**Hull
Design**

**Hydrostatics**

**Approach**

Hydrostatics is all about the forces and moments generated by the buoyancy
of the boat at zero speed and in flat water. The trends in these static
forces and moments are indicative of the load carrying ability and the
stability of the yacht under sail. I use the hydrostatic calculations to
determine:

displacement - determines load carried by the hull

wetted surface area - important for hull resistance calculations

transverse center of buoyancy - a measure of the heeling trim and stability

longitudinal center of buoyancy - a measure of the pitching trim and stability

pitching moment - longitudinal trim and stability at large rotations

rolling moment - transverse trim and stability at large rotations

waterplane area - the sensitivity of the displacement to changes in immersion

moments of inertia of the waterplane area - stability of the hull for
small rotations

A key prerequisite to doing these calculations for any degree of immersion
or rotation is the ability to interpolate the internal table of offsets
to determine where the waterplane is, and what parts of the hull are above
and below the waterline. I do this by running through the points for each
section, testing to see if they are above or below the waterline. This
is easy when the points are transformed to the earth axis system - positive
Ze values indicate that the point is below the water, and negatvie Ze values
indicate that it is above the water. When successive points are above and
below the waterplane, the segment can be interpolated to find the point
at the waterplane.

The points below the waterline are integrated to calculate the cross
section area and the girth. The centroid of the sectional area is also
calculated. Integrating the wetted girth along the length yields the wetted
area. Integrating the cross sectional area gives the displacement, and
integrating the area times its distance from the reference center yields
the pitching and rolling moments. Dividing the moment by the displacement
gives the location of the center of buoyancy. Computation of the waterplane
area and moment of inertia is similar.

As mentioned a the previous section, once the hulls are rotated, their coordinates are no longer arranged in convenient stations, perpendicular to the waterplane. This makes the calculation of hydrostatics more complicated. Three general approaches suggest themselves:

- rotate the coordinates into the earth axis system, and integrate them as they are, which requires a complex integration scheme,

- interpolate the hull coordinates in the earth axis system to produce "nice" stations, which requires a great deal of computation as the inerpolation is redone for every rotation,

- compute the hydrostatics in the hull coordinate system, taking into account that gravity no longer acts along the hull's Z axis. These forces and moments are then transformed back into the boat or earth coordinates, as desired.

**Forces and
Moments**

I've chosen to use the third method, and will be computing all the forces
and moments in the hull's reference system, and about the individual hull's
reference center. These are then transformed to the boat coordinate system.
All the quantities in this section are in the body axis system of the original
hull, unless otherwise stated. The gravity vector, relative to the hull,
is in the direction:

The points around the stations are unevenly distributed, but there are
a lot of them, so I've chosen to use simple, low order methods for approximating
the integrals needed to compute sectional properties. In order to get the
properties for the whole hull, these sectional properties need to be integrated
along the length of the hull. Here I can use the fact that the stations
are evenly spaced for an even number of intervals, and use Simpson's rule.

The cross section area is computed by a discrete approximation derived
from Green's Theorem:

The Y's and Z's in this case are only those points which lie at or below
the waterplane. The moments of the cross section area about the Y and Z
axes are given by:

And the centroid of the section is found by dividing the moments by
the area:

The displacment is found by integrating the cross section areas using
Simpson's rule (m must be odd and the stations evenly spaced DX apart):

The force on the hull is equal to the displacement, in the opposite
direction to vertical down:

The moments are found by computing the center of buoyancy in each axis,
and applying the total forces at that location:

The hull's contribution to the boat's forces and moments are computed
by rotating to the boat axis system and transfering the moments to the
boat reference center:

Once the forces and moments for an individual hull are computed in the
body axis system, they can be summed with the forces and moments for the
other hulls to obtain the forces and moments acting on the whole boat.

**Waterplane
Parameters**

The area and moments of inertia of the waterplane are useful because
they indicate the sensitivity of the forces and moments. These parameters
are computed in a similar manner to the hull forces and moments. However,
in this case, it isn't necessary to do all the rotations back and forth
between the earth and hull axis systems, since the orientation of the waterplane
and the forces acting on it are understood. The forces and moments can
be found in the earth axis system, and then transformed to the boat axis
system, if desired.

The cross sections intersect the waterplane in straight lines. Because
of the hollow in some of the cross section shapes, it is possible that
there will be multiple segments where the waterplane intersects the cross
section. Each segment has the endpoints E1 and E2, and the length of the
segment is Bwp . These points around the waterplane can be transformed
to the waterplane system and the waterplane quantities computed there.
Note that the segments will not be parallel to the Ywp axis if the hull
is toed in or out.

The lines where the section plane intersects the waterplane are equally
spaced, but compared to the spacing of the sections in the hull axis system,
the spacing will be closer. There will be nj such segments (typically one
or two) for each station, according to the number of patches of the waterplane,
with a zero distance used for sections that are entirely above or below
the waterplane.

Let the independent parameters t and u represent the directions along the station segments and perpendicular to them, such that t = 0 at point E1, t = 1 at point E2, and u=1 at one end of the waterplane patch and nj at the other end. An elemental area of the waterplane is then:

And the waterplane areas can be found by integrating the area along
the t and u directions:

The area of the waterplane is found by integrating the waterplane beams
using Simpson's rule:

The centroid of the waterplane is found by taking the moments of the
waterplane beams, and dividing by the area, just as was done for the section
areas. Once the centroid is found in waterplane axes, it can be transformed
back into boat axes if desired:

The centroid of the waterplane can be transformed back to the boat body
axis system, remembering that, by definition, the vertical location of
the waterplane in the earth axes is zero:

The moments of inertia, or second moments, of the waterplane area is
computed similarly. These are used to calculate the stability for small
motions.

**Wetted Surface
Area**

The area of the wetted surface is important for calculating the resistance
of the boat. Since the wetted area is independent of the choice of axis
system, I've chosen to avoid transformations and use coordinates in the
hull axis system. I've chosen to use the method of computing the wetted
girths at each section, and adjusting them to account for the slope of
the hull by the secant method. These adjusted girths are then integrated
along the length using Simpson's rule.

As with the displacement calculation the portion of the hull below the
waterplane is found by testing each point around a section to see if it
is above or below the waterplane. The distance between points at or below
the waterplane are added into the girth, and points above the waterplane
are ignored. The girth is computed by taking the straight-line distance
between successive points, weighted by the secant of the longitudinal surface
slope:

The secant weighting is a correction for the fact that the girths will
be integrated along the length of the hull to get the wetted area, but
the actual distance along the surface is greater than the station spacing,
depending on how steeply the hull tapers at a given section. The secant
is taken by projecting the vector normal to the surface onto the plane
of the section, and the normal vector is found by taking the cross product
of the line segment and a vector connecting the corresponding points on
the adjacent sections:

In the case of the end sections, only one sided differences are used
to compute the normals:

Finally, the wetted girths are integrated along the hull to get the
wetted area of the hull:

**Hydrostatics
of the Complete Multihull**

Once the hydrostatic quantities are found for each hull, it only remains
to ocmbine them together to get the characteristics of the whole boat.
The displacement and wetted area can simply be summed up over the hulls
to get the displacement and wetted area for the whole boat.

The forces and moments were already computed in the boat axis system,
and so X, Y, Z components of the forces and the moments (L, M, N) can also
be summed for all the hulls. Once the forces and moments are in hand, they
can be transformed from the boat axis system to any other desired axis
system.

The waterplane areas were computed in the waterplane axis system, as
were the moments of inertia of the waterplane area, so these, too, can
be summed over the hulls. The longitudinal and transverse centers of buoyancy
are perhaps less of interest for the complete multihull, since the forces
and moments are already known. But if desired, the location of the complete
center of buoyancy can be found by transforming the moments to the waterplane
axis system and dividing by the displacement.