DA.IHOC Quac GIA TP.HCM
TRUONG DA.IHOC KHOA HOC nJ NHIEN
NGUYEN PHU VINH
, Ap DUNG PHu'(iNG PHAp PHAN TV HUU HAN GIAI MOT
A' , - 'A ? A ~ , A"
SO BAI TOAN TINH VA DONG CUA V~ T RAN CO BIEN
D~NG PHUC T~P
Chuyen nganh: ca HOC V~T THE RAN BIEN DA.NG
Ma S6:1.02.21
TOMTAT LU~N ANTIEN si ToAN HOC
, ' ;~
\:}".~H. nr Nt-HEN
THI1\lIEN
THANH PHa HO CHi MINH-2005
C6ng trlnh du<;choan thanh t<:tiKhoa Tmln Tin, Truong
B<:tiH9C Khoa H9C Tif Nhien, B<:tiH9C Quac Gia Thanh Pha H6 Chi Minh
Nguoi huang din khoa h9C:
PGS.TS Ng6 Thanh Phong
Truong B<:tiH9C Khoa H9C Tif NhienTp.HCM
Phan bi~n 1: GS.TSKH. Ng6 Van Lu<;c.
Lien hi~p HQiKhoa H9C Ky Thu~t. Ba Ria- Vung Tau.
Phan bi~n 2: PGS.TS. Nguy€n Luong Dung.
Truong B<:tiH9CBach Khoa. BHQG.Tp.HCM.
Phiin bi~n 3: PGS.TS. Nguy€n Dung.
Vi~n Co H9C Ung D\lng. Tp.HCM.
Lu~n an sf: du<;cbaa v~ truac hQi d6ng chilli lu~n an cip nha nuac h9P t<:ti
Truong B<:tiH9C Khoa H9C Tif Nhien Tp.HCM.
Vao h6i gio ngay thang nam 2005
C6 th€ tlm hi€u 1u~nan t<:ti:
Thu Vi~n Truong B<:tiH9C Khoa H9C Tif Nhien Tp.HCM.
Thu Vi~n Khoa H9c T6ng H<;pTp.HCM.

roan c.ach gitia mi~n dan h6i va mi~n deo clitia bier tru'dc va du'QCxac
Ginn trong qua trlnh giai sa bai roan dan-deo, tren bien roan cach cua 2
mi~n, cac thanh ph~n tenxd ling sua't, bien d'.lng, chuy~n vi phai thoa
man di~u kit%:nlien tlJc, do la nQi dung cat 16i cua thu~t giiii ann X'.lqui
h6i. TIm nghit%:mgiai rich cac phu'dng trlnh nay, con g~p noting kho khan
khong th~ khclc phlJc du'Qcv~ m~t roan hQc. Dai vdi v~t lit%:uma tr'.lng
thai ling sua't phlJ thuQc cii bien d'.lng va tac dQbien d'.lng, ta co ma hlnh
moi tru'ang dan-nodi va ht%:phu'dng trlnh se la phu'dng trlnh rich roan
2
Voltera. VI the' cac bai roan bien phi tuye'n a trong lu~n an cling chua
giiii ho~c chi giiii duQc mQt phgn tuong ling vdi s6 h,~tngphi tuye'n Clfth~.
4. T6ng quaD v~ dng d1.J.ngPPPTHH trong v~t dn hie'n d~ng
Bai roan dan-deo, dan-nhOt-deo dii khai thac tri~t d~ v~ m~t ly thuye't
tren the' gidi nhu: Bedukh6p.N.I, Khachan6p.L.M va A.Ilyushin,
Timasenka, Zeinkiewicz, George E.Mase, Hill.R, Bao Huy Bich, B~c
bi~t cac cang trlnh cua Sima.J.C, Hughes va Owen.J, Hinton.E dii duc
ke't thu~t giiii cho cac bai roan nay c6 ten la cac thu~t giai return-
mapping ma lu~n an t~m dich la "anh x~ qui h8i". Nhung cae vi dlf s6 Clf
th~ hgu nhu hie'm c6, va l<,tithu~t giiii nay tuong d6i phlic t<,tpd ph<,tmvi
mQt ehi~u va hai chi~u. Lu~n an sU'dlfng PPPTHH cho thu~t giai anh x<,t
qui h8i a chuang 2 va chuang 3. Xet cae bai roan Clfth~, c6 loi giai s6
d~ so sanh vdi nghi~m chinh xae, vi dlf a chuang 3, trong truong hQp
deo ly tuang a Khachan6p.L.M dii c6 Wi giai giai rich. Cling trong ph~m
vi PPPTHH eho cae bai roan c6 bie'n d~ng phUc t~p, lu~n an dii xet them
hai bai roan ti:nhva dQng a chuang 4 va chuang 5. Lien quail tdi llnh v\fe
nay nhu: Lions.J.L, Johnson.Claes, NT.Long. Cac tac gia nay khao sat
s\f t8n t~i va duy nha't nghi~m mQt caeh tri~t d~. d day lu~n an ma
phOng hai bai roan Clfth~, dung PPPTHH, ke't hQp vdi phuong phap giai
rich ham, sai phan hil'u h<,tn.xa'p xi nghi~m trong khang gian hil'u h~n
ehi~u, chung minh s\f t8n t<,tiva duy nha't nghi~m cua bai rain xa'p xi v~

£u E
1.3. M(it vai khai ni~m va ky hi~u cong thU'c hie'n phan daD cleo.
D~ c<i.pde'n cac cong CtJ, ky hi~u cua giai rich ham, cong thuc bie'n
phan, d~o ham suy r(jng, d~o ham Frechet, Gateaux. D~ ti~n vi~c trlnh
bay, ta dung ky hi~u tenxo h~ t<?ad(j thing tn;1'cgiao va qui uac Einstein.
.Phuong trlnh bie'n phan Euler-Lagrange: Tli vi~c c\;1'cti~u phie'm ham
nang luQng rc(ui ) = fH {W(Eij)- hi ui }dx, Uj=O V xi E 3B , ta thu
B
dt(Qcphuong trlnh can b~ng trong v<i.tth~ ran bie'ndang () U,j + hi =O.
1.4. Xay dt!ng d~.lllghie'n phan hai tmin daD h6i t6ng quat trong R3
Chuy~n vi u=(Uj), trong mi~n Q, truong ling sua't d6i xung a=(aij),
aij=aji tenxo bie'n d<:tng E=(Ejj)i,j=I,2,3. Tli phuong trlnh can b~ng:
()(j,j + ii =0 trong Q, vai di~u ki~n bien Uj=0 tren f2 va (Jij nj = gi
tren fl, i=1,2,3. Ta c6 th~ phat bi~u d<:tngbie'n phan nhu sau: TIm UiE V
sao cho : a(uj,vi)=L(v;) '\I Vi E V, trong d6
a(u, v) = Hf(Adiv(u)div(v) + IlEij (U)EU(v),ktx
Q
L(Vi) = ffffiVidx + ffgiVids, V={V=(Vi) E [HI(Q)]3: Vi=0 tren f2}.
Q rl
Ta c6 th~ ki~m chung r~ng V la khong gian Hilbert vai rich vo huang
4
(e, e) tuong ling vai chuifn II-IIv , va aCe, e)y 1a d?ng song tuye'n tinh
tren VxV, thoa di~u ki~n V-elliptic a(vi, Vi) ~ a Ihll~ \iVi EV, trong do
Ilvill~ =11vII~'(Q)=lhll~'(Q)' Ihll~'(Q) = gs(vl +(~::)2)dX.
Chu'dng 2: MO HINH PHI TUYEN M(n cHrEu
2.1. Dflll deo m(}t chi~u
e Cac phuong trlnh ling xU',dinh 1li(~tchay deo cua mo hlnh deo 1:9tuCing
duQc mo ta nhu bang 2.1, mo hlnh cho v~t 1i~u tai b~n d~ng huang nhu
bang 2.2.
Bang2.]

.
o B
[0 T]
eo: a =
E
H)
E, leu len c ay eo: =, = trong x , .
+(K+ .
cr = EE cac traCinghQp con l?i.
5
.D,~tngy€u (bi€n phan) eua bai loan BTBD mQt ehi~u
Dinh nghla tru'ong ehuy€n vi dQng hQe kha dI:
St ={u(8,t):B~IR, u(8,t)lauB =~(8,t)}, co St CHI(B)
vai m6i t e6dinh,vai H I(B) 1akhong gian Sobo1evcap 1tren B.
ta dinh nghla: V ={17Eel (B) : 17(0)= a} la khong gian cae ham co
d'.loham tren B va tri~t lieu khi x
=O.Do 1akhonggiancae hamthii',
hay bi€n phan dQng hQe khii dI. Khi do 11= M E HI (B) :11(0)=a}
Vai cae dinh nghla tren, phat bi€u d'.lng y€u eiia bai tOaDBTBD nhu' sau:
TIm ham U(8,t) E St sao eho:
fpv17dx+G(o-,17) =0, V17EV, VtE[O,T],
B
f
/
f
- / a17
trangdo G(o-,17)= 0-17dx- pb17dx-o-17la B va 17 =
B B ()" ax
Chu yding u(x,t) la ham a"n trang &= Ux va v=Utt.
Chung minh du'Qeslf tu'ong du'ong eua hai d'.lng vi phan va bi€n phan. Slf

set. n+1=set. n+! va exIt
ELSE (tinh deo)
I"trial
f:,.y= ~ > 0
E+K
[
1 f:,.y.E
1
trial
ern+l = -Ier~-~~ll ern+l
p - P A .
(
trial
)
Gn+l - Gn + uy sign (J n+l
an+! =an +f:,.y
END
a
a
an
ay
trial
CJ"n :-1
an+!
En~l ~n
I 1
~En <:0
1 I
: :
1

p 1
Gn+1 :
I
I
En+! .
l1li
f
trial <
0
Hlnh2.1a. n+! -
.Bid toaD qui hOl)ch H~irOi rl)c
Ta xet phiern ham hai bien X(a-, a) :
1 .
I 1
.
I
1
X
(er a)=-
(
ertna -er
)
E-
(ertna -er)+-(a -a)K
(
a -a
)
.
, 2 n+l n+l 2 n n
Hlnh 2.1

mQt khi ling xU'cua v~t Mu niim trang giai do~n tal b~n ding huang ma
trang [61] da ma ta. Cac ke't qua so' cho thily ma hint phu h<;ipvai tht!c
nghi~m. Ta co th6 ling dlfng trang cang ngh~ san xuilt v~t Mu d6 tang
module dan h6i biing cach lam cho v~t li~u chay deo truac de'n mQt ling
suilt mong mu6n. Khi cho cac tham so' tal b~n tri~t tieu ta thu du<;icke't
qua bai roan ba thanh deo ly tudng nhu LM. Khachanc/p [19].
Ma hlnh mQt chi~u la mQt co sd t6t d6 phat tri6n cac ma hlnh 2
chi~u, 3 chi~u. So d6 kh6i:
I B5't I
8
Nh~p dli li~u xae dinh cae kieh thu'de hlnh hQe, tal trQng tae
d\lllg, cae di6u ki~n bien,d~e tinh v~t Mu,
T'.lOcae array ban diiu zero
~
.
I
_.A " , pint
Dti'hyubandaut'.llv~tnxEB: n' an,dn,Fn ,Cn+l
Tinh roan ma tr~n de)eti'ng, veeW tal phiin tti'
Yang l~p gia tang tai
Up rap ma tr~n phiin eti'ng phgn tU'va veetd tal phgn tU'
d~ tinh ma tr~n de)eti'ng t6ng th~ va veeW t6ng th~.
Tinh roan veetd ehuy~n vi tri nut gia tang
l'.dn+! ti'ng vdi gia tal (F;:tl - FneXI)
Yang l~p
nghi~m
ehu'a hQi t\l
Tinh ehuy~n vi tri nut t6ng: dn+l= dn+ l'.dn+l
Tinh roan tru'Cing bie'n d'.lng t6ng t'.li di~m XEB: En+l=Bedn+l
Trong do Be: Veetd ehua cae; d'.lo ham eua ham d'.lng.

E
.
(
trial
)J
a n+\ =Lan+\ - DoY sIgn a n+\
P P A .
(
trial
)
tn+\ =tn + DoYSIgnan+\
an+l=an+~y
I
Tinh lOan vectd 11,I'cnQi fjnt (a n+l) cua tUng ph~n tt'!'va Hip rap tinh lOan
. n. n
1
A.':;
F
illt
A
f.
illt
( )
T d
'
A T
' ?
I
"
h

hlnh (2.3, 2.4, 2.5). Khi cho cac thalli so tai b~n trit;t lieu ta thu dU<;lCke't
qua bai loan ba thanh cleo 1y tudng nhu LM. Khachan//p [19].
I-fmh 2.3. Quan ht; p-8
Sai
Xac dinh ~dn+1ling vai gia tai (F::~ (O"n+l)- Fne:;)
L
K
J
-I
[
in!
(
exl
]
~dn+1 = - n+l Fn+1 0"n+l) - Fn+'
vai Kn+' = A keln+' : Ma tr<\indQ cling t6ng th~
e=l
keln+' = fB:Cn+,Bedx:Matr<\indQclingph~ntt'!'
Nghit;mcua bai loan
p p
a=an+\,t =tn+l,
a = Cln+l
[ Ke't thuc
p
T
Hlnh 2.4. Quan M crl- El
2 ~"-~-~Ir~ cr,-'t
i . cr
cr,
1.

0
0, .'0
cr, OJ
.1
.2
.1
.O.fi
0
O.fi 1.fi
x103
.1fi
,.~
'"
E, x11f3
2.3. Bai tmin dan- nhot- cleo mQt chi~u
ta xet mG hlnh d~ln-nhot-deo mQtchi~u, trong d6 c6 mG hlnh do Prezyna
d~ nghi nghla 1a ti'ng swlt phat sinh cua mG hlnh con phI;!thuQc VaGt6c
dQ chay deo. Ung suilt phI;!thuQc VaGt6c dQbie'n d<;lngva san khi d<;lt
nguong deo thl thai gian 1a dQc l~p, ling suilt khOng d6i nhung bie'n d<;lng
v~n tang. £>~tie'p c~n bai to~n nay, chung ta gioi thi~u mQt khai ni~m
hfc gia, duQc xU'dl;!ngtrong nhling buoc tinh toan khi hi~n tuQng dan-
nhot-deo xay fa. Xem hlnh 2.6. Ma sat khO khGng hO<;ltdQng khi
CTp<Y. Bie'n d<;lngt6ng cQng tren mG hlnh 1a: E =Ee + Eyp, CTe= E Ee, va
CT =CTd +CTp, Evp =y(cr-Y),trongd6:CTp =CT ne'u CTp<Y,CTp=Y,
ne'u CT=CTp;::Y. Ung suilt d bQph~n giiim chiln nhot 1a CTdlien h~ voi bie'n
dE
d h
'
d
?


' 4>2
HI ~O ~ 0"A + (0"A - 0"Y ht
E
. Vi dl;! biing s6: Bai toan 1: v~ stf bie'n d":lngdan-
nhdt-deo cua mQt thanh don gian (Hlnh 2.9a) chiu tai
tn;mg keG khong d6i, chi€u dai thanh L = 10 don vi,
co thallis6 gidi h(;lnchay cleo1a:O"y=10,tai trQng
P = 10 don vi, E = 10.000, tie't di~n thanh A=l don vi
di~n rich, h~ s6 nhdt y =0.001 va thong s6 tai b~n
bie'n d":lng HI=5000, Ke't qua chuy€n vi Hlnh 2.7 phu
h<;1pvdi hlnh 2.10.
Hlnh 2.7
]
Hlnh 2.8
Q.Jan ~ chu~n vi, lhOigian
~ ~
cj>,
l'
QlliI ~ ~ \oj,dit .,;on
'4> - T i-iT i"""T I
~, j ~.H~-1
.=
, z
-e., ~
I
+~ +-
;:
,~
"

qua la hlnh 2.8 v~n phil hQp vai hlnh 2.10 nhutrong ly thuye't bie'n d':lng
tuye'n tinh v~t Mu teEb~n, duang cong chuy6n v~rhea thdi gian m6i 19n
gia tai, bi6u di6n dung cong thU'cbie'n d':lng, U'ngsullt thong d6i nhung
bie'n d':lng v~n tang, do hi~n tuQng chay cMm nhat-deo.
Chu'dng3: MO HINH DAN-DEO HAl CHn~:U
3.1. Ly thuye't dan-deo hai chi~u : Nghien cU'umo hlnh dan-deo hai va
ba chi~u. Trinh bay ly thuye't chay deo, thong gian bie'n d':lng va thong
gian U'ngsullt, cac phuong trinh U'ngxU' dan-deo thong thu~n ngh~ch,
cac di~u ki~n chay deo, module tie'p tuye'n dan-deo, thong gian bie'n
d':lng vai cac di~u ki~n d~t tai va cilt tai, ke' do trinh bay v~ ly thuye't
bie'n d':lng phiing van Mises, deo ly tudng, deo tai b~n ding huang dQng
h9C, tai b~n chay ke't hQp trong thong gian U'ngsullt. Trinh bay nguyen
ly qtc d':li haG tan nang luQng deo la di~u ki~n dn cho d~nh lu~t chay
ke't hQp trong thOng gian U'ngsullt, va tinh 16i cua mi~n dan h6i. Minh
h9a d~nhlu~t chay deo ke't hQp nhula mQt hilt ding thU'cbie'n phiin.
3.2. Mo phOng so'bai toaD daD deo hai chi~u: Ung d1?-ngthu~t roan anh
X':lqui h6i cho mo hlnh bai roan dan-deo hai chi~u: Ong trlJ dai, co ban
kinh trong r=lOOmm, ban kinh ngoai R =200mm, ch~u ap Ilfc d~u
xuyen tam tU bell trong hlnh trlJ, xet bai roan trong tr<;tngthai bie'n d':lng
phing, rhea quail di6m chay deo van Mises. Ke't qua so phan anh dung
slf lam vi~c cua v~t li~u d tr':lng thai dan-deo tai b~n. E= 2.1 xl04
dN/mm2, M so Poissions v =0.3, nguong U'ng sullt deo cry =24.0
dN/mm2, thong so tai b~n ding huang HI=0.00. Mo hlnh doi xU'ngnen
ta chi xet mQt phgn tumo hlnh nhu hlnh ve tren, ta phan rich luai phgn
tU'huu h':lnnhu hlnh 3.1(576 phgn tU')va hlnh 3.2(1080 phgn tU').
.Ke'tlu~nv~ cac ke'tqua so
Ke't qua so phil hQp vai slf lam vi~c cua v~t Mu v~ m~t d~nhtinh. Do la
vi dlJ kinh di6n v~ bai roan bie'n d':lng phing thea tieu chuin chay deo
13
eua yon Mises. Truy~n ling suitt tie"p tuye"n thay d6i rhea rIa (jr tang nSi

i
\00 "" ,. ,. '"
rnn kinhr
Hlnh 3.4, P= 12dN/mm2
"~li
.
~:~
p
'~~
H
-T '~L
,
~I
H
-"
,
1
'" "
I
.
~t-
'
'
O
! , +- ', -
-t'+
, ," I
I
" 'I ",1 I I
cr' -

l
-ll: -, ,
, i ILi i 1
rn '" w m ,.
Iimkinhr
2
Hinh 3,5, P=14dN/mm
Om;r!r~[ ,
,
i
'~=+
,
I ;-~
'<I-' ' I -'" '-'
t
'Y
, -~ +-~ + l
I ,/ I I ,
:.' -+ t£- +~ l
l
[
+7~
ffi

,
1 = j
'<:: - t ~r I I '
" L- L +~~
-fI-~ =f~+ +-
,

15
thie't l~p bai roan bien phi tuye'n cho u(x) bi€u di~n goc giil'a tie'p tuye'n
va du'ong dan h6i vdi tn:lc oy nhu' hlnh 4.1, tren cd sa mQt s6 ke't qua cua
V.Valcovici du'a ra v~ du'ong dan h6i cua thanh bie'n d~ng trong moi
tru'ong chao khong vao nam 1971 nhu'sau:
-d I
-M(x,u (x)) + g(x)sinu(x) =0, 0 < x < L,
dx
u(O) = 0, (4.2)
M(L,ul (L))+YIG(L)sinu(L) = 0, (4.3)
trang do A. la mQt hiing s6 du'dng bi€u th~ h~ s6 lIfc nen d9C tn,lc cua
thanh, g(x) = -A, + (YO - Yl)F(x) + GI (L) la ham cho tru'dc co y nghia
cd hQc nao do, G(x)la moment u6n t~i di€m x, M(x,ul (x)) la d6i ng~u
lIfc ma sat anh hu'ang tren do~n thanh cong tu d~u thanh de'n v~tri x Trang
cong tdnh cua mlnh [63]. Tucsnak xet slf phan nhanh cua phu'dng tdnh vi
phan ma ta goc l~ch t~i m6i v~tri cua thanh, nhu'ng ang cling chi dung l~i
trang mQt s6 tru'ong hQp rieng cua ma hlnh. Vao nam 1992 mQt s6 tac gia
trong [30] dii tie'p tl;!Ckhao sat vdi tru'ong hQp moment u6n cua thanh chi
phl;! thuQc vao to~ dQ cua di€m dang xet tren thanh. Trang lu~n an nay,
khao sat t6ng quat hdn, do la ngoai slf phl;! thuQc vao v~ tri tQa dQ,
moment u6n con phl;! thuQc van goc giil'a tie'p tuye'n cua du'ong dan h6i
vdi trl;!cc6 d~nh, va thanh ph~n d~o ham rhea x du'Qcthem mQt Ilfc dQc
trl;!c N(x,u(x)), d€ lam t6ng quat hoa bai roan hdn nhu' sail:
- d [M(x, ul (x)) + N(x, u(x))] + g(x) sin u(x) = f(x), 0 < x < L ,(4.4)
dx
va di€u ki~n bien t~i hai d~u thanh la: u(O) = 0, (4.5),
M(L, ul (L)) + N(L,u(L)) + bl sin u(L) = b2, (4.6)
trong do L > 0 bl, b2 la cac hiing s6 cho tru'dc, cac ham s6
M,N:[O,L]xR~R, j,g:(O,L)~R la cho tru'dc thoa cac di~u
ki~n ma ta se d~t sail. Nhu' v~y bai roan (4.4) -(4.6) la ma hlnh bai roan

(H5) T6n t<;lih~ng so C4 > 0 sao cho: \iYI, Y2 E IR vdi h~u he't
x E [O,L], ta co: (M(x'YI) - M(x'Y2» (YI - Y2) Z C41YI - Y212
(H6) :JKI >O:IN(x'YI)-N(x'Y2)I~KIIYI-Y21 \iYI'Y2 EIR, vdi
h~u he't x E [0, L]. Nghi~m ye'u cua bai roan (4.4)-(4.6) duQc thanh
l~p tu bai roan bie'n phan: Bai roan (P): TIm UE V sao cho:
<M(x,u/) + N(x,u), vi) + <g(x)sinu, v) + (bi sinu(L) - b2)v(L) = <f, v)
, \iv E V(4.9). Ta x1lpXl bai roan (P) bdi hQ cac bai roan hil'u h<;ln
chi~u (Pm) nhu sail: Bai roan (Pm) : TIm um E Vm sao cho
<M(x,u~) + N(x, um)' W) + <g(x)sinum' IV;)+
17
+(blSinum(L)-b2)Wj(L)=<f,wj> Vi, l~j~m.
Dinh 1:91:Cho bl,b2 E IR. Cia sa (H1)-(H4) dung. Khi do:
i) Bili loan (Pm) co nghifm Um E Vm.
ii) Bili loan (P) co nghifm UE V.
Hun nrla nlu ta thay cae giG thilt (H1)-(H4) Mi (H1), (H3) -(H6)
vil(H7)nhusau: (H7) (Kl +lbll+llgIILdL <C4. Khido:
(4,10)
iii) Bili loan (Pm) co duy nhtft m(jt nghifm Um E Vm.
4i) Bili loan (P) co duy nhtft m(jt nghifm UE V.
5i) um ~ u hQi tl,ltrang CO(0) (h(ji tl,uliu tren dO{;m[0, L]),
D€ chung minh co xiI dl,lllgblSd~ Brouwer ke't hcjp vdi mQt s6 ba't dhg
thuc danh gia nQi suy da thuc.
4.3. Thui;Hgiai s6: Vdi L = 1, Nghi~m bai loan (4.4)-(4.6) du'cjcxa'p xi
m
bdi mQtday hQitl,l {um}: um = I Cm)Wj, Khi do Vm la mQtkhong
)=1
gian can huu h~n chi~u cua V sinh bdi m ham cd sd Wj(x) ham d~ng
hlnh rang cu'a,1~ j ~ m nhu'sau:
{
(x-xj)/h, ne'u Xj-l ~X~Xj'

b2 =-(1 + -v3)+ -[1 + z](2 + SIn 1) (4.44),
8 36 (36)
lex) = ~sin(2asin ax) - 2a2[1 + 4a4 cos2 (ax)] cos(ax) cos x
4
+ 2a3[1 + I2a4 cos2 (ax)]sin(ax)(2 + sin x)
+ ~a[sin(ax) + ax.cos(ax)]sin(2ax.sin ax), a =77:/ 6 (4.45).
2
Nghi~m chinh xac cua bib loan (4.41)-(4.45) Ia Vex (x) =~sin(~).
4.4. Philo tich cae ke't qua s6: EHnh gia sai s6 giG'a nghi~m xap Xl
Urnva nghi~m chinh xac: Ilurn- ullHI(O,L) ::;; ~. Dung thu~t giai
Newton-Raphson cho bai loan (4.41)-(4.45). Chung toi thu duQc cac ke"t
qua tinh loan va so sanh voi nghi~m chinh xac Vex (x) =~sin(~)
tticlng ling voi: m =20 xem hlnh 4.2, m =30 xem hlnh 4.3. Tinh loan
lftn lUQtvoi m = 5,10,15,20,30,50, cho ta thay sai s6
E
(k) _
II
(k)
U
II
I
(k)
U
(
)1
.?
d
"
kh
' -

cho ra ke't qua s6 va bi~u d6 so sanh. Ynghia thlfc ti~n cua mo hlnh co
th~ ap dl:Ing cho cac bai to<lnthlfc te' nhu' xac dinh dQ sai l~ch cua mlii
clan khoan d~u tren m~t bi~n, hay lien quan de'n va'n d~ xac dinh vi tri
cua mQt thanh dai du'a vito moi tru'ong cha't long, ching h,!-nnhu' bai roan
nhling 6ng dftn vito moi tru'ong bi~n khi Hip d~t du'ong 6ng.
Chu'dng 5: !"fO PHONG SO BAI TOAN D,AO DONG CUA THANH
DAN HOI YOI RANG BUOC DAN HOI NHOT (j MA T BEN
Khao sat bai toan mo ta slf va ch,!-mnhu' la dao dQng, CI;1th~:
5.1. Md dftu: Ta xet mQt v~t dn kh6i lu'<;5ngM chuy~n dQng vdi v~n
t6c ban d~u, khi va ch,!-m vao thanh co chi~u dai L qua bQ ph~n giam
cha'n g.ln d d~u thanh co dQ cling k . D~u Kia cua thanh ti,tatren mQt n~n
cling. Gia thie't d m~t ben, thanh chili mQt IlfCma sat dan h6i nhdt va
mQt ngo,!-iIlfc phl;1j(x,t). Llfc ma sat dan h6i nhdt du'<;5cxa'p Xlnhu' mQt
llfc kh6i. Khi do dQ dich chuy~n dQc u(x,t) thOa phu'dng trlnh song:
2
utt-a uxx+Ku+AUt=f(x,t), O<x<L,t>O, (5.1.1)
trong do a = .J(A + 2G)/p la v~n t6c truy~n song dan h6i dQc cua
thanh; K = Klr /F, A = Air/F, vdi r va F l~n lu'<;5tla chu vi va di~n
rich cua thie'tdi~n ngang; KI, Al l~n lu'<;5tla h~ s61lfCcan dan h6i va M
s6 nhdt (im~t hen. Ngoai ra con co cac di~u ki~n bien
crx
=Eux(0,t) = -pet), t,!-id~uthanhx = 0, (5.1.2)
u(L, t) = 0, t~icu6i thanh x = L, (5.1.3)
va cac di~u ki~n d~u: u(x,O) = uO(x), Ut(x,O)= ul (x). (5.1.4)
Llfc dan h6i P(t) lac dl;1ngten d~u thanh x = 0 thoa man bai roan
Cauchy cho phu'dngtrlnh vi phan thu'ong[30]
II k k
P (t)+-P(t)= Utt(O,t), t>O,
M F
~ I ~

(af t) a
5.3. Khao s:lt bai tmin (5.2.6)-(5.2.8): Nghi~m yell cua bili roan (5.2.6)-
(5.2.8) dU<;icthilnh l~p tU bili roan bien phan sau:
Bilitocln (~):TIm ui EV={VEHl(O,L):v(L)=O} saoeho:
a(ui,v)=(Li'v), VVEV, vdi (5.3.1)
I
f
I I
a(ui'v)= [ui(x)v (x)+aui(x)v(x)]dx+~ui(O)v(O) (5.3.2)
0
I
(Li, v) = fFi(X)V(x)dx - Giv(O).
0
Nha dinh 1;' Lax-Milgram ta co:
Dinh ly 1: Bili roan (Pi) t6n t<).ivil duy nhat mQt nghi~m ui E V.
5.4. Xftp XIbili toaD (Pi) b~ng phu'dngphap phiin to' hU'uh:;tn:
Bay gia ta xap xi bili roan (Pi) bdi hQ cac bili roan huu h<).nchi~u
(p/m) nhu sau: Bili tocin (p/m): TIm u;m) E Vm sao cho
(5.2.6)
(5.2.7)
(5.2.8)
(5.3.3)
21
a(ujm) ,Wj) = (Li' Wj), Vi, 0::; j ::;m -I. (5.4.3)
Nha dinh ly Lax-Milgram ap dl,lng eho d'.lng song tuye'n tinh a(.,.) va
d'.lng tuye'n tinh Li tren khong gian huu h'.lnehi~u Vm, ta co:
fJjnh ly 2: Bai roan (p/m» t6n t'.li va duy nha't mQt nghi~m u}m) E Vm.
fJjnh ly 3: (i) m~~IHm)- Ui!lv = O.
(ii) Ne'u nghi~m ui E V n H2, thl ta co daub gia sai s6:
max

p(m) =_~u(m)
(
O
)
-G(m)
1 L !J 1'1 """I I'
j=O
G~m) = G
(
(m)
(
0
)
(m)
(
0
)
p(m) p(m»
I UI-I' UI-2 ' I-I' 1-2 .
D~ daub gia sai s6 ta quail sat cae sai s6 sau day khi eho N (bu'de thai
gian i = O,I, ,N,), m (bu'dekhong gianj = O,I, ,m,) tang d~n. Chli y
khi tang mIen, d6ng thai v§:n eho N tang len ( tu'dng ling !1t = ]/ N
be) "khong h<;1ply" thl ke't qua tho du'<;1ekhong t6t.
E(N,m) = m.ax max
I
c~m) -uex(Xj,ti)
l
,
2'5,I'5,NO'5,I'5,m-) .
E(N,m) = max

Tuong tlf cho mi:j.tchay cleo cua bai roan hai chi~u, thu~t roan dua ling
sua't thO' n?im ngoai mi:j.tchay cleo v~ dung mi:j.tchay cleo b?ing each
EJ;1u,;;;
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va duy nhat nghi~m cua bai roan d d~ng vi philo. Lu~n an di:'ichung
minh d<iydu cac chi tie't, k€ d b6 d~ 3 la ti~n d~ cho slf danh gia sai so'
cap mQt theo budc chia khong gian dQc theo chi~u delicua thanh. Chung
minh thu~t roan hQi tl,ltheo luQc d6 sai philo. Cu6i cling minh hQa thu~t
giai tren mQt bai roan Cl,lth€. f)<lnh gia sai so' hQi tl,lcua thu~t roan, cac
ke't qua thu duQc d dily di:'ilam sang to hon cac cong trlnh truoc do.
5/ D€ thu duQc cac Wi giai so', lu~n an di:'i t~n dl,lng slf h6 trQ cua
Maple6 d€ l~p trlnh cac mo mnh tinh roan rat thu~n ti~n, ke't hQp voi
l~p trlnh tinh roan b~ng MatlabR12 cho ra cac d6 thi bi€u di~n ke't qua