Steady flight, unaccelerated flight, or equilibrium flight is a special case in flight dynamics where the aircraft's linear and angular velocity are constant in a body-fixed reference frame. Basic aircraft maneuvers such as level flight, climbs and descents, and coordinated turns can be modeled as steady flight maneuvers. Typical aircraft flight consists of a series of steady flight maneuvers connected by brief, accelerated transitions. Because of this, primary applications of steady flight models include aircraft design, assessment of aircraft performance, flight planning, and using steady flight states as the equilibrium conditions around which flight dynamics equations are expanded.
Reference frames
Steady flight analysis uses three different reference frames to express the forces and moments acting on the aircraft. They are defined as:
Earth frame (assumed inertial) Origin - arbitrary, fixed relative to the surface of the Earth xE axis - positive in the direction of north yE axis - positive in the direction of east zE axis - positive towards the center of the Earth Body frame Origin - airplane center of gravity xb (longitudinal) axis - positive out the nose of the aircraft in the plane of symmetry of the aircraft zb (vertical) axis - perpendicular to the xb axis, in the plane of symmetry of the aircraft, positive below the aircraft yb (lateral) axis - perpendicular to the xb,zb-plane, positive determined by the right-hand rule (generally, positive out the right wing) Wind frame Origin - airplane center of gravity xw axis - positive in the direction of the velocity vector of the aircraft relative to the air zw axis - perpendicular to the xw axis, in the plane of symmetry of the aircraft, positive below the aircraft yw axis - perpendicular to the xw,zw-plane, positive determined by the right hand rule (generally, positive to the right) The Euler angles linking these reference frames are:
Earth frame to body frame: yaw angle ψ, pitch angle θ, and roll angle φ Earth frame to wind frame: heading angle σ, flight-path angle γ, and bank angle μ Wind frame to body frame: angle of sideslip β, angle of attack α (in this transformation, the angle analogous to φ and μ is always zero)
Force balance and the steady flight equations The forces acting on an aircraft in flight are the weight, aerodynamic force, and thrust. The weight is easiest to express in the Earth frame, where it has magnitude W and is in the +zE direction, towards the center of the Earth. The weight is assumed to be constant over time and constant with altitude. Expressing the aerodynamic force in the wind frame, it has a drag component with magnitude D opposite the velocity vector in the −xw direction, a side force component with magnitude C in the +yw direction, and a lift component with magnitude L in the −zw direction. In general, the thrust can have components along each body frame axis. For fixed wing aircraft with engines or propellers fixed relative to the fuselage, thrust is usually closely aligned with the +xb direction. Other types of aircraft, such as rockets and airplanes that use thrust vectoring, can have significant components of thrust along the other body frame axes. In this article, aircraft are assumed to have thrust with magnitude T and fixed direction +xb. Steady flight is defined as flight where the aircraft's linear and angular velocity vectors are constant in a body-fixed reference frame such as the body frame or wind frame. In the Earth frame, the velocity may not be constant since the airplane may be turning, in which case the airplane has a centripetal acceleration ( V cos γ ) 2 R {\displaystyle {\frac {(V\cos {\gamma })^{2}}{R}}} in the xE-yE plane, where V {\displaystyle V} is the magnitude of the true airspeed and R {\displaystyle R} is the turn radius. This equilibrium can be expressed along a variety of axes in a variety of reference frames. The traditional steady flight equations derive from expressing this force balance along three axes: the xw-axis, the radial direction of the aircraft's turn in the xE-yE plane, and the axis perpendicular to xw in the xw-zE plane,
T cos α cos β − W sin γ − D = 0 ( x w -axis ) , {\displaystyle T\cos {\alpha }\cos {\beta }-W\sin {\gamma }-D=0\quad (x_{w}{\text{-axis}}),}
C cos μ + L sin μ + T ( sin α sin μ + cos α cos μ sin β ) = W g ( V cos γ ) 2 R ( x E - y E plane radial direction ) , {\displaystyle C\cos {\mu }+L\sin {\mu }+T(\sin {\alpha }\sin {\mu }+\cos {\alpha }\cos {\mu }\sin {\beta })={\frac {W}{g}}{\frac {(V\cos {\gamma })^{2}}{R}}\quad (x_{E}{\text{-}}y_{E}{\text{ plane radial direction}}),}
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