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Phan tich tTnh va dao dong t y do cua tam Mindlin SLH dung phu'ang phap phan tip chuyen dong
Ngay nhan bai: 20/9/2014 Ngay sij^ bai: 5/10/2014 Ngay chap nhan dang: 10/10/2014
T O M TAT:
Trong nghien ciiu nay, mpt phuong phap vifa mdi dupc phat trien gSn day do chinh la phUPng phap phan tilf chuyin dong se dUpc ap dung de phan tich Ung xii tinh va dao d^ng tu do c i a tam day Mindlin phat trien dpa vao ly thuyet bien dang cat cua tSm Mindlin. Theo phuong phap nay, tam se duoc chia nho thanh nhCng "phan tCf chuyen dpng" NhiJIng phan tfi nay khong phai chuyen dong that so vdi tam dtfng yen ma la chuyen d^ng gia tudng cung vdi lUc di chuyen tren ket cau tam. Do do, phUPng phap nay se tranh dUpc viec cap nhit vec tp tai trpng tUdng Ung vdi mo hinh tam. Tat ca cac phuong trinh chuyen dong cung nhu cac ma tran ket cua phan ti5 tam dupc xay dpng tren mgt he true toa dp chuyen dpng vdi van tde khong doi. Cac vi du so lien quan den phan tich tam chiu tai trpng tinh va dao dpng tU do dupc triln khai. Cac ket qua sd dUpc so s i n h vdi cdc phUdng phap so khac de nhara the hien dp chinh xac cao va tinh kha thi cua phuang phap dupc At xuat.
Tif khoa: PhUdng phap phan tijf chuyen dong, tam Mindlin, dao dpng t p do, phan tich tinh.
ABSTRACT
In this paper, a recently developed method, namely moving element method (MEM) is employed to analyze the static a n d free vibration response of the Mindlin plate based on the Mindlin shear deformation theory. By using the method, a plate is ciiscretised into "moving element". These moving element are not physical elements fixed to the plate but are conceptual elements that "flow" with the moving load through the plate.
Thus, the proposed method elemmates the need of keeping track the location of moving load with respect to the plate. The governing equations of motion as well as structural matrices of moving element are formulated in a relative coordinate system travelling at a constant speed. Numerical examples related to static and free vibration analysis of the plate are carried out. Numerical results are compared with the other numerical results in order to present high accuracy and applicability of the proposed method.
Keywords: Moving element method, Mindlin plate, free vibration, static analysis.
KS. Vo H o a n g Nhi - Hpc vi6n Cao hoc
Khoa Ky Thu^t Xay Dpng, TrUdng Dai Hoc Bach Khoa - Dai Hoc Quoc Gia TP.HCM TS. LtJong \ ^ n Hai
Khoa Ky Thu^t Xay Dung, Tnl6ng D?i Hpc Bach Khoa - Dai Hoc Quoc Gia TRHCM Email: [email protected]. vn
Di dpng: 0944 282 090 NCS. Tran Minh Thi
Khoa Ky Thuat Xay DUng va Moi Tnfdng, Dai Hpc Qu6c gia Singapore
Vd Hoang Nhi, Lu'dng Van Hai, Tran Minh Thi
1. Gidi thieu
DUa veio ntiUng phit tnen cua khoa hpc va c6ng nghe, ket cSu tam da dUpc sir dung rong rai trong cuoc s6ng 6 rat nhi^u linh vUc nhu hang khong, giao thong, dan dung ...
Trong cac Unh vuc nSy ket cau tam thudng duoc m6 ph6ng tUa tren nen dan h6i chju t3i trpng dpng. Kim (2004) [1] da thUc hien phan tfch phSn Ung dong cda tam tren n^n 6in nhcrt Winkler chiu tai trong dpng sU dung phUOng phap Fourier transform theo thdi gian va khong gian, dong thcJi cung da kheio sat mUc do go ge cua mat dUdng den dao dpng ciia vat chuyen dong. Kim va Rosset (1988) [2] da nghien cUu den trang thai dng xUcCia mpt tam vo han tren nen dan hoi chiu t^i trpng chuyen dong dieu hoa khong doi. Fryba (1999) [3]
nghien cUu dao dong rieng ciia mot tcim vo han hoac hflu han cho nhieu dieu kien bien khac nhau. Huang vSThambiratnam (2001) [4] da trinh bay phuong phap bang each sU dung phuang phap Ai\ hUu han de xU ly Ung xU cCia ket cau tam co dinh tren nen dan h6i chiu t5i trpng dpng vdi van toe bien doi. Javad (2013) [5] sCf dung phuong phap Eigenfunction Expansion Method (EEfW) de nghien cUu sUon dmh va Ung xCf dong Ipc hpc cua tam Mindlin chiu cCia t^i trong di dong.
Trong thpc te, phuong phap phan tU hflu han FEM da dUoc sU dung rong rai de giSi quyet nhieu bai toan phUc tap. A K. Gupta va cpng sU (2009) [6] da sU dung phuong phap phan tif hflu han de phan tich dao dpng t u do cua tam ngam tUa tren nen dan nhdt vdi su bien doi chieu day tam theo ham mu bac 2.
Trong phuong phap phan tU hflu han, tat ca cac ma tran ket cau se dUoc thUc hien tren mpt he true tpa dp co dinh. Khi tai trong di chuyen tU phan tU nay sang phan tU khac thi vec to tai trong phSi dupc cap nhat sau moi budc thdi gian. Dong thdi, tai trong co the tien tdi bien va vucft ra khoi bien bai to^n. Tat ci cac nhupc diem tren duoc minh hoa 6 Hinh 1.
De khSc phuc nhCfng nhupc dilm cua phuong phSp phan tfl hCfu han, Koh vi cdng sU(2003) [7] da de xuat sU dung phUOngphap
phan tU chuyen dpng (MEM) trong vi^c khSo sat Ung xU ddng cua tciu - ray PhUcmg phap nay da g i i i quyet nhflng kho khan cOa phUOng phap FEM.Tran va cpng sU (2014) 18] da ap dung MEM d ^ khdo sdt den Ung x U d d n g ciia tau cao tdc- ray khi thu tang toe hoac gidm tde Koh va cdng su (2005) [91 da khdo sdt den dao ddng cilia nen ban khdng gian ddn hoi bSng phuong phdp MEM. Xu va edng sp (2009) [10] da dp dung trong viee phdn tich dao ddng ngau nhien cua tam Kirchhoff tr^n nen Kelvin chju tdi trpng di ddng sU dung phSn tCf t f l giac. Phuong trinh chuyen dpng ciia k^t cau tam da dUpc thiet lap tr&n mdt h€ true toa d d tUOng ddi c h u y i n ddng cUng vdn tde vdi lUc d i c h u y i n . l/u diem ciia MEM duoc minh hoa d Hinh 2.
Hinh 1 Mfl lilnh phuang phap FEM truyen thong
ate
Hinh 2: Mfl hinh phuVng phap MEM Trong bai bao ndy, phuang phdp MEM duoc sCr dung de phdn tfch t i n h vd dao ddng t u do cCia tam tUa tren nin ddn hdl Winkler chiu tdi trpng ddng. Cdc phUdng trinh chuyen ddng cOa tdm, cdc ma tran ket eau eung dUOc thiet ldp trong he toa dp tuang ddi chuyen ddng eCing van tde ciia lUc Md hinh tam dat tren n4n dan hdi Winkler va tam khdng dat tr^n nen Winkler dupc khdo sat de x^t dnh hudng cua nen Winkler. Odng thdi, edc thdng sd nhU dieu kien bien, chi^u ddy, ti so chieu dai/chieu rdng dnh hudng den tan sd dao ddng t u nhien cua ket cau tam cung dupe phdn tfch. Cae ket qud t h u dupc se Id tdi lieu hflu ieh cho viec nghien cOfu vd t h i l t k l cdc k i t cau t i m chju tdi trpng ddng trong thUc ti^n.
2. Ccfsdly thuyet
Xet tdm Mindlin chju b i l n dang udn bdi cdc Ipc vudng gdc vdi mat phdng tam, he true tpa dd Oxyz duoc chon sao cho mat p h I n g toa do Oxy trung vdi mat trung binh Qcz R^ vd true z vudng gdc vdi mdt p h I n g t a m . Tam dUa tren cdc gid t h i l t Mindlin, vdi w Id d d vdng tam, p_, p^ lan lucrt Id cdc gdc xoay cCia phdp t u y i n eCia mat trung hda quanh true Oy vd Ox cOa he toa d d dia p h u o n g vdi qui Ude c h i l u d u a n g cho 6 Hinh 3, Q la mat trung hda ciia tam va t Id d d ddy cOa t i m . Cdc thdnh phan u, v vd w tuong
Ung Id chuyen vi theo phUong x, phuung y vd cdc nut
phUdng z; w" la chuyen vj tai mat trung hda (gid u = Nd (g t h i l t bien dang mdng- u° = v° - 0). trong do N la ma trdn cdc hdm d^ng ehuyfn
^ vj vd duoc xac djnh bdi "
N, 0 0 N, 0 0 N, 0 0 0 N, 0 0 Nj 0 0 N, 0 0 0 N, 0 0 N, 0 0 N, i - • " , „ vd d la vecta c h u y i n vi nut
HinhJ:Quiii(icchieudif[ingci3achuyenviwva2chiiyenvi ' ^ " L ^ ' P"' P" ^ ' P'= Pvi - w , p p^J Koay cua ta'm Reissner-Mindlm. B i l n dang ciia tam bao gdm b i l n dang ufln
Phan t U t U gidc 9 mit (Q9) dupe sU dung vd b i l n dang cSt.Cacthdnh phan b i l n d a n g n ^ trong nghien cflu nay. Tat ed cae phdn t f l deu dUdc cho bdi cdc edng thUe sau: „.
duac gdn vdo he tryc cd dinh (x,y) va dupe danh e = E, + EJ, = 2K,, + y (s) sd tU 1 -9 n h u d u p c the hien tren Hinh 4 trong do
't ' " - ' va
' t(b = Bt,u, Y = B , u [O N^ 0
0 0 N,, 0 N^ N_,
(6)
, , J N . N 01 (,
[N,^0
NJ Quan h f giCra Ung s u i t vd b i l n dang theo dinh luat Hooke nhU sau:^ , f l ol
Hinh 4' Phan tU 09 Irong toa Hi tong the.
Tat cd cae phdn t f l Q9 trong toa d d t h d n g thudng dUpc quy ve he toa dp t u nhign {i,T\) nhU Hinh 5,
o = a . + o =DzK.+Gk
1
II
9i0 0)
I-li Irfi -li it! If
.'(
6il Ol
-li HinhS Phin tli 09 trong toa do tu nhien Cac ham dang ciia phan tU Q9 trong he toa dp t p n h i e n duoe eho bdi:
trong d d E Id module ddn hdi, v la h^ sfl poison, G la module dan hdi truat vd k = - la h^
sd hieu ehinh cSt.
Nguyen 1;^ cdng ao dupe dp dyng de thiS ldp phuang trinh c h u y i n d d n g cua bdi todntim tren nen dan nhdt. Cdng ndi do dUOC tinh todn theo edng thde sau
W t - J K ^ D ^ K ^ d i i + j v ' ^ D . y d n ('"' Thay (6) vao (10) p h u a n g trinh duoc wet l?i VX = { 5 d f n B ; o , B , d n - H J B , ' D , B , d n l { d } (11)
N . = ^ a - 1 ) ( T 1 - 1 ) ^ T | , N. = : ^ a + l ) ( T | + l H i l .
Bdng each sU dung cdc hdm dang, vecto c h u y i n vi tai mdt d i l m b i t ki u = [ w p_ p i ' se dupc ndi suy t f l hdm dang vd chuyen vi cfla
1 trong d d ma trdn D,^ la ma trdn vat li&u N, = - ( 4 +1 ) ( i l -1)^11 Ung vdi b i l n dang udn, va duoc cho bdi
N ^ I f l - T i ^ V E - l k
4 V ' /yr> ^^ trong d d t la chieu day ciia tdm. Ma trjn DJ Id ma tran vdt lieu Ung vdi b i l n dang cSt, vd duoc xde dinh bdi
_ f f l ^ r , on 20+v)[o l j
c a n g ngoai 5o 3adc tfnh n h u sau
841
WE = J " b 5 u d n - j " ( 5 u ) m u d f i - J ( 5 v / ) k , w d f l
-J(sw)c,wdn 'i'^' t r o n g d d b = [p{x,y) 0 0]^ la tai trong phdn
bfi tren t a m , k^ c, ISn lupt Id d d cUng dan hdi vd hi sd cdn nhdt, m Id ma trdn khdi luong vd duac xde dinh bdt:
0 0 — 12.
Chuyen vi dUng w dUOC npi suy t U chuyen vj nOt phdn tfl".
w = N „ d (16) trong d d N^ la ma tran chfla cac ham dang
N „ = [ N , 0 0 N3 0 0 ... N, 0 0](17) Gid sU tdi t r p n g d i d d n g theo phudng x vdi van tde khdng doiV. Bang cdch s f l d u n g phuang phdp (MEM), m d t he toa d d (r,s) gdn lien vdi tdi trong di d d n g duoc thiet lap. Mdi quan he gifla hai true tpa d d dUpc xde dinh n h u sau-
r = x - V t (18) y = s (19) trong d d (x,y) Idn lupt Id h^ toa d d cd d m h ;
(r,s) l l n lupt Id he tpa d p c h u y i n ddng;V v a t lan lupt Id van tde vd t h d i gian di c h u y i n cua tdi.
Khi d d t r u d n g chuyen vi va cac dao ham rieng trong he toa d d c h u y i n d d n g dupe bieu di§n n h u sau:
... • i u ( x , t ) _ „ a j ( r , t ) flu(r,t)
3t
a
(20).. 8 ' u ( x , t ) y . a ' u ( r , t ) _,^d'u(
5t= ar^ 5r5t d
^ a»(>.t)_ ^aw(r.i) a»(r.t) j ^ ^ at 9r 3t
^ a-w(x.t)_^.a'w(,,t) ,ya'w(r,t)^c dt' ar" arat trong d o
u = [ w p . p , ] \ u = [ w p, p , J Thay (20) vd (21) vdo (14) phUcmg trinh dupe v i l t lai
W , = J b S u d f i - J ( 5 u ) m [ ' u - 2 v | i + V ' 0 ] c
- J ( S w ) k , w d i i - J ( 5 w ) e , j ' w - v | ! ^ ] d £ ;
Bdng each sfl d u n g phUdng phap Galerkin, cdc ma tran khoi lucmg, cdn vd d o cdng cfla phdn t f l t a m lan lUpt cho bdi
M , = | N ^ m N d n (25)
C, = j - 2 m V N ' N j d n -i- j " c , N > „ d n (26)
K = f B l D A d Q + f B ! D , B , d i i
i i (27) J m V ' N ' N „ d n - 1 c , V N ^ N „ ^ d a + J k ^ N ^ ^ d O
t r o n g d d ( ), Id dao ham bde nhat theo r va ( ) ^^ Id dao ham bdc hai t h e o r
Sau khi tdng hop edc ma tran k i t eau va v^c t o tdi trong cho todn b d t a m , phuong trinh d d n g lUc hpc ciia he tren cd dang
Mu+Cu+Ku=F (28) trong d d M, C vd K l l n lUdt la cac ma trdn
khdi luang, cdn va d d cUng t d n g t h i ciia he va F Id vec t o tdi trong t o n g the ciia he.
Phuang trinh phan tfch tinh eiia t a m Mindlin cho bdi
Ku = F (29) trong d o v e c t o t d i trpng d u p c x d c d i n h :
F = f pN^dn
i (30) Tam dao dpng vdi t a n sd thi phuong trinh
can bang t r d thanh
( K - O ) ' M ) U = 0 (31) Gidi hai phUOng trinh (29) va (31) t h u dUdc
bdi toan tinh vd dao d d n g t u do cfla hS.
3. C a c v f d u s o
o l chflng minh sU t i n cdy ciia phuang phap dupe de xuat, cdc v i d u sd v l phdn tich tTnh vd dao d d n g t U d o se lan lupt duoc thue hien t h d n g qua viec so sdnh vdi cac phuong phap khde sfl dung cac phan t f l nhU FEM-3 (FEM sfl d u n g phdn t f l 3 mit), FEM-9 (FEM sfl d u n g phan t U 9 nut) va SAP (phan t f l 4 nOt).
3.1 Phdn tich tinh
Trong phan tich ndy, m d hinh tdm chfl nhat Mindlin tua va khdng tUa tren nen dan vdi dieu kien bien Id ngam 4 canh (C-C-C-C) dupc khao
jt^ 3.1.1 Tdm chit nhdt
K i t eau t a m vdi bi&n ngdm c d cdc kich thudc n h u sau: dai L^ = 20m, rdng L = 10 va ddy t = 0.5. Cac t h d n g sd vat lieu cua t a m dUOC
^'wCr.t") '^^o ^^' ^ ^ 1.516.10'" N/m^ vd he sd
""ar (23) Poison v = 0.35. Tam chju tac dung cua tdi t d p trung P = 2000N tai giCfa tam, Ket cau tam se dUoc ehia thdnh eae phan t f l c d kich thudc NxN vdi N = 6,10,16,20,30,60...
cho den khi k i t qua dat duac sp hdi t u . Hinh 6 t h i hien sp hdi t u cua c h u y i n vj tai diem dat lUe vdi cdc ludi chia phdn t f l cfla k i t eau t a m . Cd t h e nhdn thay rang khi lUdl 1 cdng dupe ehia mm t h i cac cac phfldng phap khac nhau cho nghiem chuyen vj ' ' edng gan nhau vd nghiem dan t i l n tdi hdi
t u (sai sd gifla ludi ehia 30x30 va 60x60 eua MEM Id 0.1%). Cung t f l k i t qud nay cho thay nghidm ciia MEM va FEM-9 id hodn todn g i o n g nhau vl chung ciing xCr d u n g phan t f l 9 m i t . Odng thdi tU hinh 6 ta t h I y rang sU hdi t u ciia phuong phdp MEM nhanh h o n cdc phuang phap cdn lai vi MEM s f l d u n g p h a n t U e d s d n u t Idn h a n cdc phUdng phap cdn lai.
Hinh 6: Sir hoi tg cQa chuyin vi tai diem dat lUC ci)a tam Mindlin 4 canh ng^m (C-C-C-C).
Hinh 7 t h i hien c h u y i n vj cua tam tai diem dat lUc theo eae phUOng. T i m duac ehia vdi lUdi 60x60. TU k i t qud cho thay rdng k i t qud ciia MEM hoan todn dang t i n cdy, sai sd gifla phUOng phdp MEM so vdi cac phUdng phdp khdc Id rat nhd, trong d d phuong phdp MEM hau nhU cho ket qua hoan todn gidng vdi p h u a n g phap FEM-9 vi chflng deu sfl d u n g phan t f l 9 nflt.
Ddng thdi, nhfl dUac m o n g dpi e h u y i n vi gan bien theo p h u o n g x n h d h o n c h u y i n vj t h e o phuong y vi c h i l u dai t h e o p h u a n g x Idn hem phuong y nen dnh hudng cua Ipc t d p t r u n g dat tai t i m t d m ra ngodi bien cfla phuong x ft hcAi phuong y.
Hinh 7: Chuyin i/i tai diem dat luc (a) doc tiuc X, (b) doc tnjc y 3.J.2 TamchOnhdttrennendanhoi Trong bdi todn ndy, t h d n g sd t a m van dUOc sCr d u n g t u a n g t U vi d u 3.1 1 va cd t h e m he sd
nin ddn hdi k, = 9.5.1 CN/fflj.
Hinh 8 t h i hien sp hdi t u cfla chuyen vj tai d i l m dat Ipc vdi cac lUdi chia p h i n t d cua k i t cdu tdm.Tucmg t u n h u k i t qud dat dupe tU v i d u 3.2.1, khi l l m dat trdn n i n ddn hdi thi phuong phdp MEM cho Idi gidi hodn todn gidng vdi FEM-9 va hdi t u nhanh han cdc phUdng phdp dupe khdo sat cdn lai.
Hlnti B: Ket qua hoi tu cua tim (C-C-C-C) &en nen dan hoi.
Hinh 9 the hien c h u y i n vi cfla tam tai d i l m ddt lUc theo cae phuong x. Tuong t p nhu k i t qud dat dupc t f l vi du 3.2.1, khi t i m dat tr&n nen ddn hdl thi phuang phdp MEM triing khdp vdi cdc phuong phap cdn lai vdi bat ky ludi p h i n t f l dupc chia nao thi phuong phdp MEM d i u cho nghiem chinh xde han cac phuong phdp khdc vd sU hdi t u cOa MEM deu nhanh hon cdc phuang phap khdc chflng t d Uu t h i cfla phuong phdp MEM.
Hinh 10 t h i hi&n su so sdnh c h u y i n vi tai d i l m ddt lUc gifla tam ddt trSn nen ddn hdi va tdm khdng dat tren n i n ddn hdi. Nhu duoe mong dOi, chuyen vj tam tren nen dan hdi nhd han tam khdng tren n i n ddn hdi 3.63 Idn, c h u y i n vi gidm t f l 8.84.10^ xudng -2.43.10. Ket qud ndy cd t h i ap dung trong thuc t l t h i l t k l vd thi cdng: mudn gidm Ung xfl cua tam thi c i n t h i l t phdi gia cd nen vdi dd cutig Idn nhat cd t h i .
p h i n tich tan sd dao d p n g t p n h i ^ cOa kit d u tam.
Bdng 1: Bdng so sdnh t i n sd dao dong tuf n h i l n ciia 6 mode d i u tien ciia t i m bien SS-SS- SS-SS^
Phuong phdp
Hinh 10: So sanh giQa diuyen ui tai dilm dat l^c doc tme X cua t3m tren nen dan hoi va tam khong dat tren nen dan h6i.
phuang phdp MEM se dupe so sdnh vdi k i t qud duac cdng bd trong bdi bdo cfla tde gia Leissa (11 ] va phUcmg phap FEM. Odng thdi vi&c phan tich su dnh hudng eua chieu ddy tam va t i sd c h i l u ddi/ehilu rdng cfla tam d i n t i n sd dao ddng t u nhien cua tam eung da dUOC thpc hien.
3.2.1 Dao dong tU do cOa tdm Mindlin vdi cdc dieu fcien biin khdc ntiau.
T i m Mindlin tUa ddn bon canh bi^n tUa ddn (SS-SS-SS-SS) dupc khdo sdt. Thdng sd khdo sdt cfla tam L , - l(m), L^ = l(m), t = 0.01m, thdng sd vdt lieu tam: he sd Poison v = 0.166, E = 1.17-10'(kN/m').Tlm dUoe ehia vdi cdc h& ludi n h u 6x6, 10x10, 16x16, 20x20, 30x30 d e khdo sdt trong bdi todn.
Hinh 11 the hien sU hdi t u cfla tan sd dao ddng t y nhien ung vdi mode dao ddng thU n h l t vdi edc ludi ehia p h i n t f l cfla k i t cau tam K i t qud cho thay kht ludi cdng dflOc chia min thi cd MEM vd FEM-3 edng eho ldi giai tien den ldi gidi gidi tieh dUoc dUa ra trong bai bdo [11 ] . Odng thcri tU hinh 11 ta thdy rdng sfl hdi t u ciia phUdng phdp MEM nhanh hon FEM-3 vi MEM sfl dung phan t f l cd sd nut Idn hon phflOng phdp FEM3.
Hinh 9: Oiuyen vi tim tai diem dat lUc doc tme x 3.2 Midn tieh dao ddng tUdo Trong mue ndy, viec khdo sdt tinh chinh xde vd sU hflu dung eua phuong p h i p MEM trong vi^c phdn tich tan sd dao ddng t u nhien cua k i t cdu tdm dupe thuc hi&n. Hai dieu kien bien duoc dp dung cho t i m Id bdn bien tUa don (SS-SS- SS-SS) va bdn bien ngdm (C-C-C-C). 7am dUOe chia vdi n h i l u he ludi khac nhau. K i t qud cfla
Hinh 11: Suhw Ui lan so dao dong tu nhien ihg voi mode dau lien cua tam bien SS-SS-SS-SS
Bdng 1 va Hinh 12 t h i hien tan sd dao ddng t u nhien Ung vdi 6 mode dao ddng dau t i l n cfla tdm vert bien khdp. K i t qud t h u dupe cho t h I y MEM eho ldi gidi sdt vdi ldi giai gidi tich (11 ] vd eho ket qua chinh xde han so vdi MEM-3 vdi ciing mdt he ludi. O i l u ndy chflng t d rang phuong phap MEM t d ra hieu qud trong viec
Leissa [11]
FEM- _ 3node' 19.7349
49.3343 49.3343 78.8879 98.6830 98 6830
19.7392 49.348 49.348 78.9568 98.696 98.696
19.7892 49.61 tw 49.7030 79.7595 99.9194 99.9236
Hinhl2:Taiisadaodpngti/nhiSnciia6inodedaodai)^
tiln ling vdi Itidi diia 30x30 ciia tam bien SS-SS-SS-SS.
Odng t h d i t f l Hinh 12 ta thay rang 6 ciing mpt he ludi phan t f l thi tan sd dao dong tif nhign d cdc mode t h i p se hdi t u nhanh hdn c k mode cao. Vi p h u o n g phdp MEM cho Idi gidi h^
tu r I t nhanh nen MEM se Id phUong phdptiiich h p p d l phan ti'ch tan sd dao dpng eua edc mode cao.
3.2.2 Sl/ anh ht/dng cOa chieu day tam I6n tdn so dao dong tif nhien
Trong bdi todn ndy, thdng s6 tUdng tunhil vi du 3.2.1 n h u n g vdi sU b i l n ddi cfla chilu diy tam C h i l u day tam dUde khdo sdt se thayd^i tii 0.001 m d i n 0 3m. Tam duoc k h i o sdt vdi J A i
kien bien la bien ngdm. , , . « Bdng 2 va Hinh 13 the hien t i n sd daoddrig
t u nhien Ung v d i 6 m o d e dao ddng dau ti^n cua t i m bien ngdm vdi cdc chieu day tam khac nhau. K i t qua t h u dupe cho t h I y khi ehieu ddy cfla tam t d n g dan thi tan sd dao ddng tU nhiln;
eiia tdm gidm dan va tan s6 dao ddng t u nhl&l,, cfla mode dao ddng cdng Idn thi dd gidm t S | sd cdng n h a n h h o n so vdi cdc mode thdp hdn, Khi c h i l u day t a m rat n h d (0.001 m) thi khi tang e h i l u ddy t a m t i n sd dao ddng t u nhi^n giim it (tang t f l 0.001m len 0.01 m t i n sd dao dong t u nhien gidm t f l 36.16Hz xudng 36.06 Hz), khi e h i l u ddy t a m cdng Idn t h i dp gidm ciia tan sd dao d p n g t u nhien ddng k l khi tang chieu ddy t a m (chieu day t f l 0.2m \in 0.3m thi tin sd dao d d n g t u n h i ^ n gidm t f l 24.45Hz xuSng 22.54Hz).
86|B
Bdng 2: Tan sd dao ddng tfl nhien Ung vdi 6 mode dao ddng dau tien cua tam bien C-C-C-C (L/
1,-1).
1 2 3 4 5 6
130 DO tlDOO
s f. ""^
» m K M
0 05
- . ^ M o d t l 0.001 36.167 74.117 74.117 109.323 133.948 134.558
1 0 1 ! 0> 0 TJ . a i L . - ^ M « « :
0.01 36.064 73.719 73.719 108.636 132.685 133.307
^^.^^^
"^^^
~^~~*
~ *
2S O i OJ!- . - M o d . ) t/L,
0.1 32.967 63.340 63.340 89.177 105.394 106.342 Bdng3:Ta (t/L,=0.01).
UU 0.66
1.5
^
2.5
0.2 27.452 48.432 48.432 65.327 74.639 75.651
0.3 22.548 37.540 37.540 49.666 55.826 56.683 so dao ddng t p nhien Ung vdi 6
P.P MEM Leissa MEM Leissa MEM Leissa MEM Leissa
1 14.24 14.25 19,73 19.73 32 08 32.07 71,57 71.55
2 27 38 27.41 49 39 49.34 61.74 61.68 1012 101.1
mode dao ddng ddu t i l n cfla t i m bien Mode
3 43.85 43.86 49.39 49.34 98.83 98.69 150 9 150.5
4 49.36 49.34 78.96 78.95 111.4 111.0 221.2 219.5
5 56.93 57.02 99.01 98.69 128.3 128.3 257.0 256.6
SS-SS-SS-SS
6 78.80 78.95 99.01 98.69 177.8 177.6 286.5 286.2 Hinhl3:Quanli@gii^tlnsddaod6nglirnhiencua6mode
dao dong diu tien va tl sd chieu day/chieu dai (t/L) cDa tim 32.3 Anh hudng ciia tl IS chieu ddi/chieu rong (L/L) eua tdm din tdn so dao ddng tiJ nhiin Trong bdi todn ndy, thdng sd tUdng tu nhu vidu 3.2.1 nhUng ti le chieu ddi/chieu rdng dupe thay ddi tfl 0.4-2.5.Tlm duoe khdo sdt trong bdi toan cd dilu kien bien la bien khdp vdi he lUcri phin tfl 20x20. Tdn sd dflpc so sanh vdi phUdng phap gidi tich [11] d l chung td tinh dung dan cfla MEM.
Bdng 3 t h i hidn tan sd dao ddng tu nhien ling vdi 6 mode dao ddng dau ti§n cua tdm bidn khdp vdi cac ti 1& chilu ddi/chilu rdng eua tam khae nhau ciia MEM va ldi gidi gidi tfch [11]. Kit qua thu duoc cho thdy rang kit qud md phuang phdp MEM rlt gdn vdi ldi gill gidi tieh vdi sai sd rat nhd {sal sd d mode 1 la 0.008%). Oilu nay chflng td tinh chinh xde cfla MEM trong phan tieh dao ddng tp do.
Hinh ding dao ddng cua tdm vdi cac tin so dao ddng tu nhiln khde nhau 6\iac the hien trong Hinh 15.
4.K^tlu|n
Trong bai bao nay viee phdn tich Ung xiJr tinh va dao ddng tu nhien cfla tam sfl dung phuong phdp MEM dd duoc thue hien. Thdng qua cdc kit qud nghien cuU, mdt sd kit ludn cd the rut ranhusau:
- Phuang phip MEM cd tinh ehinh xac cao vd tinh khd thi trong viee phdn tfch tTnh va dao ddng tu do cua kit cdu tim. Cdc kit qud thu duoc tfl MEM dd duoe kilm chOng vdi FEM va phin mim thUdng mai SAP.
- Kit qud dat dUoe tfl phuong phdp MEM gin nhu trflng khdp vdi cac ket qud dat dUOc tfl edc phuang phip sd khdc. Hdn nfla phuang phdp MEM cho nghi&m ehinh xde hdn so vdi eac phuong phap FEM-3 nut vd SAP vdi cflng mdt
Mode 1 Mode 2
Mode 3 Mode 4
Mode 5
Hinh 15: Hinh dang 6 mode dao dong dau ben cOa tam bien khcip.
Mode 6
he ludi p h i n t f l .
- Trong viee phdn tfch bdi toan tfnh va dao ddng t u do thi flu d i l m ndi bat cfla phuong phdp MEM la phUOng phdp ndy chl c i n mdt he ludi tUOng ddi eung du d l cho nghidm gdn nhU ehinh xde.
LOI CAM ON
Nghidn eUu ndy dupe tdi tro bdi Oai hpc Qudc gia Thdnh phd H d ' c h i Minh (VNU-HCIW) trong khudn khd d l tdi md sd C201S-20-xx:
'Phdn tieh ddng lue hpc tdm tr&n nen dan nhdt ehiu tdi trong di ddng sfl dung p h i n t f l 2-D c h u y i n ddng"
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