Dynamics of steady-state autorotation

O.E. Polyntsev

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

134, Lermontova st., Irkutsk, 664033, RUSSIA

e-mail: olegpolyntsev@mail.ru

A mathematical model of motion of an autorotating rotor (AR) has been considered for the case of level flight.

With a view to assess effect of parameters upon dynamical features of a windmilling rotor the mathematical model including two differential equations of the second order is used. These equations are flapping one and autorotation one.

Application of numerical methods allows one to solve majority of tasks, including investigation of unsteady processes. In scope of engineering practice it is necessary to evaluate rapidly a number of the most general properties of an AR, for instance, rotor angular velocity in the mode of steady-state autorotation, aerodynamic forces, flapping angles etc. Application of numerical methods is concerned with time expenses and demands to use computational devices with great capacities.

Thus, it is necessary to derive approximate analytical solution of above-mentioned equations.

The AR under consideration has got two blades and a hub of cardan joint type. As against works devoted to autogiro theory and works indirectly concerned with this problem, an approximate analytical solution is effectuated, allowing for a varying induced flow as function of a rotor operational mode. A number of geometrical features has also been taken into account.

The case of steady-state autorotation of the AR with untwisted untapered blades (or equal untapered untwisted blades) has been considered.

For the mode of steady-state autorotation it is assumed that averaged angular acceleration of a rotor is equal to 0, for change of rotor speed per one turn of rotor is negligible. Ignoring effects of viscosity and compressibility of flow, the simplifications necessary to determine the elementary forces have been introduced.

The flapping angle has been expressed as the Fourier series as a function of the azimuth angle, terminating the series after terms of first harmonic.

The quadratic equation of steady-state autorotation in respect of a normed axial component of speed of motion has been derived.

The expressions allowing one to estimate integral characteristics, which include rotor thrust, lateral and longitudinal forces have also been obtained.

Based on the expressions obtained an approach to evaluate the AR characteristics has been developed.

The applicability of the approximate analytical solution has been studied in comparison with numerical computations.

The computation of characteristics of the rotor whose parameters are close to that of the A-002 gyroplane has been performed. As the results of calculations main conclusions regarding special features of steady-state regimes of the AR have been made.

The mathematical model composed has been applied to estimate dynamic properties of the AR of the A-002 gyroplane.