Vibrations of rotary-wing aircraft may derive their energy from the rotation of the rotor rather than from the air forces. A theoretical analysis of these vibrations is described and methods for its application are explained herein.nThe theory includes the effects of unequal stiffness of the pylon for deflections in different directions and the effect of damping in the hinges and in the pylon. Both the derivation of the characteristics equation and the methods of application of the theory are given. In particular, the theory predicts the so-called "odd frequency" self-excited speed range as well as the shaft critical speed. Charts are presented from which the shaft critical and the self-excited instabilities can be predicted for a great variety of cases. The influence of each physical parameter upon the instabilities has been obtained. The comprehensive treatment applies to a rotor that has any number of blades greater than two. Only a brief discussion and the formula for shaft critical speed are given for the one or two-blade rotor.nThe use of complex variables in conjunction with Lagrange's equations has been found very convenient for the treatment of vibrations of rotating systems.
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