Generalized modeling of a single bent leaf flexure

2 Ph.D I'hesis Gen がaliがd Modeling of a Single-Bent L e が の ex u が G化 dimte School of Yeungnam Univが sity Meじhaniじal Eỉnaineerina Department Meじhanical Ensineerins Major Nguyen Huu Nghia Advisor : 1 Dong-Yeon しee Chan Byon 2000005986 A ugust 2015 Ph.D Thesis GenげaUzed Modeling of a Single^Bent Leaf Flexure Advisor : Do 打g-Yco打 Lee Chan Byo 打 Prese が ed as Ph.D Th が is A ụgu^2015 ĨÍH/0ぶr!f宿 Hạ：卿 議 ， ド， * •■•íTt** い ^ THƯ V皆n Graduate School GI 仍ungtiam University Mechanical Engineering Department Mだchanicai E打ginだermg Mの or 0001 Nguyen Huu Nghia Nguyen Huu Nghỉa，s PhD Thesis わ approved Committee memb が _ 0| Committee member ^ Committee member 01 リ 给e sÌRĩiẩture 、 ぶ ‘ , Committee member Committee member Augu な 2015 Gradimte School of Yeungnam Univerおty Acknowledgements I would Uke to expr*ess my special appreciation and ứianks to r Yeon Lee and Prof Chan Byon, you have been a tremendous mentor advisor Prof Dong• me I would like to thank you for encouraging my research and for allowing me to grow as a research scientist Your useful advice on both research as we りas on my career ! wo 山d also like to thank my committee members Prof Dong-Yeon しee， Prof DongKil Shin, Prof Byoung-Duk しim Prof Dong-Joo しee and Prof Chan Byon for serving as my committee members ! also want to thank you for letting my defense be an enjoyable moment, and for your brilliant comments and suggestions, thanks to you I would Hke to acknowledge 化e support from the Yeungnam University and tile Faculty o f Mechanical Engineering I would especially like to thank Professors for giving me helpful scientific know ledge.1 would like to greatly thank the s化びs at the office for ữieir friendly help during Ph.D course I would also like to thank a り o f my lab ma化s Ji-Soo Kim, Ju-Hee Lee and Cheol Hong at Nano Syなem Labonưory ibr sharing ữieir knowledge and helping me in using 比e eq山pmenし A speciaUhanks わ my family Words cannot expreが how grateful in law, fa化er-in-law, my mo 化び，and も化ぴ and especial my beloved 化 y motheraり of ứie sacrifices that you’ve made on my behalf August 2(H N 呂uyen Huu Nghia Table of CoiUent Acknowledgemen わ C hapter Introduction 1.1 Guide mechanisms .8 1.2 Leaf flexure 1.3 Beam and torsion theory 12 し3 Be打ding analysis 13 し3.2 Torsion analysis 16 し3.3 Compliance matrix 19 C hapter Generalized Modeling of the SELF 21 2A General theoretical equations 22 2.1.1 Castigliano，s second theorem 22 2.1.2 Higher - order beam theory of Levinson 23 •し3 Governing equがion of twiがangle wUh warping effeな 25 2.1.4 Partially restrained warping 26 2.2 Strain energy analysis 2.2.1 A おalSE ( の 28 2.2.2 Shear SE 脚 29 2.2.3 Bending SE ( 的 ） 2.2.4 Torsion SE ( の 30 31 2.2.5 Total strain energy (U) 36 2.3 Displacements 2.3.1 Translational displacements 37 2.3.2 Rotational displacements 37 37 2.4 Complia打ce matrix of 化e SB L F 38 Chapter Verification 42 3.1 Finite element analysis .42 3.2 Results 43 3.2.1 Comparison results amongthree beam theories 43 3.2.2 Comparison resulじ for 化e fUly，partially， freely reなrained warping 49 3.2.3 Displacement results under6-axis loads 53 Chapter Conclusions 68 References 70 Appendix 76 Abstract 94 Chapter Introduction 1.1 Guide mechanisms Guide mechanism many applications in precision machine， espがially in MiぴoElectro-Mechanical Systems (MEMS) and NanoElectの-Mechanical Systems (NEMS) devices or where 化e mechanisms are required 化 provide a motion 化が is conがrained to lie in a single plane [1] Guide mechanism can be classi巧ed in many types, in genei*al have two mainly typだ has been widely used， such as sliding and flexure guidじ A sliding guide includes kinemがic, elaなic, hydroながic， aが0ながỈC and magnetic guide， as shown in Fig 1.1 And hinge and leaf'-spring are two kinds of flexure guide, as presenkd in Fig 1.2 Many guide mechanisms have studied and designed in recent years to respond the actual demands For example, a novel low-coが nano-manipulator which uses a six-axis compliant mechanism which is driven by electromagnetic actuators was designed [2]; a small-scale nano-positioner which is comprised of a six-axis compliant mechanism and three pairs of two-axis thermo-mechanical micro­ actuators preserưed in [3]; a novel piezo-driven，paralleレkinemがic， mi びo-positioning stage was reported in [4] with high scanning performance and the resolution is 20打m; a novel Ion呂-travel piezoelectric- driven linear nano-positioning stage capable of opががing in either stepping mode or in a sea打ning mode was proposed in [5] Kinematic guide b Hydroがatic guide c Magnetic guide F ig 1.1 Examples of syがems in kinds of sliding guide Among them， the guides thが were constructed from hinge and leafspring we化 widely used, such as a novel piezo-driven, paralleレkinematics XYZ nano-positioning s化ge hお developed in [も 7]; a new pie之oactimted XY がage with int;egrおed paralleし decoupled， and なacked kinematics なrucUire for micro-/nano-positioning application was designed [8]; a linear and angular compliance models for a claが of statically ind別erminate symmetric flexure siTUcUires were pĩ*eseがed [9]; the 幻exural hinge guided motion nano-positioning stages were designed for use in precision ma じhilling [iO-12]; a displacement reduction mechanism based on torsional leaf spring hinges have developed in [13]; a para りel leaf-spring flexure wkh increasing displacement was designed in [14]; rotがional motion and bending analysis of leaf flexure were itưroduced in [15，Ibj; In additionaし 化e flexures 化at combine 化e leaf sprin呂and hinge were presented [17-20] showed the large working range and high accuracy し Hinge ílexui^e I Leaf spring flexinで F ig し2 Examples of systems in Kinds of flexure guide In generaしsliding guide has long workin呂range (greater than millimeter) and the resolution is in order of micromけer The flexure guide has short working range (from micro to millimel:er)， no rnction and the resolutio打is in order of nanomけer Therefore, it is evidence 化が overall 化en 曰IS finding the new flexure guides which large working range and nano-resolution in 化e recent yearんBecause ứie leaf spring has smaller size and minimum use of m がerials in comparison with hinge at the same deflection and it is expe なed to make 化e new leaf spring flexure with large working range and nano-resolution Thus, die leaf spring flexuiでis focused in this study 1.2 Leaf flなlire As preseがed in [21]， the leaf flexure (beam suspension) can be divided int:o three kinds of single, singlたbent and double-bent leaf, as shown in Fig 1.3 Single leaf has two degrees of freedom with the features those are coupled error and small range of traveしSingle bent and double bent flexures have degrees of freedom and largぴ range of traveしThe double beが flexures can create the larger range than single bent， but it shows the low なiffnだs が Single leaf b Single-bent leaf c Double-bent leaf Fiもし3 Three kinds of leが flexure [21] C2 Deflection L み （mm) (mm) under F 、 Theory FEA 0.05416 0.05809 6.5 0.12053 0.12759 Error (%) 6.77 5.54 の 713 0.23785 4.51 9.5 0.38309 0.39 の 3ぶ 11 0.59772 0.61825 つ 12.5 0.88008 0.90 の 2.99 14 し24019 1.27453 し68 の 1.の ％ 17 2.23170 2.281 の 18.5 2.88072 15.5 一 • - 2；69 3.64572 — 12.20 觀 :な.かふヴ1簿« 驚 2.94080 2.04 3.71567 6"^— '• , '^:み\’ ' ;':;:.'お V 0.325 3.58562 3.71567 し69129 し63 け 0.87 の 0.9ÍT720 0.475 0.52188 0.54178 0.55 0.33604 0.34901 ぶ 25 0.22890 0.23785 痛 帳 ; 3ぶ 3.54 ’ - ト t ;い が けドZ *- -84- U8 r - - r ' '',f :鸿ぶ， V ソ 心 を か :.:、 ,.;' ミ こ マ t 0.4 ぃ’ 賓 每 '游 t (mm) 0.25 ， 封 蘇 ’ 20 バ - — ぶ3 k 論 f こ ‘ ， 3.71 3.76 0.7 0.16285 0.16の 3ぶ 0.775 0.11993 0.12477 3.87 0.85 0.09085 0.09458 3.94 0.925 0.07045 0.07339 ん 01 0.05571 0.05809 4.10 0.90959 0.92903 2.09 2ぶ 0ぶ ％ 38 0.71464 2.55 3.2 0.56320 0.58064 3.00 3.8 0.47167 0.48896 3.54 4.4 0.40567 0.が 0.35533 0.37161 5ぶ 0.31580 0.33180 6.2 0.28412 0.29%9 6.8 0.25793 0.27324 7.4 0.23627 0.25109 0.21772 0.23226 mm) ろ（ -85- の 3.93 ■1 — パ 4.38 。 ん 82 、 い， 柄V タが—， 5.20 / -5 ぶ0 ’ り ぶ 90 一 巧 ご * €16 C3 Dが が tio n を （mm) und が F 之 L (mm) Error (%) Theory FEA 0.の 228 0.02113 5.16 6.5 0.05328 0.05 じ 3.77 0.10449 0.10146 2.90 9.5 0.18106 0.17672 2.40 11 0.28792 0.28210 Z02 12.5 0.430 け 0.42263 し76 14 0ぶ 1283 0.60338 1.54 0.84095 0.82938 し38 17 し 119が し 10が し24 18.5 し が 377 し43の 1.13 20 し84858 し829が L03 0.25 し61116 し58055 1.90 0.325 0.74486 0.の 006 し99 0,4 0.40628 0.39777 2.10 0.475 0.24698 0.24151 2.22 0.55 0.16213 0.15 が ' — こ’ ， 2.36 0.625 0.11270 0.10988 け Í (mm) - 86 - V 2.51 0.7 0.08191 0.07972 2ぶ7 0.775 0.06168 0.05992 2ぶ5 0ぶ5 0.04781 0.04636 3ぶ3 0.925 0.03797 0.03675 3,22 I 0.03078 0.02973 3.42 0.52844 0.51994 1.61 2ぶ 0.37214 0.3 が 66 3.2 0.28413 0.27863 し94 3.8 0.22765 0.22270 / 4.4 0.18828 0.1の の へ， ‘2.43 0.15933 0.15497 バ V 5.6 0.13716 0.13290 6.2 0.11959 0.11542 6ぶ 0.10533 0.10じ6 7.4 0.09358 0.08956 0.08371 0.07975 ろけnm ) -87- し74 t -ぃ2.74 U ふが >ん 30 4.の C4 民otation 钱 （ rad) L (mm) undが T义 Theory FEA Error (%) 0.00115 0.00108 ん 24 6.5 0.0(H53 0.00M5 4.73 0.00190 0.001が 3.77 9.5 0.00 の 0.00221 3.16 11 0.00266 0.00258 2-70 12.5 0.00303 0.00296 2.35 14 0.00341 0.00334 15.5 0.00379 0.00371 1,87 17 0.00416 0.00409 1.70 18.5 0.00が 0.00447 1.51 20 0.00491 0.004が 1.39 0.25 0.01808 0.017の 2.51 0.325 0.00837 0.00815 0.4 0.00が 0.004が 0.475 0.00279 0.00270 0.55 0.001が 0.00178 3.07 ぶ25 0.00128 0.00 じ 3.18 ぶ9 ぃ t (mm) ■88- 'な パ 三.叩 r;* ぶ7 '' ぶ 1, 0.7 0.00093 0.00090 : 羡 0.775 0.00070 0.00068 0.85 0.00054 0.00052 3.43 0.925 0.00043 0.00042 3.49 0.00035 0.00034 黑 336 片 • ろ（ mm ) 0.00570 3ぶ 产 ’ •> 一 ， *-* -3.28 r 一 0.00561 1.67 公一 0.00405 0.00397 2.10 3.2 0.00314 0.00306 2.49 ： •ん 3.8 0.00256 0.00248 4.4 0.00215 0.00208 0.00185 0.00179 ぶ6 ■ 3.21 3.58 ♦ 5.6 0.00163 0.00156 6.2 0.00145 0.00138 6.8 0.00130 0.00124 7.4 0.00118 0.00112 , 3.96 ‘! / 4:34 画 が 0: な ぶ W、 ，5 ぶ8 / か 0.00108 -89- 0.00102 ‘ "5.44 C5 Rotation L み （ rad) (mm) under M少 FEA Error (%) Theory 0.00089 0.00084 5.75 6.5 0.00127 0.00122 ん 09 0.00164 0.00159 3-10 9.5 0.00202 0.00197 2ぶ 11 0.00240 0.00235 12.5 0.00277 0.00272 'け 14 0.00315 0.00310 15.5 0.00353 0.00348 17 0.00390 0.00385 18.5 0.00428 0.00423 20 0.00466 0.00461 ■ i ぶ3 補 1.60 か ぉ ị ぇ も * h u も； * Is ， ん ’ —‘ ！ W〜 W / (mm) ぃ 0.25 0.01614 0.01が 0.325 0.00747 0.00732 0.4 0.00408 0.00399 / ^ 0.475 0.00249 0.00243 —、 ， ， み■•ム 0.00164 0.00159 ぶ 25 0.00114 0.00111 -90- 2.04 謂 2,17 '2.33 い 0.55 1.93 、 ー ィ み 4ぅ ， —"な お 接 な 蘇 皆 な ;:… み ；ぶ 2.50 >*2.68 0.7 0.00083 0.00081 ぶ9 0.775 0.00063 0.00061 3.10 0ぶ5 0.00049 0.00047 3.32 0.925 0.00039 0.00037 3.55 0.00032 0.00030 3.79 0.00542 0.00533 しが 2ぶ 0.00378 0.00371 1ぶ3 3.2 0.00287 0.00281 2.04 3ぶ 0.00229 0.00224 4.4 0.00189 0.00185 0.00160 0.00155 、 5ぶ 0.00じ 0.00133 ; i 。 3.24 6.2 0.00120 0.00116 ぶす 6.8 0.00106 0.00101 4.01 7.4 0.00094 0.00090 4.45 0.00084 0.00080 ん 89 ち （mmj '、‘ 229 2,55 ぶ7 C6 Rotation ^ rad) u n d ^ yV/ご L FEA (mm) Lheorv E rro r (% ) 0.00331 0.00348 4.95 6.5 0.00が 0.00が 3.91 0.00539 0.00557 3.24 9.5 0.00644 0.00662 2.の 11 0.00748 0.00766 2.36 12.5 0.00853 0.00871 2.08 14 0.00957 0.00975 ぶ7 15.5 0.010 の 0.01080 L70 17 0.01166 0.01184 1：56 18.5 0.01271 0.01289 1.41 20 0.01375 0.0 じ 93 133 ■ 心: 0.25 0.05433 0.05573 2.51 0.325 0.02473 0.02537 2,53 0.4 0.01326 0.01361 2.54 0.475 0.007の 0.00813 2>が 0.55 0.00510 0.00523 0ぶ25 0.003が 0.00357 t (mm) -92- ん み く ？ が；、 " • ： 2• ぶ 一る 2.61 0.7 0.00247 0.00254 ぶ4 0.775 0.00182 0.00187 ぶ7 ぶ5 0.00 じ 0.00142 2.71 0.925 0.00107 0.00110 2.75 0.00085 0.00087 2ぶ 0.01373 0.01393 1.48 2.6 0.01052 0.01072 1.81 3.2 0.00 が 0.00871 2.13 3.8 0.00715 0.00 の 2.46 4.4 0.00616 0.00 の ぶ0 0.00540 0.00557 3.09 5ぶ 0.00481 0.00498 6.2 0.00が 0.00449 3.72 6.8 0.00393 0.00410 ん 04 7.4 0.00360 0.00377 4.32 0.00332 0.00348 ん 60 b (mmỉ -93- 祕 ' •が Generalized Modeling of a SinglたBent Leaf Flexure Nguyen Huu Nghia Graduate School o f Yeungnam University Mechanical Engineering Departmeが - Mechanical En 邑ineering Major (Advisor Dong-Yeo 打 Lee and Chan Byon) Ab がraけ A flexure guide has many applicがions in Micro-Electro-Mechanical Systems (MEMS) and NanoElectro-Mechanical Syなems (NEMS) devices, and plays an importaが role i打 nano-scannぴ s There have been many studies I*ela化 d to 化e prediction of the elaなic deformが ions of flexure However， majority of previous rだ earch u が d Euler-Bemoulli and TTimoshenko beam theory in bending analysis The variable shear deformation and warping effect were not considered in bending Ana in torsion analysis, the warping effeな was also not considered and the partially restrained warping was not inveなigおed simultaneously 中ecたd th が new flexure with large working range and 打ano-resolution is propoが d ，化 us - 94- 化e singlた bent k a f flexure (SBLF) is inveがigat:ed in this sUidy by using advanced beam ữieories In this study, we present a compliance m が rix for a SBLF 化が shows the relationships between l;he deform が ions and l:he six-axis loads applied to the SBLF Highei*-order beam 化 eory is applied that considers the variable shear and warping effect in bending The warping and partially re がrained wmping at 化 e junction between elem ent are also considered in torsion The to1:al なĩ*ain energy is calculが ed， and by using Ca が i呂liano，s second uが d to verify the 化 化 化 e complが e compliance mが rix is (krived eorem The 巧nite element analysis (FEA) is eoKtical resulじ Sensitivけy analyses over the compliance elements are performed The results show th が 化 e d な ived compliance elements are in good agreement with FEA, with errors of less tha打 770/) We suggest that theoretical compliance elements considering variable shear and w iping in bending and partially T*e；strained warping in torsion highly accurおe design eqimtions repi'esenting t:he compliant mechanism of the SBLF The prese打t work could be used in a modal analysis of a single bent leaf flexure -95- Gen がalized Modeling of a Single-Bent Leaf Flexure N gu yen H uu N g h ia 嶺南义學校大學院 巧 淵, 術 吗 叫 ブ ト ち ロ 互 过 ス ト フ け イ と 吗 (M EM S ) 奠 イ と 哨 (NEMS ) 呂 t ì i o i l 擅 ち資日印しト 对み 9HLÌ 01 養 谷 岂 与 誓 言 Ề í ữ 智对I 圳 乃 書 马 叫 马 巧 乂 坤 夸 叫 哥 を ち 干 外 裡叫巧郑 巳這司- 刮 巧ス1巧 〇Ì S お 子 马 巧 す 昔 苦 昔 譯 刮 勺 イ 却 吾 斗 与 互 句 1立 昔 智 イ フ ト 苗 S 巧 增 讀 斗 -96- 叫 吾 巧 喜 普司■巧み 豆斗 马計スÌ 结 巧 ： n， 叫看谭 斗項 却脅唱刮斗しトt 請 气 马 ス Ì 结巧斗 テ耳 刮タ岳 リ互 つ か - 舊 直 計 7] 邱 吾 叫 タ 音 迎 ち 晉 吗 叫 が B L F ) 巧 ザ 暑 却 巧 斗 巧テ叫 SBLF ト吾斗背弓4叫勾召:刘叫哥を 里 召書坤叫おと叫ミ马と暑巧晉巧可 过 巧 苗牛 斗 斗 马 を 斗暑 昔 則 却资斗.叫看皂 習をす坤づ台斗马斗 子 今 斗 马 〇 ] 立•司司巧畔.毒话弓 し 1イ 河 せ 司 五 ， 每 过 を 召 喜 坤 叫 巧 と ロ ！ ] ミ 马 と ち Castigliano 1■每み 巧刮 左 喜 召 斗 己 召 を 鲁 刮 刮 弁 を 在 么 刮 叫 げ E A )0| 吾 句 召 却 斗 乂ト菩召畔 巧な豆き] ! 叫 召暑坤り Ì 巧 と 直 么 暑 召 を 計 : Ĩ 召 を 召 斗 暑 旦 巧 在 み 營 6.770/〇 を FEA 量 坤 0]巧 と 公 と 7ト 章 芒 ト 坤 な じ} • 其お子■ 诗 旦昔 音 爸 吉 爸 d 司タ鲁お 乂Ì■ 菩 皂 牛 $1坤 - - SBLF ... those are coupled error and small range of trave? ?Single bent and double bent flexures have degrees of freedom and largぴ range of traveしThe double beが flexures can create the larger range than single. .. travel， the single- bent or double bent leaf flexure were chosen to replace For instance, a single- bent leaf scanner was designed in a pure ro化tional scanner th が has an accural:ely defined rotation... wi化 minimal parasitic errors [18]， a double -bent leaf spring was used in the new pla打ar flexure [21]? ?a? ??d it could gな large travel and nano-resolution guide The SLBF has remarkable advantages for