Nonlinear frequency response analysis of structural vibrations

Transcription

1 Comput Mech mauscrpt No. (wll be serted by the edtor) Nolear frequecy respose aalyss of structural vbratos O. Weeger U. Wever B. Smeo Receved: date / Accepted: date Abstract I ths paper we preset a method for olear frequecy respose aalyss of mechacal vbratos of 3- dmesoal sold structures. For computg olear frequecy respose to perodc exctatos, we employ the wellestablshed harmoc balace method. A fudametal aspect for allowg a large-scale applcato of the method s model order reducto of the dscretzed equato of moto. Therefore we propose the utlzato of a modal projecto method ehaced wth modal dervatves, provdg secod-order formato. For a effcet spatal dscretzato of cotuum mechacs olear partal dfferetal equatos, cludg large deformatos ad hyperelastc materal laws, we use the sogeometrc fte elemet method, whch has already bee show to possess advatages over classcal fte elemet dscretzatos terms of hgher accuracy of umercal approxmatos the felds of lear vbrato ad statc large deformato aalyss. Wth several computatoal examples, we demostrate the applcablty ad accuracy of the modal dervatve reducto method for olear statc computatos ad vbrato aalyss. Thus, the preseted method opes a promsg perspectve o applcato of olear frequecy aalyss to large-scale dustral problems. Keywords Nolear vbrato model reducto modal dervatves harmoc balace sogeometrc aalyss Olver Weeger Utz Wever Semes AG, Corporate Techology Otto-Hah-Rg 6, Much, Germay E-mal: olver.weeger.ext@semes.com, E-mal: utz.wever@semes.com Olver Weeger Berd Smeo TU Kaserslauter, Faculty of Mathematcs, P.O. Box 3049, Kaserslauter, Germay E-mal: weeger@rhrk.u-kl.de E-mal: smeo@mathematk.u-kl.de 1 Itroducto Vbrato aalyss ad olear structural aalyss both play a mportat role the dustral mechacal egeerg process, but so far there are o effcet methods avalable for a olear structural frequecy respose aalyss o a large scale. For olear frequecy respose aalyss we use the harmoc balace method (HBM) [1 3], whch trasforms ad solves the uderlyg equato of moto the frequecy doma. I prevous work we have already vestgated olear structural vbratos wth sogeometrc fte elemets ad harmoc balace, ad have demostrated ts applcablty usg the olear Euler-Beroull beam structural model [4]. Now we exteded ths method to 3- dmesoal olear structural mechacs wth large deformatos ad hyperelastc materal laws [5, 6]. Though harmoc balace s a well-establshed method for olear frequecy aalyss, for example the cotext of tegrated crcut smulatos [7,8], t s so far hardly used mechacs, oly for lower dmesoal structural models such as beams ad plates [9 13]. Commercal fte elemet aalyss (FEA) software such as ANSYS, Nastra or ABAQUS do ot provde ay methods dedcated to olear frequecy respose. Ths s maly due to the trucated Fourer expaso HBM uses for frequecy doma approxmato of each degree of freedom (DOF) of the spatal dscretzato, whch produces a blow-up of total DOFs: the sparse lear system to be solved the ed s ot oly m- tmes bgger, but also wth m-tmes as may o-zero etres per row as the spatal dscretzato, where m s the umber of Fourer coeffcets. Therefore we eed model order reducto (MOR) of the spatal dscretzato to reduce the sze of the lear system sgfcatly ad make a effcet umercal soluto of the system arsg from HBM eve possble.

2 2 O. Weeger et al. Whle modal reducto, where the equato of moto s projected oto a subspace spaed by a selecto of egemodes, s a well-establshed techque lear FEA ad vbrato aalyss [14], more advaced methods are eeded the olear cotext [5,15]. For example [16] a sgle- DOF reducto o olear modes was troduced. We propose to use a modal reducto wth modal dervatves [17,18], whch are a secod-order ehacemet of lear egemodes. The method has bee successfully appled olear dyamc aalyss by tme-tegrato before [19 22], ad we show that t s especally sutable our olear vbrato framework wth harmoc balace. I cotrast to most other commo reducto methods so far used olear tme-tegrato, t does ot requre a curret state of deformato of the system ad cotuous bass updates, thus the projecto bass ca be fully pre-computed based o the lear system. Furthermore there are also smlar well-establshed techques of secod-order ehacemets other felds of computatoal egeerg such as ucertaty quatfcato [23]. For the spatal dscretzato we rely o the sogeometrc fte elemet method, but ote that the proposed olear frequecy aalyss method wth modal dervatve reducto could be appled usg ay spatal dscretzato method. Isogeometrc aalyss (IGA) was troduced by Hughes et al. [24] 2005 ad ams at closg the gap betwee computeraded desg (CAD), umercal smulato ad maufacturg (CAM) by usg the same geometry represetato throughout the whole egeerg process. As sple fuctos, such as B-Sples ad o-uform ratoal B-Sples (NURBS) [25], are typcally used for geometry desg CAD software, IGA these fuctos are also employed for dscretzato of geometry ad umercal soluto a soparameterc fasho. Ths cocept has already bee successfully appled to several umercal dscretzato methods, such as boudary elemets, collocato, fte volumes, ad, of course, sogeometrc fte elemets [24, 26 28]. A detaled troducto to IGA ad collecto of umercal aalyss, propertes ad applcatos of the method ca be foud the moograph [29]. It has bee show that sogeometrc fte elemets have substatal advatages over classcal Lagraga fte elemets the cotext of lear vbrato aalyss,.e. soluto of egevalue problems, where so-called optc ad acoustc braches are avoded, whch leads to a much hgher accuracy especally hgher egefrequeces [30]. I geeral, IGA provdes hgher accuracy per DOF for umercal soluto of lear ellptc, parabolc ad hyperbolc partal dfferetal equatos (PDEs) tha stadard fte elemet methods due to hgher cotuty of sples, whereas rates of covergece are the same [29, 31]. The method has also bee appled olear cotuum mechacs, where the advatages of the approach could be verfed [32 34]. The further structure of ths paper after after ths troductory Secto 1 s as follows: We cotue wth a summary of cotuum mechacs equatos of large deformato hyperelastcty ad ther sogeometrc fte elemet dscretzato Secto 2. The we gve a bref revew of modal aalyss ad drect frequecy respose as meas of lear vbrato aalyss Secto 3, before we troduce the harmoc balace method applcato to olear structural frequecy respose. Secto 4 s dedcated to model order reducto methods, wth a revew of commoly used methods followed by a detalled troducto to modal reducto wth the cocept of modal dervatves. Wth the computatoal examples preseted Secto 5, we prove the fuctog of our framework for olear structural vbrato aalyss ad show that t sutable for large-scale applcatos. The we coclude wth a short summary of the work preseted ad gve a outlook o future research drectos Secto 6. 2 Isogeometrc fte elemet dscretzato of large deformato hyperelastcty I ths work we address the umercal smulato of dyamc behavour of mechacal structures descrbed by geometrcal ad materal oleartes. Therefore ths secto we gve a summary of the theory of cotuum mechacs wth large deformato kematcs ad costtutve laws of hyperelastcty. The we derve the spatal dscretzato of goverg equatos usg sogeometrc fte elemets. 2.1 Kematcs ad costtutve laws Frst we wat to gve a bref revew of the Total Lagraga formulato of kematcs ad costtutve relatos of solds subject to large deformatos ad hypererlastc materal behavour, based o the moographs [5, 6]. I the Total Lagraga pot of vew, moto ad deformato of a body over tme are descrbed wth respect to ts tal cofgurato gve by the doma Ω R 3. At every tme t the terval of terest [0,T ], the curret posto x Ω t R 3 of each pot X Ω ca be expressed terms of ts tal posto ad a dsplacemet feld u R 3 (see also Fgure 1): x(x,t) = X + u(x,t). (1) For the descrpto of the deformato process we eed the deformato gradet,.e. the spatal gradet of curret w.r.t. to tal posto of each pot: F(X,t) = dx du (X,t) = I + (X,t) = I + u(x,t). (2) dx dx

3 Nolear frequecy respose aalyss of structural vbratos 3 Ω u(x,t) Ω t ad for the olear Neo-Hooke materal t holds ψ(c) = λ 2 (lj)2 µ lj + µ (tr(c) 3), 2 S = λ lj C 1 + µ (I C 1 ). (10) O X x(x,t) Fg. 1: Moto of the body wth doma Ω The Jacoba determat J = detf s a measure of the volume chage of the body. For compressble materals, whch are ot subject of ths work, t holds J = 1. Furthermore we eed a stra measure defed the tal cofgurato, the Gree-Lagrage stra tesor E(X,t) = 1 2( C(X,t) I ). (3) It s defed usg the rght Cauchy-Gree tesor C(X,t) = F T F = I + u T + u + u T u, (4) whch s a quadratc expresso terms of dsplacemets resp. the deformato gradet. I lear elastcty theory the hgher order term s ommtted ad the lear stra measure s used: e(x,t) = 1 ( u T + u ). (5) 2 Velocty ad accelerato of a pot the referece cofgurato are gve as: v(x,t) = ẋ(x,t) = dx (X,t) = u(x,t), dt (6) a(x,t) = ẍ(x,t) = d2 x (X,t) = ü(x,t). dt2 As stress measure the materal cofgurato we use the secod Pola-Krchoff stress tesor S, whch s related to the true Cauchy stress σ the curret cofgurato by the followg equato: S = JF 1 σf T. (7) I hyperelastcty the costtutve relato of stra ad stress s defed by a stra eergy fucto ψ: S = dψ de = 2dψ dc. (8) I ths work we refer to two partcular choces of stra eergy fuctos. For the lear St. Veat-Krchhoff materal law, whch s also used lear elastcty, t s ψ(e) = λ 2 tr(e)2 + µ tr(e 2 ), S = λ tr(e)i + 2µ E, (9) For learzato wth the later descrbed soluto process we are gog to eed the costtutve 4th order tesors C SE = ds de = 2 ds dc (11) for both materal laws. For the St. Veat-Krchhoff materal t s C SE jkl = λ δ jδ kl + µ ( δ k δ jl + δ l δ k j ), (12) ad for the Neo-Hooke materal C SE jkl = λ C 1 j C 1 kl + (µ λ lj) ( C 1 k C 1 jl + C 1 l 2.2 Strog ad weak form of goverg equatos C 1 k j ). (13) Wth the kematc quattes troduced the precedg Secto 2.1, we ca follow [5, 6] formulatg the local balace dfferetal equatos. I the strog form these must hold for all materal pots X Ω ad tmes t [0, T ]. The coservato of mass the Lagraga cofgurato reads as ρj = ρ 0, (14) where ρ 0 s the tal ad ρ the curret mass desty. Furthermore we eed the coservato of lear mometum, volvg volume forces ρ 0 b: dv F S + ρ 0 b = ρ 0 ü. (15) Local balace of agular mometum yelds the symmetry of the secod Pola-Krchhoff stress tesor S = S T, (16) ad the frst law of thermodyamcs,.e. coservato of eergy, reads: ρ 0 u = S Ė dv Q + ρ 0 R, (17) where u s the specfc teral eergy, R the heat source ad Q heat flux. I addto to these equlbrum equatos, we eed boudary codtos for dsplacemets ad tractos: u = u d o Γ u, t [0,T ], F S N = t o Γ, t [0,T ], (18) where Γ u,γ Ω are the parts of the boudary of the doma Ω where prescrbed dsplacemets ū ad t tractos

4 4 O. Weeger et al. act, ad N s the outer surface ormal of a boudary pot. For the tme-depedet dyamc problem we also eed the tal codtos of dsplacemets ad veloctes u(x,0) = û, v(x,0) = ˆv X Ω. (19) I order to fd a approxmate soluto of the exact dsplacemets u, we oly demad that the equlbrum equatos are fulflled a weak sese. Thus the resdual remag (15) s multpled wth a test fucto δu, the socalled vrtual dsplacemet fulfllg the boudary codto δu = 0 o Γ u, ad the tegrated over the doma Ω. After a few mapulatos ths prcple of vrtual work yelds the weak form of our problem: Ω ρ 0 δu T ü dx + δe S dx Ω = ρ 0 δu T b dx + Ω δu T t da. Γ 2.3 Isogeometrc fte elemet dscretzato (20) For the spatal dscretzato ad soluto of the vrtual work equato (20) we use the sogeometrc fte elemet method, whch was troduced [24]. I addto to the soparameterc cocept [5, 14], whch meas that the same fucto spaces are used for the mathematcal descrpto of geometry X ad dsplacemet soluto u, the dea behd sogeometrc fte elemets s to employ the same class of fucto the umercal method as already used to defe the geometry for example a CAD program,.e. B-Sples ad NURBS. For a detalled troducto to sple fuctos we refer to [25] ad for a collecto of results ad applcatos of sogeometrc aalyss to [29]. Startg pot for the umercal soluto of (20) s a trvarate NURBS volume parameterzato of materal coordates,.e. the geometry fucto mappg a parameter doma Ω 0 R 3 oto the materal coordates X Ω R 3 : X(ξ ) = =1 N p (ξ ) C, ξ Ω 0. (21) Here, p, should be uderstood as 3-dmesoal multdces = ( 1, 2, 3 ), p = (p 1, p 2, p 3 ) ad = ( 1, 2, 3 ), gvg the umber, degree ad dex of trvarate NURBS fuctos N p (ξ ) wth parameters ξ = (ξ 1,ξ 2,ξ 3 ), ad C R 3 are the cotrol pots of the NURBS volume. The parameter doma Ω 0 s also a tesor product of 1-dmesoal tervals for each parameter drecto, gve by kot vectors. Elemets the parameter doma are defed as kot tervals, wth the total umber of elemets deoted by l = (l 1,l 2,l 3 ). Followg the soparameterc cocept, dsplacemets, veloctes ad test fuctos are dscretzed usg the pushforward of NURBS fuctos oto the materal doma: u h (X,t) = v h (X,t) = δu h (X) = =1 =1 =1 ˆN p (X) d (t) = ˆN p (X) ḋ (t) = ˆN p (X) δd = =1 =1 =1 N p (ξ (X)) d (t) N p (ξ (X)) ḋ (t), N p (ξ (X)) δd. (22) Here d (t) R 3 express the dsplacemets of cotrol pots X ad ξ (X) s the verse of the geometry mappg (21). The the kematc quattes descrbed Secto 2.1 ca be derved depedece of the dscretzed dsplacemets from (22). For the deformato gradet ths meas: F(X,t) = I + u h (X,t) = I + = I + =1 d (t) dn p dξ =1 (ξ (X)) d (t) ˆN p (X) ( ) dx 1. dξ (23) Cauchy-Gree ad Gree-Lagrage stra tesors ca the be computed from F ad the 2d Pola-Krchhoff stress be evaluated. Swtchg to the Vogt vector otato for matrces E ad S (see [5]) E = (E 11, E 22, E 33, 2E 12, 2E 23, 2E 13 ) T, S = (S 11, S 22, S 33, S 12, S 23, S 13 ) T, C SE = ds de, the vrtual Gree-Lagrage stra tesor reads (24) δe = B (X) δd, (25) where the matrx B R 6 3 s B (X) ˆ= 1 2 ( ˆN p (X)T F(X) + F(X) T ˆN p (X)). (26) The the etres of the teral force vector l { } r = B T S dx, (27) Ω e e=1 exteral force vector l { } f = ρ 0 ˆN p b dx + ˆN p t da, (28) Ω e Γ,l e=1 ad mass matrx l { M j = I ρ 0 ˆN p ˆN p j }, dx (29) Ω e e=1

5 Nolear frequecy respose aalyss of structural vbratos 5 ca be assembled elemet-wse o elemet domas Ω e, just as stadard fte elemet methods [5, 14], ad fally the dscretzed equato of moto M d(t) + r(d(t)) = f(t) (30) eeds to be solved for the ukow vector of cotrol pot dsplacemets d. For the soluto of (30) we wll also eed the tagetal stffess matrx K T = dr dd = Kgeo + K mat, (31) wth the assembled geometrc taget matrx: K geo l j = e=1 { I ˆN p T S ˆN p j }, dx (32) Ω e ad materal taget matrx: K mat l j = e=1 { } B T C SE B j dx. (33) Ω e 3 Nolear aalyss of structural vbratos Havg troduced the problem formulato of olear structural dyamcs ad the spatal dscretzato usg sogeometrc fte elemets the prevous Secto 2, we ow target the topc of frequecy aalyss. Therefore we start wth a bref revew of methods for lear frequecy aalyss,.e. modal aalyss of egefrequeces ad egeforms, ad drect frequecy respose to harmoc exctatos Secto 3.1. The we troduce the method of harmoc balace, whch allows to compute olear steady-state frequecy respose to perodc exctatos, Secto Lear frequecy aalyss I lear elastcty stras are restrcted to small deformato theory, compare (5), e(x,t) = 1 2 ( u T + u ), (34) ad for the costtutve relato the lear St. Veat-Krchhoff law s used, see (9), σ = λ tr(e)i + 2µ e. (35) The (sogeometrc) fte elemet dscretzato aalogous to Secto 2.3 leads to the followg sem-dscretzed, tme-depedet problem of lear elasto-dyamcs [14, 29]: M d(t) + K d(t) = f(t) t [0,T ], (36) Fg. 2: Frst sx egemodes of the so-called TERRIFIC demostrator, computed wth IGA from a mult-patch model wth 15 blocks where M ad K are the mass ad lear stffess matrx, d s the vector of cotrol pot dsplacemets ad f s the vector of exteral forces. Furthermore we have perodcy codtos for the dsplacemet vectors: d(0) = d(t ), v(0) = v(t ). (37) Modal aalyss of egefrequeces From (36) oe ca derve the well-kow egevalue problem ωk 2 M φ k + K φ k = 0, k = 1,...,, (38) for the lear atural frequeces ω k ad correspodg egemodes φ k [14, 29]. I [30] the propertes of sogeometrc fte elemet dscretzatos the cotext of lear egevalue problems such as (38) were examed already. For oe-dmesoal rods ad beams, t has bee show aalytcally ad umercally that sple-based fte elemets are more accurate tha Lagraga fte elemets. Whle C 0 -cotuous Lagraga FE of hgher degrees p > 1 exhbt optcal ad acoustcal braches the frequecy spectrum wth huge errors ad o p-covergece hgher egefrequeces, C p 1 -cotuous sple-based FE shows hgh accuracy ad p-covergece over the whole spectrum. These results have also bee umercally verfed [30] for 2- ad 3-dmesoal lear egevalue problems ad thus motvate the use of sogeometrc fte elemets vbrato aalyss. Furthermore the depedecy of covergece of IGA o the type of parameterzato - lear or uform - was vestgated [35]. A example for modal aalyss usg sogeometrc fte elemets s show Fgure 2 wth the TERRIFIC demostrator part [36], whch cossts of 15 NURBS patches. Ths applcato s gog to be addressed more detal Secto 5.3.

6 6 O. Weeger et al Drect frequecy respose Aother meas of lear frequecy aalyss, whch s also avalable most FEA software, s the so-called drect frequecy respose (DFR) method. Gve a harmoc exteral loadg of the form f(t) = f c cosωt + f s sωt, (39) the steady-state repose of the structure s assumed as d(t) = q c cosωt + q s sωt. (40) A trasformato of (36), cludg a addtoal dampg term C ḋ(t), oto the Fourer doma the yelds a lear system of equatos for the ukow cose ad se ampltudes of the dsplacemet q c ad q c : ω 2 M q c + ω C q s + K q c = f c, ω 2 M q s ω C q c + K q s = f s. 3.2 Nolear frequecy respose: the Harmoc Balace Method (41) Whle the soluto of egevalue problems ad drect frequecy respose are two stadard methods egeerg practse for the frequecy ad vbrato aalyss of structures, olear vbrato aalyss s a much harder task whch s lackg effcet umercal methods. Steady-state vbrato respose of a structure subject to oleartes such as large deformatos ad hyperelastc materal models (see Secto 2.1) s typcally solved by tme-tegrato of fte elemet models, also the IGA cotext [5, 29]. A more elegat approach s the Harmoc Balace Method (HBM) [1 3], whch has bee employed so far maly for FE-dscretzatos of beam ad shell models oly [9 13]. Furthermore we have already vestgated the HBM cojucto wth IGA for olear Euler-Beroull beams [4]. We could show that IGA s more accurate tha stadard or p-fem ths settg as well, due to the hgher C p 1 - cotuty. Here we apply the method to our sogeometrc fte elemet dscretzato of large deformato hyperelastcty (see Secto 2.3). Therefore we start from the equato of moto (30) wth perodcy codtos: M d(t) + r(d,t) = f(t) t [0,T ], d(0) = d(t ), v(0) = v(t ). (42) Now we cosder oly perodc exteral exctatos of the structure, wth frequecy ω (perod T = 2π/ω) ad a fte umber m of hgher harmocs: f(ω,t) = 1 2 f 0 + m k=1 cos(kωt) f k + s(kωt) f 2m k+1. (43) We expect the respose to perodc exctato to be ω-perodc as well ad therefore express the dsplacemet coeffcets d(t) of the spatal dscretzato u h (X,t) (22) (ad cosequetly also the veloctes ad acceleratos) as a trucated Fourer expaso wth m m harmoc terms of frequecy ω ad ampltudes q = (q 0,...,q 2m ): d(q,ω,t) = 1 2 q 0 + ḋ(q,ω,t) = d(q,ω,t) = m k=1 m k=1 m k=1 cos(kωt) q k + s(kωt) q k, kω s(kωt) q k + kω cos(kωt) q k, k 2 ω 2 cos(kωt) q k k 2 ω 2 s(kωt) q k, (44) wth the abbrevato k = 2m k + 1. Whe we substtue the asatz from (44) to (42), we get a resudal vector: ε(q,ω,t) = M d(q,ω,t) + r(q,ω,t) f(ω,t). (45) I the ext step we apply the Rtz procedure by projectg the resdual ε oto the temporal bass fuctos, order to obta a Fourer expaso of the resdual wth 2m + 1 coeffcet vectors that have to be evaluated to 0 (balace of the harmocs): ε j (q,ω) = 2 T ε j(q,ω) = 2 T T 0 T 0 ε(q,ω,t)cos jωt dt! = 0, j = 0,...,m, ε(q,ω,t)s jωt dt! = 0, j = 1,...,m. (46) The olear system of (2m+1) equatos gve by (46) eeds to be solved order to determe the ampltudes q for gve ω. Note that the exstece ad accuracy of a soluto may hghly deped o the umber of harmocs m, sce olear effects, such as teral resoace ad couplg of modes, typcally cause respose hgher harmocs m > m tha the hghest excted harmoc. For computatoal purposes the abovemetoed equatos may be trasformed oto a o-dmesoal tme τ = ωt. The the doma of tegrals (46) becomes [0,2π] ad for computato we ca use dscrete Fourer resp. Hartley trasform (DFT, DHT), where the resdual ε(q,ω,τ) has to be sampled at 2m + 1 equdstat tmes τ j = 2π j 2m+1, j = 0,...,2m. Furthermore, for solvg (46) wth a Newto s method we eed the Jacobas of resdual coeffcets ε j wth respect to ampltudes q k, k = 0,...,2m: dε j (q,ω) = 2 dq k T dε j (q,ω) = 2 dq k T T 0 T 0 dε dq k (q,ω,t)cos jωt dt, j = 0,...,m, dε dq k (q,ω,t)s jωt dt, j = 1,...,m.

7 Nolear frequecy respose aalyss of structural vbratos 7 (47) The tegrals are aga evaluated by dscrete Fourer trasform ad thereby we also eed the Jacobas of the resdual ε wth respect to ampltudes q k : dε dq k (q,ω,t) = k 2 ω 2 coskωt M + coskωt K T (q,ω,t), k = 0,...,m, dε (q,ω,t) = k 2 ω 2 skωt M dq k + skωt K T (q,ω,t), k = 1,...,m, (48) wth K T = dd dr from (31). Respose curves (RC) are a wde-spread meas of vsualzg the frequecy respose of steady-state vbratg systems, also for lear DFR. Typcally, the total ampltude A k ad phase φ k of oe or more harmocs k are evaluated at a specfc pot o the structure for fxed ω ad plotted over a certa rage of frequecy. They ca be computed as follows: A k = q 2 k + q2 k, φ k = arcta q k, (49) q k q k cos(kωt) + q k s(kωt) = A k cos(kωt + φ k ). (50) The smplest method for geeratg respose curves ad fuctos s smply to start from a fxed ω, compute the respose va HBM or DFR, evaluate ampltude ad phase at the evaluato pot, ad the cremet the frequecy step by step. However, for complex vbratoal behavour, bfurcatos ad turg pots mght occur ad make t ecessary to use arc-legth cotuato methods, whch we have ot doe for our umercal examples preseted Secto 5. 4 Model order reducto for olear vbrato aalyss As metoed Secto 3.2, the harmoc balace method for olear vbrato aalyss requres the soluto of a lear system of equatos of sze (2m+1) each step of a Newto terato, where s the umber of spatal DOFs ad m the umber of harmocs the Fourer expaso of each DOF. Ths system s ot oly (2m + 1)-tmes bgger tha the uderlyg statc system (.e. the taget stffess matrx K T ), but also much more desely populated. Each row of dε j dq also has (2m + 1)-tmes the umber of o-zero k etres of the correspodg row of K T ad s ot symmetrc aymore. Ths s a severe draw-back regardg the soluto process of the system usg sparse lear solvers, whch are desged for the soluto of large systems wth oly few o-zero etres per row. Whle samplg tme for Fourer trasform,.e. (2m + 1)-tmes assembly of force vector ad taget stffess, creases learly wth m, ad tme for Fourer trasform tself by mlogm, soluto tme of the system creases more rapdly. For complex egeerg structures wth hudreds of thousads or eve mllos of DOFs the fte elemet model, a harmoc balace aalyss becomes eve mpossble. Thus we are lookg for a sutable model order reducto method (MOR) order to decrease the computatoal effort for solvg the lear system wth harmoc balace ad allowg a olear frequecy aalyss eve for large-scale applcatos. 4.1 Overvew of model order reducto methods Havg detfed the eed for model order reducto, we are gvg a bref revew of dfferet kds of model reducto methods wth applcatos olear structural mechacs ad dyamcs [5, 14, 15]. There s a wde rage of projecto based reducto methods, where the physcal coordate vector d R ca be expressed by a lear trasformato of reduced coordates ˆd R r : d = Φ ˆd. (51) Φ R r s the trasformato matrx, wth rak(φ) = r. I case of r = (51) s a bass trasformato, but the teto s to chose r ad project oto a smaller subspace of the orgal soluto space. The most commo projecto method s modal reducto or trucato [14, 15], where the trasformato matrx Φ s composed from a subset of lear egevectors, see (38). It s wdely used lear structural dyamcs, but there s oly a lmted applcablty olear aalyss (see also later examples Secto 5.1). Taget modes [21, 22] are the egemodes oe obtas from the soluto of a updated egevalue problem wth the taget stffess K T for the curret deformato state. I olear tme-tegrato a updated modal bass ca be determed from these taget modes every tme step, or after a sutable umber of tme steps. I [21] also a method for drect update of egevectors each tme step s descrbed. But as we compute the ampltudes for oe whole perod of vbrato harmoc balace, there s o specfc curret state of deformato our settg ad these methods seem ot very applcable. A olear couterpart of lear egemodes are olear ormal modes (NNMs), whch have show good results olear frequecy aalyss terms of self-excted vbratos before [16,37 39]. However, the computatoal effort of umercally determg the NNMs seems very hgh, sce for example oe method s solvg a autoomous harmoc balace problem for the full system for each mode.

8 8 O. Weeger et al. Aother meas of geeratg a projecto bass are Rtz vectors [21,22]. Rtz vectors are developed from load or dsplacemet of a curret state; for olear aalyss bass updates ad dervatves may also be cluded, provdg a good approxmato of exact solutos [21]. However, aga ths method reles o a fxed curret state of deformato ad load ad seems ot sutable for applcato harmoc balace. For Proper Orthogoal Decomposto (POD) a set of sample dsplacemet vectors has to be geerated as part of preprocessg, from whch a optmal bass s created [21, 40, 41]. The method leads to good results structural tme tegrato, but samplg requres a pror kowledge of loads. Sce uexpected resoaces ad states of deformato are to be foud olear vbrato aalyss wth HBM, the POD method mght ot be sutable for those. Our choce of reducto method s modal reducto resp. trucato (MR) wth modal dervatves [17 22], whch we preset detal the ext Secto 4.2. Modal dervatves (MD) are a secod order ehacemet of the modal bass, whch accouts for quadratc terms as they appear large deformato theory ad Gree-Lagrage stras. The modal dervatves ca be computed from the lear stffess matrx ad egevectors as part of preprocessg ad do ot requre bass updates durg the computato. 4.2 Modal reducto wth modal dervatves As t s our method of choce for the use olear vbrato aalyss, we gve a troducto to the cocept of modal dervatves ad ther computato, usg refereces [17, 18, 20]: I a lear modal trucato the dsplacemets are expressed terms of egemodes of the lear problem φ ad modal coordates dˆ. Sce the taget stffess matrx depeds o dsplacemets a olear settg, also the (taget) modes deped o dsplacemets: d = =1 φ (d) ˆ d. (52) Now d s developed as secod-order Taylor seres aroud the tal cofgurato of zero dsplacemets d = 0 ( ˆd = 0): d = 0 + = =1 =1 ( ( d ˆ (ˆd = 0) dˆ + d ( φ j ˆ φ (0) dˆ + j=1 j=1 2 d (ˆd = 0) ˆ ) d dˆ j j 2 ) ) d ˆ ˆ (0) + φ (0) d j d j 2. (53) For computg the modal dervatves φ j oe eeds to dfferetate the egevalue problem (38) wth K = K T (0) w.r.t. to the modal coordates: [ (K ω 2 j M ) ] φ j = ( K ω 2 j M ) φ j + ( K ˆ ω2 j M d ) φ j = 0. (54) I [18] three dfferet approaches for the soluto of (54) are preseted: aalytcal, aalytcal excludg erta effects ad purely umercal usg fte dffereces of the re-computed taget egevalue problem. We have mplemeted the ehaced modal bass approach usg modal dervatves wth the aalytcal approach excludg mass cosderato, whch leads to the soluto of the followg lear system for φ j : φ j 1 K = K φ j, (55) wth a fte dfferece approxmato of the dervatve of the taget stffess matrx K K T ( dˆ φ ) K. (56) dˆ Thus we ca compute approxmatos symmetrc modal dervatves φ j, whch we also ortho-ormalze. A exteded reducto bass wth r d egemodes ad the correspodg modal dervatves s: ( ) Φ = φ 1,...,φ rd, φ 1, φ 1,..., φ 1,..., φ r d 1 2 rd ˆ d rd, (57) whch the has bass legth r = r d + r d (r d + 1)/2 ad the lear projecto for reducto s: d = Φ ˆd = r d =1 φ ˆ d + r d r d =1 j= φ dˆ j. (58) j Note that from the quadratc Taylor expaso (53) wth depedet quadratc coeffcets dˆ dˆ j we have geerated a reduced lear expaso (58) wth depedet coeffcets dˆ j. Curretly there are o theoretcal error estmates avalable for the accuracy of reducto wth the modal dervatve approach ad the umber ad choce of modes ad dervatves to be cluded the bass (57) has to be maually selected. However, the umercal results preseted Secto 5, especally the statc covergece study Secto 5.1, dcate covergece wth respect to the umber of bass vectors used ad acceptable accuracy already for a small umber of modes ad dervatves. Remark that the computatoal effort for computg all r d (r d + 1)/2 symmetrc modal dervatves s maly computg the r d lear egevectors φ j, assembly of r d -tmes a taget stffess matrx (56) ad the solvg the lear system (55) r d (r d + 1)/2-tmes. If the problem sze s ot too bg ad oe ca solve (55) by LU-decomposto of K, the computato becomes very effcet!

9 Nolear frequecy respose aalyss of structural vbratos Applcato of reducto to olear vbrato aalyss Here we descrbe the applcato of the modal projecto method wth egemodes ad modal dervatves, as troduced the precedg Secto 4.2, to the harmoc balace method, see Secto 3.2. The dsplacemet vector d the equato of moto of harmoc balace (42) s projected usg modal coordates ˆd: d(t) = Φ ˆd(t), (59) where Φ s the projecto matrx wth a selecto of r lear egemodes φ ad correspodg modal dervatves φ j as colums, see (58). Now the equato of moto (42) s trasformed oto modal coordates by left-multplcato wth Φ T : Φ T MΦ ˆd(t) + Φ T r(d(t)) = Φ T f(t). (60) Ths equato may be re-wrtte usg the otatos ˆM = Φ T MΦ, ˆr = Φ T r, ˆf = Φ T f, (61) as ˆM ˆd(t) + ˆr(d(t)) = ˆf(t). (62) The spatal dscretzato of the olear equato system has bee reduced from a sparse to a dese r r oe, but ote that the olear term ˆr(d(t)) (ad also the correspodg taget ˆK T (d(t)) = Φ T K T (d(t))φ) stll deped o the physcal dsplacemet vector d ad have to be assembled the usual way wthout ay speed-up. Therefore dmeso reducto methods such as Dscrete Emprcal Iterpolato [42] mght be appled the future. The further procedure of applyg the harmoc balace method s equvalet to the steps descrbed Secto 3.2: Fourer expaso of modal coordates ˆd usg ampltudes ˆq, substtuto to (62) for a resdual vector ˆε, Fourer trasform of the resdual ad ts Jacoba, ad soluto of the equato system for ˆq, whch has ow sze r (2m+1). Physcal dsplacemets ca the be recovered wth each samplg or Newto step usg d(t) = Φ ˆd(t). A schematc overvew of the algorthm for olear frequecy aalyss wth modal dervatve reducto s preseted Fgure 3. 5 Computatoal applcatos ad examples 5.1 Modal reducto 3D olear elasto-statcs Before applyg the modal reducto techque from Secto 4.2 to olear vbrato aalyss, we wat to study the Compute reducto bass (pre-processg): Assemble M, K Solve ω 2 j Mφ j + Kφ j = 0, j = 1,...,r For = 1,...,r: Assemble K T ( dˆ φ ) Compute K ˆ For j = 1,...,r: Solve φ j ˆ d K T ( dˆ φ ) K d dˆ = K 1 K φ j Bass Φ = (φ 1,...,φ rd, φ 1,..., φ 1 1,..., φ r d rd ˆ Frequecy respose for ω: Whle ˆε > 0 (Newto terato): For j = 0,...,2m (samplg): d rd ) Dsplacemet d(τ j ) = Φ ( 1 2 ˆq 0 + m k=1 cos(kτ j) ˆq k +s(kτ j ) ˆq k ) Assemble r(τ j ), K T (τ j ) Evaluate ε(τ j ), dε dq k (τ j ) Reducto ˆε(τ j ) = Φ T ε(τ j ), d ˆε d ˆq (τ j ) = Φ T dε k dq (τ j ) Φ k Fourer trasform ˆε(τ j ) ˆε j, d ˆε d ˆq (τ j ) d ˆε j k d ˆq k Solve ˆq = ( d ˆε d ˆq ) 1 ˆε Update ˆq ˆq + ˆq Fg. 3: Algorthm for olear frequecy aalyss wth modal dervatve reducto accuracy of modal reducto wth modal dervatves for olear large deformato hyperelastcty a statc 3-dmesoal case,.e. solvg Φ T r(d) = Φ T f, d = Φ ˆd, Φ R r. (63) We use a 3D geometry wth o symmetres subject to large deformatos ad lear St. Veat-Krchhoff materal law. Oe boudary face of the object s fxed by coupled Drchlet boudary codtos ad the egevalues ad egevectors are computed for settg up the modal matrx Φ wth frst r egemodes as colums. For reducto wth modal dervatves we compute the modal dervatve bass wth r d egemodes ad correspodg modal dervatves. For statc computatos a Neuma boudary codto s appled to the opposte face wth force cotrbutos all 3 dmesos, forcg a large deformato up to 100% of the object s dmesos. The geometry of the object together wth results of subsequet computatos s vsualzed Fgure 4.

10 10 O. Weeger et al. Fg. 4 Geometry of the 3D object (grey), lear dsplacemet (blue), olear dsplacemet (red), olear dsplacemet wth modal reducto r = 50 (gree). For a qute coarse sogeometrc dscretzato wth p = (2,2,2), l = (2,2,4), = (4,4,6), N = 288, we compare the accuracy of modal reducto (MR) ad modal reducto wth dervatves (MD) wth the full, ureduced olear computato. As crtera we use the relatve errors of x-, y- ad z-dsplacemet, evaluated o the ceter pot of the surface where the load s appled, e.g. u full x u MD x / u full x, as well as relatve errors L 2 - ad H 1 -orms, e.g. u full u MD L 2/ u full L 2. I Fgure 5 we compare the relatve L 2 - ad H 1 -errors of the full ad reduced solutos for the lear case wth MR ad the olear case wth MR ad MD. Whle o sgfcat mprovemet of accuracy wth creasg bass legth (umber of modes) s otcable for MR the olear case, MD provdes a smlar covergece behavour as MR the lear case. Note that for r = 240 we are already cosderg the full set of dsplacemet modes ad for r = N the trasformato s bjectve ad thus must reproduce the results of the full system. Furthermore we have also vestgated the behavour of modal reducto ad modal dervatves for dfferet load factors. Fgure 6 shows the dsplacemet at the evaluato pot for load facor 1 to 1000 (100 correspods to the load level of prevous results). Whle there s o vsble devato from full results for MD, MR shows large errors. I Fgure 7 the relatve errors of the dsplacmet at evaluato pot ad relatve L 2 - ad H 1 -orm errors are show over creasg load factor. Whle errors for MD are small ad roughly stay costat up to very large load factors ad thus dsplacemets, relatve errors for MR grow fast ad to a very hgh, urelable level. Wth ths umercal study we have examed the approxmato propertes of our modal reducto method for large deformatos a statc settg. We ca coclude that modal reducto s usutable for reducto of large deformato problems, whle modal reducto wth modal dervatves provdes a hgh accuracy the olear statc problem settg ad opes a perspectve for the use olear vbrato aalyss. 5.2 Large ampltude vbrato of a thck cylder Mathse et al. already studed the use of sogeometrc aalyss compressble ad compressble hyperelastcty wth Neo-Hooke materals [34]. We pck up the example, wth a geometry that ca be exactly represet usg a NURBS volume, wth a compressble Neo-Hooke materal law for olear vbrato aalyss. The dmesos of o eght of the cylder, materal parameters ad loads ca be foud Fgure 8. The surface Neuma loads are perodc. For the sogeometrc dscretzato we chose p = (3,3,3), l = (4,4,1), = (7,7,4), N = 588, ad therefore compute the frst lear egefrequecy as f1 h = Hz. Now we perform a harmoc balace frequecy respose aalyss wth m = 3 (HBM) ear the frst egefrequecy wth the frequecy rage of 0.85 < f / f1 h < 1.15 ad compare the results wth lear drect frequecy repose (DFR). A sapshot of the deformed vbratg cylder for f / f1 h = 0.95 ca be see Fgure 9. Although very large deforma-

11 Nolear frequecy respose aalyss of structural vbratos 11 relatve error bass legth r lear, MR L 2 -error H 1 -error ol., MR L 2 -error H 1 -error ol., MD L 2 -error H 1 -error Fg. 5 Covergece of modal reducto w.r.t. bass legth r relatve L 2 - ad H 1 -orms. Poor results for modal reducto (MR) the olear case, whle ehaced bass (MD) for olear problem performs as good as modal bass lear case dsplacemet full u x u y u z MR r = 50 u x u y u z MD r d = 10 u x u y u z relatve error load factor load factor MR r = 50 L 2 H 1 u x u y u z MD r d = 10 L 2 H 1 u x u y u z Fg. 6 Dsplacemet at evaluato pot for full computato (full), modal reducto wth 50 modes (MR r = 50) ad modal dervatves for 10 modes (MD r d = 10) for creasg load factor Fg. 7 Relatve error of dsplacemet ampltudes, L 2 - ad H 1 -orm of MR (r = 50) ad MD (r d = 10) w.r.t full computato for creasg load factor

12 12 O. Weeger et al. l = 0.15 m, ν = 0.33, r = 0.08 m, ρ = 2800 kg/m 3, t = 0.02 m, p 1 = cosωt N/m 2, E = 74.0 GPa, p 2 = cos2ωt N/m 2. Fg. 8: Geometry, materal parameters ad loads of the vbratg thck cylder Fg. 12: Geometry of the TERRIFIC Demostrator CAD system. For a comparso of reducto methods wth the exact HBM solutos, we have chaged the refemet of sogeometrc parameterzato to p = (2,2,2), l = (4,4,1), = (6,6,3), N = 324 ad the evaluato pot to the top rght corer of the frot surface E 2, compare Fgure 8. We compare the frequecy respose of the cylder at f / f1 h 1.0 for full harmoc balace (HBM), modal reducto (MR) wth r = 50 ad modal dervatves (MD) wth r d = 10 (r = 65) Fgure 11. For modal reducto we have o meagful reproducto of the results whatsoever, whle the exteded bass wth modal dervatves reproduces the ampltudes of the full harmoc balace wth hgh accuracy up to a level where strog resoace occurs. 5.3 Large-scale applcato: the TERRIFIC Demostrator Fg. 9: Sapshot of deformato of thck cylder at τ = 0 for vbrato wth f / f1 h = 0.95, colored by vo Mses stress. tos occur, the sogeometrc harmoc balace stll shows a good covergece behavour, wth 4-5 Newto teratos per frequecy step. I Fgure 10 the z-ampltudes evaluated at the ceter pot of the frot surface of the cylder E 1 are plotted. We ca fd a typcal olear resoace behavour wth two braches. For the left oe we have o more covergece at f / f1 h = 0.97, probably due to a turg pot that we ca ot detected wth smple frequecy cremets. Further away from the resoace at 1.0, where a 1 from lear DFR teds to, ampltudes of DFR ad HBM correspod qute well, but the resoace behavour becomes dfferet ad we ca also detect strog cotrbutos of other harmocs a 0, a 2 ad a 3. The so-called TERRIFIC part s a structure whch was troduced wth the Europea project TERRIFIC [36] as a demostrator for the sogeometrc CAE workflow from desg, over aalyss to maufacturg. It was desged a CAD system (Fgure 12), a IGA-sutable NURBS volume parameterzato was geerated for mechacal smulato (Fgure 13), ad other models for dp-pat smulato ad computer-aded maufacturg were derved. Here we wat to use t as a realstc large-scale applcato for our olear frequecy aalyss framework. The sogeometrc volume parameterzato costs of 15 patches of quadratc B-Sple volumes, wth a total of 6,474 cotrol pots ad 19,422 DOFs. Icludg terface costrats o the patches, the sogeometrc fte elemet dscretzato of the model has 22,914 DOFs. The materal parameters of the part, usg the St. Veat- Krchoff materal law, are the chose as follows: E = 74.0 GPa, ν = 0.33, ρ = 2800 kg/m 3. (64) As boudary codtos we take a clampg of the rght hole Fgure 13 by a zero Drchlet codto ad the perodc

13 Nolear frequecy respose aalyss of structural vbratos DFR a 1,z a 2,z HBM a 0,z a 1,z a 2,z a 3,z a/t a/t f / f h f / f h 1 Fg. 10 Frequecy respose curves of z-ampltudes at ceter pot of frot surface E 1 for m = 3. Comparso of DFR ad HBM HBM a 0,z a 1,z a 2,z a 3,z MR r = 50 a 0,z a 1,z a 2,z a 3,z MD r d = 10 a 0,z a 1,z a 2,z a 3,z Fg. 11 Frequecy respose curves of z-ampltudes of vbratg cylder at top rght pot of frot surface E 2 for m = 3. Comparso of full HBM ad HBM wth MR ad MD Wth ths spatal dscretzato, materal parameters ad boudary codtos we start the aalyss by computg the frst four egefrequeces ( f = 2π/ω) of the part: f1 h = Hz, f 2 h = Hz, f3 h = Hz, f 4 h = Hz. (66) Fg. 13: IGA-sutable mult-patch volume parameterzato of the TERRIFIC Demostrator. exctato acts as a surface tracto o the left hole (Neuma boudary codto): u = 0 o Γ u, t = (60.0, 42.0, 0.0) T 10 6 cosωt N/m 2 o Γ. (65) We are terested carryg out a frequecy respose aalyss of the part aroud the frst two egefrequeces,.e. the rage of 150 Hz < f < 450 Hz. The surface tractos specfed (65) cause a large deformato of the part ad thus we expect a sgfcat dfferece betwee the results of a lear DFR ad the olear HBM. Takg a Fourer seres legth of m = 3, the problem sze of harmoc balace grows ths case to a total of 160,398 DOFs ad due to ts low sparsty t s ot solvable o a persoal computer. Therefore we eed to apply the reducto proposed Secto 4 to make the problem resp. the lear system solvable.

14 14 O. Weeger et al. full r d = 5 r d = 10 abs. val. abs. val. rel. err. abs. val. rel. err. u x 1.42E E % 1.41E % u y 6.57E E % 6.59E % u z 1.05E E % 1.05E % L E E % 1.53E % H E E % 1.47E % Table 1: Nolear statc aalyss of TERRIFIC Demostrator. Comparso of full problem ad reducto wth modal dervatves. As part of pre-processg we compute the olear statc dsplacemet caused by a statc load of the same magtude. The we compute the reducto bass wth 5 resp. 10 lear egemodes ad all correspodg modal dervatves, ad solve the reduced versos of the olear statc problem. A comparso of absolute values ad relatve errors of dsplacemets at a evaluato pot o the very left of the structure (whch we also take for plottg frequecy respose curves), L 2 - ad H 1 -orms Table 1 reveals that r d = 5 s ot suffcet to capture the olear dsplacemet behavour, whereas r d = 10 provdes a suffcet accuracy of u f ull u rd L 2/ u f ull L 2 < 1.0%. We proceed wth the harmoc balace frequecy respose aalyss cojucto wth modal dervatve reducto wth r d = 10,.e. the frst 10 lear egemodes ad the r d (r d + 1)/2 = 55 correspodg modal dervatves. I Fgures 14, 15, 16 we have plotted the frequecy respose curves of x-, y- ad z-ampltudes evaluated at evaluato pot E o the left outer boudary of the TERRIFIC part for the frequecy rage 150 Hz < f < 450 Hz, togehter wth correspodg ampltudes computed from lear DFR. Aroud f = Hz = f2 h /2 there s a remarkable sub-harmoc respose a 2, whch ca ot be determed wth lear frequecy aalyss. I the vcty of the frst egefrequecy f1 h = Hz we otce that the olear respose behavour z-ampltudes s dfferet from the lear oe obtaed from DFR. The resoace behavour aroud f2 h = Hz s very strog ad we have covergece problems wth our method for 335 Hz < f < 355 Hz. There are strog cotrbutos from hgher harmocs here, whch lead to much more realstc deformatos as we dscuss more detal below. Rapdly growg z-ampltudes at resoace dcate that there mght by turg pots the frequecy respose curves here, whch we could oly follow usg cotuato methods. Fgure 17 shows the vbratg structure at a frequecy of f = Hz,.e. ear the frst egefrequecy of f h 1 = Hz, where resoace wth very large deformato occurs. Four sapshots are take at tmes τ = 0, π/2, π, 3π/2, dsplayg the deformed structure from the HBM-MD computato colored by vo Mses stress Pa ad as refereces the deformed structure from DFR lear frequecy aalyss ad udeformed structure both gray. It becomes obvous that the olear results lead to a much better coservato of volume of the structure ad thus much more realstc states of deformato. Furthermore t s terestg that the bedg of the structure s stroger the olear case tha the lear case for τ = π. Ths ca as well be observed Fgure 18, where we have plotted the x-, y- ad z-dsplacemet at the evaluato pot over oe vbrato perod of τ [0,2π] for f = Hz for both HBM-MD ad DFR. Altogether, the results we preset for the TERRIFIC Demostrator show that a modal reducto wth modal dervatves makes harmoc balace olear frequecy respose aalyss feasble eve for larger applcatos. 6 Summary ad outlook The am of ths paper s to preset a advaced method for olear frequecy respose aalyss of large-scale applcatos sold mechacs. We have proposed to use the harmoc balace method for olear steady-state frequecy respose of the dscretzed equato of moto the frequecy doma. I cojucto wth a modal projecto method usg egemodes ad secod order modal dervatves as reducto bass, the method ca be appled eve to realstc applcatos wth large spatal dscretzatos. For a effcet spatal dscretzato of the olear partal dfferetal equatos arsg from 3- dmesoal large deformato hyperelastcty we employ the sogeometrc fte elemet method, but our approach could be appled usg ay spatal dscretzato method. As our umercal examples show, the reducto method provdes a good accuracy of frequecy respose ampltudes ad resoace behavour, although sgfcatly reduces the effort for umercal soluto of the harmoc balace equato system. For large-scale applcatos harmoc balace becomes feasble oly usg model order reducto. Eve though the proposed reducto method makes olear frequecy respose aalyss feasble applcato to 3-dmesoal structural problems, t stll remas a tmecosumg task. Especally for very large applcatos a speedup of the samplg process s ecessary, where full resdual ad taget stffess have to be assembled for every sample. A complexty reducto mght there be acheved by methods such as Dscrete Emprcal Iterpolato [42]. For further dustral problems we pla to exted the method to materals wth olear vscoelastc propertes such as rubber ad cotact problems. We also am at combg olear frequecy aalyss wth shape optmzato.

15 Nolear frequecy respose aalyss of structural vbratos DFR a 1,x HBM a 0,x a 1,x a 2,x a 3,x a,x [m] f [Hz] Fg. 14 Frequecy respose curves of x-ampltudes of vbratg TERRIFIC Demostrator for DFR ad HBM wth reducto DFR a 1,y HBM a 0,y a 1,y a 2,y a 3,y a,y [m] f [Hz] Fg. 15 Frequecy respose curves of y-ampltudes of vbratg TERRIFIC Demostrator for DFR ad HBM wth reducto DFR a 1,z HBM a 0,z a 1,z a 2,z a 3,z a,z [m] f [Hz] Fg. 16 Frequecy respose curves of z-ampltudes of vbratg TERRIFIC Demostrator for DFR ad HBM wth reducto

16 16 O. Weeger et al. Fg. 17 Sapshots of vbratg TER- RIFIC Demostrator for f = Hz at τ = 0, π/2, π, 3π/2. Nolear deformato from HBM-MD s colored by vo Mses stress Pa, lear DFR deformato ad udeformed cofgurato are gray

17 Nolear frequecy respose aalyss of structural vbratos 17 u [m] DFR u x u y u z HBM u x u y u z π π 3 2 π 2π τ = 2π f t Fg. 18 Dsplacemet of evaluato pot o TERRIFIC Demostrator for f = Hz for HBM-MD ad DFR Ackowledgemets Ths work s supported by the Europea Uo wth the FP7-project TERRIFIC: Towards Ehaced Itegrato of Desg ad Producto the Factory of the Future through Isogeometrc Techologes [36]. The TERRIFIC part was desged by Stefa Boschert (Semes AG, Germay) ad the sogeometrc parameterzato provded by Vbeke Skytt (SINTEF, Norway). Refereces 1. A.H. Nayfeh ad B. Balachadra. Appled Nolear Dyamcs: Aalytcal Computatoal, ad Expermetal Methods. Wley Seres Nolear Scece. Joh Wley & Sos, A.H. Nayfeh ad D.T. Mook. Nolear Oscllatos. Wley Classcs Lbrary. Joh Wley & Sos, W. Szemplska-Stupcka. The Behavour of Nolear Vbratg Systems. Kluwer Academc Publshers, Dordrecht Bosto Lodo, O. Weeger, U. Wever, ad B. Smeo. Isogeometrc aalyss of olear euler-beroull beam vbratos. Nolear Dyamcs, 72(4): , P. Wrggers. Nolear Fte Elemet Methods. Sprger, T. Belytschko, W. K. Lu, ad B. Mora. Nolear Fte Elemets for Cotua ad Structures. Joh Wley & Sos, Cadece Desg Systems Ic. Rf aalyss vrtuoso spectre crcut smulator xl datasheet. Techcal report, M. Scheder, U. Wever, ad Q. Zheg. Parallel harmoc balace. VLSI 93, Proceedgs of the IFIP TC10/WG 10.5 Iteratoal Coferece o Very Large Scale Itegrato, Greoble, Frace, 7-10 September, 1993, pages , R. Lewadowsk. No-lear, steady-state vbrato of structures by harmoc balace/fte elemet method. Computers & Structures, 44(1-2): , R. Lewadowsk. Computatoal formulato for perodc vbrato of geometrcally olear structures, part 1: Theoretcal backgroud; part 2: Numercal strategy ad examples. Iteratoal Joural of Solds ad Structures, 34(15): , P. Rbero ad M. Petyt. No-lear vbrato of beams wth teral resoace by the herarchcal fte elemet method. Joural of Soud ad Vbrato, 224(15): , P. Rbero. Herarchcal fte elemet aalyses of geometrcally o-lear vbrato of beams ad plae frames. Joural of Soud ad Vbrato, 246(2): , P. Rbero. No-lear forced vbratos of th/thck beams ad plates by the fte elemet ad shootg methods. Computers ad Structures, 82(17-19): , T.J.R. Hughes. The Fte Elemet Method: Lear Statc ad Dyamc Fte Elemet Aalyss. Dover Publcatos, Meola, New York, Z.-Q. Qu. Model Order Reducto Techques wth Applcatos Fte Elemet Aalyss. Sprger, Malte Krack, Lars Pag vo Schedt, ad Jörg Wallaschek. A method for olear modal aalyss ad sythess: Applcato to harmocally forced ad self-excted mechacal systems. Joural of Soud ad Vbrato, 332(25): , S. R. Idelsoh ad A. Cardoa. A reducto method for olear structural dyamcs aalyss. Comput. Methods Appl. Mech. Egrg., 49: , P. M. A. Slaats, J. de Jogh, ad A. A. H. J. Saure. Model reducto tools for olear structural dyamcs. Computers & Structures, 54(6): , J. Barbc. Real-tme Reduced Large-Deformato Models ad Dstrbuted Cotact for Computer Graphcs ad Haptcs. PhD thess, Carege Mello Uversty, J. Barbc. Fem smulato of 3d deformable solds: A practtoer s gude to theory, dscretzato ad model reducto. part 2: Model reducto. I SIGGRAPH 2012 Course Notes, H. Spess. Reducto Methods Fte Elemet Aalyss of Nolear Structural Dyamcs. PhD thess, Uverstät Haover, J. Remke ad H. Rothert. Ee modale reduktosmethode zur geometrsch chtleare statsche ud dyamsche fteelemet-berechug. Archve of Appled Mechacs, 63(2): , Oleg Roderck, Mha Atescu, ad Paul Fscher. Polyomal regresso approaches usg dervatve formato for ucertaty quatfcato. Nuclear Scece ad Egeer, 164(2): , T.J.R. Hughes, J.A. Cottrell, ad Y. Bazlevs. Isogeometrc aalyss: Cad, fte elemets, urbs, exact geometry ad mesh refemet. Computer Methods Appled Mechacs ad Egeerg, 194(39 41): , L.A. Pegl ad W. Tller. The Nurbs Book. Moographs Vsual Commucato. Sprger, R.N. Smpso, S.P.A. Bordas, J. Trevelya, ad T. Rabczuk. A two-dmesoal sogeometrc boudary elemet method for elastostatc aalyss. Computer Methods Appled Mechacs ad Egeerg, :87 100, F. Aurccho, L. Berão da Vega, T.J.R. Hughes, A. Real, ad G. Sagall. Isogeometrc collocato methods. Mathematcal Models ad Methods Appled Sceces, 20(11): , Ch. Herch, B. Smeo, ad S. Boschert. A fte volume method o urbs geometres ad ts applcato sogeometrc flud structure teracto. Mathematcs ad Computers Smulato, 82(9): , 2012.