Seastead Drogue and Stabilizer Analysis

1. Introduction

This analysis addresses the questions posed regarding the seastead design, focusing on the range of orientations achievable with a trailing drodrogue, recommended drodrogue sizes for moving at 6 knots in various wind speeds, thoughts on adjustable drogues, and the feasibility of using stabilizers to lift the seastead at high speeds. Key assumptions are made based on the provided description and typical engineering approximations.

Key Assumptions:

• Seastead weight (W): Estimated at 27,360 lb based on buoyancy from three legs (each leg 19 ft long, NACA 0030 foil, 10 ft chord, 3 ft thickness, 50% submerged). This is a rough estimate and should be refined with detailed modeling.
• Water density (ρ): 64 lb/ft³ (seawater), yielding mass density of 2.0 slug/ft³ for dynamics.
• Air density (ρ_air): 0.0024 slug/ft³ for wind force calculations.
• Wind force estimation: Based on frontal area of 200 ft² (triangle frame + living area) and drag coefficient Cd_air = 1.0.
• Thruster capability: Total thrust estimated at 3,000 lb (6 RIM drives × 500 lb each, approximate).

2. Range of Orientation Off Downwind with Trailing Drogue

The proposed system uses two winches at the back corners, each connected to a drodrogue via ropes, forming a bridle. This allows adjustment of the drodrogue's position relative to the seastead's centerline. The range of angles off directly downwind depends on:

• Length of ropes from each winch to the drodrogue.
• Position of the drodrogue attachment relative to the seastead's pivot point.
• The geometry of the bridle (e.g., the angle between the two ropes).

Typical sliding bridle systems on vessels can achieve orientations of 30° to 60° off downwind. For this seastead, with the back corners 35 ft apart (width of triangle back) and assuming ropes of moderate length (e.g., 50-100 ft), a reasonable estimate is:

Note: The actual range should be verified with simulation or scale model testing, considering the flexibility of the ropes and drodrogue behavior.

3. Drogue Size Recommendations for 6 Knots in Various Winds

The goal is to move at 6 knots (20.25 ft/s) even with the drodrogue deployed. The thrusters must overcome the combined drag of the seastead and drodrogue. Wind forces add to the load, but the drodrogue drag should balance the wind force to maintain stability.

Wind Force Estimation:

Wind force F_wind = 0.5 * ρ_air * v_wind² * Cd_air * A_air, with A_air ≈ 200 ft², Cd_air = 1.0.

Required Drogue Drag at 6 Knots:

Assuming the drodrogue drag must equal the wind force to counteract it, and using the drag formula D_drogue = 0.5 * ρ * v² * Cd_drogue * A_drogue, with ρ = 2.0 slug/ft³, v = 20.25 ft/s, and assuming Cd_drogue = 1.0 (typical for cone drogues), we solve for A_drogue.

Recommendation:

A drodrogue with an adjustable frontal area of 4–6 ft² (e.g., a cone with diameter 2.5–3 ft) would be suitable for the range of winds, with the ability to reduce drag in lower winds. Consider a system with multiple cones (e.g., Jordan Series Drogue) for adjustability.

4. Thoughts on Adjustable Drogues

Jordan Series Drogue:

This consists of multiple small cones in series, allowing incremental deployment. By pulling the collapse line, cones can be disabled, reducing drag. For our needs, a Jordan Series Drogue with about 20 cones of 6–8 inch diameter could provide the required drag area. Each cone contributes to drag, and the total can be adjusted by adding or removing cones from the water. This is a good option for fine-tuning drag.

Galerider-Style Perforated Drogues:

These use a perforated fabric that allows water flow, reducing shock loads. They are typically not as adjustable in drag area as Jordan drogues, but some models have adjustment mechanisms. They may come in sizes suitable for this application, but availability should be checked.

Adjustable Parachute/Basket Drogues with Collapse Line:

These drogues have a purse-string that reduces the open diameter, lowering drag. This is directly adjustable on the fly, which is beneficial for changing conditions. For the required drag range (1–5 ft² area), a parachute drodrogue with an initial diameter of 3–4 ft could be scaled down to 1–2 ft. This seems very suitable.

5. Stabilizer Lift Analysis for Running from Storms

The concept is to use the stabilizers (little airplanes) to lift the seastead partially out of the water at high speeds (e.g., 12 knots), reducing drag from the legs and allowing faster movement to escape storms. We analyze the required size of each stabilizer wing to take half the seastead weight (i.e., total lift from stabilizers = 0.5W).

Lift Equation:

L = 0.5 * ρ * v² * S * Cl

where:

• L = lift force per stabilizer
• ρ = 2.0 slug/ft³
• v = 12 knots = 20.25 ft/s
• S = planform area of stabilizer wing
• Cl = lift coefficient (assumed 0.7 for a well-designed foil at moderate angle of attack)

Scenario A: Each stabilizer takes half the seastead weight (L = 0.5W)

Scenario B: All three stabilizers together take half the weight, each providing one-third (L = 0.5W/3 = W/6)

6. Structural Thickness Analysis for Stabilizers

We determine the required thickness of the stabilizer wing to withstand bending loads at high speeds. The load is from lift; we use the lift from Scenario B (L = 4,560 lb per stabilizer) as a realistic design load.

Bending Stress:

Root bending moment M = L * (span/2) = 4,560 * (12/2) = 27,360 ft-lb = 328,320 in-lb.
Assuming aluminum with allowable stress σ_allow = 15,000 psi (with safety factor), required section modulus S_required = M / σ_allow = 21.888 in³.

Approximate Wing Thickness:

Assuming a rectangular cross-section with width = chord = 18 inches, height = thickness t. Section modulus S = b * t² / 6 => t = sqrt(6S/b) = sqrt(6*21.888/18) = sqrt(7.296) = 2.7 inches.
This is about 15% of chord, which is reasonable for a thin wing structure.

7. Conclusion

• Drogue Orientation: A trailing drodrogue with a sliding bridle can likely achieve orientations up to 45° off downwind on either side, providing good control.
• Drogue Size: For moving at 6 knots in winds up to 60 mph, a drodrogue with an adjustable frontal area of 4–6 ft² (e.g., cone diameter 2.5–3 ft) is recommended.
• Adjustable Drogues: Both Jordan Series Drogues and adjustable parachute drogues are suitable. The parachute type with collapse line is simpler for on-the-fly adjustment.
• Stabilizer Lift: The existing stabilizers (12 ft span, 1.5 ft chord) can likely provide sufficient lift for half the seastead weight at 12 knots when all three are used. Individual sizing for half weight would require larger wings (~48 ft² area per stabilizer), which is not practical. Thus, the cooperative lift approach is more feasible.
• Structural Thickness: The stabilizers would need to be about 3 inches thick to withstand bending loads, which is achievable with current design (likely thicker).