## Flight Dynamics

Flight dynamics is the study of the performance, stability, and control of vehicles flying through the air or in outer space. It is concerned with how forces acting on the vehicle influence its speed and attitude with respect to time. In fixed-wing aircraft, the changing orientation of the vehicle with respect to the local air flow is represented by two critical parameters, angle of attack (“alpha”) and angle of sideslip (“beta”). These angles describe the vector direction of airspeed, important because they are the principal source of modulations in the aerodynamic forces and moments applied to the aircraft.

Spacecraft flight dynamics involve three forces: propulsive (rocket engine), gravitational, and lift and drag (when traveling through the earth’s or any other atmosphere). Because aerodynamic forces involved with spacecraft flight are very small, this leaves gravity as the dominant force. Aircraft and spacecraft share a critical interest in their orientation with respect to the earth horizon and heading, and this is represented by another set of angles, “yaw”, “pitch”, and “roll”, which angles match their colloquial meaning, but also have formal definition as an Euler sequence. These angles are the product of the rotational equations of motion, where orientation responds to torque, just as the velocity of a vehicle responds to forces. For all flight vehicles, these two sets of dynamics, rotational and translational, operate simultaneously and in a coupled fashion to evolve the vehicle’s state (orientation and velocity) trajectory.

Three right-handed, Cartesian coordinate systems see frequent use in flight dynamics. The first coordinate system has an origin fixed in the reference frame of the Earth:

- Earth frame
- Origin – arbitrary, fixed relative to the surface of the Earth
*x*axis – positive in the direction of north_{E}*y*axis – positive in the direction of east_{E}*z*axis – positive towards the center of the Earth_{E}

In many flight dynamics applications, the Earth frame is assumed to be inertial with a flat *x _{E}*,

*y*-plane, though the Earth frame can also be considered a spherical coordinate systemwith origin at the center of the Earth.

_{E}The other two reference frames are body-fixed, with origins moving along with the aircraft, typically at the center of gravity. For an aircraft that is symmetric from right-to-left, the frames can be defined as:

- Body frame
- Origin – airplane center of gravity
*x*axis – positive out the nose of the aircraft in the plane of symmetry of the aircraft_{b}*z*axis – perpendicular to the_{b}*x*axis, in the plane of symmetry of the aircraft, positive below the aircraft_{b}*y*axis – perpendicular to the_{b}*x*,_{b}*z*-plane, positive determined by the right-hand rule (generally, positive out the right wing)_{b}

- Wind frame
- Origin – airplane center of gravity
*x*axis – positive in the direction of the velocity vector of the aircraft relative to the air_{w}*z*axis – perpendicular to the_{w}*x*axis, in the plane of symmetry of the aircraft, positive below the aircraft_{w}*y*axis – perpendicular to the_{w}*x*,_{w}*z*-plane, positive determined by the right hand rule (generally, positive to the right)_{w}

Asymmetric aircraft have analogous body-fixed frames, but different conventions must be used to choose the precise directions of the *x* and *z* axes.

The Earth frame is a convenient frame to express aircraft translational and rotational kinematics. The Earth frame is also useful in that, under certain assumptions, it can be approximated as inertial. Additionally, one force acting on the aircraft, weight, is fixed in the +*z _{E}* direction.

The body frame is often of interest because the origin and the axes remain fixed relative to the aircraft. This means that the relative orientation of the Earth and body frames describes the aircraft attitude. Also, the direction of the force of thrust is generally fixed in the body frame, though some aircraft can vary this direction, for example by thrust vectoring.