Hull Design

Axis System Conventions

Approach

My first step in designing the hull is to define some axis systems I can use throughout the design process, including defining the shape of the hull, calculating its performance, and analyzing the dynamics of the boat's motion.
 
 

Body axis systems are always fixed to the body - they move and rotate with it. The reference point for the hull body axis system is the midpoint of the hull, at the design waterline. The X, Y, and Z axes form a right-handed coordinate system. The X axis is measured positive forward of midships, and negative aft. The Y axis is measured positive to starboard, and negative to port. The Z axis is measured positive down, and negative up. The position of a hull is given by the position of the midpoint of the hull relative to the reference point for the whole boat. For defining the geometry of monohulls, trimarans, and proas, this is the reference point of the main hull. For catamarans, this is the midships point, mid way between the two hulls, at the waterline. The body axis system will be relocated to the boat's center of gravity for dynamic analyses.
 
 

I've chosen the midpoint as the origin for the body axis system because it is near the center of gravity and center of buoyancy, and it avoids having all the X coordinates be negative numbers. An aft-starboard-up coordinate system starting at the forward end of the design waterline would be just as good for the geometry, but the boat's velocity would be negative, which is also awkward.
 
 

Forces applied to the hull are also positive when acting forward, starboard, and down, as are the velocity components, U (surge), V (sway), W (heave). The moments about the X, Y, and Z axes, L, M, and N, and the angular rates, P, Q, and R, are positive rolling to starboard, pitching bow up, and yawing to starboard.
 
 

The earth axis system, used to orient the boat, is positive to North, East, and down. The boat's location is measured from an earth axis system that doesn't move, and a moving earth axis system that is attached to the boat but doesn't rotate.
 
 

Axes Rotations

To compute the hydrostatics and motion for a complete multihull vessel, it is necessary to assemble the hulls in relation to each other, as they will be assembled in the complete boat. This will involve translation - lateral, longitudinal, and vertical - to position the hulls, and rotation. Rotation is necessary to consider hulls that are canted so as to be vertical when the boat is heeled, but also for amas on a trimaran that may be pitched up or hulls that are toed in or out, as with flip-tackers. Rotation of the coordinates is also necessary to compute the hydrostatics when the boat is in a heeled condition, or trimmed up or down by the bow.
 
 


 
 

Rotating the points is done by multiplying the X-Y-Z vector of coordinates by a transformation matrix. There is a transformation matrix for rotations about each axis, and these can be combined into a single transformation matrix. These transformations are necessary to change from one axis system to another, say from the body axis system (forward, starboard, keelward) to the earth axis system (North, East, down). The order of the rotations is important, since each successive rotation is done about an axis in an intermediate axis system. These transformation matrices turn out to have a special mathematical property - the inverse of the matrix is equal to its transpose. This makes it easy to change from going from earth to body axes to going from body to earth axes - just swap elements across the diagonal and reverse the order of the matrix multiplications.
 
 

I have chosen to use the same convention as is used in aeronautical practice in order to make it easy to use existing derivations for the equations of motion when flying on hydrofoils. Going from the earth to the body axis one rotates first in yaw about the Z axis- a change in the boat's heading. Then one rotates in pitch about the new Y axis, and finally one rotates in roll about the final X axis. There is some evidence that boats tend to pitch in the heeled plane of symmetry, rather than in the vertical plane, which would make a yaw-roll-pitch order more natural. However, I will be sticking with the yaw-pitch-roll convention.
 
 
 
 
 
 
 
 


 
 

The individual rotations are: Earth to intermediate system 1
 
 


 
 

Intermediate system 1 to intermediate system 2
 
 


 
 

Intermediate system 2 to body axes
 
 


 
 

These can be combined together into one single rotation matrix (here's where the order comes in):
 
 


 
 

Going from body to earth axes:
 
 


 
 

The procedure for assembling the hulls in an arbitrary arrangement is to first rotate the hulls in toe (t, positive bow to starboard), incidence (i, positive bow up), and cant (k, positive rolled to starboard), using the expression above. Then position the hull by adding to all the coordinates the position of the reference center relative to the boat's reference center.
 
 

Note that this can play havoc with the station definitions. If toe in/out or incidence is used, the points for a given station will no longer be all at the same longitudinal station for the whole boat (X coordinates will not be the same for a given hull cross section). This doesn't cause any fundmental difficulty, as long as one carries along the X coordinate as well as Y and Z for each point.
 
 


 
 


 
 

I've found it useful to have a special case of the earth axis system, which is aligned with the waterplane and the boat's X axis. Thus, the heading angle, Y, is zero, but the pitch and roll angles are the same as for the earth axis system. The origin of the waterplane axis system is also at the mean water surface, so after rotating from body to waterplane orientation, the coordinates are translated vertically to account for the height of the boat's reference center relative to the waterplane.
 
 


 
 

A similar axis system is the stability axis system. The stability axis system will be used to linearize the equations of motion to analyze the boat's dynamics. The stability axis system is a body axis system, fixed to the body at the boat's reference point, and moving and rotating with the body. However, at some time, typically then the boat is in steady trimmed motion, the orientation is frozen to be parallel to the waterplane axis system:
 
 


 
 
 

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