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Equations of Aircraft Motion Force Diagram Conventions θ line Chord T α α V T ght path li F θ M D Horizontal W Definitions V≡ θ≡ α≡ W ≡ L≡ D≡ M ≡ T≡ αT ≡ flight speed angle between horizontal & flight path angle of attack (angle between flight path and chord line) aircraft weight lift, force normal to flight path generated by air acting on aircraft drag, force along flight path generated by air acting on aircraft pitching moment propulsive force supplied by aircraft engine/propeller angle between thrust and flight path To derive the equations of motion, we apply K K (1) ∑ F = ma Note: we will not be including the potential for a yaw force. Applying (1) in flight path direction: dV ∑ F& = ma& = m dt and examining the force diagram ∑F & = T cos αT − D − W sin θ ⇒ T cos αT − D − W sin θ = m dV dt (2) Equations of Aircraft Motion Now applying (1) in ⊥ − direction to flight path ∑ F⊥ = ma⊥ = m V2 rc where rc ≡ radius of curvature of flight path ∑F ⊥ = L + T sin αT − W cos θ ⇒ L + T sin αT − W cos θ = m V2 rc (3) Equations (2) & (3) give the equations of motion for an aircraft (neglecting yawing motions) and are quite general. One important specific case of these equations is level, steady flight with the thrust aligned w/ the flight path. ⇒ dV = 0, rc → ∞, αT = 0, θ = 0 dt ⇒ T =D L =W Level, steady flight Moment definitions The pitching moment must be defined relative to a specific location. The two typical locations are: • leading edge • 1 c , quarter of mean chord 4 16.100 2002 2 Equations of Aircraft Motion Force & Moment Coefficients Typically, aerodynamicists use non-dimensional force & Moment coefficients. 1 ρ ∞V∞2 S 2 3D Drag/Lift coefficients D CD ≡ 1 2 ρ ∞V∞ S 2 L CL ≡ where ρ ∞ is freestream density V∞ is freestream velocity (flight speed) S is a reference area (problem dependent) q∞ ≡ 1 ρ ∞V∞2 Freestream dynamic pressure 2 The moment coefficient requires another length scale: CM ≡ A ref M 1 ρ ∞V∞2 S A ref 2 ≡ reference length scale (problem dependent) For 2-D problems, such as an airfoil, the forces are actually forces/length. So, for example 3D force L 2D force/length L′ D D′ Similarly, M → M ′ . The non-dimensional coefficients for 2-D are defined: 16.100 2002 3 Equations of Aircraft Motion Cl ≡ L′ 1 ρ ∞V∞2cref 2 D′ Cd ≡ 1 ρ ∞V∞2cref 2 M′ Cm ≡ 1 2 ρ ∞V∞2cref 2 where cref is a reference length such as the chord of an airfoil. Forces on Airfoils The forces & moments on airfoils are normalized by the chord length. So, Cl ≡ L′ 1 ρ ∞V∞2c 2 , Cd ≡ D′ 1 ρ ∞V∞2c 2 , Cm ≡ M′ 1 ρ ∞V∞2c 2 2 Force coefficients data is generally plotted in 2 forms: Lift curve Cl Cl max α 16.100 2002 4 Equations of Aircraft Motion Drag polar Cl Cd Cd min Cl Forces on Wings Cd min Cd x Wing planform y c(y) b c( y ) ≡ b≡ S≡ chord distribution wing span b 2 planform area = ∫ cdy −b 2 b2 S We can think of the 3-D or total lift on the wing as being the sum (i.e. integral) of the 2-D lift acting on the wing. A ≡ aspect ratio ≡ ⇒ L= b 2 ∫ L′ ( y ) dy where L′( y ) = lift distribution −b 2 16.100 2002 5 Equations of Aircraft Motion The average 2-D lift on the wing L′ can be defined: b 1 2 L L′ ≡ ∫ L′dy = b −b b 2 Plugging that into CL : b 2 CL ≡ ∫ L′dy L 1 ρ ∞V∞2 S 2 = − b 2 1 ρ ∞V∞2 S 2 = L′b 1 ρ ∞V∞2 S 2 But, the average chord or mean chord can be defined as: c≡ b 2 1 S cdy = ∫ b −b b 2 ⇒ CL ≡ L L′ = 1 1 ρ ∞V∞2 S ρ ∞V∞2c 2 2 In other words, we can think of the 3-D lift coefficient as the mean value of the 2D lift coefficient on the wing. The same is true for drag and moment: D CD ≡ = D′ 1 1 ρ ∞V∞2 S ρ ∞V∞2c 2 2 M M′ = CM ≡ 1 1 ρ ∞V∞2 S A ref ρ ∞V∞2c 2 2 2 where A ref = c is used. A Closer Look at Drag The drag coefficient can be broken into 2 parts: C2 C D = C D ,e + L N N ΠeA parasite drag 16.100 2002 induced drag 6 Equations of Aircraft Motion where e = span efficiency factor (more on this when we get to lifting line). The parasite drag contains everything except for induced drag including: • skin friction drag • wave drag • pressure drag (due to separation) It is a function of α , thus, we can also think of CD ,e as being a function of CL . The parasitic drag can be well-approximated by: CD ,e = CD0 + rCL2 where CD0 ≡ drag at CL = 0 , r = empirically determined constant. 1 2 ⇒ CD = CD0 + r + CL ΠeA Finally, we can re-define e to include r : ⇒ CD = CD0 + where e → 1 CL2 ΠeA e . 1 + rΠeA This re-defined e is known as the Oswald efficiency factor. We will refer to CD0 as the parasite drag coefficient from now on. 16.100 2002 7