AUTOROTATION, SELF‐STABILITY, AND STRUCTURE OF SINGLE‐WINGED FRUITS AND SEEDS (SAMARAS) WITH COMPARATIVE REMARKS ON ANIMAL FLIGHT
Authors: NORBERG R. ÅKE 
Published: January 2008

SUMMARY

A samara is a winged fruit or seed that autorotates when falling, thereby reducing the sinking speed of the diaspore and increasing the distance it may be transported by winds. Samaras have evolved independently in a large number of plants.
Aerodynamical, mechanical, and structural properties crucial for the inherent self‐stability are analysed, and formulae for calculation of performance data are given.
The momentum theorem is applied to samaras to calculate induced air velocities. As a basis for blade element analysis, and for directional stability analysis, various velocity components are put together into resultant relative air velocities normal to the blade's span axis for a samara in vertical autorotation and also in autorotation with side‐slip.
When falling, a samara is free to move in any sense, but in autorotation it possesses static and dynamic stability. Mainly qualitative aspects on static stability are pre sented. Simple experiments on flat plates at Reynolds numbers about 2000 as in samaras, showed that pitch stability prevails when the C. M. (centre of mass) is located 27–35 % of the chord behind the leading edge. The aerodynamic c.p. (centre of pressure) moves forward upon a decrease of the angle of attack, backward upon an increase. In samara blades the c.m. lies ca. one‐third chord behind the leading edge, and hence the aerodynamic and centrifugal forces interact so as to give pitch stability, involving stability of the angles of attack and gliding angles.
Photographs show that the centre of rotation of the samara approximately coincides with its c.m.
The coning angle (blade angle to tip path plane) taken up by the samara is determined by opposing moments set up by the centrifugal and aerodynamic forces. It is essentially the centrifugal moment (being a tangent function of the coning angle, which is small) that changes upon a change of coning angle, until the centrifugal and aerodynamic moments cancel out at the equilibrium coning angle.
Directional stability is maintained by keeping the tip path plane horizontal whereby a vertical descent path relative to the ambient air is maintained. Tilting of the tip path plane results in side‐slip. Side‐slip leads to an increased relative air speed at the blade when advancing, a reduced speed when retreating. The correspondingly fluctuating aerodynamic force and the gyroscopic action of the samara lead to restoring moments that bring the tip path plane back to the horizontal.
Entrance into autorotation is due to interaction between aerodynamic forces, the force of gravity, and inertial forces (when the blade accelerates towards a trailing position behind the c.m. of the samara).
The mass distribution must be such that the c.m. lies 0–30 % of the span from one end. In Acer and Plcea samaras the C.M. lies 10–20% from one end, thereby making the disk area swept by the blade large and the sinking speed low.
The blade plan‐form is discussed in relation to aerodynamics. The width is largest far out on the blade where the relative air velocities are large. The large width of the blade contributes to a high Re number and thus probably to a better L/D (lift/drag) ratio and a slower descent.
The concentration of vascular bundles at the leading edge of the blade and the tapering of the blade thickness towards the trailing edge are essential for a proper chord wise mass distribution.
Data are given for samaras of Acer and Plcea, and calculations of performance are made by means of the formulae given in the paper. Some figures for an Acer samara are: sinking speed 0.9 m/sec, tip path inclination 15°, average total force coefficient 1.7 (which is discussed), and a L/D ratio of the blade approximately 3.
The performances of samaras are compared with those of insects, birds, bats, a flat plate, and a parachute. They show the samara to be a relatively very efficient structure in braking the sinking speed of the diaspore.
In samaras the mass, aerodynamic, and torsion axes coincide, whereas in insect wings the torsicn axis often lies ahead of the other two. Location of the torsion axis in front of the aerodynamic axis in insects tends towards passive wing twisting and passive adjustment of the angles of attack relative to the incident air stream, the direction of which varies along the wing because of wing flapping.
Location of the mass axis behind the torsion axis may lead to unfavourable