The catamaran is the first application of torques because its geometry makes estimates relatively simple. A typical catamaran sailboat has two narrow canoe-like hulls that are separated by about half a boat length. The mast is centered in a structure rigidly connecting the two hulls. Over the ages, variations of the catamaran have been invented many times and in many places. The word "catamaran" [kattumaran = tied logs] comes from the Indian Tamil language. Polynesians colonized much of the Pacific in catamaran-like doubled canoes.
Normally, catamarans don't tip much, unless they tip over. The forces jeopardizing a catamaran on the edge of stability for roll are shown in Figure 4.2. For the boat in (a), the left hull is about to leave the water. Any increase in sail force produces the tipping shown in (b).
The rotational stability is determined by three torques (assuming the mass of the very skinny sailor is ignored). The first torque T1 is produced by the wind's horizontal force F on the sail. The center of effort is a vertical distance h1 above the center of mass, so
The negative sign applies because this torque rotates clockwise.
The second torque T2 is produced by the equal and opposite horizontal force of the water on the catamaran hull that keeps it from sliding sideways. (For simplicity, the hull was taken to be wedge shaped, so no centerboard is needed.) The center of effort of the water force is a vertical distance h2 below the center of mass. Thus,
The negative sign indicates another clockwise torque.
Figure 4.2 (a) A sailboat stays balanced when the sum of the applied torques vanish. (b) When the torque from the wind and water exceeds the compensating torque supplied by buoyancy, capsize results.
The third torque is produced by the buoyant force, which is equal to the force of gravity, Mg. If the catamaran is barely balanced, this force is applied only to the right hull section. Half the boat width (or beam), B/2 is the distance to the center of gravity. Thus,
The positive sign means the buoyancy torque is counterclockwise.
The boat will be stable only if the sum of the three torques vanishes.
Setting the sum of torques equal to zero yields a maximum value for the force of the wind.
A wind force any larger than F(max) will tip the boat over. Equation 4.7 makes intuitive sense. In order to withstand a larger wind force, the catamaran should be wider (larger B) or heavier (larger M). If the mast is taller (larger h1) or the water force is applied to a deep board (larger h2), the F(max) will be smaller.
Rough estimates of the forces and wind speeds are given here for a catamaran with properties similar to a Hobie-17 (see Figure 4.9), L = 5.2 m, B = 2.3 m, and M = 143 kg. The mast is about 8.2 m high, so h1 + h2 = 3 m is a reasonable rough approximation, which yields F(max) = 540 N. Any greater wind force would capsize the boat.
The maximum sail force F(max) can be used to estimate the wind speed needed for capsize. Assuming the wind is coming from the side, the wind force is the drag force. Using Equation 2.6 again,
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