Passive Targeted Energy Transfers and Strong Modal Interactions in the Dynamics of a Thin Plate with Strongly Nonlinear Attachments

Transcription

1 Passiv Targtd Enrgy Transfrs and Strong Modal Intractions in th Dynamics of a Thin Plat with Strongly Nonlinar Attachmnts Fotios Gorgiads Postdoctoral Rsarchr LTAS-Vibration and Idntification Group Arospac and Mchanical Eng. Dpt., Univrsity of Lig, 1 Chmin ds Chvruils (B5/), 4, Lig, Blgium, Fotios.Gorgiads@ulg.ac.b Alxandr F. Vakakis Profssor Dpt. of Mchanical Scinc and Enginring, Univrsity of Illinois, Urbana, IL 6181, avakakis@illinois.du Abstract W study Targtd Enrgy Transfrs (TETs) and nonlinar modal intractions attachmnts occurring in th dynamics of a thin cantilvr plat on an lastic foundation with strongly nonlinar lightwight attachmnts of diffrnt configurations in a mor complicatd systm towards industrial applications. W xamin two typs of shock xcitations that xcit a subst of plat mods, and systmatically study, nonlinar modal intractions and passiv broadband targtd nrgy transfr phnomna occurring btwn th plat and th attachmnts. Th following attachmnt configurations ar considrd: (i) a singl ungroundd, strongly (ssntially) nonlinar singl-dgr-of-frdom (SDOF) attachmnt trmd nonlinar nrgy sink (NES); (ii) a st of two SDOF NESs attachd at diffrnt points of th plat; and (iii) a singl multi-dgr-of-frdom (MDOF) NES with multipl ssntial stiffnss nonlinaritis. W prform paramtric studis by varying th paramtrs and locations of th NESs, in ordr to optimiz passiv TETs from th plat mods to th attachmnts, and w showd that th optimal position for th NES attachmnts ar at th antinods of th linar mods of th plat. Th paramtric study of th damping cofficint of th SDOF NES showd that TETs dcrasing with lowr valus of th cofficint and morovr w showd that th thrshold of maximum nrgy lvl of th systm with strong TETs occurd in discrt modls is by far byond th limits of th nginring dsign of th continua. W xamin in dtail th undrlying dynamical mchanisms influncing TETs by mans of Empirical Mod Dcomposition (EMD) in combination with Wavlt Transforms. This intgratd approach nabls us to systmatically study th strong modal intractions occurring btwn th ssntially nonlinar NESs and diffrnt plat mods, and to dtct th dominant rsonanc capturs btwn th plat mods and th NESs that caus th obsrvd TETs. Morovr, w prform comparativ studis of th prformanc of diffrnt typs of NESs and of th linar Tund-Mass-Damprs (TMDs) attachd to th plat instad of th NESs. Finally, th fficacy of using this typ of ssntially nonlinar attachmnts as passiv absorbrs of broadband vibration nrgy is discussd. Ky words: Nonlinar targtd nrgy transfrs, nonlinar modal intractions, shock isolation

2 1. Introduction Towards industrial applications, w study a thin plat lying on an lastic foundation with ssntially nonlinar attachmnts of diffrnt configurations attachd to it. W discrtiz th partial diffrntial quation of motion of th thin plat using a Finit Elmnt (FE) formulation to xtract th structural matrics of th systm, and thn add th ssntially nonlinar attachmnts with diffrnt configurations. For th class of systms considrd in this work (.g., a primary linar structur with an ssntially nonlinar attachmnt at its boundary), it has bn shown that at crtain nrgy lvls thr may occur passiv targtd nrgy transfr (TET) phnomna from th primary systm to th nonlinar attachmnt, which acts, in ssnc as nonlinar nrgy sink (NES) (Vakakis, 1). Whn TET taks plac, thr occurs passiv, broadband, dirctd (on th avrag) nrgy transfr from th primary, dirctly xcitd structur to th NES (that is initially at rst), whr this nrgy gts localizd (spatially confind) and locally dissipatd without major backscattring to th primary structur. In prvious works th dynamical mchanisms govrning passiv TET hav bn studid; ths includ nonlinar bat phnomna, and fundamntal and subharmonic rsonanc capturs btwn th linar part and th NES attachmnt (Gndlman t al., 1; Vakakis and Gndlman, 1; Krschn t al., 5; L t al., 5). Th prsnt work aims to systmatically study th nonlinar modal intractions and compar passiv broadband targtd nrgy transfr (TET) phnomna btwn th linar disprsiv plat and th following configurations of NES attachmnts: (i) a singl ungroundd, strongly (ssntially) nonlinar singl-dgr-of-frdom (SDOF) NES; (ii) multipl nonlinar SDOF NESs attachd at diffrnt points of th plat; (iii) a singl, multi-dgr-of-frdom (MDOF) NES with ssntial stiffnss nonlinaritis; and (iv) a singl Linar Tund Mass Dampr (TMD) attachmnt to th plat. W xamin conditions for optimal TETs from th plat to th attachd NESs, and w compar th prformanc of th diffrnt NESs to linar TMDs. In ths studis w xamind TETs with two diffrnt typs of input forc applid to th plat, ithr a singl shock that dirctly xcits at last th fiv lading plat mods, or multipl shocks that xcit quivalntly, at last thr lading plat mods. W xamin th capacity of th NESs to passivly absorb and locally dissipat broadband shock nrgy from th plat. What clarly distinguishs this work from othr TET studis is th systmatic study of transint, strongly nonlinar TET in an lastic continuum NES systm of far mor complicatd configuration that thos ths xamind in th litratur thus far towards industrial applications-, th paramtric study of TETs with th applid amplitud of th forc within th dsign limits of th linar structur, th comparativ studis of TETs btwn diffrnt NESs, TMD attachmnts and diffrnt typs of forcs that xcits highr than th thr first mods of th linar systm. In rlatd works on lastic continua with attachd NESs, Gorgiads and Vakakis (7) hav studid numrically TETs from a bam to a SDOF NES attachmnt, and found conditions for optimal TET; in that work as much as 87% of th shock nrgy of th bam could b transfrrd and locally dissipatd by th NES. Gorgiads t al. (7) hav studid TETs from a disprsiv rod (linarly lastic rod on an lastic foundation) to a SDOF NES attachmnt. In that work Wavlt transforms and Empirical Mod Dcomposition (EMD) wr usd to analyz th nonlinar rsonanc capturs that govrn TET in that systm. TET from a disprsiv rod undr shock xcitation to an attachd MDOF NES at its boundary was studid in (Tsakirtzis t al., 7). In an arlir work (Tsakirtzis t al., 5) th dynamics and TET in discrt oscillators with an attachd MDOF NES hav bn studid. Th prsnt work aims to xtnd ths prvious rsults, by studying TET and complx nonlinar modal

3 intractions of an lastic systm with mor complicatd configuration with ssntially nonlinar attachmnts.. Finit Elmnt (FE) Formulation for th Plat Dynamics and Diffrnt NES Configurations Th plat on th linar lastic foundation is dpictd in Figur 1a. It consists of a linar isotropic lastic plat with mass distribution pr unit ara M, width W, lngth L, thicknss h, and distributd proportional viscous damping pr unit ara d (this corrsponds ithr to distributd viscous damping in th foundation or to Rayligh damping in th plat with β=). Th plat is clampd on on dg only, with all othr dgs rmaining traction-fr, and is rsting on a distributd lastic foundation with stiffnss pr unit ara qual to K. Th plat is assumd to b sufficintly thin, so that its shar dformation may b nglctd (th so calld Kirchhoff assumptions). Hnc, th govrning partial diffrntial quation of motion with th associatd boundary conditions is givn by (assuming that th plat is initially at rst): w w t t 4 D w M d k w F xt,1 (x d x, y d 1 y ) F 1 xt, (x d x, y d y ) i F (t) (x b, y b ) i xi yi w(x,, t) w(x, y,) w(x,, t),, w(x, y,), x t M (, y,t) M (W, y,t) M (x,l,t) y y y Q y(, y,t) Q y(w, y,t) Q y(x,l,t) (1.1) Dpnding on th spcific forms of th forcing trms in (1.1) w will considr diffrnt modls as outlind blow: Modl 1: Simpl plat without attachmnts: Fxt,1 Fxt, (1.) Modl : Plat with SDOF NES attachmnt: Fxt,1 C w(d x, d 1 y ) v(t) w(d 1 x, d 1 y ) v(t) 1 (1.) F (1.4) xt, x1 y1 x1 y1 v(t) C v(t) w(d, d ) v(t) w(d, d ) Modl : Plat with Multi-SDOF NES attachmnt:, v(), v() (1.5) Fxt,1 C w(d x, d 1 y ) v(t) w(d 1 x, d 1 y ) v(t) 1 Fxt, C w(d x, d y ) s(t) w(d x, d y ) s(t) x1 y1 x1 y1 x y x y v(t) C v(t) w(d, d ) v(t) w(d, d ) s(t) C s(t) w(d, d ) s(t) w(d, d ) (1.6) (1.7), v(), v() (1.8), s(), s() (1.9) Modl 4: Plat with MDOF NES attachmnt: Fxt,1 C w(d x,d 1 y ) v(t) 1 (1.1) F (1.11) xt, m1 v(t) C v(t) w(d x,d 1 y ) v(t) u(t) C 1 1 v(t) u(t), v(), v() (1.1)

4 m u(t) u(t) v(t) C1 u(t) v(t) u(t) s(t) C u(t) s(t), u(), u() (1.1) m s(t) s(t) u(t) C s(t) u(t), s(), s() (1.14) Modl 5: Plat with TMD attachmnt: Fxt,1 k ln w(d x,d 1 y ) v(t) w(d 1 x,d 1 y ) v(t) 1 (1.15) F (1.16) xt, v(t) k ln v(t) w(d x,d 1 y ) v(t) w(d 1 x,d 1 y ) 1, v(), v() (1.17) z w(,,t) x y b y b x F(t) kw kw kw kw (a) 4

5 x z y k i ξ (x c,y c ) η l j (b) Figur 1. Cantilvr plat with lastic foundation: (a) Configuration of th plat, (b) th 4-nod quadrilatral finit lmnt indicating th local coordinats (, ). In ths modls F(t) is th applid xtrnal xcitation, (, ) is Dirac s gnralizd function, and th diffrntial oprator applis to both x and y dirctions. In (1) th variabls M y(, ) and Q y(, ) dnot th intrnal bnding momnts about th y-axis, and th shar forcs along th y-axis of th plat, rspctivly (cf. Figur 1). Morovr, th flxibility D in th quation of th plat is dfind as, D Eh () 1(1 ) whr E is th modulus of lasticity, and is Poisson s ratio for th matrial of th plat (Lissa, 199). Equations (1.1) and (1.) hav bn nondimnsionalisd, with th following numrical valus assignd for th plat paramtrs, W L 1, h.1, M 1, D 1,., k 1 () which ar in accordanc to th assumptions of thin plat thory; th damping cofficint is assignd th valu d=.1 for th cas of a singl forc (shock) applid to th plat, and d=.15 for th cas of multipl forcs applid to th plat, in ordr to prform comparativ studis of TET fficincy btwn various configurations of NESs and linar tund mass damprs (TMDs). In SDOF NES configurations a singl mass is attachd to th plat by mans of an ssntially nonlinar stiffnss in paralll to a viscous dampr. In MDOF NES configurations a -DOF NES systm is connctd to th plat through a linar coupling stiffnss without a dissipativ lmnt; also, two ssntially nonlinar stiffnss lmnts in paralll to two viscous damping lmnts ar usd to connct th thr masss of th MDOF NES. In all cass, th masss of th NESs ar assumd to oscillat transvrsly with rspct to th plat. Additionally, w assum that at t a singl or a st of transint forcs (shocks) F i (t) is applid to th plat. Each shock has th form of a half sin impuls: 5

6 Ai sin(t / T), t T / F i(t) (4), t T / In th cas of a singl applid shock, its position on th plat is givn by, (b x, b y ) = (1,1); whras, in th cas of multipl applid shocks, thir positions on th plat ar givn by, (b x1, b y1 )=(.6,.5) (shock 1), (b x, b y ) = (.5,.5) (shock ) and (b x, b y ) = (.4,.5) (shock ). Unlss othrwis statd, in th following computations for singl forcing th amplitud of th forc is slctd as, A=1, with priod T =.1 T 5, whr T 5 is th priod of th fifth ignmod of th linar plat (.g., th plat with no attachmnt). This rquirmnt nsurs that th applid shock xcitation has sufficintly small duration compard to th priod of th first ignmod of th plat, which can b rgardd as th charactristic tim scal of th problm; in turn, this nsurs that th applid shock dirctly xcits a sufficintly larg numbr of ignmods of th plat (in this cas th first fiv), which nabls us to study th capacity of th NES(s) to passivly absorb broadband vibration nrgy from multipl plat mods. Altrnativly, in th cas of multipl applid shocks, th forcing amplituds ar slctd as, A 1 = 5, A = -1, and A = 5 with priod T =.1 T 5, whr T 5 is th priod of th fifth ignmod of th linar plat, and xamination of th rspons at th fr cornr of th simpl plat with wavlt spctra showd that thr ar thr xcitd mods with almost quivalnt nrgy (th 1 st, 4 th and 6 th mods). Th partial diffrntial quation in (1.1) with (1.) is discrtizd using a 4- nod quadrilatral lmnt, as wll as non-conforming shap functions with cornr nods (with 1 dgrs of frdom) (Zinkiwicz and Taylor, ; Liu and Quk, ). In ach nod, w considr th transvrs displacmnt (w) and th rotations ovr th x and y axis. For this spcific finit lmnt, th xplicit forms of th matrix of shap functions, N, was drivd by Mlosh (196), and xprssd simply in trms of local normalizd coordinats (ξ,η) at ach nod as follows (Zinkiwicz and Taylor, ), 1 N p m T 1 (N p ) N p (1 )(1 ) b p(1 ) (5) 8 N p p(1 ) whr p(=i,j,k,l) is th numbr of nod and (m) is th indx dnoting to which lmnt of th vctor q p dos th shap function corrspond; th local coordinats ar dfind as, x x c, p (6) y y c, p (7) b whr p and p ar th local coordinats for nod p, and (x c, y c) dnot th coordinats of th cntr of th finit lmnt (cf. Figur 1b). Thr ar svral ways to discrtiz th partial diffrntial quation in (1.1) with (1.). In this work w found mor convnint to us th nrgy approach basd on th stimation of nrgis of a singl finit lmnt from Kirchoff s plat thory, and using th rsults to stimat th corrsponding FE mass matrics and FE displacmnts. Using a connction matrix that indicats which nods ar connctd in adjacnt lmnts w may construct th full structural matrics of th plat as follows (s dtails in Appndix 1): I,II K q Md q Dq P(t) (8) In (8) q is a column vctor containing th vrtical displacmnt and th rotations about th x and y axis at ach nod; M d is th mass structural matrix and K I,II is th 6

7 stiffnss matrix; and P(t) is a column vctor with zro lmnts, xcpt for nods whr xcitations forcs (shocks) ar applid. In quation (1) proportional distributd viscous damping is considrd; hnc, in th discrt modl (8) th damping matrix D appars, which is proportional to th mass matrix M d (s dtails in Appndix 1). Th discrt systm (8) was solvd numrically using th Nwmark Adaptiv Algorithm [for dtails s (Gradin and Rixn, 1997)]. A snsitivity analysis for th rquird numbr of discrt lmnts ndd for convrgnc showd that a total of 1 1 lmnts (1 in ach dirction) was sufficint. Initially a vrification of th accuracy of th natural frquncis was prformd for th cas of no lastic foundation (k=) using a modl in ANSYS, and comparing with th rsults rportd in (Lissa, 199). In Tabl 1 w prsnt th natural frquncis for th unforcd and undampd plat stimatd using th FE simulation, togthr with th corrsponding ignshaps and for comparison purposs thr ar th ignfrquncis arisd by Lissa (199) that shows good agrmnt with th dtrmind valus. Tabl 1. FE computations of th lading ignmods of th plat on lastic foundation (k=1), with no NES attachd. Eignmod No. 1 Eignfrquncy (Lissa 199) Eignfrquncy (Hz) % Critical damping ratios for d= % Critical damping ratios for d= Eignshaps- Nodal Lins ( ) Eignmod No. 4 5 Eignfrquncy (Rf. Lissa 199) Eignfrquncy (Hz) % Critical damping ratios for d= % Critical damping ratios for d= Eignshap W now assum that a singl ssntially nonlinar attachmnt (rfrrd to from now on as nonlinar nrgy sink NES) is attachd at position (x,y) = (d x,d y ) of th plat, modl with quations (1.1,1.-1.5). Th NES is assumd 7

8 to b lightwight of mass ε, and to possss an ssntially nonlinar (nonlinarizabl) cubic stiffnss with charactristic C, in paralll to a viscous dampr λ. Th partial diffrntial quation in (1.1) was discrtizd using th aformntiond FE formulation, and th dynamics of th NES at (x,y) = (d x,d y ) is incorporatd into th discrtizd quations of motion by xpanding accordingly th systm matrics, and adding a nonlinar stiffnss componnt (du to th ssntial cubic nonlinarity). Th fficincy of th NES to passivly absorb and locally dissipat th shock nrgy of th plat, can b studid by stimating th following nrgy dissipation masur (EDM): t w(d x,d y, ) v( ) d E damp,nes(t) (t) (9) t Ein F( ) w(b,b, )d This rprsnts th portion of th shock nrgy of th plat that is dissipatd by th dampr of th NES at tim t, and hnc, can b usd as a masur of TET fficincy. It is clar that with incrasing tim th EDM rachs an asymptotic limit, t 1 lim t1 (t) (1) which rprsnts th portion of th shock nrgy of th plat that is vntually dissipatd by th NES by th nd of th oscillation. Th portion of th input shock nrgy dissipatd by th distributd viscous damping of th plat up to tim instant t is computd by, x y plat damp,plat t Ein L W t 1 w(x, ) d E (t) (t) x y d dx dy F( ) w(b,b, )d Combining (9) and (11), th portion of th input shock nrgy dissipatd by th intgratd plat-nes systm up to tim instant t is computd as follows, (t) (t) (t) (1) total Similar formulations hold whn multipl SDOF NESs or th linar TMD and th quations of motion for th intgratd plat Multi SDOF NES systm considrd in this work ar givn by modl quations (1.1, ) and for th intgratd plat- TMD systm considrd in this work ar givn by modl 5 quations (1.1, ). Som rmarks ar now appropriat concrning th us of MDOF NES attachmnts. In (Tsakirtzis t al., 5; 7) a MDOF NES composd of thr masss coupld by ssntially nonlinar springs and damprs was attachd to a systm of linar coupld oscillators. This MDOF NES configuration was introducd du to its capacity to passivly absorb broadband shock nrgy from a linar systm with nhancd ffctivnss and robustnss compard to SDOF NESs and an asily ralizabl attachmnt in an xisting structur by a common linar spring.. Th nhancd fficincy of th MDOF NES is attributd to th complx nonlinar modal intractions (and, hnc, transint rsonanc capturs) that occur btwn multipl nonlinar mods of th NES and linar mods of th systm to which it is attachd; for xampl, MDOF NESs wr shown to b ffctiv in passiv nrgy absorption vn in low-amplitud rgims (Tsakirtzis t al., 7), in contrast to SDOF NESs which ar activatd only abov a dfinit nrgy thrshold. Ths rsults provid us with th motivation to tst th prformanc of th MDOF NES to th prsnt problm of plat vibration. plat (11) 8

9 Th quations of motion for th intgratd plat MDOF NES systm considrd in this work ar givn by modl 4 quations (1.1, ). Th MDOF NES attachmnt is consists of thr masss coupld togthr with nonlinar spring and dampr with th sam damping cofficint and th third mass coupld with th plat by linar spring. Dtails of th FE formulation and th corrsponding structural matrics of th intgratd systm can b found in Appndix 1. Th ffctivnss of th NES to passivly absorb and locally dissipat th shock nrgy of th plat (.g., th TET fficincy) can b quantitativly studid by computing th following nrgy dissipation masurs (EMDs), and E (t) 9 damp,nes1 1(t) t Ein E (t) t u( ) v( ) d F( ) w(b,b, )d damp,nes (t) t Ein t x s( ) u( ) d F( ) w(b,b, )d.g., th portion of shock nrgy dissipatd by ach of th two damprs of th MDOF NES at tim t. Th summation of ths two EMDs provids a masur of th TET fficincy of th MDOF NES. It is clar that th two EMDs rach asymptotic limits, 1, t1 lim t1 1, (t).. Mthods of Post-procssing th Computational Rsults Th rsults of th simulations ar post-procssd basd on nrgy and frquncy points of viw. From th nrgy point of viw, w xamin th nrgy dissipation in th systm and spcially th amount of nrgy that ach part of th systm (th lastic continuum and th attachmnt) is dissipating. Morovr in crtain simulations w xamin th nrgy transaction (E Trans th ratio E Trans /Δt whn Δt tnds to zro rprsnts th powr flow from th plat to th attachmnt and vic vrsa) btwn th plat and th attachmnt at any givn instant of tim. Post procssing of th numrically computd tim sris of th plat and th NES was prformd in two diffrnt ways as discussd in (Gorgiads t al., 7). First, th dynamic rspons was analyzd using a Wavlt Transform (WT) mploying a Matlab cod dvlopd in Univrsity of Lig by Dr. V. Lnarts in collaboration with Dr. P. Argoul from th Ecol National ds Ponts t Chausss. In this work, th Morlt mothrwavlt was usd for th WT computations; that is a Gaussian-windowd complx sinusoid of frquncy (in rad/sc), t / jt M (t). Th frquncy (or f in Hz) is th usr paramtr that nabls on to tun th frquncy and tim rsolution of th rsults. Using this tool w can xtract th Wavlt Transform Spctra that ar contour plots dpicting th amplitud of th WT of th signal as function of frquncy (vrtical axis) and tim (horizontal axis) (L t al., 5; Krschn t al., 5). Furthr analysis of th numrical rsponss was prformd using a combination of th Empirical Mod Dcomposition (EMD) with Hilbrt transform. EMD is a mthod to dcompos a signal in monofrquncy oscillatory mods calld Intrinsic Mod Functions (IMFs) (Huang t al., 1998; ; Vltchva and Soars, 4; Zhang t al., 5). In ssnc, IMFs ar oscillatory mods mbddd in th tim sris, and thir x y y (1) (14)

10 linar suprposition rgnrats th tim sris. By Hilbrt transforming th st of IMFs on computs thir instantanous phass and frquncis. By post procssing th transint rsponss of th plat and th attachd NES by mans of EMD and Hilbrt transform, w aim to study in dtail th complx nonlinar rsonanc intractions taking plac btwn th ssntially nonlinar attachmnt and th various mods of th plat. By combining th rsults of th Hilbrt transform with th corrsponding wavlt spctra of th transint rsponss, w can idntify th dominant IMFs of th plat and NES transint rsponss, and furthr on analyz th most important modal rsonanc intractions btwn th plat and th NES that ar rsponsibl for th nonlinar nrgy xchangs (and TET) btwn ths two subsystms. Dtails of th EMD mthod with Hilbrt transform and thir applications for th dtrmination of th rsonanc capturs btwn lastic continua and th NES attachmnts could b found in th PhD Thsis by Gorgiads (6) and also in Gorgiads t al., 7. In th following sctions all th simulations wr prformd for a sufficintly larg tim window so at last 96.5% of th nrgy of th systm has bn dissipatd by th damprs of th subsystms. This total dissipatd nrgy masur (trmd total ) nsurd that no ssntial dynamics was missd in th transint simulations du to insufficint tim of numrical intgration. All sts of simulations with SDOF nonlinar attachmnts wr prformd with using a (1 1) FE msh for th plat. In th prsntd rsults w divid th plat in y-slics corrsponding to fixd valus of y to prsnt th dynamics. Th paramtric studis of th plat with MDOF attachmnts wr prformd with attachmnts locatd all ovr x-positions on thr y-slics, namly, y = 1,.7,. of th plat, again using a (1 1) FE msh. 4. Paramtric Studis of TET from th Plat to a SDOF NES In our first paramtric study w xamin TET in a plat forcd by singl or multipl shocks, possssing a singl SDOF NES attachmnt. W prform four main sts of FE simulations with paramtr valus for th intgratd plat SDOF NES systm, modl with quations (1.1, ). In ths sts of simulations w xamin th influnc of th variation of th NES paramtrs and input nrgy on th targtd nrgy transfrs (TETs) from th plat to th NES, using as critrion of NES ffctivnss (and fficincy) th portion of total nrgy vntually dissipatd by th NES (.g., th nrgy dissipation masur EDM), t 1 lim t1 (t) dfind by (1). Hnc, from now on, TET fficincy will b judgd by th asymptotic valu t1. Unlss othrwis statd, th mass of th NES is takn as ε =.5, its nonlinar stiffnss charactristic as C=1, and its damping cofficint as λ=.1. In th first st of simulations w xamind th influnc of th nonlinar stiffnss cofficint C, and of th position of th NES on th TET fficincy from th plat to th NES. For ach simulation w dtrmind th % of instantanous total nrgy of th systm ovr th input nrgy in ordr to hav a masur of th rror of th dtrmind nrgy of th systm in cass that th maximum and th minimum of th % of th total nrgy valus is not in accptabl rang w incrasd th sampling frquncy and rpat th simulations. For th simulations with singl forcing th sampling frquncy usd is Hz that corrsponds to a Nyquist frquncy of Hz (1 tims mor than th frquncy of th 15 th linar mod which is qual to.75 Hz). Th minimum and maximum valus of th instantanous total nrgy of th systm (including th nrgy dissipatd by th damprs) at ach tim stp of th prformd simulations wr 99.7% and 1.4% of th applid shock nrgy, indicating accuracy of th computations. For th simulations with multipl applid shocks th sampling frquncy was chosn as 8 Hz corrsponding to a Nyquist frquncy of 4 Hz (18 tims mor than th frquncy of th 15 th linar mod that is.75 Hz); th corrsponding bounds 1

11 of th instantanous nrgy for all prformd simulations wr 99.97% and 1.%, which indicats that th total nrgy (including nrgy dissipatd by damping) was appliximatly consrvd at ach tim stp of th numrical simulations. In Figurs a,b w dpict th asymptotic limit of nrgy dissipation masur (EMD), t1, as function of C, and th x-position of th NES on th slic y = 1. In Figur a (with C [1,] ) th rsults of simulations with singl shock xcitation ar dpictd, whras in Figur b (with C [1,1] ) th rsults for multi-shock xcitation ar dpictd; for th simulations dpictd in Figur b th mass of NES was fixd to a smallr valu ε =.5. Judging from ths rsults w conclud that for a fixd x-position of th NES th TET ffctivnss of th NES is robust in variations of C, for C is in th rang O(1 ) - O(1 ). Morovr, th variation of th x-position of th NES on a fixd y-slic affcts strongly th TET fficincy; this snsitivity can b xplaind by th fact that crtain locations of th NES may b clos to nodal curvs of th diffrnt mods of th plat, in which cas th capacity of th NES to passivly absorb and dissipat nrgy from ths particular mods is gratly diminishd. This bcoms clar whn w dpict th TET fficincy of th NES as function of th (x,y) coordinats of th NES, for fixd valus of C, cf. Figur ; w not that th maximum fficincy of th NES dampr occurs for positions of th NES at th cornrs of th plat, with maximum valus of t1 raching lvls of 87.7% and 89.8%, for (x,y) = (,1) and (x,y) = (1,1), rspctivly. Singl SDOF NES attachd to plat at Y=1, ε=.5 Figur a 11

12 Singl SDOF NES attachd to plat at Y=1, ε=.5 Figur b. Paramtric study of TET fficincy as function of NES stiffnss C, and x-position on th plat for th y-slic y = 1: (a) Cas of singl shock with NES mass ε=.5, and (b) cas of multipl shocks with NES mass ε=.5. Th intrprtation of th rsults dpictd in this Figur (which corrspond to singl shock xcitation) must b prformd in conjunction with th rsults of Tabl 1, which dpicts th nodal curvs of th fiv lading mods of th linar cantilvr plat (with no NES attachd). Not that in th strips clos to th y-slics y =.8 and y =.9 (adjacnt to th nd of th plat and corrsponding to th nodal lins of th rd and 5 th plat mods) th NES fficincy is low; this is du to th fact that whn th NES is attachd at points in ths rgions, its capacity to passivly absorb and dissipat nrgy from ths mods is impaird. Similarly, at strips positiond clos to x =., x =., x =.7, and x =.8, th NES fficincy is rlativly low (of th ordr of 4%); again, this can b intrprtd by th fact that ths strips ar in nighborhoods of nodal curvs of th 4 th mod of th plat. Finally, thr is a strip in th middl of th plat (.g., for x =.5) whr th lowst NES fficincy is obsrvd; this is rasonabl, givn that this rgion is in th nighborhood of nods of th nd and 5 th mods of th plat. 1

13 Singl SDOF NES attachd to plat C=1, ε=.5 Figur. Paramtric study for TET fficincy as function of NES position on th plat for C=1 and NES mass ε=.5; th clampd boundary condition is at y-slic y =. Morovr, as dpictd in Figur th fficincy of th NES dcrass whn th NES is locatd closr to th clampd nd whr th displacmnts of th plat ar rducd and th nonlinar ffcts ar diminishd. In th scond st of simulations w considr singl shocks applid to th plat, and xamin th influnc of th NES mass, ε, on th fficincy of th NES, whn it is attachd at vry possibl position on th plat. For th scond st of simulations with singl forcing th sampling frquncy usd is Hz corrsponding to a Nyquist frquncy of Hz (1 tims mor than th frquncy of th 15 th linar mod that is.75 Hz); th instantanous nrgy of th systm (including th nrgy dissipatd by th damprs) is btwn 99.4% and 1.5% of th applid shock nrgy at ach tim stp of th simulations, which dmonstrats th accuracy of th numrical rsults. In addition, w xamin in mor dtail th TET fficincy whn th NES is attachd at th position corrsponding to optimal TET, in ordr to idntify th thrshold of NES mass blow which th TET fficincy starts to dtriorat; similar critical mass thrsholds wr dtctd in prvious works focusing on linar coupld oscillators with SDOF NESs attachd [s for xampl, (Krschn t al., 5)]. In Figur 4a w dpict th asymptotic valu of th EDM, 1, as function of th mass of th NES ( [.5,.1] ), and its x-position on th y-slic y=1. W not that for ach x-position of th NES, its ffctivnss appars to b robust for variations of its mass. For a SDOF NES locatd at position (x,y)=(,1), which is th optimal position for TET (away from th xcitation point), w prform an additional sris of simulations in ordr to idntify th thrshold valu of NES mass blow which th TET fficincy starts to dtriorat. In Figur 4b w dpict th plot of TET fficincy for varying NES mass; from this plot w conclud that th fficincy of th NES is high, vn for small valus of mass, and that thr is a thrshold of th NES mass btwn 1 4 and blow which th TET fficincy is lss than 65%. This 1

14 indicats that vn vry lightwight NESs (of th ordr of.4% of th plat mass) can b fficint passiv shock absorbrs with fficincy mor than 65%. As in th first st of simulations, w not that whn th NES is locatd at positions clos to nodal lins of plat mods, its fficincy is significantly diminishd. In th third st of numrical simulations, singl shocks ar applid to th plat, and w xamin th influnc of th damping cofficint λ on th fficincy of th NES, whn it is attachd at vry possibl position on th plat. For th third st of simulations th sampling frquncy usd is Hz that corrsponds to Nyquist frquncy of Hz (1 tims mor than th frquncy of th 15 th linar mod that is.75 Hz) with instantanous total nrgy bing btwn 97.49% and 1.% of th applid shock nrgy. In Figur 5 w dpict th asymptotic EDM valu 1, as function of th NES damping cofficint, λ ( [.1,.5] ), and th x-position of th NES for a rprsntativ y-slic corrsponding to y=1. W not a dtrioration of NES fficincy with dcrasing damping cofficint. This trnd, howvr, dos not ncssarily man that by indfinitly incrasing NES damping to largr valus w will achiv a corrsponding monotonic incras of NES fficincy. Indd, for sufficintly larg valus of NES damping th rlativ motion btwn th NES and th plat is xpctd to significantly dcras (as th connction btwn th plat and th NES bcoms mor rigid), which, in turn, will rsult to a dcras of th rlativ motion across th dampr of th NES, and, hnc, to a dcras of th capacity of th NES dampr to dissipat shock nrgy; this will lad to a dtrioration of NES fficincy. In th final st of simulations of this sris, w xamin th influnc of th input nrgy and th nonlinar stiffnss charactristic C on th prformanc of th NES, whn it is locatd at position (x,y)=(.1) of th plat. A singl shock is assumd to apply to th plat and th NES mass is ε=.5. For this st of simulations th sampling frquncy usd is 8 Hz that corrsponds to Nyquist frquncy of 4 Hz (18 tims mor than th frquncy of th 15 th linar mod that is.75 Hz) with instantanous total nrgy bing btwn btwn 97.67% and 1.% of th applid shock nrgy. W can dtrmin th rang of nrgis that of practical (nginring) intrst,.g., th rang of nrgis for which th vibration of th plat is of intrst from an nginring point of viw; hnc, w xamin shock amplituds in A.1, 1 (input nrgy) for which th plat without NES is vibrating th rang with maximum absolut displacmnt within th rang max[abs(w)] 6 [9.7 1,.97] (not that th linar dimnsions of th plat ar normalizd to 1). 14

15 Singl SDOF NES attachd to plat at Y=1, C=1 Figur 4a 9 Singl SDOF NES attachd to plat at (X,Y)=(,1), C=1 8 %Enrgy Dissipatd by SDOF NES mass of NES Figur 4b. Paramtric study of TET fficincy, (a) as function of NES mass and x- position of th NES on th plat for fixd y = 1; and (b) as function of NES mass whn th NES is locatd at th optimal position (x,y)=(,1). 15

16 Singl SDOF NES attachd to plat at Y=1, C=1, ε=.5 Figur 5. Paramtric study of NES fficincy as function of its damping cofficint, and its x-position on th plat for fixd y = 1. Singl SDOF NES attachd to plat at (X,Y)=(,1) Singl SDOF NES attachd to plat at (X,Y)=(,1) Figur 6. Paramtric study of NES fficincy as function of th input amplitud (nrgy) and its stiffnss C, at position (x,y)=(,1). In Figurs 6a,b w dpict th rsults of this paramtric study. Not th wak dpndnc of TET fficincy on th input nrgy: indd, TET fficincy for low input amplituds and all valus of C is of th ordr of 87%, but as th input amplitud and th nonlinar stiffnss charactristic incras th TET fficincy dcrass but within a vry small rang, lss than 7%. Similar rsults hav bn rportd by Gorgiads t al. (7) but without paying attntion to rstrict th xamination of th nrgy dissipation within a rang of input nrgy that it is of intrst in nginring point of viw. A gnral conclusion from this first paramtric study is that strong TETs can b ralisd from a plat with singl or multipl shock xcitation to a SDOF ssntially nonlinar attachmnt, spcially, whn th NES is locatd at points of th plat corrsponding to antinods of nrgtically high plat mods. Indd, it is possibl to 16

17 passivly transfr and locally dissipat at th NES as much as 87% of th shock nrgy of th plat. Morovr, th intgratd plat-nes systm can b dsignd so that TET fficincy is insignificantly affctd by th magnitud of th applid shock within a spcifid rang of dsign intrst. W now wish to study in mor dtail th nonlinar modal intractions that giv ris to TET in th plat SDOF NES systm undr considration. To this nd, w isolat a spcific cas corrsponding to a plat with paramtrs givn by (), and an NES with nonlinar stiffnss charactristic C = 1, damping cofficint λ =.1, and mass, ε =.5. W assum that thr is a singl applid shock in th form of a half-sin (as dfind in th prvious sction) at (x,y) = (1,1),.g., at on of th fr cornrs of th plat. Morovr, w assum that th SDOF NES is attachd to th plat at position (x,y) = (,.5). In this spcific cas 64.5% of th shock nrgy of th plat gts vntually transfrrd to and locally dissipatd by th NES. Th sampling frquncy for this simulation is 8 Hz that corrsponds to Nyquist frquncy 4 Hz (18 tims mor than th frquncy of th 15 th linar mod that is.75 Hz) with total instantanous nrgy varying btwn 99.86% and 1.1% of th applid shock nrgy. W xamin th transint nonlinar rsonanc intractions (transint rsonanc capturs - TRCs) btwn th plat rspons at th point of attachmnt to th NES, and th NES rspons. W focus mainly in th arly stag of th transint dynamics, whr th nrgy of th systm is at high lvls and th nonlinar ffcts ar xpctd to b mor profound; hnc, w xamin TRCs in th arly tim intrval < t < 5, during which mor than 75% of th shock nrgy is bn dissipatd by th damprs of th intgratd systm. Th rsponss of th plat and th NES ar dcomposd in trms of Intrinsic Mod Functions (IMFs) using Empirical Mod Dcomposition (EMD) (Gorgiads, 6); ths ar oscillatory mods mbddd in th rspctiv tim sris. Thn, w apply th Hilbrt transform to ach IMF to dtrmin its instantanous amplitud and frquncy. By suprimposing th instantanous frquncis of th IMFs to th wavlt spctra of th corrsponding rsponss w can dtrmin th dominant IMFs as dpictd in Figurs 7a-; this tchniqu was first dvlopd in (Gorgiads t al., 7). Th dtrmination of th dominant IMFs of th plat and NES rsponss nabls us to dtct th dominant TRCs that govrn TET in this cas. Considring th plat rspons, th 5 th and 6 th IMF ar dominant (cf. Figurs 8a,b), whras, th 1 st, nd and rd IMF of th NES rspons ar dominant as wll (cf. Figurs 8c-). By computing th ratios of th instantanous frquncis of th dominant IMFs of th plat and th NES rsponss w can idntify th possibl typs of ( k : m ) TRCs that occur in th transint dynamics, as wll as, th corrsponding tim intrvals whr ths TRCs tak plac. Considring th instantanous frquncy plots dpictd in Figur 8 w dtct th following 1:1 TRCs btwn, (a) th 5 th IMF of th plat and th rd IMF of th NES at frquncy clos to 1.9 Hz in th tim intrval.5 < t <.5 (TRC I); (b) th 5 th IMF of th plat and th 1 st IMF of th NES clos to 1.9 Hz in th tim intrval 6 < t < 1 (TRC II); and, (c) th 6 th IMF of th plat and th nd IMF of th NES clos to 1.9 Hz in th tim intrval 8 < t < 14 (TRC III). Ths TRCs ar rsponsibl for passiv TET in th spcific cas undr xamination and th prsntd analysis hlps to idntify th corrsponding nonlinar rsonanc intractions btwn th mbddd oscillatory mods in th plat and NES rsponss. 17

18 SINGLE SDOF NES ATTACHED TO PLATE PLATE RESPONSE SINGLE SDOF NES ATTACHED TO PLATE PLATE RESPONSE (a) (b) SINGLE SDOF NES ATTACHED TO PLATE NES RESPONSE SINGLE SDOF NES ATTACHED TO PLATE NES RESPONSE SINGLE SDOF NES ATTACHED TO PLATE NES RESPONSE (c) (d) () Application -NES Tim Application -Plat IMF-1 Rspons Tim IMF-5 Rspons Application -NES Tim Application -Plat IMF- Rspons Tim IMF-6 Rspons Application -NES Tim IMF- Rspons Figur 7. Dominant IMFs of th rspons of th SDOF NES, and of th plat rspons at th point of attachmnt to th NES, suprimposd to th corrsponding tim sris. 18

19 Application -NES Tim Application -Plat Tim Tim Application -NES Application -Plat Tim Application -NES I III III II I II SINGLE SDOF NES ATTACHED TO PLATE PLATE RESPONSE SINGLE SDOF NES ATTACHED TO PLATE PLATE RESPONSE Tim d 1 /dt d /dt d /dt d 5 /dt d 6 /dt (a) (b) SINGLE SDOF NES ATTACHED TO PLATE NES RESPONSE SINGLE SDOF NES ATTACHED TO PLATE NES RESPONSE SINGLE SDOF NES ATTACHED TO PLATE NES RESPONSE (c) (d) () Figur 8. Instantanous frquncis of th dominant IMFs suprimposd to th corrsponding Wavlt Transform spctra: (a,b) 6 th and 7 th IMF of th plat rspons, (c,d,,f) 1 st, nd, rd and 4 th IMF of th SDOF NES rspons; dominant 1:1 TRCs btwn IMFs ar indicatd by I, II and III. 19

20 5. Paramtric Studis of TET from th Plat to multipl SDOF NESs Th paramtric study of TET fficincy in trms of NES location on th plat carrid out in sction 4 rvald locations whr TET fficincy is low (whn th NES is attachd clos to nodal curvs of plat mods), and altrnativ locations whr th TET fficincy is high (of th ordr of mor than 7%). Th computational study carrid out in this sction aims to xamin nhancmnt of TET through th us of multipl SDOF NESs, modl with quations (1.1, ). Th paramtrs usd for th plat and th input forc in th following numrical simulations ar idntical to thos mployd in th simulations of sction 4. Two SDOF NES ar attachd to th plat, with ach possssing mass ε =.5 (.g.,.5% of th total mass of th plat), stiffnss C = 1, and damping cofficint λ =.1. A singl shock xcitation is applid to th plat, of th sam form and position as in th prvious sction. For this st of simulations th sampling frquncy usd is Hz that corrsponds to Nyquist frquncy of Hz (1 tims mor than th frquncy of th 15 th linar mod that is.75 Hz), xcpt for th final cas which was carrid out with a sampling frquncy of 8 Hz which corrsponding to a Nyquist frquncy of 114 Hz (5 tims mor than th frquncy of th 15 th linar mod that is.75 Hz); th total instantanous nrgy varid btwn 99.67% and 1.1% of th applid shock nrgy Comparison Graph Singl SDOF NES attachd to plat at position 1 Singl SDOF NES attachd to plat at position Multipl SDOF NES attachd to plat 7 6 Efficincy (.8,.8) (.5,1) Mult. (.,.8) (.5,1) Mult. (,.6) (1,.6) Mult. (,.7) (1,.7) Mult. A B C D E F G Cass (,.8) (1,.8) Mult. (,.9) (1,.9) Mult. (,.1) (1,.1) Mult. Figur 9. Comparativ study of TET fficincy whn using two singl NESs, and a st of two NESs; singl shock xcitation of th plat.

21 In th numrical simulations that follow w xamin svn spcific cass (dsignatd Cass A-G) whr two NESs ar locatd at various positions on th plat. Of spcific intrst ar cass whr th NESs ar locatd at nodal curvs of plat mods. In Figur 9 w dpict a bar diagram dpicting NES fficincis for all svn cass considrd; w compar th fficincis of singl NESs locatd in ithr on of th two locations occupid by that positions th st of NESs, to th fficincy of th st of two NESs whn thy ar both attachd to th plat at th sam tim. In ach cas w indicat th position of th two NESs. For cas A th TET achivd whn using singl (isolatd) NESs is 6.1% and 1.58%, rspctivly, whras whn using th combind st of two NESs at th sam locations TET incrass to 61.%,.g., it xcds th sum of TETs whn th two NESs ar applid in isolation. This dmonstrats a positiv synrgy ffct of th st of two NESs, which, howvr, is not xpctd to prsist in th othr cass whr NES locations mor favourabl to TET ar considrd. In cas B, th first NES is locatd at position (x,y) = (.,.8) which is a crossing point of th nodal curvs of th rd and 4 th plat mods, and th scond NES at (x,y) = (.5,1),.g., th location that ld to th lowst TET fficincy whn a singl NES was usd. Again, in this cas th fficincy of th st of two NESs incrass significantly to 6.4% (cf. Fig. 9), which again xcds th summation of th individual NES fficincis whn applid in isolation at th sam locations. Again, this dmonstrats positiv synrgy of th st of two NESs. Hnc, it appars that th minimum lvl of TET that on achivs whn using a st of two SDOF NES xcds 6%, which rprsnts an improvmnt compard to th cas whn a singl NES is usd; this rsult is furthr nhancd by th positiv synrgy achivd whn th pair of NESs is usd. In cass C, D, E and F w locatd th st of NESs at diffrnt positions of th plat to invstigat th synrgistic TET fficincy whn th NESs ar locatd at rgions of high individual TET fficincy for th SDOF NES (btwn 6-7% - such locations ar at th dgs of th plat at x = and x = 1). For cass C, D, E and F w plac th NESs at th dgs of th plat, x =, 1, with y =.6,.7,.8 and.9, rspctivly. For ths cass w not a slight improvmnt (about 1%) of th synrgistic fficincy of th st of two NESs (cf. Fig. 9). Finally, thr ar two locations whr th fficincy of th singl SDOF NES is vry high (mor than 8%); ths ar th two fr cornrs of th plat. Th numrical simulations indicat that by attaching th st of two NESs at ths locations (cas G), w obtain a combind TET fficincy of 89.9%, which can b considrd as th optimal synrgistic TET fficincy that can b achivd by th st of two NESs on th plat. W not, howvr, that th improvmnt in TET fficincy compard to using th two NESs in isolation at th sam locations is only marginal. A gnral conclusion drawn from th mntiond simulations is that th us of th st of two NESs improvs TET fficincy in rgions whr th us of singl (isolatd) NESs lads to poor TET prformanc. In such rgions thr occur positiv synrgisms btwn th two NESs of th st, which lads to TET fficincis that xcd th sum of th fficincis of singl NESs whn ths ar usd in isolation. Th us of multipl NESs, howvr, improvs only marginally TET fficincy in locations whr th isolatd NESs alrady provid good TET prformanc. 6. Paramtric Studis of TET from th Plat to a singl MDOF NES W now considr TET from th plat to a singl MDOF NES. Th quations of motion of this systm ar givn by modl 4 with quations (1.1, ), and th TET fficincy is judgd by th nrgy dissipation masurs (1) and (14) and thir long-tim asymptotic valus. Th first st of numrical 1

22 simulations of this sris was prformd in ordr to xamin th influnc of th linar coupling stiffnss btwn th plat and th MDOF NES to TET fficincy. Th paramtrs usd for th plat ar idntical to th ons usd for th cas of SDOF NES attachmnt, rlations (), and th applid shock is idntical to th half-sin xcitation usd in prvious simulations, rlation (4); th duration of th applid shock was slctd sufficintly small to dirctly xcit at last th lading fiv mods of th plat howvr, as mntiond in prvious Sctions additional plat mods may b indirctly xcitd by nonlinar coupling providd by th MDOF NES. Th thr masss of th MDOF NES ar assumd to b small, m 1 = m = m =.17 (.5/),.g., th total mass of th MDOF NES is assumd to b idntical to th smallst mass of th SDOF NES usd in th paramtric study of sction 4. In this way w wish to study th rlativ advantag of rplacing th SDOF NES by a MDOF on, without burdning th plat with additional mass. Th two nonlinar stiffnsss of th NES ar slctd as, C 1 = 5, and C =.1, whras th two damprs of th NES possss th sam cofficint, λ=.1. For this st of simulations th sampling frquncy usd is 8 Hz that corrsponds to Nyquist frquncy of 4 Hz (18 tims mor than th frquncy of th 15 th linar mod that is.75 Hz), with total instantanous nrgy varying btwn 1.% and 1.9% of th applid shock nrgy. Singl MDOF NES attachd to plat at Y=1 Figur 1. Paramtric study of TET fficincy as function of th (linar) coupling stiffnss C of a MDOF NES and its x-position, for fixd y = 1. In Figur 1 w dpict th long-tim asymptotic valu lim (t) (t) (which rprsnts th portion of shock nrgy of 1t1 t1 t1 1 th plat that is vntually dissipatd by th two damprs of th MDOF NES) as function of th coupling stiffnss C ( C [1,1] ), and th x-position of th NES for th rprsntativ y-slic y=1. A first conclusion from ths numrical rsults is

23 that TET fficincy appars to b robust as C varis abov th thrshold of C =1, for vry x-location of th NES (as dpictd in Figur 1). Morovr, strong (fficint) TET from th plat to th MDOF NES is ralizd for rlativly stiff coupling stiffnss. Maximum TET fficincy of th MDOF NES rachs lvls of 8%, which ar comparabl to maximum TET fficincy achivd by SDOF NESs. Similarly to th cas of th SDOF NES, th variation of th position of th MDOF NES appars to strongly affct TET from th plat to th NES, as dpictd in Figur 1; howvr, this will b mor vidnt in th scond st of simulations that w now procd to discuss. Singl MDOF NES attachd to plat, C = Figur 11. Paramtric study of TET fficincy for a MDOF NES locatd at vry possibl position on th plat, for coupling stiffnss C. In th scond st of simulations w xamin th influnc of th MDOF NES position on TET; for this, w xamin MDOF NES placmnt at vry possibl position on th plat (with a msh of 11 lmnts), not rstricting our study to any spcific y-slics. W us th sam st of NES paramtr valus with th prvious st of simulations, and fix th coupling stiffnss to C =. For this st of simulations th sampling frquncy usd is 8 Hz that corrsponds to Nyquist frquncy of 4 Hz (18 tims mor than th frquncy of th 15 th linar mod that is.75 Hz), with total instantanous nrgy varying btwn 1.% and 1.9% of th applid shock nrgy. In Figur 11 w dpict TET fficincy for th NES locatd at vry possibl position on th plat. Prdictably, th highst valus of fficincy of th MDOF NES ar obtaind at th fr cornrs of th plat, raching 85.85% whn th NES is locatd at (x,y) = (1,1), and 76.67% whn locatd at (x,y) = (,1); as xplaind in prvious Sctions, at ths positions th MDOF NES can intract with all plat mods, as no nodal curvs of low ordr plat mods ar locatd narby and th plat posss th maximum displacmnt. Similarly to th cas of SDOF NES, th intrprtation of th rsults dpictd in Figur 11 should b

24 carrid out in conjunction with Tabl 1, which dpicts th nodal curvs of th first fiv mods of th linar, uncoupld plat with no disprsion (th plat mods ). Morovr, th fficincy of th NES dcrass whn th NES is locatd closr to th clampd nd, whr th displacmnts of th plat ar small and th nonlinar ffcts ar lss profound. Sinc passiv TET is th rsult of nonlinar rsonanc intractions (rsonanc capturs) btwn th plat and th NES it is rasonabl to xpct that in low-amplitud rgims th ffctivnss of th MDOF should dtriorat. As dmonstratd, howvr, in (Tsakirtzis t al., 7) it is possibl (undr crtain forcing conditions and at dfinit rangs of NES paramtrs) to achiv fficint TET from a dirctly forcd linar systm to a MDOF NES, vn at low amplitud rgims; such a cas, howvr, was not ralizd in th simulations considrd hrin. Singl MDOF NES attachd to plat at (X,Y)=(,1) Singl MDOF NES attachd to plat at (X,Y)=(,1) Figur 1. Paramtric study of MDOF NES fficincy as function of th shock amplitud (nrgy) and coupling stiffnss C. Th third st of numrical simulations of this sris was prformd in ordr to xamin th influnc of th magnitud of th applid shock and th (linar) coupling stiffnss C to TET fficincy. Th paramtrs of th systm usd for this st of simulations ar idntical to th first st of simulations rportd in this A.1,1. For th third st sction, for varying shock amplitud in th rang of simulations th sampling frquncy usd is 8 Hz that corrsponds to Nyquist frquncy of 4 Hz (18 tims mor than th frquncy of th 15 th linar mod that is.75 Hz) with total instantanous nrgy varying btwn 1.% and 1.% of th applid shock nrgy. In Figurs 1a,b w dpict th rsults of this paramtric study. W not that TET fficincy dos not dpnd significantly on th variation of shock input; indd, th variation of TET fficincy for th rang of shocks xamind is lss than 6%, lying within th rang %. Additionally, TET fficincy is uniformly mor than 7% in ths simulations, which assurs satisfactory NES prformanc in th ntir rang of paramtrs considrd. Th fourth st of numrical simulations of this sris was prformd in ordr to xamin th optimum valus of th othr stiffnss paramtrs C 1,C whn th NES is locatd at th fr cornr (x,y)=(,1) of th plat (highst fficincy away from th xcitation point) with th optimum coupling stiffnss C =. Th paramtrs of th 4

25 plat and damping cofficints wr idntical to th othr sts of simulations of this sction. For th fourth st of simulations th sampling frquncy usd was 1 Hz that corrsponds to Nyquist frquncy of 5 Hz (45 tims mor than th frquncy of th 15 th linar mod that is.75 Hz), with total instantanous nrgy varying btwn 1.% and 11.7% of th applid shock nrgy. Th rsults for this st of simulations ar dpictd in Figur 1. W not that strong TET is ralizd ovr a broad rang of valus of C 1,C, with optimal stiffnss valus bing ralizd for C 1 and C 1. W not that incrasing th valu of th nonlinar stiffnss cofficints dos not ncssarily rsult in TET nhancmnt; this is du to th fact that vry stiff nonlinar connctions in th MDOF prclud th ralization of larg rlativ motions btwn th masss of th NES, and as a rsult to smallr amounts of nrgy dissipatd by its damprs. Morovr, w not that optimal TET occurs in highly asymmtric MDOF NES configurations, a rsult that agrs with th findings of a prvious work on discrt oscillators with MDOF NESs attachd (Tsakirtzis t al., 7). This is du to th fact that in such highly asymmtric MDOF NESs, a part of th NES (th on in closr proximity to th plat corrsponding to th stiffr stiffnss C 1 ) acts as broadband rsonator that ngags mods of th plat in transint rsonanc capturs (TRCs), whras th complmnt of th NES (th on furthr away from th plat corrsponding to th wakr stiffnss C ) acts as dissipatr of shock nrgy. Figur 1. Paramtric study of MDOF NES fficincy as function of nonlinar stiffnss C 1 and C. Th analysis of th complx nonlinar modal intractions and of th corrsponding transint rsonanc capturs that govrn TET in this systm can b prformd by applying th prvious combind Wavlt/EMD postprocssing mthodology. As th analysis bcoms quit involvd (sinc thr th transint rsponss of thr diffrnt masss of th NES that must b considrd in this cas), th radr is rfrrd to (Gorgiads, 6) for a dtaild prsntation of th rsults of this analysis for spcific cass. In th nxt sction w considr th 5

26 altrnativ of using a linar vibration absorbr in ordr to dmonstrat th qualitativ diffrncs of th proposd ssntially nonlinar dsigns. 7. Th Linar Altrnativ: Tund Mass Dampr (TMD) Comparativ Study In this sction w prsnt th rsults of a paramtric study of th plat with a linar Tund Mass Dampr (TMD) attachd, modl 5 with quations (1.1, ). W assss th capacity of th TMD to absorb and locally dissipat shock nrgy from th plat, by varying th TMD paramtrs and its position on th plat. W prform a sris of simulations considring singl and multipl shock xcitations applid to th plat. In ach of ths sts, th fficincy of th TMD to passivly absorb and locally dissipat shock nrgy from th plat is stimatd by th following limit, t w(d x,d y, ) v( ) d E damp,nes (t) t 1 lim t1 (t), (t) (15) t Ein F( ) w(b,b, )d whr v(t) is th rspons of th TMD. This rprsnts th portion of th shock nrgy of th plat that is vntually dissipatd by th dampr of th TMD, and is similar to th nrgy masurs dfind prviously to assss th fficincy of SDOF and MDOF NESs. In th first st of simulations w xamin th fficincy of th TMD to absorb and dissipat shock nrgy by varying its stiffnss and its location on th plat, for singl shock xcitation [as dfind by (4)]. Th systm paramtrs of th plat proprtis ar dfind in (), th TMD mass is ε =.5, and its damping cofficint is λ =.1; ths wr idntical to th paramtrs usd for th simulations with th SDOF NES attachmnt (sction 4), so th two sts of simulations can b dirctly compard. For this st of simulations th sampling frquncy usd is 6 Hz that corrsponds to Nyquist frquncy of Hz (1 tims mor than th frquncy of th 15 th linar mod that is.75 Hz), with total instantanous nrgy bing consrvd (within an rror of lss than.1% of th applid shock) whn nrgy dissipatd by th damprs of th systm is takn into account. In Figur 14 w dpict th fficincy of th TMD dpnding on its stiffnss and on its x-position on th plat for th fixd y-slic y = 1 (.g., at th fr nd of th cantilvr plat). This rsult should b compard to th plot of Figur a for th SDOF NES. W not that th variation of th location of th TMD strongly affcts its fficincy, in a similar mannr to th SDOF and MDOF NES attachmnts xamind prviously. Indd, whn th TMD is locatd at positions clos to nodal lins of th plat, th TMD can not intract with th corrsponding plat mods, and thrfor th absorption of shock nrgy from th plat dtriorats. Morovr, whn th TMD is tund to th i-th plat mod,.g., whn its stiffnss is qual to kln i (undampd cas), whr ω i is th i-th natural frquncy of th (uncoupld and linar) plat, its fficincy in xtracting nrgy from that mod is high. It must b notd that th diffrnc in th fficincy btwn th two xtrm cornrs (x=, and x=1) in th attachd potions (lft part and right part of Figur 1) is du to th fact that th xcitation forc is on th right cornr (x=1) and thrfor thr is an ffct that th TMD is absorbing nrgy from th xcitation forc dirctly. Howvr, for rlativly high stiffnss valus of th TMD,.g., whn it is dtund from th lading plat mods, its fficincy dtriorats; this rsult is in agrmnt with similar rsults rportd in th litratur (Frahm 1911, Dn Hartog 1947). Comparing to plot of Figur a w not that th fficincy of th SDOF NES dos not show such dpndnc on stiffnss, and hnc, its prformanc is mor robust to stiffnss variations. 6 x y

27 Singl TMD attachd to plat at Y=1 Figur 14. Efficincy of th linar vibration absorbr-tmd as function of its stiffnss and its x-position on th plat for th y-slic y = 1, singl shock xcitation. In th scond st of simulations w xamin th fficincy of th TMD for variation of its stiffnss and location of th TMD for multipl shock xcitations. Th paramtrs usd ar idntical to th first st of simulations, but th damping cofficint of th plat was slctd as, d = 15, and th TMD mass as, ε =.5; ths systm paramtrs ar idntical to th ons usd in sction 5, whr th fficincy of th SDOF NES for multipl shock xcitations was xamind. For th scond st of simulations th sampling frquncy usd is Hz that corrsponds to Nyquist frquncy of 8. Hz (.5 tims mor than th frquncy of th 15 th linar mod that is.75 Hz) and again th total nrgy of th systm bing consrvd for th ntir durations of th simulations whn dissipativ trms ar includd. In Figur 15 w prsnt th rsults for this st of simulations. Again, th stiffnss and position of th TMD strongly affct its fficincy. Comparing ths rsults to th plot of Figur b w again not th insnsitivity of th prformanc of th SDOF NES to stiffnss variations for multipl shock xcitation. In th third st of simulations w xamin th ffct of varying th TMD mass on its nrgy absorption fficincy. Singl shock xcitation is considrd for this st of simulations. For this st of simulations th sampling frquncy usd is Hz that corrsponds to Nyquist frquncy of 15 Hz (6 tims mor than th frquncy of th 15 th linar mod that is.75 Hz). 7

28 Singl TMD attachd to plat at Y=1 Figur 15. Efficincy of th linar vibration absorbr-tmd as function of its stiffnss and its x-position on th plat for th y-slic y = 1, multipl shock xcitations. In similarly to th cas of SDOF NES w aim to idntify th thrshold of mass valu of th TMD abov which th TMD is capabl of absorbing at last 65 % of th shock nrgy of th plat. This thrshold was computd to ε =.4,.g., slightly highr than th cas of th SDOF NES attachmnt (cf. Figur 4b). In ordr to compar th prformanc of th various nonlinar and linar attachmnt configurations considrd in this work, w prformd an additional st of simulations for fixd attachmnt placmnt at (x,y) = (,1) on th plat, and away from th sourc of th (singl) input shock at (b x, b y ) = (1,1)]. For this st of simulations th plat paramtrs ar dfind in (). All SDOF attachmnts considrd possss mass qual to.5 (or.5% of th plat mass), and viscous dampr cofficints qual to λ =.1. For th simulations with SDOF attachmnt th sampling frquncy usd is 1766 Hz that corrsponds to Nyquist frquncy of 88 Hz (8 tims mor than th frquncy of th 15 th linar mod that is.75 Hz) and th minimum and maximum valus of th % instantanous total nrgy all ovr th simulations ar 99.9% and 1.% corrspondingly, thrfor th rror is lss than.1% in th dtrmind valus of nrgy. For th MDOF NES ach mass was chosn qual to.5/, and th two viscous damping cofficints whr slctd as λ =.1; for simulations whr th coupling stiffnss (C ) varis th two nonlinar stiffnss cofficints ar C 1 = and C = 15; whn th stiffnss C 1 varis th othr stiffnss cofficints ar C = and C = 15; whras, whn C varis stiffnss th othr stiffnss cofficints ar C = and C 1 =. For th simulations with MDOF attachmnt th sampling frquncy usd is 1 Hz that corrsponds to Nyquist frquncy of 5 Hz (45 tims mor than th frquncy of th 15 th linar mod that is.75 Hz); th total instantanous nrgy varis btwn 1.% and 11.75% of th applid shock nrgy. 8