The key Laboratory of Mechanical Equipment Manufacturing & Control Technology, Guangdong University of Technology, Guangzhou, , China

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1 Smultaneous Optmal Desgn of Topology an Sze for a Flexure-Hnge-Base Gung Mechansm to Mnmze Mass uner Stffness an Frequency Constrants Zhjun Yang *, Youun Ba, Xn Chen The key Laboratory of Mechancal Equpment Manufacturng & Control Technology, Guangong Unversty of Technology, Guangzhou, , Chna * Corresponng Author: Zhjun Yang, Professor, Ph.D., IEEE member, Emal:yangzj@gut.eu.cn The gung mechansm, base on flexure hnges (FHs), s wely use n mcro/nano manufacturng technology. Both ts stffness an frequency play a sgnfcant role n ynamc performance, so the esgn task of such a structure s to fn an optmal topology an corresponng sze of the FHs uner the stffness an frequency constrants. However, the exstng optmzaton methos pay more attenton to the stffness rather than frequency constrant ue to the ffcultes n ynamc topology optmzaton. In ths paper, wth the symmetrc layout assumpton of the FHs an the analytcal equvalent stffness an mass expresson of a sngle FH, the smultaneous topology an sze optmzaton problem s converte to an analytcal optmzaton formula wth both screte an contnuous varables. Fnally, the tenson stffenng effect s use to compensate the manufacturng errors. A esgn case s use to llustrate the effcency of the propose metho. Key works: smultaneous optmzaton, flexure hnge, gung mechansm, topology an sze, stffness an frequency constrants 1

2 1. Introucton The FH base gung mechansm, such as fast tool servo (FTS), s wely use n mcro/nano precson engneerng, ue to ther goo lnearty of stffness. The major esgn avantage of FHs for ths applcaton s the absence of backlash an frcton n the recton of the moton (Karnopp 1985). Ths proves very smooth, hgh precson operatng characterstcs wthout nucng event wear whch s commonly assocate wth hgh spees or contnuous operaton (Smth 000; O'Bren 005). In orer to obtan the optmal esgn matchng the esre goal, many optmzaton methos have been nvestgate an mplemente n the lterature (Zhu et al. 011; Zhuang, Xu, an Xong 01; Jn et al. 01; Zhu, Zhang, an Fatkow 014a, 014b; N et al. 014; Torstenfelt an Klarbrng 007; Zhou et al. 004). Many optmal esgn methos of the gung mechansm have been propose. One metho s sequental optmzaton of topology an sze (Lum et al. 015). The optmal esgn s lmte by the fxe topology at the frst stage. Another one s topometry optmzaton (Mozumer, Renau, an Tovar 01) n a two-mensonal area, where the thckness of the shell elements can be optmze, an the thnnest shell elements are remove from the fnal topology. However, t s very ffcult to utlze the varable thckness for FHs. The typcal routne s to frstly perform topology optmzaton, an then optmze the cross-secton sze of the FHs. Recently, many ntegrate sze an topology optmzaton methos have been propose. The objectve s to fn the optmal topology an sze at the same tme. Integrate sze an topology optmzaton uner frequency constrants wth the objectve of mnmzng the total structural weght has been use to obtan an optmal skeletal structure (N et al. 014). Furthermore, the smultaneous optmzaton of sze, shape, an topology s presente n some papers, such as the metho propose by Torstenfelt for esgnng moular car famles (Torstenfelt an Klarbrng 007). Zhou presente an ntegrate approach of sze, shape, an topology optmzaton n a sngle teratve process wthn the commercal coe Altar/OptStruct an extene t nto a general structural optmzaton tool that proves sze, shape, an topology optmzaton capabltes (Zhou et al. 004). In gung mechansm, especally use n precson engneerng, both the stffness an frequency of a gung mechansm play a sgnfcant role n ts ynamc performance. When takng frequency nto conseraton, the ffculty of optmzaton arses as the local moes of the elements wth low ensty materal exsts. In aton, optmzaton metho can only hanle some orer moes, but the moal shapes are changng wth structural mofcaton all the tme. Moreover, n the optmal esgn, wth fferent orgnal structural szes, the corresponng topology may be fferent, whle the exstng sequental optmzaton of topology an sze cannot hanle ths kn of problem. Therefore, smultaneous optmzaton formula nees to be set up. Fortunately, n many stuatons, the FHs are symmetrcally layout bese the platform. The structural esgn of the gung mechansm can be shown n Fgure 1,

3 where the work platform s rven by a voce col motor, an supporte by the FHs n the esgn oman. On the other han, the analytcal methos are presente for calculatng the complance or stffness matrx of flexure hnges (Schoenen, Ivanov, an Corves 015), whch enable ones to bul the analytcal moellng of the optmzaton problem. Note that the stffness s the hgh orer functon of geometrc sze, therefore, the manufacturng error wll verse the performance of the gung mechansm (Hu et al. 015). We wll ajust the stffness of platform by usng tenson stffenng effect (Yang et al. 016). The goal of ths paper s to obtan the smultaneous optmal esgn of topology an sze of a FH base gung mechansm. Frstly, we escrbe the optmal esgn problem of gung mechansm an convert t nto a smple esgn by etermnng how many groups of FHs are neee base on symmetrc layout assumpton. Seconly, the commonly use FHs are scusse an selecte to esgn the gung mechansm. Thrly, the analytcal expresson of equvalent stffness an equvalent mass of a sngle FH s erve, such that the analytcal optmzaton formula s bult as a mxture of screte an contnuous varables. Fnally, a relaxaton metho s presente to solve the complex optmzaton formula. For more precse stffness an frequency constrants, the tenson stffenng effect s use to compensate the errors cause by manufacturng. A esgn case s employe to emonstrate the effcency of the presente metho.. Problem escrpton In engneerng applcatons, a gung mechansm s usually rven by a pezoelectrc or voce col motor (shown n Fgure 1(a)). The stffness n the rvng recton (from left to rght) s etermne by the maxmum rven force over the maxmum stroke range requre. The requre resonant frequency s etermne by the ynamc amplfcaton factor uner the exctaton frequency, an the eformaton n the payloa recton (From outse to nse) shoul be lmte, or the corresponng stffness must be greater than the prescrbe value. So the common esgn task s to fn the optmal topology an sze of the FHs to meet the prescrbe stffness an frequency n the rvng recton, as well as the stffness n the payloa recton, whle mnmzng the total mass, that s: fn(topology an sze) to mnmze subject to : k f k p m k f * k * * p (1)

4 where subscrpts an p represent rvng an payloa rectons, superscrpt * ncates the gven objectve value, m, k an f are mass, stffness an frequency, respectvely. As mentone n the frst secton, t s ffcult to fn optmal topology an sze soluton uner stffness an frequency constrants at the same tme. Fortunately, the layout of FHs s usually symmetrcal on both ses of the work platform. Therefore, the esgn task s to etermne the amount of FH groups an cross-secton sze of each FH (shown n Fgure 1(b)) n the esgn oman. Fgure 1 Gung mechansm rven by voce col motor Usually, there are two types of translatonal FHs (shown n Fgure ) use n a gung mechansm, namely leaf sprng type an notch type, respectvely (Zhu, Zhang, an Fatkow 014a). Fgure Two prmtve types of translatonal FHs Most esgns aopt the notch type FH for the gung mechansm (Lu et al. 01; Zhu, Jun, an Altntas 001; Tan, Shrnzaeh, an Zhang 009; Ma, Hu, an 4

5 Zhang 005; Wang an Yang 01; Luwck et al. 1999; Km, Km, an Song 004; Km et al. 009; Km an Km 00; Cuttno, Mller, an Schnstock 1999). For example, the mechansm esgne by Lu uses four parallel FHs stanng on the base as the gung mechansm an the tool holer assemble on the top (N et al. 014). However, as the notch type FHs are prone to stress concentraton an fatgue eformaton (Chen, Wang, an Lu 014). Rakuff an Cuttno esgne a mechansm usng a curve leaf flexure as a gung mechansm. When the length of the leaf flexure s approxmately 50mm, the maxmum splacement range of the cuttng tool can reach mm (Rakuff an Cuttno 009). The leaf sprng type of FH (regare as beams) has many avantages, such as no stress concentraton, unform eformaton, an hgh rato of payloa stffness to rvng stffness. The authors have esgne a ynamc-feature-ajustable mechansm usng the leaf sprng type FHs as the gung mechansm an frequency ajustng mechansm (Yang et al. 01), whch s able to elmnate the effect of manufacturng errors. When we choose leaf sprng type FH to esgn the gung mechansm, the optmal esgn problem becomes: How many groups of FHs (screte topologcal varable) are requre? What s the cross secton sze of each FH (contnuous szng varable)? Fortunately, the objectve an constrants functons of such a structure can be analytcally expresse accorng to materal mechancs.. Smultaneous optmzaton of topology an sze Snce the gung mechansm can only move n the recton of the elastc eformaton, t can be regare as a one egree of freeom sprng an mass ynamc system. The work platform can be regare as a rg boy, an all the FHs are n parallel layout (as shown n Fgure ). The total stffness can be calculate by the sum of the equvalent stffness of each FHs, an the total mass s the sum of equvalent mass of each FHs, the mass of col assemble an work platform. Conserng a sngle leaf sprng type of FH, t can be regare as a beam uner clampe-gue constrants (shown n Fgure ), an the equvalent stffness an mass are calculate accorng to the eformaton of the clampe-gue beam. 5

6 Fgure. Deformaton of the leaf sprng type FH n moton state.1 Moelng of a sngle FH uner clampe-gue constrants.1.1 The equvalent stffness Assume that the length s L, when the concentrate force F s apple to the gue en (shown n Fgure 4), the benng eformaton shape functon y x an the maxmum eformaton ymax are (Yang et al. 015) y x F x Lx EI, y max FL () 1EI wt I () 1 where E s the elastc moulus of the use materal, I s the moment of nerta of the secton, t an w are the thckness an wth of the cross-secton area, respectvely. Therefore, the equvalent stffness n the rvng recton s k F Ewt t Ew ymax L L (4) Accorngly, the equvalent stffness n the payloa recton s 6

7 k p Etw w Et L L (5) x F y L Fgure 4. The benng eformaton of a clampe-gue beam.1. The equvalent mass Usually, the frequency s obtane by moal analyss wth the structural stffness an mass. However, both the stffness an mass are the functons of the topology an sze parameters of the gung mechansm, an t s qute ffcult to perform moal analyss urng topology optmzaton. Fortunately, f we know the moal shape s, we have the frequency ( ) expresson accorng to the Raylegh-Rtz analyss T T sks sms (6) where K an M are stffness an mass matrces, respectvely. The splacement of the gung mechansm s euce by the benng eformaton of the leaf sprng type FHs. If we ntrouce the equvalent stffness sks an equvalent T mass sms of the moton stage, respectvely, we have the followng two equatons T sks ky (7) T e max sms my (8) T e max where k e an m e are equvalent stffness an equvalent mass, respectvely. 7

8 y x If we substtute the corresponng benng eformaton shape functon F x Lx EI shown n Eq.() (Yang et al. 015) to Eq.(8), we have e 7 L L F x Lx 1wtF L max ( ) t x 0 0 (9) my wtyx x w EI 5040E I From Eq.(9), we can see that the equvalent mass s proportonal to the total mass of the leaf sprng type FH. Assumng that the coeffcent factor s c, then we have m cm cwtl (10) e Substtutng Eq.(10) nto Eq.(9), we have wtf L wtf L c E wtly E FL wtl 1EI I max 5040 I 5 (11). The analytcal optmzaton moel For the symmetrcal layout of FHs, assume that the number of the FH groups s n, the total equvalent stffness woul be n tmes of a sngle FH, so that the stffness n rvng recton s k newt (1) L Smlarly, the stffness n the payloa recton s k new t p (1) L The resonant frequency n rvng recton s f 1 k m (14) t 8

9 However, the total mass m t s fferent from equvalent mass of n FHs. It s the sum of the mass of voce col assemble, work platform an the equvalent mass of all FHs, respectvely. mt m0 ( cnlt a) w (15) where m 0 s the mass of col assemble n the voce col motor, c s the coeffcent factor of equvalent mass, a s the equvalent cross-sectonal area of the work platform. For nstance, f the cross-secton of the work platform s a sol cubo wth length Ls an wth Bs, then a Ls Bs. If t s hollow rectangle wth the wall thcknessts, then a (( LsBs) Ts Ts Ts). As a result, the analytc optmzaton moel becomes fn( nwt,, ) to mnmze m ( cnlta) w subject to: 0 0 newt * k L k ( f ) m ( cnlta) w netw L k * p * (16) Wth analytcal moellng, the smultaneous optmzaton of topology an sze s converte nto a complex moel combne wth screte topology (group number of FHs) an contnuous sze (cross-secton sze) varables.. Determne the cross-secton sze from equalty constrants Usually, the rvng stffness an frequency nee to be precsely satsfe, whle the payloa stffness shoul be hgher than the gven value. Solvng Eqs.(1) an (14), we have the wth w an thckness t of the cross secton. / LcBC AB C E L 1 4 t (17) EnB ABC 1 w 4 B f Lcnt a (18) 9

10 Where, A 9a 7Ea k 64L c n f k 108E a f m 6 E k 4 f m0 0 B k 4 f m 0 C k n E f FHs. Wth fferent group number of FHs, we have fferent cross-secton sze of.4 Choose the group number of FHs Accorng to Eq. (4) an (5), the stffness rato of rvng an payloa s proporton to the square of the rato of thckness to wth. p k t t k k w w k p (19) Whch means the rato tw can be etermne by the square root of the rato of rvng stffness to the payloa stffness. Wth fferent group number of FHs n, we have corresponng secton parameters w an t from Eq. (17) an (18). The relatonshp among n, w an t s complcate. Fortunately, w ncreases whle t ecreases wth n, so t can be evaluate wth a fferent group number n. For the feasblty of the mcro moton stage, the mnmum group number of FHs s, but smaller than a maxmum value n max teraton proceure s as follows: Step 1: Let n n. O Step : Calculate t an w usng Eqs.(17) an (18)., so the Step : Calculate k p an m t usng Eqs.(1) an (15). Step 4: If ( k p k ), nn 1, go to Step. * p Step 5: If ( mmn mt ), mmn mt, no n; nn 1, go to Step. Step 6: Output the optmal result of n O, t, an w by Eqs.(17) an (18). 10

11 .5 Correcton of manufacturng errors by tenson stffenng There are some uncertantes n engneerng applcaton, such as materal propertes, manufacturng errors, an some fllets whch are ae to reuce the stress concentraton, whch lea to the change of the stffness an frequency. However, the change can be ajuste by means of tenson stffenng. For preloa beams, the relatonshp between the stffness an axal force of clampe-clampe beam (Yang et al. 015) s 19EI 1 N 4EI 1 N k (0) 5 L L 70L L The stffness of one en clampe an the other en gue wth axal force can be regare as half of clampe-clampe beam, so the corresponng stffness s k T 1 1EI 1 N k (1) L L 140 Therefore, the total stffness wth axal force s 1EI 1 N k nkt n L 140L () Accorngly, the resonant frequency at rvng recton s 1EI 1 N n 1 L 140L f m0 ( cnlta) w () As a result, the uncertantes can be compensate by the axal force apple to the leaf sprng type flexble hnges..6 Summary of the progress of the presente optmzaton metho In a wor, the progress of the smultaneous optmzaton of topology an sze for flexure-hnge-base gung mechansm s as follows: 11

12 Analytc statc moelng of a sngle FH Deformaton functon of FH Raylegh-Rtz Analyss The equvalent stffness an mass The equvalent ynamc system of gung mechansm Layout assumpton of gung mechansm Assemble of equvalent stffness an mass of each FH Optmzaton solvng The optmzaton (topology an sze ) formula wth both screte an contnuous varables Solve the contnuous varable expresson wth equalty constrants Fn the screte number accorng to the other constrants an objectve functon Compensaton for manufacturng error Correct the stffness wth tenson stffenng effect Ajust the frequency by ang or removng ump mass Fgure 5. The flowchart of the presente metho 4. Example In orer to show the effcency of the presente metho, a esgn case s gven, an the optmal results are compare wth both numercal results by fnte elements analyss (FEA) an experment results. 1

13 4.1 Optmal esgn of a FH-base moton stage The work platform s 100mm n length an 50 mm n wth, the length of each leaf sprng type FH s 50mm. It s rven by a voce col motor, an the mass of the col assembly s 0g. The esgn objectve s to mnmze the total mass uner constrants of k 1 N / m, f 00Hz, an k 50 N / m. p The materal use s alumnum, wth elastc moulus 78GPa, mass ensty 700 kg/m, Posson s rato 0.. The platform s a sol cubo, wth 100mm n length an 50 mm n wth. Therefore, equvalent cross sectonal area a s mm. Usng the Eq.(17) an (18), we have the cross secton sze of wth w an thckness t wth fferent group number of FHs n, an the results wth n from to 9 are lste n Table 1. Table 1 Desgn value for sol work platform Group Wth Thckness Drvng Payloa Drvng Total number (mm) (mm) stffness(n/μm) stffness(n/μm) frequency(hz) mass(g) If we plot the mass an the payloa stffness n Fgure 6, we fn that when the group number n s greater than, the payloa stffness (blue ash lne n Fgure 6) s N/μm, whch s greater than esre value (black ot lne n Fgure 6) 50 N/μm, an t means that the constrant s satsfe. Wth the ncrease of group number, the stffness of payloa gets hgher, an t woul be better for applcaton. However, the total mass also ncrease wth the group number, therefore the optmal results of n, w, t are, 18.40mm,.5mm. The fnal optmal esgn s shown n Fgure 7. 1

14 Fgure 6 Determne the optmal result Fgure 7 Optmal esgn 4. Comparson wth numercal results by fnte element analyss In orer to prove the effectveness of the presente metho, the fnte element moels of each optmal esgn wth fferent group number (n) of FHs are bult, an both the stffness an resonant frequency n the rvng recton are obtane an lste n Table. Compare wth the results obtane by the presente metho, we can see that the maxmum relatve error of stffness s.669%, an the relatve error of resonant frequency s wthn 1.80%. However, t s stll acceptable for the engneerng applcaton. Table Comparson wth numercal results by FEA Group Equvalent stffness Resonant frequency number FEA(N/μm) Relatve Error (%) FEA(Hz) Relatve Error (%) Experment results In orer to verfy the optmal esgn, the frst three optmal mcro moton stages wth sol frame are manufacture. The thcknesses of the three mcro moton stages were set to.68,.5 an.14 mm, respectvely. The prototypes are shown n Fgure 8. 14

15 (a) Two groups (b) Three groups (c) Four groups Fgure 8. The optmal mcro moton stages wth fferent topology an sze The stffness an frequency are all measure by an acceleraton sensor as shown n Fgure 9. Excte by a pulse, the acceleraton response of the mcro moton stage s shown n Fgure 10(a), an usng FFT, the resonant frequency s shown n Fgure 10(b). Wth an wthout col assembly of the VCM, we can get two fferent, resonant frequences. Solvng the basc vbraton equatons f k m 1, we get both the equvalent stffness an ntal equvalent mass of the mcro moton stage. The results of average thckness are shown n Table. The stffness an resonant frequency are obtane by the acceleraton sensor (shown n Fgure 9) wth fferent mass an lste n Table 4. Table Test results of the esgne mcro moton stage Group number Average thckness (mm) Frequency wthout col assembly(hz) Frequency wth col assembly(hz) Equvalent stffness (N/μm) Equvalent mass (g) Fgure 9 Measurement of resonant frequency of mcro moton stage 15

16 (a) Acceleraton response (b) FFT of acceleraton response Fgure 10 Measurement of resonant frequency n the rvng recton From Table 4, we can see that both the stffness an frequency are lower than the esgn value, but we can ajust the stffness to the esre value by changng the axal force. We can also ajust the resonant frequency to an exact value by removng or ang a lump mass. In the case of the frst experment moton stage, the average thckness s.67 mm, an the resonant frequency s 6.4 Hz wthout col assemble. When the col assemble s nstalle wth the work platform by bolts, the resonant frequency s only 98.1Hz. Calculatng the two prescrbe vbraton equatons, we get the equvalent stffness (0.98N/μm) an mass (98.6g), respectvely. Both the stffness an resonant frequency are slghtly smaller than the esre values, an t s essental to compensate these errors n ultra-precson engneerng. Table 4 Experment results of the esgne mcro moton stage Group number Average thckness (mm) Frequency wthout col assembly(hz) Frequency wth col assembly(hz) Equvalent stffness (N/μm) Equvalent mass (g) Compensaton for errors by manufacturng If axal force s apple to the leaf sprng type flexure hnges, both the stffness an frequency are change (Yang et al. 016; Yang et al. 015). At frst, we ajust the axal force to change the stffness to the esre value, the frequency also change. If the frequency s not satsfe, we can further change t by removng or ang lump mass to the workng platform. As a result, both the stffness an frequency are precsely satsfe. Table 5 Compensaton for errors by manufacturng 16

17 Group number Desre equvalent stffness(n/μm) Frequency after stffness ajuste(hz) Desre equvalent mass (g) Mass neee to be change(g) Frequency after mass ajuste(hz) Concluson In ths work, a smultaneous optmzaton metho of topology an sze of a gung mechansm s presente by convertng nto an analytcal optmzaton moel wth mxture of screte an contnuous varables. A relaxaton metho s presente to solve the contnuous varables from equalty constrants, an the fnal screte varable s chosen accorng to the nequalty constrant an objectve functon requrement. The equvalent stffness an mass expresson of a sngle FH s erve by means of Raylegh-Rtz analyss. Numercal results show that the presente metho s able to fn the optmal topology (fferent group number of FHs) an sze (cross-secton sze) to meet the objectve of mnmzng mass uner precse stffness an frequency constrants n the rvng recton. When compare wth the results by fnte element analyss, we can see the maxmum relatve error s wthn the allowe range of engneerng applcaton. For more precson engneerng applcaton, the errors by manufacturng can be compensate by changng the preloa of the leaf sprng type FHs. The presente metho s able to optmze the topology an sze of the gung mechansm at the same tme, an t s very easy to compensate the errors by manufacturng. Acknowlegements Ths work was supporte by the Natonal Key Basc Research Program of Chna (Grant No. 011CB01104), Natonal Natural Scence Founaton of Chna (Grant Nos. U114004, , 51704), Guangong Natural Scence Founaton (Grant Nos. 015A001008, 015A ), Guangong Scence an Technology Plan (Grant Nos. 015B , 015B , 01B ), an Scence an Technology Program of Guangzhou (Grant No ). References Chen, G. M., J. L. Wang, an X. Y. Lu "Generalze Equatons for Estmatng Stress Concentraton Factors of Varous Notch Flexure Hnges." Journal of Mechancal Desgn 16 ():8. 17

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