关于在中学里研究棱柱截面所遇到的函数

在中学里研究棱柱問題时,一般要解一系列的习题,其中需要求出巴知棱柱的某种确定的截面面积的习題。这类习题,对于和数学其它各科间的联系的建立,提供了丰富的材料。 1.例如,我們研究H.A.格拉哥列夫的教科书里的一个习题。底为正方形的长方体,被通过底的棱的平面所截,如果截面与长方体的軸的交点距离底面1.5m,而底面的边长为4m,求这截面的面积。答案:20m~2。长方体的底可以根据题目所給出的条件而作出“长方体”和“底是正方形”由“长方体”和“底是边长为4m”的条件确定了没有上界的“柱子”(它的側棱是垂直底平面的射线)。截面的位置则由题目的条件所确定(图1①)。     

环境规制强度影响能源效率的门槛效应研究——基于PSTR模型

利用非径向-SBM和CCR模型测算了我国30个省市(暂不包括西藏和港澳台地区)2007-2016年间的环境规制强度以及能源效率,构建面板平滑转移回归模型(PSTR),分析了环境规制强度对我国能源效率的连续非线性变化的影响.研究结果显示,全国环境规制强度对我国能源效率存在非线性效应,二者之间呈倒“U”型关系.环境规制强度小于门槛值0.6376时,环境规制对能源效率产生创新补偿效应,当环境规制强度高于门槛值0.6376时,环境规制对能源效率产生遵循成本效应.而我国东西部地区的环境规制强度与地区能源效率间却存在着与之相反的“U”型效应.     

基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论,将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量,描述为位移分量偏导数的表达式.根据最小势能原理,推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程,得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算,研究了Timoshenko微梁的挠度、转角和应力的尺寸效应,分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时,Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高,表现出明显的尺寸效应;当截面高度与材料特征长度的比值大于10时,Timoshenko微梁的刚度与强度均趋于稳定,尺寸效应可以忽略;材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素,Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型,能有效描述Timoshenko微梁的力学行为及尺寸效应,可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.     

考虑表面效应时孔边均布径向多裂纹Ⅲ型断裂力学分析

理论研究了纳米尺度孔边均布径向多裂纹的Ⅲ型断裂性能.基于Gurtin-Murdoch表面弹性理论和保角映射技术,获得了孔和裂纹应力场的解析解,给出了裂纹尖端应力强度因子的闭合解.基于解答分析了应力强度因子的尺寸效应,讨论了裂纹数量、裂纹/孔径比和缺陷表面性能对应力强度因子的影响.结果表明:当孔和裂纹尺寸在纳米量级时,无量纲应力强度因子具有显著的尺寸效应;应力强度因子随裂纹数量的变化规律受裂纹/孔径比的影响;裂纹/孔径比对应力强度因子的影响受到缺陷表面性能的制约,同时表面性能对应力强度因子的影响也受限于裂纹/孔径比;表面效应对应力强度因子的影响与裂纹数量无关.     

铁电颗粒中的尺寸效应

利用Landau唯象理论研究了球形铁电颗粒的表面效应和尺寸效应.计算了不同尺寸下二级相变和一级相变铁电体的Curie温度和自发极化强度.给出了铁电颗粒的外推长度与尺寸的关系.理论计算结果与实验符合.     

由余弦定理所想起的

余弦定理除了能“由三角形的两边长及其夹角求第三边长”及“由三角形的三边长求三内角”以外,还能解“已知三角形的两条边长及其中一边的对角求第三边”. [例1]如图1,作△ABC,使BC=4,CA=3,∠B=π/6,并求AB边的长. 作法(1)作线段BC=4; (2)以C为圆心,作半径为3的圆;     

中考求周长问题赏析

例1(2011年德州市中考题)图1是一个边长为1的等边三角形和一个菱形的组合图形,菱形边长为等边三角形边长的一半,以此为基本单位,可以拼成一个形状相同但尺寸更大的图形(如图2),依此规律继续拼下去(如图3),……,则第n个图形的周长是().     

绕轴线自转悬臂梁的局部限制失稳分析

建立了任意位置限位器约束下绕轴线自转悬臂梁的非线性模型.采用Ritz法分析系统的稳定性,获得了限位器无摩擦情形下系统的限制失稳临界值、分岔模式、后屈曲解以及致稳限位器的最佳配置位置.采用有限元法对失稳临界值与致稳限位器的优化位置进行了验证,获得了一致的结果.进一步分析了限位器夹紧力和支撑力摩擦效应对系统稳定性的影响规律,获得了有益的认识.研究表明,在限位器约束下,绕轴线自转悬臂梁存在临界转速,当转速超过临界值时,梁的零挠度平衡位置将发生叉式分岔而失去稳定性;限位器夹紧力摩擦效应将使失稳后的系统在转速回复时出现明显的滞后效应,以比失稳临界值更低的转速回到原平衡位置;绕轴线自转悬臂梁系统致稳限位器的最优配置位置在梁长距固支端的78%左右等.这些成果对提升绕轴线自转悬臂梁的局部限制失稳性能的认识和指导限位器的配置具有实际意义.     

勾股定理验证方法赏析

同学们在学习了勾股定理的知识后,已经知道勾股定理是描述直角三角形三边之间数量关系的一个重要定理,体现了数与形的和谐统一,是数形结合思想的典范.课本上已经进行了生动的验证,下面再列举几种通俗易懂的验证方法,供同学们参考. 方法一:旋转法 用两个全等的直角三角形纸板拼成如图1(a),使两条直角边a、b在同一直线上.将△ABC绕着A点旋转到△AFG的位置,△BDE绕着E点旋转到△FHE的位置,由图1(b)中不难看出:由以b为边长的正方形与以a为边长的正方形面积之和等于以c为边长的正方形面积,从而得出a2+b2=c2.     

“扩中”与“减贫”视角下的新生代农民分化影响因素研究

为进一步辨别“扩中”和“减贫”路径,形塑与建构未来农民分化图景,研究以新生代农民为研究对象,根据2016中国劳动力动态调查数据(CLDS),利用广义多层线性模型(HGLM)跨层次研究新生代农民分化影响因素.研究结果表明:个体层面来说,个人人力资本投入如健康、教育、周工作时间、基本医疗保险与养老保险、流动性与非农生产等预示着较高的进入中等收入群体(相对于低收入群体)的发生比,社会支持系统如稳定的婚姻关系、父辈非农化代际传承与人际支持等则显著增加了进入高收入群体(相对于中等收入群体)的发生比.宏观层次上,城镇化率和地区社会保障支出对新生代农民分化有显著影响.还就性别的分化效应、离土与离乡、不同的分化路径、社会保障的“减贫”与“扩中”效应、城镇化率-教育的跨层次交互效应等问题进行了进一步分析与讨论.研究的现实意义在于指出,在“扩中”与“减贫”的背景下,政府应继续推进城镇化建设,带动农村居民收入增长;提高地区社会保障支出,强化社会保障的“减贫”和“扩中”效应;多手段综合提高新生代农民社会公平感;关爱女性,确保女性享有平等的受教育机会和获得更多的人际支持.     

高烈度下双裂缝主控结构面危岩的断裂破坏机制分析

灰岩地区危岩具有典型类砌体结构特征,结构面发育复杂,其破坏本质是主控结构面在荷载作用下的断裂扩展失稳问题.基于地貌演化分析了灰岩地带陡崖复杂结构面形成与演化,获得了双裂缝主控结构面危岩的地质模型.构建了此类危岩的力学模型和断裂力学模型,按照最大周向应力准则,基于危岩自重、裂隙水压力及地震荷载作用,构建了双裂缝主控结构面危岩的断裂稳定系数表达式.通过案例分析验证了其合理性;各烈度对应的断裂稳定系数随贯通度增加而下降,基本地震烈度为Ⅷ度和Ⅸ度时,危岩处于不稳定状态;危岩稳定性受到主控结构面的裂隙贯通度与地震诱发的耦合作用影响,临界失稳贯通度随地震烈度的提高而降低;Ⅶ度地震的临界失稳贯通度理论值为25.8%,略小于贯通度实际值27.7%,略偏安全,其建立的断裂稳定系数表达式具有较好的适用性.研究结果可为推动此类危岩合理化防治提供重要的理论支撑.     

考虑地震力方向的倾倒式危岩可靠度分析

侧向卸荷作用导致高陡边坡发育大量危岩体,危岩体在降雨、地震作用下易发生失稳破坏,判断其失稳的概率对危岩防治具有重要意义.该文以倾倒式危岩体为例,建立了考虑地震力作用方向下最危险方向的物理力学模型,利用函数极值理论建立了最危险地震力作用方向的表达式,结合可靠度理论建立了倾倒式危岩体可靠度指标、失稳概率表达式及判断标准.通过对重庆南川金佛山危岩体案例的分析表明:工况1的最危险地震力作用方向与水平方向的偏转角θ在5°范围内,工况2的最危险地震力作用方向与水平方向的偏转角θ在10°左右;危岩体最危险作用方向不是一个固定角,其值与危岩体形态、裂隙水作用力大小、岩腔深度等有关.当主控结构面裂隙长度较小时,最危险地震力作用方向与水平夹角很小,随主控结构面裂隙长度增大,最危险地震力作用方向与水平夹角显著增大;危岩体失稳概率随主控结构面裂隙长度增加而增大,工况2较工况1增大幅度更明显.该研究成果对危岩的防灾减灾具有重要意义.     

Diamond-Fluoroplastic Composites for Abrasive Tools

Composite materials based on polytetrafluoroethylene (PTFE) and natural technical diamond powders from Yakutia diamond deposits are developed. It is shown that the compositions based on PTFE and a technical diamond powder at a content of up to 60 wt.%, due to their good physicomechanical characteristics, low friction coefficient, and good wetting of diamond particles by polymer, make is possible to create abrasive tools for polishing and grinding hard metals and semiprecious and precious stones with high serviceability and operational life combined with a considerable increase in the quality of treated surfaces and operational stability of the tools. It is found that PTFE, being a more elastic and softer matrix than the traditional ones, exhibits a self-sharpening effect of diamond grains upon grinding hard surfaces, when the grains go deep into the elastic matrix, the matrix wears out, and the working part of the tool becomes enriched with the diamond powder. These conclusions are confirmed by electron microscopic investigations. It is shown that the introduction of ultradisperse fillings (up to 2 wt.%) into such compositions allows us to improve the characteristics of abrasive tools considerably, especially for grinding hard semiprecious stones. The physicomechanical and frictional characteristics of the compositions and specific examples of their application in the jewelry industry and in stone working are discussed.     

壁面马赫数分布规律可控的新型内收缩基准流场设计方法

根据有旋特征线理论,设计出了沿程马赫数下降规律可控的轴对称基准流场,分析了基准流场的几何参数(前缘压缩角及中心体半径)的影响规律,发现选取较小的前缘压缩角和中心体半径有利于得到性能优良的基准流场;然后在设计状态Ma=6时研究了三种典型的马赫数下降规律对这种轴对称流场性能的影响。最后考虑了粘性的影响,并进行了粘性修正探索,结果表明,采用附面层位移厚度修正方法后,基准流场的壁面压力分布和无粘情况吻合良好。      

On the Influence of High-Frequency Excitation on Systems with Dynamic Friction

By imposing high-frequency vibrations to a system, the characteristics of dry friction for low sliding velocities can be smoothed and, consequently, undesired friction induced phenomena such as stick-slip motion can be quenched. Many studies have been published so far, most of them focussing on the reduction of friction between metal surfaces and using classical Coulomb friction models. Within this contribution the effect of high-frequency excitation on dry friction taking into account dynamic friction models will be discussed. To this end, the friction law suggested by Dahl is used and the resulting friction characteristics are compared to those obtained for the classical Coulomb friction model. Using Dahl's friction model, a reduction of the smoothing effect is observed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Friction properties of highly elastic materials at high contact pressures

A newly designed high-pressure tribometer has been employed to investigate the pressure dependence of the friction force of SKN-40 crosslinked butadiene-nitrile rubber in contact with a steel surface on the pressure range to 1200 kgf/cm² (20°C). Over the entire range of contact pressures the friction process is molecular-kinetic in nature and characterized by a linear dependence of the friction force on the logarithm of the sliding velocity. In the region of normal pressures up to 200 kgf/cm², where the effect of pressure on the friction force reduces to the formation of the actual contact area, the friction constant (proportionality factor relating the friction force and the actual contact area) is practically independent of the pressure. At pressures above 200–300 kgf/cm² the increase in the friction force at fixed actual contact area is attributable to the effect of pressure on the friction constant. The nature of this effect is related not with an increase in the chain-surface interaction energy (the activation energy does not increase), but with an increase in the forces of adhesion owing to the greater number of polymer chain-steel surface contacts on the actual contact area (increase in contact density).Lenin Moscow State Pedagogical Institute, Laboratory for Problems of Polymer Physics. Translated from Mekhanika Polimerov, No. 1, pp. 140–146, January–February, 1971.     

弹流润滑条件下表面形貌对摩擦噪声的影响

研究了弹流润滑状态下表面形貌对摩擦噪声的影响.通过激光微加工方法在金属圆盘试件表面上制造了两种沟槽型织构表面形貌,在双盘摩擦磨损试验机上对不同表面形貌进行了摩擦噪声和摩擦特性试验,分析了线接触弹流润滑条件下不同工况和表面形貌影响摩擦噪声的机理,并结合有限元分析对结论加以验证.结果表明:载荷和转速的变化对线接触摩擦噪声有明显影响,由于线接触副工作在部分膜弹流润滑状态下,所以摩擦噪声的特性与干摩擦时类似,摩擦因数较大的表面会辐射出更强的摩擦噪声;特定结构的表面形貌能改善表面润滑特性,有效降低摩擦噪声声压级,沟槽型织构的存在可以打断接触区域应力分布,减轻接触面微凸体碰撞作用,从而降低了表面自激振动,同时合适的表面形貌结构也有利于润滑油膜的形成,减小了系统的摩擦能量,达到降低摩擦噪声的效果.     

Numerical simulation of seismic activity by the grid-characteristic method

Seismic activity in homogeneous and layered enclosing rock masses is studied. A numerical mechanical-mathematical model of a hypocenter is proposed that describes the whole range of elastic perturbations propagating from the hypocenter. Synthetic beachball plots computed for various fault plane orientations are compared with the analytical solution in the case of homogeneous rock. A detailed analysis of wave patterns and synthetic seismograms is performed to compare seismic activities in homogeneous and layered enclosing rock masses. The influence exerted by individual components of a seismic perturbation on the stability of quarry walls is analyzed. The grid-characteristic method is used on three-dimensional parallelepipedal and curvilinear structured grids with boundary conditions set on the boundaries of the integration domain and with well-defined contact conditions specified in explicit form.     

Block s‐step Krylov iterative methods

Block (including s‐step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s‐step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right‐hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s‐step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Frictional Contact of Rubber on rough Surfaces

The aim of this work is to investigate the frictional behaviour of rubber on rough rigid surfaces. Rubber friction is composed of hysteretic and adhesional effects. The main focus lies on the hysteresis theory which motivates rubber friction with the internal damping inside the viscoelastic material. It is necessary to perform a multiscale analysis to regard the roughness of the surface on all important length scales. The fractal surface is approximated by a superposition of harmonic functions. The simulations will start at the lowest wavelength and proceed to larger scales by replacing the smaller roughness with the ascertained friction law. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)