• 대한수학회논문집
• /
• 제28권2호
• /
• pp.319-333
• /
• 2013

Using the fixed point method, we prove the generalized Hyers-Ulam-Rassias stability of the following functional equation in multi-Banach spaces: $${\sum_{1{\leq}i_<j{\leq}n}}\;f(\frac{r_ix_i+r_jx_j}{k})=\frac{n-1}{k}{\sum_{i=1}^n}r_if(x_i)$$.

• Structural Engineering and Mechanics
• /
• 제47권3호
• /
• pp.307-329
• /
• 2013

The bending problem of Euler-Bernoulli discontinuous beams is dealt with, in which the discontinuities are due to the loads and eventually to essential constrains applied along the beam axis. In particular, the loads are modelled as random delta-correlated processes acting along the beam axis, while the ulterior eventual discontinuities are produced by the presence of external rollers applied along the beam axis. This kind of structural model can be considered in the static study of bridge beams. In the present work the exact expression of the response quantities are given in terms of means and variances, thanks to the use of the stochastic analysis rules and to the use of the generalized functions. The knowledge of the means and the variances of the internal forces implies the possibility of applying the reliability ${\beta}$-method for verifying the beam.

• 한국전산유체공학회:학술대회논문집
• /
• 한국전산유체공학회 1999년도 춘계 학술대회논문집
• /
• pp.186-191
• /
• 1999

An Euler solver is developed to predict accurate aerodynamic data such as lift coefficient, drag coefficient, and moment coefficient. The conservation law form of the compressible Euler equations are used in the generalized curvilinear coordinates system. The Euler solver uses a finite volume method and the second order Roe's flux difference splitting scheme with min-mod flux limiter to calculate the fluxes accurately. An implicit scheme which includes the boundary conditions is implemented to accelerate the convergence rate. The multi-block grid is integrated into the flow solver for complex geometry. The flowfields are analyzed around NACA 0012 airfoil in the cases of $M_{\infty}=0.75,\;\alpha=2.0\;and\;M_{\infty}=0.80,\;\alpha=1.25$. The numerical results are compared with other numerical results from the literature. The final goal of this research is to prepare a robust and an efficient Navier-Stokes solver eventually.

• Structural Engineering and Mechanics
• /
• 제63권2호
• /
• pp.161-169
• /
• 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.

• Steel and Composite Structures
• /
• 제26권6호
• /
• pp.663-672
• /
• 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.

• Structural Engineering and Mechanics
• /
• 제53권6호
• /
• pp.1105-1126
• /
• 2015

Many vibrating mechanical systems from the real life are modeled as combined dynamical systems consisting of beams to which spring-mass secondary systems are attached. In most of the publications on this topic, masses of the helical springs are neglected. In a paper (Cha et al. 2008) published recently, the eigencharacteristics of an arbitrary supported Bernoulli-Euler beam with multiple in-span helical spring-mass systems were determined via the solution of the established eigenvalue problem, where the springs were modeled as axially vibrating rods. In the present article, the authors used the assumed modes method in the usual sense and obtained the equations of motion from Lagrange Equations and arrived at a generalized eigenvalue problem after applying a Galerkin procedure. The aim of the present paper is simply to show that one can arrive at the corresponding generalized eigenvalue problem by following a quite different way, namely, by using the so-called "characteristic force" method. Further, parametric investigations are carried out for two representative types of supporting conditions of the bending beam.

• Journal of applied mathematics & informatics
• /
• 제26권1_2호
• /
• pp.15-27
• /
• 2008

We present and discuss an algorithm for the numerical solution of initial value problems of the form $D_*^\alpha$y(t) = f(t, y(t)), y(0) = y0, where $D_*^\alpha$y is the derivative of y of order $\alpha$ in the sense of Caputo and 0<${\alpha}{\leq}1$. The algorithm is based on the fractional Euler's method which can be seen as a generalization of the classical Euler's method. Numerical examples are given and the results show that the present algorithm is very effective and convenient.

• 대한수학회지
• /
• 제34권3호
• /
• pp.515-531
• /
• 1997

In this paper, finite difference method is applied to approximate the generalized solutions of Sobolev equations. Using the Steklov mollifier and Bramble-Hilbert Lemma, a priori error estimates in discrete $L^2$ as well as in discrete $H^1$ norms are derived frist for the semidiscrete methods. For the fully discrete schemes, both backward Euler and Crank-Nicolson methods are discussed and related error analyses are also presented.

• Coupled systems mechanics
• /
• 제3권3호
• /
• pp.247-265
• /
• 2014

This article discusses the dynamic response of Bernoulli-Euler straight beam with angular elastic supports subjected to moving load with variable velocity. A new engineering approach for determination of the dynamic effect from the moving load on the stressed and strained state of the beam has been developed. A dynamic coefficient, a ratio of the dynamic to the static deflection of the beam, has been defined on the base of an infinite geometrical absolutely summable series. Generalization of the R. Willis' equation has been carried out: generalized boundary conditions have been introduced; the generalized elastic curve's equation on the base of infinite trigonometric series method has been obtained; the forces of inertia from normal and Coriolis accelerations and reduced beam mass have been taken into account. The influence of the boundary conditions and kinematic characteristics of the moving load on the dynamic coefficient has been investigated. As a result, the dynamic stressed and strained state has been obtained as a multiplication of the static one with the dynamic coefficient. The developed approach has been compared with a finite element one for a concrete engineering case and thus its authenticity has been proved.

• International Journal of Precision Engineering and Manufacturing
• /
• 제3권4호
• /
• pp.54-63
• /
• 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.