Ballistics - External Ballistics and Equations - lect 6

Workshops Reminder

• Ballistics workshops are this week on Thursday.

• The workshops are two hours long, with teaching in the first hour and questions in the second hour.

• The workshops will cover key module-related teaching and recap key maths from the previous year.

• Opportunity to work through exam type questions and ask questions.

External Ballistics - Recap

• External ballistics dictates how projectiles fly through the air and what keeps them stable.

• Key concepts:

  • Spinning a bullet using rifling provides gyroscopic stability because of the relative positions of the center of mass and center of pressure.

  • The center of mass is the balancing point of the bullet (like a seesaw pivot).

  • The center of pressure is the sum of all aerodynamic forces acting on the projectile.

  • If the center of pressure is far from the center of mass, it creates a seesawing effect, causing the bullet to tumble.

  • Spinning the bullet stabilizes it.

Projectiles That Don't Need Spinning

• Rockets:

  • Have fins at the back, which changes the center of pressure to the rear of the projectile.

  • Fins create more drag, but rockets have their own propulsion to overcome this.

  • Military rockets use aerodynamics and electronic mechanisms to change direction in flight.

• Drag Stabilization:

  • An alternative to gyroscopic stability, but only effective if the projectile has its own propulsion.

  • Free-falling projectiles using drag stabilization don't go very far.

  • May be used when short distance or spin is not required.

Equations

• Units are important, and you must be able to rearrange equations.

Drag Coefficient

• Relates various factors to the drag force experienced by a projectile.

• Drag Force Equation: $F<em>D = \frac{1}{2} C</em>D A V^2 \rho$

  • $F_D$ = Drag Force

  • $C_D$ = Drag Coefficient

  • $A$ = Cross-sectional area of the projectile (m^2) - typically $\pi r^2$ for round projectiles

  • $V$ = Flow velocity (m/s) of air or projectile

  • $\rho$ (rho) = Air density at sea level (≈ 1.2 kg/m^3)

Flow Velocity

• Aerodynamically, it doesn't matter if the projectile is moving through the air or the air is moving over the projectile.

• Wind Tunnels:

  • Used to hold the projectile still and blow air over it.

  • Easier for experimental measurement than firing projectiles at high velocities.

  • Low-velocity wind tunnel data can be used to predict high-velocity performance.

Velocity Squared Factor

• Drag force is proportional to the square of the velocity ($V^2$).

  • Doubling the velocity results in four times the drag force.

• Trade-offs:

  • Increasing velocity requires more energy, but also increases drag force significantly.

  • A balance must be found.

Drag Coefficient Details

• The drag coefficient ($C_D$) indicates the effectiveness of the projectile's profile and shape.

• Different designs have different drag coefficients, which vary with velocity.

• Independent of size and mass, allowing for scaled models in wind tunnels.

• Formula One uses scaled models to reduce costs.

• Generating laminar flow in wind tunnels is complex and expensive.

Calculating Drag Coefficient

• Rearrange the drag force equation: $C<em>D = \frac{2F</em>D}{A V^2 \rho}$

• Measure drag force, cross-sectional area, flow velocity, and air density.

• Wind Tunnel Experiments:

  • Use scaled-up models of projectiles to measure drag force and calculate drag coefficients.

Wind Tunnel Experiment Details

• Air blown over an object at a specific velocity is aerodynamically equivalent to the object moving through the air at that velocity.

• Scaled models can be used to measure drag force and calculate drag coefficients.

Drag Coefficient Calculation

• Rearranged Equation: $C<em>D = \frac{2F</em>D}{A V^2 \rho}$

• Projectile cross-sectional area: $\pi r^2$ (ensure units are in meters).

• Flow velocity measured with an anemometer.

• Fluid density: Density of air.

• Drag force: Measured using a force gauge.

• Units:

  • Force in Newtons.

  • Area in square meters.

  • Velocity in meters per second.

  • Density in kilograms per cubic meter.

• The drag coefficient is dimensionless.

Typical Drag Coefficients

• Examples of drag coefficients for different shapes:

  • Flat plate: 1.28

  • Prism: 1.14

  • Typical bullet: ~0.3

  • Sphere: Varies with velocity (up to 0.5)

  • Aerofoil: Very low

• Air management around the projectile significantly affects the drag coefficient.

Drag Coefficient Variation with Velocity

• The drag coefficient is not constant; it varies with velocity.

• Example: Spitzer-type bullet.

  • Significant change occurs around Mach 1 (speed of sound, approximately 340 m/s).

  • Sonic boom and pressure effects occur as the bullet goes supersonic.

  • Subsonic ammunition is quieter because it doesn't break the sound barrier.

  • A massive increase in drag coefficient is seen as the bullet goes supersonic, followed by a tail-off at higher velocities.

Sectional Density

• Bullets with high sectional density carry more impact energy at a given range.

• Relates to the concept of an elephant vs. a mouse. Higher mass has more impact.

Sectional Density Equation

• $S = \frac{m}{d^2}$

  • $S$ = Sectional Density

  • $m$ = Mass

  • $d$ = Diameter

• Increasing mass increases sectional density.

• Decreasing diameter increases sectional density.

• Depleted uranium example: small, heavy bullets.

• Sectional density is important for impact and penetration.

Sectional Density Definition Note

• Alternative definition (mass/cross-sectional area) exists, but in ballistics, diameter is preferred.

Ballistic Coefficient

• A measure of aerodynamic forces exerted on a bullet in flight (CB).

• Specific to individual bullet design and size.

• Can be used to calculate real-time trajectory values.

• Relates bullet's sectional density to its drag coefficient.

• Combines design, mass, diameter, air density, and velocity.

Ballistic Coefficient Standardization

• Compares the drag coefficient against a standard bullet.

• G1 projectile (drag coefficient = 0.5191) is used for comparison.

• Other standards exist (e.g., G7), and the choice affects values.

Calculating Ballistic Coefficient

• $CB = \frac{C<em>G}{C</em>D} * S$

  • $C_G$ = Drag coefficient of the G1 bullet (0.5191).

  • $C_D$ = Drag coefficient of the bullet.

  • $S$ = Sectional density.

• Units:

  • Mass in kilograms.

  • Diameter in meters.

  • Sectional density in kilograms per square meter.

  • Ballistic coefficient in kilograms per square meter.

Process Summary

• Determine drag coefficient using wind tunnel experiments.

• Plug values into equations with correct units.

• Calculate drag coefficient and ballistic coefficient.

• Vary velocity to get a range of coefficients.

• Do not average drag coefficients; they are specific to a particular velocity.

Ballistic Coefficient Values

• Commercial ammunition typically falls in the range of 50 to 500 kg/m^2.

• Depleted uranium can result in much higher values.

Workshops Reminder

• Ballistics workshops are this week on Thursday.

• The workshops are two hours long, with teaching in the first hour and questions in the second hour.

• The workshops will cover key module-related teaching and recap key maths from the previous year.

• Opportunity to work through exam type questions and ask questions.

• Ensure you bring your calculators and formula sheets to make the most of the session.

Additional Notes

• These workshops are designed to help reinforce the concepts covered in the lectures.

• Past exam papers will be used, so familiarize yourself with the format.

External Ballistics - Recap

• External ballistics dictates how projectiles fly through the air and what keeps them stable.

• Key concepts:

  • Spinning a bullet using rifling provides gyroscopic stability because of the relative positions of the center of mass and center of pressure.

  • The center of mass is the balancing point of the bullet (like a seesaw pivot).

  • The center of pressure is the sum of all aerodynamic forces acting on the projectile.

  • If the center of pressure is far from the center of mass, it creates a seesawing effect, causing the bullet to tumble.

  • Spinning the bullet stabilizes it.

Stability Explained

• Gyroscopic stability is crucial for long-range accuracy.

• Without spin, the bullet would quickly lose its orientation and become highly unstable.

Projectiles That Don't Need Spinning

• Rockets:

  • Have fins at the back, which changes the center of pressure to the rear of the projectile.

  • Fins create more drag, but rockets have their own propulsion to overcome this.

  • Military rockets use aerodynamics and electronic mechanisms to change direction in flight.

Fin Design

• Fin design is critical for maintaining stability and direction.

• Different fin shapes and sizes affect the rocket's trajectory.

• Drag Stabilization:

  • An alternative to gyroscopic stability, but only effective if the projectile has its own propulsion.

  • Free-falling projectiles using drag stabilization don't go very far.

  • May be used when short distance or spin is not required.

Applications of Drag Stabilization

• Commonly used in mortars and short-range artillery.

• Simplifies the design and reduces manufacturing costs.

Equations

• Units are important, and you must be able to rearrange equations.

• Always ensure you are using consistent units across all calculations.

Drag Coefficient

• Relates various factors to the drag force experienced by a projectile.

• Drag Force Equation: $F<em>D = \frac{1}{2} C</em>D A V^2 \rho$

  • $F_D$ = Drag Force

  • $C_D$ = Drag Coefficient

  • $A$ = Cross-sectional area of the projectile (m^2) - typically $\pi r^2$ for round projectiles

  • $V$ = Flow velocity (m/s) of air or projectile

  • $\rho$ (rho) = Air density at sea level (≈ 1.2 kg/m^3)

Importance of Variables

• Understanding the influence of each variable on drag force.

• Slight changes in velocity can significantly impact the overall drag.

Flow Velocity

• Aerodynamically, it doesn't matter if the projectile is moving through the air or the air is moving over the projectile.

• Wind Tunnels:

  • Used to hold the projectile still and blow air over it.

  • Easier for experimental measurement than firing projectiles at high velocities.

  • Low-velocity wind tunnel data can be used to predict high-velocity performance.

Wind Tunnel Techniques

• Ensures precise and repeatable conditions for testing.

• Provides valuable data for projectile design and optimization.

Velocity Squared Factor

• Drag force is proportional to the square of the velocity ($V^2$).

  • Doubling the velocity results in four times the drag force.

Implications of Velocity

• High-velocity projectiles experience significantly greater drag.

• Understanding this relationship is crucial for trajectory calculations.

• Trade-offs:

  • Increasing velocity requires more energy, but also increases drag force significantly.

  • A balance must be found.

Optimal Velocity

• Identifying the point where increased velocity yields diminishing returns.

• Factors include energy consumption and projectile stability.

Drag Coefficient Details

• The drag coefficient ($C_D$) indicates the effectiveness of the projectile's profile and shape.

• Different designs have different drag coefficients, which vary with velocity.

• Independent of size and mass, allowing for scaled models in wind tunnels.

• Formula One uses scaled models to reduce costs.

• Generating laminar flow in wind tunnels is complex and expensive.

Laminar Flow Considerations

• Essential for accurate drag coefficient measurements.

• Turbulence can distort results and affect the reliability of the data.

Calculating Drag Coefficient

• Rearrange the drag force equation: $C<em>D = \frac{2F</em>D}{A V^2 \rho}$

• Measure drag force, cross-sectional area, flow velocity, and air density.

• Wind Tunnel Experiments:

  • Use scaled-up models of projectiles to measure drag force and calculate drag coefficients.

Experimental Setup

• Proper calibration of instruments is necessary for accurate readings.

• Precise measurement of each parameter influences the final drag coefficient value.

Wind Tunnel Experiment Details

• Air blown over an object at a specific velocity is aerodynamically equivalent to the object moving through the air at that velocity.

• Scaled models can be used to measure drag force and calculate drag coefficients.

Model Scaling

• Ensures the results are applicable to full-scale projectiles.

• Scaling factors must be carefully considered to maintain accuracy.

Drag Coefficient Calculation

• Rearranged Equation: $C<em>D = \frac{2F</em>D}{A V^2 \rho}$

• Projectile cross-sectional area: $\pi r^2$ (ensure units are in meters).

• Flow velocity measured with an anemometer.

• Fluid density: Density of air.

• Drag force: Measured using a force gauge.

• Units:

  • Force in Newtons.

  • Area in square meters.

  • Velocity in meters per second.

  • Density in kilograms per cubic meter.

Unit Consistency

• Accurate unit conversion is vital for correct calculations.

• Ensure all values are converted to the appropriate base units before use.

• The drag coefficient is dimensionless.

Dimensionless Nature

• Allows comparison across different scales and conditions.

• Simplifies analysis and interpretation of results.

Typical Drag Coefficients

• Examples of drag coefficients for different shapes:

  • Flat plate: 1.28

  • Prism: 1.14

  • Typical bullet: ~0.3

  • Sphere: Varies with velocity (up to 0.5)

  • Aerofoil: Very low

Shape Influence

• Different shapes exhibit varying degrees of air resistance.

• Aerodynamic designs minimize drag and improve performance.

• Air management around the projectile significantly affects the drag coefficient.

Airflow Management

• Streamlining and reducing turbulence are key considerations.

• Boundary layer control techniques manage airflow around the projectile.

Drag Coefficient Variation with Velocity

• The drag coefficient is not constant; it varies with velocity.

• Example: Spitzer-type bullet.

  • Significant change occurs around Mach 1 (speed of sound, approximately 340 m/s).

  • Sonic boom and pressure effects occur as the bullet goes supersonic.

  • Subsonic ammunition is quieter because it doesn't break the sound barrier.

  • A massive increase in drag coefficient is seen as the bullet goes supersonic, followed by a tail-off at higher velocities.

Supersonic Effects

• Shockwaves and wave drag become significant at supersonic speeds.

• Aerodynamic design adaptations are needed to mitigate these effects.

Sectional Density

• Bullets with high sectional density carry more impact energy at a given range.

• Relates to the concept of an elephant vs. a mouse. Higher mass has more impact.

Impact Energy

• Sectional density directly influences a projectile's ability to penetrate a target.

• High sectional density bullets retain more energy over long distances.

Sectional Density Equation

• $S = \frac{m}{d^2}$

  • $S$ = Sectional Density

  • $m$ = Mass

  • $d$ = Diameter

Variable Effects

• Understanding how mass and diameter interact to determine sectional density.

• Optimizing these parameters enhances projectile performance.

• Increasing mass increases sectional density.

• Decreasing diameter increases sectional density.

• Depleted uranium example: small, heavy bullets.

• Sectional density is important for impact and penetration.

Material Selection

• Choice of material affects both mass and density.

• Depleted uranium's high density makes it ideal for armor-piercing rounds.

Sectional Density Definition Note

• Alternative definition (mass/cross-sectional area) exists, but in ballistics, diameter is preferred.

Definition Context

• Using diameter provides a more practical measure for ballistic calculations.

• Cross-sectional area is more commonly used in other engineering fields.

Ballistic Coefficient

• A measure of aerodynamic forces exerted on a bullet in flight (CB).

• Specific to individual bullet design and size.

• Can be used to calculate real-time trajectory values.

• Relates bullet's sectional density to its drag coefficient.

• Combines design, mass, diameter, air density, and velocity.

Trajectory Prediction

• Ballistic coefficient is essential for accurate trajectory modeling.

• Incorporates various factors affecting a bullet's flight path.

Ballistic Coefficient Standardization

• Compares the drag coefficient against a standard bullet.

• G1 projectile (drag coefficient = 0.5191) is used for comparison.

• Other standards exist (e.g., G7), and the choice affects values.

Standard Projectiles

• Different standards cater to various bullet shapes and sizes.

• Selecting the appropriate standard enhances prediction accuracy.

Calculating Ballistic Coefficient

• $CB = \frac{C<em>G}{C</em>D} * S$

  • $C_G$ = Drag coefficient of the G1 bullet (0.5191).

  • $C_D$ = Drag coefficient of the bullet.

  • $S$ = Sectional density.

Equation Components

• Precisely defining each parameter ensures correct BC calculation.

• Units must be consistent for accurate results.

• Units:

  • Mass in kilograms.

  • Diameter in meters.

  • Sectional density in kilograms per square meter.

  • Ballistic coefficient in kilograms per square meter.

Unit Conversion

• Accurate unit conversion is critical for valid calculations.

• Consistent units simplify the equation and prevent errors.

Process Summary

• Determine drag coefficient using wind tunnel experiments.

• Plug values into equations with correct units.

• Calculate drag coefficient and ballistic coefficient.

• Vary velocity to get a range of coefficients.

• Do not average drag coefficients; they are specific to a particular velocity.

Iterative Calculation

• Calculating across a range of velocities provides a more complete data set.

• Avoid averaging to maintain accuracy at specific velocities.

Ballistic Coefficient Values

• Commercial ammunition typically falls in the range of 50 to 500 kg/m^2.

• Depleted uranium can result in much higher values.