Aim of the Course:

Aim of the course is to obtain knowledge in external ballistics of projectiles of various kinds – spin stabilized and fin stabilized, with or without rocket motor. It is a basic course. The lectures covers classical theory of projectile motion. Numerical examples are run to illustrate the typical values of aerodynamic characteristics. The attendees will obtain copy of the Matlab program EulerModel.m which will be extensively used throughout the course. Besides, one example showing motion of projectile around the center of mass will be ran by the program ProMoS-6DoF (see ProSTools section).

Who should attend?

The course is designed for students, engineers – researchers in the research institutions. It is assumed that attendants of course have good knowledge in general projectile aerodynamics (for basic knowledge see course Classical Aerodynamics of Projectiles and Rockets), mathematics and computer skills.


Duration is two weeks (twelve working days); 48 lectures (one lecture duration 45min), but other arrangement is possible.

Course Outline

1. Review of projectiles and rockets
Small caliber bullets. Gun projectiles/ shells. Mortar shells/ bombs. Light anti-tank rockets. Artillery rockets. Fin and Spin Stabilization.
2. Review of Basic Aerodynamics
Type of aerodynamic forces: shear/friction and normal forces. Boundary layer and free flow. Flow classification – Mach number. Nature of drag and lift force on bodies and wings. Body angle of attack and local angle of incidence. Shock waves. Based drag. Influence of Mach and Reynolds number.
3. Projectile Aerodynamic Coefficients
Aerodynamic configurations. Body velocities and angular rates. Aerodynamic angles. Definition of aerodynamic coefficients. Drag, Lift, Axial and Normal force and their coefficients. Pitching moment due to angle of attack and due to pitching rate. Center of pressure and static stability. Rolling moments. General nonlinear and linear presentation of aerodynamic coefficients.
4. Drag Minimization
Importance of drag. General rules in shaping for drag minimization. Wave drag and base drag reduction.
5. Atmosphere Characteristics and Modeling
Basic atmospheric quantities and their influences on aerodynamic forces. Vertical equilibrium. And structure of calm atmosphere. Standard/normal atmosphere. Virtual temperature. Simulation of atmospheric quantities for flight dynamics.
6. Earth Models - Gravity and Coriolis Force
Flat spherical and ellipsoidal earth models. Earth rotation. Spherical coordinates. Gravity. Coriolis force. Influence of earth curvature to the range.
7. Rocket Motor Characteristics Modeling
Rocket motor thrust. Influence of ambient conditions. Total and specific impulse. Effective exhaust speed. Thrust misalignment.
8. Euler’s Point Mass Model
Motion of projectile in vertical plane - Point mass model. Derivation of the Euler’s equations. Ballistic coefficient. Examples of trajectory calculations by Matlab program TwoDoFSim.
9. Analytical Solution of the Euler Differential Equations
Tsiolkovsky formula. Changing of independent variable. Simplification of the equations for the flat trajectories and analytical solution. Influence of drag and gravity.
10. SixDoF Equations of Projectile Motion
Disadvantages of point mass model. Six degree of freedom (SixDoF) model. Earth fixed and body fixed coordinate system. Transformation matrix. Review of forces and moments. SixDoF equations presentation and discussions. Examples of trajectory calculations by program ProMoS_6DoF.
11. Linearization of the Equations
Assumptions for the linearization. Review of linearized equations. Equilibrium linearized equations and equilibrium yaw angle.
12. Motion around Centre of Mass
Roll equation. Equations of lateral motion around centre of mass. Small amplitude motion of slowly spinning finned projectile. Linear motion of spin stabilized projectile.
13. Stability of Motion
State of the problem. Static stability. Gyroscopic stability. Dynamic stability and stability diagram. Magnus instability.
14. Modified Point Mass Model
Disadvantages of SixDoF model. Modified point mass model - review of the equations. Advantages and disadvantages.
15. Differential Coefficients and Basis of Calculating of Dispersion
Differential coefficients and adjustments/corrections. Review of differential coefficients. Method of calculation of differential coefficients. Examples of differential coefficients calculation by Matlab program TwoDoFSim. Dispersion of trajectories. Causes of dispersion. Mean point and probable errors of coordinates at impact point.
16. Basic Solution of Firing Problem
Fire problem definition. Algorithm for the solution of equations. Examples of firing elements calculation by Matlab program TwoDoFSim.

Lecturer: Dr Miodrag Curcin