Moment Curves -- a flight of theoretical fancy
The engineer in me won't let this go. There's so much more we could do! In computer engineering, we create a state space description of simultaneous equations, which places all the terms of the equations in a matrix. We can then apply standard matrix operations and calculus techniques to solve the equations. I'll describe in theory what could be done. Coming up with the terms of the actual partial differential equations would be devilishly difficult in practice.
Continue by creating equations to account for the direction and magnitude of the static (i.e. constant) forces from the sails as a function of pitch and roll. Adding these equations to the state space solution means we can solve for pitch and roll as a function of the direction and magnitude of sail force. The enhanced state space solution now accounts for sail forces, buoyancy, and changes to the position of the center of gravity.
Next, use aerodynamics to approximate the forces generated on the sails by a steady wind. These equations output forces as a function of relative wind velocity (i.e. relative speed, relative direction) and an assumed sail shape. Including these equations in state space allows the calculation the boat's pitch and roll based on relative wind speed and relative wind direction.
Of course, those numbers are still not terribly useful since the boat is not moving and the forces are constant. One needs to increase the number of dimensions in the state space solution to account for water flow, so one needs equations based on hydrodynamics that describe drag forces as a function, of pitch, roll, yaw, and hull velocity. The new matrix will output static pitch, roll, yaw, and hull velocity as a function of a given constant wind velocity. A secondary effect would be to show water movement in wakes. Would we have to include the bernouli effect of water flow on buoyancy forces?
Increase the dimensions in state space again to account for wave action. This would produce time-varying perturbations in pitch, roll, and yaw as a function of wave height, relative direction, and period. I would make the simplifying assumption that all waves are sinusoidal.
If all that were taken into account, we could then create a good mathematical model from which to calculate dynamic pitch, roll, yaw, lee way, and velocity as a function of relative wind velocity and wave height, relative direction, and period. It would be enough that we could even show a real time graphic representation on a computer screen. Dial in relative wind, see the boat heel.
Finally, we would have to add higher order moments to all the equations to account for variations in wind velocity, with simplifying assumptions about effects on sail shape.
Pragmatically, I am wondering how much of this can be done by the software naval architects use for boat design?
T. P. Donnelly
S/V Tranquility Base
1984 Islander 30 Bahama