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Bending of a Tapered Rod: Modern Application and Experimental Test of elastica Theory

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World Journal of Mechanics 2018年第7期8卷 272-300页

作者： M. P. Silverman Joseph Farrah Department of Physics Trinity College Hartford USA

A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to micro-electromechanical sensors. Although the bending of a uniform rod is well studied and gives rise to mathematical shapes described by elliptic integrals, no exact closed form solution to the nonlinear differential equations of static equilibrium is known for the deflection of a tapered rod. We report in this paper a comprehensive numerical analysis and experimental test of the exact theory of bending deformation of a tapered rod. Given the rod geometry and elastic modulus, the theory yields virtually all the geometric and physical features that an analyst, experimenter, or instrument designer might want as a function of impressed load, such as the exact curve of deformation (termed the elastica), maximum tip displacement, maximum tip deflection angle, distribution of curvature, and distribution of bending moment. Applied experimentally, the theory permits rapid estimation of the elastic modulus of a rod, which is not easily obtainable by other means. We have tested the theory by photographing the shapes of a set of flexible rods of different lengths and tapers subject to a range of impressed loads and using digital image analysis to extract the coordinates of the elastica curves. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the equations of equilibrium or neglect tapering of the rod. Agreement between the measured deflection curves and the exact theoretical predictions was excellent in all but several cases. In these exceptional cases, the nature of the anomalies provided important information regarding the deviation of the rods from an ideal Euler-Bernoulli cantilever, which thereby permitted us to model the deformation of the rods more accurately.

关键词： elastica Deflection of Tapered Cantilever Euler-Bernoulli Beam Elastic Modulus Flexure Formula

Peeling of Magneto-responsive Beams with Large Deformation Mediated by the Parallel Magnetic Field

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Acta Mechanica Solida Sinica 2024年第3期37卷 457-467页

作者： Gongqi Cao Zhangna Xue Shiyang Liu Yuchen Jin Jianlin Liu College of Pipeline and Civil Engineering China University of Petroleum(East China)Qingdao266580China

Elasto-capillarity phenomena are prevalent in various industrial fields such as mechanical engineering,material science,aerospace,soft robotics,and *** this study,two typical peeling processes of slender beams driven by the parallel magnetic field are investigated based on experimental and theoretical *** first is the adhesion of two parallel beams,and the second is the self-folding of a long *** these two cases,the energy variation method on the elastica is used,and then,the governing equations and transversality boundary conditions are *** is shown that the analytical solutions are in excellent agreement with the experimental *** effects of magnetic induction intensity,distance,and surface tension on the deflection curve and peeling length of the elastica are fully *** results are instrumental in accurately regulating elasto-capillarity in structures and provide insights for the engineering design of programmable microstructures on surfaces,microsensors,and bionic robots.

关键词： Magneto-responsive beam Adhesion and peeling Self-folding elastica Transversality condition

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

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Acta Mechanica Sinica 2007年第3期23卷 287-296页

作者： BoonchaiPhungpaingam SomchaiChucheepsakul Department of Civil Engineering Faculty of EngineeringKing Mongkut's University of Technology ThonburiPracha-utid RdBangmodToongkruBangkok10140Thailand

This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The non-linear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge-Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance （10^-5）. The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.

关键词： Postbuckling elastica Concentrated moment Elliptic integral Shooting method

AN ANALYSIS OF THE LARGE DEFLECTION OF AN ELASTIC-PLASTIC CANTILEVER SUBJECTED TO AN INCLINED CONCENTRATED FORCE

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Applied Mathematics and Mechanics(English Edition) 1991年第6期12卷 547-555页

作者：栾丰余同希 Dept. of Mechanics Peking University Beijing

Based on the Plastica theory (see ref. [12]), the large deflection of an elastic-perfectly plastic cantilever subjected to an inclined concentrated force at its tip, before the unloading in the plastic region occurs, is analyzed in this paper. The emphasis of the analysis is put on the effects of the angle of inclination of the concentrated force upon the deformed shape, the load-deflection relationship and the length of the plastic region. Both analytical and computed results are given.

关键词： Large deflection inclined concentrated force elastic-plastic cantilever elastica Plastica

THE COMPLETE PROCESS OF LARGE ELASTIC-PLASTIC DEFLECTION OF A CANTILEVER

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Acta Mechanica Sinica 1986年第4期2卷 333-347页

作者： Wu Xiaoqiang Yu Tongxi (Department of Mechanics,Peking University) Department of Mechanics Peking University Beijing China

An extension of the elastica theory is developed to study the large deflection of anelastic-perfectly plastic horizontal cantilever beam subjected to a vertical concentrated force at *** entire process is divided into four stages:Ⅰ.elastic in the whole cantilever;Ⅱ.loading anddeveloping of the plastic region;Ⅲ.unloading in the plastic region;and Ⅳ.reverse *** stages Ⅰ and Ⅱ are presented in a closed form.A combination of closed-form solution andnumerical integration is presented for stage Ⅲ.Finally,stage Ⅳ is qualitatively *** are given and compared with those from small-deflection theory and from the elastica theory.

关键词： large deflection elastic-plastic cantilever beam flexible bar geometrical and material non-linearities elastica Plastica

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印度橡皮榕与紫叶橡皮榕叶片的解剖结构

仲恺农业工程学院学报 2010年第3期23卷 31-35,45页

作者：张艳霞朱必娟肖德兴仲恺农业工程学院生命科学学院广东广州510225

采用光学显微镜和扫描电镜,对印度橡皮榕（Ficus elastica Roxb. ex Hornem.）与其栽培变种紫叶橡皮榕（F. elastica cv. Decora Burgundy）的叶片解剖结构特征进行了对比观察和分析.结果表明,印度橡皮榕、紫叶橡皮榕均为异面叶,上、下... 详细信息

采用光学显微镜和扫描电镜,对印度橡皮榕（Ficus elastica Roxb. ex Hornem.）与其栽培变种紫叶橡皮榕（F. elastica cv. Decora Burgundy）的叶片解剖结构特征进行了对比观察和分析.结果表明,印度橡皮榕、紫叶橡皮榕均为异面叶,上、下表皮均由3~4层细胞组成复表皮,并具有较厚的角质层,上表皮由排列紧密、形状较规则的细胞组成;栅栏组织排列紧密,由2~3层长柱形细胞组成,海绵组织排列较疏松;气孔器仅分布于下表皮,且极度下陷.印度橡皮榕、紫叶橡皮榕在叶片厚度、表皮厚度、栅栏组织厚度、气孔密度等方面均存在差异,上述指标对比结果表明紫叶橡皮榕抗旱性强于印度橡皮榕.

关键词：印度橡皮榕(Ficuselastica Roxb. ex Hornem.) 紫叶橡皮榕(F. elastica cv. Decora Burgundy) 解剖结构

顶点受到转角控制的屈曲三角形结构的失稳回跳研究

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Acta Mechanica Sinica 2023年第5期39卷 96-104页

作者：王强邹海林邓子辰 Department of Engineering Mechanics Northwestern Polytechnical UniversityXi’an710072China Beijing Electro-Mechanical Engineering Institute Beijing100074China

本文研究了包含三个固支-固支弹性梁的屈曲三角形结构的失稳回跳问题.当屈曲三角形结构的一个顶点受到转角控制时,可以利用三个固支-固支弹性梁的刚度-曲率曲线得到屈曲三角形结构的变形与弯矩-转角曲线.根据弯矩-转角曲线的三个分支,... 详细信息

本文研究了包含三个固支-固支弹性梁的屈曲三角形结构的失稳回跳问题.当屈曲三角形结构的一个顶点受到转角控制时,可以利用三个固支-固支弹性梁的刚度-曲率曲线得到屈曲三角形结构的变形与弯矩-转角曲线.根据弯矩-转角曲线的三个分支,可以确定失稳回跳对应的临界点,包括两个鞍点分叉点和两个边界点.失稳回跳后,移除外部弯矩,屈曲三角形结构仍然是对称的.此外,根据弯矩-转角曲线,手性形状是不稳定的.本文成果有助于研究多约束弹性梁的失稳回跳机制.

关键词： Triangular structure Snap-back elastica Buckling

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基于柔性梁的太阳能电池板清洁力研究

可再生能源 2018年第9期36卷 1290-1295页

作者：杭茂尧高德东王珊青海大学机械工程学院青海西宁810016

太阳能电池板表面的积灰严重降低了光伏系统的发电效率。在宏观力理论分析的基础上,假设灰尘颗粒为刚性小球,刷丝为柔性梁,分析推导出柔性梁与灰尘颗粒之间的力学模型。依据elastica理论,分析出柔性梁形变与受力之间关系的超越方程,结... 详细信息

太阳能电池板表面的积灰严重降低了光伏系统的发电效率。在宏观力理论分析的基础上,假设灰尘颗粒为刚性小球,刷丝为柔性梁,分析推导出柔性梁与灰尘颗粒之间的力学模型。依据elastica理论,分析出柔性梁形变与受力之间关系的超越方程,结合椭圆积分表计算柔性梁的受力情况,得到了灰尘颗粒对刷丝的正压力与刷丝末端偏转角、轴向位移、径向位移三者之间的关系。根据作用力与反作用力,刷丝对灰尘颗粒的正压力等于灰尘颗粒对刷丝的正压力,依据刷丝对灰尘颗粒的正压力计算刷丝对灰尘颗粒的清洁力。结果表明:刷子对太阳能电池板作用力以及清洁力大小均取决于刷丝直径和轴向位移,刷丝直径越大,清洁力越大;刷丝长度为20 mm,轴向位移在14～18 mm时,清洁力较大,清洁效果好。通过刷丝对灰尘颗粒清洁力大小的研究,可以为太阳能电池板机械除尘提供理论依据。

关键词：柔性梁太阳能电池板灰尘颗粒清洁 elastica 椭圆积分表

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基于柔性梁模型下的太阳能电池板清洁研究

基于柔性梁模型下的太阳能电池板清洁研究

作者：杭茂尧青海大学

学位级别：硕士

在能源结构问题日益突出的今天,光伏产业越发受到重视,但是太阳能电池板积灰严重制约光伏产业发展,迫切需要解决积灰问题。机械除尘通过机械结构施加机械力来去除电池板表面的积灰,目前对机械除尘研究主要通过分析清洁前后的发电效率来... 详细信息

在能源结构问题日益突出的今天,光伏产业越发受到重视,但是太阳能电池板积灰严重制约光伏产业发展,迫切需要解决积灰问题。机械除尘通过机械结构施加机械力来去除电池板表面的积灰,目前对机械除尘研究主要通过分析清洁前后的发电效率来判断清洁效果,没有分析清洁过程中清洁力的大小。本文在宏观力理论分析的基础上,假设积灰颗粒为刚性小球,尼龙6刷丝为柔性梁,分析推导出柔性梁与颗粒的相互作用模型。依据elastica理论,分析出单个柔性梁形变与力之间关系的超越方程,结合椭圆积分表计算了柔性梁所受压力,得出了灰尘对刷丝的压力与刷丝末端偏转角、径向位移与轴向位移三者之间的关系。根据作用力与反作用力,刷丝对灰尘颗粒的正压力等于灰尘颗粒对刷丝的正压力,依据刷丝对灰尘颗粒的正压力计算刷丝对灰尘颗粒的清洁力。搭建清洁机构用于验证分析的准确性,实验验证分为清洁力实验验证和清洁效果实验验证。清洁力实验时在太阳能电池板表面粘贴压力传感器,用于检测在清洁过程中刷子对太阳能电池板的压力,将采集的压力数据分析验证刷丝于灰尘颗粒模型。将理论与实验综合分析,两者误差为9.2676%。清洁效果实验验证时用刷子去除太阳能电池板表面的积灰,分析验证清洁前后的发电功率,将清洁效果于理论分析清洁力相验证。结果表明:刷子对太阳能电池板作用力取决于刷丝直径和轴向位移,清洁力大小也取决于这两者,刷丝直径越大清洁力越大;刷丝长度为20 mm,轴向位移在13-17 mm时,清洁力较大,清洁效果好。通过刷丝对灰尘颗粒清洁力大小的研究,可以为太阳能电池板机械除尘提供理论依据。

关键词：柔性梁太阳能电池板灰尘颗粒清洁 elastica 椭圆积分表