• 대한기계학회논문집A
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• 제21권2호
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• pp.337-345
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• 1997

An analysis is presented for the response of a beam constrained by a nonlinear spring to a harmonic excitation. The system is governed by a linear partial differential equation with a nonlinear boundary condition. The method of multiple scales is used to reduce the nonlinear boundary value problem to a system of autonomous ordinary differential equations of the amplitudes and phases. The case of the third-order subharmonic resonance is considered in this study. The autonomous system is used to determine the steady-state responses and their stability.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 1996년도 추계학술대회논문집; 한국과학기술회관, 8 Nov. 1996
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• pp.275-281
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• 1996

In order to examine the validity of an asymptotic solution obtained from the method of multiple scales, we investigate a third-order subharmonic resonance response of a bar constrained by a nonlinear spring to a harmonic excitation. The motion of the bar is governed by a linear partial differential equation with a nonlinear boundary condition. The nonlinear boundary value problem is solved by using the finite difference method. The numerical solution is compared with the asymptotic solution.

• International Journal of Fluid Machinery and Systems
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• 제9권4호
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• pp.338-353
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• 2016

Information on the surge behaviors and stall stagnation boundaries for a nine-stage axial flow compressor are summarized on the basis of analytical data in comparison with those for a single-stage one, with attention to the pressure ratio effect. The general trends of the surge loop behaviors of the pressure-mass flow are similar for both compressors including the fact that the subharmonic surges tend to appear very near the stall stagnation boundaries. With respect to the nine-stage compressor, however, the mild loops in the subharmonic surges tend to be very small in size relative to the deep loops, and at the same time, insufficient surge recovery phenomenon, which is a kind of subharmonic surge, appears also far from the stagnation boundary for relatively short delivery flow-paths. The latter is found to be a rear-stage surge caused by unstalling and re-stalling of the rear stages with the front-stages kept in stall in the stalled condition of the whole compressor, which situation is caused by stage-wise mismatching in the bottom pressure levels of the in-stall multi-stage compressor. The fundamental information on the stall stagnation boundaries is given by a group of normalized geometrical parameters including relative delivery flow-path length, relative suction flow-path length, and sectional area-pressure ratio, and by another group of normalized frequency parameters including relative surge frequencies, modified reduced resonance frequencies, and modified reduced surge frequencies. Respective groups of the normalized parameters show very similar tendency of behaviors for the nine-stage compressor and the single-stage compressor. The modified reduced resonance frequency could be the more reasonable parameter suggesting the flow-induced oscillation nature of the surge phenomena. It could give the stall stagnation boundary in a more unified manner than the Greitzer's B parameter.

• International Journal of Fluid Machinery and Systems
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• 제7권4호
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• pp.160-173
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• 2014

Surge phenomena in multi-stage axial flow compressors were studied with attention to the frequency behaviors. A new parameter "volume-modified reduced surge frequency" was introduced, which took into consideration the essential surge process, i.e., emptying and filling of the working gas in the delivery plenum. The behaviors of the relative surge frequencies at the stall stagnation boundaries, compared with the corresponding duct resonance frequencies, have demonstrated the existence of two types of surges; i.e., a near-resonant surge and a subharmonic surge. The former, which has fundamentally a near-resonance frequency, occurs predominantly at the stall stagnation boundary for the short -and-fat plenum delivery flow-path and the long-and-narrow delivery duct flow-path, and possibly in the intermediate conditions. The latter, which has a subharmonic frequency of the fundamental near-resonant one and occurs mainly in the intermediate zone, is considered to be caused by the reduced frequency restricted to a limited range. In relation with those dimensionless frequencies at the stall stagnation boundary, the surge frequency behaviors in more general situations away from the boundaries could be estimated, though very roughly.

• Structural Engineering and Mechanics
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• 제84권6호
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• pp.767-782
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• 2022

In this paper, the nonlinear vibration behavior of the spiral stiffened multilayer functionally graded (SSMFG) cylindrical shells exposed to the thermal environment and a uniformly distributed harmonic loading using a semi-analytical method is investigated. The cylindrical shell is surrounded by a nonlinear viscoelastic foundation consisting of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness. The distribution of temperature and material constitutive of the stiffeners are continuously changed through the thickness direction. The cylindrical shell has three layers consisting of metal, FGM, and ceramic. The interior layer of the cylindrical shell is rich in metal, while the exterior layer is rich in ceramic, and the FG material is located between two layers. The nonlinear vibration problem utilizing the smeared stiffeners technique, the von Kármán equations, and the Galerkin method has been solved. The multiple scales method is utilized to examine the nonlinear vibration behavior of SSMFG cylindrical shells. The considered resonant case is 1:3:9 internal resonance and subharmonic resonance of order 1/3. The influences of different material and geometrical parameters on the vibration behavior of SSMFG cylindrical shells are examined. The results show that the angles of stiffeners, temperature, and elastic foundation parameters have a strong effect on the vibration behaviors of the SSMFG cylindrical shells.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2003년도 추계학술대회논문집
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• pp.504-514
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• 2003

We investigate global bifurcation in the subharmonic motion of a circular plate with one-to-one internal resonance. A system of autonomous equations are obtained from the partial differential equations governing the system by using Galerkin's procedure and the method of multiple scales. A perturbation method developed by Kovacic and Wiggins is used to find Silnikov type homoclinic orbits. The conditions under which the orbits occur are obtained.

• 대한전자공학회논문지
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• 제23권1호
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• pp.91-100
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• 1986

The nonlinear rate equations are solved analytically by using the singular perturbation method to study effects of the spontaneous emission factor and the photon lifetime on the primary resonance and the first subharmonic generation(i.e., the onset of the periocd-doubling route to chaos). By large signal modulation of Hitachi CSP laser HLP 1400, the resonance frequency shift than 100 ps with 1 GHz repetition rate are generated. The experimental observations are in reasonable agreement with the theoretical results obtained using measured parameters of the rate equations.

• 한국소음진동공학회논문집
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• 제13권4호
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• pp.317-322
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• 2003

Torsional oscillations of a system incorporating a Hooke's joint are investigated by adopting a nonlinear 2-degree-of-freedom model. Linear and Van der Pol transformations are applied to obtain the equations of motion to which the method of averaging can be readily applied. Various subharmonic and combination resonances are identified with the conditions of their occurrences. Applying the method of averaging leads to the reduced amplitude- and phase-equations of motion, of which constant and periodic solutions are obtained numerically. The periodic solution which emerges from Hopf bifurcation point experiences period doubling bifurcation leading to infinite solution rather than chaotic solution.

• Structural Engineering and Mechanics
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• 제69권2호
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• pp.131-143
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• 2019

In this study, the nonlinear vibration analysis of the composite nanoplate is studied. The composite nanoplate is fabricated by the functional graded (FG) core and lipid face sheets. The material properties in the FG core vary in three directions. The Kelvin-Voigt model is used to study the viscoelastic effect of the lipid layers. By using the Von-Karman assumptions, the nonlinear differential equation of the vibration analysis of the composite nanoplate is obtained. The foundation of the system is modeled by the nonlinear Pasternak foundation. The Bubnov-Galerkin method and the multiple scale method are used to solve the nonlinear differential equation of the composite nanoplate. The free and force vibration analysis of the composite nanoplate are studied. A comparison between the presented results and the reported results is done and good achievement is obtained. The reported results are verified by the results which are obtained by the Runge-Kutta method. The effects of different parameters on the nonlinear vibration frequencies, the primary, the super harmonic and subharmonic resonance cases are investigated. This work will be useful to design the nanosensors with high biocompatibility.