Ground and air resonance of bearingless rotors in hover and forward flight

Ground and air resonances in hover and forward flight are examined for a bearingless rotor using a finite element formulation based on Hamilton's principle. The blade configuration consists of a single flexbeam with a wrap-around type torque tube with a vertical offset of the cuff snubber attachment point (Boeing-ITR model). The outboard main blade, flexbeam and torque tube are all assumed as elastic beams undergoing flap bending, lead-lag bending, elastic twist and axial deflections, and these are discretized into beam finite elements. Quasi-steady strip theory is used to evaluate aerodynamic forces and unsteady aerodynamic effects are introduced approximately through a dynamic inflow model. The fuselage is modeled as a rigid body undergoing five degrees of motion: longitudinal, lateral and vertical translations, and pitch and roll rotations. For the hover analysis, the nonlinear equations of motion are solved for steady blade deflections through an iterative procedure including the calculation of the pitch link displacement to obtain the desired pitch angle. The equations of motion for the rotor/fuselage dynamics are linearized about the vehicle trim state and the blade steady-state deflected position, and transformed to modal space using coupled vibration modes. These normalized equations consisting of the blade equations of motion, the fuselage equations of motion and the dynamic inflow equations, are then transformed into the nonrotating frame using the multiblade coordinate transformations. The complex eigenvalue analysis is done for the stability. For the analysis of forward flight, the periodic response is calculated using a finite element method in time after the nonlinear finite element equations in space are transformed to normal mode equations. The linearized period perturbation equations in the nonrotating frame are solved for the stability using Floquet transition matrix theory as well as constant coefficient approximation. Satisfactory correlation of predicted ground and air resonance results is carried out with data obtained from the measurements on 1/8th Froude-scaled dynamic model. Systematic parameteric studies are carried out to examine the effects of several design parameters on ground and air resonance stabilities. Lag frequencies has a substantial influence on ground resonance stability, whereas pitch-lag coupling (vertical location of cuff restraint), and pitch link stiffness have powerful effects on air resonance stability in hover and forward flight.