Ổn định đàn dẻo của kết cấu chịu tải phức tạp phụ thuộc vào X và Y : Đề tài NCKH. QT.07.03

DAI HQC QUOC GIA HA NOI TRlTOfNG DAI HOC KHOA HOC TlT NHIEN ON DINH DAN DEO CUA KET CAU CHIU TAI PHirC TAP PHU THUOC VAC X VA Y Ma so: QT- 07 - 03 Chu t n de tai : PCS TS Dao Van Dung Can bo tham gia : GS TSKH Dao Huy Bich CN Bui Tin Tliuyet HaN6i-2007 I BAO CAO TOM T A T KET QUA THl/C HIEN DE TAX NAM 2007 _ - '^ A f f Ten de tai: On dinh dan deo cua ket cau chiu tai phuc tap phu thugc vao X va y Mas6:QT-07-03 Chu tri d§ tai: PGS TS Dao Van Dung, Can bo tham gia: GS TSKH Dao Huy Bich, can bg Iruang DHKHTN CN Bui TIti Thuyit, hgc vien cao hgc truang DHKHTN Muc tieu va noi dung nghien cuu: Trong thirc te, nhieu ITnh vuc ky thuat ung dung cac phuang phap va ket qua cua ly thuyel on djnh va dao dgng cua cac he dan hoi va deo Nhieu f f f cong Irinh xay dung, giao thong va cong nghiep c6 dang ket cau lam va vo.Vi vay nghien cuu ben, sir on dinh va dao dgng cua cac he nhu vay khong chi c6 y nghTa khoa hgc ma c6 y nghTa thirc lien Khi nghien cuu on dinh khia canh chinh ma la can quan lam la xac dinh tai trgng tai han, nhat la ket cau chiu quy luat tai phuc tap phu thugc vao toa O nuac la cac nghien cuu van de bat dau tir nam 2000 Ira lai day De tai QT - 07 f ^ 03 CO muc lieu di vao van de dang thai sir va nam 2007 d§ dat dugc r f mot so ket qua sau day: X f > */ On dinh dan deo cua lam mong chu nhat lam bang vat lieu nen dugc, f > chiu tai phuc tap khong ihuan nhat > *• > */ Tinh loan lam dan deo chU nhat bang phuang phap phan lu huu han */ Nghien cuu hien lugng mat on dinh dgng bang phuang phap nghiem giai lich gan dung ciia phuang Irinh Vander Pol r \ f f •> */ Cac ket qua tinh loan bang so cho mot so vat lieu cu the f Cac ket qua dat dugc a, Bai toan ve on dinh dan deo cua tdm mong chiv nhat lam bang vat lieu nen duv'c chiu tai phuc tap khong thudn nhat Da xay dung dugc he phuang Irinh on djnh din ddo X6l hai \6p bai f \ f loan vai bien tira ban le va bien ngam tren bon canh Ap dung phuang phap Bubnov - Galerkin va phuang phap tham so tai dan den he thuc cho phep tim lire tai han \ f y Da trinh bay thuat loan, tinh loan bang so, xay dung thi mo la anh f f huang ciia manh, ciia tinh nen dugc den lire tai han cua lam b Bai toan ve tinh todn tarn dan deo chit nhat bang phwong phap phan tie hitu han Da nghien cuu lam theo mo hinh tuang thich dira tren ly thuyet qua f \ y \ trinh dan deo Bai loan dugc giai bang phuang phap bien the nghiem dan hoi > • ^ f Qua trinh giai dugc chia k giai doan, moi giai doan gom n buac lap Ket • f f qua cua buac lap truac la ca sa de tinh loan buac lap tiep theo Tam dugc chia cac phan tii chu nhat Chuang trinh dugc thuc hien bang phan mem Matlab 6.5 Hinh anh mien deo dugc mo ta cu the sau cac giai doan, cho ^ y f \ f Ihay dugc mem deo xuat hien tir bien dat lire Ian dan vao ben tam c Nghien civu hien tuvng mat on djnh dong bang phuang phap nghiem giai tich gan dung cua phuvng trinh Vander Pol Bai loan dan den viec giai phuang trinh Vander Pol vai he so phu thugc vao tan so ciia lire kich dgng Cac tac gia da tim nghiem giai tich gan dung ciia phuang Irinh nay, thong qua nghiem Ihu dugc da phan tich sir phu thugc / ^ \ f f f \ f /• rat nhay ciia dang dieu ciia nghiem vao cac he so ciia phuang trinh vao sir f f f luang tac giua cac yeu to phi tuyen va lire kich dgng Da chi nhung hieu f • •> r ung dac biet lien quan den chuyen dgng xoay, den hien tugng Galloping r -y -* Cac ket qua nghien ciiu cua de tai duac the hien tren cac bai bao va bao cao khoa hoc Dao Van Dung, Bui Thi Thuyet Elastoplastic stabibity of thin rectangular plates made of compressible material under nonhomogeneous complex loading (to appear in VNU Journal of Science, 2007) Dao van Dung, Nguyfin Cao San Tinh loan tam dan deo chu nhat bang phuang phap phan tii huu han Tuyen tap cong trinh hoi nghi ca hgc loan quoc Ian thu 8, Ha Ngi 6-7 thang 12 nam 2007 Dao Huy Bich, Nguyen dang Bich, Nguyen Anh Tuan Nghien cuu f t y ^ hien tugng mat on dinh dgng bang phuang phap nghiem giai tich gan f f y f \ f f diing phuang trinh Vander Pol Hoi nghi quoc te Ian thu nhat ve thiet ke, xay f f dung va bao tri cac ket cau, - 1 thang 12 nam 2007, Ha Ngi, Viet Nam Tinh hinh kinh phi + Cac bai bao, bao cao khoa hgc va thu lao chuyen mon: 12.000.000d H- Hoi thao va xemina khoa hgc: 4.000.000d + Chay chuang trinh va che ban: 1.600.000d + Quan ly ca sa SOO.OOOd + Van phong pham va cac chi khac 1.600.000d Tong cong 20.000.000d Nhan xet va danh gia ket qua thyc hien de tai */ De tai vai thai gian thuc hien mot nam da hoan vugt muc ke hoach so vai chi lieu dat ve so lugng bai bao va bao cao khoa hgc Da c6 f \ 01 bai dang tuyen tap hoi nghj ca hgc loan quoc Ian thu 8, nam 2007 va f 01 bai nhan dang a tap chi khoa hgc Dai hgc quoc gia Ha Ngi va 01 bao cao f f khoa hgc a hoi nghi Quoc te nam 2007 */ Cac van de nghien cuu c6 y nghia khoa hgc va hgc thuat, gop phan t f f djnh huang ung dung viec xem xel sir on dinh ciia cac ket cau */ De tai gop phan nang cao chuyen mon ciia can bg, cao hgc va nghien cuu sinh ciing nhu sinh vien nganh ca hg^ Thong qua cac xemina va hoi thao ' f khoa hgc da ciing c6 va trang bi them nhiing kien thuc chuyen sau cung nhu huang ung dung ciia Bg mon Ca hgc, Khoa Toan - Ca - Tin hgc, Truang dai hgc Khoa hgc Tu nhien, Dai hgc Qu6c gia Ha Ngi */ Da huang dan cao hgc, sinh vien theo huang de tai */ Nhom de tai kien nghi thai gian tai se dugc nang cap de tai theo phuang huang Ha Noi thang nam 2008 Xac nhan cua ban chu nhiem khoa • Chu tri de t^i • CH TS Uc.^c K-^ D^ PGS.TS Dao Van Dung Truang Dai hoc khoa hoc Tu uhien I^U tPUONC* -.i^^^.^^^:^iv^^^ n BAO CAO TOM TAT BANG TIENG ANH Title: Stability of elastoplastic structures subjected to complex loading depending on x and y Project's code: Q T - - ; Duration: 2007 Head of research group: Assoc Prof Dr Dao Van Dung Paticipants: Prof Dr Sc Dao Huy Bich B Sci Bui Thi Thuyet Duration: 2007 Resume on the aim and main contents of project In practice, there is a lot of technical domains applying the research methods and results of the theory of stability and oscillation of the elastic systems and elastoplastic systems Many structures in contruetion, transportation and industry are the form of the plate and shell Therefore, the investigation on the durability, stability and oscillation of these systems is not only scicntillc sens but also practical sens The main aim of stability problem is to find critical loads, in particular for structures subjected to the complex loading defending on x and y In our country, the studies on this orientation have been beginning since 2000 up to now The project QT-07-03 has the purpose to investigate these hot current problems In this project, our staff have investigated the following topics: a Elastoplastic stability of thin rectangular plates -made of compressible material under nonhomogeneous complex loading b Calculating elastoplastic rectangular plates by finite element method c Investigation of Aerodynamical instability phenomena by using a method for the approximated analytical solution of the Vander Pol equation Results a Scientific activities: -01 research paper accepted to publication in VNU Journal of science, 2007 -01 research paper have been pulished in the proceedings of 8-th National Conference on Mechanics, 6-7, December, 2007 -01 scientific report in the 1-th Intemational Conference on Modem Design, construction and Maintenance of structures, 10-11, December, 2007, Hanoi, Vietnam b Training activities: 01 M.Sci and 01 B.Sci c Scientific papers and reports - Dao Van Dung, Bui Thi Thuyet Elastoplastic stability of thin rectangular plates made of compressible material under nonhomogeneous complex loading (to appear in VNU Journal of Science, 2007) - Dao Van Dung, Nguyen Cao Son Calculating elastoplastic rectangular plates by finite element method Proceedings of the Eigth National Conference on Mechanics, Ha Noi - , December, 2007 - Dao huy Bich, Nguyen Dang Bich, Nguyen Anh Tuan Investigation of Aerodynamical instability phenomena by using a method for the approximated analytical solution of the Vander Pol equation The 1^* International Conference on Modern design Construction and Maintenance of structures, 10 - 11, December, 2007, HaNoi, VietNam III NQI DUNG CHINK CUA BAO CAO M a dau f \ •' •> f f ^ Van de on dinh dgng va on dinh ciia cac ket cau dan deo chiu tai, dac biet la chiu tai phuc tap dugc quan tam nghien cuu vi khong nhGng c6 y nghTa khoa hgc ma c6 y nghTa thuc tiln » ^ ^ •» De giai quyet bai toan on dinh dan deo can phai xay dung mo hinh phu f t s y f hgp, thiet lap cac phuang trinh on djnh va dieu kien bien, de xuat phuang phap giai, tim bieu thuc de xac djnh lire tai han Lap chuang trinh may tinh ' f f va tinh loan cho mot so dang ket cau tir phan tich ca sa khoa hgc va y f nghTa co* hgc ciia cac ket qua f •» \ •> ist f De nghien cuu van de on djnh dgng ta dan den phuang trinh phi ' f ' tuyen, cho nen tim nghiem giai tich rat kho Vi vay ngoai phuang huang giai ^ > f gan dung bang phuang phap giai tich so, nguai ta di tim nghiem dang giai tich gan diing •» Ngoai hien sir phat trien manh me ciia tin hgc, nen ca hgc ung dung nhieu phuang phap phan tii hiiu han de giai cac bai toan xac f f \ f f djnh trang thai ung suat va bien dang va mien deo ket cau > y f Da CO nhieu cong trinh nghien cuu on djnh dan deo ciia tam va v6 chju tai thuan nhat Tuy nhien quy luat tai phu thugc vao toa thi bai toan rv r f dan den nghien cuu he cac phuang trinh dao ham rieng vai he so la ham ciia x va y Vi vay lop bai toan hien it dugc nghien cuu * \ f f f y y De tai nham giai quyet mot so khia canh nhung van de on dinh dan deo, on djnh dgng nhu vay Noi dung chinh a Van de on dinh dan deo cua tdm mong cliir nhat lam bang vat lieu nen dwac chiu taiphivc tap khong thudn nhat Bai toan vai vat lieu khong nen dugc da dugc tac gia VG Cong Ham nghien cuu nam 2003 O day cac tac gia de tai nghien cuu vai vat lieu la nen dugc Da thiet lap phuang trinh on dinh cho tam Khao sat hai 16'p bai toan voi bien tira ban le bon canh va ngam bon canh, Ap dung phuang phap Bubnov - Galerkin va phuang phap tham so tai Xay dung he thuc tim lire tai han Det(Hik) =0 Da trinh bay thuat toan va tinh bang s6 cho k6t ciu cu thi: - Xet tai trgng phu thugc vao tham s6 tai t oQi n i p, = 283 + 0.1/, p^= ' (283 + 0.1// ^—^ 450- - Xet tai trgng phu thugc vao / va toa , (283.0 - ^ ^ ^ p, =(283 + 0.1/)(l + 0.35f), P2^ ZV (1-0.25-) b ^' 450' a Xet lai vai quy luat ji;, =(246 + ( + ^ ) , b p,=20t\- a Nghiem cua bai toan tren tim duai dang chuoi lugng giac, - Xet tai vai quy luat p, =(283 + 0.10(1 + 0.35^), ' b (283 + 0-'0 ( i _ 25J^) ~ 450' a Nghiem a day tim duai dang chuoi luy thira Tinh toan va xay dung thi mo ta anh huang ciia tinh nen dugc, ciia f f manh den lire tai han ciia tam f f f r Ket qua cho thay tam chiu tai phuc tap thi lire tai han nho han tam chju tai dan gian Da chi tinh nen dugc c6 anh huang dang ke den lire tai f han ciia tam ^ > r s •y b Van de ve tinh toan tdm dan deo elm nhat bang phuang phap phdn tir hivu han > y f Phuang phap phan tu huu han, hien dugc sii* dung nhieu va rat eo hieu qua, nhat la giai cac bai toan c6 hinh dang phuc tap hoac cac bai toan dan den cac phuang trinh khong the tim dugc nghiem duai dang giai tich When E(k) is a small parameter, the equation (1.1) can be solved by the small parameter method In [4] an investigation of the aerodynamical instability phenomenon was performed by using the non-linear Vander Pol model of the form x-(2y-3ax^)x + co^x + /x^ =qcos/Jt (^-^^ Eq.(1.2) was solved by using the quasi-harmonic equilibrium method in the cases coefficients depend on the frequency of excited force a)2v=CT, 3a=0,2a, co^=l, y=0,005p, q=p^ b)2v=a, 3a=ap, 0)^=1, -r0,005p, q=p^ The nonlinear Vander Pol model will be investigated in this paper is of the form X -\- {2y -^3ax^)x + a^x^ +aqx^ +kx = pcoscot (*-^) The difference of equations (1.2) and (1.3) is that in eq.(1.3) there exists a nonlinear item of second degree relatively with unknown x The aim of the present paper is to formulate a method based on the idea of the extremum principle for finding an approximated analytical solution of the equation (1.3) and to discuss on the behaviors of the obtained solution Formulation of the method The idea of the method is to separate the equation (1.3) into two first-order ordinary differential equations To obtain an exact analytical solution of the initial equation (1.3) if two derived equations are satisfied simultaneously But of course it is difficult, we intend to seek an approximated analytical solution by means of minimizing one derived functional and considering obtained two first-order ordinary differential equations as constrained conditions 2.1 Separation eq.(1.3) into two first-order ordinary differential equations Introduce a new unknown function y relative with unknown x by equation (2-1) X — 4-ax +V = y , X from that ,2 X X X X Y + ax = y, X 2 • ^ - ^^ + a^x^ + y^ ^-^ 2ax + 2y^ — ^+- _2vax = y\ A-' X Summing up side by side obtained relations yields —+ a^x^ +2yax + 2v- + 3ax-\-y^ -y~y^ X (2.2) =0, X Subtraction of equation (1.3) and equation (2.2) after multiplying the last with x leads to 3(7x'x-3axx-\-iq-2y)ax'-^(y + y'+k-y')x-pcoscot = (2.3) Consequently equations (2.1) and (2.3) are two first-order ordinary differential equations separated from the initial equation (1.3) by introducing the new unknown y 2.2 Derived functional and constrained conditions From equation (2.1), (2.3) we can obtain a functional equation f ^ax'-yx'^-x\x + -{q^v)x'^ -— _^^ ^^ 2^_.l/'.,-u./^r2 J - r v + v {y^y'+k-y')x '-H^-vhx ^cosfy/U=0, 3a /7 3a (2.4) J - where I - is a certain value of t r'^]\^r.Ani\ Now it is necessary to minimize the left hand side of eq.(2.4) with the constrained conditions (2.1) and (2.3) •or this purpose a corresponding function (j) is formulated such as , 1Hq-^y)x^^—iy-^/^k-'' /^_.,Av-2 _L,'V;+ v^ + ^ - v ' U ^ = ax'-yx'+-x'x ^ „2 + ) ^ —cos^j/ 3^^"2a' ' 3a a -^ X,{x + ax^ -^-vx - yx) + /l,[3crx\v - 3a.Yx + {q- 2vW +{y+y^+k-v^)x- (2.5 pcoscol] where ^i, y^i -Lagrange multipliers Applying the minimum condition to functional similarly [6] S^^ d_ dt -'-l-O, dx d_^d(^) dt [dy leads to dd dy = 0, 3ax' - 2yx + 2a\x + A,v - A^y + 2a{q - 2v)X,x ^^i^^y)x ^ + {l,-^^){y^y'^k~y^) 3a X \ —+ (2.6) = Q, 2>' = 0, (2.7) X where X| = const, Xi = const are assumed By solving (2.1) and (2.3) the exact solution of (1.3) can be obtained, similarly, by solving (2.6), (2.7) the approximated solution of (1.3) can be obtained However, these sets of equations can not be solved separately By relymg on the four equations (2.1), (2.3), (2.6) and (2.7) it is possible to change and choose suitable constants leading to the solvable sets of equations such as an approximated analytical solution for the initial equation (1.3) can be obtained 2.3 Seeking a first approximated analytical solution Assuming Xi = const, X2 = const and choosing for them appropriate values we restrict a solution class of eq.(1.3), but it is possible because it only serves a purpose to seek a first approximated analytical solution Deducting the respective sides of (2.7) and (2.1) we have \ ~a x + -v-y=0 (2.8) ^ 3a ^ 3a Equation (2.6) can be analyzed and rewritten in the form 'x > — + ar-f V - j ^ {AX + X ) -Ak-B- B)^ a ^-f x-h 3a \ ~v-y [Cx-\-D) 3a + {B + D-A,)y + X 2aA^ + 2a{q -2y)A^ + - ( ^ ãƠy)-aB-yA -a D3a -yB ^ -y C ^ 3a D-f-Av/ + ( A , + — ) ( > ' - y ' + ^ - v ' ) = 0, ^^h 3(3 where denote '"^-Ya C = - ^ + 3a (2.9 ' " 2^-— 2X, ^ 3a ^ 3a In use of equations (2.1), (2.8) the obtained above equation can be reduced -Ai-B- + {l,+—\{q-2v)ax + y-y^+k-v^^ =Q (2.10) with conditions B+D-Ai=0, ^4 5" -vB~ Z) + /l^^' = , A,-!^ 3a ^ 2aA,+2a(^ - 2v)/l j-i (g + y)~aB~-yA / \ - a D- ^ + 3a / ^ + \ -V 3a ; = [^ j{g-2u)a From these relations and relations (2.9) the constants are chosen A, 3a 2v _ qX B= 4v A , — I (2.11) £) = and the equation (2.10) can be rewritten in the following form y -5\ V + — ^J ^ qA, + 2i/ — + (g - 2v^)ar + y + ; ' ^ + / t - v ' ^ = , y^ (2.12) X with A-i - an arbitrary constant Deducting the respective sides of (2.3) and (2.12) we have 4v 5^qX, a ' la XT + 3crx'x + + 2v x~ pcoscot = f y\ ( v\ Integration of the just obtained equation yields y X + 7a f HV / ^ ^ V 4v] a J -3a X qA, + + 2y X + -E^ZOS{Ot ãƠ)ãƠ CQ = ao) ^^ ^ -~\ (2.13) "J where Co - an integral constant Root of the algebraic equation x (2,13) can be represented as L If 7t ^ l_ x = o c o s - cot-\- — A if satisfying the following relafions (2.14) 2) / 5V^ 3a (2.15; 20t/3 (2.16) aco J 5^ 5^- 4y\ qA, ^-3a V oj \ r i\^ •¥2y (2.17) + -6^=0, a/ ^~lb'+C.=0 (2.18) vjy Thus b is defined by (2.16) X, - by (2.17) and Co - by (2.18) and the first approximated analytical soluti on (2.14) was determined 2.4 Seeking a sequent approximated analytical solution In use of representation X2 through X, according to (2.11) the equation (2.8) can be rewritten in the fo rm (2y ^ x + y-y = U Performing some calculations: multiplication of equation (2.1) to — ^ -(-2K, (2.19) equation (2.19) to 2c, 2(A ) a summation together with equation (2.12) and then substitution x according (2.14) we obtain qA, y + y- CO ~y -c , n k, + p, cos o)t ãƠ -\-3y (2.20) where k ^ (2.21) c - an arbitrary constant Putting qX^ y / -y-c u = - (2.22) X,— equation (2.20) can be rewritten — + — (^, +/?, cos6') = 0, u (2.23) where ~-io)t + — -^3y) Equation (2.23) contains a constant c, which can be chosen arbitrarily such as this equation gives a best approximated analytical solution By use of (2.22) we rewrite equation (2.1) 5-47 qX^ u —-H dt[x (2.24) + c — -a x a)'' _ Consequently, the pair of equations (2.23), (2 24) allows to seek an approximated analytical solution to the initial equation (1.3), herein equation (2.23) has the form of a Mathieu equation and equation (2-24) is a first-order ifferential equation, solution to which can be obtained easily ordinary linear differential easilv once to know solution of the equation (2.23) L V 3, Solution expression of the initial equation (1.3) First of all we need to seek a solution of the Mathieu equafion (2.23) with a kernel function g(t)=ki^picose According [7] an approximated analytical solution of equation (2.23) with kernel function g(t) can be found such an exact analytical solution of an auxiliary Mathieu equation with a kernel function h(t) which is equal approximately to the kernel function g(t), where 9aP 3X W £ -1 p 9ap CO' P' + — p' his) = CO X a ~ +cos^ — -i-cosy 9aP p p J The auxiliary Mathieu equation with kernel function h(t) M3 \0- -4-—/7(/) = 0, u gives an exact analytical solution as follows u= 0)^ 9ap X p ^ I —+ cos^ + —-f-COS^ C, + f f —-I-cosy p TT'/' (3.1) + — + cos^ p where C| - an integral constant From here if can be calculated — + cos^ p CO' 21aP s\n9 CO X 9aP p + zos6 N2 — H-COSt/ X a 9aP + —+ P COS0 (3.2) ^^ co' — + C0S6' P C,+ ^ " » f ^ + i + cos^ 19a/? P Suppose ( | o = «o "(01.0 = "o, from (3.1) and (3.2) if follows ' + —+ cos(—+ / ) 9aP p ^ X ,n ' — + cos(- + r) P sin(—+ ;K) 21 ap u, \ X /^ V (3.3) - + cos(^ + / ) ^"c P A (3.4) + — + cos(— + y) — + cos(— + y) 9a/? P , 9a/? p (> P After determination of function u, substituting which to the equation (2.24) we define its solution S^B a X = ue (3.5) jweVz-i-C where C2 - an integral constant and qX s = —^-^ + c 4(^ ) a For formulating expression of x equation (2.24) is rewritten in the form x= +5 x-ox , where — calculated according (3.2) and x-according (3.5) For determination of integral constants Ci, Cj it is necessary to satisfy initial conditions of the initial equation x{t)l ^=x„ i{t)l ,=x Putting t=0, from (3.5), (3.6) it follows axr c.' ^0-^0 > and then combining with (3.3), (3.4) we can determine integral constants A 7t — + cos(— + y) P ^ (0•^ • X A ,71 + •- + cos(— + v) c, (A \f T9ap p ^^ ,71 , CO A ,^ +— + cos(~ + ^/) —+ cos(—+ x) p ^9ap p ,^ A ' Tlap (3.7) X —+ cos(—-)-y) C,= p ^^ co' X n ;') (i„+W-«o)+ —+ cos(—+ —+ cos(—+ / ) 27a/7 9a/? /? 6 In the expression of solution figure parameters — and ^7-;;, which according to [8] are defined from following P 9aP relations I (k:-Pl-K)j-^p.-p-('/£-pj^ 1 J y*^*^^ ^ ^ t>^ 4 - •2 (derived parameters X,=2.9333, c=-1.0518, h,,= 1.012196,hs2=1.017899) Fig.4a Graph relating x(t)-x(t) 35i 2e 51 05 -0 -1 -1 5^ /\ :}fi'^-^:: ' ,1 J (derived parameters X|=2.9333, c=0.49995) Fig.4b Graph relating x(t)-x(t) 4.3 Discussion about the results According to the description about aerodynamical instability phenomena in the section 1.1 and the obtained graphs relating x, x some conclusions can be given - With first parameters combination: case a) describes the separated eddy phenomenon; case b)-galloping phenomenon; - With second parameters combination: both cases a) and b) describe the galloping phenomenon; - With third parameters combination: both cases a) and b) describe the eddy phenomenon; - With fourth parameters combination: both cases a) and b) describe the phenomenon of interaction between eddy and galloping excitations Conditions for occurring eddy and galloping excitations can be based on the values hsi, hs2 - When hsi, Ki are greater or nearer than 1, there exists the eddy excitation; - When hsi is smaller and nearer -1, hs2 is greater than 12, there occurs the galloping excitation These conditions reduced from data of considered examples of course are not general By help of obtained conditions the critical wind velocity can be calculated from formulations presented in this paper In considered examples there exists values of ^, satisfying (2.17) and respectively one can choose c satisfying or almost satisfying (3.9) for each combination there exists only two cases a) and b) where conditions (3.9) are satisfied 5, Conclusions - The proposed method allows to seek an approximated analytical solution of the Vander Pol equation with coefficients not being small values; - The obtained approximated solutions presented in the graphs have characteristics of the eddy excitation, galloping and interaction between eddy galloping and excitation For the first time an analytical expression describing the eddy, galloping phenomena and interaction between them is obtained; ^^i of ^ "iclhod for seeking an approximated analytical solution, proposed in present paper, is based on the idea cxiremum principle and its applied results can describe the eddy, galloping excitations and interaction of mem, that are the phenomena which were observed by experiments; The solving steps are codified such as the final results can be obtained automatically in the form of numerical tables and graphs; - The solving program can be used to study the infiuence of parameters variation on the solution of the nonlinear Vander Pol equation Acknowledgements: This paper is completed with the financial support from the National Council for Natural Sciences and the Project QT-07-03 References [1]R H Scalan Flutter Derivatives at Vortex Lock-in Struc Eng[M] ASCE April 1998 [2]G V Parkinson Aeroelastic Galloping in One Degree of Freedom [C] Proceeding of Symposium on Wind Effects on Buildings and Structures Vol J National Physical Laboratory, Teddington UK., 1963.1: 581 -609 (3]Eurocode 1: Actions of structures-?arr 1-4: General actions - Wind actions [M] BS EN 1991-1-4:2005 [4] Dao Huy Bich, Nguyen Dang Bich - Application of the Vander Pol equation for investigating aerodynamical instability phenomenon [C] Proceedings of the Seventh National Congress on Mechanics Hanoi December 18-20, 2002.1: 83-94 [SjE.Simiu, R.H Scalan [M] Wind Effects on Structures A Wiley-lnterocience Publication John Wiley and Sono, 1986.1:509 [6] Korn G A., Korn T M [M] Mathematical handbook for scientists and engineers McGraw Hill Book Company, 1968 [7] Dao Huy Bich Nguyen Dang Bich On the lateral oscillation problem of beams subjected to axial load [J] VNU Journal of Science Mathematics-Physics T.XX,No4 2004: 1-10 [81 Nguyen Dang Bich, Ngo Dinh Bao Nam Conditions for the approximated analytical solution of a parametric oscillation problem described by the Mathieu equation[J] VNU - Journal of Science Mathematics-Physics T.XXll.No2.2006 5sa DAI HOC QUOC GIA HA NOI TRUING DAI HOC KHOA HOC TlT NHIEN Bui Thi Thuyet ON DINH CUA TAM DAN DEO CHIU TAI PHtfC TAP PHU THUOC VAO TOA DO Chuyen nganh: Ca hoc Vat the rdn Ma so: 60.44.21 LUAN VAN THAC SI KHOA HOC NGudi HU3NG DAN KHOA HOC PGS TS DAO VAN DUNG Ha Noi- 2007 BAI HOC QUOC GIA HA NOI TRUING DAI HOC KHOA HOC TV N H I £ N KHOA : TOAN - CO - TIN HOC ^ca^ Nguyen Cao Son TINH TOAN TRANG THAI DAN DEO CUA BAN CHlT NHAT BANG PHUONG PHAP PHAN TlTHUtJ HAN KHOA LUAN TOT NGHIEP HE CLrNHAN KHOA HOC TAI NANG Nganh: Co hoc Can bo h\x6ng dhn: PGS.TS Dao Van Dung Ha Noi- 2007 SCIENTIFIC PROJECT Branch: Mathematics - Mechanics Title: Stability of elastoplastic structures subjected to complex loading depending on x and y Project's code: QT - 07 - 03 Head of research group: Assoc Prof Dr Dao Van Dung Paticipants: Prof Dr Sc Dao Huy Bich B Sci Bui Thi Thuyet Duration: 2007 Resume of main contents In this project, our staff have investigated the following topics: */ Dao Van Dung, Bui Thi Thuyet Elastoplastic stability of thin rectangular plates made of compressible material under nonhomogeneous complex loading (to appear in VNU Journal of Science, 2007) */ Dao Van Dung, Nguyen Cao Son Calculating elastoplastic rectangular plates by finite element method Proceedings of the Eigth National Conference on Mechanics, HaNoi - 7, December, 2007 */ Dao huy Bich, Nguyen Dang Bich, Nguyen Anh Tuan Investigation of Aerodynamical instability phenomena by using a method for the approximated analytical solution of the Vander Pol equation The l" International Conference on Modem design Construction and Maintenance of structures, - 1 , December, 2007, HaNoi, VietNam A9 PHIEU DANG KY KET QUA NGHIEN ClTU KH - CN Ten de tai (hoac du an): ON DINH DAN DEO CUA KET CAU CHIU TAI PHlTC TAP PHU THUOC VAO X VA Y Ma so: QT - 07 - 03 Ca quan chu tri de tai (hoac du an): Truang Dai hoc Khoa hoc Tu Nhien Dia chi: 334 duong Nguyin Trai Tel: 8.585.277 Ca quan quan ly de tai (hoac du an): Dai hoc Quoc Gia Ha Noi Dia chi: 144 duong Xuan Thuy Tel: 8.340.564 Tong kinh phi thuc chi: Trong do: - Tu ngan sach Nha nuac: 20.000.000d (hai muai trieu dong) Thai gian nghien cuu: Nani 2007 Ten cac can bo phoi hgp nghien cuu: GS TSKH Dao Huy Bich, Truang DHKHTN, DHQGHN CN Bui Thi Thuyet, hoc vien cao hoc Truang DHKHTN So dang ky So chung nhan dang ky ket qua nghien ciiu: de tai Bao mat: Ngay: b Pho bien han che; a Pho bien rong rai: c Bao mat: Tom tat ket qua nghien cuu: Dg tai QT - 07 - 03 da nghien cuu cac van dS sau: On djnh cua tim mong chu nhat lam bang vat lieu nen dugc, chiu tai phuc tap khong thuan nhat Tinh toan tam dan deo chu nhat bSng phuang phap phan tu hiiu han Nghien cuu hien tugng mk 6n dinh dong bSng phuang phap nghiem giai tich gSn diing cua phuang trinh Vander Pol 5o */ Da dugc nhan dang bai bao a tap chi khoa hgc DHQG, 2007; bai dang tuyen tap hoi nghi ca hoc toan qu6c ISn thu 8, 2007; bao cao hoi nghi Quoc le lAn thu nhit, 2007 Gop phan dao tao thac sT, cu nhan khoa hoc tai nang ca hgc Kien nghi ve quy mo va doi tugng ap dung nghien cuu: */ De tai gop p h k dao tao va nang cao chuyen mon */ Cac phuang phap nghien cuu c6 y nghTa khoa hgc va dinh huang ung dung xay dung kien true.v.v */ Kinh de nghi Truang xem xet cho nang cap hgp d6ng Chu tich Hoi Thu truang co Chu nhiem de tai quan chu tri dh tai dong danh giS chinh thuftji.' lo ten Dao Van Dung Hgc ham hoc vi 51 Thu truang ca quan Muc luc • • Trang I Bao cao torn tat ket qua thuc hien dh tai II Bao cao torn tat bang ticng anh III Phan chinh cua bao cao Mo dm Ngi dung chinh 3.Katluan '1 Cac cong innh cong bo ^^ IV Phu luc( cac bai bao va bao cao khoa hgc ) 13 V Scientific project ' ^^ VI Phiau dang ky kk qua NCKH 50