🌊 Tensegrity Semi-Submersible Seastead
Froude Scale Model Design

1. Determining the Scale Ratio

Given: Full-scale leg diameter = 4 feet = 48 inches

Given: Model cylinder diameter available = 5 inches

Scale Ratio (λ) = Full-Scale Dimension ÷ Model Dimension

λ = 48 in ÷ 5 in = 9.6

Scale Factor: 1 : 9.6

Froude Scaling Laws Summary

Froude scaling preserves the ratio of inertial to gravitational forces, which is essential for wave interaction and dynamic behavior of floating structures.

Quantity	Scaling Relationship	For λ = 9.6
Length, Width, Height, Diameter	÷ λ	÷ 9.6
Area	÷ λ²	÷ 92.16
Volume & Displacement	÷ λ³	÷ 884.7
Mass / Weight	÷ λ³	÷ 884.7
Force (wave, cable tension)	÷ λ³	÷ 884.7
Time / Wave Period	÷ √λ	÷ 3.098
Velocity (wave, drift)	÷ √λ	÷ 3.098
Pressure	÷ λ	÷ 9.6

2. Full-Scale Geometry Recap

FRONT VIEW SIDE VIEW ┌──────────────┐ ┌──────────────────────────────┐ │ BODY │ │ BODY │ │ 14' wide │ │ 60' long × 8' high │ └──┬────────┬──┘ └─┬──────────────────────────┬─┘ ╱ ╲ ╱ ╲ ╱ 45° 45° ╲ ╱ 45° down 45° ╲ ╱ ╲ ╱ & away ╲ ╱ LEG LEG ╲ ╱ LEG LEG ╲ ●─ ─ ─ ─ cable ─ ─ ─ ─● ● ● (submerged cross-cable) (front leg end) (back leg end) ╲ ╱ cross ╲─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─╱ cross cables (cables to opposite center) cables Legs: 4' diameter × 35' long, ~60% submerged Each leg angles outward at 45° (front view) AND downward at 45° (side view)

Component	Full-Scale Dimension (feet)	Full-Scale Dimension (inches)
Body Length	60 ft	720 in
Body Width	14 ft	168 in
Body Height	8 ft	96 in
Leg Diameter	4 ft	48 in
Leg Length	35 ft	420 in
Cross-Cable (front triangle base)	Calculated below
Cross-Cable (rear triangle base)	Calculated below
Diagonal Restraint Cables	Calculated below

3. Leg Geometry – 3D Vector Analysis

Each leg leaves the body at a compound 45° angle. We need to find the true 3D direction. The legs depart from the front centerpoint (or rear centerpoint) of the body.

Coordinate System

X-axis: Athwartships (side to side)
Y-axis: Longitudinal (bow to stern)
Z-axis: Vertical (up/down)

Interpreting the 45° Angles

Viewed from the front (YZ plane projected onto XZ), each leg makes 45° from vertical toward the side. Viewed from the side (XZ plane projected onto YZ), each leg makes 45° from vertical going away longitudinally. This means:

If we define the unit direction vector of a front-right leg as (dx, dy, dz):

Front view: dx/dz = tan(45°) = 1 → dx = dz
Side view: dy/dz = tan(45°) = 1 → dy = dz

So the direction vector is (1, 1, 1) / √3 (normalized), meaning each leg actually descends at an angle of arccos(1/√3) ≈ 54.74° from vertical, not 45°. The 45° is the apparent angle seen in each projected view.

Leg Endpoint Positions (Full Scale)

With leg length L = 35 ft and direction (1/√3, 1/√3, 1/√3):

Horizontal offset (sideways) per leg: 35 / √3 = 20.21 ft
Horizontal offset (fore/aft) per leg: 35 / √3 = 20.21 ft
Vertical drop per leg: 35 / √3 = 20.21 ft

Cable Lengths – Full Scale

Front Triangle Cross-Cable: The two front legs spread outward (port and starboard) by 20.21 ft each. They start from the same front center point, so the tips are separated horizontally (athwartships) by:

Front cross-cable = 2 × 20.21 ft = 40.41 ft = 485.0 in

(The tips are at the same depth and same longitudinal position, so the cable is purely horizontal/athwartships.)

Rear Triangle Cross-Cable: Same geometry:

Rear cross-cable = 40.41 ft = 485.0 in

Diagonal Restraint Cables (from each leg tip to the center underside of the body at the opposite end):

A front-right leg tip is at position (relative to body center):

(+20.21, +30 + 20.21, −20.21) = (+20.21, +50.21, −20.21) ft

The rear-center underside of the body is at:

(0, −30, −8) ft

(taking body center at origin, body top at z=0, bottom at z=−8, front at y=+30, rear at y=−30)

Distance = √(20.21² + 80.21² + 12.21²) = √(408.4 + 6433.6 + 149.1) = √6991.2 = 83.61 ft ≈ 1003.4 in

Note: By symmetry all four diagonal restraint cables are the same length: ≈ 83.6 ft (1003 in) full scale.

Cable	Full-Scale Length (ft)	Full-Scale Length (in)
Front cross-cable	40.41	485.0
Rear cross-cable	40.41	485.0
Each diagonal restraint cable (×4)	83.61	1003.4

4. Scaled Model Dimensions (1 : 9.6)

All full-scale dimensions divided by 9.6:

Component	Full Scale (in)	Model Scale (in)	Model (ft-in approx.)
Body Length	720	75.00	6 ft 3 in
Body Width	168	17.50	1 ft 5.5 in
Body Height	96	10.00	10 in
Leg Diameter	48	5.00	5 in (as given)
Leg Length	420	43.75	3 ft 7.75 in
Vertical Drop per Leg	242.5	25.26	2 ft 1.3 in
Sideways Spread per Leg	242.5	25.26	2 ft 1.3 in
Fore/Aft Spread per Leg	242.5	25.26	2 ft 1.3 in
Front Cross-Cable	485.0	50.52	4 ft 2.5 in
Rear Cross-Cable	485.0	50.52	4 ft 2.5 in
Each Diagonal Restraint Cable	1003.4	104.52	8 ft 8.5 in

Overall Model Footprint

Dimension	Model (in)	Model (ft-in)
Total Width (tip to tip)	50.52	4 ft 2.5 in
Total Length (front tip to rear tip)	75.00 + 2 × 25.26 = 125.52	10 ft 5.5 in
Total Depth (body top to leg bottom)	10.00 + 25.26 = 35.26	2 ft 11.3 in

5. Mass / Weight Scaling

Under Froude scaling with the same fluid (freshwater or seawater), masses scale as λ³. We must first estimate full-scale weights.

5a. Full-Scale Weight Estimates

Body (Living Area)

Body volume: 60 × 14 × 8 = 6,720 ft³

For a steel/aluminum habitable structure, a reasonable average density might be around 8–12 lb/ft³ overall (structure + interior + equipment + furnishings).

Estimate at 10 lb/ft³ average: 67,200 lbs (≈ 33.6 tons)

Each Leg (Flotation Column)

Leg volume (cylinder): π × 2² × 35 = 439.8 ft³

If these are hollow steel/aluminum floats with ~10% solid fraction and ballast water capability:

Structural weight estimate: ~3 lb/ft³ average → 1,320 lbs each

4 legs total: 5,280 lbs

Cables & Hardware

Estimate: 1,500 lbs

Total Full-Scale Weight

Full-Scale Total ≈ 67,200 + 5,280 + 1,500 = 73,980 lbs
(approximately 37 tons)

5b. Buoyancy Check (Full Scale)

Each leg: π × 2² × 35 = 439.8 ft³ total volume

At 60% submerged: 0.60 × 439.8 = 263.9 ft³ submerged per leg

4 legs submerged volume: 4 × 263.9 = 1,055.6 ft³

Seawater buoyancy: 1,055.6 × 64 lb/ft³ = 67,558 lbs

Body should ride above water, but some hull immersion contributes buoyancy too. The numbers are in the right ballpark for a semi-submersible at this draft.

5c. Model Weights (Froude Scaled)

Divide full-scale weights by λ³ = 9.6³ = 884.7:

Component	Full Scale (lbs)	Model Scale (lbs)	Model (oz)
Body	67,200	75.96	1,215
Each Leg (×4)	1,320	1.49	23.9
All 4 Legs	5,280	5.97	95.5
Cables & Hardware	1,500	1.70	27.1
Total	73,980	83.62	1,338

Practical note: The model body needs to weigh about 76 lbs and the total model about 84 lbs. You will likely need to add ballast (lead shot, sand, steel weights) inside the model body to achieve the correct Froude-scaled mass. The 5-inch PVC/plastic cylinders for the legs should be partially flooded or ballasted to hit ~1.5 lbs each.

6. Model Leg Submersion Check

Model leg: 5 in diameter, 43.75 in long

Volume per model leg: π × (2.5)² × 43.75 = 858.7 in³

At 60% submerged: 515.2 in³ per leg

4 legs submerged: 2,060.9 in³

Freshwater buoyancy: 2,060.9 in³ × 0.0361 lb/in³ = 74.4 lbs

This must support the total model weight of ~84 lbs. The body will need to be partially immersed or the legs slightly more than 60% submerged, which matches the semi-submersible behavior. The system is close to equilibrium, confirming the scaling is consistent.

7. Two 55-Gallon Barrels as Model Body

Standard 55-Gallon Plastic Barrel Dimensions

Property	Typical Value
Height	~35 inches
Diameter	~23 inches
Volume	55 gallons (12,705 in³)

Two Barrels Connected End-to-End (Lengthwise)

Model body from barrels:

Length: 2 × 35 = 70 inches
Diameter/Width: 23 inches
Height: 23 inches (circular cross section)

What This Represents at Full Scale (× 9.6)

Dimension	Model (2 barrels, in)	Full Scale (in)	Full Scale (ft)	Design Target (ft)	Comparison
Length	70	672	56.0	60	93% ✦ Close
Width	23	220.8	18.4	14	131% ✦ Wider
Height	23	220.8	18.4	8	230% ✦ Much taller

⚠ Mismatch Warning: The two 55-gallon barrels create a body that is reasonably close in length (56 ft vs 60 ft target — only 7% short), but the barrel's circular cross-section is far too wide and far too tall compared to the design's 14 ft × 8 ft rectangular cross-section. The barrel represents an 18.4 ft diameter cylinder instead of a 14 × 8 ft rectangle.

What You Would Need for the Correct Model Body

Ideal model body dimensions:

Length: 75 inches (6 ft 3 in)
Width: 17.5 inches (1 ft 5.5 in)
Height: 10 inches

This is a flat, barge-like rectangular box — very different from a barrel's round profile.

Using the Barrels Anyway? Options:

Option	Description	Pros	Cons
A. Accept the mismatch	Use barrels as-is; understand they represent a 56 ft × 18.4 ft diameter cylindrical hull	Easy, cheap, fast	Cross-section wrong; wave interaction & stability won't match rectangular body
B. Cut & reshape barrels	Slice barrels lengthwise and flatten/reshape to ~17.5″ W × 10″ H rectangular section	Correct shape	Difficult, may leak
C. Build a plywood box	75″ × 17.5″ × 10″ box from marine plywood, sealed with fiberglass or epoxy	Exact dimensions, easy to ballast	More labor
D. Use barrels for flotation testing only	Verify buoyancy and cable tension behavior; don't extrapolate wave/stability data to rectangular hull	Quick feasibility test	Limited data applicability

8. Complete Model Parts List & Specifications

Part	Qty	Model Dimension (in)	Model Weight Target (lbs)	Notes
Body (rectangular box)	1	75.0 L × 17.5 W × 10.0 H	~76 (with ballast)	Plywood/foam, sealed, ballasted
Legs (cylinders)	4	5.0 dia × 43.75 long	~1.5 each	PVC pipe, capped, partially ballasted
Front cross-cable	1	50.5 long	—	Braided line or wire
Rear cross-cable	1	50.5 long	—	Braided line or wire
Diagonal restraint cables	4	104.5 long each	—	Braided line or wire, low stretch
Attachment hardware	—	—	~1.7 total	Shackles, eye bolts, turnbuckles
TOTAL MODEL			~84 lbs

9. Summary Reference Card

SCALE: 1 : 9.6 \| λ = 9.6
Scaling	Formula	Value
Lengths	÷ 9.6
Areas	÷ 92.16
Volumes / Masses	÷ 884.74
Time periods	÷ 3.098
Velocities	÷ 3.098
Forces	÷ 884.74

Quick Conversions (Full → Model):

1 foot full scale	=	1.25 inches model
1 ton full scale (2000 lb)	=	2.26 lbs model
1 second full scale wave period	=	0.323 seconds model
1 knot full scale	=	0.323 knots model

10. Notes & Recommendations

✓ Cable Stiffness: Under Froude scaling, cable axial stiffness (EA) should scale as λ³. Use flexible braided line rather than stiff wire to better represent the scaled elasticity. Turnbuckles help fine-tune pretension.

✓ Ballasting: The most critical aspect of Froude model testing is getting the mass, center of gravity, and moments of inertia correct. Weigh each component carefully. Place ballast weights (lead shot in bags) to match the scaled CG position. The body CG should be at approximately mid-length, mid-width, and mid-height of the body.

✓ Leg Buoyancy Tuning: Cap each 5″ PVC leg and add measured water inside to adjust displacement and achieve the target 60% submersion. Mark waterlines on each leg for quick verification.

⚠ Viscous Effects: Froude scaling does NOT preserve Reynolds number. At 1:9.6 scale, viscous drag on the small cylinders will be proportionally higher than at full scale. This means the model may show more damping of motions than the full-scale structure would experience. This is a known limitation of Froude model testing and is typically accepted for preliminary design work.

⚠ Barrel Body: As analyzed above, two 55-gallon barrels create a body that represents approximately a 56 ft long × 18.4 ft diameter cylinder at full scale — substantially different from the 60 × 14 × 8 ft rectangular body. For meaningful hydrodynamic data, building a correctly proportioned rectangular model body is strongly recommended.

— Tensegrity Seastead Scale Model Design Document —
Froude Scale Ratio 1 : 9.6

🌊 Tensegrity Semi-Submersible SeasteadFroude Scale Model Design