Nose cone design

General dimensions

In all of the following nose cone shape equations, L is the overall length of the nose cone and R is the radius of the base of the nose cone. y is the radius at any point x, as x varies from 0, at the tip of the nose cone, to L. The equations define the two-dimensional profile of the nose shape. The full body of revolution of the nose cone is formed by rotating the profile around the centerline C⁄L. While the equations describe the "perfect" shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons.^[1][2]

Conic

Conic nose cone render and profile with parameters shown.

$${\displaystyle {\begin{aligned}y&={\frac {xR}{L}}=x\tan(\phi )\\\phi &=\arctan \left({\frac {R}{L}}\right)\\\end{aligned}}}$$

Spherically blunted conic

Spherically blunted conic nose cone render and profile with parameters shown.

$${\displaystyle {\begin{aligned}x_{t}&&&={\frac {r_{n}L^{2}}{R}}{\sqrt {\frac {1}{R^{2}+L^{2}}}}\\y_{t}&={\frac {x_{t}R}{L}}&&=r_{n}L{\sqrt {\frac {1}{R^{2}+L^{2}}}}\\x_{o}&=x_{t}+{\sqrt {r_{n}^{2}-y_{t}^{2}}}&&=r_{n}\left({\frac {L^{2}}{R}}{\sqrt {\frac {1}{R^{2}+L^{2}}}}+{\sqrt {1-{\frac {L^{2}}{R^{2}+L^{2}}}}}\right)\\x_{a}&=x_{o}-r_{n}&&=r_{n}\left({\frac {L^{2}}{R}}{\sqrt {\frac {1}{R^{2}+L^{2}}}}+{\sqrt {1-{\frac {L^{2}}{R^{2}+L^{2}}}}}-1\right)\\\end{aligned}}}$$

Bi-conic

Bi-conic nose cone render and profile with parameters shown.

$${\displaystyle {\begin{aligned}L&=L_{1}+L_{2}\\\phi _{1}&=\arctan \left({\frac {R_{1}}{L_{1}}}\right)\\\phi _{2}&=\arctan \left({\frac {R_{2}-R_{1}}{L_{2}}}\right)\\y&={\begin{cases}{\frac {xR_{1}}{L_{1}}}=x\tan(\phi _{1}),&0\leq x\leq L_{1}\\R_{1}+{\frac {(x-L_{1})(R_{2}-R_{1})}{L_{2}}}=R_{1}+(x-L_{1})\tan(\phi _{2}),&L_{1}\leq x\leq L\end{cases}}\end{aligned}}}$$

Tangent ogive

Tangent ogive nose cone render and profile with parameters and ogive circle shown.

$${\displaystyle {\begin{aligned}\rho &={\frac {R+{\frac {L^{2}}{R}}}{2}}\\y&={\sqrt {\rho ^{2}+x(2L-x)-L^{2}}}+R-\rho \\\end{aligned}}}$$

Spherically blunted tangent ogive

Spherically blunted tangent ogive nose cone render and profile with parameters shown.

$${\displaystyle {\begin{aligned}x_{o}&=L-{\sqrt {\left(R-r_{n}\right)\left({\frac {L^{2}}{R}}-r_{n}\right)}}\\y_{t}&={\frac {r_{n}\left(L^{2}-R^{2}\right)}{L^{2}+R^{2}-2Rr_{n}}}\\x_{t}&=L-{\sqrt {\left(R-r_{n}\right)\left({\frac {L^{2}}{R}}-r_{n}\right)}}-r_{n}{\sqrt {1-\left({\frac {L^{2}-R^{2}}{L^{2}+R^{2}-2Rr_{n}}}\right)^{2}}}\\\end{aligned}}}$$

Secant ogive

Secant ogive nose cone render and profile with parameters and ogive circle shown, ogive radius larger than for equivalent tangent ogive.

For a chosen ogive radius ρ greater than or equal to the ogive radius of a tangent ogive with the same R and L:

$${\displaystyle {\begin{aligned}\rho &\geq {\frac {R+{\frac {L^{2}}{R}}}{2}}\\\alpha &=\arctan \left({\frac {R}{L}}\right)-\arccos \left({\frac {\sqrt {R^{2}+L^{2}}}{2\rho }}\right)\\y&={\sqrt {\rho ^{2}-(x-\rho \cos \alpha )^{2}}}+\rho \sin(\alpha ),&&0\leq x\leq L\\\end{aligned}}}$$

Alternate secant ogive render and profile which show a bulge due to a smaller radius.

A smaller ogive radius can be chosen; for ${\textstyle {\frac {1}{2}}\left(R+{\frac {L^{2}}{R}}\right)>\rho >{\frac {L}{2}}}$, you will get the shape shown on the right, where the ogive has a "bulge" on top, i.e. it has more than one x that results in some values of y.

Elliptical

Elliptical nose cone render and profile with parameters shown.

$${\displaystyle y={\frac {R{\sqrt {x(2L-x)}}}{L}}}$$

Parabolic

A parabolic series nosecone is defined by ${\displaystyle r={\tfrac {2x-Kx^{2}}{2-K}}}$ where ${\displaystyle 0\leq x\leq 1}$ and ${\displaystyle K}$ is a series-specific constant.^[3]

Half (K′ = 1/2)

Three-quarter (K′ = 3/4)

Full (K′ = 1)

Renders of common parabolic nose cone shapes.

For ${\displaystyle 0\leq K'\leq 1}$, ${\displaystyle y=R\left({2\left({x \over L}\right)-K'\left({x \over L}\right)^{2} \over 2-K'}\right)}$

K′ can vary anywhere between 0 and 1, but the most common values used for nose cone shapes are:

Power series

A power series nosecone is defined by ${\displaystyle r=x^{n}}$ where ${\displaystyle (0\leq x\leq 1)}$. ${\displaystyle n<1}$ will generate a concave geometry, while ${\displaystyle n>1}$ will generate a convex (or "flared") shape.^[3]

Half (n = 1/2) Three-quarter (n = 3/4)

For ${\displaystyle 0\leq n\leq 1}$: ${\displaystyle y=R\left({x \over L}\right)^{n}}$

Common values of n include:

Haack series

LD-Haack (Von Kármán) (C = 0) LV-Haack (C = 1/3)

A Haack series nosecone is defined by:^[3] $${\displaystyle {\begin{aligned}r(\theta )&={\sqrt {\frac {\theta -{\frac {1}{2}}\sin(2\theta )+C\sin ^{3}\theta }{\pi }}}\\\theta &=\arccos \!\left(1-2x\right)\\0&\leq x\leq 1\Rightarrow 0\leq \theta \leq \pi \\\end{aligned}}}$$ where

• r is the radius divided by the maximum radius at a given θ or x,
• x is the distance from the nose divided by the total nose length.

Parametric formulation can be obtained by solving the θ formula for x (here, x is now distance from the nose, separated from the total nose length L, and y is the radius). $${\displaystyle {\begin{aligned}x(\theta )&={L \over 2}\left(1-\cos(\theta )\right)\\y(\theta ,C)&={R \over {\sqrt {\pi }}}{\sqrt {\theta -{\sin(2\theta ) \over 2}+C\sin ^{3}(\theta )}}\\\end{aligned}}}$$

Special values of C (as described above) include:

Aerospike

An aerospike can be used to reduce the forebody pressure acting on supersonic aircraft. The aerospike creates a detached shock ahead of the body, thus reducing the drag acting on the aircraft.

Influence of the general shape