This paper investigates the resistance to a change in wing shape due to the aerodynamic forces. In particular, the work required by an airfoil to overcome the aerodynamic forces and produce a change in lift is examined. The relationship between this work and the total aerodynamic energy balance is shown to have significant consequences for transient changes in airfoil shape. Specification of the placement of the actuators and the actuator energetics is shown to be required for the determination of the airfoil shape change requiring minimum energy input. A general, simplified actuator model is adopted in this study which assigns different values of actuator efficiency for negative and positive power output. Unsteady thin airfoil theory is used to analytically determine the pressure distribution and aerodynamic coefficients as a function of time for a ramp input of control deflection. This allows the required power and work to overcome the aerodynamic forces to be determined for a prescribed change in the airfoil camberline. The energy required for a pitching flat plate, conventional flap, conformal flap, and two variable camber configurations is investigated. For the pitching flat plate, the minimum energy pitching axis is shown to be dependent on the pitch rate and the initial angle of attack. The conformal flap is shown to require less actuator energy than the conventional flap to overcome the aerodynamic forces for a prescribed change in lift. The energy requirements of a variable camber configuration are shown to be sensitive to the layout of the variable camber device.
Nomenclature
A n,b= Fourier coefficients defined in Eq. (3.6), n = 1,2,.., and b is the same as defined for T a,b c = chord length C L,n = lift coefficient, n = 0, 1, and 2 correspond to the quasi-steady, apparent mass, and wake-effect terms C M,n = quarter-chord pitching moment coefficient, n represents the terms defined with C L C P = power coefficient for the power required to overcome the aerodynamic forces (P) C Pa = power coefficient for the required power input to the actuator (P a ), (related to C P in Eq. (4.14)) C Wa = energy coefficient for the input energy required by an actuator (W a ) D = drag (the barred quantity represents the time-average) E = energy dissipated to the wake per-unit time (the barred quantity represents the time-average) k = ratio of the initial lift to the change in lift defined in Eqs. (5.17) and (6.2) P = power required to overcome the aerodynamic forces (the barred quantity represents the time-average) P a = required power input to the actuator (related to P in Eq. (2.7)) Q n = defined in Eqs. (4.8 -4.13), where n = 1,2,...,5 q = dynamic pressure t = time t 0 = the time at which P is zero (as shown in