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h} QW |DavY x=oWv=O 'l}v�t |UOvyt |xOmWv=O

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C=a]k w =yxR=U |UOvy O=a@= |mJwm R= |W=v C=QF= VWwB x@ QO=k x} Q_v u}= "CU= xDiQo=@ w xOW G=QNDU= QwDwQwQm}t xawtHt |[Qa C=v=Uwv Q@ sm =L pB wm CqO=at =OD@= "CU=pw= Ot=U@ wO |=Q@ |D=Q=@a '|WtN pw= Owt wO uDiQo Q_v QO w u}mQro VwQ uOQ@ Q=m x@xOW x�=Q= CiW u=QwO =@ CyH hqN w CyHsy |vRnvr Cq=L R= l} Qy QO |a}@]

"CU= xOW G=QNDU= u=R}t=v xawtHt Q=O}=B C=W=aDQ= |xvt=O |=Q@ |}x]@=Q 'xwqax@ "CU=xO=iDU= =@ C=W=aDQ= |xvt=O w |a}@] |=yOt=U@ Q@ l}UqmQ}e |O=t C=}YwYN C=QF=xOW x�=Q= |Y=N |=yp=Ft QO R}v |OOa |H}=Dv "CU= |UQQ@ p@=k |r}rLD G}=Dv u}= R=

"OwW XNWt |OOa CQwYx@ l}UqmQ}e |O=t C=}YwYN xHwD p@=k C=Q}F-=D =D CU=

mehdihashemi@mech.sharif.irasghari@sharif.edu

C=W=aDQ= '|rB wm VvD |x} Q_v '|a@=D OvtiOy O=wt 'QwDwQwQm}t %|O}rm u=oS=w"|v=R}t=v '|Q=@H= w O=R;

xtOkt "1u}vJty "OW=@ OvtOwU X=N |=yOQ@ Q=m |=Q@ xm OQm |L=Q] |}xvwox@ u=wD|t 'u;u; |xOvyOp}mWD O=wt |tHL QUm |Ov@xHQO j} Q] R= X=wN |xDUw}B C=Q}}eDRmQtD |R=Uxv}tm Q}_v �� |Y=N |=}=Rt Qo}O K]U x@ K]U l} R= O=wt u}= QOx@ �� xQ}e w |WvD |=yGwt R= uDU=m w lQDWt Kw]U QO VvD Vy=m 'VvDQO |O=} R |xwkr=@ |=yOQ@ Q=m |a@=D OvtiOy O=wt 'wQ u}= R= [13w12]"OWN@|t u;

"OvQ=O x=Qtyx@ |DavY |=yVN@ w |UOvyt |=yxv}tRxO=iDU= x@ u}kkLt xmCU= xOW?@U|a@=D OvtiOy O=wt OQix@ QYLvt X=wNl}v=mtwQDmr=wv=v w wQm}t |=ysDU}U uwJty |}xDiQW}B |=ysDU}U QO O=wt u}= R=

[17�14]"OvQw; |wQQ=DiQ xm xOQm C@=F Q}N= |y=oW}=tR; C=k}kLD w C=ar=]t 'Qo}O |wU R=|m}v=mt |x} Q_v \UwD |DUQOx@ u=wD|tv =Q lJwm T=}kt QO =yxR=U |m}v=mtu=}@ =OYl} C=k}kLD u}= G}=Dv [20�18]"OQm |v}@V}B l}Uqm |xDUw}B \}LtK} QWD w |v}@V}B |}=v=wD l}Uqm |xDUw}B \}Lt |m}v=mt |x} Q_v xm OvQ=O|tu; Q} R w wQm}t O=a@= QO xOW xDN=U |QwD=}v}t |=yxR=U QO T=}kt x@ xDU@=w Q=DiQuDiQo Q_vQO =@ =yQwDwQwQm}t |m}v=mt Q=DiQ QDK}LY p}rLD 'wQ u}= R= "OQ=Ov =Q|=yx} Q_v T=U=Q@ =Q =yxR=U u}= xm CU= u; OvtR=}v 'xR=Ov= |mJwm x@ \w@ Qt C=QF=|R=UpOt |rB wm VvD |x} Q_v Ovv=t xOW O=yvW}B l}UqmQ}e |xDUw}B \}Lt

"OQm

|rY= |=RH= R= 'u=NQJ lU}O � CiW uwo =vwo |=yxawtHt pt=W =yQwDwQ `=wv=?wULt |DavY hrDNt |=yVN@ QO xO=iDU= OQwt Q=wO C=R}yHD w Cq;u}W=tu}kkLt xHwD OQwt R=@ Q}O R= =yxR=U u}= |m}v=mt Q=DiQ |UQQ@ wQ u}= R= "OvwW|t"CU= xDW=O RwQt= x@ =D l}t=v}OQwDwQ sra |xaUwD QO |}=RUx@ Vkv w xOw@=yQwDwQ u}= |D=W=aDQ= Q=DiQ XwYNQO |v=w=Qi C=k}kLD w C=ar=]t RwQt= =D xJQo ='OW=@ wQm}t T=}kt QO =yQwDwQ u}= |UOvy O=a@= xm |v=tR [5�1]'xDiQo CQwYpt=W u} wv C=R}yHD QO =yQwDwQwQm}t "O@=}|t CQwQ[ QDW}@ |=y|UQQ@ s=Hv=QO xm OvQ}o|t Q=Qk xO=iDU= OQwt =yQwDwtwQm}t w =yQwUQBtmwQm}t '=yu}@ QwDwQm}t[9�6]"OvQ=O Q=Qk R}v q=@ |=tO =@ p=}U CQw=Ht QO =yu}@ QwDwQm}t uwJty |OQ=wt|}xR=U Q=DN=U ^iL ut[ OvQO=k xm |DQ=QL |=yQBU |L=Q] |=Q@ =[=kD|iQat x@ QHvt 'Ovvm ptLD X=N \}Lt l} QO =Q |}q=@ |}=tO u=}O=Qo 'OwN[11w10]"OvwW|t xDN=vW �|a@=D OvtiOy O=wt� s=vx@ xRwQt= xm OW |O}OH O=wt,qwtat �� xO=t wO R= xm OvDUy |voty ?mQt O=wt |a@=D OvtiOy O=wt |rm Qw]x@|=DU=Q QO C}akwt R= |a@=D \UwD u; X=wN w xOW p}mWD �� Rri w l}t=QUQ}}eD xO=t wO X=wN R= Q_v OQwt w xDUw}B CQwYx@ 'XNWt C=yH =} CyH l}|xOvyOp}mWD O=wt Q_v OQwt X=wN R= |OvtxQy@ |x]U=wx@ =Q O=wt u}= "O@=}|t

pw�Ut xOvU} wv �"1395 1 24 VQ}PB '1394 12 22 x}LqY= '1394 8 20 Ci=} QO %M} Q=D

"""wu=R}

t=v|=yQw

DwQwQm}t

|D=W=aDQ=

p}rLDCiW \Uw QO CUQO xm CU= ?rY lU}O l} w 'OvtiOy O=wt R= pmWDt

'R Q@=Q@ u; `=aW 'L Q@=Q@ sDU}U x@ \w@ Qt CiWwQm}t pw] "CU= xDiQo Q=Qk|=Q=O R}v lU}O Qo}O |wU R= "CU= A CL=Ut x@ |}xQ}=O u; `]kt K]U w|@]k |UQv}= u=tt w Id |Q]k |UQv}= u=tt w M sQH 'Rd |oQR@ x@ |a=aWQO xDUw}B Qw]x@ CiWwQm}t |O=t X=wN xm OwW|t ZQi u}vJty "CU= Jdu; K]U CtU x@ CiWwQm}t RmQt R= CmQL =@ l}t=QU x@ Rri R= |a=aW |=DU=Q`=aW R= |a@=D 'R}vu; |r=oJ xrtHR= 'CiW |m}v=mt X=wN ?}DQDu}O@ "Ovm Q}}eD|rw] QwLt pwL C@=F |v=QwO CaQU =@ QwmPt QwDwQwQm}t "CU= �(r) CiWX � Y � Z C=YDNt x=oDUO 'u; Q@ xwqa "CU= u=QwO p=L QO CiWwQm}tx=oDUO xm |r=LQO '�1 pmW� CU= C@=F x=oDUO =} |UQv}= `HQt x=oDUO hQat C@=F |}x} w=R CaQU =@ xm CU= |rLt C=YDNt x=oDUO Qou=}@ x� y � zQO QwDwQwQm}t CiW xm OwW|t ZQi u}vJty "ONQJ|t X |rw] QwLt pwL

"OW=@ s}kDUt \N l} CQwYx@ (t = 0) wQW u=tRu2(x; y; z; t) 'u1(x; y; z; t) |}=Hx@=H |=yxir-wt '|r}=Q Q}D pOt T=U=Q@xDWwv u}vJ Q=wO `HQt x@ C@Uv CiW C=QP |}=Hx@=H Q=OQ@ R= u3(x; y; z; t) w

%OwW|t

u1 = u(x; t)� z @ �w(x; t)@x

� y @�v(x; t)@x

;

u2 = �v(x; t);

u3 = �w(x; t); (1)

x=oDUO x@ C@Uv xm CU= |v=tR `HQt QO C=QP C}akwt Qou=}@ (x; y; z) u; QO xmCiWwQm}t |RmQt\N OwOLt pmW Q}}eD hQat R}v �w w �v &OwW|t |Q}oxR=Ov= Q=wO`HQt x@ C@Uv u=wD|t =Q pmW Q}}eD |=yxir-wt u}= "OvDUy Q=wO `HQt x@ C@Uv

%OQm x@U=Lt Q} R \@=wQ j} Q] R= C@=F

�v = v cos'+ w sin';

�w = w cos'� v sin'; (2)

X � Y �Z x=oDUO x@ C@Uv x� y� z x=oDUO VNQJ |x} w=R ' u; QO xmCmQL Q@ xwqa CiW |=yu=tr= xm CU= XNWt "�1 pmW� CU= _' = =@|=yxir-wt u}}aD Qw_vtx@ wQ u}= R= "OvQ}o|t Q=Qk R}v |v=QwO CmQL CLD '|r=kDv=' z(x; t) |va} |r=wDt Qr} w= |x} w=R xU '?rY sUH CmQL u}= |}x} w=R CaQUpmW Q}}eD R= xm =Hv; R= "�2 pmW� OwW|t xDiQo Q_v QO x(x; t) w y(x; t)

" x(x; t) = t %u}=Q@=v@ 'xOW Q_vhQY Q=DWwv u}= QO VJ}B x@ \w@ QtCiWwQm}t u=tr= l} |}x} w=R CaQU Q=OQ@ 'Qr} w= |=}=wR T=U=Q@ '?}DQDu}O@

[40w39]%OwW|t u=}@ u}vJ

! =!1e1 + !2e2 + !3e3 =�

� _ z sin y�

e1

+�

_ z cos y sin t+ _ y cos t�

e2

+�

_ z cos y cos t� _ y sin t�

e3; (3)

OvDUy X3�Y3�Z3 C=YDNt x=oDUO |xm} |=yQ=OQ@ e3 w e2 'e1 u; QO xm4 |x]@=Q j@=]t Qr} w= pw= |x} w=R wO "Ov=xOW pYDt Q_v OQwt u=tr= x@ ,qt=m OwN xm

|iQat |]N u=UWm |xO=t OQwtQO Q=@u}rw= |=Q@ |rB wm VvD |x} Q_v|O}OH pmW 'QwmPt |x} Q_v QO |D=Q}}eD p=ta= =@ 'Oa@ p=U u}OvJ [23�21]"OW"Ov} wo|t xOWKqY= |rB wm VvD |x} Q_v u; x@ xm [24]OW O=yvW}B x} Q_v u}= R=|m}v=mt Q=DiQ |xar=]t |=Q@ |rB wm VvD |x} Q_v R= xO=iDU= 'Q}N= |=yp=U QOXwYN u}= QO "CU= xDW=O u}kkLt u}@ QO |Q}otWJ V}=Ri= =yxR=UwQm}t� Qr} w= pOt uDiQoQ_vQO =@ =Q =yQ}DwQm}t |m}t=v}O w |m}D=DU= MU=B [26w25]u}kkLt|x} Q_v T=U=Q@ 'u}= Q@ uwRi= "OvO=O Q=Qk|UQQ@ OQwt x} Q_v u}= R= xO=iDU= =@ w|rwvQ@R= xO=iDU= =@ [28w27]=yQ}DwQm}t Q=DiQ QO xR=Ov= |mJwm x@ \w@ Qt C=QF= '|rB wm VvDO=a@= QO T=}kt x@ xDU@=w |D=W=aDQ= Q=DiQ u}vJty "OW |UQQ@ wmvWwt}D Q}D pOt[32�29]"CiQo Q=Qk xar=]t OQwt |a@=D OvtiOy O=wt R= pmWDt |=yQ}DwQm}t lJwmw |m}D=DU= Q=DiQ QO =yxLiYwQm}t O=a@= |mJwm x@ \w@ Qt C=QF= 'u}= R= V}B[35�33]"CiQo Q=Qk|UQQ@ OQwt xOWKqY=|rB wmVvD |x} Q_vT=U=Q@ R}v|m}t=v}OpmWDt |=yjQwwQm}t lJwm T=}kt QO xR=Ov= x@ xDU@=w |m}v=mt Q=DiQ u}vJty

[38�36]"CiQo Q=Qk u}kkLt |@=} RQ= w xar=]t OQwt OvtWwy O=wt R=|Q=@H= w O=R; |W=aDQ= Q=DiQ Q@ xR=Ov= |mJwm x@ \w@ Qt C=QF= 'Q=DWwv u}= QOVvD |x} Q_v T=U=Q@ |a@=D OvtiOy O=wt R= pmWDt |=yCiW =@ =yQwDwQwQm}tX=wN xm CU= u; Q@ ZQi "CU= xDiQo Q=Qk |UQQ@ OQwt |r}rLD CQwYx@ |rB wmCtU x@ CmQL =@ �� l}t=QU x@ Rri R= p=Ft |=Q@ �� |a=aW CQwYx@ CiWwQm}tQwDwQwQm}t sDU}U CiW |xv=}t QO `k=w lU}O |=Q@ "Ovm|t Q}}eD CiW K]U=@ "OwW|t sDU}U |Q=@H= C=W=aDQ= ?@U xm xOW xDiQo Q_v QO |tQH |v=R}t=v l}QwDwQwQm}t CmQL Q@ sm =L |xOWpB wm |Oa@xU CqO=at 'uwDr}ty pY= R= xO=iDU=QwDwQwQm}t |Uv=vwRQ |=yOt=U@ |=Q@ |r}rLD C=Q=@a TBU "CU= xOW G=QNDU=QO w u}mQr=o VwQ |Q}oQ=m x@ R= xO=iDU= =@ wQTB w wQV}B |vRnvr Cr=L QOC=W=aDQ= |xvt=O 'u; Q@ xwqa "CU= xOW pY=L |WtN pw= Owt wO uDiQo Q_v|=Q@ "CU= xOW u}}aD lU}O QO RmQt R= GwQN OwHw QF= QO Q_v OQwt QwDwQwQm}t|D=W=aDQ= C=YNWt Q@ lJwm T=}kt QO xR=Ov= |=yQDt=Q=B Q}F-=D |ovwoJ K} QWD

"CU= xOW x�=Q= |OOa p=Ft OvJ QwDwQwQm}t

uw}UqwtQi x@ \w@ Qt x}=B s}y=it w xr�Ut h} QaD "2l} |rY= |=RH= u=wvax@ lU}O � CiW pt=W sDU}UwQm}t l} 1 pmW QOl} Q=@ CiWwQm}t l} pt=W QwDwQwQm}t u}= "CU= xOW xO=O V}=tv QwDwQwQm}t

=@ u=QwO p=L QO w |a@=D OvtiOy O=wt R= pmWDt CiW =@ QwDwQwQm}t l} "1 pmW" C@=F |}x}w=R CaQU

64

1|xQ=t

W'33

�3|xQ

wO'�13

96Q=y@

�h}QW

l}v=mt

|UOvyt sDU}U |R=UpOt "3

O=wt R= pmWDt CiW |w=L QwDwQwQm}t CmQL Q@ sm =L CqO=at G=QNDU= Qw_vtx@|WvQm |SQv= QO C=Q}}eD CU= sRq =OD@= 'uwDr}ty pY= R= xO=iDU= =@ |a@=D OvtiOyw OwW x@U=Lt u; |x}rw= Cr=L x@ C@Uv pmW Q}}eD u}L CiW |W@vH |SQv= wT=U=Q@ "OQ}o Q=Qk x@U=Lt OQwt R}v |HQ=N |=ywQ}v \UwD xOW s=Hv= Q=m u}vJty|=Q@ mij w �ij '�ij '"ij |=yxir-wt =OD@= QO p@k CtUk QO xOWKQ]t C=}[QiQO 1 |x]@=Q |Q=Po}=H =@ "OW Oy=wN x@U=Lt 12 |x]@=Q j@=]t CiWwQm}t

%OwW|t pY=L VvQm QwUv=D QiYQ}e |xir-wt =yvD 7 |x]@=Q

"11 = �y @2�v@x2 � z @

2 �w@x2 : (12)

VNQJ Q=OQ@ QiYQ}e |=yxir-wt '10 w 9\@=wQ QO 1 |x]@=Q |Q=Po}=H =@ u}vJty"OwW|t pY=L 14 w 13 \@=wQ R= ?}DQDx@ =vLv= uQ=kDt |=yxir-wt w

�2 = �@ �w@x

; �3 = @�v@x: (13)

�12 = �12 = �12@2 �w@x2 ;

�13 = �31 = 12@2�v@x2 : (14)

13 \@=wQ |Q=Po}=H =@ R}v w 6 |x]@=Q QO 12 |x]@=Q |Q=Po}=H =@ 'u}= Q@ uwRi=w 15 \@=wQ R= m QiY Q}e |=yxir-wt w �11 |xir-wt ?}DQDx@ '8 |x]@=Q QO 14 w

"OvwW|t x@U=Lt 16

�11 = E(r)"11 = �E(r)�y @

2�v@x2 + z @

2 �w@x2

�; (15)

m12 = m21 = �l2(r)�(r)@2 �w@x2 ;

m13 = m31 = l2(r)�(r)@2�v@x2 : (16)

%O=LD= x@ xHwD =@ p=L

(@i�v=@xi)2 + (@i �w=@xi)2 = (@iv=@xi)2 + (@iw=@xi)2

sDU}U |WvQm |SQv= uDWwv w 11 |x]@=Q QO 16 =D 12 \@=wQ |Q=Po}=H =@ wK]U x@ C@Uv |Q}op=QoDv= =@ R}v w 'U =

R�udV =

Rx

RA �udAdx CQwYx@

%s} Q=O CiW `]kt

U = (EI)eq + (�l2A)eq2 �

Z L

0

�(@

2v@x2 )2 + (@

2w@x2 )2

�dx; (17)

%u; QO xm

(�l2A)eq =ZA

�(r)l2(r) dA;

(EI)eq =ZA

z2E(r) dA: (18)

"Qr} w= |r=wDt |x} w=R xU x@ \w@ Qt |=yVNQJ "2 pmW[40w39]%OvwW|t \@DQt |}=Hx@=H |=yxir-wt x@

z = sin�124 @v

@xq(1 + @u

@x )2 + ( @v@x )2

35 ; y = � sin�1

24 @w@xq

(1 + @u@x )2 + ( @v@x )2 + ( @w@x )2

35 : (4)

=@ l}B wQDwR}= |xO=t l} |=Q@ |WvQm |SQv= |r=oJ '|rB wm VvD |x} Q_v T=U=Q@[24]%OwW|t x@U=Lt 5 |x]@=Q j@] lJwm |=ypmW Q}}eD

�u = 12 (�ij"ij +mij�ij); (i; j = 1; 2; 3) (5)

%u; QO xm�ij = �(r)"kk�ij + 2�(r)"ij ; (6)

"ij = 12 (ui;j + uj;i); (7)

mij = 2l2(r)�(r)�ij ; (8)

�ij = 12 (�i;j + �j;i); (9)

QwUv=D 'VvD QwUv=D |=yxir-wt hQat ?}DQDx@ �ij wmij '"ij '�ij q=@ \@=wQ QO"CU=vLv= uQ=kDt QwUv=D w |rB wm VvD QwUv=D |rY= Q]k GQ=N VN@ 'VvQm|x]@=Q \UwD xm OvDUy sr l}Uqm C@=wF � w � \@=wQ u}=QO 'u}vJtyE nv=} pwOt x@ � = E= [2(1 + �)] w � = E�= [(1� 2�)(1 + �)]

'OQ=wt u}= Q@ xwqa "�l}Uqm |x} Q_v x@=Wt� OvwW|t \@DQt � uwU=wB C@Uv wxO=t l} X=wN <RH l QDt=Q=B &CU= |rw] T=}kt Qou=}@ |rw] Oa@ =@ l QDt=Q=B�i u}vJty "OwW|t u}}aD XNWt |xO=t l} |=Q@ V}=tR; j} Q] R= xm CU=u |}=Hx@=H Q=OQ@ =@ Q} R |x]@=Q j} Q] R= xm CU= VNQJ Q=OQ@ |=yxir-wt Qou=}@

[24]%OvwW|t \@DQt�i = 1

2 [curl(u)]i ; (10)

?Q[pY=L T=U=Q@ u}vJty u=wD|t =Q �u|WvQm |SQv=|r=oJ xmCU= QmP u=}=W[24]%OQm ?=UL 11 |x]@=Q j} Q] R= mij QO �ij w �ij QO "ij |=yxir-wt

�u = 12 (�ij"ij +mij�ij) : (11)

65

"""wu=R}

t=v|=yQw

DwQwQm}t

|D=W=aDQ=

p}rLDlU}O sQH uOw@ RmQt R= GwQN u=R}t hQat ?}DQDx@ |rw] Oa@ =@ ez w ey u; QO xm

|SQv= wtHt xm CU= QmP u=}=W "OvDUy z w y |=yQwLt x@ C@Uv u; RmQt R='CiQo Oy=wN Q=Qk xO=iDU= OQwt uwDr}ty pY= QO w xt=O= QO xm 'QwDwQwQm}t |W@vH

"O};|t CUO x@ 24 |x]@=Q R=

T = T1 + Tecc (24)

u; x@ QwDwQwQm}t |R=Ht pmW Q}}eD u}L xm |HQ=N |=ywQ}v Q=m 'Qo}O |wU R=%OwW|t x@U=Lt 25 |x]@=Q j} Q] R= OwW|t p=ta=

�W =hVy�v + Vz�w +My�(

@v@x

) + Mz�(@w@x

)�����x=L

x=0; (25)

Q@ xOQ=w |=yVvD pY=L w 'z w y C=yH QO |WQ@ |=ywQ}v Vz w Vy u; QO xmQO xOQ=w |=yQw=DWo Ov};Q@ Mz w My u}vJty "OvDUy CiWwQm}t `]kt K]U'|a@=D OvtiOy R= pmWDt CiW =@ QwDwQwQm}t |=Q@ p=L "OvDUy z w y |=y=DU=Q

"OwW|t xO=iDU= uwDr}ty pY= R=t2Zt1

(�T � �U + �W ) dt = 0; (26)

w 26 |x]@=Q QO �W w T 'U |=Q@ ?}DQDx@ 25 w 24 '17 \@=wQ |Q=Po}=H =@|a}@] |RQt \}=QW w CmQL CqO=at 'C=Q}}eD ?=UL x@ \w@ Qt C=@U=Lt s=Hv=

"OwW|t pY=L u}vJ |QwQ[ w

[(EI)eq+(�l2A)eq]@4v@x4 +

�(�A)eq +M�(x� L

2 )�

�v

� @@x

��(�I)eq + Id�(x� L

2 )���@�v@x

+ 2@ _w@x

��= M2 [ey cos t� ez sin t] �(x� L

2 ); (27)

[(EI)eq+(�l2A)eq]@4w@x4 +

�(�A)eq +M�(x� L

2 )�

�w

� @@x

��(�I)eq + Id�(x� L

2 )���@ �w@x� 2@ _v

@x

��= M2 [ey sin t+ ez cos t] �(x� L

2 ); (28)��[(EI)eq+(�l2A)eq]@3v@x3 + (�I)eq(

@�v@x

+ 2@ _w@x

)

�Vy����x=0;L

= 0;or �vjx=0;L = 0; (29)��[(EI)eq+(�l2A)eq]

@3w@x3 + (�I)eq(

@ �w@x� 2@ _v

@x)

�Vz����x=0;L

= 0;or �wjx=0; L = 0; (30)�

[(EI1)eq + (�l2A)eq]@2v@x2 � My

�����x=0; L

= 0;

or ��@v@x

�����x=0; L

= 0; (31)

[40w39]"OwW|t x@U=Lt 20 w 19 \@=wQ j} Q] R= R}v CiWwQm}t |W@vH |SQv=Tshaft =Ttran + Trot = 1

2Z L

0

ZA

�(r)_r � _r dA dx

+ 12Z L

0

ZA

�(r)f!gT [I]f!gdAdx

=12Z L

0

h(�A)eq

�_v2 + _w2�+ (�J)eq!21

+(�I)eq�!22 + !23

�idx (19)

%u; QO xm(�A)eq =

ZA

�(r)dA;

(�I)eq =ZA

z2�(r)dA =ZA

y2�(r)dA;

(�J)eq =ZA

(y2 + z2)�(r)dA =ZA

r2�(r)dA: (20)

"OwW|t x@U=Lt 21 |x]@=Q j} Q] R= lU}O |W@vH |SQv= u}vJtyTdisk =1

2Z L

0

hM�

_v2 + _w2�+ Jd!21

+Id�!22 + !23

�i�(x� L

2 )dx: (21)

QwDwQwQm}t |W@vH |SQv= ?}DQDu}O@ "CU= l =Q}O |=DrO `@=D hQat �(�) u; QO xm|=yxir-wt 'lJwm pmW Q}}eD \}=QW QO "CU= T1 = Tshaft + Tdisk =@ Q@=Q@ z Q}O=kt u}=Q@=v@ 'Ow@ Oy=wN lJwm Q=}U@ x x@ C@Uv =yu; jDWt w |}=Hx@=Hw @v=@x Q@=Q@ ,=@} QkD ?}DQDx@ w xOw@ lJwm OW |iQat 4 |x]@=Q QO xm y wQ_v QO wO |x@DQt =D |}=ysQD '|SQv= |xrO=at QO u}=Q@=v@ "Ow@ Oy=wN �@w=@xlU} Q w CiW `]kt K]U uOw@ |wQ}=O p}rOx@ Qo}O |wU R= "OW xDiQo Oy=wNu; pY=L w 21 w 19 \@=wQ ?}mQD "Jd = 2Id w (�J)eq = 2(�I)eq %s} Q=O

"OW Oy=wN sDU}U pm |W@vH |SQv= |=Q@ 22 |x]@=Q x@ QHvt QwmPt OQ=wt =@

T1 =12

LZ0

��(�A)eq +M�

�x� L

2���

_v2 + _w2�� dx

+ 12Z L

0

��(�I)eq + Id�(x� L

2 )�

�

4@ _v@x

@w@x

+�@ _v@x

�2+�@ _w@x

�2!#dx

+ ((�J)eq + Id) 2: (22)

|W@vH |SQv= x@ |Qo}O sQD 'lU}O QOC} RmQt R= GwQN OwHw ZQi =@ 'u; Q@ xwqa%OwW|t x@U=Lt 23 |x]@=Q R= u; Q=Okt xm (Tecc) OW Oy=wN xi=[= sDU}U

Tecc=�M�_v (ey sin t+ez cos t)+ _w (�ey cos t+ez sin t)

� 2�e2y + e2

z

������x=L=2

; (23)

66

1|xQ=t

W'33

�3|xQ

wO'�13

96Q=y@

�h}QW

l}v=mt

|UOvyt |U} wvR=@ Z ?ULQ@ 35 =D 33 \@=wQ QO xOW xO=O |RQt \}=QW 'u}= Q@ uwRi=

%OwW|tZj~x=0; 1 = 0;�((EI1)eq + (� l2A)eq)

@2Z@~x2

����~x=0;1

= 0: (40)

pL VwQ "4CiW =@ QwDwQwQm}t |[Qa CmQL Q@ sm =L CqO=at pL Qw_vtx@ CtUk u}= QO|x]@=Q QO xOW xO=O |RQt \}=QW =@ �38 |x]@=Q� |a@=D OvtiOy O=wt R= pmWDt

%OwW|t xO=iDU= Z(~x; �) `@=D |=Q@ 41 |x]@=Q R= '40

Z(~x; �) =1Xk=1

�k(~x)Zk(�) (41)

xO=U |y=ox}mD \}=QW QO Q}D l} |=Q@ xS} w `@=D �k =p2 sin(k�~x) u; QO xm

l} 'u}mQr=m VwQ |Q}oQ=mx@ w 38 |x]@=Q QO 41 |x]@=Q uOQm OQ=w =@ "CU= �qwr�pY=L u=tR x@ C@Uv 2 |xDUQ R= C@=F ?}=Q[ =@ |rwtat p}Uv=Qi}O |xrO=at

%OwW|t

(1 + n2�2 ~R2)@2Zn@� 2 � 2�i(n� ~R)2 @Zn

@�+ (n�)4 ~SZn

+ 2M� sin n�21Xk=1

@2Zk@� 2 sin k�2 + 2n�I�d cos n�2

�1Xk=1

k��@2Zk@� 2 � 2�i@Zk

@�

�cos k�2

= M��2 (~ey + i ~ez) sin n�2 ei�� ; (42)

Q_v QO u}vJ Zn(�) |=Q@ u=tR x@ xDU@=w MU=B VN@ '42 |x]@=Q x@ xaH=Qt =@%OwW|t xDiQo

Zn(�) = Znei�� ; (43)

%OwW|t pY=L 44 |x]@=Q R= w CU= V=aDQ= |xvt=O Zn u; QO xm��2(1+n2�2 ~R2)Zn;+2(n� ~R)2�2Zn + (n�)4 ~SZn

� 2�2M�1Xk=1

Zk sin k�2+ 2n��2 ~R2I�d

1Xk=1

Zkk� cos k�2= M��2 (~ey + i~ez) sin n�2 ; (44)

Qo}Om} Q@ hrDNt |=yOwt |=yxvt=O u; QO xm CU= |Q@H |x]@=Q l} 44 |x]@=Qpw= |=yOwt x@ \w@ Qt xm pw= sQD wO =yvD ?} QkD u=wvax@ 'xt=O= QO "OvQ=Po|t Q}F-=D

%CQwY u}= QO "OwW|t xDiQo Q_v QO CU= swO w%n = 1 |=Q@

Z1 = M��2 ~ey + i~ez�4 ~S � (1 + 2M� � �2 ~R2)�2

(45)

�((EI1)eq + (�l2A)eq)

@2w@x2 � Mz

�����x=0; L

= 0;

or ��@w@x

�����x=0; L

= 0; (32)

w 'CiW u=QwO R= |W=v |UQv}= sy OwW|t xOy=Wt 28 w 27 \@=wQ QO xm u=vJQw[L QwDwQwQm}t CmQL CqO=at QO l}B wmUwQ} S C=QF= x@ \w@ Qt |=ysQD sy|=yu=k=D=} |wQ QwDwQwQm}t CiW |=yDv= wO Qy xm OwW|t ZQi xt=O= QO "OvQ=OCQ=@ax@ "Ovvm|t |Q}owrH hrDNt C=yH QO u; CmQL R= xm OvQ=O Q=Qk ?rY

%s} Q=O CiWwQm}t |=Q@ Qo}Ov jx=0;L = w jx=0; L = 0: (33)

=Q =yDv= wO QO xOQ=w |WQ@ |wQ}v Ov=wD|t =yvD QwmPt |=yu=k=D=} xm s}vm|t ZQiCiWwQm}t |=yDv= wO Qy QO Mz w My |=yQw=DWo Qo}O CQ=@ax@ "Ovvm ptLD\}=QW xm CiQo xH}Dv u=wD|t '32 w 31 \@=wQ |x_Lqt =@ "OvQ=Ov OwHw Q_v OQwt

%OW=@ Q=QkQ@ O}=@ Q} R |RQt��(EI)eq + (�l2A)eq

� @2v@x2

�����x=0;L

= 0; (34)��(EI)eq + (�l2A)eq

� @2w@x2

�����x=0;L

= 0; (35)

Q@ sm =L|�RH jDWt =@ |r}Uv=Qi}O |xrO=at wO 's = v+ iw\rDNt `@=Dh} QaD =@|r}Uv=Qi}O |xrO=at l} x@ u=wD|t =Q �28 w 27 \@=wQ� QwDwQwQm}t |[Qa CmQL

"OQm p}O@D \rDNt |�RH C=kDWt =@((EI)eq+(�l2A)eq)

@4s@x4 +

�(�A)eq +M�(x� L

2 )�

�s

� @@x

��(�I)eq + Id�(x� L

2 )���@�s@x� 2i @ _s

@x

��= M2 (ey + iez) �(x� L

2 )ei t; (36)

%37 |x]@=Q Oa@|@ |=yQDt=Q=B R= xO=iDU= =@ u}vJtyZ = s

R; ~ey = ey

R; ~ez = ez

R; ~x = x

L;

~! =��L

�2s

(EI)eq(�A)eq

; � = ~!; � = ~!t; (37)

CiW |w=L QwDwQwQm}t l} CmQL |=Q@ |�RH C=kDWt =@ |r}Uv=Qi}O |xrO=atxOW Oa@|@ CQwYx@ lU}O QO RmQt R= GwQN =@ |a@=D OvtiOy O=wt R= pmWDt

%OwW|t pY=L~S @

4Z@~x4 +

�1 +M��(~x� 1

2 )�@2Z@� 2 � @

@~x

��~R2 + I�d�(~x� 1

2 )�

��

@3Z@� 2@~x

� 2�i @2Z

@�@~x

��= M��2 (~ey + i~ez) �(~x� 1

2 )ei�� ; (38)

%u; QO xm~S = 1

�4 (1 + (� l2A)eq(EI)eq

); ~R = R2L;

M� = M(�A)eqL

; I�d = Id(�A)eqL3 : (39)

67

"""wu=R}

t=v|=yQw

DwQwQm}t

|D=W=aDQ=

p}rLD

'Em = 70 GPa pt=W u; Rri R=i |O=t X=wN xm CQwY u}O@ "CU= xOWl}t=QU R=i |O=t X=wN w �m = 2700 kg=m3 w �m = 26 GPa

�c = 3960 kg=m3 w �c = 157 GPa 'Ec = 393 GPa pt=W u;j@=]t |rO=at |O=t X=wN '20 w 18 \@=wQ QO 50 \@=wQ |Q=Po}=H [29]"OvDUypY=L Q_v OQwt OvtiOy O=wt R= pmWDt CiW =@ QwDwQwQm}t |=Q@ 51 |x]@=Q

"OwW|t

(�A)eq = p�m + 2�cp+ 2 A

(�I)eq = p�m + 4�cp+ 4 I

(�J)eq = 2p�m + 4�cp+ 4 I (51)

(�l2A)eq = p�m + 2�cp+ 2 Al2;

(EI)eq = pEm + 4Ecp+ 4 I: (52)

pY=L u}vJ Q_v OQwt QwDwQwQm}t |=Q@ I�d wM� ' ~S Q=Okt 39 |x]@=Q R= u}vJty%OwW|t

~S = 1�4 (1 + 4 p�m + 2�c

pEm + 4Ecp+ 4p+ 2 ( l

R)2);

M� = (p+ 2)M(p�m + 2�c)AL;

I�d = (p+ 2) Id(p�m + 2�c)AL3 : (53)

|=Q@ |rw] Oa@|@ QDt=Q=B l} CQwYx@ l=R C@Uv '53 |x]@=Q x@ xaH=Qt =@l=R = 0 Cr=L QO xm OQm xQ=W= O}=@ u}vJty "OwW|t h} QaD |Oa@ |=yp}rLDQmP u=}=W "O@=}|t Vy=m xDUw}B \}Lt l}Uqm |x} Q_v R= pY=L G}=Dv x@ G}=Dv20 x@ 1 Q@=Q@ R}v CiWwQm}t pw] x@ `=aW C@Uv wQW}B |=yp=Ft QO xm CU=

"OwW|t ZQiM� = 0�CiWwQm}tl} !n |a}@] |=yOt=U@C@Uv C=Q}}eD '3 pmW QO|a}@] |=yOt=U@ x@ |a@=D OvtiOy O=wt R= pmWDt w u=NQJ Q}e �I�d = 0 w

M� = 0� CiWwQm}t l} !n |a}@] |=yOt=U@ C@Uv C=Q}}eD "3 pmW|a}@] |=yOt=U@ x@ |a@=D OvtiOy O=wt R= pmWDt w u=NQJQ}e �I�d = 0 w!�n l}Uqm |x} Q_v j} Q] R= xOW x@U=Lt w |Rri TvH =@ x@=WDt |DiWwQm}t

"p hrDNt Q}O=kt |=R=x@ w l=R |rw] Oa@|@ QDt=Q=B hrDNt Q}O=kt ?ULQ@

%n = 2 |=Q@ wZ2 = 0; (46)

C=W=aDQ= Cr=L QO |va} 'OW=@ QiY lU}O RmQt R= GwQN xm |DQwY QO u}vJty%OwW|t x@U=Lt u}vJ u=tR x@ xDU@=w CtUk 'QwDwQwQm}t O=R;

Zk(�) = Akei!k� ; (47)

Ot=U@ !k w 'xOW x@U=Lt x}rw= \}=QW R= xm CU= |]rDNt C@=F `@=D Ak u; QO xm|x]@=Q |Q=Po}=H =@ u=wD|t =Q =yOt=U@ u}= "CU= s=k Owt |xQ=tW Q}_v Tv=vwRQR= Oa@ ?}DQDu}O@ "OQm x@U=Lt |y}O@ Q}e |=y?=wH uDiQo Q_v QO =@ w '42 QO 47\w@ Qt |Uv=vwRQ |=yOt=U@ 'pw= Owt wO uOQm ^=Lr =@ w QwmPt C=@U=Lt s=Hv=CyH hqN x@ \w@ Qt |vRnvr w 1wQW}B =} CiW u=QwO =@ CyHsy |vRnvr x@

%OwW|t pY=L pw= Owt wO |=Q@ QwDwQwQm}t |=Q@ 2wQUB =} CiW u=QwO =@

!f1 = �2 � ~R2 +q

�2 ~R4 + ~S(1 + �2 ~R2 + 2M�)1 + �2 ~R2 + 2M� ;

!b1 = ��2 � ~R2 �q

�2 ~R4 + ~S(1 + �2 ~R2 + 2M�)1 + �2 ~R2 + 2M� : (48)

w

!f2 = 4�2

1 + 4�2 ~R2 + 8�2I�d

h�( ~R2 + 2I�d )

+q

�2( ~R2 + 2I�d )2 + ~S(1 + 4�2 ~R2 + 8�2I�d )�;

!b2 =� 4�2

1 + 4�2 ~R2 + 8�2I�d

h�( ~R2 + 2I�d )

�q

�2( ~R2 + 2I�d )2 + ~S(1 + 4�2 ~R2 + 8�2I�d )�: (49)

|OOa G}=Dv "5QwDwQwQm}t |D=W=aDQ= Q=DiQ |wQ Q@ T=}kt |mJwm Q}F-=D |ovwoJ K} QWD Qw_vtx@"OwW|t x�=Q= |OOa G}=Dv CtUk u}= QO '|a@=D OvtiOy O=wt R= pmWDt CiW =@QO |O=t X=wN `} RwD |=Q@ |v=wD pOt l} 'xar=]t OQwt Cr=L l} u=wvax@

"CU= xOW xDiQo Q_v QO Q} R CQwYx@ CiWwQm}t�(r) = �m +

� rR

�p(�c � �m);

E(r) = Em +� rR

�p(Ec � Em);

�(r) = �m +� rR

�p(�c � �m); (50)

&CU=l}t=QU w Rri QO Q_v OQwt X=wN hQat?}DQDx@ c wm |=yT}Ov= u; QO xmX=wN OW=@ p = 0 xm |DQwYQO "CU= O=wt X=wN `} RwD u=wD hQat p u}vJtyX=wN C}=yv|@ QO O@=} V}=Ri= p xm |DQwYQO w 'pt=m l}t=QU x@ CiWwQm}tQDt=Q=B xm OwW|t ZQi =Hu}= QO u}vJty "Ow@ Oy=wN |Rri ,qt=m CiWwQm}t|xOvyOp}mWD QY=va |=Q@ "CU= u=Um} Ct=N[ |=DU=Q QO l |rw] T=}ktxDiQo Q_v QO CiWwQm}t l}t=QU w Rri R=i |=Q@ |Y=N O=wt '|a@=D OvtiOy |xO=t

68

1|xQ=t

W'33

�3|xQ

wO'�13

96Q=y@

�h}QW

l}v=mt

|UOvyt

QwDwQwQm}t l} |WtN swO w pw= Owt |=Q@ |a}@] |=yOt=U@ C=Q}}eD "5 pmW|v=QwO hrDNt |=yCaQU QO p = 1 =@ w |a@=D OvtiOy O=wt R= pmWDt CiW =@

"I�d = 0 001 w M� = 0 5 |=Q@ l=R |rw] Oa@|@ QDt=Q=B ?UL Q@

V}=Ri= wQV}B |vRnvr x@ \w@ Qt |=yOt=U@ |v=QwO CaQU V}=Ri= =@ "CU=Qw]x@ "Ci=} Oy=wN Vy=m wQTB CmQL Q}_v |=yOt=U@ xm |r=L QO 'O@=}|tV}=Ri= xDUw}B CQwYx@ l=R V}=Ri= =@ R}v |Ot=U@ Q}O=kt 'QDV}B OQ=wt x@=Wt

"O@=}|t|a}@] |=yOt=U@ C=Q}}eD sUQ 'l}t=v}O QwDwQ sra QO syt |=yQ=Owtv R= |m}u}}aD |=Q@ Q=Owtv u}= R= "Ov} wo|t �pBtm Q=Owtv� u; x@ xm CU= QwO ?ULQ@ QwDwQ|=Q@ Q=Owtv u}= "OwW|t xO=iDU= hrDNt |=yl}vwtQ=y |=R=x@ |v=QL@ |=yCaQUl=R xOW Oa@|@ |rw] QDt=Q=B hrDNt Q}O=kt |=Q@ 6 pmW QO Q_v OQwt QwDwQwQm}tsUQ |WtN pw= Owt wO |=Q@ I�d = 0 001 w M� = 0 5 'p = 1 Cr=L QO wV}=Ri= =@ sDU}UwQm}t |a}@] |=yOt=U@ OwW|t xOy=Wt xm Qw]u=ty "CU= xOWwQUB Vy=m w wQW}B |a}@] |=yOt=U@ V}=Ri= ?@U R}v QwO V}=Ri= w V}=Ri= l=R

"OwW|tGwQN OwHw |=R=x@ QwDwQwQm}t |xOW p=tQv |xvt=O C=Q}}eD Q=Owtv 7 pmW QOu=Wv l=R?ULQ@ w ez = 0 w ~ey = e Q=Oktx@ lU}O QO |v=R}t=v =} C} RmQt R=u=vJ "CU= xOW ZQi M� = 0 5 w p = 1 R}v Cr=L u}= QO &CU= xOW xO=Oxvt=O |xv}W}@ xm |}=H 'CiW |v=QL@ CaQU l=RV}=Ri= =@ 'OwW|t xOy=Wt xmsDU}U |Uv=vwRQ |=yOt=U@ V}=Ri= Qt= u}= Cra "O@=}|t V}=Ri= 'OwW|t Q=O}OB

"CU= l=R V}=Ri= QF=Q@

QwDwQwQm}t l} |WtN swO w pw= Owt |=Q@ |a}@] |=yOt=U@ C=Q}}eD "4 pmWOa@|@ QDt=Q=B?ULQ@ p = 1 =@ w|a@=D OvtiOy O=wt R= pmWDtCiW =@ u=NQJQ}e

"I�d w M� hrDNt Q}O=kt |=Q@ l=R |rw]

l}Uqm |x} Q_v j} Q] R= xOW x@U=Lt w |Rri,qt=mTvH =@ =t= 'x@=WDt|DiWwQm}t`} RwD hrDNt |=yu=wD |=R=@ w l=R |rw] Oa@|@ QDt=Q=B hrDNt Q}O=kt |=Q@ '!�nC@Uv u}= OwW|t xOy=Wt xm u=vJ "CU= xOW xO=O V}=tv 'p u; |O=t X=wNVvD |x} Q_v h=QLv= Q=Okt u}= V}=Ri= =@ xm |Qw]x@ O@=}|t OWQ l=R V}=Ri= =@!n C@Uv 'p Q=Okt Vy=m =@ u}vJty "O@=}|t V}=Ri= l}Uqm |x} Q_v R= |rB wm

"O@=}|t V}=Ri= !�n x@QwDwQwQm}t l} |WtN pw= Owt wO |=Q@ |a}@] |=yOt=U@ C=Q}}eD 4 pmW QO?ULQ@ p = 1 ZQi =@ w |a@=D OvtiOy O=wt R= pmWDt CiW =@ u=NQJQ}e"CU= xOW xO=O V}=tv I�d w M� hrDNt Q}O=kt |=Q@ l=R |rw] Oa@|@ QDt=Q=BswO Owt Q}_v Ot=U@ w M� =@ =yvD pw= Owt Q}_v Ot=U@ '49 w 48 |=yx]@=Q j@=]tOwt Q}_v |=yOt=U@ ?}DQDx@ I�d w M� V}=Ri= =@ "O@=}|t Q}}eD l�d =@ QwDwQwQm}tV}=Ri= =@ R}v |Ot=U@ Q}O=kt xm CU= QmP u=}=W "Ov@=}|t Vy=m |WtN swO w pw=

"Ov@=}|t V}=Ri= xDUw}B CQwYx@ l=RCiW =@ QwDwQwQm}t l} |WtN pw= Owt wO |=Q@ |a}@] |=yOt=U@ C=Q}}eD|v=QwO hrDNt |=yCaQU QO p = 1 ZQi =@ w |a@=D OvtiOy O=wt R= pmWDt5 pmW QO I�d = 0 001 wM� = 0 5 |=Q@ l=R |rw] Oa@|@ QDt=Q=B ?ULQ@nvQ =@ wQV}B CmQL Q}_v |=yOt=U@ pmW u}= QO "CU= xOW xO=O V}=tvxOW XNWt (BW) RtQk nvQ =@ wQTB CmQL Q}_v |=yOt=U@ w (FW) |@;

69

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t=v|=yQw

DwQwQm}t

|D=W=aDQ=

p}rLD

O=wt R= pmWDt CiW =@ QwDwQwQm}t xOW p=tQv |xvt=O C=Q}}eD Q=Owtv "7 pmWx@ lU}O QO |v=R}t=v OwHw |=R=x@ M� = 0 5 w p = 1 =@ w |a@=D OvtiOy

"l=R xOW Oa@|@ |rw] QDt=Q=B ?ULQ@ w ez = 0 w ~ey = e Q=Okt

|Q}oxH}Dv "6R= xOW xDN=U |DiW R= pmWDt |=yQwDwQwQm}t |D=W=aDQ= Q=DiQ Q=DWwv u}= QO|x} Q_v T=U=Q@ '|tQH |v=R}t=v =@ u; |xv=}t QO |mU}O w |a@=D OvtiOy O=wtsm =L CqO=at "CU= xDiQo Q=Qk xar=]t OQwt xOW KqY= |rB wm VvD l}UqmQ}e\@=wQ "CU= xOW G=QNDU= uwDr}ty pY= R= xO=iDU= =@ QwDwQwQm}t sDU}U CmQL Q@pmW Q}}eD x@ \w@ Qt pw= |a}@] Ot=U@ wO |=Q@ 49 w 48 CqO=at QO |r}rLD|xvt=O |=Q@ 45 |xrO=at QO R}v w 'wQTB w wQV}B CmQL wO Qy QO |WtN\@=wQ u}= R= xO=iDU= =@ "CU= xOW x�=Q= |v=R}t=v l} QLD QF= Q@ |Q=@H= C=W=aDQ=C=YNWt Q@ sDU}U hrDNt |=yQDt=Q=B Q}F-=D |UQQ@ |=Q@ |OOa G}=Dv '|r}rLD

"CU= xOW x�=Q= |}=yp=Ft QO |W=aDQ=

p = 1 =@ w|a@=D OvtiOy O=wt R= pmWDtCiW =@ QwDwQwQm}t pBtm Q=Owtv "6 pmWw M� = 0 5 Cr=L QO w l=R xOW Oa@|@ |rw] QDt=Q=B hrDNt Q}O=kt |=Q@

"|WtN swO w pw= Owt |=Q@ I�d = 0 001

=yCWwv=B1. forward2. backward

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