Discrete-Time, Closed-Loop Aeromechanical Stability Analysis of Helicopters with Higher Harmonic Control

Abstract

This paper presents an aeromechanical closed-loop stability and response analysis of a hingeless rotor helicopter with a higher harmonic control system for vibration reduction. The analysis includes the rigid body dynamics of the helicopterandblade flexibility.Thegainmatrixisassumedtobe fixedandcomputedoffline.Thediscreteelementsof the higher harmonic control loop are rigorously modeled, including the presence of two different time scales in the loop. By also formulating the coupled rotor-fuselage dynamics in discrete form, the entire coupled helicopter higher harmonic control system could be rigorously modeled as a discrete system. The effect of the periodicity of the equationsofmotionisrigorouslytakenintoaccountbyconvertingthesystemintoanequivalentsystemwithconstant coefficients and identical stability properties using a time-lifting technique. The most important conclusion of the present study is that the discrete elements in the higher harmonic control loop must be modeled in any higher harmonic control analysis. Not doing so is unconservative. For the helicopter configuration and higher harmonic control structure used in this study, an approximate continuous modeling of the higher harmonic control system indicates thatthe closed-loop, coupled helicopter higherharmonic control systemisalways stable,whereas the more rigorous discrete analysis shows that closed-loop instabilities can occur. The higher harmonic control gains must be reducedtoaccountforthelossofgainmarginbroughtaboutbythediscreteelements.Otherconclusionsofthestudy are 1) the higher harmonic control is effective in quickly reducing vibrations, at least at its design condition; 2) a linearized model of helicopter dynamics is adequate for higher harmonic control design, as long as the periodicity of the system is correctly taken into account, that is, periodicity is more important than nonlinearity, at least for the mathematical model used in this study; and 3) when discrete and continuous systems are both stable, the predicted higher harmonic control control harmonics are in good agreement, although the initial transient behavior can be considerably different.