Nonlinear Vibrations of Rotating 3D Tapered Beams With Arbitrary Cross Sections

Abstract eng:
The geometrically nonlinear vibrations of 3D beams that rotate about a fixed axis are investigated by the p-version finite element method. The beams are considered to be tapered, i.e. with variable thickness along its length, and with arbitrary cross sections. The beam model is based on Timoshenko’s theory for bending and Saint-Venant’s theory for torsion, i.e. it is assumed that the cross section rotates about the longitudinal axis as a rigid body but may deform in longitudinal direction due to warping and the torsion is not considered to be uniform. The warping function is obtained preliminarily by the finite element method. For the case of tapered beams, the warping and torsional constants, the cross sectional area, the second moment of inertia and all cross sectional properties are expressed as functions of the longitudinal local coordinate. Geometrical nonlinearity is taken into account and derived from Green’s strain tensor. Linear elastic and isotropic materials are considered and generalized Hooke’s law is used. The rotation is included in the model through the inertia terms, two coordinate systems are considered: one fixed and one rotating about the fixed one. The equation of motion is derived by the principle of virtual work in the rotating coordinate system but the influence of the rotation of the coordinate system on the displacements of the beam is included in the equation of motion. A setting angle and hub radios are considered, and their influence, as well the influence of the speed of rotation, on the natural frequencies is examined. Forced vibrations in time domain are presented for various setting angles.

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