• 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 the Korean Society for Railway
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• v.3 no.3
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• pp.131-138
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• 2000

The design of current collection system of high speed train requires the fundamental understandings for the dynamic characteristics of catenary system and pantograph. The stiffness of catenary system of high speed train has the varying characteristics for the change of contact point with pantograph, since the supporting pole and hanger make the different boundary conditions for the up-down stiffness of a trolley wire. The variation of stiffness results in Mathiue equation, which characterizes the stability of the system. However, the two-term variation of the stiffness due to span length and hanger distance cannot be solved analytically. In this paper, the stiffness variations are calculated and the physical reasoning of linear model and one term Mathieu equation are reviewed. And the numerical analysis for the two-term variation of the stiffness is done for the several design parameters of pantograph.

• Steel and Composite Structures
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• v.31 no.3
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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.

• Structural Engineering and Mechanics
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• v.73 no.6
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• pp.685-698
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• 2020

The dynamic stability of a functionally graded polymer microbeam reinforced by graphene oxides subjected to a periodic axial force is investigated. The microbeam is assumed to rest on an elastic substrate and is subjected to various immovable boundary restraints. The weight fraction of graphene oxides nanofillers is graded across the beam thickness. The effective Young's modulus of the functionally graded graphene oxides reinforced composite (FG-GORC) was determined using modified Halpin-Tsai model, with the mixture rule used to evaluate the effective Poisson's ratio and the mass density. An improved third order shear deformation theory (TSDT) is used in conjunction with the Chebyshev polynomial-based Ritz method to derive the Mathieu-Hill equations for dynamic stability of the FG-GORC microbeam, in which the scale effect is taken into account based on modified couple stress theory. Then, the Mathieu-Hill equation was solved using Bolotin's method to predict the principle unstable regions of the FG-GORC microbeams. The numerical results show the effects of the small scale, the graphene oxides nanofillers as well as the elastic substrate on the dynamic stability behaviors of the FG-GORC microbeams.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.18 no.2
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• pp.160-168
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• 2008

This paper presents the study on the stability of nonlinear oscillations of a thin cantilever beam subject to harmonic base excitation in vertical direction. Two partial differential governing equations under combined parametric and external excitations were derived and converted into two-degree-of-freedom ordinary differential Mathieu equations by using the Galerkin method. We used the method of multiple scales in order to analyze one-to-one combination resonance. From these, we could obtain the eigenvalue problem and analyze the stability of the system. From the thin cantilever experiment using foamax, we could observe the nonlinear modes of bending, twisting, sway, and snap-through buckling. In addition to qualitative information, the experiment using aluminum gave also the quantitative information for the stability of combination resonance of a thin cantilever beam under parametric excitation.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.20 no.9
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• pp.849-855
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• 2010

Damping may change the response characteristics of nonlinear oscillations due to the parametric excitation of a thin cantilever beam. When the natural frequencies of the first bending and torsional modes are of the same order of magnitude, we can observe the one-to-one combination resonance in the perturbation analysis depending on the characteristic parameters. The nonlinear behavior about the combination resonance reveals a chaotic motion depending on the natural frequencies and damping ratio. We can analyze the chaotic dynamics by using the eigenvalue analysis of the perturbed components. In this paper, we derived the equations for autonomous system and solved them to obtain the characteristic equation. The stability analysis was carried out by examining the eigenvalues. Numerical integration gave the physical behavior of each mode for given parameters.

• Proceedings of the KSR Conference
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• 2000.05a
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• pp.598-605
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• 2000

The design of a current collection system of high speed train requires the fundamental understandings fer the dynamic characteristics of a catenary system and pantograph. The stiffness of the catenary system of high speed train has the varying characteristics for the change of the contact point with a pantograph, since the supporting pole and hanger make the different boundary conditions for the updown stiffness of a trolley wire. The variation of stiffness results in Mathiue equation, which characterizes the stability of the system. However, the two terms variation of the stiffness due to span length and hanger distance cannot be solved analytically. In this paper, the stiffness variations are calculated, and the physical reasoning of linear model and one term Mathieu equation are reviewed. And the numerical analysis for the two term variation of the stiffness is done for the several design parameters of the pantograph.

• Ocean Systems Engineering
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• v.2 no.4
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• pp.239-255
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• 2012

Analysis of ship parametric roll has generally been restricted to simple analytical models and sophisticated time domain simulations. Simple analytical models do not capture all the critical dynamics while time-domain simulations are often time consuming to implement. The model presented in this paper captures the essential dynamics of the system without over simplification. This work incorporates various important aspects of the system and assesses the significance of including or ignoring these aspects. Special consideration is given to the fact that a hull form asymmetric about the design waterline would not lead to a perfectly harmonic variation in metacentric height. Many of the previous works on parametric roll make the assumption of linearized and harmonic behaviour of the time-varying restoring arm or metacentric height. This assumption enables modelling the roll motion as a Mathieu equation. This paper provides a critical assessment of this assumption and suggests modelling the roll motion as a Hills equation. Also the effects of non-linear damping are included to evaluate its effect on the bounded parametric roll amplitude in a simplified manner.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.38 no.10
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• pp.845-851
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• 2014

Dielectrophoresis (DEP) is defined as the motion of suspended particles in solvent resulting from polarization forces induced by an inhomogeneous electric field. DEP has been utilized for various biological applications such as trapping, sorting, separation of cells, viruses, nanoparticles. However, the analysis of DEP trapping has mostly employed the period-averaged ponderomotive forces while the dynamic features of DEP trapping have not been attracted because the target object is relatively large. Such approach is not appropriate for the nanoscale analysis in which the size of object is considerably small. In this study, we thoroughly investigate the dynamic response of trapping to various system parameters and its influence on the trapping stability. The effects of particle conductivity on its motion are also focused.

• 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.