随机结构动力学及其进展

第4卷第3期福建工程学院学报Vol 4 No. 32000年6月Journal of Fujian University of TechnologyJun. 2006文章编号:1672-4348(2006)03-0283-05随机结构动力学及其进展林幼堃(美国佛罗里达大西洋大学应用随机研究中心,佛罗里达33431)摘耍:简述了随机结构动力学的发展历程,给出了日益复杂的随机结构动力学问题的解析方法,并以实例说明随机结构动力学在土木和航空工程中的应用。讨论了工程结构的各种随机损伤,包括运动失氇、首通问题和疲劳损伤。关键词:随机过程;动力学;结构;微分方程中图分类号:0211.6:TU3113献标识码:AStochastic structural dynamics and some recent developmentsLin youkunCenter for Applied Stochastics Research, Florida Atlantic University, Boca Raton, FL 33431 U.S.A.)Abstract: A brief survey is first presented of the history of stochastic dynamics, followed by techniques ofobtaining analytical solutions for increasingly more complicated problems. Practical exemples are given ofcivil and aeronautical engineering applications. Also discussed are various failure modes of engineeringstructures, including motion instability, the firsr-passage type failure, and fatigue failureKeywords: stochastic process; dynamics; structure; differential equation(4)Civil engineers- winds, earthquakes, roadHistorical development(1)Physicists-Brownian motionHousner(1941)Einstein(1905), Omstein-Uhlenbeck(1930)Solution formsWang-Uhlenbeck (1945)(2)Electrical engineers-generalized harmonicObjective: obtain correlation functions or spanalysis for communication systemstral densities)of the response from those of excitationsWeiner(1930), Khintchine (1934),Rie(1)Input statistical properties-Output statistical(1944);3)Mechanical and aerospace engineersPossible if(a)system is linear, and(b)inputslence, night vehicles excited by turbulenceareket noise2) Input probability distribution- Output pro-Rayleigh ( 1919), Pontryagin, Andronov, Vitt bability distribution(1933), Taylor(1935),C.C.Lin(1944),Possible if (a)system is linear, and(b)inputsdditive and gaussian中国煤化工收稿日期:2005-05-07CNMHG作者简介:林幼堃(1923-),男(汉),福建泉州人,美国工程院院工,W旯刀向限饥绍們明力学及其应用研究福建工程学院土木工程系吴国荣博土根据林幼整院士在纪念福建工程学院办学10周年学术讲座摘录整理,文稿未经本人审阅。)284福建工程学院学报第4卷Early works on exact probability solutions forY+h(Y,Y)+u(Y)=8(y,Y)W(t)(4)multi-dimensional nonlinear systems(restricted where h(Y, y)is damping term, u(r)is stiffnessto Gaussian white noise excitations)term,and W, (i)denotes Gaussian white noises,forwhich cross-spectral density isAdditive excitation only(a) Nonlinear stiffness, linear damping( PontryaE[W()W(t+r)]=2πK26(r),etal1933)In which EI. I means ensemble average(b)Additent for MDF systems-The FPK equation for joint stationary probabilityequipartition of kinetic energydensity P(xI, x2) of Y(u)and Y(r)is()hdm:,需+a1,x)-)+]小stant damping coof totalrk, a-6 6p)=02. Adding multiplicative excitations(a)First success( Dimentberg 1982)where x, the state variable of Y (t); *, is the state vari(b)Detailed balance( Yong-Lin 1987)able of r(o);TKas ar- 8) is the Wong-Zakai correc(c)Generalized stationary potential (Lin-Cetion term (it occ1988).Zakai correction term if exist(d)Removing the restriction of equipartition en-Into two partergy( Cai- Lin-ZhuKh(x1,x2)+(x)(6)Markov random processThe FPK equation can be rearranged as follows:The process is said to be markov if we haveProb[ X(tn)≤xX(t.:)=x1,…,X(t1)x2a2-[a(x)+a(x)132,-ax;[a(x1,x1]=Pob[X(tn)≤x|x(nt)=x。]=x2)+(x,x2)]p+πK4842P|=0(7)F(x,|xn,t-1)(>t1)(1)here X(s)is the multi-dimensional MarkovReplacing the FPK equation by the sufficient conditionsvector,and F(x, t*o, to)is the transition probaap.-[u(x1)+(x)]2=0(8a)bility distribution. This is one of the most widely studhich transition probabili[(x1,x2)+(x1,x)]p+πKg0density g(x, tIzo, fo)satisfies the following Fokker-(8b)Planck- Kolmogorov( FPK) EquationSolving for(8a):(a;q)-2a2(b4q)=0(2)[-中λ)]Wherewhere x, is the jth component of X: a, drift coefficients:ba diffusion coefficients. For the StationaryU(UMarkow process, it can be reduced toRestriction [from(8b)(bnP)=0(3)x1,x2)Kag,gn2where p(x)=g(x, t xo, to)eneralized stationary potential)中国煤化工A single-degree-of-freedom systemRcNMHGitythe controling equation of stochastic dynamic(1)Averaging techniques(generalization of Kry第3期林幼堃:随机结构动力学及其进展285stems1. Stochastic stability concepts(a) Stochastic averaging( Stratonovich, 1963;(1)Lyapunov stability with probabilityone( sam-Khasminskii,1966)-linear or weakly nonlinear stiff. pleness terms1,y.gex()1≥1≤(b) Quasi-conservative averaging--strongly non-provided‖x(t0)‖=‖xo‖≤8linear stiffness term(2) Stability in probabilityer avePob[‖x(t)‖≥ε1]≤ε2, provided(2)Slaving principle(Haken‖x(t0)‖=‖xo‖≤δMaster-slow motion(3) Stability in LSlave- fast motionExample 1: A column excited by horizontal andE[‖X(t)‖‘]≤ε, providedvertical earthquakes( Fig. 1)‖x(to)‖=‖xl≤Example 2: Column under fluctuating axial loadPo p, cos orFig. 2 A column under fluctuating axial loadThe dynamic equationFig 1 A column exited by earthquakesm++EW+[P。-P1csot0Consider one dominant mode(12)Y+2u+a2[1+61()]Y=62()(10)LetW(,t)sx()sn,toyildAfter transformationx +25wox +(wo+ ecos wt )x=0(13)Y= A(r)cos 0, 0= wo(:)+(r),andY=-A(two sin 8wherea=(2)(1)P.-P=()thd 5A =-2Ewo Asin'0 o Asin Acos OE, (t)Equation(13)is the famous Mathiew-Hill Equation-sin AE2(r)(11a) for the random perturbation, which can be replaced by:x+2如u0x+ab[1+(t)]X=0(14)2Ewo sin Acos 0 +w.,(t)where A(t): wide-band stationary random processAw cos A52(r)(1lb)Condition for stability in probability isFor systems with strongly nonlinear stiffness>φ1(2a)A(i)is replaced by total energy U(or more generally The results are shown in Fig3Hamiltonian)Example 3: Bridge in turbulent windTwo torsional and two vertical modes( Fig 4)ystem failures中国煤化工System failure can be classified, in general, intoCNMHG as system failsthree categories, namely(1)first-passage failure, (2) when : )reaches B, for the first time( Fig. 5).B,fatigue failure, (3) motion instabilityunion of failure states, which is an absorbing bound286福建工程学院学报第4卷ary.When a sample function reaches B,, it must be where to is initial time, xo =Ix,o, x2, " xoI isremoved.B,: safe space. T is the random time when tial state. R(i)satisfiesthe first-passage failure occursR+2(xn,4)aR+习(xR=0Coefficients a,, b, can be obtained from equat22[R(E1,ixCB(17)2Boundary conditions4[R(,BB(18)IR(I, B, i fo,]o)]= finite, if ](15)(1-)01|)(6)=-MRgx,co(0.-|0)第3期林幼堃:随机结构动力学及其进展287where v s restitution coefficientH=20The results for average toppling time are plotted inig. 7, in which n=(M Rg/Io),k, spectraldensity of xc, K,= spectral density of yc, K,/K,B/H=0.50. 5, and ur =average toppling timeK=0.00050.050.100.501.00B/H=Fig8 Average toppling time vs. size scaleConcluding remarks(1)The present review is focused on analytical000050.0010005000100olutions. The important Monte Carlo simulation techniques are not covered, such as the works by ShinozuFig7 Average toppling time vs, base excitation levelka, Schueller, and Pradlwarter, etcSolld line: horizontal exctation only dotted Hne: com-(2)Recent works by Amold and his associates onbined horizontal and vertical excitationsdynamical systems are not covered(3)Numerical solutions, such as those given byNaess, and Johnson, Bergman and Spencer, Kloeden etal. etc. also are not covered本文作者简介林幼堃教授,我校1941届校友,美国国家工程院院士、做国国家工程院院士。现任美国佛罗里达大西洋大学“希密德杰出学者讲座”教授,福建工程学院客座教授。曾多次被意大利帕维亚大学、美国土木工程师协会、美国机械工程师协会等机构、组织授予各类荣誉奖章,曾荻德国洪堡高级科学家奖,并被编入世界名人录、美国名人录、国际教育名人录。是随机结构动力学创始人之一,其主要著作《结构动力学的概率理论》为本科目最常引述的经典,他创建并担任主任的“美国佛罗里达大西洋大学应用随机图为林幼篁教授在母校—一福建工学研究中心”,被国际公认为这一学科最权威的研究机构之一。程学院办学110周年庆典大会上发言中国煤化工CNMHG