Domains of dynamic instability of FGM conical shells under time dependent periodic loads
Introduction
When in service, the different shell structures or their components may be subjected to periodic loads that somehow change some characteristics of the system. These include, for example, aircrafts with pulsating thrust; underwater pipelines subjected to external pressure and transmitting pulsating flows of oil and gas, mine hoists, which periodically interact with the pit–shaft, etc. The stability problems of such systems under various loads are of interest for reasons of accident prevention. A theoretical analysis of these issues, generally involves solution of nonlinear transient dynamic problems for elastic systems and analysis of various waveforms and scenarios for the emergence of irregular modes. The important dynamic problem of identifying the domains of the instability may be solved in the linear case. This got a boost after Bolotin’s [1] contribution to the literature. Bolotin [1] developed the general theory of dynamic instability of elastic systems of deriving the coupled differential equation of the Mathieu-type and the determination of the domains of instability by seeking periodic solution using Fourier series expansion. A comprehensive review of early developments in the dynamic instability of shell elements was presented in the review work by Simitses [2]. Recently, an extensive bibliography of earlier works on dynamic stability/dynamic instability/parametric excitation/parametric resonance of plates and shells was presented by Sahu and Datta [3] from 1987–2005. The dynamic instability of cylindrical shells was adequately studied in Refs. [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. A considerably fewer publications on the dynamic instability are concerned with homogeneous conical shells [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. For example, the approximate solution of the dynamic stability of truncated conical shells under pulsating pressure applying Galerkin and Bolotin method to Donnell type basic equations considering only the transverse inertia term was proposed by Kornecki [14]. The dynamic instability of truncated conical shells under periodic axial loads was studied by Tani [15], [16] using the Donnell equations. The dynamic stability of clamped conical shells with variable modulus of elasticity under pulsating torsion excitations was studied using Bolotin’s method to obtain the principal instability regions by Massalas et al. [18]. The effects of boundary conditions on the parametric instability of truncated conical shells under periodic edge loading utilizing the generalized differential quadrature (GDQ) method was examined by Ng et al. [19]. The dynamic instability analysis of truncated circular conical shells under periodic in-plane load, was investigated using C0 two nodded shear flexible shell element by Ganapathia et al. [20]. The WKB (Wentzel, Kramers and Brillouin) method combined with the method of multiple time scales was employed by Kuntsevich and Mikhasev [21] to study the parametric instability for a thin conical shell subjected to non-uniform pulsating pressure. Recently, the parametric resonance of a truncated conical shell subjected to periodic axial loads and rotating at periodically varying angular speed based upon the Love’s thin shell theory and generalized differential quadrature (GDQ) method was studied by Han and Chu [22], [23], respectively.
Recently, a new class of composite materials known as functionally graded materials (FGM) has received considerable attention because of the increasing demands of high structural performance requirements, especially in extreme high-temperature environments and high-speed industries. FGMs are designed to achieve a functional performance with gradually variable properties in one or more spatial directions [24], [25]. Noteworthy works considering various aspects of FGM have been published in recent years; see, e.g. [26], [27], [28]. In recent years, functionally graded (FG) conical shells are widely used in space vehicles, aircrafts, nuclear power plants and many other engineering applications. Therefore, many studies have been conducted to investigate the static and dynamic behaviors of FG conical shells. A review of the literature shows that the majority of these studies related to the static stability or free vibration analyses of FG conical shells using various shell theories [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45].
The studies on the dynamic instability analysis of FG shells are much less in comparison to the vibration and static stability. An overview of the works on the subject of dynamic instability of FG shells of various types is mentioned below. Ng et al. [46] examined dynamic instability analysis of FG cylindrical shell subjected under periodic axial loading within the classical shell theory (CST). Ansari and Darvizeh [47] presented a general analytical approach to investigate dynamic behavior of temperature-dependent FG shells under different boundary conditions within the CST. Bespalova and Ursova [48] studied the influence of alternating curvature on the dynamic instability domains of inhomogeneous shells under combined static and dynamic loadings. Ovesy and Fazilati [49] studied a dynamic stability analysis of moderately thick FG cylindrical panels is conducted by employing finite strip formulations. Lei et al. [50] presented a first-known dynamic stability analysis of carbon nanotube-reinforced FG cylindrical panels under static and periodic axial forces by using the mesh-free kp-Ritz method. Torki et al. [51] studied dynamic stability of FG cantilever cylindrical shells under distributed axial follower forces. Sofiyev and Kuruoglu [52] studied the dynamic instability of sandwich cylindrical shells containing an FGM interlayer within classical shell theory (CST). To the best of the authors’ knowledge, there is no literature for the solution of dynamic instability analysis of FG truncated conical shells subject to static and time dependent periodic axial load and the authors attempt to fill these apparent voids.
Section snippets
Formulation of the problem
The schematic configuration of an FGM truncated conical shell and coordinate system are shown in Fig. 1, where the S-axis in the direction of the generator of the cone, the -axis in the direction normal to the reference surface of the cone, and -axis in the direction “perpendicular” to the plane. and indicate the radii of the cone at its small and large ends, respectively, denotes the semi-vertex angle of the cone, is the length and is the thickness of the truncated
Basic equations
In the present study, the conical shells are assumed to be thin and the classical shell theory based on Love’s hypotheses is used to investigate the dynamic instability of FGM conical shells under static and time dependent periodic axial load. The stress–strain relations of FG truncated conical shells within the classical shell theory are given as [43]:where and are the strains and stresses of the FG conical shell,
Solution of basic equations
The FG truncated conical shell is assumed to be simply supported at and , thus the solution of Eq. (14) is sought in the following form as [54]:where f(t) is time dependent unknown function and the following definitions are introduced:
Substituting Eq. (17) into Eq. (14) and by applying the Superposition principle can be obtained as:where the following definitions apply:
Results and discussion
In this section, two comparative studies and the new numerical analysis for the dynamic instability of FG truncated conical shells are presented using Maple 14 software.
Conclusion
The present study deals with the theoretical analysis of parametric instability characteristics of functionally graded (FG) conical shells subjected to harmonic axial loading based on the classical shell theory (CST). The basic relations and equations are derived using the Donnell shell theory. Appling Galerkin’s method, the partial differential equations are reduced into a Mathieu type differential equation describing the dynamic instability behavior of the FG conical shell. Following
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