General Dynamical Equations of Motion for Elastic Body Systems

JOURNA GUIDANCEF LO , CONTROL DYNAMICD AN , S Vol. 15, No. 6, November-December 1992

General Dynamical Equation f Motioso n for Elastic Body Systems

Shui-Lin Weng* and Donald T. Greenwood! University Michigan,of Arbor,Ann Michigan 48109

A modeling technique capable of determining the time response of a single body (rigid or flexible) that is, in general, undergoing large elastic deformations, coupled with large, nonsteady translational and rotational motions, is presented. The derivations of the governing equations of motion are based on Lagrange's form of d'Alembert's principle generae Th . l dynamical equation f motioo s expressee nar termn di f stresso straid san n tensors, kinematic variables, the velocity and angular velocity coefficients, and generalized forces. The formulation of these equations is discussed in detail. Numerical simulations that involve finite elastic deformations coupled with large, nonsteady rotational motions are presented for a beam attached to a rotating base. Effects such as centrifugal stiffening and softening, membrane strain effect, and vibrations induced by Coriolis forces are accommodated. The effects of rotary inertia as well as shear deformation are also included in the equations f motiono . Although discussions her restrictee ear singla o dt e body formulatioe th , n allow capabilite sth a f yo general dynamical formalis handlinr mfo g multibody (rigi flexibler do ) dynamics.

I. Introduction and torques do not appear and 2) the resulting equation set is OOKER and Margulies1 as well as Roberson and Witten- of minimum dimension. H burg2 wrote early e papermid-1960th n i s s that dealt modelinA g technique capabl f determinino e time - gth ere primarily with spacecraft dynamics. The methodologies they sponse of a rigid or flexible body that is, in general, under- presented were applicable onl systemo yt f rigiso d bodies with going large elastic deformations, coupled with large, non- simple joint restrictea n si d configuration. Later numbea , f o r steady translationa rotationad an l l motions considereds ,i e Th . studies based on these methods were published.3'4 A typical derivations of the governing equations of motion are based on modern spacecraft consists of structural subsystems, some es- Lagrange's for f d'Alembert'mo s principle generae Th - . dy l sentially rigid and others extremely flexible, frequently inter- namical equation f motioo s expressee nar termn di f stresso s connected in a time-varying manner. Formulation dealing with straid an n tensors, kinematic variables velocite th , angud yan - this category of spacecraft is a procedure that employs discrete r velocitla y coefficients generalized an , d forces. These equa- coordinate o describt s e unrestricteth e d motion f thoso s e derivee b tion n sca d systematically. structural subsystems idealized as rigid bodies, in combination wels i t I l known that when flexible structural elemente sar with distributed or modal coordinates to describe the time- attache rotatina o dt g base apparene th , t stiffnes struce th f -o s varying deformation f thoso s e structural subsystems idealized tural elements varies wit magnitude hth inertiae th f eo l angular as flexible elastic appendages.5'6 Advances also have been velocity of the spinning base. Also, in linear structural theory, made concernin coupline gth g effects between gross transla- the transverse vibration of a beam is calculated without consid- tional or rotational motion and the elastic deformation of ering axial forces. But in some cases, e.g., in rapidly rotating elastic bodies.7'11 turbine or helicopter blades, it is not possible to ignore the Repeated numerical simulations are needed to establish a effect of axial forces on the bending vibration of blades. When satisfactory design for the prototype of a spacecraft. In an the beamlike blade is spinning, so-called centrifugal stiffening effor facilitato t t simulatione eth f multiplo s e interconnected presence th effect o t e axiaf eo s du thale (centrifugalar t ) forces bodies, analysts have come to rely more and more on general come into play. Coupling betwen centrifugal forces and bend- multibody dynamics formalisms dynamicaA . l e systeb y mma ing moments make a rapidls y spinning beam stiffer thas i n translatin spinninr expectee go b y wholma n g di par n d i r an te o predicte lineay db r theory lighe th f thi o tn I s. situations i t i , to undergo large changes in inertial position and orientation. important that a multibody formulation correctly reflects mo- It has become necessary to devise methods of dynamic analysis tion-induced stiffness. that combine the generalities of nonlinearities and large mo- resolvo T difficultiese eth singla , e generalized formalism, tions with the computational efficiency afforded by the use of the general dynamical equations of motion, is introduced. It is modal coordinate describinn si vehicle gth e deformations. distinguished from a method using the shortening effect16'18 Mathematical modeling tool usee ar sanalyzo d t probe eth - explicitly in that here the centrifugal stiffening terms enter the lems and to derive the dynamical equations of motion for a final equations through the potential energy rather than multibody system. Comparative studies12 suggest that Kane's through the kinetic energy. Also, we investigate the mecha- method13 'somr 14o e related generalizatio Lagrange'f no s form nism of motion-induced stiffness variations in various types of of d'Alembert's principle15 most closely combines the two com- elastodynamic structures undergoing large overall motions. putational advantages: 1) the nonworking constraint forces Effects such as centrifugal stiffening and softening, membrane strain effect d vibrationan , s induce Corioliy db s forcee ar s accommodated. The effects of rotary inertia as well as shear deformatio alse nar o include equatione th n di f motionso . Received Nov , 199015 . ; revision received Nov , 199115 . ; accepted In Sec. II, we include the finite displacement theory of elas- for publication Nov , 199127 . . Copyrigh Americae 199th © t y 2b n ticity, the principle of virtual work, and the preliminaries to Institute of Aeronautics and Astronautics, Inc. All rights reserved. actuae th l dynamical proble e purposth r mf fo completeo e - * Graduate Research Assistant, Department of Aerospace Engineer- ing; currently, Researcher, Cente r Aviatiofo r Spacd an n e Technol- ness Secn I . . Ill generalizea , d formalis - analyzmo t dy e eth ogy, ITRI, Hsinchu, Taiwan. Member AIAA. namical system generae th , l dynamical equation f motioo s n fProfessor, Departmen f Aerospaco t e Engineering. Associate Fel- base Lagrange'n do s for f d'Alembert'mo s principle devels i , - low AIAA. oped analysin a Secn , I .singla IV .f so e flexible bea mpres i - 1434 WEN GREENWOODD GAN : GENERAL DYNAMICAL EQUATION MOTIOF SO N 1435

sented to illustrate the formulating procedures of the general dynamical equations of motion. Section V applies the flexible beam theorseverao t V I f Secyo l . problems. Sectio I prenV - sent conclusionse sth . II. Modeling of Flexible Bodies In this investigation, we assume that a rigid or deformable body may experience large translational or rotational displacements relative to an inertial coordinate system. Although a body-axis system that is rigidly attached to a point on the body is commonly employed as a reference for rigid components, ther e manar e y arrangement e bodth yr fo axes f flexiblo s e components.19'20 The origin of this reference frame does not rigidlhave b o et y attache deformabl e poina th o dt n o t e body, t frequentlbu choseno s s i wile t yi W .l attac bode hth y axeo st smala l rigi dbeame volumth f o . d e en elemen e on t a t Fig. 2 Geometry of body axes and inertial axes. Analysi f Straiso n presene th n I t section shale w , l trea finite th t e displacement In general, the lattice vectors are neither unit vectors nor are theor f elasticityo rectangulan yi r Cartesian coordinated an s they mutually orthogonal. emplo e Lagrangiayth n approach whicn i , e coordinatehth s Next, let us express the position vector of the point P as defining a point of the body before deformation are employed for locating the point during the subsequent deformation. r = r<°> + d (5) Let a rectangular Cartesian coordinate system xlx2x3 be assigne relativeace o dt th ht bodyle e d positioan , n vectof o r where d is the elastic displacement vector expressed in the an arbitrary point P(0) of the body before deformation be j-frame. From Eqs. (4) and (5), we have representey db (6) where 6* is the Kronecker symbol. The summation conven- as shown in Fig. 1, where the superscript (0) means that the tion will be employed hereafter. Therefore a Greek letter index quantity refers to the state before deformation. The base vec- r Kjito ,tha , fx , t appears e samtwicth en ei term indicatea s tor thin si s coordinate syste givee mar y nb summatio . Consequently3) n , ove2 , r(1 , Green's strain tensor can be calculated in terms of the displacement components as (2) follows:

wher notatioe eth n( ) >M denotes differentiation with respeco t (7) e baseJCMTh . vector e uniar s t vector e directioth e n th i s f no coordinate axes and are mutually orthogonal. Analysis of Stress and Stress-Strain Relations The body is now assumed to be deformed into a strained (0) (0 (0) The force equation for the equilibrium of the deformed configuration pointe Th . s P , Q \ R<®, ST^d an , move parallelepiped is given by21 positionw tone s denote P,y db Q, R, T,d S respectivelyan , , infinitesimae th d an l rectangular parallelepipe s deformedi d into a parallelepiped that, in general, is no longer rectangular. = 0 (8) Let us denote the position vector of the point P relative to where a^ is the second Piola-Kirchhoff stress vector acting on bode th y framy eb the jLtth face of the deformed parallelepiped, and F is the body l 23 M r(x= r , x) ,x (3) force actin thin gi s parallelepiped note W .e that adefines i d per uni s definei t F r areauni d pe d tan , volume, both with where (xl,x2 ,xe actuall3ar ) e Cartesiayth n coordinatef o s respect to the undeformed state. P(0). Introduc lattice eth e vectors definey db Assume that the material is isotropic. The stress-strain rela- 21 8r tion is G*=l,2,3) (4) Ev (9)

where Young's modulus E, Poisson's sheae ratith ord v,an modulu 2( relate e = v)G.1 + ar E y sG d b

T(O) Dynamic Elastin a f so c Body Le rectangulaa t r Cartesian coordinate system X2X e l b 3 fixe inertian di correspondine th lt spacele d an , g unit vector syste iie m b / 2 1*3origibodye e shows th a ,Th f . n o -2 Fig n ni .

axis system xxx is located at a position rrelative to the G

2 3 origin of the inertial l frame. Then the inertial position vector r representativa (sef o e ) 2 Fig. elastie eth poinf o c bodP t s yi

rr<°d G= + >+ r (10)

wher recale ew l that r<®(x positio,s xit , s x)i n relative th o et

2 3 body frame before l deformation elastis it s i cd deformad an , - Fig1 Geometr. infinitesiman a f yo l parallelepiped. tion vector relative to the body frame. The vector rG is ex- 1436 WENG AND GREENWOOD: GENERAL DYNAMICAL EQUATIONS OF MOTION

where Z is the nonorthogonal matrix:

2 l+

-",3

Genera! Case: Elastic Body Wwritn followine . (14eth ca eEq n i ) eace g se hfor d man term more explicitly:

d2r* P 2 Fig 3 . Geometr rigia f ydo body. V dt

pressed in terms of its inertial components, but r(0) and d are (19) expressed iny-frame components; that is, and (11) ^e can a*so rewrite Eq. (15) as follows:

From Eq. (10), we obtain (20) -framL e th et e {ab skewe Le £) d component e t b { le o} f d san the /-frame components of the same stress vector a*. Then

where the absolute angular velocity of they -frame is (13) No revernotatioe e ww th o t n {a £=) {writd a^an } e Eqs. (19) (20d matrin an i ) x fors ma wherd an e (dd/dt) rate th e s i changviewes ra d f deo from this rotating fram whicn ei j^e hth uni t vector constante sar . ([A]T{F})T(dr}dV+ ([A]T[B})Tldr}dS The dynamical equations of motion are obtained by using s, d'Alembert's principl s Lagrangiait n ei n form involving virtual work. Here we lump the inertia force -p(d2r/dt2) with the body force F, eac unir hpe t volume considee W . elasn a r - l 2 3 tic body tha s subjeci t o bodt t y forces F(x ,x 9x ,t) dis- l 2 3 tributed throughout the body, surface forces B(x 9x 9x 9t) applie surface th n do e specified Sian , d surface displacements (22) l 2 3 r(x , Jc , x , t) on the surface S2. The virtual work expression dynamicae foth r l problems i and T = idr G[A]+ ] [5d]+ (23) d2A -p—- -6rdK v\ dt2) Here [60] is a 3 x 3 skew-symmetric matrix representing the vector cross-product. In addition to the preceding two expressions stile w ,l nee knoo dt w [6r>/x). Note thaa t t {rno s Gi } 0 = dS 5r -B (14) functio f bodno y axis variables obtaie w ; n

e firsTh t integral represent e firssth t variatio e storeth f ndo (24) elastic energy, whereas the second integral represents the vir- tua lbod e worth yf k o force inertid san a thir e forcesth dd an , Suppose we have n generalized coordinates, and those corre- integral represents the virtual work of the surface forces acting spondin elastio gt c displacement associatee sar d with assumed on Si. As the motion of the surface S2 is prescribed, it does not deflection forms. Then, in terms of x and q, we have enter intvirtuae oth l work expression virtuae Th . l displacement dr is given by = (d(x9q)} (25)

dr = 5rG + WyM + 69 x (r<°> + d) (15) where 60 is the virtual rotation of the body axes; i.e., the j- frame. III. General Dynamical Equations of Motion begis u t n Le wit matrie hth x equation relatin unie gth t vectors of the body-axis j -frame and the inertial /-frame, where orthogonan a s i A l matrix:

thae Usin relatee se t. /(6)d e ar rL gw Eq an x,y db

(17) Fig. 4 Rotating cantilever beam. WENG AND GREENWOOD: GENERAL DYNAMICAL EQUATIONS OF MOTION 1437

where x represents the spatial variables (x\ x2, x3). The abso- Now we are ready to consider the first integral of Eq. (22). lute velocity of the point G is Whe substitute nw . (23eEq ) into this integral usd ean , j -frame components, we obtain

(v } = \r ] = yJLrelative velocity, and we find that

d}=p^ (d]qj (29) and

The relative velocity coefficient { yj } due to QJ is fcr5 _ f) cr \ fl&\ [ o j — [ o j vyo/

iyj}^r= [d]/= - [d] (30) B ya simila r procedure obtaie w , e seconnth d integraf o l d

(«rf^£ -= ) Wbq t = [yj}dqj (31) wher e superscripeth denoteS t n appliea s d external surface y=i d] lr«»}

(34) (42)

where we note that [^4] = - [<£] [A ] = [w] T[A ] . Substitute Eqs. (42) and (23) into the third integral of Eq. (22) thed an ,n substitut virtuae eth l displacement Eqsf so . (28) and (31) and the virtual rotation of Eq. (34) into the former result, obtaining

j (43)

where the asterisk denotes an inertial force. Using j -frame components,

(44)

Finally shale w , l conside fourte th r h integra . (22)Eq ,f o l which involve strese sth s tensor substitute w f I . . (24eEq ) into Fig 5 . Flexible bea rigima f cantileveredo wheelm ri e .th t da the fourt derivative th h e integra us . (22) virtuaf d eEq o an ,f lo l 1438 WEN GREENWOODD GAN : GENERAL DYNAMICAL EQUATION MOTIOF SO N

0.005 o.i

0.0 0.000 -0. 1

>-0.005 IT -0.2

f -0.3 3-0.010

-0.4

-0.015 -0.5

-0.020 -0.6 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0 5. 10.0.00 15.0 20.0 25.0 30.0 t (sec) t (sec)

Fig. 6 The x displacement u vs time t. Fig. 7 The y displacement v vs time t.

displacemen virtuaf o . (32d Eq ) an l f rotatioto . (34)e Eq ,w f no Next, let us make the further assumption that the various obtain, after switching toy-frame components, e independentar q . Then each coefficien j vanishesdq f o tn i obtaie w . (48d nEq an ) dV QJ + Qf + Qf = o, (50) greaten i r o r detail, (45) 7=UJJ V where the superscript E denotes an elastic force and = -[Z]Ti^} (46) Note that no elastic energy is stored due to a rigid body displacement, so only elastic displacements are considered here. Next accordancn i , e with d'Alembert's principlen ca e w , su virtuae mth lappliee worth inertiaf d ko d an l generalized forces to obtain C/ = l,2,...,n) (51) Equation (51) is the matrix form of the general dynamical equations of motion for a system that is not subject to holo- nomi nonholonomid can c constraint generalizee th n so d coordinates. s takNowu t a closee le , re problem looth t ka f o s (47) representing the dynamical equations of constrained systems.

where bq conform constraintsy an o t s . Constrained Systems e generalizeth e b f t Qj,Q dLe d externaQf,an l applied Consider a system whose configuration is given by n gener- forces generalizee th , d inertia generalize e forcesth d an , d elas- alized coordinates q\,q2>--,q- Suppose there are m inde-

tic forces, respectively, and we obtain the following expres- pendent constraint fore th m f so n sions: (52) Qj = where these expressions are not integrable for the case of non- holonomic constraints. If the constraints are actually holo- Qf = nomic and of the form

(53)

then, upon differentiation with respect to time, they have the Hence, we see that d'Alembert's principle leads once again to form of Eq. (52) but are integrable. In either case, let us introduce a set of (n - m) independent velocity parameters wy , £(Q G; + e/)«« ;+ 0 = / (48) known as generalized speeds,14 which are consistent with the y=i equatione constraintth y relatee b ar q sd o dt s an More explicitly, however, with the aid of Eqs. (36), (39), (43), and (45) we obtain (54)

where u may represent true velocities or quasivelocities; i.e., therintegrabilito n s ei y requiremen . (52) Eq f Eqs I .n o t . (52) and (54) are solved for q in terms of u , one obtains

dVJdqj = 0 (49) 4f/="xfMtf>OH; + */fte,0, (i = 1,2,..., /i) (55) WEN GREENWOODD GAN : GENERAL DYNAMICAL EQUATION MOTIOF SO N 1439

r thiFo s constraine . (55 Eq elimio t ) e us -d n systemca e on , IV. Analysi f Flexiblso e Beams -m)n ( f o nat independent ese favon qa i f o rvelocits a u t y When flexible structural element attachee sar rotatina o dt g parameters. The . (51nEq ) still applies definee on veloce f i sth - base, as shown in Figs. 4 and 5, the apparent stiffness of the ity coefficients and angular velocity coefficients with respect to structural elements varies with the magnitude of the inertial these new velocity parameters, that is, angular velocit spinnine th f yo g base r somFo . e base-element attachment configuration e stiffnes e th Fign i th , 4 s f . a o ss elements increases with base angular speed,23 wherea othr sfo - ers, such as Fig. 5, the stiffness decreases.24 In this section we analyze the deflection of a flexible beam with large overall motions, which performs a prescribed planar rotational motion aroun i'e 3-axidth s (or yr axis) usiny genera,b e gth l dynamical equations of motion. Now let us derive the differential equations of motion for Thus, we obtain (n -m) dynamical equations this rotating beam (see Fig , whos.4) e motio confines ni e th o dt Y- plane.X choos e bods e a W origith e z ys e th a xy axe d nG san referenc e beam th poin r .fo t Generalized coordinates QJ 1,2,(= j ... ,n) wil usee b l represeno dt configuratioe th t f no the rotating beam at any time /. The velocity of the reference point G is zero. This leads to the velocity coefficients (yf ] being equa zero o t le angula Th . r velocit f they-framyo s i e prescribe functioa s da correspondin e timef no th o s , g angular velocity coefficients l vanish[fy]al . Also e assumw , e that applieo thern e ear d forcesgeneralizee th o s , d external applied force velocite s zeroth e QJt ar Bu .y coefficients (7^) resulting Cl,2,...,w-/n= / ) (56) from elastic deformations and their spatial derivatives are nonzero. wher representeQ s generalized forces associatedd witan hu maks Nextu followine t eth le , g assumptions. e Firstth y b , where choice of axes and from the definition of the central line as the lin f centroideo crose th f sso sections have w , e yj] = [A](yf] (57) 0 = dz dy yz dz= dy z dyy dz= (60) Special Case: Rigid Body Now, conside speciae th r l casrigia f eo d body motione Th . where A is the area of the cross section. Second, the centroidal

origin G is the reference point and we assume that this point is displacement vector {D} {uv= w}q isd functioa an x f no bode fixeth yn di (se bod e mase th e md s yf Th Figi s o . ,an 3) . . Third t w=0 se assume e w , w only d e an tha , strese th t s com- T its center mass position relativ ps i c . G Also o e knot e ,w w that ponents ayy, neglectee ab zz ,y ad yzxz ma comparisoan ,n d i n wit othee hth r stress components thed e stress-straian ,nth n d an m p= dV (58) relation f interesso e ar t and a** 2Ge= (61) After some algebraic manipulations generae th , l dynamical x 22 equatio f motiono n (51) becomes, vecton i r form, Fourth, we employ the hypothesis that the cross sections per- pendicula e centroidath o t r l locus before bending remain plane. Also, the shape of the cross section does not change during bending. (j = l,2,. ..,/i) (59) Assume that the displacement in the x direction is

where independenthern e ear t generalized coordinates. Notice (62) that the inertia dyadic / is taken about its reference point at G . where ij(x)assumen a s i d deflection form ,generala q\js i - ized coordinate, and pi is the number of spatial functions representing u(x,q). e lateraTh l displacemen y displace( t - ment), which is due to bending and shear deformations, is

v(x,q) v= b (x,q) v+ s(x,q) = + £ faj(x)q3j (63) where fajW and faj(x) are assumed deflection forms due to bendin sheard gan , respectively corresponde th ,qe d 3Jqar 2 an j - generalizeg in d coordinates corresponde th e fi d ar /x3d an 2 an , - ing numbers of spatial functions representing bending and shear displacements vb(x,q) vd s(x,q).an We shall now consider a large deflection of an elastic beam. However, we will be satisfied to limit the problem by assuming that, although the deflection of the beam is no longer small in comparison wit heights hit stils i t li , smal comparison i l n with the longitudinal dimension of the beam. We may then employ the following expressions for total elastic displacements dx, dy, z and thirdo t d smale ordeth n l i rquantitie sv d y,b:an u,, v Fig. 8 Rotating pinned-pinned beam. (64) 1440 WEN GREENWOODD GAN : GENERAL DYNAMICAL EQUATION MOTIOF SO N

wher beae eth mvs i bbendin o t slop e edu g deformationd an , V. Numerical Simulations ( )' represent ( )/dx.sd Recently it has been shown that the geometric nonlinearities The strains e^ can be calculated in terms of y, u, v, and vb, arising from the coupling of longitudinal and transverse defor- thiro t d order, mations have a considerable effect on the deformation of beams with large, nonsteady translationa rotationad an l - mo l e u l 2 l 2 25 32 xx — ' — yv£ + /2(u') + /2(v ') — yu' vb tions. " In this section we simulate some rotating beam systems by using the general dynamical modeling method dis- l l 2 e/2xy= (v' - v b- u' V&- /2 v' (Vfc ) ) cussed in the previous seciton. The angular velocity w of the spinning system as a function of time t for the first two exam- *0 «= (65) ples is

Since the shear deformation is small in our examples, the ex- \us/Ts) [t- 0 < t < Ts pressio simplifiee b n r exyca nfo o dt t>Ts

ex = (66) (68) which shows tha straie th t n e equas xyi one-halo t l beae th f m wher time e th reac o Tet s i ssteady-stat e hth e angular velocity slope due to shear deformation. coy. This motio sometimes ni s calle dspin-ua p motion since eth Finally obtaie w , e nonlineanth r differential equationf o s speed smoothly increases fro coo t t represent 5mI .0 particusa - motion: lar exampl f generaeo l overall motion geometricae Th . d an l

— 03V— co (x + M) — 2cov + ii 2

2 dv_ PA\ cb(;c +w)-w dx + Imvi dx + cb Im L JL dqj 0 0

EAu' + l/2(v ')2 + 3/2(w 'Y + Viu '( dvr d Vh EA(U'V' V2(v')*+ y2(u'+ I EI dx\ kGAv+ vS—^ + dx ='^dx 0 z L i s dqj ™ o 0 (l,2,...,fiy= ) (67) where we retain terms up to third order for the first five numerical data use r simulatiodfo systee th listef e nmo ar n di terms, which involve u, vb, and v, and to first order for the last Table 1. term resulting from e termvsTh . s wite rotarth h y inertia Im = \\Apyi dy dz treated as second order are included in the equation motiof so n durin analysise gth determininn I . g orders Example 1 of magnitude, we consider area A = Jf^ dy dz to be of zero A cantilever beam attached to a rigid base is shown in Fig. 4. 2 order momend an , f secono f inertie o tb do t a z jf^7I= d z y d The rigid base performs a prescribed spin-up rotational e expressioorderth e facton i Th . k rs appende i n o takdt e motion co [Eq. (68)] around a vertical axis, assuming co5 = 6.0 account of the nonuniformity of the exy over the cross section. rad/ bead san m lengt 10. = sho. Figure7 h 0L m e d wth an s6

coj = 2.219 0 0)6. 5= 12.0 0.05

a.o -

6.0 - -0.05

4.0 - -0.10 - 2.0 -

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 t (sec) t (sec)

Fig 9 . Midpoin displacementx tims v eu t t. Fig. 10 Midpoint y displacement v vs time t. WENG AND GREENWOOD: GENERAL DYNAMICAL EQUATIONS OF MOTION 1441

Tabl e1 Beam properties (solid line) physicae Th . l meanin existence th reside f go th f e-o

3 3 l lateraua l displacemen Fign i 0 . 6. 1cor tha s 0i trans= 5e fo t tth - Density, kg/m p =3.0xl0 verse centrifugal inertial force, created by the steady-state an- Young' =7.OsE modulusx 101,0 N/m2 10x 15 0-SheaG 2. r modulus, N/m2 gular velocity, increases rapidly enough o causwitt v he Length, m L buckling. Nonlinear membrane force importane ar s detern i t - Cross section area, m2 A -4.Ox 10~4 minin finae gth l steady displacemen. v n i t 4 7 Area moment of inertia, m Iz = 2.0 x 10~ Example 3 Spin-up time, s 7^=15.0 Steady-state angular velocity, rad/s u ___ Figure 5 shows a beam cantilevered at the rim of a rigid s wheel, which perform constana s t rat f rotationaeo l motion around a vertical axis. By assuming the initial conditions of nonzero displacements zer d (q)an o velocities (-0.004 diverges. The physical meaning of the simulation result is that buckling occurs when the centrifugal inertial force, created by the steady-state angular velocity s strongei , r tha e elastinth c 3-o.ooe restoring force, and remains so with increasing deflection. VI. Conclusions -0.008 generaA l formalis s beemha n presented- , baseLa n o d grange's for f d'Alembert'mo s principl includind ean g nonlinear strain relations. This theory can be used to determine the -0.010 time response of flexible structures undergoing large elastic 0.0 20.0 40.0 60.0 80.0 100.0 t (sec) deformations coupled with large, unsteady rotational motions. Various assumptions, suc plans ha e sections remaining plane, Fig. 11 The x displacement u vs time t. are added when appropriate t include no orige e th ar -n di t bu , inal formulation. Several example includee sar d that involve dynamice th rotatinf so g beams same .Th e formulation applies to cases emphasizing centrifugal stiffening, membrane effects, d bucklingan o speciaN . l assumption e needear s n thesi d e cases formulatioe Th . suitabls ni elastir efo c systems with prescribed forces or displacements as functions of time, and it can be extended to the analysis of multibody systems. References looker Margulies. W.d W ,an , , G., 'The Dynamical Attitude Equations for an n-Eody Satellite," Journal of the Astronautical Sci- ences, , 19654 . . 123-128 Volpp , No , .12 . 2 0.5 Roberson Wittenburg d . E.R ,an , , DynamicaJ.A " , l Formalism foArbitrarn a r y Numbe f Interconnecteo r d Rigid Bodies, with Refer- Problee encth o et Satellitmf o e Attitude Control," Proceedingse oth f Third Congress International ofthe Federation of Automatic Control, 0.0 Butterworth, London, 1967. 3Hooker, W. W., "A Set of r Dynamical Attitude Equations for an Arbitrary n -Body Satellite Having r Rotational Degrees of Freedom," -0.5 0.0 20.0 40.0 60.0 80.0 100.0 AIAA Journal, , 19707 . 1205-1207. Volpp , No , 8 . . 4 t (sec) Larson, V., "State Equation «-Bodn a r sfo y Spacecraft," Journal of the Astronautical Sciences, Vol. 22, No. 1, 1974, pp. 21-35. Fig. 12 The y displacement v vs time t. 5Likins, P. W., "Dynamic Analysis of a System of Hinge-Con- nected Rigid Bodies with Non-Rigid Appendages," International Journal of Solids and Structures, Vol. 9, 1973, pp. 1473-1487. plots of the x displacement u and the y displacement v of the 6Likins, P. W., "Finite Element Appendange Equations for Hybrid cantilevee th f o fred ren e beam. Coordinate Dynamic Analysis," International Journal f Solidso d an Structures, Vol , 1972.8 . 709-731pp , . 7 Example 2 Boland, P., Samin ,Willems d J.an , , P., "Stability Analysif o s Interconnected Deformable Bodie Closed-Looa n si p Configuration," beaA m pinne ends t rigibota dit a o f st dh o bod shows yi n AIAA Journal, , 19757 . 864-867. pp Vol, No , .13 . in Fig. 8. The rigid body performs a rotational motion around 8Hooker, W. W., "Equations of Motion for Interconected Rigid verticaa l axis passin beame th r thi f go Fo .s througd en e hon Elastid an c Bodies," Celestial Mechanics, Vol , 19753 . . . 11pp , No , system, the same type of prescribed spin-up motion is given as 337-359. in Eq. (68). Three cases of final steady-state angular velocities 9Hughes, P. C., "Dynamics of a Chain of Flexible Bodies," Jour- 2.219(jo= s rad/0 , 6. e considere2.0 d sar ,an d wit e samhth e nal of the Astronautical Sciences, Vol. 27, No. 4, 1979, pp. 359-380. beam firs e lengt 20.Th t = . cas 0hm L e give criticae sth l angu- 10Singh . P.R ,, Vandervoort Likins d . J.R an , . W.P , , "Dynamics lar velocity, and a comparison of the last two cases shows the of Flexible Bodies in Tree Topology—A Computer Oriented Ap- existenc residuaf eo l lateral displacement angulae th f si r veloc- proach," Journal of Guidance, Control, and Dynamics, Vol. 8, No. 5, 1985 . 584-590pp , . it larges yi r tha criticae nth l angular velocity.0 Figure1 d an s9 HKeat Turnerd . E.J , an , . D.J , , "Dynamic Equation f Multio s - show the plots of the x displacement u and the y displacement body Systems with Applicatio o Spact n e Structure Deployment," v, respectively, of the midpoint of the pinned-pinned beam vs Charles Stark Draper Lab., Rept. CSDL-T-822, Cambridge, MA, ur 2.2\9tims= fo et (dashed line) (dotte0 2. , 0 d 6. line) d an , May 1983. 1442 WENG AND GREENWOOD: GENERAL DYNAMICAL EQUATIONS OF MOTION

12Kane, T. R., and Levinson, D. A., "Formulation of Equations of C., "Vibration Mode f Centrifugallso y Stiffened Beams," Journalf o Motion for Complex Spacecraft," Journal of Guidance and Control, Applied Mechanics, Vol. 49, March 1982, pp. 197-202. , 19802 . 99-112. pp ,Vol No , 3 . . 24Peters, D. A., and Hodges, D. H., "In-Plane Vibration and Buck- 13Kane Levinson. R.d T , ,an A.. D , ,Kane'f "Tho e eUs s Dynamical ling of a Rotating Beam Clamped off the Axis of Rotation," Journal Equations in Robotics," International Journal of Robotics Research, of Applied Mechanics, Vol , Jun47 . e 1980 . 398-402pp , . Vol. 2, No. 3, 1983, pp. 3-21. 25Banerjee Dickens. d K.A ,an , . M.J , , "Dynamic Arbitrarn a f so y 14Kane, T. R., Likins, P. W., and Levinson, D. A., Spacecraft Flexible Bod Largn yi e Rotatio Translation,d nan " Journal f Guid-o Dynamics, McGraw YorkHillw Ne , , 1983. ance, Control, and Dynamics, Vol. 13, No. 2, 1990, pp. 221-227. n Lagrange'Passerellod 15Huston"O an , , L. E. . . R sC , Forf mo 26Banerjee, A. K., and Lemak, M. E., "Multi-Flexible Body Dy- d'Alembert's Principle," Matrix and Tensor Quarterly, Vol. 23, 1973, namics Capturing Motion-Induced Stiffness," Journal of Applied pp. 109-112. Mechanics, Vol. 58, Sept. 1991, pp. 766-775. 16Greenwood . T.D ,, "Helicopter Rotor Dynamics," Univf o . 27Hanagud, S., and Sarkar, S., "Problem of the Dynamics of a Michigan, Dept f Aerospaco . e Engineering Rept. Arborn , An , MI , Cantilever Beam Attached to a Moving Base," Journal of Guidance, Jan. 1980. Control, and Dynamics, Vol. 12, No. 3, 1989, pp. 438-441. 17 KvaternikKazaV.. d R ,an . ,K G.. R , , "Nonlinear Flap-Lag-Axial 28Ider AmiroucheK.. d S , an , . L.M , . "Th,F e Influenc Geometf eo - Function Rotatina f so g Beam," AIAA Journal, , 19776 . , No Vol , 15 . ric Nonlinearitie Dynamice th n si Flexibld san e Tree-Like Structures," . 871-874pp . Journal f Guidance,o Control, Dynamics,d an , 19896 . , No Vol , 12 . 18Vigneron, F. R., "Comment on 'Mathematical Modelling of pp. 830-837. Spinning Elastic Bodie r Modasfo l Analysis'," AIAA Journal, Vol. 29Ider Amirouche. d K.S , an , . L.M , . "NonlineaF , r Modelinf go 13, No. 1, 1975, pp. 126-128. Flexible Multibody Systems Dynamics Subjected to Variable Con- 19de Veubeke F.. B ,, "The Dynamic Flexiblf so e Bodies," Interna- straint," Journal of Applied Mechanics, Vol. 56, 1989, pp. 444-450. tional Journal f Engineeringo Science, Vol , 1976 . 14 895-913. pp , . 30Kane . R.T , , Ryan Banerjeed . R.R ,an , . K.A ,, "Dynamica f so 20Cavin Dustod . K.R ,an , . R.A , , "Hamilton's Principle: Finite Cantilever Beam Attache Movina dto g Base," Journal Guidance,of Element Method d Flexiblan s e Body Dynamics," AIAA Journal, Control, and Dynamics, Vol. 10, No. 2, 1987, pp. 139-151. Vol. 15, No. 12, 1977, pp. 1684-1690. 31Simo, J. C., and Vu-Quoc, L., "On the Dynamics of Flexible 21Washizu, K., Variational Methods Elasticityn i Plasticity,d an Beams Under Large Overall Motions—The Plane Case: Part I and II," 3rd. ed., Pergamon, London, 1982. Journal of Applied Mechanics, Vol , Dec53 . . 1986 . 849-863pp , . Greenwood, D. T., Principles of Dynamics, 2nd. ed., Prentice- 32Christensen Lee. d R. E . ,W. an ,S , , "Nonlinear Finite Element

Hall22 , Englewood Cliffs , 1988NJ , . Modeling of the Dynamics of Unrestrained Flexible Structures," 23Wright, A. D., Smith, C. E., Thresher, R. W., and Wang, J. L. Computers Structures,d an , 19866 . 819-829. pp , No Vol , 23 . .

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