Fa2004 16.33312-2 Equations of Motion Analysis to date has assumed that the atmosphere is calm and fixed Rarely true since we must contend with gusts and winds Need to understand how these air motions impact our modeling of the aircraft Must modify aircraft equations of motion since the aerodynamic forces and moments are functions of the relative motion between the aircraft and the atmosphere and not of the inertial velocities Thus the LHS of the dynamics equations(f= ma)must be written in terms of the velocities relative to the atmosphere If u is the aircraft perturbation velocity(X direction ), and ug is the gust velocity in that direction, then the aircraft velocity with respect to the atmosphere is Now rewrite aerodynamic forces and moments in terms of aircraft velocity with respect to the atmosphere(see 4-11) OX OX OX △X a-a aU )+am(0-y)+ OX OX 0X9 (q-qg)+…+ 0+..+-6+△XC 0Q The gravity terms ae and control terms △Xe=X66e+X820p stay the sameFall 2004 16.333 12–2 Equations of Motion • Analysis to date has assumed that the atmosphere is calm and fixed – Rarely true since we must contend with gusts and winds – Need to understand how these air motions impact our modeling of the aircraft. • Must modify aircraft equations of motion since the aerodynamic forces and moments are functions of the relative motion between the aircraft and the atmosphere, and not of the inertial velocities. – Thus the LHS of the dynamics equations (F� = m�a) must be written in terms of the velocities relative to the atmosphere. – If u is the aircraft perturbation velocity (X direction), and ug is the gust velocity in that direction, then the aircraft velocity with respect to the atmosphere is ua = u − ug • Now rewrite aerodynamic forces and moments in terms of aircraft velocity with respect to the atmosphere (see 4–11) ∂X ∂X ∂X ΔX = (u − ug) + (w − wg) + ∂ ˙ ( ˙ w˙ g) ∂U ∂W W w − ∂X ∂X ∂Xg + (q − qg) + . . . + θ + . . . + θ + ΔXc ∂Q ∂Θ ∂Θ – The gravity terms ∂Xg and control terms ∂Θ ΔXc = Xδeδe + Xδpδp stay the same