Reference / Paper
Apogee Components' Peak of Flight newsletter, issue 18. Date: undated; published between 2000 and 2005 (from the neighbouring dated issues).
Apogee Components' Peak of Flight newsletter, issue 18. Date: undated; published between 2000 and 2005 (from the neighbouring dated issues).
I S S U E N E W S L E 1 8 T T E R The Cardboard Cut-out CP Method By Tim Van Milligan Many modelers have heard about the cardboard cut-out method for approximating the location of a model's Centerof-Pressure (CP). But why does this method work, what are it's limitations, and how does it compare to the Barrowman method of calculating stability? In this article, I'd like to explore all of these questions, so that we can design better models which will be safe when launched. First, what is the Center-of-Pressure? By definition, it is the point on the rocket where all the aerodynamic forces balance. As shown in Figure 1, when a fin is placed in an airflow, a lift force is created along the entire surface of the fin. Since it would be a nightmare to try to determine the lift force at each point on the surface of the fin, we just combine them all into one 'big' force. But where do we put the location of this force. That is what the CP of the fin is: the location where we can place this force. Lif tg ern era ted by w Lift gernerated by wing ing Air Flow CP Air Flow Fig. 1 As you know, the location of the CP is very critical when it comes to the stability of the rocket. So we need a method to find out where it is. Without a location, we can't compare the relationship of the CP to the CG, and we won't know if the rocket will be stable prior to launch. To keep the discussion simple, lets assume our rocket has straight rectangular fins. For starters, lets ignore everything on the model; except one fin. Lets mount this fin on a thin CP CG CG CP Unstable Stable Fig. 2 dowel running along the span, so that it is free to pitch up or down. Now lets assume the fin is in an airflow, so the wind is blowing toward the leading edge of the fin. When air flows over the fin, it creates forces (lift and drag) that act on the fin. Because our fin is free to rotate, it will pitch up or down in response to the forces. First, lets position the dowel at the midpoint of the fin as shown in Figure 3. This location is the 1/2 chord line. Now we can do mind exercises and see what happens when we move the location of the CP (the point where we assume all the forces act). If the CP is behind the 1/2 chord line, the fin will pitch up in response to the forces acting on the fin. If it is in front of the line, it will pitch downward. And if the CP is on the 1/2 chord line, the fin will not rotate either up or down. The same situation occurs no matter where we place the location of the dowel along the chord of the fin; as shown in Figure 4. What is most important for us to recognize is that when the CP is positioned on the dowel line, the fin will not rotate either up or down in response to a force. This is the location we will be looking for. You might have heard that the "Cardboard Cut-out 1130 Elkton Drive, Suite A Colorado Springs, CO 80907 USA www.ApogeeRockets.com orders @ApogeeRockets.com phone 719-535-9335 fax 719-534-9050 Page 1 P E A K O Point Force 1/2 Ch F F L Lin G Fig. 3 1/4 Ch Pivot Axis Fig. 4 Lin e Air Flow Pivot Axis Leading edge pitches up Leading edge pitches up Point Force Point Force CP CP Air Flow Air Flow Pivot Axis Leading edge pitches down Pivot Axis Leading edge pitches down Point Force Point Force 1/2 Air Flow T CP ord e Air Flow H Point Force CP ord I 1/4 Ch ord CP Lin e Pivot Axis Fin does not rotate when CP is on pivot axis Method" assumes that the model is at a 90° angle of attack to the airflow. For our fin, this is shown in Figure 5. The full face of the fin is being blasted by the airflow. As each air particle hits the surface of the fin, it creates a tiny force that is trying to push the fin backward. By simple intuition, we can combine all the tiny forces and substitute one big force acting in the exact center of the fin area. So now we know where the location of a fin's CP is for You also remember that a model will rotate around its Center-of-Gravity (CG). As long as the CG location is in front of the CP, the model will be stable. From this we can define "stability" by which direction the rocket pitches when a force is applied. Going back to our single rectangular fin, the CG Ch ord CP Lin Air Flow e Pivot Axis Fin does not rotate when CP is on pivot axis will be along the 1/2 chord line; which is also the CP location from the cut-out method. This is the same situation as shown in Figure 3. Since we want the fin to be stable, the CG must be moved ahead of the CP, similar to what is shown if Figure 4. By looking at Figure 3, we can define a stable rocket as one that will pitch upward when a downward force is applied to the CP of the fin. If we change the direction of the force to "up" (like a LIFT force), then the fin will pitch down. I mention this as a definition of stability only. How does this cardboard cut-out CP location compare to the location calculated by the Barrowman equations? If you look at the Barrowman equations, you can do a quick calculation for a single rectangular fin. Lets assume the chord length About this Newsletter You can subscribe “FREE” to receive this e-zine at the Apogee Components web site (www.ApogeeRockets.com), or sending an email to: [email protected] with "SUBSCRIBE" as the subject line of the message. A P O G E E Page 2 R O C K E T S PAGE 2 P E A K O F Fig. 5 CP 1/2 Air Flow ch ord lin e Pivot Axis Uniform force over entire surface is 3 inches. The equation for the location (Xf) of the CP as measured from the leading edge is: Xf = Xr/3 * ((C r + 2Ct )/(Cr + Ct)) + 1/6((Cr + Ct) - ((Cr*C t)/ (Cr+Ct))) Where: Xr = Leading edge sweep Ct = Tip chord length Cr = Root chord length For our fin, Xr= 0, and Cr = Ct = 3 inches. The equation now becomes: Xf = 0 + 1/6((Cr + Ct) - ((Cr*Ct)/(Cr+Ct))) Xf = 1/6(2*Cr) - Cr2/2*Cr) Xf = 1/6(2*3 - 32/2*3) Xf = 1/6(6 - 9/6) Xf = 1/6(6 - 1.5) Xf = 1/6(4.5) Xf = .75 inches F L I G H T tube does not affect the CP position of the rocket at all. The only thing the body tube does is to move the CG forward. But in the cut-out method, the extra area greatly changes the CP location; moving it too far forward. The longer the tube, the further forward the CP moves. So because the cut-out method's CP has moved so far forward of the fins, when a comparison is made between the locations of the CP and CG, we may find that the model is "unstable!" So to compensate, we add an even longer tube or more nose weight to move the CG forward. Because of this, we say that the cut-out method is "overly conservative." Being conservative in stability is not a bad thing. We want our models to fly stable, but you can take it too far. When a model is "too stable," it weathercocks into the wind. This can be a dangerous situation; particularly if you assumed the model will go straight up and have selected a long delay rocket motor. In this case, the model will arc over and eject the recovery device on a downward trajectory; or even impact the ground without ejecting the parachute. That is not good. The best situation is to use the Barrowman equations to design your rocket and check its stability. It is more reliable in predicting the location of the CP. But the equations can be tedious if you had to do them long-hand, and fortunately there are computer programs that use them. The "RockSim" program from Apogee Components (for Windows95) is particularly nice for designing rockets. It not only calculates the CP, but it also calculates the CG of the rocket and checks for stability. The newest version of RockSim also has both the Barrowman equations and the cardboard cut-out method for CP location determination. This lets you check both methods at the same time, and you can see the differences in the CP locations. There are limitations to the Barrowman equations. For example, you are limited to rockets with the "classical" shape. As soon as you make one fin bigger than the other, or strap on Fig. 6 To find the chord percentage that Xf is, we just divide by the chord length (3 inches). This tells us that the CP for a rectangular fin as measured by the Barrowman equation is on the 1/4 chord line. Therefore, the Barrowman method predicts the CP will be at the 1/4 chord line, while the cardboard cutout method predicts it will be further back at the 1/2 chord line. Based on this, you'd think that the cut-out method overestimates the model's stability, because the CP is further back (which we think is a good thing). But things really start to change when you add a body tube to the simulation. In the Barrowman method, the body A P O G E E Balance point = C.P. Cardboard “cutout” of rocket Locating the rocket’s approximate “center-of-pressure” R O C K E T S PAGE 2 Page 3 P E A K O F side pods, the method can't be used. That is why the cardboard cut-out method is still around; because you can still use it to design odd shaped rockets. I hope this article answered some of your questions about the cardboard cut-out method for calculating rocket stability About the Author: Tim Van Milligan is the owner of Apogee Components (http://www.apogeerockets.com) and the curator of the rock- F L I G H T etry education web site: http://www.apogeerockets.com/education. He is also the author of the books: "Model Rocket Design and Construction," "69 Simple Science Fair Projects with Model Rockets: Aeronautics" and publisher of the FREE ezine newsletter about model rockets. You can subscribe to the e-zine at the Apogee Components web site, or sending an email to: [email protected] with "SUBSCRIBE" as the subject line of the message. that they add strength to your rocket s body tube. Fig 1-13 Equipment Pods and Air Scoops: Equipment pods are used to store and protect delicate cargo, and air scoops allow air inside the aircraft. Adding either of these features to your rocket can dramatically change its appearance. Construction of these components is fairly easy, as they can be made out of cardboard, plastic, or balsa wood. What thickness of wood should you use for fins a rocket powered by a D motor? What are the nine types of fin construction? What are the other five different recovery methods besides: parachute, streamer, glider, and helicopter recovery? What size wing do you need for a rocket glider? How does high power construction differ from small rockets? Forward Fins and Stabilizing Vanes: In addition to having fins at the rear of the rocket, they can be added to the nose, middle, or tail of the rocket. Vanes are the same a fins, but are usually much longer. Although they add a distinctive look to the model, forward fins and vanes should be small in size because they can cause the rocket to become unstable. Always perform a stability check before flying the model equipped with forward fins or vanes. Fin Geometry and Shape: By changing the geometry, the number, or the size of the fins, you can completely change the overall appearance of the model. Fins are usually made out of flat balsa sheet, but you don t Fig 1-17 Forward fins Forward fin should be kept small Fig 1-18 Vanes Vanes are similar to tunnels except that they are made from thin sheets of wood Fig 1-14 Scoop Air scoop made from flat cardboard Scoop Air scoop made from short body tubes Fig 1-15 Fig 1-16 Fig 2-11 Fig 2-11 Pod Simple pods can be CP constructed from body tubes Scoop Frontal air scoop is a great decoration for missiles CP Old CP have to limit yourself to that material or to New CP CP a flat shape. You can make stabilizing fins out of plastic, paper, wood dowels or even hollow body tubes. The model does not even have to use any fins if you use cone stability. This is described in Chapter 2. Fig 2-12 centerline of the model, the fins will have to Extraneous Details:beLike muchadding largeratodifferprovide the restoring force ent paint pattern or using a to lot of decals, needed overcome the unbalanced force of the engine. adding three dimensional components to the This is why you outside of the rocketkeep can the enhance look of should always thrustthe forces concentric around the centerline a basic rocket. These details orof the components model. Otherwise, the model can be simple pieces of wood, they can be may or become unstable as soon as it extra plastic parts you have a the plastic clears thefrom top of launch rod. airplane kit. Adding them to the rocket is Besides moving theas location of the CP or thetoCG, easy as gluing them the side the body you of might be able to correct instability by causing the tube. rocket to spin on the Canted motor Keep way up. This is called motor causes rotation spin stabilization. The straight spinning Body Tube Shape: You can vary not only action creates angular momentum , which the diameter but the shape of the body tube provides inherent stability (think of a as well. Why nottoy change the top shapeorfrom a spinning a gyroscope). The spinning also cancels but there is a photo out any unbalanced on page (18). forces acting on a rocket, Fig 1-19 The drawback to spinning such as that is the amount produced by one fin of drag it produces. This creating more lift that increase in drag the other fins. will reduce the overall height that the model Spinning the rocket rocket is capable of can be accomplished achieving. in three ways. You Rockets with small can cant the fins on fins have a harder the body tube at some time producing the small angle; you can necessary forces to add small canted tabs toA model s appearance restore the model to the correct the bottom of existingis flight path. fins; or you can sand easily anged modifying To help increase these each of the by forces you can fins so they have a cambered airfoil increase the velocity. fin shape geometry shape.and More air flowing over Another way of inducing a spin to the fins will produce the rocket is to use larger restoring a 15 You can spin tower-launcher. increase speed by reducing forces. These are very rare, drag or Fig 2-13 selecting an engine that produces more Fig 2-14 Fig 2-15 Spin tabs Canted fins Cambered airfoil on each fin 22 How did you do? If you couldn’t answer them, you’ll be happy to know the answers are in the book Model Rocket Design & Construction. It was written for modelers that want to build their own designs. For more information, or to order your own copy, see our web site at: www.ApogeeRockets.com/design_book.asp A P O G E E Page 4 Call Us at: 719-535-9335 R O C K E T S