Impact Responses by FEM for a Living Finger Protected by Shock Absorbers Using Nonlinear Complex Springs Connected in Series

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1 Ipact Responses y FEM for a Lvng Fnger Protected y Shock Asorers Usng Nonlnear Coplex Sprngs Connected n Seres Toru Fukusha, a, Takao Yaaguch,, Yusaku Fuj,c, Shnch Maruyaa,d, Akhro Takta,e and Hsanor Tota,f Guna Unversty, -5- Tenjn-cho, Kryu, Guna, Japan a <t4866@guna-u.ac.jp>, <yaage3@guna-u.ac.jp>, c <fuj@guna-u.ac.jp>, d <aruyaa@guna-u.ac.jp>, e <takta@guna-u.ac.jp>, f <t83254@guna-u.ac.jp> Keywords: fnte eleent ethod, nonlnear vraton, vscoelastc ateral. nonlnear coplex sprng Astract. Ths paper deals wth dynac responses of a fnger protected y vscoelastc asorers under pact forces. Restorng forces of a fnger and asorers are easured usng Levtaton Mass Method proposed y Fuj. In ths paper, we carry out nuercal analyss of dynac responses for the fnger protected y the asorers under sae condtons wth the experent usng LMM. The asorers and the fnger are odeled y nonlnear concentrated sprngs usng power seres of ther elongatons. We propose that the sprngs have nonlnear coplex sprng constants to represent changes of the hysteress as ther elongatons progress. A fnte eleent for the nonlnear coplex sprng s newly expressed usng the relatve dsplaceent etween the nodes on the ends of the sprng. The nonlnear coplex sprng for the fnger s nserted etween the nonlnear coplex sprngs for the two asorers. These nonlnear coplex sprngs are connected n seres. Further, these nonlnear coplex sprngs are attached to the levtated lock, whch s odeled y three-densonal fnte eleents. By colldng the lock wth the fnger nserted etween asorers, transent responses of ths syste are coputed. The experental data are copared wth the calculated ones usng our proposed FEM.. Introducton We often use solaton usng concentrated sprngs to protect structures (e.g. electronc apparatus, edcal apparatus and precson apparatus) fro undesrale pacts and external vratons. But, these lghtweght structures soetes do not have hgh rgdty. In such cases, these structures are regarded as elastc odes. Soe concentrated sprngs as solatng eleents have nonlnear relatons etween load and dsplaceent. Thus t s portant to clarfy dynacs for the coupled prole etween elastc odes and nonlnear sprngs. Many researchers have nvestgated for the nonlnear vratons of concentrated asses wth concentrated sprngs. For nstance, Feeny studed ths type of syste usng the proper orthogonal odes (POM) technque []. The dynac responses for large-scale proles, whch conssted of eas supported y nonlnear concentrated sprngs were studed y Kondou. Further, Kondou also proposed an effcent dentfcaton ethod for the nonlnear stalty n large-scale proles [2]. Shaw proposed nonlnear odal analyss, and he appled t to a sply supported ea connected to a nonlnear concentrated sprng [3]. Prevously, we proposed a fast nuercal ethod to calculate the nonlnear vratons n an elastc lock or a vscoelastc lock wth a nonlnear concentrated sprng. In ths study, one end of the nonlnear sprng s attached to the ground [4,5]. Moreover, we extended the proposed ethod to deal wth the dynac phenoena of vscoelastc locks connected wth a nonlnear concentrated sprng havng a restorng force expressed as functons of the relatve J. Tech. Soc. Sc., Vol., No.3, 27

2 dsplaceent etween the two ends of the sprng [6]. In the nuercal analyss, the dscrete equatons n physcal coordnate are transfored nto the nonlnear ordnary coupled equatons usng noral coordnate correspondng to lnear egenodes. In ths calculaton, odal dapng s also transfored. The transfored equatons are ntegrated nuercally n extreely sall degree-of freedo. Soe shock asorers often have nonlnear hysteress n ther dynac characterstcs. For exaple, we oserved phenoena that the hysteress ncreases as the elongaton of the shock asorers ncrease. However, there are few reports to study coupled vraton etween elastc structures and the shock asorers havng nonlnear hysteress. We extend our proposed ethod to deal wth vraton analyss usng fnte eleent ethod for elastc structures ncludng nonlnear concentrated sprng wth nonlnear hysteress [7]. The restorng force of the sprng s expressed as power seres of ts elongaton. The restorng force also nvolves nonlnear hysteress dapng. Thus, coplex stffness s ntroduced for not only the lnear coponent ut also nonlnear coponents of the restorng force. Fnte eleent for the sprng s expressed and s attached to elastc structures odeled y lnear sold fnte eleents. The dscrete equatons n physcal coordnate are transfored nto the nonlnear ordnary coupled equatons usng noral coordnate correspondng to lnear egenodes. In ths calculaton, odal dapng s also transfored. The transfored equatons are ntegrated nuercally n sall degree-of freedo. Ths nuercal ethod s appled for a fnger protected y asorers under pact forces as follows. To protect huan odes such as ars [8,9], fngers and legs fro pacts n echancal apparatuses, uldngs or transporters when doors are closed, vscoelastc ruers are soetes nserted etween the doors and the fraes around the doors. The ruers have rolls of shock asorers to decrease the pacts usng vscoelastcty. These ruers have any knds of shapes to prove the perforance of shock asorers. In one of the, there are thn ruers havng hollow cross sectons. For these types of the shock asorers, ucklng phenoena are soetes utlzed to decrease pacts effcently. Thus, vscoelastc asorers have nonlnear restorng force under relatvely large load. The vscoelastc asorers soetes have nonlnear hysteress n ther dynac ehavors. Thus, t s of portance to clarfy nonlnear dynac characterstcs of vscoelastc shock asorers wth elastc structures under pact load. We apply our proposed ethod usng Fnte Eleent Method wth the nonlnear coplex sprngs to ths prole. In the nuercal ethod, the vscoelastc asorers are odeled as the nonlnear sprngs wth nonlnear hysteress. The restorng force of the sprng s expressed as power seres of ts elongaton. The restorng force also nvolves nonlnear hysteress dapng. By colldng a lock wth a par of vscoelastc shock asorers wth/wthout a lvng fnger, veloctes of the lock are oserved usng Levtaton Mass Method proposed y Fuj [7]. The experental data [7] are copared wth the calculated results usng our proposed FEM. 2. Outlne of Experental Results Usng the Levtaton Mass Method, velocty of the lock s oserved y colldng a lock wth a par of vscoelastc shock asorers. A lvng fnger s set etween the two asorers. And the velocty of the lock s also oserved. Usng the veloctes, we studed the restorng forces. Fg. A par of vscoelastc shock asorers J. Tech. Soc. Sc., Vol., No.3, 27

3 Fg.2 Experent setup [7] Fgure shows the cross sectons of the par of the vscoelastc asorers. The shape of the asorer s tue wth a thckness of 3[]. As shown n Fg., the cross sectons of the asorer have D-type shapes. The asorers are ade of foa ruer. Fgure 2 shows outlne of the experental setup usng Levtaton Mass Method proposed y Fuj [7]. Ths experental syste has a lock, whch can soothly ove along a gude n z drecton usng a pneuatc lnear earng. Therefore, the lock s levtated y ar fl havng 8[ ] thckness at the nterfaces etween the gude and the lock. By usng ths syste, the lock can travel toward z drecton under low frcton. One of the vscoelastc shock asorers s connected wth a rgd ase. Another vscoelastc asorer s attached to the levtated lock. The lock s pushed n negatve z drecton fro the ntal poston z= y a huan hand. Then, wth ntal velocty v, the lock s collded wth the par of the vscoelastc shock asorers. After the collson, velocty of the transent response for the lock s oserved usng an optcal nterferoeter. A corner cue s farcated on the levtated lock to receve laser ea for the nterferoeter. Acceleraton of the lock can e calculated y dfferentaton of the oserved velocty. By ntegratng the easured velocty, poston of the lock s also deterned. Inerta force can e calculated y product of the ass of the lock and acceleraton. And the nerta force s converted to restorng force y changng ts sgn. Then, hysteress curve can e otaned usng the relaton etween the poston and the restorng force. Fgure 3 shows oserved restorng forces. The ordnate s restorng force, and ascssa shows dsplaceent. The red and green lnes show the results of the asorers wthout the fnger. Under sall ntal velocty, the restorng force n for the red lne has a characterstc of softenng type sprng. And ts hysteress n the restorng force s ncreasng as the dsplaceent changes fro [] to 5[]. Under condton of large ntal velocty, the slope of the green curve suddenly changes to have harder sprng constant when the dsplaceent s 9[]. Whle the dsplaceent progresses fro 9[] to [], the hysteress n the green lne decrease. These ehavors are caused y the ucklng phenoena of the shock asorers. Note that there s no peranent deforaton when the hollow parts of the D-shaped asorers are uckled. The lue lne represents the result of the shock asorers wth the huan fnger. The slope of ths curve also changes when the dsplaceent s aout 9[]. The lue curve also shows soft-hardenng characterstcs n the restorng force. Whle the hysteress decreases fro 9[] to 4[], the hysteress n the lue lne ncreases fro [] to -9[]. The dynac characterstcs of the asorers wth/wthout the fnger are slar. But, the restorng force for the vscoelastc asorers wthout the fnger has J. Tech. Soc. Sc., Vol., No.3, 27

4 sharper change than that for the asorers wth the fnger. We regard that ths phenoenon s due to elastc deforatons of the lvng fnger. Fg.3 Experental results of restorng forces for vscoelastc shock asorers wth/wthout a fnger [7] 3. Nuercal analyss Consderng the experental results n the prevous chapter, the followng ehavors for the restorng forces of the vscoelastc asorers wth/wthout the lvng fnger are suarzed. (a) There are ehavors as nonlnear sprngs. () The hysteress ncreases when the deforaton of the sprngs ecoe large. The hysteress nversely decreases after the deforaton ecoe larger. (c) Near the regular deforatons, the slopes of the curves n the restorng forces ncrease. The phenoena (a) and () are coon ehavors for the asorers wth/wthout the fnger. The phenoenon (c) appears the vscoelastc asorers wth/wthout the fnger under large deforaton. The change of the stffness for the asorers wth the fnger s saller than that for the vscoelastc asorers wthout the lvng fnger. To sulate these phenoena appeared n the experent, we use the FEM odel as shown n Fg.4. We use two nonlnear concentrated sprngs havng nonlnear hysteress for the par of the vscoelastc shock asorers and one nonlnear sprng for the fnger to sulate the restorng forces n Fg.3. We propose a fnte eleent for sprngs havng nonlnear coplex sprng constants. The nonlnear coplex sprng of the fnger s connected n seres etween the two nonlnear coplex sprngs of the asorers. These sprngs are attached to the levtated lock usng three-densonal sold fnte eleents. Detal nuercal procedures are shown n the followng sectons. Fg.4 Sulaton odel J. Tech. Soc. Sc., Vol., No.3, 27

5 3. Dscrete Equaton for Nonlnear Coplex Concentrated Sprngs (Vscoelastc Shock Asorers and a Lvng Fnger) The par of the vscoelastc shock asorers s odeled y usng two concentrated nonlnear sprngs wth nonlnear hysteress. For the fnger, we also use another nonlnear sprng wth nonlnear hysteress. We assue that the -th nonlnear concentrated sprng wth vscoelastcty for the has prncpal elastc axs n z drecton. The -th nonlnear concentrated sprng s connected etween the nodal ponts and.we denote dsplaceents as U n z drecton at the nodal U and z z ponts and, respectvely, Nonlnear restorng force R of the -th sprng usng power z seres s gven for nodal force at the pont. Thus, the restorng force R s expressed usng the z relatve dsplaceent U z U z as 2 3 R z z U z U z 2z U z Uz 3zU z Uz, R R R R. Further, lnear hysteress dapng s ntroduced as,, x y x z R z R y z z j z. z s the real part of z, and z s the ateral loss factor of the concentrated sprng. j s the agnary unt. Moreover, nonlnear hysteress dapng s also ntroduced as 2z 2z j2z, 3z 3z j3zand 4z 3z j4z n the sae anner. 2 z, 3z and 4zare the real part of 2 z, 3zand 4z, respectvely. 2 z,3z and 4zare the nonlnear coponents of ateral loss factor for the concentrated sprng, respectvely. These relatons can e rewrtten n the atrx for as: R U d () Where, R s the nodal force vector at the node. U s s the nodal dsplaceent vector at the node. s the coplex stffness atrx ncludng only lnear ter of the restorng force. d s the vector nvolvng nonlnear ters of the restorng force. 3.2 Dscretzed equatons of an elastc structure s It s assued that equatons of oton for the levtated lock are expressed under sall deforaton. We consder that dapng of the levtated lock usng coplex odulus of elastcty E E j ). ( The real part E of the E represents the Young s odulus of elastcty. s the ateral loss factor of the lock. All eleents related to the structure are superposed, the equatons n the entre doan of the levtated lock are otaned as: F,U,M and M U K U F (2) Where, K are the force vector, the dsplaceent vector, the ass atrx and coplex stffness atrx, respectvely. We anly use soparaetrc hexahedral eleents wth non- conforng odes [,] for the later nuercal coputaton. 3.3 Dscrete equaton for coned syste etween a lock and vscoelastc sprngs The -th restorng force R n Eq. () for the nonlnear coplex sprngs are added to the correspondng nodal forces at the nodal ponts and. We also add the restorng force at the attached node etween the nonlnear concentrated sprng for one of the vscoelastc shock asorers and the levtated lock. The followng equaton n the gloal syste can e otaned: M U K U d ˆ F (3) J. Tech. Soc. Sc., Vol., No.3, 27

6 Where, U, F, M and K are the dsplaceent vector, the external force vector, the ass atrx and the coplex stffness atrx n the gloal syste, respectvely. dˆ s the nonlnear coplex restorng force of the nonlnear sprngs n seres for the asorers and the fnger. dˆ has the dentcal vector sze to degree- of- freedo of Eq.(3). 3.4 Converson fro Dscrete Equatons n Physcal Coordnate to Nonlnear Equatons n Noral Coordnate Long coputatonal te requres to copute Eq. (3) n physcal coordnate, drectly. In ths secton, a nuercal procedure s explaned to decrease the degree-of-freedo for the dscrete equatons of oton. () Frst, t s assued that lnear egenodes { } of vraton can e approxated to the real ( ) egenodes { }. Next, y ntroducng noral coordnates correspondng to the lnear ( ) egenodes { ( ) }, the nodal dsplaceent vector can e expressed usng oth { } and [2]. U (4) Susttutng of Eq.(4) nto Eq.(3), the followng nonlnear ordnary sultaneous equatons as to noral coordnates can e expressed as: ( ) ( ) 2 D E P,,j,k,l =,2,3, tot jk j k jkl j k l F 2z j k j T P, x y z x y,, x, y, z,,,,, 2, 2 x, y, z, D, jk E jkl 3z k l T z z z z jz kz lz ( ) jz kz jz kz lz jz kz jz kz lz jz kz jz kz lz jz kz lz tot jz kz jz kz lz jz kz lz jz kz lz In Eq. (5), s the -th natural frequency. s the -th odal loss factor. Because Eq. (5) has uch saller degree-of -freedo than that of Eq. (3), we can save coputatonal te. In Eq. (5), dot stands for partal dfferentaton wth respect to te t. z s the z-coponent of the egenode at the attached nodes of the nonlnear coplex sprngs. 3.5 Nonlnear Ipact Response Nonlnear pact responses are coputed usng Runge-Kutta-Gll ethod to Eq.(5). In the nuercal ntegraton, we gve an pact for the force vector F n Eq. (5). (5), 4. Nuercal results and dscusson The sae ntal veloctes as the experental values n Fg.3 are gven to the sulaton odel of the levtated lock. We coputed the restorng forces usng the sulaton odel as shown n Fg.4 y colldng the lock wth the vscoelastc shock asorers wth/wthout the fnger. In ths fgure, the levtated lock s odeled as an elastc ody usng fnte eleents. The par of the vscoelastc shock asorers and the fnger are odeled usng nonlnear coplex sprngs to consder nonlnear dapng. We set an orgn of ths odel on the poston where the levtated lock egns to contact wth the par of the asorers wth/wthout the fnger. Frstly, the real part of the nonlnear sprng J. Tech. Soc. Sc., Vol., No.3, 27

7 constants are dentfed usng ackone curves fro the experent. Next, we deterne the lnear/nonlnear loss factors n the proposed nonlnear coplex sprng constants y fttng curves of the nonlnear restorng forces. 4. Nuercal Results for a Par of Shock Asorers wthout a Fnger The followng nonlnear coplex sprng constants are dentfed for the restorng forces of each asorer of two wthout the fnger. We use the nonlnear sprng constants fro the frst to the seventh coponents of power seres. 3 6 z. [N/], z. 2 [-], 2z.5 [N/ 2 ], 2z. 5 [-], 9 3z 2.2 [N/ 3 ], 3z. 6 [-], 4z 6.8 [N/ 4 ], 4z. 6 [-], 3 5z 3.3 [N/ 5 ], 5z. 5 [-], 6z. [N/ 6 ], 6z. [-], 8 7z.5 [N/ 7 ], 7z. 9 [-] Fgure 5 shows the calculated restorng forces of the par of the vscoelastc shock asorers wthout the fnger n coparson wth the experental ones. In the fgure, the red curves show the results under sall ntal velocty, whle green curves show the results under the large ntal velocty. Both results are consstent qualtatvely etween the experental curves and the calculated curves. In specal, the slopes of the experental/calculated curves ncrease near the deforaton 9[]. The changes of the hysteress n the curves can also e coputed. Note that these characterstcs cannot e reproduced usng the only lnear ter of hysteress dapng. We can confr the consstency etween the experent and the nuercal analyss y ntroducng the nonlnear coplex sprngs n seres.. Fg. 5 Hysteress curves of a par of vscoelastc shock asorers wthout a fnger fro FEM 4.2 Calculated Results for a Par of Shock Asorers wth a Fnger Next, to deterne the paraeters for the lvng fnger, we copute the restorng forces of the shock asorers wth the fnger y varyng the nonlnear coplex sprng constants of the fnger. For ths coputaton, we used the prevously deterned nonlnear coplex sprng constants for the asorers. We connected these nonlnear coplex sprngs n seres as shown n Fg.4. Fnally, we dentfed the followng nonlnear coplex sprng constants for the restorng forces of the lvng fnger. We use the nonlnear sprng constants fro the frst to the seventh coponents of power seres as follows. 2 z 7.5 [N/], z. 2 [-], 2z. [N/ 2 ], 2z. [-], 3z. [N/ 3 ], 3z. [-], 4z. [N/ 4 ], 4z. [-], J. Tech. Soc. Sc., Vol., No.3, 27

8 2 5z 4.5 [N/ 5 ], 5z. [-], 6z. [N/ 6 ], 6z. [-], 7 7z 3.4 [N/ 7 ], 7z. [-] Fgure 6 shows coparson etween the experental restorng curve of the shock asorers wth the lvng fnger under an ntal velocty =-.249[/s] and calculated ones usng these paraeters. As can e seen n the Fg 6, the coputed and the experental curves agree well. Consequently, we confr that our proposed ethod s vald. The changes of the slopes appear oth n the experental and calculated curves near the deforaton = -9.[]. These changes of the slopes are less than those n the restorng forces of the asorers wthout the fnger dscussed n the prevous secton 4.. We thnk ths phenoenon s due to the elastcty and vscoelastcty of the lvng fnger. The rgdty of the fnger s larger than the vscoelastc shock asorers under sall deforaton. Intally, the par of the soft vscoelastc asorers defors, and then the asorers are copressed. Due to the copresson, the stffness of the asorers ecoe larger than the stffness of the fnger. Then, the fnger defors. Fg. 6 Hysteress curves of a par of vscoelastc shock asorers wth a fnger fro FEM 4.3 Nuercal analyss to study dynac errors of experental syste usng LMM In ths secton, to check dynac errors of the levtated lock, we study accuracy of the easureent syste usng the Levtaton Mass Method. To study undesrale dynac ehavors for ths syste, we nvestgated consstency etween dsplaceent D at the corner cue and the dsplaceent D at the center of gravty of the lock. Fro the vewpont of hgh precson g easureent, the dsplaceents Dc and D g ust have the sae values. If rgd otons except the otons n z drecton or elastc deforatons of the lock ncrease, the undesrale dfferences etween D and D wll ncrease. We otaned the rato D / when the c g c D g dsplaceents reach up to the local axal values at t=.6 [sec]. Fro ths rato, we can regard that the undesrale dynac coponents of the levtated lock are neglgle sall. Thus, we confr that all of the restorng forces n chapter 4 are free fro the dynac errors due to the undesrale ehavors of the lock n the easureent syste. 5. Concluson By ntroducng the proposed coplex nonlnear sprng eleents connected n seres nto nuercal analyss usng FEM, we copute pact responses of a huan lvng fnger protected y vscoelastc shock asorers. We can reproduced the experental restorng forces of the par of the shock asorers wth/wthout the fnger. We can sulate the change of hysteress due to ther deforaton. c J. Tech. Soc. Sc., Vol., No.3, 27

9 Acknowledgeents Journal of Technology and Socal Scence (JTSS) Ths work was supported y Grant-n-Ad for Scentfc Research (C) (KAKENHI ). References [] B. F. Feeny and R. Kappagantu, "On the Physcal Interpretaton of Proper Orthogonal Modes n Vratons", Journal of the Sound and Vraton, Vol.2, pp.67-66, 998. [2] T. Kondo, T. Sasak, and T. Ayae, "Forced vraton analyss of a straght-lne ea structure wth nonlnear support eleents", Transactons of the Japan Socety of Mechancal Engneers, Vol.67, No.656C, pp.94-92, 2 [3] E. Pesheck, N. Bovn, C. Perre and S. W. Shaw, "Non-Lnear Modal Analyss of Structural Systes Usng Mult-Mode Invarant Manfolds", Nonlnear Dynacs, Vol.25, No., pp.83-25, 2. [4] T. Yaaguch, Y. Fuj, K. Naga and S. Maruyaa "FEM for vrated structures wth non-lnear concentrated sprng havng hysteress ", Mechancal Systes and Sgnal Processng, Vol. 2, pp , 26. [5] T. Yaaguch, T. Sato, K. Naga, S. Maruyaa, Y. Kurosawa and S. Matsuura, "Analyss of daped vraton for a vscoelastc lock supported y a nonlnear concentrated sprng usng FEM ", Journal of Envronent and Engneerng, Vol. 2, No.3, pp , 27. [6] T. Yaaguch, K. Naga, S. Maruyaa, T. Sato, Y. Kurosawa and S. Matsuura, " Analyss of daped Vraton for a vscoelastc locks connected y a nonlnear sprng wth lnear hysteress usng FEM descred y noral coordnate", Transactons of the Japan Socety of Mechancal Engneers, Vol.7, No.73C, pp , 25. [7] T. Yaaguch, Y. Fuj, T. Fukusha, T. Toota, A. Takta, K. Naga and S. Maruyaa "Dapng response analyss for a structure connected wth a nonlnear coplex sprng and applcaton for a fnger protected y asorers under pact forces", Mechancal Systes and Sgnal Processng, Vol. 42, pp.88-96, 24. [8] T. Yaaguch, T. Kana, Y. Fuj, K. Naga and S. Maruyaa, " Nuercal analyss of dynac responses for a huan ar that receves pact forces", Transactons of the Japan Socety of Mechancal Engneers, Vol.76, No.772C, pp.92-98, 2. [9] Y. Fuj and T. Yaaguch, "Ipact response easureent of a huan ar". Experental Technques, May/June, pp.64-68, 26. [] O. C. Zenkerwcz and Y. K. Cheung, The Fnte Eleent Method n Structural and Contnuu Mechancs, McGraw-Hll, 967. [] E. L. Wlson, R. L. Taylor, W. P. Doherty and J. Ghaouss, Incopatle Dsplaceent Methods, n Nuercal and Coputer Methods n Structural Mechancs, Acadec Press, 973. [2] T. Yaaguch and K. Naga, "Chaotc vraton of a cylndrcal shell-panel wth an n-plane elastc support at oundary", Nonlnear Dynacs, Vol.3, pp , J. Tech. Soc. Sc., Vol., No.3, 27