External ballistics.

The program is intended for simple simulation of the firing of various types of bullets. The program solves the following tasks:

  1. Definition of the bullet.
  2. Selection from the list of typical bullets.
  3. Calculation and selection of typical ballistic coefficients.
  4. Calculation of the forces, velocities, trajectories, angles and other external ballistic parameters.
  5. Graphic representation of the results of one or two bullets.

Data, procedures, algorithms and informations from professional literature and the catalogues of the ammunition manufacturers are used in the calculation.

Control, structure and syntax of calculations.

Information on the syntax and control of the calculation can be found in the document "Control, structure and syntax of calculations".

Information on the project.

Information on the purpose, use and control of the paragraph "Information on the project" can be found in the document  "Information on the project".

Theory.

The following formulas are used in the calculation:

Air pressure related to the elevation:

p [kPa] = p0 / (1 - (0.0065 * h) / (t+0.0065 * h + 273.15)) (-5.257)

p0 ... Air pressure at sea level (101.32) [kPa]
h ..... Altitude [m]
t ...... Temperature [°C]

Air density:

Q [kg/m^3] = (p * 1000 * 1 * 0.02895) / (8.314 * (273 + t))

Aerodynamic resistance:

Fd [N] = 0.5 * Q * T * Cx(M) * S * v^2

Q ... Air density [kg/m^3]
T ... Bullet coefficient of form
Cx(M) ... The air resistance coefficient depends on the velocity and type of the bullet
S ... Facing [m^2]
v ... bullet velocity [m/s]

Ballistic coefficient:

BC [lb/in2] = m / (d^2 · T)

m ... Bullet mass [lb]
d ... Bullet diameter [in]
T ... Coefficient of form

Coefficient of form:

T = m / BC / d^2

m ... Bullet mass [lb]
d ... Bullet diameter [in]
BC ... Ballistic coefficient [lb/in^2]

Gravitational force:

Fg [N] = m * g

m ... Bullet mass [kg]
g ... Gravitational acceleration [9.81m/s^2]

Air-flow:

It is included in the calculation of aerodynamic resistance and position of the bullet.

Coriolis force:

Not included in the calculation.

Derivation of the bullet (Magnus effect):

Not included in the calculation.

The calculation is based on simple composition of forces, velocities and gradual calculation of parameters over a given interval dt. The interval can be set manually (calculation runs in 10,000 steps) and/or select an estimate for the entered bullet, entered distance and elevation.

Process of calculation.

  1. Select the calculation units and environmental parameters [1.0]
  2. Select the sample bullet, or define your own parameters [2.1-2.10]
  3. Enter the angle [2.12] or distance and evaluation [2.13, 2.14]
  4. Run the calculation of the calculation of the ballistic curve.

Setting of environment units and parameters. [1]

In this paragraph, select the units, temperature , altitude and wind speed.

1.1 Calculation units.

Select the desired calculation unit system from the selection list. Upon switching units, all the input values shall change immediately.

1.3 Air temperature.

Select air temperature, and upon pressing the button on the right side, the temperature value is entered according to MSA (simplified model of the Earth’s atmosphere). At the ideal sea level, the air temperature is 15°C and pressure 1013.2hPa. Up to 11km, the temperature drops with increasing height at the rate of 0.0065°C/m).

1.4 Air pressure.

Upon picking the button, you can enter any value independent of [1.1-1.3].

1.7 Wind speed.

Define wind action as:

1.The wind speed and angle in relation to the firing axis.
2.Wind speed at the defined points on axes x, y, z.

Use the switch on the right side to select the input method.

The buttons are used for conversion and transfer of the values from one system to another.

Note: The wind speed in the calculation is considered as a constant for the given section (e.g. x1-x2).

1.8 Number of points.

It states the number of defined points, which shall be used in the calculation. The last entered value is valid until the end of the flight path.

Calculation of the ballistic parameters. [2]

In this paragraph, you select/enter the firing parameters (diameter, weight, initial velocity, BC) and the required parameters for calculation of the ballistic curve (firing angle/distance, super-elevation). The calculation is executed by pressing the “Calculate” button above the graph of values.

Note: After changing the units, it is necessary to restart the calculation.

2.1, 2.2 Selection of the bullet.

From the selection list, pick the sample bullet (weapon). The parameters can be modified after selecting the button on the right side of the selection list.

2.8 Form factor.

It gives the degree to which bullet is worse (better) than the sample bullet from the selected aerodynamic resistance function [2.10].

2.9 Ballistic coefficient.

The BC is related to the given aerodynamic resistance function, most often to G1. The BC value can usually be obtained from the manufacturer.

2.10 Drag function.

The drag coefficient depends on the velocity of the bullet. The most commonly used function C(M) is G1, for which most manufacturers also state the ballistic coefficient. Select the aerodynamic resistance function from the list for bullet A and B. You can define the actual progress of function C(M) on sheet "DragC(M)". The velocity is given in the table and graph in relation to the speed of sound.

The value of the aerodynamic resistance function related to the bullet velocity (M) is given in the graph.

2.11 Selection of calculation for:

From the list, select if the solution and curve for only bullet A and/or the curves of both bullets should be visualised.

2.12 Firing angle.

You have two potential solutions. You make your selection with the switch on the right side.

1. When selecting the impact angle, the complete path of the bullet is calculated for 10,000 points with selected calculation step dt [2.16]. By setting dt, you set the length of the trajectory, for which the parameters are calculated.
2. When selecting the sighting distance, the value of dt is estimated and the results are shown and cut to use a flat graph.

2.13 Zero range.

Enter the sighting distance and sighting elevation [2.14].

Note: It is suitable to select automatic estimate of dt per line [2.16]

2.15 Calculation step.

You can enter the calculation step in [ms] using the slide, or by directly entering the value. The calculation step determines the time interval for which the flight of the bullet shall be calculated.

Note: When doing the calculation for the given angle, you can change the scope of the calculation of the bullet trajectory by suitable selection of dt.

2.18 Number of calculation steps.

During the calculation for the given sighting distance [2.13, 2.14], the number of steps used is displayed. Calculation for the given firing angle [2.12] is done for 10,000 steps.

2.19 Bullet parameters.

Lines [2.20, 2.21] show the input parameters for bullet A and B.

2.22 Results graph.

Select the graph you desire to visualise from the list.

The numerical values of the graph are shown on the sheet "Data1".

Note: After changing the units, it is necessary to restart the calculation.

Setting calculations, change the language.

Information on setting of calculation parameters and setting of the language can be found in the document "Setting calculations, change the language".

Workbook (calculation) modifications.

General information on how to modify and extend calculation workbooks is mentioned in the document "Workbook (calculation) modifications".