Optimal Placement of Piezoelectric Actuated Trailing-Edge Flaps for Helicopter Vibration Control

Abstract

This study aims to determine the optimal locations for dual trailing-edge flaps on a helicopter blade in the presence of actuator hysteresis. An aeroelastic analysis based on a finite element approach in space and time is used in conjunction with an optimal control algorithm to determine the actuator control input for vibration minimization. The reduced hub-vibration level and the flap power are the two optimization indices considered in this study. The location of the flaps along the blade are the design variables. The hysteresis in the piezo-actuators is modeled using a dynamic hysteresis model based on an extension to the classical Preisach model. It is found that second-order polynomial response surfaces based on the central composite design of the theory of design of experiments describe both objectives adequately. Numerical studies for a four-bladed hingeless rotor show that both objectives are more sensitive to outboard flap location compared with inboard flap location. Optimization studies indicate that the dualflap configuration for the least hub-vibration level is different from the configuration for the least flap power. The Pareto front between the two objectives is found to be discontinuous. However, a reasonable tradeoff configuration is obtained by careful inspection of the Pareto front. This configuration yields about a 70% reduction in hub-vibration levels from the baseline conditions at an advance ratio of 0.30, while requiring about 21% more flap power from the initial configuration of the optimization study. Nomenclature C T = rotor thrust coefficient c = blade chord c f = trailing-edge flap chord EI y = flap bending stiffness EI z = lag bending stiffness F x = vibratory hub longitudinal shear force F y = vibratory hub lateral shear force F z = vibratory hub vertical shear force GJ = torsion stiffness J = objective function M h = trailing-edge flap hinge moment M x = vibratory hub roll moment M y = vibratory hub pitch moment M z = vibratory hub yaw moment m f = trailing-edge flap mass per unit length m 0 = blade mass per unit length P f = power required by a single trailing-edge flap, averaged over one revolution P t = total actuation power required by all flaps on all rotor blades, averaged over one revolution R = rotor radius u = input to the hysteresis transducer/voltage applied to the piezostack, V X f g = trailing-edge flap center of gravity (after hinge) x 1 , x 2 = nondimensional location of the inboard and outboard flap, respectively = output of the hysteresis transducer/flap deflection angle, deg = hysteresis operator = lock number = advance ratio/Preisach distribution function 0 , 1 , 2 = Preisach distribution functions for the dynamic hysteresis model = blade solidity ratio = blade azimuth angle = rotor angular speed Subscripts St = steady component Nc = Nth cosine harmonic Ns = Nth sine harmonic Superscript 4p = 4/rev component

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