Executive Summary: Comfort in rolling seas depends on the relationship between a vessel's natural roll period, the wave encounter period, and available damping mechanisms. Solar yachts face a "triple disadvantage": no sail aerodynamics, insufficient speed for hydrodynamic stabilization, and limited power for active systems.
Rolling motion is governed by the second-order differential equation of a damped harmonic oscillator:
Where:
The natural roll period determines how quickly a vessel rocks in calm water when disturbed. It is calculated by:
Where k is the radius of gyration (√(I/m)), and g is gravity.
GM represents the distance between the Center of Gravity (G) and the Metacenter (M):
Where:
Waterplane Area (Awp): A wide waterplane (high Iwp) creates high "restoring stiffness" (high GM), leading to quick, jerky rolls. A fine waterplane rolls slower but deeper. Catamarans have extreme waterplane inertia due to hull separation—high GM with high damping.
When the wave encounter period (Te) matches the natural roll period (Tn), amplitude increases dramatically:
Where λ is wavelength, V is boat speed, μ is heading angle, and c is wave celerity.
| Vessel Type | Te Manipulation | Resonance Avoidance |
|---|---|---|
| Sailboat | Sails provide aerodynamic damping (Naero ≈ 15-30% of total) | Damping prevents build-up even near resonance |
| Fast Powerboat | High V shifts Te far from Tn (Te << Tn) | Averages many waves; hydrodynamic lift reduces coupling |
| Trawler | Moderate V + stabilizers (fins/tanks) | Active/passive damping at 6-10 knots |
| Solar Yacht | Low V (3-6 knots) means Te ≈ Twave | Often sits exactly in resonance with 6-10s harbor swells |
Damping is the dissipation of roll energy. Sources include:
Energy radiated away as waves generated by the rolling hull. Increases with roll velocity squared.
The sail acts as a flat plate with drag coefficient Cd ≈ 1.2-1.5:
This nonlinear velocity-squared damping is highly effective at reducing roll acceleration peaks.
Requires forward speed. Lift L = ½ρV²A·CL·α. At 5+ knots, fin stabilizers become effective. Below 4 knots, lift ∝ V² becomes negligible.
The Roll Acceleration Damping (RAD) comfort criterion quantifies seasickness and comfort:
Or more practically, the Motion Sickness Incidence (MSI) correlates with:
Bilge Keels: Extend 3-5% of beam, length 30-40% of LWL. Can increase damping by 40-60% but add drag (3-5% speed penalty). Critical for solar.
Flume Tanks (Anti-Roll Tanks): U-tube or free-surface tanks sized at 2-3% of displacement. Phase-shifted water movement opposes roll. Effective at all speeds, zero energy cost, but complex to tune.
Gyroscopic Stabilizers: Heavy flywheels (spinning at 5,000-10,000 RPM) precess to counter roll. Power draw: 1.5-4kW (manageable for solar arrays). Effective at zero speed. Weight penalty: 2-4% of displacement.
At 4-6 knots, active fins can generate 20,000-50,000 N·m of damping moment. Power: 2-5kW. Only viable if solar array > 3kW continuous.
Interceptor Trim Tabs: Vertical drops at transom change pressure distribution. Effective at moderate speeds, lower drag than fins.
For a 12m solar monohull (15,000 kg displacement):
| Concept | Relevance to Solar Yacht | Design Action |
|---|---|---|
| GM (Metacentric Height) | Keep 0.6-1.0m for 10-12s period | Lower battery placement; keep panels light |
| Waterplane Inertia | Higher = stiffer roll | Consider moderate flare (increase Iwp without weight) |
| Natural Period Tn | Avoid 6-10s (common wave period) | Aim for 11-14s using weight distribution |
| Damping (N) | Critical gap (no sail, low speed) | Bilge keels + gyro or flume tank |
| Center of Gravity | High panels raise KG | Counterweight in keel or deep batteries |
| Underwater Foils | Ineffective at 4 knots | Use for directional stability, not roll damping |
| RAD Index | Target < 0.2g RMS | Prioritize damping over stiffness |
Solar yacht comfort is achievable not by模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模仿模范传统帆船或快艇，而是通过优化阻尼和避开共振频率来实现。