(fng dung phiFcyng phap nguyen ly CLFC trj Gauss trong c a hoc
Ngay nhan bai: 16/12/2013 Ngay siia bai: 20/12/2013 Ngay cliap nhan dang: 27/12/2013
Dao Cong Binh
TOIVI TAT:
Cac sach ca hoc hien nay chi de cap den he co hpc cd hen ket giii.
Do vay, Idii gap cac bai toan cO co Uen Icet Ichong gifl la ichong co ldi giai. Phflong phap nguyen ly cilc tri Gauss do GS.TSKH Ha Huy Cflong de xuat xem lien ket gifl la trfldng hop rieng cia Uen ket khong gifl. Do vay, phan ldn cac bai toan co khi flng dung phflong phap nay diu co dfloc ldi giai giai tich.
Bai bao trinh bay Phfldng phap nguyen ly ale tri Gauss va dng dung cua no trong cO hpc v?t ran bien dang nham giai cac bai toan CO hpc noi chung va cac bai toan dpng Iflc hpc, tfldng tac kit cau vdi moi trfldng noi rieng.
Til khoa: Ha Huy Cflong; nguyen ly cflc tri Gauss; Uen ket khong gifl; tflong tac.
ABSTRACT
Most of the present mechanical documents deal with two-way links systems, therefore, they are not suitable for one-way one.
The method of Gauss's extremum principle, which are proposed by Pro. Dsc. Ha Huy Cuong, considered mechanical systems with both type of links. Thereby, it could be appUcable to a larger class of mechanical problem, and even derive analytic solution.
In this article, this method was briefly presented and applied in mechanics of solids subject to deformations, including general mechanics, dynamic problems and interaction between structure and mecUa.
NCS.Th.S Dao Cdng Binh
Master of Civil Engineering and Industry The MiUtary Technical Academy Di dpng: 091 338 1697 Email: [email protected]
1. Dat van de
Khi gicii cac bai toan ket cau nam trong moi trudng chju tac dung cCia tSi trpng, de don gian hoa mo hinh tinh, nhieu tac gi^ thudng thay the moi trudng bSng lien ket nhU: 16 xo, si lanh- pit tong, v.v... Tuy nhien, vi^c tim cac gia trj do curng cua 16 xo, hoac gia trj ein nhdt, v.v... de xet dieu kien bien la kho xac dinh. Do vay, khong co duoc Idi giSi giai tich. Viec xet dieu kien bien trong cac bai toan dpng la kho khan.
Phuong phap nguyen ly cUc trj Gauss do GS.TSKH Ha Huy Cuong de xuat, CO the xet hoac khong can xet den dieu kien bien. Chinh vl vay, sCr dung Phuong phap nguyen ly cue trj Gauss, ta co the gl4i duoc cac bai toan cd lien ket giCTva khong giii.
2. Cac lien ket cd hoc
Cac rang bupc, cac han che ddi vdi chuyen dpng cCia he chat diem dang xet do sir tdn tai cua cac chat diem khac trong khong gian duoc goi la lien ket CO hpc.
Cac lien ket cd hpc thudng dung dUdc bleu thj dudi dang cac ham, bat phUdng trinh hoac phUdng trinh vi phan.
Cac lien ket khdng giCr (lien ket mpt chieu) la cac lien ket dupc bieu thi bang cac bat phiTdng trinh hoac cac bat phuong trinh vi phan. Cac lien ket giCr (lien ket hai chieu) diTpc bleu thj bang cac phuong trinh hoac cac phirong trinh vi phan [4].
Cac tai lieu va giao trinh co hoc giJl tich hien nay chi nghien cufu cd he CO lien ket giuf.
3. PhUcmg phap nguyen ly ale trj Gauss
Cac lien ket co hoc thudng dUdc bleu dien dudi dang cac ham, phirong trinh hoac bat phuong trinh. Trong do, cac lien ket giii dupc bieu thj bang cac phirong trinh. Cac sach co hpc hien nay chi de cap den he cd hpc cd lien ket giiif hay dupc hieu la lien ket hai chieu.
Nha toan hpc ngirdl Dilc Gauss K. F (1777 -1855) khi trinh bay nguyen ly cCia minh da xet lien ket khong giii va xem lien ket giCr la trudng hdp rieng cCia lien ket khdng giii.
Nguyen li cUc tieu Gauss dupc xay dung ddi vdi he co cd lien ket khdng giOrva dupc bieu dien dudi dang bat dang thurc.
Theo Nguyen li circ tieu Gauss, ddi vdi he cd fipc cd cac lien ket khong giuf, thi tong cdng cCia cac lire tac dung thuc hien tren cac chuyen vj So la cac dai lupng khdng di/dng. VI vay, dieu kien can va dii de b e d trang thai can bang trong trudng hpp lien ket khong giuf la:
V(Xi6ui-(-yi5Vi-i-Zi6Wj)<0
(1) Trong do:
X, Y, Z - cac luc trong he toa dp Be Cac tac dung len chat diem 1;
u,, y, w^ - cac chuyen vj tuong ufng theo phUdng cac lire tac dung len chat diem i.
Bieu thilc tren do Fourier (1798), Gauss, va Ostrogradsky (1834) ddc lap dua ra [5, tr.887] va con dupc gpi la bat dang thijfc Fourier
Dpa tren c o s d c u a bat d i n g thurc (1), Gauss da ehufng minh nguyen ly cLia minh [2], cho nen bat d i n g thurc tren sau nay ta gpi la bat dang
01.2014BIDniEnSl 1 1 1
thufc Gauss.
De giai cac bai toan cd lien ket khdng giO thi phJi dirng bat dang thiifc Gauss, lien ket giuf la trudng hpp rieng khi bat dang thufc trd thanh dSng thufc.
Phat bleu ciia nguyen ly cue tieu Gauss (1829) ddi vdi co hpc chat diem: chuyen ddng thpc ciia mpt he chat diem co lien ket tuy y, chju tac dung ciia mdt lire bat ky, 6 moi thdi diem xiy ra mpt each triing hpp nhat cd the, vdi chuyen ddng ciia he do khi hoan toan t u do, ndi each khac chuyen ddng thUc xJy ra vdi liTpng cudng bufc tdi thieu neu nhu sd do ciia lUpng cUdng biic lay b i n g tong cac tich khdi lupng chat diem mi vdi blnh phudng dp lech vj tri chat diem so vdi vj t r i khi chiing hoan toan tU do [2].
Tai mpt thdi diem nao dd, gpi Bi la vj trf thuc ciia c h i t diem, C Ja vj tri khi nd hoan toan tU do, ml la khdi lupng ciia c h i t diem, BiC, la khoang each giiifa hai vj tri. Gauss viet lupng cirdng bufc Z nhu sau:
Z = ^ rnjBjCj —> min ,^.
Do he c i n tinh va he hoan toan tir do deu chju lire tac dung gidng nhau, nen trong cong thufc (2) lupng eudng bufc khdng xuat hien Ipc tac dung. Ve hinh thufc toan hpc, lupng cUdng biJc Z cd dang binh phuong tdi thieu.
PhUdng phap nguyen ly cue trj Gauss la phudng phap so sanh. Si tim gia trj nhd n h i t ciia lupng cifdng bufc giufa chuyen ddng eiia he c i n tinh, vdi ehinh he do khi he hoan toan t p d o , tren co sd bat dang thirc Gauss.
Xet CO he chat diem cd Hen ket b i t ky, chju tac dung ciia Ipc bat ky tai thdi diem t nao dd.
Gpi f la Ipc tac dung len c h i t diem i. Neu nhu gicii phdng lien ket, c h i t diem t u do van cd Ipc tac dung f, can bang vdi luc quan tinh f^^^ = f. Cho nen theo bat dang thirc Gauss ta cd:
5Z=2](fi-foi)5i-iSO
(3)Trong dd 6r la bien phan ciia chuyen vj ao c h i t diem. Chuyen vi Jo pheii thda man eae lien ket da cho eiia he c i n tinh, va nd la dai lupng dpc lap (vl la So) ddi vdi luc tac dung.
Tir cdng thufc (3), ta t h i y 6Z la dai Ippng khdng dUdng cho nen ta cd:
--Y,^frfoi)'r
(4)Neu nhP chuyen vj So r^ thda man cac dieu kien lien ket da cho ciia he can tinh, thi ta cd the diing van tdc ao Tj lam dai lupng bien phan, nghTa la:
6 Z = ^ ( f i - f o i ) 6 f i < 0 Hay
Z=X''i-foi)^i-
(5)
(6) Trong bleu thufc (5) va (6) van tdc ciia chat diem la dai Ippng bien phan. Khi chuyen vj ao
r thda man cac dieu kien lien ket da cho cua he c i n tinh, thi ta cd the dirng gia tdc So Ij lam dai lupng bien phan, khi nay:
6 Z = ^ ( f i - f o i ) 5 V < 0 Hay
Z = V ' ( f i - f o i ) ^ ->min
(7)
(8) Ta viet eac bien doi t h u i n tiiy toan hoc sau day ddi vdi edng thiic (8):
Z = ^ ( f | - f o i ) C ^ -roi)->min
Z = y ( f i - f o i ) ( — - — ) - > m i n
^—1 mi m.
z = y ^(frfoi)^ -
" r H i
Z=ymi(J-fe)^
^—1 m.
(9)
(10) Hai edng thufc (9) va (10) cung chinh la cac edng thiifc thudng dirng ciia nguyen ly cue tieu Gauss vdi dai lupng bien phan la gia tdc.
Cac bleu thUc (4), (6), (9) va (10) la tUdng duong cho nen ta gpi la lupng cUdng bUc ciia chuyen dpng ciia cd he c i n tinh.
4. Ong dung PhUc^ng phap nguyen ly cUc trj Gauss
Viet phUdng trinh chuyen dpng ciia c h i t diem cd khdi lupng m, dudi tac dyng cua trpng trudng gia tdc g, tren dudng cong y = bx^ trong mat phang (xoy). Chuyen dpng khdng cd lUc ma sat (hinh 1).
•*" y=bx2
Hinh 1. Chat diem tren Jircing cong y = bx' Lien ket y - bx' = 0 doi vdi chuyen dpng chat diem la lien ket giO ddi vdi mpi thdi diem ciia y(t) va x(t) vdi t la bien thdi gian. VI vay, cac d i u b i t dang thUc trong cac cdng thUc tren t r d thanh dang thUc
Cac luc tac dung len c h i t diem:
-I- Theo chieu y la F = my -i- mg + Theo chieu X la F^ = mx
a. Khi chuyen vj la dai lupng bien phan:
Theo bleu thUe (4), lupng cudng bUc se la:
Z=(Fy)y-t-(Fx)x -> min (11) y va X la dai lupng bien phan. Vi y = bx= nen:
(12) TU dieu kien — =0, ta cd phUdng trinh dZ chuyen ddng: 3"
Z=(Fy)2bx-l-(FJ=0
Thay F^= mic va F = m y + m g vao taed:
Z = (my+mg)2bx-i-(m)()=0 (13)
Tinh y:
y = bx^
y=2bxx
y=2bxx+2bx^ " ' "
Thay y tU bieu thUc (14) vao bleu thdc (13), ta nhan dupc phuong trinh chuyen ddng eiia chat diem:
m(2bxx-i-2bx 4-g)2bx+(mx)=0 (15) Phuong trinh (15) la ket quS c i n tim.
b. Khi van tdc la dai lUpng bien phan:
Theo bleu thUc (16), lupng cUdng bdc dupe xac djnh bang:
Z=(my-i-mg)y-i-(mx)x -> min (16) Sai lupng bien phan d day la y va x. Thay tU bleu thufc (14) vao bleu thUc (16), ta cd:
Z=(my+mg)2bxx-i-(m)c)x -> min (17) Tir dieu kien — =0, ta cd:
5x
(my-i-mg)2bx+(mx)=0 (^g) Ta thay bieu thUe (18) giong n h u bleu thUc
(13), nen khi thay y tU bleu thUe (14) vao ta nhan dupc phudng trinh chuyen ddng ciia c h i t d i l m gidng nhu bleu thufc (15).
c. Khi gia tdc la dai lupng bien phan:
Theo bleu thUc (18), lupng cUdng biic Z dupc viet n h u sau:
Z = (my + mg)y-t(mx)x —» min (19) Thay y tU bleu thdc (14) vao bleu thUc (19), taed:
Z=(my-i-mg)(2bx)c-i-2bx^)-i-(mx)x - » min (20) TU dieu kien — =0, ta cd:
• 5x'
(my-l-mg)2bx+(m)<)=0 (21) Ta t h i y phUOng trtnh (21) gidng vdi phUdng
trinh (13), cho nen sau khi ta thay tU bieu thUc (14) vao ta cting se nhan dupe phuong trinh chuyen ddng (15).
5. Ket luan
PhUdng phap nguyen ly eUc trj Gauss la phUdng phap so sanh chuyen ddng ciia he nay so vdi he b i t ky nao khac, khi hai he cd cirng luc tac dung. Oay la phUdng phap hoan toan mdi trong cd hpc.
Dua tren md hinh bai toan neu tren, cho t h i y khS nang Ung dung ciia PhUdng phap nguyen ly cue tri Gauss trong co hpc, va md ra phUdng phap mdi de giSi mpt Idp cac bai toan CO hpc vat ran bien dang cd lien ket b i t ky.
TAI LlEU THAM KHAO
1. Ha Huy Ci/ong (1984), Sif dung nguyen ly circ trj Gauss vao cac bai toan mat during ciJng san bay va dudng 6 to, Luan an tien si ky thuat, Dai hoc MADl-Matxcova.
2. Ha Huy Cuong (IV/2005), Phuong phap nguyen ly cue trj Gauss,Tap chi Khoa hgc vaky thuatTr112H-118.
3.UnaosC.(1952),ThevariationalprindplesofmechanlG, University of Toronto Press Toronto.
4. M. A. AiiaepMaH (1980), KnaccuiecKan mama, MocKBa.
5. R C Ponair (1959), BapuauuoHHue npuHuunw MexaHMKU, MocKBa.
