Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

不同结构性黄土的强度规律及传统强度准则适应性分析

在真三轴试验基础上揭示了多种不同结构性原状黄土的强度变化规律,分析了在岩土工程实践中常用的Mohr-Coulomb准则和Matsuoka-Nakai准则对不同结构性原状黄土的适应性。结果表明:当结构性黄土的强度越大时,其在π平面上的破坏线越接近于抹圆曲边三角形,反之其在π平面上的破坏线越接近于抹圆角三角形;结构性黄土随着结构性的增大在π平面上的强度破坏线从抹圆角三角形逐渐向抹圆曲边三角形发展,最终趋近于Mises圆;两个强度准则所描述的强度都比结构性原状黄土的实际强度值小,Mohr-Coulomb准则的误差比Matsuoka-Nakai准则的误差大,并且随着结构性的增强其误差也逐渐增大。     

复数域总体最小二乘平差

在复数域最小二乘的基础上提出了复数域总体最小二乘平差方法,推导了复数域总体最小二乘和复数混合总体最小二乘的相关公式。通过算例比较分析了复数观测值的残差的模的平方和最小(平差准则1)下及残差的实部和虚部的平方和分别最小(平差准则2)下的复数最小二乘、复数观测值和系数矩阵的残差的模的平方和最小(平差准则3)下及残差的实部和虚部的平方和分别最小(平差准则4)下的复数总体最小二乘方法的优劣。试验结果表明:平差准则1下复数最小二乘较平差准则2下得到的结果更加合理,平差准则3下复数总体最小二乘较平差准则4下得到的结果更为准确;当顾及系数矩阵误差时,平差准则3下复数总体最小二乘要优于平差准则1下复数最小二乘。     

基于梯度塑性理论的岩样在压剪条件下变形及稳定性分析

考虑了正应力及剪应力联合作用在岩样的端部,而且,在正应力被视为变量,剪应力被视为常量这一特殊条件下,对剪切带的变形特征及剪切带-带外弹性体系统的失稳判据进行了分析.根据剪切虎克定律及梯度塑性理论,提出了剪切带内部弹性应变、塑性应变及总应变的解析式,对上述三种应变进行积分,提出了剪切带错动弹性位移、塑性位移及总位移的解析式.利用压缩虎克定律及几何关系建立了剪切带外弹性体的刚度的表达式.根据刚度理论,建立了剪切带-带外弹性体系统的失稳判据.考虑岩样端面的剪切应力后,在流动剪切应力相同时,剪切带内部的弹性剪切应变(均匀分布)将增加;剪切带内部的塑性剪切应变(不均匀分布)将减小;剪切带的弹性剪切位移(线性分布)将增加;剪切带的塑性剪切位移(非线性分布)将减小;剪切带错动位移要小一些.另外,剪切应力对剪切带-带外弹性体系统的失稳判据没有影响.失稳判据仅和岩石的本构参数、剪切带倾角及结构尺寸有关.     

爆炸应力波对新浇混凝土衬砌的影响研究

爆炸应力波作用下新浇混凝土衬砌的破坏模式及安全振动速度是水电、铁道工程隧洞建设中需要解决的重要问题。根据弹性P波及SV波在介质中的传播规律,讨论了P波及SV波在薄层混凝土衬砌中的传播及近似处理方法,依照莫尔-库仑准则、极限拉应力准则、极限拉应变准则,研究了爆炸应力波作用下新混凝土衬砌的破坏模式及安全振动速度。结果表明,入射应力波及衬砌中的折射波对衬砌的破坏作用最主要,折射波在衬砌内产生的反、折射波将降低衬砌中的应力,具有卸载效果;混凝土衬砌的安全振动速度随着龄期的增大而增大,P波入射时,安全振动速度随着入射角的增大而增大,入射角较大时,胶结面容易发生剪切破坏,较小时,容易发生拉伸破坏,SV波入射时,较小角度下容易发生剪切破坏,较大角度时容易发生拉伸破坏;围岩弹性模量越高,衬砌的安全振动速度越小。     

冲刷桥墩基础的HHT振动频谱特性分析与判读研究

尝试以移动车流为振动源,在明显受冲刷与未受冲刷桥墩系统上方一侧布设讯号传感器,接收桥梁下部结构因振动源所产生的振动讯号。再以经验模态分解法EMD与希尔伯特黄转换HHT作为讯号分解与振动频谱解析的方法,针对桥墩系统（桥梁下部结构）的Y方向振动讯号进行解析与判读。于EMD分析过程中导入间断性准则进行振动讯号分解。针对完整（未裸露）桥墩系统（P1）,与跨河冲刷桥段桥墩系统（P2）的Y方向,进行频谱解析与比对探讨。由研究结果可知,EMD讯号分解过程可能产生模态混合现象,导致HHT图形中产生频率损失以及可判读区间缩短的现象。借由间断性准则的应用,可有效解决这些问题,获得判读性较高的HHT频谱图与边际频谱图。此外,HHT频谱图所得结果证实,桥墩系统Y方向的瞬时频率变化为一频宽带,其区间范围约在1～4 Hz之间。而由边际频谱图中可知,P1、P2的频率能量分别集中于2.5 Hz与1.7 Hz,可知P2的整体系统刚度小于P1。据此,明显受冲刷桥墩系统P2,其承载能力弱化的现象得以由HHT方法中明显呈现。     

锦屏一级水电站左岸坝肩边坡施工期稳定分析

锦屏一级水电站左岸坝肩高陡边坡地质条件复杂、主要发育有f5、f8、f42-9断层,煌斑岩脉X,顺坡卸荷裂隙及深部裂缝等结构面,其中断层f42-9、煌斑岩脉X和深部裂缝SL44-1组合形成的左岸坝头拉裂变形体,在边坡施工期的稳定性问题突出,特别是当边坡开挖至1 780 m高程时拉裂变形体前沿剪出口阻滑岩体被完全挖除,f42-9断层在开挖坡面出露,拉裂变形体稳定条件恶化,稳定性急剧下降。采用考虑开挖过程的有限元强度折减法分析开挖边坡在施工期的稳定性,并提出了一种新的边坡失稳判据：通过滑体内关键部位测点水平位移随折减系数的变化率,即水平位移增量与强度折减系数增量之比值 和强度折减系数 关系曲线的突变点来确定边坡的临界失稳状态。通过与其他边坡失稳判据相比较,验证了该判据的合理性和实用性,同时基于该判据计算了边坡开挖加固至不同高程下的稳定安全系数,合理评价左坝肩边坡施工期的整体稳定与安全性。     

基于SLFT边坡双折减系数强度失稳判据

基于滑移线场理论（SLFT）,提出了一种边坡双折减系数强度折减法失稳判据：当由滑移线场理论计算得到的极限状态下的边坡坡面（简称\     

江苏省小麦播种期的干旱研究及应用

本文确定了以降水量和蒸散量为参数的干旱指数计算方法,以小麦适宜播种期及前后各一句的时间范围内综合考虑各旬干旱指数,确定干旱指标,并讨论了江苏省小麦播种期干旱的发生规律。     

Morphological differentiation characteristics and classification criteria of lunar surface relief amplitude

Lunar landforms are the results of geological and geomorphic processes on the lunar surface. It is very important to identify the types of lunar landforms. Geomorphology is the scientific study of the origin and evolution of morphological landforms on planetary surfaces. Elevation and relief amplitude are the most commonly used geomorphic indices in geomorphological classification studies. Previous studies have determined the elevation classification criteria of the lunar surface. In this paper,...