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National Advisory Committee for Aeronautics, Technical Notes - Theoretical Analysis of Oscillations in Hovering of Helicopter Blades with Inclined ad Offset Flapping and Lagging Hinge Axes

The frequency and damping characteristics of the coupled flapping
and lagging oscillations of helicopter blades in hovering are derived
for the general case in which the lagging (vertical) hinge axis is off—
set from the flapping hinge axis, while both hinge axes are inclined.
An offset of the flapping hinge axis from the rotor axis of rotation is
also considered. An inclination of the hinge axes causes pitch change
of a blade section due to angular displacements about these axes. It
is found from this analysis that an inclination of the flapping hinge
axis chiefly affects the values of the natural flapping frequencies and
the damping factors of the lagging motion of a blade. An inclination
of the lagging hinge axis, which causes considerable flapping motion
during the lagging oscillations, mainly affects the natural frequencies
and damping factors in lagging. The analysis and the numerical examples
worked out here indicate that significant increases in the damping of
the lagging motions, which ordinarily border on instability, can be
obtained. by suitable inclinations of the hinge axes, especially of the
lagging axis. Offsetting the flapping and [lagging hinge axes tends
especially to increase the natural lagging frequency.

The aim of this investigation is to determine theoretically the
characteristics of the rigid—body oscillations of helicopter rotor
' blades with inclined and offset flapping (horizontal) and lagging
(vertical) hinge axes. For this purpose a simple method which requires
the solution of a fourth-degree algebraic equation ‘for obtaining the
natural frequencies and damping decrements of the small coupled flapping
and lagging oscillations of a blade in hovering about a state of static
equilibrium is developed here“ Approximate formulas for these charac-
teristics, based on_ the quartic equation, are, moreover, given.

The equations of motion of a blade are derived by means of the
Lagrange equations for a system having two degrees of freedom, Inamely,
rotation about an inclined lagging hinge axis and rotation about an ‘
inclined flapping hinge axis. The flapping axis is assumed as offset
from the rotor axis of rotation, while the lagging axis~is further
assumed as offset from the flapping axis. The aerodynamic loads in
hovering are derived here on the basis of quasi—stationary flow, with
the assumption of constant downwash throughout the rotor disk.