Non-linear free vibrations of a rotating flexible arm

The non-linear, moderately large amplitude flexural free vibrations of an arm clamped with a setting angle to a rigid rotating hub are studied. The shear deformation and rotary inertia effects are assumed to be negligible, but account is taken of axial inertia, non-linear curvature and the inextensibility condition. The Lagrangian approach in conjunction with the assumed modes method, assuming constant hub rotation speed, is used in a consistent manner to obtain the third order non-linear uni-modal temporal problem. Because of the strength of the non-linearities in the temporal problem, which includes elastic and inertial geometric stiffening as well as inertial softening terms, a time transformation method is employed to obtain an approximate solution to the frequency-amplitude relation of arm free oscillation. Results in non-dimensional form are presented graphically, for the effect of hub rotation speed, blade setting angle, and hub radius on the variation of the natural frequency with vibration amplitude.