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MUC
LUC
Trang
Muc
luc
L6i cam on
M6
DAU
1
Chuong 1 PHUONG TRINH
DAO DONG CUA DAM DAN HOI
3
DU6l
TAC
DUNG
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
h4ng so
di dong
2.3.
Xac
djnh
he
so
dong
lire
hoc
ciia
dam cau khi co vat the di 22
dong
Chuong 3 XAC DINH HE SO DONG
LUC
HOC
TlTSO
LIEU 38
DO DAO DONG
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
3.1.3 Cac yeu
cdu
doi v6i thu nghiem dong 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

Ldl
CAM ON
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.
MO
DAU
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
• Tinh
toan va nghien cuu

tinh
chat cua he so dong luc hoc trong trudng
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

CHUONG1
PHUONG TRINH DAO DONG CUA DAM DAN
HOI DUOl
TAC DUNG
CUA VAT THE DI
DONG
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
Mo men uon
M-M(x,t)
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.
p(x,t)
K4
^^
A
M
+
dx
dx
R
^
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
dx dx
'
df
dw
phu thuoc vao van tdc R =
(3(pF
—
)dx,
(3
= const.
tacd:
Theo nguyen ly
D'Alembert,
tir
di^u
kien can bang cac luc theo phuong z
dO
d'
w dw

•Q-^^dx + Q +
p(x,t)dx^pF—rdx-
p(pp
—
)dx
=
0
dx df dt
hay
-f-
+
pF(—^
^P'^) = P(''>0
dx df dt
(1-1-1)
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
dx
:>
M =
EJ( —r +

a——
)
(1.1.2)
GX
exet
Tinh todn xdc dinh he
so
dong
ciia
dam can dudi tdc dong
ciia tdi
di dong
Tir
di^u kien can
bang cac mo men, bo qua cac dai luong bac cao cua dx ta cd:
dM
Q =
dx
Thay
M
tir (1.1.2) vao bieu
thiic
tren ta duoc:
^
c.,.5V
d'w
Q
=
EJ(— ^-a—T—
)

^
dx'
dx'dt
Thay Q vao
(1.1.1)
ta nhan duoc phuong trinh:
dx
dx dt dt dt
(1.1.3)
Day la phuong trinh mo ta dao dong uon cua dam dan h6i vdi tai trong phan bd
bat ky.
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)
coc^x = a
cu the:

b(x -a)
= lim bjx
- a)
<=>
|x-a|
<e
0
o
\x-a\
> €
i-*0
I
bjx-a)
=
e>0
va cd
tinh
chat:
CO
l8(x-a)dx=I
—00
\f(x)8(x-a)dx=f(a)
-co
Cd
the'
xem tai trong tap trung
P(,
di dong lac dong len dam nhu la mot luc phan
bd/;(.v.;j
cd cudng do

P„
tren khoang
[vf-i:,
M+c]inc
la tai vi
irf lure ihdi
cua tai
trong di dong tai thoi
diem
/.
Sir
dung ham Delta-Dirac ta c6
p(xj)-
F„d(x-\t}.
Tinh
todn
\ac
dinh he so dong
ciia
dam can
diccn
tdc dong
ciia
tdi di dong
That
vay,
theo tinh
ch^^t
ham Delta-Dirac:
P

= p(x,t)-2€ =
1
=
P.5(x-vt)-2£
= limP,SJx-vt)-2e^PJim^-2E =
P^
c-*0
£-*0
2£
Tir do, phuong trinh dao dong cua dam vdi tai trong hang sd di dong nhu sau:
d^w d^w d'w
dw
EJ(^^<^^)^f>^(^^P^) = P(^'0-PoS(x-vt)
(1.2.1)
dx
dx dt df dt
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
2.Chuy^n
dich tuong ddi
ciia
vat so vdi
n^n,
ky hieu la
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
Khi do, luc tac dung len
n^n
bao
gdm:
iuc dan hdi trong
16
xo:
P^
=
-kz
-\-
mg
luc can nhdt:
P^
=
-cz
vay
tong luc tac dung len dam tai vi tri x=vt la:
P
=
P-\-P=
-kz
-\-mg-cz
Ne'u
tinh

da'n
phuong trinh (1.3.4) ta cd:
P
= mg-kz-cz =
mg -
m'x^
-
m'z
= m(g-x„-z)
(1.3.6)
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

^ ,
d'w(vt,t)

X (t)
=
=
wta
se duoc
df-
m
d w d w d'w dw
EJ(
+a-—) +
pF(^
+
l3—)
= m(g-w-z).5(x-vt)
(1.3.8)
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
^
m
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,/)
+
B
+
Cu{xj)^q(xj)
(1.4.1)
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:

BO{U(XJ)1__^=BM^^OI^,=0
(1.4.2)
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
10
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
{B(PJ,<P,) =
{C(Pj,(p,) =
m-\\
=
n
.
neu j
~k
0
neuj ^
k
\(p\
= /,. neu j
=
k
\\~
f

Wc
^
0
neuj ^
k
(1.4.5)
khi do thay (1.4.4) vao (1.4.1)
tadupc:
J
nhan hai ve phuong trinh tren vdi
(Pi^
(A)
ta duoc:
X[(^^,,^J-^'(o+K,,^J-^,(/)+(c^,.^J-^,(o]=^,(0
J
L
^k(0= j(p^(x).qixj)dx
0
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
11
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
12
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:
w(x,t)
. ^
>V/A)
Neu hieu diin do vong dong
w(x,t)=WQ(x)+ w(x,t)
tuc la
w(x,t)
la phan do vong
them vao do tdc dong cua hieu
i(ng

dong
life
hoc cua tdi trong thi:
fi(x,t) =
}
+ fi(x,t)
vai
fi(x,t) = ——
Id he so dong
lire
thong thudng cua dam cau vd noi chung la
VV/A>
ham cua cd x va t.
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
13
2.2.Xac djnh
he
sd dong
lire
hoc cua dam cau vdi tai trong hang sd di dong
Di xac dinh
he
sd dong luc hoc cua dam cau vdi tai trong hang sd di

dOng,
ta phai giai phuong
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
d'w
EJ-—-
=
P,.S(x-xJ
tinh
wjx)
dx
(2.2.2)
roi giai tiep:
d w d w d'w dw
^•^^^^^"hw,^^P^(^^PT^
=
P''-5(x-x„-vt)
(2.2.3)
OX
dx dt df dt

de tinh
VV^A,V,/^
Chiing
minh
w/x)
=
lim max w(x,v,t)
\-*0
I
Va cudi cung
w(x,v,t)-w(x,0,t)
w(x.Oj)
*Trudc
tien,
ta
ap dung phuong phap Bupnov-Galerkin dua phuong trinh dao
ham rieng:
d'w
d w d'w dw
EJ(^^a^^)+pF(^+^—) =
P,Mx-vt)
dx'
dx'dt'
' df
dt^
' ^
vi
he
phuong trinh
vi

phan thudng.
Ta tim nghiem cua phuong trinh dudi dang:
rTDC
w(x,t)
=
Y,qJt)sin
I
(2.2.4)
vdi nghiem nhu tren
ta
cd:
^/-Tr^
dj^_f
dx'
r=\
I
rTDC
q^
sin
dw
Yt
I'/.
rirc
sin
•-/
Tinh todn xdc dinh he so dong
ciia
ddm can dudi tdc dong
ciia
tai

di
dong
14
d'w
^ rTDC
—-
=y
a
sin
dt'
t',
I
d^w
_^
dx'dt
~
^
Thay vao
(2.2.1)
ta cd:
rwc
q.sin^
EJ
z
+ pF
v/y
.
rjix ^
q^sin—-
+

a^
V ' y
q^
sin
rjDc
2.q,sm— \-(32^q^sin~—
r=l / r=/ /
p,.(^rx-vr;
-^
.
rTDC
r=l
I
EJ
I)
'
rir^
q^
+aEJ
q,+
pF.q^
+
&pF.q^
V '
y
« z
A-;D:
sin
—
/

^.+
£7
a.—
'ri:^
V
'
J
+ 0
qr +
EJfrir^
pF
V
' J
^r
pF
P,rS(X-Vt)
b(x - vt)
STDC
Nhan hai ve vdi
sin rdi
lay tich phan tir 0 den / va de y rang:
rTDC
.
STDC
\sin .sin
—
i
I I
dx
0 neu r

^
s
I
neu r = s
Tacd:
rTDC .
STDC
Sin sin
—
0
r=l
/ /
Qr^
a
pF
I
/
+ y9
9.+
EJ(rn^
pF
V ' /
<l.
dx
O
q^
+
0_
a
\sin df

A-v^^A
r\
I
+
P
Ej(s7t^
pF
V
/ J
EJ
(],
+ —
pF
ds
2^
ipFi
r
.
STDC
^ . ^
,
\sin d(x-vt
)ax
n *
vdi s = I,n
Xet
tich phan:
jsin
S(x-vt)d.
STD:

^,
r
>/i.v
,, ,
,.
f I S7VC
sin (> (X
- vt
kix
= urn
— sin
dx
I ,
I
STDC
—
lim
—
cos
Tinh todn xdc dinh he so dong
ciia
ddm can
dim
tdc dong
ciia tdi
dt dong
15
1
,. } { STt(vt->r£)
STt(vt-~£)\

= lim
—
{cos
-
cos >
=
STT '-^
2e[ / /J
]
J.
I .
STn^t
.
STtE ^ ^ ,
7
,
/
STTVt STt£
= lim
—
sin . sin
(-2) = —
lim
—
sin . sin
STT ^ ^ 2s I I sTt '^
2e
I
I
. STtE

sin
.
STtVt ,
/
=
sm lim
— =
I '^^^ SK£
.
STtVt
=
sm
I
Dan ra he phuong trmh vi phan
^>
EjfsTT^
a
pF
I
/
+
>^
^s +
EjfsTT^
kyhitu:2h
=a—
'
pF
EJfsir^'
I

pF
2/1
. STTVt
Sin
V '
y
EJ(STT^'
IpF
V '
y
pF
Kl
)
:Po
2P.
-•^
STDC
IpF ' I
(2.2.5)
Tacd
he:
q^
+2h^q^
+^5^5
=PfjSinvJ
vdi s=I
,n
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
/.
rTDC
Nghiem can
tim
cd dang w(x)
=
^q/r)sin
r=I
vdi nghiem nhu tren ta cd:
/
d'w
dx
7
=

Z
^rx^'
v/
y
rTDC
q^
sin
Thay vao (2.2.2) ta duoc:
EJT
^rx^
q^
'^'"
—
~
K'
^{-^
~
-^.)
\ ' y
Tinh todn xdc dinh he so dong
ciia
ddm cau
du&i
tdc dong
ciia tdi
di dong
16
STtX
Nhan hai ve vdi
sin

va tich phan
tir
0 den / ta nhan duoc:
EJ
^
sir^
2'c
V '
y
^s
=-\PrMx~xJsin—-dx
i
0
I
Nhu da biet, tich phan d ve phai cd gia tri bang:
fo
STDC
.
STDC
jPo6(x-xJsin—~dx
=
P,sin—^
0
II
^
EJ
-J
q^=P,sin—^
<is
2P^

sin -—^
I
lEJ
v/y
EJ
Neu ky hieu
co^
=
pF
'STT^
I
J
(2.2.6)
— 2P
va
/I
= —- va nhan ca tir va
miu
cua
/pF
(2.2.6) vdi pF thi (2.2.6) se la:
^•sm'-^
CD'
I
Do
vay
W(,(x) =
/^,2.~T'^''^~7^'^'"^~
s=l
0)^

I I
(2.2.7)
Ket qua nay se
diing
de xac dinh he sd dong luc hoc sau khi tinh duoc
w(x,v,t)
*Tinh
wfx,v,/^tir:
^,.d'w
d''w
,
r-/^'^
o^^'
i r>
c/
^A av or
df dt
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:
5St
7
=
1
^VT^^
dx r=l 'v /
rTDC
q.

^"»'"
Tinh todn xdc dinh he so dong cua dam cau
du&i
tdc dong cua
tdi di
dong
17
dw
^ ,
.
rTDC
— =
/ Qr Sin
dt
tt^'
I
d^w
^ rTDC
dt'
t!^'
I
d'w
^y
dx'dt ~ h
n
A^^>
/
rTDC
q^sin
V

'
y
/
Thay vao phuong trinh tren rdi bien doi ta cd:
2,
sin
r=l /
^.+
EjfrTT^
a
pF
V
'
J
+
P
Ejfrrt^
pF
V
'
J
=
-^S(x-x„-vt)
pF
STDC
Nhan hai
ve
vdi
sin
roi

la'y tich
phan hai ve
tir
0 den / ta nhan duoc:
^,+
a
Ejfsn^
V
/
J
+
P
^s +
EJ_
pF
' STT
^
vdi
5=/,n
Tich phan d ve phai cd gia tri bang:
\5(x
-Xg-vt) sin
dx
=
P^
n t
\
'
y
2P^

V
.
STDC
„^
^,
q^
=——\sin-—S(x-x^-vt)d}
IpF i I
sin
SK(
Xfj +vt)
I
Dan ra he n phuong trinh vi phan thudng:
^.+
a
Ejfsir^'
pF
vais=Ln
K t
y
+
^
EJ
pF
'
sir^
\ '
y
2P,
.

sTr(x,+vt)
q^
=—-sin
IpF I
(2.2.8)
ky hieu:
2h^
=
a
EJ_
pF
'sic^'
I
+
ft-
cj;
Ej(sTc\
^ 2P
\ t /
pF
V i
y
p
=zi±-
IpF
STDC
,
SK(X^
^-Vt)
V

= ,"
0
=
Tacd
he:
cj^
-^^h^q^
'^^l^s =PoSin<f)
(2.2.9)
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'"
ov H
fi/.
'• i
j
Tinh todn xdc dinh he so dong cua ddm can dudi tdc dong cua
tdi
di dong
\'-UlJ04
18
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:
l(a?:-v-:)A-2v^Bh^=P^
\(co;-v-)B-h2vJh^
=0

Giai he
(2.2.11)
de tinh he
sdy\,
B
B
2vh ,
.' \A
CO'
-v;
(2.2.11)
A =
P,
A =
B
CI.
(co;
-v-y +4h;vi
2hvA
(co] -v]
)'
+4h-v]
(oy]-v])P,
((o]-v\y ^4hlv]
sin
(j>^
''-'"
cos^,
(CO]
-vj-

+4h;v;
Viet
7^dudi
dang
q^=asin(<f>^
-^cp^)'^
ci=yJA'
-\-B'
;
<p^
= arctg
B
<=>
Qs
P„sin(<t>^
-\-(pJ
^J(co;
-V-
)'
-^4h;v~
2h
V
voi ^
=
-arctg
CO'
-
v;
Tinh todn xdc dinh he
sddpng

cua ddm can
du&i
tdc dong
ciia tdi
di dong
19
'^x,v,t) =
Y,qJt)sin
S=I
STDC
_^
P,sin((l>^+cpJ
.
STDC
—
/ I sin
*Tim
lim
w(x,v,t)
lim
w
v-*0
(x,vJ) =
lim±-,^^Mi,l£j=sin^
'^'
^->^(co:wy+4h:v:
i
Ta
cd:
lim

v, =
lim =
0
v-frO
'-^0 I
V->0
V-¥0
lim
cp
=
lim
2hv^
arctg
—,
co:
-V
0
" / S
TDC
V TTX
lim w(x,v,t)=
P^Y^^sin
-sin
= vv/A^da
xac dinh trong (2.2.7)
s=i
co^
I I
v-*0
Nhu vay, bieu thire

do
vong
tinh
cd
the'
xac
dinh duoc
tir
bieu
thirc
do
vong
dong bang each
cho van tdc cua tai
trong
v -> 0.
w.
. .
j^^
I .
STDC,.
.
STDC
,.
ix)
~
r.y
—-
sin sin
=

lim
w(x,
v,
/)
'tto)'
I I -^
*Tiep
theo, ta phai xac dinh
h^
sd dong
lire
hoc cua dam cau vdi tai trong hang
sd di dong.
w(x,vj)-w(x,0,t)
M(xJ) =
Mx.Oj)
^
P^sin(6-\-(p)
.
STDC —^
J .
STDCQ
. STDC
2_
. \ '
=
sin Po2^~T sm sin
s=i ^(col -v] y ^4h;v] i
s=i CO] I I
Poll

\
1 .
STDCn
.
STDC
1 CO
-sin sin
—
'
/ /
s = l
sin(
STDC^
^^^^^=^^= Sin
yj(co] -v] y +4lvv]
CO'
STDC
sin
^
I .
STDC,.
.
STDC
> —-sin sin
—
ttco I I
(2.2.12
Tinh todn xdc dinh he
sddpng ciia
dam cdu dudi tdc dong

ciia
tdi di dong
20
vdi:
2h.=a
EjfsTT^'
pF
2 _EJ_f
I
^P;
\ '
y
\4
STt
"
IpF'
STtV
STt(Xf^
-k-Vt)
1
V =
*Tinh chat cua he sd dong trong trudng hop tai di dong
Tir (2.2.12), de don gian ta xet he sd:
^(x.t)
=
l-h^(xj)
Y,^(o),,h^,v^
)sin(vj +
cp^
)sin

^ I
. STDC,. . STDC
>
—-
Sin sin ——
1^1
0)-
I I
STDC
voi
cp^=(p^
+
STDC,
I
H(co^,h^,vJ
=
4{oy:-v:y -4hivi
va dimg lai d dang dao dong
thu"
nhat, tu'c
s=L
Khi dd:
1
sin(Vjt -^-cp,)
l
+ fu(x,t)
=
\i-f.y
4h]7]
sin

co\
TO,
I
hay
i^K
V-r,)
4h]y]
Ta.
sin
0)\
(2.2.13)
vdi
Yt'
= —T - d<*i lugng
nay thuc te
ludn
nhd hon 1
CO,
Theo bieu thirc (2.2.13) ta thay, he
sddong
phu thuoc vao vi tri tren dam.
tu"c
la d
nhtrng
vi tri khac nhau, he sd dong khac nhau. He sd dong cang
ktn
d
nhimg vi tri cang gan vdi gdi va he sd dong nhd nhat d giira dam va cd
gia tri
lii:

Tinh
todn
xdc
dinh
he sddpng
ciia ddm
cdu
du&i
tdc
dong ciia tdi di dong
21
^^
=^
+
>",
'y
\^-f^
4^j]
(2.2.14)
(o\
Trudc khi xem xet su phu thuoc cua he sd dong vao cac dac trung khac
ciia
dam, ta
ki^'m
tra dieu kien:
7
7+27
=
I
max

>1
Vr])
4h\Y]
CO.
That vay,
di6u
kien cudi tuong duong vdi:
[l-f,y
+^^<}h^yy:-2y]+^^<0
CO,
1-2
yj<2(I-2—^):
v6\
/r
nhd
diau
kien nay
luon
thoa man vi
y;<7.
Nhu
co:
vay
/J^>I
hay
//,
r
7\
u
>o

Cong thu'c (2.2.14) cho thay: He sd dong phu thuoc vao tdc do di dong
cua tai, tan sd rieng va he sd can. He sd dong cang tang neu tdc do cua tai di
dong tang va cang gan vdi tan sd rieng. Nhu vay tai di dong cang nhanh thi he
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
22
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):
d'^w d^w
d'w dw
EJ{—T + cc~j-)+QF{ ^^P—) = m(g-w).5(x-vt)

ox ox dt
Of
dt
Sau khi giai (1.3.10) thu duoc ket qua
w(x,v,t),
chov
->•
0 ta cd:
WQ(x,t) =
lim max w(x,v,t)
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:
rTDC
M^J)

=
Y.^r(Osin
r=l
7
De don gian cho cong viec tinh toan ta lay gidi han n=l va
bi^u
thu'c gan dung
cd dang:
TDC
w(x,t) = q(t
)sin
I
(2.3.3)
vdi nshiem nhu tren ta c6
d'w
dx'
'TT^
I
TDC
qsin
V
' /
dw
TDC
— =
qsin
—
dt
^ I
d'w TDC

—- = q
sin
—
df
^ I
a-
vv
dx'dt
1/
TDC
qsin
Tinh todn xdc dinh he so dong cua ddm cdu dudi tdc dong
ciia
tdi di dong