• Smart Structures and Systems
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• v.21 no.3
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• pp.279-286
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• 2018

The free longitudinal vibration of a circular truncated nanocone is investigated based on the nonlocal elasticity theory. Exact analytical formulations for tapered nanostructures are derived and the nonlinear differential governing equation of motion is developed. The nonlocal small scale effect unavailable in classical continuum theory is addressed to reveal the long-range interaction of atoms implicated in nonlocal constitutive relation. Unlike most previous studies applying the truncation method to the infinite higher-order differential equation, this paper aims to consider all higher-order terms to show the overall nonlocality. The explicit solution of nonlocal stress for longitudinal deformation is determined and it is an infinite series incorporating the classical stress derived in classical mechanics of materials and the infinite higher-order derivative of longitudinal displacement. Subsequently, the first three modes natural frequencies are calculated numerically and the significant effects of nonlocal small scale and vertex angle on natural frequencies are examined. The coupling phenomenon of natural frequency is observed and it is induced by the combined effects of nonlocal small scale and vertex angle. The critical value of nonlocal small scale is defined, and after that a new proposal for determining the range of nonlocal small scale is put forward since the principle of choosing the nonlocal small scale is still unclear at present. Additionally, two different types of nonlocal effects, namely the nonlocal stiffness weakening and strengthening, reversed phenomena existing in nanostructures are observed and verified. Hence the opposite nonlocal effects are resolved again clearly. The nano-engineers dealing with a circular truncated nanocone-based sensors and oscillators may benefit from the present work.

• Steel and Composite Structures
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• v.46 no.3
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• pp.319-334
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• 2023

Localizing damages is an essential task to monitor the health of the structures since they may not be able to operate anymore. Among the damage detection techniques, non-destructive methods are considerably more preferred than destructive methods since damage can be located without affecting the structural integrity. However, these methods have several drawbacks in terms of detecting abilities, time consumption, cost, and hardware or software requirements. Employing artificial intelligence techniques could overcome such issues and could provide a powerful damage detection model if the technique is utilized correctly. In this study, the crack localization in flat and folded plate structures has been conducted by employing a Backpropagated Artificial Neural Network (BPANN). For this purpose, cracks with 18 different dimensions in thin, flat, and folded structures having 15⁰, 30⁰, 45⁰, and 60⁰ folding angle have been modeled and subjected to free vibration analysis by employing the Classical Plate Theory with Finite Element Method. A Four-nodded quadrilateral element having six degrees of freedom has been considered to represent those structures mathematically. The first ten natural frequencies have been obtained regarding healthy and cracked structures. To localize the crack, the ratios of the frequencies of the cracked flat and folded structures to those of healthy ones have been taken into account. Those ratios have been given to BPANN as the input variables, while the crack locations have been considered as the output variables. A total of 500 crack locations have been regarded within the dataset obtained from the results of the free vibration analysis. To build the best intelligent model, a feature search has been conducted for BAPNN regarding activation function, the number of hidden layers, and the number of hidden neurons. Regarding the analysis results, it is concluded that the BPANN is able to localize the cracks with an average accuracy of 95.12%.

• Steel and Composite Structures
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• v.26 no.4
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• pp.513-531
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• 2018

In this article, static, buckling and free vibration analyses of a sinusoidal micro composite beam reinforced by single-walled carbon nanotubes (SWCNTs) with considering temperature-dependent material properties embedded in an elastic medium in the presence of magnetic field under transverse uniform load are presented. This system is used at micro or sub micro scales to enhance the stiffness of micro composite structures such as bar, beam, plate and shell. In the present work, the size dependent effects based on surface stress effect and modified strain gradient theory (MSGT) are considered. The generalized rule of mixture is employed to predict temperature-dependent mechanical and thermal properties of micro composite beam. Then, the governing equations of motions are derived using Hamilton's principle and energy method. Numerical results are presented to investigate the influences of material length scale parameters, elastic foundation, composite fiber angle, magnetic intensity, temperature changes and carbon nanotubes volume fraction on the bending, buckling and free vibration behaviors of micro composite beam. There is a good agreement between the obtained results by this research and the literature results. The obtained results of this study demonstrate that the magnetic intensity, temperature changes, and two parameters elastic foundations have important effects on micro composite stiffness, while the magnetic field has greater effects on the bending, buckling and free vibration responses of micro composite beams. Moreover, it is shown that the effects of surface layers are important, and observed that the changes of carbon nanotubes volume fraction, beam length-to-thickness ratio and material length scale parameter have noticeable effects on the maximum deflection, critical buckling load and natural frequencies of micro composite beams.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.14 no.5
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• pp.1023-1031
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• 1994

The closed forms of consistent mass matrix with rotational inertia matrix are developed for free vibration analysis in space sutructures containing linearly tapered members with cross section of thin-walled I-sections. The exact displacement functions are used for formulating mass matrices. The very small slopes of the tapered member are used in usual practice, such that the series expansion forms of these are also developed to avoid numerical failure in vibration analysis. Significant improvements of accuracy and efficiency of free vibation analysis are achieved by using the mass matrices developed in this study. Frequencies of free vibation of tapered members are compared with solutions based upon stepped representation of beam element in the ANSYS. The mass matrices presented in this study can be used for the free vibration analysis of tapered and prismatic members.

• Magazine of the Korean Society of Agricultural Engineers
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• v.38 no.3
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• pp.50-57
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• 1996

A numerical method is presented to obtain the natural frequencies and mode shapes of the arbitrary tapered beams with static deflection due to arbitrary distributed dead loads. The differential equation governing free vibration of such beams is derived and solved numerically. The double integration method using the trapezoidal rule is used to solve the static behaviour of beams loaded arbitrary distributed dead load. Also, the Improved Euler method and the determinant search method are used to integrate the differential equation subjected to the boundary conditions and to determine the natural frequencies of the beams, respectively. In the numerical examples, the various geometries of the beams are considered : (1) linearly tapered beams as the arbitrary variable cross-section, (2) the triangular, sinusoidal and uniform loads as the arbitrary distributed dead loads and (3) the hinged-hinged, clamped-clamped and hinged-clamped ends as the end constraints. All numerical results are shown as the non-dimensional forms of the system parameters. The lowest three natural frequencies versus load parameter, slenderness ratio and section ratio are reported in figures. And for the comparison purpose, the typical mode shapes with and without the effects of static deflection are presented in the figure. According to the numerical results obtained in this analysis, the following conclusions may be drawn : (1) the natural frequencies increase when the effects of static deflections are included, (2) the effects are larger at the lower modes than the higher ones and (3) it should be betteF to include the effect of static deflection for calculating the frequencies when the beams are supported by both hinged ends or one hinged end.

• Journal of Advanced Marine Engineering and Technology
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• v.10 no.4
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• pp.41-49
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• 1986

With the trend towards high propulsive level, increasing ship's dimensions and heavier shaft systems supported by the hull structure of relatively stiffness in modern ships, transverse vibrations of propulsion shaft system have become one of the problems that should be predicted in the early design stage. Regarding transverse vibrations, coupling terms such as oilfilm, gyroscope and hydrodynamic effect of the propeller exist between the vertical and horizontal vibration, furthermore for the shaft system with strut and bossing its physical properties incorporated with hull structure must be considered. In order to predict the transverse vibratory condition of the propulsion shaft and take some appropriate countermeasures, it is necessary to make a fairly strict estimation of the vibratory behaviours of it. In this paper, theoretical approach using the finite element method is investigated to calculate natural frequencies and vibration modes for coupled free transverse vibrations of shaft system in two planes. Based on the method investigated a digital computer program is developed and is applied to calculate the above-mentioned vibrations of an experimental model shaft system. The results of the calculation are compared with those of the experimental measurements and they show an acceptable agreement.

• International Journal of Aeronautical and Space Sciences
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• v.18 no.2
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• pp.255-269
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• 2017

Free vibration analysis is presented for a simply-supported, functionally graded piezoelectric (FGP) nanobeam embedded on elastic foundation in the framework of third order parabolic shear deformation beam theory. Effective electro-mechanical properties of FGP nanobeam are supposed to be variable throughout the thickness based on power-law model. To incorporate the small size effects into the local model, Eringen's nonlocal elasticity theory is adopted. Analytical solution is implemented to solve the size-dependent buckling analysis of FGP nanobeams based upon a higher order shear deformation beam theory where coupled equations obtained using Hamilton's principle exist for such beams. Some numerical results for natural frequencies of the FGP nanobeams are prepared, which include the influences of elastic coefficients of foundation, electric voltage, material and geometrical parameters and mode number. This study is motivated by the absence of articles in the technical literature and provides beneficial results for accurate FGP structures design.

• Structural Engineering and Mechanics
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• v.52 no.4
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• pp.663-686
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• 2014

This paper deals with free vibration analysis of bidirectional functionally graded annular plates resting on a two-parameter elastic foundation. The formulations are based on the three-dimensional elasticity theory. This study presents a novel 2-D six-parameter power-law distribution for ceramic volume fraction of 2-D functionally graded materials that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. Various material profiles along the thickness and in the in-plane directions are illustrated by using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori-Tanaka scheme. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Some new results for natural frequencies of the plates are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D functionally graded materials.

• Structural Engineering and Mechanics
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• v.64 no.3
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• pp.381-390
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• 2017

Present paper deals with the large amplitude flexural vibration of carbon nanotube reinforced composite (CNTRC) plates. Distribution of CNTs as reinforcements may be uniform or functionally graded (FG). The equivalent material properties of the composite media are obtained according to a refined rule of mixtures which contains efficiency parameters. To account for the large deformations, von $K{\acute{a}}rm{\acute{a}}n$ type of geometrical nonlinearity is included into the formulation. The matrix representation of the governing equations is obtained according to the Ritz method where the basic shape functions are written in terms of the Chebyshev polynomials. Time dependency of the problem is eliminated by means of the Galerkin method and the resulting nonlinear eigenvalue problem is solved employing a direct displacement control approach. Results are obtained for completely clamped and completely simply supported plates. Results are first validated for the especial cases of FG-CNTRC and cross-ply laminated plates. Afterwards, parametric studies are given for FG-CNTRC plates with different boundary conditions. It is shown that, nonlinear frequencies are highly dependent to the volume fraction and dispersion profiles of CNTs. Furthermore, mode redistribution is observed in both simply supported and clamped FG-CNTRC plates.

• Composites Research
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• v.12 no.1
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• pp.47-58
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• 1999

This paper describes the results of experiments to analyze the free vibration of the laminated composite and hybrid composite plates with various shapes and boundary conditions. The materials of specimens were the carbon fiber reinforced plastic (CFRP), the glass fiber reinforced plastic (GFRP), the GFRP-Aluminum hybrid composite and the CFRP-CFRP hybrid composite. The natural frequencies and nodal patterns of plates with various shapes were experimentally obtained by impact exciting test using an impact hammer and an accelerometer. The experimental results were presented with normalized frequency parameters. The effects of composite material properties, fiber orientation angles, various geometrical shapes and boundary conditions on the vibration characteristics of composite plates were evaluated. To compare and verify these experimental results, the finite element analysis was carried out, and was well agreed with experimental results.