Fin Design – Case Study 1

One of my early rockets (Flight C-34) was fitted with fins fabricated from plexiglass (acrylic) sheet. Prior rockets had aluminum fins. Not being aware of the phenomenon of flutter, this seemed like a good idea at the time. These plastic fins were lighter and easier to cut to shape. When the rocket was recovered after its flight, all four fins were found to have fractured. Figure 1 illustrates the rocket and a sketch from my notebook that shows the location of the fracture (all four fractured alike). Perform a flutter analysis to determine if flutter was a likely cause of the fin fracturing.

Figure 1 – Author with C-34 rocket launched July 1984 (with my Dad); Fin fracture location.

The dimensions of the fin are shown in Figure 2. The method of NACA Technical Note 4197 will be used to calculate the flutter velocity (V_f) where flutter velocity is given by: (Ref. Introduction to Rocket Design – Fins )

where:

As the fins for the C-34 rocket are non-symmetric, we’ll use the method of Resource 28 ( Calculating Fin Flutter Velocity for Complex Fin Shapes by John K. Bennett ) to calculate the value of epsilon. As the fin has a polygon-based shape, the method of geometric decomposition is used to divide the fin into a set of triangles. The first step is to assign x,y coordinates to each of the fin vertices formed by the four triangles (dimensions are millimetres). This is illustrated in Figure 3 (it is purely a case of serendipity that the fin shown in the Res.28 example is nearly identical to the C-34 fin).

Figure 2 – C-34 Fin dimensions (note that the 3 holes are for attachment of the fin to an L-bracket)

Figure 3 – C-34 fin divided into triangles and x,y coordinates of each vertex.

The area of each of the four triangles is given by:

giving:

Area (T1) = ½ [0 (0 – 117.7) + 105.66 (117.7 – 0) + 89.22 (0 – 0)] = 6218 mm²

Area (T2) = ½ [105.66 (9.7 – 117.7) + 127.12 (117.7 – 0) + 89.22 (0 – 9.7)] = 1343 mm²

Area (T3) = ½ [127.12 (105.2 – 117.7) + 144.32 (117.7 – 9.7) + 89.22 (9.7 – 105.2)] = 2739 mm²

Area (T4) = ½ [144.32 (117.7 – 117.7) + 144.32 (117.7 – 105.2) + 89.22 (105.2 – 117.7)] = 344 mm²

The total fin area is simply the sum of the area of the four triangles:

Area (fin) = 6218 + 1343 + 2739 + 344 = 10644 mm²

The centroid (C_x) of any triangle, with respect to the x-axis, is given by the average of the x-coordinates of the three vertices:

Which gives the following for the four triangles:

To calculate C_x of the entire fin, we calculate a weighted average by adding the products of C_x and the area of the that triangle for each triangle, and then divide the result by the total fin area, as follows:

giving

Now we can calculate ԑ using the formula from Res.28, where ԑ is a measure of distance expressed as a fraction of the whole root chord:

where C_r is the fin root chord length, taken as 105.66 mm, giving

The value of Y may now be calculated, noting that k = 1.4 and P_o = 98.3 kPa.

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

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

λ = 55.1/105.66 = 0.521

For the pressure ratio and speed of sound, we’ll simply assume an altitude of 1000 metres. From this table:

P = 89.8 kPa a = 336 m/s

P/P_o = 89.8/98.3 = 0.914

The shear modulus of acrylic plastic is G = 1,151,500 kPa. From my notebook, the fin thickness is 0.075 inch, or 1.91mm.

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

My software app EzAlt.xls was used to estimate the maximum velocity of the C-34 rocket. This was determined to be approximately 200 m/s. This leaves little doubt that the fins fractured as a result of fin flutter.

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