�1393

R}}=B�'h

}QWl}

v=mt|U

Ovyt�| :v

iCW=O

O=}�'14

5�139

"X'2/

1|xQ=t

W'30

�3|xQ

wO

Res

earc

hN

otes}kDUt |v=UQRwQx@ =@ xR=U |DNU w sQH u}tND

xR=U |wor=�OWQ= T=vWQ� � �|OtQU uUL

�Q=}O=DU=� u}Or=sQm T=@a�|QDmO |wHWv=O� |t=_Dv= =[Q}ra

OyWt |UwOQi x=oWv=O 'u=Qta |UOvyt xwQo

'|DNU w sQH |=yT} QD=t pt=W xR=U |m}t=v}O |=yQDt=Q=B u}tND 'VywSB u}= R= hOyC=aq]= uDiQo Q_v QO =@ '=DU=Q u}= QO "CU= xR=U |wor= |v=UQRwQx@ VwQ R= xO=iDU= =@

pmW w |a}@] |=yTv=mQi uwJty p=Owt |=yxO=O R= xO=iDU= w |Q_v |wor= |x}rw=hOy `@=wD '|R=Ux]@=Q VN@ QO "OvwW|t |}=U=vW |DNU w sQH |=yT} QD=t '=yOwt

\U@ R= |Q}oxQy@ =@ w 'h} QaD |@ QHD w |Q_v |=ywor= u}@ hqDN= CQwYx@"OwW|t |R=Ux]@=Q xR=U |m}t=v}O |=yQDt=Q=B u}tND '=yOwt Ot=aD w xS} w Q=Okt|xr�Utu}}aD xk@]xU ?=k l} |wQ p=Owt |@ QHD V}=tR; |] p=Owt |y=oW}=tR; |=yQDt=Q=B

=@ TBU w '|UQQ@ |OOa|xr�Ut pL =@ =OD@= xOWx�=Q= |=yx]@=Q |DUQO "CU=xOW=@ 'C}=yv QO "OwW|t G=QNDU= xk@]xU ?=k |DNU w sQH '|y=oW}=tR; |=yxO=O

u=wD|t '|y=oW}=tR; |wor= =@ xOW xOR u}tND |wor= |a}@] |=yTv=mQi |xU}=ktu=v}t]= |m}t=v}O |=yQDt=Q=B u}tND |DUQO R= w OQm xOy=Wt =yu; u}@ |@U=vt |}=Qoty

"OQm pY=L

hassan.sarmadi@stu-mail.um.ac.irakaramodin@yahoo.comentezami@stu-mail.um.ac.ir

VwQ 'p=Owt |k}kL |=yxO=O '|DNU w sQH |=yT} QD=t u}tND %|O}rm u=oS=w"xR=U |wor= s}kDUt |v=UQRwQx@

xtOkt "1|xr}Uwx@ xR=U sQH T} QD=t u}vJty [1]"OW |}=U=vW |DNU w sQH |=yT} QD=t|Q_v |wor= hqDN= CQwYx@|vRw `@=D |R=Uxv}tm w xOW |Q}oxR=Ov=xS} w |=yQ=OQ@xR=U |Dq=YD= \}=QW hPL ut[ '|Oa@ C=k}kLD QO [2]"OW |v}@V}B |@ QHD wT} QD=t |}=U=vW OQwtQO 'xOW|Q}oxR=Ov= p=Owt |=y|}=Hx@=H R= |Q}oxQy@ =@ wT} QD=t u}tND VwQ '|Q_v |wor= Ow@y@ VwQ O=H}= =@ [3]"OW |R=Ux]@=Q sQHBerman VwQ T=U=Q@ =Q |DNU T} QD=t u}vJty u}kkLt [4]"Ci=} Ow@y@ sQH|wor= |Ot=aD C}U=UL VwQ |x�=Q= =@ 'xt=O= QO w OvOQm |v=UQRwQx@ w K}LYDsQH C=}YwYN |=y|DU=m u=tRty Qw]x@ =yu; VwQ QO [5]"OW KqY= xR=U"OW ^iL =yT} QD=t |ak=w |xR=Ov= xm|Qw]x@ 'Ci=} Ow@y@ xOW wor= |DNU w|}=U=vW xR=U |wor= s}kDUt |v=UQRwQx@ VwQ <=kDQ= =@ R}v |Oa@ C=ar=]t QO=@ xm ZQi u}= =@ Qw_vtu}O@ [6]"OW |UQQ@ =yxR=U |DNU w sQH |=yT} QD=t'xOW RwQx@ xR=U |ak=w sQH T} QD=t '|Q_v |wor= p=Owt |=yQDt=Q=B R= xO=iDU=|wor= |v=UQRwQx@ |xv}tR QO =yVwQ Qo}O R= "OW xOR u}tND |DNU T} QD=t|DNU w sQH |=yT} QD=t u}tND XwYNQO xOWs=Hv= VywSB x@ u=wD|t 'xR=Uu}kkLt xOWO=} |UQQ@ QO "OQm xQ=W= [7]p=Owt xOW |Q}oxR=Ov= |=yxO=O \UwDpt=m =v |=yxO=O |=Q@ |R=UTwmat x@W VwQ w hOy `@=D h} QaD R= xO=iDU= =@OvJ R= u}vJty "Ovvm |}=U=vW =Q xR=U |DNU w sQH T} QD=t OvDUv=wD 'p=Owt

'p=Owt |m}t=v}O V}=tR; CiQW}B =@ xDWPo |xyO OvJ QO �sDU}U |}=U=vW�xDiQo Q=Qk |W=aDQ= |=yQ=@ QF= CLD =yxR=U |@=} RQ= |=Q@ O}it |=yVwQ R= |m}'Q_v OQwt sDU}U QO OwHwt |=y|oS} w w C=aq]= uDW=O uwO@ xm |Qw]x@ "CU=w p}rLD|xr�Ut Q@ uwRi= "OQm |@=} RQ= =Q u; |m}t=v}O Q=DiQ |DUQOx@ u=wD|tvw =yGQ@ '=yOU uwJty |t}_a |=yxR=U xRwQt= '=yxR=U |m}t=v}O Q=DiQ |@=} RQ='=yxR=U xvwou}= QO "OvDUy |Q=OQ@xQy@ p=L QO w xOW xDN=U u=yH K]U QO '=ypBQO xR=U OQmrta |@=} RQ= |=Q@ |r=tDL= |=y?}U; |}=U=vW w CtqU x@ uOQ@ |Bh=Oy= u}= x@ uO}UQ u}=Q@=v@ "CU= Q=OQwNQ@ |Q=}U@ C}ty= R= '|Q=OQ@xQy@ u=tR"Ow@ Oy=wNv |UQDUO p@=k =yxR=U |m}t=v}O |=yQDt=Q=B u}tND w |}=U=vW =@ RHpt=W xR=U |m}t=v}O � |m} R}i |=y|oS} w |rm Qw]x@ |m}t=v}O |=yQDt=Q=B u}=

"OvwW|t xO}t=v |DNU w |}=Q}t 'sQH |=yT} QD=t|=yQDt=Q=B |}=U=vW w u}tND |=Q@ |OOaDt |=yVwQ 'Q}N= |xyO OvJ QOxO=iDU= =@ '=DU=Q u}ty QO "CU= xOW |R=Ux]@=Q p=Owt |=yxO=O \UwD |m}t=v}O�FRF� |Uv=mQi MU=B |=y`@=D R= |Q}oxQy@ w xR=U |wor= |v=UQRwQx@ VwQ R=

pw�Ut xOvU} wv �"1393 3 20 VQ}PB '1393 3 18 x}LqY= '1392 3 2 Ci=} QO %M} Q=D

"""s}kDU

t|v=UQ

RwQx@=@

xR=U|DN

UwsQH

u}tNDQO 'p=Owt |=yQDt=Q=B |}=U=vW |}=v=wD OwHw =@ "OvDUy |@ QHD |wor= w |Q_v |wor=

V}=tR; s=Hv= QO OwHwt CqmWt uwJty |}=yC}OwOLt p}rOx@ p=Owt V}=tR;"Ov=pt=m =v xOt; CUO x@ |=yxO=O 'p=Owt C=aq]= s=tD G=QNDU= uOw@v |rta wu=wD|t =yu; ltm =@ xm OvDUy |}=yVwQ xrtHR= wor= Vy=m w VQDUo |=yVwQxHQO N CQwYx@ =Q �� |W=aDQ= |=yOwt T} QD=t XwYNx@ �� p=Owt |=yxO=O|=Q@ |@Uv |}=Q}t ZQi =@ 'xS} w Q=Okt|xr�Ut uDiQo Q_v QO =@ [15]"OQm u=}@ |O=R;

"Ci=} CUO xOWv |Q}oxR=Ov= |=yOwt x@ 1 |x]@=Q VQDUo R= u=wD|t xR=U�K � !2

iM��i = 0 (1)

|=yxO=O u=wD|t |DNU w sQH |=yT} QD=t |R=U=OH =@ w 1 |x]@=Q T=U=Q@%OQm G=QNDU= pt=m CQwYx@ =Q p=Owt

K =

"K11 K12K21 K22

#; M =

"M11 M12M21 M22

#(2)

|Q}oxR=Ov= w xOW |Q}oxR=Ov= |=yOwt Qou=}@ ?}DQDx@ 2 w 1 |=yT} wvQ} R u; QO xm2 |x]@=Q QO |DNU w sQH |xOWl}miD |=yT} QD=t uO=O Q=Qk =@ "OvDUy xOWv

"OQm u}}aD =Q xOWv |Q}oxR=Ov= |=yOwt u=wD|t "K11 K12K21 K22

#� !2

i

"M11 M12M21 M22

#!(�i1�i2

)= 0 (3)

w xOW |Q}oxR=Ov= |=yOwt |xOvvmu=}@ xm 5 w 4 |x]@=Q wO 3 |x]@=Q VQDUo =@|=yOwt pmW x]@=Q wO u}= R= |m} pL =@ "OwW|t pY=L OvDUy xOWv |Q}oxR=Ov=

"OW Oy=wN pt=m |W=aDQ=�K11 � !2

iM11��i1 +

�K12 � !2

iM12��i2 = 0 (4)�

K21 � !2iM21

��i1 +

�K22 � !2

iM22��i2 = 0 (5)

CUO xOWv |Q}oxR=Ov= |=yOwt pmW x@ u=wD|t 5 =} 4 |=y x]@=Q R= xO=iDU= =@"Ci=}

�i2 =�K22 � !2

iM22��1 �

K21 � !2iM21

��i1 (6)

|=yxO=O 'xS} w Q=Okt|xr�Ut VwQ x@ |OOa |=yxr�Ut pL QO OwW|t u=WvQ]=Nx@ p=Owt V}=tR; QO =yvD =yOwt uOw@ pt=m =v C}OwOLt w Ov=pt=m |Q}O=kt p=OwtV}=tR; OQwt |xR=U Cy@=Wt R}v w uOw@ xO=U p}rOx@ Qo}O |wU R= "O};|t OwHw|=yxO=O '|W=aDQ= |=yOwt |Q}oxR=Ov= C}r@=k w |m}t=v}O |xDUUo |=ysDU}U x@'xOW x�=Q= C=L}[wD x@ xHwD =@ "Ov=pt=m |}=yxO=O |Q_v |=yVwQ Ovv=ty p=Owt`@=Dh} QaD CQwYx@ xR=U |DNU w sQH |=yT} QD=t u}tND |=Q@ |R=Ux]@=Q |=v@t"OwW|t xDiQo Q_v QO|@ QHD w |Q_v |wor=|m}t=v}O |=yQDt=Q=B u=}thqDN= 'hOy[�M ] = [Mx]� [Ma]|tQH hqDN=T} QD=t %R= Ov=CQ=@a hOy |=y`@=D u}=Q=}DN= QO ZQi =@ u}vJty "[�K] = [Kx]� [Ka] |DNU hqDN= T} QD=t w|x]@=Q \UwD =yu; VQDUo w |y=oW}=tR; |wor= |W=aDQ= |=yOwt pmW uDW=O|x]@=Q CQwYx@ |@ QHD w |Q_v |wor= |=yOwt hqDN= T} QD=t x@ u=wD|t '6|=yT} QD=t u}}aD xm CU= lQO p@=k "Ci=} CUO [��] = [�x] � [�a]

|DNU w [Mx] sQH |=yT} QD=t OwHw sOa p}rOx@ |DNU w |tQH hqDN=|rY= hOy VwQ u}= QO u}=Q@=v@ [15]"CU}v Q}PBu=mt= 'xR=U |ak=w |wor= [Kx]

u} wv |WwQ OvDUv=wD w xOQm xO=iDU= xR=U |wor= |v=UQRwQx@ VwQ QO hOy`@=D[8]"OvyO O=yvW}B p=Owt |=yxO=O \UwD |DNU w sQH |=yT} QD=t u}tND |=Q@VwQ R= |Q}oxQy@ w p=Owt \rDNt |=yxO=O uDiQo Q_v QO =@ [10w9]Qo}O |WywSB QO|m} "OW xOQw;Q@ |m}t=v}O |xDUUo |=ysDU}U QO |DNU w sQH 'Cr=L |=[i=@ u=QoVywSB "CU=yu; uOw@ pt=m =v p=Owt |=yxO=O R= xO=iDU= |=yC}OwOLt R=|=yT} QD=t |}=U=vW x@ |Uv=mQi MU=B |=y`@=D R= xO=iDU= =@ w VwQ l} O=yvW}Bp=Owt OwOLt |=yxO=O R= |Q}oxQy@ =@ u}vJty [11]"OvDN=OQB |}=Q}t w |DNU 'sQH=Q OwN |O=yvW}B VwQ u=v; [12]"OvDN=OQB |DNU w sQH |=yT} QD=t |}=U=vW x@"OvO=O Q=Qk |UQQ@ OQwt |m}t=v}O |xDUUo sDU}U u=wvax@ |WQ@ |=y?=k |=Q@|DNU w sQH |=yT} QD=t [13]p=Owt pt=m =v |=yxO=O R= xO=iDU= =@ Qo}O |wU R=pY=L |]N |=yMU=B R= xO=iDU= =@ u}vJty "OW xOQw;Q@ |R=Uxv}y@ VwQ \UwD|}=U=vW |WQ@ |=y?=k |DNU T} QD=t [14]u=tR |xRwL QO |]}Lt |=yQ=@ R=

"OW|=yT} QD=tpt=W xR=U|m}t=v}O |=yQDt=Q=Bu}tND 'VywSBu}= s=Hv= R= hOyx@ |@=}CUO |=Q@ "CU= xR=U |wor= |v=UQRwQx@ VwQ R= xO=iDU= =@ |DNU w sQH|W=aDQ= |=yOwt pmW w |a}@] |=yTv=mQi pt=W p=Owt |=yQDt=Q=B R= h=Oy= u}=hqDN= CQwYx@ hOy `@=wD h} QaD T=U=Q@ VywSB |Q_v |=v@t "OwW|t xO=iDU=|Q}oxQy@ w |Owt Ot=aD |=y\QW VQDUo =@TBU "CU=|DNU w sQH |=yT} QD=t"OwW|t |R=Ux]@=Q xR=U |m}t=v}O |=yQDt=Q=B '|v=UQRwQx@ Ov}=Qi QO hOy `@=wD R=s}kDUt VwQ Q=}at T=U=Q@ [6]u}kkLt \UwD xOW x�=Q= \@=wQ 'Q[=L VywSB QOu}}aD '=yu; VywSB hqNQ@ "O@=}|t <=kDQ= w xOW xv}y@ xR=U |wor= |v=UQRwQx@"OQ=Ov |DQwQ[ |DNU T} QD=t u}tND R=}vV}B u=wvax@ xR=U |ak=w sQH T} QD=t|=yxO=O w |Q_v |wor= R= x}rw= C=aq]= uDW=O =@ '|O=yvW}B VwQ QO `k=w QO|DNU w sQH |=yT} QD=t xv=o =OH Qw]x@ u=wD|t ,=iQY 'p=Owt |xOW |Q}oxR=Ov=|=yT} QD=t u}tND |=]N Vy=m u}vJty "OQm x@U=Lt =Q xR=U |ak=w |wor=|wor= =@ xOW xOR u}tND |wor= |m}t=v}O Q=DiQ |}=Qoty R}v w |DNU w sQHVwQ \@=wQ xm OwW|t u=WvQ]=N "CU= VywSB u}= |=}=Rt Qo}O R= |y=oW}=tR;s�qa Q}}eD =@ 'Qo}O |DQ=@ax@ "Ov=xO}J}B w T=UL Q=}U@ xR=U |wor= |v=UQRwQx@"OvwW|t wQx@ wQ =]N =@ |yHwD p@=k Qw]x@ G}=Dv '=yu; QO OwHwt |=yQDt=Q=B w|v=UQRwQx@ VwQ QO =yx]@=Q |xOvyOp}mWD |=RH= w C=aq]= Cy@=Wt u}=Q@=v@"Ow@ Oy=wNv |=y|R=Ux]@=Q Cy@=Wt =} G}=Dv uOw@ u=Um} |=vatx@ 'xR=U |wor=|OOa CQwYx@ xk@]Q=yJ |WQ@ ?=k l} =OD@= '|O=yvW}B |=yVwQ |@=} RQ= |=Q@R= xO=iDU= =@ xk@]xU |y=oW}=tR; ?=k l} |DNU w sQH TBU "OwW|t |UQ=w

"OvwW|t xOR u}tND p=Owt |y=oW}=tR; |=yxO=O

|DNU w sQH u}tND |=Q@ |[=} Q |R=Ux]@=Q "2=OD@= �� |DNU w sQH |=yT} QD=t �� xR=U |m}t=v}O |=yQDt=Q=B u}tND |=Q@\UwD pwtatQw]x@ =yQDt=Q=B u}= "OwW|t h} QaD |Q_v |wor= R= |}x}rw= C=aq]=xJv=vJ T=U= u}= Q@ "OQ}o|t s=Hv= |Q_v |R=Uwor= w OwOLt |=RH= |O=}v@ s}y=it=@ u=wD|t OvW=@ xR=U |Q_v |DNU w sQH |=yT} QD=t ?}DQDx@ [Ka] w [Ma]

|W=aDQ= |=yOwt pmW w f!ag xR=U |a}@] |=yTv=mQi 'xS} w Q=Okt |xr�Ut pLu}vJty "OQw; CUO x@ xR=U x}rw= C=aq]= R= p=Owt |=yxO=O u=wvax@ =Q [�a]

QO [�x] w f!xg xOW |Q}oxR=Ov= |=yxO=O 'p=Owt |m}t=v}O V}=tR; s=Hv= =@=@ \@DQt |=yxO=O ?}DQDx@ x w a |=yT} wvQ} R xOWu=}@ |=yCQ=@a QO "OvQ=}DN=

140

"""wu}O

r=sQmT=

@a'|Ot

QUuU

L |tQH |=]N T} QD=t u}}aD w [cm] xR=U |tQH ?}=Q[ T} QD=t uDiQo Q_v QO =@T=U=Q@ |@ QHD |wor= sQH T} QD=t CQwYx@ =Q xR=U sQH T} QD=t u=wD|t [�M ]

"OR u}tND 14 |x]@=Q[Mx] = [Ma] +

�[Ma] [�a] [cm] [�a]T [Ma]

�(14)

|W=aDQ= |=yOwt pmW w xR=U |x}rw= sQH uDW=O =@ \ki OwW|t xOy=Wt xm u=vJQO p=Owt |=yxO=O s=tD uDW=Ov "OQm G=QNDU= =Q xR=U |ak=w sQH T} QD=t u=wD|t|O=R; xHQON xR=U sQH T} QD=t u}tND QO C}OwOLt =yvD '|O=R; |=yxHQO s=tD

"CU=*pt=m *pmW x@ |Owt VQDUo |=yVwQ =@ u=wD|t xOW xQ=W= QDV}B xm u=vJxR=U |x}rw= Owt OvJ R= \ki xJv=vJ Qo}O |wU R= "Ci=} CUO |W=aDQ= |=yOwt=Q =yT} QD=t |t=tD '|U} QD=t C=}rta QO =]N R= |Q}owrH |=Q@ 'OwW xO=iDU=w Q]U 'OvDU}v |Q}oxR=Ov= p@=k xm |}=yOwt |=Q@ =yvD w xOQm h} QaD pt=m CQwYx@G=QNDU= Q@ xwqa VwQ u}= =@ "OvwW|t xO=O Q=Qk QiY =@ Q@=Q@ =Q =yu; =@ Q_=vDt uwDUQO u}vJty "OQm xO=iDU= xR=U |W=aDQ= Owt OvJ R= u=wD|t =yvD sQH T} QD=t pt=mR= xO=iDU= |=Hx@ CQwY u}= QO "CU= sm =L xR=U Q@ pw= Owt \ki =yxR=U R= |NQ@T} QD=t 'T=U= u}= Q@ "OQ@ xQy@ xR=U pw= Owt R= =yvD u=wD|t |O=R; |=yxHQO s=tD

%OwW|t u}}aD Qr=mU= Q}O=kt CQwYx@ [cm] |tQH ?}=Q[cm = �f��gT [Ma] f�ag � f�agT [Ma] f��g (15)

h} QaD N � 1 |vwDU |=yQ=OQ@ CQwYx@ |W=aDQ= |=yOwt pmW x]@=Q u}= QOCQwYx@ sQH hqDN= T} QD=t x@ u=wD|t '|tQH ?} Q[ h} QaD =@ "OwW|t=@ \ki u=wD|t =Q xR=U sQH T} QD=t xH}DvQO "Ci=} CUO [�M ] = cm[Ma]

%OQm x�=Q= xR=U |W=aDQ= pw= Owt uOQm^=Lr[Mx] = (1 + cm) [Ma] (16)

sQH T} QD=t u}tND VQDUo x@ \w@ Qt 16 |x]@=Q 'OwW|t xOy=Wt xm u=vJu}= "OW=@ sm =L u; |m}t=v}O Q=DiQ Q@ xR=U pw= Owt \ki xm |t=ovy CU= xR=U|OQ@ Q=m Q=}U@ 'Q_v OQwt |y=oW}=tR; ?=k Ovv=ty 'xO=U |=yxR=U |=Q@ QDW}@ Ov}=Qi|x]@=Q R= xO=iDU= w |W=aDQ= pw= Owt OvJ uDiQo Q_v QO ZQi =@ u}vJty "CU=

"OQw; CUO x@ sQH T} QD=t |}=U=vW R= |v=v}t]= p@=k G}=Dv u=wD|t 16

xR=U |DNU T} QD=t u}tND "2"2|=yxO=O w |Q_v |wor= |x}rw= C=aq]= T=U=Q@ R}v |DNU T} QD=t u}tND |=Q@hOy "OQm xO=iDU= |DNU |=Q@ |Owt Ot=aD \QW R= u=wD|t |y=oW}=tR; p=Owt=@ xm CU= K[=wQB "CU= [�K] |DNU hqDN= T} QD=t |x@U=Lt s=o u}= R=uO=O Q=Qk =@ "Ci=} CUO xR=U |ak=w |DNU T} QD=t x@ u=wD|t T} QD=t u}= u}}aDxR=U |ak=w |DNU w [�x] = [�a] + [��] xR=U |ak=w |=yOwt pmW T} QD=tT} QD=t u=wD|t u; VQDUo w 17 |x]@=Q QO [Kx] = [Ka] + [�K] CQwYx@|=yOwt s=tD xm ZQi u}= =@ R}vVN@ u}= QO "OQm x@U=Lt =Q [�K] |DNU hqDN=

"CU= Q=QkQ@ xQ=wty 17 |x]@=Q Owt Qy |=Q@ 'Ov=xOW T=}kt |W=aDQ=[�x]T [Kx] [�x] = �x (17)

([�a] + [��])T ([Ka] + [�K]) ([�a] + [��]) = �x (18)

|Q_v |wor= |x}rw= C=aq]= T=U=Q@ |DNU w |tQH hqDN= |=yT} QD=t u}}aD|=yQDt=Q=B '=yT} QD=t u}= u}}aD R= TB "CU=yu; R= pY=L p=Owt |=yxO=O w'OvDUy |y=oW}=tR; |wor= |DNU w sQH u=ty `k=w QO xm xR=U |ak=w |m}t=v}O|=yQDt=Q=B |Q}oxR=Ov= w |Q_v |wor= R= x}rw= C=aq]=h} QaD =@ "OwW|t xOR u}tND|rm Cr=L QO xR=U |DNU w sQH T} QD=t |x@U=Lt x@ xt=O= QO 'p=Owt |@ QHD

"s} R=OQB|t

xR=U sQH T} QD=t u}tND "1"2|=yxO=O w |Q_v |wor= |DNU w sQH |=yT} QD=t uwJty |D=aq]= uDW=O =@|x@U=Lt x@ xOW h} QaD hOy `@=D R= xO=iDU= =@ u=wD|t '|@ QHD |wor= p=Owt'OwHwt |=yxO=O R= xO=iDU= =@ O}=@ CUNv s=o QO "CN=OQB xR=U sQH T} QD=t|=yOwt pmW 'T} QD=t u}= x@ uO}UQ |=Q@ "OQm x@U=Lt =Q [�M ] u}at=v T} QD=tCQwYx@ =Q xR=U |ak=w sQH w [�x] = [�a] + [��] CQwYx@ =Q xR=U |ak=w

%OwW|t u} Ro}=H 7 |x]@=Q QO w xDiQo Q_v QO [Mx] = [Ma] + [�M ]

[�x]T [Mx] [�x] = I (7)

([�a] + [��])T ([Ma] + [�M ]) ([�a] + [��]) = I (8)

"Ov=xOW T=}kt |W=aDQ= |=yOwt pmW |R=Ux]@=Q pL=Qt |t=tD QO OwW|t Qw;O=}u=wD|t 8 |x]@=Q VQDUo =@ "CU= Q=QkQ@ Owt Qy |=R=x@ 7 |x]@=Q Qo}O CQ=@ax@

%OQm x�=Q= u}vJ =Q u;[�a]T [Ma] [�a] + [�a]T [�M ] [�a] + [��]T [Ma] [�a]

+ [��]T [�M ] [�a] + [�a]T [Ma] [��] + [�a]T [�M ] [��]

+ [��]T [Ma] [��] + [��]T [�M ] [��] = I (9)

|=Q@ |}=yv |x]@=Q '=yv; uOW QiY =} w Q_v OQwt |=yCQ=@a R= |NQ@ hPL =@%R= Ow@ Oy=wN CQ=@a |tQH hqDN= T} QD=t u}tND

[�a]T [�M ] [�a] = � [��]T [Ma] [�a]� [�a]T [Ma] [��] (10)

CU= N �N T} QD=t l} 10 |x]@=Q CU=Q CtUk 'OwW|t xOy=Wt xm u=vJ|Q_v |wor= R= pY=L |W=aDQ= |=yOwt pmW w |Q_v |wor= |x}rw= sQH x@ xmu}= u=wD|t |R=Ux]@=Q CrwyU |=Q@ u}=Q@=v@ "OQ=O |oDU@ |y=oW}=tR; |wor= w

%OQm h} QaD sQH ?}=Q[ T} QD=t CQwYx@ =Q =yCQ=@a[cm] = � [��]T [Ma] [�a]� [�a]T [Ma] [��] (11)

%OwW|t xO=O VQDUo [�a] w [�a]T ?Q[ TB w V}B =@ 10 |x]@=Q[�a] [�a]T [�M ] [�a] [�a]T = [�a] [cm] [�a]T (12)

|x]@=Q |Owt Ot=aD \QW R= xO=iDU= w|W=aDQ= |=yOwt pmW uOw@ T=}kt x@ xHwD =@pmW ?Q[pY=L u=wD|t '[�a]T [Ma][�a] = I CQwYx@ |Q_v |wor= |=Q@ 7T=U= u}=Q@ "CUv=O xR=U sQH T} QD=t Twmat =@ Q@=Q@ =Q |Q_v |wor= |=yOwtu}at=v T} QD=t w xOW u} Ro}=H 12 |x]@=Q QO [�a][�a]T = [Ma]�1 |x]@=Q

"OwW|t u}}aD [�M ] |tQH hqDN=[�M ] = [Ma][�a][cm][�a]T [Ma] (13)

141

"""s}kDU

t|v=UQ

RwQx@=@

xR=U|DN

UwsQH

u}tND"(Ton) xk@]Q=yJ |WQ@ ?=k |=Q@ C=k@] |ak=w sQH "1 pwOH

4 xk@] 3 xk@] 2 xk@] 1 xk@] [M ]

150 200 200 200 |ak=w sQH"(KN/m) xk@]Q=yJ |WQ@ ?=k |=Q@ C=k@] |ak=w |DNU "2 pwOH

4 xk@] 3 xk@] 2 xk@] 1 xk@] [K]

20 25 30 35 |ak=w |DNU

"xk@]Q=yJ |WQ@ ?=k "1 pmW

|y=oW}=tR; |=yxO=O u=wvax@ =yu; R= xOt;CUO x@ |=yOwt w |a}@] |=yTv=mQiC=aq]= 2 w 1 |=ypwOH QO "OvwW|t xDiQo Q_v QO p=Owt V}=tR; |R=Ux}@W =@"CU= xOW x�=Q= C=k@] |ak=w |DNU w sQH pt=W xk@]Q=yJ |WQ@ ?=k |x}rw=�RmQtDt sQH sDU}U� |m}t=v}O |xDUUo sDU}U l} CQwYx@ Q_v OQwt ?=kw sQH |=yT} QD=t 'xk@] Qy |m}t=v}O |=yQDt=Q=B uDW=O =@ "OwW|t |R=Uwor=?=k R= |}=tv 1 pmW "OvwW|t p}mWD xDUUo |=ysDU}U sD} Qwor= \UwD |DNU

"OyO|t u=Wv =Q Q_v OQwt |WQ@xS} w Q=Okt|xr�Ut pL w Q_v OQwt |wor= |=Q@ p=Owt V}=tR; |R=Ux}@W =@xR=U |xOWT=}kt |=yOwt pmW w |a}@] |=yTv=mQi |Q]k T} QD=t u=wD|tT=}kt |W=aDQ= |=yOwt pmW Qou=}@ ?}DQDx@ 4 w 3 |=ypwOH "OQm |}=U=vW =Q

"Ov=|a}@] Tv=mQi Q}O=kt w xOW|m}t=v}O |=yQDt=Q=B x@C@Uv Cw=iD|tm =@ 'xR=U|Q_v |wor= |x}rw= C=aq]=|xk@] w uD 150 Q@=Q@ swU =D pw= C=k@] sQH xJv=vJ "OvwW|t h} QaD xR=U |ak=w|Q_v |wor= QO C=k@] |DNU R}v w 'OwW xDiQo Q_v QO uD 112 5 Q@=Q@ sQ=yJxR=U |a}@] Tv=mQi |Q]k T} QD=t x=ov; 'OvW=@ xOW ZQi |ak=w |=yxO=O hYv|=yx]@=Q x@ xHwD =@ "OW Ovy=wN x@U=Lt |Q_v |wor= QO |W=aDQ= |=yOwt pmW w

�x = diag(!2x)i CQwYx@ xS} w |=yQ=Okt |Q]k T} QD=t 18 |x]@=Q QO

VQDUo =@ "CU= xR=U |@ QHD |wor= |a}@] |=yTv=mQi QwPHt |xOvQ}oQ@ QOh} QaD 19 |rm |x]@=Q =yu; R= |NQ@ uOWQiY w =yCQ=@a hPL w 18 |x]@=Q

"OwW|t[�a]T [�K] [�a] =�x � �a � [��]T [Ka] [�a]

� [�a]T [Ka] [��] (19)

pmW uOw@ |U=}kt ZQi =@ w 'sQH T} QD=t u}tND |=Q@ xOW x�=Q= VwQ uwJtyu=wD|t 19 |x]@=Q QO [�a] w [�a]T ?Q[ TB w V}B =@ '|W=aDQ= |=yOwt

%OQm x@U=Lt =Q [�K] |DNU hqDN= T} QD=t[�K] = [Ma] [�a] [ck] [�a]T [Ma] (20)

p=Owt |=yxO=O pt=W T} QD=t u}= "CU= |DNU ?}=Q[ T} QD=t [ck] u; QO xm"CU= xR=U |Q_v |wor= |DNUT} QD=t x=Qty x@ |@ QHD |wor= w |Q_v |wor= R=[ck] = �x � �a � [��]T [Ka] [�a]� [�a]T [Ka] [��] (21)

'|Q_v |wor= |x}rw= C=aq]= uDW=O w |DNU hqDN= T} QD=t u}}aD =@ s=Hv=QU"OQm |}=U=vW =Q xR=U |ak=w |DNU T} QD=t u=wD|t

[Kx] = [Ka] + [Ma] [�a] [ck] [�a]T [Ma] (22)

R= xO=iDU= =@ xR=U |ak=w |DNU T} QD=t u}tND |=Q@ OwW|t xOy=Wt xm u=vJxR=U |ak=w sQH T} QD=t R= xO=iDU= x@ |R=}v Qo}O 'xR=U |wor= |v=UQRwQx@ VwQx@ 'u; =@ Q_=vDt p=Owt |=yxO=O w |Q_v |wor= |x}rw= C=aq]= =@ ,=iQY w CU}v|DNU T} QD=t u=wD|t '|@ QHD |wor= R= p=Owt xOW |Q}oxR=Ov= |=yxO=O x=QtyQ=DiQ Q@ pw= Owt OvJ uOw@ sm =L CQwY QO "OQm u}}aD ?wr]tCkO =@ =Q xR=U |ak=w|=yxO=O T} QD=t |rm pmW 'sQH T} QD=t u}tND VwQ Ovv=ty xR=U |m}t=v}O|Q}oxR=Ov= |=yOwt x@ \w@ Qt uwDU w Q]U |x}=QO s=o u}= QO "Ovm|tv Q}}eD p=Owt"OQ}o|t s=Hv= pt=m CQwYx@ |U} QD=t C=}rta w xOW xO=O Q=Qk QiY Q@=Q@ xOWvQr=mU= CQwYx@ [ck] ?}=Q[ T} QD=t 'OW=@ sm =L xR=U Q@ pw= Owt \ki xJv=vJOwt uDiQo Q_v QO =@ =yvD =Q xR=U |DNU T} QD=t u=wD|t xH}Dv QO "OwW|t u}}aD

"OR u}tND xR=U pw= Owt Tv=mQi wck = (!x)2 � (!a)2 � f��gT [Ka] f�ag � f�agT [Ka] f��g

(23)

"OwW|t h} QaD N � 1 Q=OQ@ CQwYx@ |W=aDQ= |=yOwt pmW R}v 23 |x]@=Q QO|R=Ux]@=Q xR=U pw= Owt R= xO=iDU= =@ xR=U |DNU T} QD=t ck ?} Q[ u}}aD =@

%OwW|t[Kx] = [Ka] + ck [Ma] (24)

|y=oW}=tR; |UQQ@ w |OOa|xr�Ut pL "3|OOa VwQ x@ xk@]Q=yJ ?=k |DNU w sQH u}tND "1"3

|UQQ@ xk@]Q=yJ |WQ@ ?=k l} 'xOW s=Hv= |R=Ux]@=Q CLY |@=} RQ= |=Q@'xR=U u}= |ak=w |DNU w sQH |=yT} QD=t uOw@ OwHwt ZQi =@ =OD@= "OwW|t

142

"""wu}O

r=sQmT=

@a'|Ot

QUuU

L |v=UQRwQx@ VwQ R= xO=iDU= =@ xR=U|ak=w|DNU w sQH |=yT} QD=t|}=U=vW |=Q@"CU= xR=U |wor=

|y=oW}=tR; |xk@]xU ?=k |DNU w sQH u}tND "2"3|=yxO=O R= xO=iDU= =@ xk@]xU ?=k l} |DNU w sQH |=yT} QD=t VN@ u}= QO|UQQ@ OQwt ?=k "OwW|t xOR u}tND p=Owt |@ QHD V}=tR; R= xOt; CUO x@ p=Owt"CU= QDt|Dv=U 65 ZQa w pw] =@ w QDt 2 1 `=iDQ=x@ |y=oW}=tR; ?=k l}4 5 Ct=N[ w QDt|r}t 30 O=a@= =@ |tw}v}twr; |=y|W@v R= ?=k u}= |=yuwDUCt=N[x@ |Oqwi |xLiYl} CQwYx@ xk@] QyhkU "CU=xOW xDN=U QDt|r}tpYDt =yuwDU x@|J}B p=YD= w|W@v\UwD QDt|Dv=U 65�65 O=a@= w QDt|r}t 4

"CU= xOW xO=O u=Wv V}=tR; OQwt |xR=U R= |} wor= 2 pmW QO "CU= xOW|y=oW}=tR; |=yxO=O [16]'p=vo}U VR=OQB w Q_v OQwt |xR=U l} QLD =@9 w 8 pw=OH j@=]t |W=aDQ= |=yOwt pmW w |a}@] |=yTv=mQi pt=W p=Owtp=Owt V}=tR; R= pY=L |W=aDQ= |=yOwt pmW OwW|t u=WvQ]=N "OwW|t x�=Q=|=yxO=O 'p=Owt |=yQDt=Q=B G=QNDU= Ov}=Qi QO u}vJty [17]"Ov=xOW T=}kt|}=yxO=O|twywt VN@ 'xR=U sm |}=Q}t OwHw p}rOx@ "OvOW u}}aD \rDNt xOW |Q}oxR=Ov=

"|y=oW}=tR; |xk@]xU ?=k |wor= "2 pmW

"xk@]xU ?=k |y=oW}=tR; |=yOwt pmW T} QD=t "8 pwOH3 Owt 2 Owt 1 Owt [�x]

0 508 �0 633 0 282 1 xk@]�0 633 0 282 0 508 2 xk@]0 282 �0 508 0 633 3 xk@]

"(rad/s) xk@]xU ?=k |a}@] Tv=mQi |y=oW}=tR; Q}O=kt "9 pwOH3 Owt 2 Owt 1 Owt [!x]

10 874 7 525 2 685 |a}@] Tv=mQi

"xk@]Q=yJ ?=k |W=aDQ= |=yOwt pmW |ak=w Q}O=kt "3 pwOH4Owt 3Owt 2Owt 1Owt [�x]

0 044 �0 039 �0 035 0 014 1 xk@]�0 047 �0 003 �0 042 0 030 2 xk@]0 026 0 048 �0 003 0 043 3 xk@]�0 010 �0 036 0 050 0 051 4 xk@]

"(rad/s) xk@]Q=yJ ?=k |a}@] Tv=mQi |ak=w Q}O=kt "4 pwOH4Owt 3Owt 2Owt 1Owt [!x]

142 032 17 615 11 922 4 520 |a}@] Tv=mQi

"(Ton) xk@]Q=yJ |WQ@ ?=k |=Q@ C=k@] sQH u}tND "5 pwOH4 xk@] 3 xk@] 2 xk@] 1 xk@] [Mx]

142 614 190 379 190 197 190 252 C=k@] sQH

"(KN/m) xk@]Q=yJ ?=k |=Q@ xOW xOR u}tND |DNU T} QD=t "6 pwOH0 0 �26 535 57 482

[Kx]0 �22 127 48 641 �26 535

�17 684 39 809 �22 127 017 674 �17 684 0 0

"(KN/m) xk@]Q=yJ |WQ@ ?=k |=Q@ C=k@] |DNU u}tND "7 pwOH4 xk@] 3 xk@] 2 xk@] 1 xk@] [Kx]

17 674 22 127 26 535 30 947 C=k@] |DNU

w 5 pwOH j@=]t sQHT} QD=t 'p=Owt pt=m C=aq]= uDiQo Q_v QO =@ w '22 w 14"OwW|t xOR u}tND 6 pwOH j@=]t |DNU T} QD=t

C=k@]|DNU u=wD|t �6 pwOH� xOW xOR u}tND|DNUT} QD=t |R=U=OH =@"OQm x�=Q= 7 pwOH j@=]t =Q

C=k@] |ak=w sQH =@ Q@=Q@ |@} QkD Qw]x@ u=wD|t =Q C=k@] |=Q@ |v}tND sQH|ak=w sQH w |D=@U=Lt |=yVwQ R= xOt; CUO x@ sQH u}@ `k=w QO "CiQo Q_v QOx@ |=]N x@ C@Uv =]N Q=Okt u}= "OwW|t xOy=Wt =]N OYQO 4 81 OwOL xR=Uu}tND R= u=wD|t xH}DvQO "CU= R}J=v Q=}U@ Chen w Yang VwQ R= xOt; CUO

"OQm pY=L u=v}t]= sQH T} QD=t CUQO|ak=w Q}O=kt =@ xOW x�=Q= |=yx]@=Q R= xOt; CUO x@ |DNU Q}O=kt |xU}=ktu}tND |O=yvW}B VwQ xH}DvQO "CU= OYQO 11 OwOL QO |iqDN= Qou=Wv =yu;"CU= xH=wt |QDsm |=]N =@ Chen w Yang VwQ x@ C@Uv |DNU T} QD=t|v}tND |wor=|a}@] |=yTv=mQi hqDN= |xU}=kt =@ xR=U|m}t=v}O Q=DiQ|@=} RQ=u}= `k=w QO "OyO|t u=Wv =Q OYQO 3 5 |@} QkD |=]N '|y=oW}=tR; |wor= w|O=yvW}B |R=Ux]@=Q |DUQO Qou=}@ |}=yvDx@ '|a}@] |=yTv=mQi ?wr]t |}=Qoty

143

"""s}kDU

t|v=UQ

RwQx@=@

xR=U|DN

UwsQH

u}tND"(rad/s) |v}tND |wor= |a}@] Tv=mQi Q}O=kt "14 pwOH

3 Owt 2 Owt 1 Owt [!x]

10 404 7 231 2 645 |a}@] Tv=mQi"(Kg) pw= Owt T=U=Q@ C=k@] sQH u}tND "15 pwOH

3 xk@] 2 xk@] 1 xk@] [Mx]11 348 1 348 1 348 C=k@] sQH

|=yTv=mQi VHvU x@ 'xOW xORu}tND |=yQDt=Q=B |DUQO x@ uOQ@ |B |=Q@14 pwOH "s} R=OQB|t xOW xOR u}tND |wor= w |y=oW}=tR; |wor= u}@ |a}@]xOR u}tND |wor= |DNU w sQH |=yT} QD=t R= pY=L |a}@] |=yTv=mQi Qou=}@

"CU= xOW|a}@] |=yTv=mQi R}J=v Q=}U@ hqDN= R= |m =L 14 w 9 |=ypwOH |xU}=ktu}@ ?wr]t |}=Qoty Qou=Wv xm CU= xOW |}=U=vW |wor= w |y=oW}=tR; |wor=|=yx]@=Q \UwD xOW G=QNDU= |DNU w sQH |=yT} QD=t xH}Dv QO "CU=ywor= u}=

"Ov=xR=U |m}t=v}O |=yQDt=Q=B Qou=}@ |DUQOx@ 22 w 14=yxR=U |m}t=v}O OQmrta Q@ |W=aDQ= pw= Owt OvJ =yvD =yxR=U R= |NQ@ QO|Q}oxQy@ =@ ,=iQY u=wD|t =yOwt pt=m pmW R= xO=iDU= |=Hx@ xH}DvQO "CU= Q=PoQF=|wor= QO T=U= u}=Q@ "OQm |}=U=vW =Q |m}t=v}O |=yQDt=Q=B 'xR=U pw= Owt OvJ R==@ C=k@] sQH 'xR=U pw= Owt uOw@ sm =L uDiQo Q_v QO =@ 'xk@]xU |y=oW}=tR;

"OwW|t u}}aD 16 |x]@=Q R= xO=iDU=R= xO=iDU= =@ xR=U sQH T} QD=t u}tND Qou=}@ ?}DQDx@ 15 w 11 |=ypwOHu=wD|t 'OwW|t xOy=Wt xm Qw]u=ty "Ov=xR=U pw= Owt =yvD w p=Owt pt=m C=aq]=xO=iDU= =@ =yvD 'p=Owt |=yQDt=Q=B pt=m C=aq]= uDW=O Q@ xwqa =Q xR=U sQHT} QD=t\UwD xOW G=QNDU= sQH T} QD=t Q}O=kt `k=w QO "OQm u}}aD R}v xR=U pw= Owt R=|=yxO=O uOw@ pt=m =v CQwY QO 'Qo}O CQ=@ax@ "Ov=u=Um} 16 w 14 |=yx]@=Qpw= Owt T=U=Q@ u=wD|t w OQ=Ov OwHw sQH T} QD=t u}tND |=Q@ |rmWt p=Owt|}=U=vW Ov}=Qi 'OwHw u}= =@ "CN=OQB xR=U sQH |x@U=Lt x@ T} QD=t |xR=U"OwW|t xH=wt =]N =@ |W=aDQ= pw= Owt =yvD R= xO=iDU= T=U=Q@ |DNU T} QD=t|oDU@=w w |a}@] |=yTv=mQi s=tD uOQmv ^=Lr |}=Qo =w u}= |rY= p}rO `k=w QO|DNU T} QD=t G=QNDU= |=Q@ xH}Dv QO "CU=yxO=O u}= x@ |DNU T} QD=t u}tNDxQy@ p=Owt pt=m C=aq]= XwYNx@ '|rY= Owt OvJ R= |rm Qw]x@ O}=@ xR=U

"CUH

|Q}oxH}Dv "4pt=W xR=U |m}t=v}O |=yQDt=Q=B u}tND w |}=U=vW 'VywSB u}= s=Hv= R= hOyxR=U |wor= |v=UQ RwQx@ VwQ R= Qw_vt u}= |=Q@ "Ow@ xR=U |DNU w sQH T} QD=thqDN= |=yT} QD=t CQwYx@ hOy `@=wD h} QaD =@ VwQ u}= QO "OW xO=iDU='=yOwt Ot=aD |=y\QW VQDUo w |y=oW}=tR; w |Q_v |=ywor= QO |DNU w sQHu}= x@ uO}UQ |=Q@ "OR u}tND =Q xR=U |ak=w |DNU w sQH |=yT} QD=t u=wD|t|DNU w sQH |=yT} QD=t pt=W |Q_v |wor= |x}rw= C=aq]= uDW=O 'h=Oy=R= �� |W=aDQ= |=yOwt pmW w |a}@] |=yTv=mQi Q}_v �� p=Owt |=yQDt=Q=B x=Qtyx@|R=Uwor= =@ '|DNU w sQH |}=U=vW Ov}=Qi QO "OvDUy |R=Ux]@=Q |rY= |=yVN@u}tND |=Q@ |O=yvW}B |R=Ux]@=Q '|OOa VwQ x@ xk@]Q=yJ |WQ@ ?=k l} |Q_v

'T=U= u}=Q@ w CU= R}J=v Q=}U@ |Q}O=kt '|k}kL VN@ x@ C@Uv \rDNt |=yxO=O|=yQDt=Q=B '|twywt w |k}kL |=yVN@ C=a@ Qt QwPHt QPH VwQ R= xO=iDU= =@

"OvOW G=QNDU= |k}kL |}=yxO=O CQwYx@ p=Owt'|UOvy \}=QW R= xk@]xU ?=k |Q_v |wor= |x}rw= C=aq]= h} QaD |=Q@"OwW|t xO=iDU= |@ QHD |wor= QO xOW xDiQo Q=m x@ Kr=Yt w =[a= '|Dq=YD=xHwD =@ "CU= xOW xO=iDU= |Oqwi |xLiY R= |y=oW}=tR; ?=k u}= hkU |=Q@|tQH 'xk@] Qy |=Q@ u=wD|t Oqwi XwYNt sQH w xLiY u}= OwHwt O=a@= x@p=YD= |=y|W@v sQH R= x}rw= u}tND QO "CiQo Q_v QO sQowr}m 1 352 OwOL QOw =y|W@v u}= R= xO=iDU= p}rOx@ u}vJty "OwW|t Q_vhQY |Oqwi |xLiY x@ZQi QO u=wD|t |Oqwi |xLiY x@ uwDU uOQmpYDt |=Q@ |J}B p=YD= O=H}=

"OQm ZQi 10 pwOH j@=]t =Q xR=U |Q_v |DNU 'x}rw=Q=Okt|xr�Ut pL ut[ u=wD|t '|y=oW}=tR; ?=k |x}rw= C=aq]= h} QaD =@QO x}rw= ZQi p}rOx@ OwW|t Qw;O=} "OQm |}=U=vW =Q |Q_v p=Owt |=yQDt=Q=B xS} wxR=U |ak=w |DNU w sQH R}v w p=Owt |=yQDt=Q=B '|y=oW}=tR; ?=k |Q_v |wor=|x}rw= C=aq]= uO=OQ=Qk =@ `k=w QO "CW=O Ovy=wN hqDN= |ak=w |=yQ=Okt =@|wor= |v=UQRwQx@ Ov}=Qi QO p=Owt |y=oW}=tR; |=yxO=O R= xO=iDU= w xR=U |wor=Q=DiQ |UQQ@ =@ s=Hv=QU "Ci=} CUO =yxR=U |ak=w |DNU w sQH x@ u=wD|t xR=UxOW xOR u}tND |wor= |a}@] |=yTv=mQi |xU}=kt j} Q] R= xR=U |m}t=v}Ou=v}t]= xR=U |m}t=v}O |=yQDt=Q=B |DUQO R= u=wD|t '|y=oW}=tR; |=yxO=O =@|=Q@ xk@]xU ?=k |ak=w |DNU w sQH |=yT} QD=t =OD@= VN@ u}= QO "OQm pY=L'xOW x�=Q= |=yx]@=Q x@ xHwD =@ pw= Owt |=Q@ \ki TBU w 'p=Owt pt=m |=yxO=O'|@ QHD w |Q_v |=ywor= R= xOt; CUO x@ |=yxO=O T=U=Q@ "OvwW|t |}=U=vWxOR u}tND 22 w 14 |=yx]@=Q R= xO=iDU= =@ xR=U |ak=w |DNU w sQH |=yT} QD=t|DNU w xOW |R=U=OH 12 pwOH |DNU T} QD=t "�12 w 11 pw=OH� OvwW|t

"OwW|t x�=Q= 13 pwOH j@=]t xk@] Qy"(N/cm) |y=oW}=tR; xk@]xU ?=k C=k@] |Q_v |DNU "10 pwOH

3 xk@] 2 xk@] 1 xk@] [Ka]

74 626 74 626 74 626 C=k@] |DNU

"(Kg) |y=oW}=tR; xk@]xU ?=k xOW xOR u}tND sQH "11 pwOH3 xk@] 2 xk@] 1 xk@] [Mx]

1 352 1 352 1 352 C=k@] sQH

"(N/cm) |y=oW}=tR; xk@]xU ?=k |DNU T} QD=t "12 pwOH0 �49 257 98 522

[Kx]�49 268 98 522 �49 25749 266 �49 268 0

"(N/cm) |y=oW}=tR; xk@]xU ?=k C=k@] |DNU "13 pwOH3 xk@] 2 xk@] 1 xk@] [Kx]

49 266 49 268 49 257 C=k@] |DNU

144

"""wu}O

r=sQmT=

@a'|Ot

QUuU

L |@=} RQ= |y=oW}=tR; |wor= =@ |v}tND |wor= |a}@] |=yTv=mQi '|y=oW}=tR;|v}tND |wor= |a}@] |=yTv=mQi j}kO w ?wr]t |}=Qoty 'CUNvCr=L QO "OWT=U=Q@ |DNU w sQH |=yT} QD=t u}tND |DUQO Qou=Wv |y=oW}=tR; |wor= =@pt=m |=yxO=O Cr=L QO sQH T} QD=t uDW=O =@ u}vJty "Ow@ p=Owt pt=m C=aq]==@ \ki xm s}@=}|tQO |y=oW}=tR; |=yxO=O =@ xR=U |m}t=v}O Q=DiQ |UQQ@ w p=Owt"OQm u}}aD |@ wN CkO =@ =Q xR=U sQH T} QD=t u=wD|t |W=aDQ= pw= Owt R= xO=iDU='xR=U |m}t=v}O |=yQDt=Q=B |}=U=vW Ov}=Qi pL=Qt |t=tD QO OwW|t u=WvQ]=Nx@ C=aq]= |rm Qw]x@ "OQm xO=iDU= p=Owt |xOWT=}kt w s=N C=aq]= R= O}=@|R=Ux}@W |=yVwQ R}v w =y=]N uDiQo Q_v QO uwO@ p=Owt V}=tR; R= xOt; CUO

"Ov=s=N w xOW T=}kt |Q}O=kt 'p=Owt V}=tR;

|=yxO=O =@ xOW xOR u}tND |=yQDt=Q=B |}=Qoty "OW C=@F= |m}t=v}O |=yQDt=Q=B"Ow@ |DNU w sQH T} QD=t u}}aD |R=Ux]@=Q |DUQO Qou=}@ =yu; =@ Q_=vDt |ak=w|a}@] |=yTv=mQi Cw=iD Q=}at T=U=Q@ xR=U |m}t=v}O Q=DiQ hqDN= u}vJty|=yQDt=Q=B xH}DvQO w xOw@ R}J=v Q=}U@ |y=oW}=tR; |wor= =@ |v}tND |wor= QO

"OvOW |}=U=vW |@ wr]t CkO =@ |m}t=v}O|=yxO=O '|y=oW}=tR; |xk@]xU ?=k l} |wQ p=Owt |@ QHD V}=tR; s=Hv= =@|wor= |DNU w sQH 'OwHwt C=aq]= R= xO=iDU= =@ "OvOW |}=U=vW p=OwtxO=iDU= p=Owt pt=m C=aq]= R=CUNv s=o QO "OW u}}aDCr=L wO QO |y=oW}=tR;w sQH 'xR=U |m}t=v}O Q=DiQ Q@ pw= Owt C}tm =L uDiQo Q_v QO =@ TBU w 'OW?=k |DNU w sQH |}=U=vW |DUQO x@ uOQ@|B |=Q@ "OvOW xOR u}tND |DNU

�References� `@=vt1. Visser, W.J. \Updating structural dynamics models us-

ing frequency response data", Degree of Doctor of Philos-ophy, Imperial College of Science, University of London(1993).

2. Baruch, M. \Optimization procedure to correct sti�nessand exibility matrices using vibration tests", AIAAJournal, 16, pp. 1208-10 (1978).

3. Berman, A. \Mass matrix correction using an incompleteset of measured modes", AIAA Journal, 17, pp. 1147-8(1979).

4. Berman, A. and Nagy, E.J. \Improvement of a large an-alytical model using test data", AIAA Journal, 21(8),pp. 1168-73 (1983).

5. Guo, H. \An orthogonality sensitivity method for ana-lytical dynamicmodel correction usingmodal test data",Journal of Sound and Vibration, 187, pp. 771-80 (1995).

6. Yang, Y.B and Chen, Y.J. \A new direct method forupdating structural models based on measured modaldata", Engineering Structures, 31(1), pp. 32-42 (2009).

7. Lee, E.T. and Eun, H.C. \Update of corrected sti�nessand mass matrices based on measured dynamic modaldata", Applied Mathematical Modelling, 33, pp. 2274-2281 (2009).

8. Lee, E.-T. Rahmatalla, S. and Eun, H.-C. \Estimationof parameter matrices based on measured data", AppliedMathematical Modelling, 35, pp. 4816-4823 (2011).

9. Lu�s, H., De Angelis, M., Betti, R. and Longman, R.\Constructing second-order models of mechanical sys-tems from identi�ed state space realizations. Part I: The-

oretical discussions", Journal of Engineering Mechanics,129, pp. 477-488 (2003).

10. Lu�s, H., De Angelis, M., Betti, R. and Longman, R.\Constructing second-order models of mechanical sys-tems from identi�ed state space realizations. Part II:Numerical i2nvestigations", Journal of Engineering Me-chanics, 129, pp. 489-501 (2003).

11. Lin R. and Zhu, J. \Model updating of damped struc-tures using FRF data", Mechanical Systems and SignalProcessing, 20, pp. 2200-2218 (2006).

12. Chakraverty, S. \Identi�cation of structural parametersof multistorey shear buildings from modal data", Earth-quake Engineering And Structural Dynamics, 34(6), pp.543-554 (2004).

13. Yuen, K.-V. \Updating large models for mechanical sys-tems using incomplete modal measurement", MechanicalSystems and Signal Processing, 28, pp. 297-308 (2012).

14. Omrani, R., Hudson, R.E. and Taciroglu, E. \Story-by-story estimation of the sti�ness parameters of laterally-torsionally coupled buildings using forced or ambient vi-bration data: I. Formulation and veri�cation", Earth-quake Engineering & Structural Dynamics, 41, pp. 1609-1634 (2012).

15. Ewins, D.J., Modal Testing: Theory and Practice andApplication, Second Edition ed.: John Wiley & Sons, Inc.(2000).

16. Avitabile, P. \Structural dynamic test and experimen-tal modal analysis tutorial notes", Modal Analysis andControl Laboratory, University of Massachusetts, Lowell(2004).

17. Harris, C.M. and Piersol, A.G., Shock and VibrationHandbook, Fifth ed.: McGraw-Hill Companies (2002).

145