Gemilerin manevra denklemlerinin bilgisayarla çözümü

Abstract

Manevra hareketleri, 3 serbestlik derecesine sahip, x ekseninde ileri-geri hareket, y ekseninde yan öteleme ve z ekseni etraf ında rotadan sapma hareketleridir. Bu hareketler sonucu, X,Y kuvvetleri (x,y eksenlerinde) ve N momenti (z ekseni etrafında) oluşur. Gemiye bağlı ek sen takımına göre bu hareket incelenirse, X,Y ve N'den oluşan hareket denklemleri elde edilir. Bu kuvvetler ve moment Taylor açılımı yapılarak açılır ve birinci merte beden terimler alınır, diğer terimler ihmal edilirse li neerleştirilmiş hareket denklemleri elde edilir. Lineer leştirilmiş hareket denklemleri çözülerek bir geminin doğrusal denge kriterleri bulunabilir. Denklemlerin çözümü incelendiğinde, doğrusal denge şartı: A > 0, B > 0 ve C>0 olmasına bağlıdır. Ge nel olarak gemiler için A ve B > O'dır. Bundan dolayı C > 0 şartı doğrusal denge için gerekli tek şart olmak tadır. Bu durum, dümen kullanmaksızın gerekli olan doğru sal denge şartıdır. Hareket denklemlerine dümen kuvveti nin etkisi eklenerek doğrusal" denge için gerekli olan şart şu duruma gelir. Oı 02 <0 (denge kökleri) şartı açısal hı zı sürekli bir yapıya ulaştırır. Gemi doğrusal dengeli değil ise, açısal hız sürekli bir yapıya ulaşmayacaktır. Lineer yaklaşım, küçük dümen açıları için iyi sonuç lar vermektedir. Ancak, büyük dümen açıları ve zor olan dönüşler için lineer olmayan hareket denklemlerinin çö zülmesi gereklidir. Hareket denklemleri, 3. mertebe Taylor açılımı yapılarak açılır ve diğer terimler ihmal edilir se, lineer olmayan hareket denklemleri elde edilir. Bu yaklaşım, manevraların simülasyonunda iyi sonuçlar ver mektedir. X kuvveti incelendiğinde, çeşitli değişkenlere bağlı polinom tarzında bir çift fonksiyon olduğu görüle bilir. Y kuvveti ve N mom enti ise, aynı değişkenlerin polinom tarzında tek fonksiyonları olacaktır. Lineer ol mayan bu hareket denklemleri bilgisayar yardımı ile ko layca çözülebilir. Geliştirilen bilgisayar programında, denklemlerdeki hidrodinamik katsayılar çeşitli model deneylerinden elde edilerek çözüm yapılmaktadır. Bu çözüme göre, "Dönme Dai resi", "Zig-zag Manevra ve "Spiral Manevra `lar ın simü lasyonu yapılmaktadır. Dikkat edilmesi gereken en önemli husus ise kuşkusuz hidrodinamik katsayıları bulmak için yapılan deneylerin doğru ve hassas olarak yapılmasıdır. Çeşitli kuru luş ve kişilerce yapılan bu deneylerden Hy^A(Hydrc>-Aerodynamics La boratory) tarafından yapılan deneylerin sonuçları simülasyon hesabı için en uygun olanıdır. Bu çalışmada bu kuruluşun deney sonuçları kul lanılmıştır.

Nowadays, research is carried out in Naval Architecture as in many fields. Many improvements appear and are pre - sented to the shipbuilders. In particular, computers in creasingly aid in the research so that studies can be concluded in a short time. This is saving of time and work power. One of the important subjects in Naval Architecture is the ship motions through the waves. These are roll, yaw, sway, heave, surge and pitch motions. Thus, the ship moving in the water encounters fluid actions. These are drug forces applied by the water to the ship and the forces and moments caused by the motions. All of these depend on properties of the motion and properties of the fluid and properties of the ship's hull. The shi p motions may be classified as controlled and uncontrolled motions. The controlled motions are the ship manoeuvres. The ship manoeuvring is the controlled change or retention of the direction of motion of a ship and its speed in that direction. For surface ship, it includes all forces, moments and motion vectors acting in any direction in the horizontal plane. Furthermore for submarines, manoeuvring embraces the control of the path of the motion in three-dimensional space. After so many centuries in which the ships have set out to sea, reliance, is still placed on the evolutionary processes of design, backed up in some cases by adhoc mo del testing, in order to decide the control arrangements. A further factor is the greater use of automatic control, requiring more detailed information on the handling charac teristics of ship at an early stage in the design process. Through the development of techniques to meet these de - mands, a better understanding of the problems involved has been obtained which is of value for all ship types. There are distinct advantages in terms of benefits in safety, and fuel consumption or speed, as well as the more obvious ones, in ensuring that a new ship has good, rather than adequate or acceptable handling qualities. fhe forces and moments acting on a body which in turn cause the ship to move, need now be studied in order to analyse the motion of a body. -V11İ- Through the dependence of various phenomena on the properties of the body, properties of the motion and pro perties of the fluid, the relationship for the forces and moments (in unresticted water) become Forces Moments = f ' -{Properties of body, properties of motion, properties of fluid } For a given ship, in a given fluid, in the absence of excitation forces, one can express the general func - tion as - ¦ j* } - f (x0,yo,zo,^,e,f, u,v,w,p,q,r,u,v,w,p,q,f,6,6,6) and F m X* + Y^ + Z £ m s Rt +. Mt + N* 1 3 K The Taylor expansion for the forces and moments act ing on the ship would be expressed as I }= [eAXoDx0+ A^o Dvo f...+ Av Dv +-"+ Ar Drl f ['Vo' "-V- rol m and the expansion of the power series into the actual functional form would indeed give an extremely long, cum bersome^ and almost impossible to handle expression for F and m. In this expansion, only first order terms are taken. In the manoeuvring, this involves the three degrees of motion freedom of translation along the x and y axes and rotation about z axis. Under this limitation only the follow! ng variables will appear in the motion equations: xo' Yo'^'U'V^'û'V'* and the linearised equations of the motion become as (xu-m)ü + xuAu + x.v + xv + x^r + xrr o 0 Yuû + YuAu + (Y^-m)v + Yvv f (Yr-mxG)f f (Yr-muo)r=0 N.u + n Au + (N.-mxr)v + (Na-I_)£ + (N -mxrurt)r - 0 -ix- The test for any type of stability is to establish an equilibrium situation and determine whether the sys tem returns to the original condition of equilibrium after a disturbance of the smallest amount (infinitesim al disturbance). If it returns or tends to return to the original equilibrium condition when the disturbance is removed, it is stable. If it departs or has »tendency to depart from the original equilibrium condition, the ori ginal equilibrium condition is unstable. The usual case for a body which is unstable in a given condition of equ ilibrium i s to depart from that condition until it re aches another equilibrium condition (not the original one) which is a stable one. The ship is disturbed slightly from its uupright equilibrium condition and when the dis turbance is removed the tendency to return to the original upright position is observed. If it returns, it is stable, if it departs, it is unstable. A ship which is dynamically unstable in straight line motion cannot maintain straight line motion when there is no rudder deflection. The unstable ship will go into a starboard or port turn without any rudder deflec tion. The ship which is dynamically unstable in straight line motion can maintain a straight course (on the average) only by continuous use of the rudder. With both A and B positive for ships, the criteria for dynamic stability in straight line motion becomes whether C is positive. C is called the stability criterion. The linear equations are useful with limitations for steering, stability and manoeuvring. It was indicated that certain important information in the prediction of certain ship manoeuvres would require the use and solu - tion of non-linear equations. These will be developed and discussed from the point of view of certain hydrodynamic aspects. The mathematical model at present in use at HyA (Hydro- Aerodynamics Laboratory) for simulation of steering and > manoeuvring characteristics of surface ships is based on the general equations of motion of a rigid bod y moving in the horizontal plane. For a body having freedom in surge, sway and yaw but restricted in heave, pitch and roll, the equations developed for a coordinate system fixed in the symmetry plane of the body are } 2 X = m(u - rv - xGr ) Y = m(v + ru +. xGr) N = mx_(v + ru) + I r Where terms on the right-hand side describe mass and inertial responses and the left-hand side expresses the x- the external hydrodynamic forces and moments acting on the body. The hydrodynamic forces and moments are functions of body geometry, motions and orientation. For a given body with a single control surface moving in unrestict- ed water, they may be expressed as the general functions j X Yj= f (û,v,f,u,v,r,6) N The functions describing the hydrodynamic forces and moments have been developed into a useful form for analysis purposes with the aid of a Taylor expansion of the functions. If the Taylor expansion is limited to the first order terms, the linearised equations are obtain ed previously. The present stage of development which enables realistic simulations of ship manoeuvres to be made is based largely on a third order Taylor expansion of the functions. Introducing the assumptions that ; 1. Forces and moments have appropriate port and starboard symmetry except for a constant force and mo ment caused by propeller. 2. There are no second or higher order acceleration terms and that cross-coupling between acceleration and velocity parameters is negligible. It is seen that the equation in the mathematical model are comprised of numerous coefficients such as Y, N etc.. These coefficients in general depend on tK¥ particular geometry and design of a ship and they must be known with reasonable accuracy before manoeuvres can be simulated by solving the mathematical model with the aid of a computer. Ideally, numerical values for the coefficients would be evaluated by theoretical means, but some of the hydro- dynamic coefficients can be calculated approxiametly, the only reliable way, at this time, of obtaining values with the accuracy needed for simulations is to conduct captive model tests. Jn a captive model test, the models forced to per form precisely controlled movements, one or two different motion and rudder parameters being assigned values si multaneously. The resulting hydrodynamic forces and mo ments acting on the model are measured as functions of the parameters and the coefficients are subsequently ob tained from these measurements. In the execution of the captive model tests, parameters are first explored one at a time, all other parameters being zero. The result ing forces and moments can then be expressed by -xi- coefficients which ar e functions of the single para - meter. The next step is in the captive model tests is to vary two parameters simultaneously and if the resulting forces and moments differ from the superimposed results of the individually measured values, then the difference is expressed as a two-variable function of the parameters and the coefficient representing the cross-coupling ef fect can be determined. The range of motion and rudder parameters explored during testing should in principle, cover the range of subsequent simulation. Surge, sway and yaw accelerations, speed loss, drift angle, yaw velocity and rudder angle should be varied systematically up to the values corres ponding to maximum rudder manoeuvres for the free- sail - ing ship. Captive model tests in which measurements are made of hydrodynamic forces and moments resulting from drift angle, from rudder angte and from combinations of drift and rudder angles can be conducted with relatively simple equipment in a conventional towing tank. Many methods have been used to measure forces and moments, due to angular Velocity, amongst them being such devices as curved models, curved-flow channels and freely decaying oscillators. All of these systems have major disadvantages, being either in accurate or unwieldy or both. The use of a rotating arm in a circular tank is at present by far the most widely used approach and is furthermore, a most satisfactory means of obtaining cross- coupling terms in angular velocity, drift angle and angu lar velocity, rudder angle. The disadvantages, of a ro tating arm, apart from its high capital costs are that it is not practicably possible to measure acceleration de rivatives and that it is not well suited to the genera - tion of small angular velocities. The Planar-Motion Mechanism system conceived and de veloped by Gertler and Goodman, provides a means of con - ducting captive model tests in which angular and straight line motion can be imposed on a model in a conventional to wing tank. Developed as a technique for submerged body research, the original mechanism generated motions of the body in the vertical plane. For application to surface ships, a Planar-Motion Mechanism must operate in the ho rizontal plane. This stystem can be used in two different modes of operation, designated static and dynamic. In the staticmode, the model is constrained to travel along a straight path at constant velocity and the mechanism is used to set discrete drift angles. The unique feature of the Planar-Motion Mechanism is its ability to generate oscillatory motions which are pro duced in the dynamic mode of its operation. Sinusoidal -xxi- motions are imposed on the model with sway and yaw phas ed in such a way as to produce conditions of pure sway and pure yaw. In the pure yaw test, bow and stern are oscillated with phase angle chosen such that pure angular velocity and acceleration result. As a means of experimentally measuring acceleration as well as angular velocity, drift angle and rudder angle derivatives. The Planar-Motion Mechanism system constitutes an almost ideal method of obtaining all the linear terms needed for course stabi lity studies. All of the coefficients in the non-linear equations can be measured by iMPMM. The use of non-dimen sional coefficients in the equations provide the solu - tion to other non-linear equations. Research shows that Hy-A (Hydro Aerodynamics Laboratory) test results are the most agreeable with trial results. These results were used in this study to predict the ship manoeuvres. There are four manoeuvre tests for ships. These are; 1. Turning Test 2. Zig-zag Test 3. Spiral Test 4. Pull-out Test Turning test consist of a series of runs in which the ship makes a straight course approach at a fixed speed (constant throttle setting). The rudder is then put over to a specified angle and held there whilst the ship completes a circle. The trajectory of the ship is measur ed by a variety of methods, usually employing an optical tracking system and measurements of tide are taken so that the path of the ship or model in calm water can be plott ed. Consequently, turning circle parameters are obtained. These are j Advance : The distance the ship travels forward in the direc tion of the approach course from the instant the rudder is put over until the vessel has made a 90 degree change of heading. Transfer : The distance to the L.C.G. of the ship normal to approach course which the vessel achieves at 90 degrees change of heading. Tactical Diameter : The distance normal to approach course in which ship achieves 180 degrees change of course. -xiii*- Steady Turning Radius : The radius of the steady circle path the ship takes up. Steady Drift Angle t Angle between the ship's head and the tangent to the path on the circle. Rate of Change of Heading : This is normally quoted for the steady circle. Speed on Circle : With fixed throttle settings there is a drop in speed on the circle compared with approach speed. For closer investigations, records, of the transient motion of the ship in to a circle may be taken. Thus the rate of change of heading is found to build up rapidly and usually overshoots the steady value on the circle. At much the same time the drift angle also increases, rapidly so that though the heading has changed, the ship may per sist for some time in the same direction as the approach course before going into the circle. The drift is normally in advance of the heading, so that the ship initially moves to the opposite side of approach track to the circle. This test is used to compare the new design with previous successful designs of the same type. The second important definitive manoeuvre is the zig zag manoeuvre. The results of this manoeuvre are indica - tiv'e of the ability of a ship's rudder to control the ship. So the results depend some what on the stability charac - teristics of the ship as well as on the effectiveness of the rudder. The rudder is put over to 20 degrees and the ship is allowed to begin a turn. When the heading is changed by 20 degrees, the rudder is put over to 20 degrees opposite angle and held there while the ship changes direc tion. At the 20 degrees heading the other side of base course, the rudder is again reversed and the procedure repeated for about 5 cycles. The normal records taken are of the ship's heading and rudder angle time history. Again the results are used for comparison with previous vessels, the important features being the heading overshoot at helm reversal and the period of the cycle. The spirals are tests to evaluate the directional stability of the ship, i.e. its ability or otherwise to maintain a straight path with the rudder amidships, if there are no other disturbances. The spiral was originally proposed by Admiral Dieudonne and is known by his name. With the fixed throttle setting the rudder is put over -xiv - to 15 degrees starboard and the ship is allowed to settle to a steady rate of change of heading. When steady the rate is recorded, the helm is eased to 10 degrees star - board and again the rate is allowed to steady and is re corded. The helm is then eased to 5 degrees starboard and then in smaller steps taken through to 5 degrees port and then on out to 15 degrees port again in steps. Prom here the whole procedure is repeated back to 15 degrees star - board. The result s obtained are plotted as heading rate against rudder angle. If the ship is directionally stable the curve so plotted will 'be a single valued function passing through or near the origin, depending on whether there is any steering bias and usually S shaped. If the ship is unstable, there is a different appearance to the curve near zero rudder angle. It will be found that the ship persists in turning to starboard when the rudder has been brought back to amidships, and may still persist to starboard for a few degrees of port helm. There is then a sudden jump to port turnina as the helm is put further over to port. On' the reverse procedure the ship continues to turn to port for zero and small starboard helms before suddenly reverting to starboard turning. Thus the curve has the appearance of a hysteresis loop. The larger the loop the more unstable the vessel and the more difficult it is to handle. In this form, the spiral test takes up a large amount of sea room and time and also tells us nothing about how the ship steers a straight path or does a small turn. For these reasons, Bech proposed what is known as the reverse spiral in which specified rates of turn are demanded and the rudder is operated to maintain each steady rate. Thus with this test, it is possible to specify zero turning rate and find what the rudder has to do to achieve this. For unstable ships the rudder cannot stay at zero but will os cillate about a mean of zero or the bias angle. This re quires a statistical analysis of the rudder movements, but as well as giving information about the are inside the loop ittalso shows what rudder activity can be expected to main tain the ship on a straight course even in calm water. The last test is Pull-out manoeuvre. The test is for the determination of the directional stability of a ship and is a definitive test, in that it asks the ship to per form the motion which defines directional stability. To do this, the ship is put into a steady turn by the rudder and when a steady rate of turn is achieved the rudder is return ed to amidships and the subsequent motion of the ship is recorded. If the ship is directionally stable then the mo tion will be one of decaying rate of turn until the ship is again on a straight path. On the other hand if the ship is unstable then, though there may be some decay in the rate of turn, a residual rate of turn will persist. The im portant feature of this test is that the time constant of the decay curve can be used as a measure of the directional stability of the ship. -x v- The simulation of the ship manoeuvres provide not only important facilities but also difficulties appear in the simulation. The mathematical model used should be accurate. The results obtained of the mathematical model must agree with full-scale trial results. In this study, the mathematical model presented by Str0m-Tejsen and Chis- lett is used. According to this mathematical model, the forces and moments in the equations of the motion include non-linear terms. The use of non-linear terms complicate the solution of the equations. But this case is solved with the aid of the numerical procedure in a computer. The inclusion of the non-linear terms increase the accuracy of the solution. An important point is that the hydrody - namic coefficients used to solve equations in the computer programme must be accurate. They are obtained by the cap tive model tests in a towing tank. The test conditions (water and tank conditions) must suit the sea conditions. The measurement instruments and other equipment must be fitted in the model with precision. It has been found that the resonance with a system of standing waves which builds up in the towing tank, pre cludes the possibility of oscillating models above a certain critical frequency. The resonance is easily avoid ed, however, at the low frequency of oscillation that is recomended for experiments. It is finally concluded that the semi-theorotical technique combining Planar-Motion Mechanism model testing and computer prediction of manoeuvres allows scaling prob lems to be treated satisfactorily. Large models can be utilized in the testing, whereby scale effects in general are reduced. Tests can be carried out at the ship propul sion point by applying a towing force via the rigid con nection to the mechanism, the correct propeller r.p.m. being obtained by taking the particular engine type and control arrangement into account. The indivual measure ment of the various coefficients gives an insight into the hydrodynamic phenomena involved and consequently allows corrections to coefficients for difference in Reynolds ' number for model and ship to be considered.