第4卷第3期福建工程學(xué)院學(xué)報(bào)Vol 4 No. 32000年6月Journal of Fujian University of TechnologyJun. 2006文章編號(hào):1672-4348(2006)03-0283-05隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)及其進(jìn)展林幼堃(美國(guó)佛羅里達(dá)大西洋大學(xué)應(yīng)用隨機(jī)研究中心,佛羅里達(dá)33431)摘耍:簡(jiǎn)述了隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)的發(fā)展歷程,給出了日益復(fù)雜的隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)問題的解析方法,并以實(shí)例說明隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)在土木和航空工程中的應(yīng)用。討論了工程結(jié)構(gòu)的各種隨機(jī)損傷,包括運(yùn)動(dòng)失氌、首通問題和疲勞損傷。關(guān)鍵詞:隨機(jī)過程;動(dòng)力學(xué);結(jié)構(gòu);微分方程中圖分類號(hào):0211.6:TU3113獻(xiàn)標(biāo)識(shí)碼: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中國(guó)煤化工收稿日期:2005-05-07CNMHG作者簡(jiǎn)介:林幼堃(1923-),男(漢),福建泉州人,美國(guó)工程院院工,W旯刀向限饑紹們明力學(xué)及其應(yīng)用研究福建工程學(xué)院土木工程系吳國(guó)榮博土根據(jù)林幼整院士在紀(jì)念福建工程學(xué)院辦學(xué)10周年學(xué)術(shù)講座摘錄整理,文稿未經(jīng)本人審閱。)284福建工程學(xué)院學(xué)報(bào)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)中國(guó)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:隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)及其進(jìn)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中國(guó)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福建工程學(xué)院學(xué)報(bào)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+習(xí)(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:隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)及其進(jìn)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本文作者簡(jiǎn)介林幼堃教授,我校1941屆校友,美國(guó)國(guó)家工程院院士、做國(guó)國(guó)家工程院院士?，F(xiàn)任美國(guó)佛羅里達(dá)大西洋大學(xué)“希密德杰出學(xué)者講座”教授,福建工程學(xué)院客座教授。曾多次被意大利帕維亞大學(xué)、美國(guó)土木工程師協(xié)會(huì)、美國(guó)機(jī)械工程師協(xié)會(huì)等機(jī)構(gòu)、組織授予各類榮譽(yù)獎(jiǎng)?wù)?曾荻德國(guó)洪堡高級(jí)科學(xué)家獎(jiǎng),并被編入世界名人錄、美國(guó)名人錄、國(guó)際教育名人錄。是隨機(jī)結(jié)構(gòu)動(dòng)力學(xué)創(chuàng)始人之一,其主要著作《結(jié)構(gòu)動(dòng)力學(xué)的概率理論》為本科目最常引述的經(jīng)典,他創(chuàng)建并擔(dān)任主任的“美國(guó)佛羅里達(dá)大西洋大學(xué)應(yīng)用隨機(jī)圖為林幼篁教授在母?！桓＝üW(xué)研究中心”,被國(guó)際公認(rèn)為這一學(xué)科最權(quán)威的研究機(jī)構(gòu)之一。程學(xué)院辦學(xué)110周年慶典大會(huì)上發(fā)言中國(guó)煤化工CNMHG