Catamaran Dynamics

If you enjoy math and investigating physics (simple mechanics) you will want to know about the simple equations that model the main behaviors of many boats (but not hydroplanes). Because these behaviors have to do with the possible motions your boat can make, these equations are called "equations of motion".

For a simple sail boat of any size or hull design, the equations of motion consist of equations that play each main force against the other in a constant struggle for balance. Mathematicians call this ability to become "unbalanced" a "degree of freedom". Perhaps they realized how we as individuals have a tendency to become unbalanced in those areas where we are given freedoms. Then just like the laws of physics, God has given us laws to bring us back into balance with Him and His creation once again when we become "unbalanced".

Each of the equations of motion for a simple sail boat are "staged" on a different degree of freedom. In order to make the equations a bit simpler, it is assumed that your boat has been corrected for forward tilt. That is, your boat tips up in front when floating with no wind and levels out under the force of your breath.

Dynamic cycles

The type of boat examined from this page is a catamaran or other type of boat that sails with its hull in the water. When one of these boats is raced in a raingutter, it usually takes several breaths to blow it across the 10 foot length. Each time, breath speeds the boat up then the boat relaxes as the captain catches another breath.

Because of this cycle of acceleration and deceleration, these two motions need to be modeled along the central roll axis. Tipping and drifting to the side are also modeled around the roll, pitch and yaw axies. If we thought a current flowing across the raingutter was a factor, we could add a fifth equation for motion along the pitch axis. A sixth could be added along the yaw axis if water was being added during the race; but it's not!

To apply the acceleration and deceleration models, one must supply particulars about the boat, how strongly one blows and how long each breath lasts and how long it is before the next breath begins. The final speed computed from the acceleration model is used as the initial speed in the deceleration model. likewise, the final speed of deceleration model is used as the initial speed for the next breath in the acceleration model. This cycle continues until the boat reaches the length of the trough.

1. Initial and Subsequent Breath Acceleration
2. Deceleration

Modeing Hull Drag

Hull drag for regatta boats is essentially the pressure drag of the water diverted around the hull. We know this because the Reynold's number for the fastest regatta catamaran of raingutter size lies in the laminar range; there is no possibility of water turbulence nor cavitation (tearing of the water creating evacuated bubbles).

A quick evaluation of the Reynolds number for a fast boat begins with,

RN = vLp/ug for p/ug = 64.27 s/in2

The fastest catamaran regatta boat travels about 2 feet per second and that's probably generous. So let v = 24 in/s. By the usual rules, the hull length is 6 inches and that does nicely for the scale length, L. Put these factors together to get,

RN = 24*6*64.27 = 9255 which is much less than 100,000, the upper limit for guaranteed smooth water flow.

We can do a similar calculation for the top side air resistance. When we do, the Reynolds number comes to about 2/3rds of the one in the water. So the air flow is also a smooth ride, all pressure drag.

Pressure drag is easy to model but not always easy to find the forces and constants in the expression. The general expression for any type of pressure drag operating through a center of pressure is,

Pressure Drag = cpAv2/2

In the models above for the motion of catamarans, you will see terms that look like this general one but they will use different symbols that relate more closely with the parts experiencing pressure drag.