Stability

A rocket's stability is critical for achieving high altitude flights - an unstable rocket will go up about 50 feet (max.) and then flutter back down to the ground. There are two ways to change the stability - change the surface area of the fins, or add ballast to the nose. Clearly the overall weight of the rocket must also be considered, so increasing the surface area of the fins is usually the preferred method. This however increases drag obviously and so a stabile rocket is generally a compromise between drag and weight.

A rocket, once it is in flight, has eight degrees of freedom; in other words there are eight different directions in which the rocket can move. Two of these are linear - thrust and drag. The other six are rotational and are yaw left, yaw right, pitch up, pitch down, roll right and roll left.

The eight degrees of freedom of a water rocket

According to Newton's First Law of Motion, an external force exerted upon a body will cause that body to accelerate in the same direction as the force applied. Clearly there will always be some form of external force exerted upon a rocket in flight (for example a gust of wind). Now any body in space will tend to rotate about its center of gravity (CG) - the point where all of the mass of the body is balanced. If the rocket has no means of stabilisation whatsoever, the external forces will cause the rocket to go entirely off course. For this reason fins are used; they create a restoring force that helps to keep the rocket going as straight as possible. It is important however to note that the restoring force created by the fins causes the rocket to rotate around its center of pressure (CP) rather than its CG. This is the point about which all aerodynamic forces act.

Fins work by using the difference in pressure between the two largest surfaces (on either side of the fin). During the coast phase, an unmodified bottle (i.e. no fins, nose cone etc) would be unstable and would tumble neck over bottom through the air. This is much draggier than if the rocket maintains an orientation so that the longest length of the rocket is in the same direction as it is travelling (the case for a stable rocket). Now if fins are added to the bottle, the rocket will attempt to behave in exactly the same manner. However, since the rocket is travelling upwards, an area of high pressure will be created on one side of the two fins in the pitch or yaw axis. A corresponding area of low pressure will be created on the other side of the same two fins. This pressure difference between the two sides creates a lift force which acts about the CP (towards the low-pressure side), causing the rocket to pitch back in the opposite direction and hence restoring the rocket to its flight path. The diagram below attempts to illustrate this.

How fins work, using air pressure differences to generate a restoring force

The final consideration for producing a stabile rocket are the positions of the CG and the CP. First of all it is important that the CP is behind the CG (known as positive stability) - if it is in front of the CG (known as negative stability), then the rocket will be unstable (it will try to fly tail first!). But how far behind the CG should the CP be positioned? In model rocketry one caliber stability is generally used. That means that the CP is one body diameter behind the CG. However model rockets tend to be much narrower than water rockets, making this rule less applicable to water rockets. If the CP is too far back, then the rocket will suffer from weathercocking (overstability that leads to the rocket essentially over-compensating the pitch or yaw motion). Usually in water rocketry it's safe to have the CP a minimum of several centimeters (5-10) behind the CG. This does vary however, so do not assume that you must always build your rockets with that dimension - as long as you test your rockets stability before you launch it (ways of doing this are discussed below) then you'll be fine. A useful way of remembering the positions is a rule I found in The Handbook of Model Rocketry (6th edition) by G. Harry Stine. The rule is known as the alphabetical stability rule and states that G comes before P (in the alphabet). From the top of the rocket, the CG comes before the CP.

The three types of stability: (a) positive, (b) neutral and (c) negative

To make things slightly more complicated, it turns out that the CP actually changes position depending upon the angle of attack. At an angle of attack of 90 degrees (when the body of the rocket is parallel to the plane tangent to the Earth's surface directly below the rocket) the CP will be at its most forward point. This is why the CP must be behind the CG rather than the two points being at the exact same place (which is known as neutral stability); if there is a small gust of wind for example which blows the rocket slightly off course then the angle of attack will increase and the CP will move forward, past the CG and towards the nose of the rocket, causing the rocket to become unstable. This does however lead to the interesting possibility of having a stabile rocket on the way up, but at apogee when the rocket becomes horizontal (while flipping to begin its descent) and the angle of attack is at 90 degrees, it becomes unstable and descends in a 'stable' unstable manner (the rocket remains at a high angle of attack and so is unlikely to become stable again and lawn dart). 'Stable' because it remains unstable throughout the descent (the angle of attack remains at 90 degrees). This is known as backsliding or sidegliding and has been used by various members of the model rocketry community ever since it was first observed by Robert H. Goddard in 1938 (he also built the first liquid-fueled rocket); it tends to work best with long, slender rockets. The Alway brothers wrote a paper about the phenomenon and Robert Youens has used backsliding very effectively with his FTC rockets. Backsliding requires the CG to be positioned between the CP and CLA (this is essentially the same as the CP at maximum angle of attack). This is why it is easiest to design a backsliding rocket when the rocket is long and slender (the CP and CLA are farther apart than on a short, fat rocket and so it is easier to position them correctly).

So how do you design a rocket with the CP and/or CLA where you want? First of all there is the cardboard cutout method. This involves cutting out a cardboard side view of the rocket and balancing this cutout to locate its CG. This point corresponds to the CLA on the rocket. Using this method ensures that even in the worst condition (when the angle of attack is 90 degrees) the rocket will still be stable, however this isn't really necessary (it's extremely unlikely that a rocket will ever reach that position during its ascent). I have written a program that calculates the position of the CLA which you can download, or there is Ulrich Hornstein's online CLA-calculator. Knowing the location of the CLA is important when attempting to construct a backsliding rocket, but in a typical water rocket when you only care about the rocket being stable during the ascent it is not necessary to know the position of the CLA. Instead, it is much more important that you know the location of the CP and that you ensure that the CG is located in front of the CP. I have also written a program to calculate the position of the CP of a rocket, again available at the programs page listed on the left. There are some other programs available on the web, such as VCP, a downloadable CP calculator or there is this online calculator. The calculations for the CP (known as the Barrowman equations after James S. Barrowman who worked for NACA (now NASA) and developed them in the 1960's) are fairly long and, if you are not very comfortable with algebra, not very easy to peform, so I would recommend using a program to calculate its position. Click here to download the original report written by Barrowman (pdf format, zipped; 687Kb; copy obtained from www.ApogeeRockets.com). Using my CP calculator you can work out exactly where the CP is and then make sure that the CG is at least a few centimeters in front of that position; using this approach more or less ensures that your rocket will be stable, without even having to test it (but still do so - just in case!).

To conclude, there is a way of testing the stability of your rockets known as the swing test. Simply tie a length of string (the longer the better) around the rocket body, making sure the string is located at the center of gravity (when the rocket dangles from the string it should be horizontal). Then swing the rocket around your head via the string. If the rockets nosecone points in the direction that you swing the rocket and remains in that position, then it's stable. If not, then you need to either replace the fins with larger ones or add ballast to the nose (obviously the easier option).

One final tip: if you do need to make the fins larger, increase the span dimension (the dimension perpendicular to the rocket body at the bottom of the fin). This is because this increases the restoring force instead of increasing the distance between the CP and the CG, which in turn improves the dynamic damping.