A NEW FLEXIBLE BODY DYNAMIC FORMULATION FOR BEAM STRUCTURES UNDERGOING LARGE OVERALL MOTION IIE THREE-DIMENSIONAL CASE. W. J.

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1 A NEW FLEXIBLE BODY DYNAMIC FORMULATION FOR BEAM STRUCTURES UNDERGOING LARGE OVERALL MOTION IIE THREE-DIMENSIONAL CASE W. J. Haering* Senior Projet Engineer General Motors Corporation Warren, Mihigan R. R. Ryan+ Exeutive Vie President and. Chief Operating Offier Mehanial Dynamis, Inorporated Ann Arbor, Mihigan R. A. Sott Professor, Department of Mehanial Engineering and Applied Mehanis The University Of Mihigan Ann Arbor, Mihigan A previously developed flexible body dynami formulation for simulating beam strutures undergoing large overall motion, restrited to two-dimensional presribed motion, is extended to the study of both three-dimensional presribed motion and nonpresribed motion. The formulation, alled the Augmented Imbedded Geometri Constraint (AIGC) approah, is restrited to small elasti deformations of the beam struture. The overall motion is haraterized by six degrees of freedom, while the elasti deformation is haraterized by the superposition of a number of assumed global shape funtions, developed using substruturing tehniques. This extension allows the approah to be applied to the study of a broader spetrum of problems involving systems with applied fores/ torques where the overall motion is not known. Thus, the formulation an be used to study the two-way interation between overall motion and loal deformation whih is fundamental to a general purpose flexible body dynami formulation. The AIGC approah desribes the elastodynami behavior with a set of differential equations linearized in terms of the deformation degrees of freedom (but not in the degrees of freedom desribing the overall motion), and a number of, possibly nonlinear, onstraint equations desribing the physial attahments of the ends of the beam. The beam haraterization is extended to inlude the effets of torsion, rotatory inertia, and an offset between the entroid and shear enter of the beam (eentriity). Transverse shear deformation is negleted as it is generally not of importane in flexible beam dynamis. The validity of the extended formulation is demonstrated by omparison of solutions for two validation problems to independently obtained solutions. Introdution Interest in flexible body dynamis, the oupling between large overall motion and loal deformation in smtural dynamis, has inreased signifiantly sine the early 1960's. Initially motivated by problems seen in the aerospae industq, the interest has spread to other industries, inluding robotis and ground transportation. Efforts to develop general purpose analysis tools to address flexible body dynamis an be broken into two general lassifiations, nonlinear finite element approahes, and rigid body dynami approahed modified to allow loal flexibility. The former strategy is used by Simo and Vu Quo* and Christensen and Lee 2, while the later is the approah adopted by Kane, Ryan and Banerjee3. In the work *Member AIAA +Member AIAA in referene [3], the authors disussed the need to develop element-speifi approahes in order to inlude interrelationships between the omponents of loal deformation. They developed a beam speifi formulation, often referred to as the Imbedded Geometri Constraint (IGC) approah. A similar formulation was developed by Y004, referred to as the Nonlinear Strain Displaement (NSD) approah, to overome the inability of the IGC approah to aurately solve problems where the lateral deformation of the beam struture is dominated by membrane stiffness. However, the NSD approah does not reliably solve problems where the lateral deformation of the beam struture is dominated by bending stiffness. The inability of these two approahes to aurately solve both lasses of problems prompted the authors 5 to develop another formulation, whih is apable of aurately solving both lasses of problems. This new formulation is referred to as the Augmented Imbedded Geometri Constraint (AIGC) Approah. In this paper, the AIGC approah is extended to the solution of general beam dynamis problems, with the assumption of small elasti loal deformation. This is aomplished by allowing for three dimensional motion and deformation, and also allowing the overall motion to be unknown. The removal of the latter restrition allows the AIGC approah to study the two-way interation between overall motion and loal deformation, whih is a fundamental issue in flexible body dynamis. As in the two-dimensional presribed motion work in referene [5], the development presented in this paper involves the dynamis of a single beam. The extension of the AIGC approah presented in this paper losely follows the original IGC development in referene [3], and a subsequent extension6 of that work to non-presribed motion. As a result, this paper will onentrate on the major differenes between the two approahes, and will leave out many of the details overed in referenes [3] and [6]. Interested readers an also obtain a detailed presentation in the Ph.D. Dissertation of Haering7, whih overs the work presented in this paper and in referene VI. As disussed in referene [5], the primary differene between the AIGC and IGC approah is that the differential equations of motion from the IGC approah, referred to as the system differential equations in the AIGC approah, have onstraint equations added to enfore the physial boundary onditions for the beam struture. This set of differential equations with algebrai onstraints forms the equations of motion for the AIGC approah. In addition, a set of general modal funtions is employed in the AIGC approah to ensure the ability to Copyright by the Amerian Institute of Aeronautis and Astronautis, In. All Rights reserved. 1117

2 satisfy any boundary onditions and prevent inadvertently imposing boundary onditions whih are not orret for any given problem. The plan of the paper is as follows. First, the physial system to be studied is introdued, the basi mehanis of the problem are introdued, and the system differential equations are given. Seond, a disussion of the onstraint equations and hoie of modal funtions is presented. Third, the equations of motion are presented and the tehnique for their solution is disussed. Finally, solutions for two verifiation problems, obtained with the AIGC approah, are presented and ompared with independently obtained solutions. The three-dimensional beam model is shown in Figure 1. A flexible beam B is attahed to a rigid base A at point 0. The mass of body A is given by ma, and the enter of mass is loated at a point, A* (not shown in Figure 1). A dextral set of unit vetors, ;i,g2,;3, is fixed in A, with the 21 diretion aligned with the undeformed elasti axis of the beam. Following Kane et al [3], z2 and 23 are aligned with the prinipal area moment of inertia axes of the beam. Prior to deformation, a point on the elasti axis ontained within a generi ross setion, db, is loated at point Co, at a distane x measured along the undeformed elasti axis. After deformation, that point on the elasti axis within ross setion db, is loated at point C, at a distane of x+s measured along the deformed elasti axis. An additional set of dextral unit.+---t vetors, bi,b2,b3, are fixed in ross setion db and are aligned 4-4 with, al,a2,a3, respetively, when the beam is undeformed. The entroid of setion db is defined by point P whih is offset from the point C on the elasti axis by the eentriity vetors given by e = e2 b2 +e3 b3 Allowing general three-dimensional deformation, the position vetor from the attahment point, 0, to the entroid P of the generi setion db is given by (1) f / Rigid Body@ / 4 / Figure 1 - Three-Dimensional Beam Model x I 1 F,I where u1 21, u2 ;2, and u3 23 represent orthogonal omponents of the beam deformation. The orientation of setion dei relative to body A an be desribed by three suessive rotations of amounts 01, 02, and 63 about lines parallel to unit vetors 21, h, and 23, respetively. It is one of 24 possible desriptions, and is hosen to failitate the introdution of beam torsion, and rotatory inertia. The relative orientation of db and A is desribed by the following diretion osine matrix where s1 s2 3 - s3 CI s1 s2 s s1 2 ACiy I= zi. bj, (ij=1,2,3) and 1 s2 3 + s3 s1 1 s2 s3-3 s1 I 1 2 (3) (4) The angular measure 61 is introdued to aount for twist, and 62 and 63 omplete the desription of the arbitrary orientation of db with respet to body A. The latter two angular measures allow shear deformation to be introdued (see Kane, et al 131 or Haering [7]). Shear deformation is generally negligible in long beams and it will not be inluded in this development. With shear defletion negleted, the angles 62 and 63 an be related to the spatial derivatives of the deformations u2 and u3 as -= 03 ax Thus the symbols Si and Ci (i=2,3) are now given by s2 =-sin u; s3 = sin u; (9910) 2 = os u3 3 = os u2 (11,12) where a prime denotes a derivative with respet to x. 1118

3 The deformation measures s, u2, u3, and 8 1 are represented as follows: V j=l V j=l, where $ij (x) (i=1,..., 4, j=l,aa., V) are modal funtions, the seletion of whih will be disussed later. The qj's, ( qj, j=1,2,,..,v ) are the first v generalized oordinates. The system differential equations are developed using Kane's8 approah. The ontribution to the generalized ative fore from internal fores ating in the beam, an be obtained from the following strain energy funtion, V, whih takes into aount axial strething, transverse bending, and torsion: +I E12[ %l2dx+i E13[ $]2dxj (17) 0 0 Where E, Ao, G, K', 12, and I3 are the modulus of elastiity, ross setional area, shear modulus, effetive torsional onstant, and the area moments of inertia about the & and g3 axes, respetively. Four measures of the beam deformation have been introdued, s, u1, u2, and u3, but due to the nature of beam deformation, only three are independent. The following relationship exists between the four deformation measures (see referenes [4 or 71 for details): %.. J o where (T is a dummy variable of integration. This relationship an be redued to the following more useful form [see referene [71). Note that (Pim)ij (i,j=l,2,...ev, k,m=2,3) are indefinite integrals. For the shape funtions that will be introdued latter, they an be determined expliitly. In this paper, only external fores applied to body A will be onsidered. It should be noted that referene [6] ontains a development inluding external fores applied to the beam struture itself and gravitational fores, and that development is diretly appliable to the AIGC approah. Fores and torques applied to body A an be replaed by an equivalent set, inluding a fore ating through the mass enter, (E)A', and a torque, (?)A as follows The disussion above highlights the essential mehanial ingredients of the theory. The derivation of the system differential equations is quite lengthy and details will not be given here. As disussed earlier, the details an be found in referenes [3] and [6] (transverse shear deformation inluded), and also in referene [7] (both with and without transverse shear deformation inluded). The system differential equations for non-presribed motion are given by the form j=1 j= 1 j= 1 h = Fi, i=1,2,..., v+6 The fiist v qj's, ( Sj, j=la..,v )were disussed above. The last six q's, ( qj, j=v+l,..., v+6 )are angular and translational measures used to desribe the orientation and loation of the rigid base, body A, in the Newtonian referene frame. The first v generalized speeds ( uy, j=1,2,..., v ) are the time derivatives of the orresponding generalized oordinates. The remaining six generalized speeds ( u;, j=v+l,..., v+6 ) desribe the overall motion of the beam. The v+6 unknowns u;, ( u;, j=1,2,..., v+6 )are the time derivatives of the generalized speeds. The matries,a, and 2 ontain, in addition to onstants (resulting from integrals over the domain of the beam), terms involving the generalized oordinates and/or generalized speeds. The olumn matrix? ontains additional nonlinear terms in the generalized oordinates and generalized speeds. It should be noted that beause of the h h h nature of the matries M, G, K, and ^F the set of differential equations in equation 23 are nonlinear. where (Pim)ij is defined as The use of onstraint equations to enfore the beam boundary onditions, and proper seletion of the orresponding global shape (modal) funtions, are the primary differenes between the AIGC approah and the IGC and NSD approahes (referenes [3,4,and 61. As demonstrated in referene [5], the use of onstraints and proper modal funtion seletion, allows the AIGC approah to aurately solve beam dynamis problems where the lateral deformations of the beam are dominated by either bending or membrane stiffness. This 1119

4 stems from the ability of the AIGC approah to enfore boundary onditions whih annot be expliitly defined in the deformation measures hosen. In partiular, the IGC approah uses the s, u2, and u3 deformation measures and an only ensure satisfation of boundary onditions expliitly defined in those deformation measures. As a result, the IGC approah fails to aurately solve beam problems where the lateral deformations are dominated by membrane stiffness, whih inludes a boundary ondition expliitly desribed in terms of the u1 deformation measure. Conversely, the NSD approah uses the u1, u2, and u3 deformation measures and therefore an only ensure satisfation of boundary onditions expliitly defined in those deformation measures. Thus, the NSD approah fails to aurately solve some beam problems where the lateral deformations are dominated by bending stiffness, whih inludes boundary onditions expliitly desribed in terms of the s deformation measure. The AIGC approah overomes these limitations by enforing boundary onditions whih are not expliitly defined in terms of the hosen deformation measures through the use of onstraint equations. Furthermore, the general nature of the AIGC approah allows the solution of problems where the dominant elasti effets are not known before hand. The physial boundary onditions for the beam an be related to the deformations at the beam ends; this is aomplished diretly with the relationships in equations 13-16, or 19 for deformation onditions, or with spatial derivatives of those relationships for fore or moment onditions. Speifially, onsider a beam antilevered to a rigid mass, as the one shown in Figure 1. The boundary onditions for that beam are those for a antilever beam with the built-in end loated at x=o, and the free end at x=l, and are given by the following equations. no axial deformation at the built-in (24) end %Ix=, no axial torque at the free end = 0 no bending moment in the <3 = 0 31x=L diretion at the free end no bending moment in the 22 diretion at the free end Corresponding to these 12 boundary onditions are 12 onstraint equations relating the deformation generalized oordinates, Sj(t) 6=1,..., V) V j=l $kj(l) qj = 0, k=2,3 (33) (34) (35) (37) (39) For simpliity, the 12 onstraints expressed in equations are represented as no lateral deformation in the g2 diretion at the built-in end no lateral deformation in the g3 diretion at the built-in end no twist at the built-in end no bending slope in the & diretion at the built-in end no bending slope in the g2 diretion at the built-in end no axial load at the free end no shear fore in the 22 diretion at the free end no shear fore in the g3 diretion at the free end (25) (26) (31) = 0, k=1,2,..., 12 The modal (x) (i=l,.., 4, j=1,..., V), are developed using the substruturing tehniques developed by Craig and Bamptong and are disussed in depth in referenes 14 and 71. This tehnique subdivides the modal funtions into two subsets alled dynami and stati modal funtions. For the sake of ompleteness, that proedure will be briefly disussed here. The same modal funtions are used to desribe the lateral deformation measures, u2, and u3. On the other hand, the axial streth, s, and the twisting deformation, 61, are desribed by a different single set of modal funtions The same modal funtions are used for the axial streth and twist, beause for a simple rod axial and twisting behavior is governed by wave equations of the same basi form. The lateral dynami modal funtions, the dynami funtions are developed from an eigenanalysis for lateral vibration of a non-rotating beam with boundary onditions of zero displaement and zero slope at both ends. The lateral stati modes are obtained by applying unit displaements ( or rotations) in the diretions held fixed in developing the dynami modes. While enforing eah unit displaement or rotation, the other displaements or rotations are held fixed. For example, one suh shape funtion satisfies the onditions $(o) = 1, and 1120

5 , For the StretWtwist modal funtions, the dynami funtions are developed from an eigenanalysis for axial vibration of a non-rotating rod with boundary onditions of zero displaement at both ends. The stati modes are obtained by applying unit displaements at eah end while holding the other end futed. This method of seleting the modal funtions serves two purposes. First, the set of modal funtions are general enough to satisfy any boundary ondition. Seond, they do not satisfy any partiular end ondition, thus they prevent satisfying any boundary ondition whih is inorret for the problem at hand. Eauations of Motion and Solution Tehniaue The final equations of motion are obtained by ombining the system differential equations (equation 23) with the onstraint equations desribing the appropriate boundary onditions (equation 41 for the bending stiffness dominated problem being onsidered) to form a set of DAEs. Thus, for the ase being onsidered (non-presribed motion and bending dominated lateral deformations), the equations of motion are vi6 vi6 h Gij u; + Gij u; ij qj = Fi, i=1,2,..., v+6 j=l j=l j=l (42) Referene [7] desribes the development of equations of motion for the ase of presribed motion, whih for the same boundary onditions yield the following equations of motion: The three-dimensional presribed motion problem is illustrated in Figure 2, and the onstants haraterizing the beam are given in Table 1. Links L1 and L2 are assumed rigid (length of 8 m), link L3 (undeformed length 8 m), possessing a hannel setion, is haraterized as a flexible beam with the AIGC approah. The motion of the joints is haraterized by the following desriptions of their respetive angles of rotation. Y2(0 = Y30) = [;-%[t -(f-) sin(?)] E (rad) E-- ;[ ($1 '(TI1 (rad), if 0 I t 2 T,if t>t t- - sins (rad),ifolt<t 0 (rad),if t>t where T = 15 seonds. (45) = F; +F;*, i=1,2,..., v Note that there are six less system differential equations for the presribed motion ase beause the overall motion is assumed to be known. It is also worth noting that the system differential equations are linear (but with time varying oeffiients) for the ase of presribed motion. For the results whih are about to be disussed, the equations of motion were solved using Baumgarte'slO approah. The values of the onstraint stabilization parameters a, and p were hosen suh that no signifiant onstraint drift was notied over the range of simulation. The validity of the equations of motion just presented will be demonstrated by omparing simulations for two distint problems. First, a three-dimensional presribed motion problem for a beam with an offset of the shear enter and entroid, ommonly referred to as non-ompat, will be investigated. This will demonstrate the ability to apture the multi-axial oupling not present in the two-dimensional work in referene [5], and desribe the oupling introdued by the offset of the shear enter and entroid. Seond, a nonpresribed motion problem will be investigated, thus demonstrating the ability of the AIGC approah to apture the two-way oupling between overall motion and loal deformation assoiated with this type of problem. Figure 2 - Physial System for the Three-Dimensional Presribed Motion Problem This problem, similar to one studied by Kane, el al [3], was simulated using the AIGC approah (Equation 43) and a disrete ADAMS" model. The results for the axial, twisting, and lateral deformations of the beam tip are given in Figures 3, 4,5, and 6. The solutions are similar, although some differenes (primarily peak magnitudes) exist. As no additional independent solutions are available, the differenes are 1121

6 aepted. Thus, this simulation demonstrates the ability of the AIGC approah to aurately solve a three-dimensional problem involving torsion and eentriity. Mass per Unit Length Area Moments of Inertia Length Cross Setional Area Elasti Modulus Shear Modulus Eentriity Measms Effetive Torsional Constant p kgl, i 12 = x lo9 m4 I3 = x lo9 m4 L3=8 m &=1.3 x lo5 m2 E = 1.0 x 10 N/m2 G = 5 x 109 N/,2 e2 = 0 e3 = D i = x m e - p 0.0 il n i AIGC and ADAMS Lateral (u2) Defletion Solutions of the Beam Tip for the Three- Dimensional Presribed Motion Problem Table 1 - Flexible Beam Charaterization for the Three- Dimensional Presribed Motion Problem Figure 6 - AIGC and ADAMS Lateral (u3) Defletion Solutions of the Beam Tip for the Three- Dimensional Presribed Motion Problem Figure 3 - AIGC and ADAh4S Axial Defletion Solutions of the Beam Tip for the Three-Dimensional Presribed Motion Problem.,-. e. -u o.- - OI L f -0.5 VI The ability to aurately desribe non-presribed motion problems is demonstrated by simulating the system shown in Figure 7. This problem was studied by Ryan [6], and is analogous to ones arising for some satellites. The mass and inertia properties of the rigid base, and the harateristis of the beam are given in Table 2. The applied torque (defined in equation 22) is given by the following relationship: i 0.1 Nm, O<t<S(se) T3A = (47) 0 Nm,if t > 5 (se) Figure 4 - AIGC and DAMS Axial Twist Defletion Solutions of the Beam Tip for the Three- Dimensional Presribed Motion Problem Figure 7 - Physial System for the Non-Presribed Motion Problem 1122

7 Mass of Rigid Base 5.0 Mass Moments of Inertia of Base Flexible Beam Length Mass per Unit Length Elasti Modulus Cross Setional Area Area Moments of Inertia Effetive Torsional Constant Shear Modulus Eentriity Measws Jzz-50 kgm2 J33 = 130 kg m2 Jlz = J13 = J23 = 0 L =20 rn p =O.2 kgl E = 1.0 x 1O'O N/m2 AJ = 9.30 x lo2 rn2 Iz=I3=5x 1010 m4 K' = 1.2 x 109 m4 G = 5 x 109 N/,z e2 = e3 = 0 CI 'z h s < 0.0 Q.% - rr) Figure 9 - Ryan's IGC Solution of the Overall Beam Angular Veloity for the Non-Presribed Motion Problem Table 2 - Rigid Base and Beam Charaterization for the Non- Presribed Motion Problem The results of Ryan's work (referene [6]) are ompared to simulation results from the AIGC approah, generated using equation 42. The z 3 measure of the angular veloity, is shown in Figures 8 (AIGC) and 9 (Ryan's IGC solution). The loal deformation, desribed by the lateral (22) measure of the tip defletion, is given in Figures 10 (AIGC) and 11 (Ryan). Exellent agreement is seen between the AIGC and IGC (Ryan's) approahes. Also the effet of the loal deformation on the overall motion is learly seen by omparison of the rigid and flexible beam results for the angular veloity. This seond simulation demonstrates the ability of the AIGC approah to aurately solve simultaneously for overall motion and loal deformation when foreshorques are applied. T 0.5 W e.- 0 ll g o = a, n h Figure 10 - AIGC Solution of the Lateral Beam Tip Defletion for the Non-Presribed Motion Problem Figure 8 - AIGC Solution of the Overall Beam Angular Veloity for the Non-Presribed Motion Problem T 0.5.& S g o.e- a, n Figure 11 - Ryan's IGC Solution of the Lateral Beam Tip Defletion for the Non-Presribed Motion Problem 112 3

8 All problems that have been addressed in this paper and in referene [5] have beam boundary onditions whih are time invariant, and are expliitly zero. The AIGC approah ould be easily adapted to problems where the above restrition does not apply. Consider the system shown in Figure 12; in this ase, neither the bending moment, transverse shear load, nor the axial strain are zero at the right hand end, but are related to the aeleration and angular aeleration (time varying) of the attahed mass. The mass at the right hand end has to be treated differently from the one at the left (rigid base), beause of the oordinate system employed. The AIGC approah ould be extended to suh problems by rewriting the onstraints used to enfore the boundary onditions. Kane, T. R., Ryan, R. R., and Banerjee A. K., "Dynamis of a Cantilever Beam Attahed to a Moving Base," Journal of Guidane, Control, and Dynamis,Vol. 10, Marh- April, 1987, pp Yoo, H. H., "Dynami Modeling of Flexible Bodies in Multibody Systems," Ph.D. Thesis, The University of Mihigan, 1989, University Mirofilms In. Haering, W. J., Ryan, R. R., and Sott, R. A., "A New Flexible Body Dynami Formulation for Beam Strutures Undergoing Large Overall Motion," AIAA , 33rd Strutures, Strutural Dynamis, and Materials Conferene, Dallas,TX, April (submitted for publiation in Journal of Guidane, Control, and Dynamis) Ryan, R. R., "Simulation of Atively-Controled Flexible Spaeraft," Journal of GuidQne, Control, and Dynamis, Vol. 13, July-August, 1990, pp Rigid Bnse Figure 12 - Non-Constant, Non-Zero Boundary Condition: Beam With Attahed Mass Summary In this paper, the AIGC approah has been extended to threedimensional motion and deformation, and the overall motion is no longer restrited to a known funtion of time. This extended apability has been demonstrated by investigating the following two problems: 1) a three-dimensional presribed motion problem with torsion and eentriity, 2) a problem with an applied torque (non-presribed motion), exhibiting two-way oupling between loal deformation and overall motion. Haering, W. J., "A New Flexible Body Dynami Formulation for Beam Strutures Undergoing Large Overall Motion," Ph.D. Thesis, The University of Mihigan, May 1992, University Mirofilms In. Kane, T. R., and Levinson, D. A., Dynamis, Theory and Appliations, MGraw-Hill Book Co., New York, N.Y., Craig, R. R., Jr., and Bampton, M. C. C., "Coupling of Substrutures for Dynami Analysis," AIAA Journal, Vol 3, 1968, pp Baumgarte, J. W., "Stabilization of Constraints and Integrals of Motion in Dynamial Systems," Computational Methods in Applied Mehanis and Engineering, Vol. 1, 1972, pp ADAMS 6.0 User's Manual, Mehanial Dynamis In., Ann Arbor, MI, April This paper, in ombination with referene [5], has shown that the AIGC approah an be applied to general flexible body dynami problems involving beam strutures (restrited to small elasti loal deformations). Referene [5] demonstrated the ability of the AIGC approah to solve problems where the lateral deformation of the beam is dominated by either bending or membrane deformation, whih no other lmown approah, using assumed global shape funtions, is apable of. The development in referene [5] was limited to two-dimensional problems with known overall motion as a funtion of time. That restrition has been removed through the development in,this paper. Referenes 1 Simo, J. C., Vu Quo, L., "A Finite Strain Beam Formulation. The Three-Dimensional Dynamis Problem. Part I,," Computer Methods in Applied Mehanis and Engineering, Vol. 49, 1985, pp Christensen, E. R., and Lee, S. W., "Nonlinear Finite Element Modeling of the Dynamis of Unrestrained Flexible Strutures," Computers and Strutures, Vo1.23, 1986, pp