Assessment of classical, advanced, and layer-wise theories for the vibration of rotating composite anisotropic blades

Abstract

The present paper aims at studying composite cambered structures, tracking the seminar work “Rotating Blade Vibration Analysis Using Shells” presented by Leissa in Journal of Engineering for Power, in 1982, devoted to homogeneous metallic blades. A refined unidimensional (1D) formulation is here implemented to overcome the limitations of classical beam theories. With an appropriate choice of cross-sectional expansions, it is possible to make the 1D model suitable for analyzing shallow blades. This approach enables one to generate both layerwise (LW) and equivalent single layer (ESL) descriptions of the problem unknowns. Furthermore, it is possible to implement classic theories as special cases. Natural frequencies are determined for isotropic and composite blades, showing the effects of changing the fiber lamination angle of symmetric and unsymmetric configurations. Besides, this study investigates the effects of thickness and rotational speed over the structure. Significant differences between classic and high-order theories are found, concerning the accuracy and the computational costs. The causes of these differences are discussed, and the results can be used as a benchmark for future studies.

Introduction

Vibration analysis of rotating blades is a central problem in the rotordynamics and aerospace field, and they have been extensively carried out in recent decades. The commercial interest of the rotating blade is well-known, and they find many applications in several components such as compressors, turbines, propellers, and helicopter rotors. Several investigations have been understanding to reduce noise, increase efficiency, and avoid catastrophic failure. To achieve these goals knowing the dynamic characteristics of these structures is essential, and a correct approach is required to describe the modal aberrations correctly. A considerable number of references can be found in the literature; many of them have been focused on beam formulations. Rao [1] defined the governing equations in terms of a single coordinate corresponding at the main axis of the structure. Putter and Manor [2] established a finite element method (FEM) to study the natural frequencies of radial beams mounted on a rotating disk. This FEM model incorporated shear effects, rotary inertia, and varying centrifugal forces. Chandra and Chopra [3] carried out an analysis of composite box beams utilizing a Newtonian approach along with the Galerkin method. Hodges et al. [4], [5] proposed a variational asymptotic beam sectional analysis (VABS), in which the blade analysis is divided into a linear two-dimensional (2D) problem over the cross-section and a nonlinear analysis along the coordinate of the beam axis. Beam formulations can be highly inaccurate in the case of thin-walled structures, small aspect ratio, and large chordwise chamber. Many authors overcame these limitations by employing shell formulation [6], [7], [8]. Yu et al. [9], extended VABS to 2D problems, in which the analysis were performed through an equivalent single layer (ESL) approach. The multi flexible-body code DYMORE is mentioned in the literature [10], and it finds numerous applications in the rotorcraft field [11], [12]. Leissa and Ewing [13] presented an accurate survey of the one- and two- dimensional theories and made quantitative comparisons of frequencies obtained for turbomachinery blades. Moreover, Leissa, with his coworkers, conducted several studies by developing both beam and shell formulations [14], [15]. For composite structures a shallow-shell theory was presented by Qatu and Leissa in Ref. [16], in which the first known natural frequencies and mode shapes of laminated pre-twisted cantilever plates were calculated. In recent years Sun et al. developed a two-dimensional model for multilayer rotating blades using a quadratic layerwise theory. A comprehensive description of this model can be found in Ref. [17], in which results of numerical simulations are compared to the full three-dimensional model showing an excellent agreement.

Many papers have been recently published to extend Carrera Unified Formulation (CUF) to the rotordynamics analysis. Carrera et al. [18] compared, with published solutions, the natural frequencies of rotating beams with compact cross-sections and open profiles made of either isotropic or orthotropic materials. Carrera and Filippi [20], [19] investigated the dynamic behavior of metallic and composite rotors by utilizing both TE and LE beam elements. The formulation encompassed all contributions due to the rotation, namely the Coriolis term, the spin-softening matrix, and the stress-stiffening matrix. The several comparisons presented in those papers demonstrated that beam CUF models were a viable alternative to FE solutions of higher dimensionality. Moreover, an accurate nonlinear dynamics analysis was presented in Ref. [21], showed relevant discrepancies when structures with deep curvatures were considered. Despite the considerable amount of papers devoted to the dynamics of rotating blades, it seems that there is a gap in the vibration analysis of evaluating the accuracy of theories for rotating composite blades incompatible with the growing popularity of composite structures. The purpose of this paper is to providing reference results for composite rotating blades that can be used as a benchmark for future studies. The present research is inspired by Leissa’s paper [15] in which the effects of the thickness, of the angular velocity, and the kinematic theories on the natural frequencies of the blade were evaluated. This study is done in the framework of CUF approach that permits to adopt two different models to the focus on the composite blades, namely the equivalent single layer (ESL) theory and the layer-wise (LW) models.