• International Journal of Aeronautical and Space Sciences
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• v.4 no.2
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• pp.1-8
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• 2003

Satellite formation flying is the placing micro-satellites with the same mission into nearby orbits to form a cluster. Clohessy-Wiltshire equations are used to describe the relative motion and control strategies between satellites within a cluster, which are known as Hill's equations. Even though Hill's equations are powerful in determining initial conditions for the satellite formation flying, they can not accurately express the relative motion under J2 perturbation. Some methods have been developed for the determination of initial conditions to avoid limits of Hill's equation. This paper gives a new method of determining initial conditions using mean elements. For this research mean elements were transformed to osculating elements using Brouwer's theory and the orbit was propaeated with the consideration of J2-J8 to get a relative position. The results show that satellites within a cluster are maintained in the desired boundary for long period and the method is effective on a fuel saving for satellite formation flying.

• Structural Engineering and Mechanics
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• v.52 no.3
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• pp.525-540
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• 2014

The Lyapunov exponent and moment Lyapunov exponents of Hill's equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged $It\hat{o}$ differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.

• Journal of KSNVE
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• v.7 no.3
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• pp.489-498
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• 1997

In this paper, chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonliear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which has the external and parametric excitation with a same frequency. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Numerical simulations are performed to demonstrate theoretical results and show the strange attractor of the chaotic motion.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1996.10a
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• pp.288-294
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• 1996

In this paper, Chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonlinear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which have the parametric and external excitation. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Poincare maps numerically demonstrate theoretical results and show transverse homoclinic orbit of the chaotic motion.

• International Journal of Fluid Machinery and Systems
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• v.8 no.3
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• pp.169-182
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• 2015

When simulating the dynamic behaviour of a hydro power plant, it is essential to have a good representation of the turbine behaviour. The pressure transients in the system occurs because the flow changes, which the turbine defines. The flow through the turbine is a function of the pressure, the speed of rotation and the wicket gate opening and is, most often described in a performance diagram or Hill diagram. In the Hill diagram, the efficiency is drawn like contour lines, hence the name. A turbines Hill diagram is obtained by performance tests on scaled model in a laboratory. However, system dynamic simulations have to be performed in the early stage of a project, before the turbine manufacturer has been chosen and the Hill diagram is known. Therefore one have to rely on diagrams for a turbine with similar speed number. The Hill diagram is drawn through measured points, so for using the diagram in a simulation program, one have to iterate in the diagram based on curve fitting of the measured points. This paper describes an alternative method. By means of the Euler turbine equation, it is possible to set up two differential equations which represents the turbine performance with good enough accuracy for the dynamic simulations. The only input is the turbine's main geometry, the runner blade in- and outlet angle and the guide vane angle at best efficiency point of operation (BEP). In the paper, simulated turbine characteristics for a high head Francis turbine, and for a reversible pump turbine are compared with laboratory measured characteristics.

• Transactions of Materials Processing
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• v.6 no.4
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• pp.279-290
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• 1997

The sectional forming analysis of stamping pocesses for aluminum alloy sheet metals was investigated. For the modeling of the anomalous behavior of aluminum alloy sheet. the Barlat's strain rate potential and Hill's 1990 non-quadratic yield theory with an isotropic hardening rule were employed. The rigid-viscoplastic FEM formulation which solves equilibrium equation for plane-strain stage with mesh-normal geometric constraints was derived. A new method to determine the Barlat's anisotropic coefficients was also suggested. To verify the validity of the formulation, the stretch and draw forming processes of a square cup were simulated.

• Journal of KSNVE
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• v.6 no.2
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• pp.233-244
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• 1996

In this paper chaotic mothions of a straight pipe conveying oscillatory flow and being subjected to external forces such as earthquake are theoretically investigated. The nonlinear partial differential equation of motion is derived by Newton's method. In this equation, the nonlinear curvature of the pipe and the thermal expansion effects are contained. The nonlinear ordinary differential equation transformed from that partial differential equation is a type of Hill's equations, which have the parametric and external exciation term. This original system is transfered to the averaged system by the averaging theory. Bifurcation curves of chaotic motion of the piping system are obtained in the general case of the frequency ratio, n by applying Melnikov's method. Numerical simulations are performed to demonstrate theorectical results and show strange attactors of the chaotic motion.

• Structural Engineering and Mechanics
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• v.61 no.4
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• pp.497-510
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• 2017

A framed structure may be composed of two sub-structures, which are linked by a hinged joint. One sub-structure is the primary system and the other is the secondary system. The primary system, which is subjected to the periodic external load, can give rise to an auto-parametric resonance of the second system. Considering the geometric-stiffness effect produced by the axially internal force, the element equation of motion is derived by the extended Hamilton's principle. The element equations are then assembled into the global non-homogeneous Mathieu-Hill equations. The Newmark's method is introduced to solve the time-history responses of the non-homogeneous Mathieu-Hill equations. The energy-growth exponent/coefficient (EGE/EGC) and a finite-time Lyapunov exponent (FLE) are proposed for determining the auto-parametric instability boundaries of the structural system. The auto-parametric instabilities are numerically analyzed for the two frames. The influence of relative stiffness between the primary and secondary systems on the auto-parametric instability boundaries is investigated. A phenomenon of the "auto-parametric internal resonance" (the auto-parametric resonance of the second system induced by a normal resonance of the primary system) is predicted through the two numerical examples. The risk of auto-parametric internal resonance is emphasized. An auto-parametric resonance experiment of a ${\Gamma}$-shaped frame is conducted for verifying the theoretical predictions and present calculation method.

• Journal of KSNVE
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• v.10 no.5
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• pp.849-858
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• 2000

In this paper the chaotic out-of-plane vibrations of the uniformly curved pipe with pulsating flow are theoretically investigated. The derived equations of motion contain the effects of nonlinear curvature and torsional coupling. The corresponding nonlinear ordinary differential equation is a type of nonhomogenous Hill's equation . this is transformed into the averaged equation by averaging theorem. Bifurcation curves of chaotic motion are obtained by Melnikov's method and plotted in several cases of frequency ratios. The theoretically obtained results are demonstrated by numerical simulation. And strange attractors are shown.

• International Journal of Naval Architecture and Ocean Engineering
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• v.8 no.1
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• pp.13-21
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• 2016

Parametric pitch instability of a Deep Draft Semi-submersible platform (DDS) is investigated in irregular waves. Parametric pitch is a form of parametric instability, which occurs when parameters of a system vary with time and the variation satisfies a certain condition. In previous studies, analyzing of parametric instability is mainly limited to regular waves, whereas the realistic sea conditions are irregular waves. Besides, parametric instability also occurs in irregular waves in some experiments. This study predicts parametric pitch of a Deep Draft Semi-submersible platform in irregular waves. Heave motion of DDS is simulated by wave spectrum and response amplitude operator (RAO). Then Hill equation for DDS pitch motion in irregular waves is derived based on linear-wave theory. By using Bubnov-Galerkin approach to solve Hill equation, the corresponding stability chart is obtained. The differences between regular-waves stability chart and irregular-waves stability chart are compared. Then the sensitivity of wave parameters on DDS parametric pitch in irregular waves is discussed. Based on the discussion, some suggestions for the DDS design are proposed to avoid parametric pitch by choosing appropriate parameters. The results indicate that it's important and necessary to predict DDS parametric pitch in irregular waves during design process.