Richard Nakka’s Experimental Rocketry Web Site

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Introduction to Rocket Design

6. Fins

Introduction

Rocket fins had long been something of a puzzle to me, a puzzle with a few missing pieces. For example, why is it that just about every rocket I have seen in my (rather huge) collection of books, journals and photos have fins of such different shapes and sizes? (see Figure 1) One might suppose that there is, as engineers like to think, a single ideal shape and size that is optimum. Certainly, it was always clear to me that fins work like feathers on an arrow, and without fins, a rocket (or similar free-flying projectile) would simply tumble end-over-end when catapulted through the air. As I learned about the science of fins, specifically their role in stabilizing a rocket, many of the missing pieces of the puzzle were filled in. As I built and flew rockets over the years, more pieces of the puzzle were filled in, as I learned the reason why some shapes are more suitable than other shapes. In this webpage, I am endeavouring to present my take on rocket fins, based on both my knowledge of the science of fin makeup and behaviour and based on my experience with fitting my rockets with fins. In other words, this webpage endeavours to present the engineering of EX (Experimental) rocketry fin design.

Figure 1: Several sounding rockets of the 50’s and 60’s
Ref. “Small Sounding Rockets, A Historical Review of Meteorological Systems”, Richard B.Morrow

A starting point in the discussion of fins is to define the basic terms associated with describing the profile of a fin, which is shown in Figure 2.

Figure 2: Fin nomenclature

A set of fins (typically 3 or 4, but may be more) is known as a finset. A set of fins manufactured as a single unit, mounted on a cylindrical sleeve that slides over the rocket body, is known as a fincan.

Fins and Rocket Stability

The stability of a fin-stabilized rocket is dependent upon the fin normal force generated when a rocket tends to deviate from a straight flight path, in other words, flying at a non-zero angle-of-attack. This force, acting normal or perpendicular to the fin plane, imparts a rotational (corrective) moment to the rocket, about its Centre of Gravity (C.G.), which tends to counteract this deviation and to restore the desired straight flight path. At a small angle-of-attack, not only the fins, but other components of the rocket, such as nosecone, conical shoulders and boattail also generate a normal force. In the case of the nosecone and boattail, the forces generated by these components cause a destabilizing rotational moment. This is illustrated in Figure 2. The resultant of all of these normal forces acts at a point on the rocket body deemed the Centre of Pressure (C.P.). This resultant force, which diminishes to zero as the angle-of-attack is corrected to zero, generates a stabilizing moment about the C.G. It is clear from this discussion that the fins must generate enough force to overcome the destabilizing forces to the extent that the C.P. lies aft of the C.G. by a certain distance. This distance (measured in body diameters, or calibres, for convenience) is deemed the Stability Margin (SM). If the fins are too small, and as such generate a relatively small fin formal force, the resultant of the forces will lie ahead of the C.G. It is obvious, looking at the figure on the right, that the rocket will become unstable and tend to flip over.

Figure 3: Concept of normal forces due to angle-of-attack and C.P.

Stability Margin (SM) is defined as:

Fins should be one of the last components of the rocket to be designed, or more specifically, sized. This is because the fins are sized to provide the rocket with the desired Stability Margin. As explained earlier, Stability Margin must be a positive value (as determined by Eqn.1). A rocket with a Stability Margin that is negative will be unstable, and will either tumble after leaving the launch rail, or will fly off in a random direction. Most rockets, including EX rockets, tend to have a Stability Margin in the range of 1.0 to 3.0). A rocket with a Stability Margin that is too high (e.g. >3) may tend to veer severely into the wind (weathercock). A small amount of weathercocking can actually be desirable, as this can reduce downwind range of the rocket, at the cost of slightly reduced peak altitude. I have used this approach with my Xi rockets. Due to the nature of the methods used to calculate C.P. (see Appendix C), as well as construction details of a rocket, it is advisable to have a Stability Margin no less than 1.5. Based on my own experience, a good target value lies in the range of 1.5 to 2.5. Leaning toward a larger stability margin (within this range) has an additional benefit. The frequency at which the rocket oscillates (in pitch) when disturbed increases with an increase in static margin. This is because the corrective moment coefficient is greater. The net effect is that the rocket more quickly returns to it straight and true flight path with a minimum of oscillation. This topic is covered in greater detail in the Dynamic Stability webpage.

The logic behind sizing the fins last of all is that the exact location of the Centre of Gravity (C.G.) of the fully-assembled rocket must be known (usually with a loaded motor). The C.G. depends on the mass distribution of all the components of the rocket. Once the C.G. location (x_cg) is known , the desired location of the C.P. (x_cp) is obtained from Equation 1, based on the desired Stability Margin. The fins are then sized to attain such a Stability Margin. Note that the location of the rocket's C.G. will change during its flight. As propellant is consumed, the rocket's mass will decrease. This will result in a forward shift in the rocket C.G. which has a positive effect on the Stability Margin. After burnout, the mass of the rocket is constant and the stability margin will likewise remain constant for the remainder of the flight to apogee. As such, the minimum Stability Margin exists at liftoff. This is the design condition.

Centre of Pressure is dependent solely on the geometry of the rocket. Figure 4 shows an example of a 3-finned rocket designed using AeroLab. The distances from the nose tip to the C.P. and C.G. are shown. For this rocket, the Stability Margin is
SM = (1332.8 - 1225)/74.8 = 1.44

Figure 4: Example of rocket showing dimension that relate to stability

In order to determine the static Stability Margin for a proposed design, it is necessary to know the location of the C.P. and the location of the C.G. The C.P. is readily determined using software such as AeroLab, RASAero or BARROWMAN.XLS. Or, if one really wants to learn by doing, perform the calculations by hand using the Barrowman Method (see Appendix C). The "fly in the ointment", when it comes to designing a rocket fins, is knowing or predicting the location of the C.G. The location of the C.G. is usually determined by measurement of the balance point of the completed rocket (including propellant, or a dummy mass representing the actual propellant) laid horizontally. Unfortunately, this is obviously not an option during the design phase when the rocket exists solely on paper (or pixels). Although it is certainly possible to calculate an estimation of the C.G. by knowing the masses and locations of the individual rocket parts (such as nosecone, fins, motor, parachute, payload, etc.) this can be a bit thorny (Appendix D provides an example of how this is done). A state-of-the-art solution is to design your rocket, including all components, using 3D CAD software. Individual components can be modeled with great accuracy. Assigned suitable density values, the masses of each part and the rocket as a whole (and the C.G. location) can be readily obtained. This is the approach I have taken with my newest rocket, which is now in the final stages of construction.

An alternate, perhaps more pragmatic, approach is to simply assume a C.G. location and go from there. This may seem odd, but in fact, making such assumptions is a standard engineering approach to dealing with such a quandary. For a typical EX rocket, a useful assumption is that C.G. lies in the range of 65 to 70% of the rocket length. Armed with a tentative location for the C.G., the fins may be sized to provide for a Stability Margin of SM = 1.5. This will most likely result is slightly oversized fins once the actual location of the C.G. of the finished and fully-loaded rocket is measured.

As the resulting SM may be greater than desired, the fins should be designed to be readily clipped to bring the Stability Margin to 1.5 (or whatever value is desired). Note that the effectiveness of a fin set is more greatly influenced by its span than by its chord. As such, clipping a fin to reduce its span is a good way to "move" the C.P. forward as required to reduce excessive SM.

An alternative means to move the C.P. forward is to simply mount the fins slightly forward. Figure 5 shows both of these techniques.

Figure 5: Moving Centre of Pressure forward

Another important thing to note is that the outcome of the standard Barrowman equations is a constant value for C.P. This is because one of the Barrowman assumptions is subsonic flow. In fact, C.P. varies with Mach number. As Mach number increases, C.P. may move forward, thereby reducing the stability margin. This is important to bear in mind when designing a supersonic rocket. Barrowman does provide a means for calculating C.P. for supersonic flow (see Res. 22). MIL-HDBK-762 (Res. 23) contains graphs that plot C.P. of rectangular and swept fins at supersonic velocities. And of course, software such as AeroLab and RASAero give C.P. for supersonic rockets.

Fin Planform Profile

What is the best shape for your fins? Fins can be just about any shape as long as they achieve the goal of moving the rocket C.P. aft by a suitable amount to give the SM required while keeping fin drag to a minimum. There are also other considerations, such as structural, and practical concerns. Figure 6 illustrates several popular fins shapes and my own assessment of each. The swept designs are popular for model rocketry, however, are less suitable for EX rockets. Model rockets are, by necessity, lightweight. As such, swept fins generally handle the impact of ground contact at touchdown without overdue concern. EX rockets are typically a lot heavier than model rockets and the likelihood of swept fins being damaged (bent or broken) upon landing is great. This I know from personal experience. There are ways to mitigate this issue if swept design is preferred, such as mounting the fins a short distance forward such that the aft end of the rocket body contacts the ground first. A minor drawback to this approach is that the fin effectiveness is slightly reduced, necessitating a correspondingly larger fin to achieve the same Stability Margin.

The tapered swept shapes are more susceptible to flutter due to less torsional stiffness than the other fin shapes.

Figure 6: Fins shapes and assessment

One study concludes that rectangular fins are most efficient (produce most lift with minimum drag) for subsonic rockets, particularly fins with a lower aspect ratio. For transonic rockets, elliptical fins may be the optimum shape, as the rounded profile at the fin tip region minimizes the formation of shockwaves and consequently minimizes drag. As well, tip vortex shedding, another form of drag, is reduced. For supersonic rockets, Clipped Delta and Taper Swept profiles generate less drag than the other shapes. These profiles reduce the strength of shockwaves by gradually redirecting airflow around the fins.

Fin Restoring Force

The finset on a rocket serves to provide stability by generating a restoring force, that is, a force normal (perpendicular) to the fin surface when a fin is deflected relative to the airflow. This occurs when the rocket is flying at a non-zero angle-of-attack, denoted by the symbol alpha (a). This angle-of-attack may result from a flight disturbance such as wind gust. The restoring force is a function of angle-of-attack, such that when a=0, no restoring force is created (nor needed). In equation form, the finset restoring force (or normal force) is given by

where:
N_F = normal force, lbf or Newtons

q = dynamic pressure = ½ ρ V₂, lbf/in² or N/m²
A = reference area = ¼ π D² where D = nosecone base diameter, in. or m.
V = vehicle velocity, ft/sec or m/s
(C_N α)= slope (or gradient) of the normal force coefficient at α = 0 (∂C_N/∂α | α=0) , per radian
α = effective angle-of-attack, radians

The slope of the normal force coefficient at α = 0, derived by Barrowman, is given by

Where the various terms are defined below:

N = number of fins in finset (3, 4 or 6). Theta (θ) and l are the mid-chord sweep angle and mid-chord length, respectively. Interference coefficient f=1 for 3 or 4 fins, f=0.5 for 6 fins. And d is the reference diameter, in this case, the nosecone base diameter.

The above equations are valid for rocket subsonic velocities (approaching Mach one). MIL-HDBK-762 (Res. 23) contains graphs that plot the normal force coefficient gradient for rectangular and swept fins at supersonic velocities.

Span and Chord

Regarding fin design, there is one notable thing to bear in mind. As alluded to earlier, fin effectiveness, for a given shape, is more influenced by span width than by chord length. In other words, fin effectiveness (in shifting C.P. aftward) is not solely a function of fin surface area. The greater effectiveness of span width, compared to chord length, is a consequence of the dominant S/d squared term in the Barrowman equation for fin force coefficient (C_N)_F slope. Along those same line, a particular fin shape influences its effectiveness in generating fin (restoring) force. Let’s compare two fins of the same basic profile and same wetted area (and therefore same mass). Consider a baseline rectangular fin of dimensions 100mm × 100mm. We wish to increase the stability margin by enlargening the fins. We can make the fin wider by increasing the span by 10mm or make the fin taller by increasing the chord by 10mm. This example is shown in Figure 7, where the hatched area represents the baseline fin.

.

Figure 7: Enlargening fins to increase stability margin (example)

In this example, d = 75mm, R = 37.5mm and N = 4. Using Eqn.2 to calculate (C_N)_F we obtain the following:

Baseline fin (100 × 100) (C_N)_F = 15.0 (per radian)

Taller fin (100 × 110) (C_N)_F = 15.4 (per radian)

Wider fin (110 × 100) (C_N)_F = 17.4 (per radian)

The taller fin increases the normal force by 2.7%. Increasing fin span results in a 16% increase in finset normal force.

Fin Drag

Fin skin friction and fin pressure drag are the primary drag forces generated by a finset. For a LoPER class EX rocket that flies exclusively in the subsonic range, fin pressure drag force is generally not significant compared to the overall drag force acting on the rocket, as shown in Figure 1 of the Aerodynamics and its Role in Experimental Rocket Design web page. Skin friction drag accounts for 20-25% of the total drag.

As was discussed in the Aerodynamics and its Role in Experimental Rocket Design web page, interreference drag is developed at the interface where a fin meets the rocket body. Interference drag can be reduced significantly by adding a rounded fairing at the junction of the fins and body.

Fins can generate fin-base interference drag due to disturbance of the airflow caused by the fins. The presence of fins on a rocket affects the external flow characteristics and results in increased base drag (see Res.23). The pertinent parameters are fin thickness, fin longitudinal position, as well as number of fins and Mach number. This effect is more significant at lower Mach numbers (< M=1). As such, for LoPER and MiPER class rockets, it may be beneficial to locate the finset a short distance forward of the rocket body aft end. The addition of a boattail may also alleviate the increase in base drag due to fins.

Fins can also generate additional drag due to fin tip vortices that are generated at a non-zero angle-of-attack. Energy is lost generating such vortices. This form of drag can be reduced by making the cross-section of fin tip tapered to a wedge-shape.

Fin Cross-section Profile

As mentioned earlier, fin pressure drag is not significant for LoPER class rockets. As such, the fin cross-section profile is not that important. However, as it is simple enough to do, I round off the leading edge and put a taper on the trailing edge of my sheet metal or plywood fins, such as that shown in Figure 8 (where LE=Leading edge; TE=Trailing Edge).

Figure 8: Profiling of a simple subsonic fin

If larger sized or thicker fins are called for, an airfoil profile may be beneficial (see NACA Report No.824, Res.43). A good choice is the NACA 0005 symmetric profile (Figure 8). This is the airfoil shape I used for my Frostfire 3 rocket. It is important to note that an airfoil profile should be avoided for a rocket that will achieve transonic velocity. As explained in my Frostfire 3 report, dynamic instability, in particular, Transonic Shock Oscillations could occur due to a rocket being fitted with airfoiled finset. Worst case results could be structural failure as happened to Frostfire 3 as its velocity approached Mach one.

Figure 8: Fin cross-sectional profile used for Frostfire 3

For an EX rocket designed to fly in the supersonic range (MiPER or HiPER class) or hypersonic range (ViPER class), the most commonly used cross-sectional profiles are Double-Wedge (a.ka. Diamond) or Modified Double-Wedge (a.k.a. Hexagonal) shaped. Two other options are Biconvex and Blunt Trailing Edge (see Res. 42). The latter offers reduced fin drag of equal section modulus combined with better stability characteristics due to the tendency for flow separation at the aft end to be minimized. These shapes are shown in Figure 9.

Figure 9: Fin cross-section profiles suitable for supersonic rockets

Figure 76 and Figure 77 of NSWC TR 81-156 (Res. 42) present the supersonic wave drag of a Double-Wedge airfoil and Biconvex airfoil finset, respectively. Using the data from these two graphs, a comparison of the wave drag for Double-Wedge and Biconvex was generated, presented in Figure 10 for fins of the same thickness-to-chord ratio (10%) and Aspect Ratio (AR), and with 45 degree sweep.

Figure 10: Wave drag for Double-Wedge and Biconvex finset

The above plot suggests that Double-Wedge is a better choice than Biconvex. A more fair comparison would consider the two shapes being of equivalent section modulus, as the section modulus of the fin shape determines its bending strength and stiffness.

Flutter

When designing a fin particular care should be taken to ensure the fin is resistant to catastrophic fin flutter. Flutter, as applicable to rocket fins, is a rapid oscillation of a fin involving combined elastic bending and torsion which extracts energy from the air stream. Flutter results from insufficient stiffness, with the driving force (excitation) being high speed air flow over the fin surface. When this excitation matches the natural resonance frequency of the fin, flutter will occur. Worst case, flutter can be catastrophic and can cause fins to snap off. Interestingly, flutter can be heard as a distinctive buzzing sound. I have heard this eerie sound myself. The phenomenon of catastrophic flutter is rarely an issue with metallic fins, due to the inherent stiffness and strength of metal. However, even if a fin is strong enough to remain intact, flutter displacement could increase fin drag appreciably. Materials such as plastic and wood are particularly susceptible to flutter due to their relatively low shear modulus. .

One of my early rockets experienced partial loss of all four fins due to flutter. These particular fins were fabricated of thin plexiglass. I was fortunate at the time as the fins did not break off at the root, rather, they fractured, diagonally, at mid-span and as such provided enough stability to maintain straight flight. If the fins had broken off at the root, the rocket would have become catastrophically unstable. My rocketry buddy Rob also experienced fin flutter with his early rockets. His fins were cut from old phonograph records bonded to the rocket body. When launched, the rockets made a shrill whining sound, and more often than not, were recovered sans fins.

In NACA Technical Note 4197 (Res. 29) the author provides a simplified formula for estimating fin flutter velocity based upon an empirical expression developed by the authors of NACA Report No.685 (Res.30). Flutter velocity (V_f) is given by the following formula:

where:

a = speed of sound at the altitude where rocket reaches maximum velocity.

G_E = effective shear modulus of fin section. For a solid (isotropic) fin, this is simply G, the shear modulus of the fin material. Note that shear modulus for isotropic materials is related to elastic modulus (E) and upon poisson ratio (ν) by the formula E = 2G(1+ν). Units of G_E are force per unit area (example: psi or kPa).

A = aspect ratio of fin = (semi-span)²/fin area

P_o = is air pressure at sea level in psi (14.696), or

kPa (101.325)

P/P_o = is the ratio [air pressure at the altitude where the

speed of sound was determined / air pressure at sea

level] (dimensionless)

t/c_r = ratio of fin thickness (assumed constant) to root chord length

λ (“lambda”) = fin taper ratio = c_t /c_r

Y is a constant whose value is dependent upon which units of measure are chosen for pressure (P_o):

ԑ (“epsilon”) is the distance of the fin center of mass behind fin quarter-chord, expressed as a dimensionless fraction of the full chord. For a symmetric fin, ԑ is 0.25. For non-symmetric fins, Res.28 provides details on how to calculate the value of epsilon.

k = ratio of specific heats for the fluid the rocket is traversing. For air, k = 1.4 (dimensionless)

It is imporant to note that units of Y are force per unit area as a result of the P_o term. Consquently, the units for P_o and G must be the same.

In the Resource 27 article, John K. Bennett provides an must-read technical discussion of rocket fin flutter and how to calculate flutter velocity for model and EX rockets based on the above formula, including a worked example. Resource 28 article, by the same author, delves into the details of flutter calculations for fins of complex shapes.

It must be noted that the formula for fin flutter velocity developed in NACA 4197 is based on certain assumptions, estimates of parameters, and data-fitting. Consequently, the calculated value of fin flutter velocity should be taken as an estimate and due diligence is called for by applying a healthy safety factor.

The software app AeroFinSim is specifically tailed for fin flutter analysis. This app is based upon the method of NACA 4197 (see Res.31)

Fin Materials and Construction

Fins for model rockets are made from either balsa wood or injection-moulded plastic. Balsa is ideal for model rockets due to its being very lightweight combined with decent strength. As mentioned earlier, model rockets need to be very lightweight due to the limited power of model rocket motors. This is less true for EX rockets, whereby motors can be tailored to deliver needed performance. A more important factor for EX fins is robustness and aerodynamic considerations, especially for transonic or supersonic rockets. Being mounted at the extreme aft end of a rocket, fin mass can have an appreciable influence on the rocket C.G. Heavy fins should be avoided, not only to keep the rocket weight to a minimum, but to minimize the aftward (destabilizing) influence of the fin’s mass on the rocket C.G.

Suitable material for EX fins are:

· Plywood (such as 1/8” birch plywood)

· Resin-glass laminate sheet (such as G10/FR4 or Garolite)

· Aluminum alloy sheet

· Composite-sandwich (such as balsa or foam core with glass or carbon skins)

· 3D printed plastic (such as PLA or fibre-reinforced filament)

Each of these materials is discussed below.

Plywood, in particular birch or other hardwood plywood, has reasonable strength and stiffness (historically, aircraft-grade birch plywood was used for fabricating early airplanes), and is a good option for LoPER class rockets. Due to its relatively low stiffness, plywood fins may be susceptible to flutter. As such, fins profiles with good torsional rigidity should be chosen for plywood fins, such as rectangular, swept or trapezoidal. Birch plywood can be purchased at better hobby shops, as it is used in the making of model airplanes. Note that plywood has different bending strength and stiffness properties in the lengthwise and crosswise directions. When making fins from plywood, the spanwise direction of the fin should correspond to the direction with higher strength (typically lengthwise).

Resin-glass laminate (Garolite) has considerable strength and stiffness and is suitable for LoPER and MiPER class rockets. With careful design and choice of material type, this material can also be suitable for ViPER class rockets. Garolite comes in various grades such as G10 (glass/epoxy) and G11 (glass/phenolic or hi-temperature epoxy). G11 is the strongest grade of Garolite and has the additional advantage of being heat-resistant. The strength of Garolite is similar to that of 6061 aluminum alloy and has the advantage of having 1/3 less weight. The drawback, compared to aluminum, is that the stiffness is about 1/3 that of aluminum. As such, fins made of Garolite may be more susceptible to flutter. This should be borne in mind when selecting fin profile shape. Note that Garolite has different bending strength and stiffness properties in the lengthwise and crosswise directions. When making fins from Garolite, the spanwise direction of the fin should correspond to the direction with higher strength.

For HiPER and ViPER class rockets, carbon (graphite) laminate is the best choice as fin composite material.

Composite-sandwich is a high-tech solution to fin design and produces fins of outstanding strength and high stiffness combined with exceptional lightweight, truly the Cadillac of fins. My Frostfire3 rocket featured composite-sandwich fins with hybrid Kevlar-carbon skins and syntactic foam core, with an integral foot, with the foot being bonded to the rocket body with structural epoxy. These fins were skillfully made by fellow rocketeer Roman Lev. The fins for MiniSShot and DoubleSShot (DSS) rockets, of the Sugar Shot to Space (SS2S) program, were similarly of composite-sandwich construction, consisting of balsa core and carbon-fibre skins, as shown in Figure 11. The DSS fins had an integral aluminum foot that allowed the fin to be epoxy-bonded to the rocket body tube.

Figure 11: DSS composite-sandwich fins. Top: balsa core, aluminum foot
Bottom: Completed fins

These fins were skillfully made by SS2S member Marco Torriani. A set of DSS fins were proof load tested followed by ultimate load test to failure. The fins were supported solely at the tips. Weights were applied to the airframe to the required proof load, then increased to fin failure. The mounted fins exhibited exceptional strength and stiffness as these photos patently show:
Photo_A
Photo_B
Photo C

Corecell is an excellent structural foam core material. I have used this material for fins and other composite sandwich structures. Corecell comes in various densities and shear strengths. Note that, for sandwich panel in bending, the skins react bending load and the core reacts shear load.

Aluminum alloy sheet is the material that I have used for most of my fins. Aluminum is available in alloys of various strengths and are available in a wide range of thicknesses. Aluminum fins are suitable for all classes of rockets including ViPER and HiPER class. Historically, aluminum fins have been used for nearly all professional sounding rockets. Aluminum has particularly high strength (dependent upon the alloy) combined with relatively high stiffness. Aluminum is the ‘heaviest’ of the various fin material options. This necessitates selecting the thinnest sheet that will meet the strength and stiffness required. If used for hypersonic rockets where aerodynamic heating is a factor, the leading edge of aluminum fins can be fitted with a shroud, or cap, fabricated of steel, stainless steel or titanium. Note that aluminum has different bending strength in the grainwise and cross-grain directions. When fabricating fins from aluminum sheet, the spanwise direction of the fin should correspond to the direction with higher strength, which is in the grainwise direction (indeed, aluminum sheet has a grain direction, clearly visible using a magnifier as fine lines or striations).

3D printing has definite potential for EX fins, particularly LoPER class. I have not, to date, utilized 3D printing for fins, but have produced many other rocket structural components such as nosecones, couplers and deployment pistons and have found that 3D printed parts, with good design, are remarkably well-suited for such applications. The material that I have used exclusively is PLA plastic. Interestingly, PLA (Polylactic acid) plastic can be heat-treated to further increase its strength and robustness. This would be a good option to consider for an application such as fins. In addition to appreciable strength, PLA has a relatively high stiffness (elastic modulus), at least in comparison to other plastics. Same is true for hardness, PLA is not easily gouged or scratched. 3D printed fins can be made as a quasi-sandwich construction, using honeycomb infill and solid skins. Parts made by filament 3D printing have different strength properties in the filament and cross-filament directions. As such, fins should be printed with the filament extruded in the fin spanwise direction. 3D printed parts have a rather unsmooth surface. Using progressively finer grits of sandpaper, the ‘as-printed’ rough surface of 3D printed PLA parts, after painting, can be mirror-smooth.

Figure 12 presents a table of tensile and bending strength for the various materials suitable for fins. The values in this table are typical values and present values for tensile/bending in the stronger direction.

Figure 12: Fin materials properties (click for metric version of table)

Mounting Fins

Over the many years of building rockets, I’ve tried several different techniques of mounting fins on a rocket body. For my very first amateur rocket (A-Rocket), which was an all-metal rocket, the fins were bent approximately 90 degrees near the root, and attached with sheet-metal screws, as shown here. This is a simple and effective means of securely attaching fins. Note that the bend angle should be slightly more than 90 degrees and depends upon the fin thickness and body diameter. Resource 34 (fin_rootbend.xls) can be used to readily determine this bend angle. Only softer aluminum alloys can be bent tightly, such as 1100 series. Harder alloys, such as 2000 or 7000 series aluminum will crack unless given a healthy bend radius. This table gives the recommended bend radius for various aluminum alloys. This figure illustrates allowable bends for various fin thickness and bend radii. For illustrative purpose, the bends shown are at ninety degrees.

Another technique for mounting fins, used for many of my early C-Series rockets, is to use a pair of aluminum angle brackets, one on each side of the fin. The concept is illustrated in this figure. This concept is similar to that used on many sounding rockets, except a single machined bracket is used instead of a pair of discrete angle brackets. For my rockets, the idea behind this concept is to allow for easy swapping of fins, in case a fin gets damaged, or to readily change the fin shape or size. For example, to tailor the stability margin, a larger or small fin may be installed. The C.G. of a rocket may be different for each flight, depending on the motor being used, or perhaps the payload is different. This allows the stability margin to be maintained (or adjusted) by selecting suitable-sized fins. The fins are attached to the pair of angles with small machine screw and nut combination (for example). The angles can be attached to the rocket body with blind rivets.

My Skydart rocket had aluminum fins mounted on a PVC body tube. The fins were fabricated with a pair of integral tabs that protrude along the base chord and that fit into slots cut into the body tube. These tabs were bonded securely to the rocket body using epoxy impregnated with chopped glass fibres, as shown in this photo. To provide lateral strength to the fins, I bonded on a pair of “quarter-rounds” made from an aluminum tube cut lengthwise into four. These quarter-rounds additionally served as fairings to reduce fin-body interference drag.

A similar approach was used for my Cirrus rocket and for my Arrow rocket, which is my newest rocket currently under development. The difference with the tabs on these fins is a slot cut into the tabs. The slot width is equal to the body tube thickness. This allows the fins to be retained by the slotted tabs, rather than solely by bonding. Made from one of the stronger aluminum alloys (6061-T6, 2024-T3 or 7075-T6), these slotted tabs have appreciable strength able to carry the inertia and drag loading with a good margin. Resource 35 (fin-tab-strength.xlsx) can be used to readily determine the fin tab strength. Two examples of tabbed fins are shown in Figure 7.

Figure 13: Cirrus fin (left) and Arrow fin (right)

Fins may also be attached to the rocket using a pair of dowel pins or spring pins. The method is well suited to composite fins. The pins are embedded (and bonded) into the core of the fin, of suitable depth, with the other end of the pins bonded to centering rings internal to the rocket body. These centering rings typically also serve to centre the rocket motor. This method of fin mounting was used for UMSATS Winnie 3.0 rocket. Spring pins, which are made of hardened alloy steel, have high shear strength (click for table).

As mentioned earlier, a fin made with an integral foot can be bonded directly to the rocket body using structural-grade epoxy. J-B Weld epoxy

adhesive works well for a LoPER class EX rocket. For high performance EX rockets, an aerospace grade epoxy adhesive such as Scotch-Weld 2216 or Scotch-Weld 1751 is better suited.

Although I have not (yet) tried this method, Tip-to-tip reinforcing of the fins is an effect means of strengthening and stiffening fins for HiPER and ViPER class rockets. This method is particularly suited to minimum-diameter rockets that cannot accommodate through-the-body mounting of fins. The fins are first tacked to the body, solely to position and align them. Using a suitable epoxy paste, fillets are formed at the fin-body interface. Sheets of either fibreglass or graphite cloth are then layed over the each pair of fins and the body between the fins, i.e. tip-to-tip, as detailed in this webpage. This creates a solid fincan. This Youtube video provides a good demonstration of how this is done.

Fin Structural Strength

Rocket fins and their mounting to the rocket body (or tube or other support structure, if fin can is utilized) must be capable of handling the following loads:

1) F_inertia : In-plane fin mass inertia load due to maximum acceleration of the rocket

2) F_drag : In-plane aerodynamic drag load at Max-Q or maximum angle-of-attack

3) F_alpha : Out-of-plane loading due to fin restoring force due to non-zero angle-of-attack

4) F_handling : Handling load, typically out-of-plane

Figure 14 illustrates the four loading conditions. The inertia and drag load resultants act at the fin C.G. and fin C.P. respectively in a aft direction parallel to the fin plane. The force resultant due to angle-of-attack acts at the fin C.P. in a direction normal (perpendicular) to the plane of the fin. The handling load can be applied anywhere. Conservatively it is usually assumed to act at the fin tip. It may be assumed that the inertia, drag and restoring force act simultaneously. The handling load is a “ground” loading condition that acts alone and is the result of any condition that results in loading other than flight conditions. Such as transporting, bumping or landing impact.

Figure 14: Fin loading

Location of an individual fin C.G. (for a flat plate fin of uniform thickness) may be readily obtained using Resource 36 ( fin-C.G.xls).

Location of a finset C.P. is calculated as shown in Appendix B. However, location of the C.P. of an individual fin is of interest here, as this is the location of the force resultant. For subsonic flow, the C.P. of a fin is located at the intersection of the ¼ Chord line relative to the fin leading edge (X̅ ) and the mean aerodynamic chord (Y̅ ) (see Barrowman, Res. 20). This is illustrated in Figure 15.

Figure 15: Location of a fin C.P.

The location of the fin C.P. is found using the following equations:

For fins in supersonic flow environment, the chordwise location of the C.P. can be found using Figures 36 and 37 of MIL-HDBK-762 (Res.23). The spanwise location of the C.P. (X̅ ) can be considered to be the same as for subsonic flow conditions.

The maximum bending moment (at the fin root) for each of the four cases results from the loading shown in the figures below:

Figure 16: Cases 1-4 fin loading details

The maximum (fin root) bending moment (M) for the four cases are:

1) M = F_Inertia × Ycg

2) M = F_drag × Y̅

3) M = F_Alpha × Y̅

4) M = F_Handling × YH

The first two cases are considered to be critical solely for the fin attachment, and the third and fourth cases are critical for fin bending. The fin and its attachments are also subject to shear loading, however, this loading is not critical for most EX rockets and may be neglected.

The method of assessing the attachments depends on the specifics of the fin attachment methods (e.g. fasteners, adhesive). For the case of three attachments screws, as an example:

Figure 17: Fin attachment reactions

The applied moment (M) is reacted as a force-couple (R) by the top fastener in tension (T) and the lowest part of the fin in bearing against the rocket body. It may be conservatively assumed that solely the top fastener reacts the applied moment (if additional fasteners are to be considered effective in reacting the moment, a linear distribution of reactions over the remaining fasteners may be assumed).

Fastener tension is given by:

T = R = M /s where s the distance indicated

For Cases 3 and 4, the root bending moment generates bending stress (f_b) at the fin root, given by:

where Z is the section modulus of the fin
cross-section at the fin root

Section modulus is dependent upon geometric shape of the cross-section. Figure 18 presents the formulas for section modulus for four of the more common fin cross-sections. When calculating bending stress, it is important to use consistent units of measure. For example, to obtain stress in psi (pounds per square inch, or lb_f /in²), units of moment, M, are lbf-in and section modulus is in³. To obtain stress in MPa or N/mm², units of M are N-mm and section modulus is mm³.

Figure 18: Section modulus for various fin cross-sections

Bending stress is then compared to the material strength and the resulting Safety Factor determined. Safety Factor should be 2 or higher.

Based on yield, or permanent deformation

Based on ultimate strength, or breakage

Fins to Induce Roll

In some cases it may be desirable to impart a roll motion (spin) to the rocket along its longitudinal axis in order to enhance stability or to minimize trajectory dispersion. In the first case, inducing roll provides for some degree of gyroscopic stabilization. This may be important for upper stages of high-flying rockets where the thin atmosphere provides marginal fin restoring force. Or for rockets that fly at a relatively leisurely pace such as a rocket powered by an end-burner motor. In the second case, a rolling rocket is less subject to dispersion by horizontal winds or fin misalignment than a non-rolling rocket. Dispersion is the horizontal displacement of the rocket at its apogee, compared to a perfectly straight-line ascent. A rolling rocket distributes any disturbing forces in a radially symmetric fashion about the intended axis of flight.

Two potential drawbacks of imparting roll to an EX rocket need to be considered. The potential for tangling of the parachute recovery system lines, and the potential for dynamic instability due to pitch-roll coupling.

The three most common methods of inducing roll are:

1) Canted fins

2) Fins with spinneron (or tabs)

3) Asymmetrically airfoiled fins

These are illustrated in Figure 19. Canted fins are symmetrically profiled fins mounted at a slight angle relative to the rocket longitudinal axis. Spinnerons, or tabs, are features of the trailing edge of the fins that are bent at a small angle. An asymmetrical (airfoiled) fin is profiled to induce a small amount of lift at zero angle-of-attack, and thereby induce roll to the rocket. An example is NACA 2204 airfoil, used by UMSATS for their Winnie 3.0 rocket. My Cirrus One rocket was fitted with sheet metal fins that were wedge-shaped on one side to generate a small amount of lift with subsequent roll with the objective being to minimize trajectory dispersion.

Figure 19: Fins for imparting roll motion to a rocket
Ref. Topics in Advanced Model Rocketry, Mandel et al

Another means of imparting spin to a rocket is to use a helical launch tower such as for the Super Loki Dart rocket. The fins of the rocket fit within spiral guides such that when the rocket departs the launcher, a roll-rate of 8.5 rotations per second (or greater) is achieved.

Alternatives to Fins

Planar fins are nearly always the best option as the stabilizing device for an EX rocket, providing maximum restoring (stabilizing) moment combined with minimum weight and minimum drag, and of course, are simple to fabricate and mount. However, it is interesting that there exists alternative stabilizing devices. Two examples are shown in Figure 20: Conical Flair and Ringtail.

Figure 20: Alternatives to planar fins

A Ringtail will produce, at both subsonic and supersonic speeds, approximately twice the restoring moment of planar fins with equal total span and chord. The offset is greater drag at subsonic velocity. Interestingly, at supersonic velocities, there is a favourable interference between the Ringtail and exhaust plume which serves to reduce base drag. This benefit will only be favourable for longer burn times, such as end-burner motor.

A Conical Flair may have an advantage over planar fins at hypersonic velocity. On the basis of projected planform area, a Conical Flare will produce more than twice the normal force of planer fins at hypervelocity speeds. However, the drag force of a Conical Flare usually greatly exceeds that of fins providing an equal restoring moment. For the EX rocketeer, the Conical Flare could be an interesting choice for a tube launched rocket. A tube launcher is simply a long tube closed at the bottom end and open at the top end (a.k.a. closed-breech launcher). A tube launcher provides the same initial trajectory guidance as a standard rail (or rod) launcher, but differs in one key aspect. The rocket exhaust gases pressurize the tube, which gives an initial boost to the rocket by the piston action of the trapped gases. This results in a greater launch velocity which is beneficial in reducing weathercocking during the initial course of trajectory, when wind effect is most pronounced in generating undesirable angle-of-attack with resulting trajectory dispersion. The Arcas sounding rocket (although it was a finned rocket) utilized this launching technique. Flare angle should be less than 8° to avoid flow-separation which could reduce the stabilizing effectiveness. Calculation of normal force coefficient gradient and location of Centre of Pressure for a Conical Flare, for subsonic velocities, is given in Appendix B (Conical Flare is equivalent to a conical shoulder). For velocities from Mach 1.5 to Mach 5, graphs of normal force coefficient gradient are provided in MIL-HDBK-762 (Resource 23).

Examples

Example 1: A LoPer class rocket is being designed. Clipped Delta profile has been chosen as a suitable fin shape. The maximum altitude is targeted to be 3300 feet (1000 metres) and maximum velocity is expected to be 550 feet/second (170 m/s). The fins are to be 3D printed of PLA plastic as a fin can and bonded to the rocket body. Size the fins for stability, strength and flutter-resistance.

Example 2 : A supersonic HiPER class rocket is being designed. Three fin cross-sections are being considered:Diamond, Hexagon and Biconvex. Compare the bending strength of the three fin types assuming each have the same span-to-thickness (b/H) ratio.

Case Study 1: The four plexiglass fins mounted on the rocket for Flight C-34 broke off during flight. Perform a flutter analysis to determine if this was the likely cause.

Resources

Res.19   Fins for Rocket Stability webpage

Res.20   Barrowman Equations for Computing the Rocket Center of Pressure

Res.21   The Practical Calculation of the Aerodynamic Characteristics of Slender Finned Vehicles (James.S.Barrowman)

Res.22   The Theoretical Prediction of the Center of Pressure (James S.Barrowman, Judith A. Barrowman)    This report is particularly interesting and useful as it shows the derivations of the normal force coefficients used for calculating C.P.

Res.23 MIL-HDBK-762 Design of Aerodynamically Stabilized Free Rockets

Res.24   MS Excel spreadsheet for calculating C.P. by Barrowman method

Res.25   TIR-30 Centuri technical report: Stability of a Model Rocket in Flight (James Barrowman, 1970)

Res.26   TIR-33 Centuri technical report: Calculating the Center of Pressure of a Model Rocket (James Barrowman)

Res.27   Fin Flutter Analysis (Revisited), John K. Bennett (see also Peak of Flight Newsletter, Issue 615, December 19, 2023)

Res.28   Calculating Fin Flutter Velocity for Complex Fin Shapes, John K. Bennett (see also Peak of Flight Newsletter, Issue 617, January 16, 2024)

Res.29 Summary of Flutter Experiences as a Guide to the Preliminary Design of Lifting Surfaces on Missiles, NACA Technical Note 4197, Dennis J.Martin

Res.30 Mechanism of Flutter – A Theoretical and Experimental Investigation of the Flutter Problem, NACA Report No.685, Theodore Theodorsen & I.E.Garrick

Res.31 https://www.aerorocket.com/finsim.html

Res.32 Fin Flutter (good article, but beware of error in V_f equation)

Res.33   Aeroelastic Fin Flutter Calculation (Jeffrey M. Hopkins) (good article, but beware of error in V_f equation. This error is also present in Peak of Flight #291 article by Z.Howard)

Res.34   Estimating the dynamic and aerodynamic parameters of passively controlled high power rockets for flight simulation, Simon Box, Christopher M. Bishop, Hugh Hunt (February, 2009)

Res.35 https://www.aerorocket.com/finsim.html

Res.36 Corecell structural foam data sheet

Res.37 Standard Fins for Nike Rocket Motor

Res.38 fin_rootbend.xls Excel spreadsheet for calculating root bend angle of a fin

Res.39 fin-tab-strength.xlsx Excel spreadsheet for calculating strength of a slotted fin tab

Res.40 fin-C.G.xls Excel spreadsheet for calculating C.G. of a flat plate fin

Res.41 Missile Configuration Design, S.S.Chin, McGraw-Hill Inc. (1961)

Res.42 Aerodynamic Design Manual for Tactical Weapons, NSWC TR 81-156, Naval Surface Weapons Center

Res.43 Summary of Airfoil Data, NACA Report No.824, I.H.Abbott, A.E. Von Doenhoff & L.S. Stivers, Jr.

Next--  Dynamic Stability

Originally posted January 4, 2024

Last updated June 14, 2024

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