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Chapter 4- The Equations of Motion Aircraft Performance Aircraft performance is defined as how the aircraft responds (its motion) to the four forces of flight. It is considered to be a branch of the Flight Mechanics discipline. We have already reviewed aerodynamics and propulsion. We use the following information in performance: AE 3310 Performance aerodynamics drag polar propulsion thrust or power, SFC Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion The Four Forces of Flight Lift, Drag, Weight, Thrust Lift and Drag are for complete airplane L perpendicular to V by definition not necessarily in the flight direction T ε D parallel to V by definition W always acts towards the center of the earth Steady, Level Flight AE 3310 Performance V always in the direction of the local flight of the aircraft. Shows flow velocity relative to the airplane Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Four Forces in Climbing Flight flight path T L V ε θ local climb angle D θ W earth AE 3310 Performance Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Now, Bank the Aircraft φ Bank (roll) angle φ L T sin ε φ Wcosθ AE 3310 Performance Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion The Equations of Motion Based on Newton’s Second Law: F=ma note this is vector form In scalar form, for arbitrary direction in space, s Fs = m as General, Formal Derivation rotating spherical earth acceleration of gravity with distance from center of the earth AE 3310 Performance Less Formal, more Physical Derivation flat, stationary earth Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Climbing, Banking Flight Replace aircraft with point mass at its center of gravity (because we are only concerned with translational motion). + radius of curve r1 Flight Path s L cos φ V T sin ε cos φ θ instananeous flight path direction T cos ε D θ AE 3310 Performance W center of gravity of the airplane Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion First Equation of Motion Take components parallel to the flight path The force is F The acceleration is = T cos ε - D - W sin θ dV a = dt Therefore, Newton’s Second Law parallel to the flight path is dV m dt AE 3310 Performance First Equation of Motion = T cos ε - D - W sin θ ma = F Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Second Equation of Motion Take components perpendicular to the flight path The force is F = L cos φ + T sin ε cos φ - W cos θ The radial acceleration is V a = r1 Therefore, Newton’s Second Law perpendicular to the flight path is m AE 3310 Performance V r1 2 Second Equation of Motion 2 = L cos φ + T sin ε cos φ - W cos θ ma = F Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Forces on Horizontal Plane Now look at flight path from a “top” view D cos θ T cos ε cos θ V cos θ T sin ε sin φ L sin φ projection of flight path r2 + AE 3310 Performance Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Third Equation of Motion Take components perpendicular to the flight path in the horizontal plane (2) The force is F2 = L sin φ + T sin ε sin φ The radial acceleration is (V cos θ) a2 = r2 Therefore, Newton’s Second Law perpendicular to the horizontal flight path is 2 Third Equation of Motion 2 m AE 3310 Performance (V cos θ) = L sin φ + T sin ε sin φ r2 ma = F Dr. Danielle Soban Georgia Institute of Technology Chapter 4- The Equations of Motion Summary The three Equations of Motion are simply statements of Newton’s Second Law. The three Equations of Motion describe the translational motion of an airplane through three-dimensional space over a flat earth. There are three additional equations of motion that describe the rotational motion of the aircraft about its three axes. Final note: the three equations of motion here do not assume a yaw component. The free stream velocity vector is assumed always parallel to the symmetry plane of the aircraft. AE 3310 Performance Dr. Danielle Soban Georgia Institute of Technology