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. (C) 2001 Academic Press.