Classical dynamics in deformed spaces
非线性动力学入门-西安交通大学教师个人主页
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另一方面梁的轴向应变的表达式也会因变形大小的不同而采用不同的表达式比如小变形时应变而当考虑大变形时可能采用的应变表达式就是进而得到的梁的振动方程将会是一个含有高度非线性项的偏微分方程组
非线性动力学入门
张新华
西安交通大学 工程力学系 2011 年 07 月
前 言
─1687 年，牛顿(Isaac Newton, 1643 ～ 1727)发表了《自然哲学之数学原 理》(Mathematical Principles of Natural Philosophy)，标志着经典力学(亦即牛 顿力学)的正式诞生。牛顿力学主要研究自由质点系的宏观运动规律。 ─1788 年，拉格朗日(Joseph Louis Lagrange, 1736 ～ 1813)发表了分析力 学教程(Analytical Mechanics)，标志着拉格朗日力学的诞生。Lagrange 力学属 于分析力学的主要内容之一，在位形空间中研究带有约束的质点系动力学。 ─1833 年，哈密尔顿(William Rowen Hamilton, 1805 ～ 1865)对 Lagrange 力学进行了改造，引进了相空间(2n 维空间)，对系统内在的对称性(辛对称， Symplectic)进行了刻画。狭义上的哈密尔顿力学只适用于保守系统，而广义 的哈密尔顿力学在适用于非保守系统。哈密尔顿力学也属于分析力学的主要 组成部分。在此后发展起来的量子力学中 Hamilton 力学发挥着巨大的作用。 目前在天体力学、计算 Hamilton 力学，量子力学，甚至弹性力学(即所谓的 辛弹性力学)中哈密尔顿力学依然发挥着重要作用。 ─1927 年，Birkhoff(George David Birkhoff, 1844 ～ 1944)发表了“动力系 统”(Dynamical Systems)，标志着 Birkhoff 动力学的正式问世。Birkhoff 动力 学建立了研究非完整力学的框架。 ─1892 ～ 1899， 彭加莱(Henri Poincaré, 1854 ～ 1912)发表了三卷本的“天 体力学中的新方法”(New Methods of Celestial Mechanics)，系统性地提出了 研究动力学系统的定性方法，即几何方法。经典力学的目标之一就是设法求 得系统的解析解，而 Poincaré意识到对于大多数非线性系统而言，求其解析 解是不可能的，而必须发展新的研究方法。他超越了他的时代，极富远见地 预测到了非线性系统混沌现象(系统的解对初始条件具有极端敏感依赖性)的 存在。更为重要的是，Poincaré开创了研究非线性动力系统的几何方法，当之 无愧地被誉为非线性科学之父，其影响是划时代的。 ─1892 年，李亚普诺夫(Aleksandr Mikhailovich Lyapunov, 1857 ～ 1918)在 他的博士论文“运动稳定性的一般问题”(General problem of the stability of motion )中，系统地探讨了非线性动力学系统的稳定性问题。他提出了两种研 究稳定性的方法：李亚普诺夫第一方法(间接方法)和李亚普诺夫第二方法(直 接方法)。他从代数角度出发，对动力学系统的研究开创了一个崭新的领域。 彭加莱与李亚普诺夫，前者从几何角度，后者从代数角度，开拓了非线 性科学的研究疆域和研究手段。 ─1963 年，Lorenz(Edward Norton Lorenz, 1917 ～ 2008)发表了“确定性 非周期流”(Deterministic Nonperiodic Flow)的论文，认为大气系统的性态对 初值极为敏感，从而导致准确的长期天气预报是不可能的。该文标志着人类 首次借助于计算机发现了混沌(Chaos)现象的存在。 ─1757 年，欧拉(Leonhard Euler, 1707 ～ 1783)发表了压杆稳定性的论 文，首次探讨了力学系统的分岔现象。作为分岔理论重要分支的突变理 论(Catastrophe Theory)则主要由法国数学家托姆(René Thom, 1923 ～ 2002)于 上个世纪 60 年代创立，由齐曼(Christopher Zeeman，1925 ～)在 70 年代大力 推广普及。 ─1834 年，英国的罗素(John Scott Russell, 1808 ～ 1882)骑着马在 Union 运河上散步时，发现了现在称之为孤立波(又称作孤波，Solitary wave)的 i
学数学的必看GTM经典著作下载三
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206 Problems in Analytic Number Theory,M.Ram Murty 207 Algebraic Graph Theory,Godsil,Royle(代数图论)镜像下载(4062KB，英文版，DJVU格式，支持关键词检索，点击打开下载页面，支持迅雷、快车下载)Godsil是加拿大滑铁卢大学的教授，代数组合图论的权威。
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208 Analysis for Applied Mathematics,Ward Cheney 209 AShort Course on Spectral Theory,William Arveson(谱理论简明教程)镜像下载(4366KB，英文版，DJVU格式，支持关键词检索，点击打开下载页面，支持迅雷、快车下载)本书给读者提供谱论-被称之为解决算子理论基本问题的基本工具，并主要计算了无限维空间特别是希尔伯特空间算子的谱。
2003-孪晶Al
DOI: 10.1126/science.1083727, 1275 (2003);300 Science et al.Mingwei Chen Deformation Twinning in Nanocrystalline AluminumThis copy is for your personal, non-commercial use only.clicking here.colleagues, clients, or customers by , you can order high-quality copies for your If you wish to distribute this article to othershere.following the guidelines can be obtained by Permission to republish or repurpose articles or portions of articles): April 22, 2014 (this information is current as of The following resources related to this article are available online at/content/300/5623/1275.full.html version of this article at:including high-resolution figures, can be found in the online Updated information and services, /content/300/5623/1275.full.html#related found at:can be related to this article A list of selected additional articles on the Science Web sites /content/300/5623/1275.full.html#ref-list-1, 3 of which can be accessed free:cites 24 articles This article 293 article(s) on the ISI Web of Science cited by This article has been /content/300/5623/1275.full.html#related-urls 10 articles hosted by HighWire Press; see:cited by This article has been/cgi/collection/mat_sci Materials Sciencesubject collections:This article appears in the following registered trademark of AAAS.is a Science 2003 by the American Association for the Advancement of Science; all rights reserved. The title Copyright American Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by the Science o n A p r i l 22, 2014w w w .s c i e n c e m a g .o r g D o w n l o a d e d f r o mthat are normally FCC,such as gold,to maintain local,close-packed arrangements that are compatible with the constraints imposed by the surrounding environment. This result establishes a physically useful connection between dislocations and grain boundaries,the two most fundamental mi-crostructural elements underpinning the be-havior of polycrystalline materials.References and Notes1.A.P.Sutton,R.W.Balluffi,Interfaces in CrystallineMaterials(Clarendon Press,Oxford,1995).2.J.Schiøtz,F.D.Di Tolla,K.W.Jacobsen,Nature391,561(1998).3.H.Van Swygenhoven,M.Spaczer,A.Caro,Acta Ma-ter.47,3117(1999).4.S.X.McFadden,R.S.Mishra,R.Z.Valiev,A.P.Zhilyaev,A.K.Mukherjee,Nature398,684(1999).5.H.Van Swygenhoven,Science296,66(2002).6.V.Yamakov,D.Wolf,S.R.Phillpot,A.K.Mukherjee,H.Gleiter,Nature Mater.1,45(2002).7.Y.Wang,M.Chen,F.Zhou,E.Ma,Nature419,912(2002).8.J.D.Rittner,D.N.Seidman,K.L.Merkle,Phys.Rev.B53,R4241(1996).9.W.Krakow,D.A.Smith,Ultramicroscopy22,47(1987).10.K.L.Merkle,Colloq.de Phys.51,C1-251(1990).11.F.Ernst et al.,Phys.Rev.Lett.69,620(1992).12.U.Wolf,F.Ernst,T.Muschik,M.W.Finnis,H.F.Fischmeister,Phil.Mag.A66,991(1992).13.K.L.Merkle,J.Phys.Chem.Solids55,991(1994).14.J.D.Rittner,D.N.Seidman,Phys.Rev.B54,6999(1996).15.D.L.Medlin,G.H.Campbell,C B.Carter,Acta Mater.46,5135(1998).16.D.L.Medlin,S.M.Foiles,D.Cohen,Acta Mater.49,3689(2001).17.T.Radetic,nc¸on,U.Dahmen,Phys.Rev.Lett.89,85502(2002).18.M.L.Jenkins,Philos.Mag.26,747(1972).19.T.J.Balk,K.J.Hemker,Phil.Mag.A81,1507(2001).20.R.D.Heidenreich,W.Shockley,in Report of a Con-ference on the Strength of Solids(Physical Society,London,1948),pp.57–75.21.F.C.Frank,J.H.van der Merwe,Proc.R.Soc.LondonA198,205(1949).22.N.Thompson,Proc.Phys.Soc.B66,481(1953).23.G.H.Campbell,D.K.Chan,D.L.Medlin,J.E.Angelo,C.B.Carter,Scripta Mater.35,837(1996).24.Here,we specify the boundary using the CSL notation(25),where⌺refers to the reciprocal density ofcoincident sites of two interpenetrating lattices re-lated by a rotation,.25.H.Grimmer,W.Bollmann,D.H.Warrington,ActaCrystallogr.A30,197(1974).26.D.G.Brandon,Acta Metall.14,1479(1966).27.C.B.Carter,D.L.Medlin,J.E.Angelo,ls,Mater.Sci.Forum207-209,209(1996).28.This work is supported by the U.S.Department ofEnergy,in part by the Office of Basic Energy Sciences,Division of Materials Sciences,under contract DE-AC04-94AL85000.26February2003;accepted15April2003Deformation Twinning inNanocrystalline AluminumMingwei Chen,1*En Ma,2Kevin J.Hemker,1Hongwei Sheng,2Yinmin Wang,2Xuemei Cheng3We report transmission electron microscope observations that provide evi-dence of deformation twinning in plastically deformed nanocrystalline alumi-num.The presence of these twins is directly related to the nanocrystalline structure,because they are not observed in coarse-grained pure aluminum.We propose a dislocation-based model to explain the preference for deformation twins and stacking faults in nanocrystalline materials.These results underscore a transition from deformation mechanisms controlled by normal slip to those controlled by partial dislocation activity when grain size decreases to tens of nanometers,and they have implications for interpreting the unusual mechanical behavior of nanocrystalline materials.The extremely high strength and hardness of nanocrystalline materials relative to their coarse-grained counterparts suggest that normal dislocation activity—the dominant plastic deformation mode of ductile coarse-grained materials—is suppressed in nanocrys-talline grains.The resulting high strength of nanocrystalline materials may give rise to unique plastic deformation mechanisms that are not seen in coarse-grained materials(1–7).Twinning in aluminum,which has been suggested by recent molecular dynamics (MD)simulations(7,8),is an interesting case.Deformation twins have been observed in face-centered cubic(fcc)metals,such as copper and nickel(9–11),when deformed at subambient temperatures and/or high strain rates,but deformation twins have not been experimentally confirmed in single or poly-crystalline pure aluminum(7,8,11,12),evenwhen shock-loaded at low temperatures(13,14).The appearance of Shockley partial dis-locations at a crack tip of aluminum has beenregarded as an indication of deformationtwinning(15),but supporting evidence forthis claim has not been forthcoming.Nanocrystalline aluminum films withthickness ofϳ200andϳ400nm were pre-pared by physical vapor deposition of purealuminum(99.9%)onto oxidized(001)sili-con and sodium chloride substrates that werecooled with liquid nitrogen.The depositionwas performed at a pressure ofϳ5ϫ10Ϫ8torr and a deposition rate of5to10nm/min.To suppress columnar grain structures,weinterrupted the deposition for1min atϳ20-nm thickness intervals during film growth.Texture was negligible in these films,as in-dicated by x-ray diffraction analysis.The as-deposited films had grain sizes in the range of10to35nm(Fig.1A).High-resolution trans-mission electron microscope(HRTEM)ob-servations with a point-to-point resolution of0.2nm revealed that most of the nanograinswere separated by high-angle boundaries andthat no grain boundary phases were present inthese samples.Occasionally,growth twinswere observed with the twin boundary ap-pearing as a perfectly flat interface(Fig.1B).Such sharp interfaces have been previously1Department of Mechanical Engineering,2Depart-ment of Materials Science and Engineering,3Depart-ment of Physics and Astronomy,Johns Hopkins Uni-versity,Baltimore,MD21218,USA.*To whom correspondence should be addressed.E-mail:mwchen@Fig.1.Microstructure of as-prepared nanocrys-talline aluminum.(A)Bright-field TEM micro-graphsh owing nanocrystalline grains withsizesranging from10to35nm.No dislocations ordeformation twins can be seen.(B)HRTEMimage of a growth͚3{111}twin boundary(marked by a white arrow).R E P O R T S SCIENCE VOL30023MAY20031275observed in deposited or annealed aluminum and have been referred to as the ͚3{111}twin boundary (16,17).The nanocrystalline aluminum films were deformed by microindentation and by manual grinding to allow the introduction of large plastic strains and to facilitate TEM observa-tions of the deformation defects.We chose these methods because the thin films cannot be tested in compression,and they fail with little plastic deformation in tension (18).The indented samples were thinned from the back using tripod polishing,dimpling,and finally low-temperature ion milling.The insert in Fig.2A shows a bright-field TEM micro-graph of a micro-indent with the fourfoldgeometry.No cracks can be seen,even at the corners of the indent,indicating that the for-mation of the indent is fully plastic.Around the edges of the indent,planar defects with two parallel flat boundaries were seen in a number of grains and were identified as de-formation twins.The width and orientation of the twin bands vary from grain to grain.The twinning seems to occur preferentially in smaller grains and propagates across the en-tire grain.An example is shown in the HRTEM micrograph of Fig.2B,together with a fast Fourier transform (FFT)pattern.The deformation twin can be recognized by the mirror symmetry between the band and the matrix in the atomic resolution image,in which the bright spots correspond to individ-ual atomic columns.The twin boundaries are determined to be parallel to one set of the {111}planes.In addition,the HRTEM image reveals that the deformation twin boundaries are several atomic planes thick,caused by twinning dislocations that are tilted relative to the electron beam and not clearly imaged.A large number of dislocations rather than de-formation twinning were found in coarse-grained aluminum deformed under the same indenter as shown in Fig.2C,confirming the grain-size dependence of the twinning mechanism.To rule out the possibility that deforma-tion twinning can only be induced under the very high pressures generated by a diamond indenter (19),we used a different deforma-tion scheme.Freestanding nanocrystalline aluminum films were released from sodium chloride substrates and manually ground into small fragments with an agate mortar and pestle in pure methanol for ϳ1min.This procedure is generally used to crush materials for TEM observations by applying complex stresses.These stresses are much lower than those generated by a sharp diamond indenter.The original film thickness was ϳ200nm,whereas the thickness of the fragments used for subsequent TEM observations was mea-sured to be less than 100nm by electron energy loss spectroscopy.The reduction in thickness appears to result from large plastic deformation,and the bright-field TEM mi-crograph in Fig.3A illustrates the features of the heavily deformed nanocrystalline alumi-num.The high density of planar defects that is shown in this figure is not seen in the as-deposited samples and is found to appear preferentially in smaller grains.Some of the defects were confirmed to be deformation twins by HRTEM.In these cases,multiple narrow twins often resided in a single grain,and the twinning planes were determined to be of the {111}type (Fig.3B).The FFT pattern of Fig.3B demonstrates the twin re-lationship among the bands and the matrix (Fig.3C).The multiple twins are highlighted in the Fourier-filtered image (Fig.3D)foraFig.2.(A )TEM micrographof deformation twins around an indent in nanocrystalline alu-minum.The inset shows the indent with the fourfold geometry.(B )HRTEM micrograph showing a deformation twin in (A)with parallel boundaries.This atomic resolution image cor-responds to the [110]direction and illustrates the mirror symmetry between the twin and the matrix.The morphological feature of the grain boundary at the twin band indicates that the twinning results in plastic deformation.The inset shows a FFT pattern confirming the twin relationship between the band and the matrix.(C )Dislocations witha grain boundary (GB)around an indent in coarse-grained pure alumi-num and no evidence of deformationtwinning.Fig.3.TEM micrographs of nanocrystalline alu-minum deformed by manually grinding.(A )A high density of planar defects in a bright-field image.Suchdefects are not observed in th e as-deposited nanocrystalline aluminum.(B )Multiple deformation twins and stacking faults in a [110]-oriented nanograin.(C )The FFT pat-tern showing the twin relationship among the narrow bands and matrix.(D )A Fourier-filtered image from inside the white box in (B)for a close-up view of the deformation twins (T)and stacking faults (S).R E P O R T S23MAY 2003VOL 300SCIENCE 1276selected region in Fig.3B.Stacking faults,dislocations,and microbands were also ob-served in the samples.The TEM observations provide direct experimental evidence confirming the MD predictions of twinning during plastic defor-mation of nanocrystalline aluminum (7,8).Regarding the twinning mechanism of fcc materials,several models have been proposed in which deformation twins are created by stacking faults led by 1/6Ͻ112ϾShockley partial dislocations (20–23).The preference for twinning and stacking fault formation in nanocrystalline grains can be understood by comparing the critical shear stress needed to nucleate a perfect dislocation (1/2Ͻ110Ͼ)with an approximation of the source size equal to the grain size (D ),N ,with that required to initiate the Shockley partial (1/6Ͻ112Ͼ)twinning dislocation to generate stacking faults and deformation twins,p .Such a comparison can be made using for-mulations given by classical dislocation the-ory (22,23),whereN ϭ2␣b ND(1)andP ϭ2␣b P D ϩ␥b P(2)Here is the shear modulus (ϳ35GPa for aluminum),␥is the stacking fault energy [104to 142mJ/m 2for aluminum (7)],and b N and b P are the magnitudes of the Burgers vectors of the perfect dislocation and the Shockley partial dis-location,respectively.The parameter ␣reflects the character of the dislocation [␣ϭ0.5and 1.5for edge and screw dislocations,respectively (23)]and contains the scaling factor between the length of the dislocation source and the grain size.The grain boundaries are taken as dislocation sources,as predicted by computer simulations for nanocrystalline grains (6–8,12,24).When the grain size becomes smaller than a critical value,D c ,determined by equating Eqs.1and 2D c ϭ2␣(b N Ϫb P )b P␥(3)p becomes smaller than N .Taking ␣ϭ1,the estimated D c is approximately 10to 15nm for aluminum.For simplicity,this model for nanocrystalline materials does not include the influence of elastic anisotropy,the small Peierls-Nabarro stress,localized stress con-centrations,and the interactions of disloca-tions with grain boundaries.Nevertheless,the predictions given by this simple model shed light on the experimentally observed trends in several ways.First,twinning becomes a pre-ferred deformation mode in aluminum with a grain size on the order of 10nm,which is consistent with our HRTEM observations of deformation twins in the grains with sizes ofϳ10to 20nm.The model also provides a physical explanation of the preferential gen-eration of partial dislocations,which results in the formation of stacking faults and defor-mation twins in nanocrystalline grains as sug-gested by computer simulations (6–8,12,24).Second,the p estimated from Eq.2is much higher than N in large aluminum grains,which is in agreement with the dom-inance of the normal dislocation plasticity in conventional aluminum.Third,for other fcc materials with lower ␥and higher ,the D c values are much larger and can be used to interpret the observation that deformation and growth twins are found in nanocrystalline copper and nickel (9,25–27)at D c values that are more than twice that of aluminum.Fourth,the generation of twin interfaces and stacking faults offers an alternative interpre-tation to dislocation pile-up at grain bound-aries to explain the continuous grain-size strengthening,as suggested by Eq.2,and the strain hardening of nanocrystalline materials (25–27).References and Notes1.J.R.Weertman et al .,Mater.Res.Soc.Bull.24,44(1999).2.I.A.Ovid’ko,Science 295,2386(2002).3.H.Van Swygenhoven,Science 296,66(2002).4.S.X.McFadden,R.S.Mishra,R.Z.Valiev,A.P.Zhilyaev,A.K.Mukherjee,Nature 398,684(1999).5.M.Murayama,J.M.Howe,H.Hidaka,S.Takaki,Science 295,2433(2002).6.J.Schiøtz,F.D.Di Tolla,K.W.Jacobsen,Nature 391,561(1998).7.V.Yamakov,D.Wolf,S.R.Phillpot,A.K.Mukherjee,H.Gleiter,Nature Mater.1,45(2002).8.V.Yamakov,D.Wolf,S.R.Phillpot,H.Gleiter,Acta Mater.50,5005(2002).9.J.Y.Huang,Y.K.Wu,H.Q.Ye,Acta Mater.44,1211(1996).10.T.H.Blewitt,R.R.Coltman,J.K.Redman,J.Appl.Phys.28,651(1957).11.J.A.Venables,in Deformation Twinning ,R.E.Reed-Hill,J.P.Hirth,H.C.Rogers,Eds.(Gordon &Breach,New York,1964),pp.77–116.12.K.W.Jacobsen,J.Schiøtz,Nature Mater.1,15(2002).13.G.T.Gray III,Acta Metall.36,1745(1988).14.G.T.Gray III,J.C.Huang,Mater.Sci.Eng.A145,21(1991).15.R.C.Pond,L.M.F.Garcia-Garcia,Inst.Phys.Conf.Ser.No.61(1981),p.495.16.M.Shamzuzzoha,P.A.Deymier,D.J.Smith,Philos.Mag.A 64,245(1991).17.Y.Champion,J.Bigot,Nanostruct.Mater.10,1097(1998).18.M.A.Haque,M.T.A.Saif,Scripta Mater.47,863(2002).19.E.B.Tadmor,ler,R.Phillips,M.Ortiz,J.Mater.Res.14,2233(1999).20.J.A.Venables,Philos.Mag.6,379(1961).21.J.B.Cohen,J.R.Weertman,Acta Metall .11,996(1963).gerlo ¨f,J.Castaing,P.Pirouz,A.H.Heuer,Philos.Mag.A 82,2841(2002).23.J.P.Hirth,J.Lothe,Theory of Dislocations (KriegerPublishing,Malabar,UK,ed.2,1992).24.H.Van Swygenhoven,P.M.Derlet,A.Hasnaoui,Phys.Rev.B 66,024101(2002).25.C.J.Youngdahl,J.R.Weertman,R.C.Hugo,H.H.Kung,Scripta Mater.44,1475(2001).26.F.Ebrahimi,G.R.Bourne,M.S.Kelly,T.E.Matthews,Nanostructured Materials 11,343(1999).27.K.S.Kumar,S.Suresh,M.F.Chisholm,J.A.Horton,P.Wang,Acta Mater.51,387(2003).28.We thank F.R.N.Nabarro for valuable discussion.This work was partially supported by the NSF (Divi-sion of Material Researchgrant 210215)and was conducted at the Electron Microscopy Center at Johns Hopkins University.The Johns Hopkins Electron Microscopy Facility is made possible by grants from the W.M.Keck Foundation and NSF.21February 2003;accepted 4April 2003Published online 24April 2003;10.1126/science.1083727Include this information when citing this paper.n -Type Conducting CdSe Nanocrystal SolidsDong Yu,Congjun Wang,Philippe Guyot-Sionnest*A bottleneck limiting the widespread application of semiconductor nanocrystal solids is their poor conductivity.We report that the conductivity of thin films of n -type CdSe nanocrystals increases by many orders of magnitude as the occupation of the first two electronic shells,1S e and 1P e ,increases,either by potassium or electrochemical doping.Around half-filling of the 1S e shell,a peak in the conductivity is observed,indicating shell-to-shell transport.Introducing conjugated ligands between nanocrystals increases the conductivities of these states to ϳ10Ϫ2siemens per centimeter.During the past decade,it has become appar-ent that solids of monodispersed nanocrystals provide the opportunity for developing mate-rials with novel properties (1,2).In particu-lar,semiconductor nanocrystals (3)offergreat promise for fabricating optoelectrical devices (4–6).In these “artificial atoms,”the inorganic cores allow precise tuning of the discrete electronic states by size confinement.To stabilize against sintering,retain solubili-ty,and maintain good optical properties,the surfaces are capped by organic ligands,as in the prototypical CdSe system (7).However,these ligands and traps on the nanocrystal surfaces are thought to inhibit electronicJames Franck Institute,University of Chicago,5640SouthEllis Avenue,Ch icago,IL 60637,USA.*To whom correspondence should be addressed.E-mail:pgs@R E P O R T S SCIENCE VOL 30023MAY 20031277。
lecture9
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
4
16.61 Aerospace Dynamics
Spring 2003
This analysis extends to rigid body case
F4
F1
Assume system given small, arbitrary displacements in all directions. Called virtual displacements - No passage of time - Applied forces remain constant
δ W = ∑ F jδ x j
j =1
3N
Note use of δx and not dx. Note: • There is no passage of time • The forces remain constant.
In vector form:
δ W = ∑ Fi • δ ri
i =1
δ W = ∑ ( R i + Fi − mi"" ri ) • δ ri = 0
i =1
N
⇒ ∑ ( Fi − mi"" ri ) • δ ri = 0
i =1
N
! So we can apply all of the previous results to the dynamics problem as well.
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
RecurDyn
IntegratedMulti-DisciplinaryDynamicsConcept of RecurDyn Free input and outputPre-Defined Process AutomationControl System ModelingApplication Development Environment
Robust Optimization(for Six Sigma Design)
MBD/FEA(MFBD)Eigen-SolverLarge Problem SolverMulti-CPU SolverTribologyNew Contact FrictionLubricationGeneral CFD
HighPerformanceInterface for CAE
Plug-in ToolkitsControlCustomization
Macro & API
DesignOptimizationAutoDesign
ProcessNetCoLinkApplication Toolkits
Communicators
INTEGRATED MULTI-DISCIPLINARY DYNAMICS
Example:Windshield Wiper Simulation
Flexible LinksGear Creation (with Contacts)
Customized Model BuildingMotor / Controller
RecurDyn is Computer-Aided Engineering (CAE) software that is focused on Multi-Body Dynamics (MBD) while also offering multi-physics solutions. The technology behind RecurDyn has superior calculation efficiency because it is based on a recursive formulation. The core is Multi- Flexible Body Dynamics (MFBD) that combines a multibody dynamics solver and a nonlinear finite element method solver at the integrator level. RecurDyn supports other multi-disciplinary CAE technologies such as control system modeling, nonlinear system-level optimization, and CFD analysis. RecurDyn provides the efficiency gains of process automation through a wide variety of application toolkits. Users can create their own custom applications using the bundled ProcessNet environment.
研究生机械工程专业英语考试必背单词
研究生机械工程专业英语考试必背单词在研究生机械工程专业的学习中，英语考试是必不可缺的一部分。
为了顺利通过考试，背诵和掌握一些重要的专业词汇是至关重要的。
本文将为大家介绍一些研究生机械工程专业英语考试必背单词，助您在考试中取得好成绩。
1. Mechanics（力学）Mechanics is a branch of physics that deals with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effect on their motion.（力学是物理学的一个分支，研究物体在受力或变位作用下的行为及其对运动的影响。
）2. Dynamics（动力学）Dynamics is the branch of mechanics that deals with the motion of objects and the forces that cause the motion.（动力学是研究物体运动及引起运动的力的力学分支。
）3. Kinematics（运动学）Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion.（运动学是研究物体运动而不考虑引起运动的力的力学分支。
）4. Thermodynamics（热力学）Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy.（热力学是研究热能与其他形式能量之间关系的物理学分支。
《机械工程专业英语教程》课文翻译
Lesson 1 力学的基本概念1、词汇：statics [stætiks] 静力学；dynamics动力学；constraint约束；magnetic [mæɡ＇netik]有磁性的；external [eks＇tə:nl] 外面的, 外部的；meshing啮合；follower从动件；magnitude [＇mæɡnitju:d] 大小；intensity强度，应力；non-coincident [kəu＇insidənt]不重合；parallel [＇pærəlel]平行；intuitive 直观的；substance物质；proportional [prə＇pɔ:ʃənəl]比例的；resist抵抗，对抗；celestial [si＇lestjəl]天空的；product乘积；particle质点；elastic [i＇læstik]弹性；deformed变形的；strain拉力；uniform全都相同的；velocity[vi＇lɔsiti]速度；scalar[＇skeilə]标量；vector[＇vektə]矢量；displacement代替；momentum [məu＇mentəm]动量；2、词组make up of由……组成；if not要不，不然；even through即使，纵然；Lesson 2 力和力的作用效果1、词汇：machine 机器；mechanism机构；movable活动的；given 规定的，给定的，已知的；perform执行；application 施用；produce引起，导致；stress压力；applied施加的；individual单独的；muscular [＇mʌskjulə]]力臂；gravity[ɡrævti]重力；stretch伸展，拉紧，延伸；tensile[tensail]拉力；tension张力，拉力；squeeze挤；compressive 有压力的，压缩的；torsional扭转的；torque转矩；twist扭，转动；molecule [m likju:l]分子的；slide滑动; 滑行；slip滑，溜；one another 互相；shear剪切；independently独立地，自立地；beam梁；compress压；revolve (使)旋转；exert [iɡ＇zə:t]用力，尽力，运用，发挥，施加；principle原则, 原理，准则，规范；spin使…旋转；screw螺丝钉；thread螺纹；2、词组a number of 许多；deal with 涉及，处理；result from由什么引起；prevent from阻止，防止；tends to 朝某个方向；in combination结合；fly apart飞散；3、译文：任何机器或机构的研究表明每一种机构都是由许多可动的零件组成。
二阶非完整性水下机器人的路径跟踪与协调控制研究
华中科技大学博士学位论文二阶非完整性水下机器人的路径跟踪与协调控制研究姓名：向先波申请学位级别：博士专业：控制理论与控制工程指导教师：黄心汉2010-06-02华 中 科 技 大 学 博 士 学 位 论 文摘要21世纪是海洋的世纪，日益增长的海洋开发需求，以及自主式水下机器人(Autonomous Underwater Vehicle, AUV)技术的快速发展，使得欠驱动水下机器人已成为AUV家族的重要成员和海洋探测开发的重要工具。
欠驱动水下机器人属于二阶非完整性系统，且具有本质上的强非线性，其运动控制问题非常具有挑战性。
因此，二阶非完整性欠驱动水下机器人的非线性运动控制研究具有重要的理论价值和实用价值，已经成为水下机器人控制领域的热点问题。
本文深入分析了国内外关于二阶非完整性水下机器人运动控制方面的研究成果，并总结了所存在的问题，在此基础上完成了二阶非完整性水下机器人的路径跟踪控制，并对基于路径跟踪的机器人编队协调控制进行了研究。
主要研究内容如下：首先，简要介绍了非完整性系统，比较了路径跟踪与轨迹跟踪各自的特点，选择了建立路径跟踪误差动力学的合适方案。
以一阶非完整性移动机器人为研究对象，深入探讨机器人路径跟踪控制的设计方法，在Serret-Frenet移动坐标系下，提出基于虚拟参考目标点“协作式”路径跟踪的控制策略，完成具有“时空解耦”特性的控制器设计。
其次，建立水下机器人运动学、动力学模型，基于一阶与二阶非完整性系统运动控制的部分相似性，借鉴一阶非完整性机器人路径跟踪的控制策略，实现二阶非完整性水下机器人的三自由度全状态欠驱动路径跟踪控制器，并解除了普遍存在的角速度持续激励的路径跟踪限制条件。
通过引入建立在Serret-Frenet移动坐标系且可随路径曲率自适应变化的视线角，将其由传统的船舶直线路径跟踪，扩展到水下机器人曲线参考路径的跟踪控制。
采用“视线角导航”和“位置”分步设计，简化了路径跟踪控制律。
拉格朗日力学
人物简介
拉格朗日约瑟夫·拉格朗日（Joseph-Louis Lagrange，1736~1813）全名为约瑟夫·路易斯·拉格朗日， 法国著名数学家、物理学家。他在数学、力学和天文学三个学科领域中都有历史性的贡献，其中尤以数学方面的 成就最为突出。
坐标
拉格朗日力学在矢量力学中，约束的存在体现于作用于系统的约束力。约束力引入额外的未知量，通常使问 题变得更为复杂。但若能选取适当的s个完全满足约束条件的独立坐标，则约束不再出现于问题中，只需要求解关 于s个未知变量的方程，使问题得以大大简化。这样的s个坐标不再局限于各质点的位置坐标，而可以是任何能描 述系统的几何参量，因此称为“广义坐标”。
谢谢观看
拉格朗日力学
拉格朗日建立的一种分析力学
01 定义
03 坐标 05 概念拓展
目录
02 人物简介 04 拉格朗日方程
拉格朗日力学，分析力学中的一种，由拉格朗日在1788年建立，是对经典力学的一种的新的数学表述。经典 力学，最初的表述形式由牛顿建立，它着重分析位移，速度，加速度，力等矢量间的关系，又称为矢量力学。拉 格朗日引入了广义坐标的概念，运用达朗贝尔原理，得到和牛顿第二定律等价的拉格朗日方程。但拉格朗日方程 具有更普遍的意义，适用范围更广泛。并且，选取恰当的广义坐标，可以使拉格朗日方程，于1788年由约瑟夫·拉格朗日所创立。拉格朗日力学是对经典力学的一 种的新的理论表述，着重于数学解析的方法，是分析力学的重要组成部分。
拉格朗日力学力学系统由一组坐标来描述。比如一个质点的运动（在笛卡尔坐标系中）由x，y，z三个坐标 来描述。一般的，N个质点组成的力学系统由3N个坐标来描述。力学系统中常常存在着各种约束，使得这3N个坐 标并不都是独立的。力学系统的独立坐标的个数称之为自由度。对于N个质点组成的力学系统，若存在m个约束， 则系统的自由度为
《经典力学》札记
《经典力学》札记本文写作，有点模仿梅凤翔老先生的《分析力学及理论力学札记》的味道。
他的文章在一些刊物上发表，我不这么做。
这些短小的讨论，又类似周国平的著作。
他的散文有个特点，即都是一些短小的类似格言一样的刊物。
我写这些片段式讨论和总结，尽量做到言之有物。
我希望这些可以作为经典力学教材的补充和总结。
我的主要参考教材是Landau的《力学》和Arnold的《经典力学的数学方法》。
札记指的是读书时摘记的要点和心得，故取名札记，是很合适的。
01经典力学的重要性经典力学是理论力学或者理论物理的第一门课，学好经典力学是学好物理的关键。
理论力学的思想——最小作用量原理等，会用在后续几乎所有的物理课程中。
这门课有一条非常清晰的主线，即这个原理，理清这根脉络，是学好理论力学的根本。
如果学好了，这门课的所有内容可以写在半页纸上。
但是这门课学起来会非常困难，有几个原因：第一，大二学生刚学完微积分和微分方程，但不熟练；第二，大量的近似，让人眼花缭乱；第三，大量的抽象的公式推导，很多公式的物理意义也难以理解（或缺乏明确的图像）。
那么，如何才能学好呢？两个建议。
第一，多证明一些基本的结论，多推导；第二，多用Mathematica 的 NDSolve 和 Plot 等（也可以用其它软件）做数值模拟，碰到不懂的，就先模拟画图看看。
这样可以对这些问题积累第一手的感觉，对提高自己的理解力和直觉能力有很大帮助。
物理图像很重要。
在物理中，每个公式应该都有明确的物理意义和图像。
在讲课的时候，我会尽量注意这些细节。
学生们在学习的时候也要做到这一点。
第一个作业题是数值模拟双摆 (double pendulum) 的运动过程。
学生需要用Mathematica 求解其运动方程，并观察简谐振动和混沌过程。
这个摆是最简单的混沌摆。
在WSU物理系门口，就有这样的一个混沌摆，我以前每次进楼，都会摇一下它并观察其复杂的运动过程。
积累一些生活经验对于理解物理的运动会非常有帮助。