• Wind and Structures
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• 제23권1호
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• pp.59-73
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• 2016

In this paper, a hyperbolic shear deformation beam theory is developed for free vibration analysis of functionally graded (FG) sandwich beams. The theory account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The material properties of the functionally graded sandwich beam are assumed to vary according to power law distribution of the volume fraction of the constituents. The core layer is still homogeneous and made of an isotropic material. Based on the present refined beam theory, the equations of motion are derived from Hamilton's principle. Navier type solution method was used to obtain frequencies. Illustrative examples are given to show the effects of varying gradients and thickness to length ratios on free vibration of functionally graded sandwich beams.

• Wind and Structures
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• 제21권3호
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• pp.273-287
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• 2015

In this paper, a free vibration analysis of functionally graded beam made of porous material is presented. The material properties are supposed to vary along the thickness direction of the beam according to the rule of mixture, which is modified to approximate the material properties with the porosity phases. For this purpose, a new displacement field based on refined shear deformation theory is implemented. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. Based on the present refined shear deformation beam theory, the equations of motion are derived from Hamilton's principle. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

• Geomechanics and Engineering
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• 제9권3호
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• pp.361-372
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• 2015

In this paper, a refined exponential shear deformation theory for free vibration analysis of functionally graded beam with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a new displacement field based on refined shear deformation theory is implemented. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. Based on the present refined shear deformation beam theory, the equations of motion are derived from Hamilton's principle. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

• Structural Engineering and Mechanics
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• 제13권3호
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• pp.277-298
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• 2002

In this study, free vibration analysis of Reissner plates on Pasternak foundation is carried out by mixed finite element method based on the G$\hat{a}$teaux differential. New boundary conditions are established for plates on Pasternak foundation. This method is developed and applied to numerous problems by Ak$\ddot{o}$z and his co-workers. In dynamic analysis, the problem reduces to the solution of a standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and bending and torsional moments, transverse shear forces, rotations and displacements are the basic unknowns. The element performance is assessed by comparison with numerical examples known from literature. Validity limits of Kirchhoff plate theory is tested by dynamic analysis. Shear locking effects are tested as far as $h/2a=10^{-6}$ and it is observed that REC32 is free from shear locking.

• Steel and Composite Structures
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• 제18권1호
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• pp.187-212
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• 2015

In this paper, various four variable refined plate theories are presented to analyze vibration of temperature-dependent functionally graded (FG) plates. By dividing the transverse displacement into bending and shear parts, the number of unknowns and governing equations for the present model is reduced, significantly facilitating engineering analysis. These theories account for parabolic, sinusoidal, hyperbolic, and exponential distributions of the transverse shear strains and satisfy the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Power law material properties and linear steady-state thermal loads are assumed to be graded along the thickness. Uniform, linear, nonlinear and sinusoidal thermal conditions are imposed at the upper and lower surface for simply supported FG plates. Equations of motion are derived from Hamilton's principle. Analytical solutions for the free vibration analysis are obtained based on Fourier series that satisfy the boundary conditions (Navier's method). Non-dimensional results are compared for temperature-dependent and temperature-independent FG plates and validated with known results in the literature. Numerical investigation is conducted to show the effect of material composition, plate geometry, and temperature fields on the vibration characteristics. It can be concluded that the present theories are not only accurate but also simple in predicting the free vibration responses of temperature-dependent FG plates.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 1997년도 추계학술대회논문집; 한국과학기술회관; 6 Nov. 1997
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• pp.173-176
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• 1997

A refined plate theory including the effects of transverse shearing is used to predict the free vibration frequencies, mode shapes and stress distributions in spinning laminated composite plates. In this theory, the displacements are expressed by trigonometric series representation through the thickness. In the series for the displacements only the first few terms are retained. The model is validated by comparing the results for isotropic plates with those available in the literature.

• Structural Engineering and Mechanics
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• 제40권5호
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• pp.689-703
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• 2011

A new perturbation method is introduced to study the undamped free vibration of a non-prismatic Timoshenko beam for its natural frequencies and vibration modes. For simplicity, the natural modes of vibration of its corresponding prismatic Euler-Bernoulli beam with the same length and boundary conditions are used as Ritz base functions with necessary modifications to account for shear strain in the Timoshenko beam. The new method can transform two coupled partial differential equations governing the transverse vibration of the non-prismatic Timoshenko beam into a set of nonlinear algebraic equations. It significantly simplifies the solution process and is applicable to non-prismatic beams with various boundary conditions. Three examples indicated that the new method is more accurate than the previous perturbation methods. It successfully takes into account the effect of shear deformation of Timoshenko beams particularly at the free end of cantilever structures.

• Journal of the Korean Wood Science and Technology
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• 제16권3호
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• pp.48-64
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• 1988

This study was carried out to investigate the free vibrational properties of traditional and replaceable species for sounding board, using piezoelectrical accelerometer and signal analyzer. In the study, the frequency equations of free-free beam carrying a concentrated mass in the transverse vibration and of free-mass beam in the longitudinal vibration were used. The results obtained were as follows. 1. Average values of dynamic modulus of rigidity of Korean commercial species measured were as follows. Paulownia tomatosa (Thunb.) Steudel: $5.590\times10^9\;dyne/cm^2$ Tilia amurensis Rupr.: $5.414\times10^9\;dyne/cm^2$ Macckia amurensis Rupr.: $10.044\times10^9\;dyne/cm^2$ Fraxinus mandshurica Rupr.: $8.876\times10^9\;dyne/cm^2$ Ulmus davidiana var.japonica Nakai: $8.677\times10^9\;dyne/cm^2$ Pinus rigida Miller: $6.33\times10^9\;dyne/cm^2$ Fraxinus rhynchophylla Hance: $4.666\times10^9\;dyne/cm^2$ 2. The ratio of dynamic transverse Young's modulus to dynamic modulus of rigidity, $E_T/G_{LT}$, was 24.922 for Fraxinus rhynchophylla Hance, which was the highest, 7.317 for Paulownia tomentosa (Thunb.) Steudel, which was the lowest among Korean commercial species measured. 3. The value of longitudinal dynamic Young's modulus was about 3.7% and 19.8%, respectively, higher than those of transverse dynamic and static Young's modulus. The value of transverse dynamic Young's modulus was about 15.5% higher than that of static Young's modulus. 4. Predicted value of MOR in terms of correlation coefficient by transverse dynamic Young's modulus was slightly higher than that of static Young's modulus, but no significance was found out. 5. Transverse dynamic Youne's modulus was $(2.002\pm0.288)\times10^{11}\;dyne/cm^2$ for Fraxinus mandshurlca Rupr., which was the highest, $(0.601\pm0.100)\times10^{11}\;dyne/cm^2$ for Paulownia tomentosa (Thunb.) Steudel, which was the lowest among Korean commercial species measured. The sound velocity of sitka spruce was 5,685 m/sec, which was the highest. 6. Internal friction of Paulownia tomentosa (Thunb.) Steudel was shown the lowest value among Korean commercial species, $(8.795\pm1.795)\times10^{-3}$, but was higher than that of sitka spruce, $(7.331\pm0.991)\times10^{-3}$. Internal friction was shown negative correlation with density and dynamic Young's modulus, respectively. 7. K value was affected largely by internal friction and was the highest, $2.225\times10^8$ for Paulownia tomentosa (Thunb.) Steudel and was the lowest, $0.550\times10^8$ for Fraxinus rhynchophylla Hance. K value of Paulownia tomentosa (Thunb.) Steudel was higher than that of sitka spruce and K values of melapi and cottonwood, which have been considered to be replaceable species with sitka spruce in the piano industry, were lower than those of Paulownia tomentosa (Thunb.) Steudel and mill amurensis Rupr.

• Steel and Composite Structures
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• 제20권2호
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• pp.227-249
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• 2016

The objective of this work is to present a zeroth-order shear deformation theory for free vibration analysis of functionally graded (FG) nanoscale plates resting on elastic foundation. The model takes into consideration the influences of small scale and the parabolic variation of the transverse shear strains across the thickness of the nanoscale plate and thus, it avoids the employ use of shear correction factors. Also, in this present theory, the effect of transverse shear deformation is included in the axial displacements by using the shear forces instead of rotational displacements as in available high order plate theories. The material properties are supposed to be graded only in the thickness direction and the effective properties for the FG nanoscale plate are calculated by considering Mori-Tanaka homogenization scheme. The equations of motion are obtained using the nonlocal differential constitutive expressions of Eringen in conjunction with the zeroth-order shear deformation theory via Hamilton's principle. Numerical results for vibration of FG nanoscale plates resting on elastic foundations are presented and compared with the existing solutions. The influences of small scale, shear deformation, gradient index, Winkler modulus parameter and Pasternak shear modulus parameter on the vibration responses of the FG nanoscale plates are investigated.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 2002년도 춘계학술대회 논문집
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• pp.647-651
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• 2002

This study dealt with the free vibration of two identical rectangular plates coupled with fluid. In order to investigate the vibration characteristics of fluid-coupled rectangular plates, an analytical method based on the finite Fourier series expansion and Rayleigh-Ritz method was suggested. A commercial computer code, ANSYS was used to perform finite element analysis and we investigated the vibration characteristics with mode shapes and natural frequencies. As a result, the transverse vibration modes, in-phase and out-of-phase, were observed alternately in the fluid-coupled system. The effect of fluid bounding and plate boundary condition on the fluid-coupled natural frequency were investigated. It was shown that the mode numbers increased, the normalized natural frequencies monotonically increased.