UNDERSTANDING ABOUT THE OGIVES OF THE PROJECTILES

November 22, 2024

UNDERSTANDING ABOUT THE OGIVES OF THE PROJECTILES

UNDERSTANDING ABOUT THE OGIVE OF THE PROJECTILES

The ogive in the context of projectiles refers to the curved, tapered shape of a projectiles nose or front section. It is a crucial design feature that influences the aerodynamic performance, stability, and efficiency of a projectile as it moves through air or other media.

An ogive is a curved that connects the cylindrical body of a projectile to its pointed tip. It is designed to reduce drag by allowing air flow smoothly around the projectile, minimizing turbulence and energy loss.

In engineering and ballistics, the ogive refers to the curved or conical shape of the forward nose section of a projectile, such as a projectile, missile, or artillery shell. The design and characteristics of the ogive play a crucial role in determining the projectiles aerodynamic efficiency, stability, and overall performance.

Definition and Geometrical Basis: The term ogive originates from architecture, referring to a pointed, arch-like shape. In projectiles, the ogive is the curve or contour that transitions form the projectiles cylindrical body to its tip. This shape reduces drag by providing a smooth airflow around the projectile during its flight.

To find the ogive number generally we can use the formula

Ogive number = R / D

R is the radius of curvature and D is the diameter of the projectile, for military projectile the ogive number falls between 0.5 to 1.0 and for sporting and long range projectiles the ogive number falls between 1.0 to 2.0

Ogive profiles are usually defined mathematically using geometric curves like:

Circular or tangent ogive: Formed by a portion of a circle.

Elliptical ogive: Part of an ellipse.

Parabolic ogive: Derived from a parabola for better supersonic performance.

Secant ogive: A circular curve that intersects the cylindrical body rather than being tangent to it, reducing drag further.

The ogive shape significantly impacts the projectiles drag coefficient; in aerodynamics a sharper ogive minimizes drag but can reduce structural strength. And blunter ogives may produce more drag but provide better structural integrity and stability.

Supersonic Flight Stability: Pointed ogives like parabolic and secant designs perform well at high speeds.

Range and Accuracy: A well-designed ogive reduces aerodynamic drag, extending the range and improving accuracy.

Mach Number Adaptation: The optimal ogive shape depends on the speed range of the projectile, particularly whether it operates in subsonic, transonic, or supersonic regimes.

Here are the essential mathematical equations for an ogive-shaped nose in projectile design.

Round Nose Ogive: A classic, semi-circular curve that is less aerodynamic than other ogives, the main advantages are reliable feeding in lever action and semi-automatic firearms and high energy transfer on impact.

For a round nose ogive, the curve is a segment of a circle. To calculate the length of the curve, you can use the formula for the arc length of a circle.

Arc length S = R x theta

S = is the arch length (the length of the round nose ogive)

R = radius of the circular arc.

Theta = is the central angle of the arc (in radians)

The angle theta can be calculated based on the dimensions of the ogive (like the desired nose length) and the overall design. For simplicity, the round nose ogive will generally be a small segment of the full circle.

You can calculate the height h of the ogive (form the tip of the base of the curve) using the following relation, assuming the circular arc is symmetric.

Height of the round nose

h = R. (1- Cos (theta / 2)

h is the vertical height from the center of the circle to the tip of the projectiles nose.

R is the radius of the circle.

Theta is the central angle in radians (the same angle as in the arc length formula)

Spitzer Ogive: Spitzer ogive refers to a pointed nose shape that is more aerodynamically efficient compared to round nose or blunt designs. The term spitzer is derived from the German word spitz meaning sharp or pointed. The ogive is the curved section of the projectile nose, and in a spitzer design, this curve is often a secant ogive, which has more pronounced, sharper point compared to a tangent ogive. The curve of a spitzer ogive typically intersects the projectile body at an angle, leading to a more gradual transition between the nose and the cylindrical body. The sharp point at the tip of the projectile allows for reduced air resistance as the projectile travels through the atmosphere. The angle of the point is critical in determining the projectiles aerodynamic performance. A sharper point reduces drag but might increases the tendency to tumble upon impact unless designed for controlled expansion for fragmentation. The main advantages of the spitzer shape is Lower drag, High Ballistic Co efficient and Improve accuracy.

To calculate the height of the ogive that is base to the tip of the projectile.

h = R. (1 – cos (theta / 2)

h is the height of the ogive.

R radius of the circle defining the ogive.

Theta is the central angle of the arc in radians.

To calculate the arc length of the projectile is

L = R. Theta

L is the length of the ogive from the tip to the transition to the cylindrical body.

Theta is the central angle of the arc in radians.

To calculate the central angle of the projectile is

Theta = 2. arccos (1- h/R)

Tangent Ogive: The tangent ogive shape is the most familiar in projectiles, the profile of this shape is formed by a segment of a circle such that the projectile body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. The popularity of this shape is largely due to the ease of constructing its profile, as it is simply a circular section.

The radius of the circle that forms the ogive is called the ogive radius, ρ, and it is related to the length and base radius of the nose cone as expressed by the formula:

The radius y at any point x, as x varies from 0 to L is:

The nose cone length, L, must be less than or equal to ρ. If they are equal, then the shape is hemisphere.

L is the overall length of the nose cone

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 center line ^C⁄_L.

While the equations describe the "perfect" shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons.

Spherically Blunted Tangent Ogive or Soft Point: A tangent ogive nose is often blunted by capping it with a segment of a sphere. The tangency points where the sphere meets the tangent ogive can be found from:

L is the overall length of the nose cone

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.

where r_n is the radius and x_o are the center of the spherical nose cap.

Secant Ogive: The profile of this shape is also formed by a segment of a circle, but the base of the shape is not on the radius of the circle defined by the ogive radius. The Projectile body will not be tangent to the curve of the nose at its base. The ogive radius ρ is not determined by R and L (as it is for a tangent ogive), but rather is one of the factors to be chosen to define the nose shape. If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same R and L, then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated.

Secant ogive nose cone render and profile with parameters and ogive circle shown above

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

Then the radius y at any point x as x varies from 0 to L is:

If the chosen ρ is less than the tangent ogive ρ and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. A classic example of this shape is the nose cone of Honest John.

Elliptical Ogive: The profile of this shape is one-half of an ellipse, with the major axis being the centreline and the minor axis being the base of the nose cone. A rotation of a full ellipse about its major axis is called a prolate spheroid, so an elliptical nose shape would properly be known as a prolate hemispheroid. This shape is popular in subsonic flight (such as model rocketry) due to the blunt nose and tangent base. [further explanation needed] This is not a shape normally found in professional rocketry, which almost always flies at much higher velocities where other designs are more suitable. If R equals L, this is a hemisphere

An elliptical ogive is a nose cone generated by the segment of an ellipse where, the base of the ogive lies along the minor axis of the ellipse and the curve extends from the base to the tip, forming a smooth continuous shape.

Descriptive Geometry of the Spitzer Projectile:

Ogive: The ogive shape forms the front of the bullet. The ogive shape is formed from the arcs of two circles. The ogive may or may not be tangent at the point of intersection to the cylindrical portion of the bullet. When the circles are tangent to the cylinder portion, we call say this is a tangent ogive. When the circles are not tangent to the cylinder portion, we say we have a secant ogive. The rationale behind the use of the term "secant" can be seen in Figure 6, where there are two horizontal reference lines (brown colour) that are both secant lines.

Cylinder: The cylindrical portion of the bullet is what engages the rifling of the barrel.

Frustum: The back of the bullet (aka "boattail") geometrically is in the shape of the frustrum of the cone Tapering the back of the bullet reduces drag, particularly at speeds less than supersonic.

Projectile Mass Calculation:

We will compute the volume of the spitzer bullet as follows.

Compute the volume of the frustum portion (V_Frustum)

Compute the volume of the cylinder portion (V_Cylinder)

Compute the volume of the ogive portion (V_Ogive)

Compute the total volume by summing all the volumes of the pieces