# Why is a paper plane stable?

1. Apr 28, 2015

### Jon Steensen

I am going to hold a lecture on aerodynamics for a group of high school students, after which a competition on constructing the best glider is held. I think the planes will be made of sheets of balsa wood, and they are therefore likely to have flat wings. To help them succeed, I plan on telling them about stability in the lecture. According to what I have read, and a little reasoning, I think it is not possible to have a plane plane with symmetric wings and tail plane fly with the same setting angle on both surfaces. Then I came to think about a paper plane which has no tail surface, and which obviously can fly, I just do not understand how
My argumentation is as follows:

For a symmetric profile the moment around the center of gravity will be zero, and the lift will always be located at the center of gravity. Three conditions must be full filled for a plane to fly stably:

1) the moment around the center of gravity must be zero, in the flying attitude
2) the aerodynamic center must be located behind the center of gravity
3) the lift must be(approximately) equal to the weight of the plane

As I see it all three conditions cannot be fulfilled at the same time. If the aerodynamic center is behind the center of gravity to fulfill condition 2, in order to fulfill condition 1 the lift must be zero, and condition 3 is not fulfilled. Condition 1 and 3 can be fulfilled at the same time, if the center of gravity is at the same point as the aerodynamic center, but then condition 2 is not fulfilled.

...when theory and practice does not agree it is seldom practice that is wrong. Practice clearly shows that a paper plane can fly, so there must be something wrong with my theory but what?

2. Apr 28, 2015

### Staff: Mentor

Last edited by a moderator: May 7, 2017
3. Apr 28, 2015

### HuskyNamedNala

A paper airplane will be pretty different from a regular plane with regards to those rules of thumb. More generally stability is simply the response of the aircraft to a disturbance. IE, if a gust of wind comes out of nowhere, the plane make shake but will return to equilibrium. An unstable aircraft might actually oscillate more until it crashes! There are two types of stability: static stability, which describes the sum of the moments equaling zero as you stated, and dynamic stability, which describes how the aircraft responds to a change in flight conditions. A simplified expression with some good figures is here: http://www.differencebetween.com/difference-between-static-stability-and-vs-dynamic-stability/

Here is an interesting real life example: the T-Tail (see http://en.wikipedia.org/wiki/T-tail ). Generally, the T-Tail is superior to a conventional tail for aerodynamic reasons except at high angles of attack! Angle of attack is defined as the orientation between the oncoming wind velocity and the chord of the wing/airfoil. At high angles of attack the wake of the main wing will fall directly in the path of the t-tail wing, this will cause a rapid loss of lift, and longitudinal control via stall and cause the aircraft to suddenly pitch forward and lose altitude. The pilot will not have any control over the aircraft until the wake of the main wing is away from the t-tail.

Personally, I wouldn't bother trying to use a paper airplane as an example. The principles described above are the same but the aerodynamics is completely different. Why not purchase some cheap foam gliders, and have the kids add some weight to the nose or tail and see what happens? I think a real neat project would be to have the kids throw the gliders through a disturbance (maybe have the glider fly past a box fan) and video tape the glider response.

Last edited by a moderator: Apr 28, 2015
4. Apr 28, 2015

### Jon Steensen

Oops instead of: For a symmetric profile the moment around the center of gravity will be zero, It should be: For a symmetric profile the moment around the aerodynamic center will be zero

5. Apr 30, 2015

### stedwards

Most of your students will be very disappointed with their non-flying models without some recommendations.

Pitch Roll and Yaw

1) To obtain self correcting roll stability the wings should have some dihedral.

2) A vertical stabilizer for yaw stability.

3) The concern you have brought up is pitch stability. For a plane flying horizontally to the right, there will be a true center of lift and a clockwise acting force-couple that *increases* with angle of attack. To obtain self correcting pitch stability you need a force-couple that counteracts the force couple of the main wing. In fact, it must overcompensate to some amount to obtain stability. Too much compensation and it will oscillate pitch-up and pitch-down in ever increasing amplitude.

a) In the normal airplane configuration this can be accomplished with a horizontal stabilizer having negative pitch with respect to the main wing.

b) In a canard configuration with the horizontal stabilizer in the front, the angle of attach of the horizontal stabilizer is greater than the main wing.

c) A delta wing such as a hang glider accomplishes this dual-angle-of-attack business by what is called "wash out"--the angle of attack of the wing tips is less than at the center and also further behind the center of lift. (Small private planes may have about 2 degrees washout. This lessens wing tip stall.)

d) There is also a reflex wing, and variations and combinations.
Finally, I would recommend your students keep the aspect ratio low for better dynamic stability against poor launch angles and wind gusts. This is a trade-off. The flight path will be steeper.

Last edited: Apr 30, 2015