• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.26 no.12
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• pp.579-587
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• 2014

This paper deals with precise analytical state equation modeling of a variable speed refrigeration system (VSRS) for optimum control in state space. The VSRS is described as multi-input and multi-output (MIMO) system, which has two controlled variables and two control inputs. First, the Navier-Stokes equation and mass flow rate were applied to each component of the basic refrigeration cycle to build a dynamic model. The dynamic model, represented by a differential equation, was transformed into the state equation formula. Next, a full-order state observer was built to estimate all of the state variables to compose an optimum control system. Then, an optimum controller was designed to minimize an evaluation function that has input energy and control error. Finally, simulations and experiments were conducted to verify the validity of the proposed modeling and designed optimum controller to regulate target temperature and superheat in a 1RT oil cooler system. The results show that the proposed method, state equation modeling and optimum control, is efficient to ensure optimal control performance of the VSRS.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1085-1100
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• 2023

For any given function f, we focus on the so-called prescribed mean curvature problem for the measure e^-f(|x|2)dx provided thate^-f(|x|2) ∈ L¹(ℝⁿ⁺¹). More precisely, we prove that there exists a smooth hypersurface M whose metric is ds² = dρ² + ρ²d𝜉² and whose mean curvature function is ${\frac{1}{n}}(\frac{u^p}{{\rho}^{\beta}})e^{f({\rho}^2)}{\psi}(\xi)$ for any given real constants p, β and functions f and ψ where u and ρ are the support function and radial function of M, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere 𝕊ⁿ, $${\sum_{i,j}}({{\delta}_{ij}-{\frac{{\rho}_i{\rho}_j}{{\rho}^2+|{\nabla}{\rho}|^2}})(-{\rho}ji+{\frac{2}{{\rho}}}{\rho}j{\rho}i+{\rho}{\delta}_{ji})={\psi}{\frac{{\rho}^{2p+2-n-{\beta}}e^{f({\rho}^2)}}{({\rho}^2+|{\nabla}{\rho}|^2)^{\frac{p}{2}}}}$$ under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)={\frac{t}{2}}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

• Geophysics and Geophysical Exploration
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• v.26 no.3
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• pp.114-125
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• 2023

The physics-informed neural network (PINN) has been proposed to overcome the limitations of various numerical methods used to solve partial differential equations (PDEs) and the drawbacks of purely data-driven machine learning. The PINN directly applies PDEs to the construction of the loss function, introducing physical constraints to machine learning training. This technique can also be applied to wave equation modeling. However, to solve the wave equation using the PINN, second-order differentiations with respect to input data must be performed during neural network training, and the resulting wavefields contain complex dynamical phenomena, requiring careful strategies. This tutorial elucidates the fundamental concepts of the PINN and discusses considerations for wave equation modeling using the PINN approach. These considerations include spatial coordinate normalization, the selection of activation functions, and strategies for incorporating physics loss. Our experimental results demonstrated that normalizing the spatial coordinates of the training data leads to a more accurate reflection of initial conditions in neural network training for wave equation modeling. Furthermore, the characteristics of various functions were compared to select an appropriate activation function for wavefield prediction using neural networks. These comparisons focused on their differentiation with respect to input data and their convergence properties. Finally, the results of two scenarios for incorporating physics loss into the loss function during neural network training were compared. Through numerical experiments, a curriculum-based learning strategy, applying physics loss after the initial training steps, was more effective than utilizing physics loss from the early training steps. In addition, the effectiveness of the PINN technique was confirmed by comparing these results with those of training without any use of physics loss.

• Journal of the Korean Society for Precision Engineering
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• v.22 no.7 s.172
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• pp.112-121
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• 2005

This paper is concerned with smooth trajectory generation of biped robot which has inverted pendulum type balancing weight. Genetic algorithm is used to generate the trajectory of the leg and balancing weight. Balancing trajectory can be determined by solving the second order differential equation under the condition that the reference ZMP (Zero moment point) is settled. Reference ZMP effect on gait pattern absolutely but the problem is how to determine the reference ZMP. Genetic algorithm can find optimal solution under the high order nonlinear situation. Optimal trajectory is generated when use genetic algorithm which has some genes and a fitness function. In this paper, minimization of balancing joints motion is used for the fitness function and set the weight factor of the two balancing joints at the fitness function. Inverted pendulum type balancing weight is very similar with human and this model can be used fur humanoid robot. Simulation results show ZMP trajectory and the walking experiment made on the real biped robot IWR-IV.

• Steel and Composite Structures
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• v.20 no.3
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• pp.513-543
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• 2016

Third order shear deformation theory is used to evaluate electro-elastic solution of a sandwich plate with considering functionally graded (FG) core and composite face sheets made of piezoelectric layers. The plate is resting on the Pasternak foundation and subjected to normal pressure. Short circuited condition is applied on the top and bottom of piezoelectric layers. The governing differential equations of the system can be derived using Hamilton's principle and Maxwell's equation. The Navier's type solution for a sandwich rectangular thick plate with all edges simply supported is used. The numerical results are presented in terms of varying the parameters of the problem such as two elastic foundation parameters, thickness ratio ($h_p/2h$), and power law index on the dimensionless deflection, critical buckling load, electric potential function, and the natural frequency of sandwich rectangular thick plate. The results show that the dimensionless natural frequency and critical buckling load diminish with an increase in the power law index, and vice versa for dimensionless deflection and electrical potential function, because of the sandwich thick plate with considering FG core becomes more flexible; while these results are reverse for thickness ratio.

• Geomechanics and Engineering
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• v.19 no.5
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• pp.463-472
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• 2019

The mode perturbation method (MPM) is suitable and efficient for solving the eigenvalue problem of a nonuniform soil deposit whose property varies with depth. However, results of the MPM do not always converge to the exact solution, when the variation of soil deposit property is discontinuous. This discontinuity is typical because soil is usually made up of sedimentary layers of different geologic materials. Based on the energy integral of the variational principle, a new mode perturbation method, the energy-based mode perturbation method (EMPM), is proposed to address the convergence of the perturbation solution on the natural frequencies and the corresponding mode shapes and is able to find solution whether the soil properties are continuous or not. First, the variational principle is used to transform the variable coefficient differential equation into an equivalent energy integral equation. Then, the natural mode shapes of the uniform shear beam with same height and boundary conditions are used as Ritz function. The EMPM transforms the energy integral equation into a set of nonlinear algebraic equations which significantly simplifies the eigenvalue solution of the soil layer with variable properties. Finally, the accuracy and convergence of this new method are illustrated with two case study examples. Numerical results show that the EMPM is more accurate and convergent than the MPM. As for the mode shapes of the uniform shear beam included in the EMPM, the additional 8 modes of vibration are sufficient in engineering applications.

• Journal of Energy Engineering
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• v.8 no.2
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• pp.333-340
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• 1999

A numerical study of turbulent condensing vapor jet submerged in subcooled liquids has been conducted. A physical model of the process is presented employing the locally homogeneous flow approximation of two phase flow in conjunction with a $\kappa$-$\varepsilon$-g model of turbulence properties. In this model the turbulence is represented by differential equations for its kinetic energy and dissipation. A differential equation for the concentration fluctuations is solved and a clipped normal probability distribution function is proposed for the mixture fraction. Effects of steam mass flux, pool temperature and nozzle internal diameter on the condensing vapor jet are also analyzed. The model is evaluated using existing data for turbulent condensing vapor jets. The agreement between the predictions and the available experimental data is good.

• Journal of the Korean Institute of Navigation
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• v.4 no.1
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• pp.31-50
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• 1980

It does not seem necessarily practicable to keep the system always in optimal condition, athough the control system of the follow-up mechanism on the most marine gyro compasses is to be adjusted by the operator through the gain adjustment. Sometimes a sustained oscillation or an incorrect gyro reading occurs to the system. For such a system any systematical research or theoretical basis of the guide for the optimal gain adjustment has not been reported yet. As a basic investigation of the theoretical system analysis to solve the problems concerned, the author attempts in this paper to express the system in a mathematical model deduced from the results of the theoretical approach and the experimental observation of each element contained in the follow-up mechanism of Hokshin D-1 gyro compass, and to constitute an over-all closed loop transfer function. This funciton being reverted to a fourth orderlinear differential equation, the first order simultaneous differential equations are obtained by means of the state-variables. The latter equations are solved by the Runge-Kutta method with digital computer. By comparing the characteristic of the simulated over-all output with that of the experimental result, it is shown that both outputs are nearly consistent with each other. It is also expected that the system representation proposed by this paper is valid and will be a prospective means in a further study on the design and optimal adjustment of the system.

• Journal of the Korean Society for Marine Environment & Energy
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• v.12 no.3
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• pp.165-172
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• 2009

A set of equations for description of transformation of harmonic waves is proposed here. Velocity potential function and separation of variables are introduced for the derivation. The continuity equation is in a vertical plane is integrated through the water so that a horizontal one-dimensional wave equation is produced. The new equation composed of the complex velocity potential function, further be modified into. A set up of equations composed of the wave amplitude and wave phase gradient. The horizontally one-dimensional equations on the wave amplitude and wave phase gradient are the first and second-order ordinary differential equations. They are solved in a one-way marching manner starting from a side where boundary values are supplied, i.e. the wave amplitude, the wave amplitude gradient, and the wave phase gradient. Simple spatially-centered finite difference schemes are adopted for the present set of equations. The equations set is applied to three test cases, Booij's inclined plane slope profile, Massel's smooth bed profile, and Bragg's wavy bed profile. The present equations set is satisfactorily verified against existing theories including Massel's modified mild-slope equation, Berkhoff's mild-slope equation, and the full linear equation.

• Journal of Advanced Research in Ocean Engineering
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• v.1 no.3
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• pp.157-167
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• 2015

In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean's stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.