International Journal of Solids and Structures

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1 Interntionl Journl of Solids nd Structures 46 (29) Contents lists ville t ScienceDirect Interntionl Journl of Solids nd Structures journl homepge: Periodic stedy stte response of lrge scle mechnicl models with locl nonlinerities C. Theodosiou, K. Sikelis, S. Ntsivs * Deprtment of Mechnicl Engineering, Aristotle University, Thessloniki, Greece rticle info strct Article history: Received 9 April 29 Received in revised form 2 June 29 Aville online 13 June 29 Keywords: Periodic stedy stte response Lrge scle dynmicl models Nonliner properties Multi-level dynmic sustructuring Long term dynmics of clss of mechnicl systems is investigted in computtionlly efficient wy. Due to geometric complexity, ech structurl component is first discretized y pplying the finite element method. Frequently, this leds to models with quite lrge numer of degrees of freedom. In ddition, the composite system my lso possess nonliner properties. The method pplied overcomes these difficulties y imposing multi-level sustructuring procedure, sed on the sprsity pttern of the stiffness mtrix. This is necessry, since the numer of the resulting equtions of motion cn e so high tht the clssicl coordinte reduction methods ecome inefficient to pply. As result, the originl dimension of the complete system is sustntilly reduced. Susequently, this llows the ppliction of numericl methods which re efficient for predicting response of smll scle systems. In prticulr, systemtic method is pplied next, leding to direct determintion of periodic stedy stte response of nonliner models sujected to periodic excittion. An pproprite continution scheme is lso pplied, leding to evlution of complete rnches of periodic solutions. In ddition, the stility properties of the locted motions re lso determined. Finlly, respresenttive sets of numericl results re presented for n internl comustion cr engine nd complete city us model. Where possile, the ccurcy nd vlidity of the pplied methodology is verified y comprison with results otined for the originl models. Moreover, emphsis is plced in compring results otined y employing the nonliner or the corresponding linerized models. Ó 29 Elsevier Ltd. All rights reserved. 1. Introduction There re mny occsions in engineering prctice, where there is strong interest in determining nd studying the long term ehvior of structurl or mechnicl system, which is sujected to periodic excittion. In such cses, the min interest lies mostly in locting nd investigting periodic stedy stte response. A typicl exmple is the production of response spectr, which re used in order to detect structurl or cousticl resonnces (Crig, 1981; Kropp nd Heiserer, 23) or to predict ftigue filure of criticl structurl prts (Lutes nd Srkni, 1997; ensciutti nd Tvo, 25). When the dynmicl system resulting fter the modeling possesses liner chrcteristics, there re stndrd methods tht cn led to this informtion (Ro, 199). In cses where the dimension of the models considered ecomes excessive, due to the mny geometricl detils tht need to e included in order to cover criticl design needs, the volume of the clcultions cn ecome prohiitive quite frequently. In such cses, ppliction of pproprite dimension reduction methods in either the time or the frequency domin ecomes necessry (Crig, 1981; Cuppens et l., 2). * Corresponding uthor. E-mil ddress: ntsiv@uth.gr (S. Ntsivs). However, more difficulties rise when nonlinerities re present. In prticulr, for smll scle nonliner systems sujected to periodic externl excittion there is lot of nlyticl nd numericl work referring to their long term response (Doedel, 1986; Nyfeh nd lchndrn, 1995). Among other things, it is well known y now tht these systems cn exhiit mny types of periodic motion s well s more complex ehvior, including qusi-periodic nd chotic response. On the other hnd, little is still ville on cpturing long term dynmics for lrge scle nonliner systems (Fey et l., 1996; Verros nd Ntsivs, 22). The min ojective of the present work is to develop nd pply systemtic methodology leding to direct determintion of periodic stedy stte response of periodiclly excited complex mechnicl systems. Here, the term complex refers to two chrcteristic properties of the clss of systems exmined. The first level of complexity is relted to the lrge numer of the corresponding equtions of motion. In fct, the detiled geometricl discretiztion of some of the structurl sustructures, sed on the ppliction of the finite element method (Zienkiewicz, 1986), leds to lrge numer of equtions of motion. As consequence, in mny cses it my not e fesile even to crry over the evlution of the dynmic quntities needed for the trnsformtions leding to the clssicl dimension reduction methods (Fey et l., 1996; Chen /$ - see front mtter Ó 29 Elsevier Ltd. All rights reserved. doi:1.116/j.ijsolstr

2 3566 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) et l., 1998). The second level of complexity is relted to the nonlinerities ssocited with the system response. This poses severe restrictions in the pplicility of most of the ville nd commonly employed methods. On the positive side, good feture of the clss of systems exmined is tht the nonliner chrcteristics re ssocited with reltively smll numer of degrees of freedom of the clss of dynmicl systems exmined. sed on the loclized nture of the nonliner ction, the sic ide of this work is to first reduce the dimension of ll the liner components of the structurl system exmined y pplying n pproprite coordinte trnsformtion. In prticulr, the reduction method pplied is sed on n utomtic multi-level sustructuring (ennighof et l., 2; ennighof nd Lehoucq, 24). Aprt from incresing the computtionl efficiency nd speed, the reduction of the system dimensions mkes menle the susequent ppliction of numericl techniques for determining the dynmic response of complex systems, which re pplicle nd efficient for low order systems. For instnce, this method hs lredy een pplied successfully to the solution of the rel eigenprolem nd the prediction of periodic response of lrge scle liner models with nonclssicl dmping (Kim nd ennighof, 26; Pplukopoulos nd Ntsivs, 27), lrge order gyroscopic systems (Elssel nd Voss, 26) nd rodnd viro-coustic simultions of vehicle models (Kropp nd Heiserer, 23). In ddition, the sme method hs lso fcilitted determintion of the trnsient response of lrge scle nonliner models (Pplukopoulos nd Ntsivs, 27; Theodosiou nd Ntsivs, 29). In the present work, the sme multi-level sustructuring method is coupled with n pproprite numericl procedure in order to determine periodic stedy stte motions of the models exmined, resulting in response to periodic excittion in direct mnner (Doedel, 1986). This coupling tkes into ccount nd exploits the chrcteristics of the generl clss of mechnicl systems considered. Moreover, suitle method is lso pplied, sed on clssicl Floquet theory nd leding to determintion of the stility properties of the locted periodic motions (Nyfeh nd Mook, 1979). Finlly, the methodology developed is complemented y continution method, yielding complete rnches of periodic motions over specified frequency intervl (Rgon et l., 22). The finl outcome is expected to provide vlule informtion nd insight to engineers deling with the nlysis nd design of complex mechnicl systems. The vlidity nd effectiveness of the methodology developed is illustrted nd verified y presenting selected set of numericl results. More specificlly, some typicl results re presented first for detiled finite element model of crnkshft, elonging to n internl comustion engine of commercil cr. Then, response spectr for quite involved city us model re lso determined nd presented. The results otined re useful in ssessing the dynmic response of the mechnicl systems exmined. In oth cses, prticulr emphsis is plced on identifying nd evluting effects cused y the nonliner ction in the dynmics, in comprison with similr predictions of the liner theory. The orgnistion of this pper is s follows. First, the chrcteristics of the clss of mechnicl models exmined re presented in the following section. Then, the sic steps of the methodology developed, including oth the ppliction of the coordinte reduction nd the direct determintion of the periodic stedy stte response, re summrised in the third nd fourth section, respectively. Next, the dynmic response of two exmple models sujected to periodic externl excittion is investigted. Where possile, the ccurcy nd effectiveness of the present methodology is estlished y comprison of results otined for the reduced nd the corresponding complete dynmicl models. The work is completed y summrizing the highlights in the lst section. 2. Clss of mechnicl models exmined Accurte prediction of the dynmic response of mechnicl systems requires frequently the development nd exmintion of geometriclly detiled dynmicl models. On the one hnd, this leds to quite lrge set of equtions of motion. The sitution ecomes more complicted when the systems re forced to operte in conditions involving ctivtion of nonliner chrcteristics. On the other hnd, the informtion extrcted from the systemtic prediction of the dynmic response is essentil nd vlule in performing efficiently other useful studies, s well, relted to ftigue, coustics, identifiction, optimiztion nd control of system. Typiclly, complex mechnicl system is composed of severl structurl components. sed on strict design requirements on ccurcy, these components re usully discretized geometriclly y reltively lrge numer of finite elements (the, 1982; Hughes, 1987). In mny cses, this gives rise to dynmicl system with n excessive numer of degrees of freedom. For ll prcticl purposes, the model of ech mechnicl sustructure cn e ssumed to possess liner properties. However, the elements connecting the system sustructures re typiclly chrcterized y strong nonliner ction. Tking ll the ove into ccount, the equtions of motion of the generl clss of dynmicl systems considered in this work cn e put in the compct mtrix form M _ x þ C x þ K _ x þ p _ ðx; _xþ ¼f _ ðtþ; where ll the unknown coordintes re included in the vector xðtþ ¼ðx 1 x 2... x n Þ T ; while M _ ; C _ nd K _ re the mss, dmping nd stiffness mtrix of the system, respectively. These quntities include contriution from ll the structurl components of the system. Moreover, the elements of vector p _ ðx; _xþ include the contriution of the nonliner terms rising from the ction of the coupled dynmicl system, while vector f _ ðtþ represents the ction rising from the externl forcing. esides the lrge numer of degrees of freedom, the level of difficulty in determining the dynmic response of the clss of systems exmined increses considerly when nonlinerity effects ecome importnt. However, n ttrctive feture of these systems is tht their nonlinerities usully pper minly t reltively smll numer of plces, involving smll portion of the degrees of freedom. This mkes possile the ppliction of specil techniques, which re pproprite for systems with locl nonlinerities. Nmely, for such systems it is possile to reduce significntly the numer of the originl degrees of freedom y pplying suitle coordinte reduction methods (Crig, 1981; Fey et l., 1996; Verros nd Ntsivs, 22). Aprt from incresing the computtionl efficiency nd speed, the reduction of the system dimensions mkes menle the ppliction of severl numericl techniques, which re pplicle nd efficient for low order dynmicl systems. The dimension of the clss of systems exmined in the present work cn e so high, tht ordinry coordinte reduction methods my not e numericlly efficient to pply. For this reson, specil coordinte reduction method is pplied insted (ennighof et l., 2; ennighof nd Lehoucq, 24), whose sic steps re presented in the following section. This reduction leds to sustntil ccelertion of the susequent clcultions. More specificlly, the emphsis of the present study is plced on locting periodic stedy stte motions, when the mechnicl models exmined re sujected to periodic externl excittion. In generl, the long term response of nonliner dynmicl system to periodic excittion cn e either regulr (periodic or qusiperiodic) or irregulr (chotic) (Nyfeh nd lchndrn, 1995). Typiclly, such motions re determined y direct integrtion of the equtions of motion, strting from some selected set of initil ð1þ

3 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) conditions (Crig, 1981; Hughes, 1987). However, ppliction of such method cn possily led to determintion of stle periodic solutions only. Difficulties rise when unstle periodic motions or mny, periodic or nonperiodic, motions coexist for the sme set of technicl prmeters. As consequence, this rute force method cn not crete glol picture of the system resonnces nd dynmics. In the present study, the emphsis is plced on developing systemtic method leding to direct determintion of complete rnches of periodic stedy stte motion, including their stility properties. More detils on this method re presented in the fourth section. 3. Sustructuring method esides the clssicl methods, which re ville nd hve een employed successfully in the pst for reducing the originl numer of degrees of freedom of complex mechnicl system, new clss of coordinte reduction methods hs lso een developed recently, which presents certin computtionl dvntges (Kropp nd Heiserer, 23; Kim nd ennighof, 26; Elssel nd Voss, 26; Pplukopoulos nd Ntsivs, 27). The sic steps of these methods re riefly illustrted in this section. First, the dmping nd nonliner forces re neglected temporrily from the equtions of motion. Then, tking into ccount the sprsity pttern of the stiffness mtrix, the originl set of equtions of motion of the system exmined is reordered nd split into numer of mthemticl sustructures (Krypis nd Kumr, 1995). As result, the equtions of motion for the ith sustructure lone pper in the following liner form: M i x i þ K i x i ¼ f i ðtþ ð2þ At the sme time, the displcement vector of the sustructure considered is split in the form! x i ðtþ ¼ ; xðiþ I x ðiþ where the vectors x ðiþ I nd x ðiþ include the so clled internl nd oundry degrees of freedom of the ith sustructure (Fey et l., 1996). Then, Eq. (2) is prtitioned ccordingly nd is put in the form " #! " #! M ðiþ II M ðiþ I x ðiþ I þ KðiÞ II K ðiþ ðiþ I x I ¼ f! ðiþ I ðtþ M ðiþ I M ðiþ x ðiþ K ðiþ I K ðiþ x ðiþ f ðiþ ðtþ : Next, through suitle coordinte trnsformtion with form x i ðtþ ¼T i q i ðtþ; the originl set of equtions (2) is replced y considerly smller set of ordinry differentil equtions, expressed in terms of the generlized coordintes q i. More specificlly, ppliction of the Ritz trnsformtion (3) into the set of equtions (2) yields the smller dimension set _ M i q i þ K i qi ¼ f iðtþ; where _ M i ¼ T T i M it i ; _ K i ¼ T T i K _ it i nd f i ¼ T T i f i: Following the clssicl Crig mpton pproch, the first columns of the trnsformtion mtrix T i re selected to coincide with numer of fixed interfce norml modes, complemented y numer of sttic correction modes (Crig, 1981). Consequently, Eq. (3) cn e expnded in the form x ðiþ I x ðiþ! ¼ U i W i g i I ðiþ x ðiþ! ; ð3þ ð4þ where the columns of sumtrix U i re determined y solving the eigenvlue prolem K ðiþ II U i ¼ M ðiþ II U ik i nd include the modes corresponding to the lowest nturl frequencies of the component up to specified level. The squres of these frequencies re plced t the digonl of the digonl mtrix K i. On the other hnd, the columns of sumtrix W i represent sttic correction modes nd re determined y solving the following system of liner lgeric equtions: K ðiþ II W i ¼ K ðiþ I ; while I ðiþ is n identity mtrix with pproprite dimension. y pplying n nlogous tretment, similr set of equtions of motion is otined for ll the components of the system. In fct, the multi-level sustructuring method pplied is generliztion of the clssicl Crig mpton method. Its min dvntge is tht the order of the eigenvlue nd liner prolems, defined y Eqs. (5) nd (6), is much smller thn the dimension of the originl system. For lrge order systems, this mkes fesile nd efficient the clcultion of the coordinte reduction trnsformtion. In prticulr, fter performing the necessry lger ssocited with the trnsformtion nd synthesis stges, the liner undmped terms of the equtions of motion cn eventully e put in the form ¼ M q þ Kq f ðtþ; with 1 2 g 1 g 2 q ¼. C ; M ¼ A 4 x sym 2 3 K 1 K 2 K ¼ : sym I 1 l 1;2 l 1; I 2 l 2; K ;..... M ; nd The upper prt of the trnsformed stiffness mtrix is digonl, while the corresponding digonl locks of the mss mtrix re occupied y identity mtrices. Also, from the off digonl locks of the mss mtrix, only those involving coupling etween the sustructures re nonzero. Finlly, the lst prt of vector q, represented y x, includes ll the oundry degrees of freedom of the system. The corresponding prts of the mss nd stiffness mtrix, represented y M, nd K,, respectively, re full. However, the dimensions of these sumtrices re usully much smller thn the dimensions of mtrices M nd K, which in turn re much smller thn the dimension of the mss nd stiffness mtrices of the originl system. When dmping effects re present, the trnsformed dmping mtrix hs form similr to tht of the trnsformed mss mtrix. However, the digonl locks re full nd ecome digonl only when the corresponding sustructure exhiits clssicl dmping. In prcticl pplictions, the devition from clssicl dmping is cused frequently y the existence of reltively smll numer of discrete dmpers. This sitution hs lredy een hndled efficiently in previous work, y including the degrees of freedom involved with these discrete dmpers in the set of the oundry degrees of freedom (Kim nd ennighof, 26; Pplukopoulos nd Ntsivs, 27). In similr fshion, the set of the oundry degrees of freedom x is selected in this study so tht it includes ll the interconnection points s well s the points where nonliner ð5þ ð6þ

4 3568 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) ction is present. In this wy, the exct nonliner chrcteristics of the system re preserved in the reduced system. This leds to sustntil numericl improvements, which hve lredy een demonstrted in previous work on direct integrtion of mechnicl systems with smooth nd nonsmooth chrcteristics (Pplukopoulos nd Ntsivs, 27; Theodosiou nd Ntsivs, 29). Therefore, fter dding ll these effects, the equtions of motion cn eventully e cst in the generl form expressed y M q þ C _q þ Kq þ pðq; _qþ ¼f ðtþ; As consequence of the pplied trnsformtions, the order of the finl set of the equtions of motion is reduced sustntilly, while mintining numericl ccurcy up to suitly selected forcing frequency level. Moreover, since the individul trnsformtions re performed on mny smll dimensionl systems insted of one lrger dimensionl system, drstic reduction in the computtion time is chieved. esides, this pproch leds to other importnt numericl enefits, since it is ssocited with much smller volume of dt trnsfer nd cuses tremendous reduction in the computer memory required for the execution of the overll computtions. Finlly, prt from incresing the computtionl efficiency nd speed, the reduction of the system dimensions resulting fter employing the multi-level sustructuring process descried mkes menle the ppliction of severl numericl techniques, which re efficient for low order dynmicl systems. Here, this is exploited in ccelerting the determintion of periodic motions of the complex mechnicl models exmined, resulting from externlly imposed periodic excittion, s explined in the following section. 4. Direct determintion of periodic stedy stte response The drstic reduction of the originl model dimension order mkes possile the ppliction of specil methods, which re pplicle for cpturing periodic stedy stte motions of low order systems (Doedel, 1986; Nyfeh nd lchndrn, 1995). Moreover, the specil coordinte reduction technique pplied in the present study mkes menle the ppliction of these methods to systems of much lrger dimension thn those investigted in the pst in the re of engineering pplictions (Ntrj nd Nelson, 1989; Fey et l., 1996; Verros nd Ntsivs, 22). In prticulr, shooting method hs een developed for the purposes of the present study, leding to direct determintion of periodic stedy stte response of the clss of dynmicl systems exmined. For this purpose, the velocity vector _q ¼ v; is first introduced nd the reduced set of equtions of motion (7) is rewritten in the first order form _u ¼ hðu; tþ; ð8þ with u ¼ q v nd hðu; tþ ¼ When the externl forcing is periodic, tht is f ðt þ T E Þ¼f ðtþ;! v M 1 : ½f ðtþ Cv Kq pðq;vþš it is resonle to expect tht, mong the different possile types of long term response, the system my lso rech periodic stedy stte, with uðt þ TÞ ¼uðtÞ: The period T of the response is in generl commensurte multiple of the forcing period T E, with the most common cse eing T = mt E, ð7þ corresponding to hrmonic (m = 1) or suhrmonic (m =2,3,...) response. The min ojective of the present study is to pply suitle method for determining such motions in direct nd systemtic wy. In rief, this method is sed on the fct tht if uðþ ¼ q T _q T T u corresponds to set of initil conditions leding to periodic motion of the dynmicl system represented y (8), then the following condition is lso stisfied: uðtþ ¼u : The method employed tkes into ccount tht n ritrry initil estimte of vector u will not stisfy in generl the lst condition to within specified numericl tolernce. Therefore, in order to improve this choice, the following vectoril function is introduced: gðu Þ¼uðTÞ u : ð9þ ð1þ This converts the originl initil vlue mthemticl prolem exmined to two point oundry vlue prolem with unknowns included in vector u. Clerly, the correct set of initil conditions must led eventully to stisfction of the following system of lgeric equtions: gðu Þ¼: ð11þ This lst system is nonliner nd n pproprite Newton Rphson type method is needed for its numericl solution (the, 1982; Pozrikidis, 1998). All these methods re sed on n itertive process defined y the scheme u ½iþ1Š ¼ u ½iŠ þ Du½iŠ : In prticulr, the correction Du ½iŠ imposed on the ith itertion is determined y solving the liner system of lgeric equtions J u ½iŠ Du ½iŠ ¼ g u½iš where J u ½iŠ u ; ð12þ is the corresponding Jcoin mtrix. Numericlly, the computtion of vector g u ½iŠ in Eq. (12) is otined y performing direct integrtion of Eq. (8) over response period. In ddition, the Jcoin mtrix of the dynmicl system exmined is lso needed in the clcultions nd computed y employing Eq. (1) in the form @u ¼ UðTÞ I; ð13þ where I is n identity mtrix with pproprite dimension, while UðTÞ ðtþ is the so-clled monodromy mtrix of the dynmicl system represented y Eq. (8). The evlution of this mtrix strts with the definition of the corresponding trnsfer mtrix UðtÞ ðtþ: Then, employing this definition nd the equtions of motion in the first order form Eq. (8), it turns out tht the elements of this mtrix cn e evluted from _UðtÞ ðtþ ðu; @u or eventully y solving the system ðtþ;

5 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) _UðtÞ ¼AðtÞUðtÞ; with ðu; ð17þ ð16þ Therefore, the determintion of the trnsfer mtrix U(t) nd consequently of the monodromy mtrix U(T) nd of the Jcoin mtrix J, defined y Eq. (13), is chieved y integrting the system of liner ordinry differentil equtions (16), suject to the initil conditions UðÞ ¼I: ð18þ In order to void costly opertions, relted to determintion nd inversion of the Jcoin mtrix, suitle qusi-newton method hs een developed for solving Eq. (11) nd consequently for locting periodic motions of Eq. (8), or equivlently of Eq. (7) (royden, 1965; Golu nd vn Lon, 1996). The numericl process ends when the vector u is computed with sufficient ccurcy. Once periodic motion is locted, it is eqully importnt to determine its stility properties, since only stle motions re oservle in prctice. As usul, the stility nlysis of locted periodic motion, sy u p (t), strts y introducing smll perturtion into this motion. The ide is to exmine how this perturtion evolves with time. More specificlly, let uðtþ ¼u p ðtþþeyðtþ; with e 1. Sustituting the lst expression in Eq. (8), Tylorexpnding round u p (t) nd keeping only up to first order terms, it turns out tht the perturtion introduced stisfies the following liner set of equtions: _yðtþ ¼AðtÞyðtÞ; ð19þ where the mtrix A(t) is defined y Eq. (17). It is esy to verify tht this mtrix stisfies the periodicity condition AðtÞ ¼Aðt þ TÞ: Therefore, tking into ccount the lst condition nd pplying clssicl Floquet theory in Eq. (19), it turns out tht the stility properties of the periodic motion u p (t) depend on the mgnitude of the eigenvlues of the monodromy mtrix U(T) (Nyfeh nd Mook, 1979). In prticulr, if ll the eigenvlues of this mtrix hve mgnitude less thn one, the originl perturtion dies out grdully nd the motion exmined is stle. However, if there exists t lest one eigenvlue of the monodromy mtrix with mgnitude lrger thn one then the originl perturtion grows with time nd the locted periodic motion is unstle. Finlly, for prmeter comintions leding to t lest one eigenvlue of mtrix U(T) with mgnitude equl to one ifurctions occur, signling qulittive chnges in the system response (Wiggins, 199). In mny prcticl pplictions, it is frequently required to locte complete rnches of periodic motions of mechnicl system, s n importnt prmeter of the system is vried. For instnce, in periodiclly excited dynmicl systems, typicl such prmeter is the fundmentl forcing frequency. In such cses, it is useful to pply proper continution technique. In the most commonly employed ctegory of pth following methods, the complete process is split into prediction nd correction stge (Fried, 1984; Wlker, 1999). For instnce, ccording to the rclength continution scheme, oth the solution nd the system prmeter, sy l, re expressed s function of new prmeter, representing the rclength, sy s, of the solution pth in the corresponding phse spce, so tht u ¼ uðsþ nd l ¼ lðsþ: Then, if (u m,l m ) is the m-th locted solution of the rnch, n initil estimte of the next solution is determined from the scheme u mþ1 ¼ u m þ u g;m Ds m nd l mþ1 ¼ l m þ w m Ds m ; where the tngentil vector u g,m is evluted from u ðu m; l m Þ; while Ds m nd w m re weighting sclrs selected in n pproprite mnner for ech prolem. Introduction of prmeter l in the set of unknowns necessittes the considertion of n dditionl lgeric condition. This extr eqution is provided in the correction step. For instnce, when the Riks Wempner scheme is pplied (Wempner, 1971; Riks, 1979), the corresponding new eqution is expressed in the form u T g;m Du½iŠ m þ w2 m Dl½iŠ m ¼ : ð2þ This reflects the fct tht the correction imposed to the solution is perpendiculr to the prediction. Addition of the lst condition to the originl system of Eq. (1) leds to new system of lgeric equtions, equl in numer to the unknowns, including the elements of vector u nd prmeter l. In generl, ddition of this extr eqution cuses difficulties in the convergence of Newton s method. Therefore, cre should e exercised so tht this is done only in the vicinity of some specil points, like ifurction points, of frequency response digrm. 5. Numericl results In this section, some typicl numericl results re presented for two selected mechnicl models. Specificlly, the first exmple focusses on periodic stedy stte response of the crnkshft of cr engine, sujected to periodic gs loding. Likewise, the second exmple investigtes periodic response of the complete structure of city us sujected to hrmonic excittion. In oth cses, the min ojective is to explore the ccurcy nd efficiency of the methodology developed. At the sme time, the effect of some importnt technicl prmeters on the dynmics of the exmple systems is lso investigted. For convenience in the presenttion, these two exmples re treted seprtely in the following susections Results for cr engine crnkshft The first exmple model is the crnkshft of four-cylinder inline internl comustion engine, elonging to cr. In fct, esides the shft, the model exmined includes the engine pulley t the left end nd the flywheel t the right end, s shown in Fig. 1. A finite element model of this system ws otined fter discretizing its geometry y solid (mostly hexhedrl) finite elements, leding to mechnicl model with more thn 16,8 degrees of freedom. The crnkshft intercts with the engine lock t five positions (indicted y Ltin numering in Fig. 1) through oil journl erings, possessing strongly nonliner chrcteristics. In prticulr, the qulittive form of the restoring forces developed t those erings is depicted in Fig. 1. The corresponding dmping forces exhiit similr chrcteristics lso. In ddition, the forcing is cused y the gs pressure developed within the engine cylinders nd is periodic function of time, s shown in Fig. 1c. In the lst figure, the numers in prentheses correspond to the position of the cylinders of the engine, which re ssemled through the connecting rods with the crnkshft t the positions depicted in Fig. 1. Therefore, they indicte the firing order of the cylinders. Moreover, in ll the susequent clcultions the forces pplied re ssumed to hve the sme form for ll the vlues of the fundmentl forcing frequency. Finlly, the inerti forces developed due to the pure rigid ody rottion of the shft were neglected (Morit nd Okmur, 1995). More specificlly, the rigid ody rottion of

6 357 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) (1) (3) (4) (2) c Restoring Force Force Displcement T/2 T Fig. 1. () Geometry of the crnkshft. () Chrcteristics of the restoring force developed t the erings. (c) Force due to the gs pressure developed within the engine cylinders over forcing period. the shft ws restrined elsticlly y plcing n pproprite system of springs t its right end (on the flywheel side). During the coordinte trnsformtion phse, the highest response frequency of interest ws set equl to 7 Hz, due to the high frequency content of the gs forcing, while the cutoff frequency for the sustructure eigenvlue prolems ws selected s 3. As result, the originl model ws divided into 31 sustructures, lying on four levels nd the dimension of the sustructures rnged from 567 to 11. Eventully, this trnsformtion led to reduction in the size of the dynmicl model from the originl 16,822 degrees of freedom to only 28 degrees of freedom. Moreover, 54 of these were oundry degrees of freedom. Since the forces pplied to the mechnicl model exmined re periodic, it is nturl to expect tht the corresponding dynmicl system will exhiit periodic stedy stte motions fter sufficiently long initil time intervl. Indeed, Fig. 2 shows frequency response digrms otined directly y pplying the methodology developed, for periodic forcing corresponding to n effective gs pressure equl to p eff = 1 r. In prticulr, the effective (root men squre) vlue of the periodic ccelertion history signls recorded t two selected positions re displyed in Fig. 2, over frequency rnge extending from 6 to 6 rpm. First, in Fig. 2 re shown ccelertion spectr otined t the position where the first connecting rod, sed on the numering of Fig. 1, is connected to the shft. The continuous lines were otined from the nonliner model, while the roken lines represent similr results, otined y considering linerized model, insted. Specificlly, since there is no sustntil sttic loding in the cse exmined, the linerized model ws defined y considering the potentil energy of the equivlent ering springs t the given forcing level, so tht the mximum displcements t the ering loctions of the fully nonliner nd the linerized model re out the sme t 3 rpm. Direct comprison confirms tht significnt differences re oserved etween the results of the Fig. 2. Effective vlue of the periodic ccelertion history, recorded t: () the connection position of the crnkshft with the first connecting rod nd () the first journl ering of the crnkshft.

7 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) [mm] [rpm] [rpm] Fig. 3. Effective vlue of the periodic: () displcement nd () ccelertion history, recorded t the connection position of the first connecting rod nd the crnkshft. linerized nd the nonliner model. Moreover, these differences were found to e mplified considerly t other positions, especilly t points close to the ering nonlinerities. For instnce, in Fig. 2 re shown similr ccelertion spectr, otined t the position where the crnkshft is connected to the first journl ering. These results demonstrte tht the ccelertion levels re out one order of mgnitude smller thn those recorded in the previous position. Also, the nonliner effects cn hve sustntil effect on the system response nd my e completely different thn those predicted y pplying liner nlyses, something tht is done in prctice quite frequently (Morit nd Okmur, 1995). The lowest nturl frequency of the linerized model exmined ws computed to e 9876 rpm. Oviously, this frequency is well ove the mximum fundmentl forcing frequency exmined. This implies tht the peks oserved in Fig. 2 correspond to superhrmonic resonnces induced y either the periodic nture of the forcing or the system nonlinerities (Nyfeh nd Mook, 1979). In order to isolte the effects due to the ction of the nonliner forces, in Fig. 3 re shown results otined y imposing uniform increse in the originl forcing mplitude y fctor of two nd three, respectively. For instnce, Fig. 3 shows the displcement spectr otined t the position where the first connecting rod is ttched to the crnkshft. Likewise, Fig. 3 presents similr comprison for the ccelertion mplitude vlues otined t the sme point. In oth cses, the increse in the response mplitude is nonuniform nd frequency shifting occurs, which is in contrst to predictions otined sed on liner models (Crig, 1981; Ro, 199). Moreover, more resonnces ecome pprent s the forcing is incresed. Agin, the sitution ecomes more intensified t points close to the nonliner erings. For instnce, in Fig. 4 re shown similr results with those presented in Fig. 3, otined t the position where the first journl ering is connected to the crnkshft. Specificlly, Fig. 4 shows the displcement spectr, while Fig. 4 presents the corresponding ccelertion spectr. Once gin, the effect of the ering nonlinerities re quite pronounced. Finlly, similr differences were lso oserved in oth the form nd the mplitude of the periodic histories cptured for the model response quntities. For instnce, in Fig. 5 is presented the history otined for the ccelertion recorded over response period t the sme two points exmined ove, t stedy stte conditions. More specificlly, these results were otined t 3 rpm nd for the lrgest forcing level exmined, corresponding to p eff = 3 r. First, direct comprison with the results otined y the corresponding linerized model (represented y the roken curves) demonstrtes tht n cceptle greement level is reched for the ccelertion recorded t the position where the first connecting rod is connected to the crnkshft (Fig. 5). On the other hnd, sustntil differences re oserved in oth the form nd the mximum vlues for the sme signls, recorded t the position of the first oil journl ering (Fig. 5). Among other things, it is gin pprent tht much smller response mplitudes re developed in the ltter cse [mm] [rpm] [rpm] Fig. 4. Effective vlue of the periodic: () displcement nd () ccelertion history, recorded t the connection position of the first oil journl ering nd the crnkshft.

8 % Error 3572 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) nonliner liner T/2 T nonliner liner -8 T/2 T Fig. 5. One response period of the ccelertion history recorded t the connection of the crnkshft with: () the first connecting rod nd () the first oil journl ering, t 3 rpm nd for p eff = 3 r. The following set of results is included in n effort to illustrte the numericl ccurcy nd efficiency of the methodology developed. First, in Fig. 6 re presented selected results, ssessing the numericl ccurcy of the coordinte reduction method pplied. Specificlly, in Fig. 6 is shown the periodic stedy stte displcement history otined over response period, developed t the connection position of the crnkshft with the first connecting rod. This response ws cused y periodic forcing corresponding to n effective gs pressure equl to p eff = 1 r, with fundmentl forcing frequency of 3 rpm. The continuous line represents results otined from the originl model, while the roken curve corresponds to results otined from the reduced model. Since the two curves re virtully indistinguishle, Fig. 6 is lso included, illustrting the numericl error defined y e n ¼ x ori xred =jxmx j: n n In the lst expression, x ori n nd x red n represent the displcement vlue otined from the originl nd the reduced model, respectively, t the time instnt t n, while x mx corresponds to the mximum vlue of the signl. Among other things, these results verify the ccurcy of the coordinte reduction method nd especilly the exct incorportion of the nonliner terms in the reduced model. Likewise, the results of Tle 1 re presented in n effort to ssess the numericl efficiency of the methodology developed. More specificlly, these results refer to clcultion of the sme periodic stedy stte motion, otined t 3 rpm for p eff = 1 r. Due to the reltively smll size of the prolem exmined, in ddition to Tle 1 Comprison of numericl efficiency y running the reduced nd originl models. Reduction method used CPU time (m:s) Elpsed time (m:s) I/O (G) MLDS 1:12 1: CMS 2:14 2: No reduction 13:45 14:2 3.5 running the clcultions fter ppliction of the multi-level dynmic sustructuring method (MLDS), it ws lso possile to run similr clcultions fter ppliction of the clssicl single-level component mode synthesis method (CMS) s well s the originl model, without prior ppliction of ny coordinte reduction method. All the numericl results of this work were otined on worksttion (CPU: Intel Pentium 4, 3.2 GHz, RAM: 2 G, OS: GNU Linux 2.6 i586) nd the results of Tle 1 refer to the CPU time, the ellpsed time nd the volume of the input/output clcultions performed for the prticulr cse exmined. Oviously, the results otined fter ppliction of either of the coordinte reduction methods re much etter thn those otined y employing the originl model. The lst set of results referring to the crnkshft exmple re shown in Tle 2. The results presented re similr to those of Tle 1, with the emphsis plced now on the comprison of the performnce of the multi-level sustructuring method nd the single-level component mode synthesis method. The results demonstrte tht the performnce of the multi-level sustructuring [mm] Originl Reduced -.4 T/2 T 1-4 T/2 T Fig. 6. () Comprison of displcement history signls nd () numericl error e n in the clcultions, over response period t 3 rpm.

9 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) Tle 2 Comprison of numericl efficiency etween MLDS nd CMS. Method used CPU time (m:s) Elpsed time (m:s) I/O (M) MLDS :12 :14 26 CMS 1:1 1:4 223 method is superior. Moreover, this is mplified s the order of the system exmined ecomes lrger. In ddition, fter criticl size of the originl model, it my not e fesile to pply the single-level method t ll, s verified y results presented in the following exmple Results for city us model The numericl methodology developed is pplicle to lrge models, with dimension in the order of millions of degrees of freedom. For instnce, mechnicl system with much igger originl dimension is considered next. More specificlly, in Fig. 7 is shown the mechnicl model of city us. esides the us upper ody structure (superstructure) nd chssis frme, the front nd the rer suspensions re included mong the other importnt structurl susystems. The min prts of the us superstructure were geometriclly discretized y reltively lrge numer of shell finite elements, so tht the models exmined cn lso e used t lter stge in order to produce results for viro-coustics studies s well (Kropp nd Heiserer, 23). As result, the finite element discretiztion of the vehicle superstructure led to model possessing 955,866 degrees of freedom. On the other hnd, the us chssis frme ws lso discretized y shell finite elements, leding to model with 337,26 degrees of freedom. In ddition, the driver nd the pssengers s well s the tire susystems were modeled y pproprite sets of discrete msses, springs nd dmpers. Finlly, specil dded mss elements were lso employed in modeling systems like the ir-condition unit, the fuel tnks, the us floor including the pssenger sets nd the ggge store comprtment. As result, the finl model possesses totl of 1,372,699 degrees of freedom. For ll prcticl purposes, the model of the us ody cn e ssumed to possess liner properties. However, the (ushing) elements connecting the system sustructures re typiclly chrcterized y strong nonliner ction. Furthermore, strongly nonliner chrcteristics re lso encountered in the ction of the shock sorer nd spring units. In prticulr, the restoring force developed t the shock sorer unit of typicl us, connecting the ody with the wheels, cn e represented y comintion of liner nd two nonliner nd symmetric springs with hrdening chrcteristics (Verros nd Ntsivs, 22). Likewise, the dmping force developed in the suspension dmpers cn e represented y piecewise liner chrcteristics. For instnce, Fig. 7 presents the dmping force developed t the shock sorers of ll the suspension units of the vehicle exmined. Clerly, ll the min suspension dmpers exhiit different ehvior in tension thn in compression, which is lso typicl in utomotive engineering prctice (Gillespie, 1992). Moreover, the equivlent dmping coefficient (slope) is reduced s the reltive speed increses. The ove description mkes cler tht the model exmined is n idel exmple for the methodology developed. The first set of numericl results, depicted in Fig. 8, were otined for the linerized model otined round the sttic equilirium position, resulting y pplying the weight of the structure, for three reduced models of the us. More specificlly, during the coordinte trnsformtion phse, the dimensions of the reduced dynmicl models were selected to include liner modes up to 2, 5 nd 1 Hz, respectively, while the cutoff frequency for the sustructure eigenvlue prolems ws selected equl to 3. As result, the originl model ws divided into 842 sustructures, lying on nine different levels nd the dimension of the sustructures vried from 2118 to 36. This trnsformtion led eventully to reduction in the size of the dynmicl model to only 21, 38 nd 523 degrees of freedom, from which 177, 249 nd 277 were oundry degrees of freedom, respectively [N] Fig. 8. Comprison of nturl frequencies, otined for three reduced linerized us models [mm/s] Fig. 7. () Finite element model of the us structure. () Force developed in the min dmpers of the front nd the rer suspension shock sorer nd spring units.

10 3574 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) Fig. 9. Frequency response digrms: () t the driver set position nd () t selected point of the us roof, for three reduced liner models. In prticulr, the horizontl xis of Fig. 8 presents the index, while the verticl xis presents the vlue of the nturl frequencies predicted y the forementioned models. These results demonstrte tht the ccurcy of the nturl frequencies of ech reduced model depends strongly on the pre-selected frequency rnge. In ddition, rpid nd strong devition is oserved to occur in the results outside the pre-specified rnge, which is in ccordnce with similr oservtions in n erlier study (Pplukopoulos nd Ntsivs, 27). Finlly, it is worth mentioning tht in terms of numericl performnce, ppliction of the CMS method ws not possile for ny of the cses considered here, due minly to the lrge dimension of the eigenvlue prolem, defined y Eq. (5), tht needs to e solved. Moving long the sme direction, Fig. 9 presents the spectr of the ccelertion otined t two specific points of the linerized us models considered. The first set of them is shown in Fig. 9 nd refers to the point where the driver set is mounted to the us frme, while the second is selected point t the us roof. The response ws cused y pplying verticl hrmonic excittion t the front left wheel of the us, which is close to the driver position. The effective (root men squre) vlue of the ccelertion history is presented within the forcing frequency intervl 3 Hz, which is typicl for ride studies referring to ground vehicles (Ellis, 1969; Gillespie, 1992). The results demonstrte tht the ccurcy level otined is sufficient within the frequency rnge exmined, t lest for the lst two reduced models. In fct, considering models with even more degrees of freedom cuses virtully no chnge in the results otined within the selected frequency rnge. Next, Fig. 1 displys frequency-response digrms otined for the ccelertion t the sme two specific points of the us considered. In ll the results reported from here on, the reduced model employed in the clcultions is sufficiently ccurte within the rnge from to 5 Hz. For comprison purposes, the roken curves represent similr results, otined y running the model resulted y linerizing the equtions of motion round the sttic equilirium position of the us. Clerly, there pper significnt devitions etween the predictions of the fully nonliner nd the linerized model employed, t lest within certin frequency rnges. Among other things, the results of the lst figure demonstrte the fct tht the methodology developed extends the clssicl frequency response nlysis (FRA) from the liner to the nonliner domin. In this respect, the informtion extrcted from Fig. 1 is useful in ssessing the forcing frequency rnges where the response quntity exmined exhiits high level virtions. In generl, the devitions oserved etween the predictions of the nonliner nd the corresponding linerized models re mplified s the forcing mplitude is incresed. Similr digrms re lso useful in predicting the effect of the importnt system prmeters, like the horizontl velocity or the equivlent stiffness nd dmping prmeters of the suspension, on the system response. This provides the sic informtion needed in selecting these prmeters in n optimum mnner. In ddition, such informtion is lso useful in helth monitoring studies (Metllidis et l., 28). For instnce, Fig. 11 presents similr frequency response digrms, otined fter introducing dmge in one of the min suspension springs. Specificlly, ccording to the dmge scenrio dopted, the stiff- 4 nonliner liner 2 nonliner liner Fig. 1. Frequency response digrms: () t the driver set position nd () t selected point of the us roof.

11 C. Theodosiou et l. / Interntionl Journl of Solids nd Structures 46 (29) k k/2 4 k k/ Fig. 11. Frequency response digrms: () t the driver set position nd () t selected point of the us roof. ness coefficient of the front left suspension unit ws reduced to hlf of its originl vlue. As confirmed y the results of Fig. 11, the results re ffected significntly, especilly round the rnge of 1 Hz, which is the rnge ffected mostly y the wheel ction. 6. Synopsis nd conclusions A complete methodology ws presented for determining periodic stedy stte response of clss of periodiclly driven mechnicl models in direct nd computtionlly efficient wy. Specificlly, the models exmined involve reltively lrge numer of degrees of freedom nd my possess nonliner chrcteristics. However, the nonliner ction is confined to reltively smll numer of degrees of freedom. Therefore, the sic ide ws to first pply multi-level dynmic sustructuring method, in order to condenste the originl dimension of the system significntly, so tht the reduced model is sufficiently ccurte up to pre-specified level of forcing frequencies. This ws chieved y employing n pproprite sequence of coordinte trnsformtions, sed on the sprsity pttern of the stiffness mtrix. The nlysis ws then completed y systemtic method leding to direct determintion of stedy stte response of nonliner systems sujected to periodic externl excittion y exploiting the chrcteristics of the clss of systems exmined. A method for determining the stility properties of the locted periodic motions ws lso developed, in conjunction with continution technique, yielding complete rnches of periodic motions over specified frequency intervl. The ccurcy nd effectiveness of the methodology developed ws illustrted y presenting numericl results for n involved model of the crnkshft of cr engine under periodic excittion s well s for quite complex finite element model of city us under hrmonic se excittion. First, frequency spectr of severl response quntities relted to the crnkshft dynmics were constructed for stedy stte motions resulting from periodic gs excittion. Specil emphsis ws put on exmining the devitions rising etween predictions of the nonliner nd the corresponding linerized models. The effect of incresing the forcing mplitude on chnging the qulittive form of the response spectr ws lso investigted. Then, the ttention ws shifted to exmining ride dynmic performnce of detiled finite element model of city us. Initilly, comprison of response digrms otined y employing reduced models with different numer of liner norml modes nd consequently with different level of ccurcy ws performed. For the sufficiently ccurte reduced model selected, the effect of the system nonlinerities on the response ws investigted. Also, it ws illustrted tht the nlysis developed cn e useful in mny res, including optiml selection of criticl prmeters. In closing, it is worth mentioning tht gret dvntge of the present methodology my e relized y extending it to other importnt res, such s identifiction, dignostics nd vehicle control. Acknowledgements This reserch ws co-finnced y E.U. Europen Socil Fund (75%) nd the Greek Ministry of Development GSRT (25%) through the PENED 23 progrm (3-ED-524). The first uthor ws lso supported in prt y fellowship from the Ntionl Fellowship Institute (IKY). References the, K.J., Finite Element Procedures in Engineering Anlysis. Prentice-Hll, Englewood Cliffs, NJ. ensciutti, D., Tvo, R., 25. Spectrl methods for lifetime prediction under widend sttionry rndom processes. Interntionl Journl of Ftigue 27, ennighof, J.K., Lehoucq, R.., 24. An utomted multilevel sustructuring method for the eigenspce computtion in liner elstodynmics. SIAM Journl on Scientific Computing 25, ennighof, J.K., Kpln, M.F., Muller, M.., Kim, M., 2. Meeting the NVH computtionl chllenge: utomted multi-level sustructuring. In: Proceedings of the 18th Interntionl Modl Anlysis Conference, Sn Antonio, TX. royden, C.G., A clss of methods for solving nonliner simultneous equtions. Mthemtics of Computtion 19, Chen, C.S., Ntsivs, S., Nelson, H.D., Coupled lterl torsionl virtion of ger-pir system supported y squeeze film dmper. ASME Journl of Virtion nd Acoustics 12, Crig Jr., R.R., Structurl Dynmics An Introduction to Computer Methods. Wiley, New York. Cuppens, K., Ss, P., Hermns, L., 2. Evlution of FRF sed sustructuring nd modl synthesis techniques oolied to vehicle FE dt. ISMA 2, K.U. Leuven, elgium, pp Doedel, E., AUTO: Softwre for Continution nd ifurction Prolems in Ordinry Differentil Equtions. Cliforni Institute of Technology, Psden, CA. Ellis, J.R., Vehicle Dynmics. usiness ooks Limited, London. Elssel, K., Voss, H., 26. Reducing huge gyroscopic eigenprolems y utomted multi-level sustructuring. Arch. Appl. Mech. 76, Fey, R.H.., vn Cmpen, D.H., de Krker, A., Long term structurl dynmics of mechnicl systems with locl nonlinerities. ASME Journl of Virtion nd Acoustics 118, Fried, I., Orthogonl trjectory ccession to the nonliner equilirium curve. Comput. Methods Appl. Mech. Eng. 47, Gillespie, T.D., Fundmentls of Vehicle Dynmics. Society of Automotive Engineers, Wrrendle, PA. Golu, G.H., vn Lon, C.F., Mtrix Computtions. University Press, ltimore, MD. Hughes, T.J.R., The Finite Element Method. Prentice-Hll, Englewood Cliffs, NJ. Krypis, G., Kumr, V., METIS: unstructured grph prtitioning nd sprse mtrix ordering system. Technicl Report, University of Minnesot, Minnepolis.