NON-STATIONAR STOCHASTIC VIBRATION SI(NAL MOD3LIN

Transcription

NON-STATIONAR STOCHASTIC VIBRATION SI(NAL MOD3LIN
!"
#
$%&
"'(
")
'(*
+-,.
,/'021435$68789:3<;2=?>/,.,/@A?335$9:3
B2CEDGFIH$JLKMCENOF(PQ4FIH$J4R?NOFJ4S8B2T?KMCEQUKVMBGP-B?WYXZEDL[?\
Q] \$J4ZECEUQRGCD_^`PQ4FIH$J4R?NOFJ4SbacQZEDLR$J4[?CENOFJ4S8d/R?eLN.R?QQZEN.R?e
f R?N.g2QZIKMN.CMTD_^ih/J4CEZIJLKjXkmlLnLopLph/J4CEZIJLKjbXZEQQ4FQ
drqsUJ4N.Sutwv4xGy$zL{b|2}L~G€u$‚G|G|2}GƒG|2„L…_†‡z2ˆL‰YŠ:~Gy$‚4{G‹?‚?|/Š:ŒG‹
 H?Q(\?ZEDLŽ?S.QUD_^rR?DLRq‘KMCIJ4CEN.DLR$J4ZETKMCEDGFIH$JLKMCENOF(gGN.Ž?ZIJ4CEN.DLRKMN.eLR$J4SUDG’GQS.N.R?eJ4R$’J4R$J4S.T?KMNOK
NOKFDLR$KMNO’GQZEQ4’-CEH?ZEDL[?eLH%J\$J4ZIJL’GN.eLU#N.RGg2DLS.gGN.R?egGN.Ž?ZIJ4CEN.DLR“J4R$J4S.T?KMNOK(N.R%J\$QZEN.DG’GNOFJ4S.S.T
gLJ4ZETGN.R?eUJ4R?N.\?[?SOJ4CEDLZ”  H?Q\$J4ZIJ4UQCEZENOF  N.UQ•q‘’GQ\$QR$’GQRGCc[?CED2kYQeLZEQ4KEKMN.g2QPD4gGN.R?e
cg2QZIJ4eLQ(V  ckPc(WUQCEH?DG’NOK/J4\?\?S.N.Q4’–^—DLZwCEH?Q™˜šZIKMCwCEN.UQ›YN.CEH?N.RCEH?QgGN.Ž?ZIJ4CEN.DLRJ4R$J4Sœq
T?KMNOKFDLRGCEQGCjYJ4R$’%NOKKMH?D4›YRžCEDŸS.Q4JL’“CED“J  ckPcV: ?j¡o2Ws¢¤£G¥ZEQ\?ZEQ4KMQRGCIJ4CEN.DLR”  H?NOK
JLFIH?N.Qg2Q4KKM[?\$QZEN.DLZKM\$Q4FCEZIJ4SZEQ4KMDLS.[?CEN.DLRjJLK›QS.SJLKCEN.UQ•q‘’GQ\$QR$’GQRGCKM\$Q4FCEZE[?U¦J4R$’
UDG’?J4S/FIH$J4ZIJLFCEQZENOKMCENOFKCEH$J4C(J4ZEQN.RFS.D2KMQJ4eLZEQQUQRGCY›YN.CEHCEH?QN.ZCEH?QDLZEQCENOFJ4S8FDL[?RGCEQZ§q
\$J4ZECIK”žh/[?ZEQ  ckJ4R$J4S.T?KMNOKS.Q4JL’?KCED-Jž¨ES.DLR?e2©  ck(VulLª2Ws¢¤£G¥ZEQ\?ZEQ4KMQRGCIJ4CEN.DLR“›YH?NOFIH
NOKR?DLC(DLR?S.T%¨IJ›Y«G›YJ4ZI’?©Gj8Ž?[?CJ4SOKMDSOJLFI«GN.R?eN.R-CEQZEUKYD_^¬’GN.ZEQ4FC\?HGT?KMNOFJ4S/KMN.eLR?Nœ˜‡FJ4R$FQ2”
 H?QKMCE[$’GTŸ’GQUDLR$KMCEZIJ4CEQ4KCEH?QFJ4\$J4Ž?N.S.N.CEN.Q4KJ4R$’­^sJLFQCIKD_^™CEH?Q  ckPcUQCEH?DG’®^—DLZ
R?DLRq‘KMCIJ4CEN.DLR$J4ZETgGN.Ž?ZIJ4CEN.DLRKMN.eLR$J4S8J4R$J4S.T?KMNOK”
¯ ,°*
 H?NOK\$J4\$QZ(NOK(FDLR$FQZER?Q4’›YN.CEH-CEH?QR?DLRq‘KMCIJ4CEN.DLR$J4ZETŸUDG’GQS.N.R?eJ4R$’ŸJ4R$J4S.T?KMNOKD_^
CEH?QZIJ4R$’GDLUgGN.Ž?ZIJ4CEN.DLRN.RJCM›D_qsS.N.R?«\?SOJ4R$J4ZUJ4R?N.\?[?SOJ4CEDLZ”  H?QUJ4R?N.\?[?SOJ4CEDLZYgGN.Ž?ZIJ±q
CEN.DLRNOKY’G[?QCEDCEH?QKMCIJ4CEN.DLR$J4ZETZIJ4R$’GDLU²FDLU\$DLR?QRGCYD_^iJ4RQGCEQZER$J4S.S.TJ4\?\?S.N.Q4’CEDLZI³2[?Q2”
cS.CEH?DL[?eLHCEH?QKMT?KMCEQU´NOKJLKEKM[?UQ4’CEDDL\$QZIJ4CEQN.RJ(S.N.R?Q4J4ZEN.µQ4’(ZEQeLN.UQ2j2J4ZEDL[?R$’J(R?DLU¶q
N.R$J4SY\$D2KMN.CEN.DLRjCEH?QgGN.Ž?ZIJ4CEN.DLR“NOKR?DLRq‘KMCIJ4CEN.DLR$J4ZETž’G[?QCED-CEH?QKMT?KMCEQU·¡K(CEN.UQ•qsgLJ4ZETGN.R?e
FDLR˜šeL[?ZIJ4CEN.DLRjbD4›YN.R?eCEDCEH?QUDLCEN.DLRD_^`CM›DZEN.R?e_qsCMTG\$QUJLKEKMQ4KJ4S.DLR?eCEH?QUJ4R?N.\?[?SOJ±q
CEDLZYS.N.R?«?KVMB2Q4FCEN.DLRl2WI”
¸ DLRq‘KMCIJ4CEN.DLR$J4ZET¹KMN.eLR$J4SOKJ4ZEQFIH$J4ZIJLFCEQZEN.µQ4’ŸŽGTº^—Q4J4CE[?ZEQ4KCEH$J4CgLJ4ZET%›YN.CEH%CEN.UQ
J4R$’ZEQ4³2[?N.ZEQCEN.UQ•q:^—ZEQ4³2[?QR$FITUQCEH?DG’?K™^—DLZ(CEH?QN.Z(J4R$J4S.T?KMNOKVM»wDLH?QRjw¼½L½LoG¾w¿J4UUDLR$’
J4R$’žÀžH?N.CEQ2j¼½L½Ln2WI”²B2[$FIHžUQCEH?DG’?KH$Jg2QŽ$QQRmJ4CECEZIJLFCEN.R?e%KMN.eLR?Nœ˜‡FJ4RGCN.RGCEQZEQ4KMCN.R
ZEQ4FQRGCT2Q4J4ZIK”  H?NOKNOK’G[?QCED-CEH?QÁ^sJLFCCEH$J4CJSOJ4ZEeLQRG[?UŽ$QZD_^ÂgGN.Ž?ZIJ4CEN.DLRžKMN.eLR$J4SOK
J4ZEQR?DLRq‘KMCIJ4CEN.DLR$J4ZETŸN.R-R$J4CE[?ZEQ2jwJLK(›QS.SJLK(CEDCEH?QF[?ZEZEQRGCJgLJ4N.SOJ4Ž?N.S.N.CMTD_^ÃJRG[?UŽ$QZ
D_^™CED2DLSOKÂ^—DLZCEH?QN.Z(J4R$J4S.T?KMNOKVu¿J4UUDLR$’J4R$’ŸÀžH?N.CEQ2jw¼½L½Ln2WI”cÄFSOJLKEKMNOFJ4Suj8JLK›QS.SwJLK
SOJ4ZEeLQS.TQU\?N.ZENOFJ4Suj4KM[$FIHCED2DLSbH$JLKŽ$QQRCEH?QB2H?DLZEC§q  N.UQÅ?DL[?ZEN.QZ  ZIJ4R$K‘^—DLZEUÆVMB  Å/P-WI”
ÇCEH?QZ/R?DLRqs\$J4ZIJ4UQCEZENOFUQCEH?DG’?K8N.R$FS.[$’GQYh/ZEN.Q4KMCES.QTb·¡KQg2DLS.[?CEN.DLR$J4ZETKM\$Q4FCEZE[?UVuh/ZEN.Q4KMC§q
S.QT2j$¼½LªL½2WIj$ÀžN.eLR?QZ§qsÈN.S.S.Q’GNOKMCEZEN.Ž?[?CEN.DLR$KYJ4R$’CEH?QN.ZQGCEQR$KMN.DLR$KVM»wDLH?QRj¼½L½Lo2WIj$JLKY›QS.S
JLK›YJg2QS.QC§qsŽ$JLKMQ4’ŸUQCEH?DG’?K›YH?NOFIH%J4ZEQ-J4CECEZIJLFCEN.R?e-N.R$FZEQ4JLKMN.R?eŸJ4CECEQRGCEN.DLRmV ¸ Q›YSOJ4R$’bj
¼½L½LÉGj8¼½L½4 GWI”
c S.CEH?DL[?eLHD‡QZEN.R?eJL’GgLJ4RGCIJ4eLQ4KKM[$FIHJLKZEQ\?ZEQ4KMQRGCIJ4CEN.DLR\$J4ZIKMN.UDLRGT2j?H?N.eLHJLFF[q
ZIJLFTJ4R$’ZEQ4KMDLS.[?CEN.DLRj?J4R$’šQGN.Ž?N.S.N.CMT(N.RJ4R$J4S.T?KMNOKj$KMTGRGCEH?Q4KMNOKjG\?ZEQ4’GNOFCEN.DLRjGJ4R$’FDLRGCEZEDLSuj
\$J4ZIJ4UQCEZENOFUQCEH?DG’?KH$Jg2QZEQ4FQN.g2Q4’ŸFDLR$KMNO’GQZIJ4Ž?S.TŸS.Q4KEKJ4CECEQRGCEN.DLRžJ4R$’%J4ZEQKEFJ4ZIFQS.T
[$KMQ4’ŸN.R“gGN.Ž?ZIJ4CEN.DLR%J4R$J4S.T?KMNOK”“cUDLR?eCEH?QUj/JL’?J4\?CEN.g2QUQCEH?DG’?KV QS.SOJ4R?eLQZjY¼½Lª LWIj
Ž$JLKMQ4’[?\$DLRZEQ4F[?ZIKMN.g2QQ4KMCEN.UJ4CEN.DLRKEFIH?QUQ4Kj?J4ZEQJ4\?\?ZEDL\?ZENOJ4CEQ–^—DLZKMS.D4›YS.TgLJ4ZETGN.R?eKMN.e_q
R$J4SOKJ4R$’%N.RGg2DLS.g2QJ-CEZIJL’GQ•qsD Ž$QCM›QQR%CEZIJLFI«GN.R?eŸJ4Ž?N.S.N.CMT“J4R$’%Q4KMCEN.UJ4CEN.DLR%JLFF[?ZIJLFT2”
B2CEDGFIH$JLKMCENOF\$J4ZIJ4UQCEQZYQg2DLS.[?CEN.DLRUQCEH?DG’?KjG\$D2KMCE[?SOJ4CEN.R?eKMN.eLR$J4S8ZEQ\?ZEQ4KMQRGCIJ4CEN.DLR$K›YN.CEH
CEN.UQ•q‘’GQ\$QR$’GQRGC/FD2Q FN.QRGCIK8DLŽ$QTGN.R?eKMCEDGFIH$JLKMCENOFKMUD2DLCEH?R?Q4KEKFDLR$KMCEZIJ4N.RGCIKV N.CIJ4e2J›YJ
J4R$’XQZIKEFIHjb¼½L½Ln2WIjbN.U\$D2KMQ(JL’?’GN.CEN.DLR$J4SY¨IKMCEZE[$FCE[?ZEQ4©DLR\$J4ZIJ4UQCEQZQg2DLS.[?CEN.DLRjG›YH?NOFIH
UJT2jR?Qg2QZECEH?QS.Q4KEKjŽ$QYKMCEN.S.S2N.R$KM[ FN.QRGC$N.RFQZECIJ4N.R(J4\?\?S.NOFJ4CEN.DLR$K”  H?Q`^—[?R$FCEN.DLR$J4SGKMQZEN.Q4K
UQCEH?DG’?KwV QRPZIJL’ ; ¼½L½Lª2W8\$D2KMCE[?SOJ4CEQKMN.eLR$J4S?ZEQ\?ZEQ4KMQRGCIJ4CEN.DLR$K›YN.CEH\$J4ZIJ4UQCEQZIK
Ž$QS.DLR?eLN.R?eCEDJ¶^—[?R$FCEN.DLR$J4SKM[?Ž$KM\$JLFQ2j/J4R$’-N.U\$D2KMQ–^—[?ZECEH?QZ¨IKMCEZE[$FCE[?ZEQ4©DLRŸ\$J4ZIJ4UQ•q
CEQZQg2DLS.[?CEN.DLRjwCEH?QFJ4\$J4Ž?N.S.N.CMT-D_^ÂCEZIJLFI«GN.R?e-J4Ž?ZE[?\?CFIH$J4R?eLQ4Kj\?S.[$KJH?N.eLHž’GQeLZEQQD_^
\$J4ZIKMN.UDLRGT2”
 H?Q ([?\$DLRŸ›YH?NOFIH-CEH?NOK(›DLZE«NOK(Ž$JLKMQ4’-NOKCEH$J4Cj/N.RŸKM\?N.CEQD_^™CEH?QUN.R?N.UJ4S
J4CECEQRGCEN.DLR-CEH$J4CN.C(H$JLKZEQ4FQN.g2Q4’bjbCEH?Q ^—[?R$FCEN.DLR$J4SKMQZEN.Q4KJ4\?\?ZED2JLFIHŸNOKŽ$DLCEH“\?HGT?KMNOFJ4S.S.T
UDLCEN.gLJ4CEQ4’J4R$’-J4\?\?S.NOFJ4Ž?S.Q–^—DLZ(R?DLRq‘KMCIJ4CEN.DLR$J4ZET-gGN.Ž?ZIJ4CEN.DLRŸJ4R$J4S.T?KMNOK”  H?Q ! ; D_^¬CEH?Q
KMCE[$’GTCEHG[$KNOKCED’GQUDLR$KMCEZIJ4CEQN.CIKJ4\?\?S.NOFJ4Ž?N.S.N.CMT2j$J4R$’CEDQG\?S.DLZEQN.CIKQ ‡Q4FCEN.g2QR?Q4KEKYN.R
Ž$DLCEHUDG’GQS.N.R?eJ4R$’J4R$J4S.T?KMNOKCEH?ZEDL[?eLHCEH?Q(UJ4R?N.\?[?SOJ4CEDLZYgGN.Ž?ZIJ4CEN.DLR\$J4ZIJL’GN.eLU”
 H?Q\$J4ZECENOF[?SOJ4Z ^—[?R$FCEN.DLR$J4S(KMQZEN.Q4KUQCEH?DG’“[$KMQ4’ ^—DLZCEH?NOK\?[?ZE\$D2KMQ-NOKCEH?QDLR?Q
Ž$JLKMQ4’[?\$DLRR?DLRq‘KMCIJ4CEN.DLR$J4ZET  N.UQ•q‘’GQ\$QR$’GQRGCbc[?CED2kYQeLZEQ4KEKMN.g2QwPD4gGN.R?eYcg2QZIJ4eLQYV  c™q
kPc(WbUDG’GQSOK”  H?Q4KMQJ4ZEQFDLR$FQ\?CE[$J4S?QGCEQR$KMN.DLR$K8D_^ CEH?QN.Z/FDLRGg2QRGCEN.DLR$J4SbVuKMCIJ4CEN.DLR$J4ZET$W
ckPc¹FDL[?RGCEQZE\$J4ZECIKjGN.RCEH$J4CYCEH?QN.Zw\$J4ZIJ4UQCEQZIKwJ4ZEQ(QG\?S.NOFN.Cr^—[?R$FCEN.DLR$KwD_^ CEN.UQ(V QR
PZIJL
’ ; ¼½L½Lª2WI” ] Q\$QR$’GN.R?e([?\$DLRCEH?Q™^—DLZEU´D_^ CEH?Q4KMQ¬^—[?R$FCEN.DLR$Kj  ckPc%UDG’GQSOK
J4ZEQFJ4\$J4Ž?S.Q(D_^iZEQ\?ZEQ4KMQRGCEN.R?eQN.CEH?QZKMUD2DLCEHDLZJ4Ž?ZE[?\?C(KM\$Q4FCEZIJ4SQg2DLS.[?CEN.DLRj$J4R$’CEHG[$K
UDG’GQS.N.R?eJ›YNO’GQFSOJLKEK(D_^™R?DLRq‘KMCIJ4CEN.DLR$J4ZET%KMN.eLR$J4SOK” "wQCj8›YN.CEH-g2QZET®^—Q›´Q?FQ\?CEN.DLR$K
CEH$J4CYH$Jg2Q(Ž$QQRS.N.UN.CEQ’CEDCEH?Q(\?[?ZEQ  ckmFJLKMQVuÅ?DL[$KM«GN.CIJ4«GNO#K ; j¼½L½Lª2WIj  ckPc
UDG’GQSOKwH$Jg2QR?DLCŽ$QQR[$KMQ4’N.RR?DLRq‘KMCIJ4CEN.DLR$J4ZETgGN.Ž?ZIJ4CEN.DLRKMN.eLR$J4S/J4R$J4S.T?KMNOK”
 H?QKMCE[$’GTCEHG[$K™^—DGF[$KMQ4Kjš^—DLZCEH?Q˜šZIKMCCEN.UQ2jGDLRCEH?Q  ckPcmUQCEH?DG’b·¡K`^sJLFQCIK
J4R$’-FJ4\$J4Ž?N.S.N.CEN.Q4K™^—DLZ(gGN.Ž?ZIJ4CEN.DLRŸKMN.eLR$J4SwUDG’GQS.N.R?eJ4R$’ŸJ4R$J4S.T?KMNOK”cÄFDLU\$J4ZENOKMDLR›YN.CEH
CEH?Q\?[?ZEQ  ckÄUQCEH?DG’-NOKJ4SOKMD-UJL’GQ2j/J4R$’%FQZECIJ4N.R“JL’GgLJ4RGCIJ4eLQ4KD_^ÃCEH?Q\$J4ZIJ4UQCEZENOF
UQCEH?DG’?KJ4ZEQ’GNOKEF[$KEKMQ4’b”
,°)
'(*
#
"
&%83140(' 0G3 )*4,9 +149:5$7w,Ÿ H?QCM›D_qsS.N.R?«-\?SOJ4R$J4ZUJ4R?N.\?[?SOJ4CEDLZNOK’GQ\?NOFCEQ4’N.R%Å8N.eL[?ZEQ
¼L”dwJLFIHS.N.R?«NOK(FDLR$KMNO’GQZEQ4’CEDŽ$QJZEN.eLNO’bjb[?R?Nœ^—DLZEUjFTGS.N.R$’GQZD_^¬S.QR?eLCE.
H -J4R$’-UJLKEK
/ ”  H?Q¶˜šZIKMCS.N.R?«NOKFDLR?R?Q4FCEQ4’-CEDCEH?QŽ$JLKMQŽGT-UQ4J4R$KD_^ÂJ4R“QSOJLKMCENOFKM\?ZEN.R?e_qsH?N.R?eLQ
D_^ÃCEDLZIKMN.DLR$J4SKMCE0N ‡R?Q4KE2K 1(3j/›YH?N.S.QCEH?QKMQ4FDLR$’ŸNOKFDLR?R?Q4FCEQ4’-CEDCEH?Q¶˜šZIKMCŽGTŸJKMN.UN.SOJ4Z
KM\?ZEN.R?eD_^™KMCE0N ‡R?Q4KE4K 15”c´ZEN.R?e_qsCMTG\$QUJLKEKNOKJ4S.S.D4›Q4’CEDUD4g2QJ4S.DLR?eQ4JLFIH-S.N.R?«b¾CEH?Q
N.R$KMCIJ4RGCIJ4R?QDL[$K(’GNOKMCIJ4R$FQD_^¬CEH?Q–˜šZIKMC(DLR?Q,V 673IW™^—ZEDLU²CEH?QŽ$JLKM9Q 8§DLN.RGCNOK(’GQ4KMN.eLR$J4CEQ4’-JLK
’ 8§DLN.RGCYNOK’GQ4KMN.eLR$J4CEQ4’JLK : 5>” =$Q>C ?@3j
: 3IjbJ4R$’CEH$J4CD_^iCEH?QKMQ4FDLR$’-,V 6;5W`^—ZEDLUCEH?QKMQ4FDLR$<
?5’GQ4KMN.eLR$J4CEQQGCEQZER$J4S/CEDLZI³2[?Q4KJ4\?\?S.N.Q4’-J4CCEH?9
Q 8§DLN.RGCIKj/J4R$B
’ A;3Cj A&5(CEH?QS.N.R?«J4R?eL[?SOJ4Z
\$D2KMN.CEN.DLR$KY›YN.CEHZEQ4KM\$Q4FCYCEDCEH?Q(\$D2KMN.CEN.g2QH?DLZEN.µDLRGCIJ4S8J4GNOK”
 H?QUJ4R?N.\?[?SOJ4CEDLZ8NOK8JLKEKM[?UQ4’CEDDL\$QZIJ4CEQJ4ZEDL[?R$’JR?DLUN.R$J4S2\$D2KMN.CEN.DLRV A;3,D@EA&5DWI”
 H?QJLFCE[$J4S$S.N.R?«\$D2KMN.CEN.DLR$KJ4ZEQCEH?QRQG\?ZEQ4KEKMQ4’JLFK A;3HGIA;3,DJA;3K3/J4R$’ A&5FGIA&5DJA&53Ij
$
Å8N.eL[?ZEQ¼LtBGFIH?QUJ4CENOFYD_^rCEH?Q(CM›D_qsS.N.R?«\?SOJ4R$J4ZUJ4R?N.\?[?SOJ4CEDLZ”
›YN.CEH A;3K3j(A&53w’GQ4KMN.eLR$J4CEN.R?eCEH?QS.N.R?«J4R?eL[?SOJ4ZY’GNOKM\?SOJLFQUQRGCIK8ZEQSOJ4CEN.g2QCEDCEH?QN.ZwR?DLUN.R$J4S
\$D2KMN.CEN.DLR$K” f R$’GQZYJKMUJ4S.SbJ4R?eL[?SOJ4Z’GNOKM\?SOJLFQUQRGCJLKEKM[?U\?CEN.DLRj?JS.N.R?Q4J4ZEN.µQ4’UDG’GQS$D_^
CEH?QKMT?KMCEQUJ4ZEDL[?R$’CEH?Q(R?DLUN.R$J4Sb\$D2KMN.CEN.DLRNOKt
VE¼4W
V IW J V IW J + G /V IW
›YN.CEH <N.R$’GNOFJ4CEN.R?eFDLRGCEN.RG[?DL[$K(CEN.UQ2j G A;3K3 A&53 ŸCEH?QV:ZEQSOJ4CEN.g2QLWJ4R?eL[?SOJ4Z’GNOK‘q
\?SOJLFQUQRGCwg2Q4FCEDLZV:N.R :
WIj$J4R$’ V IWIj V IWIj + CEH?Q(UJLKEKj?’?J4U\?N.R?e?j$J4R$’KMCE0N $q
R?Q4KEKUJ4CEZENOFQ4Kw›YH?NOFIHJ4ZEQ(D_^iCEH?QÂ^—DLZEUKt
5
5
Vul2W
V IWCG V 5CJB37JB6;5 67: 35:-uW 3 JB6;5 V - ADW V 5CJB6;5 ¥H: JB5-uW 6;5 : 55 V ADW
l@673 : 3 : 3
6;5 : 5V ADW
6;5 : 5V ADW
l@6;5 : 5 : 5
+ G 1(37J 15 15 1?1l 5
 H?Q(N.R?\?[?CYg2Q4FCEDLZNOKD_^iCEH?QÂ^—DLZEUt
/V IWCG ?@3IV IW?5V IW?5V IW
V IWCG
! "
$#
#
#
J
+
VuÉ2W
V: GW
Vuo2W
›YH?N.S.Q 3 G ¥ j 5 G 5 j ¥ G ¥ j AD G A;3,D A&5D”%cSOKMDR?DLCENOFQCEH$J4CCEH?Q
’?J4U\?N.R?eUJ4CEZEN. Q4³2[$J4CEN.DLRVuÉ2W GN.R$FS.[$’GQ4K8J4RQGCEZIJ\?ZEDL\$DLZECEN.DLR$J4Sb’?J4U\?N.R?eCEQZEUj›YN.CEH
ZEQ\?ZEQ4KMQRGCEN.R?eŸCEH?Q-FD2Q FN.QRGCD_^–\?ZEDL\$DLZECEN.DLR$J4S.N.CMT2”  H?Q-RG[?UQZENOFJ4SYgLJ4S.[?Q4KD_^–CEH?Q
KMT?KMCEQU\$J4ZIJ4UQCEQZIKJ4ZEQ(N.R$’GNOFJ4CEQ4’N.R  J4Ž?S.Q¼L”
80 80G5 *40214,9 )2A 9 b78A 8A *4A )L14(0 *49:314,9 )23L,Y H?Q(UJ4R?N.\?[?SOJ4CEDLZYgGN.Ž?ZIJ4CEN.DLRJ4ZEDL[?R$’
CEH?QR?DLUN.R$J4S\$D2KMN.CEN.DLRV A;3,D@EA&5D
W G¦V: 2o EIp IWJ4R$’ž[?R$’GQZJŸŽ?ZED2JL’GŽ$J4R$’)KMCIJ4CEN.DLR$J4ZET
CEDLZI³2[?Q(Q?FN.CIJ4CEN.DLR ?@3IV IWwNOKYFDLR$KMNO’GQZEQ4’ ?5V IW mp u”  H?QKMS.NO’GN.R?eZEN.R?e_qsCMTG\$Q(UJLKEKMQ4FK 673j
6;5YUD4g2QJ4S.DLR?eCEH?QN.ZYZEQ4KM\$Q4FCEN.g2QS.N.R?«?K`^—DLS.S.D4›YN.R?eKMN.RG[$KMDLNO’?J4SCEZIJ 8§Q4FCEDLZEN.Q4KwD_^iCEH?QÂ^—DLZEUt
l
V Gm¼ EIl2W
Vun2W
: V IWHG : DHJ
: 4KMN.R
&(
%
'*),+ -
1
&$
%
),+ -
.(
%
),+ - /
%
0#
32
32
54
76
54
32
98
; &
;
. ; 98
,:
=<
9>
?A@
 J4Ž?S.Q¼LtB2T?KMCEQU\$J4ZIJ4UQCEQZIK”
h/ZEDL\$QZECMT
B2TGUŽ$DLS ÈJ4S.[?Q
=$QR?eLCEHD_^iS.N.R?«?K¼J4R$’l
¼U
PJLKEKYD_^rS.N.R?«?K¼(J4R$’l
(
¼
«Ge
/
B2S.NO’GN.R?eZEN.R?e_qsCMTG\$Q(UJLKEKMQ4K
673 E6;5
pG”¡lLo«Ge
B2CE0N ‡R?Q4KEKYD_^rCEDLZIKMN.DLR$J4S/KM\?ZEN.R?e¼
1(3
¼ pLp ¸ UZIJL’
B2CE0N ‡R?Q4KEKYD_^rCEDLZIKMN.DLR$J4S/KM\?ZEN.R?el
15
Lª p ¸ UZIJL’
h/ZEDL\$DLZECEN.DLR$J4Sw’?J4U\?N.R?eFD2Q FN.QRGC
Gp ”¡pLpLo
Spectrum (db)
140
100
60
20
5
4
3
2
Frequency (Hz)
1
20
10 ec)
s
e(
0 0
m
i
T
Å8N.eL[?ZEQlGt  ?H QDLZEQCENOFJ4SbCEN.UQ•q‘’GQ\$QR$’GQRGCKM\$Q4FCEZE[?UD_^iCEH?QKMN.eLR$J4SwVuFDLRGCEN.RG[?DL[$K‘qsCEN.UQLWI”
›YN.CEH G²l N.R$’GNOFJ4CEN.R?eCEH?QUDLCEN.DLRŸ\$QZEN.DG’bj : DG²p¡o / jwJ4R$’ : G²p¡É /
V G¼EIl2WI”  H?QKMT?KMCEQUÆZEQ4KM\$DLR$KMQD_^™N.RGCEQZEQ4KMCNOKCEH?QgGN.Ž?ZIJ4CEN.DLRŸ’GNOKM\?SOJLFQUQRGC A&53D_^
CEH?QKMQ4FDLR$’S.N.R?«b”
] [?QCEDCEH?Q\$QZEN.DG’GNOF(UDLCEN.DLRD_^`CEH?QZEN.R?e_qsCMTG\$QUJLKEKMQ4KYCEH?QKMT?KMCEQUNOK(V:\$QZEN.DG’GNœq
FJ4S.S.T$WwR?DLRq‘KMCIJ4CEN.DLR$J4ZET›YN.CEH\$QZEN.DG’ G)l VuFDLZEZEQ4KM\$DLR$’GN.R?e ^—ZEQ4³2[?QR$FT±¢ G)p¡o
WI”  H?QCEH?QDLZEQCENOFJ4SwCEN.UQ•q‘’GQ\$QR$’GQRGC(KM\$Q4FCEZE[?UV¨§^—ZEDLµQR$©KM\$Q4FCEZE[?UWD_^™CEH?Q A&53
gGN.Ž?ZIJ4CEN.DLRKMN.eLR$J4S$[?R$’GQZKMCIJ4CEN.DLR$J4ZET[?R$FDLZEZEQSOJ4CEQ4’Q?FN.CIJ4CEN.DLR ?@3IV IWIjGJ4R$’FDLU\?[?CEQ4’gGNOJ
CEH?QKMT?KMCEQU²Q4³2[$J4CEN.DLR“VE¼4WIjNOK(’GQ\?NOFCEQ4’N.R-Å8N.eL[?ZEQlJLK(J¶^—[?R$FCEN.DLRD_^¬Ž$DLCEH-CEN.UQJ4R$’
^—ZEQ4³2[?QR$FT2”  H?Q–^—DLS.S.D4›YN.R?eZEQUJ4ZE«?KUJTŽ$QUJL’GQ ^—ZEDLUCEH?NOKFDLRGCEN.RG[?DL[$K‘qsCEN.UQKM\$Q4F q
CEZE[?Ut
A H?QCEH?QDLZEQCENOFJ4SwCEN.UQ•q‘’GQ\$QR$’GQRGCKM\$Q4FCEZE[?UÆNOKj8QG\$Q4FCEQ4’GS.T2jb\$QZEN.DG’GNOFLj›YN.CEHŸCEH?Q
Q?JLFCYKEJ4UQ(\$QZEN.DG’JLKYCEH?QKMT?KMCEQUÆV G¹l WI”
 H?QKMN.eLR$J4Su·¡KUJ4N.RQR?QZEeLTFDLR$FQRGCEZIJ4CEN.DLRNOKN.RCEH?Qp %n ^—ZEQ4³2[?QR$FTŽ$J4R$’bj
›YN.CEH%CEH?QCM›D-R$J4CE[?ZIJ4SÃ^—ZEQ4³2[?QR$FN.Q4K 3J4R$’ 5Ž$QN.R?eŸN.R%CEH?QŸVup¡ª4 EIp¡½Ll2WJ4R$’
VuÉ 2(ª EIÉ¡½Lp2W ZIJ4R?eLQ4Kj/ZEQ4KM\$Q4FCEN.g2QS.T2”  H?Q 3Â^—ZEQ4³2[?QR$FTNOK’GDLUN.R$J4RGCj›YH?N.S.Q
CEH?Q(CM›D’?J4U\?N.R?eZIJ4CEN.DCEZIJ 8§Q4FCEDLZEN.Q4KJ4ZEQŽ$DL[?R$’GQ4’J4Ž$D4g2Q(ŽGTpG”¡p 2”
9 b78A ,9 '-6 :AG149:5$7w, YJLKMQ4’Ÿ[?\$DLRžCEH?Q\?ZEQgGN.DL[$KZEQUJ4ZE«?KjwCEH?QCEDLZI³2[?QQ?FN.CIJ4CEN.DLR
?@3IV IWNOK(KMQS.Q4FCEQ4’-JLKJµQZED_qsUQ4J4RS.D4›¬qs\$JLKEK˜šS.CEQZEQ4’“VuF[?C§qs
D J4Cn W[?R$FDLZEZEQSOJ4CEQ4’
?
;
;
?
?
3#
76
54
4
Angular
Displacement (rad)
0.04
0
−0.04
0
50
100
Time (sec)
150
200
Å8N.eL[?ZEQÉGt  H?Q(gGN.Ž?ZIJ4CEN.DLR’GNOKM\?SOJLFQUQRGCKMN.eLR$J4SUDG’GQS.Q4’V )
G ¼l¡o WI”
MK N.eLR$J4SbKEJ4U\?S.Q4’J4ClLo ”  H?QKMT?KMCEQUQ4³2[$J4CEN.DLRNOK8N.RGCEQeLZIJ4CEQ4’[$KMN.R?eJkY[?R?eLQ•q [?CECIJ
4oUQCEH?DG’›YN.CEHgLJ4ZENOJ4Ž?S.Q(N.RGCEQeLZIJ4CEN.DLRKMCEQ\j$J4R$’CEH?Q(eLQR?QZIJ4CEQ4’gGN.Ž?ZIJ4CEN.DLRZEQ4KM\$DLR$KMQ
NOKS.D4›¬qs\$JLKEKؚS.CEQZEQ4’J4C Gn J4R$’ZEQ•q‘KEJ4U\?S.Q4’J4C Gļl¡o ”  DUN.R?N.UN.µQ
CEZIJ4R$KMN.QRGC(Q ‡Q4FCIKj/J4RŸN.R?N.CENOJ4SKMQeLUQRGCD_^™CEH?QZEQ4KM[?S.CEN.R?eKMN.eLR$J4SwNOK’GZEDL\?\$Q4’bj/›YN.CEHŸCEH?Q
ZEQUJ4N.R?N.R?eKMN.eLR$J4S8Ž$QN.R?e G¹lLÉLn Vu(l EI½LoG¼KEJ4U\?S.Q4KIW/S.DLR?e?”
 H?QKMN.eLR$J4S/NOK(’GQ\?NOFCEQ4’N.RŸÅ8N.eL[?ZEQÉGj ^—ZEDLUƛYH?NOFIHCEH?QR?DLRq‘KMCIJ4CEN.DLR$J4ZEN.CMT-NOKQgGNœq
’GQRGC” uRJLFFDLZI’?J4R$FQ›YN.CEHCEH?Q\?ZEQgGN.DL[$KZEQUJ4ZE«?Kj$JFS.D2KMQZYQ?J4UN.R$J4CEN.DLRZEQg2Q4J4SOKCEH?Q
’GDLUN.R$J4CEN.DLRD_^iCEH?Q 3`^—ZEQ4³2[?QR$FTN.RCEH?QKMN.eLR$J4Su”
, &
)
Æ
!
80ž
)
)021 85
w, c’GNOKEFZEQCEQ  N.UQ•q‘’GQ\$QR$’GQRGCc[?CED2kYQeLZEQ4KEKMN.g2QPD4gGN.R?e
cg2QZIJ4eLQV  ckPc(W/UDG’GQSbNOKD_^iCEH?QÂ^—DLZEUt
A V IW G V IW KV LW
£(D
£(D
›YN.CEH G<p(E¼EIl(E H?QR$FQ•^—DLZECEH¹N.R$’GNOFJ4CEN.R?ež’GNOKEFZEQCEQ-CEN.UQ2j CEH?Q“KEJ4U\?S.Q4’%R?DLRq
KMCIJ4CEN.DLR$J4ZET-gGN.Ž?ZIJ4CEN.DLR“KMN.eLR$J4SUDG’GQS.Q4’bj 5JKMCIJ4CEN.DLR$J4ZET-N.R?R?D4gLJ4CEN.DLR$KV:[?R$FDLZEZEQSOJ4CEQ4’$W
KMN.eLR$J4S8›YN.CEHµQZEDUQ4J4RJ4R$’gLJ4ZENOJ4R$FQ &j A V IWV Gm(p E E WCEH?Q(UDG’GQSu·¡Kc[?CED2kYQ•q
eLZEQ4KEKMN.g2QVuckW\$J4ZIJ4UQCEQZIKjbJ4R$’ V IWV G)(p E E / WYCEH?QUDG’GQSu·¡KPD4gGN.R?ecg2QZIJ4eLQ
VuPc(W8\$J4ZIJ4UQCEQZIK” ¸ DLCENOFQCEH$J4C AD D )¼LjG›YH?N.S.QCEH?QZEQ4KMCD_^ CEH?Qck 4Pcž\$J4ZIJ4U¶q
QCEQZIK/J4ZEQJLKEKM[?UQ4’CED(Ž$QS.DLR?e(CEDJÃ^—[?R$FCEN.DLR$J4S$KM[?Ž$KM\$JLFQD_^ ’GN.UQR$KMN.DLR$J4S.N.CMT ‡jGKM\$J4R?R?Q4’
ŽGT-JKMQC(D_^™S.N.R?Q4J4ZES.TN.R$’GQ\$QR$’GQRGCÂ^—[?R$FCEN.DLR$Kv >D4V IWE #3V IWE E ™!¢ 3V IWE„VuKM[?Ž$KM\$JLFQ
Ž$JLKMNOK¬^—[?R$FCEN.DLR$KIWIjGCEH$J4CNOKt
!¢ 3
!¢ 3
A #G "
G V IW
Vuª2W
£$ D &% $ $ V IW
(
#" $ (£ D &% $ $
›YN.CEH &% $ J4R$’ &% $ ZEQ\?ZEQ4KMQRGCEN.R?eCEH?QckžJ4R$’PcjLZEQ4KM\$Q4FCEN.g2QS.T2jLFD2Q FN.QRGCIKD_^ \?ZED 8§Q4F q
CEN.DLR”  H?QwUDG’GQS2’GQ•˜šR?Q4’ŽGTQG\?ZEQ4KEKMN.DLR$K/KV LWbJ4R$’Vuª2W$NOKbZEQ•^—QZEZEQ4’YCEDJLKJ ('*)+,'V E / Ws¢
UDG’GQS/V QRPZIJL’ ; ¼½L½Lª2WI”
 ckPc)UDG’GQS.N.R?eNOKŽ$JLKMQ4’[?\$DLR-CEH?QUDG’GQS.N.R?e ^—ZIJ4UQ›DLZE«D_^ QRPZIJL’ E
V
¼
; ½L½Lª2WIj›YH?NOFIHJLFFDL[?RGCIK¬^—DLZ\$J4ZIJ4UQCEQZYQ4KMCEN.UJ4CEN.DLRj?JLK›QS.S8JLKKM[?Ž$KM\$JLFQ’GN.UQRq
KMN.DLR$J4S.N.CMT2jGŽ$JLKMNOK¬^—[?R$FCEN.DLRj$J4R$’UDG’GQSbDLZI’GQZKMQS.Q4FCEN.DLR-” uR?N.CENOJ4Sb\$J4ZIJ4UQCEQZYQ4KMCEN.UJ4CEN.DLR
NOKJLFIH?N.Qg2Q4’-gGNOJCEH?Qh8DLS.TGR?DLUNOJ4Sœq‘cS.eLQŽ?ZIJ4NOFVuhbc(WUQCEH?DG’bjw›YH?NOFIH%D4g2QZIFDLUQ4K’G0N q
F[?S.CEN.Q4KKM[$FIH%JLKCEH?D2KMQJLKEKMDGFNOJ4CEQ4’%›YN.CEH“S.DGFJ4SYQGCEZEQUJ\?ZEDLŽ?S.QUKJ4R$’“CEH?QR?QQ4’­^—DLZ
32
2
)
; ;
;
; ; ; :
;
;
:
;
;
;
;
;
;
LJ FF[?ZIJ4CEQ(N.R?N.CENOJ4SbeL[?Q4KEKY\$J4ZIJ4UQCEQZgLJ4S.[?Q4Kj?›YH?QZEQ4JLKY\$DLCEQRGCENOJ4S8h/ZEQ4’GNOFCEN.DLRd/ZEZEDLZVuhwdW
ZEQ•˜šR?QUQRGCwNOKYJ4SOKMDFDLR$KMNO’GQZEQ4’b”wÅ?DLZ’GQCIJ4N.SOKCEH?Q(N.RGCEQZEQ4KMCEQ4’ZEQ4JL’GQZYNOKYZEQ•^—QZEZEQ4’CED QR
PZIJL’B ; VE¼½L½Lª2WI”
)
9 *4AG149:5$79 b78A ()
5 80 :9:7 8, Å?DLS.S.D4›YN.R?eKEJ4U\?S.QUQ4J4R-KM[?Ž?CEZIJLFCEN.DLRj
CEH?Q(l EI½LoG¼KEJ4U\?S.Q•qsS.DLR?egGN.Ž?ZIJ4CEN.DLRŸKMN.eLR$J4SwNOKKM\?S.N.CN.RGCEDCM›D’GNOK 8§DLN.RGC(KMQCIKt  H?Q G
l EIªLoG¼KEJ4U\?S.Q4K(V > GlLlLª WYS.DLR?B
(
e ; ! = @/J4R$’CEH?Q2GļpLpKEJ4U\?S.Q4K
V G²ª W(S.DLR?e
; >2; ! = @:”  H?Q ^—DLZEUQZNOK[$KMQ4’ ^—DLZUDG’GQSQ4KMCEN.UJ4CEN.DLRj
›YH?QZEQ4JLKCEH?Q-SOJ4CECEQZNOKjJLFFDLZI’GN.R?e%CED“CEH?QŸFZED2KEK‘qsgLJ4S.NO’?J4CEN.DLRm\?ZEN.R$FN.\?S.Q2jYZEQ4KMQZEg2Q4’º^—DLZ
UDG’GQSbgLJ4S.NO’?J4CEN.DLR\?[?ZE\$D2KMQ4K”
YJLKMQ4’[?\$DLRCEH?Q\$QZEN.DG’GNOFR$J4CE[?ZEQD_^rCEH?Q(R?DLRq‘KMCIJ4CEN.DLR$J4ZETgGN.Ž?ZIJ4CEN.DLRKMN.eLR$J4Suj?\?ZED_q
8§Q4FCEN.DLR$KYD_^¬CEH?Qck 4Pc)\$J4ZIJ4UQCEQZIKYDLRGCEDJKM[?Ž$KM\$JLFQKM\$J4R?R?Q4’ŽGTKMN.R?QJ4R$’-FD2KMN.R?Q
^—[?R$FCEN.DLR$KD_^iCEH?QÂ^—DLZEUt
Vu½2W
>DLV IWCG)C¼ E LV IWCG¹KMN.R 1 E 4V IWCG¹FD2K 1 J4ZEQFDLR$KMNO’GQZEQ4’b” uR-CEH?QJ4Ž$D4g2QQG\?ZEQ4KEKMN.DLR$#K 1G¼ EI(l E E m¼Ljb›YH?N.S.Q G(l EIªLoLpGj
J4R$’ HG¹(p E¼ EI(l E Vu’GNOKEFZEQCEQCEN.UQLWI”
B2[?Ž$KM\$JLFQ’GN.UQR$KMN.DLR$J4S.N.CMT2j2Ž$JLKMNOK¬^—[?R$FCEN.DLR$Kj$J4R$’UDG’GQSbDLZI’GQZIKJ4ZEQ’GQCEQZEUN.R?Q4’
ŽGT“CEH?QÁ^—DLS.S.D4›YN.R?e-CEH?ZEQQ•q‘KMCEQ\%\?ZEDGFQ4’G[?ZEQ2t“VuJ2W uR%CEH?Q®˜šZIKMCKMCEQ\ž\?[?ZEQ  ckUDG’GQSOK
D_^˜šGQ4’)V¨ES.DLR?e2©2WDLZI’GQZJ4R$’¹KM[?Ž$KM\$JLFQ“’GN.UQR$KMN.DLR$J4S.N.CMT­˜šGQ4’žCED GÉ N.RžDLZI’GQZ
CEDŸN.R$FDLZE\$DLZIJ4CEQ >D4V IWJ4S.DLR?e“›YN.CEHžJŸ\$J4N.ZD_^–KMN.R?Q4FD2KMN.R?Q®^—[?R$FCEN.DLR$KN.RžCEH?QN.ZŽ$JLKMNOK
J4ZEQKM[$FFQ4KEKMN.g2QS.TQ4KMCEN.UJ4CEQ4’b” YJLKMQ4’[?\$DLRŸCEH?Q4KMQZEQ4KM[?S.CIKjbCEH?Q¨EŽ$Q4KMCI©\$DLCEQRGCENOJ4S/Ž$JLKMNOK
^—[?R$FCEN.DLR$KV 8§[$’GeLQ4’ŽGTCEH?QJLFIH?N.Qg2Q4’UDG’GQS/³2[$J4S.N.CMT qsD_^Oq:˜šCWJ4ZEQKMQS.Q4FCEQ4’J4R$’ZIJ4R?«2Q4’b”
V:Žb*W uR-CEH?QKMQ4FDLR$’ŸKMCEQ\  ckUDG’GQSOKD_^Y˜šGQ4’%V¨ES.DLR?e2©2W(DLZI’GQZŽ?[?CN.R$FZEQ4JLKMN.R?eKM[?Žq
KM\$JLFQ’GN.UQR$KMN.DLR$J4S.N.CMTJ4ZEQFDLR$KMNO’GQZEQ4’ŽGTKMQ4³2[?QRGCENOJ4S.S.TJL’?’GN.R?eCEDCEH?QN.ZKM[?Ž$KM\$JLFQCEH?Q
Ž$Q4KMCjKMQ4FDLR$’-Ž$Q4KMCjJ4R$’%KMDDLRjwŽ$JLKMNOK ^—[?R$FCEN.DLR$Kjw[?RGCEN.SwR?D-KMN.eLR?Nœ˜‡FJ4RGCN.U\?ZED4g2QUQR2C
N.RUDG’GQS8³2[$J4S.N.CMT qsD_^Oq:˜šCNOKJLFIH?N.Qg2Q4’b”  H?QJL’GQ4³2[$J4CEQKM[?Ž$KM\$JLFQ’GN.UQR$KMN.DLR$J4S.N.CMT ®J4R$’
FDLZEZEQ4KM\$DLR$’GN.R?eŽ$JLKMNOKÂ^—[?R$FCEN.DLR$K(J4ZEQCEHG[$K(’GQCEQZEUN.R?Q4’b”VuF4W uR-CEH?QCEH?N.ZI’KMCEQ\ŸUN.GQ4’
 ckPcmUDG’GQSOKYD_^`gLJ4ZEN.DL[$KDLZI’GQZIKjbŽ?[?C›YN.CEHCEH?Q\?ZEQ4’GQCEQZEUN.R?Q4’KM[?Ž$KM\$JLFQ’GN.UQRq
KMN.DLR$J4S.N.CMTJ4R$’-Ž$JLKMNOKÂ^—[?R$FCEN.DLR$Kj/J4ZEQQ4KMCEN.UJ4CEQ4’[?RGCEN.SwR?DN.U\?ZED4g2QUQRGCYN.R-CEH?QUDG’GQS
³2[$J4S.N.CMT qsD_^Oq:˜šCNOKJLFIH?N.Qg2Q4’bj8J4R$’ŸCEH?QZEQ4³2[?N.ZEQ4’DLZI’GQZIKJ4ZEQCEHG[$K’GQCEQZEUN.R?Q4’b” uRŸCEH?NOK
FDL[?ZIKMQCEH?Q\?ZEQgGN.DL[$KMS.TKMQS.Q4FCEQ4’KM[?Ž$KM\$JLFQ’GN.UQR$KMN.DLR$J4S.N.CMTJ4R$’-Ž$JLKMNOKÂ^—[?R$FCEN.DLR$KUJT
Ž$QJ4SOKMDgLJ4ZEN.Q4’J4R$’FDLRGCEZIJLKMCEQ4’J4e2J4N.R$KMCw\$DLCEQRGCENOJ4S$J4S.CEQZER$J4CEN.g2Q4KbŽ$Q•^—DLZEQJ™˜šR$J4S$KMQS.Q4FCEN.DLR
NOKUJL’GQ2”
PDG’GQS³2[$J4S.N.CMT qsD_^Oq:˜šCNOKjGN.RCEH?Q\?ZEQ4KMQRGCYFDLRGCEQGCj 8§[$’GeLQ4’N.RCEQZEUK/D_^ CEH?Q(UDG’GQS
kYQ4KMNO’G[$J4SB2[?U´D_^iBG³2[$J4ZEQ4KVukB?B?WYVuKM[?UÄD_^ CEH?QKE³2[$J4ZEQ4KwD_^ CEH?QUDG’GQSœqsŽ$JLKMQ4’DLR?Q•q‘KMCEQ\q
J4H?Q4JL’\?ZEQ4’GNOFCEN.DLRQZEZEDLZIKIWIj?N.CIKU[?S.CEN.\?S.Q•q‘KMCEQ\q‘J4H?Q4JL’\?ZEQ4’GNOFCEN.g2QJ4Ž?N.S.N.CMT2j?JLKY›QS.SJLKYCEH?Q
c«LJ4N.«2QFZEN.CEQZENOJVuc E»´J4R$’ E»YWV N.CIJ4e2J›YJ-J4R$’žXQZIKEFIHjY¼½L½LnG¾ QR%PZIJL’ ; j
¼½L½Lª2WI”
 H?QŸJ4\?\?S.NOFJ4CEN.DLR¹D_^–CEH?QŸJ4Ž$D4g2Q-\?ZEDGFQ4’G[?ZEQ2j[$KMN.R?e  ck(VulLª2Ws¢¤£G¥UDG’GQSOK›YN.CEH
>D4V IWJ4R$’JgLJ4ZEN.QCMTD_^iDLCEH?QZFJ4R$’GNO’?J4CEQ ^—[?R$FCEN.DLR$Kj?S.Q4JL’?KCED 1 G¹lLlLªFIH$J4ZIJLFCEQZEN.µN.R?e
CEH?QŽ$Q4KMCKMN.R?QJ4R$’“FD2KMN.R?QŽ$JLKMNOK ^—[?R$FCEN.DLRŸ\$J4N.Z”-PQ4J4R?N.R?e_^—[?S.S.T2jCEH?NOKFDLZEZEQ4KM\$DLR$’?KCED
CEH?QQ?JLFCKMT?KMCEQU ^—ZEQ4³2[?QR$FTD_^ ±¢ G¹p¡o VuKMCEQ\J2WI”wcYCECEQU\?CEQ4’KM[?Ž$KM\$JLFQ’GN.UQRq
KMN.DLR$J4S.N.CMTN.R$FZEQ4JLKMQ4K(S.Q4JL’ŸCEDR?DN.U\?ZED4g2QUQRGCN.R“CEH?QUDG’GQS³2[$J4S.N.CMT qsD_^Oq:˜šCVuKMCEQ\“ŽbWI”
 H?QQ4KMCEN.UJ4CEN.DLR%VuhbcÄUQCEH?DG’$WD_^ÃUN.GQ4’  ckPcV:Rj UWs¢¤£G¥(UDG’GQSOK›YN.CEHŸCEH?Q¶^—DLZEQ•q
76
54
6
4
5>
5>
#
=
*
Å8N.eL[?ZEQ ?tkB?B“V WJ4R$’ŸcE»´V
KMQC¾?hbcmUQCEH?DG’$WI”
,5
)
AIC
(9
,5
)
(1
0,
5)
TARMA model orders (n,m)
(8
,5
)
(7
(6
(5
(4
(4
(4
(4
(4
(4
(4
(4
,5
)
−3.08
,5
)
0.04
,8
)
−3.04
,7
)
0.05
,6
)
−3
,5
)
0.06
,4
)
−2.96
,3
)
0.07
,2
)
0.08
,1
)
RSS
4
x 10
−2.92
WD_^¬gLJ4ZEN.DL[$K  ckPc´UDG’GQSOKV:Q4KMCEN.UJ4CEN.DLR
eLDLN.R?eKM[?Ž$KM\$JLFQŽ$JLKMNOK(J4R$’-D_^™gLJ4ZEN.DL[$K(DLZI’GQZIK E / V JLK(CEH?QKMT?KMCEQUÆNOKÃ^—DL[?ZECEH
DLZI’GQZW8S.Q4JL’?K/CEDJ(Ž$Q4KMC  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQS?J4R$’JKMS.N.eLHGCES.T(N.R^—QZEN.DLZVuKMQ4FDLR$’Ž$Q4KMCW
 ckPcV: ?j¡É2Ws¢¤£G¥YUDG’GQS/VuKMCEQ\FL¾?\$J4ZECENOJ4SZEQ4KM[?S.CIKN.RÅ8N.eL[?ZEQ( GWI”
h/ZEQ4’GNOFCEN.DLRq‘d/ZEZEDLZVuhwdW8Ž$JLKMQ4’UDG’GQS?ZEQ•˜šR?QUQRGCV:N.R?N.CENOJ4S.N.µQ4’(›YN.CEHCEH?QhbcžQ4KMCENœq
UJ4CEQ4KJ4R$’[$KMN.R?e =$Qg2QRGŽ$QZEe_q‘PJ4ZI³2[$J4ZI’GCYDL\?CEN.UN.µ4J4CEN.DLR›YN.CEHJUN.GQ4’³2[$JL’GZIJ4CENOF(J4R$’
F[?Ž?NOFYS.N.R?Q(KMQ4J4ZIFIHbW/NOKJ4SOKMDQ?J4UN.R?Q4’b” uCNOK`^—DL[?R$’CEDS.Q4JL’CED–^—[?ZECEH?QZN.U\?ZED4g2QUQRGC/N.R
CEH?Q  ckPcV: ?j¡o2Ws¢¤£G¥FJLKMQ2j8Ž?[?C(CEDR?DN.U\?ZED4g2QUQRGCN.R-CEH?Q  ckPcV: ?j¡É2Ws¢¤£G¥FJLKMQ2”
 H?Q ˜šR$J4S8UDG’GQS.N.R?eZEQ4KM[?S.CIKj?N.R$FS.[$’GN.R?eCEH?Q  ck(VulLª2Ws¢¤£G¥Ljbhbc™q  ckPcV: ?j¡É2Ws¢¤£G¥j$hbc™q
 ckPcV: ?j¡o2Ws¢¤£G¥J4R$’hwdrq  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQSOKj?J4ZEQKM[?UUJ4ZEN.µQ4’N.R  J4Ž?S.QlG”
Å?ZEDLUCEH?Q4KMQ(ZEQ4KM[?S.CIKwN.CYNOKDLŽ$KMQZEg2Q4’CEH$J4CYCEH?Qhwdrq  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQSbNOKYKM[q
\$QZEN.DLZCED-J4S.SJ4S.CEQZER$J4CEN.g2QZEQ\?ZEQ4KMQRGCIJ4CEN.DLR$KÂ^—ZEDLU#CEH?QkB?B$jwc E»J4R$’ E»\$DLN.RGCIKD_^
gGN.Q›(” uCNOKJ4SOKMD›DLZECEHŸUQRGCEN.DLR?N.R?eCEH$J4CCEH?QŸ¨ES.DLR?e2©  ck(VulLª2Ws¢¤£G¥UDG’GQSV:›YN.CEHŸN.CIK
¨EQ?FQ4KEKMN.g2Q4©\$J4ZIJ4UQCEZEN.µ4J4CEN.DLRbWJLFIH?N.Qg2Q4KKMDLUQ›YH$J4C8UDLZEQYJLFF[?ZIJ4CEQ¼ q CED(ɱq‘KMCEQ\q‘J4H?Q4JL’
\?ZEQ4’GNOFCEN.DLR$KY›YN.CEH?N.RCEH?QgLJ4S.NO’?J4CEN.DLR-KMQCjb›YH?N.S.QCEH?Qhwdrq  ckPcV: ?j¡o2Ws¢¤£G¥(UDG’GQS/’GD2Q4K
CEH$J4Ci^—DLZCEH?QYS.DLR?eLQZV: _qrCED½±q‘KMCEQ\q‘J4H?Q4JL’$WwH?DLZEN.µDLR$K”  H?Qhwdrq  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQS
NOK8CEHG[$KwKMCIJ4CENOKMCENOFJ4S.S.T2j2JLK/›QS.S$JLK/\?HGT?KMNOFJ4S.S.TVu’G[?QCED(N.CIK/’GZIJLKMCENOFJ4S.S.TS.D4›QZ/\$J4ZIJ4UQCEZEN.µ4J±q
CEN.DLR-J4R$’U[$FIHUDLZEQ¨I’GN.ZEQ4FCI©ZEQSOJ4CEN.DLR›YN.CEHCEH?Q[?R$’GQZES.TGN.R?e ^—DL[?ZECEH-DLZI’GQZ\?HGT?KMNOFJ4S
KMT?KMCEQUWIj?\?ZEQ•^—QZIJ4Ž?S.Q2jbJ4R$’NOKJL’GDL\?CEQ4’JLKCEH?Q ˜šR$J4SwKMN.eLR$J4S/UDG’GQSV:UDG’GQS\$J4ZIJ4UQCEQZIK
N.R  J4Ž?S.QÉ2WI”
, !m
&
#
$
 H?Qhwdrq  ckPcV: ?j¡o2Ws¢¤£G¥YUDG’GQSbNOKYKM[$FFQ4KEK‘^—[?S.S.TgLJ4S.NO’?J4CEQ4’ŽGTJLKEKMQ4KEKMN.R?eZEQ4KMNO’ q
[$J4S/›YH?N.CEQR?Q4KEKJ4CCEH?Q Gp¡pLoS.Qg2QS›YN.CEH?N.RCEH?QgLJ4S.NO’?J4CEN.DLR-KMQC”PDG’GQSœqsŽ$JLKMQ4’DLR?Q•q
KMCEQ\q‘J4H?Q4JL’\?ZEQ4’GNOFCEN.DLR$KYJ4ZEQ2j$›YN.CEH?N.RCEH?Q(gLJ4S.NO’?J4CEN.DLRKMQCj?\?ZEQ4KMQRGCEQ4’N.RÅ8N.eL[?ZEQoG”  H?Q
\?ZEQ4’GNOFCEN.DLR$Kr^—DLS.S.D4›žCEH?Q(KMN.eLR$J4Sbg2QZETJLFF[?ZIJ4CEQS.T2j2›YN.CEHCEH?QSOJ4CECEQZ·¡KwJLFCE[$J4SbgLJ4S.[?Q4KwŽ$QN.R?e
›YN.CEH?N.RCEH?QN.ZQ4KMCEN.UJ4CEQ4’½Lo ´\?ZEDLŽ$J4Ž?N.S.N.CMTS.N.UN.CIK”
 H?QŸQ4KMCEN.UJ4CEQ4’¹hwdrq  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQS(Ž$JLKMQ4’¹gGN.Ž?ZIJ4CEN.DLR)KMN.eLR$J4SCEN.UQ®q
’GQ\$QR$’GQRGCV¨§^—ZEDLµQR$©2WKM\$Q4FCEZE[?UÆNOKKMH?D4›YRŸN.R-Å8N.eL[?ZEQnJLK(J¶^—[?R$FCEN.DLR-D_^¬Ž$DLCEH-CEN.UQ
J4R$’ ^—ZEQ4³2[?QR$FT2”  H?NOKQGH?N.Ž?N.CIKCEH?QQ?JLFCKEJ4UQ-FIH$J4ZIJLFCEQZENOKMCENOFKJLKN.CIKCEH?QDLZEQCENOFJ4S
VuFDLRGCEN.RG[?DL[$K‘qsCEN.UQLW/FDL[?RGCEQZE\$J4ZEC(VuÅ8N.eL[?ZEQl2WIj?›YN.CEH›YH?NOFIHN.CNOKYN.RFS.D2KMQJ4eLZEQQUQRGC”
 H?QQ4KMCEN.UJ4CEQ4’UDG’GQSœqsŽ$JLKMQ4’(CEN.UQ•q‘’GQ\$QR$’GQRGCbR$J4CE[?ZIJ4S^—ZEQ4³2[?QR$FN.Q4K 3IV IWIj 5V IW
J4R$’Ÿ’?J4U\?N.R?eZIJ4CEN.D2K 3IV IWIj 5V IW wJ4ZEQFDLU\$J4ZEQ4’CEDCEH?QN.Z(CEH?QDLZEQCENOFJ4SwFDL[?RGCEQZE\$J4ZECIK
 4J Ž?S.Q-lGtŸ»wDLU\$J4ZENOKMDLRžD_^ CEH?QQ4KMCEN.UJ4CEQ4’  ck(VulLª2Ws¢¤£G¥4jhbc™q  ckPcV: ?j¡É2Ws¢¤£G¥jhbc™q
 ckPcV: ?j¡o2Ws¢¤£G¥jYJ4R$’žhwdrq  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQSOKN.R%CEQZEUKD_^–kB?B$jcE»jHE»
V:Q4KMCEN.UJ4CEN.DLRKMQCWIjb\$J4ZIJ4UQCEZENOFFDLU\?S.QGN.CuT2j?JLK›QS.S/JLKJLFIH?N.Qg2Q4’\?ZEQ4’GNOFCEN.DLRJLFF[?ZIJLFT
V:gLJ4S.NO’?J4CEN.DLRKMQCW UN.R?N.UJ4S?gLJ4S.[?Q4KYN.RŽ$DLSO’ ^sJLFQFIH$J4ZIJLFCEQZIK u”
0
*
bh c™q
hbc™q
hwdrq
 ck(VulLª2W  ckPcV: ?j¡É2W  ckPcV: ?j¡o2W  ckPcV: ?j¡o2W
kB?B
pG”¡pLoLp4 2½
pG”¡pLo 2¼n
pG”¡pLoLn4 2É
, cE»
q‘ÉG¼Lj¡pLl4 ?”¡nG¼ q‘ÉLpGj 4½ 2”¡lLl
q‘ÉLpGj¡ªLlG¼L”4l ¯:¯ , ¯
E»
q‘lLoGj¡pLoLpG”¡pLn q‘lLoGj.@¼ 4ÉG” 4o
q‘lLoGj.¼nG¼L”.¼l $$ , $b¯
h/J4Z”»wDLU\?S.QGN.CMT
ª4 l
l
$b¯
h/ZEQ4’GNOFCEN.DLRd/ZEZEDLZ
V W
¼ q‘KMCEQ\
ÉG”¡É4 2o
ÉG”.¼½Ln
ÉG”¡pLÉG¼
$ , o±q‘KMCEQ\
? ”¡½Ll4 oG”¡oLÉLª
oG”.¼ 2ª
, ¯
ª±q‘KMCEQ\
oG”¡nLp4 oG”¡½4 L oG”¡½Lp4 , "!$#%#&!'"(#)&+*,-.)'#/0*21//34#/65879#67:"17:6;#/1(<=
N.RÅ8N.eL[?ZEQ#2”  H?QJ4eLZEQQUQRGC/Ž$QCM›QQRCEH?QZEQ4KM\$Q4FCEN.g2QR$J4CE[?ZIJ4S‡^—ZEQ4³2[?QR$FTCEZIJ 8§Q4FCEDLZEN.Q4K
NOK(³2[?N.CEQeLD2DG’bjJ4R$’ŸKMDNOKCEH$J4CD_^¬CEH?QZEQ4KM\$Q4FCEN.g2Q’?J4U\?N.R?eZIJ4CEN.D2KV:R?DLCENOFQCEH?QKMUJ4S.S
gLJ4S.[?Q4K/D_^ CEH?QSOJ4CECEQZwCEH$J4C\$D2KMQ(KMQZEN.DL[$KwQ4KMCEN.UJ4CEN.DLRJLFF[?ZIJLFT\?ZEDLŽ?S.QUKIWI”uCKMH?DL[?SO’Ž$Q
[?R$’GQZES.N.R?Q4’CEH$J4CCEH?QFJ4\$J4Ž?N.S.N.CMTD_^`\?ZED4gGNO’GN.R?e[?R$J4UŽ?N.eL[?DL[$KYQ4KMCEN.UJ4CEQ4KD_^iCEH?Q\?HGT?KMNœq
FJ4S.S.T(UQ4J4R?N.R?e_^—[?SGUDG’?J4S?\$J4ZIJ4UQCEQZ/CEZIJ 8§Q4FCEDLZEN.Q4K8NOK8DLR?QD_^ CEH?QYN.U\$DLZECIJ4RGCwJL’GgLJ4RGCIJ4eLQ4K
D_^`CEH?QUN.GQ4’  ckPcJ4R$J4S.T?KMNOK”  H?NOKNOKU[$FIHUDLZEQ’G0N F[?S.C›YN.CEHCEH?Q\?[?ZEQ  ck
J4R$J4S.T?KMNOK(Ž$Q4FJ4[$KMQD_^™CEH?QQ?FQ4KEKMN.g2QS.TH?N.eLH“DLZI’GQZUDG’GQSOKVuFIH$J4ZIJLFCEQZEN.µQ4’ŽGTŸJSOJ4ZEeLQ
RG[?UŽ$QZwD_^iR?DLRqs\?HGT?KMNOFJ4SUDG’GQ4KIWwCEH$J4CJ4ZEQ(R?Q4FQ4KEKEJ4ZET2”
, *#!)
+
 H?Q\?ZEDLŽ?S.QUD_^`R?DLRq‘KMCIJ4CEN.DLR$J4ZETgGN.Ž?ZIJ4CEN.DLR-KMN.eLR$J4S/UDG’GQS.N.R?eJ4R$’-J4R$J4S.T?KMNOK›YJLK
KMCE[$’GN.Q4’CEH?ZEDL[?eLHŸJUJ4R?N.\?[?SOJ4CEDLZgGN.Ž?ZIJ4CEN.DLR\$J4ZIJL’GN.eLU”(h/J4ZIJ4UQCEZENOFUN.GQ4’  ckPc
 4J Ž?S.QÉGt  H?QQ4KMCEN.UJ4CEQ4’hwdrq  ckPcV: ?j¡o2Ws¢¤£G¥UDG’GQS?FD2Q FN.QRGCIK8D_^ \?ZED8§Q4FCEN.DLRV
¼oGj >>G¹oN.RCEH?QhbcmUQCEH?DG’$WI”
5
3
%$
%$
¥ %$
'
%$
c k¹FD2Q FN.QRGCIK
8 Gp
8 G¼
q¤¼L”¡l 2ª pG”¡pLo4 2ª
pG”¡ªLoLªLÉ q‘pG”.¼oG¼½
q‘pG”¡½LÉLlL½ pG”.¼ªL½Ll
pG” 4½LnLª q‘pG”¡pLªLÉG¼
8 Gl
q‘pG”¡lLpLpG¼ 3 % $
pG”¡ÉG¼( 5 % $
q‘pG”.¼oLpG¼ I¥ % $
q‘pG”¡pLÉ 4½ % $
?
'
%$
P cmFD2Q FN.QRGCIK
8 Gp
8 G¼
pG”¡pL½LÉLl pG”¡pLo4o
q‘pG” 2o 2¼ q‘pG”¡pG¼ÉLn
pG”¡lLpLlLl pG”¡pLªLp q‘pG”.¼ 2½G¼ q‘pG”¡pLpLnLo
pG”¡pL½LpLl q‘pG”.¼pLlLª
8 Gl
q‘pG”¡p4 L 2p
q‘pG”¡pLÉLÉLÉ
pG”¡pLÉLnLª
q‘pG”¡pG¼pG¼
q‘pG”¡pLl G
1−step ahead predictions
0.04
0.02
0
−0.02
−0.04
231.5
232
232.5
233
Time (sec)
233.5
234
Å8N.eL[?ZEQYoGt/hwdrq  ckPcV: ?j¡o2Ws¢¤£G¥/UDG’GQS2Ž$JLKMQ4’DLR?Q•q‘KMCEQ\q‘J4H?Q4JL’\?ZEQ4’GNOFCEN.DLR$K›YN.CEH?N.RCEH?Q
gLJ4S.NO’?J4CEN.DLRKMQC(V:D?tJLFCE[$J4S8KMN.eLR$J4Suj?bt/\?ZEQ4’GNOFCEN.DLR$Kj tw½Lo Ä\?ZEDLŽ$J4Ž?N.S.N.CMTS.N.UN.CIKIWI”
Spectrum (db)
200
160
120
20
80
5
4
3
2
Frequency (Hz)
1
10 ec)
(s
e
0 0 im
T
Å8N.eL[?ZEQ(nGt  N.UQ•q‘’GQ\$QR$’GQRGCKM\$Q4FCEZE[?UD_^iCEH?Qhwdrq  ckPcV: ?j¡o2Ws¢¤£G¥KMN.eLR$J4SUDG’GQSu”
4J R$J4S.T?KMNOKY›YJLKJ4\?\?S.N.Q4’¶^—DLZCEH?Q ˜šZIKMCCEN.UQ›YN.CEH?N.RCEH?QgGN.Ž?ZIJ4CEN.DLRJ4R$J4S.T?KMNOKFDLRGCEQGCj$J4R$’
JFDLU\$J4ZENOKMDLR›YN.CEH-\?[?ZEQ  ck´J4R$J4S.T?KMNOK›YJLK(J4SOKMDUJL’GQ2”  H?QUJ4N.R-FDLR$FS.[$KMN.DLR$KD_^
CEH?NOKYKMCE[$’GTJ4ZEQ2t
A H?Q  ckPcmUQCEH?DG’S.Q4’CEDJR?DLRq‘KMCIJ4CEN.DLR$J4ZET  ckPcV: ?j¡o2Ws¢¤£G¥KMN.eLR$J4S/ZEQ\?ZEQ•q
KMQRGCIJ4CEN.DLR›YN.CEHŽ$JLKMNOK ^—[?R$FCEN.DLR$K8CEH$J4CwN.R$FS.[$’GQYJFDLR$KMCIJ4RGCJ4R$’J(KMN.R?Q4FD2KMN.R?Q\$J4N.Z
FIH$J4ZIJLFCEQZEN.µQ4’ŽGTCEH?QKMT?KMCEQU ^—ZEQ4³2[?QR$FTD_^ ±¢ G¹p¡o ”
 H?Q  ckPcV: ?j¡o2Ws¢¤£G¥8UDG’GQS2›YJLK8KM[$FFQ4KEK‘^—[?S.S.TgLJ4S.NO’?J4CEQ4’bjLJ4R$’(S.Q4’(CEDUQ4J4R?N.R?e_^—[?S
J4R$’“JLFF[?ZIJ4CEQKMN.eLR$J4SYJ4R$J4S.T?KMNOK” uCIK(CEN.UQ•q‘’GQ\$QR$’GQRGC(KM\$Q4FCEZE[?Uj/JLK›QS.SJLKN.CIK
R$J4CE[?ZIJ4S ^—ZEQ4³2[?QR$FTJ4R$’’?J4U\?N.R?eZIJ4CEN.DCEZIJ 8§Q4FCEDLZEN.Q4KjG›YJLKKMH?D4›YRCEDŽ$Q(N.RFS.D2KMQ
J4eLZEQQUQRGCw›YN.CEHN.CIKCEH?QDLZEQCENOFJ4SFDL[?RGCEQZE\$J4ZEC”
) H?Q  ckPcUQCEH?DG’b·¡KKM[?\$QZEN.DLZKM\$Q4FCEZIJ4SZEQ4KMDLS.[?CEN.DLRžNOK›DLZECEHžQU\?H$JLKMN.µN.R?e?”
uR$’GQQ4’bjGKMN.eLR$J4SbQ4KMCEN.UJ4CEN.DLRNOKwŽ$JLKMQ4’[?\$DLRDLR?S.TlLoKEJ4U\?S.Q4K/\$QZYKMT?KMCEQUÄ\$QZEN.DG’b”
cS.CEH?DL[?eLH%R?DLC\?ZEQ4KMQRGCES.TŸN.S.S.[$KMCEZIJ4CEQ4’bjwR?DLRqs\$J4ZIJ4UQCEZENOFUQCEH?DG’?KH$Jg2QKMQZEN.DL[$K
’G0N F[?S.CMTN.RKM[$FIHJKMQCECEN.R?e?j$JLKCEH?Q[$KMQD_^iJ¨IKMH?DLZECI©›YN.R$’GD4›¹S.Q4JL’?KwCEDZEQ4’G[$FQ4’
JLFF[?ZIJLFTJ4R$’ZEQ4KMDLS.[?CEN.DLRjb›YH?QZEQ4JLKCEH$J4CD_^¬J-¨ES.DLR?e2©DLR?QCEDZEQ4’G[$FQ4’CEZIJLFI«GN.R?e
J4Ž?N.S.N.CMT2”
Å?DLZ(CEH?Q  ckZEQ\?ZEQ4KMQRGCIJ4CEN.DLRD_^¬CEH?QKMN.eLR$J4Suj8JŸ¨ES.DLR?e2©  ck(VulLª2Ws¢¤£G¥UDG’GQS/›YJLK
[$KMQ4’b>” QT2DLR$’N.CIKYH?Q4JgGTD4g2QZ§qs\$J4ZIJ4UQCEZEN.µ4J4CEN.DLRjGCEH?NOKYUDG’GQSNOK¨IJ›Y«G›YJ4ZI’?© ^—DLZ
Frequency (Hz)
Frequency (Hz)
0.92
0.9
0.88
0.86
0.84
0
5
Time (sec)
10
3.8
3.7
3.6
3.5
0
5
Time (sec)
10
5
Time (sec)
10
0.08
Damping
Damping
0.04
3.9
0.02
0
0.06
0.04
0
5
Time (sec)
10
0
Å8N.eL[?ZEQ 2t  N.UQ•qsgLJ4ZETGN.R?eR$J4CE[?ZIJ4Si^—ZEQ4³2[?QR$FN.Q4KYJ4R$’-’?J4U\?N.R?eZIJ4CEN.D2KD_^¬CEH?QgGN.Ž?ZIJ4CEN.DLR
KMN.eLR$J4SwV štwCEH?QDLZEQCENOFJ4Suj qwq/q¤twQ4KMCEN.UJ4CEQ4’$WI”
J4R$J4S.T?KMNOKj ^sJ4[?S.C’GQCEQ4FCEN.DLR-J4R$’“FDLRGCEZEDLSuj8J4R$’ŸSOJLFI«?KCEH?Q’GN.ZEQ4FC\?HGT?KMNOFJ4SKMN.eLR?Nœ˜?q
FJ4R$FQD_^iN.CIKUN.GQ4’  ckPcmFDL[?RGCEQZE\$J4ZEC”
!@Y!!!(
QS.SOJ4R?eLQZjLPŸ” X”.jG¼½Lª2j >2; @
F K ;2=?> =?; =?; j2PJ4ZIFQS ] Q«G«2QZj
¸ Q› "wDLZE«b”
QRPZIJL’bjLk(”.j4Å$JLKEKMDLNOKjGB$” ] ”J4R$2
’ =$QgGN.CECj ?”¡c”.jL¼½L½LªGj¨Ic“h8DLS.TGR?DLUNOJ4Sœq‘cS.eLQŽ?ZIJ4NOF
PQCEH?DG’º^—DLZ ¸ DLRq§B2CIJ4CEN.DLR$J4ZET  ckPcÆB2N.eLR$J4ScR$J4S.T?KMNOKt-h/J4ZECIK J4R$,
’ Ij¡© =?; ! = Lj?nLoGVE¼4WIj$\?\”Y¼ q‘ÉLªG”
»wDLH?QRj&=8”.j¼½L½LoGj 7 = =?; j$h/ZEQRGCENOFQ•q‘¿J4S.Sbh  k(”
Å?DL[$KM«GN.CIJ4«GNOKj2X” ¸ ”.j4BGJ4«2QS.SOJ4ZEN.DL[j ?” B$”4J4R$’Å$JLKEKMDLNOKjGB$” ] ”.j2¼½L½LªGj¨ ¸ DLRq§B2CIJ4CEN.DLR$J4ZET
B2CEDGFIH$JLKMCENOFPDG’GQS.N.R?eJ4R$’ŸB2N.U[?SOJ4CEN.DLRD_^ÃB2CEZEDLR?edwJ4ZECEH$³2[$J4«2QXZEDL[?R$’-PDLCEN.DLRtcR
d/G\?S.DLZIJ4CEDLZETB2CE[$’GT2j¡© ! > = @H! ; ! =?; ! = ! = ;2= j uD2J4Rq
R?N.R$JGjbXZEQQ4FQ2j2\?\” 2l±q 2½G”
¿J4UUDLR$’bj ?” ”8J4R$’-ÀžH?N.CEQ2j8h8”¡k(”.j¼½L½LnGj¨  H?QcR$J4S.T?KMNOKD_^ ¸ DLRq§B2CIJ4CEN.DLR$J4ZET
B2N.eLR$J4SOK f KMN.R?e  N.UQ•q‘Å?ZEQ4³2[?QR$FTYPQCEH?DG’?Kj¡© ! =?; ! ! =?>;2=?> ; ! = j2¼½LpGVuÉ2WIj
\?\”/ ?¼½±qs L( 2”
N.CIJ4e2J›YJGjwX”8J4R$’“XQZIKEFIHjÀ)”.jw¼½L½LnGj 2
!@! = ! =?; ! 7 ,jbB2\?ZEN.R?eLQZ§qsÈwQZESOJ4e?j ¸ Q› "wDLZE«b”
¸ Q›YSOJ4R$’bj ] ”¡dY”.j¼½L½LÉGj ;2=?> ! ; ! = ; Y;2=?> ; @
=?; > > ! = j$c’?’GNOKMDLRÀ-Q4KMS.QT =$DLR?eLUJ4R”
¸ Q›YSOJ4R$’bj ] ”¡dY”.j¼½L½4 ?j¨EÀŸJg2QS.QCcR$J4S.T?KMNOKD_^¶ÈN.Ž?ZIJ4CEN.DLRtžh/J4ZECIK-¼ŸJ4R$’)lGj¡©
!
=?; ! ; ! =;2=?> ! j¼L¼nGj$\?\”w 2pL½±qs 2lLoG”
h/ZEN.Q4KMCES.QT2jbPŸ” ”.j8¼½LªL½Gj #! (= = ; ;2=?> #! = ; ! =?; 7 , =?; j
cFJL’GQUNOFYh/ZEQ4KEK”
"
!
$#
%
'& "(
"
*) %+
" '
,
$-
1.2 #65
) %+
" .
'
0/
43
& "(
) %+
"
7
%
8

Similar documents