Coupling of Substructures for Dynamic Analyses

Transcription

1 Coupling of Substuctues fo Dynamic Analyses Roy Caig, Mevyn Bampton To cite this vesion: Roy Caig, Mevyn Bampton. Coupling of Substuctues fo Dynamic Analyses. AAA Jounal, Ameican nstitute of Aeonautics and Astonautics, 1968, 6 (7), pp < /3.4741>. <hal > HAL d: hal Submitted on 12 Jun 2017 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés. Distibuted unde a Ceative Commons Attibution 4.0 ntenational License

2 Coupling of Substuctues fo Dynamic Analyses RoY R. CRAG JR.* The Univesity of Texas at Austin, Austin, Texas AND MERVYN c. c. BAMPTONt The Boeing Company, Seattle, Wash. A method fo teating a complex stuctue as an assemblage of distinct egions, o substuctues, is pesented. Using basic mass and stiffness matices fo the substuctues, togethe with conditions of geometical compatibility along substuctue boundaies, the method employs two foms of genealized coodinates. Bounday genet alized coodinates give displacements and otations of points along substuctue boundaies and ae t elated to the displacement modes of the substuctues }{nown as "constaint modes." All constaint modes at e geneated by matix opeations fom substuctue input data. Substuctue nomal-mode genealized coodinates ae elated to fee vibation modes of the substuctues elative to completely t estained boundaies. The definition of substuctue modes and the equiement of compatibility along substuctue boundaies lead to coodinate tansfomation matices that ae employed in obtaining system tnass and stiffness matices font the mass and stiffne.o;;s matices of the substuctue.<;;. Povision is made, though a Rayleigh Ritz pocedue, fo educing the total numbe of degees of feedom of a stuctue while etaining accuate desciption of its dynamic behavio. Substuctue boundaies may have any degee of edundancy. An example is pesented giving a fee vibation analysis of a stuctue having a highly indeteminate substuctue bounday. D.( ) D.W T u {a l<) { ab} () [k] () fm]() {F}c> {<i>/}c> = [<i)<] () [<i)n]() {p}() {pb} () {pn}() [ili] () P}() { ~~ () {Q} [a] () lq} lqb} Nomenclatue vitual change in quantity ( vitual wok kinetic enegy stain enegy physical, o genealized, displacements of points in th substuctue physical, o genealized, displacements of points on bounday between th substuctue and adjoining substuctues physical, o genealized, displacements of "inteio" points of th substuctue, i.e., points not held in common with othe substuctues stiffness matix of th substuctue fo all feedoms in lo} () mass matix (diagonal) fo th substuctue in coodinates {a l () substuctue foces in the {a} () coodinate system spatial dependence of {li 1 ) C> when substuctue 1s vibating in the jth substuctue nomal mode matix of constaint modes of substuctue matix of etained nomal modes of substuctue final genealized coodinates fo th substuctue final genealized coodinates fo bounday displacements of th substuctue final genealized coodinates fo amplitudes of nomal modes of th substuctue with totally constained bounday stiffness matix of th substuctue in { p} () coodinates mass matix of th substuctue in { p} > coodinates substuctue genealized foces in the { p} C> coodinate system statically equivalent applied foces genealized foce in { q} coodinate system tansfomation matix elating { li l () and { p} () system genealized coodinates system genealized coodinates fo bounday displacements * Associate Pofesso of Aeospace Engineeing. Membe AAA. t Dynamics Reseach Enginee, Commecial Aiplane Division. qn} [K] [M] ll} system genealized coodinates elated to nomal modes of constained substuctues stiffness matix fo total stuctue in q} genealized coodinate system mass matix in { q} system coodinates a co a(2) O(s) Pc1> PC2) collection of substuctue vectos, ll} c> { p} - = collection of substuctue vectos, { p l C> Pc > [/3] tansfomation matix elating { p l and { q} l. ntoduction THE fundamental poblems in the analysis of stuctual systems subjected to time-dependent foces ae the detemination of the displacement esponse of the system and the intenal stesses. One appoach to the solution of these poblems is to idealize the system as an assemblage of discete stuctual elements, theeby obtaining sets of equations which ae conveniently teated by the opeations of matix algeba. n ode to obtain the detailed displacement and stess infomation equied in the analysis of a complex stuctue, it may be necessay to divide the stuctue into a vey lage numbe of elements, theeby intoducing matices of lage ode. To etain such lage matices and analyze the stuctue as a whole may equie compute facilities of geate capacity than those that ae available. At any ate, it fails to take into account the pocedues usually employed in the design o modification of complex stuctues wheeby majo stuctual components, o substuctues, ae often designed by diffeent engineeing goups o at diffeent times. t is desiable, theefoe, to use a substuctue appoach so that such designs and modifications may poceed as independently 1

3 A substuctue is consideed to be composed of inteio degees of feedom and bounday degees of feedom. f the numbe of bounday feedoms is geatm; than the numbe of igid-body feedoms of the substuctue, the fome ae sepaated by Huty into "statically deteminate constaints" and "edundant constaints." Displacements of a substuctue ae defined in tems of genealized coodinates that ae elated to specified sets of nomalized displacement functions. The modes ae consideed, by Huty, 3 in thee categoies as follows: Fig. Typical suhstuctm e anangement. as possible with due consideation being given to the final coupling of substuctues to fom the complete stuctue. At the same time, the ode of the final matices may be educed by incopoating a Rayleigh-Ritz pocedue into the substuctue method. The hieachy of stuctues Lo be consideed in this pape consists of the stuctual system, the substuctue, and the element. t is the pocess of dividing the system into substuctues and coupling these again to fom the system that is teated hee. t is assumed that analysis of the individual substuctues by discete element idealization has been caied out to obtain the necessay substuctue stiffness matices and mass matices. n the pesent discussion, a lumped-mass fomulation is used, although othe pocedues fo obtaining substuctue mass matices could be employed. 1 The displacement method is employed in the final coupling of substuctues. Substuctue methods that ely upon the displacement method, i.e., upon geometic compatibility along substuctue boundaies, fo coupling of the substuctues have been employed in analysis of both static 2 3 and dynamic 4 5 behavio of stuctues. Each of the latte employs a Rayleigh-Ritz pocedue to educe the ode of the final system mass and stiffness matices. The "banch-modes" technique discussed by Gladwell 6 has the advantage that, fo a special choice of banches o substuctues, the system stiffness matix is diagonal and is easily fomed along with the nomalmode analysis of the banches. Fo othe divisions of the system, the banch-modes method equies fomation of a stiffness matix in the oiginal system coodinates, with subsequent eduction to final system coodinates by the Rayleigh-Ritz method. The two pocedues of Gladwell 6 fo choosing the banches ae: 1) "by demanding that one of the components vibates with distotion while the othe vibates as a igid body attached to the distoting component;" 2) "by demanding that one of the components vibates with distotion while the othe emains fixed." n eithe case it may be seen that the bounday between two substuctues is assumed to be igid, a estiction that cannot be accepted fo cases of highly edundant boundaies between substuctues. The pesent pape may be consideed as,a genealization of banch-modes pocedues to stuctues with such edundant substuctue boundaies. Huty 4 5 has developed a pocedue fo the analysis of stuctual systems in which the displacement behavio of each substuctue is descibed by a set of genealized coodinates consisting of the following thee types: 1) igidbody coodinates, 2) constaint coodinates, and 3) nomalmode coodinates. Since the pesent pape diffes fom Refs. 4 and 5 pincipally in the selection of genealized coodinates fo substuctues, the significance of the genealized coodinates used by Huty equies futhe elucidation hee. Fist ae igid-body displacements in which the component (substuctue) is displaced without defomation. f no fixed extenal constaints ae imposed on the component, it will have six degees of feedom as a igid body; hence thee may be as many as six igid-body modes fo the component. This numbe will be educed if extenal constaints ae pesent. n the second categoy ae modes that will exist only if the system of constaints on the component is indeteminate. These modes ae defined by poducing a unit displacement on each edundant constaint in tun, with all othe constaints fixed. Fo bevity these modes ae called "constaint modes," and they ae equal in numbe to the numbe of edundant constaints. The thid categoy includes modes that define displacements elative to the constaint system. They may be, fo example, the nomal modes of vibation of the component with all constaints fixed. As igid-body modes Huty~ suggests the use of igid-body tanslations and small otations. Hence, the displacements of the deteminate constaints must be expessed in tems of these igid-body motions, since it is the fome that will appea in the equations expessing compatibility of displacements along substuctue boundaies. The total numbe of genealized coodinates to be used in descibing a stuctual system may be educed, in the Huty method, by using only a potion of the total possible nomal modes of the constained substuctues in descibing the displacement of inteio points elative to the constaints. A numbe of basic simplifications occu if all of the bounday feedoms ae teated alike, athe than being sepaated into deteminate and indeteminate constaints. Since, fo a highly edundant bounday connection it is not at all clea which feedoms should be teated as "statically deteminate" and which as "edundant," it is highly advantageous to have a method that does not equie the enginee to make such distinctions. n addition, it will be shown that the fomulation of igid-body modes and the establishment of equations elating igid-body genealized coodinates to statically deteminate constaint displacements is unnecessay when all bounday feedoms ae teated alike. Theefoe, in the pesent pape all bounday displacements will be teated in the same manne that Huty uses fo "edundant constaints." To teat the displacements of substuctues as being composed of "constaint modes" and nomal modes does not peclude the possibility of analyzing, by the pesent substuctue pocedues, a stuctual system that has igidbody feedom, povided that constaint modes ae defined fo all bounday feedoms. n the pesent analysis the constaint modes of a substuctue ae defined by poducing a unit displacement of each bounday feedom in tun, with all othe bounday feedoms fixed and with all inteio feedoms unconstained. The nomal modes of a substuctue descibe the motion of inteio feedoms elative to the fixed boundaies in tems of the nomal modes of fee vibation of the substuctue with all bounday feedoms fixed. Equations of compatibility elate the displacements of adjoining substuctues. The compatibility equations ae used to detemine a set of system bounday genealized coodinates which is equal in numbe to the total numbe of substuctue bounday feedoms minus the numbe of compatibility equations. The final set of system genealized coodinates consists, then, of the system bounday genealized coodinates plus genealized coodinates fo the substuctue nomal modes of all of the substuctues. Lagange's equations ae used to deive the 2

4 equations of motion fo the system. The system stiffness matix [K] is found to have a paticulaly simple diagonalbanded fom. n this espect, the pesent analysis is the logical mathematical extension of the banch-modes method to systems having highly edundant substuctue boundaies. 2. Substuctue Constaint Modes and Nomal Modes; Substuctue Mass Matix and Stiffness Matix Figue 1 shows a typical th substuctue and its boundaies with substuctues ( - 1) and ( + 1). Each substuctue is connected to one o moe othe substuctues. Fo the pesent analysis it is necessay to isolate the th substuctue fom all connecting substuctues when the bounday feedoms ae totally constained. This may necessitate incopoation of some dummy bounday feedoms. This is illustated by Fig. 2, in which a cantileve plate is divided into two substuctues. The oiginal coodinates of the poblem ae taken to be the tansvese displacements { ~} of the nodal gid points. n ode that the bounday feedoms will allow complete isolation of the substuctues it is necessay to intoduce dummy bounday feedoms, in this case the otations at the bounday nodal points as shown. t is assumed that the stiffness matix [k lcj is available. Patitions of [k]cj ae fomed as shown in Eq. (1): { -~;-} () = [ ~~; i-~;;-1) {!; L) (l) whee supescipts B and efe to bounday and inteio, espectively. The coodinates { o 8 } <l ae physical displacements of bounday points. n the pesent analysis { o 1 } cl ae the physical displacements of inteio points, although fo othe substuctue idealizations, e.g. Ref. 1, { o 1 } cl may be any othe set of specified genealized coodinates descibing the defomation of the inteio of the substuctue. Constaint modes will be defined as the mode shapes of the inteio feedoms due to successive unit displacement of bounday points, all othe bounday points being totally constained. To detemine constaint modes the foces at all inteio feedoms ae set equal to zeo. Equation (1) gives o (2) The subscipt () will be omitted when no confusion aises theeby. The matix [qi 0 ] is the desied matix of constaint modes. The matix patition [kii] gives the foces at inteio feedoms due, successively, to unit displacement of inteio feedoms with all bounday points totally constained. Let the diagonal mass matix, obtained fom a lumped-mass fomulation, be patitioned consistent with Eq. (1): The substuctue nomal modes will be defined as the nomal modes of the substuctue with totally constained bounday. These ae obtained fom the equations The eigenvectos of Eq. (4) ae the nomal modes of the constained substuctue. These eigenvectos fom the espective columns of the matix [pn]. A fundamental assumption of the pesent analysis is that each { oi} Cl contains a easonably lage numbe of elements, i.e., inteio degees of feedom, and that the contibution of the substuctue nomal modes to the displacement of these inteio feedoms can be appoximated by a significantly smalle set of coodinates {pn} () Let [ N]c) be patitioned accoding to whee the modes to be etained ae collected in [q)n]c) Let the elements of { pn} <l be the amplitudes of the espective modal vectos contained in [q)n], and let { p} () = { ]!.~. } pn () The coodinate tansfomation elating the substuctue final coodinates to the substuctue initial coodinates is, then, whee (3) (4) (5) (6) (7) (8) z Fig. 2 Oiginal substuctue coodinates and substuctue bounday displacement coodinates. foce esultants a l bounday displacement feedoms e~, J ', ', ',,_-.., _. second substuctue...,..,. 3

5 The unit matix occuing in [a]() implies that The stiffness matix in the final set of substuctue genealized coodinates 1 p} () is [k] = [a]'[k][a] = _[E~-~[ -_,9 J 0 [lcnn (9) (10) The matices [k 8 N] and [knb] ae null due to the way in which [a] has been defined in Eq. (8). Futhemoe, Finally, (11) (12) which is diagonal due to the othogonality of the substuctue nomal modes. The nullity of the off-diagonal submatices is fundamental to the simplicity of the final system stiffness matix. The mass matix fo the substuctue is given by [m] = [a]'fmj[a] = [~;;:-~;: J (13) will be poduced only at the inteio points of the substuctues adjacent to the displaced bounday point and along the boundaies of these substuctues. Fo example, a unit displacement of point B in Fig. 1 will cause foces only within, and along the boundaies of, substuctues () and ( + 1). Compatibility conditions ae needed to insue that the displacements on the bounday of substuctue () match those of its adjoining substuctues. No compatibility conditions exist among the nomal mode coodinates { pnj <J of the vaious substuctues. Theefoe, let (20) The compatibility equations elating all bounday displacements may be witten, symbolically, in the fom (21) whee the {pas) ae dependent vaiables and {pu 8 ) ae independent, o genealized, coodinates. [Ad] is a squae matix. Let (22) The constituent submatices ae (fiibbj = fmbb J + (<ficj'fmhj[<ficj (fiibnj = [fiinbj 1 = [<ficj 1 [mil j[<finj [fiinn J = [<fin]' [mil j[<finj (14) Then, fom Eq. (21) Since (23) The matix [fiinn J is diagonal due to the othogonality popeties of the substuctue nomal modes. The elements of [fiinn J ae elated to coesponding elements of [knn j by the equations (15) whee w; is the natual fequency of the jth substuctue nomal mode. Let fl W be the vitual wok done by foces {F) in an abitay vitual displacement { M). The genealized foces in the { p) coodinate system ae, by definition, those foces that do an amount of vitual wok equal to fl W when- the { p} coodinates undego a vitual displacement consistent with {flo}: flw = {P)'{flp) = {F)'(M) Substitution of Eq. (7) into Eq. (16) gives Since { flp) is abitay, o {P)'{ flp) = {F)'[al{ flp) {P} 1 = {F} '[a] {P} = [a]'{f} (16) (17) (18) (19) (24) is not, in geneal, aanged accoding to the patitions shown in Eq. (21), the tansfomation matix in Eq. (23) expesses in essence, although possibly not in exact fom, the elationship between { pb) and { q 8 }. The geneal fom of the compatibility equation may be witten [BJ{pB) = {0) (25) and the geneal fom of the coodinate tansfomation is then (3(~)B] =. {qb} [ f3(s)b (26) Finally, the geneal fom of the coodinate tansfomation fom {p} to {q) is o {p) = [(3J{q) (27) 3. Geometical Compatibility of Substuctue Boundaies Figue 1 illustates typical substuctue aangements fo an aiplane. As indicated peviously, each substuctue is connected to one o moe othe substuctues along boundaies that can be totally constained. Thus, if points along the bounday of a given substuctue ae totally constained, foces applied at inteio points cause eactions only along the bounday of the given substuctue. On the othe hand, if a given bounday point is subjected to a unit displacement while all othe points ae totally constained, foces Let {p} () (28) and ( p(l)b l 1--~~~i_ {p) = i. (29) ~~(~~- l P< >"' J 4

6 Then [.8] has the fom 0 i 0 : ; 0 1,8(1) 8.., (J_)_jv! --o-; i--o--! --o- ~ o-! --o-- ~ !--0-!~~;)-s ---- : i/<2lni i 0 i , ---- ~ [,8] = o 1 (30) 1 i f i l 1 : :...! : and [,8]<,> has the fot m 0 : 0 i i 0 i,b(s) B : , , L --o--! -o-! ~j~s~n~--0- (,B]() = [ ~---: J Q [---~---~----~ J~i~~~ ]... :l()nl...! : 0 () (s) 4. System Equations of Motion _j (31) The coodinate tansfomation of Eqs. (27) and (28) allows the mass and stiffness matices fo the system to be obtained. These may then be used in deiving the equations of motion though the usc of Lagange's equations. The system mass matix and system stiffness matix ae obtained by summing, espectively, the kinetic enegies of the substuctues and the stain enegies of the substuctues. Let [fiij = ' '. : : ' ! fii()bb! fii()bn! ' ! fii()nb! fii()nn : ! : ] ]. : : : l : : ' ' ' l :...J (32) ffii j is a diagonal-banded matix with the substuctue mass matices along the diagonal. Because of the diagonal chaacte of ffii ], the following elationship holds: [M] = L (ij]c/[fii)<> [,B] () = LB] 1 [fiij(,8] (33) whee [M] is the final system mass matix. let [k](t)i -----: ! ' ' k -] = i : ' - J ' l -!!!(k ]cs)_j Then, due to the diagonal chaacte of [k ], i Futhemoe, (34) [K] = L (,8Jc,) 1 (k]() (,B]() = (/3] 1 [kj[,b] (35) whee [K] is the system stiffness matix in { q) coodinates. When Eq. (31) is combined with Eq. (35), the final stiffness matix is found to exhibit the diagonal chaacte shown below: whee [K ] = fknn](l)l i l.., : ~ ~ ] ' 1 ' ' ' ' ' ' ' : ~: -..:: : :! flcn N J<a)l , L! l![kbbj..j (36) (37) When [K] is fomed by Eqs. (36) and (37), the handling of lage numbes of zeos, as equied by Eq. (35), is avoided. The system mass matix may likewise be found in a moe efficient manne than that given in Eq. (33). When Eq. (31) is combined with Eq. (33), [M] is found to have the fom [M] whee and ffiinn ] (1) i i i i (M NB]() : : !- :! : - l j ~---~ i i! -----~ ' l : fin,nn J<) :! (M NB )() l : : ~ : : : :. l :!. :. :! ! _! ~ L [MBNl<tJ i! [M BN]() i! [MBB ].J (38) (39) The total stain enegy of the stuctual system is given by U =!{q) 1 [K]{q) (41) and the system kinetic enegy by T =! {q) 1 [M]{q ) (42) The equations of motion fo the system may be deived fom Lagange's equations (d/ dt) (ot/ oqj) - (ot/ oqj) + (ou/ oqj) Qj (j = 1,..., n) (43) whee Qi is the genealized foce coesponding to the jth genealized coodinate, exclusive of the foce contibution deivable fom the elastic stain enegy potential U. Thus, [M]{ q) + [K]{q) = {Q) (44) Fo dynamic esponse poblems it is necessay to detemine the time-dependent genealized foce vecto { Q} coesponding to the time-dependent applied foces. This is obtained by examining the vitual wok done on the system when vitual displacements { ~ q ) ae given to the system. The vitual wok of the system may be expessed as a sum of the vitual wok of the substuctues: ~W = :E {P) < l' {~P) <l = {P) 1 {~p) = {Q) 1 {~qj (45) The last equality equies that the { ~p) and { ~q} be consistent, which is guaanteed by Eq. (27). Then, and since { ~q) { p} [/3 ]{ ~ q) = { Q) { ~q l (46) is completely abitay, { Q} = [{J]'{P} (47) Equations (19) and (47) may be combined to give {Q) = (fj ] 1 (a] 1 {F) (48) Howeve, since the substuctue foce vecto {F) contains the unknown bounday foces {FE}, Eq. (48) does not seve fo the initial detemination of { Q}. At this point a set of substuctue applied foces will be intoduced. n addition, the substuctue bounday coodinates { o 8 ) (l must now be sepaated into { o8r) ( ) whee extenal loads exist and { o8d) (), which ae dummy coodinates that may h ave been intoduced in ode to povide fo complete bounday constaint. Thus, extenally applied foces will exist fo the co- 5

7 s=no d e num bes 16" _l 8" substuctue bounday 0~0 \ \1 -ZM typ. 0 l 2" typ. 0 ~ f 0,,.,,_._ 2024-T) alum. E=l0.5xl0 15 psi. JolcO.)) f'' lb/in'. t.. O.l25 in. Fig node cantileve ectangula plate. odinates {o} ( ) and { obr l ( ). Let { 0 l<) be odeed and patitioned accoding to {o}<l ={;;} 0 () (49) and let the set of statically equivalent applied foces be given by (50) The genealized foces { Q} of the system may be detemined fom these statically equivalent applied foces by equating the vitual wok done by these foces in going though vitual displacements {.M} ( ) and the vitual wok done by { Q} in going though vitual displacements { t.q} consistent with { t.o}. Thus, {Q} = L ([i3br)()'{ ;JBR} + ({J](/(</)C\</)N)(l'{ :Jl} ()) (51) whee [{JBR]<l is the potion of [{:lejc) associat ed with theetained bounday feedoms { obr} ( ) 5. Fee Vibation Example n ode to check out the pocedues outlined in the peceding sections, the fee vibmtion of a cantileve plate was consideed. A compaison was made between mode shapes and fequencies of the plate calculated using the stiffness and mass matices of a finite-element epesentation of the full plate with coesponding fequencies and mode shapes calculated by the peceding substuctue method. The plate had the dimensions and popeties shown in Fig. 3. The substuctue bounday was located so that it would not lie along the line of symmety of the full plate. To povide fo complete estaint along the substuctue bounday, otations have been intoduced as "dummy" bounday feedoms. Each sustuctue, theefoe, has 18 bounday degees of feedom. A diect stiffness pogam was used to geneate stiffness matices fo the full plate and fo the two substuctues. The full-plate stiffness matix was used in a vibation analysis to obtain modes and fequencies. The substuctue stiffness matices wee used as input fo the coupling pogam. The fequencies and mode shapes of the cantileve plate, obtained by using the substuctue coupling pocedue, wee compaed with the esults obtained using stiffness and mass matices of the full plate. n both analyses, the mass was lumped at the gid points. No otatoy inetia was included at the dummy bounday otation feedoms. Table 1 gives the fequencies obtained by the two methods indicated peviously and the amplitudes of seveal node points fo modes 3 and 6. The coupled-plate solution employed the 18 bounday feedoms plus 5 nomal-mode feedoms fo each substuctue. Figue 4 shows a constaint mode of plate 2, and Fig. 5 gives the shape of a nomal mode of the same substuctue. Table 2 shows some of the node-point displacements fo the sixth mode, which is the thid anti- symmetic mode of the plate. t may be seen that the coupled-plate solution gives acceptable mode shape esults when compaed with the full-plate solution. The effect of Fig. 4 A constaint mode of plate 2. l<'ig. 5 A nomal mode of pfate 2. 6

8 Table Some numeical esults fo cantileve plate fequencies and mode shapes Feq., cps. 1 2 Full-plate Coupled-plate %Diffeence Node no., amplitude (t.) Full Mode numbe / Mode numbe 3 Coupled 6 Full 6 Coupled the delibeate location of the substuctue bounday away fom the line of symmety of the plate may be seen by compaing the displacement of symmetically located node points. Fo instance, node points 12 and 72 have the same numeical amplitude, , in the full-plate solution, wheeas the values and wee obtained in the coupled-plate solution. Simila compaisons may be made fo othe symmetically located points, and it may be concluded that the accuacy of the mode shapes is not affected advesely by the choice of substuctue bounday location. Amplitudes of othe modes up to and including the eighth mode indicate that the coupling technique is capable of poducing satisfactoy mode shapes as well as fequencies. Futhe esults of this investigation ae given in Ref. 9. Table 2 Mode-shape data, mode 6 (198 cps) Node Full Coupled Node Full Coupled c b b b b b b a Nodes in this column ae symmety nodes fo points in left-hand column. b Nodes on line of symmety of plate. c Both solutions nomalized on node Concluding Remaks This pape has descibed a method of substuctue analysis simila to that of Huty 4 5 but diffeing in one impotant espect; namely, it simplifies the teatment of igid-body modes of substuctues by eliminating the sepaation of bounday foces into statically deteminate and statically indeteminate eactions. This enables all modes associated with bounday feedoms to be teated alike. Futhemoe, all such modes may be geneated by matix opeations pefomed by the compute on input substuctue matices. The advantages of the method pesented hee ae that it leads to easie fomulation of substuctue poblems, moe compact pesentation, simplified compute pogamming, and, in all likelihood, shote compute times. Refeences 1 Agyis, J. H., "Some Results on the Fee-Fee Oscillations of Aicaft Type Stuctues," Revue Fancaise Mecan., Vol. 3, 1965, pp Pzemieniecki, J. S., " Matix Stuctual Analysis of Substuctues,'' AAA Jounal, Vol. 1, No.1, J an. 1963, pp Taig,. C., " Automated Stess Analysis Using Substuctues," Oct. 1965, pape pesented at Confeence on Matix Methods in Stuctual Mechanics, Wight-Patteson Ai F oce Base, Ohio. 4 Huty, W. C., " Dynamic Analysis of Stuctual Systems Using Component Modes," AAA Jounal, Vol. 3, No.4, Apil 1965, pp Huty, W. C., "Dynamic Analysis of Stuctual Systems by Component Mode Synt hesis," Rept , 1964, Jet Populsion Lab., Pasadena, Calif. 6 Gladwell, G. M. L., "Banch Mode Analysis of Vibating Systems," Jounal of Sound Vibation, Vol. 1, 1964, pp Huty, W. C. and Rubinstein, M. F., Dynamics of Stuctues, Pentice-Hall, Englewood Cliffs, N. J., Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeoelasticity, Addison-Wesley, Reading, Mass., Caig, R. R. and Bampton, M. C. C., " Coupling of Substuctues fo Dynamic Analyses," Rcpt. D TN, 1!)66, The Boeing Company, Commecial Aiplane Div., Seattle, Wash. 7