• Structural Engineering and Mechanics
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• 제3권3호
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• pp.273-287
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• 1995

Stochastic response of systems to random excitation can be estimated by direct integration methods in the time domain such as the stochastic central difference method (SCDM). In this paper, the SCDM is applied to compute the variance and covariance in response of linear and nonlinear structures subjected to random excitation. The accuracy of the SCDM is assessed using two-DOF systems with both deterministic and random material properties excited by white noise. For the former case, closed-form solutions can be obtained. Numerical results also are presented for a simply supported geometrically nonlinear beam. The stiffness of this beam is modeled as a random field, and the beam is idealized by the stochastic finite element method. A perturbation technique is applied to formulate the equations of motion of the system, and the dynamic structural response statistics are obtained in a time domain analysis. The effect of variations in structural parameters and the numerical stability of the SCDM also are examined.

• Journal of applied mathematics & informatics
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• 제37권5_6호
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• pp.357-382
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• 2019

In this paper, we consider boundary value problem for a weakly coupled system of two singularly perturbed differential equations of convection diffusion type with discontinuous source term. In general, solution of this type of problems exhibits interior and boundary layers. A numerical method based on streamline diffusiom finite element and Shishkin meshes is presented. We derive an error estimate of order $O(N^{-2}\;{\ln}^2\;N$) in the maximum norm with respect to the perturbation parameters. Numerical experiments are also presented to support our theoritical results.

• Kyungpook Mathematical Journal
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• 제61권3호
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• pp.591-612
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• 2021

In this paper, a uniformly convergent numerical scheme is designed for solving singularly perturbed reaction-diffusion problems. The problem is converted to an equivalent weak form and then a Galerkin finite element method is used on a piecewise uniform Shishkin mesh with linear basis functions. The convergence of the developed scheme is proved and it is shown to be almost fourth order uniformly convergent in the maximum norm. To exhibit the applicability of the scheme, model examples are considered and solved for different values of a singular perturbation parameter ε and mesh elements. The proposed scheme approximates the exact solution very well.

• Journal of applied mathematics & informatics
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• 제40권1_2호
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• pp.25-45
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• 2022

In this study, the non-standard finite difference method for the numerical solution of singularly perturbed parabolic reaction-diffusion subject to Robin boundary conditions has presented. To discretize temporal and spatial variables, we use the implicit Euler and non-standard finite difference method on a uniform mesh, respectively. We proved that the proposed scheme shows uniform convergence in time with first-order and in space with second-order irrespective of the perturbation parameter. We compute three numerical examples to confirm the theoretical findings.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.141-154
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• 2023

We study the stability of traveling waves of a certain chemotaxis model. The traveling wave solution is a central object of study in a chemotaxis model. Kim et al. [8] introduced a model on a population and nutrient densities based on a nonlinear diffusion law. They proved the existence of traveling waves for the one dimensional Cauchy problem. Existence theory for traveling waves is typically followed by stability analysis because any traveling waves that are not robust against a small perturbation would have little physical significance. We conduct a numerical nonlinear stability for a few relevant instances of traveling waves shown to exist in [8]. Results against absolute additive noises and relative additive noises are presented.

• 대한기계학회논문집B
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• 제31권9호
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• pp.780-789
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• 2007

Lock-on phenomenon in the wake of a circular cylinder is investigated at the Reynolds number of 360 using direct numerical simulation (DNS). To induce lock-on, a streamwise velocity perturbation with a frequency of twice the natural shedding frequency is superimposed on the free stream velocity. The Reynolds stress distributions are investigated to analyze the streamwise force balance acting on the recirculation region and the results are compared with the previous experimental result. When the lock-on occurs, the pressure force on the recirculation region is shown to increase mainly due to the reversal of the Reynolds shear stress distribution, which is consistent with our previous results using PIV measurement. It is also shown that, with the lock-on, the strength of the primary vortices increases whereas that of the secondary vortices decreases significantly. Further, under the lock-on condition the wavelength of the secondary vortices increases by as much as 2.5 times.

• International Journal of Aeronautical and Space Sciences
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• 제7권2호
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• pp.26-32
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• 2006

This paper targets a direct and quantitative prediction of characteristics of unstable waves in a combustion chamber, which employs the governing equations derived in terms of amplification factors of flow variables. A freshly formulated nonlinear acoustic equation is obtained and the analysis of unsteady waves in a rocket engine is attempted. In the present formalism, perturbation method decomposes the variables into time-averaged part that can be obtained easily and accurately and time-varying part which is assumed to be harmonic. Excluding the use of conventional spatially sinusoidal eigenfunctions, a direct numerical solution of wave equation replaces the initial spatial distribution of standing waves and forms the nonlinear space-averaged terms. Amplification factor is also calculated independently by the time rate of changes of fluctuating variables, and is no longer an explicit function for compulsory representation. Employing only the numerical computation, major assumptions inevitably inherent, and in erroneous manner, in up to date analytical methods could be avoided. With two definitions of amplification factor, 1-D stable wave and 3-D unstable wave are examined, and clearly demonstrated the potentiality of a suggested theoretical-numerical method of combustion instability.

• 수산해양기술연구
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• 제34권3호
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• pp.328-338
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• 1998

It is main objective of this approach to present a method to analyse stochastic design sensitivity for problems of structural dynamics with randomness in design parameters. A combination of the adjoint variable approach and the second oder perturbation method is used in the finite element approach. An alternative form of the constant functional that holds for all times is introduced to consider the time response of dynamic sensitivity. The terminal problem of the adjoint system is solved using equivalent homogeneous equations excited by initial velocities. The numerical procedures are shown to be much more efficient when based on the fold superposition method : the generalized co-ordinates are normalized and the correlated random variables are transformed to uncorrelated variables, where as the secularities are eliminated by the fast Fourier transform of complex valued sequences. Numerical algorithms have been worked out and proved to be accurate and efficient : they codes whose element derivative matrices can be explicitly generated. The numerical results of two cases - 2-dimensional portal frame and 3/4-cylindrical shell structure - for the deterministic and stochastic sensitivity analysis illustrates in this paper.

• KSTLE International Journal
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• 제2권2호
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• pp.93-102
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• 2001

Ball bearings have been widely used for the spindle motor bearing in various kinds of information storage devices. Recently many researchers have been trying to replace ball bearings with fluid film bearings because of their superior NRRO(non-repeatable runout) characteristics. In this study, a numerical analysis has been conducted for the complicate bearing system composed of herringbone groove journal bearing and spiral groove thrust bearing for the spindle motor of the information storage device. At first, spindle motor bearing is modeled as journal bearing part and thrust bearing part separately, and then observed various influences of geometric parameters. Previous studies had considered only the translational motion of the journal bearing. However, this study takes the additional 2-degree of freedom rotation into consideration to attempt to describe the real motion of the spindle bearing. As a result, rotational stiffness coefficients and rotational damping coefficients are obtained. In addition, a spindle bearing system made up of four bearings is modeled and interpreted at once and coefficients of dynamic characteristics of each bearing are obtained. Finally, an eigen analysis of bearing system is made with these results. Through this analysis, it is possible to estimate an unstable condition of the system for given geometric parameters and to propose a method which is able to avoid the unstable condition by a proper adjustment of geometric parameters.

• 대한기계학회논문집
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• 제11권1호
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• pp.132-144
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• 1987

본 논문에서는 Isida의 이론해석 방법을 참고로 하여 유한한한 폭을 갖는 두 께가 변화하는 판재에서의 응력확대계수를 2차원 이론해석한 것이다.