Luận án phó tiến sỹ Sử dụng phương pháp số vào một số bài toán cơ học

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I BO GIAO DUCVADAO TAO
I . .' .
I. TRUONG D~ H<;>CT6NG HQP THANH PHO HO CHi MINH
I
TRAN VAN LANG
sir DI)NG PHUONGPHA?56 VAo M9T 56 BAI rOAN CO HQC
Chuyen nganh : Cd'HQCV~TRAN81(HD~G
}riaso : 1.02.21
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TOM TAT LU!N AN
Ph6TienSiKhoaHQcTDanLy -
Thanhph6H~ChiMinh
- 1995 -
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\ LuAnan nay duoc ho3n thanh tai Khoa Toan-Tin hoc
" .' .' . .
Twang D~i Hc;>cT6ng Hqp Thanh ph6 H6 Chi Minh
Ngum hu(mg dAn
- Ph6 Giao su Ph6 Ti€n 81Ng6 Thanh Phong
-Ph6 Ti€n 81Tran Thanh Trai
Ngum nh~n-K-et-l
Ngum nh*n xct 2
Ca quaD nh~n ~ct

Lu~n an se duqc bite v~ ~i H(>idbng cham lu~ an Nha nu6'e
hc;>p~i: Tw<mgD~i Hc;>cT6ng Hqp Thanh ph6 H6 Chi Minh
vao hie giG ,ngay thang flam 1995
C6 th€ tlm hi~u lu~ an ~i cae Thu vi~n
-Tw<mg D~i Hc;>cT6ng Hqp T19.H6Chi Minh
- Khoa Hc;>cT6ng Hqp Tp.If6 ChIMinh .
- Trung Tam Khoa Hc;>cTg Nhien va C6ng Ngh~ Qu6c
Gia Vi~t Nam (Van Phong 2).
LOIN6IDAU
Ngay nay, vm nhUng phuong pMp loan hQc UnIt tmin hi~n d~i, s1,f
pIlat trien ciia may Hnh ngay cang nhanh, fir d6 giup nhung ngHai lam (XJ
hQc c6 the giro quyel mQt htgng 1611cac biii loan cURminh. Bhng Sl!k(}thgp
ba lInh vl!c Toan hQc -Tin HQc -Co hQc, mQt hu6ng mm duqc roo ra cho
nganh Co hQc trong thai dl;\ingay nay - nganh Co Tin hQc.
KhOngngoai ml!c dicIt d6, trong lu~ an n'!\ychUng Wi muOnk~l
hqp hM hoa ca ba Jinh V1!C,de giai quy<!tmQt sObM toan Co hQc~ Ihe.
Chung wi sit dl;1OgmQt sO phuong phap sO nhu phuong phap sai pllan,
phuong pMp pllan ra luan hu6ng Ian ~ mQt chieu, phuong pM.pphan 1"a
thco qua trlnh v~lly, phuong pMp Ga1crkin,phuong pMp phau ttr hituh~n,
phuong phap khai trien ti~m ~ theo tham sO be, de khc\osat 1D<?ls6
phuong trinh trong co hQc.DOngthai, blingngOnngft tl1U~ltolin, chun!~loj
da ma boa thanh chuong trlnh boi cae ngOnngu FORTRAN 4 (chl;\YIr~n
may ffiM 360/501), FORTRAN 77, C, PASCAL (chl;\ytr~n cac may vi
Unh).Qua vi~ tfnh loan tr~n may tfnh, chung Wi diiIdem nghi~m v/1.m.~t
dinh tfnh cua roOhinb, cling nhu m~t dinh hrqng cua phuong phap. Ngoili
ra, c6 mQtsOvan <Th,do duge mOhi dum d~ng mQtphuc1i1gtrlllh loan hl?G
hoWlchinh, n~n chUngWi ding dii kiem nghi~m tru6c, sau d6 mm <TUI!C
Hnhloan Il;\ibbtlgmay tfnh de baa dam bai toan d~t ra, ding nhu 1mgiiUli\
chap nh~ duq~.
Lu~ an gOm4 chuong,

Chlldng 1: T6ng qUaDve mOhlnh va phuong phap giro mQts(fhi\i
tmlh co hQc.
Chlldng 2: MQtsObili loan dao dQngva bi<!ndl,ll1gcUathanh dan
hOi.
Chl(dng 3: MQt sO bai toan dl)ng Il!c hQc mO ta bCliphlfO'ng tdnh
parabolic phi tuy<!n.
Chuang 4: Ml)t sOkel qua Hnh tOtin.
Cuoi C\)ngla phau tai li~u tham khao cUa lu~ ~n. .
'/, I . i-"
";' ,. '
-1- iHLf\:.i[;,~:'-
'
CHlJONGI
T6ng quaDv~m()blnb va pbll<mgpbap giaim()lso bai
toaD cO' bQc
Trong chuang n~y chUngWi triOObay mQt s6 kl!t qua nghien CUll
hen Tht gi6i va hong nuercv~ cac bAiloan d~tfa hong lu~n an. Cling nlll!
mQts6 ktt qua ciiachung wi diid~t dugc so veriOOUngkef qua cii~1.1cgh\
hong va ngoai mt6c. Nhihlg bAitmin chUngtOikMo sat trong lu~ an nay
bao gbm:
1. Bai loan bien d~g eua mQtthanh dan hbi phi luytn dugc nhung
trong moi fIuemgcha:tlong. Cae kef qua n~y, cluing Wida'dl1og!rong [l]l2]
[21][22][23][30].
2. Bai loan thief kt bua may d6ng C9c.Cae kef qua dii dugc dl1ng
trong t24][25][26][27][28] [29]. I'
3. Bai loan dQngl,!chQcbien t,!a I-chien. .Cac kef qua etadugc d~
~p dtn trong [3][4].
4. Bat loan etQngl"c hQcbien va d~i duang. Car kef qua cua mO
h1OObai loan n~y da dugc dilng trong [5][6][7][8][9][10]fll][12]ft3]f14]
[15].

5. Mo h1nh dQng l"c hQct"a phtlang triOOSaint-Venant l-chi~u.
ktt qua ciiabai loan n~y chUngtoi c6ng b6 trong [16] I
6.Bi'tiloan Iantnly~n va khu~h tan ciia ngubn gay ~ nhiem. . MOl
86ktt qua Hnhloan chUngWietatriOObay trong [18][19](20].
7. Bai fOliov~ 51!Ian truy~n ngubn eMt ban trong mr6e dtl6i dat
ChUngwi da tfob loan cho mQt86 fIuemghgp tuy theo IlugngnhiCmban
ban ~u va ngubn0 OOiembO sung hen bien, mQt sO kef qua c6 dtlgc,
chUngtoi ~ ~p Mn hong [17].
-2-
CHUONG II
M~t s6 bai toan dao d~ng va bie'nd~ng cua thanh d~mht}i
I. DiU toan u6n thanh dan hoi phi tuyd!'nnhung trong JI1tli
trtrang long. Trongph1lnn~ychung Wixet sv bien d~ngdla m~\tthanh dun
hbi phi tuyCn c6 kh6i luqng rieng r 0 dlt<;1Cnhunl~hoan toan trong moi
truaog chat long c6 kh6i luqng rieng r I. Xuat pMt h'ily lhuy!t.c6 (liencua
Bemoulli va Euler v~ cac xtfp xi dflDhbi mQt chil!u, lIen CCIsa gia !J,i,!t
Kirchhoff va di~u ki~n lien ket figaro cua thanh, sao chI) ducrngdan hl}j
ntun trong cung mOtm~t phflng,cluIngta rut fa dltqc phuong trlnh (1anhl)i
Euler cua Ihanh d~ctrong mOitruang chao khOng.
(l.1) - .!_-M(x,8'(x») + ,r:(x.0(x»)sin9(x)
-=lex),
dx
dieu ki~n bien
\:.Ixc:(O,L)
0(0) = 0, M(L,0'(L») I blsin0(L) =b:~
Bili toan nhy (lttqc giai bimg each dua v~ d<;mgbien ph:ln. V(~imilt
so ghi thi!t 1Ien cae ham M, g, f va tren cae h~ s6, bang xap xi Gale-rIcin,
chUng wi da ch(tng minh (htqc mOt 86 tinh ch11'tlien quan Mn sl! tOn t~i va
duy nMt lai &jiUeuabili loan (1.1), (1.2). San <16b1mg each rai r~c: hOi1bi'ti
toan theo plHI("Jngphlip phhn tit him ht.'o, chUng toi da chUng minh dtl,~cI;'!

hQi t\! cua lai giai xap xi v~ nghi~m ctla phuClng trinh xuat phat. C:~ckef
qua da dttqc dAng trong [21][22][30].
(1.2)
San d6 chUngwi da xet sl! ph\! thuQcclla lo-igiai vao cac dit kien
cho ban dhu bJ,b2,/,g cila bai toan va da.chti'ngminh duqc sv ph\! lhu<>c
nftylitlien tl!cva dClndi~u,van loi giMduy nh11'1cua bi'tiloan (1.2), (1.3).
II. Bi\i toan boa may dong c<;'c.Co h~ ky thu~t cua bUa may d5ng
c9c c6 t116xcI mQt each t6ng quat nlnt san: M<,\tbUa c6 kh6i htqng ,17/(,
dltqc n6i liCn v6i C9c va de c6 kh6i 1ttqng n72' bhng Ie)xo v6i h~ 86 oan h()j
- 3-
Ct. Gina bUa va c<,)ccon dllgc gaB them mQt bQ giarn cMn' c6 h~ s6 ma sat
Iii bl" Khmmg cach ban d'au (& trlpl.g tllai ct\n bhng) giUa bua va c9C Hi <\.
L1,£cngoai (tuan bean) taG dQng len bua ml la F. G<,)iUI'u2 Ian lugt la dQ
dich chuyen clla bua va C9C.L1,£ccan clla dat sinh fa trong qua trlnh chuy~n
dQng baa gbm l1,£ema sat kilo gifra <hItva c<,)cP'nsd'I1,£ccan 1<1'aucge ~. Cae
I1,£cn'ay n6i chung, phI,!thuQe VaGdQ dich chuy~n eua e9c. v~n t6e Gilacqc
va cac d~c trung cua Mt v.V
Dl!a VaGnguyCn uk rung va, voi cac gh\ thic"t ma sM gifta C~)cvii
<1:1'1.la ma sat khO, c9c drug hly~t d6i, va ch~m gilla boa viI c9c 111lue 1110i,
chUng tOt xet ba giai do~ co the Kayfa tfang qua trlnh dong C9Cnhl1sail:
- Giai do~n rung, day la giai dolpl.elma xtiy ra va chl;\mgilla Ma va
C<,)c,00 h~ dao dQng d\10i taG d\lng eua h;rcngo1\i tu'an hoim F, Ivc ma sal
giUa bua va daL H~ phl1O11gtrinh dao dQng co d~g nlllr saIl:
(2.1) 11z.~+ bt(if.l - uJ+ Ct(u, - It'}) =F
(2.2) ~~- bl(i4 - "2)- CI(UI-'~) = -Frn.<d
De ket thUGgiai <1o~nrung nay chung ta co di~uki~:nv~sl!va ch~m
gillaMa vac9c.
(2.3) '4- 'uz=80
- Giai dOff1Jva dqp, thl!c ra day chi IiimQt 8\1tac dl}l1gchu khOl1g11\
mOt giai d9an, hat qua t1"1l1hva chl;\tnx~y fa 1.(1Cthai, co ngh!a Hithai gian

va ehl;\1llciing nh1151!dieh ehuy~n ella bua va cge khOng dang ke. Cho n£\n,
chllng ta e6 the eoi dQ dich chuyCn Ill'112khOng thay <16isaIl khi va ch;~m,
chi e6 v~ t6e eua bOa va C9Cthay d6i mQt cach dang ke thee COngthuG va
eh~n:
-4-
(2.4)
U{=u (X+l) 11/2 (UI-l~),
11l1+m2
Ui=I~+(X+l) ml (UI-~)'
n;+n7.
tTOngdo, it;,u' 111 v~ tOc cua bUa va cqc saIl khi va chlpll, Xlii h? sOva
ch~m, phV thuQc vao d~ng v~t chM cua bua va cqc.
(2.5)
- Giai dqan LItH,day la giai dqan xiiy ra saIl qua trlnh va chl.lfil
Phuong trlnh 1110ta sv chllyCn dQng cua htJa va cc;>ctrong giai do1,llln~ly
ltWr~gt~!phucmg tdllh (2.1), (2.2) cua giai <19anrung, tuy nhi~n c6 xet the "
uk d<jngcua Ivc can dfill Cqc. Cae GOngtl1lreva ehlpll (2.4), (2.5) dtf<;1ccoi
nh!! Ii'!v~nt6e\1anef:\11de gi;\j110plllrong Irlnh ehlly~ndQngclIa giai doall
ni.ly.
KN thue giai <1qanrung 111di~n ki~n v~n tOe eua eqc tri~t ti(~u(cc.1c
kbOng <IixuOng nila)
(?R) ill =o.
Trongtnrang h<;1pxay ra sVva ch~m giila bua va cqc tlwo dibl ki~n
(2.3), nhung dura iliaa di~lI ki<;n (2.8), qua trlnh chuyen I~i giai do~n va
dl;\p,neu di~ukt~n (2.8) xay fa, quatrinh chuyen den giai do~nrung.
Cae kef qua eu.a
bili toan nny <Iiiehr<;1ctrlnh bay trong 124][25][26]
127][28][29].
Mohlnh hili loan bua may dung cqc duc;>cxay dvng nlnr [lh11ntrUl'lc
di\n dC'nvi~c gi<\ibai toan:

11m hai thai diCm1»12be nhtIt, 10< II < 12va hai ham vector
;;(t),
10S I S It,
U(I),
10s I ~; II
sao cho
(2.12)
du=Au+~,
dl
'0 < It < '2.
- 5 -
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
-
( )
-0
u to =u ,
tl =min{t > to / Ul(t)- uit) =80}
dv ~
- = A"+ Fx+ F2' to < tl < t2
dt
V(tl) = BU(II)
'2= min{t > tl
/ "4(1).(111(1)-1I2(t) 8J =O}.
TrltO"nghf/P khOng va dljp "1(t2)- "2(t2):f:-80
Khi do chUng ta giai bili loan (2.12) - (2.14) vui
(2.18) to=tv ;;0=V(t2)'

TrltiJnghf/Pva drip "1(t2)-"2(t2) =80
Khi do chUng ta giro bi'titoan (2.15) - (2.17) v6i
(2.19)
/1=t2, ;;(tt)=V(t2)'
Bhng cach khao sat cae gia tr! rieng cua ma tr~n h~ s6, chUngWi
Omdugc cac rang buQctren dO'li~u.Tren 00 s6 cae gia hi rieng do, chUng
toi cOdugc k!1 qua ve nghi~m tOng quat cua cae h? (2.12) - (2.14) va h?
(2.15) - (2.17).
"
-6-
ClnJONG III
M~ts6 bililoan d~ng.'!ch~cm6 ta bmphtJ(m~:trluh parabolic
phi tuy~n
I. M6 hlnh bai toan d(!ng h.rchC!cbiln t1!al-chj~u. Chung la xcI
m(\t mien (2
C R3 bi ch~n,vai h~tIl!cto~dQDescartes°xyz, e6 tr~1COx,0)'
huang theo vi luy~n, kinh tuy~n trl!c Oz huang vao t.am uai dfit [JI{4].
Moi tnrang dU<;1ccoi Hidi hu(mg, khOng nen du<;1c,h{~86 nhat th(:o
tIl!COz ky hi~u Hivz' Trong tang m~tphang song song vai m~t ph~IWOx)'
gia lhi~l dong chay 6n d!nh, hi~u lIng rOi lheo chieu ngang khOng dang kC.
nai toan xac dinh pMn b6 twang v~n t6c a tung I(Jpc6 d~g:
011 011 1 &V ~ .,
_of w . , \1" I" - I Fir
, 1 2' r K
;// (}Z P Dz
d I~Uki~n bi~n. Tren be m~t (hoang, cho bi~t gifi tT! cua truOng ":,\n We va-
eho ap 8U1(tkhf quy&t:
(1.1 )
-
V. V -: (J

. -,
(1.2) V(x,y,z,t)= V/nI(x,y,t),
(1.3) p(x,y,z, t) = ~q (x,y, t)
Pl1\n duai day bi~n, giii su co ma sat 16nva mrac khOng ngam xu6ng, trong
tn! emghgp nlly <lieuki~n bien sc11\:
(1. 1) V(x,y,z,t) = O.
Ph \1]bien xung quanh, c6 th~ cho dieu ki~n nu0c khOng tach khoi bo:
. oV
(1.5) - =0,
. an
ho~(;dieu ki~n ma sat lao
(1.6) V(x,y,z,t)= O.
Die\] ki~n d1\u
(1.7)
- -
V(x,y,z,O):= Vo(x,y,z),
p(x,y,z,O)= Po(x,y,z)
- 7-
De girobai to~n n~y, chUngtoi dii xet sI!hQit\l ciia nghi~m phuong
trlnh sai phan xap xi cap hai tl1cothai gian va khOnggian. Tuy theo m(,\!s6
truOnghgp, chUngtoi ClIngch(rngminh duqc sI!6n djnh da Jai gi:\idp xi
tit h~phuong trlnh (l~is6.
II. Bai toaD dqng h.rch«:,cbi~n va d~i duong. Ngoai gi:\ thi~t v~
cMt long nInt di'ineu trong bai loan n~utr~n, d6i v6i bi\itoan n~y ch~ng (Oi
xet tMm cae hi~u 1h1gr6i thoo phUC1Ilgnfun ngang Ox, OJ, ding nhu die
thi\nhphM nh6t theo ba Inr6ng.Cac Il!Cbell ngoai baa g<lml,!c Coriolis va
lllc tr<;mgtruOng"cOnthi\nh pMn v~ t6e gi6 thi hi~n qua di~u ki~n biCn.
M~t thming b trl!-llgthili ban dhu duqc gic\thi~t c6 dao dQngso v6i mQtm~!
chuan nao d6 (cae k~t qua cila bai loan nay chUngtOjdiitrlnh bay trong Ill]
[12][13][14][15]).H~pln1ongtrinh mOta bai loan c6 d~ng;

(2.1)
V.v=o,
011 ~ ~ I ~ 02\1 ~ -~
+(V.V)V= V
p
+v ~V-,-v +F J F,
at P xy 1 f)z2 j' c 8
di~u ki~n bien va di~u ki~n d~u
- Tren m~t thoang cho phan b6 ap su~t kIll quy8n
(2.2) p(x,y,z,t)=~q(x,y,t),
thi\nh phM v~ t6c gi6
au. Ix Ov 1y
(2.3) vza; = p"( , V1 8z = - p"( ,
va dao dQng cila m~t thoang
(2.4)
(2.5)
(2.6)
Or)+ U
I
Or)-+-V
I
0"1 =W
I
.
or -1J ox -1J Oy -1J
-Di~u ki~n dfnh do challop.g nh6t 6 du6i day
u = v = w = o.
- Dr~uki~nclla b bien long b Kungquanh
unx +vny =O.
- Di~uki~ndhu ( cho vao thai diem t = 0 )

-8-
- -
V(x,y,z,O) = Yo(x,y,z), p(x,y,z,O)= po(x,y,z)
Sit dl.mg phl1ang pMp phfin ra theo t<;>adQ, bhng each klu10 sat !.fuh
nita xac d!nh duang va hennit. cia tOaDtit vi phnn. ChUng tOi Om nghi~m
thco tang IlltOOg,GOngthC1ichUng minh dltgC nghi~m elm biii lotio tl1cOqufi
trlnh pMn ra ehfnh Iiinghi~m eua phuang trinh xuat pMt (D!nh Iy 1,MI!G II,
Chuang 3 cia Lu~n an).
Buoc tiep thoo, b~ng each sai phan theo thOng gian, chUng toi nh~n
dllgc cae xlfp xi h~e n cua phuong trinh sai pMn so voi phuang triah x\l1ft
phill ban dhu
III. Mo hinh d9ng h9C tt.ra ph.tong tring Saint- Venant. 1 chn~QI.
'I110ng thuong, dtl.Hnh dong eh?y khOng 6n dinh tren h~ thong sOng ktnh
ciia vung anh hui':1ngthl1y trieu, ngtf()i ta su dl!ng phuong trlnh Saint- Venwl(
I-chiCu, trong lntemg hqp 1111y?lnh 11ltal1gcua ma sat nh6l b! (Iii bo qua 'lit
xcm SI.tma sat clJa eMt long vii th1'mhr:\n 11'1nan!,-kf (j (T1\y,do mu()n ~d
hi~lI (cngnh6l LacdQng I~n dong eMy, u\jng t1J~jjmuOn xua1 pIlat lir phlJ'OTlg
trinh dQng h!e l19c Navier-Stokes, phuong trlnh baa to1\n kh6i lugng, chung
16i dua fa dUllCmOt mO hinh ty:a pllltO'11gtrinh Saint- Vcnant l-chi~u, trang,
<16 e6 s~r tham gia elm thal1hph110nhc)ttrong pluto'ng trInh, ket qlla cluing wi
cIatrlnh bay trong [16].
V6i cae gia thiet san day:
- Chlft lr'mg dOng ehftt, khOng nen lhlc;1e,dAnghu6ng,
- Ap suit li\ tll"y Hnh.
Khi eto fir h~ phuong trlnh chuyCn etQng t6ng quat baa g'orn die
tensor (tng suft'tnhot 't~ cluing ta e6:
(2.7)
(3.1)
Olt au au au 1 op ]
(

m~ m~ m~.
)
at + u ax + v oy + Wa; =- pax+ p ox + ay + -Oz + Lv,
(3.2)
011-I It 01' + v ~ -j IV ~~- :00 }- 017+!
(
~t + m; +-~~-
)
-Iu ,
Of ox oy m (10' r ax 0' oz
-9-
op
(3.3) - =-pg
Oz
va phttC1.tlgtrlnh bao toan kh6i Itt<1ng
ou ov Ow
(3.4) -+-+-= O.
Ox Oy Oz
Di~uki~ntren m~tHlOangva dtt6i day dtt<1ccho nhtt san:
(3.5)
.
"
wi =m, + ul m, +vl m"
11 at 11ax 11Oy
ah ah
W
I
=U
I
-+v

l

-h -hax -h Oy
ChUng ta gh\ thWl them rhng dQ sftu 1/ =11+ h cua m,!c ntt6c
khOng dang ke so v6i m~t phang nhm ngang, khi d6 chUng ta c6 th~ (ltta V8.0
cac df;liItt<1l1gd~c trung cho sv phftn bO v~n tOe trung blnh theo chi~u thang
dUng:
(3.6)
(3.7)
11
V(x,y,t) =J u(x,y,z,t)dz,
-h
11
V(x,y,t) = J v(x,y,z,t)dz.
:-h
va cae gia thief atia M~nh d~ 1 (Mvc III, Chttcmg3 eua Lu~n an) va C1'13
M~nh ~ san, chUngWinh~ dttqe h~ phttC1.tlgtrlnh lien h~cchuycindQngde
xac dinh ehi~u cao Ii cling nhtt phftn b6 Wong v~ Wc V, V trong m~\t
phang Oxy. H~ cac phttong trlnh n'a.ykhac h~ phttong trlnl1Saint-Venant hai
ehi~u(jeM, c6 tbam gia eae tbanh ph~ d~oham b~ehai cua v~ t6c.
M~nh~ 2: Gia sit
I
(i) Moi Wemg 1ftdang htt6ng 't~ = 't;,
(3.8)
"
- 10-
(ii) H~ s6 nh6\: eua eMt long thee cae hl1(mg 11\ nhl1 nhau
v.=v.xy=v,
(Hi) Tensor N} duClcxap xi dltai dl;Ulg:
(3.9)

. oU 2 8V.
(
OU OF
)
N.=2vp-, N2=2vp-, N2=vp -+- ,
ox ay By ax
(iv) lrng sual teenbe m~t Unhthee I1'ngsuat gi6, ,
(3.10)
I
2 2
1". =1
9
f\W
g
cos(1 ,
.1/ g
L
I
=,,2
p
W;'sinH
21/ Ig a g l:
(v) (Jng sual ma sa(l,~i day (!nh Ihea ma sat eua dong 611djnh,
(3.11)
Lll h = ~ P0l!? ~ \/2)}S
C 1f2
L2Lh = g2P V{U2 :':\t ')Y
c III

Khi do, h~ phur111gtrluh chuy611dl)ng mrac n()ng hai chieu cung v6i plllrur'g

trlnh Jit:nt\lc c6 dC;lng:
au a u2 a UV 8rl a
(
au ()V'
)
(3.12) ot -+-fy-;/i-+- By II = gH ax -+-lV-l- v~U -+-v a; ax -I [~1;,I
g ~(~:_:~_V~1~+1g2P~~2.coseg,
1 It- p .
c
av iJ UV iJ \/2 8rl iJ
(
iJU <11/
)
(3.13) &-+- ij-; H -+- By H :co -gl! B.y-LU - v~V -+-v By Ox -I-~?}; -
)
1/
V
(
u7 I v2 /7 - 2 Fa W2sin8g
g +"(1'. 8
(/ [(J. p
- I I -
all aU oV
(3.14) + + =0
at ox 8y
San d6, gi<\thiel d6y cua long dan c6 d~g ntta hloh tang tl1;l,SI!thay
d6i b'e mM va da
y
theo true O
y

khOn
g
dan
g
ke. Gii\sa V =0, -~ 11\ham
II II
cla x va t, h,reCorilolis [= 0, h,rcgi6 ~ = O. Chung Wi nh~n (htqc he
phuong trlnh mOhi chuyen dQngdong cMy mQtchien:
(3.15) B Or} tJQ
+ 0
ot ax-
(3.16)
~ 8Q +~~ Q2 + Or}+ QIQI =2v o2q
gA at gAox A ox c2A2R gAOX2
trong d6 , . 1 '
-A H\di?nHehm~tej\teua long dan, B 111ehi(;lJ rQng m~t thoang,
-'r.' 1
R
A "A-
= B li'tbankfnhthuyh,recualong dull,
De giro h~ (3.14),(3.15) chUngtOisa dl;lngphuong pMp khai trien
I
tj~m ct,\ntheo tham s6 be, bhng each d~t g = 2fi ,gh\ siltdiehamQ va'1
duqc khai trieD theo lUythira ctla 8 runt san:
(3.17)
m
Q(x,t,g) = LQ",(x,t)!>"',
",=0
(3.18)
m

l1(X,t,&) = L l1";(X,t)!>'"
",=0
-12-
v6i E all nho lit!c6 thl! cui Sign(Q) nhula Sign(Q)J. Trong .:16die h~ 86
Qo' 110thou h~ phuong trlnh:
(3.19) B~llo+ ~ = 0
at ox '
1 0{1 1 a ,~:! Urlo QoIQ,1
+ + + =,0
gA at gAox.l1 at c2A2R
con cae h~ 86 Q"" 11"" f11= 1, 2, thoa h~ ph\1t:mgtrlnh
(3.20) B ?J", + ~g", - 0
at ox ,
~~_.?f2'!!+.3_!. ~q",.,: <7r.b.+~~q~:: l~'I(Q(I,QI, ,Q",I)
gA at gAox A ax cAR
IV. Hai foan Ian fruY4!:nva khulch tan cua l.gu;~11ga)/ ~i}II1Ihij~l1l.
Chllng fa ghi sl'r(p(x,y,z,t) bi6u dj~n luqng nhi~m ban du<;rcIan truy:~n vit
!JlU~ch t{mc!(?ctheo quy d~o cua de h<;1tm{)i tntang chuy(!n d<>ngv6'i v~n
t6c V(x,y,z,t). Baj loan m() l<\51!Ian truy'en va khut!ch t{in (lJa ngul'in gay
(\ nhiCm c6 o,.\ngsan:
(4.1)
o<p O(P 8<p - O(P 0 o<p
+ /1-+V + w -" J-tLl<p+ -v-+ f
at ox oy oz 8z 8z
\I(x,y,z) EO, \It E(O,T]
Dr~u ki~nullu
(4.2) <p=<Po trong 0, khi t = 0
Di'eu ki~n bien
(4.3)
<p:=<Pstr~n L, t E( O,TJ

- 13 -
(4.4)
?!.=a<p 1r~nLo, tE(O,T]
oz
(4.5)' i3<p= ° 1renLH' t 'E(O,T]
oz
\'
trong d6 bien i3O=L: u L:o u L:II
V6i bai loan di;l.tfa, chung Wi nh~n dH9'Csl;t du~ nM't nghi?m cua
bai loan trong trttang hqp cac h? s6 khuecp tan, h? s6 {Hang Hic v6i mOi
Wang Mn tren 11\khOng fun. '
Su d~ng plnrang phap phftn ra theo qua trinh v~t 1:9,chUng wi tach
bai loan thanh hai d~g: bai loan Ian trUyen va bai loan khuech tan. Khi d6,
chung tei cling ch(rng minh 'dH9'e,nghi?m dp xi t1'cbili toan khue.ch tan
tren tirng d9an thai gian phfin ho~ch du9'e tach fa, clIng!than phuong trlnh
xuttt pMt. San d6, bhng cach phfin ra thee t9a dQ,tirng bi'liloan du9'Cdtta ve
d~g mQt chie.n v6i cae loan tu nun xac dinh dl1ang. ChUng wi cling kMo
sat s,! 6n dinh cua lai giro b) h? phl1ang trinh d~i s6 tuye.n Hnh. Cae ke.t qlla
n~y chUng wi da dang trong [18][19][20].
V. Bal toan v(l.sy Ian truy~n va khu€ch tart ngu'on clult bftn
~rong mtac dual diU.ChUngWithie.tl~pmQth? phHangtrinh bao gom hai
phl1angtrInh:
- phttong trlnh Bussine,sqme hi m~ttv <ioCURmr6e du6i dat, va
- phuong trlnh bi~u di~n s,! Ian tnly~n va khu6ch tan clIa nOngdQ
chAthoa tan trong mr6e.
c6 d~g nhu gnu:
(5.1)
0 Oz~ = V.[(z,- zet)KVz,]+F
ot
- 14-

as V.[(Zt-z(J/~KIVz,IVS]
(5.2) 0 == . + KVzSS+ Q
, Ot Zt Zd
tren <X1sa (tinh lu~t th:1m Darcy
(5.3) V(x,y,r) =-KVz,(x,y,t)
va mQt s6 gic\thi(ll [17J: m~t tlJ do cua mtoc duoi dtI't z,(x,y,t) n;!m thap
han so voi m~t dtI't,chuyen dQng cua nuoc dttoi d:1tgan m~t (%1:Iii chuy&n
dQng khOng ap, lOp dtI'tsetz(,(x,y)dttoi mi~n chuy~n dQng cua mr6e thtlm
thay d6i ft, nttoc dttoi dtI'tla GMt long GOngeMf, khOng ncn dugc:, 1:1:11: 1:\
mOi tntang kllOng ntn dttqc va dang htt6ng.
ChUng tOi pMn ra pai plltrang trlnh tr~n 111Qtcckh lil~ntWp dbng
thoi trCn cling mQt kh<n\nglhhi ginn. 'f'{nhIlIla xac dinh dllctng din die: lolin
ta vi phfin xuff'thi~n trong p)ntctng trlnh dIng dugc khan 8M d€n. Sa" rf6
bang phuong pMp sai phfin fin, chUng Wi dua v~ h? phl1ong tflnh ft~j :><5
tuyen Hnh, d~ gic\ih~ n'!\ycMng wi con khan sat tht'h1 !fnh 6n dinh clla lai
gic\ivoi die raub bu(!c v~ v~n tOed()ng thatn cling n]n( cua bl1~5Chr6i,
- 15 -
CHtJONGIV
M<'ts6klt quaHnhto{m
Trang chuang n1\ychUngwi neu ml>ts6 ket qua Hnh s6 du6i d~g
dOthi cua cac mOhlnh bai Loan,rung nhu phuong pMp tfnh Loan,da d~t ra
!rang Chttdngl va Chlfdllg/ll. Cac ket qua n1\ydt!aLIenmQts6 s6 li~uOWe
te ding nhu ghi dinh, de qua d6 danh gia ve m~t dinh tfnh cua mOhlnh va
phttong pharo
I. Tlnh toan dao d(mg va biln d~.mget'm f.hanl. d~lI1hoi. CluIng
Wi11\nluc;tf.Hnhloan cae bai loan gall:
1. Bd; loan
u6fJ lhafJh dafJ /Wi phi luyefJ, Cqung wi tinb loan cho ml;)ts6
truem.ghgp neu trong Chttdng/l, M~4CI nhtt du6i dlly, cae ket qua (15du<;lC
trlnhbay!rang [2][22].Gia su caeham g, M dtt<;lcch911t11baeae yell Cftll

cUanhih1g dinh ly ve st! ton ~i, duy nhal eua.lai giai, rung nhu S\fd{mh girt
sai 86 tren 1Mgiro. Khi d6 bai loan xffp xi tuang duong v<'1ih~ phuang trlnh
phi tuyln. Trong d6 ma tr~ li~ s6 c6 d1;U1g3 duang chen. Tit dlly, chung la
c6 the su dl!ng phuong phap truy (lu6i de Hm nghi~m.
2. Bd;loan bUamayd61lgcqc.ChungWidaHnhloandvatrenmOh}nhva
phuong phap rua bai loan d~t ra trong Chudng/l, M,!c fl, v6i cae diI li~u
thoa cae dieu ki~nda khao sat nhu sau:
-Kh6iIttongbuatir40Mn 1O0kg, ,
- Kh6i lu<;lI1gC9CdIng de (bQga Hip)nh~ gia Ifj lir30 dCn90kg,
-Loxo c6 gia I.titrong khoang 50000N/m,
-Khoang caeh ban dl\ll giUa bua va de c6 gfa 1rj Ui"1 Mn 9mm,
- Thn s6 quay eua dQngeCitit 110 uCn190rad/s
va mQt s6 86 li~u ve dAtdung trang quy ph~m v~ d6ng C9Cgia c6 n~n
mongo
Cac torang hgp chUng wi da khao sat baa gOm: sl! phI! thuQc giITa
dQ sAu d~t duge v6i c<;>cva de, s\! ph,! thul;)cgiUa dQ sau d~t du<;Icv6i bun,
st! ph,! thuQc giO'adQsftu dl;\tdll<;1C,v6i khoang cach ban dftu eiia bua va dc,
S\!ph,! thuQc giO'adQsau dl;\tdu<;1cv6i t1\ns6 quay rua m;iy bUa,
- 16-
Tren (;(j so cae k~t qua s6 (Hi nhl~l1 dtt<;1e, cluing wi nit ra l}}i?1Stj'kE~t
lu~n (cae kef qua Hnh loan chung Wi da trlnh bay trong [24]1[2,!j][2:6][2'71
[28]129]):
-MO hl11hHnh loan, cung nhtt phttC1l'\gphap giiu cho kef qiia phil
hqp ve m~t d~nh Hnh so vOi bUa may Uwe 16da (11«;1(:tlwc nghi~m.
-Hi~u qua cua va d~p Ii\r11't16'n.Trang nhi'eu tnt<mg hilI' dQsau cae
d611gd11<;1Cntu c6 va d~p, 16'ng11'pdoi so v6'i qua trlnh J~b()ngc6 va (I:~p.
Khoang each ban <.filucua bUava de rat c6 ynghla, dij sfiu GilaO?Cgh\m dhn
khi t<lngkhoang cacho
- Khi t<lngkh6i 111I,:IIlgcua bUa den mf)t ngttiJng nao <16,se kMng
can ynghia Wi l1U.D6i v6i de va c«Ccung cha ktl qua r~ul v~y.

II. Tinh toan ID9ts6 bai toan d9ng life h9C.
1. BiJi roan a{Jng llfc hqc bii'll wa i-chicll. Bi\i loan nfty dtf<;1Cdi)ng df; tJnh
loan khOi plwc I~i dl)ng dulY b cae thng stiu khae nhau dw~i bien khi bi{rt
s6 li~u v~ dong dulY va pIlau b6 Ir11ongv~n We tfen m~t Ihoting (~ic ktrt
(l\\a da d~ng trfmg [3][4]).
Thtl~1loan da dulle kjCmnghi~m IJennuly tfnh, va ~mld6 Sttdl,mg
Hohtoan cho khu V\fCphfa Nam bien DOng,vai cae 86 Ii~u do ('[<,1cd\1OC
clla Phong V~t Iy bi~n, Phan vj~n Dftu kbf phfa Narn.
Cae btrde ltt6'i khOng giao Ihco Irt;leOx, Oy dlUJCIa'y c6 d!nh, CC!D
thea tn,lc Oz dlt9'c Iftythay d6i tily thea df) sau.
. Ax=~y =112000m, 1m ~ ~Zk ~ 100m,
trang d6 ~z/'lftn 111<;11nh~n cae gia tr~: 1m, 1m, 1m, 1m, 1m, Sm, Sm. 5m,
5m, 25m, 25m,,;.25m,50m, 50m, 50m, 5Om, 50m, 100m, 100m, 100m,
100m, 100m., '
BUde Itt61thbi gian t\t = 360CB.
Cae s6 li~u ciia mOi truong du<;1cla'y 0011sau:
- M~t (If) n116'ebi~n p = 1025kg f m3,
- He s6nhdt v== 10-2m2fs,
. z
- V~n 16cg6c quay cua trai d:I'1()) =7,29 x 10-5 /S
-Vi d0 t~i (ti(!mkhao sat <D=0,1745
- 17 -
-Gia t6e tT<,mgwang g = 9,81m 182,
Dl!atxen ibQ s6li~u (eua 2 ngay) v~ v~ t6e M m~L,eho chung La
khoi ph,!e I~i 00 eM dong eMy elm cae tMg san Mn du<'1ithoo LunggiG
mQt.ChUngLoi<Hitinh LoanIan hrql.txong Lnrangh'lP 1.1faG 16peach bb
m~t 1m,2m, 3m, 4m vao gia thu 24 va vao gia thu 48.
2. Edi toan
dqng lifc hqc bi!n va dqi dlldng. Cae sOli~u eua moi Inlang clln
biliLm'innay du'le I~ynhu san [5][6][7]:

I'
-M~L dQ nu6e bien p == 103kg/m3,
'~ 7'
- H~ s6 nh6t thoo phuong nl1mngang vxy == 10' III ~s,
-H~ s6 nh6t theo phuong thang dUng Vz==1O-2m7/s,
-Lve Coriolis c6 d~g I =-2(0 sin (D, trong (16 w 7,27 ,< 10 1 S 1
va <I>.lavi dQtrong vung vjnh n~e bQtit 1i) d~n 220, I
-Gia tOetrc,mgtxuang g = 9,81m/s7, .
-
TTItang gi6 ill gem hai thanh phlln (1:x,1:)1)du'le tfnh thoo GOng
I
thueil/ = (31"~I~,trong d6 ~ Ia vectorv~ tOegi6, c6 g6eh'lP v6i t~c
Ox 11\81/' Thy thoo v~n tOe gi6 m1,lnhhay y~lI, gh! trj f3dU<.1e111yntH!san:
f
O,98448 x 10-2 ne'u IWgl< 6,6m/s2
p= ~
3,10956 X 10-2 ne'u IWgl;:::6,6m/s2
- Ap su11'tkhf quyen 11lygia trj h~ng sO 11\97820 Mb
Tuy thco dja hmh, cl1ling tOi d5 tfnh Loan eho hai Lruang h~}psall [8][9][10J:
3. Eai toan tan truyelz W1khuech tan cua ngu1Jngtiy 6 nhi/;n. Chung (hi cia
tfnh Loan eho hai tTItang hqp ri~ng Ie [18][19Jr20J:
- Truang hqp nguOn () nhiCm t~p trung Lr~nbi~n
- 18 -
"-' '.'
- Tnrong h<;>pnguOn 0 nhiem phful bOu.en bien va diem chinh giCra
mien.
Trong ca hai tn101lg hY'P, chung ta nh{111tl1iiy nong d~ chat. gay
nhi0m brill Ian tnlyen d<;>ethee hl16ng v~ tOe, gia tf! Hnh Loan lubnlll{}n
cho k~l qua th:fp hon gia tf! trtn bien.
TfCn C<.1so nhftng kiem nghi~m ve m~t Iy thuye"t tfong vill,c (](;fa

phuong phap giai cling nhtl cae ket qua Hnh Loan, chUng tOi nh~n 1My lnl~
hinh va phl1ong pMp dan Mn ket. qua phil hqp ve m~t dinh Huh.
4. Bai tOlln Ian truyen va khutch trIll cua nguon chdt bJn trong nUde £!l{m
ddt. ChUng wi <15Hnh Loancho 6 trl10ng hqp kMc nhau cURdiet! ki~~nbi{',u.
Ke"tqua Hnh Loancho th1I"ymve mr6c Ian truyen dlln vao Mn trong mn~ll, do
h~ s6 thllm K Illy kM be, nen mvc nunc 0 Mil tfong tha'p hon gl1nbien. D6i
v6i nbng d~, chung ta cling e6 ke"tqua tuong tv, khi eho gia tT!tren biCn
tang cfllnleu thi Mn trong mibn gia t1'!nh~n dtl<;>ccOng U\ng <l!lnI~~n.Cae
ke"tgila dllqc t1'lnhbay trong [17J.
.' ' '
- 19 -
TAl LIttUTHAMKHAo
I
[1] N.T.BANG,T.V.LANG,al al., A nonlinear differential equation relating to
the buckling of a nonlinearly elastic bar ~ersed in a fluid, Proceedings
of the HCMC Mathematics Consortium 1st Conference, Vol.1,1993, p.5-
12.
[2] T.l.CU'ONG,H.B.LAN,T.V.LANG,Giro s6 mQt phuC1ngtrlnh phi luy~n
lien kef v6i toan tu Bessel, Proceedings of the 4th Workshop on Applied
Mechanics, 4/l994,TpHCM,1994, fr. XVIII-7.
[3] T.V.LANG,LV.THIEM, Mo hlnh t,!a mQt.chien M Huh loan e0 eM dong
char 1.1cae fang sau dufli bi~n, f/tJi nglzt Cd hqc toi}n q1"5Cfan tlllr lV, Hi)
nQi, 20 - 22/01/1988.
[4] T.V.LANG,LV.THIEM, Mo hloh t,!a mOt ehieu M tlnh loan co che' c\i'H1g
char a cae mug san dtt6i bien, B(IO cao Khoa lu?c stY87.202, TrunR Tam
Tl{(fD&TH, Vi~nKHVN, TpHCM, 1987, 19fr.
[5] T.V.LANG,Mo 111nhdQng l,!e hge bien 3-chieu de tfnh loan dhng cl1<\Y(\
mQt khu v'!c c6 dQsan dang k~, 1'6111tat Cong Trinh Khca Hqc - lltJi Ngh!
Khoa Hqc, PMn vifll Khoa Hqc Vi~tNam, TpHCM, 1988, tf. 105.
[6] T.V.LANG,Mo hlnh dQng l,!e h9C bi~n 3-ehicu de Hnh loan dong eh,\y (~

mQtkhu vl!c c6 dQsan d{mgke, IitJiNgIltKlioGHqc Co'Hqc 1']1HCM Ian
thu I, Tp HCM, 01 - 03/03/1989.
[7] T.V.LANG,Bai loan 3-chfeu Hnh dong chay va m,!c Mac do gi6 gay fa.
C1Uldngtrinh 48.B, Ha nQi, 1988.
[8] T.V.LANG,PhUC1l1gphap s6 cho bai loan dQng Ivc h9c bien 3-ehj~u.I1(;i
Thdo CdHqc PMp Vift, TpHCM, 19-22/07/1994.
[9]T.V.LANG,T.T.TRAI,PhuC1ngphap s6 cho bbi loan dQng h!e hge biCn 3
chien. Proceeding of the 5th Work hop on Applied Meclwnics, 4/1995,
HCMC, 1995, p. 10.1- 10.5.
[10]T.V.LANG,T.T.TRAI,PhUC1l1gphap sO eho bi\i toan dQng l~rchQc bien
3-ehieu.T{lp cM Khoa hqc & Gong ngh~, BK & SPKT, So 9, (1995), I-lanQi.
(Gia'y nh~ dMg s6 15/NCKH ngay 26/06/1995).
- 20-
[11] T.V.LANG,D.M.NGQC,LV.THIEM,Xfiy dV11gmO hlnh toan hqG giiu s6
(rang khOng gim1 3-chi~u cho vi?c mO ta, dt! bao hii' hlqng va ch~fLIlft;1ng
mcoc cua hb chUa 11111Ydi~n 'If! An 1tllU<)Cnh6m d~ tili "OJ.!baa ch~lthlt;mg
nuoc cua hb ch(ta Thi'IYdi~n 1'rj An va roO h1nh toan", De tai Nh/l 11l(('h:
"Nghien cuu m6i truiJng sinh tluli h6 clll(a nU'<}'cTMty dien Trt An", Tp
HCM,1985.
[12] T.V.LANG,D.M.NGQC,LV.TtIlEM,Xay d1.!DgmO hlnh toan hqc ghH 86
trong khOng gian 3-chi~u cho vi~c mO ta, dt! baa 11ii'hlqng va chill htr;mg
nuoc cua hb chUa Thuy di~n Tri An, liqi nght TOlIn hqc Vift Nam Lan thti
Ill, Ha n<>i,07/1985.
[13]T.V.LANG,D.M.NGQC,LV.THIEM,X:\y dt!ng mO hlnh toan hqc gi:'li:~:(\
Lrong khOng gian 3-chi~u eho vi9c mO ta, dt! baa 11ii'luqng va GMt ltcr;mg
mroe cua hb chtia 'Thuy di9n Trj An, llqi thdo Cd hqc chat lOng, Nha trang,
07/1985.
[14] T.V.LANG,D.M.NGQC,LV.HIIEM, XAy d~r11gmO hl11htoan h9c giiUr,:6
lrong khOng gi:U13-ehiCu cho vi9c mO ta, dt! baa trii luqng va eh11LIHong
mro,c cua hb chua ThUy di~n Trj An, l-lqi ngh( Khoa l'oan Dqi hqc Tdng

lz(jpTpHCM, 06/1985.
[15] T.V.LANGva;c~cI~cgia.Nghien c{cudong cMy ngoai bi~n, IMi light
TOIIIIhqc Vi~t Na17lLantluHll, Ha n<)i,07/1985. 1'6m t~Ltr. 133.
[16] T.V.LANGva c~clac gia, Xc! anhlnrbngnhol tTenbili toan dc)ng c:h:\y
khOngon djnh m<)tchi~u,Hqi nght Khoa hqc Lanthu V, TruiJngDIIBK II}
HCM, 06/1990, cr. 42-43.
[17] T.V.LANG,B.M.NGQC,CAnbhng va GMt hcq'J1gmtoc dum o1'\tlrong
mila khO,Bao do klwa hqc stf 83.203, Trung Tam l' oan Hqc (}"/1gDlfl18 WI
Tin Hqc, Vi~n KHVN, TpHCM, 1983,21 cr.
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