Optimal Takeoff Trajectories of a Heavily Loaded Helicopter

Optimal control theory has been applied to the STOL takeoff of a heavily loaded helicopter. The object of the analysis was to determine constrained and unconstrained extremal trajectories that maximize the terminal vertical distance for a given fixed horizontal distance. The resulting optimal trajectory consists of an initial acceleration segment followed by a decelerating segment in which most of the terminal altitude is gained. If the problem was constrained to nondecelerating flight, the resulting optimum consisted of a maximum acceleration segment followed by a constant velocity climbout segment. The slightly decreased performance of the constrained optimum was more than offset by its ease of implementation. Nomenclature A — area of rotor disc Ci-s = momentum theory expansion coefficients CPL = power loss eg = helicopter center of gravity / = wetted drag area of HLD helicopter /o = the derivative of the payoff function with respect to time g = acceleration of gravity HLD = heavily loaded hg = vertical distance measured from the ground plane J = payoff function m = mass of the helicopter Pd = power difference in steady level flight R = radius of rotor R/C = rate of climb ST = reference distance T = thrust t = time Vio = induced velocity V = velocity of the helicopter Vm = smallest velocity where equilibrium flight of the helicopter out of ground effect is possible Vr = reference velocity: induced velocity at hover out of ground effect Vsi = static best climb angle speed Vs2, — static best rate of climb speed W = weight xg = horizontal distance measured on the ground plane Z = vertical height of the rotor above the ground plane CLTP = angle of attack of the rotor tip path plane /3 = dy/dt = control variable rj = dV/dt = control variable 7 = flight path angle dvio = effective reduction of induced velocity of the rotor due to ground effect = inflow through the rotor plane nondimensionalized by SIR