Variable Kinematic One-Dimensional Finite Elements for the Analysis of Rotors Made of Composite Materials

Abstract

This paper deals with the dynamic response o f rotors made o f anisotropic, laminated composite materials. It is a sequel to the authors' previous work, which was devoted to the rotordynamics o f metallic structures. The used variable kinematic one-dimensional models describe any cross-sectional deformation o f the rotor and go beyond the plane strain assumptions o f classical Euler-Bernoulli and Timoshenko beam theories. Refined theories are obtained by applying the Carrera unified formulation, which is extended here to the rotordynamics o f multilayered composites. The displacement variables over the rotor cross section x-z plane are approximated by x,z polynomials o f any order N. Thin-walled cylindrical shafts and boxes are analyzed. These structures are made o f uni directional layers, whose fiber orientation can vary with respect to the rotor-axis as well as in the x-z plane. Several analyses have been carried out to determine the vibrational response as a function o f the rotating speed. Classical beam theories are obtained as par ticular cases and results available in the literature, including shell results, are used to assess the presented theory. The proposed refined models are very effective in investigat ing the dynamic behavior o f laminated composite rotors. I n t r o d u c t io nIt is well known that composite materials present excellent me chanical properties, such as high specific stiffness and strength, ease of formability, a wide range of operating temperatures, and many others. These properties justify their extensive use in many applications, among which the design of rotors, whose dynamic behavior is worthy of study. For example, interesting experiments were carried out in Refs. [1-3] on either graphite or boron/ epoxy shafts. The advantages of orthotropic materials over their isotropic counterparts were pointed out, and useful references were provided for analytical formulations. Despite their limita tions, classical beam theories have been extensively used to inves tigate the critical speeds and instabilities of composite shafts. For instance, with the purpose of proposing an optimization algorithm to define the best lamination scheme, a first shear deformation theory, including the rotatory inertia, was used in Ref.[4], The optimization was direct in order to maximize the first bending fre quency, thus ensuring a sufficient torsional strength by imposing that the lamination angle of a certain number of plies was equal to 45 deg. Later, Bert and Kim developed one-dimensional models, based on Euler-Bemoulli [5] and Bresse-Timoshenko [6] theo ries, which included bending-twisting coupling. The critical speeds were in good agreement with the results obtained through both shell and experimental approaches. Naturally, when shear effects become important, the Euler-Bernoulli theory does not lead to accurate results. In Ref.[7], Chen and Peng investigated the stability of composite spinning cylinders subjected to com pressive loads using the Timoshenko model and the equivalent modulus beam theory...

Flutter analysis of fixed and rotary wings through a one-dimensional unified formulation

“…The TE models have been adopted for the study of mechanical behaviors of thin-walled structures [42,43] and non-conventional cross-sections made of isotropic, composite [39,40] and functionally graded [41] materials. Other encouraging results have also been obtained in the dynamics of rotors by analyzing both blades [44] and spinning shafts [45][46][47]. Furthermore, aeroelastic analyzes were performed on fixed-wing configurations by combining the TE elements with aerodynamic panel methods [48,49] and unsteady strip theories [50].…”

“…For the sake of the brevity, the explicit forms of the coefficients of theC matrices are shown in [46]. The finite element method is used in order to consider arbitrary cross-section profiles.…”

Finite element models with node-dependent kinematics for the analysis of composite beam structures

“…whereC, is the matrix of the material coefficients defined in the general physical system (x, y, z) rotated from the material coordinate system (1,2,3), and their explicit expressions can be found in [53].…”

Recent developments on refined theories for beams with applications

“…2.6.9. Rotors and Rotating Blades 1D CUF models have been extended to rotor dynamics analyses in Carrera and Filippi (2014) and Carrera et al (2013c,i). Coriolis and centrifugal stiffening were considered for the free vibration analysis of compact and thin-walled rotating structures.…”