• Title/Summary/Keyword: Mathieu

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Stability Analysis of Mathieu Equation by Floquet Theory and Perturbation Method (Floquet 이론과 섭동법에 의한 Mathieu Equation의 안정성해석)

  • Park, Chan Il
    • Transactions of the Korean Society for Noise and Vibration Engineering
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    • v.23 no.8
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    • pp.734-741
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    • 2013
  • In contrast of external excitations, parametric excitations can produce a large response when the excitation frequency is away from the linear natural frequencies. The Mathieu equation is the simplest differential equation with periodic coefficients, which lead to the parametric excitation. The Mathieu equation may have the unbounded solutions. This work conducted the stability analysis for the Mathieu equation, using Floquet theory and numerical method. Using Lindstedt's perturbation method, harmonic solutions of the Mathieu equation and transition curves separating stable from unstable motions were obtained. Using Floquet theory with numerical method, stable and unstable regions were calculated. The numerical method had the same transition curves as the perturbation method. Increased stable regions due to the inclusion of damping were calculated.

Effect of Parametric Excitation on Lateral Vibrations of Long, Slender Marine Structures (장주형 해양구조물의 횡방향 진동에 대한 파라메트릭 가진의 효과)

  • Park, Han Il
    • Journal of Ocean Engineering and Technology
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    • v.7 no.1
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    • pp.73-80
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    • 1993
  • 본 연구에서는 장주형 해양구조물의 횡방향 진동에 대한 파라메트릭 가진 효과를 고찰하였다. 먼저, 장주형 해양구조물의 횡방향 운동에 대한 4계 편미방지배방정식을 비선형 Mathieu 방정식으로 유도하였다. 비선형 mathieu 방정식의 해를 구하여 장주형 해양구조물의 동적 반응 특성을 해석하였다. 유체 비선형 감쇠력은 불안정 조건하에 있는 파라메트릭 진동의 반응크기를 제한 하는데 중요한 역활을 한다. 파라메트릭 진동의 경우 가장 큰 반응크기는 Mathieu 안정차트의 첫번째 불안정 구간에서 일어난다. 반면에, 파라메트릭 진동과 강제진동의 결합 진동인 경우, 가장 큰 반응 크기는 두번째 불안정 구간에서 발생된다. 파라메트릭 가진으로 인한 장주형 해양구조물의 횡방향 운동은 동적조건에 따라 subharmonic, superharmonic 또는 chaotic 운동이 되기도 한다.

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GALOIS THEORY OF MATHIEU GROUPS IN CHARACTERISTIC TWO

  • Yie, Ik-Kwon
    • Journal of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.199-210
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    • 2007
  • Given a field K and a finite group G, it is a very interesting problem, although very difficult, to find all Galois extensions over K whose Galois group is isomorphic to G. In this paper, we prepare a theoretical background to study this type of problem when G is the Mathieu group $M_{24}$ and K is a field of characteristic two.

MATHIEU GROUP COVERINGS AND GOLAY CODES

  • Yie, Ik-Kwon
    • Journal of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.289-317
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    • 2002
  • We associate binary codes to polynomials over fields of characteristic two and show that the binary Golay codes are associated to the Mathieu group polynomials in characteristics two. We give two more polynomials whose Galois group in $M_{12}$ but different self-orthogonal binary codes are associated. Also, we find a family of $M_{24}$-coverings which includes previous ones.

MATHIEU-TYPE SERIES BUILT BY (p, q)-EXTENDED GAUSSIAN HYPERGEOMETRIC FUNCTION

  • Choi, Junesang;Parmar, Rakesh Kumar;Pogany, Tibor K.
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.789-797
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    • 2017
  • The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and its associated alternating version whose terms contain a (p, q)-extended Gauss' hypergeometric function. Certain upper bounds for the two series are also given.

Parametrically Excited Vibrations of Second-Order Nonlinear Systems (2차 비선형계의 파라메트릭 가진에 의한 진동 특성)

  • 박한일
    • Journal of Advanced Marine Engineering and Technology
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    • v.16 no.5
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    • pp.67-76
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    • 1992
  • This paper describes the vibration characteristic of second-order nonlinear systems subjected to parametric excitation. Emphasis is put on the examination of the hydrodynamic nonlinear damping effect on limiting the response amplitudes of parametric vibration. Since the parametric vibration is described by the Mathieu equation, the Mathieu stability chart is examined in this paper. In addition, the steady-state solutions of the nonlinear Mathieu equation in the first instability region are obtained by using a perturbation technique and are compared with those by a numerical integration method. It is shown that the response amplitudes of parametric vibration are limited even in unstable conditions by hydrodynamic nonlinear damping force. The largest reponse amplitude of parametric vibration occurs in the first instability region of Mathieu stability chart. The parametric excitation induces the response of a dynamic system to be subharmonic, superharmonic or chaotic according to their dynamic conditions.

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THE KERNELS OF THE LINEAR MAPS OF FINITE GROUP ALGEBRAS

  • Dan Yan
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.45-64
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    • 2024
  • Let G be a finite group, K a split field for G, and L a linear map from K[G] to K. In our paper, we first give sufficient and necessary conditions for Ker L and Ker L ∩ Z(K[G]), respectively, to be Mathieu-Zhao spaces for some linear maps L. Then we give equivalent conditions for Ker L to be Mathieu-Zhao spaces of K[G] in term of the degrees of irreducible representations of G over K if G is a finite Abelian group or G has a normal Sylow p-subgroup H and L are class functions of G/H. In particular, we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if K is a split field for G.

Mathieu stability of offshore Buoyant Leg Storage & Regasification Platform

  • Chandrasekaran, S.;Kiran, P.A.
    • Ocean Systems Engineering
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    • v.8 no.3
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    • pp.345-360
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    • 2018
  • Increasing demand for large-sized Floating, Storage and Regasification Units (FSRUs) for oil and gas industries led to the development of novel geometric form of Buoyant Leg Storage and Regasification Platform (BLSRP). Six buoyant legs support the deck and are placed symmetric with respect to wave direction. Circular deck is connected to buoyant legs using hinged joints, which restrain transfer of rotation from the legs to deck and vice-versa. Buoyant legs are connected to seabed using taut-moored system with high initial pretension, enabling rigid body motion in vertical plane. Encountered environmental loads induce dynamic tether tension variations, which in turn affect stability of the platform. Postulated failure cases, created by placing eccentric loads at different locations resulted in dynamic tether tension variation; chaotic nature of tension variation is also observed in few cases. A detailed numerical analysis is carried out for BLSRP using Mathieu equation of stability. Increase in the magnitude of eccentric load and its position influences fatigue life of tethers significantly. Fatigue life decreases with the increase in the amplitude of tension variation in tethers. Very low fatigue life of tethers under Mathieu instability proves the severity of instability.

Propagation and Radiation Characteristics of Elliptical Corrugated Waveguide (타원형 커루게이트 도파관의 전파 및 복사 특성)

  • 고욱희;백경훈
    • The Journal of Korean Institute of Electromagnetic Engineering and Science
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    • v.9 no.5
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    • pp.614-620
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    • 1998
  • In this paper, we analyze theoritically the propagation and radiation characteristics for elliptical corrugated waveguides. The solutions of wave equations in an elliptic cylinder system are obtained in terms of Mathieu functions of 1st and 2nd kind. The electromagnetic fields in the elliptical corrugated waveguide can be represented by series and products of angular and radial Mathieu functions. By using impedence matching at the boundary between the inner region and the slot region, characteristic equations are derived. Then the characteristic equation is solved for $HE_{11}$ mode which is dominant mode in the elliptical corrugated waveguide and the fields in the aperture is calculated. And the propagation pattern for the elliptical corrugated waveguides is calculated through the field equivalence principle.

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