Reference / Paper · 2016
Peak of Flight #411: Trapezoidal Fin Flutter Analysis Revisited
Apogee Components' Peak of Flight newsletter, issue 411 (2016-02-23).
- peak-of-flight
- apogee-components
- model-rocketry
Reference / Paper · 2016
Apogee Components' Peak of Flight newsletter, issue 411 (2016-02-23).
Apogee Components' Peak of Flight newsletter, issue 411 (2016-02-23).
ISSUE 411 FEB 23rd 2016 IN THIS ISSUE Trapezoidal Fin Flutter Analysis Revisited www.ApogeeRockets.com/Rocket_Kits/Skill_Level_2_Kits/Diamondback_Rocket_Kit Trapezoidal Fin Flutter Analysis Revisited By John D Sahr 1 Introduction Recently I have begun co-designing a two stage rocket that will exceed Mach 1 for the “high altitude” version and will approach Mach 2 for the “high speed” version. At such speeds it is important to have a good idea of fin strength requirements to defend against fin flutter. To this end I was grateful to find an article by Zachary Howard on the Apogee website: https://www.apogeerockets.com/education/ downloads/Newsletter291.pdf Howard’s article provides a very useful, semi-empirical model of fin flutter derived from more elaborate developments¹. The key result is estimation of the “flutter velocity” Vƒ for a particular fin design. If a fin is flown faster than its flutter velocity, is is likely to fail, leading to subsequent unstable flight. In working my way through Howard’s article, I found a few things to be a little awkward. In particular the article employs imperial units rather than SI units. Also, the formula for the flutter velocity Vƒ can be rearranged in a way to simplify its application and illuminate important dependences that will aid in fin design. The next section of the note explains my reasoning and method for manipulation of Howard’s model. You may wish to skip immediately to section 3 to see the new, simpler (and hopefully more useful) formula that I present. 2 Recasting the flutter velocity equation Howard’s version of the Flutter Velocity equation can be rearranged to have the following completely equivalent form: ¹c.f. http://www.ewp.rpi.edu/hartford/users/papers/engr/ernesto/kapust/ EP/Other/ References%20-%20Rocket/Jon%20Champion%20docs%20 from%20JMRC/Example of Fin Flutter.pdf About this Newsletter You can subscribe to receive this e-zine FREE at the Apogee Components website www.ApogeeComponents.com, or by clicking the link here Newsletter Sign-Up Page 2 In these expressions, L corresponds to a length unit, e.g. inches or mm; R corresponds to pressure units, psi or Pa, and V corresponds to velocity units, mph, fps, m/s. You can choose the units of the answer by selecting the sound speed in whatever units you prefer; you can use any length units for L as long as you always use the same length units; you can use any pressure units R as long as you use the same units for both P and G. The fin dimensions are illustrated in Figure 1 on Page 3. I’ll rewrite the formula once more and add some annotation: Here P0 indicates the (constant) sea level pressure expressed in R pressure units. The underbrace a indicates that the term depends only upon properties of atmosphere; ƒ indicates that the term depends only upon fin geometry and material. While it might be tempting to cancel the Continued on page 3 Newsletter Staff Writer: John D Sahr Layout/CoverArtist: Chris Duran Proofreader: Michelle Mason Issue 411 | February 23rd, 2016 Trapezoidal Fin FinFlutter FlutterAnalysis AnalysisRevisited Revisited Continued from page 2 P0 factor, the formula, as written, is almost completely free of dependence upon the system of units chosen by the analyst. Here H is the atmospheric scale height at sea level, which is about 8077 m or 26500 ft (in cold weather). Also, CS0 is the sound speed at sea level, and P0 is the pressure at sea level. Notice that for rockets which don’t exceed the altitude H, the two expressions are nearly constant. Furthermore, using the following theorem from calculus, we can rewrite the sounds speed and pressure formulas, Figure 1: Definition of dimensions of a trapezoidal fin. The remaining dimension is the fin thickness. The simplicity of the pressure function is no ac2.1 The atmospheric model for CS and P cident, as this represents the exact solution for pressure versus altitude of an isothermal atmoReferring to Howard’s article, and combining his expressions for temperature and pressure and sphere². sound speed vs height, we can re-arrange a bit to ²https://en.wikipedia.org/wiki/Scale height come up with two new formulas for pressure and sound speed versus altitude alone: Minimum Diameter Motor Retainers! Apogee is your one stop shop for your minimum diameter rockets projects! We Have: • Minimum Diameter Retainers • Motor Extenders • Fly High • Threaded Forward Closures • Fly Fast • Adapters for • Impress Your Cesaroni Cases www.apogeerockets.com/RockSim/RockSim_Information Join Tripoli.org Mention Apogee Components Friends! www.ApogeeRockets.com/Building_Supplies/Motor_Retainers_Hooks Page 3 Continued on page 4 Issue 411 | February 23rd, 2016 Trapezoidal Fin Flutter Analysis Revisited Continued from page 3 1. It handles all systems of units. As long as all lengths and pressures use the same units, the formula will yield the correct answer. You can choose the units of the answer by putting in the appropriate value for the sea level sound speed C0S. 2. It shows a gradual increase in the fin flutter velocity with altitude, because the pressure drops more quickly than the sound speed. 3. It shows fairly weak dependence on fin aspect ratio B and taper ratio λ — that term will always have a value near √2 for reasonably shaped rocket fins. Figure 2: (Left) a plot of Zach Howard’s fin flutter velocity (solid), with an overlay of his formula refactored eq. 1, and the new expression eq. 20. (Right) a plot of the difference between the new formula (eq. 20) and Zach Howard’s fin flutter velocity formula. The difference is less than 1 mph below 10000 ft. 3 A simpler, more general formula Returning to the fin flutter formula we can put it all together: Howard tests his model with a particular fin with root chord 9.75”, tip chord 3.75”, height 4.75”, and thickness 0.125”, and with a shear modulus of 380000 psi. I was able to duplicate his results quite closely especially when using a scale height H = 26500 ft. A comparison is shown in Figure 2. This formula, which is nearly identical to Howard’s, has the following nice properties: 4. It shows very strong dependence on the fin thickness T. 5. It should be straightforward to incorporate this formula into rocket modeling software such as RockSim³ and OpenRocket4, to cause warnings when fin flutter velocity is exceeded; it is straightforward to add such a check in spreadsheet analysis of RockSim and OpenRocket numerical output. The limitations of this formula are the same as of Howard’s presentation: 1. The formula is only valid for trapezoidal fins, although one would expect the formula to work for “nearly trapezoidal” fins. Continued on page 5 Experienced HPR Builders Use Thrust Plates • Eliminates Shear Forces on Centering Rings • Mates with AeroPacks Flanged Engine Retainers • Fits Standard HPR Tubes, Blue Tubes, and Fiberglass Tubes • Made from Aircraft Grade Aluminum https://www.apogeerockets.com/Building_Supplies/Thrust_Plates Page 4 Issue 411 | February 23rd, 2016 Trapezoidal Fin Flutter Analysis Revisited Continued from page 4 2. The formula is only valid in the troposphere, for h < H. 3. Even for trapezoidal fins the formula is not exact, but rather an attempt to capture the main physics without doing a full blown numerical model of the fin. One should not be surprised if a rocket travelling at 0.9Vƒ loses its fins due to flutter — and of course fins can fail other ways. You get: (4) AT 29/13 (4) AT 41/18 (2) AT 56/18 (2) AT 66/18 (1) AC-56 (1) AC-66 John D. Sahr is a professor of Electrical Engineering at the University of Washington in Seattle. He dedicates this note to the memory of Professor R. D. Middlebrook of CalTech. 3 https://www.apogeerockets.com/Rocket Software/RockSim 4 http://openrocket.sourceforge.net You get: (6) AT 13/18 (6) AT 18/18 (6) AT 24/18 (6) AT 33/18 www.ApogeeRockets.com/Building_Supplies/Body_Tubes https://www.apogeerockets.com/TARC_Supplies www.youtube.com/apogeerockets Page 5 Issue 411 | February 23rd, 2016