DYNAMIC ANALYSIS OF FLEXIBLE BODY WITH DEFINITE MOVING ATTITUTE

Applied Mathematics and Mechanics (English Edition), 2006, 27(1):133- -140CEditorial Committee of Appl. Math. Mech., ISSN 0253-4827DYNAMIC ANALYSIS OF FLEXIBLE BODY WITHDEFINITE MOVING ATTITUTE *YANG Yuan-ming(杨元明)12,ZHANG Wei(张伟) ,SONG Tian-xia(宋天霞)，CHEN Chuan yao(陈传尧)'(1.College of Civil Engineering and Mechanics, Huazhong University of Science andTechnology, Wuhan 430074, P.R.China;2.Department of Civil Engineering,Nanyang Institute of Technology, Nanyang 430074,Henan Province P. R. China)(Communicated by YUE Zhu- feng)Abstract: The nonlinear dynamic control equation of a fAexible multi-body system withdefinite moving attitude is discussed. The motion of the aircraft in space is regarded asknown and the infuence of the fexible structural members in the aircraft on the motionand attitude of the aircraft is analyzed. By means of a hypothetical mode, the defor-mation of flexible members is regarded as composed of the line element vibration in theaxial direction of rectangular coordinates in space. According to Kane' 8 method in dy-namics, a dynamic equation is established, which contains the structural stiffness matrixthat represents the elastic deformation and the geometric stifness matrix that representsthe nonlimear deformation of the deformed body. Through simplification the dynamicequation of the influence of the planar flexible body with a windsurfboard structure onthe spacecraft motion is obtained. The numerical solution for this kind of equation canbe realized by a computer.Key words: flexible body; line element vibration; dynamic equation; Kane's methodChinese Library Classification: 0312000 Mathematics Subject Classification: 70k75; 70E55Digital Object Identifier (DOI): 10.1007/s 10483-006-0117-1IntroductionThe methods of dynamic modeling using the Kane method [1- 7 matured with each passingday. These methods all used the elastic deformation of discrete points of a continuum areaas generalized coordinates, with the dynamic stiffness matrix obtained using the finite elementmethod. This kind of stifness matrix is essentially the geometric stifness matrix of the fexiblemulti-body system’ s rotation characteristics. It has been proved in practice that this kindof method is not applicable to large scale spatial structures. In contrast, from the point ofview of the steady-state theory, the theory of modality is capable of more effectively solvingthe problem of motion. -induced stiffness. Reference [8] used the substructures method andLagrange multipliers, but the dimension of the equation is increased. With the constrainttaken into consideration, the simulation susceptibility to constraint is increased. Later, inRefs. [9- 13], its most general motion was represented by thegenmetrir sifnoss matrix, involvingthe approximate expression related to the rotational中国煤化工with the modalstifness matrix to obtain the stiffness matrix of arbit:CN M H G a combinationof the above-mentioned methods, with the fexible multi-body system with definite moving* Received Mar.6,2004; Revised Aug. 16,2005Project supported by the Natural Science Foundation of Henan Province (No.0311011100)frending author YANG Yuan-ming, Associate professor, Doctor, E-mail:yym7823@sina.com134YANG Yuan-ming, ZHANG Wei, SONG Tian-xia, et al.attitude in mind, the authors have made an analysis of the infuence of the fexible structuralmembers of an aircraft on its attitude. Taking into account elastic deformation, so deformationis description by employing the method of hypothetical modality and regarding the fexiblemembers as made up of the line element vibration in three orthogonal directions in space andperforming dynamic modeling with the Kane method in dynamics, dynamic modeling is alsoperformed for the influence of the flexible members of a spacecraft with definite attitude on itsmotion. After obtaining the geometric nonlinear stiffness and structural modal matrices usingthe finite element method and the theory of modality, the differential equation for motion isobtained.1 Dynamic Modeling for Spatial Flexible Body1.1 Description of deformationAs shown in Fig.1, R is a coordinate system fixed to a known moving body, with 2] ,X2,and x3 as mutually perpendicular coordinate axes fixedP(5.5253)to R, passing through 0, and parallel to unit vectorsr1,r2,r3. Let P'(x1,Xz,x3) be the point of R of spa-P(x1,x*x)tial flexible body when it is not deformed P(51,52, ξ3)be the point of R of spatial fexible body when it isdeformed. Then the position of P can be expressed ink^Bterms of displacements u1, U2, U3 as的好u =u1(x1,x2,x3,t)r1 + u2(x1, x2, X3, t)rz+ u3(I1,Z2, T3, t)rg.Fig.1 The spatial flexible bodyThe defornation of an arbitrary elastic deformedbody of B along the straight line length can be shown asx= 6*{1+ Z(K; u (xu1,x,1B}/2o.(1)j=1Let(7),z,x3,t)兰1+ sz(xm+(un,x,zs,UI,where Kj is a constant correlating with j,i,j= 1,2,3,ifi= j, then Kj = 0,rij = σ; otherwise, .Kj=1,xj=xiThe coordinates ξ1, ξ2 and ξ3 relative to xI, $2,I3 and U1, u2, u3 have the following relation-ship:ξi=x; + u(x1,I2,x3,t)(i= 1,2,3).(3)Diferentiating equations (1) and (3) with respect to t yieldsi =- (xu,(x,)-I/2[u(x1,z,1)-1/n2 (x,zN,t,)do},(4).(一1/。1Dt'where i is the derivative of xi respect with t, 2中国煤化工of the functionJ,i,j= 1,2,3,ifi= j, then x;i=ξi;xjl = σ, otherw:YHCNMHG1.2 Line element vibration modeWe shall now consider the description of deformation: let ψ represent the vibration modeof an clastic-body, qj the generalized coordinate to which the vibration mode corresponds, andthen the deformation of the corresponding line elements can be expressed as follows (11:Dynamic Analysis of Flexible Body with Definite Moving Attitude135nDirection x1: ui(σ,x2,x3,t)= > 4ij(σ,xz,x3)q;(t) (i=2,3), .(5)Direction 22: u4(x),0,23,t)= 2 4Pis(x1,0,x3)gs(t) (i= 1,3),(6)j=1Direction xz: u;(x1,T2,σ,t)=,Pir(x1,J2; o)g;(t) (i=1,2).1.3Generalized partial velocity, generalized partial acceleration and generalizedinertial forceUsing formulae (5)-(7), the formula (4) can be witten as the fllowing form:n=- [(xnxsx,t),1/{2x_ [[J(xu, x21zxs,t)]-1/2n.n_322 i汇Kkfk(x,x1)Hx(xu, x21, x)ido},(8)i=1 j=1 k=1where (71,2,50)=2说KxkPhe(x1t, x21, x3)}4;(21,2,2, x)], andi,j,h= 1,2,3.i=1j=1 k=1Ifi= k, then K'k = 0,xi = σ; otherwise Kk = 1,xil = xi; ifi=l, then xi =σ, 2x't = 5i;otherwise xi=Ti = Ti.“p!”represents the derivative φ respect with σ.The velocity of P in the Newtonian reference frame N can be written asVβ =V^ +wK x {[x1+ 2pus(1,x2, x3)g(t)]rn+ [x2 + 2 92j(x),2, x)qi(t)]rnz +[x3+ 2 (3(x,2, xs)(t)]r3}j=3+ i1r1 + i2r2 + i3r3,(9)where V represents the velocity of 0 in the inertial reference frame N, and wA; the angularvelocity ofO in N. As the spacecraft fies in a specified orbit, VN and wh can be consideredas known motion, which can therefore be expressed by a set of special functions asVW = Vi()r1 + V:(t)r2+ Vx()x; wR = wr(t)r1 + w2(t)r2 + w3()r3，(10)Using Gi1, q2,... ,9n as the generalized velocities and taking into account Eqs.(9) and (10),we have the generalized partial velocity of P.vN =- r(J(x1>1x,tes, t)-1/{xjKk$ke(ru; TaL ran)ho; .1... Po1. x3)y; }do},j=1 k=1中国煤化工TYHCNMHGwhere i,j = 1,2,3...,n, if j = k then Kk = 0; otherwise k- 1, 1i= i, ihen Til = σ,Ii = 5q; otherwisexu =xil = Ti.In order to solve linearity 80 that a theory good for small motion of the body relative to Rcan be generated, we perform such linearization and place a tilde over 9r and 9j. Substitution of136YANG Yuan-ming, ZHANG Wei, SONG Tian-xia, et al.(5)-(7) into (2) leads to J:(x11, x21, X3i,t)= 1; wherei= 1,2,3; ifi= I then Ii = σi; otherwisexil = xi. Now we introduce the notationsaijm(z1,2,x3) =/。”它Kx9k;(xrt, x21, x()(1,2I, x)]da,where m = 1,2,3, and Qij1 = Qaij,Qij2 = Bi,Qij3 = ij,ifm= k, then 2xml =σ,Kk = 0;otherwise, Kk= 1,xml = Tm.The linearized partial velocity of P can be written asnV: =-rn(> Qijqi-x1)-r2( BjQi-x2)-r3(> ; 7ij9i - x3).j=1The linearized velocity of P becomesVp =rn{Vi + u>{[r3+ > z 43g:()] -w3[x2 +'，42j9j]+ > 4ljG}+r2{Vz +u3[x1+ 2 φ1j9;] -w1[x3 + 2 P3j9s] + 242z9}+ r3{Vs +wr[x2+ > 42jq}] -w2[1 +41jq;]+ > P3j9j}. .(11)The acceleration of P in the inertial reference frame can be obtained by diferentiating (11),that is, a% = NdVp\/dt = RdVN /dt +uX x Vp, where NdVH /dt is a derivative of Vp~ withrespect to in the Newtonian reference frame N, RdVp /dt is a derivative of VpN with respectto in the movement reference frame R.1.4 Generalized inertial force, generalized active force and dynamic equation 11,12]1The generalized inertial force corresponding to the i -th generalized velocity is given byF* = vNF*, where F* =- fg pa}dxrdxzdxg. Thus F* = - Je pVN afdx1dxzdx3.Now we consider error due to nominal motion- induced stifness. The resulting error will becompensated by taking the nonlinear stiffness into consideration while finding the active force.For a description of the nonlinear deformation of the body, we shall analyze the stifness matrixof the following body.By taking into account an arbitrary point of an arbitrary element e of body j, the displacement-strain relation can be expressed asEa,8= (Wa,+ Wg,a+ 2 W,aW.,s),(12)γ=1where Wa,3 = 8Wa/8xp, Wa represents displacement component while xβ is the positionalcoordinate component. The strain array matrix made up by Wa,; isε= [e11, E22, E33, 2ε12, 2ε23, 2Ee13](13)中国煤化工Using Eq.(12), we have3MHCNMHGea.g= jlWa,+ Wa,a+ 2 (W.,aW7,. + Ww.waW.3), .(14)| e= LW,Dynamic Analysis of Flexible Body with Definite Moving Attitude137where L= L°+L'; L',L0, is the 6 x 3-order matrixes, L0 represents the partial derivativewith respect to the positional coordinate Trx, that is,[81 0a2 0|r0=) 03|8] 0|'0830203 0 81」W1,181W2,181W3,181W;,202W2,202W3,202W1,303Wz,383W3,303I'=W,182+ W1,281 W2,182 + W2,281 W3,182 + W3,201W1,203+ W1,302 W2,203+ W2,302 W3,283 + W3,382[W1,381 + W1,103 W2,381 + W2,103 W3,381 + W3,103]where 8a = 8()/8xa，I' contains the quadratic of the arithmetic product of a and Wa,and the stress is denoted by σ = [011, 022,033,012,023,031」. Considering the stross strainrelation of linear-elastic material (the initial stress and resistance neglected), σ = Hε, whereH is an elastic matrix. Suppose N represents the element shape function, then the elementnode displacement satisfies W = Nu (u represents the element node displacement).Now strain energy is generated by the elastic body as follows:Pe=| e'odB= fu"(LN)T[H(LN)u]dB = uT Ku.where K = Jg, (LN)TH(LN)dB.Substitution ofL= L°+ L' into K yields: K = Koo+ Ko1 + K1o + K11, whereKoo= | (L°N)TH(L°N)dB， Ko1= | (L°N)TH(L'N)dB,JB,K1o=(L'N)TH(L°N)dB，K11 = .(L'N)T H(L'N)dB.B;LetKe= Koo,Kc= Kou+ K1o+ K1,(15)where Koo is an elastic deformation stiffness matrix (or structural stifness matrix), KG is thecorresponding nonlinear deformation stiffness matrix, or geometric nonlinear stiffness matrix.The generalized active force thus induced can be written as 14]F=-2ijqj(i= 1,2...n),(16)j=1where入ij is a generic element of the modal stiffness matrix φ Kφ (中is the characteristicvector matrix) of structural analysis.According to Kane's equation, the dynamic equation is written as中国煤化工F" +F= 0.(17)CNMHG2 Dynamic Equations of Planar Board Flexible BodyThe influence of structures with fexible members on the spacecraft was discussed earlier.Actually, these fexible members often have planar board structures. If we restrict our analysis138YANG Yuan-ming, ZHANG Wei, SONG Tian-xia, et al.to a plate whose middle surface is inextensible, then theP(52)P(J12)lengths of these line elements remain constant during defor-mation of the plate. Because the motion of the spacecraftcan be regarded as known, and a known vibration within thesurface or it can be said that in this direction the vibrationfunction possesses consistency. Section one will be simplifiedand revised with respect to such a case (see Fig.2).Let the unit vectors of R be r1,r2,rs, P' be the point ofR when it is unreformed, P be the point of R when it is de~Fig.2 Plate flexible bodyformed. Then ub = u1(x], x2)r1 + u2(x1, x2)r2 + u3(I1, x2)rs.The relations of the coordinates 51, 62 with X1, x2 and u1,U2 are61=x1 +u[(x1,c2,), 62 =x2 + u2(21,x2,t)while the generalized inertial forceF?=- pV:"a%dxrdx2.(18)Neglecting all nonlinear terms in Gi,主;, one can derive the generalized inertial force using Eq.(18)as follows:F=-> Esijj- >(w1 Ezij - w2E1j)4;j=1- 2I-( +u2V% -usk)Cij - (wrw2 - u的)Cas + (u经+ u吲)CIis- (% + w3Vi -wnV3)Dj - (i3s + w1w2)D1j + (w弓+ w?)D2ij+ (山1 + w2us})Ezj + (w3W1 -的2)E1ij - (wi + w引)Esij]qj- (vs + w1Vz - w2V1)A; - (的+ w2zw3)B2i - (w3W1 - i的2)B1i(k= 1,2,3;i= 1,2...n)(19)where A: = js43ipdx1dx2; Bki =_ Js TkP3ipdx1dx2; Cij_ = Js Qijpdxndx2; Ckij. =Js xkQijpdxrdx2; Dij = js Bxjpdx1dx2; Dkij = Js xeB:jpdx]dx2; Ekij = Is 4P3iPkjPdx1dx2 (h =1.2,3;i,j= 1,2....; S is the area of the plane board).Each coefficient of qj in Eq.(19) is called dynamics stiffness coefficient. The generalizedactive force F is written as Eq.(16). Substitution of (19) and (16) into (17) yields: Mxai +Cxd9i + Kxdqi = Qrd, where Mxd,Cxd,Kxd are the n X n order matrixes. (Mxd, Cxd, Kxd,analogous to the mass matrix, damping matrix, rigidity matrix of the stucture separately), Qxdis nx 1 matrix (analogous to the generalized force), q, q and q are n X 1 matrixes. Observing thecofficient of qj, we can have the coeficient ofq1 :-21=1 E31j=-E311- E312-...- E31n; the .cofficient of2:- 21=1 E32j= - E321 - Es22 -..- E32n; the cofficient ofqn:- 2}=1 E3nj=- E3n1 - E3n2-...- E3nn.Mxd can be written as following forms:中国煤化工「-E311 - E321 ... - E3n1Mxd=-E312 - E322 ... - E3n2 |MHCNMHG=- [Esji](ij= 1,2...n.l-E31n -E32n ... - Eann」The rest may be deduced by analogy, we have“Dynamic Analysis of Flexible Body with Definite Moving Attitude139[ (山1 E211 -w2E11) (w1 E221 -w2E121) ... (w1 Exn1 - w2Ern1)(w1E212 - w2E112) (w1 E222 - w2E122)(w1E2n2 - w2EIn2)Czd=-[(w1 E2in -w2B11n) (w1 E22n -w2E12n) ... (w1 E2nn - w2Ernn)]=- [(w1 E2ji - wurEiji)],Kxd=-[-(V + w2V3 - w3V2)Cj - (u1w2 -的3)C2ji + (w经+峙)Cjt- (2 + wsVi - wnVg)Dji - (的+ wrw2)Djti + (w号+ w})D2ji .+ (wr + w2u3)Erzji + (w361 - u2)Erji - (w2 + w岭)Exzji + Ajil,Qxd = [(V3 +w1Vz - w2Vi)Ai + (w1 + w2u3)Bri + (W3W1 - i的2)Br] (i,j = 1,2...,.n).3 Examples of Numerical SimulationTwo rectangular board systems (see Fig.3), of length a1 = a2 = 2.00m, widthb1 = b2 =1.20 m, thickness h1 = h2 = 0.002 m, density P1 = P2 = 2766 kg/m' , Young's modulus for allmembers is E] = E2 = 70GPa, simulation timet=80s. The angular velocities of the rectangular boardin the inertial coordinate system of R are given as fol-啄↓茂lows:aw1 =01 = 4rad/s,w2 =02 = 0.1(t - 20/π sin(π/20)t) rad/s.①②bThe vibration modes of the rectangular board sys-tems were obtained by means of the finite elementprogram. Figures 4 and 5 show respectively the defor-mation of the central point each of rectangular board1 and 2. It can be seen from the figures that the defor-Fig.3 Rectangular board systemmation of the uniformly rotating rectangular board 1is much smaller than the variable rotating rectangular board 2. This is because the tangentialinertial effect (tangential acceleration) of the rectangular board has augmented the deforma-tion of the rectangular board. At the same time, as board 1 and board 2 revolve in the samedirection, the revolution has a positive superposition effect on the deformation of board 2, thusthe deformation is increased.0.20-0.1-0.2F0.104-0.40.05--0.6--0.8-" -0.052040260/st/sFig.4 The deformation of the centralFig.5 The deformation of the centralpoint of rectangular中国煤化工ard4 ConclusionMHCNMHGFor the fexible multi-body system with definite moving attitude, an analysis is made of theinfuence of the fexible structural members of an aircraft on the attitude of the aircraft. Bymeans of the method of hypothetical modality, the flexible structural members are regarded140YANG Yuan-ming, ZHANG Wei, SONG Tian-xia, et al.as composed of linear element vibration in three orthogonal directions. Dynamic modeling isperformed with the Kane method in dynamics. By employing the finite element method andthe theory of modality, a dynamic analysis is made of the infuence of the fexible structuralmembers of a spacecraft with definite attitude on its motion. The motion-induced geometricnonlinear stiffness and structural modal matrices have been obtained.Dynamic simulation computation has been made for an aircraft with planar board struc-tures. The results show that by decreasing the inertial effect of the fexible body and inverselysuperposing the relative movement of all flexible bodies, the deformation of the flexible bodycan be reduced.References[1] Banerjee A K, Kane T R. Dynamics of a plate in large overall motion [J]. ASME J of AppliedMech, 1989, 56(1); 887- 892.[2] Kane T R, Ryan R R, Banerjee A K. Dynamics of a cantilever beam attached to a moving basel[].J of Gridance, Control and Dymamics, 1987, 10(2): 135-151.[3] Kane T R, Ryan R R, Banerjee A K. Reply by authors to K W London [J] J of Guidance, Controland Dymamics, 1989, 12(2):286 287.[4] Levinson D A, Kane T R. Autolev- a New Approach to Multibody Dynamics [M]. Schiehlen W(ed). Mutlibody Systerms Handbook. Springer-Verlag, Berlin, 1990, 81- -102.[5] Roberson K E, Schwertassek K. Dynamics of Muti-body System[M] . Springer- Verlag, New York,1998,122 -156.[6] Rosenthal K E, Shermand M A. High performance multi-body simulations via symbolic equationmanipulation and Kane's method [J]. J of the Astronautical Sci, 1986, 34(3):223 239.[7] Likins P W. Geometric stifness characteristics of a rotating elastic appendage[J]. InternationalJ Solid and Structures, 1974, 10(2):161-167.[8] Wu S C, Haug E J. Geometric nonlinear substructuring for dynamics of flexibly mechanical Bys-tem[J] . J for Numerical Method in Engineering, 1989, 44(3):135-146.[9] Zeiler T Buttrill C. Dynamics analysis of an unrestrained rotating structure through nonlinear sim-ulation[C] . AIAA 29th Structure Structural Dynamics and Materials Conference. Williamsburg,Va, 1988,18 20.[10] Banerjee A K, Dickens J M. Dynamics of an arbitrary flexibly body in large rotation .and .trans-lation[J]. ASME J of Mech,1990, 13(2):221 -227.[11] Banerjee A K, Lemak M E. Multi-flexibly dynamics capturing motion induced sifness[J] .Trans-action of the ASME, 1991, 58(4):113 -121.[12] Yang Yuanming, Guo Jiansheng . Dynamics modeling of the fexibly body with determined move-ment position[J]. J of Huaxhong University of Sci and Tech, 1999, 19(7):103- -105.[13} Yang Yuanming, Zhang Wei. Dynamics modeling of the flexibly multi-body[J]. Acta MechanicaSolid Sinica, 1999, 20: 153-158.14] Kane T R, Likins P W, Levinson D A. Spacecraft Dynamics[M]. McGraw- Hill,New York,1983,247.中国煤化工MYHCNMHG