• The Journal of the Acoustical Society of Korea
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• v.18 no.7
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• pp.39-44
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• 1999

An analysis of the frequency parameters and vibrational modes is described for a rectangular plate. Double Fourier sine series is used as a modal displacement functions of a rectangular plate and applied to the free vibration analysis of a rectangular plate under various boundary conditions. The frequency parameters obtained by the double Fourier sine series method are compared with those obtained by the theory of finite element method and Ritz method. Frequency parameters are presented for the various aspect ratios for plate. The first four modal shapes for the rectangular plate under various boundary conditions are accurately described.

• Structural Engineering and Mechanics
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• v.8 no.4
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• pp.343-360
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• 1999

This work applies a structural analysis method based on an analytical solution from the Fourier series which transforms a half-range cosine expansion into a static solution involving plated structures. Two sub-matrices of in-plane and plate-bending problems are also formulated and coupled with the prescribed boundary conditions for these variables, thereby providing a convenient basis for a numerical solution. In addition, the plate connection are introduced by describing the connection between common boundary continuity and equilibrium. Moreover, a simple computation scheme is proposed. Numerical results are then compared with finite element results, demonstrating the numerical scheme's versatility and accuracy.

• Kyungpook Mathematical Journal
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• v.52 no.4
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• pp.383-395
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• 2012

In this paper, we use a computational algorithm for the inversion of the one and two-dimensional $\mathcal{L}_2$-transform based on the Bromwich's integral and the Fourier series. The new inversion formula can evaluate the inverse of the $\mathcal{L}_2$-transform with considerable accuracy over a wide range of values of the independent variable and can be devised for the functions which are not Laplace transformable and have damping motion in small interval near origin.

• Journal of the Korean Society for Precision Engineering
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• v.23 no.9 s.186
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• pp.76-83
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• 2006

Machining accuracy is closely related with tool deflection induced by cutting forces. In this research, cutting forces and tool deflection in end milling are expressed as a closed form of tool rotational angle and cutting conditions. The discrete cutting fores caused by periodic tool entry and exit are represented as a continuous function using the Fourier series expansion. Tool deflection is predicted by direct integration of the distributed loads on cutting edges. Cutting conditions, tool geometry, run-outs and the stiffness of tool clamping part are considered together far cutting forces and tool deflection estimation. Compared with numerical methods, the presented method has advantages in prediction time reduction and the effects of feeding and run-outs on cutting forces and tool deflection can be analyzed quantitatively. This research can be effectively used in real time machining error estimation and cutting condition selection for error minimization since the form accuracy is easily predicted from tool deflection curve.

• Journal of Mechanical Science and Technology
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• v.19 no.spc1
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• pp.452-460
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• 2005

Based on an introduced optimal trajectory planning method, this paper mainly deals with the accuracy analysis during the function approximation process of the optimal trajectory planning method. The basis functions are composed of Hermit polynomials and Fourier series to improve the approximation accuracy. Since the approximation accuracy is affected by the given orders of each basis function, the accuracy of the optimal solution is examined by changing the combinations of the orders of Hermit polynomials and Fourier series as the approximation basis functions. As a result, it is found that the proper approximation basis functions are the $5^{th}$ order Hermit polynomials and the $7^{th}-10^{th}$ order of Fourier series.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2005.10a
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• pp.781-785
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• 2005

Cutting forces and tool deflection in end milling are represented as the closed form of tool rotational angle and cutting conditions. The discrete cutting forces caused by tool entry and exit are continued using the Fourier series expansion. Tool deflection is predicted by direct integration of the distributed loads on cutting edges. Cutting conditions, tool geometry, run-outs and the stiffness of tool clamping pan are considered for cutting forces and tool deflection estimation. Compared to numerical methods, the presented method has advantages in short prediction time and the effects of feeding and run-outs on cutting forces and tool deflection can be analyzed quantitatively. This research can be effectively used in real time machining error estimation and cutting condition selection for error minimization since the ferm accuracy is easily predicted by tool deflect ion curve.

• Journal for History of Mathematics
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• v.19 no.2
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• pp.115-124
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• 2006

Functions of bounded variation were discovered by Jordan in 1881 while working out the proof of Dirichlet concerning the convergence of Fourier series. Here, we investigate Waterman's study with respect to bounded variation and its application on a closed bounded interval. The value of his study is whether Dirichlet-Jordan theorem holds in which function classes or not and summability method is what modifies its Fourier coefficients to make resulting series converge to the associated function. We have a view that the directions of future research with respect to bounded variation are two things; one is to find the function spaces which are larger than HBV and smaller than ${\phi}BV$, and the other is to find a fields of applications.

• Korean Journal of Mathematics
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• v.21 no.3
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• pp.285-292
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• 2013

B. C. Berndt computed the Fourier series of a class of generalized Eisenstein series, which gives an analytic continuation to the generalized Eisenstein series. In this paper, continuing his work, we consider generalized non-holomorphic Eisenstein series and give an analytic continuation to the $s$-plane.

• Journal of Advanced Marine Engineering and Technology
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• v.14 no.4
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• pp.46-56
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• 1990

Stress analysis of axi-symetric body with non-symetric loading and non-symetric given displacements is investigated in this paper using the finite element method. As the non-symetric load and non-symetric given displacements of axi-symetric body are generally periodic functions of angle .theta., the nodal forces and nodal displacements can be expanded in cosine and sine series, that is, Fourier series. Furthermore, using Euler's formula, the cosine and sine series can be converted into exponential series and it is prooved that the related calculus become more clear. Substituting the nodal displacements expanded in Fourier series into the strain components of cylindrical coordinates system, the element strains are expressed in series form and by the principal of virtual work, the element stiffness martix and element load vector are obtained for each order. It is also showed that if the non-symetric loads are even or odd functions of angle ${\theta}$ the stiffness matrix and load vector of the system are composed with only real numbers and relatively small capacity fo computer memory is enough for calculation.

• Bulletin of the Korean Mathematical Society
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• v.21 no.2
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• pp.127-135
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• 1984

In this paper, we find the Fourier series of X(t, .omega.).mem. $L^{2}$$_{s.a.p.}$ and the Parseval relation of X(t, .omega.).mem. $L^{2}$$_{s.a.p.}$. In section 2, we investigate some basic properties of X(t, .omega.).mem. $L^{2}$$_{s.a.p.}$ In section 3, we show that the mean of X(t, .omega.).mem. $L^{2}$$_{s.a.p.}$ exists and in section 4, after showing the existence of Fourier exponents and Fourier coefficients of X(t, .omega.).mem. $L^{2}$$_{s.a.p.}$. we give the Parseval relation of X(t, .omega.).mem. $L^{2}$$_{s.a.p.}$. For convenience we will denote X(t, .omega.) as X(t) in what follows.hat follows.