Dynamic stability analysis for rotating pre-twisted FG-CNTRC beams with geometric imperfections restrained by an elastic root in thermal environment

Highlights

• Dynamic stability of geometrically imperfect rotating pre-twisted FG-CNTRC beams is investigated.

• Influence of elastic root on the critical load and instability regions are investigated.

• Imperfection sensitivity of instability regions to various imperfection modes is studied.

• Studying the effects of temperature-dependence of material properties.

• Using the differential quadrature method to deal with elastic boundary conditions.

Abstract

This paper deals with the problems of free vibration, buckling, and dynamic stability of rotating pre-twisted functionally graded carbon nanotube reinforced composite (FG-CNTRC) imperfect beams in thermal environment. The imperfect beam contains different modes of geometric imperfections such as sine, global, and local modes, and it is restrained by an elastic root. Three types of CNTs distributions including FG-X, UD, and FG-O distributions are considered and the material is temperature-dependent. First, bending–bending coupled governing equations are established through the Hamilton’s principle based on the Euler–Bernoulli beam theory. By setting different parameters, the governing equations can solve the problems of free vibration, buckling, and dynamic stability of the beam. Then, the differential quadrature method (DQM) is employed to get the discrete equations and numerical solutions of the natural frequency, critical buckling load, and instability region. Finally, parametric studies are carried out to present the effects of hub radius, rotating speed, material properties, geometric imperfections, and rigidity of the elastic root on the natural frequencies, critical buckling load, and instability regions. Results show that the elastic root and imperfection mode have obvious influence on the instability regions.

Introduction

With the development of aerospace and energy technology, rotating pre-twisted beams become increasingly important in industry. Several applications including gas turbines, turbo-machinery and compressors, helicopters and wind turbines rely on the rotating pre-twisted beams [1], [2], [3]. In the last few decades, advanced materials are introduced in design of rotating pre-twisted beams. Compared with traditional materials, these advanced materials have more outstanding mechanical properties such as, low density, high strength, resistance to electrochemical corrosion, and other unique features [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. And FG-CNTRC materials have become a good choice. When operating, the rotating beams in complex working conditions are probably to be subjected to in-plane static and harmonic excitation. Thus, the problems of buckling and dynamic stability occur which are essential to the safety of the structure.

Previous researchers have investigated the static and dynamic behaviors of rotating pre-twisted structures in recent years. Based on the Euler–Bernoulli beam theory, Adair and Jaeger [1] introduced a numerical method named Adomian modified decomposition method (AMDM) to analyze the bending–bending coupled free vibrations of rotating pre-twisted cantilever beams. Taking the tendon-induced axial loading into consideration, Ondra and Titurus [3] explored the bending–bending–torsion vibration of pre-twisted cantilever beams. Coupled frequencies and critical buckling load were calculated by DQM. In addition, Leung and Fan [2] discussed the coupled influence of more kinds of initial stresses including compression, moments and torque on the free vibration and buckling of rotating pre-twisted beams based on the Timoshenko beam theory. Under the influence of locally distributed Kelvin–Voigt damping, axially-loaded pre-twisted Timoshenko beams were introduced by Ref. [14] to show the influence of the damping amount, size and location of damped segment, and restraint types on the vibration characteristics of the beam. As advanced materials are widely used in rotating structures, there exists some study about the influence of material properties on the structures. On the basis of the Rayleigh–Ritz method, Chen and Li [15] investigated the vibration behaviors of composite laminated pre-twisted blades considering the effects of the Coriolis. Besides the ply angle of laminated structures, the aspect ratio and stagger angle were discussed. In Ref. [16] functionally graded materials (FGMs), size effects and thermal effects were employed. Based on the Chebyshev–Ritz method, influence of these parameters on the vibration behaviors were displayed. On the other hand, the temperature can affect the static stability of the beam. High temperature may lead to the buckling of the beam. Shenas et al. [17] dealt with the buckling problem of pre-twisted FGM beams with temperature-dependent material properties. According to the third-order shear deformation theory and Chebyshev–Ritz method, critical temperature was calculated. As to dynamic stability analysis, Sabuncu and Evran [18], [19] obtained the instability regions of rotating pre-twisted beam subjected to lateral parametric excitation and axial periodic force, respectively.

Benefit from the development of advanced manufacturing technology, the implementation of carbon nanotubes (CNTs) has been widely used as polymer reinforcement by advantage of their remarkable mechanical properties [4], [20], [21], [22]. And the FG-CNTRC as a new kind of composite material is developing rapidly and many researchers have focused on it. Khosravi et al. [23] investigated the effects of uniform temperature elevation on the vibration characteristics of rotating FG-CNTRC beams based on the Timoshenko beam theory with general boundary conditions. Influence of uniform distribution and two symmetric gradient distributions of CNTs on the mode shapes were compared in their research. In Ref. [24], the problems of interactive thermal and inertial buckling of rotating temperature-dependent FG-CNT beams were studied based on DQM. Taking the pre-twisted angle and third-order shear deformation into consideration, Shenas et al. [25] researched the free vibration behavior of pre-twisted FG-CNTRC beams in thermal environment with the help of the Chebyshev–Ritz method. Results showed that effects of the pre-twisted angle relied on the boundary conditions and the mode number.

All the studies mentioned above focus on the perfect structures. In fact, limited to the manufacture technology, the structure in practice always has initial geometric imperfections, which may affect the mechanical properties of the structure [26]. Sah et al. [27] studied the impacts of geometrical and loading imperfections on the transverse vibration characteristics at the same time. The pre-buckling and post-buckling phenomena of geometrically imperfect composite beams were analyzed in Ref. [28]. Emam presented free vibrations in the postbuckling domain and acquired an analytical solution to the static response affected by geometric imperfections. Barati and Zenkour [29] additionally showed effects of the nonlinear hardening foundation on the post-buckling behavior of geometrically imperfect beams. Under the influence of high order shear deformable, the nonlinear resonant phenomenon of imperfect FG-CNTRC beams was studied in Ref. [30] based on the generalized differential quadrature (GDQ) method. And the dynamic responses of perfect and imperfect cantilever beams under harmonic primary resonant base excitation were compared in Ref. [31].

These studies about the geometric imperfection assume that the shapes of the initial geometric imperfection are similar to the vibration mode or buckling mode of the beam. However, the assumption is not always accurate in real situations, the geometric imperfections may exist in other shapes. Recently, some general geometric imperfections are investigated by few researches. Li and Qiao [32] presented three different modes of initial geometric imperfections, including sine, global, and local mode. Effects of these geometric imperfections on the postbuckling behaviors of imperfect composite beams were studied, according to higher order shear deformation beam theory. Chen et al. [33] explored the effects of geometric imperfection on the flapwise vibration of rotating composite laminated microbeams based on re-modified couple stress theory. When the coupling of thermo-electro-mechanical fields occurred, the postbuckling behavior of FG-CNTRC beams with geometric imperfections was studied by Ref. [34] using DQM. In case of the nonlinear vibration problems, Wu et al. [35] calculated the nonlinear frequencies of FG-CNTRC beam with global and localized geometric imperfections using the Ritz method.

To the authors’ best knowledge, there is little research focusing on the rotating pre-twisted FG-CNTRC imperfect beams. In this paper, a new dynamic model of rotating pre-twisted FG-CNTRC imperfect beams is established for the first time. Combined influence of the thermal environment, rotating effect, geometric imperfection, and pre-twisted angle are investigated. Additionally, the elastic root has significant influence on the rotating structures and it is rarely researched. Another novelty of the present work is that elastic boundary conditions are carried out, showing the effects of elastic root on the dynamic stability of the rotating beam.

This paper researched the natural frequencies, critical buckling load, and dynamic stability of rotating pre-twisted FG-CNTRC imperfect beams restrained by an elastic root in thermal environment. Three different modes of geometric imperfections including sine, global, and local modes of the imperfect beam are considered in this paper. The critical buckling load and instability region of the system are investigated. According to the Hamilton’s principle, bending–bending coupled governing equations of the system are derived. Then, discrete equations and approximate solutions of the critical buckling load and dynamic stability are obtained with the help of DQM. Finally, influence of some factors including the hub radius, rotating speed, temperature variation, rigidity of the elastic root, geometric imperfections, and material properties on the natural frequencies, critical buckling load, and instability regions are discussed.