andindependent of wave frequency. To derive the expression of the added inertia matrix, MA, and the thematrix of the hydrodynamic Coriolis and centripetal terms, CA(ν), the fluid kinetic energy approach interms of Kirchhoff’s equation is used. Fluid kinetic energy is the energy needed for the fluid to move asideand close behind the moving vessel. The expression of fluid kinetic energy may be expressed as[12]

TA =1

2νTMAν, (4.17)

where the added inertia matrix, MA, is defined as a 6x6 matrix

MA =

Xu Xv Xw Xp Xq Xr

Yu Yv Yw Yp Yq Yr

Zu Zv Zw Zp Zq Zr

Ku Kv Kw Kp Kq Kr

Mu Mv Mw Mp Mq Mr

Nu Nv Nw Np Nq Nr

. (4.18)

The matrix expressions follow the SNAME (1950) notation[16]. As an example; the hydrodynamic addedmass force XA along the x-axis due to the acceleration u in the x-direction is written as XA = Xuu. Anothercommon way to express the notation is Aij = −MAij . This gives the notation A11 = −Xu for the sameexample as above. However, bodies that are used for underwater vehicles are usually symmetric in multipleplanes. With a dorsal fin at the maximum diameter of the body and four fins located at the rear end body,a finned prolate spheroid is a good approximation for a submarine body. This would then give the reducedexpression[10][9]

MA =

Xu 0 0 0 0 00 Yv 0 Yp 0 Yr

0 0 Zw 0 Zq 00 Kv 0 Kp 0 Kr

0 0 Mw 0 Mq 00 Nv 0 Np 0 Nr

. (4.19)

The added force and moments can be then be derived through potential flow theory, which describes thevelocity field as a gradient of a scalar function. This method is based on the assumptions that the fluid isinviscid, has no circulation and that the body is completely submerged in an unbounded fluid. However,these assumptions are not valid near submerged objects or at the surface. To solve this problem double-bodytheory[6] is applied with the assumption that MA = MT

A , which gives

2TA =−Xuu2 − Yvv

2 − Zww2 −Kpp

2 −Mqq2 −Nrr

2

− 2Krrp− 2p(Ypv)− 2q(Zqw)− 2r(Yrv). (4.20)

Through the kinetic energy of the fluid, the forces and moments are derived through the application ofKirchhoff’s equation. This equation relates the fluids energy forces and moments acting on the vessel.Applying Kirchoff’s equation in component form will then give

13

d

dt

∂TA

∂u= r

∂TA

∂v− q ∂TA

∂w−XA,

d

dt

∂TA

∂v= p

∂TA

∂w− r ∂TA

∂u− YA,

d

dt

∂TA

∂w= q

∂TA

∂u− p∂TA

∂v− ZA,

d

dt

∂TA

∂p= w

∂TA

∂v− v ∂TA

∂w+ r

∂TA

∂q− q ∂TA

∂r−KA,

d

dt

∂TA

∂q= u

∂TA

∂w− w∂TA

∂u+ p

∂TA

∂r− r ∂TA

∂p−MA,

d

dt

∂TA

∂r= v

∂TA

∂u− u∂TA

∂v+ q

∂TA

∂p− p∂TA

∂q−NA. (4.21)

Substituting Equation 4.20 into Equation 4.21 will gives the following added mass terms

XA = Xuu+ Zwwq + Zqq2 − Yvvr − Yppr − Yrr

2,

YA = Yvv + Ypp+ Yrr − Zwwp− Zqpq +Xuur,

ZA = Zww + Zqq −Xuuq + Yvvp+ Ypp2 + Yrrp,

KA = Ypv +Kpp+Krr − vw(Yv − Zw)− wr(Yr − Zq)− Ypwp− vq(Yr − Zq) +Krpq − qr(Mq −Nr),

MA = Zq(w − uq) +Mqq − wu(Zw −Xu) + Ypvr − Yrvp−Kr(p2 − r2) + rp(Yp −Nr),

NA = Yrv +Krp+Nrr − uv(Xu − Yv) + Ypup+ Yrur + Zrur − Ypvq − pq(Kp −Mq)−Krqr. (4.22)

These six equations can be written in component form according to Equation 4.15, where MA is accordingto 4.19 and the hydrodynamic Coriolis and centripetal matrix can thus be written as[6]

CA(ν) =

0 0 0 0 −a3 a2

0 0 0 a3 0 −a1

0 0 0 −a2 a1 00 −a3 a2 0 −b3 b2a3 0 −a1 b3 0 −b1−a2 a1 0 −b2 b1 0

, (4.23)

with the matrix elements

a1 = Xuu,

a2 = Yvv + Ypp+ Yrr,

a3 = Zww + Zqq,

b1 = Ypv +Kpp+Krr,

b2 = Zqw +Mqq,

b3 = Yrv +Krp+Nrr. (4.24)

The coupling cross-terms in the added mass terms, Equation 4.22, can for simplicity according to Ridley &Fontan[15] be written as shown in Table 4.2.

14

Table 4.2: Cross-terms for added mass couplings derived from added mass terms in the CA(ν) matrix.

X Y Z K M NZw = Xwq −Zw = Ywp −Xu = Zq −(Yv − Zw) = Kvw −Zq = Mq Yr = Nr

Zq = Xqq −Zq = Ypq Yv = Zvp −(Yr + Zq) = Kwr −(Zw −Xu) = Mw −(Xu − Yv) = Nv

−Yp = Xpr Xu = Yr Yp = Zpp −Yp = Kwp Yp = Mvr Yp = Np

−Yr = Xrr Yr = Zrp (Yr + Zq) = Kvq −Yr = Mvp Zq = Nwp

−Yv = Xvr Kr = Kpq −Kr = Mpp −(Kp −Mq) = Npq

−(Mq −Nr) = Kqr Kr = Mrr −Kr = Nqr

(Kp −Nr) = Mrp

The added mass terms can be calculated with both numerical and semi-empirical methods, which would beStrip Theory and Planar Motion Mechanism respectively. However, this thesis will not cover that part sinceall hydrodynamic coefficients and derivatives are given for the task. A description of both the numerical andthe semi-empirical method is described in Appendix A.

4.2.2 Damping

In the general case, see Equation 4.16, four terms give contributions to the damping. However, for a fullysubmerged vessel, only two of the terms are effective:

• The quadratic and non-linear skin friction, DS(ν).

• The viscous damping due to vortex shedding, DM(ν).

This will reduce the damping term to D(ν) = DS(ν) +DM(ν), which are the viscous terms. Generally,the motion of an underwater vessel moving in six DOF will be highly non-linear and coupled. Therefore,the damping is considered non-linear as well and is described as a third order function. For an example thedamping in the x-direction for a velocity u is expressed as[11]

XD = Xuu+Xu|u|u|u|+Xuuuu3. (4.25)

In practice, see Fossen[6] and Ridley & Fontan[15], the third order term is often simplified down to thesecond order. The third order term could be used if motions are highly non-linear, which occurs due to thespeed dependence for Reynolds number. Like the added mass terms, the damping terms can be calculatedeither semi-empirically or numerically with the strip theory through Morison’s equation[6]

F (U) = −1

2ρCDAU |U |, (4.26)

where U is the velocity of the vehicle, A is the projected cross-sectional area and CD is the drag coefficient.

4.2.3 Hydrostatics

The hydrostatics term g(η) is the balance between the gravitational and buoyancy forces as explained byArchimedes, also known as the restoring forces. The gravitational force W is acting through the vesselsCG, with its coordinates rG = [xG, yG, zG]. In the same way the buoyancy force B acts in CB, with itscoordinates rB = [xB, yB, zB]. The forces are shown for the xz- and yz-plane in Figure 4.2.

15

(a) xz-plane. (b) yz-plane.

Figure 4.2: Gravity and buoyancy forces acting on a submarine.

Resolving the forces onto the submarines body frame with the Euler angles gives the restoring forces andmoments in the six DOF as[6]

g(η) =

(W −B) sin(θ)

−(W −B)) cos(θ) sin(φ)−(W −B) cos(θ) cos(φ)

−(yGW − yBB) cos(θ) cos(φ) + (zGW − zBB) cos(θ) sin(φ)(zGW − zBB) sin(θ) + (xGW − xBB) cos(θ) cos(φ)−(xGW − xBB) cos(θ) sin(φ)− (yGW − yBB) sin(θ)

. (4.27)

4.3 Non-linear cross flow drag

The manoeuvring characteristics of a submarine depend on, as shown in previous sections, the hydrodynamicforces and moments that are developed at the hull. In a turning manoeuvre the hydroplane lift force willcreate a moment around the submarines centre axis and thus develop a yawing angular velocity. As thesubmarine is turning it will also move with a drift angle β. This will create a lifting force on the hull,pushing the submarine towards the centre of the turn, Figure 4.3.

Figure 4.3: Projected turning radius of a submarine and its drift angle, β.

This kind of turning manoeuvre results in cross-flow separation that generates large hydrodynamic forces inform of drag. In such a turn the components of velocity vary along the hull due to the angular velocity. Theprevious sections has only covered the linear terms of the equation of motion. However, if the turn is sharp

16

with a large drift angle the term become non-linear. The non-linear cross flow may, according to Feldman[5] and Bohlmann [2], be calculated using strip theory i.e. the hull is split into multiple sections where thedrag force is a function of the local velocity. The total forces are then calculated by integrating over thelength of the vessel

YCF = −ρ2Cd

∫l

h(x)v(x)√w(x)2 + v(x)2 dx,

ZCF = −ρ2Cd

∫l

b(x)w(x)√w(x)2 + v(x)2 dx,

MCF =ρ

2Cd

∫l

xb(x)w(x)√w(x)2 + v(x)2 dx,

NCF = −ρ2Cd

∫l

xh(x)v(x)√w(x)2 + v(x)2 dx, (4.28)

where x is the axial location on the hull, Cd is the drag coefficient, h(x) and b(x) are the height and beamrespectively for each strip. The local cross-flow velocities, v(x) and w(x), at each strip are calculated as

v(x) = v + xr,

w(x) = w − xq. (4.29)

This model is assuming that the drag coefficient is constant over the whole length (and direction) of thebody. However, there are models that use separate drag coefficients for better cross-flow predictions. Oneof the models is[6]

YCF = −ρ2

∫l

[Cdyh(x)v(x)2 + Cdzb(x)w(x)2

] v(x)

Ucf(x)dx,

ZCF = −ρ2

∫l

[Cdyh(x)v(x)2 + Cdzb(x)w(x)2

] w(x)

Ucf(x)dx,

MCF =ρ

2

∫l

[Cdyh(x)v(x)2 + Cdzb(x)w(x)2

] w(x)

Ucf(x)x dx,

NCF = −ρ2

∫l

[Cdyh(x)v(x)2 + Cdzb(x)w(x)2

] v(x)

Ucf(x)x dx, (4.30)

where Ucf(x) =√v(x)2 + w(x)2 is the cross-flow velocity.

4.4 Horse-shoe vortex

On the submarine’s slender body it carries a couple of appendages. The shapes on these independentappendages are usually, if not always, streamlined. However, when they are assembled the shape is degradedand the flow around these appendages becomes more complex. One of these complex flow phenomena thatoccur is the horse-shoe vortex at the sail-body junction of the submarine. In the area around the body-appendage junction an adverse pressure gradient is induced so that the upstream flow begins to form atransverse vortex. The transverse vortex is blocked by the sail which causes it to bend into a horseshoe, seeFigure 4.4. The side flow is speeded by the lateral of the appendage similar to an aerofoil, the longitudinalvortex is drawn into the horse-shoe vortex[14].

17

Figure 4.4: The horse-shoe vortex at the sail-body junction[14].

As the submarine turns, lift is created on the bridge fairwater causing this vorticity of varying strength toshed and convect downstream along the hull as a trailing vortex. This vorticity causes circulation aroundthe hull and therefore also lift on the hull. This lift at a location after the sail is considered a product ofthe sail’s lift coefficient, Cl, the local lateral velocity component, w(x) or v(x), and the local lateral velocity,vFW(t − τ [x]), which was developed at the sail at the time the vortex was shed from the sail. With otherwords, the lateral velocity created at the sails quarter chord travels downstream along the hull and thus, ata location on the hull, the vortex velocity will be velocity created at the sail a certain time before, Figure4.5.

Figure 4.5: Illustration of vortex starting point x1 with the lateral velocity at the time t and the detachmentpoint x2 with lateral velocity at the time t− τ [x2].

The time at a location x from the vortex starting point x1, can be calculated according to Feldman[4] as∫ t−τ(x)

t

u(t)dt = x1 − x. (4.31)

The total force acting on the hull is integrated along the hull from the vortex starting point to the vortexdetachment from the hull. This gives the following five terms[2][4]

18

YV = −ρ2Cl

∫ x1

x2

w(x)vFW(t− τ [x]) dx,

ZV = −ρ2Cl

∫ x1

x2

v(x)vFW(t− τ [x]) dx,

KV = KiuvFW(t− τT ) +ρ

2l2z1Cl

∫ x1

x2

w(x)vFW(t− τ [x]) dx,

MV = −ρ2lCl

∫ x1

x2

xw(x)vFW(t− τ [x]) dx,

NV = −ρ2lCl

∫ x1

x2

xw(x)vFW(t− τ [x]) dx. (4.32)

In the KV term the Ki contribution is the interaction between the trailing vortex and the stern controlsurfaces. The lateral velocity components are calculated as mentioned in Equation 4.29 and the velocity ofthe hull-bound vortex is calculated as[4]

vFW(t− τ [x]) = v(x) + x1r − zFW p(x), (4.33)

where zFW is the z-coordinate of the 42% span of the bridge fairwater.

4.5 Propulsion

The most common way to propel a submarine is to use a propeller propulsion system. The propeller willwith a rotating motion of its aerofoil shaped blades create a high pressure behind the vessel and at the sametime a low pressure at the stern generating a forward motion. In this section, the equations and variablesare based on Garme (2007)[7]. The propeller will propel the vessel with a thrust T, calculated as

T = KTρD4n2. (4.34)

With that thrust follows a torque Q as

Q = KQρD5n2. (4.35)

In Equation 4.34 and 4.35 D is the propeller diameter, n is the propeller revolution per second and KT andKQ are the thrust and torque coefficients, respectively. The thrust and torque coefficients are taken fromgraphs made by the manufactures through tests, and are presented as a function of the advance number J,as seen in Figure 4.6.

19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

Advance ratio J

Propeller coeffcients

KT

&η0

KQ

KT

η0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.05

0.1

KQ

Figure 4.6: Thrust and torque coefficients as a function of J.

The advance ratio is a dimensionless ratio between the distance the propeller moves forward during onerevolution and the diameter of the propeller:

J =u(1− wf)

nD, (4.36)

where wf is the wake fraction, taking into consideration that the relative fluid velocity will decrease dueto the wake created from the vessel body. The low pressure created at the stern of the submarine by thepropeller gives an increases resistance. This effect is called the thrust deduction factor, tdf

tdf =∆R

T, (4.37)

where ∆R is the added resistance. The relation between the drag resistance R and the propellers thrustforce can be described as T = R + Ttdf . Assuming that the propeller is mounted in the body frame x-axisand the force is parallel to x, the force in the x-direction is given by

XP = T (1− tdf). (4.38)

Following the propellers mount assumption the torque will only give a pure contribution in the roll momentequation i.e.

KP = −Q. (4.39)

4.6 Control surfaces

Like most water vehicles, submarines are steered with rudders. However, different from surface vessels,the submarine is manoeuvred in space instead of in the plane. The rudders and fins on a submarine areconsidered as hydroplanes and a typical submarine has six hydroplanes. Most common is to have fourstern hydroplanes and two bow/sail hydroplanes. The bow/sail hydroplanes are used for upward/downwardcontrol, whereas the stern hydroplanes are used for both horizontal and vertical movement as rudders andpitching fins. The stern hydroplanes has two common configurations, the cruciform (+-configuration) andthe cross-form (×-configuration) as seen in Figure 4.7.

20

(a) +-configuration. (b) ×-configuration.

Figure 4.7: The cruci- and cross-form stern hydroplane configuration, where α is the hydroplane mountingangle from the submarines vertical line.

The main difference between the two configurations is that for the ×-configuration, all four hydroplanes areused for every movement whether it is horizontal or vertical. Because of this, there is a built in redundancy,meaning that if one or even two hydroplanes breaks down, the vessel is still manoeuvrable in contrast tothe +-configuration. The fact that all four hydroplanes are used for every movement indicates that thehydroplanes does not have to be as big to give the same amount of force to create the rotating moment,and thus creating less drag. The ×-configuration allows the vessel to come closer to the sea bed than a+-configuration without damaging the hydroplanes.

The hydroplanes work as a wing, creating two force components: Drag D, parallel to the incoming flow, andlift L, perpendicular to the incoming flow. The lift and drag can be calculated with the following equations[1]

D =1

2ρu2ACD, (4.40)

L =1

2ρu2ACL. (4.41)

In these equations A is considered the surface area or the hydroplane. The drag coefficient CD is calculatedas the sum of the parasitic drag CD0 and the lift induced drag CDi as

CD = CD0 +C2

L

πaee︸ ︷︷ ︸CDi

. (4.42)

Here, e is the Oswald efficiency factor, which depends on the lift distribution and is in the range of 0.9-0.95for low aspect-ratio hydroplanes. ae is the aspect ratio of the hydroplane and CL is the lift coefficient. Theparasitic drag is often very small compared to the induced drag and can be neglected. The lift coefficientdepend on the angle of attack α as

CL = CLαα, (4.43)

where the slope of the lift curve CLα for a hydroplane in a flow field can be approximated empiricallyaccording to Whicker[17] as

21

CLα =1.8πae

1.8 + cos(Ω)(

a2ecos4(Ω) + 4

)1/2, (4.44)

where Ω is the hydroplanes the quarter chord sweep angle. With these equations the hydroplane forces andmoments can be calculated by knowing their placements with respect to the rotation centre. As an example,the force in the x-direction due to the bow-plane can be deduced from Equation 4.40 to 4.43 as

Xδbδb =ρ

2

(CLδbδb)2

πaeeu2A. (4.45)

4.7 Equations of motion

Newton’s second law of motion is described in Equations 4.9 - 4.10, which are reintroduced here for reference:

f0 = m(v0 + ω × v0 + ω × rG + ω×(ω × rG))),

m0 = I0ω + ω×(I0ω) +mrG × v0 + ω × v0,

where f0 and m0 are the external forces and moments, respectively. The right hand side of the equationis the rigid body dynamics explained in section 4.1. The external forces and moments are described as thesums

∑X = XHD +XHS +XP +XC,

∑Y = YHD + YHS + YCF + YV + YC,

∑Z = ZHD + ZHS + ZCF + ZV + YC,

∑K = KHD +KHS +KV +KP +KC,

∑M = MHD +MHS +MCF +MV +MC,

∑N = NHD +NHS +NCF +NV +NC, (4.46)

where the index HD refers to the hydrodynamic forces/moments as described in section 4.2, HS refers tohydrostatics as described in section 4.2.3, CF refers to the non-linear cross flow drag as described in section4.3, P refers to propulsion as described in section 4.5, V refers to the horse-shoe vortex as described in section4.4 and C refers to the control surfaces as described in section 4.6. Putting the external forces and momentstogether with the rigid body dynamics would then give the six EOM’s. The following six equations are basedon the equations given by Feldman[4] and with some updates given by Lennart Bosser, senior scientist, FOI.These equations are for a submarine vessel moving with a forward speed. In the following paragraph allcoefficients are introduced in dimensionless form, indicated with a prime index. In order to give the correctdimension of force (N) or moment (Nm), the terms are multiplied with an appropriate factor ρ

2 ln, n = 1...5.

22

Force equation in x-direction

m[u− vr + wq − xG(q2 + r2) + yG(pq − r) + zG(pr + q)] =

+ρ

2l4[X ′qqq

2 +X ′q|q|q|q|+ (X ′rr +X ′rrξξ)r2 +X ′rprp]

+ρ

2l3[X ′uu+ (X ′vr +X ′vvξξ)vr +X ′wqwq +X ′w|q|w|q|]

+ρ

2l2[X ′uuu

2 + (X ′vv +X ′vvξξ)v2 +X ′www

2 +X ′w|w|w|w|]

+ρ

2l2[X ′δrδru

2δ2r +X ′δsu

2δs +X ′δsδsu2δ2

s +X ′δbu2δb +X ′δbδbu

2δ2b]

+ρ

2l2[X ′δrδrηu

2δ2r (η − 1

C)C +X ′δsδsηu

2δ2s (η − 1

C)C]

− (W −B) sin(θ) + T (1− tdf) (4.47)

23

Force equation in y-direction

m[v − wp+ ur − yG(r2 + p2) + zG(qr − p) + xG(qp+ r) =

+ρ

2l4[Y ′r r + Y ′r|r|r|r|+ Y ′pp+ Y ′pqpq]

+ρ

2l3[(Y ′r + Y ′rξξ)ur + Y ′v v + Y ′pup+ Y ′v|r|v|r|+ Y ′|v|r|v|r]

+ρ

2l3[Y ′wpwp+ Y ′wrwr + Y ′|r|δru|r|δr]

+ρ

2l2[Y ′∗u

2 + (Y ′v + Y ′vξξ)uv + Y ′v|v|R|(v2 + w2)1/2|+ Y ′vwvw]

+ρ

2l2[Y ′δru

2δr + Y ′δr|δr|u2δr|δr|+ Y ′δrηu

2δr(η −1

C)C]

− ρ

2Cd

∫l

h(x)v(x)[w(x)2 + v(x)2]1/2dx

− ρ

2Cl

∫ x1

x2

w(x)vFW(t− τ [x])dx

+ (W −B) sin(θ) sin(φ) (4.48)

24

Force equation in z-direction

m[w − uq + vp− zG(p2 + q2) + xG(rp− q) + yG(rq + p) =

+ρ

2l4[Z ′qq + Z ′q|q|q|q|+ (Z ′rr + Z ′rrξξ)r

2]

+ρ

2l3[Z ′ww + Z ′quq + Z ′vpvp+ Z ′vrvr + Z ′w|q|w|q|+ Z ′wqwq + Z ′|q|δsu|q|δs]

+ρ

2l2[(Z ′∗ + Z ′ξξ + Z ′ξ|θ|ξθu

2 + Z ′wuw + Z ′vwvw + (Z ′vv + Z ′vvξξ)v2]

+ρ

2l2[Z ′|w|u|w|+ Z ′ww|w(v2 + w2)1/2|]

+ρ

2l2[Z ′δsu

2δs + Z ′δs|δs|u2δs|δs|+ Z ′δbu

2δb + Z ′δb|δb|u2δb|δb|+ Z ′δsηδsu

2(η − 1

C)C]

− ρ

2Cd

∫l

b(x)w(x)[w(x)2 + v(x)2]1/2dx

− ρ

2Cl

∫ x1

x2

v(x)vFW(t− τ [x])dx

+ (W −B) cos(θ) cos(φ) (4.49)

25

Roll moment equation, K

Ixxp+ (Izz − Iyy)qr − (r + pq)Izx + (r2 − q2)Iyz + (pr − q)Ixy

m[yG(w − uq + vp)− zG(v − wp+ ur)] =

+ρ

2l5[K ′pp+K ′rr +K ′qrqr +K ′p|p|p|p|+K ′r|r|r|r|]

+ρ

2l4[K ′pup+K ′rur +K ′vv +K ′wpwp]

+ρ

2l3[K ′∗u

2 +K ′vRuv +K ′iuvFW(t− τT ) +K ′v|v|v|v|]

+ρ

2l3[K ′δr1u

2δr1 −K ′δr2u2δr2 −K ′δr3u

2δr3 +K ′δr4u2δr4 ]

+ρ

2l3[K ′δr1 |δr1 |

u2δr1 |δr1 | −K ′δr2 |δr2 |u2δr2 |δr2 | −K ′δr3 |δr3 |u

2δr3 |δr3 |+K ′δr4 |δr4 |u2δr4 |δr4 |]

+ρ

2l2z1Cl

∫ x1

x2

w(x)vFW(t− τ [x])dx

+ (yGW − yBB) cos(θ) cos(φ)− (zGW − zBB) cos(θ) sin(φ)

−Q (4.50)

26

Pitch moment equation, M

Iyyq + (Ixx − Izz)rp− (p+ qr)Ixy + (p2 − r2)Izx + (qp− r)Iyz

m[zG(u− vr + wq)− xG(w − uq + vp)] =

+ρ

2l5[M ′qq +M ′rprp+M ′q|q|q|q|+ (M ′rr +M ′rrξξ)r

2]

+ρ

2l4[M ′ww +M ′quq +M ′vrvr +M ′wqwq +M ′w|q|w|q|+M ′|q|δsu|q|δs]

+ρ

2l3[M ′∗ +M ′ξξ +M ′ξθξθ +M ′ξ|θ|ξ|θ|u

2 +M ′wuw + (M ′vv +M ′vvξξ)v2]

+ρ

2l3[M ′vwvw +M ′|w|u|w|+M ′ww|w(v2 + w2)1/2|]

+ρ

2l3[M ′δsu

2δs +M ′δs|δs|u2δs|δs|+M ′δbu

2δb +M ′δb|δb|u2δb|δb|+M ′δsηu

2δs(η −1

C)C]

+ρ

2Cd

∫l

xb(x)w(x)[w(x)2 + v(x)2]1/2dx

− ρ

2lCl

∫ x1

x2

xv(x)vFW(t− τ [x])dx

− (xGW − xBB) cos(θ) cos(φ)− (zGW − zBB) sin(θ)

− ρ

2l2X ′δbδbu

2δ2b · dbp (4.51)

27

Yaw moment equation, N

Izzv + (Iyy − Ixx)pq − (q + rp)Iyz + (q2 − p2)Ixy + (rp− p)Izx

m[xG(v − wp+ ur)− yG(u− vr + wq)] =

+ρ

2l5[N ′r r +N ′r|r|r|r|+N ′pp+N ′pqpq]

+ρ

2l4[N ′pup+ (N ′r +N ′rξξ)ur +N ′vv +N ′v|r|v|r|+N ′|v|r|v|r +N ′wrwr +N ′|r|δru|r|δr]

+ρ

2l3[N ′∗u

2 + (N ′v +N ′vξξ)uv +N ′v|v|Rv|(v2 + w2)1/2|]

+ρ

2l3[N ′δru

2δr +N ′δr|δr|u2δr|δr|+N ′δrηu

2δr(η −1

C)C]

− ρ

2Cd

∫l

xh(x)v(x)[w(x)2 + v(x)2]1/2dx

− ρ

2lCl

∫ x1

x2

xw(x)vFW(t− τ [x])dx

+ (xGW − xBB) cos(θ) sin(φ)− (yGW − yBB) sin(θ) (4.52)

28

Chapter 5

Simulation method

The mathematical model for a submarine is a system of six differential equations, Equations 4.47 - 4.52, thatdescribes the motion in six degrees of freedom. The different coefficients and dimensionless derivatives in theEOM’s are given in a submarine coefficient text file from a program provided by FOI. The following chapterdescribes the method of how the equations are used to extract the velocity and position of a submarine.

5.1 Assumptions

The following assumptions are used for this method:

• The submarine is modelled as a single rigid body

• The submarine is fully submerged.

• The mass and mass distribution are constant.

• The submarine is modelled in a calm environment i.e. this model is neglecting wave interactions.

• Rotational effects of the earth are negligible, making the transformation between earth-fixed and body-fixed reference frame valid.

• Fluid properties are considered constant.

• Local variances of the gravitational field are ignored.

• One propeller propulsion system mounted in the body frame x-axis only contributing with XP =T (1− tdf) and KP = −Q

• The submarine is moving with enough velocity and is large enough to yield Reynolds number over5.5 · 105.

• The drag coefficient Cd is considered constant and not dependent on orientation

• The lift coefficient Cl is considered constant and not dependant on orientation

• Symmetry in the xz -plane with a conventional submarine design.

29

5.2 Non-linear cross flow drag

The non-linear cross flow is, as described in section 4.3, calculated using the strip method where the totalforce is computed by integrating each two dimensional section over the length of the submarine. The dragforce in the y-direction is

YCF = −ρ2Cd

∫l

h(x)v(x)√w(x)2 + v(x)2︸ ︷︷ ︸f(x)

dx.

The integral is solved by using the numerical implementation of the trapezoidal rule[3]:

I =h

2

N∑k=1

−1 (f(xk+1) + f(xk)) , h =L

N − 1. (5.1)

The widths, heights and longitudinal locations x of each two dimensional segment are given in the coefficientfile. However, the longitudinal location is given with its origin at the aft of the ship. The longitudinallocations must be transformed to the body reference system by subtracting xap, see in Figure 5.1

Figure 5.1: Location of the aft longitudinal position with respect to centre of rotation, xap.

5.3 Horse-shoe vortex

The horse-shoe vortex is, as described in section 4.4, calculated by integrating the lift over a span betweenthe vortex starting point x1 to the position x2 where the vortex leaves the body. If the drift angle β is small,the vortex will not leave the body until it reaches the aft of the vessel. However, as the drift angle increases,detachment from the hull occurs earlier. The location of the detachment from the hull is calculated accordingto Feldman[4] and Bohlmann[2] as

x2 =

xap for|β| ≤ βST

x1 − (x1 − xap)(S1 + S2|β|) for|β| > βST,(5.2)

where the drift angle is calculated as

β = −tan−1( vu

), (5.3)

where S1, S2 and βST are constants that are obtained from experimental measurements. βST is the driftangle where the detachment of the vortex begins to travel from the aft towards the tower. As explainedin section 4.4, the local lateral velocity vFW at the sail generates a vortex, which travels down along thehull with the flow velocity u. That means, at a certain location on the hull, the lateral velocity will be the

30

velocity at the sail some time offset τ back in time. This offset can be calculated with Equation 4.31. Sincethe velocity u is constant over the span of the hull the time offset can be expressed as

τ =x1 − xu(t)

, (5.4)

where x is a vector of locations between x1 and x2, giving a range of time offsets for each location. Thelocal lateral velocity for a location x is then calculated as stated in Equation 4.33, which is dependent onthe lateral velocity component v and the rotational velocities r and p. However, the velocity components areneeded at specific times t−τ , and the integrator is solving with a time-step that is either constant or adaptivedepending on the preferred algorithm and error limit. A one-dimensional spline is used to interpolate theprevious time-step solutions to render the lateral velocity as a continuous function of time. When the correctlateral velocity is interpolated for all the locations between x1 and the detachment at x2, the integral is solvedusing the numerical trapezoidal method in Equation 5.1.

5.4 Control surfaces

In the coefficient file, the stern hydroplanes are calculated assuming a cross-configuration (×−configuration),and each coefficient is considered for one hydroplane. Therefore, the coefficients regarding the stern hy-droplanes must be multiplied with 4 before using in Equations 4.47 - 4.52. As presented in section 4.6,when using the cross-configuration all four stern hydroplanes are used to perform a manoeuvre, but for acruci-configuration only two are used as shown in Figure 5.2, where the submarine is initiating a horizontalturn. Positive hydroplane deflection is with its trailing edge towards port side as the figure is portraying.

(a) +-configuration (cruci) seen from behind. (b) ×-configuration (cross) seen from behind.

Figure 5.2: The cruci- and cross-form hydroplane forces due to their deflection Fδ and their force componentsin z- and y-direction, Fδz and Fδy respectively.

31

Assuming that the cross-configuration has a mounting of 45 degrees from the submarines vertical line, thehydroplane force for one blade for each configuration will be

Fδy× =Fδ√

2,

Fδy+ = Fδ. (5.5)

One blade on the cruci-configuration will thus create a force of√

2 times larger than a blade with a sweep of45 degrees. If a cruci-configuration model submarine is used, each of the stern hydroplane coefficients shouldbe multiplied with a factor of 2

√2 since the cruci-configuration only uses two hydroplanes instead of four

for a turn or a dive.

In the EOM’s, Equations 4.47 - 4.52, the stern hydroplanes are treated with two variables, except in therolling moment equation, Equation 4.50, where each plane has its own variable. The horizontal manoeuvringvariable is the rudder deflection δr calculated as

δr× =(δ1 + δ2 + δ3 + δ4)

4, (5.6)

δr+ =(δ2 + δ4)

2, (5.7)

where δ1−δ4 are the angles of the four different hydroplanes as seen in Figure 5.2. The vertical manoeuvringvariable is the stern-plane deflection δs calculated as

δs× =(−δ1 + δ2 − δ3 + δ4)

4, (5.8)

δs+ =(−δ1 − δ3)

2. (5.9)

For this thesis, the deflection of the stern hydroplanes are defined as positive when the trailing edge is to-wards port side of the submarine and negative towards starboard.

On a submarine there are some dynamic movements that has to be considered to be able to get as good amodel as possible. Such dynamics is considered with the hydroplanes. As the controller orders a deflectionof the hydroplanes to execute a turn or dive, the planes will be rotating with a limited rate until the desireddeflection is reached. This rotation dynamic is described with a first order differential equation:

δi =(δc − δi)

Tr, i = 1, ..., 5, (5.10)

where i = 1 − 4 are the stern rudders and i = 5 is for the bow plane. δc is the commanded deflection andδi is the current deflection. Tr is a time constant for the change of the hydroplanes. To control that therotational velocity is not changing to fast and that the deflection does not overshoot, they are controlledwith the following two statements:

|δ| ≤ |δ|max,

|δ| ≤ δmax. (5.11)

5.5 Propulsion

Section 4.5 explains how the torque and thrust of the propulsion affects the EOM’s. To calculate the thrustand torque from the propeller, the thrust and torque coefficients are needed. The coefficients are given inthe coefficient file as a fourth order polynomial dependent on the advance ratio, J . The advance ratio iscalculated with Equation 4.36 as

32

J =u(1− wf)

nD,

where wf and D are given in the coefficient file. When the advance ratio is known the thrust coefficient iscalculated as1

KT = kT0 + kTJJ + kTJ2J2 + kTJ3J3 + kTJ4J4. (5.12)

The torque coefficient is calculated with the same method as

KQ = kQ0 + kQJJ + kQJ2J2 + kQJ3J3 + kQJ4J4. (5.13)

Here, kQ0, kQJi , kT0 and kTJi are polynomial multipliers from the input file. With Equation 4.34 and 4.35the thrust and torque are calculated, respectively, as

T = ρKTn2D4,

Q = ρKQn2D5.

Just like the hydroplanes cannot move instantly to the commanded deflection, the engine needs some time tochange to a new desired rotational speed. Therefore, the rotational speed rate is controlled by a first orderdifferential equation

n =(nc − n)

Tn, (5.14)

where nc and n are the commanded and current revolutions per second, respectively. Tn is a time constantfor the change of the rotational speed. The rotational speed and its time derivative are limited accordingthe following two statements:

|n| ≤ |n|max,

|n| ≤ nmax. (5.15)

The propeller moment must not exceed the maximum allowed propeller moment. If, during an acceleration,the propeller moment exceeds the maximum allowed moment, the rate of change of the propeller rotationalspeed is reduced by setting the time derivative to

n =

(Qmax

Q− 1

)n. (5.16)

If the propeller is going in reverse the torque will be negative instead and therefore the moment must nottranscend the negative maximum allowed propeller moment, −Qmax. If this occurs, the rate of change ofthe propeller rotational speed is limited to

n =

(−Qmax

Q− 1

)n. (5.17)

5.6 Model

The mathematical model can, as described before, be expressed with Newton’s second law of motion

Aν = F , (5.18)

where A is the inertia matrix containing the total mass, the added mass and the moment of inertia. Thismatrix is multiplied with an acceleration vector containing the linear and angular accelerations for six degreesof freedom. Applying this model to the EOM’s in Equations 4.47 - 4.52 will thus yield

1This interpolation of the thrust and torque coefficients is chosen for compliance with the existing program.

33

m− ρ2 l

3X′u 0 0 0 zGm −yGm

0 m− ρ2 l3Y ′

v 0 −yGm− ρ2 l4Y ′

p 0 xGm− ρ2 l4Y ′

r

0 0 m− ρ2 l3Z′

w yGm −xGm− ρ2 l4Z′

q 0

0 −zGm− ρ2 l4K′

v yGm Ixx− ρ2 l5K′

p −Izy −Izx− ρ2 l5K′

r

zGm 0 −xGm− ρ2 l4M ′

w −Ixy Iyy− ρ2 l5M ′

q −Iyz−yGm xGm− ρ2 l

4N ′v 0 −Izx− ρ2 l

5N ′p −Iyz Izz− ρ2 l

5N ′r

uvwpqr

=

XYZKMN

. (5.19)

At this form the equation can be re-written to be solved for the linear and angular accelerations, i.e.

ν = A−1F . (5.20)

This form will thus give six ordinary differential equations that can be written as

u = A−111 X +A−1

12 Y +A−113 Z +A−1

14 K +A−115 M +A−1

16 N,

v = A−121 X +A−1

22 Y +A−123 Z +A−1

24 K +A−125 M +A−1

26 N,

w = A−131 X +A−1

32 Y +A−133 Z +A−1

34 K +A−135 M +A−1

36 N,

p = A−141 X +A−1

42 Y +A−143 Z +A−1

44 K +A−145 M +A−1

46 N,

q = A−151 X +A−1

52 Y +A−153 Z +A−1

54 K +A−155 M +A−1

56 N,

r = A−161 X +A−1

62 Y +A−163 Z +A−1

64 K +A−165 M +A−1

66 N. (5.21)

To get the translation and rotational motions, the ordinary differential equations in Equation 5.21 are solvedby numerical integration from an initial state vector, using the Adams’ method[3]. The Adams’ methodis based on both forward and backward Euler method, using a multi-step approach with Adams-Bashforthmethod as a predictor and Adams-Moulton as a corrector. The Adams’ method is described in AppendixB. With the same method, the earth-fixed coordinates and Euler-angles are extracted. From Equation 4.2the following differential equations are extracted:

φ = p+ q sin(φ) tan(θ) + r cos(φ) tan(θ),

θ = q cos(φ)− r sin(φ),

ψ = [q sin(φ) + r cos(φ)] / cos(θ),

x0 = u cos(θ) cos(ψ) + v(sin(φ) sin(θ) cos(ψ))

+ w(sin(φ) sin(ψ) + cos(φ) sin(θ) cos(ψ)),

y0 = u cos(θ) sin(ψ) + v(cos(φ) cos(ψ) + sin(φ) sin(θ) sin(ψ))

+ w(cos(φ) sin(θ) sin(ψ)− sin(φ) cos(ψ)),

w0 = −u sin(θ) + v cos(θ) sin(φ) + w cos(θ) cos(φ). (5.22)

In this model there are eighteen derivatives and therefore the same number of state variables. Meaningthat there are eighteen variables that change with time. So, after integrating the differential equations for acertain time interval, the following state vector will be updated at each time step:

x =[u v w p q r φ θ ψ x0 y0 z0 δ1 δ2 δ3 δ4 δ5 n

].

34

5.7 Post Processing

With the state variables calculated for each time step, more components can be calculated for presenting,analysing and verifying purposes. The following section is covering all the post processing variables that arecalculated. Many of the variables and equations in this section has been shown earlier in the report, but arementioned again for the readers convenience.

From the translational velocity components u, v and w, the translation speed V is calculated as

V =√u2 + v2 + w2. (5.23)

The composite stern hydroplane angles δr (turn) and δs (dive) for an ×-configuration are calculated fromindividual hydroplane angles with Equations 5.6 and 5.8 as

δr =(δ1 + δ2 + δ3 + δ4)

4,

δs =(−δ1 + δ2 − δ3 + δ4)

4.

The drift angle β is a key component during turns and has a significant influence of the vessels turningability. The drift angle is calculated with Equation 5.3 as

β = −tan−1( vu

).

From post processing, it is of importance that the propulsion variables can be analysed. Therefore, theadvance ratio J, the thrust T and torque Q are calculated with Equations 4.34, 4.35 and 4.36 as

J =u(1− wf)

nD,

T = ρKTn2D4,

Q = ρKQn2D5.

Adding these variables and the time steps to the integrated state variables will give a matrix with thefollowing 26 components to be used for analysis of the submarine:

x =[t u v w p q r φ θ ψ x0 y0 z0 δ1 δ2 δ3 δ4 δb n V δr δs β J T Q

]

35

5.8 Tests

With the model and post processing described in the previous sections, different tests can be simulated toverify if the submarine design is fulfilling the requirements or if the design has to be altered. The followingtests are currently implemented in the program:

• Acceleration test.

• Stern plane deflection.

• Turning circle test.

• Meander test.

• Straight line test.

• Hydroplane failure.

• Control-system failure.

• Zig-zag test.

• Step response test.

The tests will be elaborated in detail in Chapters 6 and 7.

36

Chapter 6

Validation and verification

Validation is the process of assuring that a product or system meets the stakeholders’ needs. In this casethe complete method, with towing-tank experiments with scale models to measure the hydrodynamic deriva-tives and applying these in a system of linearized differential equations, should be compared to the dynamicbehaviour of a full-scale submarine. Such validation is beyond the scope of this thesis. The fact that thetank-test and simulation approach has been used for decades as a standard method for analysing of bothsurface and submarine ships is real-life evidence that the method is useful for its purpose.

Verification of the results from the tests listed in section 5.8 is a very important step, due to the fact thatthese are the results that the design of the submarine is verified on. The best way to verify the model is bytaking full-scale submarine test data and compare it to the simulated results. However, due to the fact thatsubmarine designs are sensitive information as are their manoeuvring capabilities, this verification processcannot be done. The new software is therefore compared to FOI’s current simulation software. This softwarecreates an output file containing a matrix with the following vectors:

x =[t x0 y0 z0 ψ δ1 δ2 δ3 δ4 u v φ δs δr V −ψ −δs β θ r δb Q n J

]This means that every result from the current simulation software can be directly compared to resultssimulated in this thesis, see section 5.7. However, due to the fact that the programs are using differentintegration methods, the time steps will differ and a visual comparison of the time solutions is deemedsufficient. The verification can be done by comparing all variables for each and every test. However, thisis a bit time consuming and not very effective. During development of the software a structured approachto verification was needed. This approach, described below, focused on a subset of the available test cases,namely:

• Straight line test.

• Bow-plane step response.

• Acceleration test.

• Turning Circle test.

• Stern-plane deflection test.

The strategy behind the selected test sequence is to start with cases having a minimum coupling between theEOM’s, then gradually proceed to more complex cases with more cross-coupling between the equations. Forinstance, the straight line test has the minimum coupling with no hydroplane deflection and just propulsionacting on the vessel. As remembered from section 4.5 the propulsion has two terms acting in the X - andK -equations, thrust and torque respectively. The roll moment will create cross-coupling in other EOM’s,but it will be insignificant and if there is a major difference between the results the error probably lies in thepropulsion model. The bow-plane step response has the same number of couplings as the straight line test.

37

The hydroplane force from the bow plane creates the least moment on the hull since its location is close tothe CG, restricting the coupling to be mostly between the X - and Z -equations. The bow-plane step responsealso gives a chance to see that the hydroplane dynamics works. The acceleration test has the same couplingas a straight line test and gives an opportunity for further verification of the propulsion dynamics. Afterthese tests, the amount of coupling is increased by starting to use the stern hydroplanes. The turning circletest will increase the coupling mainly between the X -, Y - and N -equations, while the stern-plane deflectionis exercising the coupling between the Z - and M -equations.

Special concern is used when isolating the effects of the non-linear cross flow drag and the horse-shoe vortex.This is done by manually manipulating the coefficient file and setting CL = CD = 0. The verification is thusdone in two steps for the above described tests. The two steps being:

1. Without contribution from non-linear cross flow drag and horse-shoe vortex.

2. With contribution from non-linear cross flow drag and horse-shoe vortex.

With this two-step approach it is possible to first check the linear part of the equations to verify a correctimplementation and then proceed to verification of the non-linear contributions. As mentioned in sections4.3 and 4.4, the non-linear cross flow drag and horse-shoe vortex are calculated with strip theory, whichintegrates two dimensional strips along the hull to get the three dimensional effect. For the non-linear crossflow drag, the number of two dimensional sections is given in the coefficient file. For the horse-shoe vortexthe number of sections for integration is set by the designer of the program. If the data points are too few,the trapezoidal method might over/under estimate the force on the hull. However, too many points willincrease the computational time. Therefore, a verification of the necessary number of locations along thehull to be integrated is performed. Increasing the amount of locations until the change is low enough in theresult yields the accurate number of data points needed. The effect of the horse-shoe vortex is determinedto be with 20 data points spaced with equal distance along the hull. The determination of the number ofdata points is shown in Appendix C.

The following sections presents the five chosen tests and their two-step comparison between the programcreated for this thesis and the program currently used by FOI. The current program is referenced to as”Current”, while the program written for this thesis is referenced as ”New”. Each test is simulated withthe Flipper Topolino submarine1. The submarine general arrangement is shown in Appendix D and itsderivative, mass properties and miscellaneous inputs are listed in Appendix E. The coded program steps andimplementations are explained in the program structure in Appendix F.

6.1 Straight line test

The aim of the test is to see how the vessel is behaving without any hydroplane induced forces. The per-formance of the submarine is measured by its loss/gain in depth z0, its pitch θ and yaw ψ angles. Thesimulation begins with the vessel moving at a constant velocity of u0 = uc for a user defined simulation time.The test is simulated with the following input variables:

uc = 5 knotstsim = 200 s

The simulated result is shown in Figure 6.1.

1The Flipper Topolino is a submarine concept developed as a student task during a course in submarine design at FOI2009-2010. It was selected for the tests and verification since it does not represent any current Swedish or foreign navy submarine.

38

0 20 40 60 80 100 120 140 160 180 200

−0.6

−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Yaw

&Pitch[]

t ime [s ]

0 20 40 60 80 100 120 140 160 180 200

197.2

197.4

197.6

197.8

198

198.2

198.4

198.6

198.8

199

199.2

199.4

199.6

199.8

200

Depth

[m]

YawPitchDepth

Figure 6.1: Yaw, pitch and depth as a function of time resulting from constant speed and no hydroplanedeflection.

The figure shows how the vessel behaves without any hydroplane activities. Initially the pitch is decreasingto a stable value of -0.45. This is a rather small angle, but as the depth plot shows, it has the effect ofdecreasing the depth with almost 3 m. This shows that the vessel has a natural pitch moment as it moveswith a forward velocity, which is a result of the drag from the sail and bow-plane hydroplanes. The vesselalso has a tendency to ”pull” towards one side as seen from the yaw angle that is increasing with a almostconstant slope. This means that the vessel will perform a slow turn in the horizontal plane as well. Thisis probably a result of the propeller torque giving the vessel a slight roll angle combined with the effects ofdrag on the sail.

Step 1.

The linear comparison between the two software’s using the same inputs as in the previous simulation areshown in Figure 6.2.

0 20 40 60 80 100 120 140 160 180 2009.227

9.228

9.229

9.23

9.231

t ime [s ]

Torq

ue,Q

[kNm]

0 20 40 60 80 100 120 140 160 180 2000.6355

0.6356

0.6356

0.6357

0.6357

t ime [s ]

Advancera

tio,J

[-]

CurrentNew

(a) Propulsion.

0 50 100 150 200 250 300 350 400 450 500−0.4

−0.3

−0.2

−0.1

0

0.1

x 0 [m]

y0[m

]

CurrentNew

0 50 100 150 200 250 300 350 400 450 500

197

197.5

198

198.5

199

199.5

200

x 0 [m]

z0[m

]

(b) Path.

Figure 6.2: Straight line test comparison between Current (blue) and New (red) simulation programs.The left two graphs present a comparison in the propulsion terms J and Q. The right two graphs show acomparison of the vessel path both in the horizontal (top) and vertical plane (bottom).

39

From the comparison in the figures it can be seen that the forward motion correlates very well with the currentprogram. There is a small difference in the torque that is constant over the whole simulation. This might bedue to different precision and libraries for floating-point calculations. For an example, the current programand the new have small difference in the initial velocity. The EOM’s are handling its variables according themetric system so as the speed is transformed from knots to m/s the rounding gives: u0,Current = 2.5720 m/swhile u0,New = 2.5722 m/s. There is also a small uncorrelated phenomena in the vertical plane at the end ofthe simulation. This can be due to floating-point arithmetic’s, but can also indicate that there is somethingwrong with the Z - or M - equations.

Step 2.

Adding the non-linear terms for the simulation gives the results shown in Figure 6.3.

0 20 40 60 80 100 120 140 160 180 2009.227

9.228

9.229

9.23

9.231

t ime [s ]

Torq

ue,Q

[kNm]

0 20 40 60 80 100 120 140 160 180 2000.6355

0.6356

0.6356

0.6357

0.6357

t ime [s ]

Advancera

tio,J

[-]

CurrentNew

(a) Propulsion.

0 50 100 150 200 250 300 350 400 450 500−0.4

−0.3

−0.2

−0.1

0

0.1

x 0 [m]

y0[m

]

CurrentNew

0 50 100 150 200 250 300 350 400 450 500

197

197.5

198

198.5

199

199.5

200

x 0 [m]

z0[m

]

(b) Path.

Figure 6.3: Straight line test comparison between Current (blue) and New (red) simulation programs withnon-linear terms. The left two graphs present a comparison in the propulsion terms J and Q. The right twographs show a comparison of the vessel path both in the horizontal (top) and vertical plane (bottom).

From the comparison in the figures it can be seen that the forward motion correlates very well with thecurrent program. The same difference that was discussed in the previous test, see Figure 6.2, is seen in thistest as well. This is expected since the effect from the non-linear terms are insignificant due to the low lateralvelocities and drift angle in a straight line test. The error in the vertical path is however smaller with thenon-linear terms as shown in Figure 6.3(b).

6.2 Bow-plane step response

The step-response is a test where the rudder is set at a certain deflection at certain times. Assuming thatthe vessel has a pure velocity in the surge direction and that the initial value of the velocity is equal to thecommanded velocity, u0 = uc. All the other initial values are set to zero. The first hydroplane executionδc = δ1 at t = 0 is held until the next time step, t = t1, where the second execution takes place, δc = δ2.This is done for at least four different hydroplane deflections, see Figure 6.4.

40

Figure 6.4: Example of a step-response test with rudder angle as a function of time.

The bow-plane step response is simulated with the following inputs:

δb0 = 10 at t0 = 0 sδb1 = 5 at t1 = 100 sδb2 = −5 at t2 = 200 sδb3 = −10 at t3 = 300 su0 = uc = 5 knotstsim = 400 s

Results form the simulation is shown in Figure 6.5.

0 50 100 150 200 250 300 350 400−11−10−9−8−7−6−5−4−3−2−1

0123456789

1011

δb&

Pitch

[]

t ime [s ]

0 50 100 150 200 250 300 350 400

183

185

187

189

191

193

195

197

199

201

Depth

[m]

δ bPitchDepth

Figure 6.5: Bow-plane deflection, pitch and depth as a function of time during a bow-plane step responsetest.

From this simulation, it is seen that the bow-planes are placed so that a diving manoeuvre can be performedwith minimal influence of the pitch angle, with just the bow-planes.

41

Step 1.

Using the same inputs as the previous simulation, a comparison of the two software linear terms is done andillustrated in Figure 6.6.

0 50 100 150 200 250 300 350 400−15

−10

−5

0

5

10

15

t ime [s ]

δb[]

CurrentNew

(a) Bow-plane deflection.

0 100 200 300 400 500 600 700 800 900 1000−2

−1.5

−1

−0.5

0

0.5

x 0 [m]

y0[m

]

CurrentNew

0 100 200 300 400 500 600 700 800 900 1000

180

185

190

195

200

x 0 [m]

z0[m

]

(b) Path comparison.

Figure 6.6: Bow-plane step response comparison between Current (blue) and New (red) simulation programs.The left graph show the bow-plane deflection over the simulation. The right two graphs show the horizontal(top) and vertical (bottom) paths of the submarine.

The comparison in Figure 6.6(a) indicates that the hydroplane dynamics are working as intended. The pathsmatch very well in the horizontal plane as seen in the upper graph of Figure 6.6(b). However, the verticalplane is uncorrelated. This indicates that there might be something wrong with the X - and/or Z -equations.Using the bow-plane reduces the effect of the pitching moment but it does not exclude it so the error mightbe in the M -equation as well. It should be noted that the error is under 1 m and could well be caused byfloating-point differences.

42

Step 2.

Adding the non-linear terms gives the result shown in Figure 6.7.

0 50 100 150 200 250 300 350 400−15

−10

−5

0

5

10

15

t ime [s ]

δb[]

CurrentNew

(a) Bow-plane deflection

0 100 200 300 400 500 600 700 800 900 1000−2

−1.5

−1

−0.5

0

0.5

x 0 [m]

y0[m

]

CurrentNew

0 100 200 300 400 500 600 700 800 900 1000

180

185

190

195

200

x 0 [m]

z0[m

]

(b) Path comparison

Figure 6.7: Bow-plane step response comparison between Current (blue) and New (red) simulation programswith the non-linear terms. The left graph show the bow-plane deflection over the simulation. The right twographs show the horizontal (top) and vertical (bottom) paths of the submarine.

From the figure it is visible that the calculations correlates well, both in the horizontal and vertical plane.The small error in the vertical plane, see Figure 6.6(b), is reduced as the non-linear terms are introduced,see Figure 6.7(b).

6.3 Acceleration test

This test illustrates how the vessel moves during acceleration. It can be seen how different propeller dynamicscan affect the time it takes for the vessel to reach a constant velocity. The vessel starts with initial velocityu0 = v0 = w0 = 0 and a commanded propeller revolution speed nc and will be simulated for a simulationtime tsim. To control the propellers dynamics, the maximum torque Qmax and rotational acceleration nmax

is set by the user. The resulting revolution velocity n and the submarine’s velocity V are then plotted asfunctions of time.

The result from the test shown in Figure 6.8. was simulated with the following input variables:

u0 = 0 knotsn0 = 0 rpmnc = 100 rpmnmax = 25 rpm/sQmax = 50 kNmtsim = 200 s

43

0 20 40 60 80 100 120 140 160 180 20005

101520253035404550556065707580859095

100105

t ime [s]

Revolutions[rpm]

0 20 40 60 80 100 120 140 160 180 20000.511.522.533.544.555.566.577.588.599.51010.5

Velocity[knots]

Revolut ionVe loc ity

Figure 6.8: Revolution speed and submarine velocity as a function of time from an acceleration test simulatedfor 200 s.

In the figure the revolution speed of the propeller and the submarines velocity is plotted as a function oftime. The submarine velocity is increasing towards a constant velocity, 9.7-9.8 knots. The revolution speedhas a steep slope during the first few seconds of the simulation following the maximum revolution rate limit.From 5 to 80 seconds the speed of the revolution is limited by the maximum allowed torque of the propulsion,which is achieved at 85 rpm.

Step 1.

The linear terms are compared to the current software simulating with the same inputs as previous. Figure6.9 presents the results of the comparison.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

t ime [s ]

Surg

evel.

u[kn]

CurrentNew

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

t ime [s ]

Revolutionvel.

n[rpm]

(a) Velocity comparison

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

t ime [s ]

Advancera

tioJ

[-]

CurrentNew

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

t ime [s ]

Torq

ue,Q

[kNm]

(b) Propulsion comparison

Figure 6.9: Acceleration test comparison between Current (blue) and New (red) simulation programs. Theleft two graphs show the surge velocity (top) and the propeller rotational velocity (bottom). The right twographs show the propulsion terms J (top) and Q (bottom).

Figure 6.9(a) gives evidence that the propulsion dynamics are correct and that the commanded revolutionspeed is reached. At the same time the maximum torque is not exceeded as seen in Figure 6.9(b). All thecurves are matching the current program very well.

44

Step 2.

Adding the non-linear terms results in the comparison shown in Figure 6.10.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

t ime [s ]

Surg

evel.

u[kn]

CurrentNew

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

t ime [s ]

Revolutionvel.

n[rpm]

(a) Velocity comparison.

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

t ime [s ]

Advancera

tioJ

[-]

CurrentNew

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

t ime [s ]

Torq

ue,Q

[kNm]

(b) Propulsion comparison.

Figure 6.10: Acceleration test comparison between Current (blue) and New (red) simulation programs withthe non-linear terms. The left two graphs show the surge velocity (top) and the propeller rotational velocity(bottom). The right two graphs show the propulsion terms J (top) and Q (bottom).

The non-linear terms have no significant influence on the acceleration test and the good correlation fromFigure 6.9 is maintained.

6.4 Turning circle

The turning circle test is a measure of the ability to turn the vessel. The test is simulated for a certaincommanded rudder deflection δrc and commanded velocity uc. At the start of the turn, the submarine ismoving with constant forward speed u0 = uc. All the other initial conditions are set to zero. The test resultsare then plotted in a graph in the earth-fixed horizontal reference frame, meaning that the coordinates forthe x0(t) and y0(t) are plotted for each time step to illustrate the turning circle of the submarine, see Figure6.11.

45

Figure 6.11: Example of a turning circle test.

The essential information to determine the turning ability of the submarine is the tactical diameter, advanceand turning radius. As seen in Figure 6.11, the tactical diameter is when the change of heading is 180 degreesand the advance is at 90 degrees. The different velocities during the simulation are of interest. This to knowhow much of velocity is decreasing/increasing during the turn. This test should be performed with bothpositive and negative rudder deflection.

The result illustrated in Figure 6.12 was simulated with the following input variables:

u0 = uc = 5 knotsδr = −20

tsim = 300 s

−20 0 20 40 60 80 100 120 140 160 180

−20

0

20

40

60

80

100

120

y0 [m]

x0[m

]

Figure 6.12: Submarine horizontal path during a turning circle test with consecutive plots of the submarinestop view at an interval of 20 seconds illustrating the drift angle.

46

The plot is showing the horizontal path of the vessel going into a turn. The submarine is showing goodstability since it performing a stable turn that ultimately does not spiral into a smaller or bigger turn radiusimplying that the yaw rate has reached a stable constant value. The submarine has a turning ability withan advance of 120 m and tactical diameter of 150 m.

Step 1.

The same input variables are used when comparing the two software’s linear terms as shown in Figure 6.13.

−20 0 20 40 60 80 1000

20

40

60

80

100

y0 [m]

x0[m

]

CurrentNew

(a) Horizontal path.

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

t ime [s ]

Yaw

rate

,r[/s]

CurrentNew

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

t ime [s ]

Surg

e,u

[m/s]

(b) Yaw rate and course.

Figure 6.13: Turning circle test comparison between Current (blue) and New (red) simulation programs.The left graph show the horizontal path of the submarine. The two right graphs show the surge velocity(bottom) and the yaw rate (top).

The path of the vessel is matching very well between the two programs as seen in Figure 6.13(a) as does theyaw rate and the surge velocity in Figure 6.13(b). This indicates that drift angle is also matching very welland that the X - and N -equations are correct for rudder executions.

47

Step 2.

Adding the non-linear terms gives a different result as shown in Figure 6.14.

−20 0 20 40 60 80 100 120 140 160 180

−20

0

20

40

60

80

100

120

y0 [m]

x0[m

]

CurrentNew

(a) Horizontal path.

0 50 100 150 200 250 3000

0.5

1

1.5

2

t ime [s ]

Yaw

rate

,r[/s]

CurrentNew

0 50 100 150 200 250 3001

1.5

2

2.5

3

t ime [s ]

Surg

e,u

[m/s]

(b) Yaw rate and surge velocity.

Figure 6.14: Turning circle test comparison between Current (blue) and New (red) simulation program withnon-linear terms considered. The left graph show the horizontal path of the submarine. The two right graphsshows the surge velocity (bottom) and the yaw rate (top).

The turning circle test matched well without the horse-shoe vortex and non-linear cross flow drag, see Figure6.13(a). With the two terms added to the EOM’s there is a visible difference, see Figure 6.14(a). Thisindicates that the two programs interpretation and calculation of these two terms are not identical. Thedifference in turning radius is approximately 3 m. Since the turning radii was almost identical between thetwo programs without the vortex and drag, the difference must lie in how these two terms are solved andinterpreted between the two programs. Excluding the horse-vortex by manually setting Cl = 0 in the inputfile, the non-linear cross flow drag is compared. The comparison is illustrated in Figure 6.15.

−20 0 20 40 60 80 100 120 140 160 180

−20

0

20

40

60

80

100

120

y0 [m]

x0[m

]

CurrentNew

(a) Horizontal path.

0 50 100 150 200 2500

0.5

1

1.5

2

t ime [s ]

Yaw

rate

,r[/s]

CurrentNew

0 50 100 150 200 2501

1.5

2

2.5

3

t ime [s ]

Surg

e,u

[m/s]

(b) Yaw rate and course.

Figure 6.15: Turning circle test comparison between Current (blue) and New (red) simulation programswith non-linear cross flow drag considered. The left graph show the horizontal path of the submarine. Thetwo right graphs show the yaw (bottom) and the yaw rate (top).

48

From the figure it is seen that the non-linear cross-flow drag term is showing a good correlation with adifference of 1 m in the radii. The major difference is thus in the implementation of the horse-shoe vortex.Such a difference is expected when doing a new implementation, where different interpretations, solutionmethods and integrators are used.

6.5 Stern-plane deflection

This test is performed to see the behaviour of the vessel during a stern-plane diving/surfacing manoeuvre.First of all, to see if the vessel is capable to perform a manoeuvre with its stern hydroplanes or if the sub-marine is experiencing the ”reverse rudder” effect. This is when the submarine is travelling at a too lowvelocity to create a large enough trim moment for the submarines lift force to be larger than the hydroplanes.This gives the reverse effect from the hydroplanes than what is expected, causing the submarine to translatewith the hydroplanes lift force. If a dive/surfacing has commenced, is it a stable? Does the vessel find aconstant pitch angle or is it constantly increasing? How much is the velocity decreasing? Also, how muchhas it decreased/increased in depth.

The simulation is done by having the vessel initially moving with a constant forward velocity u0 = uc. Atthe beginning of the simulation, the submarine immediately starts moving its stern-planes towards the com-manded stern-plane deflection δsc. The vessel will perform a diving manoeuvre with its stern hydroplanesuntil the simulation time tsim is reached. The essential variables to look for in the results are the trim angle,depth, and submarine velocity. These variables are plotted against the time.

The result shown in Figure 6.16 is simulated with the following input variables:

u0 = uc = 5 knotsδs = 10

tsim = 200 s

0 20 40 60 80 100 120 140 160 180 200

−19

−17

−15

−13

−11

−9

−7

−5

−3

−1

1

3

5

7

9

11

Angle

[],

Velocity[kn]

t ime [s ]

0 20 40 60 80 100 120 140 160 180 200

200205210215220225230235240245250255260265270275280285290

Depth

[m]

Ve loc ityδ s

PitchDepth

Figure 6.16: Stern-plane deflection, δs, as a function of time. Plotted together with the submarine depth,pitch and velocity.

It is seen that the stern-plane deflection is increasing towards the commanded angle as governed by Equation5.10. The submarine pitch angle is initially increasing at a high rate and after a slight overshoot it flattensout to a stable pitch angle at approximately -18 at 120 seconds. During the whole simulation time thevessel has done a controlled dive to a depth of 290 m, which is an increase in depth by 90 m. The velocity

49

has a little bit of a decrease in the beginning of the dive but stabilizes to a constant velocity just under 5 knots.

Step 1.

The same input variables as in the previous simulation are used when comparing the two software’s linearterms. The results from the comparison can be seen in Figure 6.17.

0 50 100 150 200 250 300 350 400 450 500

180

200

220

240

260

280

300

x 0 [m]

z0[m

]

CurrentNew

(a) Vertical path.

0 20 40 60 80 100 120 140 160 180 2002.2

2.3

2.4

2.5

2.6

t ime [s]

Surg

evel.,u

[m/s]

0 20 40 60 80 100 120 140 160 180 200−0.5

−0.4

−0.3

−0.2

−0.1

0

t ime [s]

Heav

evel.,w

[m/s]

CurrentNew

(b) Heave and surge velocity.

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

t ime [s]

δs[]

CurrentNew

0 20 40 60 80 100 120 140 160 180 200−25

−20

−15

−10

−5

0

t ime [s ]

Pitch

,θ[]

(c) Hydroplane deflection δs and pitch angle.

Figure 6.17: Stern plane deflection test comparison between Current (blue) and New (red) simulationprograms. Graph (a) show the vertical path of the submarine, (b) show two graphs of heave (top) and surgevelocity (bottom) and (c) show two graphs of the stern-plane deflection δs (top) and the submarines pitchangle (bottom).

As seen from the results in Figure 6.17, the only thing matching with the current program is the hydroplanedeflection. The vessel has not the same path, velocity or pitch angle. This indicates that there are differencesin the implementation of the M -, Z - or X -equations. The higher velocity indicates that there is less dragin the simulation from the New program. Since the the previous test cases shows much better correlation,the drag force due to the stern-plane deflection is the suspected culprit. See Equation 4.47, the terms withcoefficients Xδs , Xδsδs and Xδsη.

50

Step 2.

Adding the non-linear terms gives the results shown in Figure 6.18.

0 50 100 150 200 250 300 350 400 450 500

180

200

220

240

260

280

300

x 0 [m]

z0[m

]

CurrentNew

(a) Vertical path.

0 20 40 60 80 100 120 140 160 180 2002.2

2.3

2.4

2.5

2.6

t ime [s ]

u[m

/s]

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.3

−0.2

−0.1

0

t ime [s ]

w[m

/s]

CurrentNew

(b) Heave and surge velocity.

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

t ime [s ]

δs[]

CurrentNew

0 20 40 60 80 100 120 140 160 180 200−20

−15

−10

−5

0

t ime [s ]

Pitch

,θ[]

(c) Hydroplane deflection δs and pitch angle.

Figure 6.18: Stern plane deflection test comparison between Current (blue) and New (red) simulationprograms with non-linear terms. Graph (a) show the vertical path of the submarine, (b) show two graphs ofheave (top) and surge velocity (bottom) and (c) show two graphs of the stern-plane deflection δs (top) andthe submarines pitch angle (bottom).

Comparing these tests with the ones made without the two added terms, see Figure 6.17, it is seen thatthe error is almost identical implying that the two added terms are not the fault in this case. The erroris identified to the stern-plane drag in the surge equation, since the surge velocity is too high compared tothe current program, see Figure 6.17(b) and 6.18(b). The coefficients in question could thus be: Xδs , Xδsδs

and Xδsη, see Equation 4.47. Manually setting these coefficients to zero in the input file and run it throughboth programs again would confirm or disclaim these coefficients as being the difference between the twoprograms. Doing this resulted in Figure 6.19.

51

0 100 200 300 400 500 600

180

200

220

240

260

280

300

x 0 [m]

z0[m

]

CurrentNew

(a) Vertical path.

0 20 40 60 80 100 120 140 160 180 2002.45

2.5

2.55

2.6

t ime [s]

u[m

/s]

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.2

0

t ime [s]

w[m

/s]

0 20 40 60 80 100 120 140 160 180 200−30

−20

−10

0

t ime [s]

θ[]

(b) Heave and surge velocity.

Figure 6.19: Stern plane deflection test comparison between Current (blue) and New (red) with non-linearterms, but without stern-plane drag coefficients. The left graph is the vertical path of the submarine. Theright three graphs are the heave (top) and surge velocity (middle) together with the pitch angle (bottom).

As seen in the figures, when the stern-plane coefficients i x-direction are cancelled from the equations theagreement between the programs are much better and since the Xδs and Xδsη already are zero, the faultis narrowed down to Xδsδs . That implies that within the source code of the current program the Xδsδs isincreased giving it a higher drag due to stern-plane deflection. In an attempt to raise the stern-plane dragin the ”New” program, the coefficient is multiplied by 4. This gives the result depicted in Figure 6.20.

0 50 100 150 200 250 300 350 400 450 500

190

200

210

220

230

240

250

260

270

280

x 0 [m]

z0[m

]

CurrentNew

(a) Vertical path.

0 20 40 60 80 100 120 140 160 180 2002.2

2.4

2.6

t ime [s]

u[m

/s]

0 20 40 60 80 100 120 140 160 180 200−0.4

−0.2

0

t ime [s]

w[m

/s]

0 20 40 60 80 100 120 140 160 180 200−20

−10

0

t ime [s]

θ[]

(b) Heave and surge velocity.

Figure 6.20: Stern plane deflection test comparison between Current (blue) and New (red) with non-linearterms. The New program stern-plane coefficient Xδsδs is multiplied by 4. The left graph is the vertical pathof the submarine. The right three graphs are the heave (top) and surge velocity (middle) together with thepitch angle (bottom)

Comparing Figure 6.19 with 6.20 it is seen that the difference between the two programs stays approximatelythe same. The correlation is significantly improved, which can be seen by comparing Figure 6.20 with Figure6.18, when the stern-plane drag term is multiplied by 4 in the ”New” program. This implies that thedifference between the two programs is a factor 4 on the stern-plane drag coefficient. No basis for this

52

multiplication has been found in either theory or documents. The error is suspected to be mathematicalwhere the factor 4 multiplication concerning the four hydroplanes as

...+Xδsδsu2δ2

s · 4 + ...

has been coded as

...+Xδsδsu2(δs · 4)2 + ...

Since all other hydroplane tests, the bow-plane and rudders, correlates well between the programs, the de-duction is that an unknown implementation has been made to the stern-plane drag coefficient in the surgeequation of the ”Current” program. Therefore, a dialogue has commenced between FOI and the manu-facturer of the current program to solve the issue. Until the error is solved, the results from tests usingstern-plane deflection should be used with caution and with the knowledge that there are uncertainties in-volved2.

2June 2015, the mathematical error that was proposed in this thesis was confirmed by the manufacturer. They were awareof the error and it was corrected in the updated version of the software.

53

Chapter 7

Results

The tests that are not used in the verification process are presented in this section. Each test is simulatedwith the Flipper Topolino submarine. The submarine general arrangement is shown in Appendix D. Thecoded program steps and implementations are explained in the program guide in Appendix F.

7.1 Meander test

The goal of the meander test is to investigate if the submarine is returning to a stable state after a disturbanceof the pitch angle. The stern- and bow-planes get an opposite deflection to create the pitch angle withoutinfluencing the other conditions too much. After the manoeuvre, the submarines behaviour is examined tosee if the vessels pitch angle returns to, or close to, 0. This test should be performed for different velocitiesup to maximum speed.

The submarine starts travelling at a constant velocity of u0 = uc, and will have commanded stern-plane andbow-plane deflections, δsc and δbc, respectively. The hydroplanes start moving towards their commandeddeflection as soon as the simulation begins, and will keep this deflection until the commanded trim angle θc

is reached, from where they will return to their neutral position δs = δb = 0.

The results presented in Figure 7.1 was simulated with the following input variables:

u0 = uc = 5 knotsδb = −10

θc = 5

δs = 5

tsim = 200 s

54

0 20 40 60 80 100 120 140 160 180 200

−10−9−8−7−6−5−4−3−2−1

012345

δs,δb&

Pitch[]

t ime [s ]

0 20 40 60 80 100 120 140 160 180 200

198

199

200

201

202

203

204

205

206

207

208

209

210

Depth

[m]

δ sPitch

δ bDepth

Figure 7.1: Bow-plane, δb, and stern-plane deflection, δs, as a function of time causing the resulting pitchand depth.

The pitch disturbance is effected with simultaneous deflection with both the stern-plane and bow-planehydroplanes as seen in Figure 7.1. The hydroplanes are restored to neutral, 0, after the submarine hasreached the commanded pitch angle of θc = 5. The submarine is then seen returning to a stable 0 pitchangle. Likewise, the depth of the submarine is increasing with the pitch angle and finally returns to a stablepath with constant depth.

7.2 Control-system failure

For a submarine, the bow-plane together with the stern-plane is used, in a control system, to keep the subma-rine at a constant depth. This test simulates a control-system failure that sets the bow- and stern-plane at adiving/surfacing manoeuvre with a certain deflection δbc and δsc, respectively. When the failure is detected,a counter manoeuvre is initiated by shifting the deflection to its failed negative, −δbc and −δsc. Some timeafter the counter manoeuvre the hydroplanes are reset to their neutral position at 0 and simulated untilt = tsim.

The goal of the test is to investigate the effects from a failed control-system and the routine counter ma-noeuvre, in order to create a requirements specification for the submarine. The interest lies in the amountof trim and change of depth at different velocities to see how close to the maximum depth or bottom thevessel can travel without the risk of damaging the hull in case of a failure. Also if the vessel can return to astable position as the hydroplanes return to their default deflection and if the counter manoeuvre can bringvessel back to its original depth.

The results depicted in Figure 7.2 was simulated with the following input variables:

u0 = uc = 5 knotsz0 = 100 mδb = −35

δs = 35

tcount = 11 strest = 40 stsim = 200 s

55

0 50 100 150 200 250 300 350 400 450−35

−30

−25

−20

−15

−10

−5

0

5

10

15

20

25

30

35

δs,δb&

Pitch[]

D istance [m]

0 50 100 150 200 250 300 350 400 450

82

84

86

88

90

92

94

96

98

100

102

104

106

108

Depth

[m]

δ sδ b

PitchDepth

Figure 7.2: Hydroplane, pitch and depth as function of distance travelled in the x0 direction.

The figure is showing how the control-system has failed and put the stern- and bow-plane hydroplanes tomaximum deflections. After 11 seconds, a counter manoeuvre is performed to restore the pitch and depthto their initial values. The pitch will return close to zero by just setting the hydroplanes to their neutralpositions since the vessel is stable in forward motion. The question is, how long of a counter manoeuvreshould be performed to return to the initial depth? As seen in the figure, the counter manoeuvre is too longand the vessel overshoots and stabilizes at a depth of around 86 m. The counter manoeuvre should thus bedecreased to a trest closer to 30 seconds instead. This submarine should never, at this speed, travel closerthan 7 m to the bottom or an object in case there is a control-system failure, assuming that it takes thecrew 11 seconds to perform a counter manoeuvre.

7.3 Hydroplane failure

This test simulates that one of the four stern hydroplanes has failed while the submarine is moving at con-stant velocity u0 = uc. The chosen failed hydroplane is set to a user defined deflection, δic. Some time afterthe failure, a counter manoeuvre is initiated by setting the other stern hydroplanes to the negative deflectionof the failed one, −δic. After some user defined time, all the hydroplanes are reset to their neutral positionat 0 and simulated until t = tsim.

The goal of the test is to see the effects from a failed hydroplane and the routine counter manoeuvre. Theinterest lies in the amount of trim and change of depth during these manoeuvres. Also if the vessel canreturn to a stable position as the hydroplanes return to their default deflection. Like the control-systemfailure test, this test is done to create a requirements specification for the submarine.

The results illustrated in Figure 7.3 was simulated with the following input variables:

Failed hydroplane: δ3 = 35

u0 = uc = 5 knotsz0 = 100 mtcount = 11 strest = 40 stsim = 200 s

56

0 50 100 150 200 250

−31

−26

−21

−16

−11

−6

−1

4

9

14

19

24

29

34

δi&

Pitch[]

D istance [m]

0 50 100 150 200 250

68707274767880828486889092949698100

Depth

[m]

δ 1δ 2δ 3δ 4

PitchDepth

Figure 7.3: Hydroplane, pitch and depth as function of distance travelled in the x0 direction during ahydroplane failure test.

The test simulates that the third stern hydroplane, δ3, fails and ends up at 35 as seen in the figure. Thesubmarine then tries to counter the failure by setting the other hydroplanes to maximum hydroplane negativedeflection at tcount. All hydroplanes are then restored to 0, at trest. The vessel self-stabilizes to neutralpitch and ends up at a depth of 70 m. In order to return to the initial depth one could try different countertimes or restore the hydroplanes earlier.

7.4 Zig-zag tests

A δ/ψ zig-zag test is performed with a commanded velocity as the initial velocity, u0 = uc, and applying thechosen rudder deflection δc. When the heading ψc is reached, the rudder is flipped to −δc until the headinghas changed to −ψc. The process is repeated a number of cycles.

The zig-zag test is a way to check the submarines initial turning ability, course changing ability and stability.The initial turning ability is defined by the change-of-heading response. Meaning, how long time it takes thevessel reaching its first designated heading. The yaw-checking ability is measured from when the counter-rudder is applied until the headings time derivative is negative i.e. the heading is approaching its nextheading. This is called the overshoot, see Figure 7.4.

57

Figure 7.4: Example of a 10/10 zig-zag test in the horizontal plane.

This test is done for both the horizontal plane and the vertical plane. The example above is for a horizontaltest. For a vertical test the change of heading ψc is substituted with a change of the trim angle θc and δscinstead of δrc

7.4.1 Horizontal zig-zag test

The test was simulated with the following input variables:

u0 = uc = 5 knotsδr = 10

ψc = 10 stsim = 200 s

Figure 7.5 shows the results for this simulation. The test performed is called a 10/10 horizontal zig-zag test.As seen in the figure the rudder deflection changes between positive and negative 10, for every negativeand positive course of 10. The interesting part is to see how big the overshoots are and at what overshootit becomes stable. Seen in the figure is that the overshoots tend to be stable at 23 and that starts at thethird overshoot. During this test the submarine tends to surface a little bit to 191 m after 400 seconds ofsimulation. This test should also be done for multiple submarine velocities to examine how the submarineability to change course is altered by its velocity.

58

0 50 100 150 200 250 300 350 400

−22−20−18−16−14−12−10−8−6−4−2

02468

1012141618202224

δr&

Yaw

[]

t ime [s ]

0 50 100 150 200 250 300 350 400

190

191

192

193

194

195

196

197

198

199

200

201

202

203

Depth

[m]

δ rYawDepth

Figure 7.5: Rudder deflection, yaw and depth as a function of time for a horizontal zig-zag test

7.4.2 Vertical zig-zag test

The test was simulated with the following input variables:

u0 = uc = 5 knotsδr = 5

θc = 5 stsim = 200 s

Figure 7.6 illustrates the results from the simulation. The test performed is called a 5/5 vertical zig-zag test.As seen in the figure the stern-plane deflection changes between negative and positive 5, for every negativeand positive pitch of 5. The interesting part is to see how big the overshoots are and at what overshootit becomes stable. Seen in the figure is that the overshoots tend to be stable at 6.3 and that starts at thesecond overshoot. This submarine seems to have appropriate vertical stability at this speed with no tendencyto increase the overshoot. From the depth curve it is seen that the vessel sinks slower than it rises.

0 20 40 60 80 100 120 140 160 180 200

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

δs&

Pitch[]

t ime [s ]

0 20 40 60 80 100 120 140 160 180 200

195

196

197

198

199

200

201

202

203

204

205

Depth

[m]

δ sPitchDepth

Figure 7.6: Stern-plane deflection, pitch and depth as a function of time during a vertical zig-zag test.

59

7.5 Step response

The step-response is a test where the hydroplanes are set at a certain deflection at certain times. Initially,the submarine is running straight ahead at the commanded speed, u0 = uc. The first execution at t = 0 achosen hydroplane deflection, δc = δ1, is held until the next time step, t = t1, where the second executiontakes place, δc = δ2. This is done for at least four different hydroplane deflections, see Figure 7.7.

Figure 7.7: Example of a step-response test with rudder angle as a function of time.

The test can be performed in the horizontal plane and the vertical plane for both the stern hydroplanes andbow-plane. With other words, the test can be performed for all the hydroplanes on the vessel.

7.5.1 Horizontal step response

The test was simulated with the following input variables:

δr0 = 10 at t0 = 0 sδr1 = 5 at t1 = 100 sδr2 = −5 at t2 = 200 sδr3 = −10 at t3 = 300 su0 = uc = 5 knotstsim = 400 s

As seen in Figure 7.8, there are no sudden changes in yaw rate implying that the vessel has good and smoothtransitions between rudder deflections. As the other tests also has implied the vessel tend to surface duringturns, in this case 22 m.

60

0 50 100 150 200 250 300 350 400−11−10−9−8−7−6−5−4−3−2−1

0123456789

1011

δr[]&

Yaw

rate

[/s]

t ime [s ]

0 50 100 150 200 250 300 350 400

178

183

188

193

198

Depth

[m]

δ rYaw rateDepth

Figure 7.8: Rudder deflection, yaw rate and depth as a function of time during a horizontal step responsetest.

7.5.2 Vertical step response

The results shown in Figure 7.9 was simulated with the following input variables:

δs0 = 10 at t0 = 0 sδs1 = 5 at t1 = 100 sδs2 = −5 at t2 = 200 sδs3 = −10 at t3 = 300 su0 = uc = 5 knotstsim = 400 s

0 50 100 150 200 250 300 350 400

−19−17−15−13−11−9−7−5−3−1

13579

1113151719

δs&

Pitch[]

t ime [s ]

0 50 100 150 200 250 300 350 400

200

205

210

215

220

225

230

235

240

245

250

255

260

265

270

275

280

Depth

[m]

δ sPitchDepth

Figure 7.9: Stern-plane deflection, pitch and depth as a function of time during a vertical step response test.

61

Chapter 8

Conclusion

A program with the core functions of a the new submarine manoeuvre simulation software was created, usingthe programming language Python. The program contains a number of standard tests with the possibilityto do a simulation giving a time series solution for some standard manoeuvres of special interest for theevaluation of a submarine design. The code is written so that enhancements and extensions are possible.The program correlates well with the current program, used as a reference, on almost every level exceptwhen using the stern-plane deflection.

62

Chapter 9

Further work

There are numerous of options for further work, improving the accuracy, the versatility and the usefulnessof the program. Some suggestions are outlined below:

• Verify the submerged results from this thesis against full-scale test data from real. This will givefurther knowledge about the programs weaknesses and strengths that can be improved.

• Real-world submarines operate most of the time on autopilot or with autopilot assistance. Therefore,a control system should be implemented to further increase the realistic scope of the program.

• To be able simulate surface and near-surface operation, wave interactions and wave simulations shouldbe added.

• Ballast tanks are vital components submarine. Therefore the program should be able to handle lo-cations of ballast tanks and the way the water moves during the test and how that will affect themovement of the vessel. Adding this will enable simulation of leakage and emergencies.

• Emergency blowing thermodynamics

• GUI, reporting results for printout

• For know, the coefficients and non-dimensional derivatives are extracted regression analysis from con-ventional submarines. It would be great to increase the library of the calculations of the derivatives tomore unconventional geometries thus increasing the working range of the program.

63

References

[1] John David Anderson Jr. Fundamentals of aerodynamics. Tata McGraw-Hill Education, 1985.

[2] Hans Jurgen Bohlmann. Berechnung hydrodynamischer Koeffizienten von Ubooten zur Vorhersage desBewegungsverhaltens. na, 1990.

[3] C Chapra Steven and P Canale Raymond. Numerical methods for engineers. Tata McGraw-Hill Pub-lishing Company, 2008.

[4] Jerome Feldman. Dtnsrdc revised standarrd submarine equations of motion. Technical report, DTICDocument, 1979.

[5] JP Feldman. State-of-the-art for predicting the hydrodynamic characteristics of submarines. In Pro-ceedings of the Symposium on Control Theory and Naval Applications, pages 87–127, 1975.

[6] Thor I Fossen. Guidance and control of ocean vehicles, volume 199. Wiley New York, 1994.

[7] Kalle Garme. Fartygs motstand och effektbehov. Kursparm for Fordjupningsarbete i Marina Systemoch Marin Teknik. Marina System, KTH, Stockholm, Februari, 2007.

[8] Fredrik Granholm. Fran Hajen till Sodermanland. Marinlitteraturforeningen nr 89, 2003.

[9] DE Humphreys and KW Watkinson. Prediction of acceleration hydrodynamic coefficients for underwatervehicles from geometric parameters. Technical report, DTIC Document, 1978.

[10] Frederick H Imlay. The complete expressions for added mass of a rigid body moving in an ideal fluid.Technical report, DTIC Document, 1961.

[11] Johan MJ Journee and WW Massie. Offshore hydrodynamics. Delft University of Technology, 4:38,2001.

[12] Horace Lamb. Hydrodynamics. 1932.

[13] Lennart Lindberg. U 3503: Dokumentation-Danzig 1944, Goteborg 1946. Marinlitteraturforeningen,2001.

[14] Zhi-hua Liu, Ying Xiong, Zhan-zhi Wang, Song Wang, and Cheng-xu Tu. Experimental study on effectof a new vortex control baffler and its influencing factor. China Ocean Engineering, 25:83–96, 2011.

[15] Peter Ridley, Julien Fontan, and Peter Corke. Submarine dynamic modelling. In Proceedings of the 2003Australasian Conference on Robotics & Automation. Australian Robotics & Automation Association,2003.

[16] SNAME. Nomenclature for treating the motion of a submerged body through a fluid. Technical andResearch Bulletin, pages 1–5, 1950.

[17] L Folger Whicker and Leo F Fehlner. Free-stream characteristics of a family of low-aspect-ratio, all-movable control surfaces for application to ship design. Technical report, DTIC Document, 1958.

64

Appendix A

Determination of hydrodynamicderivatives

There are a many different experimental methods that can be used to determine forces and moments asso-ciated with variations in linear-, angular velocity and acceleration. Typical facilities are:

• Straight-line

• Rotating arm

• Planar motion mechanism (PMM)

Besides these, some hydrodynamic coefficients can be determined by theoretical and semi-empirical methods.Such a method is the strip theory that has been successfully applied on ships. During this chapter the differentways of determine the hydrodynamic derivatives and coefficients are described.

A.1 Straight-line test and rotating-arm technique

The velocity dependent derivatives of a vessel can be determined through the use of a straight-line test,whereas a the rotary derivatives uses the rotating-arm technique. Both techniques are called captive modeltests. The main idea with these test is to have all parameters zero except the one that is measured. Astraight-line test is performed in a conventional tank where a model of the vessel is pulled on a straight linethrough the tank. On the model there is then dynamometer that measures the forces and moments acting onthe vessel during the test. Figure A.1 illustrates the set-up for a straight-line with the purpose of extractingforce and moments due to the sway velocity, Yv and Nv respectively.

Figure A.1: Straight-line test setup

If β is small enough then v ≈ −βV . Then by towing the vessel with different drift angles while measuringthe force and moments will give a set of data points shown in Figure A.2.

A1

Figure A.2: Force or moment as a function of v

The data points can the be used to describe a curve as a polynomial such as

Ycurve = Yvv + Yv|v|v|v|... (A.1)

Most times a second degree polynomial is sufficient. Using the absolute value is necessary in order to preservethe sign when squaring v. A similar procedure is used for measuring the influence of rudder deflection. Insteadof keeping the vessel in a drift angle, the vessel is kept on a straight path but with different rudder deflectionsas force and moment are recorded. This yields a graph as in Figure A.2 but instead of the force and momentbeing functions of v they are now functions of δr. Coupled derivatives can also be measured with the straight-line technique. For example, the coupled Y force due to sway velocity and rudder deflection, Yvδr . This isdone by doing the rudder deflection test at multiple sway velocities and then cross plot the slopes as seen inFigure A.3. The coupled curve is then fitted to a polynomial as in Equation A.1.

(a) Y or N as function of δr (b) Yδr or Nδr as a function of v

Figure A.3: Extraction of coupled terms

The rotating-arm technique is mainly used to get the rotary derivatives. This is done by having a rotationalmotion where the vessel is always tangential to the rotational path. This is important since there shouldnot be any sway velocity acting on the vessel when the rotary derivatives are measured. The set-up of therotating-arm technique is seen Figure A.4.

A2

Figure A.4: Rotating-arm technique set-up

The rotating arm the works with the same principle as straight-line test. However, to get different rotationalvelocities either the velocity, u, or the radius, R, has to change since the rotational velocity is:

r =u

R(A.2)

The measurement should be done close to the vessels operational velocity, so in order to achieve differentrotational velocities the measurement has to be performed at different radii. This becomes a problem becausethat means that the tank has to be really big. The measurement has to be completed during one revolutionsince the vessel will be travelling in its own wake after the first revolution. This will give the starting lengthof R as pretty big from the beginning and has to be increased to get multiple rotational velocities.

A.2 Planar motion mechanism (PMM)

This technique was developed by a research team at the David Taylor Model Basin in 1957[6]. This systemcan be used to extract all the derivatives and coefficients in six DOF. The PMM consists of two oscillatorsmounted at the bow and stern of the model. These oscillators are used to produce a oscillation of the movingmodel. The forces induced by oscillation are measured by two transducers. The set-up for a PMM is seenin Figure A.5.

A3

Figure A.5: PMM set-up with the oscillators a and b

The idea is that the model is oscillating harmonically and then the known phase shift between the derivativesof the harmonic oscillation is used to extract the derivatives. For an example, a pure sway test will oscillateboth a and b with the same magnitude y. The derivatives will then be expressed as

y = a0sin(ωt)

y = v = ωa0cos(ωt)

y = v = −ω2a0sin(ωt) (A.3)

Running such a test would give the graph seen in Figure A.6.

yvv

Figure A.6: PMM oscillation test data

As seen in the figure, the velocity and acceleration curve has a 90 phase shift. This means that when thevelocity is zero the acceleration is at its maximum and vice versa. Reading the forces from the transducersat those times will give the derivatives Yv and Yv.

A4

A.3 Strip theory

An estimate of the hydrodynamic derivatives can be obtained by applying this theory. The principle behindstrip theory is to divide the underwater parts of the vessel into a number of strips. Then the two-dimensionalhydrodynamic coefficient for added mass can be calculated for each strip and finally the three-dimensionalcoefficients are obtained by summing the contribution from each strip.

A5

Appendix B

Adams’ Method

The multi-step Adams’ method is a numerical integration based on both Adams-Bashforth method as apredictor and Adams-Moulton as a corrector. The following chapter will describe this method of numericalintegration and follow the explanation from Chapra and Raymond[3]. The Adams’ method is derived fromthe forward Euler method

yi+1 = yi +dyidth, (B.1)

the backwards Euler method

yi+1 = yi +dyi+1

dth, (B.2)

and the the trapezoidal rule

I = hdyi+1

dt + dyidt

2, (B.3)

where h is the step-length defined as

h = ti+1 − ti. (B.4)

The Adams’ method is based on two formulas: The Adams-Bashforth (AB), and the Adams-Moulton (AM).AB is derived from the Taylor series expansion of y(t) around t as

yi+1 = yi + fih+f ′i2h2 +

f ′′i6h3 . . . (B.5)

Here, fi is short-hand for the derivative dyidt , f ′i = d2yi

dt2 etc. As seen in Equation B.5 there is a derivative ofthe expansion. This expansion derivative can be approximated using the backwards difference

f ′i =fi − fi−1

h+f ′′i2h. (B.6)

Substitution of Equation B.6 in B.5 will the give:

yi+1 = yi + h

(3

2fi −

1

2fi−1

)+

5

12f ′′i h

3. (B.7)

Equation B.7 is called the second order Adams-Bashforth formula and is used as a predictor for the multi-step Adams’ method. Higher orders of the formula can be developed by substituting higher differenceapproximations giving the general expression as

yi+1 = yi + h

n−1∑k=0

αkfi−k. (B.8)

B1

The coefficients αk for the n:th order can be found in Table B.1.

Table B.1: Coefficients for Adams-Bashforth correctors[3]

Order α0 α1 α2 α3 α4 α5

1 12 3/2 −1/23 23/12 −16/12 5/124 55/24 −59/24 37/24 −9/245 1901/720 −2774/720 −2616/720 −1274/720 −251/7206 4277/720 −7923/720 9982/720 −7298/720 2877/720 −475/720

As seen in Equation B.7, the first order of the AB formula is equivalent to the forward Euler formula,Equation B.1. In the same the closed formula, AM, is derived by applying a Taylor series. However, for theimplicit AM method the backwards Taylor series is applied instead and solved for yi+1 as

yi+1 = yi + fi+1h−f ′i2h2 +

f ′′i6h3 . . . (B.9)

This time, using the forward difference approximation for for the first derivative as

fi+1 =fi+1 − fi

h+f ′′i+1

2h (B.10)

and substituting Equation B.10 in B.9 yields

yi+1 = yi + hfi+1 + fi

2− 1

12f ′′i+1h

3. (B.11)

Equation B.11 is called the second order Adams-Moulton formula and can be generated for higher orders as

yi+1 = yi + h

n−1∑k=0

βkfi+1−k, (B.12)

where the coefficients βk for the n:th order can be found in Table B.2.

Table B.2: Coefficients for Adams-Moulton correctors[3]

Order β0 β1 β2 β3 β4 β5

2 1/2 1/23 5/12 8/12 −1/124 9/24 19/24 −5/24 1/245 251/720 646/720 −264/720 106/720 −19/7206 475/1440 1427/1440 −798/1440 482/1440 −173/1440 27/1440

As seen from Equation B.12, the first order is equivalent to the backwards Euler formula, Equation ??, andthe second order is the trapezoidal rule, Equation B.3. The multi-step Adams’ method proceeds as follows:For each step the ”predictor” function, Equation B.8, is used to calculate the first estimate. This value isthen used in the ”corrector” function, Equation B.12, to improve the estimate. Constructing the first ordermulti-step method yields

y0i+1 = ymi + hfmi (ti, yi) Predictor ,

yji+1 = ymi + hf j−1i+1 (ti+1, y

j−1i+1 ) Corrector . (B.13)

The initial value and derivative are denoted with the notation m and the notation j indicates the iterationprocess of how the value obtained by the ”predictor” is used in the ”corrector” equation and how the result

B2

from the ”corrector” is used again in the same equation in an attempt to obtain a value closer to the truevalue, Figure B.1.

Figure B.1: The iteration process of the Adams method

If the problem becomes stiff, it can be solved using a backward differentiation formula (BDF). They arederived by forming the k -th degree interpolating polynomial approximating y(t) by using the previous solvedvalues y(tn), y(tn−1), . . . , y(tn−k), and then backwards differentiating. This yields the general expression as

k∑i=0

γiyn−i = hζ0f(tn, yn) (B.14)

where the coefficients γ and ζ are shown in Table B.3.

Table B.3: Coefficients for backwards differentiation formula. [3]

Order ζ0 γ0 γ1 γ2 γ3 γ4 γ5 γ61 1 1 -12 2/3 1 -4/3 1/33 6/11 1 -18/11 9/11 -2/114 12/25 1 -48/25 36/25 -16/25 3/255 60/137 1 -300/137 300/137 -200/137 75/137 -12/1376 60/147 1 -360/147 450/147 -400/147 225/147 -72/147 10/147

The chosen integrator in Python, scipy.integrate.odeint, automatically shifts from the Adams’ method to theBDF if the problem becomes stiff.

B3

Appendix C

Selection of vortex data points

The horse-shoe vortex is created by lateral velocities and thus a test with a small rudder deflection anda constant speed will be performed. The test should be performed in a ”worst case scenario” situation,which means running the test with high velocity, and with a rudder deflection that yields a drift angle belowthe detachment angle, β < βST. This means that the detachment occurs at the stern, which is the largestdistance for the vortex effect on the vessel. Since the number of data points are constant for a test, earlierdetachment will just give a more exact solution. The test is simulated with following inputs:

uc = 20 knotsδr = 10

tsim = 101 s

the number of x-locations for data points are varied from 5 to 60 and the effect from this is shown in thehorizontal and vertical paths in Figures C.1 and C.2, respectively.

0 50 100 150 200 250 300

−50

0

50

100

150

y0 [m]

x0[m

]

Loc=5Loc=10Loc=15Loc=20Loc=40Loc=60

(a) Horizontal path

13.55 13.6 13.65 13.7 13.75 13.8 13.85 13.9 13.95 14 14.05

108.48

108.5

108.52

108.54

108.56

108.58

108.6

108.62

108.64

y0 [m]

x0[m

]

Loc=5Loc=10Loc=15Loc=20Loc=40Loc=60

(b) Zoomed horizontal path

Figure C.1: Horizontal path result with different numbers of x-locations integrated. Graph (a) shows thehorizontal path of the submarine and (b) shows a zoomed in version of (a).

C1

0 20 40 60 80 100 120

140

150

160

170

180

190

200

t ime [s]

z0[m

]

Loc=5Loc=10Loc=15Loc=20Loc=40Loc=60

(a) Vertical path

100.75 100.8 100.85 100.9 100.95 101 101.05 101.1 101.15

143

144

145

146

147

148

t ime [s]

z0[m

]

Loc=5Loc=10Loc=15Loc=20Loc=40Loc=60

(b) Zoomed vertical path

Figure C.2: Vertical path result with number of different x-locations integrated. Graph (a) shows the verticalpath of the submarine and (b) shows a zoomed in version of (a).

The figures show how an increased number of locations considered in the integration improves the result. Asseen in the figures where the result is zoomed, Figure C.1(b) and C.2(b), the result vary a lot between 5 - 20locations and especially vertical direction, which is expected. For each amount of x-locations the simulationwas timed. The result is shown in Table C.1.

Table C.1: Number of locations and simulation time

x-locations 5 10 15 20 40 60time [s] 1.61 2.08 2.30 2.51 3.31 4.34

It is obvious that increasing the number of locations improves the results. However, this also increases thesimulation time. Picking the amount of locations is a trade-off between accuracy and time. Since this is asimulation program working on estimated values, the accuracy of the horse shoe vortex is deemed sufficientwith 20 locations.

C2

Appendix D

General arrangement

Prod

ucer

ad:

FMV

AK S

jöTi

tel:

Gen

eral

arra

ngem

ang

Konc

eptn

amn:

Flip

per T

opol

ino

Rita

d av

:G

odkä

nd a

v:

Vers

ion:

Anm

ärkn

ing:

Dat

um:

2015

-04-

29Sk

ala:

1:25

0R

itnin

g:

3 av

3

AA

BB

CC

DD

EE

FF

GG

HH

II

11

22

33

44

A-A

B-B

C-C

D-D

E-E

F-F

G-G

H-H

I-I

D1

Appendix E

Hydrodynamic derivatives

Table E.1: Dimensionless hydrodynamic force derivatives

X Y ZX ′u = −0.00048129 Y ′p = −0.00049893 Z ′q = −0.00020374X ′rp = 0.00049893 Y ′r = −0.0016656 Z ′w = −0.01334X ′qq = 0.00080214 Y ′v = −0.01744 Z ′vp = −0.01744X ′q|q| = 0 Y ′p = −0.0012749 Z ′q = −0.0040505

X ′rr = 0.001123 Y ′pq = 0.00020374 Z ′q|q| = 0

X ′vr = 0.016132 Y ′wp = 0.01334 Z ′rr = 0X ′uu = −0.0011095 Y ′r = −0.00082411 Z ′vr = 0X ′vv = 0.011293 Y ′r|r| = 0 Z ′vv = 0

X ′ww = 0.0037018 Y ′v|r| = 0 Z ′vw = 0

X ′w|w| = 0 Y ′|v|r = 0 Z ′w = −0.023841

X ′wq = −0.012339 Y ′wr = 0 Z ′|w| = 0

X ′w|q| = 0 Y ′v = −0.061469 Z ′ww = 0.0033377

X ′δrδr = −0.0015181 Y ′v|v| = −0.063314 Z ′w|w|R = 0

X ′δrδrη = 0 Y ′vw = 0 Z ′w|q| = 0

X ′δs = 0 Y ′δr = 0.0034403 Z ′wq = 0X ′δsδs = −0.0015181 Y ′δr|δr| = 0 Z ′δs = −0.0034403

X ′δsδsη = 0 Y ′δrη = 0.00034403 Z ′δs|δs| = 0

X ′δb = 0 Y ′|r|δr = 0 Z ′δsη = −0.00034403

X ′δbδb = −0.0017798 Y ′∗ = 0 Z ′|q|δs = 0

X ′rrξ = 0 Y ′rξ = 0 Z ′δb = −0.003852

X ′vvξ = 0 Y ′vξ = 0 Z ′δb|δb| = 0

Z ′∗ = −0.00029985Z ′rrξ = 0

Z ′vvξ = 0

Z ′ξ = 0

Z ′ξ|θ| = 0

E1

Table E.2: Dimensionless hydrodynamic moment derivatives

K M NK ′p = −0.000028877 M ′q = −0.00084386 N ′p = −0.000059225K ′r = −0.000059225 M ′w = −0.00055669 N ′r = −0.0010027K ′v = −0.00049893 M ′rp = 0.0009738 N ′v = 0.000064172K ′p = −0.00066585 M ′q = −0.0031448 N ′p = −0.00022142K ′p|p| = 0 M ′q|q| = 0 N ′pq = −0.00081498

K ′qr = −0.00015882 M ′wq = 0 N ′r = −0.0045527K ′i = 0 M ′w|q| = 0 N ′r|r| = 0

K ′wp = 0.00049893 M ′rr = 0 N ′v|r| = 0

K ′r = −0.00074631 M ′vr = 0 N ′|v|r = 0

K ′r|r| = 0 M ′vv = 0 N ′wr = 0

K ′v = −0.0035942 M ′vw = 0 N ′v = −0.018713K ′v|v| = −0.0072963 M ′w = 0.0044211 N ′v|v|R = 0.0080467

K ′δr1 = 0.00023426 M ′|w| = 0 N ′vw = 0

K ′δr2 = 0.00023426 M ′ww = 0 N ′δr = −0.0015498K ′δr3 = 0.00023426 M ′w|w|R = −0.0036253 N ′δr|δr| = 0

K ′δr4 = 0.00023426 M ′δs = −0.0015498 N ′δrη = −0.00015498

K ′δr1|δr1| = 0 M ′δs|δs| = 0 N ′|r|δr = 0

K ′δr2|δr2| = 0 M ′δsη = −0.00015498 N ′∗ = 0

K ′δr3|δr3| = 0 M ′|q|δs = 0 N ′rξ = 0

K ′δr4|δr4| = 0 M ′δb = 0.00073669 N ′vξ = 0

K ′4S = 0 M ′δb|δb| = 0

K ′8S = 0 M ′∗ = 0.000019606K ′∗ = 0 M ′rrξ = 0

M ′vvξ = 0

M ′ξ = 0

M ′ξ|η| = 0

M ′ξη = 0

E2

Table E.3: Mass properties and Miscellaneous

xG = 0 [m]yG = 0 [m]zG = −0.36697 [m]xB = 0 [m]yB = 0 [m]zB = −0.54848 [m]kxx = 0.04176 [ - ]kyy = 0.24012 [ - ]kzz = 0.23858 [ - ]kxy = 0.0010303 [ - ]kzx = 0.031494 [ - ]kyz = 0.0021967 [ - ]S1 = 1.17 [ - ]S2 = −1.281 [ - ]x1 = 10.825 [m]xFW = 10.825 [m]zFW = 0 [m]βST = 0.133 [rad]xap = −33.978 [m]

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Appendix F

Program structure

This chapter gives an overview of the structure and functional features of the program developed for thisthesis.

The flow chart of the program is shown in Figure F.1. The program is built with four major functionalblocks that goes from user input to visual output presentation in the form of graphs.

Figure F.1: Flowchart describing the program

The processing start with the user selecting a test from the list of available tests, see 5.8. For each test someadditional input data must be supplied. For an example, if ”Turning circle test” is chosen, the initial veloc-ity and rudder deflection has to be determined. When the user then runs the program the following happens:

The test selection and input values enter the TEST function containing the test library with all the tests andtheir sequence of operation. The EOM’s is solved with scipy.integrate.odeint in a specific sequence dependingon what test is being simulated.

The EOM’s contain the equations of the state variables as described in Chapter 5.When the state variables are calculated for the specific sequence, TEST will call the POST PROCESS func-tion that adds the vectors that are described in section 5.7 to the output matrix.

TEST uses the result from POST PROCESS and access the PLOTS function that contains a library ofdifferent plots depending on the users test choice. The function utilize the results from POST PROCESS tocreate a specific graph containing the essential results as described in Chapters 6 and 7.

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