Drop a flat piece of paper and it will flutter and tumble through the air as it falls, but a well-fashioned paper airplane will glide smoothly. Although these structures look simple, their aerodynamics are surprisingly complex. Researchers at New York University’s Courant Institute of Mathematical Sciences conducted a series of experiments involving paper airplanes to explore this transition and develop a mathematical model to predict flight stability, according to a March paper published in the Journal of Fluid Mechanics.
“The study started with simple curiosity about what makes a good paper airplane and specifically what is needed for smooth gliding,” said co-author Leif Ristroph. “Answering such basic questions ended up being far from child’s play. We discovered that the aerodynamics of how paper airplanes keep level flight is really very different from the stability of conventional airplanes.”
Nobody knows who invented the first paper airplane, but China began making paper on a large scale around 500 BCE, with the emergence of origami and paper-folding as a popular art form between 460 and 390 BCE. Paper airplanes have long been studied as a means of learning more about the aerodynamics of flight. For instance, Leonardo da Vinci famously built a model plane out of parchment while dreaming up flying machines and used paper models to test his design for an ornithopter. In the 19th century, British engineer and inventor Sir George Cayley—sometimes called the “father of aviation”—studied the gliding performance of paper airplanes to design a glider capable of carrying a human.
An amusing “scientist playing with paper planes” anecdote comes from physicist Theodore von Kármán. In his 1967 memoir The Wind and Beyond, he recalled a formal 1924 banquet in Delft, The Netherlands, where fellow physicist Ludwig Prandtl constructed a paper airplane out of a menu to demonstrate the mechanics of flight to von Kármán’s sister, who was seated next to him. When he threw the paper plane, “It landed on the shirtfront of the French minister of education, much to the embarrassment of my sister and others at the banquet,” von Kármán wrote.
Enlarge/ Flight motions of paper airplanes with different center of mass locations.
NYU Applied Mathematics Laboratory
While scientists have clearly made great strides in aerodynamics—particularly about aircraft—Ristroph et al. noted that there was not a good mathematical model for predicting the simpler, subtler gliding flight of paper airplanes. It was already well-known that displacing the center of mass results in various flight trajectories, some more stable than others. “The key criterion of a successful glider is that the center of mass must be in the ‘just right’ place,” said Ristroph. “Good paper airplanes achieve this with the front edge folded over several times or by an added paper clip, which requires a little trial and error.”
He and his team verified this by test-flying various rectangular sheets of paper, changing the front weight by adding thin metallic tape to one edge. They found that an unweighted sheet tumbled end over end while descending left to right under the force of gravity. Adding a small weight to shift the center of mass slightly forward also produced a tumbling trajectory. Overall, they found that flyers with greater front-loading produced erratic trajectories full of swoops, climbs, flips, and dives.
The next step was to conduct more controlled and systematic experiments. Ristroph et al. decided to work with thin plastic plates “flying” through a large glass tank of water. The plates were laser-cut from an acrylic plastic sheet, along with two smaller “fins” embedded with lead weights to displace the center of mass, and they also serve as aerodynamic stabilizers. There were 17 plastic plates, each with a different center of mass. Each was released into the tank by sliding it down a short ramp, and the team recorded its free-flight motion through the water.
Enlarge/ Trajectories of plates falling through water, where the different colors represent different degrees of front weighting. Only the “just right” weight distribution leads to the smooth gliding shown in blue.
NYU Applied Mathematics Laboratory
They found the same dynamics played out. If the weight was centered, or nearly so, at the center of the wing, the plate would flutter and tumble erratically. Displace the center of mass too far toward one edge, and the plate would rapidly nosedive and crash. The proverbial “sweet spot” was placing the weight between those extremes. In that case, the aerodynamic force on the plane’s wing will push the wing back down if it moves upward, and push the wing back up if it moves downward. In other words, the center of pressure will vary with the angle of flight, thereby ensuring stability.
This differs substantially from conventional aircraft, which rely on airfoils—structures designed to generate lift. “The effect we found in paper airplanes does not happen for the traditional airfoils used as aircraft wings, whose center of pressure stays fixed in place across the angles that occur in flight,” said Ristroph. “The shifting of the center of pressure thus seems to be a unique property of thin, flat wings, and this ends up being the secret to the stable flight of paper airplanes. This is why airplanes need a separate tail wing as a stabilizer while a paper plane can get away with just a main wing that gives both lift and stability.”
The team also developed a mathematical model as a “flight simulator” to reproduce those motions. Ristroph et al. think their findings will prove useful in small-scale flight applications like drones or flying robots, which often require a more minimal design with no need for many extra flight surfaces, sensors, and controllers. The authors also note that the same strategy might be at work in winged plant seeds, some of which also exhibit stable gliding, with the seed serving as the payload to displace the center of mass. In fact, a 1987 study of the flying seeds of the gourd Alsomitra macrocarpa showed a center of mass and glide ratios consistent with the Ristroph group’s optimal gliding requirements.
