无限均质弹性体中圆片裂纹受均布载荷时的超奇异积分方程封闭解

本文运用一种特殊技巧将一个受均布压力作用的圆片裂纹超奇异积分方程化为Ａｂｅｌ积分形式，从而可获得超奇异积分方程中未知位移间断的封闭解。再利用这个封闭解和应力强度因子的定义，得到了一个无限弹性体中受均布载荷分布时圆片裂纹前沿Ｉ型应力强度因子的精确表达式。所得到的结果与现有解完全相同。     

LOCAL BIFURCATION ANALYSIS OF STRONGLY NONLINEAR DUFFING SYSTEM

ＬＯＣＡＬＢＩＦＵＲＣＡＴＩＯＮＡＮＡＬＹＳＩＳＯＦＳＴＲＯＮＧＬＹＮＯＮＬＩＮＥＡＲＤＵＦＦＩＮＧＳＹＳＴＥＭＢｉＱｉｎｓｈｅｎｇ（毕勤胜）；ＣｈｅｎＹｕｓｈｕ（陈予恕）；ＷｕＺｈｉｑｉａｎｇ（吴志强）Ａｂｓｔｒａｃｔ：Ｂｙｕｓｉｎｇｃｏｏｒｄｉｎ...     

Analysis on Double Hopf Bifurcation Using Computer Algebra with the Aid of Multiple Scales

In this paper, a perturbation method based on multiple scales is used forcomputing the normal forms of nonlinear dynamical systems. The approach,without the application of center manifold theory, can be used to systematicallyfind the explicit normal form of a system described by a general n-dimensionaldifferential equation. The attention is focused on the dynamic behaviour ofa system near a critical point characterized by two pairs of purely imaginaryeigenvalues without resonance. The method can be easily formulated and implemented using a computer algebra system. Maple programs have been developedwhich can be `automatically' executed by a user without knowing computeralgebra. Examples chosen from mathematics, electrical circuits, mechanics andchemistry are presented to show the applicability of the technique and theconvenience of using computer algebra.     

多体动系统动力学方程在流形上的辛算法

多体系统动力学方程的数值方法一直是数学与力学家们的热门研 究课题.特别是多体系统动力学微分/代数方程组形式的数学模型，是所 谓的指标-3问题，它的求解是一个难题.目前流行的关于它的数值方法都 有不尽人意的地方，主要是对出现的所谓的违约问题和刚性问题未很好地 解决.多体系统动力学方程在流形上的辛算法是近几年出现的新的数值方 法，它将闭环型多体系统动力学的方程的约束部分和常微分方程部分利 用所谓的辛方法同时进行处理，其中的一些方法已证明是有效的，所以， 用它求解多体系统动力学方程前景看好.本文介绍了这些新的理论，并提 出了一些有待解决的问题.     

碳纤维板增强钢筋砼梁开裂性状试验研究

采用实验研究的方法，对碳纤维薄板增强钢筋混凝土三点弯曲缺口梁和无缺口梁的开裂性状及破坏规律进行探讨，分析其影响因子，研究结果表明：1)与无纤维板增强钢筋混凝土梁相比，碳纤维薄板增强钢筋混凝土三点弯曲梁的开裂性状和破坏模式更加复杂，开裂荷载和极限荷载提高，刚度增加；2)与无缺口梁相比，缺口梁的开裂性状大致相同，但开裂荷载和极限荷载都稍有下降，刚度增加；3)碳纤维薄板的粘贴长度对梁的开裂性状、开裂荷载、极限荷载和刚度有较大的影响；4)按照梁的不同的开裂性状，提出了相应的力学分析模型。     

Global dynamics of the cable under combined parametrical and external excitations

The chaotic dynamics and global bifurcations of the suspended elastic cable under combined parametric and external excitations are investigated. The non-linear equations of motion of the elastic cable to small vibration of one support are derived. The averaged equations are obtained by using the method of multiple scales. Based on the averaged equations, the theory of normal form and Maple program are used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. On the basis of the normal form, global bifurcation analysis of the parametrically and externally excited suspended elastic cable is given by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of the elastic cable is also found by numerical simulation.     

多重非内共振Hopf分岔的正规形

利用我们提出的求正规形的新方法［１］，推导高维非线性动力系统的多重非内共振Ｈｏｐｆ分岔正规形（系统的Ｊａｃｏｂｉ矩阵有多对比值为无理数的纯虚特征值出现的情况），给出原系统系数与其正规形系数之间的简单的关系式．     

Transient lightning responses of grounding systems buried in horizontal multilayered earth with a hybrid method

Combined with the Fast Fourier Transform (FFT), a mathematical hybrid method for accurately computing the lightning response from grounding systems buried in multilayered earth model has been developed in this paper. In the method, electrical circuit consists of “T” typical of basic elements. To accelerate calculations of the method, quasi-static complex image method and closed form of Green's function and analytical formula for mutual induction and impedance coefficients were introduced into this method. With the inverse FFT, the method can be used for studying performances of transient lightning response from the grounding systems.     

Synthesis,structure, spectroscopic (FT-IR) and density functional modelling studies of 1-[(4-ethoxyphenylimino)methyl]napthalene-2-ol

Synthesis, crystallographic characterisation, spectroscopic (Fourier transform infrared spectroscopy [FT-IR]) and density functional modelling studies of the Schiff base 1-[(4-ethoxyphenylimino)methyl]napthalene-2-ol (C₁₉H₁₇NO₂) have been reported. The molecular structure obtained from X-ray single-crystal analysis of the investigated compound in the ground state has been compared using Hartree–Fock and density functional theory (DFT) with the 6-311++G(d,p) basis set. In addition to the optimised geometrical structures, atomic charges, molecular electrostatic potential, natural bond orbital, non-linear optical (NLO) effects and thermodynamic properties of the compound have been investigated by using DFT. The experimental (FT-IR) and calculated vibrational frequencies (using DFT) of the title compound have been compared. The solvent effect was also investigated for obtained molecular energies and the atomic charge distributions of the compound. There exists a good correlation between experimental and theoretical data for enol-imine form of the compound. The total molecular dipole moment (µ), linear polarisability (α), and the first-order hyperpolarisability (β) were predicted by the B3LYP method with different basis sets 6-31G(d), 6-31+G(d,p), 6-31++G(d,p), 6-311+G(d) 150 and 6-311++G(d,p) for investigating the effects of basis sets on the NLO properties. Our computational results yield that β_tot for the title compound is greater than those of urea.     

关于微积分解题的两个注记

微积分是理工科大学生的一门非常重要的基础课.给出了微积分中求未定式极限时需注意采取的一个措施及由微分方程的通解求相应的微分方程的一般方法,并给出了算例.