Elsevier

Composite Structures

Volume 250, 15 October 2020, 112580
Composite Structures

Predicting vibration characteristics of rotating composite blades containing CNT-reinforced composite laminae and damaged fiber-reinforced composite laminae

https://doi.org/10.1016/j.compstruct.2020.112580Get rights and content

Abstract

Existing of cracks within blade structures can cause stiffness degradation and hence changes their vibration characteristics. This study investigates the vibration behaviors of rotating pre-twisted hybrid composite blades containing functionally graded carbon nanotube-reinforced composite (FG-CNTRC) laminae and damaged fiber-reinforced composite (FRC) laminae. The degraded stiffness of the cracked lamina is modeled through the self-consistent model (SCM). The blade is modeled as a shell structure that is formed by twisting a plate around its mean line. With the help of the differential geometry theory, a novel shell model has been derived to describe the kinetics of the blade. The effect of the Coriolis and centrifugal force are both presented in the formulation, which results in a damped-like free vibration system governed by a system of second-order ordinary differential equations (ODEs). Utilizing the state space technique, the system is reformulated to a system of first-order ODEs. The IMLS-Ritz method is then used for discretizing the ODEs. After carefully validating the effectiveness of the presented model through a series of comparison studies, parametric studies including CNT distribution configuration, rotating speed, geometrical parameters on the vibration responses of cross-plied composite blades are systematically examined. The vibration characteristics of angle-plied composite blades are also investigated.

Introduction

Since the uncovering of carbon nanotubes (CNTs), they have gradually been experimentally and theoretically justified with remarkable thermal, mechanical, and electrical properties [1]. Those belongings make them a kind of very promising reinforcement, which can account for the reason why such a large amount of literature has been published on the CNT reinforced composites (CNTRCs) [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Recently, the fast advance of fabrication techniques on ultralong allied CNTs, for example, Zhang et al. [12] fabricated 55-cm-long CNTs with perfect structures by optimizing the growing parameters as well as employing a “furnace-moving” technique, has provided a flexible way of adjusting the material properties of CNTRCs by properly arranging the distribution configuration of CNTs along a predetermined direction within matrix. The fabricated CNTRCs are the so-called functionally graded (FG) CNTRCs, which combine the unique properties of functionally graded materials (FGMs) and CNTRCs [13], [14], [15], [16], [17], [18], [19], [20]. Therefore, FG-CNTRCs may possess higher specific stiffness and strength over uniformly distributed (UD) CNTRCs, which are in great demand for rotating structures.

On account of the aforementioned reasons, the very recent publications have started to work on this research area. Considering the thermal effect, Rout and Karmakar [21] carried out free vibration analysis of a rotating pre-twisted FG-CNTRC shallow cylindrical shell which was used to model a turbomachinery blade. It was found by the authors that FG-CNTRC cylindrical shell with CNT of FG-X type had the largest fundamental frequency when compared shells with CNT of other types. The free vibration of a rotating Timoshenko beam reinforced with CNTs was analyzed in [22]. Bahaadini and Saidi [23] performed the aerothermoelastic flutter analysis of a tapered rotating FG-CNTRC blade under supersonic flow. Considering the geometrical nonlinear terms in strain–displacement relationships, large-amplitude vibration behaviors of a rotating beam reinforced by CNTs were investigated by Heidari and Arvin [24]. Xiang et al. [25] performed a vibration study of rotating laminated FG-CNTRC conical shells. The authors also carried out the lay-up optimization of the laminated composite shell using the genetic algorithm [25].

As the critical component of a rotating machine (as shown in Fig. 1), the rotating blades mainly work in a very tough environment and withstand high centrifugal and excitation forces [26], [27], [28], [29]. Therefore, it is prone to initiate the fatigue cracks inside or across the blade structures [30], [31], [32], [33]. And the present cracks will influence the dynamic mechanical properties of rotating blades and further reduce the safety of the aero engines. That is why the mechanical modeling of free vibration and forced behaviors of rotating structures containing cracks must be investigated. Liu and Jiang [34] developed a cracked hexahedral element for crack modeling of rotating blades. The results showed that using the element could accurately simulate the breathing effects and extract the vibration frequency of blades with various crack depths. Later, taking alternating loads into account, Liu et al. [35] applied the cracked hexahedral element to investigate nonlinear vibrations of a cracked blade in a rotor system. After an experimental analysis and finite element modeling the vibration behaviors of a cracked and perfect steam turbine blade, Shukla and Harsha [36] concluded that the cracks definitely changed the dynamic behaviors of the blade. Taking the centrifugal force effects, the breathing effects, as well as the crack effects into consideration, Xie et al. [37] put up with an efficient analytical model for vibration investigation of a blade with simple geometry and a regular crack.

Based on the above literature, the assumption of finite local cracks within the damaged isotropic blades is considered in most of the studies. Little attention has been pained to the mechanical response of damaged composite blades with a series of matrix cracks. However, in many fibrous composites under load, as stated by Dvorak et al. [38], [39], due to the mismatch of the failure strains of matrix and fibers, a lamina may develop a series of cracks that are parallel with the fiber orientation and across the ply thickness [40]. The cracks within the matrix mentioned above are named transverse cracks or ply cracks [38], which may cause the degradation of the stiffness and further affect the vibration characteristic of rotating blade structures. A lamina with matrix cracks is referred to as a damaged layer. In our previous works, with the stiffness of the damaged layer being modeled by the self-consistent model (SCM), we explored the effect of cracks on the linear [41] and large amplitude [42] free vibrations of the pre-twisted plate consisting of CNTRC laminae and damaged FRC laminae under a static state. The results demonstrate that the cracks would lower the vibration frequencies of the blade [41]. It should be pointed out that our previous model was only valid for small to moderate pre-twisted angle and the static state was assumed. With the consideration of matrix cracks, presenting an effective computational framework that can handle the vibrations of blades with a large pre-twisted angle under rotating state are in high demand from the engineering perspective [43]. To meet this requirement, a general shell theory is proposed to be used in the computational model for predicting the vibration characteristics of composite blade structures with a large pre-twisted angle.

In this study, a meshfree numerical framework is developed for vibration analysis of a rotating composite blade containing FG-CNTRC laminae and damaged FRC laminae with a series of matrix cracks. The organization of the remaining sections of the article is as follows. In Section 2, we give a clear definition of the problem that we will study in this article. Meanwhile, the basic geometric description of the blade structure, as well as the micromechanical models for evaluating the material parameters of the damaged FRC and CNTRC laminae, are presented. Section 3 presents the theoretical formulations including the shell theory for the blade, equations of motion, and their corresponding solutions. The detailed computational results and discussion are presented in Section 4. The article is finally close by listing the new findings and main conclusions obtained from this study.

Section snippets

Problem statement

As demonstrated in Fig. 2, the pre-twisted blade considered here is made up of several perfectly bonded damaged FRC laminae and CNTRC laminae. As stated in [39], [41], [42], the fatigue cracks generated within the matrix are along with fiber orientations and throughout the ply thickness. The uniform ply crack spacing and the illustration of the crack density parameter of a cracked lamina are shown in Fig. 2(b) and (c). The geometric description of the blade structure, as well as the

Shell theory for pre-twisted blade

The developed shell theory is based on the first-order shear deformation theory (FSDT) of a shell [52], [53], according to which the displacement fields are given byux,ξ,ς,t=u0x,ξ,t+ςφ1x,ξ,tvx,ξ,ς,t=v0x,ξ,t+ςφ2x,ξ,twx,ξ,ς,t=w0x,ξ,twhere u,v,w denote components of displacement at an arbitrary point in the blade; u0,v0,w0are the translational displacements at the mid-surface;φ1,φ2 are the rotations. According to the definition of the strain of a general shell structure [45], the strains in the

Discrete equations of motion

The meshless IMLS-Ritz method [58], [59], [60] is utilized for the discretization of the dynamic equations. Here, for saving space, only the main steps are provided. Detailed derivations and discussions of this method are seen in reference [58]. The IMLS-Ritz approximates the displacement field ux usinguhx=i=1mpixaix=pTxaxwhere px is a vector containing basis functions, m is the number of terms used in the approximation and ax is the coefficient vector. The coefficients are obtained by

Results and discussion

In the section, the convergence analysis and a few comparative studies are presented to test the accuracy and robustness of the proposed theoretical and numerical model, after which a thorough investigation of the vibration behaviors of rotating pre-twisted blade consisting of FG-CNTRC laminae and damaged FRC laminae are presented and discussed. For convenience, the nondimensional geometrical parameters including aspect ratio (or length-to-width ratio) (L/b), width-to-thickness ratio (b/h), and

Conclusion

In the study, based upon the differential geometry theory and the FSDT of a shell, a novel shell theory is derived and presented to model the kinetics of the pre-twisted blade. The proposed shell model is justified with the ability to model the mechanical behaviors of blades having a large pre-twisted angle. Then, taking the combined effect including Coriolis effect and centrifugal force duo to rotating, as well as the stiffness degradation because of matrix cracks into consideration, the

CRediT authorship contribution statement

Zhouzhou Pan: Methodology, Formal analysis, Writing - original draft, Investigation. K.M. Liew: Conceptualization, Methodology, Funding acquisition, Resources, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The work described in this paper was fully supported by a grant from the China National Natural Science Foundation (Grant No. 51378448).

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