fin forces at supersonic speeds

Fin Forces at Supersonic Speeds

PROBLEM: PSAS seeks to design rocket control fins that will facilitate guidance of an amateur rocket. With the success of the roll-control system, it is assumed that the main fins can be angled in a similar fashion to steer the rocket on a desired trajectory. To maintain stability, we must be able to predict the forces produced by the fins in flight. Then, when a desired amount of fin force is needed for control, the fins can be angled to produce that force.

REFERENCE DIAGRAM:

GIVEN: Fin length l, notch length ζ l = 0, leading edge angle of delta wing Λ_LE = 60°, velocity v, dynamic pressure q, Mach number M, and angle of attack ∝ (0-30°).

REQUIRED: For supersonic speeds determine (a) area of the fin A, (b) air density ρ, (c) coefficient of lift CL, (d) Lift force L, (e) and drag force D (f) Normal force N. Calculate parts a, c, d, e,and f for a fin length of 8 inches, Mach number of 1.2, velocity of 405 m/s, air density of 1.14 kg/m^3, and an angle of attack of 1°. Calculate the normal force on the fin with an AOA of 30°. Then graph Lift vs. AOA, Drag vs. AOA, and Normal Force vs. AOA.

EQUATIONS:

(a) Notch ratio ζ =0

For a notch ratio of 0. Fin width $w = \ \tan(90^\circ-\Lambda_{LE})*l$

Area of the fin A = 0.5*l*w

(b) Dynamic pressure $q = \rho*v^2$

Air density $\rho = \frac{q}{v^2}$

$CL = \ \frac{4*m}{E(k)}\ \left [\frac{\zeta}{1+\zeta}+\ \frac{1-\zeta}{(1-\zeta^2)^{3/2}}\ \cos^{-1}(-\zeta)\ \right ]\ \frac{1}{\beta}, \qquad m \le 1$

Where

$\beta = \sqrt{M^2-1}$

$m = \beta*cot(\Lambda_{LE})$

$k = \sqrt{1-m^2}$

$E\left(k\right)=\int^{\frac{\pi }{2}}_0{\sqrt{1-k^2*{\left({\rm sin}(\theta )\right)}^2}d\theta }$

(d) Lift force equation comes from NASA’s website ("The Lift Coefficient").

$L=0.5*\rho *v^{^{2}}*A*CL*\alpha$

(e) Drag force equation comes from Ashley and Landahl’s book, Aerodynamics of Wings and Bodies on page 169 (Ashley, and Landahl 279-169).

$D=\alpha *L-T$

Where

$T=\frac{\pi}{2}*\rho *v^{2}*\alpha ^{2}\frac{A^{2}}{E(k)^{2}}*\sqrt{1-m^{2}}$

(f) Normal force $N=\sqrt{L^{2}+D^{2}}$

Where α is in radians for parts d and e

CALCULATIONS:

(a) Notch ratio ζ =0

For a fin length of 8 inches l = 0.2032 m

For a notch ratio of 0.

Fin width

$w=\tan{(90^{0}-60^{0})}*(0.2032)$

w=0.117m

Area of the fin

A=0.50*(0.2032m)*(0.117m)

$A=\mathbf{0.0119m^{2}}$

(c)

Where

$\beta = \sqrt{(1.2)^{2}-1}$

$\beta = \mathbf{0.663}$

$m=(0.663)*cot(60^{o})$

$m = \mathbf{0.383}$

$0.383\leq 1$

$k=\sqrt{1-(0.383)^{2}}$

$k = \mathbf{0.924}$

$E(k)=\int_{0}^{\frac{\pi}{2}} \sqrt{1-(0.924)^{2}*(\sin(\theta)))^{2}}d\theta$

$E(k)= \mathbf{1.141}$

Coefficient of Lift

$CL=\left [ \frac{4*0.383}{1.141} \left [ \frac{0}{1-0}+\frac{1-0}{(1-0^{2})^{3/2}}*\cos^{-1}(-0) \right ] \right ]*\left [ \frac{1}{0.663} \right ]$

$CL = \mathbf{3.179}$

(d)

$\alpha (rads)=\frac{2\pi}{360}*\alpha (deg)$

$\alpha (rads)=\frac{2\pi}{360}*1^{o}$

$\alpha (rads)= \mathbf{\frac{\pi}{180}}$

Lift force at 1 degree

$L = 0.5*(1.14kg/m^{3})*(405m/s)^{2}*(0.0119m^{2})*(3.179)*\frac{\pi}{180}*{1^{o}}$

$L = \mathbf{61.84 N}$

$L(\alpha )=(61.84N/deg)*\alpha(deg)$

At 30°

$L(30^{o})=(61.8N/deg)*(30^{o})$

$L(30^{o})=1855N$

(e)

Thrust at 1°

$T=\frac{\pi}{2}*(1.14kg/m^{3})*(405m/s)^{2}*(\frac{\pi}{180}*1^{o})^{2}*\frac{(0.0119m^{2})^{2}}{1.141^{2}}*\sqrt{1-0.383^{2}}$

$T = \mathbf{9.021*10^{-3}}$

Drag force at 1°

$D = \frac{\pi}{180}*(1^{o})*(61.84N)*(1^{o})-(9.021*10^{-3})*(1^{o})^2$

$D = \mathbf{1.070 N}$

$D(\alpha) = (1.079N/deg^2)*(\alpha(deg))^2-(0.009021N/deg)*(\alpha(deg))^{2}$

At 30°

$D = 1.079*(30^{o})^2-0.009021*(30^{o})^{2}$

$D = \mathbf{962.981 N}$

Normal force with an AOA of 30°

$N = \sqrt{D^{2}+L^{2}}$

$N = \sqrt{(962.981N)^{2}+(1855N)^{2}}$

$N = \mathbf{2090N \: or \: 469.9lbs}$

$N(\alpha) = \sqrt{\left [1.079*(\alpha(deg))^{2}-0.009021*(\alpha(deg))^{2} \right ]^{2}+\left [61.84*\alpha(deg) \right ]^{2}}$

GRAPHS:

Works Cited

Ashley, Holt, and Marten Landahl. Aerodynamics of Wings and Bodies. United States of

 America: Addison-Wesley Publishing Company.Inc, 1965. 279-169. Print.

Mason, W.H. "Supersonic Aerodynamics."www.dept.aoe.vt.edu. N.p., 03 Sept 2006. Web. 12 Mar 2011.

 <http://www.dept.aoe.vt.edu/~mason/Mason_f/ConfigAeroSupersonicNotes.pdf>

The Lift Coefficient. , www.nasa.gov. Web. 12 Mar 2011.

 <http://microgravity.grc.nasa.gov/education/rocket/atmosmet.html>.