Tính toán xác định hệ số động của dầm cầu dưới tác động của tải di động

Trong thiet ke, nguoi ta chu yeu dua vao so lieu phan tich tinh, phan ung dong dudi tac dung cua cac loai tai trong khac nhau chua duoc nghien cuu mot each day du.. Khi kiem dinh, danh g

Trang Muc luc

L6i cam on

Chuong 1 PHUONG TRINH DAO DONG CUA DAM DAN HOI 3

DU6l T A C D U N G CUA VAT THE DI DONG 1.1 Phirang trinh tong quat dao dong uon cua dam dan hoi 3

1.2 Phuong trinh dao dong cua dam voi tai h^ng so di dong 5

1.3 Phuong trinh dao dong cua dam v6i vat the di dong 6

1.3.1 Phuong trinh dao dong cua vat vdi nen dao dong 6

1.3.2 Phuong trinh dao dong cua dim du6i tac dung cua vat di 7

dong 1.3.3 He phuong trinh dao dong cua dSm voi vat the di dong 8

1.4 Phirong phap Bup nov-Galerkin trong ly thuyet dao dong 9

cac he lien luc

Chuong 2 KHAI NIEM VA TINH CHAT CUA HE SO DONG 12

LUC CUA DAM CAU

2.1 Khai niem 12 2.2 Xac dinh he so dong lire hoc cua dam cau v6i tai trong 13

3.1 Thilr nghiem dong voi cau 38

3 L1 Cac phuong phap kich dong 38

3.1.2 Cac duns cu ihicl hi do "hi 41

3.2 Cac dai lirong do dac diroc 42

3.2.1 Tdnsorieng 43

3.2.2 He so can 43

3.3 Tinh he so dong cua dam cau tir so lieu do 44

KET LUAN 45 TAI LIEU THAM KHAO

Tac gia xin bay to long biet cm ch^n thanh va sau s4c doi voi thay giao hudng dSn Pho giao su Tien sy khoa hoc Nguyin Tien Khiem, nguai da het sue tan tinh huong dSn, giiip da, tao dieu kien thu£ln Icri va c6 cac chi dSn khoa hoc cho noi dung nghien cuxi cua lu|in van

Tac gia chan thanh cam on su giiip do dong vien cua cac thay c6 giao cua Trung t^m Hop tac dao tao va b6i duong co hoc Vien Co hoc Nhan day, tac gia cung xin ch^n thanh cam on su quan tarn, dong vien va tao dieu kien thuan Id cua Ban giam doc Trung t^m cong nghe thong tin Dien luc Viet nam, noi tac gia dang c6ng tac

Cong tac ki^m dinh, chan doan va danh gia cau giao thong 6 Viet nam dang

la mot va'n de duoc nhi^u ngudi quan tarn do he thong cau giao thong la he thong co

so ha tang quan trong co gia tri 16n, dam bao hoat dong cua ca nen kinh te quoc dan

nhung lai da dang phu'c tap vi k^'t c^'u va co nhieu nguyen nhan khac nhau dSn den

su xudng c^'p va hu hong Doi voi cau giao thong, truoc khi dua vao khai thac dat ra

cong tac kiim tra, danh gia, xem xet viec thi cong co dam bao diing thiet ke va cac

thong so yeu cSu hay khong Doi v6i cau cu, viec chuan doan danh gia cau phai dua

ra duoc cac danh gia v6 kha nang lam viec tiep theo v^ do bin, do cung, do on dinh,

d6 an toan va tu6i tho con lai ciia cau Ket luan dua ra la can cu vao mot loat cac so lieu ihu Ihap duoc tir Ho so cong trinh, lir kiem tra thuc nghiem va thu tai w

M6t trong nhung thong so quan trong can phai xac dinh khi ki^m tra, danh gia c^u la he so dong luc hoc He so dong luc hoc la mot dac trung mo ta anh huong cua cac tai trong dong len cau so v6i tai trong tinh duoc tinh b^ng ty so giua chuyen

vi dong tren chuyen vi tinh Trong thiet ke, nguoi ta chu yeu dua vao so lieu phan tich tinh, phan ung dong dudi tac dung cua cac loai tai trong khac nhau chua duoc nghien cuu mot each day du Khi kiem dinh, danh gia cau he so dong duoc xac dinh b4ng phuong phap thuc nghiem dua tren so lieu do Tuy nhien, 6 nuoc ta viec xac dinh he so dong b^ng thuc nghiem lai chua co mot phuong phap thong nhat va hieu qua Truoc thuc te do, tinh toan, nghien cuu tinh chat va d6 xuat phuong phap xac djnh he so dong dua vao dong thoi ca ly thuyet va thuc nghiem, chii trong den phuong phap xac dinh b^ng thuc nghiem thong qua phan tich dao dong ciia dam dudi tac dung cua vat the di dong la mot viec lam can thiet

Tir yeu cau dat ra va muc di'ch nghien cuu, luan van co nhiem vu giai quyet nhung van de sau:

• De nghien cuu tac dung cua vat the' di dong so voi tai trong tinh va tai trong h^ng so di dong len dam cau Ian luoi viet cac phuong trinh dao dong cua dam cau dudi dang tong qual, phuong trinh dao dong cua dam Ciiu dudi lac dung ciia lai irong hang so di dong va phucmg irinh dao dong cua dam Citu dudi tac dung cua vai the di dong

Tinh toan xac dinh he so dong cua dam cau ducri tac dong ciia tai di dong

hop tai trong h^ng so di dong va vat the di dong de co ket luan \6 anh

huong cua v$t th^ di dong den he so dong luc hoc

• Tim hieu v6 cong tac thu nghiem dong vdi cau, cac dai lugng do dac duoc tir viec thu nghiem dong va phuong phap xac dinh he so dong luc hoc dim ciu dang lam viec tir so lieu do dac thuc nghiem

Phuong phap nghien cuu duoc ap dung trong luan van la; phuong phap giai tich d^' giai cac phuong trinh vi phan dao dong cua dam cau va phuong phap thu nghiem dOng d^ xac dinh cac dac trung dong luc hoc cua dam cau phuc vu viec tinh toan he so dong lire hoc tir so lieu do

Noi dung luan van bao gom:

Chuong 1: Phuong trinh dao dong ciia dam dan h6i dudi tac dung cua vat th^

di dong Trong chuong nay trinh bay \6 cac phuong trinh dao dong t6ng quat cua

dim dan hoi, phuong trinh dao dong ciia dam cau dudi tac dung ciia tai trong h^ng

so di dong va phuong trinh dao dong cua dim ciu dudi tac dung cua vat the di dong

Chuong 2: Khai niem va tinh chat cua he so dong luc hoc dam cau Chuong nay dua ra khai niem ciia he so dong luc hoc dam cau, he so dong luc hoc dam cau duoc giai ra tir cac phuong trinh dao dong Tinh chat cQa he so dong cung duoc nghien cuu

Chuong 3: Xac dinh he so dong lire hoc dam cau tir so lieu do dao dong Cong lac thu nghiem dong cau, cac dai luong do duoc can cho viec tinh toan he so dong luc hoc va phuong phap xac dinh he so dong luc hoc dua vao so lieu do duoc irinh bay 6 day

Ket luan chung dua ra nhimg ket qua chinh va phuong hudng nghien cuu phat tri^n ciia luan van

Tinh todn xac dinh he so dong ciia dam edit ducn tdc dong ctia lai di dong

PHUONG TRINH DAO DONG CUA DAM DAN HOI DUOl TAC DUNG

CUA VAT THE DI D O N G

l.l.Phuong trinh tong quat dao dong uon cua dam dan hoi

Noi dung cua phin nay la thiet lap phuong trinh dao dong uon cua dam dan hoi vdi tai trong phan bo bat ky

Gia thiet r^ng:

- dam dong chat vdi do dai /

- mat do khoi luong p

- dien tich tiet dien F

- pF: khoi luong don vi dai

- mo dun dan hoi E

- mo dun quan tinh J

- EJ: do cung chong uon

p(xj}

Hucftig dao dong

Hinh I.}.1 Dao dong uon cua dam

Ta la'y true trung hoa ciia dam khi d trang thai ban dau khi chua cd tac dong

cua tai trong la true x\ true r vuong gdc vdi true x Dam duoc xet vdi dao dong uon

ve phuong :; bo qua dao dong xodn va dao dong doc true Tai mat cat A ta cd:

Do vong u=uf.v./i

Gdc xoay (p=(p(xj)

Tinh todn xdc dinh he so dong ctia dam can du&i tdc dong ciia tai di dong

Luc cat Q=Q(x,t)

Y>i thiet lap cac phuong trinh vi phan dao dong uon, ta xet mot phan to ciia

dzim CO chi6u dai dx nhu mo hinh sau day

Hinh I.J 2 He Ivtc tdc dung len mot phan to cua dam

He lire tac dung len phan to nay bao gom: p(x,t)dx; luc cat Q va

Q-\-~dx; mo menM va M + dx; luc quan tinh R = pF—rdx; luc can

Mat khac mo men uon M tai mat cat bat ky khong chi phu thuoc vao do cong x

ma con phu thuoc vao van tdc bien thien cua nd

M = EJ(x + a^)

dt d'w

1.2.Phirong trinh dao dong cua dam vdi tai h^ng so di dong

Xet tai trong hang so cd gia tri FQ

De mo ta tai hang sd P„di dong tren dam ta sir dung ham Delta-Dirac

S(x)

Theo dinh nghia, ham Delta-Dirac xac dinh bdi he thuc:

0 ox^a S(x-a)

Tinh todn \ac dinh he so dong ciia dam can diccn tdc dong ciia tdi di dong

1.3.Phirong trinh dao dong cua dam vdi vat the di dong

U.l.Phuang trinh dao dong ctia vat vol nen dao dong

m

Y '^ X

/ / / / / / / / / / / / / / Hinh 1.3.} Vat vcri nen di dong

Xet vat the cd khdi lugfng m dat tren mot 16 xo do cutig k va bo giam

cha'n he sd c Ca he dat tren n^n cd dich chuyen thing dung la x„

Lay gdc toa do la nen dung yen ban dau va dua vao cac ky hieu sau day:

Xi - do dai 16 xo khong bi nen (hang sd)

X() - khoang each ban dau tir vat den nen (hang sd)

A XQ= A'/ - Xo - do nen tinh cua 16 xo sau khi vat da dat vao vi tri ban dau

(hang sd)

A- khoang each tu'c thdi (tai thdi diem /) tir vat den nen ban dau (ham cua

thoi gian)

x„ - la djch chuyen tuyel ddi ciia nen (khoang each cua nen tuc thdi den

nen ban dau)

Nhu vay ta cd the' tinh cac dai luong sau day lai thdi diem /

1 Chuyen dich luyel ddi cua vai, ky hieu la x = xft) - x„ ~x(t)

Tinh todn xdc dinh he so dong ciia dam can didri tdc dong ciia tdi di dong

z = x-x^ ^x(t)~xjt) = x(t)-x^-xjt)

3.Tai trang thai ban dau can bang tinh cua vat ta cd:

kAx,=mg (1.3.1)

4.Luc dan hdi sinh ra trong 16 xo bang he sd k nhan vdi chenh lech do dai

ciia 16 xo so vdi trang thai tu do, tu'c la:

P^=k-(x-x^-x,) = k-(x-x^-x,^-AxJ = k(x-xJ-kAx,

^ P^=kz-mg (1.3.2)

5.Luc can nhdt sinh ra trong bo giam chan:

P^=c-(x-xJ^c-(i-xJ =c-z (1.3.3)

Dua vao cac dai luong da duoc tinh toan, ta c6 the thiet lap phuong trinh

chuyen dong ciia vat That vay, tir cac luc tac dung len vat sau day:

quan tinh ciia vat: - mx(t)

trong luc cua vat: -mg

luc dan hoi trong 16 xo: -kz-\- mg

lire can nhdt trong bp giam chan: -cz

Theo nguyen ly D'Alembert ta duoc phuong trinh:

- mx(t) - mg -kz -\- mg -cz -0

hay mx(t)-\-kz-hcz = 0

nhung vi: x = z-\-x^ nen phuong trinh dao dong cudi ciing cd dang:

m'z + cz-¥kz = -mx^ (1.3.4)

L32.Phi(ang trinh dao dong ciia dam ditai tdc dung ciia vat di dong

Trong muc 1.1 ta da thiet lap phuong trinh dao dong udn ciia dam vdi

lire tac dung phan bd bat ky va d muc 1.2 la phuong trinh dao dong ciia dam vdi

tai di dong Nhu vay, de thiet lap phuong trinh dao dong ciia dam dudi tac dung

cua vat the di dong, ta chi can tinh luc tac dung vao dam khi vat the dao dong tai

vi tri CLia vat la A=*^=V/

Trong 1.3.1 coi dam la nen dao dong tuc la:

x^=w(vtj)==xjt) (1.3.5)

Tinh todn xdc dinh he so dong cua dam can di(&i tdc dong ciia tai Ji dong

iuc dan hdi trong 16 xo: P^ = -kz -\- mg

vdi A„ cddang (1.3.5)

Nhu vay, phuong trinh dao dong cua dam dudi tac dung cua vat the di dong cudi

cijng cd dang:

d^w d w d'w dw

EJ(^ + a^)+pF(-— + p—) = m(g~x„-z).5(x-vt) (1.3.7)

dx dx dt df dt

1.3.3 He phirang trinh dao dong ciia dam vai vat thedi dong

Ghep hai phuong trinh (1.3.4) va (1.3.7) vao thanh mot he vdi ky hieu

dx dx dt df dt

mz + cz + kz = -mw (1.3.9)

TrKong hap rieng: Khi vat khong tach ra khoi dam, tu'c la vat Ian tren dam thi

z = 0 ,\c phai phuong trinh (1.3.8) chi edn la: m( g - w).S(x - vt)

luc do, ta chi can giai mot phuong trinh:

d w d w d~ w dw EJ(~^a^)^pF(^ + p^) = m(g-w).5(x-vt) (1.3.10)

dx dx dt df dt

Trudng hop tdng quat, he phuong trinh dao dong ciia dam vdi vat the di dong cd

dang (1.3.8) - (1.3.9) vdi di^u kien bien khdp tai hai dau:

w{0}=w"{0)=0

yX'(l) = H"(l)=0

Tinh todn xdc dinh he so dong cua dam can dudi tdc dong ciia tdi di dong

^

I Y

C

K

Hinh 1.3.3 Mo hinh vat thedi dong tren dam

1.4 Phuong phap Bup nov- Galerkin trong ly thuyet dao dong cac he lien tuc

Xet phuong trinh I6ng quat ciia he lien tuc cd dang:

,a^w(A,/) „5W(A,/)

dt^ dt

trong do A, B,C Xh cac toan tir vi phan tuyen tinh theo toa dp A; wfA./j-chuye'n vi

ciia he tai A va thdi diem /, q(x,t) la mat do phan bd luc ngoai; cac phuong trinh

tren xac dinh trong doan (0, L)

Dieu kien bien cua bai loan cd dang:

BQ va Bi ciing la cac loan tir tuong ung *

Ddi vdi cac ham sd (p(x), i//(x) xac dinh trong doan (0, L) ta xac dinh tich v6 hudng

nhu sau:

{g>,f^)=\(p(x)M'X)dx (1.4.3)

va

\d = {(p,(p)- = j<p(xl<p(x)dx goi la chuan cua ham (p(x)

Vdi khai niem nay, toan lir A goi la ddi xung neu

(A(p, y/) = {(p,Ay/)

va A la xac dinh duong neu {A<p, (p) > 0: (Acp (p) = 0 khi va chi khi (p=()

Phuong phap Bup nov- Galerkin ddi vdi phuong irinh (1.4.1) d6 nghi vice

lim nghiem cua (1.4.1) d dang:

Tinh todn xdc dinh he so dong ciia dam can du&i tdc dong cua tat di dong

u(xj) = f^^^(t)(pj(x), (1.4.4)

trong do |^^ (A), j = 1,2, } la mot he ham thoa man di6u kien bien (1.4.2) va true

giao ddi vdi cac toan tLri4,5, C neu tren Dieu do cd nghIa la;

0

m neu j = k neu j "^ k

Theo tinh chat cua (1.4.5) cua cac ham <p^ (A) ta duoc:

rrijl it) + n^4j (0 + ij^, (0 - ^ (0 (1 -4.6)

j=I,2,3

Nhu vay bang phuong phap Bupnov-Galerkin ta dua phuong trinh (1.4.1) v^

he cac phuong trinh vi phan thudng (1.4.6) Giai cac phuong trinh (1.4.6) ta duoc

<f^(/)va thay vao (1.4.4) ta duoc loi giai cua phuong trinh ban dau

Van de dat ra bay gid chi c6n la tim cac ham (p^ (A) - goi la cac ham co sd

Ndi chung cac ham nay thudng duoc chon la cac dang dao dong rieng ciia he, tu'c

thoa man phuong trinh:

- (o'Aipix) + C(p{x) = 0

hay {C-(o\4)p{x) = ^ (1.4.7)

va cac dieu kien bien Do tinh chat ciia cac loan lir A va C (ddi xung va xac dinh

duong) nen cac dang rieng deu true giao \i\ do do ihoa man dieu kien ciia phucrng

Tinh todn xdc dinh he so dong ciia dam can diari tdc dong ciia tdi di dong

phap Bupn6v- Galerkin Trong trudng hop nay, cac dang rieng phai gia thiet la

ciing true giao ddi vdi ma tran B

Trong thuc te, chudi v6 han trong (1.4.4) la khong thi tim duoc nen ngudi ta

han ch6' 6 mdt sd hChi han cac thanh phan, tiie la ehi tim

u{x,t) = f^^^(t)<p^{x) (1.4.8)

D'\6u nay khong lam giam y nghia ciia Idi giai vi: Mot la ta chi quan tarn den

mot dang dao dong nao dd thi chi can <Pjix) tuong irng la duoc Hai la ndi chung

cac ham q)j (x) se rat nhd khi j Idn va do dd cd the' bd qua cac sd hang bac cao

Phuong phap nay duoc ap dung trong phan tiep theo cua luan van

Tinh todn xdc dinh he so dong ciia dam can du&i tdc dong ciia tdi di dong

CHl/ONG 2 KHAI NI$M VA TINH CHAT CUA H$ SO DONG LlTC CUA DAM CAU

2.1.Khai niem

Dudi tac dung ciia tai trong tinh, dam cau chiu cac luc tap trung va do vong

do tai trong tinh gay ra duoc goi la do vong tinh Tai trong tinh phu thuoc vao vi tri dat tai va do vay do vong tinh cung phu thuoc vao vi tri dat tai Do vong tinh dat ciic dai khi tai trong tinh dat d giua nhip (ddi vdi dam cau gian don khau do / ma

chung ta dang xet thi dd la vi tri 112) Gia six xet do vong tinh ciia dam tai mat cat A

dudi tac dung cua tai trong tinh P,, tac dung d diem XQ la WQ(X) DO vong tinh tai

mat cat X phu thuoc ca vao x la vi tri dat tai

Ddi vdi tai trong dong, do vong dong con phu thuoc vao thdi gian / Gia sir

xet tai trong dong Pacoscot Dudi tac dong ciia tai trong dong, do vong tdng the ciia ddm tai mat cat A la w(x,t) Do vong nay ngoai do vong tinh con cd phan gia tri

lh6m vao do tac dong cua hieu ung dong luc hoe cua tai trong Khi dam chiu lac dung cua tai trong di dong, do vong cue dai dat duoc khong phai khi tai trong d vi tri giiJa nhjp ma sau khi tai da di qua vi tri giira dam

Su anh hudng cua tai trong dong den do vong da dua den viec xac dinh mot thong

sd rat quan trong can duoc xac dinh dd la He sd dong lire hoc He sd dong luc hoc

la dac trung mo ta anh hudng cua lai trong dong len dam cau so vdi lai trong tinh

va duoc tinh bang ty sd giira chuyen vi dong tren chuyen vi tinh Khai niem He sd dong lire hoc duoc dinh nghIa nhu sau:

Dinh nghia: He so dong lire tong quat ciia dam duac xdc dinh bang ty so:

Trong thuc te, ngudi ta thudng xet: // - ma x fi( x t) la hang sd

Tinh todn xdc dinh he so dong ctia dam can du&i tdc dong cua tdi di dong

2.2.Xac djnh h e sd d o n g lire hoc cua d a m c a u vdi tai trong h a n g sd di dong

Di xac dinh h e sd dong luc hoc cua d a m cau vdi tai trong hang sd di

dOng, ta phai giai p h u o n g tfmh:

„ , , 5 ' w d^w ^ r-yd'w ^dw ^ ^

tinh fifxj) hoactinh Ji(x,t) va ju^^^

Mudn giai duoc bai toan nay dau tien giai bai toan tinh

*Trudc tien, ta ap d u n g p h u o n g phap Bupnov-Galerkin dua phuong trinh dao

h a m rieng:

d'w d w d'w dw

EJ(^^a^^)+pF(^+^—) = P,Mx-vt)

dx' dx'dt' ' df dt^ ' ^

vi he p h u o n g trinh vi phan thudng

T a tim nghiem cua p h u o n g trinh dudi d a n g :

EJ z

+ pF

v / y

rjix ^ q^sin—- + a^

Nhu vay ta da dua phuong trinh dao ham rieng \i n phuong trinh vi phan

thudng Ta se sir dung ket qua nay cho viec giai phuong trinh dao ham

rieng \i sau

*Bay gid ta giai bai loan tinh tim w„(x)

EJ^ = P,-S{x-xJ

dx

Ta se tim nghiem gdn dung dua vao phuong phap bien doi 0 tren, ap dung

cho w khong phu thuoc /

Tim nghiem gan diing dua vao phuong phap bien ddi d phan tren

Nghiem cd dang: w(x,v.t)= ^q/t)sin

vdi nghiem nhu tren ta da cd:

IpF i I

sin SK( Xfj +vt)

I

Dan ra he n phuong trinh vi phan thudng:

^.+ a Ejfsir^'

pF vais=Ln

=zi±-V = ," 0 =

vdi s=I ,n

vi vai Iro ciia s trong timg phuong trinh trong he la nhu nhau, ta chi can xet phuong

trinh thu s Ket qua suy ra cho ca he

CA: i-*^.-'- vjo'" o v H f i / '• .-.i j

Tinh todn xdc dinh he so dong cua ddm can dudi tdc dong cua tdi di dong

\'-UlJ04

Xet q,+2h^q^^(j)]q^ =PoSin<f>^ (2.2.10) Day la phuong trinh vi phan tuyen tinh cap 2 Ta phai tim nghiem q^ cua (2.2.10)

dudi dang:

q^ (t)=A sin (t>^ + B cos^^

vdi nghiem tren la cd

q,(t)=Av^ cos(p^-Bv^ sin(t>^

qJt)=-Av] sin^^ - Bv'^ cos<j>^

Thay vao (2.2.10) thuc hien mot sd bien ddi so cap nhdm cac sd hang theo sin^^ va

cos^^ia rut ra:

(oy]-v])P, ((o]-v\y ^4hlv] sin (j>^ (CO] -vj- +4h;v; ' ' - ' " -.cos^,

Viet 7^dudi dang q^=asin(<f>^ -^cp^)'^ ci=yJA' -\-B' ; <p^ = arctg B

2_ \ ' = sin Po2^~T sm sin

s=i ^(col -v] y ^4h;v] i s=i CO] I I

vdi: 2h.=a EjfsTT^'

*Tinh chat cua he sd dong trong trudng hop tai di dong

Tir (2.2.12), de don gian ta xet he sd:

Tinh todn xdc dinh he sddpng ciia ddm cdu du&i tdc dong ciia tdi di dong

sd dong cang Idn Dieu nay phii hop vdi viec han che tdc do phuong tien giao thong chay tren cau

Mat khac cong thu:c (2.2.14) ciing chi ra rang, ta cd the xac dinh duoc he

sd dong neu biet tdc do tai di dong, tan sd rieng va he sd can cua dam cau Day

la ket luan rat bd ich de xac dinh he sd dong ciia cau bang thuc nghiem ma khong can do chuyen vi dong va chuyen vi tinh

He sd dong cung phu thuoc tuan hoan vao thdi gian vdi chu ky:

2Tr 2Td

V

21

TTSV SV

vdi s=l thi T^ =211 v Nhu vay he sd dong d giira dam cau Idn nhat vao luc tai

trong di het chieu dai ciia dam cau

Tinh todn xdc dinh he so dong ctia ddm cdu dudi tdc dong ciia tai di dong

2.3 Xac dinh he sd dong luc hoc cua dam cau khi co vat the di dong

Trong luan van nay ta chi gidi han xet trudng hop rieng trong viec xac dinh he sd dong luc hoc cua dam cau khi cd vat the di dong: Trudng hop vat

khong tach ra khoi dam, tu'c la vat Ian tren dam 'z = 0 Luc nay, d6 xac dinh he

sd dong luc hoc ciia dam cau khi cd vat the di dong, ta chi can giai mot phuong trinh (1.3.10):

va //("A,/^ duoc tinh ra tir w(x,v,t)vkWf)(x.t)

*Trudc tien, ap dung phuong phap Bupnov-Galerkin de dua phuong trinh dao ham rieng

EJ(—^ + a—^-) + pF(—^ + /3—) = m(g-w).S(x-vt)

ox dx dt dt' dt

ve he phuong trinh vi phan thudng

Ta tim nghiem cua phuong trinh nay dudi dang: