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International Journal of Mechanical Sciences 43(2001)2103–2123Quasi-static axial compression of thin-walled circular aluminium tubesS.R.Guillow a ,G.Lu a ;∗,R.H.Grzebieta ba School of Engineering and Science,Swinburne University of Technology,PO Box 218,Hawthorn,Victoria 3122,Australiab Department of Civil Engineering,Monash University,Clayton,Victoria 3168,AustraliaReceived5October 2000;receivedin revisedform 26March 2001AbstractThis paper presents further experimental investigations into axial compression of thin-walledcircular tubes,a classical problem studied for several decades.A total of 70quasi-static tests were conducted on circular 6060aluminium tubes in the T5,as-receivedcond ition.The range of D=t considered was expanded over previous studies to D=t =10–450.Collapse modes were observed for L=D 610anda mod e classiÿcation chart developed.The average crush force,F AV ,was non-d imensionalisedandan empirical formula establishedas F AV =M P =72:3(D=t )0:32.It was foundthat test results for both axi-symmetric and non-symmetric modes lie on a single prehensive comparisons have been made between existing theories andour test results for F AV .This has revealedsome shortcomings,suggesting that further theoretical work may be required.It was found that the ratio of F MAX =F AV increasedsubstantially with an increase in the D=t ratio.The e ect of ÿlling aluminium tubes with di erent density polyurethane foam was also brie y examined.?2001Elsevier Science Ltd.All rights reserved.Keywords:Axial compression;Circular tube;Foam;Plastic collapse;Thin-walledtubes0.IntroductionThe behaviour of thin-walledmetal tubes subjectedto axial compression has been stud iedfor many years.Such tubes are frequently usedas impact energy absorbers andReid[1]has pre-sented a general review of deformation mechanisms.Fig.1shows a typical force–displacement curve for quasi-static loading.Generally speaking,the axial load rises until a ÿrst buckle is formedat a characteristic maximum force value,F MAX .This initial buckling behaviour is well ∗Corresponding author.Fax:+61-3-9214-8264.E-mail address:glu@.au (G.Lu).0020-7403/01/$-see front matter ?2001Elsevier Science Ltd.All rights reserved.PII:S 0020-7403(01)00031-52104S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123NomenclatureD average diameterF AV average axial forceF MAX maximum axial force forÿrst peakg acceleration due to gravityH half-wavelength of foldL lengthM P full plastic bending moment of tube wall per unit lengthm geometric eccentricity factor—i.e.ratio of outward s foldlength to total foldlength N number of circumferential lobes(or corners)in non-symmetrical bucklingR average radiust wall thickness of tubeV Vickers hardness number(kg=mm2)e e ective crushing distancef density of foam0 ow stress0:20.2%proof stressult ultimate tensile stressFig.1.Typical load–de ection curve for an axially loaded thin-walled metal tube which collapsed by progressive folding.S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–21232105Fig.2.Examples of various collapse modes for thin-walled circular6060-T5aluminium tubes under axial load-ing(more examples shown in Fig.11):(a)axi-symmetric mode(D=97:9mm;t=1:9mm;L=196mm);(b)non-symmetric mode(D=96:5mm;t=0:54mm;L=386mm);(c)mixedmod e(D=97:5mm; t=1:5mm;L=350mm).known and will not be studied in depth here.Thereafter,depending on geometrical parameters such as the ratios of D=t(diameter=thickness)and L=D(length=diameter)and also on material properties,there are a variety of possible modes of collapse.Generally,collapse involves plastic2106S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123Fig.3.Schematic axial view of non-symmetric or diamond collapse mode.Two cases are shown,N=3and4 circumferential lobes.buckling andthe formation of progressive fold s(whether axi-symmetric or non-symmetric).The formation of these folds causes the characteristic uctuation in the axial force shown in Fig.1. This plastic collapse behaviour is of primary interest in this paper.Experimentally the following modes of collapse have been observed and Fig.2shows some typical examples:(i)axi-symmetric concertina bellowing,(ii)non-symmetric buckling(also known as diamond or Yoshimura mode),with a variable number of circumferential lobes or corners(refer to Fig.3),(iii)mixed mode(combination of the two previous modes),(iv)Euler or global buckling;and(v)other(simple compression,single fold s,etc.).Research on circular tubes in the past has generally concentratedon annealedaluminium or steel tubes with D=t ratios between10and150.It is common ind ustrial practice to use aluminium alloys in the heat treatedas-receivedcond ition,but little research appears to have addressed this particular case.Moreover,Gupta and Gupta[2]have identiÿed metal temper as one of the signiÿcant factors in determining behaviour.Hence,it was decided to undertake an experimental program to extendthe range of research up to approximately D=t=450andto test aluminium alloy tubes which were in the heat treatedas-receivedcond ition.This work is of potential application in civil,mechanical,marine andaeronautical engineeringÿeld s.1.Review of previous studiesThe following section summarises the available literature on the plastic collapse behaviour of thin-walledcircular metal tubes subject to quasi-static axial load ing.It is arrangedbroad ly in a chronological order.Theÿrst signiÿcant work to address the mechanics underlying the observed behaviour of axially load edthin-walledtubes was by Alexand er[3].He proposeda simple mod el for the axi-symmetric foldpattern(refer to Fig.4)basedon experiments with metal tubes of D=t= 29–89.At a global level,external work done was equated with internal work from bending at three stationary plastic hinges andcircumferential stretching of the metal between the hinges.S.R.Guillow et al./International Journal of Mechanical Sciences 43(2001)2103–21232107Fig.4.Axi-symmetric collapse mechanism assumedby Alexand er [3].Thus the following theoretical equation was obtainedfor average crush force (axi-symmetric folds):F AV =K o t1:5√D;(1)where K is a constant and o is the ow stress.Also,plastic half-wavelength,H (refer to Fig.4)was determined as follows:H =C √Dt;(2)where C is a constant.The experimental results observedby Alexand er were generally in agreement with the above two equations.Although simple,this model seems to re ect the und erlying physical processes involvedandmany subsequent researchers have usedit as a starting point.Pugsley andMacaulay [4]were among the ÿrst researchers to consid er the non-symmetric folding mode,their study being largely empirical.Johnson et al.[5]attempted to develop a theory for the non-symmetric mode based on the actual geometry of folding,with the tube material at the mid-surface being considered inextensional.Hence they were able to develop equations to predict average axial crush force,F AV .However,agreement between their model andtest results for P.V.C.tubes was not particularly good .In 1978Magee andThornton [6]cond ucteda review of previous work by researchers who hadcond uctedaxial crushing tests on circular metal tubes.By consid ering these collectedd ata they d evelopeda number of empirical equations which involvedthe speciÿc ultimate tensile strength of the metal.Andrews et al.[7]conducted a comprehensive series of tests on annealed aluminium alloy tubes covering a wide range of D=t (4–60)and L=D (0:2–8:8).Consequently,they developed a collapse mode classiÿcation chart which predicted the mode of collapse for any given D=t and L=D combination.2108S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123Fig.5.Axi-symmetric mod el usedby Abramowicz andJones[8,9].H is the half-wavelength of the fold. Abramowicz andJones[8,9]cond uctedaxial compression tests on a range of thin-walledcir-cular andsquare steel tubes.They analytically consid eredboth axi-symmetric andnon-symmetric modes.Abramowicz introduced the important concept of e ective crushing distance, e(refer to Fig.5),where a foldconsistedof two equal rad ii segments of length H,curvedin opposite d irections andthe material hadÿnite thickness.For axi-symmetric folds,Abramowicz and Jones[9]developed the following equation in1986 (anda similar one in1984[8]):F AV M P =[25:23D=t+15:09][0:86−0:568t=D];(3)where M P= o(t2=4):For non-symmetric fold s,in1984[8]and1986[9]Abramowicz andJones commencedwith two di erent starting relationships.Taking into account e ective crushing distance,material strain rate,etc.,resultedin two d i erent equations for average crush force.The simple relation-ship d evelopedin1984[8]appliedregard less of the number of lobes:F AV M P =86:14Dt0:33:(4)However,thisÿnding appears to have developed from work carried out by Wierzbicki and Abramowicz[10]on rectangular rather than circular tubes.On the other hand,the relationshipS.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–21232109Fig.6.Collapse mechanism assumedby Grzebieta[12]for axi-symmetric mod e. Abramowicz andJones d erivedin1986[9]was of the formF AV M P =A N1Dt+A N2;(5)where A N1and A N2are constants which were a function of the number of lobes.For further details the reader is directed to this reference.In Refs.[8,9],Abramowicz andJones observedthat reasonable agreement existedbetween pred ictions of average crush load s basedon the above notedequations andtheir experimental results for steel tubes with D=t=9–65.In a subsequent work,Abramowicz andJones[11] reportedon further tests andsummarisedtheirÿnd ings for both static andd ynamic load ing cases in two failure mode maps,adding to the previous work by Andrews et al.[7].Gupta andGupta[2]performeda series of quasi-static axial compression tests on thin-walled aluminium andmildsteel circular tubes in both the annealedandas-receivedcond itions.They combinedall results andd evelopedempirical equations for the average crushing force in terms of the Vickers hardness and D=t.Grzebieta[12–14]useda strip methodto analyse both axi-symmetric andnon-symmetric folding modes.He equated external work done with internal energy from horizontal,inclined and travelling plastic hinges as well as stretching of the metal,to produce equations for determining the instantaneous forces involved.Unfortunately,these equations do not yield simple expressions for determining the average crush force.Grzebieta’s collapse mechanism model for axi-symmetric mode(refer to Fig.6)was a mod-iÿcation of Alexand er’s.A foldconsistedof three equal lengths,two of which were curves of equal rad ius andthe thirda straight line segment.For the non-symmetric mod e Grzebieta analysedthe fold s as a half-d iamondmechanism.Grzebieta carriedout static andd ynamic tests on steel tubes with D=t=30–300.2110S.R.Guillow et al./International Journal of Mechanical Sciences 43(2001)2103–2123Fig.7.Axi-symmetric mod el usedby Wierzbicki [15]andSingace et al.[16,17].Wierzbicki et al.[15]introduced a new model for the axi-symmetric collapse mechanism (shown simpliÿed in Fig.7)which allows for both inwards and outwards radial displacement.The geometry is governedby an arbitrary geometric eccentricity factor,m ,which is deÿned as the ratio of outwardfoldlength to total foldlength.By consid ering energy rate equations Wierzbicki et al.[15]were able to develop equations for not only determining average crush load but also a representative load–de ection history.The latter helped explain the experi-mental observation that sometimes there are two force peaks during the formation of a single fold.Singace et al.[16,17]extended upon the previous work by Wierzbicki et al..For the axi-symmetric mode,they considered a global energy balance leading to an implicit equation for m ,which when solvedgave a theoretical constant value of m =0:65.In their secondpa-per [17],Singace et al.reportedgoodexperimental agreement with the pred ictedvalue of 0:ter,Singace et al.[18]re-considered the eccentricity factor,m ,for the non-symmetric mode case.From the results of experiments on a small range of circular metal tubes they de-duced that the factor m was (surprisingly)approximately constant at m =0:65for this mode also.The equations developed by Singace et al.[17,18]for calculating average axial crush force are as follows.For axi-symmetric mode:F AV M P =22:27 D t+5:632:(6)For non-symmetric mode:F AV M P =− 3N +2 2N tan 2N D t :(7)One particular problem of interpretation arises with most theoretical equations developed for non-symmetric mode collapse,for example Eq.(7).They require a knowledge of the number of lobes,N ,at a given D=t ratio.We have not foundany publishedequation entirely satisfactory in determining N .S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–21232111Fig.8.Experimental set-up.With regardto energy absorption,it has been suggestedthatÿlling metal tubes with low-density polyurethane foam(to provide wall stability)may be preferable to increasing the wall thickness.Early investigations into the e ectiveness of this foamÿlling methodwere cond ucted by Thornton[19]and Lampinen and Jeryan[20].Reid,Reddy and Gray[21]have conducted experiments on the axial compression of thin-walledrectangular metal tubes which hadbeen ÿlledwith foam.Red d y andWall[22]subsequently testedfoamÿlledcircular aluminium alloy cans.Academic opinion appears to be divided about the relative beneÿts of foamÿlling versus increasing the wall thickness.2.Test procedure and material propertiesA series of approximately70axial compression tests were conducted under quasi-static condi-tions.Tests were carriedout on a SHIMAD ZU universal testing machine which appliedthe axial loadthrough at endplatens(refer to Fig.8).Cross-headspeedwas approximately5mm=min.A LABTECH data-logger recorded the data digitally for later analysis.The tubes tested were made from commercial quality extruded6060aluminium alloy in the as-received,heat treated T5condition.Mechanical properties were determined from tensile testing of coupons cut from several tubes.Fig.9shows a stress–strain curve for a typical tensile test specimen which hada0.2%proof stress, 0:2,of180MPa,an ultimate stress, ult, of212MPa anda Vickers hard ness,V,of73kg=mm2.By averaging results from several tensile tests we were able to determine an empirical relation between the0.2%proof stress and Vickers hardness for this particular type of alloy as follows:Vg= 0:2=3:92,where g=9:81m=s2is the gravitational acceleration.2112S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123Fig.9.Typical tensile stress–strain curve for6060-T5aluminium.Fig.10.Stress–strain curves for polyurethane foams of three di erent densities.The variety of commercial tubes available was insu cient to achieve the full range of D=t values we required.To produce tubes with very large D=t ratios,the outside surface of stock tubes was machined to produce the wall thickness desired.Theÿnal thickness,mean diameter and Vickers hardness were measured for each compression testpiece.Vickers hardness read-ings were usedto quantify the mechanical properties of each tube testpiece through the above equation.A representative sample of these tube properties andthe results of our testing may be foundin the append ix.Most tests involvedempty aluminium alloy tubes.However,some tests were carriedout on aluminium alloy tubes which hadbeenÿlledwith polyurethane foam.This polyurethane foam usually comes as a two part mix(base andaccelerator)andwe usedthree d i erent d ensities (35,60and140kg=m3)during testing.Fig.10shows typical compressive stress–strain curvesfor the three di erent density foams,which were obtained from axial compression tests on 96mm diameter cylindrical foam blanks unrestrained laterally.3.Experimental results and discussionOur experimental results are summarisedon the following pages in terms of the collapse mode,average force F AV,force ratio F MAX=F AV,eccentricity factor m andthe e ect of foam ÿlling.3.1.Collapse modeFurther examples of collapse modes are shown in Fig.11,in addition to those shown in Fig.2.Of particular interest was the non-symmetric mode(refer to Fig.2b)which has multiple corners(or lobes).We observedthat for tubes with an increasing D=t ratio,the number ofFig.11.Further examples of collapse modes for axially loaded thin-walled6060-T5aluminium tubes: (a)mixedmod e(D=57:1mm;t=1:15mm;L=628mm);(b)three sided non-symmetric folding (D=57:1mm;t=1:15mm;L=628mm);(c)Euler buckling(D=58mm;t=2:0mm;L=566mm).Fig.12.Schematic axial view of spiralling non-symmetric folding with N=312lobes,from Grzebieta[13].Fig.13.Mode classiÿcation chart for circular6060-T5aluminium tubes. circumferential lobes also increasedfrom2up to5or6.At high values of D=t(¿200),the number of lobes often variedd uring testing(in one case erratically between3,4and5lobes). The number of lobes,N,was not always an integer—for example,in some cases we observed a relatively stable pattern with312lobes in a spiralling arrangement(refer to Fig.12).In other cases the lobes were simply incompletely formed.From our test results for as-received6060-T5aluminium tubes a mod e classiÿcation chart was produced,see Fig.13.This chart is divided up into areas which correspond,approximately, to the di erent modes of collapse.The general shape of our chart is similar to that produced by And rews et al.[7],who testedannealedaluminium tubes.However,there are noticeable di erences in the location of the lines delineating the various areas.For example,consider analuminium tube with D=t=50and L=D=10.From our chart we wouldexpect a mixedmod eFig.14.Plot of non-dimensional experimental average force F AV=M P versus D=t.collapse but from Andrews et al.chart an Euler collapse is indicated.Note that a logarithmic scale is usedfor D=t on our chart in order to cover the wider range of D=t values considered.It may be observed broadly from our chart that non-symmetric mode is present when D=t¿100, while axi-symmetric mode occurs when D=t¡50and L=D¡2.3.2.Average crush forceOne of the most signiÿcant parameters for quantifying the behaviour of axially compressed tubes is the average crush force F AV.This is usually expressednon-d imensionally as a ratio F AV=M P.When calculating the plastic moment,M P,di erent researchers have used various di erent measures for the ow stress, 0.Since our tests involvedonly aluminium we chose to take the value of0.2%proof stress, 0:2,as the ow stress.Thus e ectively:M P= 0:2(t2=4):Fig.14shows our test results for non-dimensionalised average axial force,F AV=M P,plotted logarithmically versus D=t.There is only a relatively small amount of experimental scatter(some points shown represent more than one test result).Note that when calculating the average axial force,F AV,results for the initial peak have been ignored.From Fig.14it can be seen that when plottedlogarithmically,all the results(whether axi-symmetric,non-symmetric or mixed modes)approximately form a straight line.Hence,we obtained the following empirical relation for6060-T5aluminium alloy tubes:F AV M P =72:3Dt0:32:(8)This equation is of similar form to Eq.(4)proposedby Abramowicz andJones in1984[8] for non-symmetric mode but quite di erent from the corresponding equation proposed by Gupta andGupta[2].parison of present experimental results for average crush force with empirical equations of Gupta and Gupta[2].parison of experiment and theory for average forceThe following paragraphs are a subjective comparison between our experimental results for average crush force,F AV,andvarious theories andempirical relationships.Fig.15shows our test results for F AV comparedwith empirical equations by Gupta andGupta [2].They usedVickers hard ness,V,to characterise material properties.These equations were determined from tests on metal tubes with a relatively small range of dimensions(D=t=10–33, L=D=2–3).In view of this,agreement for both axi-symmetric andnon-symmetric mod es is quite goodin the range D=t=10–100.For D=t¿100,their curve for axi-symmetric mode is closer to our experimental test points than their non-symmetric one,even though the actual collapse mode exhibited was non-symmetric.Fig.16shows our test results for average force,comparedwith equations d evelopedby Abramowicz andJones[8].From thisÿgure it may be seen that agreement for both axi-symmetric and non-symmetric modes is fair.Their axi-symmetric equation predicts average forces which are rather low comparedwith our test points.On the other hand,their equation for non-symmetric mode,Eq.(4),predicts average forces which are rather high compared with our test points. Nevertheless,it may be notedfrom Fig.16that the slope of the line representing Eq.(4) (non-symmetric mode)is almost the same as our test points.This is also evident from a comparison of Eqs.(4)and(8).Fig.17shows our test results for average force,F AV,comparedwith the theoretical equations d evelopedby Abramowicz andJones[9].Their axi-symmetric Eq.(3)estimates an average force which is still low comparedwith our test results,but closer to our test points than their 1984prediction.In the case of non-symmetric collapse,Abramowicz and Jones[9]developed Eq.(5),which produces a family of lines,one for each value of N.Thus,we needto know the number of lobes,N,in order to interpret Fig.17.From the appendix it will be noted that formost of the tubes we tested,N falls in the range N=3–4.Agreement between their theory andparison of present experimental results for average force with theory by Abramowicz andJones[8].parison of present experimental results for average force with theory by Abramowicz andJones[9]. our test points is goodin this range.For cases with low D=t values(¡50),where N¡3,their prediction for F AV is rather low.For high values of D=t(¿300),where N¿4,their predicted value for F AV is rather high.Nevertheless,overall it appears that this methodof pred icting F AV is satisfactory.Fig.18shows our test results for average force comparedwith the equations d evelopedby Singace et al.[17,18].Their equation for axi-symmetric mode,Eq.(6),gives values for F AV which are much too low comparedwith our test points.In the case of non-symmetric mod e, their Eq.(7),when plottedon logarithmic axes prod uces a series of very steep lines,one for each number of lobes,N.This makes the process of interpretation even more di cult. Determining the precise number of corners or lobes for each test specimen presents somepractical di culties.As has previously been noted,if D=t¿200we sometimes observedthatparison of present experimental results for average force with theory by Singace et al.papers [16–18].the number of lobes variedd uring the one test.Nevertheless,on Fig.18test points are shown for which we felt conÿdent of the lobe number.It will be noted that the lines representing the Singace et al.Eq.(7)are not inconsistent with our test points although agreement is not close.However,we observedthat when the number of lobes variedd uring testing there was not a corresponding variation in the instantaneous crush force.This observation casts doubt on the validity of Eq.(7),as the large gaps between the lines in Fig.18suggest there should be a large variation in crush force with a change in the number of lobes,N .3.4.Discussion of average forceAt this stage the following observations may be made.In general,the existing theories produce numerical predictions for average force which are reasonable only for a limited range of D=t .Comparison of our test results with these theories has revealedtwo fund amental features which remain inexplicable at present.The ÿrst feature is that all our test points,regardless of mode of collapse (axi-symmetric or non-symmetric),lie on one curve whereas the theories treat these modes quite separately.Further,most theories for non-symmetric mode predict average forces which are a function of the number of lobes but experimentally this does not appear to be the case.The second,more important,feature relates to the functional dependence of average force on D=t .Our experiments clearly show that F AV =M P is empirically dependent on (D=t )0:32.Existing theories for axi-symmetric mode,however,suggest that F AV =M P should be dependent on D=t .In the case of non-symmetric mode,a wide variety of theories have been suggested;typically F AV =M P is seen as being a linear function of D=t as for the Singace et al.Eq.(7).An exception to this is Eq.(4)d evelopedby Abramowicz andJones [9],where F AV =M P was proportional to (D=t )0:33.However,as previously noted,this equation appears to have developed from work by Wierzbicki andAbramowicz [10]on rectangular rather than circular tubes.Thus it seemsFig.19.F MAX=F AV force ratio versus D=t.that for circular tubes a rigorous theoretical explanation of the D=t exponent of13is still to be developed.3.5.Force ratio F MAX=F AVIn a previous paper,Guillow andLu[23]id entiÿedthe force ratio F MAX=F AV as being of some interest.In that paper it was notedthat F MAX=F AV variedas a function of D=t ratio(this has also been notedby other researchers).The variation in force ratio highlights the fact that the mechanics of formation of the initial and subsequent folds is substantially di erent.Fig.19 shows the results of our more recent tests at larger values of D=t.The F MAX=F AV ratio appears to be monotonically increasing up to D=t=450.Variability in the F MAX=F AV ratio increased markedly for D=t¿100.This scatter couldbe d ue to signiÿcant variation in the initial buckling force,F MAX,at large values of D=t.Incidently,the common wisdom attributes the scatter of initial buckling force to imperfection sensitivity of thin-walled shells.However,Calladine[24]has recently provided an alternative explanation basedon post-buckling consid erations.3.6.Eccentricity factor mWhen folds occur during progressive buckling,they form partly on the outside and partly on the inside of the original tube proÿle.As previously noted,Singace et al.[16–18]have investigatedthis phenomenon by consid ering the eccentricity factor,m,(refer to Fig.7for its deÿnition).We were surprised at their claim that the factor m was approximately constant at 0.65.Therefore,we decided to examine our test pieces to see if the Singace et al.ÿndings also applied to6060-T5aluminium alloy tubes.Our test results for axi-symmetric mode folding are shown in Fig.20.They appear to conÿrm that a constant value of approximately0.65alsoapplies in this case.(It is not clear why the m value shouldbe so d i erent at D=t=20.)Fig.20.Eccentricity,m,as a function of D=t,for axi-symmetric mode.Fig.21.E ect of varying density of foamÿlling in6060-T5aluminium tubes.All tubes of length196mm,average diameter97mm and thickness1:0mm.Refer to Fig.10for stress–strain curves of polyurethane foam.3.7.E ect of foamÿllingMost of our tests involvedempty aluminium alloy tubes.However,a few tests were carried out on aluminium alloy tubes which hadbeenÿlledwith polyurethane foam.Fig.21shows some of our test results for foam-ÿlledaluminium tubes andTable1presents the d ata for aver-age axial crush force.Stress–strain curves for foam only were presentedearlier in Fig.10.All of the aluminium alloy tubes usedin this stage of testing were id entical(D=97mm;t=1:0mm and L=196mm).Test results for an identical empty aluminium tube are shown in Fig.1. We expectedto observe an increase in the average crushing force,F AV,for aluminium alloy tubes which hadbeenÿlledwith foam,as comparedwith id entical empty aluminium tubes.In fact,there is a complex interaction between the metal tubes andthe foamÿlling.The foamprovides support for the thin walls of the aluminium tubes leading to an increase in the overall。
等球Packing问题的序列对称换位算法余亮;黄文奇【期刊名称】《计算机应用研究》【年(卷),期】2012(029)005【摘要】为处理等球Packing问题,在基本拟物算法的基础上设计了序列对称换位策略,形成了一个启发式的序列对称换位算法.在球形容器内装填1～50个等球时,此算法改进了其中45项当前记录.特别地,此算法成功将68个半径为1的等球装进半径小于5的球形容器.此结果证否了一个猜想,该猜想认为半径为5的球形容器至多只能装下67个半径为1的等球.其结果的质量说明了序列对称换位算法的有效性.%To deal with the equal sphere Packing problem,this paper designed a serial symmetrical relocation strategy based on the basic quasi physical algorithm to form a serial symmetrical relocation algorithm. When packing up to 50 equal spheres in a spherical container, this algorithm improved 45 current records. Especially, it successfully packed 68 equal spheres of radius 1 into a spherical container whose radius is less than 5. This result proves wrong a conjecture which states the spherical container of radius 5 can at most contain 67 spheres of radius 1. The quality of the results justifies the serial symmetrical relocation algorithm.【总页数】3页(P1695-1697)【作者】余亮;黄文奇【作者单位】华中科技大学计算机科学与技术学院,武汉430074;华中科技大学计算机科学与技术学院,武汉430074【正文语种】中文【中图分类】TP301.6【相关文献】1.输电线路不换位引起的不对称问题及其改进方法 [J], 陶凯;刘明波2.求解等球packing问题的两个策略 [J], 余亮;黄文奇3.不换位输电线路产生的不对称问题及解决方法 [J], 丁洪发;段献忠4.求解一刀切式二维矩形Strip Packing问题的混合搜索算法 [J], 郭超;王磊;尹爱华5.一种求解等圆Packing问题的柔性位置选择算法 [J], 王英聪;张领;肖人彬因版权原因，仅展示原文概要，查看原文内容请购买。
一阶电路first-order circuit三要素法three-element method for analyzingfirst-order circuitss 平面s-plane二端元件two-terminal element二端网络two-terminal network无源网络passive-terminal network有源网络active-terminal networkT 形网络T-networkΓ形网络inverted L-network, Γ-network入射波incidence wave三相three-phase三相电路three-phase circuit三相制three-phase system三相四线制three-phase four-wire system三角形连接delta-connection, △-connection 三角形网络delta-network三端网络three-terminal network端口portπ形网络π- network已调信号modulated signal支路branch支路电流法branch current method支路阻抗矩阵branch impedance matrix支路导纳矩阵branch admittance matrix分压器voltage divider分压比voltage division ratio分贝decibel(dB)分离图separated graph开路open-circuit开路阻抗open-circuit impedance开路阻抗矩阵open-circuit impedance matrix反接inversed connection, connection inopposition反射阻抗reflected impedance反相opposite in phase反向串联inverted series connection反向传输矩阵inverted transmission matrix互感mutual inductance互感应现象mutual induction phenomenon互感耦合mutual-inductance coupling互感耦合电路mutual-inductance coupled circuit 互易性reciprocity 互易定理reciprocity theorem互易网络reciprocity network中线 (零线 )neutral wire中性点 (中点 )neutral point无功功率reactive power无功功率守恒theorem of conservation of reactive 定理power无功伏安reactive Volt-Ampere无功分量reactive component无功因数reactive factor双口网络two-port network, two-port对称双口网络symmetrical two-port network不对称双口网unsymmetrical two-port network 络X 形双口网络lattice network复合双口网络composite two-port networkT 形桥式双口bridge-T two-port network网络双 T 网络double-T network双 T 选频网络double-T frequency selectionnetwork匹配matching方阵square matrix韦伯 (韦)Weber(Wb)乏var辅助分析computer-aided analysis瓦特 (瓦)watt(W)分布电感distributed inductance内部法internal approach分段线性化法piece-wise linear approximation 分布参数电路distributed circuit反射系数reflection coefficient反射波reflected wave匹配match无损耗线lossless line无损耗线的输input impedance of lossless line 入阻抗无畸变distortionless无畸变条件distortionless condition无畸变线distortionless line电路circuit电源source理想电源ideal source1 /15实际电源physical source电位potential电位差potential difference电位升potential rise电位降potential drop电位参考点potential reference point电压voltage电压圆图voltage circle diagram电压源voltage source电压控制电压voltage-controlled voltage source 源电压控制电流voltage-controlled current source 源电压反馈系数voltage feed-back factor线电压line voltage相电压phase voltage电流current电流源current source电流控制电压current-controlled voltage source 源电流控制电流current-controlled current source 源电流放大系数current amplification factor线电流line current相电流phase current电动势electromotiveforce(e.m.f.),electromotance电激流excitation current电阻resistance内电阻internal resistance自电阻self-resistance共电阻 ( 互电mutual resistance阻)电导conductance内电导internal conductance自电导self-conductance共电导 ( 互电mutual conductance导)电感inductance电容capacitance电抗reactance电纳susceptance电信号electric signal 电场能量electric field energy电场强度electric field intensity电磁场electromagnetic field电力网power network电报方程telegraphic equation正弦波sinusoidal wave正弦信号sinusoidal signal正弦函数sinusoidal function正弦响应sinusoidal response正弦交流电路sinusoidal responsealternating current circuit正序positive sequence正相序positive phase sequence正负号函数signup矢量vector节点node节点方程node equation节点电流方程node current equation节点电压法node voltage method节点关联矩阵node incidence matrix节点电导矩阵node conductance matrix广义节点Super-node对称三相电路symmetrical three-phase circuit 对称均匀链形symmetrical uniform chain network 网络对偶原理principle of duality对偶网络dual network对偶元件dual element对应端corresponding terminal对象阻抗image impedance对象参数image parameter对象传输常数image propagation constant平面网络planar network非平面网络non-planar network功率power功率因数power factor功率因数角power factor angle功率三角形power triangle功率守恒定理theorem of conservation of power 平均功率average power有功功率active power无功功率reactive power视在功率apparent power2 /15右手螺旋定则right-handed screw rule外网孔outer mesh失谐状态detuned condition小失谐状态slightly detuned condition四端网络four-terminal network, quadripole 主元pivot element, pivot平衡工作点balanced operating point龙格 -库塔法Runge-Kutta method四阶 R-K 法forth-order R-K method四分之一波长quarter-wave line线史密斯阻抗图Smith Chart网络network网络分析network analysis网络分析法method of network analysis网络方程法network-equation method网络变换法network-transformation method网络拓扑network topology网络模型network-model有源网络active network无源网络passive network线性网络linear network非线性网络nonlinear network网孔mesh网孔电流法mesh-current method网孔矩阵mesh matrix网孔阻抗矩阵mesh-impedance matrix网孔对支路关mesh-to-branch incidence matrix 联矩阵自感 (自感系self-inductance数)并联parallel connection并联谐振parallel resonance有效值effective value有源二端网络equivalent source theorem of active 的等效电源定two-terminal network理有源二端网络equivalent voltage source theorem 的等效电压源of active two-terminal network定理 (戴维南定(Thevenin's theorem)理)有源二端网络equivalent current source theorem 的等效电流源of active two-terminal network定理 (诺顿定(Norton's theorem)理 )同名端dotted terminal同相in phase回路loop回路电阻矩阵loop-resistance matrix回路电流法loop-current method回转器gyrator导纳admittance导纳角admittance angle导纳圆图admittance circle diagram自导纳self-admittance共导纳 (互导mutual admittance纳 )共轭匹配conjugate matching共轭旋转相量conjugate rotating phasor负载load负序negative sequence负相序negative-phase sequence全磁通total magnetic flux全波整流full-wave rectification全通图completely-connected graph 次级线圈secondary coil行row行阵row matrix行子阵row submatrix行矢量row vector列column列阵column matrix列子阵column submatrix列矢量column vector关联incidence关联矩阵incidence matrix正向关联positive incidence负向关联negative incidence似功率quasi-power似功率守恒定theorem of conservation of理quasi-power传递函数transfer function传播常数propagation constant传输矩阵transmission matrix传输效率transmission efficiency米勒定理Miller's theorem3 /15级联cascade connection伏特 (伏 )Volt(V)伏秒Volt-second伏安特性volt-ampere characteristic安培 (安 )Ampere(A)西门子 ( 西)Siemens(S)过渡过程transient state过电压over voltage过电流over current自由分量free component自激振荡self-sustained oscillation自然功率natural power自然频率natural frequency网络的自然频natural frequency of a network率网络变量的自natural frequency of network然频率variables零输入响应的natural frequency of zero-input自然频率response阶跃响应step response冲量响应impulse response冲量响应矩阵impulse response matrix动态电阻dynamic resistance网孔运算阻抗mesh operational impedance matrix矩阵网络函数network function网络函数的极pole-zero diagram of network零点分布图function行波travelling wave正向行波direct wave反向行波returning wave行波功率travelling wave power行波系数travelling wave ratio串联series connection串联谐振series resonance连支link连通图connected graph连续频谱continuous spectrum不连续频谱discrete spectrum初相initial phase初相角initial phase angle初级线圈primary coil角频率angular frequency均方根值root-mean-square value均匀频谱uniform spectrum均匀链形电路uniform chain circuit时变电流time-varying current位移电流displacement current运算放大器operational amplifier两瓦特表法two-wattmeter method亨利 (亨)Henry(H)时域分析time-domain analysis时域位移定理real shifting(translation)theorem 时间常数time constant初始条件initial condition初始状态initial state初值定理initial value theorem张弛振荡relaxtion oscillation阻尼系数damping coefficient均匀传输线uniform transmission line均匀传输线的primary parameters of uniform 原始参数transmission line均匀传输线的differential equations of uniform 微分方程transmission line均匀传输线的propagation constant of uniform 传播常数transmission line均匀传输线的characteristic impedance of uniform 特性阻抗transmission line均匀传输线的attenuation constant of uniform 衰减常数transmission line均匀传输线相phase constant of uniform 移常数transmission line均匀传输线输input impedance of uniform 入阻抗transmission line均匀传输线的lumped equivalent circuit of 集中参数等效uniform transmission line 电路折射波reflected wave状态state状态变量state variable状态矢量state vector状态变量法state variable approach状态方程state equation状态空间state space状态空间法state space approach状态轨迹state trajectory4 /15状态转换矩阵state transition matrix状态变量计算superposition method for computer 机辅助分析的aided analysis of state variables叠加法状态变量计算topological method for computer机辅助分析的aided analysis of state variables拓扑法极限环limit cycle极点pole步长step length延时线time-delay line线性linearity线性电阻linear resistance线性电感linear inductance线性电容linear capacitance线性网络定理linear network theorem线状频谱line spectrum周期period周期信号period signal非周期信号non-periodic signal非线性元件nonperiodic element非正弦周期电non-sinusoidal periodic current流电路circuit单位阵unit matrix单位阶跃函数unit step function单位阶跃电压unit step voltage单位冲量函数unit impluse function单位冲量电流unit impluse current单脉冲信号single pulse signal单口网络one-port network, one-port拓扑图topological graph, graph有向拓扑图oriented graph拓扑结构topology, topological construction 转移transfer转移阻抗transfer impedance转移导纳transfer admittance转移电压比transfer voltage ratio转移电流比transfer current ratio转移函数transfer function转置阵transposed matrix转移函数transfer function转移函数矩阵transfer function matrix空载状态no-load condition 空心变压器air-core transformer参考方向reference direction参考相量reference phasor参考节点reference node受控源controlled source受控源关联矩controlled source incidence matrix 阵图graph子图Sub-graph奇谐波函数odd harmonic function变比transformation ratio环流circulating current直流direct current直流网络direct current network直流分量direct current component阻抗impedance阻抗角impedance angle阻抗逆变器impedance inverter阻抗频率特性impedance-frequency characteristic 自阻抗self-impedance共阻抗mutual impedance内阻抗internal impedance输入阻抗input impedance欧姆 (欧)Ohm's欧姆定律Ohm's law广义欧姆定律generalized Ohm's law欧拉法Euler's method法拉 (法)Farad(F)微法micro-Farad(F)皮法pico-Farad(F)法拉第电磁感Faraday's law of electromagnetic 应定律induction奈培neper(Np)经典法classical method非零状态响应non-zero-state response非强制网络unforced network非线性电路nonlinear circuit非线性电阻nonlinear resistance电流控非线性current-controlled nonlinear电阻resistance电压控非线性voltage-controlled nonlinear电阻resistance范式normal form范式状态方程normal form state equations5 /15放电过程discharge过阻尼放电过Over-damped discharge程欠阻尼放电过Under-damped discharge程非振荡放电过non-oscillatory discharge程振荡放电过程oscillatory discharge临界阻尼放电critically damped discharge过程拉普拉斯Laplace拉普拉斯正变Laplace transformation换拉普拉斯反变inverse Laplace transformation 换拉普拉斯积分Laplace integral拉普拉斯象函Laplace transform数线性组合定理linear combination theorem终值定理final value theorem波长wavelength波阻抗wave impedance波腹loop波节node驻波standing wave驻波系数standing wave ratio规则信号regular signal卷积convolution Integral卷积定理convolution Integral theorem 响应response响应信号response signal相位 (相 )phase相位角phase angle相位差phase difference相位频率特性phase-frequency characteristic 相矢量phasor相矢量分析法phasor analysis相序phasor sequence信号signal信号源signal source总电导total conductance树tree树支tree branch 树余cotree星形 (Y) 连接star-connection, Y-connection星形网络star-connection network复数complex number复数平面complex plane复数阻抗complex impedance复数导纳complex admittance复数功率complex power复数导纳矩阵complex admittance matrix顺接connection in aiding顺序positive sequence独立电源independent source品质因数quality factor逆序negative sequence选择性selectivity选频特性frequency-selection characteristic 恒定分量constant component脉冲pulse脉冲幅度pulse amplitude脉冲高度pulse altitude脉冲宽度pulse width脉冲持续时间pulse duration脉冲重复周期repeating period of pulse指数衰减因子exponential attenuation factor指数矩阵exponential matrix临界电阻critical resistance临界值critical value复频率complex frequency复频率平面complex frequency plane复频谱函数complex frequency spectrumfunction复频域complex frequency domain复频域位移定complex理shifting(translation)theorem复频域等效电complex shifting equivalent circuit 路复频域中广义generalized ohm's law in the欧姆定律complex frequency domain复频域传播常complex frequency domain数propagation constant复频域特性阻complex frequency domain抗characteristic impedance复频域反射系complex frequency domain6 /15数reflection coefficient电容的复频域complex frequency domain阻抗impedance of capacitor电感的复频域complex frequency domain阻抗impedance of inductorRLC 串联电路complex frequency domain的复频域阻抗impedance of RLC series circuit 相移速度phase velocity柏德生法则Peterson's Rule容抗capacitive reactance容纳capacitive susceptance振幅amplitude振幅频谱amplitude spectrum振幅旋转相量amplitude rotating phasor效率efficiency矩阵matrix矩阵分析法matrix analysis特性characteristic特性方程characteristic equation特性阻抗characteristic impedance特性参数characteristic parameter特性损耗characteristic loss特性相移characteristic phase displacement 特勒根定理Tellegen's theorem离散性discreteness离散频谱discrete spectrum高次谐波higher harmonic高斯消去法Gauss elimination method高斯主元消去Gauss elimination with pivoting法T 形阻抗网络bridge-T impedance network浮地电感floating inductance积分电路integrating circuit积分定理integration theorem衰减attenuation衰减系数attenuation constant振荡oscillation阻尼振荡damped oscillation衰减振荡attenuated oscillation减幅振荡attenuated oscillation等幅振荡unattenuated oscillation无阻尼振荡自由振荡高阶电路特征方程特征根 (值 )特征多项式特征方程复频域形式部分分式展开法留数计算法逐步近似法预解矩阵载波调制信号被调制信号换路基本回路基本回路矩阵基本割集基本割集矩阵基本子阵基波基尔霍夫基尔霍夫方程基尔霍夫定律基尔霍夫电流定律基尔霍夫电压定律基尔霍夫电流定律的复频域形式基尔霍夫电压定律的复频域形式偶谐波函数理想元件理想激励源理想电压源理想电流源理想受控源7 /15undamped oscillationfree oscillationhigher order circuitscharacteristic equationcharacteristic root, eigenvaluecharacteristic polynomialcomplex frequency domaincharacteristic equationpartial-fraction expansionevaluation by the residuemethod step-by-stepapproximation resolvent matrixcarriermodulating signalmodulated signalswitchingfundamental loopfundamental loop matrixfundamental cut setfundamental cut set matrixfundamental submatrixfundamental harmonicKirchhoffKirchhoff's equationKirchhoff's lawKirchhoff's current lawKirchhoff's voltage lawKirchhoff's current law in thecomplex frequency domainKirchhoff's voltage law in thecomplex frequency domaineven harmonic functionideal elementideal excitation sourceideal voltage sourceideal current sourceideal controlled source理想变量器ideal transformer理想变压器ideal transformer旋转相矢量rotating phasor混合参数矩阵hybrid parameter matrix累接阻抗iterative impedance接地点ground point谐振resonance谐振状态resonance state谐振电路resonant circuit谐振阻抗resonant impedance谐振频率resonant frequency谐波harmonic谐波分量harmonic component离散化discretization常态树proper tree随机性信号random signal集中参数lumped parameter集中参数元件lumped element集中参数电路lumped circuit等效网络equivalent network等效阻抗equivalent impedance等效导纳equivalent admittance短路short-circuit短路导纳short-circuit admittance短路导纳矩阵short-circuit admittance matrix 超前lead滞后lag惠斯登电桥Wheatstone bridge割集cut set割集电导矩阵cut set conductance matrix 策动点driving point策动点阻抗driving point impedance策动点导纳driving point admittance策动点函数driving point function替代定理substitution theorem链形网络chain network晶体管电路transistor circuit插入功率比insertion power ratio插入衰减insertion loss傅里叶Fourier傅里叶级数Fourier's series 傅里叶积分Fourier's integral傅里叶积分变Fourier's integral transform 换傅里叶系数Fourier coefficient傅里叶正变换positive Fourier transform傅里叶反变换inverse Fourier transform暂态transient state暂态分量transient component强制分量forced component确定性信号regular signal输入input输入电路input circuit输入功率input power输入端口input port输出output输出电路output circuit输出阻抗output impedance输出端口output port感抗inductive reactance感纳inductive susceptance零电位点zero potential point零子阵zero submatrix数值解法numerical analysis数值积分法numerical integration愣次定律Lenz's law幅角argument频率frequency频率特性frequency characteristic频谱frequency spectrum频谱函数frequency spectrum function 频带frequency band频带宽度band width通频带pass-band频域frequency domain频域响应frequency domain response 简谐分量simple harmonic component 微分电路differentiating circuit微分定理differentiation theorem零状态zero state零状态响应zero state response零状态分量zero state component零输入响应zero-input response8 /15零输入分量zero-input component输出方程output equation输入 -输出法input-output approach数值解法numerical solution群速group velocity群时延group time-delay畸变distortion叠加定理superposition theorem磁通magnetic flux磁通链magnetic flux linkage磁耦合magnetic coupling磁场能量magnetic field energy端电压terminal voltage端线terminal wire端口port terminal模modulus缩减矩阵reduced matrix谱线spectrum line稳态steady state稳态响应steady state response赫兹Hertz(Hz)稳态分量steady state component稳定性stability静态电阻static resistance端部法terminal approach截断误差truncation error耦合系数coupling coefficient增广矩阵augmented matrix增广节点导纳augmented node admittance matrix 矩阵额定电压rated voltage额定电流rated current额定功率rated power激励excitation激励信号excitation signal激励源excitation source激励函数excitation function瞬时值instantaneous value瞬时电压instantaneous voltage瞬时电流instantaneous current瞬时功率instantaneous power电压串联电阻 A voltage in series with a resister 电源变换Source transformation 电流并联电阻 A current source in parallel with aresister双边的Bilateral叠加Superposition麻烦的Cumbersome同时发生的Simultaneous术语Terminology二维的Planar安培表Ammeter编造的Fictitious操纵（作）Manipulation相关的Pertinent运算放大器The operational amplifier二极管Diode晶体管Transistor喜好Penchant明智的Judicious求助于Invoke复制品Replica比较器Comparator运动中的电荷Charge in motion定量关系Quantitative relationship绝缘体Insulator电介质材料Dielectric material时变电场Time-varying electric field位移电流Displacement current传导电流Conduction current无源元件Passive element单位是Be measured in图形上Graphically线圈Coiled wire短路Short current跃变Change instantaneously电弧Arcing微分Differential代数的Algebraic功率是消耗能Power is the time rate of对时间的倒数expending energy字母 C Letter C金属板Conductive plate零输入响应Natural response阶跃响应；零Step response状态响应9 /15系数Coefficient一阶电路First-order circuit指数Exponent倒数Reciprocal瞬间响应Transient response稳态响应Steady-state response示波器Oscilloscope类比的Analogous初始电压的指Exponential decay of the initial 衰减voltage减率The rate of decay基本微积分Elementary calculus等幅震荡Oscillation衍生物Derivation特征根Characteristic roots谐振角频率Resonant radian frequency量纲Dimension复频率Complex frequency过阻尼Over damp欠阻尼Under damp临界阻尼Critically damp正弦的Sinusoidal提及的Allude to配线电路Distribution circuit详述Spell out领域Realm间隔时间Interval相互的Reciprocal波幅Amplitude相位角Phase angle平面三角学Trigonometry均方根Rms value直流电压Dc voltage暂态分量Transient component无穷小Infinitesimal丧失Forfeit向量Phasor括号Argument符号Notation向量变换Phasor transform时域Time domain复数域Complex-number domain 黑体字Boldface letter极坐标假设下脚标无意义的术语，命名法无源元件有源元件阻抗电抗瞬时功率有功功率；平功率无功功率视在功率通过A滞后 B120 ° A超前 B120 °相序圆柱体表面定子线圈发电机线电压相电压动态元件滤波器听得见的选频电路衰减图式均衡器低通高通带通带阻滤波器初步通带阻带频率响应曲线幅频特性曲线相频特性曲线截止频率Polar formPostulateSubscriptNonsensical NomenclaturePassive elements Active elements ImpedanceReactance Instantaneous power Average powerReactive power Apparent powerViaA lagB by 120°A lead B by 120°Phase sequence PeripheryStatorWindingGeneratorLine-to-linevoltageLine-to-neutralvoltageReactivecomponentsFilterAudibleFrequency-selective circuitsAttenuateGraphicequalizerLow pass filtershigh pass filtersBand pass filtersBand rejectfiltersPreliminaryPassbandStopbandFrequencyresponse plotMagnitude plotPhase angle plotCutoff frequency 10/ 15Chapter 1 Elements and Laws of Eletrical Circuits电路electrical circuit电路模型circuitmodel电源source负载load导线line开关switch电荷electric charge电流current电压voltage电位potential电位升potential rise电位降potential drop电位差potential difference参考点referencepoint线性电阻linear resistance磁通链magnetic flux linkage功率power能量energy电阻器resistor电阻resistance电动势electromotive force （ e.m.f ）伏安特性u-i characteristicvolt-ampere characteristic电导器conductor电导conductance电感器inductor电感inductance电容器capacitor电容capacitance欧姆定律Ohm’s Law广义欧姆定律generalized Ohm ’s Law参考方向reference direction电压极性voltage polarity正极positive polarity负极negative polarity开路open-circuit短路short-circuit理想独立电压源ideal independent voltage实际电压源physical sourcesource理想独立电流源ideal independent current理想受控源ideal dependent / controlledsource source压控电压源voltage controlled voltage压控电流源voltage controlled currentsource（VCVS ）source（ VCCS ）流控电压源current controlled voltage流控电流源current controlled currentsource（CCVS ）source（ CCCS）节点node支路branch回路loop路径path网孔mesh网络network基尔霍夫电流定律Kirchhoff ’s current law（KCL ）基尔霍夫电压定律Kirchhoff ’s voltage law（KVL ）闭合面closed surface集总参数lumped parameter集总（参数）电路lumped circuit集总（参数）元件lumped element分布参数distributed parameter分布（参数）电路distributed circuit直流direct current （DC）交流alternating current（ AC ）有源元件active element无源元件passive elementChapter 2 Analysis methods to simple resistor circuits端钮terminal串联series connection分压voltage division并联parallel connection分流current division等效变换equivalent transformation等效电阻equivalent resistance入端电阻input resistance最大功率传输定理Maximum power transfer theorem Y- 变换Wye-Delta transformation11/ 15Chapter 3 methods of Analysis节点法node analysis / node voltage method支路电流法branch current method回路电流法loop analysis / loop current method外网孔outer mesh网孔电流法mesh analysis / mesh current method自导纳self admittance互导纳mutual admittance矩阵matrix行row列column参考节点reference node平面电路planar circuit方程equation消去法elimination technique克莱姆法则Cramer’s rule代入法substitution method运算放大器operational amplifier(op amp)同向输入端noninverting input反向输入端inverting input输出端output等效电路模型equivalent circuit model开环放大倍数open-loop gain闭环放大倍数closed-loop gain入端电阻input resistance输出电阻output resistance线性工作区linear region正向饱和区positive saturation反向饱和区negative saturation同向放大器noninverting amplifier反向放大器inverting amplifier加法器summing amplifier / summer积分器integrator微分器differentiator自激振荡self-excited oscillationChapter 4 Circuit Theorems叠加原理superposition theorem齐性原理homogeneity property输入 /激励input / excitation输出 /响应output / response线性电路linear circuit代数和algebraic sum替代定理substitution theorem戴维南定理Thevenin ’s theorem诺顿定理Norton ’s theorem二端网络two-terminal circuit开路电压open-circuit voltage短路电流short-circuit current特勒根定理Tellegen ’s theorem功率平衡定理power-balancing theorem互易定理reciprocal theorem对偶原理principle of duality对偶元件dual element对偶图dual graph对偶电路dual circuitChapter 5 Analysis of Op Amp Circuits非线性电路nonlinear circuit非线性元件nonlinear element压控电阻voltage-controlled resistor流控电阻current-controlled resistor工作点operating point静态电阻static resistance动态电阻dynamic resistance小信号分析small-signal analysis小信号模型small-signal mode分段线性化法piece-wise linear method数值解法numerical analysisChapter 6 First-order Circuit动态电路dynamic circuit一阶电路first-order circuit一阶微分方程first-order differential equation过渡过程transient process/ transient 线性非时变电路linear time-invaried circuit单位阶跃函数unit step function单位冲激函数unit impulse function单位斜坡函数unit ramp function起始条件initial condition起始值initial value换路定则switch law零输入响应zero-input response12/ 15零状态响应zero-state response稳态响应steady-state response 暂态响应transient response时间常数time constant指数函数exponential function冲激响应impulse response阶跃响应step response自由响应natural response自由分量natural component强迫响应forced response强制分量forced component全响应complete response稳态值final value卷积convolution时域延迟time delay换路switching跳变现象jump phenomenon脉冲持续时间pulse duration脉冲重复周期repeating period of pulseChapter 7 Second-order Circuit常系数微分方程constant coefficients differential齐次微分方程homogeneous differential equation equation二阶电路second-order circuit特征方程characteristic equation 特征根characteristic root特征值eigenvalue特征向量eigenvector特解particular solution通解general solution自然频率natural frequency衰减系数damping factor谐振频率resonant frequency过阻尼情况overdamped case欠阻尼情况underdamped case临界情况critically damped case固有频率natural frequency衰减振荡damped oscillation无损lossless正弦响应sinusoidal response波形waveform复数complex衰减attenuationChapter 8-9 Sinusoidal Steady-State Analysis复数complex幅值amplitude / magnitude相位phase相位差phase difference角频率angular frequency周期period频率frequency正弦的sinusoidal初相角initial phase angle瞬时值instantaneous value最大值maximum有效值effective value / root-mean-square value u 领先 i φu leads i byφu 落后 i φu lags i byφ同相in phase反相opposite in phase实部real part虚部imaginary part直角坐标形式rectangular form极坐标形式polar form指数形式exponential form相量phasor参考相量reference phasor旋转相量rotating phasor电压三角形voltage triangle瞬时功率instantaneous power平均功率average power阻抗impedance阻抗角impedance angle阻抗三角形impedance triangle导纳admittance电抗reactance电纳suspectance感性inductive感抗inductive reactance感纳inductive suspectance容性capacitive容抗capacitive reactance容纳capacitive suspectance正弦稳态响应sinusoidal steady-state response时域time-domain相量域phasor-domain瞬时概率instantaneous power视在 /表观功率apparent power功率因数power factor (pf)复功率complex power功率三角形power triangle复共轭complex conjugate有功分量active component有功功率active power无功分量reactive component功率守恒定理theorem of conservation of power无功功率reactive power阻抗匹配impedence matching共轭匹配conjugate matching串联谐振series resonance并联谐振parallel resonance谐振频率resonance frequency品质因数quality factor特性阻抗characteristic impedence频率响应frequency response选择性selectivity选频特性frequency-selection characteristicChapter 10 Magnetically Coupled Circuits耦合couple互感mutual inductance自感self-inductance磁通magnetic flux线圈coil铁心线圈iron core coil匝数turn耦合系数coupling coefficient变压器transformer空心变压器air-core transformer原边primary coils / windings副边secondary coils / windings引入阻抗reflected impendence理想变压器ideal transformer全耦合unity coupling coefficient全耦合变压器perfect coupling transformer变比turns ratio / transformation ratio自耦合变压器auto transformer多绕组变压器multiple-winding transformer激磁电感magnetizing inductance右螺旋定则right-hand screw rule漏感leakage inductance同名端dotted terminalterminals of same magnetic polarityChapter 11 Three-phase Circuits对称三相电路symmetrical three-phase circuit三相电源three-phase sources中线neutral line中性点neutral point三相四线制three-phase four-wire system相电压phase voltage线电压line voltage相序phase sequence正序positive / abc sequence负序negative / acb sequence相电流phase current线电流line currentChapter 12 Steady-State Response of Periodic Excitation信号signal周期函数periodic function周期性非正弦激励nonsinusoidal periodic excitation帕斯瓦尔定理Parseval ’s theorem 指数形式的付里叶级数exponential Fourier series付里叶系数Fourier coefficient基波fundamental harmonic基波频率fundamental frequency 谐波harmonic wave高次谐波higher harmonic频谱frequency spectrum谱线spectrum line线状频谱line spectrum奇次谐波oddharmonic偶次谐波even harmonic奇对称odd symmetry。
2011年技术物理学院08级（激光方向）专业英语翻译重点！！！作者：邵晨宇Electromagnetic电磁的principle原则principal主要的macroscopic宏观的microscopic微观的differential微分vector矢量scalar标量permittivity介电常数photons光子oscillation振动density of states态密度dimensionality维数transverse wave横波dipole moment偶极矩diode 二极管mono-chromatic单色temporal时间的spatial空间的velocity速度wave packet波包be perpendicular to线垂直be nomal to线面垂直isotropic各向同性的anistropic各向异性的vacuum真空assumption假设semiconductor半导体nonmagnetic非磁性的considerable大量的ultraviolet紫外的diamagnetic抗磁的paramagnetic顺磁的antiparamagnetic反铁磁的ferro-magnetic铁磁的negligible可忽略的conductivity电导率intrinsic本征的inequality不等式infrared红外的weakly doped弱掺杂heavily doped重掺杂a second derivative in time对时间二阶导数vanish消失tensor张量refractive index折射率crucial主要的quantum mechanics 量子力学transition probability跃迁几率delve研究infinite无限的relevant相关的thermodynamic equilibrium热力学平衡（动态热平衡）fermions费米子bosons波色子potential barrier势垒standing wave驻波travelling wave行波degeneracy简并converge收敛diverge发散phonons声子singularity奇点（奇异值）vector potential向量式partical-wave dualism波粒二象性homogeneous均匀的elliptic椭圆的reasonable公平的合理的reflector反射器characteristic特性prerequisite必要条件quadratic二次的predominantly最重要的gaussian beams高斯光束azimuth方位角evolve推到spot size光斑尺寸radius of curvature曲率半径convention管理hyperbole双曲线hyperboloid双曲面radii半径asymptote渐近线apex顶点rigorous精确地manifestation体现表明wave diffraction波衍射aperture孔径complex beam radius复光束半径lenslike medium类透镜介质be adjacent to与之相邻confocal beam共焦光束a unity determinant单位行列式waveguide波导illustration说明induction归纳symmetric 对称的steady-state稳态be consistent with与之一致solid curves实线dashed curves虚线be identical to相同eigenvalue本征值noteworthy关注的counteract抵消reinforce加强the modal dispersion模式色散the group velocity dispersion群速度色散channel波段repetition rate重复率overlap重叠intuition直觉material dispersion材料色散information capacity信息量feed into 注入derive from由之产生semi-intuitive半直觉intermode mixing模式混合pulse duration脉宽mechanism原理dissipate损耗designate by命名为to a large extent在很大程度上etalon 标准具archetype圆形interferometer干涉计be attributed to归因于roundtrip一个往返infinite geometric progression无穷几何级数conservation of energy能量守恒free spectral range自由光谱区reflection coefficient（fraction of the intensity reflected）反射系数transmission coefficient（fraction of the intensity transmitted）透射系数optical resonator光学谐振腔unity 归一optical spectrum analyzer光谱分析grequency separations频率间隔scanning interferometer扫描干涉仪sweep移动replica复制品ambiguity不确定simultaneous同步的longitudinal laser mode纵模denominator分母finesse精细度the limiting resolution极限分辨率the width of a transmission bandpass透射带宽collimated beam线性光束noncollimated beam非线性光束transient condition瞬态情况spherical mirror 球面镜locus（loci）轨迹exponential factor指数因子radian弧度configuration不举intercept截断back and forth反复spatical mode空间模式algebra代数in practice在实际中symmetrical对称的a symmetrical conforal resonator对称共焦谐振腔criteria准则concentric同心的biperiodic lens sequence双周期透镜组序列stable solution稳态解equivalent lens等效透镜verge 边缘self-consistent自洽reference plane参考平面off-axis离轴shaded area阴影区clear area空白区perturbation扰动evolution渐变decay减弱unimodual matrix单位矩阵discrepancy相位差longitudinal mode index纵模指数resonance共振quantum electronics量子电子学phenomenon现象exploit利用spontaneous emission自发辐射initial初始的thermodynamic热力学inphase同相位的population inversion粒子数反转transparent透明的threshold阈值predominate over占主导地位的monochromaticity单色性spatical and temporal coherence时空相干性by virtue of利用directionality方向性superposition叠加pump rate泵浦速率shunt分流corona breakdown电晕击穿audacity畅通无阻versatile用途广泛的photoelectric effect光电效应quantum detector 量子探测器quantum efficiency量子效率vacuum photodiode真空光电二极管photoelectric work function光电功函数cathode阴极anode阳极formidable苛刻的恶光的irrespective无关的impinge撞击in turn依次capacitance电容photomultiplier光电信增管photoconductor光敏电阻junction photodiode结型光电二极管avalanche photodiode雪崩二极管shot noise 散粒噪声thermal noise热噪声1.In this chapter we consider Maxwell’s equations and what they reveal about the propagation of light in vacuum and in matter. We introduce the concept of photons and present their density of states.Since the density of states is a rather important property，not only for photons，we approach this quantity in a rather general way. We will use the density of states later also for other(quasi-) particles including systems of reduced dimensionality.In addition,we introduce the occupation probability of these states for various groups of particles.在本章中，我们讨论麦克斯韦方程和他们显示的有关光在真空中传播的问题。
作者简介:王利斌(1982-),男,山西太原人,硕士研究生,工程师,从事机载无线通信设备设计与开发工作,主要研究方向为射频功放设计与无源射频电路设计。
基于ADS 的宽带定向耦合器的设计与仿真Design and Simulation of BroadBand Directional Coupler Based on ADS王利斌(中国西南电子技术研究所,四川成都610036)Wang Li-bin (Southwest China Institute of Electronic Technology,Sichuan Chengdu 610036)摘要：该文简单阐述了定向耦合器的工作原理，通过对比分析微带线耦合器和带状线耦合器。
通过使用Keysight 公司的ADS 仿真软件，设计一款采用LCR 补偿方案可以兼顾平坦度和方向性的带状线双定向耦合器，最后给出该耦合器的实际电路模型和满足设计预期的仿真数据。
关键词:定向耦合器；LCR 补偿方案；带状线中图分类号：TN622文献标识码：A文章编号：1003-0107(2019)09-0038-06Abstract:This paper briefly describes the working principle of directional coupler,and analyzes microstrip-line coupler and stripline coupler.By using Keysight's ADS simulation sofware,a LCR compensation is designed which the scheme can give consideration to both flatness and directional of the stripline bi-directional coupler.Finally,the actual circuit model of the coupler and the simulation result satisfiying the design expectation are given.Key words:directional coupler;LCR compensation;stripline CLC number:TN622Document code:AArticle ID ：1003-0107(2019)09-0038-060引言定向耦合器的基本工作原理同和微带功率分配器一样[1]，同时有四个端口，分别是输入端、耦合端、直通端和隔离端。
Aactive circuit elements 有源电路元件active branch 有源电路alternating current 交流电amplitude （叫变量的）幅值振幅angnlar velocity 角频率 A..C.generator 交流电动机active length 有效长度anti-phase 反向accuracy 精确性准确性精度aperiodic 非周期性的approach 接近途径approximation 接近值arbitrary 任意的algorithm 算法alternative 替换的交替的比较方案a good-quality voltmeter 高质量的电压表Bbio-polar junction transistor双极性晶体管buffer amplifier 缓冲放大器Ccircuit components 电路元件circuit parameters 电路参数conductor 导体conductance 电导circuit diagram 电路图carbon-filament lamp 碳丝电灯circuit branch 支路complex circuit 复杂支路communication circuit 通信电路common base 共基极common collector 共集电极common source 共源极common drain 共栅极channel 信道频道coefficient 系数convergence 收敛conversely 相反的逆的coordinate 坐标criterion 标准规范decaying oscillatory function 衰减震荡函数cascade 串联串级constancy 不变恒定corresponding to 符合于Ddirect-current（D.C）circuit 直流电路displacement current 位移电流double subscript 双下标D.C machine 直流电机definition 定义distribution of B 磁场分布D.Cblocking capacitor 直流耦合电容器 D.C supply 直流供电电源dimension 维数Differentiate 求…的微分Dirichlet 狄里克雷条件discontinuity 心电图心动电流图Dead-hand 死区deem 认为认定diagram 图表简图Diagrammatically 利用图表dead-weight tester 净重测量仪Eelectrical device 电气设备electrical energy 电能energy source 能源energy converter 电能转换器 e.m.f 电动势external characteristic 外特性electrical-sheet stel 电工钢步epoch angle 初相角electrode 电极电焊条emitter 发射器发射机emitter follower 射极跟随器even 偶数的expansion 展开（式）FFrequeney 频率field effect transistor 场效应管foregoing前述的在前的Faithful 可靠的正确的GGenerator 发电机gain增益geometrical 几何学的Geometry 几何学HHeating applicance 电热器heat-treatment 热处理器harmonic current 正弦电流Hybrid-πmodel混合π型模型half-wave symmetry 半波对称harmonical 谐波的Harmonic 谐波hysteresis 滞后hysteresis over 滞后误差IIdeal source 理想电源ideal voltage source 理想电压源ideal current source 理想电流源Internal resistance 内阻instant of time 瞬时instantaneous value of current 电流瞬时值Induction-heading 感应发热initial phase 初相角in quadrature 正交In phase 同相insulator 绝缘体in parallel with 与…并联Isolation 隔离绝缘隔振input resistance 输入电阻identity 恒等式Instant 瞬时瞬间integrand 被积函数integrate 求…的积分Intuitively 直观的直觉的intermittently 间歇的JKLLoad characteristic 负载特性load resistance 负载电阻leakage current 漏电流Lag in phase behind （相位）落后Legendre 勒让德多项式linear 线性的Locus 轨迹linearity 线性度MMagnetic and electric field 直磁场metal-filament lamp 金属丝灯泡mechanical strength 机械强度Modulated by （由…）调制magnetic pole 磁极magnetic induction 磁感应强度Mid-frequency band 中频带main 电源电力网manifestation 表现Minimum 最小是最小化mutually 相互地mapping 映射Momentum 要素magnification 放大放大倍数measurand 被测量Measuring system deemed by expert 由专家认定的测试系统NNon-linear characteristics 非线性特性non-salient pole 隐（非凸）极notation 符号记号Negligible 可忽略的OOhm’s low 欧姆定律over-voltage 过电压output resistance 输出电阻Odd 奇数的单数的ohm 欧姆order 次序阶Origin 原点orthogonal 正交的直角的occasion 时机机会场合Offset 便宜oscilloscope 示波器PPrimary cell 原生电池passive element 无源原件P.D=potential drop 电压降passive circuit elements 无源电路元potential distribution 电位分布period 周期power transmission line 输电线periodic current 周期电流pole-pairs 极对数position reference direction 正（参考）方向peak value （交变量）最大值phase 相位pulsatance 谐振常数角频率phase shift 相位移polarity 极性偏极paralled circuit 并联电路paralled resonance 并联混振paralled series 混联peak 最高的最高峰periodicity 周期perpendicular 垂直的正交的polynomial 多项式的多项式power series 幂级数performance 特性precision 精确度put…away 拿开送走QQuasi-periodic 准周期的Rreceiving end 接收端reference point 参考点remote control 遥控rotor 电机转子radian 弧度rectifier 整流器resistor 电阻器reinforce 加强reproducibility 可再现性resolution 分辨率Ssecondary cell 再生电池self/mutual induction 自/互感storage battery 蓄电池series and parallel equipment circuit 串并联等值电路single-loop network（circuit）单回路网络（电路）symbolized by 用符号表示semiconductor oscillator 半导体振荡器stator 电机定子slip-ring 滑环source follower circuit信号跟随电路steaked up 叠装small signal amplifier 小信号放大器self-bias resistor 偏置电阻slot 槽series 展成级数级数set 集合sinusoidal 正弦的susceptible 敏感的易受影响的symmetry 对称symmetrical 对称的shift operator 移位算子shifting property 筛选特性static friction 静态摩擦sensitivity 灵敏度span 范围tolerance 公差Ttime-invariant 时不变的the dielectric 电介质terminal voltage 端电压term 术语项theorem 定理法则time domain 时域trigonometric 三角法的trigonometric identities 三角恒等式trajectory 轨道truncate 截断截断的Uunidirectional current 单方向电流underlying 基础的在下面的Vvoltage drop 电压降volt-ampere characteristics 伏安特性vector 向量矢量vice versa 反之亦然vicinity 附近Wwire 导线Walsh function 沃尔什函数XYZ。
quasi-experimental shift-share research designsQuasi-experimental shift-share research designs are used to identify the impact of a specific policy or intervention on a particular group or population. These designs are commonly used in social sciences, economics, and public policy research.The shift-share design is widely used to measure the impact of an intervention, policy, or treatment on a specific population by comparing the change observed in the population to that of a comparison or control group. In this design, the population is divided into two groups - those who receive the intervention and those who do not. Using the comparison group, researchers can evaluate the extent to which observed changes can be attributed to the intervention.The quasi-experimental design is similar to the randomized controlled trial, with one key difference. In a randomized controlled trial, participants are randomly assigned to either the treatment or control group. However, in a quasi-experimental design, the researcher does not have control over the assignment of participants to the treatment and control groups. Instead, they must use observational data to create the groups.Despite the potential for selection bias in quasi-experimental shift-share research designs, they are often preferred over randomized controlled trials, especially when the cost of the intervention is high or when ethical concerns make it difficult to allocate certain populations to the control group.There are different types of quasi-experimental shift-share researchdesigns, each with its own strengths and weaknesses. One of the most commonly used designs is the difference-in-differences (DiD) design. This involves comparing the change in the outcome of interest between the treatment and control groups before and after the intervention. The DiD design has the advantage of comparing the change over time between the two groups, making it more robust to time-varying confounders.Another commonly used design is the regression discontinuity design (RDD). This design involves using a cut-off point to assign individuals to the treatment or control group. The cut-off point serves as the threshold for eligibility for the intervention. RDD designs are best suited for situations where the intervention is allocated based on a continuous variable, such as income or test scores.The instrumental variable (IV) design is another commonly used quasi-experimental shift-share design. This involves identifying an exogenous variable that predicts treatment assignment but is not related to the outcome of interest. The IV design is useful when the assignment of the intervention is not random but is instead based on some external factor not related to the outcome.In conclusion, quasi-experimental shift-share research designs are essential tools in evaluating the impact of interventions and policies on populations. Researchers must carefully consider the research design they use, as different designs have different strengths and weaknesses. Despite the potential for selection biasin quasi-experimental designs, they are often preferred overrandomized controlled trials when allocating participants to a control group is not feasible or ethical.。
