Flight Dynamics (spacecraft) - Powered Flight

Powered Flight

Flight calculations are made quite precisely for space missions, taking into account such factors as the Earth's oblateness and non-uniform mass distribution; gravitational forces of all nearby bodies, including the Moon, Sun, and other planets; and three-dimensional flight path. For preliminary studies, some simplifying assumptions can be made (spherical, uniform planet; two-body patched conic approximation; and co-planar local flight path) with reasonably small loss of accuracy.

The general case of a launch from Earth must take engine thrust, aerodynamic forces and gravity into account. The acceleration equation can be reduced from vector to scalar form by resolving it into tangential and angular components. The two equations thus become:

where θ is the flight path angle from local vertical, α is the angle of attack, F is the engine thrust, D is drag, L is lift, r is the radial distance to the planet's center, and g is the acceleration due to gravity, which varies with the inverse square of the radial distance:

Mass, of course changes as propellant is consumed and rocket stages, engines or tanks are shed (if applicable). Integrating the two equations from time zero (when both v and θ are 0) gives planet-fixed values of v and θ at any time in the flight:

Finite element analysis can be used to numerically integrate often by breaking the flight into small time increments.

For most launch vehicles, relatively small levels of lift are generated, and a gravity turn is employed, depending mostly on the third term of the angle rate equation. But notice, when the angle is initially 0 immediately after liftoff, the only force which can cause the vehicle to pitch over is the engine thrust acting at a non-zero angle of attack (first term), until a non-zero pitch angle is attained. In the gravity turn, pitch-over is initiated by applying an increasing angle of attack (by means of gimballed engine thrust), followed by a gradual decrease in angle of attack through the remainder of the flight.

Once velocity and flight path angle are known, altitude and downrange distance are computed as:

The planet-fixed values of v and θ are converted to space-fixed (inertial) values with the following conversions:

where ω is the planet's rotational rate in radians per second, φ is the launch site latitude, and A_z is the launch azimuth angle.

Final v_s, θ_s and r must match the requirements of the target orbit as determined by orbital mechanics (see Orbital flight, below), where final v_s is usually the required periapsis (or circular) velocity, and final θ_s is 90 degrees. A powered descent analysis would use the same procedure, with reverse boundary conditions.