Track Nonlinear Energy Sinks for Impulsive Response Reduction

Transcription

1 Track onlnear Energy Snks for Impulsve Response Reducton Jngjng Wang 1, cholas Werschem, Blle F. Spencer, Jr. 3, Xln Lu 4 1,4 State Key Laboratory of Dsaster Reducton n Cvl Engneerng, Tongj Unversty, Shangha 9, Chna,3 Department of Cvl and Envronmental Engneerng, Unversty of Illnos, Urbana 6181, IL,USA ABSTRACT onlnear energy snks (ESs) are a type of passve mass damper that can reduce structural responses wth ther nonlnear restorng force. A new type of ES featurng an auxlary mass whch moves along a specally shaped track s proposed n ths paper. Followng the dervaton of the equaton of moton, an analytcal nvestgaton s performed to reveal the dynamcs of the system. umercal analyses are then carred out to optmze system performance for the case when the ES s attached to a two degree-of-freedom prmary structure that s subjected to mpulsve loadngs. Comparsons wth a tuned mass damper (TMD) show that the response reducton of the prmary structure acheved usng the track ES s compettve wth the TMD durng deal condtons, whle the ES shows superor performance when uncertanty or changes to the prmary structure are consdered. ITRODUCTIO Structural control has a long hstory of successful applcaton to response mtgaton for structures subject to varous types of dynamc exctaton, ncludng earthquake, wnd, and blast. One of the most mature structural control methods s to use passve supplemental oscllators, such as tuned mass dampers (TMDs) to act as dynamc vbraton absorbers. TMDs consst of a secondary mass, usually less than 5% of the prmary structure to whch t attaches, and ncrease the apparent dampng n a prmary structure by transferrng nput energy to the out-of-phase vbratng secondary mass (Housner et al. 1997). A key factor n the effectveness of TMD systems s the tunng of the TMD, whch s often done so that the TMD shares the same or almost the same fundamental frequency of the prmary structure (Fahm Sadek et al. 1, Graduate Student 3,4 Professor

2 1997). However, whle the TMD provdes the advantages of smplcty and cost effectveness, t s lmted n the ablty to adapt to changes n the structure. For example, when the natural frequency of the prmary structure shfts due to structural degradaton or other reasons, a lnear TMD can act nstead as a vbraton amplfer, ncreasng the response ampltude of the prmary structure (Sun et al. 13). onlnear energy snks (ESs), on the other hand, can perform well n the presence of substantal changes n the dynamcs of the structure, as the restorng force employed has an essental nonlnearty (.e., the restorng force has no lnear component and s not lnearzable). Whle an ES functons as a damper by absorbng and dsspatng energy, t also couples together the vbraton modes of the lnear structure; the result s the transfer of energy from lower to hgher vbraton modes of the prmary structure where the energy can be dsspated rapdly (McFarland et al. 5). Ths phenomenon s termed Targeted Energy Transfer (TET). The nonlnearty observed n ESs shows hgher stffness when dsplacement ncreases; thus, the hardenng effect n the system s more apparent wth ncreasng energy. Ths feature helps accommodate the dynamc change n the structure but also makes the effcacy of ES systems load-depended. ES systems behave dfferently for dfferent magntude exctatons. Several dfferent expermental realzatons of ES systems have been prevously studed. A pano wre was used n the expermental verfcaton of a two-degree-of-freedom system comprsng a damped lnear oscllator coupled to an essentally nonlnear attachment by McFarland et al. (5). o-pretenson wres also appeared n the subsequent study of a lghtweght ES coupled to an mpulsvely loaded lnear oscllator (McFarland et al. 5). Werschem et al. (13) studed the wre ES attached to a buldng structure model and dd thorough research on ts optmzaton and effect. Implementatons usng elastomerc bumpers to provde a desred nonlnear restorng force show that the bumper ES s capable of dramatcally reducng the response of a large-scale 6-storey base structure to mpulsve ground moton (Werschem et al. 13). A bumper ES system has been tested on a larger 9-storey structure under both mpulsve and blast exctatons (Werschem et al. n press; Werschem et al. 13). In the study of acoustc systems, nonlnear attachments made of membrane were frst verfed n (Cocheln et al. 6). Bellet et al. (1) presented a more complete study of a two degree-of-freedom model usng membrane absorber. In ths paper, a new type of track ES s proposed. An analytcal nvestgaton s performed to reveal the dynamcs of the ES and numercal analyses are carred out on ths ES. The performance of the ES system s optmzed for a two degree-of-freedom prmary structure when subjected to mpulsve loads. Comparsons of responses, measures, and force-dsplacement relatonshps are made wth the locked system and a tradtonal TMD system. The results show that the track ES s compettve wth the TMD durng deal condtons and s more robust effcent aganst the stffness change n the base structure. TRACK ES AD PRIMARY STRUCTURE Track ES The system studed n ths paper s referred to as a track ES. Ths system conssts of a mass movng along a vertcally curved path, as shown n Fg. 1a. One possble physcal realzaton of ths track ES s a cart that runs along a set of curved tracks (Fg. 1b). The track s specally shaped n the vertcal plane to follow a desred profle. As the ES mass moves back and forth along the track, the reacton of the ES mass on the track s transferred to the prmary structure to whch the track s attached. As comprehensvely dscussed below, the drecton of the normal force as well as the horzontal component of the normal force between the mass and the track changes dependng on the slope of the shape of the track at each

3 poston; thus, by carefully desgnng the track shape, varous possbltes for the force-dsplacement relatonshp can be acheved, ncludng essentally nonlneartes. (a) (b) Fg. 1. Confguraton of track ES In dervng the dynamcs of the track ES, the mass s assumed not to rotate, and as a result, no rotatonal nerta force s consdered n ths study. Also, the mass s assumed to reman n contact wth the track; thus, the restorng force s contnuous. v hu ( ) F z θ o mg u Fg.. Free body dagram of track ES Fg. shows the free body dagram of the ES mass. m s the mass of ES. u and v are the relatve horzontal and vertcal dsplacement of the ES to the track, respectvely. z s the dsplacement of the track. h( u ) s the shape of the track and defnes the relatonshp between the horzontal and vertcal dsplacement of the ES mass. F s the normal force between ES and track. θ s the angle of the track normal whch s defned as a functon of the track shape, h( u ). g s the vertcal acceleraton due to gravty. The EOM of track ES s derved from energy perspectve. The Lagrange s approach (Roy R. Crag and Andrew J. Kurdla 6) s appled to seek out knetc energy () and potental energy (3) separately. The EOM s formulated by assemblng those two energy equatons nto the Lagrange s equaton (1).

4 d T T V + = p t dt q q q () (1) In ths formulaton, dampng s neglected and no external force on the ES s consdered; thus, no non-conservatve force term s ncluded n the equaton. ( ) T = 1 m 1 u + z + mv () V ( ) = m gh u (3) Also, wth the followng relaton, v ( ) = h u (4) Expand each term n () and (3). T dh dh = mu ( + z ) + m u u du du (5) d T dh dh d h = m( u z) m u m u dt u du du du (6) T u du du dh d h = m u (7) V u dh = mg du (8) After combnng lke terms and droppng EOM for the ES mass s m n each term, the fnal expresson of the ( ( ) ) ( ) ( ) ( ) 1+ h u u + h u h u u + h u g = z (9) Prmary Structure The lnear model beng used throughout the paper s a two degree-of-freedom shear buldng and s supposed to move only unaxally. The mass s 4.3kg and 4.kg and the stffness s 68/m and 8/m for the frst and second floors respectvely. Ther matrx

5 forms are shown as follows. 4.3 M = kg 4. (1) K = m 8 8 (11) These values are dentfed from a physcal expermental model n the Smart Structure Technology Laboratory n the Unversty of Illnos at Urbana-Champagn (Werschem et al. 1). otce that although the columns are dentcal for both floors the stffness of the frst floor s reduced as a result of the geometrc stffness effect due to the ncreased weght on the bottom columns. The dampng of the model s set at.1% n each mode. The resultng natural frequences are 1.63Hz and 4.56Hz for the frst and second modes, respectvely. Equatons of Moton The EOMs of the prmary structure and the ES can then be combned to reflect the effect of the track ES to the prmary structure to whch t s attached on the top floor. The dampng assocated wth the track ES system s prmarly due to frcton between the wheels and the track and dampng n the bearngs; for smplcty, the dampng n the ES s assumed to be proportonal to ts velocty relatve to the structure. Addtonally, because the ES mass s relatvely small, geometrc stffness effects are neglected. The ES system s shown n Fg. 3. m 1, k 1 and m, k are the frst floor and second floor masses and stffness, respectvely and they are from the model structure. c 1 and c are the equvalent vscous dampng of the.1% modal dampng for the frst floor and second floor, respectvely. c s the vscous dampng coeffcent of the ES. x 1 and x are the dsplacements of frst and second floors relatve to the ground. x s the ground acceleraton. g u c m m x c k k m 1 x 1 1 c 1 k k 1 x g

6 Fg. 3. Free body dagram of track ES system To apply the Lagrange s approach, the knetc energy, potental energy and the work by non-conservatve forces of the system are wrtten as follows T = m1( x 1+ x g) + m( x + x g) + m( u + x + x g) + mv (1) 1 1 V = k1x1 + k( x x1) + mgh( u) (13) ( ) ( ) δw = c x δx + c x x δ x x + c x δx (14) Partally dfferentate (1), (13) and (14) by x 1, x and u, and then ncorporate correspondng terms nto the Lagrange s equaton respectvely. The EOMs of the frst, second floors and the ES mass are: ( ) ( ) mx + cx + c x x + kx + k x x = mx (15) g ( ) ( ) ( ) mx + c x x + k x x + m u + x + x = mx (16) 1 1 g g ( ) + ( ) h ( u ) h ( u ) u h u u h u g mu + cu + m = m x+ x + ( ) g (17) Rewrte the ES EOM wth separate force term (,, ) ( ) m u + c u + F u u u = m x + x (18) g The restorng force s easy to recognze. ( ( ) ( ) ( ) ( ) ) F = h u u + h u h u u + h u g m (19) As stated n Track ES Secton, the ES force-dsplacement relatonshp depends on the shape of the track, h( u ). The followng plots n Fg. 4 show the force-dsplacement relatonshps of a track ES wth dfferent polynomal track shapes along wth the force-dsplacement relatonshp of a typcal TMD.

7 TMD restorng force () TMD 1.. TMD dsplacement (m) ES restorng force () 5 h(u ) = a u 5.. ES dsplacement (m) ES restorng force () h(u ) = a u ES dsplacement (m) ES restorng force () 5 h(u ) = a u ES dsplacement (m) ES restorng force () 5 h(u ) = a u ES dsplacement (m) ES restorng force () 4 h(u ) = a u ES dsplacement (m) Fg. 4. Force-dsplacement relatonshps of TMD and ES wth dfferent track shapes A crtcal task n the desgn of an ES s the selecton of the track shape. The second order polynomal track behaves almost lnearly as a TMD, thus t s elmnated from ths study due to our desred to have a restorng force wth an essental nonlnearty. Whle all the hgher order tracks have a nonlnear restorng force relatonshp, n ths study a track wth a shape of fourth order polynomal s chosen. The reason for ths choce s that a track shape of 4 ( ) a u h u = s more practcally realzable than the other shapes due to the less severe curve n the track. Furthermore, to smplfy the analyses and optmzaton of ths system only one-term polynomal track shapes are consdered. Fg. 5 shows the dsplacement responses of the two degree-of-freedom prmary structure wth a track ES utlzng dfferent track shapes. All the tracks depcted n Fg. 5 (The aspect rato of ths fgure has been adjusted such that the horzontal and vertcal dmensons of the track shape are correctly proportoned) produce smlar response reductons; however, some tracks are easer to realze due to lower curvatures. Therefore, the track shape s assumed to be of the form ( ) 4 h u = a u.

8 nd floor dsplacement (m) 1st floor dsplacement (m) h(u ) = 15 u 3 h(u ) = e3 u 4 h(u ) = 5e4 u 5 h(u ) = 1e6 u 6 o ES Fg. 5. Dsplacement response of ES systems wth dfferent track shape orders ES track shape (m) h(u ) = 15 u 3 4 h(u ) = e3 u h(u ) = 5e4 u 5 6 h(u ) = 1e6 u ES dsplacement (m) Fg. 6. ES track shapes consdered (equal scalng on the axes). The EOM of ES mass correspondng to the confguraton usng a fourth order track shape s

9 ( ) ( ) mu + cu + auu + auu + aug m = m x+ x () g h u where a s the coeffcent n the track shape ( ) 4 OPTIMIZATIO = a u. Performance Measures To evaluate the performance of the track ES n reducng the response of the prmary structure, measurements ncludng effectve dampng and storey drft are used. Effectve Dampng measure Due to the nonlnearty n the system, the conventonal lnear dampng rato doesn t apply to the track ES system studed n ths paper. Therefore a measure referred to as effectve dampng s developed to evaluate the apparent modal dampng that results from the addton of the ES to the prmary structure. As s dscussed n Qunn et al. (13), by convertng the resultng response of the lnear prmary structure after the nput has fnshed nto modal coordnates, the mass normalzed sngle-degree-of-freedom EOM for the th mode s: q + λ q + kq = (1) q, q, and q are the th mode dsplacement, velocty, and acceleraton, λ s the nstantaneous effectve dampng n the th mode, and k s the nstantaneous effectve modal stffness of the th mode. The average effectve dampng, λ, can be derved by multplyng the equaton by q and ntegratng over the tme length, from T to T end. Tend Tend Tend q qdt + q λ q dt + q kqdt= () T T T Wth the assumpton that the changes n effectve stffness are small, ths equaton can be smplfed by usng the constant ω nstead of k, where ω s the natural frequency of the th mode. The result of ths ntegraton s: 1 1 T end 1 1 q ( Tend) q ( T) + λ q dt ω q ( Tend) ω q ( T) + = (3) T Solvng for λ yelds: λ = q q q q Tend q dt ( T ) ( T ) + ω ( T ) ω ( T ) end end T (4) The numerator of ths expresson for energy n the th mode at tme t. λ can also be wrtten n terms of E () t, the

10 E T λ = ( ) E ( T ) end Tend q T dt (5) Fnally, the effectve dampng rato can be obtaned va the followng expresson: ζ λ = (6) ω The accumulated rato measure, known as effectve dampng, s an ntegral of the entre record and results n the most representatve effectve dampng of that tme hstory; as a consequence, the value s dependent on the tme of analyss. It s necessary to make sure the length of analyzng tme s long enough to capture the major changes n the responses and the measures of effectve dampng are derved from tme of the same length when they are compared. Ths dampng measure reflects the dampng ablty of the system thus a hgher value of effectve dampng results from a more effcent response reducton. Storey drft measure The other measure used to evaluate the performance of the ES s referred to as story drft measure. Ths measure s the rato of maxmum RMS story drft n the prmary structure when the ES s unlocked and locked as s shown n Eq. (7). A lower value of the storey drft measure corresponds to a more effcent reducton n response, whch s the opposte of the measure of effectve dampng. The value of ths measure depends on the length of the tme record that s used for ts calculaton, especally when the structure s subjected to a transent loadng. Therefore t s the same as the effectve dampng measure that the tme needs to be long enough and kept at a consstent length when the measures from dfferent systems are compared. Storey drft measure max std = max std w/es w/es w/es ( x1 ),std ( x x1 ) w/oes w/oes w/oes ( x1 ),std( x x1 ) (7) Optmzaton Procedure The goal of the optmzaton procedure s to equp the structure wth a track ES wth a mass and track shape that can produce the hghest effectve dampng and smallest normalzed storey drft, when subjected to an mpulsve loadng. Another goal s that the resultng ES system s robust to changes and uncertanty n the prmary structure when the detunng of the prmary structure s taken nto consderaton. The mpulsve load used for ths numercal optmzaton s realzed by applyng an ntal velocty to both floors of the prmary structure and the ES mass, whch resembles the effect on the structure from a sudden ground moton. The prmary load for ths optmzaton s an ntal velocty of.15 m/s. Ths value s chosen so that the response of the structure s n a reasonable range, consderng the test structure on whch the numercal model s based. For ths optmzaton, the ES dampng factor s set at 1.6 s/m, a emprcal value based on prevous studes of relevant ES systems (Werschem et al. 13; Werschem et al. 1; AL-Shudefat et al. 13). The optmzaton ncludes three steps. (1) Choose the ranges of the ES mass and the coeffcent of the track shape that are reasonably realzable and have the potental to produce good performance. The number of dstnct ES systems analyzed for ths optmzaton s the product of the numbers of the ES

11 mass and the numercal value of the track coeffcent to be consdered. For ths optmzaton analyss 19 masses from 1% to 1% of the prmary structure are consdered (Table 1) and 1 track shape coeffcents from 1 to 1 are consdered (Table ), totalng 19 ES systems. Table 1. ES mass range consdered n optmzaton o Mass (kg) Rato to prmary structure 1% 1.5% %.5% 3% 3.5% 4% 4.5% 5% 5.5% o Mass (kg) Rato to prmary structure 6% 6.5% 7% 7.5% 8% 8.5% 9% 9.5% 1% Table. Track shape coeffcent range consdered n optmzaton o Track coeffcent ( 1 3 /m 3 ) () Multple analyses are performed wth each of the systems consdered, so that the effectveness of the ES can be evaluated when the ampltude of the load and frequency of the base structure are changed. By nvestgatng 6 structural stffnesses ncludng the orgnal stffness and fve modfed ones and eght ntal veloctes from.3m/s to.35m/s, there are 48 stffness-load combnatons, and, thus, 48 sets of performance measures are calculated for each ES confguraton consdered. To consder all these combnatons of structural stffness and ntal velocty n the optmzaton, the weghted-average of each measure s used. Whle some ntal veloctes and stffnesses are the prmary focus, some are less lkely to happen, such as hgher stffnesses and hgher loads, or produce relatvely small responses, such as lower loads. Accordngly, the weght of each stffness-load combnaton s confgured to account for the mportance of the combnatons. The hghest weght s gven to the.15 m/s ntal velocty and orgnal stffness combnaton. The weghts for each stffness-load combnaton are lsted n Table 3. Table 3. Weghts for stffness-load combnaton to produce weghted-average of measures used to evaluate ES effectveness Rato to orgnal stff. Intal vel. 7% 8% 9% 1% 11% 1% (m/s)

12 (3) After runnng analyses over all those ES systems, the ES mass and the track shape that produce the maxmum weghted-average effectve dampng measure and the mnmum weghted-average storey drft measure can be dentfed as the optmal values for the track ES system. ote that the optmal mass and track shape may not be the same when consderng the effectve dampng measure or storey drft measure, but they should be confned n a close regon where the these parameters can be chosen from. The weghted-average of the performance measures are shown n Fg. 7. The resultng optmal ES mass s.45 kg, whch s 5% of the prmary structure, and the optmal track 4 shape s hu ( ) = u. These parameters are not the ones that correspond to maxmum effectve dampng measure or mnmum storey drft measure but are wthn the favorable regons of both measures. Ths optmal confguraton s marked wth an X n Fg. 7. Track shape coeffcent Effectve dampng measure ES mass (kg) Track shape coeffcent Storey drft measure ES mass (kg) Fg. 7. Weghted-average of the performance measures of ES systems

13 COMPARISO To evaluate the effectveness of the optmzed track ES system, the response of the ES system s compared to the response of the prmary structure wth the ES locked and a tradtonal TMD system. When ES s locked the system remans lnear wth the mass of ES added to the top floor. TMD System The TMD consdered for ths comparson has the same mass and dampng as the ES system. To provde an accurate comparson, the TMD s stffness s obtaned by conductng a smlar optmzaton procedure as the ES n Optmzaton Procedure Secton. The dfference n ths optmzaton s that the TMD stffness s the only varable nstead of the two varables of mass and track shape for the ES. Addtonally, for ths optmzaton, only the detunng of structure s taken nto consderaton. The load level s not consdered as t has no effect on the response effcency of structures wth the TMD attached due to the lnear response of the system. The weghts assgned to detuned structures are lsted n Table 4. The hghest weght s gven to the orgnal stffness and the lowest weght s gven to the largest ncreased stffness as t s the least lkely to occur. The resultng performance measures of TMD systems are shown n Fg. 8. Table 4. Weghts for stffness Rato to orgnal stff. 7% 8% 9% 1% 11% 1% Weght Effectve dampng measure Storey drft measure TMD stffness (/m) TMD stffness (/m) Fg. 8. Performance measures of TMD systems From the peak value from ths analyss, the complete parameters of the optmzed TMD system are m c k TMD TMD TMD =.45kg = 1.6s/m = 1/m (8)

14 Response Comparsons Fg. 9 shows the dsplacement of the frst and second floors under medum-level mpulsve load (ntal velocty =.15m/s) when no change s made n the structural stffness. Both control systems make a dramatc reducton n dsplacement by 5% wthn the frst 5 seconds. The tme hstores of the mechancal energy contaned n each of the systems are shown n Fg. 1. For the locked system, the total energy decreases only due to the mechansm of the ntrnsc dampng n the lnear prmary structure and by the end of 1 seconds there s only a 1% reducton. In contrast to ths, the ES system s compettve wth the TMD system and both systems are able to reduce the energy n the system below 5% after 1 seconds. However, the ES s able to have a faster effect on the system than the TMD. As early as 3 seconds, the reducton by the ES s 7% compared to 55% made by the TMD. nd floor dsplacement (m) 1st floor dsplacement (m)..1.1 ES system TMD system Locked system Fg. 9. Dsplacements of 1% stffness structures under.15 m/s load

15 .7.6 ES system TMD system Locked system.5 Energy (m) Fg. 1. Energy of 1% stffness structures under.15 m/s load The wavelet transform of square of velocty s shown n Fg. 11 where the frst storey responses of the locked system, the TMD system, and the track ES system under.3 m/s ntal velocty load and wth no stffness changes are compared. As the knetc energy can be wrtten n terms of velocty squared, the amount of energy dspersed among the frequency spectrum s relevant to the densty of the lnes n the wavelet fgure. The hgher load s chosen because the energy transfer s more apparent at a hgher level. In all the wavelet plots, two domnant lnes at about 1.5 Hz and 5 Hz can be seen over the entre tme consdered; these lnes correspond to the natural frequences of the prmary structure. For the locked system, the lne at the lower frequency s much denser than the one at the hgher frequency showng that the structure vbrates manly at ts frst mode. Furthermore, the lnes keep almost the same densty tll the end of the record revealng that very lmted energy s dsspated. The TMD system, although vbrates at the frst frequency, can acheve a sgnfcant energy reducton. In contrast to those two lnear systems, the track ES enables the structure to vbrate more at the second mode transferrng the energy from the lower mode to the hgher mode. The energy also spreads nto frequences other than these two natural frequences showng that the ES s able to resonate wth a wde range of frequency.

16 Frequency (Hz) 1 5 Locked system Frequency (Hz) 1 5 TMD system Frequency (Hz) 1 5 ES system Fg. 11. Wavelet transform of frst storey velocty squared Fg. 1 shows the dsplacement responses under the same load as Fg. 8 and Fg. 9, but the structural stffness of the prmary structure s reduced to 7% of the orgnal stffness. Addtonally, the tme hstores of mechancal energy n each system are compared n Fg. 13. Both ES system and TMD system outperform the locked system as ther responses are less than half of responses of the locked system toward the end of the tme. However, whle the ES system stll keeps a hgh effcacy and delvers a 95% reducton by the end of 1 seconds, the TMD system reduces energy only by 85% as t s no longer n tune wth the prmary structure.

17 nd floor dsplacement (m) 1st floor dsplacement (m) ES system TMD system Locked system Fg. 1. Dsplacements of 7% stffness structures under.15 m/s load.7.6 ES system TMD system Locked system.5 Energy (m) Fg. 13. Energy of 7% stffness structures under.15 m/s load To nvestgate the trends n the response of these control systems wth modfed structural stffness and under varous levels of load, the energy tme hstores are dsplayed n a 3 by 3 matrx form of combnatons of structural stffness and load level n Fg. 14. In these matrx

18 plots, the structural stffness scatters at 5%, 1% and 1% of the orgnal stffness from left to rght n the horzontal drecton, and the load level changes from.7 m/s to.15m/s and to.3 m/s upward n the vertcal drecton. ES system TMD system Locked system 3 5% stffness 3 1% stffness 3 1% stffness.3 m/s load Energy (m) 1 Energy (m) 1 Energy (m) m/s load Energy (m).6.4. Energy (m).6.4. Energy (m) m/s load Energy (m) Energy (m) Energy (m) Fg. 14. Energy of varous stffness structures under dfferent loads Because the nature of TMD systems s lnear, for each column n Fg. 14 where the structural stffness s kept the same, the TMD system has an dentcal percentage of reducton under the dfferent levels of load. In the frst column, the TMD system s only capable of reducng the energy n the system by 4% compared to nearly 95% n the second and thrd columns at 1 seconds. Ths dfference hghlghts the fact that when the TMD s not n tune, ts ablty to effectvely control the structure s lmted. Ths lmtaton exsts especally when the structural stffness decreases whch s lkely to occur after severe damage happens. The effcency of the track ES system on the other hand s less senstve to the changes n frequency of the prmary structure but more depends on the level of load. Comparng the energy tme hstory of the ES system n the mddle row of the matrx to other rows, the mddle ones outperform the rest. Ths s because, n the prevous optmzaton, the hghest weght was gven to ths.15 m/s ntal velocty; consequently, n a sense, ths ES s

19 desgned for ths specfc load. In the upper row where the ntal velocty ncreases, the ES system keeps nearly the same effcacy as the TMD system. In the lower row where the ntal velocty decreases, the ES system outperforms the TMD system when the stffness s reduced but underperforms the TMD system when the stffness s ncreased or unchanged. The measures of effectve dampng and drft rato for varous load-stffness combnatons are shown n Fg. 15. They are consstent wth the energy plots of both control systems. Generally speakng, the performance of the TMD system s unaffected by changes to the load ampltude, but the system shows a performance decay when t s detuned. The track ES system, on the other hand, s more robust to the change of structural stffness and keeps a hgh effcacy among a wde range of stffness when t has been properly desgned. However, the performance of the ES system s load-depended. Effectve dampng Storey drft.3.3 TMD system Intal velocty (m/s) Intal velocty (m/s) Stffness percentage Stffness percentage ES system Intal velocty (m/s) Stffness percentage Intal velocty (m/s) Stffness percentage Fg. 15. Effectve dampng and storey drft measures of varous stffness structures under dfferent loads Force-dsplacement Relatonshp Fg. 16 shows force-dsplacement relatonshps of the ES and the TMD for dfferent load-stffness combnatons. As expected, the force and dsplacement relatonshp of TMD s lnear. Ths lnearty explans why TMD can t fulfll the same standard effcacy for detuned structures. For ES, however, ts force-dsplacement relatonshp s ted to the load level as well as the structural stffness. In small ranges, ES force s bascally of cubc relaton to ts dsplacement. When the dsplacement goes up, some peak behavor s ntroduced. Further

20 nvestgaton s needed on ths behavor whch s unque to the track ES system. However, based on the observaton, the peak behavor prevents the restorng force from becomng excessvely large wth a negatve stffness component. Addtonally, the most effectve response reducton for the track ES n ths study occurs when a slght peak behavor appears, as shown n the 1% structural stffness.15 m/s ntal velocty case. ES system TMD system 4 5% stffness 5 1% stffness 1 1% stffness.3 m/s load Force () Force () Force () Dsplacement (m) 5.. Dsplacement (m) 1.. Dsplacement (m).15 m/s load Force () 1 1 Force () 1 1 Force () m/s load Force ().5.5 Dsplacement (m) Dsplacement (m) Force ().1.1 Dsplacement (m) Dsplacement (m) Force ().1.1 Dsplacement (m) Dsplacement (m) Fg. 16. ES/TMD force-dsplacement relatonshp of varous stffness structures under dfferent loads COCLUSIO Track nonlnear energy snks (ESs), as a new member of ES famly, have been ntroduced n ths paper. Ths ES features an auxlary mass movng on a specally shaped track that can produce nonlnear restorng force. Track ESs are able to dramatcally reduce mpulse nduced responses by ther nonlnearty. Track ESs couple the modes of the prmary structure and transfer energy from lower modes to hgher modes through Targeted Energy Transfer and also consume energy by the movement of ther own. The optmzaton to reduce

21 response of a two degree-of-freedom prmary structure by usng a track ES s reported n ths paper. The results show that the track ES can mtgate the response of the structure consdered to some gven mpulsve loads compettvely compared to an n-tune TMD and, addtonally, reman robust aganst detunng n the structure; however, the effcacy of ES system s shown to be vulnerable to changes n the level of loads. Addtonally, evaluaton of the force-dsplacement relatonshp of the track ES helps to reveal the effect of ts nonlnearty compared to the lnear force-dsplacement relatonshp of the TMD. Ths research demonstrates the potental of nonlnear energy snks to mtgate mpulsve loads on structures. Further study s underway to demonstrate expermentally the results found n ths paper. REFERECE AL-Shudefat, Mohammad A., cholas Werschem, D. Dane Qunn, Alexander F. Vakaks, Lawrence A. Bergman, and Blle F. Spencer. 13. umercal and Expermental Investgaton of a Hghly Effectve Sngle-sded Vbro-mpact on-lnear Energy Snk for Shock Mtgaton. Internatonal Journal of on-lnear Mechancs 5 (June): Bellet, R., B. Cocheln, P. Herzog, and P.-O. Matte. 1. Expermental Study of Targeted Energy Transfer from an Acoustc System to a onlnear Membrane Absorber. Journal of Sound and Vbraton 39 (14) (July): Cocheln, Bruno, Phlppe Herzog, and Perre-Olver Matte. 6. Expermental Evdence of Energy Pumpng n Acoustcs. Comptes Rendus Mécanque 334 (11) (ovember): Fahm Sadek, Bjan Mohraz, Andrew W. Taylor, and Rley M. Chung A Method of Estmatng the Parameters of Tuned Mass Dampers for Sesmc Applcatons. Earthquake Engneerng & Structural Dynamcs 6: Housner, G., L.A. Bergman, A.G. Caughey, and Chassakos Structural Control: Past, Present, and Future. Journal of Engneerng Mechancs 13 (9): McFarland, D. Mchael, Lawrence A. Bergman, and Alexander F. Vakaks. 5. Expermental Study of on-lnear Energy Pumpng Occurrng at a Sngle Fast Frequency. Internatonal Journal of on-lnear Mechancs 4 (6) (July): McFarland, D. Mchael, Gaetan Kerschen, Jeffrey J. Kowtko, Young S. Lee, Lawrence A. Bergman, and Alexander F. Vakaks. 5. Expermental Investgaton of Targeted Energy Transfers n Strongly and onlnearly Coupled Oscllators. The Journal of the Acoustcal Socety of Amerca 118 (): 791. Qunn, D. Dane, Alexander F. Vakaks, and Lawrence A. Bergman. 13. Effectve Stffenng and Dampng Enhancement of Structures Wth Strongly onlnear Local Attachments. Roy R. Crag, and Andrew J. Kurdla. 6. Fundamentals of Structural Dynamcs. nd ed. Wley. Sun, C., R.P. Eason, S. agarajaah, and A.J. Dck. 13. Hardenng Düffng Oscllator Attenuaton Usng a onlnear TMD, a Sem-actve TMD and Multple TMD. Journal of Sound and Vbraton 33 (4) (February): Werschem,. E., J. Luo, S. Hubbard, Fahnestock, L. A., B. F. Spencer Jr, A. F. Vakaks, and L. A. Bergman. Expermental Testng of a Large 9-story Structure Equpped wth Multple onlnear Energy Snks Subjected to an Impulsve Loadng. Werschem,. E., B. F. Spencer Jr, L. A. Bergman, and A. F. Vakaks. 13. umercal Study of onlnear Energy Snks for Sesmc Response Reducton. Werschem, cholas E., Je Luo, AL-Shudefat Mohammad, Sean Hubbard, Rchard Ott,

22 Larry A. Fahnestock, D. Dane Qunn, D. Mchael McFarland, B. F. Spencer Jr, and Alexander Vakaks. 13. Smulaton and Testng of a 6-story Structure Incorporatng a Coupled Two Mass onlnear Energy Snk. Werschem, cholas E., D. Dane Qunn, Sean A. Hubbard, Mohammad A. Al-Shudefat, D. Mchael McFarland, Je Luo, Larry A. Fahnestock, Blle F. Spencer, Alexander F. Vakaks, and Lawrence A. Bergman. 1. Passve Dampng Enhancement of a Two-degree-of-freedom System through a Strongly onlnear Two-degree-of-freedom Attachment. Journal of Sound and Vbraton 331 (5) (December): Werschem, cholas E., Sean A. Hubbard, Je Luo, Larry A. Fahnestock, B. F. Spencer Jr, D. Dane Qunn, D. Mchael McFarland, Alexander F. Vakaks, and Lawrence A. Bergman. Expermental Blast Testng of a Large 9-Story Struture Equpped wth a System of onlnear Energy Snks. In press.