7-Manoeuvring - Fundamentals (2012)

Manoeuvring − Fundamentals MANOEUVRING – FUNDAMENTALS MODULE Note: These notes are drawn from those issued by Dr Jonathan Duffy to students of JEE329 Seakeeping & Manoeuvring at the Australian Maritime College Dr Duffy has used edited extracts from the main reference books for the subject and which are listed at the end of the module Introduction Ship manoeuvring is about the behaviour of a ship in the horizontal plane, in particular the way in which it responds to rudder action, and why Whilst a ship must be easy to keep on a straight line for long ocean passages it must also be sufficiently manoeuvrable to avoid collisions and to travel safely within channels restricted in width The theory of ship manoeuvring is very complex and research continues to fully understand some parts of ship manoeuvring This course will deal only with generalities and it is important to understand that there is far more to ship manoeuvring than is covered within this course Ship Manoeuvring Control Loop The process of manoeuvring of a ship is a ‘closed loop’ control system The desired heading is given as an input by the helmsman, which is compared to the actual heading, the rudder angle indicator and sometimes the rate of change of heading The helmsman (or autopilot) then sends a command signal via the steering gear to the rudder The change in rudder angle imposes a control force on the ship causing it to turn in the desired direction In addition to the rudder force the ship is also being acted upon by external forces, such as wind, waves, current and interaction with lateral channel restrictions It is the feedback of heading that makes the system a closed loop In addition to the closed loop governing the ship’s path there will be a similar closed loop governing the speed This loop will not be addressed in this course, except where changes in thrust from the propulsion devices have an influence on the ship’s path Motion Stability A ship moving forward at constant speed in a straight line may be subject to an instantaneous external disturbing force due to a wind gust, for example The way in which the ship’s path is altered due to the instantaneous external force depends upon the motion stability characteristics of the ship Some possible ship responses are illustrated in Figure Manoeuvring − Fundamentals Figure Types of motion stability (taken from Lewis 1989) The type of stability (or instability) a ship will have depends on its hull form and its control system Examples of the different kinds of motion stability are given below: Straight line stability; e.g arrow in flight Directional stability (course stability); e.g ship with autopilot (whether oscillations occur or not depends on the autopilot setting) Manoeuvring − Fundamentals Positional motion stability; e.g submerged submarine under control in the vertical plane It is important to distinguish between controls fixed stability and controls working stability With controls fixed stability the most a ship can have is straight line stability as the control loop is not complete as there is no feedback possible However, with controls working the autopilot, or helmsman, inputs a rudder angle which is proportional in some way to the path error This provides feedback on the ship heading and hence the possibility of directional stability If feedback for heading and position is available it is possible to achieve positional motion stability For a submerged submarine the position feedback is the depth of operation This is summarized in Table Type of stability Feedback required Straight line None Directional Heading Positional motion Heading and Position Table Types of motion stability 3.1 Controls-Fixed Stability We have established that a ship with controls fixed stability can at best only possess straight line stability As the controls are fixed this type of stability must be a function of hull form alone A ship is not like an arrow in that it does not have all its mass at one point and all its lateral area at another Lift is generated in a complex manner along the length of the hull and in addition the added mass of the water is not negligible The criteria for a ship to be directionally stable is that the centre of pressure in pure yaw be ahead of the centre of pressure in pure sway The centre of pressure in pure yaw is the point where a force could be applied to a ship with pure yaw which would not cause an unbalancing moment Equations of Motion 4.1 Basic Equations of Motion In order to simulate the motion of a manoeuvring ship it is necessary to develop equations to define the motions of the ship in the horizontal plane It is necessary to define the axis system and sign convention The SNAME axis system and sign convention will be used, as these are generally accepted internationally However, it should be noted that some older publications use slightly different conventions The axis system is shown in Figure In Figure x0, y0 and z0 are earth fixed axes with z0 positive downwards x, y and z are body fixed axes with z positive downwards, x positive towards the bow and y positive towards starboard The origin is taken to be at the centre of gravity Manoeuvring − Fundamentals Figure Orientation of earth fixed axes and body fixed axes (Lewis 1989) Defining the following: V = velocity vector tangent to ship’s path X = surge force x = surge 𝑥 = u = velocity in the x direction 𝑥 = 𝑢 = acceleration in the x direction Y = sway force y = sway 𝑦 = 𝑣 = velocity in the y direction 𝑦 = 𝑣 = acceleration in the y direction N = yaw moment 𝜓 = heading angle 𝜓 = 𝑟 = yaw velocity 𝜓 = 𝑟 = yaw acceleration 𝛽= drift angle 𝛿 = rudder angle m = mass of ship The path of the ship is usually defined as the trajectory of the ship’s centre of gravity Heading refers to the direction of the ship’s longitudinal axis The difference between the heading and the actual course (or direction of the velocity vector at the centre of gravity, V) is the drift angle (𝛽) When the ship is moving along a curved path the drift angle is thus the difference between the heading and the tangent to the path of the centre of gravity Manoeuvring − Fundamentals Motion of a ship in the horizontal plane is governed by Newton’s second law, which in terms of the earth fixed axes (x0,y0,z0) the surge force, sway force and yaw moment are: 𝑋0 = 𝑚𝑥0 (Surge) 𝑌0 = 𝑚𝑦0 (Sway) (1) 𝑁 = 𝐼𝑧 𝜓 (Yaw) Equations have a simple form, however the motion of a ship is more conveniently expressed when referred to the body fixed x and y axes The surge force and sway force can be transferred to the body fixed axis system with the origin at the centre of gravity by the following expressions: 𝑋 = 𝑋0 𝑐𝑜𝑠𝜓 + 𝑌0 𝑠𝑖𝑛𝜓 𝑌 = 𝑌0 𝑐𝑜𝑠𝜓 − 𝑋0 𝑠𝑖𝑛𝜓 (2) The earth fixed surge and sway velocities of the ship can be given by: 𝑥0 = 𝑢 𝑐𝑜𝑠𝜓 − 𝑣 𝑠𝑖𝑛𝜓 (3) 𝑦0 = 𝑣 𝑐𝑜𝑠𝜓 + 𝑢 𝑠𝑖𝑛𝜓 Differentiating the velocities we obtain the earth fixed accelerations of the ship: 𝑥0 = 𝑢 𝑐𝑜𝑠𝜓 − 𝑣 𝑠𝑖𝑛𝜓 − 𝑢 𝑠𝑖𝑛𝜓 + 𝑣 𝑐𝑜𝑠𝜓 𝜓 (4) 𝑦0 = 𝑢 𝑠𝑖𝑛𝜓 + 𝑣 𝑐𝑜𝑠𝜓 + 𝑢 𝑐𝑜𝑠𝜓 − 𝑣 𝑠𝑖𝑛𝜓 𝜓 When substituting the above into the original equations (Equation 1) we obtain: 𝑋 = 𝑚 𝑢 − 𝑣𝜓 𝑌 = 𝑚 𝑣 + 𝑢𝜓 (5) 𝑁 = 𝐼𝑧 𝜓 The term 𝑚𝑢𝜓 in the equation for Y and the 𝑚𝑣𝜓 term in the equation for X were not present in Equations These are the centrifugal force terms which exist for systems with body fixed axes, but not exist when the axes are earth fixed Equations (5) have been developed for the case where the origin is at the centre of gravity The forces and moments (left hand side) of the equations of motion (Equation 5) consist of four types of forces that act on a ship during a manoeuvre: Hydrodynamic forces acting on the hull and appendages due to ship velocity and acceleration, rudder deflection and propeller rotation Inertial reaction forces caused by the ship acceleration Environmental forces such as wind, waves and current External forces such as tugs, thrusters, interaction with lateral banks and so on Manoeuvring − Fundamentals The first two types of forces generally act in the horizontal plane and involve only surge, sway and yaw responses, although roll may occur for high speed manoeuvres Hydrodynamic forces fall into two categories: those arising from the hull acceleration through the water (added mass forces) and those due to the hull velocity through the water (damping forces) The effect of the rudder on turning is indirect Moving the rudder produces a moment that causes the ship to change heading so as to assume an angle of attack to the direction of motion of the centre of gravity Consequently, hydrodynamic forces are generated on the hull which, after time, cause a change of lateral movement of the centre of gravity The lateral movement is opposed by the inertial reactions If the rudder remains at a fixed position, a steady turning condition will eventuate where hydrodynamic and inertial forces and moments come into balance The forces on the right hand side of Equation correspond to the rigid body forces 4.2 Motion Stability & Linear Equations of Motion With each of the kinds of controls-fixed stability discussed earlier, there is associated a numerical index which by it sign designates whether the body is stable or unstable in that particular sense and by its magnitude designates the degree of stability or instability These indexes may be determined by changing the form of the equations of motion (Equations 5) Changing the form of the equations also assists when using the equations to model ship motion The process to change the form of the equations of motion is discussed below The force and moment components X, Y and N in Equations are assumed to be composed of several parts, some of which are functions of accelerations and velocities of the ship Generally, they also include terms dependent on the orientation of the ship relative to the earth fixed axis as well as excitation terms such as those arising from the seaway or from use of the rudder, but these will be addressed later in the notes For now they are assumed to be composed only of forces and moments arising from motions of the ship which in turn have been excited by disturbances whose details we need not be concerned with at this point in time X, Y and N can be expressed functionally as: 𝑋 = 𝐹𝑥 (𝑢, 𝑢, 𝑣, 𝑣, 𝜓, 𝜓, 𝜕) 𝑌 = 𝐹𝑦 (𝑢, 𝑢 , 𝑣, 𝑣, 𝜓, 𝜓, 𝜕) (6) 𝑁 = 𝐹𝜓 (𝑢, 𝑢 , 𝑣, 𝑣, 𝜓, 𝜓, 𝜕) The expressions in Equations can now be reduced to a more useful mathematical form to enable analysis of motion stability using a Taylor expansion of a function of several variables The process of using a Taylor expansion to define a function is illustrated in Figure Manoeuvring − Fundamentals Figure Linearization of Taylor expansion of a function of a single variable f(x) (Lewis 1989) The Taylor expansion of a function of a single variable states that if the function of a variable, x, (see Figure 3) and all its derivatives are continuous at a particular value of x, say x 1, then the value of the function at a value of x not far removed from x 1, can be expressed as follows: 𝑓 𝑥 = 𝑓 𝑥1 + 𝛿𝑥 𝑑𝑓 (𝑥) 𝑑𝑥 + 𝛿𝑥 𝑑 𝑓(𝑥) 2! 𝑑𝑥 + 𝛿𝑥 𝑑 𝑓(𝑥) 3! 𝑑𝑥 + ⋯…… + 𝛿𝑥 𝑛 𝑑 𝑛 𝑓(𝑥) 𝑛! 𝑑𝑥 𝑛 (7) Where; 𝑓 𝑥 = value of the function at a value of x close to x1 𝑓 𝑥1 = value of the function at x = x1 𝛿𝑥 = 𝑥 − 𝑥1 𝑑 𝑛 𝑓(𝑥 ) 𝑑𝑥 𝑛 = nth derivative of the function evaluated at x = x1 If the change in the variable, 𝛿𝑥 , is made sufficiently small, the higher order terms of 𝛿𝑥 in Equation can be neglected Equation then reduces to: 𝑓 𝑥 = 𝑓 𝑥1 + 𝛿𝑥 𝑑𝑓(𝑥) 𝑑𝑥 (8) It can be seen that Equation is a linear approximation to the real function f(x) at 𝑥 = 𝑥1 + 𝛿𝑥 and that Equation becomes increasingly accurate as 𝛿𝑥 becomes smaller The linearized form Manoeuvring − Fundamentals of the Taylor expansion of a function of two variables x and y is simply the sum of three linear terms: 𝑓 𝑥, 𝑦 = 𝑓 𝑥1 , 𝑦1 + 𝛿𝑥 𝛿𝑓 (𝑥,𝑦 ) 𝛿𝑥 + 𝛿𝑦 𝛿𝑓 (𝑥,𝑦 ) (9) 𝛿𝑦 Both 𝛿𝑥 and 𝛿𝑦 must be sufficiently small so that higher order terms of each can be neglected as well as the product 𝛿𝑥𝛿𝑦 The assumption that renders linearization reasonably accurate, namely, that the admissible change in variables must be very small, is entirely compatible with an investigation of motion stability, since motion stability determines whether a very small perturbation from an initial equilibrium position is going to increase with time or decay with time Thus, it is consistent with the physical reality of motion stability to use the linearized Taylor expansions in connection with Equations Using the form of Equation the linearized Y force may be written as: 𝛿𝑌 𝑌 = 𝐹𝑦 𝑢1 , 𝑣1 , 𝑢1 , 𝑣1 , 𝜓1 , 𝜓1 , 𝛿 + 𝑢 − 𝑢1 𝛿𝑢 + 𝑣 − 𝑣1 𝛿𝑌 𝛿𝑣 + ⋯ … … 𝛿 − 𝛿1 𝛿𝑌 𝛿𝛿 (10) The subscript refers to the values of variables at the initial equilibrium condition and where all of the partial derivatives are evaluated at the equilibrium condition Since the initial equilibrium condition for an investigation of motion stability is straight line motion at constant speed, we can write 𝑢1 = 𝑣1 = 𝜓1 = 𝜓1 = Also since most ships are symmetrical about their x-z plane they travel in a straight line at zero angle of attack, therefore 𝑣1 is also zero (this may not be the case 𝛿𝑌 𝛿𝑌 for ships with an odd number of propellers) Also because of symmetry = = since a 𝛿𝑢 𝛿𝑢 change in forward acceleration or forward velocity will produce no transverse force with ship forms that are symmetrical about the x-z plane Finally, if the ship is in equilibrium in straight line motion, there can be no Y force acting on it in that condition, therefore: 𝐹𝑦 𝑢1 , 𝑣1 , 𝑢1 , 𝑣1 , 𝜓1 , 𝜓1 , 𝛿 = (11) Only 𝑢1 is not zero and is equal to the resultant velocity, V, in the initial equilibrium condition With the above simplifications Equation 10 reduces to: 𝛿𝑌 𝛿𝑌 𝛿𝑌 𝛿𝑌 𝛿𝑌 𝑌 = 𝛿𝑣 𝑣 + 𝛿 𝑣 𝑣 + 𝛿𝜓 𝜓 + 𝛿𝜓 𝜓 + 𝛿𝛿 𝛿 (12) Using a linearized Taylor expansion the surge force can be expressed as follows: 𝛿𝑋 𝛿𝑋 𝛿𝑋 𝛿𝑋 𝛿𝑋 𝛿𝑋 𝛿𝑋 𝑋 = 𝛿𝑢 𝑢 + 𝛿𝑢 𝛿𝑢 + 𝛿𝑣 𝑣 + 𝛿 𝑣 𝑣 + 𝛿𝜓 𝜓 + 𝛿𝜓 𝜓 + 𝛿𝛿 𝛿 The cross-coupled terms 𝛿𝑋 𝛿𝑋 𝛿𝑋 𝛿𝑣 , , 𝛿 𝑣 𝛿𝜓 and 𝛿𝑋 𝛿𝜓 (13) are zero due to symmetry about the x-z plane Also, the influence of the rudder term on surge for small perturbations will be negligible Hence the equation for surge force can be written as: 𝛿𝑋 𝛿𝑋 𝑋 = 𝛿𝑢 𝑢 + 𝛿𝑢 𝛿𝑢 (14) Using a linearized Taylor expansion the yaw moment can be expressed as follows: Manoeuvring − Fundamentals 𝑁= 𝛿𝑁 𝛿𝑣 𝛿𝑁 𝛿𝑁 𝛿𝑁 𝛿𝑁 𝑣 + 𝛿 𝑣 𝑣 + 𝛿𝜓 𝜓 + 𝛿𝜓 𝜓 + 𝛿𝛿 𝛿 (15) This concept can be extended to any desired order of non-linearity and in the form presented assumes the equilibrium state is where the perturbations are zero, i.e 𝑢 = 𝑢 = 𝑣 = 𝑣 = 𝜓 = 𝜓 = 𝛿 = 𝛿𝑌 We introduce SNAME notation, i.e = 𝑌𝑣 and so on Equating the hydrodynamic forces and 𝛿𝑣 the rigid body forces (i.e equating 14 and for surge, equating 12 and for sway and equating 15 and for yaw) and re-arranging we have the equations of motion for surge, sway and yaw: 𝑋𝑢 (𝑢 − 𝑢1 ) + 𝑋𝑢 − 𝑚 𝑢 = 𝑌𝑣 𝑣 + 𝑌𝑣 − 𝑚 𝑣 + 𝑌𝜓 − 𝑚𝑢1 𝜓 + 𝑌𝜓 𝜓 + 𝑌𝛿 𝛿 = (16) 𝜓 𝑁𝜓 − 𝐼𝑧 + 𝑁𝑣 𝑣 + 𝑁𝑣 𝑣 + 𝑁𝜓 𝜓 + 𝑁𝛿 𝛿 = Equations 16 are based on the origin being at the centre of gravity If the origin located a distance 𝑥𝐺 from the centre of gravity, Equations 16 become: 𝑋𝑢 (𝑢 − 𝑢1 ) + 𝑋𝑢 − 𝑚 𝑢 = 𝑌𝑣 𝑣 + 𝑌𝑣 − 𝑚 𝑣 + 𝑌𝜓 − 𝑚𝑢1 𝜓 + 𝑌𝜓 − 𝑚𝑥𝐺 𝜓 + 𝑌𝛿 𝛿 = (17) 𝑁𝜓 − 𝐼𝑧 𝜓 + 𝑁𝑣 𝑣 + 𝑁𝑣 − 𝑚𝑥𝐺 𝑣 + 𝑁𝜓 − 𝑚𝑥𝐺 𝑢1 𝜓 + 𝑁𝛿 𝛿 = Using 𝜓 = 𝑟 and 𝜓 = 𝑟 the equations are: 𝑋𝑢 (𝑢 − 𝑢1 ) + 𝑋𝑢 − 𝑚 𝑢 = 𝑌𝑣 𝑣 + 𝑌𝑣 − 𝑚 𝑣 + 𝑌𝑟 − 𝑚𝑢1 𝑟 + 𝑌𝑟 − 𝑚𝑥𝐺 𝑟 + 𝑌𝛿 𝛿 = (18) 𝑁𝑟 − 𝐼𝑧 𝑟 + 𝑁𝑣 𝑣 + 𝑁𝑣 − 𝑚𝑥𝐺 𝑣 + 𝑁𝑟 − 𝑚𝑥𝐺 𝑢1 𝑟 + 𝑁𝛿 𝛿 = Equations 18 are linear equations that can be used to describe the forces and moments arising from motions of the ship and to assess motion stability characteristics of the ship It will be seen later that linear equations are useful for analyzing the influence of ship features on controls-fixed stability and on the turning ability of directionally stable ships in the linear range However, linear theory cannot be used to accurately predict the characteristics of tight manoeuvres that most ships are capable of Also, it cannot be used to accurately predict the manoeuvres of directionally unstable ships 4.3 Physical Meaning of the Derivatives The physical meaning of the derivatives in Equations 18 will be discussed in this section to help understand how the equations can be used to describe the manoeuvring properties of the ship Manoeuvring − Fundamentals 𝑌𝑣 : This is the coefficient of the side force (sway force) on the ship due to a sway velocity It is a combination of the lift of the ship acting as a low aspect ratio foil and the cross flow drag Figure shows a ship with a positive value of v It is obvious that the resulting side force is in the negative direction Therefore 𝑌𝑣 must be negative for a positive sway velocity (v) This is the case for all ships A typical Y versus v curve is given in Figure 𝑁𝑣 : This is the coefficient of the yawing moment on the ship due to a sway velocity We define that positive deflection is in the clockwise sense when looking along the axis concerned from the origin, therefore a positive yaw moment will be clockwise (see Figure 4) Its sign will depend on the hull and appendage configuration and whether the bow or stern dominates If the ship has a large skeg 𝑁𝑣 is likely to be positive Apart from that, nothing can be deduced about its sign The magnitude will be relatively small Typical N versus v relationships are given in Figure Figure Ship with a forward velocity, u, and a transverse velocity, v (Lewis 1989) Figure Typical Y versus v and N versus v relationships (Lewis 1989) (Note: 𝑌𝑣 is always negative for a positive v) 10 Manoeuvring − Fundamentals 𝑌𝑣 − 𝑚 : This is the coefficient of the side force on the ship due to sway acceleration From Newton’s second law (F = ma) it can be seen that this is equivalent to a mass term In fact, 𝑚 is the ship’s mass and 𝑌𝑣 is known as the added mass Thus this term will always be large and negative for a positive value of 𝑣 A ship with sway acceleration is shown in Figure A typical Y versus 𝑣 curve is shown in Figure Figure Ship with sway acceleration (Lewis 1989) Figure Typical Y versus 𝑣 relationship (Lewis 1989) Note that the coefficients to follow comprise two parts The first part is caused by the action of water on the hull, whilst the second part is due to the rigid body and would occur if the ship were manoeuvring in a vacuum 𝑁𝑣 − 𝑚𝑥𝐺 : This is the coefficient of yawing moment due to sway acceleration Its sign will depend on the positions of the ship’s centre of gravity and the centre of the added mass It will have a small value 𝑌𝑟 − 𝑚𝑢1 : This is the coefficient of the sway force due to yaw velocity 𝑌𝑟 will be small and its sign will depend on whether the bow or the stern dominates A ship with an angular velocity is 11 Manoeuvring − Fundamentals shown in Figure Note that r = 𝜓 Typical Y versus r curves are shown in Figure 𝑚𝑢1 is large and positive Therefore 𝑌𝑟 − 𝑚𝑢1 will be large and negative 𝑁𝑟 − 𝑚𝑥𝐺 𝑢1 : This is the coefficient of the yawing moment due to yaw velocity 𝑁𝑟 will always be large and negative for a positive value of r A typical N versus r curve is given in Figure The sign of 𝑚𝑥𝐺 𝑢1 will be uncertain Thus 𝑁𝑟 − 𝑚𝑥𝐺 𝑢1 will be large and negative Figure Ship with forward velocity, v, and an angular velocity, r (Lewis 1989) Figure Typical Y versus r and N versus r relationship.s (Lewis 1989) 12 Manoeuvring − Fundamentals 𝑌𝑟 − 𝑚𝑥𝐺 : This is the coefficient of the sway force due to yaw acceleration The sign of 𝑌𝑟 will depend on whether the bow or stern dominates and it will have a small value A ship with an angular acceleration is shown in Figure 10 The sign of 𝑚𝑥𝐺 will depend on the sign of 𝑥𝐺 𝑥𝐺 is usually small as the origin is usually taken at midships, which is usually not far from the longitudinal centre of gravity Therefore 𝑌𝑟 − 𝑚𝑥𝐺 will usually be small and of uncertain sign Figure 10 Ship with an angular acceleration (Lewis 1989) 𝑁𝑟 − 𝐼𝑧 : This is the coefficient of the yawing moment due to yaw acceleration It is equivalent to a moment of inertia term 𝐼𝑧 is the moment of inertia about the z axis and 𝑁𝑟 is often termed the added moment of inertia A ship with an angular acceleration is shown in Figure 10 𝑌𝛿 : This is the coefficient of sway force due to rudder deflection Remembering that positive deflection is in the clockwise sense when looking along the axis concerned from the origin it can be seen that a rudder deflection to port is considered to be positive Therefore 𝑌𝛿 will be positive for a positive 𝛿 The linearized sway force created by rudder deflection is 𝑌𝛿 𝛿 𝑁𝛿 : This is the coefficient of the yawing moment due to rudder deflection This will be negative for a positive 𝛿 for a rudder at the stern, i.e rudder to port (positive deflection) results in a yaw to port (negative) 𝑁𝛿 𝛿 is the linearized component of the moment created by the lateral force, 𝑌𝛿 𝛿, from rudder deflection The lateral force from the deflected rudder creates a moment to turn the ship This turning action causes the ship to develop an angle of attack with respect to its motion through the water The lateral forces then generated by the well designed ship (acting as a foil moving in a liquid at an angle of attack) create a moment, 𝑁𝑣 𝑣, that greatly augments the rudder moment 𝑁𝛿 𝛿 4.4 Non-Dimensional Equations of Motion In the previous sections the equations have been given in dimensional terms For convenience these equations, and thus all of the terms in them, can be non-dimensionalised This makes it easier to compare the manoeuvring properties of different ships The non-dimensional scheme 13 Manoeuvring − Fundamentals used in these notes is the SNAME system, as it is generally used internationally The nondimensional values are denoted by a prime Examples of non-dimensional terms are given below: 𝑚′ = 𝑌𝑣 ′ = 𝑌𝑣 ′ = 𝑚 𝜌𝐿 ; 𝑣 ′ = 𝑉𝑣 ; 𝑣 ′ = 𝑉𝑣 𝐿2 ; 𝐼𝑧 ′ = 𝐼𝑧𝐿5 ; 𝑟 ′ = 𝑟𝐿𝑉 ; 𝑟 ′ = 𝑟𝑉𝐿2 𝑌𝑣 2 𝜌𝑉𝐿 𝑌𝑣 𝜌𝐿 ; 𝑌𝑟 ′ = ; 𝑌𝑟 ′ = 𝑌𝑟 𝜌𝑉𝐿 𝑌𝑟 2𝜌𝐿 ; 𝑁𝑣 ′ = ; 𝑁𝑣 ′ = 𝑁𝑣 𝜌𝑉𝐿 𝑁𝑣 𝜌𝐿 ; 𝑁𝑟 ′ = ; 𝑁𝑟 ′ = 𝑁𝑟 𝜌𝑉𝐿 𝑁𝑟 𝜌𝐿 Equations 18 can be written as non-dimensional equations of motion: 𝑋𝑢 ′ 𝑢′ + 𝑋𝑢 ′ − 𝑚′ 𝑢′ = 𝑌𝑣 ′ 𝑣′ + 𝑌𝑣 ′ − 𝑚 ′ 𝑣′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑟 ′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ + 𝑌𝛿 ′ 𝛿′ = (19) 𝑁𝑟 ′ − 𝐼𝑧 ′ 𝑟 ′ + 𝑁𝑣 ′ 𝑣′ + 𝑁𝑣 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑣 ′ + 𝑁𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ + 𝑁𝛿 ′ 𝛿′ = It can be seen that 𝑢1 does not appear in the non-dimensional equations of motion This is due to the fact that 𝑢1 /𝑉 ≈ for small perturbations The sway and yaw expressions in Equations 19 can be used to assess the motion stability of a ship assuming linearity, which is discussed in the following section The equations can also be used to model the forces and moments arising from motions of the ship where linearity holds In these equations the derivatives are constants, the rudder angle is specified and the equations can be solved for v and 𝑟 As a ship manoeuvres 𝑣, 𝑣, 𝑟, and 𝑟 will all vary as functions of time 4.5 Non-Linear Equations of Motion When a ship executes a turn it can generally easily turn in under four ship lengths resulting in large sway and yaw velocities which involve considerable non-linearities Hence, the linear equations are not sufficient to model such manoeuvres Also, linear theory cannot be used to accurately predict the manoeuvres of directionally unstable ships Therefore, non-linear analysis is required Analysis using a complete non-linear mathematical model is very complex This is partly due to the fact that there is no general agreement on the form of the model This is because different research institutions neglect different terms and use a different form of non-linearity The final predicted path of the ship will be similar but it makes it impossible to compare coefficients from different models It should be noted that the equations in this section are valid for the origin located at the centre of gravity The equations in this section relate to the forces and moments arising from only the motions of the ship One of the basic assumptions for the linear equations was that the functions below could be approximated by a linear Taylor expansion 𝑋 = 𝐹𝑥 (𝑢, 𝑢, 𝑣, 𝑣, 𝑟, 𝑟, 𝜕) 𝑌 = 𝐹𝑦 (𝑢, 𝑢 , 𝑣, 𝑣, 𝑟, 𝑟, 𝜕) (20) 𝑁 = 𝐹𝜓 (𝑢, 𝑢 , 𝑣, 𝑣, 𝑟, 𝑟, 𝜕) 14 Manoeuvring − Fundamentals To give an idea of the complexity of the resulting equations the expansion to third order of the hydrodynamic surge force is: 𝑋 = 𝑋 + 𝑋𝑢 𝛿𝑢 + 𝑋𝑣 𝑣 + 𝑋𝑟 𝑟 + 𝑋𝑢 𝑢 + 𝑋𝑣 𝑣 + 𝑋𝑟 𝑟 + 𝑋𝛿 𝛿 + 𝑋𝑢𝑢 𝑢2 + 𝑋𝑣𝑣 𝑣 + +𝑋𝛿𝛿 𝛿 + 2𝑋𝑢𝑣 𝑢𝑣 + 2𝑋𝑢𝑟 𝑢𝑟+ .2𝑋𝑟 𝛿 𝑟𝛿 + 2! 3 2 𝑋𝑢𝑢𝑢 𝑢 + 𝑋𝑣𝑣𝑣 𝑣 + ⋯ +𝑋𝛿𝛿𝛿 𝛿 + 3𝑋𝑢𝑢𝑣 𝑢 𝑣 + 3𝑋𝑢𝑢𝑟 𝑢 𝑟 + ⋯ 3! +3𝑋𝑟 𝛿𝛿 𝑟𝛿 + 6𝑋𝑢𝑣𝑟 𝑢𝑣𝑟 + 6𝑋𝑢𝑣 𝑢 𝑢𝑣𝑢+ +6𝑋𝑢 𝑣𝛿 𝑢𝑣𝛿 (21) The non-linear equations can be simplified by considering the geometry of the ship and noting that it is symmetrical about the xz plane It is obvious from symmetry that X must be an odd function of 𝑢 but an even function of 𝑣, 𝑟 and 𝛿 Thus, all the even coefficients of 𝑢 will be zero, as will the odd coefficients of 𝑣, 𝑟 and 𝛿 in the X equation, i.e 𝑋𝑣 , 𝑋𝑟 , 𝑋𝑣𝑣𝑣 and so on In addition, the cross coupling coefficients which involve even terms in 𝑢 or odd terms in 𝑣, 𝑟 or 𝛿 will be zero also, i.e 𝑋𝑢𝑣 , 𝑋𝑢𝑟 , 𝑋𝑢𝑢𝑣 , 𝑋𝑢𝑢𝑟 and so on The Y and N equations will be even with respect to u but odd with respect to 𝑣, 𝑟 and 𝛿 and hence considerable simplification will be possible The acceleration (or added mass) terms have been shown to be highly linear and hence no second order or higher order terms in 𝑢, 𝑣 or 𝑟 are included Due to symmetry 𝑋𝑣 , 𝑋𝑟 , 𝑌𝑢 and 𝑁𝑢 are all zero Considering the above simplifications, non-linear equations of motion based on the Taylor series to the third order are: 1 1 1 𝑋 + 𝑋𝑢 𝛿𝑢 + 𝑋𝑢𝑢 𝛿𝑢2 + 𝑋𝑢𝑢𝑢 𝛿𝑢3 + 𝑋𝑣𝑣 𝑣 + 𝑋𝑟𝑟 𝑟 + 𝑋𝛿𝛿 𝛿 + 𝑋𝑣𝑣𝑢 𝑣 𝛿𝑢 + 𝑋𝑟𝑟𝑢 𝑟 𝛿𝑢 + 𝑋𝛿𝛿𝑢 𝛿 𝛿𝑢 + 𝑋𝑣𝑟 + 𝑚 𝑣𝑟 + 𝑋𝑣𝛿 𝑣𝛿 + 𝑋𝑟𝛿 𝑟𝛿 + 𝑋𝑣𝑟𝑢 𝑣𝑟𝑢 + 𝑋𝑣𝛿𝑢 𝑣𝛿𝑢 + 𝑋𝑟𝛿𝑢 𝑟𝛿𝑢 + 𝑋𝑢 − 𝑚 𝑢 = (22) 1 𝑌 + 𝑌𝑢0 𝛿𝑢 + 𝑌𝑢𝑢 𝛿𝑢2 + 𝑌𝑣 𝑣 + 𝑌𝑣𝑣𝑣 𝑣 + 𝑌𝑣𝑟𝑟 𝑣𝑟 + 𝑌𝑣𝛿𝛿 𝑣𝛿 + 𝑌𝑣𝑢 𝑣𝛿𝑢 + 𝑌 𝑣𝛿𝑢 𝑣𝑢𝑢 + 𝑌𝑟 − 𝑚𝑢1𝑟+16𝑌𝑟𝑟𝑟𝑟3+12𝑌𝑟𝑢𝑣𝑟𝑣2+12𝑌𝑟𝛿𝛿𝑟𝛿2+𝑌𝑟𝑢𝑟𝛿𝑢+12𝑌𝑟𝑢𝑢𝑟𝛿𝑢2+𝑌𝛿𝛿+16𝑌𝛿𝛿𝛿𝛿3+12𝑌𝛿𝑣𝑣𝛿𝑣2+ 𝑌 𝛿𝑟 𝛿𝑟𝑟 +𝑌𝛿𝑢 𝛿𝛿𝑢 + 𝑌𝛿𝑢𝑢 𝛿𝑢2 + 𝑌𝑣𝑟𝛿 𝑣𝑟𝛿 + 𝑌𝑣 − 𝑚 𝑣 + 𝑌𝑟 𝑟 = (23) 1 2 2 𝑁 + 𝑁𝑢0 𝛿𝑢 + 𝑁𝑢𝑢 𝛿𝑢2 + 𝑁𝑣 𝑣 + 𝑁𝑣𝑣𝑣 𝑣 + 𝑁𝑣𝑟𝑟 𝑣𝑟 + 𝑁𝑣𝛿𝛿 𝑣𝛿 + 𝑁𝑣𝑢 𝑣𝛿𝑢 + 𝑁𝑣𝑢𝑢 𝑣𝛿𝑢2 + 𝑁𝑟 𝑟 + 1 1 𝑁 𝑟 + 𝑁𝑟𝑣𝑣 𝑟𝑣 + 𝑁𝑟𝛿𝛿 𝑟𝛿 + 𝑁𝑟𝑢 𝑟𝛿𝑢 + 𝑁𝑟𝑢𝑢 𝑟𝛿𝑢 + 𝑁𝛿 𝛿 + 𝑁𝛿𝛿𝛿 𝛿 + 𝑁𝛿𝑣𝑣 𝛿𝑣 + 𝑟𝑟𝑟 𝑁𝛿𝑟𝑟 𝛿𝑟 +𝑁𝛿𝑢 𝛿𝛿𝑢 + 𝑁𝛿𝑢𝑢 𝛿𝛿𝑢2 + 𝑁𝑣𝑟𝛿 𝑣𝑟𝛿 + 𝑁𝑣 𝑣 + 𝑁𝑟 − 𝐼𝑧 𝑟 = (24) For a high speed ship it may be necessary to model the roll of the ship The equation for roll will be of a similar form Other problems arise with such equations, e.g., the non-linearities in sway force due to the sway velocity may be caused by cross flow drag Drag is a velocity squared term, not a cubic term and so some institutions prefer to force a square fit rather than a cubic fit Therefore, considering the sway force due to sway velocity only we have possible expressions: 𝑌1 = 𝑌𝑣 𝑣 + 𝑌𝑣𝑣𝑣 𝑣 𝑌2 = 𝑌𝑣 𝑣 + 𝑌𝑣 𝑣 𝑣 𝑣 These equations are only approximations to express the force as a function of v Obviously 𝑌1 must equal 𝑌2 Therefore 𝑌𝑣𝑣𝑣 must equal𝑌𝑣 𝑣 Thus, 𝑌𝑣 from the first equation must be different from 𝑌𝑣 in the second equation But those are the linear terms that are supposed to be constant 15 Manoeuvring − Fundamentals Another complication is that when a ship manoeuvres, its speed reduces significantly, yet its propeller will be working essentially at the higher self-propulsion speed Thus, the rudder will feel an increased flow over what it would be expecting at the lower speed This can considerably effect the values of 𝑌𝛿 and 𝑁𝛿 and it can also have a slight effect on the other coefficients, especially 𝑌𝑣 and 𝑁𝑣 One scheme to avoid this would be to non-dimensionalise some of the derivatives with respect to the flow velocity at the rudder rather than V This can have considerable complications For the reasons given above, and many others, Equations 22 - 24 are the simplest complete non-linear equations and slightly different versions of the equations may be used for different conditions The equations above relate to the forces and moments arising from only the motions of the ship It should be noted that the equations in most shiphandling simulators also include effects such as wind, waves and restricted water Analysis of Course-Keeping & Controls-Fixed Stability Using only linear terms, solutions to the sway and yaw expressions in Equations 19 provide linear transfer functions allowing the assessment of the stability of motion in the horizontal plane When considering controls fixed stability δ = and the sway and yaw expressions in Equations 19 become: 𝑌𝑣 ′ 𝑣′ + 𝑌𝑣 ′ − 𝑚 ′ 𝑣′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑟 ′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ = 𝑁𝑟 ′ − 𝐼𝑧 ′ 𝑟 ′ + 𝑁𝑣 ′ 𝑣′ + 𝑁𝑣 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑣 ′ + 𝑁𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ = (25) (26) Equations 25 and 26 are two simultaneous differential equations of the first order in two unknowns, 𝑣′ and 𝑟 ′ The simultaneous solution of these two equations for 𝑣′ and 𝑟 ′ yields a second order differential equation The solutions for 𝑣′ and 𝑟 ′ correspond to the standard solutions of second order differential equations which are given by Equations 27 and 28 𝑣 ′ = 𝐾1 𝑒 𝜎1 𝑡 + 𝐾2 𝑒 𝜎2 𝑡 (27) 𝑟 ′ = 𝐾3 𝑒 𝜎1 𝑡 + 𝐾4 𝑒 𝜎2 𝑡 (28) For directional stability v’ and r’ should both tend to zero as time tends to infinity From equations 27 and 28, this implies that for directional stability the real parts of σ1 and σ2 should be negative σ1 and σ2 are known as the stability indexes, which can be used to provide a measure of the directional stability of the ship If one of these indexes is positive the values of v’ and r’ will increase and hence directional instability will result In order to express the stability indexes in terms of the derivatives, Equations 27 and 28 are substituted into Equations 25 and 26 and a quadratic in σ is obtained: 𝐴𝜎 + 𝐵𝜎 + 𝐶 = (29) Where: 𝐴 = 𝑁𝑟 ′ − 𝐼𝑧 ′ 𝑌𝑣 ′ − 𝑚′ − 𝑌𝑟 ′ − 𝑚′ 𝑥𝐺 ′ 𝑁𝑣 ′ − 𝑚′ 𝑥𝐺 ′ 𝐵 = 𝑌𝑣 ′ 𝑁𝑟 ′ − 𝐼𝑧 ′ + 𝑌𝑣 ′ − 𝑚′ 𝑁𝑟 ′ − 𝑚′ 𝑥𝐺 ′ − 𝑌𝑟 ′ − 𝑚′ 𝑁𝑣 ′ − 𝑚′ 𝑥𝐺 ′ − 𝑌𝑟 ′ − 𝑚′ 𝑥𝐺 ′ 𝑁𝑣 ′ 𝐶 = 𝑌𝑣 ′ 𝑁𝑟 ′ − 𝑚′ 𝑥𝐺 ′ − 𝑌𝑟 ′ − 𝑚′ 𝑁𝑣 ′ The solution of Equation 29 is: 16 Manoeuvring − Fundamentals 𝜎1 , 𝜎2 = −𝐵/𝐴± 𝐵 𝐴 −4 𝐶 𝐴 (30) From Equation 30 it can be seen that there are two conditions to be satisfied in order that both σ1 and σ2 are negative, as follows: C/A must be positive (otherwise the expression under the square root sign in Equation 30 would be larger than B/A, hence one root would be positive) B/A must be positive (otherwise it does not matter the size of the square root term, at least one root will be positive) Since most of the derivatives are of known fixed sign, due to their physical meaning, it is possible to reduce the criteria for stability further In the table below the nature of derivatives is given, based on the data provided in the above section ‘Physical Meaning of the Derivatives’ Derivative 𝑌𝑣 ′ 𝑁𝑣 ′ ′ 𝑌𝑣 − 𝑚′ 𝑁𝑣 ′ − 𝑚′ 𝑥𝐺 ′ 𝑌𝑟 ′ − 𝑚′ 𝑁𝑟 ′ − 𝑚′ 𝑥𝐺 ′ 𝑌𝑟 ′ − 𝑚′ 𝑥𝐺 ′ 𝑁𝑟 ′ − 𝐼𝑧 ′ Typical magnitude and sign of the derivative Large and negative Small and of uncertain sign Large and negative Small and of uncertain sign Large and negative Large and negative Small and of uncertain sign Large and negative From the data in the table above we know that A in Equation 29 will always be large and positive since the first two terms are large and positive and the last two are small and of uncertain sign B in Equation 29 will always be large and positive since the first four terms are large and negative (resulting in two large positive values being added) while the last four are all small Thus, the criteria for stability reduce to: 𝐶 > 𝑜𝑟 𝑌𝑣 ′ 𝑁𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ − 𝑌𝑟 ′ − 𝑚 ′ 𝑁𝑣 ′ > (31) Note that the sign of C in Equation 29 cannot be deduced in the same manner as A and B, because 𝑌𝑣 ′ , 𝑌𝑟 ′ − 𝑚 ′ and 𝑁𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ are all large and negative and 𝑁𝑣 ′ is of uncertain sign If 𝑁𝑣 ′ were positive then directional stability would be assured, however, it is usually negative for most ships Equation 31 can be written as: 𝑁𝑟 ′ −𝑚 ′ 𝑥 𝐺 ′ 𝑌𝑟 ′ −𝑚 ′ > 𝑁𝑣 ′ 𝑌𝑣 ′ (32) Equation 32 is the requirement for directional stability; i.e the centre of pressure in pure yaw must be ahead of the centre of pressure in pure sway Turning Path of a Vessel When the rudder is deflected a lateral force occurs that creates a moment to turn the ship This turning action causes the ship to develop an angle of attack with respect to its motion through the water The lateral forces then generated by the well designed ship (acting as a foil moving in a 17 Manoeuvring − Fundamentals liquid at an angle of attack) create a moment, 𝑁𝑣 𝑣, that greatly augments the rudder moment 𝑁𝛿 𝛿 When a ship turns there are three different phases Phase Consider a ship which is initially travelling along a straight line path with the rudder fixed amidships At time t=0 the rudder is deflected to a fixed angle (δ) and held there The rudder angle will cause a side force, 𝑌𝛿 𝛿, and yawing moment, 𝑁𝛿 𝛿, which must be balanced by sway and yaw accelerations 𝑣 and 𝑟, respectively Since there were no sway or yaw velocities prior to t=0 the instantaneous accelerations at t=0 caused by the rudder angle δ can be calculated from: 𝑌𝑣 ′ − 𝑚′ 𝑣 ′ + 𝑌𝑟 ′ − 𝑚′ 𝑥𝐺 ′ 𝑟 ′ + 𝑌𝛿 ′ 𝛿′ = 𝑁𝑟 ′ − 𝐼𝑧 ′ 𝑟′ + (33) 𝑁𝑣 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑣 ′ + 𝑁𝛿 ′ 𝛿′ = (34) These accelerations will cause sway and yaw velocities v and r to be set up, however, in the initial stages of the turn they will be negligible due to the large inertial forces involved Phase As the sway and yaw velocities begin to build up they will give rise to sway forces and yawing moments which will oppose those generated by the rudder and will hence reduce the sway and yaw accelerations The equations for sway and yaw for the second phase of the turn are: 𝑌𝑣 ′ 𝑣′ + 𝑌𝑣 ′ − 𝑚 ′ 𝑣′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑟 ′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ + 𝑌𝛿 ′ 𝛿′ = 𝑁𝑟 ′ − 𝐼𝑧 ′ 𝑟 ′ + 𝑁𝑣 ′ 𝑣′ + 𝑁𝑣 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑣 ′ + 𝑁𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ + 𝑁𝛿 ′ 𝛿′ = (35) (36) Phase Eventually the accelerations will have been reduced to zero and the ship will be in equilibrium in a steady turn with the force and moment induced by the rudder angle balanced by those induced by the sway and yaw velocity In this phase of the turn the sway and yaw velocities can be calculated from: 𝑌𝑣 ′ 𝑣′ + 𝑌𝑟 ′ − 𝑚 ′ 𝑟′ + 𝑌𝛿 ′ 𝛿′ = (37) 𝑁𝑣 ′ 𝑣′ + 𝑁𝑟 ′ − 𝑚 ′ 𝑥𝐺 ′ 𝑟 ′ + 𝑁𝛿 ′ 𝛿′ = (38) The velocity and acceleration acting in each phase of the turn is summarized in the table below t times ship length) If the ship is unstable (i.e the numerator is negative) then Equation 48 implies that R will have the same sign as 𝛿 (i.e a port rudder deflection (positive 𝛿) will lead to a starboard turn (positive R)) This means that the ship will turn against its rudder, which can only be true for small rudder angles If the ship is unstable (i.e C is negative) then Equations 43 and 48 are meaningless and non linear analysis will be required 6.2 Heel Angle During a Turn When a ship turns it usually heel at first into the turn and then out of the turn To understand why this happens we must study the forces acting during each phase of the turn In the first phase the rudder is providing a side force, 𝑌𝛿 𝛿, low down at approximately half the mid span of the rudder, which is counteracted only by the mass of the ship (usually with its centre above the waterline) and the added mass of the water which has its centre about mid draught In the second phase of the turn the acceleration forces start to reduce and are replaced by velocity forces which act lower down in the ship In addition, as the ship starts to turn there will be an outward centrifugal force at the centre of gravity of the ship, which is high up Thus the heel angle will be reduced and the ship may even start to heel the opposite way In the third phase the only forces acting are the rudder force (at the mid span of the rudder), the velocity forces (approximately at mid draught) and the outward centrifugal force (at the centre of gravity) For the third phase of the turn the resulting heel angle will depend on the magnitude of the separate force components, however for a normal displacement ship it is likely to be to port (out of turn) since 𝑌𝑣 ′ 𝑣′ and 𝑌𝑟 ′ 𝑟′ must be much larger than 𝑌𝛿 ′ 𝛿′ to enforce the starboard turn It should be noted that high speed planing craft with low, highly loaded rudders and a low centre of gravity usually heel into the turn throughout For a normal displacement ship the usual sequence is an initial heel into the turn when the rudder is put over followed by a reduction in heel as the ship begins to turn The ship will then continue to roll past the upright until it has reached a maximum heel angle outward before settling on its equilibrium heel angle A typical roll angle time record is shown in Figure 13 Figure 13 Typical roll angle time record for a starboard turn (Lewis 1989) 21 Manoeuvring − Fundamentals For simplification, with a conventional displacement craft, it is possible to neglect the force on the rudder in a steady turn to develop an equation to predict the steady heel angle This would give a greater heel angle than actually would exist; however, it would be the case if the rudder was returned to amidships (which may occur sometimes if there is concern that the heel angle is too large) It should be noted that if the heel angle is too large the thing to is to slow down – not reduce the rudder angle The outward centrifugal force on a ship in a steady turn is: 𝑚𝑉 𝑅 𝑜𝑟 ∆𝑉 𝑔𝑅 (49) Where: 𝑉 is the ship tangential velocity m is the ship mass ∆ is the ship weight R is the turn radius Neglecting the force on the rudder we have a couple between the outward centrifugal force (acting at the centre of gravity) and the inward velocity force, which must have the same magnitude for equilibrium but will act at the centre of lateral resistance The heeling moment is given by: (∆𝑉 𝑔𝑅) × 𝐻 (50) Therefore, it is possible to obtain the heeling lever by dividing by the ship weight, which gives: 𝐻𝑒𝑒𝑙𝑖𝑛𝑔 𝑙𝑒𝑣𝑒𝑟 = (𝑉 𝐻 𝑔𝑅) (51) This heeling lever can be plotted on the GFZ curve and the intersection will give the steady heel angle The righting lever must equal the heeling lever for equilibrium Therefore: 𝐺𝐹 𝑍 = (𝑉 𝐻 𝑔𝑅) (52) Now, for small angles GFZ = GFMsinθ 𝐺𝐹 𝑀𝑠𝑖𝑛𝜃 = (𝑉 𝐻 𝑔𝑅 ) or 𝑉 2𝐻 𝑠𝑖𝑛𝜃 = 𝑔𝑅 𝐺 𝐹𝑀 (53) (54) If the centre of lateral resistance can be assumed to be level with the centre of buoyancy then: 𝐻 = 𝐵𝐺𝑐𝑜𝑠𝜃 (55) Substituting Equation 55 into Equation 54 the steady heel angle in a turn can be given by: 𝑉 𝐵𝐺 𝑡𝑎𝑛 𝜃 = 𝑔𝑅 𝐺 𝐹𝑀 (56) The equilibrium heel angle can be obtained by calculating the resultant moment when the ship is in a steady turn and comparing this with righting lever curves 22 Manoeuvring − Fundamentals 6.3 Coupling Between Heeling & Turning In addition to the coupling between turning and heeling there will be a coupling between heeling and turning In general, for a displacement ship a heel to starboard will cause a turn to port For normal craft this can be ignored, however for some high speed craft with low GMs it must be taken into account Acknowledgements The primary reference sources used for these notes are acknowledged below   Renilson, M.R., 2003, Ship Manoeuvring, Lecture Notes, Australian Maritime College Lewis, E (Ed), 1989, Principles of Naval Architecture, Volume III, Motions in Waves and Controllability, SNAME, New York References Clarke, D., Gedling, P., Hine, G., The Application of Manoeuvring Criteria in Hull Design using Linear Theory, TRINA, 1982 Crane, C.L 1979, Manoeuvring trials of 278,000 DWT tanker in shallow and deep waters, SNAME Transactions, vol 87 Lewis, E (Ed), 1989, Principles of Naval Architecture, Volume III, Motions in Waves and Controllability, SNAME, New York Renilson, M.R., 2003, Ship Manoeuvring, Lecture Notes, Australian Maritime College 23 Manoeuvring − Fundamentals 24 ...