Fin Design – Example 1

A LoPer class rocket is being designed. Clipped Delta profile has been chosen as a suitable fin shape. To mitigate the potential for damage on landing, the fins will be mounted somewhat forward of the aft end of the rocket body. 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.

All the rocket components have been designed except the fins. The basic design of the rocket is shown in Figure 1. The items of significant mass and their locations within the rocket are illustrated (red balls) and have been tabulated and presented in Table 1.

Figure 1 – Rocket design showing items of mass and their x-station.

Table 1 – Items of mass and x-station

Note that the data for the Fin can are shown in red, indicating that these are estimates.

Problem Statement:

1. Size the fin profile to achieve a static Stability Margin of 2.0

2. Determine the fin thickness to achieve a strength Safety Factor of 3.0

3. Verify that the fins will not experience detrimental flutter

Solution:

The static Stability Margin (SM) is given by Equation 1

Re-arranging this equation to solve for C.P. location:

The first step is to determine the location of the fully-loaded rocket C.G. (x_cg). The C.G. location can be calculated knowing the mass of the rocket components and their respective locations (x-station, relative to the nosecone tip). The C.G. location is given by:

Where subscript i refers to each individual component of mass and sigma (S) simply refers to the summation of the individual (i) terms. Data from Table 1 is expanded in Table 2 with a column showing the calculation of the numerator. The sum of the individual masses and the sum of the numerator terms is then tallied.

Table 2 – C.G. calculation

The summation terms are put into the equation and the location of the C.G. is calculated:

The required location of the C.P. can now be calculated to achieve a stability margin of 2.0:

Now that we know where the C.P. is to be located, we can use an app such as AeroLab or Barrowman.xlsx to size the fins. The results of this exercise are shown in Figures 2 and 3.

Figure 2 – AeroLab representation of rocket with C.P. location indicated

Figure 3 – AeroLab fin detail for rocket

Once the fins have been fabricated and mounted on the rocket, the true C.G. is then used to calculate the true Stability Margin. It should be close to the value based on this exercise, but if not, the fins can be clipped or a small amount of mass added to the nosecone to achieve the desired Stability Margin.

The fins are next sized for structural strength using Load Cases 3 and 4 shown in the Introduction to Rocket Design - Fins webpage.

Load Case 3 is the critical flight loading of the fins due to a restoring force generated normal to the fin surface, a result of a sudden onset of non-zero angle of attack (example: due to wind shear). Load Case 4 represents critical ground loading of the fins (example: post-landing trauma or bumping a fin while handling the rocket).

Case 3 is considered first. The location of the individual fin spanwise C.P. is found from the following equation:

Where the terms are defined in Figure 15 of the Introduction to Rocket Design - Fins webpage. For our fin, the value of these terms are:

S = 50 mm span
c_r = 110 mm root chord
c_t = 50 mm tip chord

Giving

F_alpha is equal to the finset restoring force due to angle of attack and is given by:

where:

and the slope of the normal force coefficient for a single fin^* is given by:

where the various terms are shown in the Figure below, taken from he Introduction to Rocket Design - Fins webpage.

Theta (θ) and l are the mid-chord sweep angle and mid-chord length, respectively. And d is the reference diameter, in this case, the nosecone base diameter.

We will conservatively assume an angle-of-attack of 10 degrees at Vmax.

α = 0.175 radians

We want the fin normal force to be in units of Newtons so appropriate units of measure must be used for the terms involved. For air density, we’ll use the nominal value of 1.225 kg/m³. For our rocket, the value of these terms are:

D = 63.5mm or 0.0635 metres, giving

A = ¼ π (0.0635)² = 0.00317 m²

This results in a normal force acting on a single fin of:

N_1F = 17701 (0.00317) 0.175 (2.22) = 21.8 N. or approximately 5 lbf., acting the the fin C.P. This is the critical flight loading of a fin.

Load case 4 is considered next. For robustness of the fins, we’ll assume a handling load of 50 N. (11 lbf) acting at the fin tip. By inspection, this is more critical than than the flight case. As such, this will be used to calculate the root thickness of our fin, which is the location of maximum bending stress due to the handling load.

The root bending moment is:

And the root bending stress is given by

Where Z is the section modulus of the fin cross-section at the root, which is taken to be rectangular.

As we want a Safety Factor (S.F.) of 3 based on breakage of the fin, the relationship between bending stress and fin material strength is:

Where UFS is the Ultimate Flexurual Strength of the fin. The fin is to be made of 3D printed PLA plastic. From this table;

UFS = 83 MPa

This is the strength of virgin PLA filament. To achieve maximum strength, the filament must be extruded in the spanwise direction of the fin. To account for inevitable (tiny) defects in the extrusion process, as well as strength reduction due to moisture absorption, a knock-down factor of 0.8 will be applied to this value, or UFS_F = 0.8 (83) = 66.4 MPa

As such, the fin root thickness can be determined:

which is 2.6 mm or approximately 1/10 of an inch. The thickness, if desired, can be tapered (if desired) toward the fin tip, starting at ½ span. The fin should be ‘faired’ at the root for added strength and reduced body-fin interference drag. The fin cross-section would look something like this:

Now that the fin cross-section has been designed for strength, it is important to verify that the fins will not flutter, which can result in catastrophic failure of the fins.

The method of Resource 27 Fin Flutter Analysis (Revisited) by John Bennett will be utilized. The flutter velocity is given by:

where

Since our fin is not symmetric, epsilon (ε) will need to be calculated. The first step is to calculate the chordwise centroid of the fin:

Plugging in the values for our fin:

The value for epsilon is calculated as such:

Which gives:

The value of Y may now be calculated, noting that k = 1.4 and P_o = 101.325 kPa.

The aspect ratio of the fin is given by (semi-span)²/fin area, or:

Which gives:

The fin taper ratio (lambda) is tip chord length over root chord length:

λ = 50/110 = 0.455

For the pressure ratio and speed of sound, we’ll use this table . At our predicted altitude of 1000 metres:

a = 336.4 m/s

P/P_o = 89.84/101.35 = 0.886

The shear modulus for PLA plastic is taken from here:

G = 1092 MPa but we must have it in the same units as pressure (P_o) which is kPa:

G = 1092 MPa ´ 1000 = 1,092,000 kPa

The value for the estimated fin flutter velocity may now be calculated:

The maximum predicted velocity for our rocket is 172 m/s, which gives us a Safety Factor for fin flutter failure of:

S.F. (flutter) = 261/172 = 1.52 or a 52% margin

Due to the assumptions and limitations of this flutter analysis method, it is wise to be conservative and employ a Safety Factor of 2. Repeating the analysis with a fin thickness of 3.2mm (1/8 inch) gives us the desired Safety Factor of two.

_________

* See Reference 3.

References:

1. Introduction to Rocket Design: Fins

2. Biconvex-fin.xlsx

3. The Theoretical Prediction of the Center of Pressure (James S.Barrowman, Judith A. Barrowman)   

4. NACA-TN-4197 Summary of Flutter Experiences as a Guide to the Preliminary Design of Lifting Surfaces on Missiles