Nonlinear Dynamics of Autorotating Rotor

O.E.Polyntsev

Institute of system dynamics and control theory, Russian Academy of Sciences

134, Lermontova st., Irkutsk, 664033, RUSSIA

 

In this paper for the case of steady straight level flight the mathematical model of dynamics of an autorotaring rotor (AR) has been created. The purpose of the paper is to investigate nonlinear dynamics of a two-bladed AR. The principal elements of the considered AR are the following: two blades attached to rotor head through feathering hinges allowing blades to vary their pitch. The hub is attached to an aircraft’s pylon. The rotor head is of cardan joint type. Cyclic control is effectuated via the rotor disc angular position change. One of the cardan axes is common flapping hinge. Motion of the rotor is considered which is presented by the system of absolutely rigid bodies. Aerodynamic interaction between AR units is ignored. The straight level flight mode of an entirely trimmed aircraft is studied and control perturbations are assumed to be inessential (the regime is fully equal to a rotor spin up by means of wind). Based on d’Alembert principle, the equation of absolute motion of a blade element with mass with respect to normal Earth axes’ system has been derived. With a view to describe the rotor motion the matrices of direction cosines has been used. All proper frames have been introduced. The differential equations of flapping and autorotation have been derived. These equations compose the mathematical model of the autorotating rotor. As a first approximation the induced velocities are set with taking into account the special features of the distribution upon rotor disc, including axis-symmetrical low that is typical for low tip-speed ratios and linear law that is typical for large values of tip-speed ratios. It is baffling problem to solve the system when the autorotation is unsteady process. Consequently, to study special features of the unsteady modes of the AR these equations have been solved numerically. Based on the system of the obtained equations the modeling of motion of a rotor of the A-002 gyroplane has been executed. It has been pointed out, that though the approximate analytical solution and the numerical one are qualitatively equal, there are regimes where the effect of nonlinear terms of the equations upon the solution cannot be neglected.  Special features of nonlinear dynamics of autorotating rotors have been studied. The results of numerical study illustrating these features have been presented. With a view to develop optimal design decisions the main ways of further study of the dynamics have been posed. An approach to assess autorotation margins using three normed coefficients has been proposed: regarding straight autorotation; regarding flapping stability; relatively the presence of autorotation of the AR with flapping restrictions. The mathematical model is successfully applied to estimate dynamic properties of the A-002 gyroplane’s rotor.