K H A O S A T T H i e i K e X A Y DU'NG
Str DUNG THIET BI DIEU CHINH KHOI LlTONG DE HAN CHE CHUYEN VI NGANG CUA KETCAU
ThS. PHUNG NGOC DUNG Tru-ang Dai hpc Kien true Ha Npi
Tbm tit: Ddi v6i cic kit ciu cd chiiu cao I6n chiu tii trgng ddng nhw gid hay dgng dit, di dim bio chuyin vj ngang nim trong giai han cho phip thi kit ciu phii cd mdt kich thuvrc hinh hgc nhit djnh nio dd t^o ra dd cirng du I6n vi si han chi dirge chuyin vj ngang. Vi$c ting kich thuvfc tiit diin din din nhiiu hi luy nhw: chi phi ting len, biin phip thi cdng phirc t9P han,... Ngoii ra, dw6i tic dung cua tii trgng ddng len kit ciu cao ting nhu dng khdi, thip hay nhi nhiiu ting thi dd cin ciia ban thin kit ciu (damping) cd inh hu^ng rit quan trgng t6i irng suit cung nhw chuyin vi ngang cua kit ciu. Bii bio niy di cip din hai vin di chinh: han chi chuyin vi ngang cOa kit ciu cd chiiu cao ldm nhd> sir dung mdt thiit bi diiu
chinh khdi Iwgng (Tuned Mass Damper - TMD); tim quan trgng cua do cin trong kit ciu t6i phin irng cua chiing khi chiu tii trgng ddng.
1. Gidi thieu
a. Kit ciu dwgc nghien ciru
Kit eiu Id mbt ong khbi bing thdp, thdn Id hinh trg tron. Tilt didn ngang hinh vanh khuydn vd khbng doi tren suit ehilu cao. Ong khbi dugc xem nhu mbt conson ngdm tgi mbng (cot ±0.00). V l mdt kit ciu, ong khdi duge mb phdng nhu mbt dim Bernoulli, bd qua anh hudng eua lye dpc, lye c i t Vdt lidu thdp xem nhu Idm vide trpng gidi hgn ddn hoi. Cdc tinh chit cua ong khdi cho trong bang sau:
Chiiu cao (m)
38
Dudng kinh ngoai (m)
3.6
Bang 1. Cac tinh chat cua ong khoi bing thep Dudng kinh
trong (m) 3.585
Chieu day (m) 0.015
Mo dun dan hoi E (GPa)
200
Khoi lugng rieng p (kg/m^)
7800
Mo men quan tinh (m")
2.18491
M6 men chong uon (m^)
1.21384 b. Lire tic dung
Tai trpng tdc dgng Idn k i t eiu chi Id gid. Tai gid bao gom hai thdnh phin tTnh vd dbng. Thdnh phin tTnh duge xem nhu lye tTnh khbng thay doi theo thdi gian. Vide thiit k l hay phdn tich kit c i u dudi tdc dgng eua tai tTnh ndy khd dan gian, ndn khbng d l cdp trong khubn kho bdi bdo. Thanh phin dbng cd t h i duge mb hinh theo mbt s i phuang phdp khdc nhau [4], [5]. Khi tdc dgng vdo k i t eiu mdng vd cao nhu ong khbi, ca ehl gib eubn (vortex shedding) thudng chiim uu t h i [5]. Nd ed t h i duge mb ta bing mbt hdm sine cua thdi gian vd vdn toe [5]. Vdn toe eua gib thay doi trong khoang (0:100) km/h, tO-e la (0:27.8) m/s. Ham cua tai trpng gib ndy dugc t h i hidn nhu sau:
p(t) = p,Dsmi2;rn^t) (1)[4].[5]
Trpng db: po - dp lye dbng lye chinh A =-P,.irCjU' (2) Pair = 1.2kg/m'; C, = 0.5; (p,, - ty trpng khbng khi vd Cd - hd s6); U - vdn tie trung
, . . . SU binh ciia gid; rts - tan so cua gid khi 16c - n^ ~ ~ ^ ( ^ ) [5]; vdi S = 0.4; D - kich thudc ciia v|it ch5n gid.
c. Thiit bi diiu chinh khdi Iwgng Tuned Mass Damper (TMD)
TMD Id mbt thiit bj dugc gin vdo dinh cua ong khbi nhim hgn ehl chuyin vj cua nb khi chju tai trpng dbng. Cde tinh chit eua TMD bao gom: db cirng (k), khii lugng (m) vd db can (damping-c). Trong thyc t l cac gid tri nay se duge xde djnh chinh xdc sau khi tinh toan phan ung eua kit c i u vdi sy cb mdt hodc khbng eb mgt eua TMD. Tuy nhien, trong khubn kho bdi bao ndy, eac gid tri eua khIi lugng vd ty s i can se duge gia sO trudc, sau db dya vdo vide thay dii gid tri dp cu-ng eua TMD, ta eb t h i xdc djnh duge gid tri db cu-ng hgp ly cua TMD sao cho chuyin vj ngang eua TMD cung nhu chuyin vj ngang eua todn bb ong khbi nhd han mbt gid tri cho phdp ndo db (tuang duang vdi vide tinh todn theo trgng thdi gidi hgn 2).
Gid tri khoi lugng m eua TMD thudng nhd han 300kg, vd hd s i can damping (4) khcang 5%-10% [1-3].
d. Cic yeu ciu kiim tra
Dudi tdc dgng eua tai trpng gid, ede phan u-ng cua kit ciu (nhu chuyin vj, vdn tec, gia tic, nbi lye,...) se
T9P chi KHCN Xay di/ng - so 3/2011 23
thay doi theo thdi gian va phg thubc vao van tbe gid.
Cac tieu chi se d u g c kiem tra trong bai bao;
- Xac djnh c h u y i n vj Idn nhdt cua bng khdi vdi s y eb mat hoac khbng cd mat cua TMD;
- Kiem tra k i t cau vdi cac tinh chat khac nhau cua TMD, vdi dieu kien rang bupc la: chuyen vi ngang cua TMD phai nhd han 20em va chuyen vi ngang eua k i t c i u phai nhd han 4cm;
Kiem tra ung s u i t cue dai tai day trong hai trudng hgp: khbng ke d i n dp can cua k i t cau va cd ke d i n dp can cua k i t cUf* Cfng s u i t eye dai tai chan ong khdi phai nho han 5t3lWPa.
2. Cac p h u a n g phap giai v a b u d c giai
Do tai trpng gid t h a y 4 i i theo thdi gian va van tbe dudi dang ham hinh sin, -phan u-ng eua k i t e i u la dpng hpc. De xac djnh phan ung cac diem tai mpi mat cdt cua k i t cau, cd the s u dung mpt so phuang phap sau:
a. Phwang phap 1
Xac djnh true t i i p chuyen vj tu- phuang trinh dao dpng: bng khdi cd the d u g c xem nhu mbt d i m conson Bernoulli nen phuang trinh chuyen dbng cua mpt phan tb vo ciing be (khbng cd can) la:
d'u(x,t) c
P : - r ^ + Ct cx~ El d'-v{x,t)\
= p{x,t) (4) [ 1 - 3 ]
Trong dd: p khoi lugng rieng cua dam; v chuyen vi cua d i m ; x - hudng trge eua d i m ; El - dp cung uon cua dam; t- thdi gian \iip- tai trpng phan bb phg thupc vao thdi gian va tpa dp. Rb rang r i n g , viec giai tryc t i i p phuang trinh dao dbng tren khbng phai la cbng viec d i dang vi dam cd khbi lugng phan bb nen sb bac t y do cua dam la vb han.
b. Phwang phap 2
Xac djnh phan ung cua k i t c i u nhd s u dung phan tich phuang thiPC (Modal Analysis): ehuyen vj eua dam se dugc chia thanh hai thanh phan dpc
\apu(x,t) = (t>(x)Tj(t){5); bdng tich cua ham dang
$(x) (phu thubc vao tpa dp hinh hpc cua vj tri xac dinh chuyen vj) va tung dp cua gia tri chuyen vj dd (phu thupc vao thdi gian r](t)). Phuang trinh dao dbng t y do cua dam dugc chia thanh hai phuang trinh cd hai b i i n so dpc lap (khbng tinh d i n dp can trong hai phuang trinh nay):
. T * cD(x-)
^ O ( x ) = 0 v f l ^ + «^;7(/) = 0(6)[1],[2]
Giai hai phuang trinh (5) vd (6), xae djnh dugc cae tinh c h i t dbng hpc cua d i m (cae ehu ky, t i n sb, dang dao dbng rieng). Nhd vao tinh tryc giao cua cac dang dao dbng, d i m vdi sb bac t u do vb han cb the duge c h u y i n thanh d i m vdi mbt bac t y do trong mbi dang dao dbng. Ob eung, khbi lugng hay db can damping va cac luc tac dgng se duge ehilu len mbi dang dao dbng rieng de xae djnh dp cung chuin hda, khoi lugng chuan hda, eac lye c h u i n hda. Viec giai phuang trinh tung db d l xac djnh cac gia trj cua r](t) trd nen dd dang bdi vi luc nay trong moi dang dao dbng, he t r d thanh mbt bac t y do. K i t hgp t i t ca cac dang dao dbng (hoac ehl k i t hgp mbt vai dang co ban), ta cb the xae djnh phan ung tong cbng cua kit c i u [1 - 3].
c. Phwang phip 3
Phuang phap p h i n t u hu-u han k i t hgp vdi viec phan tfch phuang thuc va phuang phap sb: ong khoi d u g c chia thanh nhieu p h i n t u rdi rac, cac phin tu nay d u g c xem nhu la cac phdn t u d i m Bernoulli. Moi phdn t u ed 6 bac t u do rang bupc bdi cac chuyin vi nut. Nhu vay, tie mbt d i m eb so bac t y do la vb han se d u g c chuyen thanh dam cd sb bac t y do hu-u han.
Dya vao phuang phap phan tich phuang thuc, ta co t h i tinh toan he cb so bac t u do hu-u han trong mpt s6 dang dao dbng c a ban ddu tien cua k i t cau. Doi vdi mbi dang, phuang phap sb (phuang phap Newmark [1-3]) se d u g c su- dung de giai phuang trinh vi phan dao dbng cua h i mpt bac t y do, t u dd se xac djnh d u g c tung dp T]k(t) cua dang dao dbng thu k nao do.
Sau dd, nhd vide to hgp tat ca cac dang (hoac tb hgp mbt sb dang c a ban d i u tien) ta cd the xac dinh dugc phan ij-ng tong cpng cua k i t c i u .
d. Phwang phip 4
Phuang phap phin tu hu-u han kit hap vdi viec dp dung thing phuang phap sb d l giai he nhiiu bac ty do: nhd viec si>a doi giai thudt Newmark ([1 - 3]) ap dung cho he nhiiu bac ty do (dang ma tran), cac phan ung cua he cb the duac xac dinh tryc tiep tu- phuang trinh dao dbng cua he nhiiu bdc tu do:
[ M ] { v } + [ c ] { v } + [ ^ ] { v } = { / ' } (6) Trong db: W - ma trdn khbi luang; C - ma tran
can; K - ma trdn db eu-ng; P vector ngogi lye tac dgng tai nut; v - vector chuyin vj tai nut cua cac phan tu. Chuyin vj ngang cua ong khbi vdi sy cd mat hay
24 Tap chi KHCN Xay dirng - so 3/2011
K H A O S A T THieT Ke X A Y DU'NG
khdng ed mat TMD se duac xac dinh theo giai thudt Hai phuang phap sb 3 va 4 se dugc su dung d l Newmark thay doi [1 - 3]. tinh toan kit cau. Cac budc de giai dugc tong kit e. Trinh tir tinh toin trong bang 2.
Bang 2. Cac phwang phap dwac sw dung Phuang
phap so Cac
budc Sa do cac cong viec c i n thuc hien trong tieng budc Ket ciu khong co TMD
3
4
1
2
3
4
5
6
7 8 9 1-4
5 6
Chia ong khoi thanh hti-u han cac doan, xac djnh cac dac trung cua phan tu-: cac ma tran dp cieng, kh6i luang, can. Noi t i t ca cac ma tran cua phin tu- de co dugc ma tran tong the cua ket ciu: dp cieng K, khoi lugng M. Ap dung diiu kien bien vao ma tran kit ciu ta co duac cac ma tran culi ciing cua kit c i u : ma tran dp cil-ng K l , ma tran khoi luang M1.
Xac djnh cac trj rieng coj, cac dang dao dpng O, cua kit c i u bing giai phuang trinh Kl - (O^MX - 0 . Xac dinh cac tinh chat dpng hpc trong mpt so dang ca ban dau tien lien quan tdi chuyin vj ngang cua 6ng khoi (trong trudng hgp nay, 4 dang diu tien co lien quan tdi chuyin vi ngang).
Xac dinh ma tran can cua he vdi gia su- rang no dugc xac djnh theo ma tran can Rayleight [1] va dugc xac djnh theo K1 va M1: C1=aK1+pM1. Cac he s6 a va p duac xac dinh vai gia sir ring he s6 can trong hai dang dao dpng dau tien la 1%.
Xac djnh ma tran luc tac dung P(U,t) tai cac niit phin tu- tir cac luc phan b l p(u,t).
Chiiu cac ma tran dp cirng, khoi lupng, can cua kit ciu va ma tran luc tac dyng tai niit len bin dang dao dpng ca ban d i u tien, ta se co dupe cac ma tran dp cirng chuan hoa ( O K\0), khoi lugng chuin hoa ( O M I O ) , can chuan hoa ( O C I O ) va ma tran lye da chuin hoa. Cac ma tran nay la cac ma tran dudng cheo.
Doi vai m6i dang, sir dung phuang phap so (phuang phap Newmark cho he mpt bac tu do) de giai phuang trinh chuyin dpng tai moi thai diim t(i) (theo miln thdi gian). Kit qua dat dupe la tung dp ri(t}
tai t i t ca cac nut cua kit ciu.
Xac dinh phan irng ciia kit ciu trong moi dang.
Xac dinh phan irng tong cpng cua kit ciu do tit ca cac dang dao dpng gay ra.
Kiim tra cac dieu kipn yeu ciu va kit luan.
Giong phuang phap 3.
Giai tryc tiip phuang trinh dao dpng nhd sir dyng phuang phap gin diing Newmark cho hp nhiiu bac tu do (dang ma tran) de xac djnh phan irng tong cpng cua kit ciu.
Kiim tra cac diiu kien yeu cau va kit luan.
Ket cau c6 TMD
3
4
Phuang phap nay khong ap dung duac cho ket cau vai TMD vi ma tran can cue ket cllu khong phai la ma tran dudng cheo nhu trudng hap khong co TMD. Do vay, phuang trinh tung do cua kit cau ( [ A / * J 7 ( 0 + [ C * } 7 ( 0 + [•'*^*J7(0= [ - ^ { ^ ' O ] ' ''•°"9 ^° f^*]' ['^1 - ^^c ma trgn duang cheo, con [C] - khong phai la ma tran dudng cheo).
1-4 5
6
7 8
Giong phuang phap 3, 4 trong kit ciu khong co TMD.
Xac djnh cac tinh chit cua TMD: k - dp cirng, c - gia trj can, m - khoi luang.
Xay dyng lai cac tinh chit dpng hpc ciia kit ciu nhd vipc noi cac K1, M1, CI trudc day vdi cac tinh chit dpng hpc k, m, c ciia TMD d l dgt dupe ma tran dp cirng, khoi lupng va dp can md rpng tong the K2, M2 va C2.
Giai tryc tiep cac phuang trinh dao dpng diing giai thuat Newmark ap dung cho hp nhieu bac tu do de xac (^nh dugc phan ung tong cpng cua kit cau vdi TMD.
Kiem fra cac dieu kien yeu cau va kit luan van de.
Tap chi KHCN Xay dimg - so 3/2011 25
3. Ket qua
3.1. Cac tinh chat cua kit cau
Todn bb ong khbi duge chia thdnh 16 phin tu hQ-u hgn rdi rac. Nhu da gia thiit, mbi phin tu gom eb hai nut vdi 6 bae ty do (2 ehuyin vj thing vd 1 chuyin vj xoay tgi moi nut) vd mbi phin tu Id mbt phin tu d i m Bernoulli. Nhu vdy, tong cbng so bdc ty do eua toan bb kit eiu Id 51.
3.I.I.Kit ciu khdng khi chwa gin thiit bj TMD a. Cic ma tran dd cirng va khdi Iwgng
Cde ma trdn db eu-ng, khoi lugng eua cae phin tu trong d i m eb t h i duge xde dinh theo mbt sb phuang phdp nhu ehuyin vj ao... Sau khi xde djnh duge ma trdn nay eua ede phin tu, ede ma trdn dp eu-ng vd khbi lugng teng t h i eua kit eiu se duge xae djnh bdng each nbi cde ma trdn phin tu. Vdi 16 phin tu hQ-u hgn, cac ma trdn khoi lugng vd db eung eua k i t eiu (K vd M) se eb kich thudc 51x51. Do ong khbi duge xem Id ngam d mbng nen ba bae ty do d i u tien cua phin tu thCi- nhit (sdt chan ong khdi) se bing khbng. Sau khi dp dgng diiu kien bien, ma trdn db cu-ng vd khoi lugng eiJa k i t eiu se trd thanh cae ma trdn eb kich thude 48x48 (Kl va Ml).
b. Cic chu ky, tin sd vi cic dang dao dgng ca bin cOa kit ciu
Ddt ehuyin vj ngang cOa bng khbi u(x,t) theo:
u{x,t) = 0(x)7j(t), d l xae djnh cae ehu ky, t i n so vd ede dgng dao dbng d i u tien ta ein xae djnh tri rieng eua he phuang trinh:
[K - coi^M) {0(jc)} = {0} vd l=2nl(o (7) Trong dd: K - ma trdn dp cu-ng cua k i t eiu; d ddy
K la K l ; M - ma trdn khoi lugng eua k i t ciu, d ddy M la M l ; <2> - t i n so dao dbng rieng eua kit ciu; T - chu ky dao dpng rieng eua k i t e i u va <b(x) - dgng dao dbng rieng u-ng vdi t i n s6 co vd ehu ky T. Dd dang nhan thiy ring, d l phuang trinh (8) thda man vdi diiu kidn la k i t e i u eb dao dbng dudi tdc dgng cua lye tu-c Id (<l)(x) #0), vdy hd phai cb nghipm khbng tim thudng, do dd, djnh thue eua ma trdn ( . ^ - o ^ M j phai bdng khbng. Tue la:
det{K-o)^M\ = \K-o)^M\ = Q. Thbng qua ngdn ngu- MATLAB vdi lenh eig, ta ed t h i xae djnh 48 tri rieng a\. S i p xIp coi < coa < coa--- vd TI=27I/Q)I>
T2=27i/co2> T3=2ii/(B3..., ta ed t h i xae djnh ede tin so vd ehu ky dao dbng rieng cua k i t eiu. Tuy nhien, vi chi quan tdm tdi ehuyin vj ngang cQa ong khbi nen ta c i n loai bdt cae gia tri rieng lien quan tdi ehuyin vj xoay vd ehuyen vj dpe trge cua kit c i u trpng cdc dang d i u tien: d ddy 4 dgng dao dpng d i u tien, lien quan tdi ehuyen vj ngang eua Ing khbi. Gid tri ehu ky va tin so eua 4 dgng dao dbng sau khi da logi bd ede dang dao dbng xoay vd dpe trge duge eho trong bang 3.
Sau khi xde djnh dugc 4 t i n so eua 4 dgng dae dbng diu tien, thay cae gid tri t i n s i co-, ndy vdo phuang trinh (8) ta se xdc djnh dugc 4 veeta ridng tuang u-ng vdi cde t i n so db. Hinh 1 eho thiy 4 dgng dao dbng ngang d i u tiert eua conson bng khbi. Cae dgng dao dbng ndy da duge ehuin hba vdi gid trj chuyin vj dinh eua conson (bdc ty do s6 46) bdng dan vj.
Dang 1/2/3/4
Bang 3. Chu ky va tan so dao dgng tw nhien cua bon dang dao dgng ngang dau tien Tan s6(Hz)
0)1 = 15.6604/(02 =98.1428/0)3 =274.8128/o)4 =538.5823
Chu ky (s)
Ti = 0.4/ T2 =0.06/ Ta =0.02/ T4 =0.01 c. Ma trin can
26 T^p chi KHCN Xay di/ng - so 3/2011
K H A O S A T
THieT Ke
XAYDU'NG
35
30
~ y^ LU
O )
« '^1
X c t 1S o
10
5
n
First Four Notmaiisec Transversal Vibration Modes
^'^
.-'' / / / {
\ \ \
\
\
\ , , , , \
Reii - First Mode Blue - Second Mode Green- Third Mode Cyan-Fourth Mode ~
^ /
/ / /
/ • '
/
1 1 1 1
, i g ' ' Tranwersal Displacements at peak • U=50kni/h • method 3
' 1 -0,B -0,6 - 0 4 J ] 2 0 0 2 0 4 0 6 O.E Nomialised Modes
Hinh 1. Son dang dao dong ngang da dwac chuan hoa diu tien
Nhu da de cap trong budc 5 cua bang 2, ma tran can cua kit c i u se dugc xem la ma tran can Rayleight [1], [2]. Ma tran can cua kit c i u C1, se dugc xac djnh tu cae ma tran khoi lugng va dp cung:
C1=aM1+pK1 (8) Cac he sb a va p se dugc xac djnh nhd gia su
rang he sb can ^ (daping ratio) trong hai dang dao dpng ddu tien se bang 1%. Chung ta da cd dugc tan so dao dbng rieng cua hai dang dao dbng ddu tien o)i va 0)2. He phuang trinh dao dbng cua kit cau khi dugc ehilu len tat ca cac dang dao dbng la:
[M'^{th[c'\j {th[K'\j {t)=[p'{u4 (9)
C dang dao dpng thu /, gia trj can chuan hda C,*(generalised damping) trong phuang trinh dao dbng dang thu i:
M;fj,{t)+C'7i,{t)+ K;?j,{t)= P,'(n) dugc xac djnh theo cbng thO-c: C , ' = [ o ^ ^ j c i l o , ] (1210)
Thay(9)vao(12)tacb:
c; = [cp,' | c i i o , ]=«M;+J3K: (i 3)
Tu (11), ta chia ca hai v l cho Mi. roi K:
. ^ _ c; a , M
2co,
ddt <y,
I
,J^ + l^^ (A) ta cdM] Ico^M] 2(0, 2
duge phuang trinh xde djnh tung db dao dpng theo thdi gian trong dang thu- i Id:
Ml 0 0 m > +
Cl + c - c
-c c
"l
u^
• +
.i 2
Red-Fir^t Mode -Blue - Second Wode
Green - Third Mode Cyan- Fourth Mode
m my
-•'•'s'.ii-iiSiF';.- . -i. : liii
m M
10 15 20 25 30 Time(s)
35 40
Hinh 2. Chuyen vi ngang tai dinh ong khoi cua bon dang dao dong diu tien U =50 km/h
riXt)+2^,co,f,,{t)+C'j'il,{l)-- M.
(1511)
Tu (14), vdi gia tri ^ trong hai dang dau tien la 1%, ta cd t h i d§ dang xac dinh hai he sb or va /?:
1 26),
1
2(S)2 CO,
2
0)^
2
O.OI 0.01
0.2701 0.0001757
(16)
Thay (16) vao (9) ta xac djnh dugc ma tran can cua kit cau. Gia tri ^ la he so can damping d dang dao dpng thu i, thudng tu 1-2% cho kit cau thep.
3.1.2. Kit ciu vdi sw cd mat ciia TMD
Khi bng khdi dugc gin them thiit bj TMD tai dinh, cac tinh chat dpng hpc sin cd trong kit cau se bj thay doi. He kit ciu tu 51 bac ty do se trd thanh 52 bdc ty do, va sau khi gdn diiu kien bien, he cu cd 48 bac ty do se trd thanh 49 bac tu do. Bac tu do thu 49 se la chuyin vj ngang cua TMD. Cac tinh chat ca hpc cua he mdi vdi sy cd mat cua TMD se dugc xac djnh vdi quan niem ring he cu (chua gin TMD) la he 1 bac ty do vdi dp cung K1, khdi lugng Ml, dp can CI se noi vdi TMD cd dp cung k, khoi lugng m, dp can c. Vdi quan niem nhu tren, phuang trinh dao dbng cua he mdi cd the vilt thanh:
K\ + k -Ic -k k
U-,
P\{t) 0
(17)
T^p chi KHCN Xay di/ng - so 3/2011 27
K2--
Ml = ; ma trdn can C2 = Cl+c - c
Trong dd: Uf - chuyin vj quy \sdc eua hd cu khbng cb TMD; Uz - ehuyen vj eua TMD; P1(f) - lye quy udc tdc dgng len hd eu.
Tu phuang trinh (17), ta cb t h i d i dang xdc djnh cdc tinh chit ca hpc eho he mdi: ma trdn db eung:
K\\k -k .^ ma trdn khoi lugng:
-k k M\ 0"
0 m
Cin nhin mgnh ring, khi TMD duge g i n vdo dinh eua ong khbi, nb khbng ehl Idm thay doi cde tinh chit ca hpc eua todn kit eiu ma cbn Idm thay doi cde dae tinh ca hpc eua phin til- tren eung (phin tu- 16) g i n vdi TMD. Do vdy, cde gid tri db eu-ng K1+/c vd db can Cl+c trong phuang trinh (5) ciia he mdi se duge tinh todn bing tong cua dp eung k, dp can c eua TMD vdi dp eij-ng Kl va dg can CI vdo bdc ty do so 46 (bdc ty do ndy la tuang u-ng vdi ehuyen vj ngang tgi dinh) thubc phin tu thu 16.
3.2. Xac djnh lye tac dung ien he a. Ddi vdi phwang phip 3
Theo phuang phdp ndy, ede lye phdn bo se duge chuyin thdnh cde lye tdp trung tgi moi nut eOa mbi phin tu, cdc lye tdp trung nay sau db se dugc ehilu len bin dgng dao dbng d i u tien vd duge gpi Id ede lye chuin hba trong moi dang dao dbng. Gid tri ede lye ehuin hba tdc dgng len k i t eiu trong mbi dgng dao dbng dugc xdc djnh theo ede budc sau:
Chuyin cdc tai phdn bo p(t) thdnh ede lye tdp trung Pc"°^^'(t) = p(t)*l tgi mbi nut phin tu (trong db: I - ehilu ddi cua phin tu). Lye tap trung tai dinh Ing khbi se bing V2 gid tri tgi ede nut khdc;
Chiiu eac lye tap trung nay len cde dang dao dbng rieng d l xde djnh lye tde dgng trong ede dgng db. Vi dg d dgng thu i:
NDOFs
P,{t)= ^ c^f'J(x)P^"°'''(t) (18) (NDOFs - tong so bac tu do ciia he; nod - niit).
b. Ddi vdi phwang phip 4
Theo phuang phap nay, cac lye phdn bb do gid se duge ehuyin thanh eac lye tap trung tgi mbi nut cua phin tu. Cde gid tri lye tap trung ndy duge tinh theo Pj"^%) = p(t)*l tgi mbi nut (I - ehilu dai phin tip). Lye tdp trung tgi dinh cung ehl bing 'A so vdi tgi cdc nut khdc. Cin luu y ring, ede lye phdn bo do gib
ndy se chi dygc chuyen thdnh eac lye tdp trung tuang irng vdi ede bae ty do cd ehuyen vj ngang trong hd. Cae lye tdp trung tde dgng theo cdc chuyin vj khae se bdng khbng.
3.3. Cac tham so cua phwang phap Newmark Cdc tham so cua phuang phdp ndy bao gom cdc hd so alpha, delta, budc thdi gian DT, s i lygng budc thdi gian Nstep [1 - 3]. Chung duge lya chpn sao cho phan u-ng eua kit eiu Id on djnh vd dam bao db on djnh eua phuang phdp so Newmark. Do db, ting thdi gian tde dgng cua lye phai Idn han T1/24 vd ty so DT/T1 phai nhd han 0.551 [1], [2]. Gid tri cua cdc tham so ndy duge chpn Id: alpha=0.25, delta=0.5;
DT=0.02s vd Nstep =2500.
3.4. Phan irng cua ket cau khong co TMD 3.4.1.Chuyin vj ngang tai dinh dng khdi a. Ddi v6i phwang phip 3
De xde djnh phan ung tong cbng eua kit eiu, ta ehilu ede ma trdn db eu-ng, khoi lugng vd db can len mbi dang dao dbng d l xae djnh cdc ma trdn db cO-ng, khoi lugng vd ma trdn can da dugc ehuin hba. Sau khi thyc hidn phdp ehilu len cde dang dao dbng, phyang trinh dao dbng se trd thdnh phuang trinh so (9). Cde gid trj [M'], [K] va [C] Id cdc ma trdn dudng cheo. Cae gid trj trdn dudng ehdo cua ede ma trdn ehuin hba ndy dygc xdc djnh eho dgng thO- ; nhu-
sau:M;={0,L[Ml]{0,L;
)nx\'
Rb rdng, hd phuang trinh so (3) Id hd gIm cdc phuang trinh dbe Idp, khbng phu tliubc vdo nhau. Ta cb t h i xde djnh veeta tung dp {ri(t)}i ciia dang thi> i bdi thudt giai so Nevt/mark eho hd 1 bde ty do. Ngdn ngu MATLAB dugc lya chpn de Idp giai thudt ndy.
Tuy nhien can nhin manh ring, tai trpng gib thay doi theo thdi gian vd vdn toe ^ib IJ, do db de xde djnh chuyin vj ngang eua kit eau, d i u tien gid tri vdn tic se duge giu nguydn, sau dd se eho vdn toe gid thay doi (U trd thdnh veeta) d l xde djnh gid tri vdn toe gi6 nguy hiem. Sau khi xdc dinh dugc gid trj {r\(%, ehuyen vj cua k i t cau se dugc xde djnh theo
u(x,t) = <I)(x);7(0 vd to hgp t i t ca ede dgng. Hinh 2 the hi^n gid tri chuyen vj dinh cua ong khbi thay a6i theo thdi gian cua bin dang dao d$ng diu tuang u-ng vdi v$n toe gid U = 50km/h=13.89m/s. Tir hinh 2 ta thay ring, ddng gdp eua dgng dao dOng dau tien vao ehuyin vj ngang eua hd Id ehilm uu thi. Ba dgng dao dbng cdn Igi anh hudng r i t it tdi phan i>ng cua h$.
Hinh 3 cho thiy ede ehuyin vj ngang Idn nhit tgi dinh cua ong khdi vdi gid tri vdn t i c gid U thay doi (tif Okm/h din 100 km/h). Hinh 4 the hidn ede gid trj chuyin vj tgi dinh ong khdi cua b i n dgng dao dbng d i u tidn tuang ung vdi vdn toe U =
28 T$p chi KHCN Xay di/ng - so 3/2011
K H A O S A T
THier Ke
XAYDU'NG
79.92km/h=22.2m/s. Nhu vdy. chuyin vj Idn nhit tgi dinh eua ong khdi Id 0.2515m tuang ung vdi gid tri vHn toe gid U = 22.2m/s (79.92 km/h). Tgi U=22.2m/s;
tin s i etja luc gib tde dung Id - SU , /
a = 2n = 15.49856, trong khi dd, t i n s i eua dgng
dao dbng dau tien Id mi = 15 6604. Cb t h i nhdn ra tgi gid trj vdn tie gid U=22.2m/s, t i n so cua lye tde dgng gin bing t i n s i dao dbng rieng nguy hilm nhit eua hd, va hidn tugng cbng hudng xay ra. Tgi gid tri vdn tie ndy, k i t eiu ndm trong dai cbng hydng.
0.35
E 0.3
Maiimum Displacements at peak with vanable velocities Transversal Displacements at peak - U=79.92knn/h - method 3
10 15 2Q Velocity U (m/s) • dU = 0.2
Hinh 3. Cic chuyen vi ngang l&n nhat tai dinh vai cac gii trj van tdc U khac nhau
b. Ddi vdi phwang phip 4
(^0'^ Transversal Displacements at peak - U=50km/h • method 4
15 20 25 30 Time(s)
Hinh 5. Gii tii chuyen vj cua dng khoi t$i dinh theo thai gian vol U =50 km/h - Phwang phip 4
Theo phuang phdp ndy, cdc phan ipng eua kit eiu se dugc xde djnh tryc tiip tO- phuang trinh dao dbng d dgng ma trdn nhd dp dgng tryc tiip giai thudt Newmark cho h$ nhiiu b$c ty do. Phuang trinh dao dbng cua kit c i u Id:
[Ml]{«} + [Cl]{«} + [^l]{«} = {i'(0} (19)
Trong dd: Cdc gid tri M1, CI, Kl vd P(t) da dugc xdc djnh d trdn.
Hinh 5 t h i hi$n ede chuyin vj eua dinh Ing khdi theo thdi gian vdi gid tri v$n t i e gid U = 50 km/h = 13.88889m/s. Cdc gid tn chuyin vj eua dinh Ing khdi
20 25 30 Time(6)
Hinh 4. Cic chuyin vj ngang tai dinh cua b6n dang dao dOng dau tien vol U = 79.92 km/h
Transversal Displacements at peak - U=79.92km/h - method 4
20 25 30 Time(s)
Hinh 6. Cic chuyen vj ngang cQa dinh 6ng khdi v&i vin t6c gid U =79.92 km/h - Phwang phip 4
vdi U = 50 km/2 trong phuang phdp 4 gin nhu giing phuang phdp 3. Hinh 6 eho thiy cdc ehuyin vj tgi dinh eua Ing khdi theo thdi gian vdi v^n tic gid U = 79.92 km/h=22.2m/s. Tuang ty nhy phyang phdp 3, ehuyin vi Idn nhit tgi dinh eua ong khdi Id 0.2515m tuang ung vdi gid tri vdn toe gid U = 22.2m/s (79.92 km/h). Tgi U=22.2m/s, t i n s i eua lye gib tde dgng Id
— SU
(0^2n = 15.49856, trong khi db, tin s i cua D
dgng dao d|ng d i u tien Id co, = 15.6604. Cd t h i d l ddng nhdn ra ring, tgi gid tr .cin tie gid U=22.2m/s, t i n so cua lye tdc dgng r wing tin s i dao d|ng ridng nguy hilm nhit cua h ^ v d hi$n tygng cOng hudng xay ra.
3.5. Phin Cmg cua ket cau v&i TMD
T9P chi KHCN Xay dyng - so 3/2011 29
Bang 4. Chuyen vj ngang cua TMD va dinh ong khoi vai cac gii trj dg cung. khdi Iwgng khic nhau cOa TMD Cac tinh chat cua TMD
Khoi lugng m
- k g
300
400 280 800
Dp cung k - E l 14412 28825 43237 57649 72061 79267 108090
96082 67257 192160
He so can ^
(%)
5
5 5 5
Q)=(k/m)^'^ tan so cua TMD
6.931 9.802 12.005
13.86 15.499 16.255 18.98 15.499 15.499 15.499
Vdn toe gid khi cpng hudng (m/s)
22.2
22.2 22.2 22.2
Chuyen vj Idn nhat t^i dinh ong khdi khi khdng cd
TMD (m)
0.2515
0.2515 0.2515 0.2515
Chuyin vj Idn nhit t^i dinh Ing khdi khi cd TMD
(m) 0.1865 0.1278 0.0764 0.0384 0.0240 0.0241 0.0551 0.0206 0.0247 0.0174
Chuyin vj Idn nhit cua
TMD (m) 0.0467 0.0829 0.1070 0.1187 0.1490 0.1471 0.1550 0.1156 0.1572 0.0644 Khi k i t c i u cd thdm thiit bj TMD, cde ma trdn db
cii-ng, khbi lugng vd db can bj thay doi. Niu dung phuang phdp 3 d l xdc djnh phan ung eua k i t ciu, khi chuyen cdc ma trdn eua hd thdnh cdc ma trdn trong phuang trinh phuang thii-e, cdc ma trdn khoi lugng vd db eung da dugc chuin hba van Id cdc ma trdn dudng cheo. Tuy nhien, ma trdn can eua hd khbng cbn Id ma trdn dudng cheo nu-a vi sy cd mdt eua hd so can do TMD. Phyang trinh dao dbng cua hd khi phdn tich phuang thue la:
[M];;+ [C]rj+[K]Tj = [P(t)](ri - tung db) (20)
Trong db: ma trdn db eu-ng ehuin hba:
M = { « > } L 4 9 [ ^ 2 L „ { ( D } , , , „ ma trdn dudng chdo;
ma trdn khIi lugng ehuin hba:
[ M * ] = { < D } ^ „ , , [ A / 2 ] , , , , , { O } , , , , , ma trdn dudng cheo; ma trdn db can ehuin hba:
[c*l={^}4'9.x49[C2L,4,{o},,,„ khbng phai la ma trdn dudng chdo; [P(t)] - ma trdn lye tdp trung tgi nOt.
Dd dang nhdn thiy, phuang trinh (20) la he phuang trinh Id thubc lin nhau va chung ta khbng the su dgng phuang phdp Newmark eho he mbt bdc ty do d l giai trong trudng hgp nay. Dp vdy, phuang phap 3 khbng dp dgng duge eho trudng hgp hd eb TMD. Vdy phuang phap 4 Id su lya chpn duy nhit d l giai hd phuang trinh phg thubc. He phuang trinh dao dbng cua hd cb TMD Id:
[M2]{«} + [C2]{«} + [/:2]{uj = {/'(0} (21)
Cde gid trj M2, C2, K2 va P(t) da dugc xde djnh d tren. Phin tinh todn hd khbng eb TMD cho thiy n i u gia tie gib bing 22.2 m/s thi se xay ra hi$n tugng cbng hudng cua hd vd dao dbng tgi dinh cua Ing khbi Id 0.2515m. Gid tri vdn toe gid nay U=22.2m/s, se duge SIP dung d l ddnh gia anh hudng eua TMD len eac phan ung cua h$. Nhu da de cap trong phin 1.3, d l xde djnh ede tinh chit cOa TMD, ta can phai thii- vdi nhiiu gid tri khdc nhau eua khoi lugng, dO eu-ng 30
vd db can cua TMD. Tuy nhien, d l dan gian hba qua trinh tinh toan, d i u tien cde gid trj ve khoi lugng vd db can se gia thiit khbng doi, sau dd db cung eua TMD se dugc xde djnh sao eho ede gid tri ehuyen vj ngang cua TMD vd cua dinh ong khbi ndm trong gidi hgn cho phep. Bang 4 t h i hidn gid tri ehuyin vj ngang cua TMD (bde ty do thu 49) vd ehuyen vj ngang eua dinh Ing khdi vdi eac gid tri khdc nhau cua db cung vd khbi lugng eua TMD. Rb rang ring, vdi m=300kg, khi ede t i n so cua TMD tdng d i n nhung nhd han ede tan so cua hd vd cua tai trpng gid thi chuyen vj ngang cua dinh Ing khdi giam d i n vd chuyin vj ngang cua TMD tdng d i n Idn. Ngugc Igi, khi t i n so cua TMD Idn han t i n s i eua hd vd tai trpng gid thi ehuyin vj ngang cua hd tang d i n len vd chuyen vj ngang eua TMD eung tdng d i n len. Khi tan so cua TMD dgt 15.499, bdng vdi t i n so eua hd va tai trpng gid
w = 2n — = 15.49856, chuyen vi ngang eua dinh ong D
khdi Id nhd nhit (0.024m=2.4em<4em) vd ehuyin vj ngang eua TMD Id 0.1490m=14,9em<2qem. Khi gid trj t i n s i eua TMD dugc giu- nguyen vd bing vdi tin s6 cua hd, vdi cdc gid trj khdc nhau eua khoi lugng (tat nhien la gia tri dp eung eua TMD eOng thay dii theo d l dam bao t i n s i eua TMD Id khbng doi) thi chuyin vj cua hd Id thay doi khbng ddng k l . Nhu vdy, tuy thube vdo ehuyin vj ngang gidi hgn cua ca hd, ta co t h i xde djnh ede tinh chat cua TMD vdi vide han ehl ehuyin vj ngang eua TMD vd ngugc Igi. Khi ta tdng khoi lugng cua TMD tdi 800kg (dO cung 192160 (1/7 dp cung cua k i t ciu)), chuyin vi eua dinh thdp vd cua TMD tuang 0-ng Id 0.0174 vd d.0644m. Diiu ndy c6 nghTa la, khi TMD rat eu-ng, nb se lam eho kit ciu trd nen eu-ng han vd chuyen vj cua kit c i u cQng nhu cua TMD se nho di. Dieu nay hodn todn logic v l mdt vdt ly. Tuy nhien trong thue t l , vide l i p dung mbt thiit bj TMD cb khoi lugng r i t Idn tgi dinh eua thdp Id khbng hgp ly.
_ydi m=300kg, k=fl>^/M = 15.49856^*300 = 72061, thi chuyin vj
dinh Ing khbi Id 0.024m < 0.04m vd eua TMD 1^
0.1490m < 0.2m. Hinh 7 the hidn mli quan h$ giO-a ehuyin vj dinh vdi TMD cd d|c tinh m=300kg, k=<y^/M = 15.49856' * 3 0 0 = 72061, ^ = 5% thay dii theo thdi gian.
T^p chi KHCN Xay dyng - so 3/2011
K H A O S A T T H i e i K e X A Y DU'NG
Displacements at peak with TtvID nnF=3aO; k=w2*m - U = Uresonance
—2
Hinh 7. Chuyen vj ngang cua dinh khi c6 mat TMD - m=300kg, k=ci) m = 15.49856 * 300 = 72061 3.6. Kiem tra irng suat tai chin cua ong
Npi lye tgi bit ky tilt didn nao cua ong khbi diu eb t h i duge tinh todn dd ddng trong moi dgng dao dbng khi da biit chuyin vj nOt tgi cde phin tu.
Chuyin vj dbng hpc cua cdc nOt da dugc xde djnh thbng qua phuang phdp 4 d trdn. Sau khi eb dygc chuyin vj nut, ta cb t h i xde djnh nOi luc tai cde nut theo [ / ] = [^] {«(/)} '•> trong dd: f - ma trdn nbi lye tgi nut; k-ma trdn db eii-ng phin tu-; u - veeta ehuyin vj nut dd xdc djnh. Mb men tgi ddy eua Ing khdi duge
cua nut thu nhit thubc phin tO thu nhit (gid tri hgng thu 3"* trong tich cua [k]{u}). Cb rit nhiiu phin tO bing khbng trong cdc ma trdn ehuyin vj nut vd db eung cua phin tu d i u tien, ndn d l dan gian ta thiy ring, mb men tgi ddy Ing khdi cd t h i dugc xde djnh bing tich eua hd s i db eung xoay kxoay vd gdc xoay 0 tai ddy ong khdi. Gid trj chuyin vj tgi dinh thdp dgt dugc khi cbng hudng vdi U=22.2m/s. Mb men tgi ddy se duge xdc djnh tgi vdn toe gid ndy. Kit qua eho cdc trudng hgp k l din hodc khbng k l din db can kit c i u duge ting hgp trong bang 5 vd 6.
xde djnh theo nbi lye tyang ung vdi chuyen vj xoay
Bang 5. Ngi Iwc vi irng suit tai day ong khoi khong ke den dg cin cua ket ciu kxoay (tu ma trdn dp
CLPng-Nm/rad) 1 2.3x10'"
6 (bdc ty do thu 3"*) (rad)
2 0.0014
Mmax (khi cpng hudng) (N-m)
(1*2) 322x10°
CTmax (khi cpng hudng) (MPa=N/mm^)
a=M/W 26.5
OlirTit
(MPa) 4 50
Afjinhmax
(m) 5 0.2515 Bang 6. Ngi luc va ung suit t$i day ong khoi khong ke den dg can cua ket ciu
kxoay (tu ma trdn db cung-Nm/rad)
1 2.3x10'"
G (bdc ty do thu 3"*) (rad)
2 0.0094
Mmax(khi cpng hudng) (N-m)
(1*2) 2162x10"
ffmax (khi cbng hudng) (MPa=N/mm^)
a=M/W 178.1
CTliiiiit
(MPa) 4 50
Atfnhmax
(m) 5 17 4. Kdt luan
Vdi ede ea hpc tinh chit v i n cd ciJa Ing khdi, ung xu cOa nd phg thubc vdo v$n toe gid. Tgi vdn tie gid U=22.2m/s, gid tri t i n s i cua lye se x i p xl bing gid tri t i n s i ty nhien eua Ing khdi, hidn tugng cbng hudng se xay ra vd ehuyin vj ngang eua dinh ong khbi Id Idn nhit (=25.15em);
- O l hgn ehl ehuyin vj ngang tgi dinh, chung ta eb thi su dgng m|t thiit bj giam chin (thiit bj diiu chinh khIi lugng) TMD. Thiit bj ndy se hip thg ndng lugng gdy ra bdi lye trong h$, tij- db Idm giam chuyen vj ngang cua dinh Ing khbi. V l ban chit, thiit bj TMD se tgo ra m|t lye md tde dbng ngugc Igi dii vdi lye tde dgng ciJa gib vd til- dd Idm giam chuyin vj dinh. Cdc tinh chit cua TMD phg thulc vao chu ky, t i n s i dao d|ng ridng eua h$, phg thu|c vdo ehuyin vj gidi hgn cua dinh kit ciu cQng nhu chuyin vj gidi hgn cua TMD;
Vdi kit c i u d l cdp d tren, u-ng suit tgi ddy vd chuyin vj dinh eua Ing khbi eb k l din db can cOa kit ciu Id chip nhdn duge. Tuy nhien, niu bd qua hd s i can cua kit c i u thi ehuyin vj tgi dinh cQng nhu irng suit d chdn cbng trinh Id vugt qud gidi hgn eho phdp.
D6 can cb y nghTa eye ky quan trpng trpng kit ciu.
TAI LIEU THAM KHAO
1. F^Y. W. CLOUGH and JOSEPH PENZIEN. Dynamics of stojctures. University of Califomia - Berkeley- 1995.
2. ANIL. K. CHOPRA. Dynamics of structures - Theory and Applications to Earthquake Engineering. University of Califomia - Berkeley- 1995
3. Calcul des stmctures JS effets dynamiques et sismiques - Lecture notes - Universiti de Liige, 2010.
4. JOHN D. HOLMES. Wind Loading of Stmctures. 2001.
Ngiynh$n bii: 18/8/2011.
T9P Chi KHCN Xay dyng - so 3/2011 31