• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2014.10a
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• pp.389-391
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• 2014

The Vibration Analysis of Rotating Inward Beams Considering The Tip-Mass is presented based on Euler-Bernoulli beam theory. The frequency equations, which are coupled through gyroscopic coupling terms, are calculated using hybrid deformation variable modeling along with the Rayleigh-Ritz assumed mode methods. In this study, resulting system of ordinary differential equations shows the effects of angular speed, and Young's modulus ratio. It is believed that the results will be a reference with which other researchers and commercial FE analysis program, ANSYS can compare their results.

• Steel and Composite Structures
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• v.33 no.3
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• pp.403-432
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• 2019

Axially functionally graded (AFG) beams are a new class of composite structures that have continuous variations in material and/or geometrical parameters along the axial direction. In this study, the exact analytical solutions for the free vibration of AFG and uniform beams with general elastic supports are obtained by using Euler-Bernoulli beam theory. The elastic supports are modeled with linear rotational and lateral translational springs. Moreover, the material and/or geometrical properties of the AFG beams are assumed to vary continuously and together along the length of the beam according to the power-law forms. Accordingly, the accuracy, efficiency and capability of the proposed formulations are demonstrated by comparing the responses of the numerical examples with the available solutions. In the following, the effects of the elastic end restraints and AFG parameters, namely, gradient index and gradient coefficient, on the values of the first three natural frequencies of the AFG and uniform beams are investigated comprehensively. The analytical solutions are presented in tabular and graphical forms and can be used as the benchmark solutions. Furthermore, the results presented herein can be utilized for design of inhomogeneous beams with various supporting conditions.

• Structural Engineering and Mechanics
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• v.35 no.6
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• pp.717-733
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• 2010

The dynamic stiffness matrix is formulated for an axially loaded slender double-beam element in which both beams are homogeneous, prismatic and of the same length by directly solving the governing differential equations of motion of the double-beam element. The Bernoulli-Euler beam theory is used to define the dynamic behaviors of the beams and the effects of the mass of springs and axial force are taken into account in the formulation. The dynamic stiffness method is used for calculation of the exact natural frequencies and mode shapes of the double-beam systems. Numerical results are given for a particular example of axially loaded double-beam system under a variety of boundary conditions, and the exact numerical solutions are shown for the natural frequencies and normal mode shapes. The effects of the axial force and boundary conditions are extensively discussed.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.161-169
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• 2017

In this paper, using consistent couple stress theory and Hamilton's principle, the free vibration analysis of Euler-Bernoulli nano-beams made of bi-directional functionally graded materials (BDFGMs) with small scale effects are investigated. To the best of the researchers' knowledge, in the literature, there is no study carried out into consistent couple-stress theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. In addition, in order to obtain small scale effects, the consistent couple-stress theory is also applied. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson's ratio are assumed to be graded in both axial and thickness directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of BDFG nano-beam. At the end, some numerical results are presented to study the effects of material length scale parameter, and inhomogeneity constant on natural frequency.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.137-146
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• 2017

The size-dependent behavior of single walled carbon nanotubes (SWCNT) embedded in the elastic medium and subjected to the initial axial force is investigated using the mixed finite element method. The SWCNT is assumed to be Euler-Bernoulli beam incorporating nonlocal theory developed by Eringen. The mixed finite element model shows its great advantage of dealing with nonlocal behavior of SWCNT subjected to a concentrated load owing to the existence of two coefficients ${\alpha}_1$ and ${\alpha}_2$. This is the first numerical approach to deal with a puzzling fact of nonlocal theory with concentrated load. Numerical examples are performed to show the accuracy and efficiency of the present method. In addition, parametric study is carefully carried out to point out the influences of nonlocal effect, the elastic medium, and the initial axial force on the behavior of the carbon nanotubes.

• Advances in aircraft and spacecraft science
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• v.6 no.1
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• pp.1-18
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• 2019

This present article represents the study of the forced vibration of nanobeam of a single-walled carbon nanotube (SWCNTs) surrounded by a polymer matrix. The modeling was done according to the Euler-Bernoulli beam model and with the application of the non-local continuum or elasticity theory. Particulars cases of the local elasticity theory have also been studied for comparison. This model takes into account the different effects of the interaction of the Winkler's type elastic medium with the nanobeam of carbon nanotubes. Then, a study of the influence of the amplitude distribution and the frequency was made by variation of some parameters such as (scale effect ($e_0{^a}$), the dimensional ratio or aspect ratio (L/d), also, bound to the mode number (N) and the effect of the stiffness of elastic medium ($K_w$). The results obtained indicate the dependence of the variation of the amplitude and the frequency with the different parameters of the model, besides they prove the local effect of the stresses.

• Structural Engineering and Mechanics
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• v.75 no.6
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• pp.737-746
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• 2020

This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

• International Journal of Precision Engineering and Manufacturing
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• v.3 no.4
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• pp.54-63
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• 2002

In recent years, mechanical systems such as high speed vehicles and railway trains moving on elastic beam structures have become a very important issue to consider. In this paper, a general approach, which can predict the dynamic behavior of a constrained mechanical system moving on a flexible beam structure, is proposed. Various supporting conditions for the foundation support are considered for the elastic beam structure. The elastic structure is assumed to be a non-uniform and linear Bernoulli-Euler beam with a proportional damping effect. Combined differential-algebraic equation of motion is derived using the multi-body dynamics theory and the finite element method. The proposed equations of motion can be solved numerically using the generalized coordinate partitioning method and predictor-corrector algorithm, which is an implicit multi-step integration method.

• Structural Engineering and Mechanics
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• v.51 no.1
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• pp.89-110
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• 2014

In this study, two beam-column elements based on the Elasto-Fiber element theory for reinforced concrete (RC) element have been developed and compared with each other. The first element is based on Elasto Fiber Approach (EFA) was initially developed for steel structures and this theory was applied for RC element in there and the second element is called as Fiber & Bernoulli-Euler element approach (FBEA). In this element, Cubic Hermitian polynomials are used for obtaining stiffness matrix. The beams or columns element in both approaches are divided into a sub-element called the segment for obtaining element stiffness matrix. The internal freedoms of this segment are dynamically condensed to the external freedoms at the ends of the element by using a dynamic substructure technique. Thus, nonlinear dynamic analysis of high RC building can be obtained within short times. In addition to, external loads of the segment are assumed to be distributed along to element. Therefore, damages can be taken account of along to element and redistributions of the loading for solutions. Bossak-${\alpha}$ integration with predicted-corrected method is used for the nonlinear seismic analysis of RC frames. For numerical application, seismic damage analyses for a 4-story frame and an 8-story RC frame with soft-story are obtained to comparisons of RC element according to both approaches. Damages evaluation and propagation in the frame elements are studied and response quantities from obtained both approaches are investigated in the detail.

• Steel and Composite Structures
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• v.26 no.6
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• pp.663-672
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• 2018

This paper contains a consistent couple-stress theory to capture size effects in Euler-Bernoulli nano-beams made of three-directional functionally graded materials (TDFGMs). These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-stress tensor is skew-symmetric and energy conjugate to the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson's ratio are assumed to be graded in all three axial, thickness and width directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of minimum potential energy. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of TDFG nano-beam. At the end, some numerical results are performed to investigate some effective parameter on buckling load. In this theory the couple-stress tensor is skew-symmetric and energy conjugate to the skew-symmetric part of the rotation gradients as the curvature tensor.