Rotor Solidity

Solidity is the fraction of the rotor disc that is actually blade. Lay the blade planforms flat and add their areas, divide by the area of the circle they sweep, and for rectangular blades that is $\sigma = N_b c R / \pi R^2 = N_b c / \pi R$. Helicopter main rotors live between about 0.05 and 0.12: a rotor is mostly empty air, and it is meant to be.

What solidity buys is thrust capacity. Spread the same thrust across more blade area and each square metre works less hard, which lowers the mean blade lift coefficient, delays retreating blade stall in forward flight, and leaves more manoeuvre margin at any given weight and altitude. What it costs is profile drag — paid on every square metre of blade, at every instant, at 200 metres per second, whether the rotor is carrying anything or not. That cost lands squarely on the hover figure of merit, and hovering is what a helicopter spends its most expensive minutes doing. An over-solid rotor is a rotor that has bought speed and manoeuvre with hover efficiency, which for some missions is exactly the right trade and for others is not.

The blade count is not a free variable in practice. It is set by the hub — a teetering two-blade head is mechanically simple and cheap, an articulated or bearingless five- or seven-blade head is neither — and by vibration, because a rotor feeds the airframe at blade-passing frequency and its multiples, and more blades means higher frequency and lower amplitude. It is also set by stowage: naval helicopters fold, and folded width is a hangar problem. So the ordinary design sequence runs radius first from the disc loading the mission allows, then blade count from the hub and vibration requirements, then chord to make the solidity come out where the blade loading needs it. That is why the inverse on this page, solving for blade count, quite properly returns a fraction. If it says 3.4 blades, it is not being unhelpful; it is telling you that no rotor has that combination and that the chord has to move.

One definitional trap. The formula above assumes rectangular blades. Real blades taper, and a tapered blade has less area out at the tip where the dynamic pressure is highest, so its aerodynamic effect is not the same as an untapered blade of the same total area. The literature therefore also uses a thrust-weighted solidity, which weights the local chord by $r^2$ across the radius and comes out lower than the plain geometric figure. Both are called sigma. Mixing them is a genuine source of confusion when comparing published rotors, so if two sources disagree by five or ten percent on a solidity, that is usually why. And blade area here means planform, counted once per blade, not the wetted area of both surfaces.