Speed & Pitch Rate Derivatives (u, q)
This non-dimensional derivative measures axial force change with forward speed and simplifies to approximately -2CD0 for a subsonic jet aircraft in gliding flight.
What is CXu (Speed Damping Derivative)?
The physical cause for downwash lag at the horizontal tail plane, represented by the time delay equation
What is the convection time required for trailing vortex wake changes shed by the wing to travel downstream to the tail?
Known as weathercock stability, this derivative must be positive (> 0) for directional static stability
What is yawing moment due to sideslip
Generated primarily across the main wingspan, this derivative is always negative (< 0) and acts as the primary aerodynamic roll damping.
roll damping derivative
The characteristic reference length used to non-dimensionalize longitudinal pitching moments (M) and pitch rates (q).
What is the Mean Aerodynamic Chord
This primary airplane component accounts for over 90% of the total pitch damping derivative (Cmq).
What is the Horizontal Tail Plane?
The dimensionless rate parameter used to normalize the rate of change of angle of attack
What is hat{\dot{\alpha}}
Commonly called the dihedral effect, lateral static stability requires this derivative to be negative (< 0)
rolling moment due to sideslip
Contributed mainly by the vertical tail, this rate derivative provides directional damping that resists yawing velocity r
yaw damping derivative
The characteristic reference length used to non-dimensionalize rolling moments (L), yawing moments (N), roll rates (p), and yaw rates (r).
What is the Wingspan (b)?
Differentiating the subsonic Prandtl-Glauert compressibility rule with respect to Mach number yields this expression for
What is CLu?
These two negative derivatives combine directly to provide the total aerodynamic damping governing the short-period longitudinal oscillation mode.
What are pitch damping and downwash lag pitching moment.(Cmq and Cm .alpha)
This primary geometric feature increases local angle of attack on the windward wing during a sideslip, providing the main stabilizing contribution to C_{l_\beta}.
What is wing geometric dihedral angle?
As an aircraft rolls at rate p, higher induced drag on the advancing wing produces this adverse cross-coupling derivative.
yawing moment due to roll rate
Defined as VH = {S_t l_t}/{S c}, this dimensionless geometric parameter directly scales horizontal tail contributions to stability derivatives
What is the Horizontal Tail Volume Ratio?
The mathematical equation for the horizontal tail contribution to pitch damping in terms of tail volume ratio VH and tail moment arm ratio.
What is Cmq,tail?
The sidewash convection delay at the vertical tail caused by a rate of sideslip generates this positive derivative, which enhances directional damping during Dutch roll.
What is yawing moment due to rate of sideslip?
A high-wing fuselage placement creates a favorable stabilizing contribution to C_{l_\beta} due to this flow distortion phenomenon around the fuselage.
What is crossflow / fuselage interference effect?
Using wing strip theory, the wing roll damping derivative C_{l_p} is evaluated by integrating this expression across the span from -b/2 to +b/2.
What is C_{l_p}
Deflecting the rudder (delta_r) to generate vertical tail side force produces this non-dimensional control derivative, which is inherently negative by sign convention.
rudder control power
Known as "Mach Tuck," this speed derivative becomes strongly negative at high subsonic Mach numbers due to shifts in aerodynamic center and airframe aeroelastic bending.
Speed Pitching Moment Derivative (Cmu)
The full mathematical formula for the tail contribution to the downwash lag pitching moment derivative
What is C_{m_{\dot{\alpha},\text{tail}}}
Unlike the vertical tail, the fuselage and engine nacelles make this type of contribution to directional static stability
What is a destabilizing contribution?
This positive cross-coupling derivative (C_{l_r} > 0) arises because the advancing wing panel in a yawing motion flies at a higher local airspeed (u_0 + r y) than the retreating wing.
rolling moment due to yaw rate
The conversion equation used to calculate the dimensional pitch rate derivative M_q from the non-dimensional coefficient C_{m_q}.
What is M_q