Mean axes provide the usual reference in maneuvering and dynamic response analyses of flexible vehicles. Attached or structural axes have also been used because the flexibility characteristics of the structure are only determined for a restrained structure. If the structural axes are employed, a relationship is required for the orientations of the structural axes relative to the mean axes, since the equations of motion determine the orientations of the mean axes. If this relationship is not considered, as it has not been in a number of publications, the solution to the equations of motion of the structural axes is not invariant with the choice of support configuration used in the calculation of the structural flexibility influence coefficients and is, therefore, incorrect. This relationship and the correct equations of motion for the structural axes are presented, and the correctness of the formulation is demonstrated numerically in studies of longitudinal maneuvering of an example forward-swept-wing airplane at constant forward velocity. Although the paper assumes constant airspeed in the basic developments, it concludes with discussions of aeroelastic speed derivatives and aeroelastic effects on drag. The paper assumes quasisteady equilibrium of the structure with its applied loads throughout. Nomenclature A= aeroelastic deflection amplification factor a = flexibility, a F is free-body flexibility a s = amplitude of motion of support reference point #0 = airfoil two-dimensional lift curve slope B = aeroelastic load amplification factor b = reference span C = generalized aerodynamic force coefficient: C/ is inertial value, C q is angular rate value, C a is incidence and control surface value, C^ is incidence rate value, C 0 is initial value C e = experimental control point force coefficient C h = complex oscillatory aerodynamic influence coefficient C hs = static aerodynamic influence coefficient C m = aerodynamic pitching moment coefficient C z = aerodynamic normal force coefficient c =mean aerodynamic chord c f = local lift curve slope EI X

No abstract

Let (M,d) be a complete metric space and T be a self-mapping of M. W.A. Kirk proved a fixed point theorem for a continuous asymptotic contraction T in [4] .Y.Z. Chen extended Kirk's theorem in [2] by assuming weaker assumptions on T. Also Chen introduced some other conditions to replace the assumption on the boundedness of the orbit. We introduce the weaker condition liminf n → ∞ (d(x,T n x)) = 0 for some x in M, and prove that this condition implies the existence of a fixed point and the convergence of the Picard iterates to this fixed point.