Michele Macovez. Ship Roll Motion Reduction by Means of the Rudder

Transcription

1 Michele Macovez Ship Roll Motion Reduction by Means of the Rudder Master s thesis, march 2008

2 Ship Roll Motion Reduction by Means of the Rudder The report has been prepared by: Michele Macovez Supervisor: Mogens Blanke DTU Electrical Engineering Automation Technical University of Denmark Elektrovej Building 326 DK-2800 Kgs. Lyngby Denmark Tel: (+45) Fax: (+45) Date of publishing: Classification: Comments: Public This report is submitted in partial fulfilment of the requirements for the Master degree at the Technical University of Denmark. The report represents 35 ECTS point. Copyright: Michele Macovez, 2008

3 Abstract This thesis focuses on the design of rudder roll stabilization (RRS) systems. Some results of the research to solve this specic ship motion control problem are presented. The modelling of the ship dynamics are discussed in the rst chapters: both the non linear model and the linear one have been obtained. Since the ship's roll motion is caused by waves, a wave model is dened, assuming the disturbance as a stochastic process. Two different feedback control approaches are analyzed and implemented. The rst method aims at nding a control law that minimizes the variance of the output. The second approach directly shapes the output sensitivity function, which relates the wave disturbance to the ship roll motion, to achieve good disturbance rejection. The non minimum phase dynamics in the rudder-to-roll response results in some fundamental limitations in control system design. A trade-off between disturbance attenuation at some frequencies and amplication at others affects the performances of the feedback controllers. Results of sea-way simulations show that both these controllers have good performances. The maximum roll damping is greater than 55% for both the analyzed controllers. The best roll reduction has been obtained by the minimum variance cheap controller with a damping of almost 70%. iii

4 Acknowledgments I would like to thank my supervisor, Professor Mogens Blanke for his patience in teaching me so many things and for his guidance during the period of my thesis. I thank Ph.D. Roberto Galeazzi for his availability and support. Special appreciation go to my Italian supervisor Professor Thomas Parisini for allowing me to come to Denmark to work on this project. I would like to thank my parents for always being there and giving me the possibility to spend these months in Copenhagen, my brother Roberto and Michela. Tender thanks go to Sara for her understanding and for staying by me. iv

5 Table of contents Table of contents v List of Figures viii List of Tables xi Introduction Motivations Relevant Literature Problem statement Objectives Main results of the Thesis Thesis Outline Ship dynamics Equations of Motion Forces and Moments acting on a ship Control Devices Forces acting on the rudder Rudder-Propeller Interaction Simulation Model v

6 Table of contents vi 2.1 Nonlinear State-Space Model Multi-Role naval vessel simulator Rudder machinery Rudder Naval vessel System Linearization Rudder to Roll transfer function Non-minimum Behavior in Ship Response Environmental Disturbances Wind generated waves Stochastic representation Nonlinear models of wave spectra Linear approximations of wave spectra Encounter frequency Waves simulation model Ship Roll Stabilization Stabilizing Methods Rudder Roll Damping Control Systems Sensitivity function Ship motion performance Reduction of Roll at Resonance-RRR Reduction of Statistics of Roll -RSR Reduction of Probability of Roll Peak Occurrence -RRO

7 Table of contents vii 5 Cheap limiting optimal control Minimum Variance Cheap Control Stability and Performance Trade-offs Output variance Numerical simulations and results Cheap controller robustness Sensitivity function-based approach control Single notch sensitivity specication Numerical simulations and results Double notch sensitivity specication Numerical simulations and results Sensitivity based approach controller robustness Conclusions Future work References A Naval vessel data A.1 Principal multipurpose naval vessel data A.2 Manoevering coefcients

8 List of Figures 1.1 Ship motion components Geometrical aspects of the rudder Lift and Drag on the rudder Rudder-Propeller interaction Block diagram of the system Rudder machinery Frequency characteristic of the open loop transfer functions Poles and zeros of the rudder to roll transfer function Non-minimum-phase effect on roll dynmamics during a turn to port Power spectrum of the linear approximation and the Pierson-Moskovitz model Incident sea description and denomination for sailing conditions Encounter frequency versus actual frequency Simulink block diagram for wave disturbances Wave disturbance simulation Examples of U-tube tanks Fin stabilizer arrangement Rudder roll stabilization (RRS) system structure viii

9 List of Figures ix 4.4 Rayleigh probability density function Bode plot of wave disturbance lter H(s) Sensitivity function for the proper and improper cheap controller Nyquist plots of L(j!) for the unstable and stable plant Cheap controller sensitivity function varying the ratio between T and T Ratio between the output variance and the white noise variance for different frequencies Cheap controller system sensitivity function Simulation at Bow seas with wave period of 6 seconds Simulation at Beam seas with wave period of 8 seconds Simulation at Head seas with wave period of 10 seconds Cheap controller roll reduction ratio as function of wave period Cheap controller roll reduction ratio as function of wave period varying the ship speed Cheap controller roll reduction ratio as function of wave period. Variation in roll damping coefcient Cheap controller roll reduction ratio as function of wave period varying the wave disturbance Bode plot of the single notch sensitivity specication Bode plot of ~ G ' (s) : Bode plot of the output sensitivity function for the single notch specication

10 List of Figures x 6.4 Simulation at Head seas with wave period of 8 seconds Simulation at Beam seas with wave period of 6 seconds Simulation at Bow seas with wave period of 10 seconds Single notch sensitivity based approach roll reduction ratio as function of wave period Double notch sensitivity specication Bode plot of the output sensitivity function for the double notch specication Simulation at Quartering seas with wave period of 8 seconds Simulation at Beam seas with wave period of 10 seconds Simulation at Head seas with wave period of 6 seconds Double notch sensitivity based approach roll reduction ratio as function of wave period Sensitivity based approach controller roll reduction ratio varying the ship speed Sensitivity based approach controller roll reduction ratio varying the roll damping coefcient Sensitivity based approach controller roll reduction ratio varying

11 List of Tables 1.1 Generalized displacements of a vessel Rudder data adopted for the multi role naval vessel Principal dimensions of multi role naval vessel Cheap controller performances for a 6 seconds period wave disturbance Cheap controller performances for a 8 seconds period wave disturbance Cheap controller performances for a 10 seconds period wave disturbance Single notch sensitivity approach controller performances for a 6 seconds period wave disturbance Single notch sensitivity approach controller performances for a 8 seconds period wave disturbance Single notch sensitivity approach controller performances for a 10 seconds period wave disturbance Double notch sensitivity approach controller performances for a 6 seconds period wave disturbance Double notch sensitivity approach controller performances for a 8 seconds period wave disturbance Double notch sensitivity approach controller performances for a 10 seconds period wave disturbance A.1 Main data for multipurpose naval vessel used in the project and for the simulation xi

12 List of Tables xii A.2 hydrodynamic parameters for multipurpose vessel used in the project and for the simulation

13 Introduction The problem of ship stabilization has widely been studied in the last decades. Perez [22] gave a detailed overview on the main efforts done in this eld in chapter 10 of his book. The idea of using the rudder as a stabilization device is quite recent and it probably emerged from observations of ship roll behavior under autopilot operation. Taggart [26] reported one particular situation observed during a Trans Atlantic voyage of a high-speed container ship in the winter of During this trip the characteristics of an autopilot controller were tested under different sea conditions, and it was pointed out that under certain circumstances, the rudder induced signicant roll motion. Motivated by these observations, in 1972 aboard the motor yacht M.S. Peggy in The Netherlands,, van Gunsteren [30] performed full-scale trials using the rudder as the only stabilizing actuator. The same thing did Cowley and Lambert [6] in 1972 using an autopilot and a roll feedback loop. During the 1980s there was an interesting contribution to the issue of rudder roll stabilization (RRS) developed in different countries: The Netherlands, Denmark, Sweden and the USA. In 1983 van der Klugt [29] and van Amerongen [28] performed full scale trials consequently the designing of a controller based on Linear Quadratic Gain (LQG) techniques but this kind of approach didn't give the results expected. Laudval and Fossen [17] proposed an improvement for such a mechanism and the performances were good close to the roll natural frequency but deteriorated at lower and higher frequencies. In Denmark the Royal Danish Navy introduced RRS on some of their fast monohull patrol vessels and the experiments were conducted by Blanke [4]. Different control approaches were considered such as LQG and H 1 techniques. Källström [15] in collaboration with the Royal Swedish Navy (RSN) implemented a RRS that became later in 1987 a commercial product known by the name of ROLL-NIX. The system was designed for use on straight courses; it switched off automatically when major manoeuvring was required, and back on when the vessel resumed a steady course. Baitis & Schmidt [1] presented in 1989 the use of rudder roll stabilization employed by the U.S. Navy on a DD-963 (Spruance) class destroyer. The rudder roll stabilizer signal was added to the signal generated by the autopilot or by manual steering. The installed systems produced reductions in roll motion of approximately 40%. During the 1990s, there was signicant research activity on the theoretical aspects of the problem. Several different control techniques were proposed, but only a few full-scale implementations were reported. Blanke and Christensen [3] studied the sensitivity of the performance of LQ control to variations in the coupling coefcients of the equations of motion. They used a linear model based on the hydrodynamic data estimated during the design stage of the SF300 vessels of the Danish Navy. In Sweden Källström and Shultz [16] continued to describe the performances of ROLL-NIX and its adaptive properties while in Denmark Hearns and Blanke [13] proposed the use of Quantitative Feedback Theory to design cascade SISO controllers for roll and yaw. Laudval and Fossen [18] took the nonlinear approach, (maybe the only reference in the literature that uses a nonlinear model for the design) and proposed the use of sliding mode control. 1

14 Introduction 2 Motivations Marine vehicles are designed to operate with acceptable reliability and economy, and in order to accomplish this, it is essential to regulate the dynamics of the ship. The motions of ships and the control of those motions have been the focal point of extensive research over the years. A ship in a seaway undergoes complex motions that may weaken the operational range of the ship and be uncomfortable and sometimes even dangerous for the crew. In certain circumstances, the captain may be obligated to alter course or slow down the ship to reduce large motions. This could produce undesirable mission limitations for military vessels and reduce prots for commercial vessels. Controlling the ship motion is of great interest to many parties in the marine eld since motion control during station keeping or low-speed maneuvers may broaden the safety range of many vessels. Over the years, many types of motion control have been devised. The majority have been aimed at reducing roll motions since the force required to reduce roll is reasonably small compared to the weight of the ship. Moreover, roll is the largest and most undesirable component of ship motion. Different methods and devices have been designed to reduce ship roll motions. For an example, anti-roll ns are used in cases of high ship speeds, and bilge keels, anti roll tanks, gyroscopic stabilizers have been employed to accomplish a good roll reduction. Although most of the devices work well, additional and external power installations will lead to the weight increase and space decrease on the ship. The hydrodynamic stability and structural strength may be changed when for example anti-roll tanks are adopted. The installation cost is also generally raised and the ship speed may be decreased due to additional components. Hence the present work focuses on the use of the rudder to reduce the roll motion of the ship. The main reasons for this are that almost every ship has a rudder (thus no extra equipment may be necessary), and also because this technique can be used in conjunction with other stabilizers. Based on frequency characteristics of the rudder inuence on yaw and roll motions, the two objectives can be separated in the frequency domain. Small frequencies are used for heading control, while high frequencies for roll reduction. A primary motivation of this project is also to provide a simulation environment to assess the feasibility of various control strategies in various sea states without the expense of model testing during design iterations. Relevant Literature The literature on marine control has grown in the last decades. A comprehensive treatment of automatic control systems for marine vehicles has been given by Fossen [9]. He provided mathematical foundations and theory needed for the designing of ship control systems. He also focused on both linear and nonlinear models to describe the dynamic and kinematic equations of motion of marine vehicles and the subject of stability and control. In his work he explored in depth the modeling of ocean vehicles, environmental disturbances, and the

15 Introduction 3 sensor and navigation systems, as well as discussing in length the applications of modern control theory. A more detailed investigation on some of these aspects has then been provided by Perez [22]. A deep coverage of hydrodynamic aspects related to control, wave induced motion modelling and roll stabilization was furnished. He studied the particular problems of control system design for course autopilots, rudder roll stabilization and combined rudder-n stabilizers, and the fundamental issues of performance limitations for the particular problems of rudder and n roll stabilization. He also reviewed the fundamental performance limitations of the closed-loop system due to the dynamic characteristics of the ship. A relevant work in this sector has been done by Blanke [2]. He concentrated his research on rudder-roll damping (RRD) autopilots. He showed through parametric investigations that cross-couplings between steering and roll might give rise to problems with performance robustness for the RRD controller. He then treated the RRD design problem from a robust control outset. He demonstrated that a separation result exists that makes it possible to make separate roll and steering specications and optimize the two controllers independently. Recently together with Yang [33] he has worked on H 1 control of the roll damping loop and with Hearns [13] on qualitative feedback theory (QFT) applied to solve the combined RRD-heading control problem with due regard of model uncertainty. Problem statement It is well-known that to fully represent the motion of a rigid body in space a six-degrees-offreedom approach is required. To determine the position and orientation six independent coordinates are necessary: the rst three and their derivatives describe the translational motion of the rigid body in terms of positions and linear velocities, while the last three coordinates and their time derivatives identify the rotational motion in terms of orientation and angular velocities. The six different motion components are dened as surge (translation along the x-axis), sway (translation along the y-axis), heave (translation along the z-axis), roll (rotation about the x-axis), pitch (rotation about the y-axis) and yaw (rotation about the z-axis). However, for a control problem as roll stabilization, it is often assumed that the ship motion can be described using only four degrees of freedom, which include surge, sway, yaw and roll. The equations of motion describing the dynamics of a ship are readily obtained from Newton's law in space-xed coordinate. However, to take advantage of the symmetry property of a ship, a ship-xed coordinate system is preferred. With the origin for the axis system taken at the center of gravity of the ship, the ship equations of motion within a xed coordinate system can be found. The vessel used for the modelling corresponds to a small monohull [3]. The data have been gathered by Blanke and Christensen at the project state and are estimated by the Danish Maritime Institute, DMI. The ship is considered to advance keeping a steady course at a constant forward speed U.

16 Introduction 4 The irregularity of ocean waves gives rise to sudden and unpredictable growth of the roll amplitude. Waves with absolute frequency! 0 and angle of encounter are are included in the model, representing the disturbance acting on the vessel. These wave disturbances are modelled as ltered white noise with constant power spectral density. The lter adopted is a second order transfer function with dominant frequency equal to! 0 : It is an aim of this thesis to design a controller capable of reducing the wave roll induced motion. Objectives Ship roll motion is caused by external disturbances, e.g. wave, wind and current, which contribute to the roll by exerting varying forces and moments on the hull. Roll motion, however, is not only caused by the action of the waves, but it can also be generated by the rudder. The rudder's main function is to change the heading of a ship; however, the rudder may also be used to produce, or correct, roll motion. An alteration of course makes the ship heel and when the ship rights itself, it turns back towards its equilibrium position in a damped oscillation. The effectiveness of RRD controls has been debated. Results from full scale evaluation on vessels have indicated very satisfactory results showing % roll reduction. By contrast, other experiences have demonstrated much less effectiveness in certain cases. Although rudder roll stabilizers have been designed since the early 80's, it still represents a challenging objective. An automatic control system is necessary to provide the rudder command based on the measurements of ship motion. A good model of the vessel is needed in order to reproduce the behavior of the real system. The objectives of the thesis are to derive a mathematical model of the system to be controlled and to design and implement a feedback control law to damp the roll motion generated by the waves. The performance limitations, due to fundamental physical constraints, have to be discussed and taken in consideration. Main results of the Thesis The work that has been done during this project led to the following results: A mathematical model for the non linear system of the ship motion in the seaway has been determined.

17 Introduction 5 The system has been linearized around a stationary point for the task of control design. The effects of the wave disturbances on ships and their stochastic representation have been illustrated. A rst controller design for the reduction of the roll motion has been implemented: minimum variance cheap controller. The maximum damping achieved was 68%. A second controller design has been employed: sensitivity function based approach controller. Two different specications have been tested: A narrow frequency range disturbance attenuation with maximum damping of 62% was obtained. A wider frequency range disturbance attenuation with maximum damping of 56% was obtained. The robustness of the system for small changes in the ship parameters has been assessed. Thesis Outline The thesis is structured as follows: In the rst two Chapters the geometrical aspects of the ship and the equations which govern the motions of the ship in the seaway are presented. A nonlinear model is obtained and the simulation environment is described. A linear state space model, obtained by linearizing the nonlinear model around a stationary point, is employed to design the controller.

18 Introduction 6 Chapter 3 introduces the environmental disturbances affecting the cruising of a vessel. Both linear and non linear methods to reproduce wind generated waves disturbances are investigated. A stochastic approach is considered and the model used for the wave disturbance is shown. Chapter 4 provides an overview of the techniques commonly used for roll stabilization giving some general informations on each method. The model used for the RRD system is introduced and the gures usually adopted to asses the motion performance of the ship are reviewed. A rst type of controller is developed in Chapter 5. It is the minimum variance cheap controller and it is designed in order to minimize the output variance of the system. Performance limitations and trade-offs are explained. Simulation results in different sailing conditions are presented and the robustness of the designed controller is investigated. Chapter 6 analyzes a different control design strategy. Output sensitivity specications are used to derive the analytical expression of the controller. Two different speci- cations are used and the respective controllers are validated. The performances of both controllers are investigated and compared. Chapter 7 presents a summary of the conclusions of this thesis along with suggestions for future developments.

19 Chapter 1 Ship dynamics A ship model is important for the design of a controller and for evaluating its performances. For the purpose of simulation, a complex model may be desirable in order to catch the characteristics of the real plant. Since a ship is a physical system moving in a seaway in 6 degrees of freedom (DOF) as it is shown in Fig. 1.1, to determine its position and orientation 6 independent coordinates are necessary. These coordinates are dened using two types of reference frames: inertial frame and body-xed frame. Fig Ship motion components. Courtesy of Perez [22]. Different inertial frames can be taken in consideration for marine vessels [22]. The n-frame (o n ; x n,y n ; z n ) is xed to the Earth; the positive x n -axis points toward the North, the positive y n -axis towards the East, and the positive z n -axis towards the centre of the Earth. This frame can be taken for inertial because the velocity of marine vehicles is small enough to consider the forces due to the rotation of the Earth negligible compared to the hydrodynamic forces acting on the vessel. The hydrodynamic frame (o h ; x h ; y h ; z h ) is not xed to the hull either; it moves at the average speed of the vessel following its path. The positive x h -axis points forward, the positive y h -axis towards starboard, and the positive z h - axis points downwards. The body-xed frame (o b ; x b ; y b ; z b ) is a moving coordinate frame 7

20 1.1 Equations of Motion 8 xed to the hull of the ship. The positive x b -axis points towards the bow, the positive y b - axis towards starboard, and the positive z b -axis points downwards. The geometric frame (o g ; x g ; y g ; z g ) is xed to the hull. The positive x g -axis points towards the bow, the positive y g -axis towards starboard, and the positive z g -axis points upwards. The origin of this frame is located along the centre line and at the intersection between the baseline and the aft perpendicular. Each of these frames has a particular use. For the problem analyzed in this work the n-frame and the b-frame will be used. For a marine vehicle the 6 motion components are called: surge, sway, heave, roll, pitch, and yaw - see Table 1.1. The most generally used notation for these quantities are: x; y; z; '; ; and respectively, while their time derivatives are denoted u; v; w; p; q, and r respectively. The rst three and their time derivatives correspond to the position and translational motion, while the other three coordinates and their time derivatives correspond to orientation and rotational motion description. The position and orientation of the ship are hence described relative to the n-frame, while the linear and angular velocities are expressed in the body-xed coordinate system. translations and rotations position and angles linear and angular velocities surge (x-direction) x u sway (y-direction) y v heave (z-direction) z w roll (x-axis) ' p pitch (y-axis) q yaw (z-axis) r Table 1.1. Generalized displacements of a vessel 1.1 Equations of Motion According to the dened frames of reference, the ship dynamics are obtained and the equations of motions involving both statics and dynamics are derived. As already shown by Fossen [9] the six DOF nonlinear dynamic equation describing the motion of a marine vehicle can be conveniently written as: M _ + C () + D () + g () = + g 0 + w (1.1) _ = J () (1.2) with M given by the sum of M RB and M A and where the other terms are as follows

21 1.1 Equations of Motion 9 = [x; y; z; '; ; ] T = [u; v; w; p; q; r] T M RB M A C () D () g () J () g 0 w position and orientation vector linear and angular velocity vector rigid-body mass and inertia matrix generalized added mass matrix Coriolis-centripetal matrix (including added mass C A ()) damping matrix vector of gravitational/buoyancy forces and moments velocity transformation matrix vector of control inputs vector used for pretrimming (ballast control) vector of environmental disturbances (wind, waves, currents) J () is the linear and angular velocity transformation matrix that relates the body xed velocity vector to the Euler angles = [; ; ] and the North-East-Down position vector. It has the form R n J () = b () 0 3x3 (1.3) 0 3x3 T () where as it can be imagined Rb n () is the linear velocity transformation matrix and T () is the angular velocity transformation matrix which can be stated as follows: 2 3 c c s c' + c'ss' s s' + c c's Rb n () = 4s c c c' + s'ss c s' + ss c' 5 (1.4) s cs' cc' where c = cos () and s = sin () sin '= tan cos ' tan T () = 40 cos ' sin ' 5 (1.5) 0 sin '= cos cos '= cos Applying Newtonian mechanics the rigid-body equations of motion in the body-xed reference frame (the vessel is assumed to be rigid) can be derived: M b RB _ + C b RB () = b (1.6)

22 1.1 Equations of Motion 10 where MRB b is the generalized mass matrix MRB b mi3x3 ms r, g b ms rg b I b (1.7) where r b g = x b g; y b g; z b g is the vector representing the distance between the origin of the b-frame O b and the center of gravity CG of the ship, I b is the inertia tensor, m = r is the mass of the ship calculated as the product of the water density and the displaced volume, and S is the skew-symmetric matrix dened as: 2 S () = S T () = ; = (1.8) The Coriolis-centripetal acceleration matrix can be expressed in different ways; one representation is CRB b ms ( (), 2 ) ms rg b S (2 ) ms ( 2 ) S rg b S I b 2 (1.9) where 2, [p; q; r] T. The components in CRB b () are added forces and moments derived by representing the equations of motions in the non inertial b-frame. The term b as it will be shown later is formed of both internal and external moments and forces acting on the ship. The internal forces and moments quantity b Int includes forces and moments that can be identied as the sum of three components: Added mass due to the inertia of the surrounding uid. Radiation-induced potential damping due to the energy carried away by generated surface waves. Restoring forces due to Archimedes (weight and buoyancy). It can be written in a vectorial form as b Int = M A _ C A () D () g () + g 0 (1.10)

23 1.1 Equations of Motion 11 The rigid body equation of motion can be rewritten in components form: m _u vr + wq x b g(q 2 + r 2 ) + y b g(pq _r) + z b g(pr + _q) = b 1 m _v wp + ur y b g(r 2 + p 2 ) + z b g(qr _p) + x b g(qp + _r) = b 2 m _w uq + vp z b g(p 2 + q 2 ) + x b g(rp _q) + y b g(rq + _p) = b 3 I b x _p + (I b z I b y)qr ( _r + pq)i b xz + (r 2 q 2 )I b yz + (pr _q)i b xy +m y b g( _w uq + vp) z b g( _v wp + ur) = b 4 (1.11) I b y _q + (I b z I b z)rp ( _p + qr)i b xy + (p 2 r 2 )I b zx + (qp _r)i b yz +m z b g( _u vr + wq) x b g( _w uq + vp) = b 5 I b y _r + (I b y I b x)pq ( _q + rp)i b yz + (q 2 p 2 )I b xy + (rq _p)i b zx +m x b g( _v wp + ur) y b g( _u vr + wq) = b 6 For the motion control problems addressed in this work (mostly roll stabilization) it is a common procedure to ignore the pitch and the heave motion components. This operation yields a model of the marine vessel in 4 degrees of freedom: surge, sway, roll, and yaw. Under this assumption Eq become m _u vr x b gr 2 y b g _r + z b gpr = b 1 m _v + ur y b g(r 2 + p 2 ) z b g _p + x b g _r = b 2 I b x _p _ri b xz + r 2 I b yz + pri b xy + m y b gvp z b g( _v + ur) = b 4 (1.12) I b y _r rpi b yz p 2 I b xy _pi b zx + m x b g( _v + ur) y b g( _u vr) = b 6 It is possible to simplify these equations by choosing the position of the origin o b of the b- frame such that the inertia products are negligible and the axes x b ; y b ; z b correspond to the longitudinal, lateral and normal direction of the vessel [9]. This is done with the choice of o b in a way that the coordinates of the center of gravity CG satisfy the following relationships: miyzx g 2 g = IxyI g xz g mixzy g g 2 = IxyI g yz g (1.13) mi g yzx 2 g = I g xzi g yz: where the superscript g means that the moments of inertia are taken with the body frame xed at CG. Hence the equations of motion become: m _u y b g _r vr x b gr 2 + z b gpr = b 1 m _v z b g _p + x b g _r + ur y b g(r 2 + p 2 ) = b 2 (1.14) I b xx _p mz b g _v + m y b gvp z b gur = b 4 I b zz _r + mx b g _v my b g _u + m x b gur + y b gvr = b 6 Noting that the term y b g is equal to zero a last simplication can be made and the ship equations of motion in the body xed coordinate system, in surge, sway, roll and yaw can

24 1.2 Forces and Moments acting on a ship 12 be written as: m _u vr x b gr 2 + z b gpr = b 1 m _v z b g _p + x b g _r + ur = b 2 I b xx _p mz b g _v mz b gur = b 4 (1.15) I b zz _r + mx b g _v + mx b gur = b Forces and Moments acting on a ship The vector b in Eq. 1.6 represents the total forces and moments acting on a surface vessel and is generated by different phenomena. This vector can be separated according to the originating effects and can be studied assuming that these forces and moments can be linearly superimposed [7]. These forces and moments driving the ship model can be split into internal and external components. b = b hyd + b hs {z } Internal + b c + b p {z } External (1.16) The internal force terms address hydrostatic (or restoring) forces and hydrodynamic forces and moments arising from moving the ship in the water. They are,as is in standard procedure in literature, modelled as linear combinations of nonlinear states and coefcients, which are essentially linear. Write, for example b hyd = f hyd ( _; ; ) (1.17) The only restoring force relevant to the manoeuvring of the vehicle is the roll restoring moment b 4hs = GZ () gr (1.18) where GZ () is the so called roll righting arm of the vessel that can be approximated with the transverse metacentric height GMt, and gr is the buoyancy. The rst term is often calculated by expanding to a series representation, and the terms used in the series are deducted from physical and hydrodynamic considerations combined with experience from model testing. Among different approaches, the one used in this work is the so called second-order modulus terms, a method proposed rst by

25 1.2 Forces and Moments acting on a ship 13 Fedyaevsky and Sobolev [8], and later by Norrbin [20]. b 2hyd = Y _v _v + Y _r _r + Y _p _p +Y j ujv jujv + Y ur Ur + Y vjvj vjvj + Y vjrj vjrj + Y rjvj rjvj +Y 'juvj 'juvj + Y 'jurj 'jurj + Y 'uu 'U 2 b 4hyd = K _v _v + K _p _p +K jujvj Ujv + K ur Ur + K vjvj vjvj + K vjrj vjrj + K rjvj rjvj +K 'juvj 'juvj + K 'jurj 'jurj + K 'uu 'U 2 + K jujp jujp (1.19) +K pjpj pjpj + K p p + K ''' ' 3 grgz(') b 6hyd = N _v _v + N _r _r +N jujv jujv + N jujr jujr + N rjrj rjrj + N rjvj rjvj +N 'juvj 'juvj + N 'ujrj 'Ujrj + N p p + N jpjp jpjp +N jujp jujp + N 'ujuj 'UjUj Equations 1.19 represent hence the hydrodynamic and hydrostatic forces and moments acting on the vessel for the sway, roll, and yaw components. They have been obtained under the assumption that the dynamics associated with the surge components of motion are much slower than the dynamics of the other motion components. Thanks to this supposition it becomes possible to decouple the surge component and to consider the variable u as a constant equal to the ship service speed u = U. The linear components in these equations are referred as the hydrodynamic derivatives and are associated to the added mass and added inertia of the water close to the ship that must be accelerated together with the ship hull. For instance, Y _p b _p ; and N p b (1.20) are the force in sway due to the roll rate derivative, and the yaw moment due to the roll rate. These coefcient sets can be very large with coefcients, as it can be seen in Son & Nomoto [25]. The concept of added mass describes, from the name, a nite amount of water connected to the vessel such that the vessel and the uid represents a new system with mass larger than the original system. The increased mass of the new system is called added mass. When the ship is rolling in calm water, there are only two moments acting on it besides the damping moment given by the added inertia term: the inertial moment I b xx _p; which can be also written as I b xx', and the righting moment GMtgr: For the principle of dynamic equilibrium their sum is equal to zero I b xx K _p ' Kp _' + GMtgr = 0 (1.21) From Eq the natural roll frequency can be obtained as s grgm t! ' = (1.22) Ixx b K _p

26 1.3 Control Devices 14 This implies that the natural roll period is s T ' = 2 I b xx K _p grgm t (1.23) It is also interesting to dene the roll damping coefcient ' given by the following expression ' = 2 p (1.24) grgmt (Ixx b K _p ) where K p is the roll moment due to the roll rate. The external force terms represent forces not included in the states terms expanded from the equations above. The rudder force, the propeller (thruster) force and the disturbances from wind, waves and current. The disturbances of the system is divided into different groups. Mainly two of them have to be considered: the multiplicative and the additive disturbances. The rst ones affects the ship system dynamics like the water depth, the load condition, trim, speed changes, etc., while the second ones are basically due to the physical environment and can be modelled as extra input signals. K p 1.3 Control Devices The motion of the ship is affected by instruments known as actuators like rudders, ns, aps, thrusters and propellers and their role is fundamental since they provide a direct link between the controller and the controlled system [22]. In the model taken in account only a rudder and a propeller are considered. It has to be mentioned that the maneuvering qualities of a ship, as well as its various characteristics, depend mostly on the rudder's type and size. A rudder presents the geometry of a trapezoidal foil and it is characterized by the following dimensions: the mean cord c; the foil Area, A r ; and the effective aspect ratio a - see Fig. 1.2 c = c R + c T ; A r = spc; a = 2sp (1.25) 2 c where c R and c T are the root and the tip cord respectively and sp is the span Forces acting on the rudder As stated by Perez [22] direction the rudder induced forces and moments in 4 DOF can be expressed in the body-xed frame as b 1c D b 2c L b 4c r r L (1.26) b 6c (LCG)L

27 1.3 Control Devices 15 Fig Geometrical aspects of the rudder. Courtesy of Perez [22]. where r r is the rudder roll arm and LCG is the longitudinal center of gravity. L stands for lift and D for drag. Fig Lift and Drag on the rudder. Courtesy of Perez [22]. When the uid moves relative to the rudder, as illustrated in Fig. 1.3, and the angle of incidence (or effective angle of attack) e is small, the ow remains attached to the surface of the rudder and there appear forces on it. One of those forces, more precisely the one directed perpendicular to the ow velocity vector, is the so called lift and it can be expressed as: L = 1 2 u2 ra r C L ( e ) (1.27)

28 1.3 Control Devices 16 is always the sea water density, A r is the area of the rudder, u r is the average ow velocity over the rudder and C L ( e ) is the non-dimensional lift. According to L.F. Whicker and L.F. Fehlner [31] and their experiments the non-dimensional lift can be estimated by the formula C L ( e ) = C L j e=0 e + C Dc e 2 (1.28) e a 57:3 but since the lift develops in an approximately linear manner with an increasing angle of attack it can be approximated using only the rst term of the last expression. The maximum lift that may be generated by a rudder, as a function of its angle of attack is limited by a series of events that cause the rudder to stall. When a rudder stalls, lift suddenly falls to very low or null values, therefore, in the design phase this possibility must be carefully studied and avoided. Stall occurs when the ow separates from the rudder low-pressure area and envelops an area of vortical ow. As previously mentioned, this separation generates an abrupt decrease in lift. The other force acting on the foil is called drag and it is directed in the same direction of the ow velocity vector. The drag is a consequence of the energy carried away by the trailing vortices emanating from the tip of the foil D = 1 2 u2 ra r C D0 + C L( e ) 2 (1.29) 0:9a where C D0 is the minimum section drag. Both forces are assumed to act on a point called centre of pressure CP. Its position varies with e, the angle of attack, representing the angle between the ow and the foil, already mentioned in the latter paragraph Rudder-Propeller Interaction In Eq and 1.29 the term u r designates the velocity of the water passing the rudder, and is in general not equal to the ship speed because the rudder is located in the race of the propeller. The propeller produces the necessary force needed for transit and it will be considered in the model as a disc that produces a sudden increase in the pressure of the water that passes from one side to the other. This generates a gradual and uniform change in the speed of the uid. For a given forward speed of the vessel u, the ow speed at the propeller, applying Bernoulli's law and according to Perez [22], can be computed as: u p = 1 2 " (1 w)u + s (1 w) 2 u 2 2(1 t) 1 X ujuj u juj A p # (1.30) and consequently the average ow velocity over the rudder as: 2 rp u r = u p (1.31) r (x)

29 1.3 Control Devices 17 Fig Rudder-Propeller interaction. where t is the so called thrust deduction number, w is the wake fraction, and r p is the propeller diameter. The term r (x) is the radius of the wake at a distance of x meters behind the propeller and is calculated as: 1:5 up x 0:14 + r 2u p u a r p up r (x) = r p 1:5 1:5 (1.32) up x 2u p u a 0:14 + 2u p u a r p with u a being the velocity of the uid ahead of the propeller (advance velocity relative to the propeller).

30 Chapter 2 Simulation Model In control design, mathematical models allow to design a controller and to perform numerical simulations in different scenarios. Due to the high cost of performing both scalemodel experiments and full-scale sea trials, an experimental assessment of the design is most of the times precluded for marine systems. In this chapter a typical state-space representation is introduced both for the non-linear and the linearized system. The controlled output which is the variable for which a specic behavior is wanted is the roll angle, while the control command is the rudder angle. 2.1 Nonlinear State-Space Model Considering the following relations _' = p and _ = r cos ('), together with the other four equations of motions 1.15, it will form six nonlinear equations in u; v; p; r; '; and. Since ' is assumed small the equation for yaw becomes _ = r: When the motion of roll and yaw is considered, the surge equation is disregarded due to the weak coupling between the two modes. Therefore the surge speed u is set to a constant value. Hence, the state vector x includes ve states and it is dened as follows: x = [v p r ' ] T (2.1) The ve nonlinear equations [22] can then be written in a state space form as: _x = M 1 f (x) + M 1 b c (2.2) where 2 3 m Y _v (mzg b + Y _p ) mx b g Y _r 0 0 (mzg b + K _v ) Ixx b K _p K _r 0 0 M = 6 mx b g N _v N _p Izz b N _r (2.3) The function f (x) can be divided in the sum of two other functions; f hyd (x) and f c (x) as it is shown below hyd mur 4hyd f(x) = f hyd (x) + f c (x) = 6 6hyd mzgur b 6 mx b gur 7 (2.4) In the expression 2.4 the terms 2hyd ; 4hyd ; and 6hyd correspond to the nonlinear hydrodynamic terms, given in the chapter before in Eq. 1.19, without the terms proportional to 18

31 2.2 Multi-Role naval vessel simulator 19 the accelerations which have been included in the matrix M. The control vector b c 2.2 includes the forces and moments generated by the rudder motion. in Eq. 2.2 Multi-Role naval vessel simulator The nonlinear model used for the simulations has been created using the Marine Systems Simulator (MSS) which is a Matlab/Simulink-based environment providing the necessary resources for implementation of mathematical models of marine systems [19]. The model consisted mainly in three blocks: the rudder machinery block, the rudder block and the vessel block. The block diagram of the nonlinear ship system can be seen in Fig Fig Block diagram of the system Rudder machinery A block diagram of the rudder machinery with its simplifying dynamics is shown in Fig This block diagram contains two limiters, one describing the limitation of the rudder angle and the other describing the limitation of the rudder speed. Usually these constraints are determined in order to provide safety and reliability of the rudder action. Imposing the slew rate constraint on the rudder an appropriate lifespan of the hydraulic actuators is ensured and saturation is avoided. The magnitude constraints are instead related to performance and economy. Rudder angles, if too large, can result in ow separation causing poor performances due to the loss of actuation. Furthermore at high speed the rudder machinery is subjected to higher mechanical loads.

32 2.2 Multi-Role naval vessel simulator 20 These limitation can be changed manually or automatically with respect to the desired performances. Fig Rudder machinery. Courtesy of van Amerongen [28] Rudder The second block used in the non linear model simulates the action of the rudder. The input, the actual rudder angle, and the velocity of the uid over the rudder are used to calculate the generated forces and moments for the ship components motion. These forces and moments will be the inputs of the next block: the naval vessel. The main structural characteristics of the rudder are given in Table 2.1. Effective area A r 2 m 2 Span sp 2 m Longitudinal distance LCG 23.5 m Roll arm r r 1.54 m Lift coefcient C L Table 2.1. Rudder data adopted for the multi role naval vessel Naval vessel The third block is the main part of the simulator and reproduces the behavior of a multipurpose naval vessel. The principal data and dimensions of the ship are reported in Table 2.2. The dynamic characteristics of the vessel in 4 DOF (surge, sway, yaw and roll) are reproduced. The inputs of the naval vessel model are the rudder generated forces and moments:

33 2.2 Multi-Role naval vessel simulator 21 X e : surge external force Y e : sway external force K e : roll external moment N e : yaw external moment. while the output is given by the following components u: surge velocity [m=s] v: sway velocity [m=s] p: roll rate [m=s] r: yaw rate [rad=s] ': roll angle [rad] : yaw angle [rad] Length over perpendiculars L pp 51 m Beam B 8.6 m Draught D 2.55 m Displacement r m 3 Service speed u 15 kts Table 2.2. Principal dimensions of multi role naval vessel

34 2.3 System Linearization System Linearization Accurate models of physical systems are nonlinear as is the case of the system just described. It is though difcult to use the nonlinear model directly in controller design. To analyze the dynamics of the ship motion from a control point of view and to be able to design a controller for the ship within small perturbations of an equilibrium point, the system must be linearized. It is easy to obtain a linear model if a nonlinear model exists. The rst step in the linearization procedure is to determine the stationary states. This is done considering the equilibrium point x given by v; r; p; '; = 0; and u = u constant. The linear model is then obtained by taking the rst term of the Taylor expansion around x. This way the system and input matrices of the linearized model are dened as: with _x = Ax + Bu (2.5) A = M 1 F and B = M 1 G (2.6) F = f(x) 2 x j x=0= (2.7) 3 Y jujv juj 0 (Y ur m) u Y 'uu u 2 0 K jujv juj K p + K jujp juj (K ur + m z g ) u K 'uu u 2 grgmt 0 6 N jujv juj N p + N jujp juj N jujr juj mx g u N ujuj u juj being the viscous force coefcient matrix, and r r G = 6 (LCG) u2 ra r C L (2.8) 0 the rudder force coefcient vector, where u r is the constant ow velocity over the rudder calculated considering the speed of the vessel u constant and equal to u. The input of the system and control variable is the rudder angle. The linearized model hence takes the form: _v a 11 a 12 a 13 a 14 0 v b 1 _p 6 _r 7 4 _' 5 = a 21 a 22 a 23 a a 31 a 32 a 33 a p 6r b 2 6b (2.9) _

35 2.3 System Linearization 23 Considering ['; ] T as the output vector, it is easy to obtain the transfer functions from to ' and to. They are the basis of design of RRD controller. The roll and the yaw output are dened as: ' = {z } x; = C roll {z } x (2.10) C yaw For the state space model 2.9 the transfer functions ' (s) = (s) = C roll (si (s) = (s) = C yaw (si A) 1 B become: A) 1 B and G ' (s) = ' (s) (s) = K roll (q s)(q 1 + s) (s + p 1 )(s + p 2 )(s 2 + 2w roll s + w 2 roll ) (2.11) G (s) = (s) (s) = K yaw (q 2 + s)(s 2 + 2w q s + wq) 2 (p 1 + s)(p 2 + s)(s 2 + 2w roll s + wroll 2 ) (2.12) Having the linearized system state space model in Eq. 2.9 and 2.10, we can use the linear system methods to analyze the characteristics of the vessel model. In Fig. 2.3 the frequency responses of the linear systems from rudder to roll angle (solid line), and from rudder to yaw angle (dash line) have been plotted. Note that the linearized model is only an Bode Diagram G φ (s) G ψ (s) Magnitude (abs) 1 0,1 0,01 0,001 0, Frequency (rad/sec) Fig Frequency characteristic of the open loop transfer functions. approximation of the nonlinear system, and the simulation of the system becomes very in-

36 2.3 System Linearization 24 accurate, when the parameters of the system are moved farther away from the linearization point. There are several open loop system properties we need to know before we design the controller. One of them is the system's open loop dynamic response characteristic that gives us not only the background information about the system performance but also the guideline for controller design Rudder to Roll transfer function As shown in Eq the transfer function mapping the rudder angle into the roll angles can be written as follows G ' (s) = 0:13291(s + 0:3493)(s 0:1808) (s + 0:04306)(s + 0:4387)(s 2 + 0:1945s + 1:074) The open-loop eigenvalues of the ship-system can be found by solving the equation: (2.13) = det (I A) = 0 (2.14) or directly looking at the denominator's zeros and were found to be: 1 = 0: = 0:4387 (2.15) 3;4 = 0:0973 1:0319i The system has four negative eigenvalues; two of them real and the other two complex. Computing the partial fraction expansion of the transfer function it is possible to obtain the residues which are: r 1 = 0:0216 r 2 = 0:0157 (2.16) r 3;4 = 0:0186 0:0624i The eigenvalues and the residues are small, which indicates that the system has a relatively slow response. If the roll motion is isolated from all other motion components what is left is the second-order system given below ~G (s) = K roll (s ' w ' s + w 2 ') (2.17) where the roll natural frequency w ' is equal to 1:0359 rad=s and the roll damping coefcient ' equal to

37 2.3 System Linearization Non-minimum Behavior in Ship Response In the rudder-to-roll transfer function there are also two zeros; q = 0:3493 (2.18) q 1 = 0:1808 one of the two zeros is on the right-hand side of the complex plane - see Fig This results in a limitation in the design of a control system [14]. In particular the positive zero determines a trade-off between reducing the roll at some frequencies and amplifying it at others. 1 Root Locus Imaginary Axis Real Axis Fig Poles and zeros of the rudder to roll transfer function. If a positive step is sent in as an input to a system with a positive real zero at s = q such as G ' (s) then the output response of the closed-loop system would be Y (s) = G '(s) G ' (s) s = T (s) 1 s (2.19) The single-sided (unilateral) Laplace transform for a signal x(t) is dened as L [x(t)] = X(s) = Z 1 0 e st x (t) dt (2.20)

38 2.3 System Linearization 26 assuming that the integral can be evaluated at the upper and lower limits to yield welldened and bounded values. Associated with this transform X(s), and equivalent to the statement that the transform exists, is the signal's Region of Convergence (ROC). If the transform exists, the ROC exists, and vice versa. The ROC is the open half of the s-plane that lies to the right of all singularities (poles) of X(s). At any point s = s 0 that lies inside the ROC, the following relationship is true. X(s 0 ) = Z 1 0 e s 0t x (t) dt (2.21) Therefore, at any point s = s 0 in the region of convergence, the form of the dening equation of the Laplace transform is still applicable, with the complex variable s replaced by the complex number s φ [deg] time [s] Fig Non-minimum-phase effect on roll dynmamics during a turn to port. In the problem being considered here, since the zero q is in the region of convergence for the transform Y (s), applying the Laplace transform to y (t) and Y (s) with s 0 = q it follows that Z 1 e qt y (t) dt = Y (q) = T (q) 1 q = 0 (2.22) 0 Since q is a real number, the exponential term inside the integral in Eq is always positive. The output signal has an initial value y(0) = 0 and a nal value y(1) = 1, so y(t) is not identically equal to 0. However, the results stated in Eq indicate that the

39 2.3 System Linearization 27 area under the curve of y(t) (weighted by a time-varying positive number) over all time is equal to 0. Therefore, the output signal y(t) must take on negative values. This means that when the open-loop system has a right-half plane (nonminimum-phase) zero, the step response spends part of its time going in the wrong direction. This is generally known as a non-minimum-phase response or an inverse response. This inverse response always exists when the closed-loop system has a right-half plane zero. Since zeros of the openloop forward transfer function G(s) appear as closed-loop zeros, then whenever G(s) has a non-minimum-phase zero, the system's step response will exhibit undershoot, taking on negative values. It can be also noticed that the slower the system response is the larger will be the initial system response. Fig. 2.5 illustrates this non-minimum-phase behavior for the rudder-to-roll transfer function. The step response to a 10 degrees rudder angle input goes negative rst, then goes positive, ending with the nal value.

40 Chapter 3 Environmental Disturbances There are several disturbances with various effects on the vessel dynamics to be taken into account. Mainly three classes of disturbances can be distinguished: Disturbances which affect the dynamics of the system, like for example the depth of the water Disturbances which cause additional signals in the system such as waves Disturbances that corrupt the measurements like sensor noise. 3.1 Wind generated waves Environmental disturbances such as waves, wind and current are the principal causes of the undesirable motion of the ship. For the problem at hand, only wind generated waves are taken in account since they represent the dominant disturbance in the task of seakeeping. There are several models in literature to describe the phenomenon of wave generation. As described by Fossen [9] the process of wave generation due to wind starts with small wavelets appearing on the water surface. These short waves starts and continue to grow until they nally break and their energy is dissipated. It has been observed that a developing sea, or storm starts with high frequencies. When a storm has lasted for a long period of time is said to create a fully developed sea. After the wind has stopped, a low frequency decaying sea is formed. The objective of this section is directed to the description of wave spectra and ship motion in waves. The response of a ship to waves is quite complex. Having a certain velocity of advance, a ship experiences the wave excitation at an encounter frequency. This frequency is not related linearly to the wave frequency, as seen from a xed point, but varies with ship speed and angle of attack from wave through a nonlinear mapping. Furthermore, forces and moments on the hull are determined by the wavelength of incident waves through a square root function of the wave frequency. The mathematical description of the motion of regular gravity waves over a free surface is classical. A two dimensional wave progressing at an angle { with respect to the inertial frame, is described by its elevation at a certain position x; y at time t (x; y; t) = sin(! 0 t kx cos({) ky sin({) + ") (3.1) 28

41 3.2 Stochastic representation 29 where k = the wave number! 0 = wave frequency = wave amplitude " = the initial phase angle The phase velocity of the wave, c, is the velocity with which the wave crest move relative to the ground. Assuming a gravity wave and innite depth of water, the following dispersion relationships hold: k =!2 0 g = 2 k c = r g 2 (3.2) where g is the acceleration of gravity, and is the wavelength. The phase velocity is inversely proportional to its frequency. In other words, long waves propagate faster than short ones. This phenomenon is crucial for simulation of wave motion. A ship advancing in a seaway will overtake some short waves, while it will be overtaken by some long ones. A motion of the ship at a certain encounter frequency can, therefore, be caused by up to three harmonic waves with three different wavelengths. 3.2 Stochastic representation When the wave amplitude and frequency become random variables the simple waves are extended to an irregular sea. Ocean waves are random in terms of both time and space. Therefore, a stochastic modelling description seems to be the most appropriate approach to describe them. It is assumed that the variations of the stochastic characteristics of the sea are much slower than the variations of the sea surface itself. Due to this, the elevation of the sea at a certain position (x; y; t) can be considered as a realization of a stationary process. Haverre and Moan [12] suggested the following simplifying assumptions about the stochastic model: The observed sea surface, at a certain location and for short periods of time, is considered a realization of a stationary and homogeneous, zero mean Gaussian stochastic process. Some standard formulae for the spectral density function S(!) are adopted. Under a Gaussian assumption, the process, in a statistical sense, is completely characterized by the power spectral density function S(!). A conceptual model to describe the elevation of an irregular sea is given by the sum of a large number of essentially independent regular (sinusoidal) contributions with random

42 3.3 Nonlinear models of wave spectra 30 phases. Then the sea elevation at a location x; y is given by: NX NX (x; y; t) = i (x; y; t) = i sin(w 0i t k i x cos({) k i y sin({) + " i ) (3.3) i=1 i=1 where i (x; y; t) is the contribution of the regular travelling wave components i progressing at an angle { and a with random phase " i. The above statements imply that the mean and the variance of the waves elevation are var[(t)] = E [(t)] = 0 (3.4) Z 1 0 S (!) d! (3.5) 3.3 Nonlinear models of wave spectra Researchers who have studied ocean waves have proposed several formulation for wave spectra dependent on a number of parameters (such as wind speed, fetch, or modal frequency) [23]. These formulations are very useful especially in the absence of measured data, but they can be subject to geographical and seasonal limitations. Many models are non linear and determine the sea spectrum using spectral estimation techniques; they are used to derive linear approximations and transfer functions for computer simulations. Based on extensive data collection, mostly in the North Atlantic ocean, a series of idealized single side local spectra have been obtained to described long-crested seas. The most widely used in maritime engineering are the Pierson-Moskowitz (PM) and the JONSWAP spectra. The former model takes the name from the developers of a two parameter spectral formulation for fully developed wind generated seas. It has the following form: where the parameters A and B are S (!) = A! 5 exp( B! 4 ); [m 2 s] (3.6) A = 8: g 2 B = 3:11 (3.7) Hs 2 with H s being the signicant wave height (mean of the one third highest waves) used to classify the type of sea and g is the gravity constant. The signicant wave height is proportional to the square of the wind speed at 19.4 meters over the sea surface and their relationship is expressed here: H s = 2:06 g V :4 (3.8) The modal frequency (peak frequency)! 0 and modal period T 0 for the PM-spectrum are found requiring that ds (!) = 0 (3.9) d!!=! 0

43 3.4 Linear approximations of wave spectra 31 and they are r r 4B 5! 0 = 4 T 0 = 2 4 (3.10) 5 4B The International Ship and Offshore Structures Congress (ISSC) and the International Towing Tank Conference (ITTC) have suggested the use of a modied version of the PMspectrum. For prediction of responses of marine vehicles and offshore structures in open sea, they recommended to use the following parameters: A = 43 H 2 s T 4 z B = 163 (3.11) Tz 4 where T z = 0:710T 0 is the average zero-crossing period. For non fully developed seas the PM-spectrum cannot be used; so it should be replaced by the Joint North Sea Wave Project (JONSWAP) spectrum. It was developed for the limited fetch North Sea and is used extensively by the offshore industry. This spectrum is signicant because it was developed taking into consideration the growth of waves over a limited fetch and wave attenuation in shallow water. Over 2,000 spectra were measured and a least squares method was used to obtain the spectral formulation assuming conditions like near uniform winds. The JONSWAP spectral density function will result in a more peaked function than those representing fully developed seas. Many other spectra have been obtained and which are not discussed: these include the Neumann, the Bretschneider, the Ochi and the Torsethaugen spectra. 3.4 Linear approximations of wave spectra A method commonly used in control for analysis and simulation is to replace the nonlinear wave disturbance model with a linear wave response approximation [9]. A linear approximation of the shape of the power spectral density function of the signal of interest is given by S yy (j!) = jh(j!)j 2 S ww (j!) (3.12) where S ww is the power spectrum of a zero-mean Gaussian white noise process which is constant and equal to one. The lter H (s) is called a shaping lter and can be implemented in several ways (different orders and structure) but the most commonly used is the secondorder lter of the form: 2! 0 s H (s) = (3.13) s 2 + 2! 0 s +! 2 0 where is a damping coefcient, is a constant describing the wave intensity, and! 0 is the dominating wave frequency. Hence, substituting s = j! into Eq H (j!) = j2! 0!! 2 0! 2 + j2! 0! (3.14)

44 3.4 Linear approximations of wave spectra 32 The square of the magnitude of the lter is then easily obtained: jh (j!)j 2 = 4 2! 2 0 2! 2 (! 2 0! 2 ) ! 2 0! 2 (3.15) Wave spectrum PM spectrum Linear spectrum S(ω) [m 2 s] ω [rad/s] Fig Power spectrum of the linear approximation and the Pierson-Moskovitz model. As it is from the Pierson-Moskowitz spectrum when! is equal to the dominating wave frequency! 0 ; the maximum value of S yy (jw) can be obtained as max! S yy(!) = S yy (! 0 ) = 2 (3.16) The dominating frequency in the linear model is also equal to the modal frequency of the PM spectrum s 4 16! 0 = 3 4 (3.17) 5Tz 4 The Power spectrum of the linear approximation (solid line) and the one of Pierson-Moskowitz model (dash) for the same signicant wave height H s = 4, period T 0 = 8s, and damping coefcient = 0:25 are plotted in Fig. 3.1.

45 3.5 Encounter frequency Encounter frequency Waves incident on the structure of a ship can be described as head seas, following seas, beam seas, quartering seas or bow seas depending on the incident direction. Fig. 3.2 illustrates the different cases. The incident angle,, is measured from the stern. Fig Incident sea description. Courtesy of NTNU [21]. The motion of a ship, forward or otherwise, affects the way incident waves are viewed by someone aboard the vessel. For example if the ship is making way in following seas with a constant velocity, u, then the waves will appear to meet the ship at a slower rate than the actual frequency of the waves. This new, or observed, frequency is called the encounter frequency,! e. If the waves are incident on the ship at some angle,, then the component of the speed of the ship in the direction of wave propagation is u w = u cos (). The wave crests move at the phase speed, c =! and the relative speed between the ship and the k waves is u r = c u w =! u cos () (3.18) k Using the dispersion relationship for waves in deep water it is possible to rewrite the equation for encounter frequency as! e =!! 2 u g cos () (3.19)

46 3.6 Waves simulation model 34 Fig. 3.3 shows a schematic representation of the transformation between! and! e for different sailing conditions and u constant. From this gure, it can be seen that when the vessel is sailing in bow or head seas the wave frequencies are mapped into higher frequencies. In beam seas, however, there is no change and both! and! e are the same. In following and quartering seas, the situation becomes more complicated as different wave frequencies can be mapped into the same encounter frequency. Fig Encounter frequency versus actual frequency. 3.6 Waves simulation model To simulate the elevation of the sea surface for a fully developed condition the Simulink block diagram reproduced in Fig. 3.4 has been used. The band limited white noise used in the simulation has a power spectral density equal to one. Figure 3.5 shows the signal representing the wave disturbance generated using the Simulink blocks. The stochastic disturbance has been plotted for a period of three minutes. The values used are: wave height H s = 1:5m, period T 0 = 8s, and damping coefcient

47 3.6 Waves simulation model 35 Fig Simulink block diagram for wave disturbances. = 0:25. The disturbance has been considered as an output disturbance and hence added to the roll angle signal of the ship system. 3 2 wave height time [s] Fig Wave disturbance simulation