Bifurcation Analysis of a Nonlinear Large Deflection Plate

华南理工大学学报(自然科学版)第29卷第3期Journal of South China University of TechnologyVol.29 No.32001年3月( Natural Science Edition )March 2001Artide ID : 1000- 565X( 2001 )03 - 0018- 04Bifurcation Analysis of a Nonlinear Large Deflection PlateWang Jing' Han Qiang2( 1. Teaching and Research Section of Eng. Graphics , South China Univ. of Tech. , Guangzhou 510640 , China ;2. College of Traffic and Communications , South China Univ. of Tech. , Guangzhou 510640 , China)Abstract : The dynamic behavior of a nonlinear large deflection plate subjected to simple harmonic exci-tation is studied. Using the Galerkin principle , the double mode model is presented in this paper. Thbifurcation behavior of the plate is examined in detail in the case of internal response. The method ofaveraging is used to derive a set of autonomous equations. The averaged differential equations are thenexamined to determine their bifurcation behavior. Finally , the results of theoretical analyses are nu-merically verified.Key words : Galerkin principle ; method of averaging ; bifurcationCLC number :O 322Document code: AFor a long time the vibration of plates hasdetail in the case of internal response. The aver-drawn great attention in the field of aeronauticsaged differential equations are then examined tcand astronautics , nuclear power and mechanicaldetermine their bifurcation behavior. Finally ,engineering. In corresponding theoretical analysissomedemonstrative examples arediscussedthe nonlinear effect should be taken into consid-through the power spectrum. Through the theo-eration in some conditions. With the nonlinearretical analysis and numerical computation it isdynamics developing and consummating gradual-observed that over the range of the load parame-ly，the research field of nonlinear science be-ters the plate exhibits the extreme complexity of acomes great abundant. The nonlinear research hasnonlinear dynamic system.been a very important branch in the vibrationfield. However , there are few archival publica-1 Governing Equationstions related to the chaotic motion and bifurcationLet us consider the deformation of a platebehavior of plates or shells.pinedasshowninFig.1，h，aandbareitsThe problem studied in this paper is the bi-thickness and widthes in x and y directions , re-furcation behavior of a plate subjected to a simplespectively. The plate is subjected to a simple har-harmonic excitation. With the nonlinear effectmonic excitation N .taken into account ，the nonlinear dynamic equa-tion is derived. By the Galerkin principle , results/N.bare presented for the double mode model of aplate. The bifurcation behavior is examined in中国煤化工Received date : April 20 , 2000MHCNMHGr1g.1、Ine plateBiography : Wang Jjing( born in 1965 ), female，lecturer ,N = Po + Prcoswt( 1)mainly researches on nonlinear dynamics and engineeringgraphics.Its lateral dynamic equation isNo.3Wang Jing et al : Bifurcation Analysis of a Nonlinear Large Deflection Plate932 WaWIt is convenient to introduce the following vari-Dv4W+ ph+ δoalableshl(φ,W)+ NW=0(2)T.( t)= h[q{ t)+ q.]ax(7)(T{1)=hqAl),τ=Vθφ+号u( W ,W)= 0One obtains the non-dimensional equations.D=Eh312(1- μ2)[$q. dq1[P0+ Prcoswτ - λ小dr2 + dτlφ,W)=a2w. ap. Wayax2-(q1 + q,)+ λAq1+q,)+(3)子q. Wλ{(q1+ q、)q之= 02 axay axayq2 + dy2a942+2-4[Po+ Prcosot - 4λ]q2 +V4 =ax2ay2T 81622q2+λ6(q1+q,Yq2=0Where μ denotes the possion ratio of the ma-terial, δo the damping coefficient ，E the elasticconstant, W the lateral deflction ，and φ thewherestress function. When the plate is simply support-Po= 0\aPo，Pr=咖(π)P,品\ aed , the boundary conditions can be written as fol-lows: when x=0 and x= a, w=z W/ax2=0;λ1=咖[()+()门wheny=0andy=b，W=a2W/ay2=0.The(9)transverse displacement of the plate is approxi-λ2=.ols[(召)+(号门]mated by using the double mode model.λ3 =‘8Eoh(A+B)((7)“()“W( x ,l)= T.( t )in"sin"Y +品TA t )sin-Txn 2uy(4)2 Two Mode Analysis Using the AveragingBy using Eq.(2 ) the stress function can be ob-Methodtained and Galerkin procedure is applied .Consider the post-buckling state of the plate(04-小{N(I)”[(“)户subjected to a static load , and let.(誓)门}r+品[(四)“+(号)门r3+" Po- λ19s =,入2(10)T1T冷=0By using Eq.( 10), a set of equations can be05(A+BX吾)“(到)“. (5)rewritten as bellow7+部。一H{N(百)”- (到)”[91 + q1+ wiq1 + λs(q1 + q.)qi+入qi +3qiq,)- Prcoswt( q1 + qs)= 0(到门}r2+ f(3)“+(引"门3+(11)92+q2+吃q2+162q3+λs(q{+.8E(A+BX()“(吾)T2T?= 02q1qs )q2 - 4PrCoswT' q2 = 0wher中国煤化工2YHCNMHG4 =F( 12)[(3)(7)个(6)(w陉=(台-4)p。+(16-到)B=The time-dependent variables are assumed to be万右数捕主+(部)门of the form20Journal of South China University of Technology ( Natural Science Edition )Vol. 29_1be obtained ，corresponding to an equilibrium11 = a1coswt + 01)(13)state of the plate , as well as non-trivial solutions ,(_ 1(q2 = a2cos( 一wt + 02)corresponding to the steady state motions due tothe nonlinearity.By using the method of averaging, a set of au-( 1 ) The trivial solution istonomous equations can be derived .a1=0，a2=0.(18).a1=。wl;[ - 2wa1 + λsa1azsir(2θ1 -(2 ) The non-trivial solutions are( aη≠0 ,202 )- 2Prasin20]( 14a)a2=0)aθ1 =一4σ1a1 + 23a1az + λsa1a2co< 201 -ai =√321-201士P)1-(号)( 19)202)+ 3λ2ai - 2Prapcos20]( 14b)sin20" = -Pra2 = -- 2oa2 + λ3azaisin( 202 -(3) Other non-trivial solutions are( a2≠0,201 )- 8P ra2sin202]( 14c)a1=0)11/;a2θ2 =→4σ2a2 + 48λ2a3 + 2:3a2ai +.σ2土2PrN4cl2/321-(4,)1'(20)λ3a2a}cod( 202 - 20)- 8Pra2cos202] ( 14d)| sin202 =- -“4 Prwhere[σ1 = w听-一w2B Numerical Simulations and Conclusions.(15)(σ2=吃一aTo support the theoretical assertions, theRunge-Kutta approach with variable step was usedThrough the following transformationto solve Eq.(8) in time domain , where the fol-x1 = a1cosθ1 ，x2 = a2cosθ2( 16)lowing parameters of the panel were regarded asy1 = a1sinθ1，y2 = a2sinθ2constantsThe set of equations( 14 ) are rewritten ash=0.01 m ,a= b=0.3 m ,μ=0.3 ,x1=一- 2ax1-22σ1 + Pr)y1-2λ( x经+ρ=2.78x 103 kg/m3 ,E=69.7 GPa δo=0.01经)y + λs(吃一经)y1 - 2λ3x1x2y2-The power spectrum diagrams are shown inFig.2 , 3 and 4. All physical quantities used in3λ6 x好+ y引)y」( 17a)plots of all the Figures are non-dimensional.j1=→[-2ay1-22σ1- Pr)x1 +2λ{(始+1.y冷)x1 +入s( x娃一y经)x1 + 2入3y1x2y2 +0.63λ6 x好+ y3)x」( 17b)。10x 0.4x2=一-2wx2-4(σ2+2Pr)y2+λs(x行一0.:yi)yz - 2λ3x1x2y1 -48λ6 x始+ y经)y2 -225(好+ y9)y2」. ( 17c)中国煤化工fx 10*b)y2 = al - 2wy2 + 4(σ2-2Pr)x2 +入{ x行x2 -YHCNMHGFig.2 The power spectrum( Pr=105 w=107 )yix2 + 2x1y1y2)+ 48λζ娃+ y经)x2 +Fig.2 shows that neither of the two modes in2λ6( x好+ y?)x2j( 17d)equation( 8 ) has 1/2 subharmonic component ,By mean万布氅握( 14a~ d)a trivial solution cani.e. a1=0 and a2=0. Fig.3 shows that only oneNo.3Wang Jing et al : Bifurcation Analysis of a Nonlinear Large Deflection Plate21a2=0). As to Fig.4 both of the two modes have2001/2 subharmonic components i.e. a1≠0 ,a2≠0.150These figures denote a bifurcation process fromthe trivial solution to the non-trivial solution of10000)the single mode , and then to the non-trivial solu-50tion of the double modes.012345;References :fxl0(a[1] LeeJ Y , Symonds P S. Extended energy approach toFig.3 The power spectrum( Pr=3x10l4 w=107 )chaotic elastic plastic response to impulsive loading[J]. IntJ Mech Sci , 1992 ,34 :139- 157.250|400[2] Moon F C , Shaw S W. Chaotic vibration of a beamwith nonlinear boundary conditions[ J ]. Non- linear1s0Mech , 1983 , 18 :230 - 240.[3 ] Paniela Dinca Baran. Mathematical models used instudying the chaotic vibration of buckled beam[ J ].00}Mechanics Research Communications , 1994 ,21 :18901245↓02458fx10'fx 10~[4] Holms P , Marsden J. A partial differential equationwith infinitely many periodic orbits : chaotic oscilla-tion of a forced beam[J ]. Arch Rat Mech and Anal-Fig.4 The power spectrum( Pr=8x104 w=107)ysis ,1981 ,76 :135- 165.mode has 1/2 subharmonic component , the trivial[5] HanQ,Hu H Y , Yang G T. A study of chaotic mo-solution is unstable and the single mode motiontion in elastic cylindrical shells[J] Eur J Mech A/takes place through bifurcation ( ay ≠0 andSolids , 1999 ,182):351 - 360.非线性大挠度矩形板中的分叉王京'韩强2( 1.华南理工大学制图教研室,广东广州,510640 ;2.华南理工大学交通学院,广东广州, 510640)摘要:研究了一个非线性大挠度矩形板在简谐激励作用下的动力学行为.利用Galerkin原理和平均法建立了这-非线性系统的双模态模型讨论了由于内共振导致的分叉行为最后利用数值分析证实了理论分析得到的结论.关键词:Galerkin 原理;平均法;分叉.中图分类号:0 322文献标识码 : A文章编号: 1000 - 565X( 2001 )03 - 0018- 04中国煤化工MHCNMHG收稿日期:200004-20作者简介坐“原( 1965- ),女,讲师,主要从事工程图学及工程非线性问题研究.