• The Journal of the Acoustical Society of Korea
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• v.18 no.4E
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• pp.20-25
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• 1999

To get accurate natural frequencies of an engine crankshafts, finite element equations of motion are developed, taking real geometries of the shaft into account. For the crankshaft with wide crank webs, a specialized rotating web element is developed. This includes the effects of rotary inertia, gyroscopic moment, and shear. After the finite element equations are constructed, eigenvalues are extracted from the system equations to get natural frequencies, based on the Sturm sequence method which exploits the banded forms of the system matrices to reduce computations. The scheme developed can be used for the free vibration analysis of any type of spinning structures which include skew symmetric gyroscopic moment matrix in the system matrices. The results are compared with experimental data in order to confirm the study.

• Steel and Composite Structures
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• v.48 no.3
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• pp.263-273
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• 2023

This paper investigates dynamic characteristics of a graphene nanoplatelets reinforced composite (GNPRC) sandwich doubly curved shell based on the first-order shear deformation theory (FSDT) and Hamilton's principle. The sandwich doubly curved shell is fabricated from a core made of honeycomb materials sandwiched by composite GNPs reinforced face-sheets. Effective materials properties of composite face-sheets are assumed to vary based on Halpin-Tsai micromechanical models and rule of mixture. Furthermore, the material properties of honeycomb core are estimated using Gibson's formula. The fundamental frequencies of the shell are computed with changes of main geometrical and material properties such as amount and distribution type of graphene nanoplatelets, side length ratio, thickness to length ratio of and side length ratio of honeycomb. The Navier's technique is presented to obtain responses. Accuracy and trueness of the present model and analytical solution is confirmed through comparison of the results with available results in literature. It is concluded that an increase in thickness to length ratio yields a softer core with lower natural frequencies. Furthermore, increase in height to length ratio leads to significant decrease in natural frequencies.

• Journal of KSNVE
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• v.10 no.6
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• pp.1075-1082
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• 2000

The paper describes the vibration and stability of tapered beams on two-parameter elastic foundations. The two-parameter elastic foundations are constructed by distributed Winkler springs and a shearing layer as of ten used in soil models. The shear deformation and the rotatory inertia of a beam are taken into account. Governing equations are derived from energy expressions using Hamilton\`s principle. The associated eigenvalue problems are solved to obtain the free vibration frequencies or the buckling loads. Numerical results for the vibration of a beam with an axial force are presented and compared when other solutions are available. Vibration frequencies, mode shapes, and critical forces of a tapered Timoshenko beam on elastic foundations under an axial force are investigated for various thickness ratios, shear foundation parameters, Winkler foundation parameters and boundary conditions.

• Structural Engineering and Mechanics
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• v.42 no.5
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• pp.715-728
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• 2012

This paper deals with determining the fundamental frequency of tall buildings that consist of framed tube, shear core, belt truss and outrigger systems in which the framed tube and shear core vary in size along the height of the structure. The effect of belt truss and outrigger system is modeled as a concentrated rotational linear spring at the belt truss and outrigger system location. Many cantilevered tall structures can be treated as cantilevered beams with variable cross-section in free vibration analysis. In this paper, the continuous approach, in which a tall building is replaced by an idealized cantilever continuum representing the structural characteristics, is employed and by using energy method and Hamilton's variational principle, the governing equation for free vibration of tall building with variable distributed mass and stiffness is obtained. The general solution of governing equation is obtained by making appropriate selection for mass and stiffness distribution functions. By applying the separation of variables method for time and space, the governing partial differential equation of motion is reduced to an ordinary differential equation with variable coefficients with the assumption that the transverse displacement is harmonic. A power-series solution representing the mode shape function of tall building is used. Applying boundary conditions yields the boundary value problem; the frequency equation is established and solved through a numerical process to determine the natural frequencies. Computer program has been developed in Matlab (R2009b, Version 7.9.0.529, Mathworks Inc., California, USA). A numerical example has been solved to demonstrate the reliability of this method. The results of the proposed mathematical model give a good understanding of the structure's dynamic characteristics; it is easy to use, yet reasonably accurate and suitable for quick evaluations during the preliminary design stages.

• Structural Engineering and Mechanics
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• v.86 no.3
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• pp.291-309
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• 2023

This paper addresses the finite element modeling of functionally graded porous (FGP) beams for free vibration and buckling behaviour cases. The formulated finite element is based on simple and efficient higher order shear deformation theory. The key feature of this formulation is that it deals with Euler-Bernoulli beam theory with only three unknowns without requiring any shear correction factor. In fact, the presented two-noded beam element has three degrees of freedom per node, and the discrete model guarantees the interelement continuity by using both C⁰ and C¹ continuities for the displacement field and its first derivative shape functions, respectively. The weak form of the governing equations is obtained from the Hamilton principle of FGP beams to generate the elementary stiffness, geometric, and mass matrices. By deploying the isoparametric coordinate system, the derived elementary matrices are computed using the Gauss quadrature rule. To overcome the shear-locking phenomenon, the reduced integration technique is used for the shear strain energy. Furthermore, the effect of porosity distribution patterns on the free vibration and buckling behaviours of porous functionally graded beams in various parameters is investigated. The obtained results extend and improve those predicted previously by alternative existing theories, in which significant parameters such as material distribution, geometrical configuration, boundary conditions, and porosity distributions are considered and discussed in detailed numerical comparisons. Determining the impacts of these parameters on natural frequencies and critical buckling loads play an essential role in the manufacturing process of such materials and their related mechanical modeling in aerospace, nuclear, civil, and other structures.

• Wind and Structures
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• v.22 no.4
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• pp.429-454
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• 2016

In this paper, a shear deformation plate theory based on neutral surface position is developed for free vibration analysis of functionally graded material (FGM) plates. The material properties of the FGM plates are assumed to vary through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. During manufacture, defects such as porosities can appear. It is therefore necessary to consider the vibration behavior of FG plates having porosities in this investigation. The proposed theory is based on assumption that the in-plane and transverse displacements consist of bending and shear components, in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The neutral surface position for a functionally graded plate which its material properties vary in the thickness direction is determined. The equation of motion for FG rectangular plates is obtained through Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. Numerical results are presented and the influences of the volume fraction index and porosity volume fraction on frequencies of FGM plates are clearly discussed.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.7 no.3
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• pp.101-109
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• 1987

The governing differential equations for the free vibration of general arch are derived including the effect of rotary inertia in addition to the usual actions. These differential equations are applied to the sinusoidal arch and the numerical methods are developed to analyze these equations. A trial eigenvalue method and the Runge-Kutta method are used to determine the natural frequencies and to perform the integration of the differential equations, respectively. A detailed studies are made of the lowest three vibration frequencies for hinged arches with the span length equal to 10 m. The effect of the rotary inertia is analyzed. And as the numerical results the frequency versus the rise of arch and the radius of gyration are presented in figures.

• Transactions of the Korean Society of Mechanical Engineers
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• v.16 no.5
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• pp.849-863
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• 1992

A very simple and computationally efficient numerical method is developed for the free vibration of isotropic and orthotropic composite triangular plates. A set of two-dimensional simple series functions is used as an admissible displacement functions in the Rayleigh-Ritz method to obtain the natural frequencies, nodal patterns and mode shapes for the plates. From the prescribed starting function satisfying only the geometric boundary conditions, the higher terms in the series functions are constructed with adding order of polynominal. Natural frequencies, nodal patterns and mode shapes are obtained for right triangular plates with three different support conditions. The obtained numerical results are presented, and the isotropic and some orthotropic cases are verified with other numerical methods in the liternature.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.5 no.3
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• pp.31-38
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• 1985

The governing differential equations and the boundary conditions for the free vibration of fixed-ended uniform parabolic arch are derived on the basis of the equilibrium equations and the D'Alembert principle. The effect of rotary inertia as well as extensional and flexural deformations is considered in the governing differential equations. A trial elgenvalue method is used for determining the natural frequencies. The Runge-Kutta method is used in this method to perform the integration of the differential equations. The detailed studies are made of the lowest three vibration frequencies for the span length equal to 10m. The effect of the rotary inertia is analyzed and it's numerical data are presented in table. And as the numerical results the frequency versus the rise of arch and the radius of gyration are presented in figures.

• Structural Engineering and Mechanics
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• v.75 no.6
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• pp.747-769
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• 2020

This paper provides a semi-analytical approach to investigate the variations of 3D displacement components, electric potential, stresses, electric displacements and transverse vibration frequencies in laminated piezoelectric composite plates based on the scaled boundary finite element method (SBFEM) and the precise integration algorithm (PIA). The proposed approach can analyze the static and dynamic responses of multilayered piezoelectric plates with any number of laminae, various geometrical shapes, boundary conditions, thickness-to-length ratios and stacking sequences. Only a longitudinal surface of the plate is discretized into 2D elements, which helps to improve the computational efficiency. Comparing with plate theories and other numerical methods, only three displacement components and the electric potential are set as the basic unknown variables and can be represented analytically through the transverse direction. The whole derivation is built upon the three dimensional key equations of elasticity for the piezoelectric materials and no assumptions on the plate kinematics have been taken. By virtue of the equilibrium equations, the constitutive relations and the introduced set of scaled boundary coordinates, three-dimensional governing partial differential equations are converted into the second order ordinary differential matrix equation. Furthermore, aided by the introduced internal nodal force, a first order ordinary differential equation is obtained with its general solution in the form of a matrix exponent. To further improve the accuracy of the matrix exponent in the SBFEM, the PIA is employed to make sure any desired accuracy of the mechanical and electric variables. By virtue of the kinetic energy technique, the global mass matrix of the composite plates constituted by piezoelectric laminae is constructed for the first time based on the SBFEM. Finally, comparisons with the exact solutions and available results are made to confirm the accuracy and effectiveness of the developed methodology. What's more, the effect of boundary conditions, thickness-to-length ratios and stacking sequences of laminae on the distributions of natural frequencies, mechanical and electric fields in laminated piezoelectric composite plates is evaluated.