THE WINGED SEEDS THAT SPIN so gracefully to the ground from ash, elm and maple trees are called samaras. Watching them in flight might arouse your curiosity about their aerodynamics. It is not as simple as it looks.

The first detailed analysis of the aerodynamics of single-wing samaras such as those of the maple was done in the early 1970's by R. Ake Norberg of the University of Goteborg. Further studies were carried out by Charles W McCutchen of the National Institute of Arthritis, Metabolism, and Digestive Diseases and by F. M. Burrows of the University of North Wales. I shall first follow the work by Norberg, who applied to samaras the aerodynamics of helicopters. Then I shall briefly review McCutchen's studies of samaras other than the maple's.

Norberg made two major assumptions. One was that the samara is a flat wing. The other was that the mass of the wing can be considered as lying on the long axis from the seed to the wing tip.

A spinning maple samara does not accelerate downward, in spite of the effect of gravity, because an aerodynamic force is applied by the air through which it falls. For a simple explanation of the force one can regard the samara as spinning about its center of mass, which falls along a vertical axis. (The center of rotation is actually slightly off the center of mass.) The wing is essentially horizontal and sweeps out a disk in which the flow of air is uniform. One can visualize the flight from either of two viewpoints: the usual one of an observer watching the seed fall or one in which the observer imagines himself falling along with the samara.

In the first viewpoint the samara drops into a column of still air. Each horizontal layer of air in the column is accelerated downward as the samara passes through it. The column of air left above the samara is moving with a final downward speed lower than the sinking speed of the samara. The air in the column reflects the effects of a downward force because it has been accelerated from a stationary state to a final downward speed. The corresponding reaction force on the samara is upward.

From the viewpoint of the observer falling with the samara the column of air below the samara moves upward with a speed equal to the samara's true sinking speed. Each layer of air is accelerated downward as it passes through the disk swept out by the spinning wing. After the acceleration a layer of air is moving upward at a lower speed. Thus the layer reflects the effects of a downward force that has decreased the layer's ascent. The corresponding reaction force on the samara is upward. In each case the samara is affected by an upward force.

This aerodynamic force cancels the weight of the samara, preventing it from accelerating downward. For the observer on the ground the samara moves downward at a constant speed. For the observer falling with the samara the upward-moving column of air approaches at a constant speed equal to the true sinking speed of the samara. The sinking speed is proportional to the square root of a ratio called the disk loading. This ratio is equal to the weight of the samara divided by the area of the disk swept out by the wing during a full spin about the vertical axis of the fall.

Different maple samaras descend at different speeds because they do not have the same disk loading. One obvious reason for the difference in disk loading is the variation in the size and weight of the samaras. A less obvious reason is that a spinning samara does not actually have a horizontal wing. The angle between the plane of the wing and the horizontal differs from one samara to another. Some seeds descend with their wing almost flat, in others the wing is tilted upward at an angle of 45 degrees or more to the horizontal. The disk loading is larger for a tilted wing than it is for a more horizontal one, and the samara falls faster.

Analyzing the aerodynamic force on a maple samara with a tilted wing is difficult for an observer on the ground, and so I shall take the viewpoint of the falling observer. It is portrayed in the illustration below; the relative motion of the air passing the wing is upward and toward the thinner trailing edge of the wing. This motion provides the aerodynamic force to counter the samara's weight.

From the viewpoint of the falling observer several types of data are needed to study the force on the wing. The vertical axis about which the samara is truly spinning is the axis of rotation. I shall assume that it passes through the overall center of mass of the samara, which is near the end containing the seed. The span of the wing is the long axis that runs from the seed to the outer tip of the wing. The chord is a shorter axis between the leading edge and the trailing edge of the wing; it is perpendicular to the span. The angle the span forms within the horizontal is called the coning angle.

The aerodynamic force can be calculated by assuming that the wing is divided into thin strips running from the leading edge to the trailing edge. The force on each strip is computed by multiplying the area of the strip by the square of the speed of the air moving past it. The net force on the wing is found by adding the forces on the individual strips.

The air moving vertically past the wing can be divided into two components, one perpendicular to the span and one parallel to it. Only the perpendicular component contributes to the aerodynamic force supporting the wing. The movement of air parallel to a chord from the leading edge to the trailing edge of the wing is more difficult to evaluate because its speed depends on the distance between the axis of rotation and the strip being considered. The air K moving along the chord of a strip near; the axis of rotation has a relatively low speed. (The reason is that such a strip moves comparatively slowly as the win' spins.) The air moving along the chord of a strip farther from the axis has higher speed. To demonstrate the effect of the movement of air along a chord shall concentrate on only one strip positioned about midway on the wing.

Air moves past this strip both vertically and backward along a chord. Of the vertical motion only the component perpendicular to the wing matters. The new velocity vector of the air moving pa the strip lies somewhere between a I perpendicular to the wing and the chord running through the strip.

If the contribution from the vertical motion of air is relatively large (if the samara is sinking fairly fast), the net velocity vector is nearly vertical. If instead it is relatively small, the net velocity vector is closer to being parallel to the chord. A subtle shift between these two possibilities is a key to some of the stability of a descending samara.

The magnitude of the net velocity vector determines the magnitude of the aerodynamic force on the strip.-Most the force on the wing as a whole arises from the strips farther out on the wing. One reason is that the chords are longer there, and so the strips have more area. Moreover, the net velocity of the passing air is larger because the speed of the air moving parallel to the chord is greater. Hence a properly designed samara is narrow in chord near the center of mass, where little aerodynamic force is available, and wider in chord farther out.

The net aerodynamic force on the wing is perpendicular to the wing and also lies in a vertical plane that passes through the span. The vertical component of the force balances the downward weight of the samara. If you toss a maple samara into the air, it must somehow adjust the orientation of its wing so that the vertical component of the aerodynamic force matches its weight. It is impressive to discover that a descending samara achieves this adjustment automatically.

Once a samara has achieved the proper orientation it must rapidly counter any chance perturbations by breezes. Most samaras are inherently stable in at least four respects: the wing's angle of attack, the coning angle, the tilt of the circle traced out by the outer tip in relation to the horizontal, and the possibility of sideways motion.

The angle between the chord of a wing and the velocity vector of the passing air stream is called the angle of attack. The bottom illustration in Figure 4 shows a cross section of a samara's wing at about the midpoint of the span. The net aerodynamic force on this section of the wing is indicated by a single force vector operating through a point called the center of pressure. The effects of the aerodynamic forces on each small part of the section are the same as is indicated by this one vector. The position of the center of pressure is determined by the shape of the section and the direction of the passing air.

The section also has a center of mass reflecting the weight of the section. The weight is insignificant, however, compared with the centrifugal force on the section. This force, resulting from the spin of the samara, also operates through the section's center of mass.

The orientation of the section with respect to the velocity vector of the passing air is stable if the center of pressure coincides with the center of mass. Thus a certain angle of attack is desirable for stability. Suppose the wing is shifted suddenly in such a way that the center of pressure moves toward the trailing edge of the wing. This nose-up orientation results in a larger angle of attack, which generates a different aerodynamic force on the wing. The vertical portion of the force no longer balances the samara's weight as is needed for a stable, prolonged flight.

The samara is designed to correct this situation by regaining the proper angle of attack. The force operating through the center of pressure creates a torque that rotates the wing section about the center of mass and back to its original orientation. The center of pressure re- turns to the center of mass and the samara is again stable. Something similar happens if the perturbation reduces the angle of attack from its optimum value, moving the center of pressure forward: from the center of mass and sending the wing into a nose-down orientation.: Again the force through the center of: pressure rotates the section about the s center of mass until the proper orientation is regained.

According to research on the aerodynamics of gliding flat plates, such an automatic adjustment of a wing's angle of attack is possible if the mass is distributed in a certain way along the chord of the wing. Consider again a cross section through the wing. The center of mass of this section must be behind the leading edge by a distance that is between 27 and 35 percent of the chord length of the section. The center of mass must also be behind the most forward position to which the center of pressure could move, otherwise the wing would not survive a nose-down perturbation The samara has evolved with just such a distribution of mass along the chord The ribbing in the wing is bundled near the leading edge and spread near the trailing edge.

Another stability requirement has to do with the angle between the horizontal and the glide path of the wing. The samara actually spirals, but the angle can still be defined if the glide path is taken as a tangent to the downward spiral at any given instant. The glide angle is important because it determines the orientation of the aerodynamic force on the wing. At the proper angle, called the natural glide angle, the force is in a vertical plane that passes through the span of the wing. If the glide path is too steep, the force vector inclines toward the leading edge of the wing. If the path is too shallow, the force vector inclines toward the trailing edge of the wing. Neither of these orientations will give rise to a stable flight.

As a samara begins its fall to the ground it must rapidly achieve the proper glide angle so that the aerodynamic force on it lies in a vertical plane. Only then is the vertical component of the force able to cancel the weight of the samara to provide a constant speed of descent. If in the early stages of descent the vertical component of the force is smaller than the weight (because the velocity of the passing air is lower than it should be), the samara must adjust its glide angle accordingly or it will accelerate all the way to the ground.

The steps in the correction are as follows. Since the samara is now accelerating downward, the passing air has a velocity vector that is more vertical than it should be. The angle of attack is too large; the wing of the samara noses down to gain the proper angle of attack. The aerodynamic force shifts from being vertical (as it should be) to being canted toward the leading edge of the wing. This leaning of the force vector propels the wing into a greater spin about the axis of rotation.

From the viewpoint of the observer falling with the samara the extra spin amounts to an increase in the speed of the air moving along a chord toward the trailing edge of the wing. The net velocity vector of the passing air (which includes both the vertically moving air because the samara is falling and the air moving along the chord because the samara is spinning) was initially too vertical. With the increase in airspeed along the chord the velocity vector becomes less vertical. Now the angle of attack is too shallow. The wing noses up. The movement restores the aerodynamic force to a vertical plane, and the spin no longer increases.

In the process of adjustment the samara has increased its speed of descent (because it initially was accelerating downward) and its rate of spin (because the tilt of the aerodynamic force during the early part of the adjustment propelled the wing around the axis of rotation). Hence the velocity of the passing air is now higher and the force is stronger, just strong enough for the vertical component of the force to match the weight of the samara. Thereafter the samara falls with a constant speed and spin, with the proper angle of attack and with a suitable glide angle.

If the initial glide path results in too high a velocity for the passing air, the vertical component of the aerodynamic force exceeds the weight. The samara executes a similar kind of adjustment to reduce the aerodynamic force. As the samara continues to fall the excessive force reduces the rate of descent. From the viewpoint of the falling observer this reduction amounts not only to a decrease in the velocity of the passing air but also to a shift in the net velocity vector so that it is more nearly parallel to the chord. The angle of attack is then wrong, being too shallow. The wing noses upward to stabilize.

This movement tilts the aerodynamic force back toward the rear of the wing. The tilt opposes the spin of the samara, and the spin begins to decrease. To the falling observer the decrease in spin amounts to a reduction in the speed of the air that moves parallel to the chord. Again the overall velocity vector of the passing air is reduced. Its direction is changed too; it is now more vertical. The angle of attack is now too steep, and so the samara noses down to reduce it. This shift brings the force back to a vertical plane through the span, and the reduction in the spin ceases.

The net result is a reduction in the rate of descent (because the initially overly large aerodynamic force slowed the descent) and a decrease in the spin (because of the backward tilt of the force during the first part of the adjustment). The overall velocity vector of the passing air is now lower and therefore so is the force. The vertical component of the force now equals the samara's weight Thereafter the samara falls with constant speed and spin and with an appropriate angle of attack and an appropriate glide angle.

A chance perturbation of the glide angle is countered by similar responses. For example, if the glide path suddenly gets too steep, the net velocity vector of the passing air is too vertical. Since the angle of attack is then wrong, the: wing noses down to correct it. The aerodynamic-force vector tilts forward. The spin rate increases and makes the net velocity vector more horizontal. Again the attack angle must be corrected; this time the wing must nose upward, moving the force vector back to the vertical plane. In principle the samara should, now have the proper angle of attack and glide angle. It probably overshoots the proper glide angle, however, and must make several progressively smaller corrections in its spin, rate of descent and angles of orientation before it finally gains the proper values.

The coning angle too is automatically adjusted against small perturbations. The angle is set by a balancing of two kinds of torque. One torque results from the centrifugal force on the strips along the span of the wing. Consider a strip about midway along the span. The centrifugal force is outward from the axis about which the samara is rotating. The torque generated by this force tends to rotate the wing about its overall center of mass to decrease the coning angle. The same force affects all the strips in the wing.

The wing does not become horizontal because another torque, arising from the aerodynamic force on each strip, tends to increase the coning angle. At one optimum value of coning angle the two opposing sets of torques balance each other. If the wing is deflected into a different coning angle, these torques bring it back to the proper angle.

In general the coning angle should be shallow, so that the wing is more nearly horizontal. Then the disk loading is lower (since the wing sweeps out a larger disk) and the sinking speed of the samara is also lower. I have two maple samaras that are approximately the same in weight and overall appearance but descend at different coning angles. R The one with the shallower coning angle falls slower.

Finally, a stability is related to the direction of travel of the samara. Most of my samaras spin while falling along a vertical axis. A falling observer could describe a plane traced out by the outer tip of the wing. The majority of my samples trace out a horizontal plane. If I blow against a samara to deflect it momentarily from the horizontal, it quickly adjusts its motion to bring the low side upward so that the plane is again horizontal.

Some of my prize samaras do something else. Even when I do not disturb their descent, they move in a large helix while still spinning in the normal way. Norberg calls this additional motion sideslip. The aerodynamic force on one of the peculiar samaras causes it to tilt the plane traced by the outer wing tip. Once the plane has been tilted it is forced not back up, as with most samaras, but rather to the side, so that the samara's center of mass follows a helical path to the ground. The sense of rotation of the center of mass in this path is opposite to the sense of spin of the samara about its axis of rotation. If from an overhead viewpoint the samara spins clockwise, the center of mass moves around in a helix in a counterclockwise direction. The helical motion is much slower than the spin.

If a maple samara is dropped with its center of mass downward and its flat wing vertical, it may not spin at all. If the wing is initially tilted from the vertical or is indented, it will begin to spin after a short drop. Consider a flat wing tilted from the vertical. As the fall begins, the aerodynamic force causes the wing to lag behind the descent of the center of mass, with the result that the wing rotates toward the vertical over the center of mass. Continuously adjusting its angles of attack, gliding and coning, the wing eventually reaches the equilibrium values. Thereafter the samara spins as it descends. When I toss a samara into the air with the center of mass leading, it almost always starts to spin once it stops moving upward and begins to fall. It is then unlikely to be exactly vertical, and spin is almost certain to begin.

In a good samara the distance of the center of mass from one end is between zero and 30 percent of the full length of the span. The distance of the center of mass of maple samaras is usually between 10 and 20 percent. The center of mass along the chord of a strip on the wing must be at a distance from the leading edge that is approximately 27 to 35 percent of the chord's length. The disk loading must be low, and so a thin wing that sweeps out a large disk is a desirable combination of properties. Most of the aerodynamic force is generated at the outer end of the wing because the relative velocity of the passing air is larger there. Since the force also depends on the area of a strip on the wing, the strips at the outer end of the wing should have the longest chords.

An astute observer might disagree with this last because the outer tips of maple samaras are somewhat tapered and flared toward the trailing edge. This natural design takes into account the vortexes shed by the outer tip as the samara spins about the vertical. The vortexes decrease the lift and increase the drag on the wing tip. The taper and the flare at the tip minimize the strength of the vortexes.

Most of my samples of maple samaras are several centimeters long in the span and about a centimeter wide in the chord. A few of them follow wide helical paths to the ground, but most fall with the center of mass moving along a vertical line. I cannot distinguish any physical difference between the two sets of samaras. The difference in flight pattern must be the result of subtle differences in the samaras.

The illustration at right summarizes my analysis of the mass distribution of a typical samara. To find the overall center of mass I balanced the samara on a sharp edge, positioning it so that the span was perpendicular to the edge. When the sama-was in balance, I drew a line on it just above the edge. Then I balanced the samara with a chord perpendicular to the edge and drew a line just above the edge. The intersection of the two lines marked the overall center of mass.

I was also interested in the distribution of mass along individual strips in q e wing. I traced the outline of the samara on a sheet of graph paper. With a razor blade I sliced the wing into strips at ran parallel to chords through the wing. The position of each strip was indicated on the traced outline. To determine the approximate center of mass of a strip I balanced it on a sharp edge with the chord perpendicular to the edge.

Having achieved a balance, I marked the position of the edge of the strip. Then I measured the distance between the mark and the leading edge of the strip. I also measured the average chord length of the strip. (Since the trailing edge of the samara curves, the strip had a small range of chord lengths.) My results are shown in the illustration of mass distribution. The distance between the center of mass of a strip and the leading edge of the wing is given as a percentage of the average chord length of the strip.

If the samara were a flat wing, the distance of the center of mass along the chord of a strip should be between 27 and 35 percent of the chord length in order for the wing to be stable as it glides. The samara I analyzed had strips in which this distance was between 26 and 38 percent.

To measure the spin rate of a sample I directed a stroboscopic light downward on the path taken by a falling seed. With the room lights off I varied the flash rate of the stroboscope until each flash revealed approximately the same orientation of the descending samara. This effective freezing of motion can be accomplished by many settings of frequency on the stroboscope, including the frequency that matches the spin frequency of the samara. For example, a flash frequency of half the spin frequency reveals the same orientation on every other rotation of the samara. A flash frequency of a third the spin frequency reveals the same orientation on every third rotation.

To determine the spin frequency I begin at a low flash frequency and gradually increase the rate. The highest frequency that effectively freezes the motion of the samara is the one that matches the spin frequency. Any higher flash frequency does not freeze the motion. Instead I see images of the samara at various stages of each rotation. I found that most of my samples had a spin frequency of between 10 and 12 hertz (cycles per second). If I want to see the samara at, say, four stages during each rotation, I set the flash rate at about 40 hertz.

A stroboscope can be employed to make a photograph of a descending, spinning samara. With the room lights off the repetitive flashing of the stroboscope will illuminate the samara at various stages of its descent, leaving a permanent record of the stages on film. McCutchen made a similar photograph with a continuous light that shone upward into the path of a falling tulip-tree samara. That type of samara not only spins about the vertical axis along which its center of mass descends but also spins about the axis of its span. As the samara fell in the beam of light it exposed to the camera alternately an illuminated bottom surface and a darker top surface. The succession of separate images on the film recorded the fall. Contrast between the two sides can be obtained if one side is painted white.

I have discussed only the common maple samara. You might want to follow up McCutchen's work with other kinds of samara. Tulip-tree and ash samaras are particularly interesting. You could also experiment with cardboard models of samaras that you could modify in various ways to see how the changes affect the flight patterns.

Bibliography

AUTOROTATION, SELF-STABILITY, AND STRUCTURE OF SINGLE-WINGED FRUITS AND SEEDS (SAMARAS) WITH COMPARATIVE REMARKS ON ANIMAL FLIGHT. R. Ake Norberg in Biological Reviews of the Cambridge Philosophical Society, Vol. 48, No. 4, pages 561-596; November, 1973.

THE SPINNING ROTATION OF ASH AND TULIP TREE SAMARAS. C. W. McCutchen in Science, Vol. 197, No. 4304, pages 691-692; August 12, 1977.

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