# Equation of Motion Derivation - ASU Professor

Equations of Motion for a High Performance Fighter A Simplified Dynamical Model Based on Elementary Physics F18 Hornet - High Performance Fighter - How do we model such a system? Simplified Aircraft

Schematic - High Performance Fighter Note: Aircraft is unstable liff 1 i.e. c.g. aft of c.p. 0 f1 produces pitch-up instability

Instability desirable for Aircraft Parameters - High Performance Fighter Fuselage damping coefficient: b0 f1l1 Lift produces cw moment: - Lift coefficient:k1 0 - Lift moment arml1 can be + f 2 (l1 l2 )

Elevator produces ccw moment: - Elevator coefficient: k2 0 - Typicallyl2 0 Aircraft Pitch Angle Dynamics - Apply Newtons 2nd Law of Motion I I f 2 (l2 l1 ) f1l1 b Moment from control surface

(elevator) deflection (CCW) Moment of inertia I 0 Fuselage rotational Moment from lift damping force (CW) moment (CW)

Aircraft Pitch Angle Dynamics - Capture effect of elevator and lift I f 2 (l2 l1 ) f1l1 b I k 2 (l2 l1 )u k1l1 b Elevator term substitution: f 2 k 2u Lift term

substitution: f1 k1 Aircraft Pitch Angle Dynamics - Rewrite 2nd Order Ordinary Differential Equation (ODE) in standard form. I k 2 (l2 l1 )u k1l1 b I b k1l1 k 2 (l2 l1 )u

Pitch Angle Dynamics - Transform pitch angle t-domain dynamics to sdomain I b k1l1 k 2 (l2 l1 )u - Assume zero initial conditions - Apply Laplace Transform Differentiation Rule 2 Is s bs s k1l1 s k 2 (l2 l1 )U s

Pitch Angle Dynamics - Express pitch angle dynamics as a transfer function 2 Is s bs s k1l1 s k 2 (l2 l1 )U s s k 2 (l1 l2 ) 2 U s Is bs k1l1 Note: Poles capture aircrafts natural modes

l1 0 How Do Aircraft Dynamics Depend On System Parameters? Dependence on Moment of InertiaI Simplified Aircraft Dynamics Ifor 0 - Approximate Pitch Dynamics

s k 2 (l1 l2 ) U s bs k1l1 first order rotational damping - Approximate Vertical Dynamics Y s k 2 bs k1l2 2 U s ms bs k1l1

double integrator and first order rotational dampin

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