# Abstract

This paper presents the vibratory behavior of a spinning composite shaft with curvilinear fibers on rigid bearings in the case of free vibrations. A *p*-version of finite element is used to define the model. A theoretical study allows the establishment of the kinetic energy and the strain energy of the shaft, necessary to the result of the equations of motion. In this model the transverse shear deformation, rotary inertia and gyroscopic effects have been incorporated. A hierarchical beam finite element with six degrees of freedom per node is developed and employed to find the natural frequencies of a spinning composite shaft with variable stiffness (curvilinear fibers). A computer code is elaborate for calculating the natural-frequencies for various rotating speeds of the composite shafts with curvilinear fibers. In the absence of publications of vibration analysis of rotating composite shafts with curvilinear fibers, the formulation is verified by comparisons with published data on rotating composite shafts reinforced by straight fibers. The influence of the physical, geometrical parameters, the boundary conditions and the curvilinear fiber paths on the first natural frequencies of the spinning composite shafts is studied by plotting various Campbell diagrams.

**Keywords:**

Spinning shaft; composite materials; curvilinear fibers; variable stiffness; *p*- version; finite element method; Campbell diagram