AN ITERATIVE METHOD FOR THE DISCRETE PROBLEMS OF A CLASS OF ELLIPTICAL VARIATIONAL INEQUALITIES

ＡＮＩＴＥＲＡＴＩＶＥＭＥＴＨＯＤＦＯＲＴＨＥＤＩＳＣＲＥＴＥＰＲＯＢＬＥＭＳＯＦＡＣＬＡＳＳＯＦＥＬＬＩＰＴＩＣＡＬＶＡＲＩＡＴＩＯＮＡＬＩＮＥＱＵＡＬＩＴＩＥＳＺｈｅｎｇＴｉｅ－ｓｈｅｎｓ（郑铁生）ＬｉＬｉ（李立）ＸｕＱｉｎｇ－ｙｕ（许庆余）（Ｄ...     

THE BOUNDED EXTERMINATION AND STABILITY OF MUTUAL INTERFERENCE SYSTEM OF FOUR－DIMENSIONAL SPECIES

ＴＨＥＢＯＵＮＤＥＤＥＸＴＥＲＭＩＮＡＴＩＯＮＡＮＤＳＴＡＢＩＬＩＴＹＯＦＭＵＴＵＡＬＩＮＴＥＲＦＥＲＥＮＣＥＳＹＳＴＥＭＯＦＦＯＵＲ－ＤＩＭＥＮＳＩＯＮＡＬＳＰＥＣＩＥＳＨｏｕＧａｎ－ｓｈｅｎｇ（侯赣生）（Ｄｅｐｌ．ｏｆＰｈｙｓｉｃｓ,Ｊｉａｎｇｘ...     

Dissipative effects of an isolated bubble in water on the sound wave

ＤＩＳＳＩＰＡＴＩＶＥＥＦＦＥＣＴＳＯＦＡＮＩＳＯＬＡＴＥＤＢＵＢＢＬＥＩＮＷＡＴＥＲＯＮＴＨＥＳＯＵＮＤＷＡＶＥＨｕｎａｇＪｉｎｇ－ｑｕａｎ（黄景泉）ＬｉＦｕ－ｘｉｎ（李福新）（ＮｏｒｔｈｗｅｓｔｅｒｎＰｏｌｙｔｅｃｈｎｉｃＵｎｉｖｅｒｓｉｔｙ,Ｘ...     

Matrix perturbation method for the vibration problem of structures with interval parameters

ＭＡＴＲＩＸＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＦＯＲＴＨＥＶＩＢＲＡＴＩＯＮＰＲＯＢＬＥＭＯＦＳＴＲＵＣＴＵＲＥＳＷＩＴＨＩＮＴＥＲＶＡＬＰＡＲＡＭＥＴＥＲＳＱｉｕＺｈｉ－ｐｉｎｇ（邱志平）ＣｈｅｎＳｕ－ｈｕａｎ（陈塑寰）Ｌｉｕｚｈｏｎｇ－ｓｈｅｎ...     

Free vibration and stability of moderate-thick cantilever plates

ＦＲＥＥＶＩＢＲＡＴＩＯＮＡＮＤＳＴＡＢＩＬＩＴＹＯＦＭＯＤＥＲＡＴＥ－ＴＨＩＣＫＣＡＮＴＩＬＥＶＥＲＰＬＡＴＥＳＺｈａｎｇＹｉｎ－ｙｉ（张音翼）（ＩｎｓｔｉｔｕｔｅｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉ,ＮａｎｃｈａｎｇＵｎｉｖｅｒｓｉｔｙ,Ｎａ...     

Non-linear vibration of rectangular reticulated shallow shell structures

ＮＯＮ－ＬＩＮＥＡＲＶＩＢＲＡＴＩＯＮＯＦＲＥＣＴＡＮＧＵＬＡＲＲＥＴＩＣＵＬＡＴＥＤＳＨＡＬＬＯＷＳＨＥＬＬＳＴＲＵＣＴＵＲＥＳＮｉｅＧｕｏ－ｈｕａ（聂国华）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉｃｓ,ＴｏｎｇｊｉＵｎｉｖｅ...     

GENERALIZED BIHARMONIC OPERATOR AND ITS APPLICATION TO THE BENDING OF ELASTIC THIN PLATES

ＧＥＮＥＲＡＬＩＺＥＤＢＩＨＡＲＭＯＮＩＣＯＰＥＲＡＴＯＲＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＴＯＴＨＥＢＥＮＤＩＮＧＯＦＥＬＡＳＴＩＣＴＨＩＮＰＬＡＴＥＳＹｕＺｈｏｎｅｔｔ－ｚｈｉ（俞中直）（Ｄｅｐｔ．ｏｆＥｎｇ．Ｍｅｃｈａｎｉｃｓ,ＤａｌｉａｎＵｎ...     

THE EXAMINATION OF TURBULENCE MODELING WITH LES DATABASE

ＴＨＥＥＸＡＭＩＮＡＴＩＯＮＯＦＴＵＲＢＵＬＥＮＣＥＭＯＤＥＬＩＮＧＷＩＴＨＬＥＳＤＡＴＡＢＡＳＥＳｕＭｉｎｇ－ｄｅ（苏铭德）（ＱｉｎｇｈｕａＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ）Ｒ．Ｆｒｉｅｄｒｉｃｈ（ＬｅｈｒｓｉｒｈｌｆｕｒＦｌｕｉｄｓｍｅｃｈａ...     

DOUBLE HIGH ORDER S－BREAKING BIFURCATION POINTS AND THEIR NUMERICAL DETERMINATION

ＤＯＵＢＬＥＨＩＧＨＯＲＤＥＲＳ－ＢＲＥＡＫＩＮＧＢＩＦＵＲＣＡＴＩＯＮＰＯＩＮＴＳＡＮＤＴＨＥＩＲＮＵＭＥＲＩＣＡＬＤＥＴＥＲＭＩＮＡＴＩＯＮ￥(杨忠华，叶瑞松)ＹａｎｇＺｈｏｎｇｈｕａ；（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｓｈａｎｇｈ...     

A rate type method for large deformation problems of nonlinear elasticity

ＡＲＡＴＥＴＹＰＥＭＥＴＨＯＤＦＯＲＬＡＲＧＥＤＥＦＯＲＭＡＴＩＯＮＰＲＯＢＬＥＭＳＯＦＮＯＮＬＩＮＥＡＲＥＬＡＳＴＩＣＩＴＹＬｉａｎｇＦｅｉ（梁非）,ＺｈａｎｇＳｈａｎ－ｙｕａｎ（张善元）（ＴａｉｙｕａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ...     

Linear Stability Analysis of Resonant Periodic Motions in the Restricted Three-Body Problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses 1 −μ and μ, 0≤μ≤ 1/2, that circle each other with period equal to 2π. When μ=0, the problem admits orbits for the massless particle that are ellipses of eccentricity e with the primary of mass 1 located at one of the focii. If the period is a rational multiple of 2π, denoted 2π p/q, some of these orbits perturb to periodic motions for μ > 0. For typical values of e and p/q, two resonant periodic motions are obtained for μ > 0. We show that the characteristic multipliers of both these motions are given by expressions of the form in the limit μ→ 0. The coefficient C(e,p,q) is analytic in e at e=0 and C(e,p,q)=O(e^|p-q|). The coefficients in front of e^|p-q|, obtained when C(e,p,q) is expanded in powers of e for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass 1 −μ.     

COLLINEAR LIBRATION POINTS WITH CONTINUOUS LOW THRUST

研究了圆型限制性三体问题的平动点在连续小推力作用下的具体位置和动力学特征的变化. 研究表明,随着小推力方向在空间中的变化,平动点的具体位置也会发生相应的变化,文章详细阐述了这些变化的特征.针对航天任务中应用较多的共线平动点L1和L2,研究了其附近运动的稳定性状态,给出了线性条件稳定解,并在此基础上,构造了条件稳定解的高阶形式,将其结果与数值积分轨道进行了比对,两者符合得很好. 最后,进一步研究了共线平动点附近周期轨道族的演化状态,由于连续小推力引入的非对称性,周期轨道族会发生分岔现象.     

Influence of micro-structural defects on post-buckling and large-amplitude vibration of geometrically imperfect gradient plate

We propose a polynomial expansion method to investigate periodic motions around both collinear and triangular libration points in the planar circular restricted three-body problem. The polynomial expansion method focuses on the approximate nonlinear polynomial relations between the two directions in the motion plane, which provide an alternative way to inspect the periodic motions. Based on the nonlinear polynomial relations, the planar two-dimensional system is decoupled. As an example, the \(3\mathrm{rd}\) order analytical solutions for the periodic motion are determined by solving the decoupled system. The efficiency of the polynomial expansion method has been validated by numerical method.     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Heteroclinic Orbits and Bernoulli Shift¶for the Elliptic Collision Restricted¶Three-Body Problem

We consider two mass points of masses m ₁=m ₂= moving under Newton's law of gravitational attraction in a collision elliptic orbit while their centre of mass is at rest. A third mass point of mass m ₃≈ 0, moves on the straight line L, perpendicular to the line of motion of the first two mass points and passing through their centre of mass. Since m ₃≈ 0, the motion of the masses m ₁ and m ₂ is not affected by the third mass, and from the symmetry of the motion it is clear that m ₃ will remain on the line L. So the three masses form an isosceles triangle whose size changes with the time. The elliptic collision restricted isosceles three-body problem consists in describing the motion of m ₃. In this paper we show the existence of a Bernoulli shift as a subsystem of the Poincaré map defined near a loop formed by two heteroclinic solutions associated with two periodic orbits at infinity. Symbolic dynamics techniques are used to show the existence of a large class of different motions for the infinitesimal body. Accepted July 6, 2000?Published online February 14, 2001     

Studying the stability of equilibrium solutions in a planar circular restricted four-body problem

The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions are stable in Lyapunov’s sense if the mass parameter satisfies the conditions μ ∈ (0, μ₀, where μ₀ is a sufficiently small number, and μ ≠ μ_j, j = 1, 2, 3. We also prove that, for μ = μ₁ and μ = μ₃, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of μ, the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations are done with the Mathematica computer algebra system. Published in Neliniini Kolyvannya, Vol. 10, No. 1, pp. 66–82, January–March, 2007.     

基于混合Lie算子辛算法的不变流形计算

平动点附近周期轨道的不变流形因其在低能轨道转移中起着重要作用而受到广泛关注.在设计低能轨道过程中不变流形要实时进行能量匹配,但利用传统数值积分方法进行积分时能量会耗散.显式辛算法具有比隐式辛算法计算效率高的优势,但其要求Hamilton系统必须分成两个可积的部分,而旋转坐标系下的圆型限制性三体问题是不可分的,因而显式辛算法难以用于求解旋转坐标系下的圆型限制性三体问题.本文通过引入混合Lie算子,成功实现了带三阶导数项的力梯度辛算法对圆型限制性三体问题的求解,并将基于混合Lie算子的带三阶导数项的辛算法与Runge-Kutta78算法和Runge-Kutta45算法进行仿真对比,仿真结果表明基于混合Lie算子的含有三阶导数项的辛算法位置精度高、能量误差小且计算效率高.利用基于混合Lie算子的带三阶导数项的辛算法计算不变流形,可以实现低能轨道转移过程中轨道拼接点的能量精准匹配.     

基于多项式约束的三角平动点平面周期轨道设计方法1）

平动点是圆型限制性三体问题中的五个平衡解.其中,三角平动点在平面问题中具有“中心×中心”的动力学特性,其附近存在着大量的周期轨道,研究这些周期轨道的构建方法在深空探测中具有理论及工程意义.本文从振动角度分析周期轨道,通过多项式展开法构建出主坐标下周期轨道两个运动方向之间的渐近关系.从新的角度分析了系统的动力学特性和平面周期运动两个方向内在关联以及物理规律.这种多项式形式的关系式,可以作为约束条件用于数值微分修正算法中,通过迭代的方式寻找周期轨道.数值仿真算例验证了方法的正确性及精确性.文章从振动的角度对周期轨道进行分析,改进了微分修正算法.提出的方法可以被拓展至圆型/椭圆型限制性三体问题的三维周期轨道构建中.     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.     

Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Stability of triangular libration points in a planar restricted elliptic three-body problem is investigated for two sets of parameters corresponding to the cases of double resonances. These positions are shown to be formally stable in the case of the fourth-order double resonance. When a weak third-order and a strong fourth-order resonances occur in the system, any motion of an approximate autonomous system is found to be bounded if initial conditions are sufficiently close to the unperturbed motion. Boundedness domain estimate is obtained.