multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1...

9
HAL Id: hal-00150263 https://hal.archives-ouvertes.fr/hal-00150263 Preprint submitted on 29 May 2007 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Multiphysic modelling of a microactuator based on the decomposition of an energetic material: application to microfluidics Gustavo Aldolfo Ardila Rodriguez, Carole Rossi To cite this version: Gustavo Aldolfo Ardila Rodriguez, Carole Rossi. Multiphysic modelling of a microactuator based on the decomposition of an energetic material: application to microfluidics. 2007. hal-00150263

Upload: others

Post on 22-Sep-2020

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

HAL Id: hal-00150263https://hal.archives-ouvertes.fr/hal-00150263

Preprint submitted on 29 May 2007

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Multiphysic modelling of a microactuator based on thedecomposition of an energetic material: application to

microfluidicsGustavo Aldolfo Ardila Rodriguez, Carole Rossi

To cite this version:Gustavo Aldolfo Ardila Rodriguez, Carole Rossi. Multiphysic modelling of a microactuator based onthe decomposition of an energetic material: application to microfluidics. 2007. �hal-00150263�

Page 2: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

������������ ���������������������������! "�#��%$&�� ������� ����'$)(� ����*�#���+�,�-�����*�.���#����/�������������0 ��1�.�-�.$������,����� ����2$�� ��43� �-5����. ��������1���"�#��*$&�6�7���������*�

8-9;:�<�=7>*?A@0B�?DCFEG?A@0HGB�IJCF=LK�?DB&H�MJND9.OQPSR�T'=QHG?DCUO-K�?&:4:VIW @0@YX*ZT\[0K0X.R�]Y=7>*O�^�9.OLBD9`_Q?DCU?D^*O�C;K�?D_Qa.O/R.b�c7d�]D]Ae'?/9*CJ?/9;:GO0T5O�B�O�fYd&g*R/h2HG=�^*_�O

i5j =�IkCmlnND=�=�H�B&IJCk=�o"Ck=�=D:7p E�H

qAr�s�t4u�vnw�tx O0y*HGO�:GOQ^&<�<Ja*OA_�?D^*_�O7y�<�IU?D^�?/E5= j IJ_zHG?{y.HGO�:|:�9.HGOY:G?/9*HG_�O

EG?DH j IJ_7HG?/ECk9.IJB�IU_/:�=Qy&y*CFIJ_�=Q<UIJ?D^2:7p*}J<'_�?D^2:GI�:�<:-IF^~b j =�IF^)y*=�H�<:Ql� IJ��=�a*O�=Q<UIk^*NAHGO�:VI�:�<U=�^*_�O5��9.IJCk<�?D^�=0B�IJO�CJO�_7<UHGIJ_ j O j �*H�=�^*O�R � IJIJ�=Q^�O�^*O7HGN&O7<�IJ_ j =�<�O7HGIF=/C��0a*IU_7a�B&O�_�? j y.?&:|O�:+N*O�^*O�H�=�<JIF^*N�=a�IJNDa�>*?DCk9 j O�?/E��*I�?D_/? j y*=�<�I��*CJO�N&=D:GR � IJIJIF��=�^�O�CF=D:�<�? j O�Hj O j �*H�=�^*O��0Ik<�a�a*IFNDa"O�Ck=S:�<�IU_�y&HG?/y.OQH�<�IJO�:7p��2�0�1X~I�:�_7a*?&:GO�^=D:�<Ja.O j O j ��HG=�^*O j =,<�OQHIJ=�C�E?DH�Ik<�:�O�CF=D:�<�IJ_+y&HG?/y.O�H�<�IJO�:7pA@�*I j O,<�=/CJCFIJ_�O�^*O7HGN&O,<�IJ_ j =Q<�OzHGIk=�C5_�? j y.?&:GOQB~?DE-��^�=/^&B�T5?�I�:_Qa.?&:VO�^��*O/_�=Q9;:VO�?/E�<�a*OAa*IJNDa�= j ?/9&^�<5?/E��*IJ?&_�? j y*=�<�I��*CJOYN&=D:CJI��*O�HG=7<�O�B�EHG? j IF<:6B&O�_�? j y.?&:GIk<�IJ?D^~�Gc7�Gpx a*O7^)<Ua*OA=/_7<J9.=Q<UIJ?D^�I�:5HGO��n9.IFHGOQB.R;<Ua*OAO�^*O7HGN&O,<UIJ_ j =Q<�O7HGIF=/C

I�:�a*O�=7<�O�B��*Oz�*?D^*B�<Ua*O~IJND^&Ik<�IJ?�^+<�O j y*O�H=Q<J9&HGO�<�?"N�Oz^*O�H�=Q<�O�=a�IJNDa"_�?D^*_�O�^n<UHG=7<�IJ?D^"?/E0N&=D:7��<Ja.O)ND=/IF^�IF^�y&HGO/:4:�9&HGO�y&HG?DBD9*_�O�:<Ja*O�B&O,E�?DH j =Q<UIJ?D^�?/E#<�a*O�O�CJ=S:�<�IJ_ j O j �*HG=�^*O/p�}�^*B&I�>*IJBD9&=/Cj ?DB&O�C�: a&=7>*O¡�*O�OQ^ B&O|>*O�CJ?/y;OQB E�?DH O�=/_za y&aD�.:VIU_Q=�Cy*a*Oz^*? j OQ^.?/^�l � IJ��<Ua*O¢a.OQ=�<JIF^*N�?/E"<�a*O�HGO�:VI�:�<�=�^*_�O�����=�^OQCUOQ_7<�HG?&ZJ<�a*O7H j =QC j ?DB&O�C�R � IUIJ�Y<Ja*O"N&=D:�N�Oz^*O�H�=Q<UIJ?D^#EHG? j <Ua*OB�O�_/? j y.?&:VIF<�IJ?D^)?/E�<�a*O0O�^*O�HGN�O7<UIJ_ j =Q<�O7HIJ=/C.�*=D:GO�B�?D^�=�^�IUB�O�=/CN�=S: j ?DB�O�C�=�^*B�= j =D:4:£<JH=�^2:�EO�H�< j ?DB&OQC�=�^*B�_�?/9&y*CUOQB~�0Ik<�a<Ja*O¤O�CJ=D:�<�IJ_ j O j �*H=�^*O¥B&O7EG?DH j =�<JIJ?D^��!Ik<Ja¦= j O�_7a.=Q^&IJ_�=/CB�O7EG?DH j =7<�IJ?D^ j ?DB�O�C�p'e�a*I�:Yy&=Qy.O7HYy&HGO�:VO�^�<:�O�=/_7a§Ik^*B&I�>*IJBD9&=/Cj ?DB&O�C¢9*^*B&O7H¦T5¨L�1X&¨ W ��9.Ck<�IFy&aD�*:GIJ_�:©<�a*=7<�a.=D:¥�*O�O�^_Q?DHGHGO�CJ=7<�O�B �0Ik<�a O�fDy*O�HGI j Oz^�<J=�CªHGO�:�9*C�<:7p«e�a*O�^*R¬<Ua*OI j y&CJO j O�^D<�=7<�IJ?�^�?/E'=�CJC j ?DB�O�C�:\IF^�<�?A?D^*O�N�CJ?��*=/C j ?DB&OQC2=�CJCJ?/�92:Y<�?�y*HGO�B&IJ_7<0<�a*O~=/_7<J9.=,<�?DH{y.O�H�E?DH j =�^*_�O�=S:�=�E�9*^*_7<�IU?D^1?/EIk^�y&9&<2O�CJO�_,<�H�IJ_�=/C;:VIJND^&=/C�p­/®"¯4° tVuQ±&²*³2wQt�´J± °� i �µX�<�O�_�a&^*?DCJ?DN&IJO�:�IF^1N�O7^.O7HG=�C0=Q^&B���IJ_7HG?/ECk9.IJB�IU_/:�Ik^

y*=�H�<UIJ_79*CJ=�H�=QHGO�?D^*O�?DEL<�a.O j ?&:�<�B&O|>2O�CJ?/y.OQB~<�O�_Qa.^*?DCJ?DN&IJO�:£IJ^<Ja*O�CJ=D:�<���O�=/H4:7p�h29&H�<Ua*O7H j Ik^&IJ=7<J9&HIJP�=Q<�IJ?�^"=Q^&B�IF^*_7HGO�=D:GO)CUO|>*O�C?nE¶:��2:�<�O j :·_/? j y.CJO�f�Ik<¸�ª=/HGO¹<Ua*O«_79*HHGOQ^D<º<UHGO7^*B»Ik^�"IJ_�HG?&:��*:�<�O j :,p�}�^·<Ua�=�<¼_�?D^�<�O�fD<�R�^*O7� <�?D?DC�:½E?DH¾<Ua*O_Q?D^._QO7y&<�IJ?D^�:�<�O7y2:\=�HGO�O/:4:GOQ^D<�IF=/C�<�?�?/y&<�I j IUP/O5<Ja.O-y.OQH�E?DH j =�^*_�O?nE�<�a*O¦:��.:�<�O j R�<�?�:V=z>*O©<�I j O�=�^*B j ?D^*O4�¿BD9*HGIF^&NÀ<Ua*OB�O4>2OQCU?/y j O�^�<*y&a*=S:VODpx Ik<Ua�<Ja*O¤Ik^._7HGO�=D:GOQB¦_Q? j y.CJO�f�IF<F��CJO7>2O�C�R j 9.C¸<�IFy&aD�*:GIJ_�:

=Qy&y&HG?D=/_�a*O�:�<�=QÁ&IF^*N"Ik^D<�?�=�_�_/?/9.^�<L<Ua*O�IF^n<�O�H=/_7<�IJ?�^2:0�*O,<��!O/O�^B�IkE�E�OQHGO�^D<#y.aD�.:VIJ_�=�CÂy.a*O7^*? j O�^&=DR+=�HGO�^.OQ_�O�:4:V=�H��½EG?DHÃ<Ua*O:��*:�<�O j :\_/?D^*_�O7y&<�IJ?D^�pX�? j O0�0?DH�Á;:5a�=z>*O��*O/O�^�HGOQ_�O�^�<�C ��HGO7y.?DH�<�O�B�?D^Y<Ja*O j 9*C�<�I�Z

y*aD�.:VIU_/: j ?DB*O�CJIJ^*N»?/E j IJ_�HG?¡=/_7<J9.=Q<�?�H4:¬�FÄ*Z�]��Gp�e�a*I�:�*I��*CJIJ?DN&HG=Qy&aD�§:Ga*?/�`:�<�a*O)B&IkE�EIJ_79*CF<J�"IF^�>*=/CFIJB&=7<�Ik^.N�_�? j y.CJO7<�ObS� j ?DB&O�C�:)O�fny.O�HGI j O�^n<�=�CJC �%��?nE�<�O�^Å:VI j y.CkIFE�IUOQB j ?DB&O�C�:)=/HGO>*=/CJIJB&=7<�O�B+O�Ik<Ua*O7HA�&�+=/^*=�C��D<�IJ_�=/C j ?DB&O�C�:�?DH�O�fny.O�HGI j O7^&<J=�CJC��%p�AO,y.O�^*B&Ik^*N�?D^�<Ua*O¢_�? j y*CJO�f&Ik<F��?nE�<�a*O�:�<�H�9._7<J9&HGO¢?DH�<Ua*Oj 9.C�<�Iky&aD�.:VIJ_�_�?/9�y.CJIJ^*N&R':GI j y.CkIkEIJO�B j ?DB�O�C�:�?DE�<�O�^�CJO�=/B&IF^�N�<�?CJIk^.OQ=�HIJP�=Q<�IJ?�^ ?/EÆa*IFNDa*C�� ^*?D^2ZJCJIF^*O7=�H¹O�fDy&HGO�:|:GIJ?D^2:·=�HGOy*HGO7EO�H�HGO�B%p

}�^���IJ_�HG?/E�CF9*IJB&IJ_�:4R%=�^�I j y*?DH�<�=�^n<�I�:4:�9.OYI�:-<Ua*OYIF^D<�O�N�H=Q<UIJ?D^?DE�HGO�CJIJ=z�*CJO�R*: j =�CJC*=�^*B`OQ=S:���<?092:4O j IJ_�HG?�=�_7<J9*=�<JIU?D^{y&=�H�<:�E�?DHy&9 j y&IF^*N�=�^*B�>*=�C�>*Ik^.N�?/E0CJIU�n9.IJB.:,p�¨�>*O�H0<Ja.O�y.=D:�<LB&O�_�=/B&O)=CJ=�HGN&O�>*=/H�IUO,<¸�)?nE j IJ_�HG?0>*=/C >*O/:�=/^*BYy&9 j y2:��*=D:GO�B)?D^Ç:VO4>2OQHG=�CO7E�E�OQ_7<:~IJ^*_�Ck9.BDIk^*N1y.IJO�P/?DO�CJO�_7<�HGIJ_¢�UÈn�R�O�CJO�_7<UHG?&:�<�=�<JIU_¢�JÉD�)=�^*B� <Ua*O�H j ?D�Ly.^*O79 j =,<�IJ_�a*=7>*O`�%O�O�^�y*HG?/y.?&:VO�B1��c,d.R0c&c,�4p�e�a*O�:GO=�HGOL?/E�<�O�^�O�fny.O�^2:VI >*OL=�^*BAB�IkE�EIJ_79.Ck<�Ik^YB&O�:VIUND^Y=�^�B�E=z�*HGIJ_�=,<�IJ?/^�p}�^�=�B�BDIF<�IJ?/^;R{<Ua*O�IkH§B�O7ECJO�_7<�IJ?D^�I�:�<�?D?�: j =/CJC)<�?Ãy&9 j y�?DH_�? j y*CJO7<�O�CF�+_�CJ?&:|O�= j IJ_�HG?/ECk9*IUB�IJ_�_�a*=�^&^*O7C�^*O�_�O�:|:G=/H���E�?DHY=N&?D?DBÊ>*=�C�>%ODp�@0^ÆO�f�_�O7y�<�IJ?D^·I�:��*=D:GOQBË?D^Ëy.^*O79 j =�<JIU_=/_7<J9.=7<�IJ?D^2p%Ì!?n�!O7>2O�H4R%<Ja*O�:GO�=/_7<J9.=Q<�?DH4:L=�HGOY92:�9.=QCUC��Í<�?&?�CJ=�HGN&O<�?��*O�E=7�*HGIJ_�=�<�O�B�IF^"=�B�Oz^2:VO�=�HH�=7�"E�?DH j =�<L=�^*B~<�?��*O)E�9*CUCk�IF^�<�O�NDHG=7<�O�B¢Ik^D<�?1<Ja.O j IJ_zHG?/E�CF9*IJB&IJ_§CJ=7�2Z�?D^2ZJ=1_7a*Iky��Gc7Ä��Gp0@�^O j OQHGN&IF^�N�_Q?D^._QO7y&<'I�:�<�?�IF^D<�O�N�H=Q<�O0y&HGO�:|:�9&HGOY:G?/9.HG_QO�:�B�IJHGO�_7<�C��IF^¿<Ja*O j IJ_zHG?/E�CF9*IJB&IJ_�y*CF=Q<UE�?DH j �D�¿IJ^&<�O�NDHG=,<�Ik^&N¿O7^.OQHGN&O7<JIU_j =7<�O�H�IJ=�C�:Ë?D^Î= j IJ_7HG?¡a*O�=7<�O�HÊE=7�*H�IU_Q=�<�O�BªIF^D<�? <Ja.Oj IJ_7HG?D_�a*=�^&^*O�C)��cQbD�Gp x a*O7^1<Ua*O�=/_7<J9.=7<�IJ?�^#I�:)HGO��D9*IFHGO�B&R-<Ja.OO�^*O�HGN�O7<UIJ_ j =Q<�O7HGIk=�C�I�:+a*O7=Q<�OQB�9&y�=Q^&B�N�O7^.O7HG=7<�O�:Â=�a&IJNDa>2?DCk9 j O�?DE�N�=S:Ï�!a*IJ_7a¼<Ja*OzHGO,E?DHGO�CJ?&_�=/CJC �ÐIJ^._QHGO�=S:VO�:#<Ja.Oy.HGO/:4:�9&HGOnpÑe�a*O j IJ_7HG?/ÒU^*=�^*?D=/_7<J9&=�<�?/HÆ�*=S:VO�B ?D^ <Ua*O�:GOj O�_Qa.=Q^&I�: j :�_�=Q^��*O+E=/IUHGC���:VI j y.CF�ÏE=z�*HGIJ_�=Q<�O�B.R�CJ?/�Ó_/?&:�<R>*O�H��"_/? j y.=/_7< ��Ô c jYj�Õ �R\E�=S:�<�R��*IJ?D_�? j y*=�<�I��*CJO�R5Ik^D<�O�NDHG=7<�=7�*CUO�0IF<�aËB�IFE�EO�HGO�^�<�<�O�_�a�^.?DCJ?DN�IUO/: � :GIJCJIJ_�?D^2R�N�CF=D:4:4RÏy.Ck=S:�<�IU_/:G�GR=/B4ÖG92:�<�=7�*CUOL=�^�B`_Q?/9.CJB0�*O�Ik^&<�O�NDH�=Q<�OQB`?/^YCJ=�HN�O!:V_�=/CUOnp

e'a*O�HGOQCUIk=z�*CJO)IJ^D<�O�NDHG=7<�IJ?D^�IF^§:�9S� j IkCUCJI j O7<�O7H`:G_Q=�CJO�R\?/E�R\=j IJ_7HG?Da*O�=�<�O7HGR)=¤:G?DCJIJB�O�^*O7HGN&O7<�IJ_ j =7<�O7HGIU=QC~=�^*B¥=�^�O�CJ=D:�<JIU_j O j �*H=�^*O�IF^D<�?"= j IJ_�HG?D_�a�=�^&^*O�CLH=/I�:GO�:):GO|>*O�HG=�C j ?DB�O�CJCUIk^*N=�^*B�<�O�_�a�^*?DCU?DN�IJ_�=�CL_7a*=�CJCUO7^*N&O/:7p'e�a*O)EIJH4:�<0?D^.O�I�:�<�?�_�CJO�=/HGC��9.^�B&O�H4:�<U=�^*B�<Ua*O�y&aD�.:VIU_/:Y?D^��0a.IJ_�a�<Ua*O�:GO)=/_7<J9.=7<�?DH4:�?/y.O�HG=,<�O<Ua�HG?/9.N�a j 9*C�<�Iky&aS�*:GIJ_�: j ?DB*O�C�:4R���9.IJCJB&IF^*N¾9&yÑIF^*B�I >*IJBn9&=�Cj ?DB�O�C�:ËE�?DHËO�=�_Qa»y*aD�.:VIU_Q=�C½y.a*O�^*? j O7^*?D^Î=�^*B¡<Ja.O7^IF^�<�O�NDHG=7<�Ik^&N+<Ua*O j Ik^&<�?+=�E�9*CUC£y.=/_7Á.=/N&O/RLO�^*=z�*CJIF^�N+<�? j =QÁ.O>*IJH�<J9&=�C½y&HG?/<�?/<F�Dy*IU^*N%p�e�a*O×:GOQ_�?D^*B»_�a*=�CJCJO7^.N�O«I�:Ê<Ja.OO�CJ=7�*?DHG=�<�IJ?�^+?/EAO�^*O�HGN�O7<UIJ_ j =�<�O7HGIF=/C�:£<Ua*=7<LÁ.O�O7y2: j O�_7a*=�^�IU_Q=�C=/_7<J9.=7<�IJ?D^~_�=Qy.=7�*IJCFIk<UIJO�:!�!a*O�^�Ik<�I�:0B�O7y.?&:VIF<�O�B�IF^�=�<Ua�IF^~CJ=7�&OQH� <Ua*IJ_7Á&^*O�:4: Ô c7d&d%Ø j �|pDe�a.O-<Ua�IFHGBY?D^*O�I�:6<�?`:VO7<29&yY<�a*O�� i �µX<�O�_�a�^.?DCJ?DN�IUO/:�E�?DH�<Ua*O¢IF^n<�OQN&H�=�<JIU?D^�=Q^.B¥IF^n<�O�H�E=/_�IF^*N�?/E"=�CJCj =7<�O�H�IJ=�C�: � O�^*O7HGN&O,<�IJ_�RÎ:�9S�2:�<JHG=�<�O�R¡a*O�=Q<�O7H4R¡y*?DC�� j OQHGIU_j O j �*H=�^*OL=�^&B�ECk9*IFB&��=Q<%:�9S� j IJCkCUI j O7<�O�H':V_�=�CJODp}�^�<Ua*I�:£y.=Qy.O7H��!O)EG?D_79�:Y?D^�<Ua*O j ?DB&OQCUCJIk^&N"I�:4:�9.O�:,p

e�a.O

EIJH4:�<Yy&=�H�<Y?/E�<�a*O�y.=Qy.O7H)I�:Ç<Ua*O�y&HGO�:VO�^D<�=7<�IJ?D^�?/E)IU^*B�I >*IJBn9&=�Cj 9*C�<�Iky&aS�*:GIJ_�: j ?DB&OQC ��9.IJCk< 9&y 92:GIk^&N T�¨L�µX�¨ W��9.Ck<�I�y&aD�*:GIJ_S:,p�e�a*O�^�9 j O7HGIJ_�=�C j ?DB&O�CJCJIF^�N¤I�:4:�9.O�:��0IJCJC��*OB&I�:V_79;:|:GO�B½=�^*BÑ_Q? j y.=�HGOQB��0IF<�aºO�fny.O�H�I j O7^D<�=�<JIJ?D^2p§}�^Ó=:GOQ_�?D^*B�y.=�H�<�Rn<Ua*O�Ik^�<�OQNDHG=7<�IU?/^�?nEm=/CJC2IF^*B&I�>*IJBD9&=/C j ?DB�O�C�:\IU^n<�?A=N&OQ^.OQH=/C j ?DB�O�C��!IJCJC��*Oºy.HGO/:GO�^�<�O�BÆ=Q^.B j 9.C�<�Iky&aD�.:VIJ_�:IF^�<�O7H�E=/_�IJ^&NAI�:|:�9.O/:��0IJCJC&�*O�B�I�:G_792:4:GOQB2pÙ&®�Ú0Û sVwSu7´JÜ*tG´J± ° ±/Ý�tGÞ ÛLß ´JwDu�±&vnw�t�³�v/tG±&u

Page 3: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

e�a*O£y&HG?/y.?&:|O�B j IJ_�HG?�=/_7<J9&=�<�?/H j =�B�OA?D^~:GIkCJIU_Q?D^�_�?D^2:GI�:�<:?nE6b j =/IF^�y&=�H�<:�=D:�IJCFCk92:�<UH�=Q<�OQBAIF^���������� �2lZ ¨0^.OÃa.OQ=�<JIF^*N�y&CU=,<UE�?DH j l)IF<+I�: j =�B�O¤?nEµ?D^*OÃ<�a*Ik^

B�IJO�CJO�_7<�HIJ_ j O j �*H=�^*O � X&IJ¨��JÒGX&Ik[����?D^��0a*IJ_za�I�:LB&O7y*?&:GIk<�O�B�=y*?DCF�*:GIJCJIU_Q?D^`HGO/:GI�:�<�?DH��Gc7g&�VpZ ¨0^.O O�CJ=S:�<UIJ_ j O j �*H�=�^*O �0a.IJ_7a O7^*_�=�y2:�9*CF=Q<�O/:

OQ^.OQHGN&O7<JIU_ j =7<�O�H�IJ=�CGl��2�0�1X � ��?DCF�n�AIF���7<�aD��CX�IUCJ?Df�=�^*O���a.=D:�*O/O�^)_�a*?&:GO�^�E?DH\<Ua*OLB&O j ?D^;:�<UH�=�<�?/H\E?DH5I�<:5O�=D:��{y.HG?/<�?/<U�ny.IJ^*N2p�Gc��n�

� ¨L^*O�O7^*O�HGN�O7<UIJ_ j =Q<�O7HGIF=/Cm�Gc7�

�����������! #"%$ ��&'�)(!*�&'+��,*-+.(���/0+��0�1&2+�����43%576���-8

�9���������;:,"�$ �<&'�)(!*=&'+��*2+.(%�>/?+��.�1&2+��@�?A&2��+�6����87&'B1*2��*�&'+�2���0/0+��<&#/e�a.O j IJ_7HG?�a*O�=Q<�OzH+=�^�B j O j �*HG=�^*O�:�<UH�9._,<U9&HGO/:+=�HGO

:Va*?/�!^·IF^C�D�<����?E� =�^*BF�D�<���,��G3/p¢e�a*OºN&O�? j O,<�H�IJ_�=/Cy*=�HG= j O,<�OQH4:\<U=QÁ.O�^�E?DH5<Ja*O`:GI j 9&CJ=7<�IJ?D^�=�HGO�NDI�>*O�^�IF^�<Ua*O�H�*�I,J<K@L p

�����������;M,"�$ #NOI���*#P Q.*�R1*�P,SAB1#*2+�-�����(�NT2+����<&2*#J1Q<IUR&-BU*-�?*�&'+02����/0+��<&#/ LV W�X1Y<�! #"�$ �<&'�)(!*=&'+��*2+.(%�>���(�NT-+�����&2*�J�Z1*2��*�NT-+�2�)/ L

x a.O7^)<Ja.O�=�_,<U9*=�<JIJ?D^�I�:-HGO��n9&IFHGO�B*R2=Y_79*HHGO7^D<�E�CU?n�`:�IJ^)<Ja.Oy.?DCk�.:VIUCJIJ_�?D^¥HGO�:GI�:�<�?DH�a*Oz=Q<�IF^�N¢9&yÅ���¢<�a*O\[4?/9*CUO1OQE�EGO�_7<)<�a.Oy.?DCk�.:VIUCJIJ_�?D^�=/^*B¦���¼<�a*O�H j =/C1_�?D^*BD9._,<�IJ?/^¿<Ua*O�OQ^.OQHGN&O7<JIU_j =7<�O�H�IJ=�C�Ik^�_�?D^D<�=/_7<��0Ik<Ua¢<Ja.O�a.OQ=7<�O�H4p x a*O7^¢<Ja.O+IJND^&Ik<UIJ?D^<�O j y*O�H=Q<J9&HGO�?/EA<�a*O�O�^*O�HGN&O,<UIJ_ j =Q<�O7HGIk=�CLI�:AHGOQ=�_�a*O�B&R�=~a&IJNDa>2?DCk9 j O0?DE5=�IJH5I�:'HGOQCUOQ=S:VO�B2p&e�a*O7^2R�<Ua*OLy&HGO�:|:�9*HGO0IF^�<Ua*O0_�=7>*IF<J�IF^*_�HGOQ=S:VO�:\=�^&B�B&O7EG?DH j :�<Ja*O�O�Ck=S:�<�IU_ j O j �*H�=�^*ODpb j ?DB&OQC�:\B�O�:G_�H�I��*O�<�a*O j IJ_�HG?D=/_7<J9.=7<�IJ?�^�lZ·<Ja.O)O�CJO�_7<�HG?&Z�<�a.O7H j =/C j ?DB&OQC5<�?�:GI j 9*CJ=�<�O�<Ua*O�a*O�=,<�Ik^&N

�&�][7?/9*CUOLOQEEO/_7<RZ·<Ja.O)N&=D:�N&O7^*O�H�=�<JIU?D^�E�HG? j <Ua*O)B&O�_Q? j y.?&:VIF<�IJ?D^"?/EL<�a.O

O�^*O�HGN�O7<UIJ_ j =�<�O7HGIF=/CUR�?/^*_�O�<Ua*OYIUND^�Ik<UIJ?D^)<�O j y.O7HG=7<J9&HGO � ÄD�/dD^7T5��Gc7��I�:�HGO�=/_za*O�B*RZ¹= j O�_Qa&=�^*IJ_�=�C j ?DB�O�C+<�?�y.HGOQB&IJ_7<+<Ua*O j O j �*H=Q^.O

B&O,E�?DH j =,<�IJ?D^Y=D:�=-E�9.^*_7<JIU?D^Y?/E�N&=D:6y&HGO�:4:�9.HGODp_&®a` v/tGÞ Û�ß v/tG´FwDv1b ß ±&² Û b�s_&®�­Tc b Û wQt�uQ±0tGÞ Û u ß vdb ß ±&² Û be�a*O�O�CJO�_7<�HG?&ZJ<�a*O7H j =�C j ?DB�O�CAIJ^*_�C¸9.B&O/:�<�a*O��0O�CJC0Á.^*?/�0^

a*O�=7<2E�CU?n��O��D9*=�<�IJ?D^Y=�^*B0<Ja.OLO�CJO�_7<�H�IJ_�y*?/<�OQ^&<JIJ=�C*O��n9.=Q<UIJ?D^2p/e�a.OCJIF^�ÁA�*O7<J�AO�O�^Y<�a*O-<U�0?YO��D9*=�<�IJ?D^2:\I�:\B�?D^.O5�&�{<�a.Oe[4?/9*CJO0O7EEGO�_7<� fghi

∇= σ �|pjlk�m1n�o?p�q�n4resto0mu o0q0v�k

( ) wx =∇∇− yzσ � c7�{|mv}o2~>k�s��

( ) ������l�

=∇∇−∂∂ ��ρ � Ä&�

���<������� �@���1�<�D�2���1����������e�a*O�O|>*?DCF9�<�IU?D^�?/E�<�a*O�O7^*_�CJ?&:GOQB`N�=S:\Ik^A<Ja*O�_�=7>*I�<F�{E�?DCJCJ?D�A:

<Ua*O¥h�IJ_7Á �U:ÏCF=Q� ?DE�B�IkE�E�9;:VI�?D^ÐE?DH#<Ua*O�_�?D^*_�O�^n<UHG=7<�IJ?D^¼?/E:�y*O�_�IJO�: ��� �7l2�Gc��&�

�]��1� �@�#¡��<¢U�d�D£;��¡��d� �@�¤�1¡ ¥��d�<�>�l¦'§9��¨© �Dd�dª c����« ÉnĬ ���­ ]�Ä® ÄD�nd¯ bnd° c7d�d

� =/�

� �*�

Page 4: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

( ) ����� =∇∇−∂∂ � � bn�

x a*O�HGO��¢I�:m<Ua*OLBDIFE�E�9;:VI�?D^�_�?DOQE�EI�_QIUOQ^&<2=�^*B��"<Ua*O�HGOQ=�_7<JIJ?D^<�O�H j p�}�^�<Ja&I�:�=�y�y&HG?D=�_Qa�^*?�_�?D^D>*O�_7<�IJ?D^"I�:�_�?D^2:VIUB�O�HGO�B%p�e�a*Oy*HGO�:4:�9&HGOYIF^�<Ja*O`_Q=z>*Ik<F��_/?/9.CJB��*OYB&IJHGO�_7<�C���HGO�CJ=Q<�OQB)<�?�<Ua*OYN&=D:_Q?D^._QO�^�<UHG=7<�IJ?D^2pDT5?D^2:GIJB&O7HGIF^�N0<Ja.OLND=D:�=D:'IkB&O�=/C��!OL?���<�=/IF^�l

� ��= � g��

�0a*O�HGO��¿I�:m<Ua*O�=/IFH j ?DCk=�H j =D:4:|p������� ����;¡,�d���l�����2��¡�����2�0�1� ���D�>�,�e�a*OY_�CJ=D:4:GIJ_�=QC j O�_�a*=�^�IU_Q=�C j ?DB�O�C'I�:\�*=S:VO�B�?D^�<Ja*O{: j =�CJC

B�O7EG?DH j =7<�IJ?D^§=Qy&y&HG?Df�I j =7<�IJ?/^ml6<Ua*O j O j �*H�=�^*O~B�O7E?DH j =Q<UIJ?D^2Z<�?&Z�<�a&IJ_7Á&^*O�:4:�HG=Q<UIJ?D^ I�:¥>*O�H�� : j =/CUC ����� ¯ Ô�Ô c7��=�^*B¾<Ua*Oj O j �*H�=�^*OAB&O,E�?DH j =,<�IJ?/^2ZJ<�?&Z�CU=,<�OQH=/C�BDI j O7^;:VIJ?�^ ����� Ä ® Ô�Ô c7�5I�:CJ?/�×�Gc7]��4p)}�^�?/9.H1_�=D:GO�R��*?/<�a¦=D:4:�9 j yD<�IJ?/^¦=QHGOÅ^*?/<�<UH�9*O����� ¯�� b�Z)�¬=�^*B � Ò�Ä ®�� d2p Ä.ZJd%p g� DR�<Ja.O7HGO7EG?DHGO�RÀ<Ua*O·�2�0�1Xj O j �*H�=�^*O B&O7EG?DH j =Q<UIJ?D^¹I�:Ñ:GI j 9&CJ=7<�O�B 9;:VIF^*NÆ=ÊCJ=�HN�OB�O7EG?DH j =7<�IJ?D^ j ?DB�O�C�pe�a*O0:�<JH=/IF^�<�O7^;:V?DH�I�:'OQfDy&HGO�:|:GO�BY=D:�E?DCJCJ?D�¤�Gc7]��zl

!!"#$$%

&∂∂

∂∂

+∂∂

+∂∂

= ' ()()'

' )) ' *+*+*+* + ,- .ε� �n�

�0a*O�HGO / HGO7y&HGO�:VO�^�<:Ã<Ua*O¼B&O,E?DH j =Q<UIJ?D^ÑIk^Ñ=¿NDI >*OQ^B�IFHGO�_,<�IJ?D^*R 0 I�:-<Ja.O�:�y.=Q<�Ik=�C\_�?D?DHGB&Ik^*=7<�O`=Q^.B)<Ja*O{:�9S��IJ^.BDIJ_�O/:132 4�265 HGO7y&HGO�:VO�^�<AO�=�_Qa1_�? j y.?D^*O�^n<`Ik^87 2:9;2=< BDIFHGO�_7<�IJ?D^2p5e�a*O^*?/^2Z�CJIF^*O7=�H¥<�O7H j >?@? ABAB ∂∂∂∂ C _�a*=�H�=�_7<�O�H�IJP�O/:¥<Ua*OCJ=�HGN�O�B�O7E?DH j =Q<�IJ?/^A�*O�a�=z>*IJ?DH7pe�a*O j O j �*HG=�^*OY:�<UHGO/:4:-_�=�^��*O�_�=/CU_,9.CJ=�<�O�B�EHG? j <�a*O£h�IFH4:�<

��IU?DCJ=DZ�DAIkHG_7a&a*?/EE':�<�HGO�:|:FEµc�Èn��=D:�lG

HJILKLMM N∂∇

∂= � ����0a*O�HGO O P

∇I�:6<�a*OLB&I�:�y*CF=/_�O j OQ^&<�NDH=/B&IJO7^D<�=Q^.BRQTSVUXW�I�:'=

:�<�H=/IF^¿O7^.OQHN/�¼E�9.^*_7<�IJ?D^¼�0a*IU_Qa�_7a*=�H�=�_7<�O�H�IJP�O/:#<Ua*O�aD�Dy*O�HOQCU=D:�<�IJ_-y&HG?/y.O�H�<�IUO/:'?/E�<Ua*O-�2�0�1X2p

e�a.O�[AO�?&Z�ÌA?�?/Á.O�=/^��"?DB&O�C�y.HG?/y.?&:VO�B~���§T5¨L�1X&¨ W��9*CF<�IJZJy&aD�*:GIJ_�:���cQÉD�'_/?D^2:GIJB&O7H4:\<Ja*OA:�<UHG=/IF^)O7^*OzHGN/��E�9.^*_7<�IJ?D^�=D:EG?DCJCJ?D��l

( ) ( ) YZ[ \] \^] \ −+−= _a`bcdJeLfLg κµ � ]��

�0a*O�HGOµ=/^&B

κ=QHGO�<Ja.O�:Ga*O�=QHA=�^�BÇ��9*CkÁ j ?DBD9.Ck92:A?/E

<Ja*OA�%�0�µX*R�HGO�:�y*O�_7<�I�>*O�CF�%pihkj�l*I�:-<Ua*OYO�CJ=S:�<�IJ_�y.=�H�<�?/E5<�a*OA<�?/<�=/C>%?DCk9 j OL_�a&=Q^.N�O�?/E�<Ua*O-�2�0�1X�l

( )mn o po pq�r ssss

ε+== � Èn�

ÌAOQHGO�RJh�HGO7y&HGO/:GO�^�<:¢<Ja*O�H�=�<�IJ?¦�*O,<U�0O/O�^�<Ja.O�_79&H�HGO7^�<>%?DCk9 j O"=Q^.B+<Ja*O"?DHIJN�IF^&=/C0>*?DCk9 j O"=Q^&B8hFt u�I�:)<Ja.O�<Ja.O7H j =/COQfDy.=Q^;:VIJ?D^`OQfDy&HGO�:|:GO�BYIF^�<�O�H j :\?/E�<Ja*O�<Ja.O7H j =�C�:�<UH�=�Ik^wv x

εp

yz{ I�: <Ja.O EIJH4:�< j ?DB�IkEIJO�B :�<UHG=QIF^ Ik^S>*=QHIJ=�^n<

l ( ) |}

|�|}�}~a~�~ −++= �������� �0a.OQHGO ���� R �V�� =�^*B �V��

=QHGO~<Ja*O j =Q<UH�IJf1_�? j y.?D^*O7^�<:)?/E`<�a*O"T5=79*_�aD�.Z�8LHGO�O�^Â<�O7^2:G?DH��� �

=R\�0a*O7HGO��¤I�:�<Ja*O�B&O7E?DH j =7<�IU?/^+NDH=/B&IJO7^�< j =�<JHIJf

_Q?D^&<�=/IF^�IF^*NAIk^&E�?DH j =Q<UIJ?D^Y=7�*?/9&<�B�O7E?DH j =Q<UIJ?D^Y=�^*B0HG?/<�=7<�IU?D^%p

�����������X������� ���X���X¡;¢a£¤¢¥���i¡e�a*O�O��n9.=Q<�IU?/^2: � c�Z�g&�#=�^*B � �&Z�È/��a*=7>*O¤�%O�O�^Ñ:G?DC�>%O�B

^�9 j O7HGIJ_�=�CkC��¥9;:VIU^*N¥<Ja.O#h�Ik^&Ik<�O i CJO j O�^D<��"O7<Ua*?DB � h i �"�|pX&O4>%O�HG=/C j =�<�O7HGIF=/C�y&HG?/y.OQH�<UIJO�:#=�HGOÃHGO��D9*IFHGO�B�<�?¦:|?DC�>*O¥<�a.OB&IkE�EGO�HGO�^n< j =,<Ua*O j =�<JIJ_�=�C\O��D9*=7<�IJ?D^;:-y&HGO/:GO�^n<�OQB�IF^�<Ja*O{:VO�_7<�IJ?D^Ä�lZ¹e�a*O O�CJO�_7<�HGIJ_�=�C _�?D^*Bn9._7<�I�>*Ik<¸� ��¦ �GR <Ua*O7H j =�C

_�?D^*BD9*_7<�I�>*IF<F� ��§ ��=/^&BÃ<Ua*O+>2?DCk9 j IJ_ j =S:|: � ρ�"?/E"=�CJC)<Ja.O:�<�H�9._7<J9&HG=�C j =,<�OQHIJ=�C�:�=Q^.B�=�IFH�lD:VO�O=¨ª©¬«i­6®�¯*pZ¹e�a*O±°A?/9.^*N j ?DBD9*CF92: �3² ��R5<Ja.O���?DI�:|:G?D^�_Q?DO�E�EIJ_�IJO�^�<

��³ ��=�^*B0<Ja.O-HGO�:GIJB/9.=QC�:�<JHGO�:4: ��¦F´ ��?/E�<�a*O5�2�0�1X2pZ¹e�a*O�=/IFH�_za�=�HG=/_7<�OzHGI�:�<UIJ_�:�l0>*?DCk9 j IJ_ j =D:4: � µª¶¸· ¹ ��=�^&B

j ?DCJ=�H j =S:|: � º �"B�O7y.OQ^.B�?/^¼»¾½À¿�=�^*B�Ik^&Ik<UIJ=�C)>*?DCF9 j O�RHGO�:�y.O�_7<�I�>*O�CF�%p

Á�ÂVÃiÄ3Å�ÆXÇÉÈiʸËVÌ ® Ê�ÍÏÎ ®�Ð ËÉÑiÍ6Ò ®¾Ó=© Í ® ÊÔÎ ©Õ­ÏÐ�Ö¤Ð×®ÕØ Ñ�ËÕÊ�ÍÏÒ ® Ñ ©�« Ê�Î3Ù © ÍÏÎ3ËVÚË�Ñ�ÍÏÒ ®ÛÓ ÎÏÙ×Ê¸Ë © Ù�Í ÖX© ÍÜËVÊ�ÝÞÀÂÉßÜÅ�àaáÜÂVÄ âãÔäÉåÜæèç éã3êëåÜæìÇ íîç ρã éiï åÜæ�ð�ç ñiòó3ô¬õÔöª÷¤ø6öiùú�Î ­ Î3Ù¬ËVÚ ûîËVÍ Ö�Ð×®¬Ø ü¬ýLý ¯�þV¯�ÿ �Lý�¯úiÎ�� ¯ ûîËVÍ Ö�Ð×®¬Ø ü Ý � ¯L¯��Lý üÕýLýXýúiÎ û�� ûîËVÍ Ö�Ð×®¬Ø ��� ¯L¯��Lý üÕýLýXýÈkË ­�XÐ Î ­ ÎÏÙ¬Ë�Ú � �������� ò � ����� �����

â��! #" $ &%'�( ü ��)�Ý þ* ¯�þ�+-,/.ü¬ýLý ¯�þV¯�ÿ �LýXý

0�Ë ­3Ø �kÝ �×ü¬ý 1 ¯ � � ü � þ�ýLý ü�¯�ÿ Ý �, ÒXÊ×Ë Ó Î ÖXÓ � Ý þ

×ü¬ý 2 � þ Ý � ��ü ( ý ���3�È�465 ú ûîËVÍ Ö�Ð×®¬Ø ý Ý ü ( �7)�( üa¯iýXý8 Î6Ê ûîËVÍ Ö�Ð×®¬Ø 97:<; = >%-�

ρ:#; = &%-� ñiò :#; = &%'�����?A@6Bª��C��J£i�ª�DC��i�ª��CÕ¢�����EÛ��F�¢GBª�AHF£�E¾�IC����KJILL�¤�NMIO�¢�B ��L¡ ��CPHi�É¢¸�#L=�J£L¢���C���£��

@A:ÃB&I�:G_792:GO�B¿IJ^Ñ:GO�_,<�IJ?D^�g*p�c � O��D9&=Q<UIJ?D^ÑÉn�1<Ua*O�N&=D:CJI��*O�HG=7<�O�BY�&��<�a.O�O�^*O�HGN&O,<�IJ_ j =Q<�O7HIJ=/C�I�:-_�? j y.?&:VO�B�?DE\[RQ�R%¨SQ=�^*B�ÌSQ�¨�>*=�y*?DH�R%<Ja.O7��=�HGOYy&HGO�:VO�^�<�IF^Í<Ua*OY=�IkH���9�<�^*?/<�IF^)<Ja.O:G= j O�y.HG?/y*?DH�<UIJ?D^2p�[AO4>*O�H�<�a*O�CJO�:|:A=D:�=)EIJH4:�<�=QyDy&HG?D=/_�a"IF^�<Ja.ON&=D:YN�O�^*OzHG=,<�IJ?D^�EHG? j <Ua*O�O�^*O7HGN&O,<�IJ_ j =Q<�O7HGIk=�CJR\�!O�_�?D^2:VIUB�O�H<Ua*OLN&=D:�N�Oz^*O�H�=Q<�OQBA=D:�=/IFH|pe�a*O�[09;:4:VO�Ck< �>TRU �5^n9 j �*O�H-a&=D:5�*O�O�^�92:GO�B)<�?)B&O7<�O�H j Ik^.O

<Ua*OL_�?D^D>*O�_7<�IJ?D^�_�?DOQEE�IU_QIUOQ^&< �#V �RDN�I >*OQ^!����lWXY[Z\ ]

= � c7d��

( )^_ `

abdcefhg ijklmn

= oprqs � c&c,�

x a.OQHGOItT½vu`=Q^.Bxw�=�HGOL_�?D^2:�<�=�^�<2B&O,y.O�^*BDIF^*NY?D^�<Ua*O5>*=�Ck9.O?DE\<�a.O£y&HG?DBD9*_7<zy�{

×»r{/p.e�a*OY8LH=D:�a*?/E � y�{z�5=�^*B��2HG=�^&Bn<UC � »r{z�

^�9 j �*O�H4:£?DE�=/IJH0_�?D^�<�=QIF^*O�B~Ik^�<Ua*O j IJ_7HG?�_�=7>*IF<J��=�HGOY�0HGIk<F<�O�^=D:�E�?DCUCJ?n�£:Ll

( )||}~r����

��� ���� ���� � �����d�

µρβ −

= � c7��

x a.OQHGO��µI�:�<Ja*O�NDH=7>*Ik<¸���0Ik<�a�=�>*=/CF9*O�?/E`É*p�È j Ò:���R �I�'� ��Rρ�G� �����D�'� ��=�HGO�<Ja.O�=/IFH¥O�fDy*=�^2:GIJ?/^Ó_Q?DO�E�E�I�_�IJO�^�<�RÂB&O7^2:GIk<F�.RBD�&^*= j IU_5>*I�:G_�?&:VIk<U�YHGO�:�y*O�_7<�I�>*O�CF�%pe�a*Oz��=�HGO-<�O j y*OzHG=,<U9&HGO0B�O7y.OQ^&B&=Q^D<�=�^*B�<Ua*O�>*=/Ck9.O�:5_Q=�^A�*O

EG?/9.^*BA=Q<2=7< j ?&:�y&a*O�H�IJ_�y&HGO�:|:�9&HGOLIF^�<Ua*O�CJI¸<�OQH=Q<F9&HGOY��c%�&�Gp

Page 5: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

e�a*O1=/IFH�I�:�_�?D^�<�=QIF^*O�B�Ik^�=Å:V�D9.=QHGO1_�=z>*Ik<F�%pA}�<:�CJ=7<�O�H�=/CB�I j O7^;:VIJ?D^"I�:�����=�^*B�a*OzIJNDa�<�I�:�� � :VO�O������� ����~=�^*B"�Gp x O_Q=�CJ_79*CU=Q<�O�<Ja*O!8-H=D:�a*?/Em^�9 j �*O�H�E�?DH�<Ua*O�a*IJNDa*O�:�<�_�?D^*BDIF<�IJ?/^;:\Ik^y*HGO�:4:�9&HGO�=Q^.B£<�O j y.O7HG=7<J9&HGOLIF^�<Ua*OL_�=7>*IF<F�Y=D:\E�9*^._,<�IJ?D^�?/E�<�I j O�R�0a*O�^¥<Ua*O j =Qf&I j =�C�O�CJO�_7<�H�IU_Q=�C�y.?/�AO�H+:�<J9.B�IUOQB�I�:"=QyDy.CkIUOQB� c7d�d j�x �Gp x O¡?���<�=/IJ^.OQB = y.O�=QÁ ?/EËy*HGO�:4:�9.HGO¡?DE»���g����2p ]D���2=�=Q^.B+=�<�O j y.O7HG=�<k9&HGO~=Q<0<Ua*O�a*O�=Q<UIF^*N�y.Ck=�<JE?DH j� ¿�� ��� � ?nE�bDg&Ä�^7T�R%<Ua*O�8�HG=D:�a*?/E�^&9 j �*O�H�I�:LO��n9.=/C�<�? ��d���=�^*B<Ja*O�=�IkH��2H�=�^*B/<�C�^�9 j �*O�H'I�:6O��D9*=�C�<�?Ad*p ]��Gc��&�Vpy�{

×»r{��Óg�Ä&g%p Ä Ô2Ô c7]�d&d�<Ja&=Q<µI�:Ï<Ua*O�CkI j Ik<1?/E�<�a*O

_Q?D^.Bn9._7<�IJ?D^YHGO�N&I j OY��c%�&�|p�e�a*O�HGO7E?DHGO-<�a*O�=�IkH'_�?D^�<�=�Ik^.OQB`Ik^�<Ua*O_Q=z>*Ik<U�+I�:�IJ^+=�y&9&HGO�C��+_�?D^*Bn9._7<�IJ?D^"HGO�N�I j O�=�^*B§_�?D^D>*O�_,<�IJ?D^_Q?&OQEEIJ_�IJO�^�<2I�:5:GO7<*<�?Ad*p�F������£�E¾����F'F-JDE����i¡e�a*O�B&IkE�E�92:GI�?D^�_�?DOQE�E�_�IJO�^�<'_�=Q^��*OA_�=/CJ_79.CJ=Q<�OQB�=D:5E�9.^*_7<�IJ?D^

?nE'<�a*OA^�=�<J9&HGOA?/E5<�a*OAN�=S:|R*<Ja.O£y&HGO�:4:�9&HGOA=/^&B�<Ua*O0<�O j y.O�H�=�<k9&HGOIk^�<Ua*O�_Q=z>*Ik<F�Y=S:Ql2�Gc��D�

� � !!""#

$% &&'()+*-, ./0/0

./+12

3456+

= � cQbD�

7 IJ^*BDIJ_�=�<JIF^*N�<Ua*O j ?DCJO�_79*CJ=�H�>*?DCF9 j O0?/E�<�a*OÅ_Q?D^;:�<�IF<J9*Oz^�<:t#=�^*B98�=�^*B º =D:�<Ja*O j ?DCJO�_79*CU=QH��0O�IJNDa&<�RnE?DH�<Ja.O-=�IkH 7 ��Ä*É�p É=Q^.B º ��Ä*È*p É;pm}�^"?/9&H j ?DB�O�C5<Ja.O)B&IkE�E�9;:GIJ?D^"_/?DOQEEIJ_�Ik<�O�^�<�_�=Q^^*?/<`�*O+_�?D^2:GIJB�O�HGO�B¢_�?D^2:�<�=�^n<�=S:Ç<Ua*O�<�O j y.O7H�=�<J9&HGO"=�^*BÂ<Ua*Oy*HGO�:4:�9&HGO�IF^Ð<Ua*O�_�=7>*Ik<U�¼_�a*=�^*N�O�:¢IJ^�<�I j O�=Q^.BÐ=�HGO¤^*?/<a*? j ?DN�O�^*O79;:\Ik^{:�y&=�_QODp�F��:�� ��E�B ��¡DH��i��¢������;�£L¢¸���i¡e�a*O�O�CJO�_7<�HG?&ZJ<Ja*O�H j =/CUR j O j ��HG=�^*O~B�O7E?DH j =,<�IJ?/^1=�^&B+N&=D:

N�O�^*O�H�=�<JIU?D^ j ?DB�O�C�:�=/HGO�I j y.CJO j O7^�<�O�BÐ9&^.B�O7H¤T5¨L�1X&¨ W��9*CF<�I¸y&aS�*:GIJ_D:7p�e'a*O)?/y&<�I j IUP/O�B j O�:�a*Ik^&N�E�?DH£<Ua*O�:GI j 9.Ck=�<JIJ?D^I�:5B�O7<�O�H j IF^*O�BA���Yy*O�H�E�?DH j IF^*NY=�^�Ik^&B�O7y.O�^*B�Oz^�<�NDHIJB�:�<J9&B��Y<�?j IF^&I j IJP�O�<Ua*O j ?DB&O�CJIF^*N¢O7HHG?DH|p x a*O7^�<Ja*O§_�a�=�^*N&O"Ik^#<Ua*O:V?DCk9&<�IJ?D^ �*O7<J�!O/O�^ :�9D�2:GO��n9.O�^�< :�<U=/N&O�: ?/E j O/:Ga*IF^*NHGO,EIJ^.O j O7^D<�I�:�_�?D^2:GIJB&O7HGO�B�<�?��*O)^*O�N&CJIJNDI��*CJO�R6<Ua*O�CJ?/�0O�H4R���9&<:�<�IUCJC�:�9�E�EIJ_�IJO�^�<R j O�:Va�HGO�:G?DCk9�<�IJ?D^)I�:5Á*O7y&<�p������< =��>`IJCJC�92:�<UHG=,<�O/:<Ja*O�y.HG?D_�OQBD9.HGO�E�?DH)<�a*O"a*O�=7<�IF^�N1y*CU=,<UE�?DH j l-<Ua*O�y.?DCk�.:VIUCJIJ_�?D^j IU_7HG?Da*O�=Q<�OzH j =/f&I j 9 j <�O j y.O�H�=Q<U9&HGO =Q^.B <�I j O ?/E:VI j 9.Ck=�<JIU?D^¹I�:�_Q? j y.=�HGOQBÆ= j ?D^*NÆBDIkE�EGO�HGO�^D<�:�<�=/N&O/:�?DEj O�:Ga*IF^�N�HGO,EIJ^.O j O7^D<4l�<Ua*OY_�? j y*HG? j I�:GO0�*O7<J�AO�O�^�:GI j 9.Ck=�<JIJ?D^<�I j O)=�^*B"=/_�_79&HG=/_4�"I�:YHGO�=/_�a*O�B�E�?DHA<Ja*O�HG?/9*^.B�O�B�y*?DIF^�<�=�^*BN�I�>*O�:6<�a*O�N�?D?DBY?/y&<�I j =�C j O�:Va.Ik^&N%ph�?DH0<Ua*O)a*O�=7<�O�HLy.Ck=�<JE?DH j R6<Ua*O)?/y&<�I j IUP/O�B j O�:Ga�_�?D^2:GI�:�<:

?nE�]�]&d&]�<�O7<�HG=�a*O�BDH�=�C�=�^�B�y&HGI�: j O�CJO j O7^�<:�?D^�?D^*O��D9.=QH�<�O�H�?/E<Ja*O j IJ_7HG?Da.OQ=�<�O7HA:�<JH�9._7<J9&HGO/p?�@����� A�CB):�a.?/�`:!<�a.O j O�:Ga�IF^*N~?/E<Ja*O1a.OQ=�<�O7H�y.CF=Q<JE?DH j :GO,y.=�H�=�<�O�BÃIF^Ã<Ua*O�:�9D�2:�<�H=Q<�O1=�^*B#<Ua*OB�IJO�CJO�_7<�HIJ_ j O j ��HG=�^*ODp

D?EGFIH�JLKNMIO�PRQ<SLTVU<W �YX[Z T �]\�^9X UI_ � T �a` Sb_ ��X�c U dX[ef�fgh�I d�YZiXL _ ^j�^j� Sk_ � cj U<WGSk_ ZiXf TlT X[gI� W

D?EGF[H�JfKNm[O�n � Sk_ � cj U�WGSk_ ZiX� TpoIq �Ir�\ _ Sb_ �Ns S�t�S c<gVg[�G� W ��e _ A��eT � T r[ S c<�Nsr tuT ��\v^<� cje�a*O�?/y&<�I j IkP�O�B j O�:Va.Ik^&N©E�?DH+<Ja.OÏ�2�0�1X j O j �*HG=Q^.O

_�?D^�<�=�Ik^2:4��c,d&g£<�O,<UH�=�a*O�BDHG=�C*O�CJO j O�^n<: � :GOQOV�@����� A�Yw/�4p

D?EGF[H�JfKNx[O�y �]\v^�� cz�XIZ _ ^<�N� WGS \ _ ��e T � T rI S c<�e�a*O"?/y&<�I j IkP�O�B j O�:Va&IF^*NÂE?DH)<Ua*O"Än�Ñ=�f�I�:�� j O7<�HGIU_�N&=D:

N&OQ^.OQH=Q<UIJ?D^ j ?DB&O�C�_�?D^�<�=�IF^2:Ac,Ämc,d.b�<JHIJ=�^�ND9.Ck=�H5O�CJO j O�^n<: � :VO�O������ d�Y{/�Gp|��~}¬¡�������Ô� Jª£�� �������X�ª�Ü��� ���L���L¡�¢¬£L¢��Ü�i¡J£i¡�����i� ��CÉ�����L¡�¢¬£i���k£i���Ô� £¤¢¸���i¡

� �*�

� =��

Page 6: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

|F� ?�� �i£L¢���¡DHT�ª��£¤¢ F���C��e�a*O HGO/:�9.Ck<:¾E?DH�<Ja*O j IJ_�HG?/a.OQ=�<�O7HÓ:VI j 9*CF=Q<UIJ?D^¹=�HGO

IJCJCk9;:�<JHG=7<�O�B~Ik^ ����h�< d���np*Ì0O�HGOY<Ua*OY<�O j y*O�H=Q<J9&HGOY?/E�<�a.OYa*O�=7<�O�HB�IJO�CJO�_7<�HIJ_ j O j �*H=Q^.O�I�:�N&I�>*O�^�=�<�<Ja*O~:�<�O�=/B��§:�<�=�<�O�=�^*B�E?DH<Ja*OAE�?DCUCJ?n�!IJ^*N�_�?D^*B&Ik<UIJ?D^2:�l%@�y�y.CkIUOQB��0?DCF<�=/N&O&l%g�����O�CJO�_,<�H�IJ_�=/Cy*?D�0O�H�lDÄ*É�p È j�x �&_�?D^D>*O�_7<�IJ?D^�_�?DOQE�EG_�IJO�^�<2a���d*ph�HG? j <Ja.O�:GI j 9.Ck=�<JIJ?D^.RÍ<Ua*O�HGO��n9.IkHGO�B�y.?/�0O�H�=�^*B©<Ua*O

HGO/:�9.Ck<JIF^*NAO��n9.I�>*=/CUO7^�<*<�O j y.O�H�=�<J9&HGO�B�I�:�<UH�I���9&<�IJ?D^A=�HGOL?���<�=�Ik^*O�B2pe�a*O�N�H=Qy&a�?/E �@����� d���©N&I�>*O�:#<�a*O�a*O�=7<�O�H j =�fDI j =/C

<�O j y.O�H�=�<J9&HGOL=S:�=0E�9&^*_7<�IJ?�^)?/E�<�a*O0O�CJO�_7<�HIJ_�=/C%y.?/�0O�H\�!Ik<�a�=�^*B�0Ik<�a.?/9�<*<�=QÁ&IF^*N�IF^D<�?�=�_�_Q?/9.^�<*<Ua*O�<�O j y.O7H=Q<J9&HGO�O,E�EO/_7<�?D^ ¦��� � p@#_�? j y.=�HGI�:G?D^{�!Ik<Ua�}�^�EHG=�HGO�B � }�K���<Ua*O�H j =�C%_�a&=�HG=/_7<�O7HGIFP/=7<�IJ?�^2Rj =�B�O��0IF<Ja1="T i �A}�� [4@L� i }}�}�� x _Q= j O7HG=�R-O�^�=z�*CJO�:{<�?O|>*=�Ck9.=Q<�O�<Ua*O j ?DB&O�CJIF^*N#OzH�HG?DH|p�h2HG? j <Ua�I�:�NDHG=7y&a2R��*OQCU?/��=j =�fDI j =�C\<�O j y*O7HG=7<J9&HGO�?/E�c&c%�D^7T¢<Ja*O j =/f&I j =�C�O7HHG?DH0Ik^�<Ua*Oj O�=S:�9&HGO j O�^n<�I�:�b&b��ÇR�=7�*?�>*OL<�a*I�:�<�O j y.O7HG=�<k9&HGOL<Ua*O j =�fDI j =/COQHHG?DHLHGO�BD9*_�O�:!<�?§c%����p2e�a*O�B�IFE�EO�HGO�^*_�OA�%O7<J�AO�O�^�<Ua*O j ?DB&O�C=Q^.BÊ<Ja*O j O�=D:�9&HGO j O�^D<:¿_�?/9*CJB �*O O�fDy*CJ=�Ik^.OQB �&�Ê<�a*O9*^*_�O�H�<�=QIF^�<F�½<�?½Oz>*=/Ck9.=Q<�O¼<Ja.O�>*=/CF9*O�?nE¤<�a*O¿O j I�:4:VI >*Ik<F�_Q?&OQEEIJ_�IJO�^�< ��� ��?/E�<�a.O#B&IkE�EO�HGO�^�< j =�<�O7HGIF=/C�:�92:GIk^&NÃ<Ua*O�}�Kj O�=S:�9&HGO j O�^n<Gp

D?EGFIH�JLK��IO�y ��\v^<� cj X[Z _ ^<�+��� SLQ ��\�� T9T � _ A��e T X�g<� W ZiXf _ ^z� S \R��Lc<�k Sk_ �X�cV� cze W �<gI� cj _ ^<�N� W ��e _ =X�� _ ^<�k T SfW[T XIg[� W|F����� �L� M CX£�¡ �ì�F�7F���C��J£L¢����i¡e�a*O�HGO/:�9.Ck<:ÂE?DH1<Ja*O¤O�CJ=D:�<JIU_ j O j �*H=�^*O¥B&O7E�?DH j =�<JIJ?D^

:VI j 9.Ck=�<JIU?D^¡=QHGO IkCJCF92:�<JH=Q<�O�B×Ik^ �����< =�����Dp�Ì0O�HGO·<Ua*OB�O7EG?DH j =7<�IJ?D^ �� �5?DE�<�a*OL�2�0�1X j O j ��HG=�^*OL�!Ik<Ja�y&=�HG= j O7<�O�H4:² ��gn���;=�� ³ �2d2p g.È�� ¦"! ��d`I�:5N�I�>*O�^�=7<�<Ua*O0:�<�O�=/B���:�<�=�<�O0=�^*B�E?DH<Ja*O�E?DCJCJ?D�0IJ^*N¦_Q?D^.B�I�<�IJ?�^2:Ql�@Ly&y&CJIJO�B�y&HGO/:4:�9*HGO&l)d2p dmc|���2=���%?/9.^*B&=QH���lD_QCU= j y.O�B�=�CJC�=�HG?/9.^*B0<Ja*O j O j ��HG=�^*ODp

D?EGFIH�JLK�#IO�P?Q�SfT U<W �YXIZ _ � TVU �k Sb_ �I A�NgI��\ _ A� rI� _ �X�c9ZiXL SB����%$ T×B&���'$ T gI�� W ��e _ A��e T � T rI S c<�+^j� Sb_ � cz U<WGSb_ ZiXL Tpo

DREGFIH�JLK�(IOIn � Sb_ �k T9SLQ � T SfW[_ � T U �k Sk_ �I A�*)I\ o � W �]e _ A�e SfW[U X&+ �b ������ d�,�'��NDI�>*O�:©<�a*O¼<Ja.OQ?DHGO7<�IU_Q=�C¢=�^*BºO�fDy*O�H�I j Oz^�<J=�C

j =/fDI j 9 j j O j �*HG=�^*O§B�O7E?DH j =7<�IJ?D^¢=S:�=+E�9*^*_7<UIJ?D^�?/E)<�a.Oy.HGO/:4:�9&HGO�IU^2:�IJB&O�<Ua*O"_�=7>*Ik<U�2pLe'a*O"_79*H�>*O�:�_�?DHGHGO�:�y.?D^*B1<�?1=bDd%Ø j <Ja.IJ_7Á#�2�0�1X j O j �*H�=�^*O�E=7�*HGIJ_�=7<�O�B�=Q< W @0@YX#=�^*Ba&=7>*IJ^&N�= °A?/9*^*N¥��?DBn9.Ck9;:+?/E"gn����= j O�=D:�9*HGO�B��&�©<�a.O-&.59.CJN�OÆ<�O�:�<0/»�kÄ&d�� =�^�B¡^&=�^*?DIk^.B�O7^&<J=�<�IJ?/^ �kÄmc7�|p�e�a.O_�? j y*=�H�I�:G?D^~:�a*?/�`:-=`B�IkE�E�OQHGO�^*_�OAIk^)<Ua*O0^*?D^.Z�CUIk^*Oz=QHGI�<F��?DE\<�a.O_79*H�>*O�:4R6O�fDy*O�HGI j O7^&<J=�C5_79&H�>*O/:0=/HGO�=�C j ?&:�<�CJIF^*O7=�H��*O�_Q=�92:GO)?DE<Ua*O�:GO,y.=/HG=7<�IU?D^�O7E�EGO�_7<�?DE5<�a*O j O j �*HG=�^*O�EHG? j <�a*O{:�9S�2:�<�HG=,<�O=D:�y&HGO/:4:�9&HGOLIF^*_7HGO�=D:GO�:,p

D?EGFIH�JLK213IO�y � T r< S c��YgI��ZiXL T9Sk_ �X�c9ZiXL S B&���%$ T×B����4$ TT � T rI S c<�+r<�L�Gcj65��'$ T _ ^<��e87 o

|��: @6Bª�AHF£�E H��X¡ª��CX£¤¢¸���i¡ £�¡�� �Ï¢ E��[���i��J�¢¥�Ô�i¡@A:\=LEIFH4:�<2=�y�y&HG?D=�_7aYIF^�<Ua*OLI j y&CJO j O�^D<�=7<�IJ?D^Y?/E�<�a*O j ?DB�O�C

B&O/:G_�HGI��*IJ^*N+<Ua*O"N&=D:�N�O�^*O7HG=7<�IU?/^2RA:V? j O§:GI j y*CFIkE=Q<�IU?/^2:)�0O�HGOj =/B&O&lZ¹@«Än�×=/fDI�:�� j�j O7<�HGIJ_ j ?DB&O�C1I�:ÃI j y&CJO j O�^D<�O�BÐ<�?

>*=/CUIJB�=7<�O�<Ua*O�_�?D^*_�O7y&<§�%O�EG?DHGO�I j y&CJO j O�^D<�IF^�N�=�E�9*CJC�b/�j ?DB�O�C�R

Page 7: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

D?EGF[H�JfK21�1fO�� X TVU<S A��\=Xfc r<� _ + �f�bcV� Q[U �k A� T �fc _�SfW�S cjg\v� T � WGSk_ �Lg ���\v� W _ \RZ�XL _ ^<�NgI��ZiX� T Sk_ ��Xhc9XIZ S B������ TVQ B&����� TT � T rI S c�� oZ @A:Ã<Ua*O�N&O�? j O,<�H��¾?DE¢<�a.O�a*O�=�<JIF^*N¿HGO�:GI�:�<J=�^*_�O�I�:

B�IkE�E�OQHGO�^D<)EHG? j <Ua*O�bS� j ?DB&O�C�RY=#_�=�CJI���HG=7<�IU?D^¥?/E�<�a*O1ÄD�j ?DB&O�C0Ik^"<�O�H j :�?/E`=Qy&y.CJIJO�B�<�O7^2:GIJ?D^+=�^*B"?���<�=�Ik^.OQB"a*O�=�<JIF^*N<�O j y.O�H�=�<J9&HGO¾�!=D: j =�B�O ��� j ?DB�IFE��*IJ^*N <Ja.OºO�CJO�_,<�H�IJ_�=/C_Q?D^.Bn9._7<�I�>*Ik<F��?/E�<�a*O�a*O�=,<�Ik^*N j =�<�O7HGIF=/CURZ e�a.O�= j ?/9*^&<�?/E�OQ^.OQHGN&O7<�IJ_ j =Q<�O7HGIk=�C�B&O7y*?&:GIk<�O�B)?D^)<Ua*O

a*O�=,<�Ik^*N¼y*CF=Q<UE�?DH j I�:�_Q=�CJ_79*CF=Q<�OQB¿E�HG? j <Ja.O©y&HG?/EIJCJ? j O7<�O�Hj O�=S:�9&HGO j O�^n< � :GO�O�:VO�_7<�IJ?D^~g*p Ä��R;<Ua*O j O�=�^)<Ja&IJ_7Á&^*O�:4:L?/E5<�a*OB�O7y.?&:VIk<�OQB j =�<�O7HGIF=/C*I�:0cQp g�� j p x O-9;:GO-<Ua�I�:�>*=/Ck9.O5<�?�_�?D^2:GIJB&O�H=Q^Y9&^&IkE?DH j B�O�:�y*?&:GIk<�?/E�<�a.O j =7<�O�H�IU=QC�RZ e�a.O�N&=D:�CJI��*O�H=Q<�O�B1E�HG? j <Ua*O"O�^*O7HGN&O,<�IJ_ j =Q<�OzHGIk=�C�I�:

_Q? j y.?*:GO�B�?DE j ?DHGOY<�a�=�^�?D^.O):�y.OQ_�IJO�: � :VO�O�:|O�_,<�IJ?D^�g2p Ä&�GR�Ik^<Ja*I�:�EIJH4:�<�=Qy&y&HG?D=/_�aY<Ja.O0N�=S:�I�:�_�?D^2:GIJB&O�HGOQB�=S:�=�IkH�R*O|>*O�^�IkE�<�a*Oy*HG?/y.?DH�<JIU?D^¢?/E�<�a*O"Ik^&B&I�>*IJBD9*=�C�:�y*O�_�IJO�:�=/HGO�^*?/<{OQf&=/_�C��1<�a*O:V= j O�RZ e�a.O½N&=D:¼N�O�^*O7H=Q<�IU?/^ËI�:¼_Q?D^;:VIJB&O7HGO�B <�? a.=,y&y.O7^

Ik^2:�<�=�^D<�C���?D^*_�O�<�a*O`IJND^�IF<JIJ?D^)<�O j y.O7HG=7<J9&HGOYI�:�HGO7=/_�a*O�B.R%IF^)<Ja&=Q<�0=���<�a*O�:GI j 9&CJ=7<�IJ?D^¦I�: j =�B�O¤Ik^�<J�!?©<�I j O�:�l)h�IUH4:�<�~=Q^:�<�=�<JIU?D^*=�H��¢:GI j 9*CF=Q<UIJ?D^¢_�?D^2:GIJB&O7HGIk^&N1<Ja*O"O7^.OQHN�O7<�IU_ j =Q<�O7HGIF=/CB�O7y.?&:VIk<�OQB�?�>*O�H6<�a.OLa*O�=7<�IF^�N�y.CJ=7<JE?DH j pD}�^Y<Ja&I�:5:GI j 9&CJ=7<�IJ?D^�<Ua*O<�O j y.O�H�=�<J9&HGO�B&I�:�<JHI���9&<�IJ?D^1I�:�?���<�=/IF^*O�B�E�?DH�=�N�I�>*O�^1=�y�y.CkIUOQB>%?DCk<�=/N&O/R�Ik^+<Ua&I�:�_�=D:GO �%p b��Yp�e\a.O7^*R5<Ja.O�=S:|:GO�:|: j OQ^&<A?/EY<�a*O�n9.=�^�<JIF<F��?/E j =Q<�OzHGIJ=�C6<Ua�=�<'HGO�=/_7a*O�:-<Ua*OYIJND^.I¸<�IU?D^)<�O j y.O�H�=�<k9&HGOI�: j =/B&OA=D:L:Ga*?/�!^�Ik^������< =� �=�*p*e�a&I�:�I�:�<�=�Á*Oz^�=D:�<Ua*OAIk^.I¸<�IF=/C_Q?D^.B�Ik<�IJ?/^YE?DH6<�a*O�^*O�fn<m:�<�O7y��2X&O�_Q?D^.B�C��.R�?D^*_�O-<�a*OL�D9*=�^�<�I¸<F��?DE<Ja*O j =7<�OzHGIJ=�C+<Ja*=7<ÅB�O�_�? j y*?&:GOD:¥IU^�N�=S:ÃI�:ÃÁ&^*?/�!^2R�<Ua*O<JHG=�^2:GIJO7^D<m:VI j 9&CJ=7<�IU?D^�?/E�<�a*O�N�=S:\O4>%?DCk9&<�IJ?D^�_�=�^A�*OL_�=/CJ_79.CJ=Q<�OQB=D:�=ÐE�9*^._,<�IJ?/^º?/E¤<�a*O¼O�CJO�_7<�HGIJ_�=�C¢=Qy&y&CJIJO�B�y*?/�AO�H7p"e�a*O?/�2:GO�H�>*=Q<�IJ?D^ y*?DIF^�<¤�0a*O7HGO¿<Ua*O¿<UH�=�^2:GIJO7^�< j O�=D:�9&HGO j Oz^�<:�0a*O�HGO j =/B&O-I�:5:�a*?/�0^�IF^ ����h�< d���k� p

D?EGFIH�JLK21��<Ojq[_�Sk_ �X�c S 0� � W ��e _ dX�� _ ^<�b T9SfW \v� T � WGSb_ ��X�c oP ) SLW � Sb_ ��XLc9XIZ _ ^<�N�Lc<�k =]� _ ��e T Sk_ �b �� SLW + � _ ^ S+_ � T9U �k Sk_ �I =�^<��^<�k _ ^ S c _ ^z�N�fc�� _ ��X�c _ � TVU �b Sk_ �I A� o� rz\=�k ) Sb_ ��X�c U XI� c _ ZiXL _ ^j� _ S cz\v���bc _ \A� T � WGSk_ �X�c

������ d� � 5�:�a.?/�A:\<�a*OLE=D:�<�_�?D^*_�O�^�<UHG=7<�IJ?D^�O4>%?DCk9&<�IJ?D^ � Ä j :G�_�? j y*=�HGOQB�<�?Y<Ja.O£<Ua*O�H j =�C�O4>*?DCk9�<�IJ?D^�?/E5<�a*O`:��.:�<�O j R�:�<�=�H�<UIk^*NEHG? j <Ua*O IF^*Ik<UIJ=�C«=�IkH»_�?D^*_�OQ^&<JH=Q<UIJ?D^ � gmcQp g&g j ?DCkÒ j� �_�=/CJ_79.CJ=Q<�OQBÆ=Q<À= j �*IJO�^D<�<�O j y*O�H=Q<F9&HGOÊ=�^*BÆ=,< j ?&:�y&a*O�H�IU_y.HGO/:4:�9&HGO½<�?�<�a.O)E�IU^�=�CL_�?D^*_�O�^n<UHG=7<�IJ?D^+?/E0<�a*O�_�=7>*Ik<F� � ��c/p�bDgj ?DCkÒ j� �|p

D?EGF[H�JfK21� <O�� S \RefX�cjef�bc _ Sk_ �X�cV� )[X W � _ ��Xhc9XLc _ ^<�YXLr�\A�b ) Sk_ �X�cU XI� c _ cj� S _ ^<���%� y q�T � T rI S cj������� d� �=>`:�a*?/�`:�<�a*OLO4>2?DCk9&<�IJ?D^�?/E�<�a.O-<�O j y*OzHG=,<U9&HGOLIF^�<Ja.O

_�=7>*Ik<U��=D:-<�a*OY=/_7<J9.=Q<�?DHLI�:-<J9&H^*O�B�?/E�E�R�Ik<'<�=QÁ.O�:0b;p �.:-<�?�HGO�=/_�a<Ua*O0:�<�=7<�IJ?�^*=�H���:�<�=�<�O�=Q<2= j �*IJO�^D<*<�O j y*O�H�=�<k9&HGODp

D?EGFIH�JLK21fM[O�� � TVU �b Sk_ �I A�N�&)IX W � _ ��X�c9Xhc _ ^��YX�rz\v�k ) Sk_ �Xfc U X�� c _��c<gI�b _ ^<����� y q T � T rI S c<������� d� ��B�:�a*?/�A:Y<�a*O~O|>*?DCF9�<�IU?D^1?/E�<�a.OÍy&HGO�:4:�9.HGO�Ik^"<Ja.O

_�=7>*Ik<U�"=D:�<�a*O�=�_7<J9&=Q<�?�HYI�:�<J9&H^*O�B�?/E�E4p'@£E�<�O�H~c j :�E�HG? j <Ja.OB&OQ_�? j y*?&:GIk<�IJ?D^�?/E~<�a*O�ND=D:4RY=#a&IJNDa¥y.O�=,Á�?/E�y.HGO�:|:�9.HGO#I�:B&O,<�OQ_7<�O�B � :GO�O)<�a*O�P�?D? j Ik^ ������ �� �fB��|p�e�a*O�=D:4:VO�:4: j O�^�<�?/E<Ua*O0O7E�EGO�_7<'?DE�<�a*I�:\y&HGO�:|:�9&HGOLy.OQ=�Á�?D^�<Ua*O0N�CU?��*=/C%y.O�H�E?DH j =�^*_�O?DE�<�a.O�:��2:�<�O j I�:{9.^*B&O7H�:�<J9.B��%p5@¼EIJH4:�<`:�<�=�<JIJ?D^&=�H���:�<�=�<�O~I�:HGO�=/_7a.OQB`=Q<�d*p d&].bS����=0?D^*_�O�<Ja*O�=/_7<J9.=Q<�?�H\I�:5:�<�IJCkC2?D^2R/<Ua*O�^�<Ja.Oy.HGO/:4:�9&HGO{E�?DCJCJ?D�A:L<�a*OY<�O j y*O�HG=7<J9&HGO�O4>%?DCF9�<�IU?D^�Ik^~<Ja*O�_�=7>*IF<F�*RHGO�=/_7a.Ik^&N£<Ua*O�E�IF^*=�C*y&HGO�:|:�9&HGOLd*p d�ÄD�/����=np

Page 8: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

D?EGF[H�JfK21LmIO �j A��\d\v�I A�N�&)IX W � _ ��X�c9XLc _ ^��YX�rz\v�k ) Sk_ �Xfc U XI� c _�<c�gI�k _ ^<���"� y q T � T r[ S c<��F� � ����M�£�� �����;�X��£KE Ea��E!E����X¡�¢e�a*O�N�?D=�CL?/E0<�a.O)E�IU^�=�C0:��*:�<�O j�j ?DB&O�CLI�:�<�?~y*HGO�B&IJ_7<-<Ua*O

= j y&CJI�<J9*B&O�?/E�<�a.O#=�_,<U9*=7<�IJ?�^�=D:�=#E�9.^*_7<JIJ?D^�?/E~<�a*O�IF^ny&9�<:VIJND^.=/CUR.<Ua*=7<�I�:|R�<Ua*O j =/f&I j =�C�B�O7E?DH j =7<�IJ?/^�?/E'<�a*O j O j �*H�=�^*O=D:¡= E�9*^._,<�IJ?D^ ?/E«<�a*O O�CJO�_7<�HGIJ_�=/CÆ=Qy&y*CFIJO�B y*?/�AO�H�� � ���=�%E ��� × 7�� ���4ph�HG? j <Ua*OY=�fDI�:�� j�j O7<�HGIU_ j ?DB�O�C�R�=Yy&HGO/:4:�9*HGOYI�:£?���<�=�Ik^.OQB=D:�=)E�9&^*_7<�IJ?D^�?/EL<�a*O�=Qy&y&CJIJO�B�y.?/�0O�H4R �����< A���LwÇ:�a*?/�`:£<�a&=Q<<Ja*O j IF^&I j =�C#O�CJO�_7<�HIJ_�=/C+y.?/�AOQH�HGO��D9*IFHGO�BÐ<�?�?/��<�=�Ik^¿<Ua*O=/_7<J9.=,<�IJ?D^¢I�:��*È�p ] j�x p�h�HG? j <Ua&I�:Çy&HGO�:4:�9.HGO"B�=�<J=SR!�!O"_Q=�^?/��<�=�Ik^Y<Ua*OLB&O7E?DH j =�<JIU?D^�?/E�<�a*O�aD�ny.O�HGOQCU=D:�<�IJ_ j O j �*H�=�^*O��0Ik<�a<Ja*O!bS� j ?DB&O�C � :VO�O!:VO�_7<�IJ?D^ ��p Ä��Gp�@����� d� �Lw~NDI�>*O�:{=�C�:G?�<�a*O j =/fDI j =�CLB&O7E�?DH j =7<�IJ?D^§?/E0<�a*OOQCU=D:�<�IJ_ j O j �*H=Q^.OL=D:�E�9*^._,<UIJ?D^Y?/E�<�a*O�OQCUOQ_7<UHGIJ_�=�C;y.?/�0O�H7p

DREGFIH�JLK21fxIO �YW Xhr SfW[T X[gI� WjS \d\v�]\=\ T �Lc _ � c \ _�Sk_ �X�c S 0�9\ _�Sb_ � o�NS )�� _ � U A�]\=\v�I A� S c<g T9SbQ � T SfW g<��ZiXL T9Sk_ �X�c9XIZ _ ^j�N� WGS \ _ �eT � T rI S c<� S \ S Z�Icje _ ��X�c9X[Z _ ^<� SbU<U<W �G�Lg�� W ��e _ A��e SLW�U X + �k

�����î�i¡ILL��J E����i¡DE}�^¢<Ja&I�:�=�H�<�IU_QCUO+�0O1a*=z>*O+y&HGO/:GO�^�<�O�B¢= j IJ_7HG?�y*HGO�:4:�9&HGO

:V?/9.HG_�O��*=S:VO�BÑ?D^¾<�a*O�B&OQ_�? j y.?*:GIk<�IJ?D^Ñ?/E�=�^½O�^*O�HGN&O,<UIJ_j =7<�O�HIJ=�C\<�?�IJCJCF92:�<JH=Q<�O�= j 9&CF<�I¸y&aS�*:GIJ_D:0_/?D^*_�O7y&<�IJ?D^�=�y�y&HG?D=�_Qa�*=D:GOQB`?D^{<Ua*O�T5¨L�1X&¨ W ��9*Ck<UIky&aD�.:VIU_/:�:|?nE�<J�!=/HGODp}�^¢<Ja&I�:�=�_7<J9*=7<�?DH�R�=�^�OQCUOQ_7<�HIJ_�=/CYy.?/�AOQH~?DE~È1�%p b j�x I�:

=Qy&y*CFIJO�B#<�?1<�a*O§a*O�=,<�Ik^*N�y&CJ=Q<JE?DH j <�?�?/��<�=/IJ^�= j O j �*H�=�^*Oj =�fDI j =�C*B�O7E?DH j =Q<UIJ?D^Y?/E-c,d&d � j ph�IUH4:�<0=�^+O�CJO�_7<�HG?&Z�<Ua*O7H j =/C j ?DB�O�CL?DEA=~a*O�=Q<�IF^�N�y.Ck=�<JE?DH j

a�=S:��*O�OQ^�I j y&CJO j O�^D<�O�B�=�^*B�>*=/CUIJBD=Q<�O�B0�D�Y}�K j O�=S:�9.HGO j Oz^�<:|p

X&I j 9*CF=Q<UIJ?D^2:1HGO�:�9.C�<:#?D^�<Ua*O j IJ_zHG?¥�*=�CJCJ?D?D^��*CJ?D�0IJ^*N:Ga*?/� =ÃHGO�=S:V?D^*=z�*CJO¢=�NDHGOQO j O�^n<~�0IF<Ja�O�fny.O�HGI j O7^�<U=/C~BD=Q<U=�pXD<�IJCFC�R2:V? j OAB�I�:V_�HGO7y*=�^*_�IJO�:5=�HGO£E?/9.^*B)IF^�<Ua*O�^*?D^2ZJCJIF^*O�=QHIk<F�)?DE<Ua*OL_79.H�>*O/:7p/��?&:7:GI��*CJO j ?DB&IJEIJ_�=Q<�IJ?D^;:\?D^Y<Ja*O j ?DB&OQC�:Ga*?/9.CJB0�*O_�?D^2:GIJB�O�HGO�B2p�@L<�<Ua*O":G= j O~<�I j O�R-<�O�_�a�^&IJ_�=/CA=D:�y.OQ_7<:4R-=S:�<Ja.O=/BDa.OQHGOz^*_�O5�*O7<J�AO�O�^{<Ua*O!:�9S�2:�<�H�=Q<�O-=�^*B0<Ua*OLO�CJ=S:�<�IJ_ j O j �*H=Q^.O�0IUCJC��*O�?/y&<�I j IkP�O�B�IF^�?DHGB&O7HL<�?�_�a*=�H�=�_,<�OQHIJP/OY<Ua*O j O j �*H=Q^.Oy.HG?/y*O�H�<�IJO�:6=�^*BAIk^._7HGO�=D:GO-<Ua*O0:��*:�<�O j �J:6y.O�H�E�?DH j =�^*_�ODp@¿EIJH4:�<Y=�yDy&HG?D=/_�a#?/E�= j ?DB&O�C�E?DH)<Ja.O�ND=S:ÍN&O7^*O�H=Q<UIJ?D^

EHG? j <Ua*O�B�O�_�? j y*?&:GIk<�IU?D^�?/E#<�a.O�O�^*O7HGN&O7<�IJ_ j =7<�O�H�IJ=�C1I�:y.HGO/:GO�^n<�OQB.RDHGO/:�9.Ck<:5:Ga*?/� �D9*Ik<�O-E=D:�<2<�HG=�^2:VIUO7^�<.�*O�a*=7>*IJ?DH'IF^�<Ja.ON&=D:�_�?D^*_�O�^�<UHG=7<�IJ?/^1B���^.= j IJ_�:�IF^*BD9*_�IF^�N§=�E=D:�<0y*O�=QÁµIk^"<Ja.O_�=7>*Ik<U��y.HGO�:|:�9.HGO/R5=�B���^.= j IJ_~O4>*=/Ck9.=Q<�IJ?/^1?/EA<�a*O j O j �*H=Q^.OHGO�:�y.?D^2:GO-<�?0<�a*I�:6y*O�=�Á j 92:�<.�*OL_/?D^;:VIJB&O�HGO�B*Rn<Ua&I�:6�0IUCJC2CJO�=/B�9;:<�?�=ÀE�9.CJC j 9.Ck<�IFy&aD�*:GIJ_�:©_�?/9&y*CUIJ^*N¿�0IF<Ja�<Ja.O j O j �*H=Q^.OB&O,E�?DH j =,<�IJ?D^Y=D:�=-ECk9.IJB.Z�:�<UH�9*_7<J9&HGO�IF^�<�O7HG=�_7<JIJ?D^2pe�a*OA=/B�>*=�^�<�=�N�OAIF^�=/_7<J9.=,<�Ik^&N��!Ik<�a)=�^�O�^*O7HGN&O7<�IJ_ j =Q<�O7HGIk=�C

I�:"OQfDy.?&:VO�B¥=S:�<�a.O#O7E�EIJ_�IJO�^*_z�¥IJ^¥<Ua*O1CJO4>*O�C�?nE�=/_7<J9.=Q<�IJ?D^IF^*_�HGOQ=S:VO�:�HGO�Bn9._�IJ^&N <Ua*O½O�CJO�_7<�HIJ_�=/C¥y.?/�0O�H¼^*O�OQB.:7p#e�a.OHGO�Bn9._�O�BÃy.?n�AO�H"_�?D^2:�9 j y�<UIJ?D^�?/E�<�a*I�:�=�_7<J9&=Q<�?�H j =�Á*O�:"Ik<�D9*Ik<�OL=Qy&y&HG?/y.Ik=�<�O�B�E?DH j IJ_7HG?/ECk9.IJB&IJ_-y.?DH�<�=z�*CJOL=�y�y&CJIU_Q=�<JIU?D^2:\?DH=Qy&y.CkIJ_�=�<JIJ?D^;:µ�!a*O�HGO¢^*?/�0=�B�=z�2:Âa.IJNDa�IF^D<�O�N�H=Q<�O�B�y&HGO/:4:�9&HGO:G?/9*HG_�O�:\=/HGO�HGO��D9&IkHGO�B��FÄDÄ��4p[AO�fn<¤=�N&CJ?��*=/C j 9.C�<�Iky&aD�.:VIJ_�: j ?DB&O�CÃIU^*_�Ck9*BDIF^*N�<Ja.O

IF^�<�O7HG=�_,<UIJ?D^�?/EY<�a*O~=/_7<J9&HG=7<�?DH��0Ik<Ua+<Ua*O�=/_7<J9&=�<�O�B+ECk9*IFB1IF^1=j IJ_7HG?D_�a*=�^*O�C*I�:�O�^D>*I�:G=/N&O�B*p� L�¡ ��ì���i�IHi���L¡�¢�Ee�a*I�:�HGO�:VO�=�HG_�a�I�:-E�9*CJC ��:�9&y�y.?DH�<�O�B�����=Yh�HGO�^*_7a�[A=Q<UIJ?D^�=�C

K�O�:|O�=/HG_�a¼@0N&O7^*_7��@L[AK N�HG=�^�<p�e�a*O¤=Q9&<Ja.?/H4:#N&H�=�<�O7E�9.CJC��=/_7Á.^*?/�0CUOQB&N&O£<�a.OY:GO�H�>*IJ_�O/:�e i @L��=�^*B�Ä&}\EHG? j <Ja.O W @[email protected]\[0K0XYE?DH6<�a*O�IkH'_�? j y.O7<�O�^*_�O�:\=�^*B�:�Á&IJCJC�:�<Ja.=,< j =�B�O�y*?&:4:GI��*CJO<Ua*I�:Â�!?DH�Á�p�e�a.OÃ=�9�<Ua*?�H4:#=�C�:V?��A?/9.CJB�CJIkÁ.OÃ<�?©<�a&=�^�Á .0pT5a�=�9*BDHGO7<�R�X2p�X�=z�*?&Z i <�IJO7^&^*O¢=�^*B¥�mpYE�=�9�EHG? j <Ja.O W T�TT\[0K0XY=�^*BY@Yp/hmp/�"IF^*N�?/<�=Q9&B.Rn�+p��"=Q9&P�=/_L=�^*B�[7p/h�Ik<JHGO j =�^&^EHG? j <Ja.O0}���K�T\��E?DH\<Ua*O�IkH�_Q?DCU=7�*?DHG=Q<UIJ?D^�IF^{<Ua*OLy&HG?GÖ4O�_,<�p�e�a.O=Q9&<Ua*?/H4:\=�C�:G?£�!I�:VaY<�?0<�a*=�^DÁY<Ua*O0T5¨L�1X&¨ W :�9&y�y.?DH�<2<�O�= j E�?DH<Ua*O�IkH�:VOzH�>*IJ_�O/:7p�T�PF¥��CÉ�L¡IL���Ec/p"TLp)��H�=�B� 7HGO�R�X2p�XD9.a�=�HGB&R W p �0OQ^.B�IJOzH4R�8£p�[4=/_�?��2R .0pT5a�=�9*BDHGO7<�R%X*p*XS=7�*?&Z i <�IJO�^&^*O�R/<�?��*O-y&9D�*CJI�:Va*O�B2p

Ä2p"T5=7�*=�C�RY@Yp � � ����R -7e�a*O�H j =�C)=�_7<J9*=7<�?DH��!Ik<Ja¤?/y�<�I j IFP/O�Ba*O�=7<�O�H6E?DH'CJIF�n9.IJBYBDHG?/y�O�Ö4O�_,<?DH4: /DR� � u�����{��V��u��rt��I� U �����3{��t �0?DC�p;c7Ä*b&Z4c,Ä&g � Ä�d&dD�n�2y�y2pU��b�c�Z'�Db&É;p

b*p�Ì0=�HG?/9*_�a*O�R�}Gp��'p�h�p)=�^&BÐXSa*=QE=/IUR~T-p -DX&I j 9.CF=Q<�IJ?�^�?/E:Ga*=7y*O�B«_�? j �×B�HGI�>*OË=S:º=Æ:�<�O7y&y*O�B«=/_7<J9&=�<�?DH�E�?DHj IJ_7HG?/<J�!O/O�P/O�H4:�=Qy&y*CJIU_Q=7<�IU?/^ /�R�� � u����3{�� ��u�� t���� U �j����{�� tè½�0?DC�p�c7Ä.b! mc7Ä�g � Ä&d&d��D�RSy&y�pU�DgDd" >�ng��*p

g2p���=/H�Á2RÛ°`p�ÌYpm=Q^.BÍ��=�H�Á.RÛD�pmT!p -,Ì!IJNDa2ZJE�IUB�O�CJIk<¸� j ?DB&O�CJIJ^*N?DE½� i �1X×HGO�:|?D^*=7<�?DH4:$#¢��=QH�<Ñ}�}zl¼T�?/9&y.CJO�B¹�*O�= j Z:�9D�2:�<�HG=Q<�O¢B/�&^*= j IJ_�:+=/^&B�>*=/CJIJB&=7<�IJ?�^ /�R&%�' º)( �P{���w � � V '��*+�]��'"�0?DCp2c�b � Ä��Gp�*p���=�^*B�I ��=/^.R [Qp � �����UR�-,��?DB�O�CJCJIU^*N-,À:VI j 9*CF=Q<�IU?/^�?/E�^*?�>*O�C<Ua�HGO�Oº=�H j � i �1XË=/_7<J9&=�<�?/H4:., Ik<:¿=�y�y.CJIJ_�=�<JIJ?D^�/DR� u�� � {!u��j� ( ��u���� º ²�º �0/1�3u32 � { � u�� � �54�4�6L½7%�� U { u@���8�92» V *+� ( �:�<;=/>��u32 � { � u�� � � � { (�� �¬½*�0?DC�p�b/g � Ä&d&d1����R;y&y�p.g*b1�" g&g�c/p

Page 9: Multiphysic modelling of a microactuator based on the ...€¦ · # / 0 1 . - .$ , 2$ 43 - 5 . 1 " # *$& 6 7 * 8-9;: *?a@0b ?dcfeg?a@0hgb ijcf=lk ?db&h mjnd9.oqpsr t'=qhg?dcuo-k

�%p��'a�92RL°�p � �@���JR�-Q@1<Ua*O7H j =�C2=/_7<J9&=�<�?/H�E?DH�^*=�^*?&:V_�=�CJO ( u � ( � Uj IU_7HG?&:V_�?/y&�#<�O�:�<�IJ^*N�lLB&O/:GIJND^�=�^�BÅ_Qa&=�H�=�_,<�OQHIJP�=Q<�IU?/^ /�R %�'º)( �P{��3w � � V ' º)( �P{�� � u��5'��0?DC�p;c%� � Ä&d&d��&�RSy&y�p�Ä&g�Ä .Ä��&b�p

]%p�@L<�HGO�Rª@Yp -�@0^*=�C��.:VI�: ?DE ?/9&<Z�?DE�Z�y.CJ=/^*O <Ja.O7H j =/Cj IU_7HG?D=/_7<J9.=Q<�?�H4:8/ %�' º)( �P{���w � � V ' º ( �7{ � � u�� ' �0?DC�p§c��� Ä&d�d����Rny&y2p�Ä�dD�! .Ä�c�b;p

È;p�e�H�9*?D^*N&R)e�a&=/I�Z��-9.=�^*N¥=�^*B¥[AND9S��O�^2R�[A= j ZJe�H�9&^*N&R -�@y*?DCF� j O�HGIJ_ y*IJO�P�?DOQCUOQ_7<�HGIU_ j IJ_�HG?/y&9 j y �*=D:GO�B ?D^CJ= j IF^&=Q<�IJ?�^½<�O�_�a�^.?DCJ?DN/��/DR %�' º)( �7{ ��w � � V ' º)( �7{ � � u7� '�0?DC�p�c7g � Ä�d&d�gD��RSy&y2p1�&bDÄ" D�.b�È�p

É;p .L?/9.HG?/9.Ik^*=�R�e�p � ������R -7�AO/:GIJN&^�=Q^&B¤:VI j 9&CJ=Q<UIJ?D^�?/E~=/^OQCUOQ_7<�HG?&:�<�=�<�IJ_ j IJ_zHG?/y�9 j y E?DH BDH�9&N.Z�B&O�CJI�>*O�H��=Qy&y*CFIJ_�=Q<UIJ?D^2: /�R:%�' º)( �7{ ��w � � V ' º)( �P{�� � u��5' �0?DC�pD] � cQÉ&ÉD]D�GRy�y�p2c�È1�" 6c7È�È*p

c,d2p .�=/O7Á;R [�p � ������R -�@¥y.^*O79 j =,<�IJ_�=QCUC���_/?D^�<UHG?DCJCF=7�*CJOYEC�O�f�I��*CUO=Q^.B�y*?DC � j O�HGIJ_ j IJ_�HG?/ECk9.IJB�IU_5>*=/C >*OLE�=z�*HGIJ_�=Q<�O�B0>*IU=0Ik^�:GIk<J9B�O4>2OQCU?/y j O�^�<0/�R %�' º)( �P{��3w � � V ' º ( �7{ � � u��5'6½ �0?DC�pÅc��� Ä&d�dD�n�Rny&y2p;c,dmc@�&Z4c,d&Ä&d*p

c�c/pzT�a&IJO�^.Z�T\a*?D^&N.R�ÌYp � � ����R -�@L^Ñ?D^2Z�_7a*I�yÑ=/IFHGZk��9.H4:�<JIF^*NB�O7<�?D^*=�<�?/H)E?DH�B&HGI�>*IF^*NÂECk9.IJB.:Í?D^ÅB�I�:�y.?&:V=z�*CJO"CJ=7�2Z�?D^2Z�=DZ_Qa.Iky :��2:�<�O j :8/SRa[7p���IJ_�HG? j O�_Qa2p���IJ_�HG?DO�^*N*p �0?DCp1c7]� Ä&d�d&]���g�c7d� *g�c7]%p

c,Ä2p�X�?D^.N*R x p�ÌYp&=�^*B W IJ_�aD<�O�^D�*O�HGN&R>[7p R -�e�a*O7H j ?&ZJy*^*O79 j =7<�IJ_�R:VIJ^&N&CJO�Z�:�<JHG?/Á.O j IJ_�HG?/y&9 j y��&R"%�' º)( �P{���w � � V ' º ( �7{ � � u��5'6½�0?DC�p�c�� � Ä�d&d��/��RSy&y2p�c|g&Ä��&Z7c7g&bDÄ%p

cQb*pGKL?&:4:GI�R5TLp�=�^&B i :�<9 4>*O/R���p�R�-7�"IJ_�HG?/yD�&HG?/<�O/_�a&^*IJ_�:4R6=�^*O7�<�O�_Qa&^*?DCJ?DNS�ªEG?DH j =QÁ.Ik^.NªO7^.OQHGN&O7<JIU_ j IJ_zHG?&:��2:�<�O j :�lHGO|>*IUO,�Ó=/^.B#y&HG?&:�y.O�_7<�I�>*O�/DR�X&O�^2:G?DH4:�=�^*B¢@0_7<J9.=7<�?DH4:�@AR�0?DC�p�c7Ä�d � Ä&d�d��D�GR/y&y2p/Ä.Én]� ;b�c|d2p

c,g2pGKL?&:4:GI�R\TLp � ������R -7K�O�=/CUIJP�=Q<UIJ?D^�=�^�B�y.OQH�E?DH j =�^*_�O�?DE-<�a*Ik^X�IU¨0ÄnÒGX&Ik[Af j O j �*H=�^*O�E?DH j IU_7HG?&Z�a*O�=7<�O�H0=Qy&y&CJIJ_�=7<�IJ?D^;: /DR� � u����3{��V��u��/t��I� U ������{��vt �0?DC�p,��g � c�É�É&ÈD��R/y&y�p�Ä�g�c�Z�Ä&gD��p

c@�*p4@0H j =�^�I�RD�YpLD�p � �?����R"-7K�O�Z�_�?D^�EIJNn9.H�=z�*CJO-ECk9.IJBY_�IJHG_79.Ik<:��&��%�0�µX�OQCU=D:�<�? j OQH j IU_7HG? j =/_7a*IF^&Ik^*N%/SR��<��� V�� u���'5/ �3u32 '+��uºë²�º � � ºë²�º ���� � �� {��i��u�����½¤� � � ��� '&Ä&Ä�Ä� &Ä�Ä&]2p

c%�2p�ÌA?DC j =�^2R1[Qp��'p RnÌ0O�=Q<;e�H�=�^2:�E�O�H4R �"_08�HG=Q�1Ì!IJCJC � cQÉ&Én]&�c,]2p4e\I j ?&:Va*O�^DÁ*?&R�X2p¢�'p�=�^*B x ?DIF^*?/�£:�Á&�2Z�DAHGIJO�N�O�H4R�X*p Re�a���?DHGIJOLB&O�:6y&CJ=/�/9.O�:\O7<�_Q?D�D9.O/: R�. ��HG=�^*N&O7H � �2=�HGI�:GR�c7É1�mc,�Gp

cQÈ*p4e\=�B j ?DH�R i p .!p � � ���JR -7�"IJf&O�B�E�IU^�Ik<�O�OQCUO j O�^D<1=�^*B=Q<�? j I�:�<JIU_ÏE?DH j 9.CJ=7<�IJ?/^�E�?DH1_�? j y.CJO�f�_�H��2:�<�=/C�: /�R » V *+��'� ��� '<8�R"�0?DC�p��&ÉDRS[A?*p�c � c�É&É�É/��RSy&y2pDÄ&b���Z�Ä�gD��p

cQÉ*pGÌA?DOQC�R�@Yp~=�^&B [G9&CJCJIUO7^*R��§p�T-p�R -,��?DB�O�CJI�:G=Q<�IJ?/^�?/E�=aD�ny.O�HGOQCU=D:�<�IJ_ y*?DC � j O�H j O j �*HG=�^*O B�O7E?DH j =Q<UIJ?D^�/DR/ �º � � º U �i� ( � V *5� ( �:� /1�3u32 � { � u�� � �54�4��L½ � »���{ ( � � ½�y�y�pbn].Z�b&É;p

Ä�d2pG��?DIUCJ=�^*O�R¥T-p � � ���UR -�@�^*=�C��*:GI�:Ð?DE�<�a*O j O�_7a.=Q^&IJ_�=/C�*OQa.=7>*IJ?/9.H�?/E�:Ga*=Qy.O j O j ?/H��"y2?DCF� j O�H j O j ��HG=�^*O�:��&�^�=�^*?DIF^*B&OQ^D<�=7<�IJ?�^2RÊ��9.CJN&Ik^*N =Q^&B y*?DIF^D< j O j �*H�=�^*OB�O7ECJO/_7<�IU?D^Â<�O�:�<: //RT¿ V�( u ����� ( �"��� ( ��w,�0?DC�p�bn].É � Ä�d&d&dD�GRy�y�p�c��1�&Z7c�����p

Ä�c/p��'a*=�^�N.R x p � �����UR -7[A?�>*OQC�HG?D? j ZJ<�O j y*O�HG=7<J9&HGOYEIJH4:�<�Z�CUO|>*O�Cy*=�_,Á.=/N&IF^�N#y&HG?D_�O�:4:�E�?DH j IJ_zHG?&:V_�=/CUO+B&O|>*I�_�O�:8/SR � � u�� ��{���3u��rt��I� U �����3{��It �0?DC�p;c,Ä.b�Zzc,ÄDg � Ä&d&d��D�RnyDy�p,�&g��.Z0���Dg*p

Ä�Ä2p4e�a*?DH4:VO�^.R e-p � � ����R -7�"IJ_zHG?/E�CF9*IJB&IJ_ CF=QHGN&O�Z�:G_�=/CJOIk^�<�OQNDHG=7<�IU?/^ /!R!��� (�� u�� � ½ �0?DC2Ä*É&È � Ä&dDd�Ä&�GR/y�y �&Ènd*p