• ETRI Journal
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• v.29 no.1
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• pp.50-58
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• 2007

Robust adaptive beamforming based on worst-case performance optimization is investigated in this paper. It improves robustness against steering vector mismatches by the approach of diagonal loading. A closed-form solution to optimal loading is derived after some approximations. Besides reducing the computational complexity, it shows how different factors affect the optimal loading. Based on this solution, a performance analysis of the beamformer is carried out. As a consequence, approximated closed-form expressions of the source-of-interest power estimation and the output signalto-interference-plus-noise ratio are presented in order to predict its performance. Numerical examples show that the proposed closed-form expressions are very close to their actual values.

• Structural Engineering and Mechanics
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• v.51 no.5
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• pp.773-792
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• 2014

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

• International Journal of CAD/CAM
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• v.6 no.1
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• pp.157-166
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• 2006

In order to model blending surfaces with curvature continuity, in this paper we apply sixth order partial differential equations (PDEs), which are solved with a composite power series based method. The proposed composite power series based approach meets boundary conditions exactly, minimises the errors of the PDEs, and creates almost as accurate blending surfaces as those from the closed form solution that is the most accurate but achievable only for some simple blending problems. Since only a few unknown constants are involved, the proposed method is comparable with the closed form solution in terms of computational efficiency. Moreover, it can be used to construct 3- or 4-sided patches through the satisfaction of continuities along all edges of the patches. Therefore, the developed method is simpler and more efficient than numerical methods, more powerful than the analytical methods, and can be implemented into an effective tool for the generation and manipulation of complex free-form surfaces.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.33 no.1
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• pp.9-16
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• 2020

The objective of this study is to propose closed-form solutions to the free vibration response of single-degree-of-freedom (SDOF) systems, as part of fundamental research on dynamic systems with Coulomb friction. The motion of a dynamic system with Coulomb friction is described by a nonlinear differential equation, and, due to the variation in the sign of friction force term with the direction of motion, it is difficult to obtain the closed-form solution. To solve this problem, the nonlinear differential equation is directly computed by numerical integration, or an approximated solution is indirectly obtained using a linear differential equation wherein the damping effect due to Coulomb friction is replaced by an equivalent viscous damping term. However, these conventional methods do not provide a closed-form solution from a mathematical point of view. In this regard, closed-form solutions to the free vibration response of SDOF systems with Coulomb friction are derived herein by considering that the sign of the friction force term is reversed in each half-cycle of motion and by expanding it to the entire time history using the power series function. In addition, for a given initial condition, both the number of free vibration half-cycles and the response at the instant when free vibration motion stops are predicted under the condition that the motion of free vibration is stopped when the amplitude of the friction force is higher than that of the restoring force due to stiffness.

• Journal of the Institute of Electronics and Information Engineers
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• v.52 no.3
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• pp.19-24
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• 2015

We introduce an optimization of the number of base station antennas in massive multiple-input multiple-output (MIMO) wireless powered communication network (WPCN). We use channel hardening property of massive MIMO system to approximate channel gain in terms of the number of base station antennas. Then, we find an optimal solution by partial differential and obtain a closed form solution by using Lambert-W function. The simulation results show that the approximation and the method of solving the optimization problem are reasonable, and the optimal solution of proposed scheme is almost identical to the optimal number of base station antennas by the exhaustive search method.

• Structural Engineering and Mechanics
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• v.43 no.2
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• pp.239-251
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• 2012

The analysis of laterally loaded piles is generally carried out by idealizing the soil mass as Winkler springs, which is a crude approximation; however this approach gives reasonable results for many practical applications. For more precise analysis, the three- dimensional finite element analysis (FEA) is one of the best alternatives. The FEA uses the modulus of elasticity $E_s$ of soil, which can be determined in the laboratory by conducting suitable laboratory tests on undisturbed soil samples. Because of the different concepts and idealizations in these two approaches, the results are expected to vary significantly. In order to investigate this fact in detail, three-dimensional finite element analyses were carried out using different combinations of soil and pile characteristics. The FE results related to the pile deflections are compared with the closed-form solutions in which the modulus of subgrade reaction $k_s$ is evaluated using the well-known $k_s-E_s$ relationship. In view of the observed discrepancy between the FE results and the closed-form solutions, an improved relationship between the modulus of subgrade reaction and the elastic constants is proposed, so that the solutions from the closed-form equations and the FEA can be closer to each other.

• Journal of the Korean Geotechnical Society
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• v.18 no.6
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• pp.33-42
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• 2002

Ground reaction curve is a very important information for evaluating the side wall displacements and installation time of the tunnle support. The ground reaction curve can be estimated by analytical closed form solutions derived on the supposition of circular section and isotropic stress condition. The conditions of stress field and tunnel configurations, however, are quite different in practice. Therefore, it is necessary to investigate the effects of stress anisotropy and tunnel configurations in order to use simply in practical design. This paper describes a study of influence factors in the ground reaction curve. In order to evaluate the applicability of analytical closed form solution in practical design, two sets of parametric studies were carried out by numerical analysis in elastic tunnel behaviour: one set of studies investigated the influence of the K and the other set investigated the influence of the tunnel configurations such as circular and horse-shoe shape. In the studies, K value varies between 0.5 and 3.0, initial ground vertical stress varies between 5～30MPa far each K values. The results indicated that the self-supportability of ground is larger in the ground having lower K value. However, it is suggested that the applicability of closed form solution may not be adequate to determine directly the installation time of the support and self-supportability of ground. It is necessary to consider stress anisotropy and tunnel configurations.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.40 no.7
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• pp.647-652
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• 2016

Guided waves can be used for the inspection of long range pipelines. Surface corrosion is often found as a major defect type in pipelines. The reciprocity relation is a well-established theorem by which one can simplify complicated mathematical expressions. The approach has been already applied to plate and half-space structures to obtain the closed-form solutions of scattered amplitude. However, results for the case of cylindrical structures have not been reported yet. In this paper, the scattering of torsional waves, which is widely used in commercial applications, is explored by the reciprocity theorem approach. Obtaining closed-form solutions of the amplitudes of propagating waves is much simplified by using the reciprocal relation. The scattered amplitudes for elliptical and rectangular defect shapes are calculated with respect to defect depth and width, at frequencies between 0 and 500 kHz. The amplitude shows the periodic result as a function of frequency. The derived closed-form solutions can play a significant role in quantitative signal interpretation.

• Structural Engineering and Mechanics
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• v.16 no.6
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• pp.731-748
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• 2003

In this paper a thick cylindrical shell with varying thickness which is subjected to static non-uniform internal pressure is analyzed. At first, equilibrium equations of the shell have been derived by the energy principle and by considering the first order theory of Mirsky-Herrmann which includes transverse shear deformation. Then the governing equations which are, a system of differential equations with varying coefficients have been solved analytically with the boundary layer technique of the perturbation theory. In spite of complexity of modeling the conditions near the boundaries, the method of this paper is very capable of providing a closed form solution even near the boundaries. Displacement predictions are in a good agreement with the calculated finite elements and other analytical results. The convergence of solution is very fast and the amount of calculations is less than the Frobenius method.

• Journal of Korean Tunnelling and Underground Space Association
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• v.4 no.3
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• pp.175-184
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• 2002

Ground reaction curve is very useful information for estimating the installation time of the tunnel support. The ground reaction curve can be estimated by analytical closed form solutions derived in case of circular section and isotropic stress condition. The nature of the ground reaction, however, depends significantly on tunnel configurations. Nevertheless, few purely analytical and experimental studies of this problem due to tunnel configurations appear to have been carried out. Therefore, it is necessary to investigate the influence of tunnel configurations in order to use simply in practical design. This paper describes a numerical study for the intial elastic displacement in the ground reaction curve due to configuration of tunnel excavation. In order to evaluate the applicability of analytical closed form solution in practical design, the parametric studies were carried out by numerical analysis in elastic tunnel behaviour. In the studies, S value, namely configuration factor, defined as the ratio between tunnel height (b) and width (a), varies between 0.5 and 3.0, initial ground vertical stress varies between 5~30 MPa for each S values. The results indicated that the self-supportability of ground is larger in the ground having low S value. It, however, is suggested that the applicability of closed form solution may not be adequate to determine directly the installation time of the support and self-supportability of ground. It should be necessary to perform the additional numerical analysis.