Fin Design – Example 2
A supersonic HiPer class rocket is being designed.
Three fin cross-sections are being considered:
·
Diamond
·
Hexagon
·
Biconvex
Compare the
bending strength (out-of-plane) of the three fin types assuming each have the
same span-to-thickness (b/H) ratio. The figure below shows this ratio.
For each of the
fins, b = 120 mm and H = 5 mm.
In reality, the
sharp tips of each of these fins would be radiused (rounded) to some extent.
This radiusing has insignficant effect on the bending strength and may be
neglected.
Bending strength
is directly proportional to the section
modulus (Z) of the cross-section.
For a given bending moment (M) the
maximum bending stress (fb max)
the fin will experience is given by:
fb max = M/Z
The section modulus
for each fin shape is calculated:
Diamond:
Z = 1/24 (120) (5)2
= 125 mm3
Hexagon:
Biconvex:
h
=
½ (5) = 2.5 mm
Yo
= 722
– 2.5 = 719.5 mm
Calculation of Z is tricky due to taking the difference of the terms which are
numerically very large values. Rounding off any of the terms can lead to
significant deviation from the true value of Z. As such, the calculation is best done using a precision app such
as Excel. Another benefit of using Excel is that the Arcsin term is, by default, given in radians, as required (not
degrees). From Excel calculation:
Summary:
Diamond, Z = 125
mm3
Hexagon, Z =313
mm3
Biconvex, Z = 229
mm3
Not surprisingly,
the Diamond shape is the weakest and Hexagon is strongest, by a factor of 2.5.
For a Diamond shaped fin of the same span, a thickness of nearly 8 mm is
required to match the bending strength of the Hexagon.
The Biconvex shape
lies in between, being 1.8 times as strong as the Diamond shape.
As pressure drag
of a fin is a function of thickness, it would appear that either Hexagon or
Biconvex may be a superior choice. However, wave drag for the three shapes
would need to be compared, over the expected velocity range, to make a rational
choice.
References: