STABILITY ANALYSIS OF NON-LINEAR STOCHASTIC DYNAMIC SYSTEM BASED ON NUMERICAL SIMULATION

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1 Jounal of Theoetcal and Appled Infomaton Technoloy th Febuay 13. Vol. 48 No JATIT & LLS. All hts eseved. ISSN: E-ISSN: STABILITY ANALYSIS OF NON-LINEAR STOCHASTIC DYNAMIC SYSTEM BASED ON NUMERICAL SIMULATION IAOHONG LI School of Scence Qha Unvesty Qha 1616 Helonjan Chna ABSTRACT The stablty of nonlnea stochastc dynamcal system s a theoetcally pomnent poblem as t s of eat theoetcal and pactcal snfcance to many pojects n ealty. Fstly the autho establshed the basc fomula. Secondly he eseaches the stablty theoy of non-lnea stochastc dynamcal system and dscusses the stablty cteon of null soluton stablty and nonlnea stochastc dynamc system. Fnally he conducts numecal smulaton analyss fo knetcs esponse and stablty status of ennee stuctues unde sesmc condtons. The smulaton esults povdes theoetcal bass fo the pomoton of eathuake esstance of ennee stuctues. Keywods: Numecal Smulaton Nonlnea Stochastc Dynamcs SmulatonStablty 1. INTRODUCTION At pesent the applcaton of nonlnea dynamcal system theoy n the felds of mathematcs physcs mechancs and economcs has become hot spot n focus. Theefoe t s necessay to et to know the ntenal mechansm of powe system nonlnea phenomena n ode to conduct theoetcal analyss and numecal analyss. But many poblems eman to be solved due to the complcty of non-lnea systems[1]. At pesent eseaches put much emphass on phenomenon of defnte systems. In fact t s unavodable that most systems ae subject to the ntefeence of stochastc noses and defnte system s dealzed system thus stochastc systems could eveal the natual laws n a moe deepon and tue way[]. Many eseaches pove that nose could detemne the evoluton of system due to the nteactons n between whch pobably totally destoy system stuctues o enende the system n ode nstead. Theefoe t s necessay to et pofound undestandn of the ntenal mechansm of nonlnea stochastc phenomena and movement pattens so as to lead t to a favoable decton. It s a eseach of eat scentfc snfcance and pactcal udance value[3]. The exstn stochastc systems can be classfed to thee cateoes: system wth stochastc stmulaton system wth stochastc ntal condtons and system wth stochastc physcal paametes[4]. In ealty thee exsts many uncetantes n nstallaton measun mateal and poducton sectos whch could be descbed by stochastc vaables wth some statstcal popetes so the numbe of the thd cateoy s the laest. Due to the nceasn demand fo pecson and accuacy of actual model stochastc system especally those wth stochastc paametes have been appled moe wdely to descbe the dynamc elatonshp between vaous objects[5]. The th centuy has wtnessed the boomn of stablty theoes. Lasalle poposed the Lasalle nvaance theoy based on Lyapunov s stablty theoy and Bkhoff s nvaant set. Moe than 5 yeas ao wth the ntoducton of stochastc nteal by Itó stablty theoy of stochastc dynamc system has also expeenced apd development but theoetcal esults about the stablty of nonlnea stochastc dynamc systems ae stll fa fom enouh especally those wth stochastc paametes. The stochastc dynamcal systems wth some known statstcs nfomaton ae much close to actual models thus the stablty study s of eat pactcal and theoetcal snfcance and deseves moe attenton[6]. The exstn methods to deal wth stochastc dynamcal systems wth stochastc paametes nclude: Monte Calo method whch s easy and wdely adopted but tme consumn; stochastc fnte element method whch s tme savn but on the pemse that the stochastc vaable s small; and othoonal polynomal appoxmaton method based on othoonal polynomal expanson theoy whch s fee fom the lmtatons of the pevous two methods and wdely appled nto the study of evoluton stochastc esponse bfucaton and chaos of stochastc systems. 18

2 . BASIC EQUATION OF NONLINEAR The dynamcs model wth multple feedom deee could be expessed as follows[7]: M & & + C + f ) = F t) 1) Whee M s mass matx; C means dampn f ) ndcates ecupeablty functon; & & epesents acceleaton; & s speed; s F s exctn foce vecto and F snfes andom vecto. F t) = MIx && t) matx dsplacement; nonlnea stochastc system model can be expessed as follows: M & & + C + T I = [ 11 L1] f ) = MIx && t) x&& Whee acceleaton of the exctaton foce. T T T Y ) = & ) epesents the the state euaton of nonlnea stochastc dynamc systems can be conveted to the follown fom[8]: Whee & Y = A Y t) + BF t) 3) 1 & 1 M C M f ) A Y t) = & 1 M B = Y = 1 1 j = f = a = 1 1 j j the mathematcal model of nonlnea stochastc system can be expessed as follows: d dt f ) + ) A t) = j j 4) Whee vecto; p t) j ) epesents the component of state snfes Makov pocess vecto s the coespondn pobablty densty functon. The contol euaton of expessed as follows: x [ f j ) p )] 1 [ x xj The pobablty densty functon subject to the follown euaton: p t) lm ) p t) = f j ± [ Gj ) p t)] lm = ± x j can be ) p )] = p t) 3. STABILITY THEORY OF NONLINEAR 3.1 Null Soluton Stablty Defnton 1: set andom vaable > t ) [1] 5) s 6) 7) s a s the soluton of euaton 5). If thee exsts neatve moment ndex the expected null soluton moment ndex of euaton 5) s stable; on the contay null soluton moment ndex of euaton 5) s unstable[9]. Fue 1 and Fue ae the coespondn schematc daam. Fue 1: Schematc Daam About The Stablty Of Null Soluton Moment Expectaton Index 19

3 f Fue : Schematc Daams About The Instablty Of Null Soluton Moment Expectaton Index If t s a andom numbe fue 1 shows that the absolute expectaton moment ndex E t ) t of zeo. Fue shows that f expectaton moment ndex t ) ) t would ncease to nfnty. would decease to the absolute E t ) of 3. Stablty Cteon Lemma: fo nonlnea stochastc dynamcal systems [1] : d t ) = F t t)) + G t t)) B t ) dt t ) = 8) Soluton of the euaton 8) s defned t) = t ) as fomula s tue:. Assume the follown b) If thee exsts a nomal numbe λ whch s subject to the follown LV t ) λv t ) fomula: follown neualty[5] s tue: Whee E x c. then the t t ) 1 λ x e c 11) LV t ) = Vt t ) + Vx t ) F t ) + 1 G 4. SIMULATION ANALYSES ABOUT THE STABILITY OF NONLINEAR t ) V Take enneen stuctues eathuake fo nstance F epesents sesmc exctaton. The defomaton of enneen stuctues s a pont of focus. Poammn smulaton poams wth MATLAB softwae the coespondn smulaton esults ae as follows: 4.1 The Dynamc Response of Enneen Stuctues Unde Sesmc Condtons Fue 3 and 4 show the dynamcs smulaton esults of enneen stuctues unde sesmc exctaton. Fue 3 s the esponse esult of enneen stuctue defomaton unde sesmc exctaton whch chanes damatcally at ntal stae and then stablzes adually. Fue 4 s the esponse esult of enneen stuctue hysteess defomaton unde sesmc exctaton whch chanes damatcally at ntal stae and then stablzes adually. xx t ) d1 x V t ) d x 9) d Whee 1 and d ae constants and eate than V t ) zeo. s a smooth functon; a) If thee exsts a nomal numbe λ whch s subject to the follown LV t ) λv t ) fomula: follown neualty[11] s tue: E x c. then the t t ) λ < x e c1 1) Fue 3: Response Cuve Of Enneen Stuctue Defomaton 11

4 safety and stablty of enneen stuctues. Theefoe t s vey pomsn n pactces. Fue 4: Response Cuve Of Enneen Stuctue Hysteess Defomaton 4. Stablty analyss Stablty smulaton of nonlnea stochastc dynamc systems s shown n fue 5 fom whch we could notce that enneen stuctues ae unstable unde sesmc condtons and they ae stable fo lon at no tme. Theefoe t s eued to enfoce the stuctue n ode to mpove the sesmc capacty of stuctues and mpove the stablty of stuctues. Fue 5: Smulaton Cuve About The Stablty Of Enneen Stuctues 5. CONCLUSION In ths pape the autho conducts systematc eseach on the stablty of nonlnea stochastc dynamcal systems. He analyses the mathematcal models of nonlnea stochastc dynamcal systems and stablty theoem and cteon. Moeove the autho conducts numecal smulaton on the enneen stuctues unde sesmc condtons. The esults eveal the stablty deee of stuctues whch povdes vald theoetcal bass fo the mpovement of sesmc capacty and ensues the REFERENCES: [1] Chen Janbn L Je oveall senstvty analyss of stuctue stochastc nonlnea esponse JCM: Jounal of computatonal mechancs Vol. 5 No. 8 pp. 169 ~176. [] u Yon Ma Shaojuan Zhan Hun Stablty analyss of Stochastc Busselato system wth andom paametes CS: Chnese Scence: Physcs Mechancs Astonomy Vol. 4 No pp. 13 ~19. [3] Wan Wenje u We Fou moment cente CN mplct dffeence decomposton of stochastc extenal exctaton nonlnea system FPK euaton JDC: Jounal of dynamcs and contol Vol. 9 No. 11 pp [4] E Guokan Yao Webn Subspace method to obtan the pobablty densty functon soluton of lae-scale nonlnea stochastc dynamc system JSM: Jounal of sold mechancs Vol. 31 No. 1 1 pp [5] Jan Guo "Towads Automatc Analyss of Softwae Reuement Stablty" AISS: Advances n Infomaton Scences and Sevce Scences Vol. No. 4 1 pp. 33 ~ 4 [6] Q Lan Suln Ja "Reseach on Two-stae Supply Chan Oden Statey Optmzaton Based on System Dynamcs" AISS: Advances n Infomaton Scences and Sevce Scences Vol. 4 No. 3 1 pp. 16 ~ 4 [7] Yan Ha Hu Wan "Reseach on QoS based Dynamc Web Sevce Systems" AISS: Advances n Infomaton Scences and Sevce Scences Vol. 4 No. 3 1 pp. 65 ~ 633 [8] Hejun Yao Zhen Yuan "Exponental Stablty Contol fo T-S fuzzy Nonlnea Stochastc Netwoked Contol Systems" IJACT: Intenatonal Jounal of Advancements n Computn Technoloy Vol. 4 No. 3 1 pp. 5 ~ 14 [9] ZHANG Shen-wen NING Tao FENG Rufan "A new smulaton method based on dscete event dynamc system" JCIT: Jounal of Conveence Infomaton Technoloy Vol. 7 No. 7 1 pp. 58 ~

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