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Engineering

Lifeboat launching

Release an enclosed free-fall lifeboat from the stern ramp of a supply vessel at dusk. It slides down the skids with Coulomb friction and leaves them at exactly sqrt(2 g L (sin theta - mu cos theta)), tips nose-down over the lip until the contact lets go, falls along a parabola and meets the dark sea where the closed form says, at the speed energy gives. It dives two metres deep, the buoyancy of its hull - nearly five times its weight - turns it up, it surfaces and the propeller drives it away while the pitch and heave die out. Read the loads at the bow, midship and stern seats: a higher drop enters faster and loads harder, a steeper ramp enters better aligned, trim by the stern lowers the lip. Every joule is followed into friction, drag, slamming and the water.

By Mykhailo Makarov

Lifeboat launching — Actual application image
Actual application image

What happens

An 8.5 m enclosed lifeboat of 8475 kg (30 persons), its centre of gravity 1 m above the keel and 0.6 m aft of the middle, radius of gyration 2.3 m, rests on a 35° stern ramp whose lip is 12 m above the sea. It slides 3.5 m with skid friction 0.1 at g (sin theta - mu cos theta), stepped exactly, and leaves at 5.81 m/s; it then tips over the lip on one sliding contact whose normal force comes from the constraint, and is released when that force vanishes, 1.72 s after the start. The flight is the exact parabola; the first hull section touches the water at 2.35 s, at 14.05 m/s on a 65.8° path with the bow 54° down. In the water the hull is 24 strips with lens-shaped sections of 39 m³ in all, 4.7 times its weight in buoyancy: buoyancy of every submerged section, added mass rho A normal to the keel with the von Karman slam force -max(dm/dt, 0) w, quadratic normal and axial drag, and 3 kN of propeller thrust once the stern is wet. RK4 steps of 1 ms; the energy ledger closes to the integrator’s order. Occupant loads are acceleration minus gravity at three seats, along the boat and through the spine; the combined index weighs them against 15 g and 7 g.

Try this

Watch the slide, the tip over the lip, the fall and the dive in slow motion; compare the exit and entry speeds with their closed forms; raise the lip and read the seat loads; try a steeper ramp and the bow loads drop; trim the ship by the stern and the lip comes down.

The model and its limits

A planar rigid boat in calm water, strip theory with a lens section, a momentum slam model, quadratic drag, no air drag, waves or roll, unfiltered loads. Loads are those of this idealized boat, not a certification test.

Equations used in the module
Equations
a_ramp = g·(sin θ − μ·cos θ),  v_exit = √(2·g·L·(sin θ − μ·cos θ))
N = (g·cos α − 2·ω·u + ω²·k) / (1/M + ξ_c·(ξ_c − μ·k)/I),  I·dω/dt = N·(μ·k − ξ_c)
(M + Σm_i)·a_n − S1·β = F_n − Σm_i·ω·u,  m_i = C_a·ρ·A_i,  f_i = −max(dm_i/dt, 0)·w_i
M·g·y0 = M·g·y + E_k + E_w + U_h + W_fric + W_drag + W_slam − W_thrust,  CAI = √((a_x/15)² + (a_z/7)²)

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Seaplane takeoff — Actual application image

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