On One Numerical Method of Integrating the Dynamical Equations of Projectile Planar Flight Affected by Wind

Common way to integrate the dynamical equations of projectile planar motion introduces two Cartesian coordinates x(t) and y(t) and attack angle ϑ(t), all depending on time t, and three coupled ordinary differential equations (ODE) each nominally of II-nd order. It leads to inevitable computational complexities and accuracy risks. The method proposed excludes the time variable and diminishes the number of functions to n = 2: the attack angle ϑ(b) and intercept a(b) of the tangent to the trajectory at the point with slope b = tan θ with the θ being the inclination angle. This approach based on Legendre transformation makes the integration more convenient and reliable in the studied case of quadratic in speed aerodynamic forces i.e. drag, lifting force, conservative and damping momenta and the wind affecting the flight. The effective dimensionality of new ODE system is diminished by 2 units and its transcendence is eliminated by simple substitution η = sin ϑ. Also the method enables to obtain easily and reliably the projectile trajectories in conditions of tail-, head-or side wind. Investigated are main ranges of aerodynamic parameters at which takes place different behavior of the attack angle ϑ vs slope b including quasi-stabilization and aperiodic auto-oscillations. In addition, it was revealed non-monotonous behavior of speed with two minima while projectile descending if launched at the angles θ0 close to 90 ∘ . The numerical method may implement into quality improvement of real combat or sporting projectiles such as arch arrow, lance, finned rocket etc.

projectile, lifting force, quadratic drag, conservative/damping momenta, attack angle, projective-dual variables, wind