VISIÓN, MISIÓN Y OBJETIVO GENERAL 5.2.1 VISIÓN

characteristics. This problem was motivated by the fact that conventional rigid body 6 DOF nonlinear simulation equations typically have quasi-static correction factors applied to the aerodynamics. Rigid body equations with QSAE corrections make use of two assumptions: (1) the airplane’s deformation state is always in equilibrium with the air loads, (2) the air loads are well approximated by their steady flow solution. When the dynamic aeroelastic degrees of freedom are added to a 6 DOF simulation, the added elastic modes bring their own QSAE effect, and unless these are removed, the QSAE effects are double book kept in the simulation. The QSAE effects supplied with conventional “rigid” model are derived from high-fidelity finite element structural

models and with higher-fidelity (for steady effects) aero codes than used to form dynamic aeroelastic models. Therefore, it is desired to retain the original QSAE effects.

In either case, the DASE integration with nonlinear simulation occurs as follows. The core of the simulation is a mathematical model represented by the equation

( , )

NL NL NL

x =F x u (3.1)

where x_NLdenotes an array of state variables that specify the aircraft motion. The

function (F x_NL,u_NL)represents the solution to the nonlinear 6 DOF equations, which are based on information derived from an aerodynamic database as well as from models of the atmosphere, aircraft mass distribution, engine performance, and actuator

characteristics. The initial conditions are computed using a constrained nonlinear optimization process that determines the equilibrium point where selected “trim-to” xNL

response values are generated from a set of “trim-with” u_NL variables. The equilibrium point is referred to as the trim point even though nonzero accelerations may be specified in the x_NL vector. Each selected output represents one degree of freedom or one

constrained equation of motion. The trim procedure may contain a maximum of 11 equations defining three linear accelerations, three angular accelerations, aircraft velocity, rate of climb, sideslip, roll rate, and pitch rate. The u_NL set may include three linear velocities, three angular velocities, pitch angle, bank angle, control surface deflections, and engine thrust.

Aircraft structural dynamics are modeled by adding the unsteady aerodynamic part of the linear equations to the nonlinear simulation in the following manner

( _NL, _NL) _{A B}

x=F x u + ∆F x+ ∆F u (3.2)

where x denotes a state vector that combines x_NL with the structural degrees of freedom. Similarly, u represents a vector of input variables. Defining the linear structural response equations in a manner that allows for efficient computation while maintaining model fidelity in “rigid” and flexible dynamic frequency range has been a major issue.

Of the two methods proposed in Reference 42, the method based on rational function approximation of unsteady aerodynamics was chosen to provide the DASE vehicle model. To accurately represent the underlying physics, the model for a class of highly flexible vehicles is derived from a more fundamental set of equations. The generalized force equation (3.3), is used to represent the equations of motion. The terms on the left hand side are rigid and elastic displacement states respectively and the terms on the right represent the generalized aerodynamic forces and moments acting on the airplane

(

)

Ms +Ds+K Χ =F =q Qξ (3.3)

where M is the generalized mass matrix, D is the generalized damping, and K is the generalized stiffness. The state vector X is a vector of rigid displacements, elastic displacements, and control deflections. The generalized coordinate ξ is in terms of rigid rates, elastic displacements, and gust velocities. The generalized mass matrix consists of vehicle mass and appropriate moments of inertia all modified by aerodynamic force and moment derivatives with respect to unsteady aerodynamic accelerations such as

( , , , , , )u v w p q r .

For a typical rigid body vehicle, the aerodynamic forces and moments are considered in steady state and the only unsteady aerodynamic terms are those associated with gust velocities. However, in an elastic aircraft all aerodynamic forces must be considered as unsteady. The generalized force, Q, premultiplied by dynamic pressure, is a complex matrix representing unsteady aerodynamic forces, which are Mach number and frequency dependent, arising from motion of the generalized coordinates, ξ. In order to express

approximated with rational functions in s. This rational function approximation, described in detail in Reference 44, explicitly introduces state dependence on control surface rates and acceleration not typically present in standard equations of motion. The acceleration terms are from inertial mass coupling of actuators to the fuselage and directly influence aircraft states.

In order to focus on the question of rigid body and structural deformations, the equations of motion were separated into longitudinal and lateral-directional parts. As is generally the case with rigid aircraft, with the exception of rapid maneuvering at very high angles of attack, the longitudinal and lateral-directional separation holds for

structural deformations of the vehicle. Thus, two different axes of motion can be studied separately without loss of important dynamics. This entire work focuses on the

longitudinal axes of motion, and any future reference to the equations of motion pertains strictly to the longitudinal axes.