Catenary Calculator: Mathematical Concepts & Equations

1. The Catenary Curve

A catenary is the curve formed by a perfectly flexible, uniform chain or cable suspended by its ends and acted on by gravity. The equation for a catenary curve is:

y(x) = a [ cosh(x/a) - 1 ]

y(x): vertical height above the lowest point (vertex) of the curve
x: horizontal distance from the vertex
a: catenary parameter, a = H / w
H: horizontal component of the tension at the lowest point
w: weight per unit length of the chain (submerged weight)

2. Chain Parameters

Chain Weight per Meter (Submerged): w = w_dry × (1 - ρ_water / ρ_steel)
Catenary Parameter: a = H / w

3. Arc Length (Chain Length)

The length of chain required to reach a vertical height y is:

s = sqrt( y (y + 2a) )

where s is the arc length from the lowest point to height y.

4. Seabed Contact and Lift-Off

Seabed Contact: When the chain is longer than the minimum required to reach the boat, part of it lies flat on the seabed. The horizontal segment is drawn as a straight line at y = 0.
Lift-Off Force: The minimum horizontal force required to lift all the chain off the seabed is:
H_lift = w · (L² - D²) / (2D)
where L is the total chain length, D is the water depth.

5. Forces

Horizontal Force (H): User input, in kg (converted to Newtons if needed).
Vertical Force: F_v = w · L
Force Down Chain: F_down = sqrt( H² + F_v² )

6. Table Calculations

For each water depth, the minimum chain length to just lift off the seabed is calculated using the arc length formula above. Scope ratio is L / D.

7. Graph Rendering

Seabed Segment: Drawn as a horizontal line from (0,0) to (s_seabed, 0).
Lifted Segment: Drawn using the catenary equation from (s_seabed, 0) to (x_boat, D).

8. Angle at Anchor

The angle at the anchor (where the chain leaves the seabed) is:

θ = arctan( sinh( x_boat / a ) )

where x_boat is the horizontal distance from the vertex to the boat.

References: