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 formu- lation 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 displace- ments 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.