We present an exact method to model the free vibration of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) beams with arbitrary boundary conditions based on first-order shear deformation elasticity theory. Five types of carbon nanotube (CNT) distributions are considered. The distributions are either uniform or functionally graded and are assumed to be continuous through the thickness of the beams. The displacements and rotational components of the beams are expressed as a linear combination of the standard Fourier series and several supplementary functions. The formulation is derived using the modified Fourier series and solved using the strong-form solution and the weak-form solution (i.e., the Rayleigh–Ritz method). Both solutions are applicable to various combinations of boundary constraints, including classical boundary conditions and elastic-supported boundary conditions. The accuracy, efficiency and validity of the two solutions presented are demonstrated via comparison with published results. A parametric study is conducted on the influence of several key parameters, namely, the L/h ratio, CNT volume fraction, CNT distribution, boundary spring stiffness and shear correction factor, on the free vibration of FG-CNTRC beams.

The paper introduces a semi-analytical approach to analyze free vibration characteristics of stepped functionally graded (FG) paraboloidal shell with general edge conditions. The analytical model is established based on multi-segment partitioning strategy and first-order shear deformation theory. The displacement components along axial direction are represented by Jacobi polynomials, and the Fourier series are utilized to express displacement components in circumferential direction. Based on penalty method about spring stiffness technique, the general edge conditions of doubly curved paraboloidal shell can be easily simulated. The solutions about doubly curved paraboloidal shell were solved by approach of Rayleigh–Ritz. Convergence study about boundary parameters, Jacobi parameters et al. are carried out, respectively. The comparison with published literatures, FEM and experiment results show that the present method has good convergence ability and excellent accuracy.

The aim of this paper is to extend the modified Fourier‐Ritz approach to evaluate the free vibration of four‐parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The first‐order shear deformation theory is employed to formulate the theoretical model. In the modified Fourier‐Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two‐dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.

This paper is concerned with the free vibration analysis of open circular cylindrical shells with either the two straight edges or the two curved edges sim

In this paper, a modified Fourier solution based on the first-order shear deformation theory is developed for the free vibration problem of moderately thick composite laminated annular sector plates with general boundary conditions, internal radial line and circumferential arc supports. In this solution approach, regardless of boundary conditions, the displacement and rotation components of the sector plate are written in the form of the trigonometric series expansion in which several auxiliary terms are added to ensure and accelerate the convergence of the series. Each of the unknown coefficients is taken as the generalized coordinate and determined using the Raleigh- Ritz method. The accuracy and reliability of the present solution are validated by the comparison with the results found in the literature, and numerous new results for composite laminated annular sector plates considering various kinds of boundary conditions are presented. Comprehensive studies on the effects of elastic restraint parameters, layout schemes and locations of line/arc supports are also made.New results are obtained for laminated annular sector plates subjected to elastic boundary restraints and arbitrary internal radial line and circumferential arc supports in both directions, and they may serve as benchmark solutions for future researches.

Acoustic black holes have good application prospects in the field of vibration and noise reduction. Based on engineering practice, this study proposes a systematic process method for the application of acoustic black hole structure in raft structure, which provides new ideas and references for improving the vibration isolation performance of floating raft system and reducing the level of ship vibration and noise. The influence law of each parameter on structural vibration and the recommended value range of each parameter are given, which provides support for the systematic method and process of the application of acoustic black holes in the raft structure. Then, the acoustic black hole process is applied to a floating raft system. According to the characteristics of the raft structure, an application scheme of the acoustic black hole in the raft structure is formed, and the vibration level drop of the floating raft vibration isolation system before and after the acoustic black hole is embedded, calculated, and analyzed. The changes further improve the vibration reduction and isolation performance of the raft system and effectively reduced the mechanical noise level of the ship’s cabin.