• Journal of applied mathematics & informatics
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• 제33권5_6호
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• pp.657-670
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• 2015

In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2007년도 추계학술대회논문집
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• pp.1212-1217
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• 2007

ATM(automated teller machine) is a machine which can deposit and withdraw money directly. For effective transfer of bills in the machine, crown belts are used. In this paper, the transverse vibration of crown belt is investigated. The equation of motion of the belt is derived using Lagrange's equation. Galerkin's method is applied to convert the partial differential equation to the ordinary differential equations. Experimental investigations are performed on the belt system with the variation of pulley type, eccentricity, and tension. The results of numerical analysis show in good agreement with the experimental results.

• 한국농공학회지
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• 제34권4호
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• pp.97-105
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• 1992

The main purpose of this paper is to present the buckling loads of tapered columns due to dynamic concept. The ordinary differential equation governing the bucking loads for tapered columns is derived on the basis of dynamic concept. Three kinds of cross sectional shape are considered in the governing equation. The Improved Euler method and Determinant Search method are used to perform the integration of the differential equation and to determine the buckling loads, respectively. The hinged-hinged, hinged-clamped, clamped-clamped and free-clamped end constraints are applied in numerical examples. The buckling loads are reported as the function of section ratio, and the effects of cross-sectional shapes are investigated. The buckling load equation, which are fitted by numerical data, are proposed as a function of section ratio. It is expected that these equations can be utilized in structural engineering field.

• 한국기계가공학회지
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• 제10권4호
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• pp.8-15
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• 2011

In this paper, the influence of a crack on the natural frequency of cracked cantilever L-shaped beam with coupled bending and torsional vibrations by analytically and experimentally is analyzed. The L-shaped beam with a crack is modeled by Hamilton's principle with consideration of bending and torsional energy. The two coupled governing differential equations are reduced to one sixth-order ordinary differential equation in terms of the flexural displacement. The crack is assumed to be in the first, second and third mode of fracture and to be always opened during the vibrations. The theoretical results are validated by a comparison with experimental measurements. The maximal difference between the theoretical results and experimental measurements of the natural frequency is less than 7.5% in the second vibration mode.

• 한국대기환경학회지
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• 제14권4호
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• pp.303-312
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• 1998

To obtain the solution to the time-dependent particle size distribution of an aerosol undergoing gravitational coagulation, the moment method was used which converts the non linear integro-differential equation to a set of ordinary differential equations. A semi-numerical solution was obtained using this method. Subsequently, an analytic solution was given by approximating the collision kernel into a form suitable for the analysis. The results show that during gravitational coagulation, the geometric standard deviation increases and the geometric mean radius decreases as time increases.

• 대한기계학회논문집A
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• 제21권2호
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• pp.337-345
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• 1997

An analysis is presented for the response of a beam constrained by a nonlinear spring to a harmonic excitation. The system is governed by a linear partial differential equation with a nonlinear boundary condition. The method of multiple scales is used to reduce the nonlinear boundary value problem to a system of autonomous ordinary differential equations of the amplitudes and phases. The case of the third-order subharmonic resonance is considered in this study. The autonomous system is used to determine the steady-state responses and their stability.

• 동력기계공학회지
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• 제9권1호
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• pp.82-88
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• 2005

This paper presents numerical study on dynamic characteristics of evaporator to control evaporator superheat and compressor capacity with optimum condition in refrigeration system. It is very important to reduce energy consumption and to keep room temperature within a very restricted range with minimum oscillation in some special applications of the refrigeration system. Heat exchange is mainly happened in the evaporator. So, making mathematical model of evaporator and analyzing evaporator characteristics are necessary in order to control the superheat and the capacity of the system. A mathematical model based on the one dimensional partial differential equations representing mass and energy conservation and a tube-wall energy is described. A set of ordinary differential equation is formulated by integrating separately over the two regions(two-phase and vapor) generally presented in a heat exchanger.

• Nuclear Engineering and Technology
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• 제49권8호
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• pp.1636-1644
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• 2017

The main purpose of this article is to describe the magnetohydrodynamic stagnation point flow of Walter-B nanofluid over a stretching sheet. The phenomena of heat and mass transfer are based on the involvement of thermal radiation and chemical reaction. Characteristics of Newtonian heating are given special attention. The Brownian motion and thermophoresis models are introduced in the temperature and concentration expressions. Appropriate variables are implemented for the transformation of partial differential frameworks into sets of ordinary differential equations. Plots for velocity, temperature, and nanoparticle concentration are displayed and analyzed for governing parameters. The skin friction coefficient and local Nusselt and Sherwood numbers are studied using numerical values. The temperature and heat transfer rate are enhanced within the frame of the thermal conjugate parameter.

• Biotechnology and Bioprocess Engineering:BBE
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• 제7권6호
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• pp.362-366
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• 2002

A mathematical model has been developed for describing the oxygen concentration during the exponential growth of microorganisms, in a static solid substrate bed supported on a tray fermentor. The model equations comprise of one partial differential equation for mass transfer and an ordinary differential equation of growth. After nondimensionlisation, analytical solution tn the model has been obtained by the method of Laplace transforms. An expression for critical thickness of bed is deduced from the model equation. The significance of the model in the design of tray fermentors is discussed. The validity of the discussion is verified by taking an illustration from the literature.

• Steel and Composite Structures
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• 제31권3호
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• pp.277-290
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• 2019

This paper investigates the parametric instability (PI) of multilayered composite conical shells (MLCCSs) under axial load periodically varying the time, using the first order shear deformation theory (FOSDT). The basic equations for the MLCCSs are derived and then the Galerkin method is used to obtain the ordinary differential equation of the motion. The equation of motion converted to the Mathieu-Hill type differential equation, in which the DI is examined employing the Bolotin's method. The expressions for left and right limits of dimensionless parametric instability regions (PIRs) of MLCCSs based on the FOSDT are obtained. Finally, the influence of various parameters; lay-up, shear deformations (SDs), aspect ratio, as well as loading factors on the borders of the PIRs are examined.