Report on Artillery Projectile Trajectory Optimization Script
Report on Artillery Projectile Trajectory Optimization Script
1. Introduction
This script is designed (listed at the end) to calculate the optimal launch parameters for an artillery projectile to hit a specified target. The objective is to determine the ideal launch angle and initial velocity required to strike a target at a given distance and height, considering real-world physical and environmental factors such as air resistance, wind, the Coriolis effect, and the Magnus effect. The script leverages mathematical models and numerical optimization to minimize the error between the simulated impact point and the target position.
2. Purpose and Objective
The primary goal of the script is to compute the best launch parameters (angle and initial velocity) for different types of projectiles used in European artillery systems. The script simulates the projectile’s flight path, adjusting for complex atmospheric and ballistic influences, to minimize the discrepancy between the simulated impact and the target location.
Key Objectives:
Accurately model the physical forces acting on the projectile during flight.
Minimize the error between the computed impact point and the target.
Adapt the solution to various projectile shapes and types used in European artillery.
Provide realistic launch conditions considering environmental factors such as wind and humidity.
3. Scope and Application
The script is applicable to a wide range of artillery systems and projectile types, including but not limited to:
Military howitzers and cannons.
Naval artillery systems.
Mortars and field guns.
Large-caliber sniper rifles and other ballistic projectiles.
The script is designed to work under typical European atmospheric conditions, which includes average humidity, temperature, and wind profiles. It can be customized for different environmental settings and projectile designs.
4. Methodology
4.1. Input Parameters
The script prompts the user to input the following data:
Mass – Mass of the projectile (in kg).
Diameter – Diameter of the projectile (in meters).
Length – Length of the projectile (in meters).
Spin Rate – Rate of rotation of the projectile (in revolutions per second).
Magnus Coefficient – Coefficient defining the Magnus effect on the projectile.
Shape – Type of projectile (sphere, cylinder, ogive, teardrop, shell, bullet).
Target Distance – Horizontal distance to the target (in meters).
Target Height – Vertical height of the target (in meters).
Latitude – Latitude of the launch site (in degrees).
4.2. Physical and Environmental Models
The script incorporates several physical and environmental models to simulate realistic projectile motion:
4.2.1. Air Density Calculation
Air density is calculated using the barometric formula, accounting for humidity and altitude variations.
4.2.2. Drag Coefficient
Drag coefficient is computed based on the Mach number (ratio of projectile speed to speed of sound) and the Reynolds number, adjusting for laminar and turbulent flow conditions.
4.2.3. Wind Influence
Wind speed variation with altitude is modeled using a power-law profile.
4.2.4. Magnus Effect
The Magnus effect introduces a lift force due to the projectile's spin, altering the trajectory in both horizontal and vertical directions.
4.2.5. Coriolis Effect
The Coriolis effect introduces a deflection due to the Earth's rotation, influenced by the latitude of the launch site.
4.3. Numerical Optimization
The script uses the Nelder-Mead method (a gradient-free optimization algorithm) to minimize the error between the simulated and target impact points.
The optimization variables are:
Launch angle (in degrees).
Initial velocity (in m/s).
The cost function computes the squared error between the simulated and target impact positions.
5. Output
After successfully computing the optimal parameters, the script provides the following output:
Optimal Launch Angle – The ideal angle of elevation for the projectile.
Optimal Initial Velocity – The necessary initial velocity to reach the target.
Example output:
Ángulo óptimo: 45.32 grados
Velocidad inicial óptima: 350.87 m/s
6. Key Features
✅ Supports multiple projectile shapes common in European artillery.
✅ Accounts for complex atmospheric effects, including wind, humidity, and air density.
✅ Models both Magnus and Coriolis forces for high-accuracy prediction.
✅ Allows easy customization for different artillery systems and environmental conditions.
✅ Provides real-time feedback on calculated parameters.
7. Assumptions and Limitations
Assumptions
The Earth is treated as a sphere for the Coriolis calculation.
The projectile is modeled as a rigid body.
Airflow around the projectile is assumed to transition from laminar to turbulent at a defined Reynolds number.
Limitations
Does not account for projectile deformation or fragmentation.
Ignores localized weather variations (e.g., gusts, turbulence).
Assumes a fixed gravitational acceleration value.
8. Recommendations and Future Improvements
Introduce real-time weather data integration for more accurate wind and humidity modeling.
Expand the range of projectile types by incorporating more complex shapes and aerodynamic profiles.
Improve the numerical solver to handle edge cases and faster convergence.
Allow for multi-stage projectiles (e.g., rockets) and controlled thrust scenarios.
9. Conclusion
This script provides a robust and accurate solution for calculating optimal launch parameters for artillery projectiles under realistic physical and environmental conditions. Its flexibility and precision make it suitable for military and defense applications across various European artillery systems. The combination of detailed aerodynamic modeling and efficient numerical optimization ensures high-accuracy targeting and improved operational performance.
Script for Python
import mathfrom scipy.optimize import minimize
# Constantes físicas
G = 9.80665
OMEGA = 7.2921e-5
R = 287.05
R_v = 461.5
GAMMA = 1.4
L = 0.0065
T0 = 288.15
P0 = 101325
MU = 1.81e-5
RE_CRIT = 5e5
K_TRANSITION = 0.00001
# Modelo de viento (condiciones europeas)
V0_WIND = 3 # Velocidad media del viento en Europa
H0 = 1000
ALPHA_WIND = 0.12
# Humedad media en Europa
HUMIDITY = 0.6
# Coeficientes de resistencia por forma (ajustado para munición europea)
FORM_COEFFICIENTS = {
'sphere': 0.47,
'cylinder': 0.82,
'ogive': 0.3,
'teardrop': 0.04,
'shell': 0.35, # Proyectil tipo obús
'bullet': 0.29 # Proyectil tipo bala
}
def saturation_pressure(temp):
return 610.78 * math.exp((17.27 * (temp - 273.15)) / (temp - 35.85))
def air_density(altitude):
temp = T0 - L * altitude
p_v = HUMIDITY * saturation_pressure(temp)
p_d = P0 * (1 - L * altitude / T0) ** ((G * 0.029) / (R * L)) - p_v
rho = (p_d / (R * temp)) + (p_v / (R_v * temp))
return rho
def speed_of_sound(temp):
return math.sqrt(GAMMA * R * temp)
def wind_speed(altitude):
return V0_WIND * (altitude / H0) ** ALPHA_WIND
def drag_coefficient(mach, reynolds, shape):
Cd_lam = FORM_COEFFICIENTS[shape]
Cd_turb = Cd_lam * 0.7 if Cd_lam > 0.1 else Cd_lam * 1.2
transition = 1 / (1 + math.exp(-K_TRANSITION * (reynolds - RE_CRIT)))
Cd = Cd_lam + transition * (Cd_turb - Cd_lam)
return Cd
def coriolis_effect(vx, vy, latitude):
lat_rad = math.radians(latitude)
ax = 2 * OMEGA * vy * math.sin(lat_rad)
ay = 2 * OMEGA * vx * math.cos(lat_rad)
return ax, ay
def magnus_force(vx, vy, rho, spin_rate, mass, s_magnus):
v = math.sqrt(vx**2 + vy**2)
Fm = s_magnus * rho * v * spin_rate
ax = Fm * (-vy / mass)
ay = Fm * (vx / mass)
return ax, ay
def distance_error(vars, target_distance, target_height, latitude, params):
theta, v0 = vars
theta = math.radians(theta)
vx = v0 * math.cos(theta)
vy = v0 * math.sin(theta)
x, y, altitude = 0, 0, 0
dt = 0.01
while y >= 0:
rho = air_density(altitude)
temp = T0 - L * altitude
sound_speed = speed_of_sound(temp)
v = math.sqrt(vx**2 + vy**2)
mach = v / sound_speed
reynolds = (rho * v * params['length']) / MU
Cd = drag_coefficient(mach, reynolds, params['shape'])
drag = 0.5 * rho * Cd * params['area'] * v**2
wind = wind_speed(altitude)
ax = -drag * vx / params['mass']
ay = -drag * vy / params['mass'] - G
magnus_ax, magnus_ay = magnus_force(vx, vy, rho, params['spin_rate'], params['mass'], params['s_magnus'])
cor_x, cor_y = coriolis_effect(vx, vy, latitude)
ax += magnus_ax + cor_x
ay += magnus_ay + cor_y
vx += ax * dt
vy += ay * dt
x += vx * dt
y += vy * dt
altitude = y
error = (x - target_distance) ** 2 + (y - target_height) ** 2
return error
def calculate_launch_parameters(target_distance, target_height, latitude, params):
initial_guess = (45, 300)
result = minimize(
distance_error,
initial_guess,
args=(target_distance, target_height, latitude, params),
method='Nelder-Mead'
)
if result.success:
theta_opt, v0_opt = result.x
return math.degrees(theta_opt), v0_opt
else:
raise ValueError("No se pudo encontrar una solución válida")
def main():
# Selección dinámica de parámetros del proyectil
params = {
'mass': float(input("Masa del proyectil (kg): ")),
'diameter': float(input("Diámetro del proyectil (m): ")),
'length': float(input("Longitud del proyectil (m): ")),
'spin_rate': float(input("Tasa de rotación (rev/s): ")),
's_magnus': float(input("Coeficiente Magnus: ")),
'shape': input("Forma del proyectil (sphere/cylinder/ogive/teardrop/shell/bullet): ")
}
params['area'] = math.pi * (params['diameter'] / 2) ** 2
target_distance = float(input("Distancia al objetivo (m): "))
target_height = float(input("Altura del objetivo (m): "))
latitude = float(input("Latitud (grados): "))
theta_opt, v0_opt = calculate_launch_parameters(target_distance, target_height, latitude, params)
print(f"Ángulo óptimo: {theta_opt:.2f} grados")
print(f"Velocidad inicial óptima: {v0_opt:.2f} m/s")
if __name__ == "__main__":
main()
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