International Journal of Small Craft Technology


USING A FORWARD RUDDER TO KEEP A LATERALLY UNSTABLE BOAT UPRIGHT D C Witt, University of Oxford, UK SUMMARY


A vessel that is laterally unstable in the static sense can be maintained upright during forward motion by use of a forward rudder, in a way analogous to the riding of a bicycle. This may permit the use of a longer and narrower hull, and hence a lower power requirement, than would otherwise be possible. experimental, are presented.


NOMENCLATURE A, b, D, E, F, M Various matrices A


Cf


CL Cp F g I


Ixx, Izz, Ixz lWL


Kp, Kr L


Lv, Lp, Lr


Lvd, Lpd, Lrd L, N, Y


m n


N Nv, Np, Nr


Nvd, Npd, Nrd p r


rd


Re s


u0 v


rV


x, y, z x


x0


xm, zm yw Y


Yv, Yp, Yr


Yvd, Ypd, Yrd zk


zmet zt  δ


Some results, both theoretical and


Area, of e.g. rudder or fin (m2) Friction coefficient Lift coefficient


Prismatic coefficient


Froude Number based on displacement Acceleration due to gravity (m s-2) Unit matrix


Moments and product of inertia (kg m2)


Length on waterline (m) Controller gains Roll torque (N m)


Roll torque velocity derivatives Roll torque acceleration derivatives Rudder derivatives Vessel mass (kg)


Exponent defining hull shape Yaw torque (N m)


Desired yaw rate (rad s-1) Reynold’s Number Laplace operator (s-1) Forward velocity (m s-1) Lateral velocity (m s-1) Relative velocity (m s-1) Coordinate axes (m)


Yaw torque velocity derivatives Yaw torque acceleration derivatives Roll rate (rad s-1) Yaw rate (rad s-1)


 


  


1.


Density of water (kg m-3) Roll angle (rad)


Heading or yaw angle (rad) Angular frequency (rad/s) Volume displacement (m3)


INTRODUCTION


Marine vehicles are, for obvious reasons, normally designed to be statically stable in roll. For a surface vessel this means basically that the mass centre has to be below the metacentre. But there are exceptions. A racing eight for instance is unstable when stationary unless its oars are in the water and close to the surface; and many windsurfers are probably unstable if their crews are regarded as passive blocks of matter fixed to the hull. In both cases stability can be maintained by the active motion of the crew.


Several small vessels have been built [1,2,3], mostly but not all in the past twenty years, in which propulsion is by pedals


driving a propeller, and roll stability is


maintained, in spite of the mass centre being well above the metacentre, by the “rider” controlling a vertical rudder-like surface with handlebars.


If this “rudder” is


positioned forward, it can not only stabilise the vessel in roll, but


State variable vector (v, p, r, )T


x-coord of zero lateral velocity in a turn (m)


x- and z-coords of mass centre (m) Waterline semi-beam (m) Side force (N)


Side force velocity derivatives Side force acceleration derivatives Draught to keel (m) z-coord of metacentre


Draught to tip of rudder or fin (m) Angle of attack (rad) Rudder angle (rad)


©2007: Royal Institution of Naval Architects


steering the vessel. Or it may be placed amidships, and other means used for steering.


If the need for hydrostatic roll stability is dispensed with in this way, it is possible to use a very narrow single hull, which may give an increased speed for a given propulsion power, as compared to a wider, stable hull or a catamaran. This is clearly significant when the power source is human, and therefore severely


also perform the normal rudder function of


restricted,


especially over long periods; but in a world in which energy conservation becomes more important, it might perhaps have other applications.


If the controlling surface is placed forward, and acts both to keep the vessel upright and to steer it, then the


B-1

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