• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.257-279
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• 2023

Applying the Lyapunov-Schmidt reduction, we consider spectral stability of small amplitude stationary periodic solutions bifurcating from an equilibrium of the generalized Swift-Hohenberg equation. We follow the mathematical framework developed in [15, 16, 19, 23] to construct such periodic solutions and to determine regions in the parameter space for which they are stable by investigating the movement of the spectrum near zero as parameters vary.

• International Journal of Naval Architecture and Ocean Engineering
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• v.11 no.2
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• pp.875-882
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• 2019

The effects of Triangle Groove Strips (TGS) on Vortex-induced Vibration (VIV) suppression of marine riser are numerically investigated using Computational Fluid Dynamics (CFD) method. The range of Reynolds number in simulations is 4.0 × 10⁴ < Re < 1.2 × 10⁵. The two-dimensional unsteady Reynolds-Averaged Navier-Stokes (RANS) equations and Shear Stress Transport (SST) k-ω turbulence model are used to calculate the flow around marine riser. The Newmark-β method is employed for evaluating the structure dynamics of marine riser. The effect of the height ratio (ε) of TGS on VIV suppression is evaluated. The amplitude responses, frequency responses, vortex patterns and the flow around the structures are discussed in detail. With the increase of the height ratio of TGS, the suppression effect of TGS on VIV suppression is improved firstly and then weakened. When ε=0.04, the suppression effect of TGS is the best. Compared with the VIV responses of smooth marine riser, the amplitude ratio is reduced by 38.9%, the peak of the lift coefficient is reduced by 69% and the peak of the drag coefficient is reduced by 40% when Re=6.0 × 10⁴. With the increase of Reynolds number, the suppression effect of TGS on VIV suppression is improved firstly and then weakened. When the Reynolds number is 7.0 × 10⁴, the amplitude ratio can be reduced by 40.1%. As to the large-amplitude vibration cases, the TGS show nice suppression effect on VIV.

• Proceedings of the Korean Institute of Navigation and Port Research Conference
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• 2006.10a
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• pp.134-143
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• 2006

Out of all types of motions the critical motions leading to capsize is roll. The dynamic amplification in case of roll motion may be large for ships as roll natural frequency generally falls within the frequency range of wave energy spectrum typical used for estimation of motion spectrum. Roll motion is highly non-linear in nature. Den are various representations of non-linear damping and restoring available in literature. In this paper an uncoupled non-linear roll equations with three representation of damping and cubic restoring term is solved using a perturbation technique. Damping moment representations are linear plus quadratic velocity damping, angle dependant damping and linear plus cubic velocity dependant damping. Numerical value of linear damping coefficient is almost same for all types but non-linear damping is different. Linear and non-linear damping coefficients are obtained form free roll decay tests. External rolling moment is assumed as deterministic with sinusoidal form. Maximum roll amplitude of non-linear roll equation with various representations of damping is calculated using analytical procedure and compared with experimental results, which are obtained form forced tests in regular waves by varying frequency with three wave heights. Experiments indicate influence of non-linearity at resonance frequency. Both experiment and analytical results indicates increase in maximum roll amplitude with wave slope at resonance. Analytical results are compared with experiment results which indicate maximum roll amplitude analytically obtained with angle dependent and cubic velocity damping are equal and difference from experiments with these damping are less compared to non-linear equation with quadratic velocity damping.

• Advances in nano research
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• v.16 no.2
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• pp.111-125
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• 2024

Recently, a lot of research has been done on the analysis of axial vibrations of homogeneous and FG nanotubes (nanorods) with various aspects of vibrations that have been fully mentioned in history. However, there is a lack of investigation of the dynamic internal resonances of FG nanotubes (nanorods) between them. This is one of the essential or substantial characteristics of nonlinear vibration systems that have many applications in various fields of engineering (making actuators, sensors, etc.) and medicine (improving the course of diseases such as cancers, etc.). For this reason, in this study, for the first time, the dynamic internal resonances of FG nanorods in the simultaneous presence of large-amplitude size dependent behaviour, inertial and shear effects are investigated for general state in detail. Such theoretical patterns permit as to carry out various numerical experiments, which is the key point in the expansion of advanced nano-devices in different sciences. This research presents an AFG novel nano resonator model based on the axial vibration of the elastic nanorod system in terms of derivation from large-amplitude size dependent internal modals interactions. The Hamilton's Principle is applied to achieve the basic equations in movement and boundary conditions, and a harmonic deferential quadrature method, and a multiple scale solution technique are employed to determine a semi-analytical solution. The interest of the current solution is seen in its specific procedure that useful for deriving general relationships of internal resonances of FG nanorods. The numerical results predicted by the presented formulation are compared with results already published in the literature to indicate the precision and efficiency of the used theory and method. The influences of gradient index, aspect ratio of FG nanorod, mode number, nonlinear effects, and nonlocal effects variations on the mechanical behavior of FG nanorods are examined and discussed in detail. Also, the inertial and shear traces on the formations of internal resonances of FG nanorods are studied, simultaneously. The obtained valid results of this research can be useful and practical as input data of experimental works and construction of devices related to axial vibrations of FG nanorods.

• Advances in nano research
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• v.9 no.3
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• pp.133-145
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• 2020

In this study, nonlinear vibrations and dynamic instabilities of a smart embedded micro shell conveying varied fluid flow and subjected to the combined electro-thermo-mechanical loadings are investigated. With the aim of designing new hydraulic sensors and actuators, the piezoelectric materials are employed for the body and the effects of applying electric field on the stability of the system as well as the induced voltage due to the dynamic behavior of the system are studied. The nonlocal piezoelasticity theory and the nonlinear cylindrical shell model in conjunction with the energy approach are utilized to mathematically modeling of the structure. The fluid flow is assumed to be isentropic, incompressible and fully develop, and for more generality of the problem both steady and time dependent flow regimes are considered. The mathematical modeling of fluid flow is also carried out based on a scalar potential function, time mean Navier-Stokes equations and the theory of slip boundary condition. Employing the modified Lagrange equations for open systems, the nonlinear coupled governing equations of motion are achieved and solved via the state space problem; forth order numerical integration and Bolotin's method. In the numerical results, a comprehensive discussion is made on the dynamical instabilities of the system (such as divergence, flutter and parametric resonance). We found that applying positive electric potential field will improve the stability of the system as an actuator or vibration amplitude controller in the micro electro mechanical systems.

• Bulletin of the Korean Chemical Society
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• v.27 no.10
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• pp.1525-1530
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• 2006

A simple oscillating reaction model subject to additive Gaussian white noise is investigated as the model is located in the dynamic region of oscillations. The model is composed of three ordinary differential equations representing the time evolutions of X, Y, and Z, respectively. Initially, a uniform random noise is separately added to the three equations to study the effect of noise on the oscillatory cycle of X, Y, and Z. For a given value of noise intensity, the amplitude of oscillation increases monotonically with time. Furthermore, the noise is added to any one of the three equations to study the impact of noise on one species on the bifurcation behavior of the other.

• Journal of Korea Water Resources Association
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• v.36 no.3 s.134
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• pp.413-422
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• 2003

Numerical analysis for the Bragg reflection due to sinusoidally varying seabeds tying on a sloping beach was performed by using a couple of ordinary differential equations derived from the Boussinesq equations. Incident waves were wane groups generated by two short waves with slightly different phases. Effects of the slope of a seabed to the reflection were investigated in detail. It is shown that the reflection of long waves enhanced by increasing the slope of a seabed. This phenomenon caused by increase of wave amplitude due to increase of nonlinearity and shoaling.

• Structural Engineering and Mechanics
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• v.61 no.2
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• pp.209-220
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• 2017

The purpose of this paper is to study the geometrically nonlinear free vibration of functionally graded nano/micro beams (FGNBs) based on the modified couple stress theory. For practical applications, some analytical expressions of nonlinear frequencies for FGNBs on a nonlinear Pasternak foundation are developed. Hamilton's principle is employed to obtain nonlinear governing differential equations in the context of both Euler-Bernoulli and Timoshenko beam theories for a comprehensive investigation. The modified continuum theory contains one material length scale parameter to capture the size effect. The variation of two-constituent material along the thickness is modeled using Reddy's power-law. Also, the Mori-Tanaka method as an accurate homogenization technique is implemented to estimate the effective material properties of the FGNBs. The results are presented for both hinged-hinged and clamped-clamped boundary conditions. The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method and then the powerful method of homotopy analysis is utilized to obtain the semi-analytical solutions. Eventually, the presented analytical expressions are used to examine the influences of the length scale parameter, material gradient index, and elastic foundation on the nonlinear free vibration of FGNBs.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.16 no.3 s.108
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• pp.263-269
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• 2006

The objective of the present work is to develop a time-domain numerical method of broadband noise in a cascade of airfoils. This paper focuses on dipolar broadband noise sources, resulting from the interaction of turbulent inflows with the flat-plate airfoil cascade. The turbulence response of a two-dimensional cascade is studied by solving both of the linearised and the full nonlinear Euler equations employing accurate higher order spatial differencing, time stepping techniques and non-reflecting inflow/outflow boundary condition. The time-domain result using the linearised Euler equations shows good agreement with the analytical solution using the modified LINSUB code. Through the comparison of the nonlinear time-domain result using the full nonlinear Euler equations with the linear, it is found that the acoustic mode amplitude of the nonlinear response is less than that of the linear response due to the energy cascade from low frequency components to the high frequency ones. Considering the merits of the time-domain methods over the typical time-linearised frequency-domain analysis, the current method is expected to be promising tools for analyzing the effects of the airfoil shapes, non-uniform background flow, linear-nonliear regimes on the broadband noise due to turbulence-cascade interaction.

• Geotechnical Engineering
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• v.6 no.3
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• pp.41-52
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• 1990

To investigate the liquefaction potential of sands, a series of untrained cyclic triaxial compression tests is carried out on the samples of Ottawa, Joomoonjin, Hn river and Hongseung sands. The constitutive equations of sands are derived to explain the mechanical behavior of sands under cyclic stresses, and are applicable to liquefaction analysis. The following results are obtainded in this study. 1. Sands with the lower confining pressure or relative density are to be easily liquefied, and when the amplitude of cyclic stress are large, liquefaction takes places over only a few cycles. 2. Stress ratio, porewater pressure ratio and cyclic shear strains are to be good criteria to evaluate liquefaction potential of sands. 3. Hongseung sands which contains some silty clay shows higher dynamic properties than other sands. 4. The dynamic behaviors of undisturbed Hongseung sand are about same as those of dense sands. It is noted that undisturbed Hongseung sand shows higher liquefaction potential than the samples made by pluviation under same relative density, 5. The constitutive equations of soils under cyclic loads are developed based on the theory of elasto-plasticity, logarithmic stress-strain rela'tionship, non-associated flow rule and the concept of the boundary surface. The derived equations is applicable to predict the behavior of sands under cyclic loads and liquefaction potential with a higher accuracy. 6. Based on results of the study it may be concluded that cracks of the foundations and dislocation of the structures at Hongseung earthquakes(Oct. 7, 1978, Richter scald 5.2) are not brought by the liquefaction process.