# Elliptical Fins Equations#

Author:

Mateus Stano Junqueira,

Author:

Franz Masatoshi Yuri,

Author:

Kaleb Ramos Wanderley Santos,

Author:

Matheus Gonçalvez Doretto,

Date:

February 2022

## Nomenclature#

• $$Cr$$ - Root chord

• $$Ct$$ - Tip chord

• $$A_{r}$$ - Reference area

• $$C_{N\alpha0}$$ - Normal force coefficient derivative of a 2D airfoil

• $$C_{N\alpha1}$$ - Normal force coefficient derivative of one fin

• $$F$$ - Force

• $$L_{r}$$ - Reference length, equals rocket diameter

• $$M_{roll}$$ - Roll moment

• $$M_{f}$$ - Roll forcing moment

• $$M_{d}$$ - Roll damping moment

• $$N$$ - Number of fins

• $$\overline{q}$$ - Dynamic pressure

• $$r_{t}$$ - Reference radius at fins position

• $$s$$ - Fin span

• $$v_{0}$$ - Rocket speed in relation to the wind

• $$\omega$$ - Angular velocity

• $$Y_{MA}$$ - Spanwise location of mean aerodynamic chord measured from the root chord

• $$\delta$$ - Fin cant angle

• $$x_{i}$$ - Distance to rotation axis

• $$\rho$$ - Ambient density

• $$C_{lf}$$ - Roll moment lift coefficient

• $$C_{lf\delta}$$ - Roll moment lift coefficient derivative

• $$C_{ld}$$ - Roll moment damping coefficient

• $$C_{ld\omega}$$ - Roll moment damping coefficient derivative

## Introduction#

In order to calculate the effects of elliptical fins in RocketPy we need to calculate:

• Geometric parameters

• Center of pressure

• Interference Factor

All non proved equations where based on [Barrowman].

## Geometrical Parameters#

An elliptical fin can be defined with two basic parameters: the root chord ($$Cr$$) and the span ($$S$$), as can be seen in Figure 1. Through them, other geometrical properties are calculated, which are then used in the computations for the center of pressure and roll coefficients.

### Chord Length ($$c$$)#

The chord length ($$c$$) at a spanwise position must be calculated through two axis: $$y$$, that begins at the fuselage wall, and $$\xi$$ that begins in the fuselages center line (Figure 1).

First we calculate $$c(y)$$ through the following elliptical equation:

$\frac{x^2}{a} + \frac{y^2}{b} = 1$

Substituting variables for the elliptical fins parameters:

$\frac{y^2}{S^2} + \frac{\Bigl(0.5 \cdot c(y)\Bigr)^{2}}{\Bigl(0.5 \cdot Cr\Bigr)^{2}} = 1$

Simplifying:

$\frac{c(y)}{2} = \frac{Cr}{2} \cdot \sqrt{1 - \Bigl(\frac{y}{S}\Bigr)^2}$
$c(y) = Cr\sqrt{1 - \Bigl(\frac{y}{S}\Bigr)^2}$

Transforming to the $$\xi$$ axis:

$c(\xi) = Cr\sqrt{1 - \Bigl(\frac{\xi-r}{S}\Bigr)^2}$

### Spanwise position of the Mean Aerodynamic Chord ($$Y_{ma}$$)#

We can find the length of the Mean Aerodynamic Chord ($$Y_{ma}$$) using the known definition [Barrowman]:

$c_{ma} = \frac{1}{A_{f}}\int_{0}^{s}c^2(y) \,dy$

Where $$A_{f}$$ is the area of the fin, in our case $$A_f = \frac{\pi \, C_r \, S}{4}$$

$c_{ma} = \frac{4}{\pi \, C_r \, S}\int_{0}^{s} \Bigl(Cr\sqrt{1 - \Bigl(\frac{y}{S}\Bigr)^2} \, \Bigr)^2 \,dy$

Solving the integral:

$c_{ma} = \frac{8 C_r}{3 \pi}$

Finally, the span wise position of the mean aerodynamic chord can be found by equating $$c_{ma}$$ with $$c(Y_{ma})$$ and solving for $$Y_{ma}$$.

$c_{ma} = c(Y_{ma})$
$\frac{8 C_r}{3 \pi} = Cr\sqrt{1 - \Bigl(\frac{Y_{ma}}{S}\Bigr)^2}$
$Y_{ma} = \frac{S}{3\pi}\sqrt{9\pi^2 - 64}$

### Roll Geometrical Constant ($$R_{cte}$$)#

For the calculation of roll moment induced by a cant angle in the fins, a geometrical constant that takes in regard the fin geometry is used in the computations.

The formula for the constant is as follows:

$R_{cte} = \int_{r_t}^{s + r_t} c(\xi) \, \xi^2 \, d\xi$
$R_{cte} = C_r\, S\ \frac{ \Bigl(3\pi S^2 + 32 r_t S + 12 \pi r_t^2 \Bigr)}{48}$

## Center of Pressure#

The position of center of pressure of a elliptical fin along the center line of a rocket can simply be calculated by the following equation, according to [Model]:

$\overline{X_f} = X_f + 0.288 \cdot C_r$

## Roll Damping Interference Factor#

According to [Barrowman], the roll damping interference factor can be given by:

$k_{R_D} = 1+ \frac{\displaystyle\int_r^{s+r} {r}^{3} \cdot \frac{c(\xi)}{\xi^2} \,d\xi}{\displaystyle\int_r^{s+r}\xi \, c(\xi) \, d\xi}$

Solving the integrals for elliptical fin geometry, using equations described at Chord Length ($$c$$):

When $$S > r$$:

$k_{R_{D}} = 1 + {r}^{2} \cdot \frac{2\cdot {r}^{2} \sqrt{s^2-r^2}\cdot \ln\Bigl(\frac{2s\cdot\sqrt{s^2-r^2}+ 2s^2}{r}\Bigr) - 2{r}^2\sqrt{{s}^2-{r}^2}\cdot \ln(2s) + 2s^{3} - {\pi}rs^{2} - 2r^2s + {\pi}\cdot {r}^3}{2\cdot {s}^{2} \cdot \Bigl(\frac{s}{3}+\frac{{\pi}\cdot r}{4}\Bigr) \cdot\Bigl(s^2-r^2\Bigr)}$

When $$S < r$$:

$k_{R_{D}} = 1-\frac{r^2 \cdot\left(2 s^3-\pi s^2 r-2 s r^2+\pi r^3+2 r^2 \sqrt{-s^2+r^2} \cdot \operatorname{atan}\left(\frac{s}{\sqrt{-s^2+r^2}}\right)-\pi r^2 \sqrt{-s^2+r^2}\right)}{2 s\left(-s^2+r^2\right)\left(\frac{s^2}{3}+\frac{\pi s r}{4}\right)}$

And by calculating the limit of the above expressions when $$S \rightarrow r$$ we have that, for $$S = r$$:

$k_{R_{D}} = \frac{28-3\pi}{4+3\pi}$

## References#

[Barrowman] (1,2,3)

Barrowman, James S.. (1967). The practical calculation of the aerodynamic characteristics of slender finned vehicles.

[Model]

Barrowman, James S.. (1970). Model Rocketry.