Phan tich dao dong ty do cua tam fgm dya tren

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Phan tich dao dong t y do cua tam fgm dya tren phiFo-ng phap khong lu'O'i va ly thuyet do-n gian bien dang cat bac nhat

Free vibration analysis of fgm plates based on the meshless method and simple first- order shear deformation theory

Ngay nhan bai: 15/11/2014 Ngay siia bai: 4/10/2015 Ngay chap nhan dSng: 10/12/2015

TOM TAT

Bii b i o niy gidi thif u mpt mo hinh so mdi phan tich dao dpng tfl do cila tam vat lieu bien d6i chflc ning vdi cac thupc tinh vat Ii?u thay d6i theo chieu day tam.

Mo hinh niy dfla trln phUcmg phap k h o i ^ ludi sfl dyng ham npi suy Moving Kriging (MK) ket hpp vdi ly thuyet bien d ^ g cit b i c nhat dem gian(S-FSD). Cac vi du so dupc thflc hien di so sanh kit q u i d?t dflpc vdi cac ket q u i cua cac nghiln cflu da cong bo nhim kiem chflng sU chinh xac cua mo hinh phan tich dupc d l xuit.

Tfl khda. Dao dpng tfl do, tam vit heu chflc ning, ly thuyet bien d^ng cat b|>c nhat ddn giin, npi suy Moving Krigmg, phfldng phap khong lUdi.

ABSTRACT

This paper presents a new numerical model for analysing the free vibration problem of the functionally graded material (FGM) plates in which material properties vary through the thickness This model employed the the mesh-free method with Moving Kriging (MK) interpolation with the simple first-order shear deformation(S-FSD) theory. Several verification numerical examples are solved and compared with the other available numerical methods showing the accuracy of the proposed method.

Keywords: Free vibrations, Functionaly graded plates, Sunple first-order shear deformation theory, Moving Kriging mterpolation, mesh-free method.

T S . V u T a n V a n

Giang vien, Khoa ky thuat Xay dfliig, Trflong Dai HQC Kiln Tnic Tp.HCM Email: [email protected]

Di.en thoai: +84 123 686 9610

KS. Nguy£n Ngpc HUng

HQC vien cao hpc, Khoa ky thuat Xay dUng, TrUOng Dai Hoc Bach Khoa - Dai Hpc Qu6c Gia Tp.HCM

Email: [email protected] Dien thofd: +84 939 272 901

Vu Tan Van, Nguyen Ngoc Hulig

1 . G i d i t h i e u

Vat lieu bien doi chflc nang (Functionally Graded Material- FGM) la m o t loai composite c6 dac tinh vat lieu b i l n doi lien tuc trong vat t h i do do se loai b d dupc h i f n tucmg t a p trung flng suat thudng gap d loai composite t h d n g thudng.

FGM thucmg duOc c h l tao t f l hon h o p gom gom va kim loai. Day 1^ loai vat lieu d i n g hudng nhung khdng dong nhat. Hien nay, FGM dUOC quan t a m vi co t h i tao ra nhflng k i t cau cd k h i nang thfch flng vdi nhflng d i l u k l f n vhn hhnh.

Thong t h u d n g , phan tich flng xU cua tam sSn xuat t f l vat li^u chflc nang [tam FGM) dUOC dUci tren bdi cac IJ t h u y l t CO bdn sau- (i) T^m co diln (CP), (ii) B i l n dang cSt bac nhat (FSD), (iii) Bien dang cSt bac cao (HSD).

Ly thuyet 0> [1] khdng xet den Snh hudng cua bien dang cSx ngang d i n flng x f l cua tam mdng.

Khi chieu day tSm tang l^n, bien dang cat ngang cd inh hudng ddng k i d i n d^p Ong cfla tam.

Ly t h u y l t FSD [2-3] xit Oin dnh hudng biln dang cat nay bSng cdch xdy dUng trUdng chuyen Vj t u y I n t i n h bdc n h i t trong mat phSng doc theo chieu day cfla t i m . Tuy vay, cdc phuong trlnh can bSng, 6n dinh dUpc xdy dUng dua tr&n 1|^

thuyet CPT va FSDT d i u khdng thda man d i l u kien bien ve sy triet ti&u flng suat d mat t r l n vi dudi cfla tSm. NhSm gidi q u y l t dupc khd khdn nay, m d t he so d i l u chlnh b i l n dang c3t d f l ^ sil dung d l dieu chinh m 6 i quan he ket hpp glOa ung suat cdt va b i l n dang cdt ngang. Gid tri h^

sfi Oieu chinh nay p h u thudc vao cac thdng sfi n h u hinh hpc, tdi trong tdc dung, dieu kien bi&n cfla t d m .

Ly t h u y l t HSD [4-21] x^t d i n dnh hUdng b i l n dang c3t ngang bdng each xay dflng cdc trudng chuyen vi bdc cao d trong mdt phSng dpc theo chieu day cfla t d m , hoac theo mdt phang ngang cfla t d m . Cac phuong trinh cdn bdng, on d i n h dua tren trudng c h u y i n vi da thda man cac tdt cd d i l u kiSn bien. Tuy vay, viec phdn tich flng x f l cua t d m dUa tren cac ly thuyet HSD nay rdt phflc tap d o sS luong b i l n sd d cdc phuong trinh cdn bSng, 6n dinh tdng len, chdng

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h^n hdm c h u y i n vi duoc xdy d u n g t r ^ n ly t h u y l t HSD de xuat b d i Pradyumna vd Bandyopadhyay [11], Neves vd c d n g sfl [13,20-21] sfl dMng 9 an so; Reddy [9], Taiha va Singh [12] sfl d u n g Idn l u o t g o m 11,13 ^n so.

Oil cho m d t sd ly t h u y l t HSD khdc sfl dung ham chuyen vi g d m 5 in sd tuong t f l n h u ly thuyet FSD chang han n h u ly thuyet b i l n dang cat bdc ba (TSD) [4], 1;/ thuyet b i l n dang cat ham sin [14], ly thuyet b i l n dang cdt hdm lupng giac [15-17]. Tuy vay, phUOng trlnh cdn bdng, dn dinh dat dupc t f l cdc \'y thuyet nay vdn phflc tap hon so vdi 1;^ t h u y l t bien dang cdt bac nh ^t (FSD).

LJ t h u y l t b i l n dang cdt bdc nhdt dOn gidn (S-FSD) dUpc d e xudt ddu tien bdi Huffington [22] vdi ham chuyen vj chi g d m 4 an sd. Khdc vdi ly t h u y l t FSD, thdnh phdn gdc xoay dupc b i l u dien t h d n g qua thanh phan udn vd cat tao nen trUdng c h u y i n vi trong mat phang, chuyen vi ngang cfla tdm

Mdt khdc, khi khdo sat flng xfl m i t dn dmh cfla tam FGM chiu tdc dyng cua t i i t r p n g phdn bd phi tuyen trong mat phdng tai cdc canh bien cCia tdm, Chen vd cdng sU [23] cOng khdng dinh rdng phucmg phdp khdng ludi-sfl dung trUdng c h u y i n vi xdy dUng dfla tren toa d d cua cdc nilt rdi rac trong cdu true se tranh dupc nhflng su phflc tap ve sd khi sfl dung cac loai phdn t f l t r o n g phuong phdp phan t f l hflu han.

Gu [24] g i d l t h i f u d a n g thflc m d i ciia p h u a n g phdp khdng ludi dua t r l n dgng y l u Galerkin ket h p p vdi hdm ndi suy M o v i n g Kriging (MK) g p i Id phUPng phdp MKG. M d t t r o n g n h f l n g Uu d i l m cfla ham n$i suy MK Id t h d a m d n t i n h chdt ciia hdm delta Knonecker, khdc phuc dupc n h f l n g t r d ngai ve dieu kien bien t r p n g y l u xdy ra d o i v d i phuang phdp khdng ludi,

Ndi dung bdi bao de xuat md hinh phdn tich dao ddng t u do cua tdm FGM dua vdo ly t h u y l t S-FSD k i t hop vdi phuang phdp MKG. Md hinh vdt l i l u chflc ndng dUOC trinh bdy d myc 2. ly t h u y l t dan gidn b i l n d^ng cdt bdc nhdt dupc trinh bay d myc 3.

Md hinh phdn tich dUpc de xuat d muc 4. Vi du sd dUoc thUc hien d l k i l m chUng d d tin cay cfla m d hinh dupc trlnh bdy d muc 5. Sau cung la cdc ket luan t h u duoc t f l md hinh duoc nghiSn cflu n^u tr^n.

2. Tam FGM

X « mdt t d m FGM dUOC c h l t?o t f l vat lieu kim loai va g d m cd c h i l u d d y h . Mat dudi vd tr§n cfla tam hodn toan Id kim loai va g d m . Mat phdng xy ndm d gifla tam Chieu dUPng cfla t r u c z hudng len tren.

Trong bdi bdo ndy, ty' sd Possion's v duac xem la hdng sd. Nguoc lai, mddun dan hdi E , mat d d khdi luong p dUOc xem Id thay ddi lien tyc theo chieu ddy tam FGM vdi ludt hdn hpp Voigt hay theo luoc d d Mori- Tanaka [4]. Theo d d . m d d u n dan hdi E { z ) , mat d d khdi lu'png p { z ) duac xdc dinh n h u sau:

E(z) = E „ + { E , - E „ ) V , (1) p{z) = p^ + ( p , - p „ ) V , (2) Trong dd chl so m vd c dai di^n cho thdnh phdn kim loai vd gdm

tuong flng;V( = 0 . 5 + - Id t h i tfch thanh phdn g d m ; n Id chi sd cfla hdm mu, t h i hi^n sU gia tang t f IS cfla phdn t h i tich; z la b i l n toa d d theo chieu ddy -0,5h < z < 0.5h .

Hinh 1 bleu d i l n sfl thay ddi cfla the tich thdnh phdn g d m y . ddi vdi ty sd c h i l u ddy t3m FGM khi tri sd n thay ddi. Ddi vdi gid t n n rat Idn n > 1 0 0 thi V( rdt be - cd the xem n h u vdt lieu cfla tdm chl bao g d m la kim loai. Odi vdi gid tr) n rdt be n < 0.01 thi V, = 1 - cd the xem nhU vdt li^u cua tam chl bao g d m Id g d m . SUthay ddi cua vi^c k i t hop gifla hai vdt lieu kim loai vd g d m Id tuyen tinh khi n = 1 .

»

1 i

10 Q .

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y y . iiii

-'--

1 = 3.

y'

^ '

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„- ,'

J ^

. - • • '

I -^

„--

y

0.5,

,''

•—"

^ ^ '•'•'l y' y

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0.3^

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0.0 0,! 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 9 1.0 Hinh 1 .Quan hi gifla V^ vd t y le e h i l u ddy ^ h theo chi sd n

3. Ly t h u y l t b i l n dang cdt bac 1 ddn gidn

Ddi vdi ly thuyet bien dang cdt bac nhat FSD [2-3], trUdng c h u y i n vj ciLatam ( u , , u , , u , ) c d t h i dUpc b i l u dien ddi vdi 5 bien s d n h u sau:

u,(x,y,z) = u(x,y)-zaWb(x,y)/ax (3) Uj(x.y,z) = v(x.y)-zawt,(x,y)/ay (4)

u,(x,y,z) = w(x,y) (s) Trong do u(x,y),v(x,y),w(x,y) Id nhflng I n sd c h u y i n vi cua mat gifla

cfla tam theo cdc p h u a n g x , y , z t u a n g flng; (p,{x,y),ip^(x,y) Id cac gdc xoay cua phap tuyen cfla mat phang gifla tam theo t m c x , y . Ly thuyet bien dang cat bdc nhdt dem gidn (S-FSD) sfl dung cdc gid t h u y l t sau de Idm dem gidn ly t h u y l t b i l n dang cdt bdc nhat (FSD): (i) c h u y i n vj theo phuang dflng gdm thanh phdn chuyen vj do udn w^ vh cdt w^ gdy ra, nghra la: w(x,y) = Wt(x,y) + w,(x,y),(ii} thdnh phdn gdc xoay chi do thdnh phdn c h u y i n vj do uon gdy ra:tp,(x,y) = - a w j { x , y ) / a x tp,(x,y)=-avwi,{x,y)/ay;.Vi vdy cac cdng thflc (3), (4) va (5) cd the v i l t lai nhusau:

u,(x,y,z) = u(x,y) + z<p,(x,y) (6) U;(x,y,z) = v(x,y)+Z(p^{x,y) (7) Uj (x, y, z) = W;, (x, y) + w , (x, y) (8) Khdng gidng vdi ly thuyet FSD, trUdng chuyen vj dUpc xac djnh

theo cong thflc (6)-(8) chl gdm 4 an sd:u(x,y),v{x,y),w^(x,y)vdw,(x,y).

Bdi vi thanh phan gdc xoay Id dao hdm bdc nhdt cfla thdnh phdn chuyen vi do udn tuang thich vdi su rdi rac cfla ly thuyet bien dang cdt bdc nhdt don gidn (S-FSD) trdnh dUpc hi#n tUcmg khda cdt (shear locking).

DUa tren gid t h i l t b i l n dang nhd, mdi quan he gifla b i l n dang vd c h u y i n vi dupc bleu dien n h u sau:

au ay

au 5w^

dx dx' dv a'wj, ex dx' dv d'vl^_

CV axay awf,

dx

~di

149

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Cdng thflc (9) v i l t dudi dang ma trdn n h u si

du

t «

5v oy

Su (V ey 5x

a^vif.

ax' a'w„

V

, 5 ' w . [ axdy

y - aw 1 aw J

ay 1

Mdi quan he k i t hpp thiet lap dua t r ^ n ludt Hooke b « phuang trinh sau:

D.(z)(e,-ZK)

D . ( z ) ( e . - a < )

T = D . ( z l v

- [ ' . '.]'

(12a,b)

{13a,b)

D.(z) =

0 0 ( 1 - v ) / 2 _kE(z)_ri O l

r'(x) = [ R ( x „ x ) , R ( x „ x ) , . _ J l ( x „ . x ) ] ' ^ 3 ' Trong d d R ( X „ X J la hdm tUdng quan gifla cdc cap

cua n nflt x,vd x , n d duoc bieu hien bdng cdc phuong sai cfla cdc t r u d n g gid tri

u(x): R(x.,x,) = c o v [ u ( x , ) , u l x , ) ] v a R(x„x) = cov[u(x,),u(x)].

Cd n h i l u each d l xac djnh hdm R(x,,x,) nhUng phUcmg phdp hdm Gauss la phucmg phap t h u d n g sfl d u n g vi t i n h d o n gian, hieu qud

Vdi: r^ = | | x , - x J I , v d 9 > 0 l a h e s d t U d n g q u a n . T r o n g baibdondysiJl d y n g p ^ ( x ) ! a m d t h d m b d c h a i n h u s a u :

p'(x) = [ l , x , y , x ' , y S x y ] '^^^

Ngoai ra, ma trdn R [ R ( X , , X , ) ] ^ dupe bieu dien dudi dang tudng minh nhusau:

R [ x „ x , ) ••- R ( x „ x „ ) "

1 - R ( x , . x J

2{1+ (15)

Trong dd A Id he sd hieu chinh cdt.

4. M d h i n h p h d n tfch d a o d d n g t i i d o cua t a m FGM 4 . 1 . Hdm dang MK

Phucmg phdp MK ducK dflng de xdy dung ham dang va cdc dao ham [24-261. Gid t h i l t ham phdn b d u ( x , ) d u p c xap xl trong m i l n con n , sao cho f l , £ 12. Gid sfl rdng cdc gid tri cua hdm sd dUpc npi suy dfla tr#n cdc gid tri t^i cdc d i l m nut x , ( i g [ l , n ] ) v d i n Id tong sd diem nflt trong m i l n il,. Hdm ndi suy MK fl" ( x ) , Vx e Q , dUoc xdc dinh nhU

u"(x) = [pMx)A + r'(x)B]u(x) hay

u''(x) = X ® i ( ' < M

Trong d d (D,(x) Id hdm dang MK. dUpc xdc dinh n h u sau

©,(x) = [p'(x)A + r'(x)B]

Ma trdn A, B dUpc dinh nghia nhusau:

A = (P'R-'P)''P'R ' B = R " ' ( I - P A )

(16)

(20) Trong d d I Id ma trdn d<m vi, v ^ t o p(x) la da thflc vdi m hdm co s d :

p'(x) - [p,(x),p,(x),p,(x)_,p,„{x)] (21) Cy t h i , ddi vdi ma tran P kich thude n x m , cac gid tri cua ham co

sd da thflc (13) duac cho bdi n h u sau:

"p,{x,) p,(x,) •• p,(x,) p,(Xj) p,(x,) -. p , ( x , )

(22) P,(x„) P,(x„) - p „ { x „ )

V ^ t d Kx) trong phuang trinh (16) dUpc cKnh nghia n h u sau:

R [ R ( X „ X , ) ] =

n-X,)

(26)

001 vdi bdi todn tdm FGM, khdng chi dao ham bac 1 dupc sfl dung md cdn dao ham bdc 2 cOa ham dang cung dUdc thiet Idp n h u sau:

< t i , , ( x ) - i ; p „ ( x ) A „ + Xr^j(x)B^, (27)

* u , ( x ) - J p , ^ | x ) A ^ + £ r ^ , ( x ) B , , (28)

Can lUu y anh hUdng cua he sd tUcmg quan 9 Ooi vdi ham dang la rd rdng. Mdt trong nhflng diem quan t r p n g nhat cua ham dang MK, dd la sd hflu tinh chdt Kronecker's delta. Dieu ndy se loai bd n h f l n g t r d ngai ddng k l nhdt cfla hdu h i t cac phuong phdp khdng ludi khi dp ddt dieu k i l n bifin de giai bai todn c d hpc. De chung minh cho dieu ndy, chflng ta khdo sdt lai ham dang MK xdc dinh bdi b i l u thflc (18)

* , ( x , ) - £ p , ( x , ) A , + X r , ( x , ) B „ (29) Hay b i l u thflc (29) cd t h i v i l t dUdi dang sau:

[ai,(x,)] = PA + RB ' 3 ° ' Trong d d ma trdn A,B,R vd P d u p c ^ n h nghia bdi cdng thflc

(19)(20) vd (22). Thay cdng thflc (20) vao (30) ta dUdc:

[(D,(x,)] = PA+RR-'(l-PA) = l *31' B i l u thflc (31) dan d i n t i n h chat Kronecker's delta xac dinh bdi bieu

thflc (32).

( I k h i i ^ j (32) ' " [ O k h i i ^ j

Ngodi ra, ham ndi suy MK sd hflu t i n h nhdt q u d n , nghia la cd the xdy dflng lai bat c f l ham cd bdc t h d p hem. Oe d o n gidn, t h u d c t i n h ndy cd the t d m tdt n h u sau: N l u u, dat duoc t f l da thflc cd b^c n h d h a n hode bdng m nghia Id

u = Pa (33) trong dd, P dUpc xdc djnh t f l cdng thflc (22) vd a Id he sd bdt ky, thi

sU xdp xi do la chinh xdc. Su xdp xi cua trUdng c h u y i n vj nhU sau:

u''(x) = p ' ( x ) a = u(x) (34)

(4)

Ode biet, n l u sfldung hdm p(x) la hdm tuyen tinh khi xdy dUng ham dang MK thi tdt cd hang sd, sd hang tuyen tinh cd the xac dinh lai hodn todn-

J 4 ^ ( x ) = 1 , j 4 . , { x ) x , = x , j ^ , ( x ) y , = y (35) Mat khac, mdt trong cdc y l u to quan trong ddi vdi phuong phdp khdng ludi la m i l n dnh hudng, trong d d ban kinh m i l n dnh hudng duoc diing de xdc dinh sd luang cdc nut rdi r?c trong pham vi mien ndi suy dang xet. Bdn kinh mien dnh hudng d ^ duoc xac dinh n h u sau:

d , = ad,^

Trong d d a Id he sd cfla mien gia dd, thdng t h u d n g a n i m trong khodng t f l 2.0 d i n 3.0. Gid tri d^ Id c h i l u dai ddc trung cho khoang cdch cdc nflt vdi d i l m dang x l t .

4.2. Cac phUdng t r i n h r d i rac

Nhflng chuyen vi trong he toa do t d n g qudt trong mat phdng gifla duoc xap xi theo b i l u thflc (17), trong d d

i/=[u^ v" w^ wlj (37)

(38) Thay b i l u thflc (17) vdo bleu thflc (11.a,b,c) nhan dupc

-ZB^^

=EBrH - Z B ; " I

*.,. 0 0 0 0 »|i,^ 0 0

[ 0 0 0 'I'l,.' [ 0 0 0 itl,,,

B^ = 0 0 41,^ 0 0 0 (h^ 0 0 0 (|i,^ 0

Vdi bdi toan dao ddng t u do, dang y l u duoc b i l u dien n h u sau:

JSe^DEdn + JSY^D'ydfi = JSu^mCidn

B D" ^ f D,[z)dz

5. K i t q u ^ 50

Trong phdn ndy, flng x f l dao ddng t u do cfla tdm FGM vdi chl s6 n suy gidm thay ddi cflng vdi cac d i l u kien bien khdc nhau dupc khao sat dua t r l n m d hinh phdn tich k i t h p p gifla Ij? thuyet S-FSD vdi phuong phdp khdng ludi MKG (S-FSD-MKG). Lupc d d bdc 2 Gauss 4 x 4 duoe sfl dung tnang phflong phdp khdng ludi MKG d l tich phdn dang y l u . Dieu kien bien eua tdm duoc ki? hieu n h u sau: gdi tua don gidn (S), ngam (C), va t u do (F). Cdc d i l u kien bien ndy dupc dp ddt t h d n g qua cac phUPng trinh nhusau [27]

(1) Canh bifin gdi tfla d o n : 3Wt,

dy aw.

ay aw.

- = 0,tai r = 0,fl.

O.tai y = 0,b.

(ii) C?nh bien gdi tUa ngam:

dx dy taix = 0,avhy=0,b.

D - . J D . t z j d z B—fzD.(z)dz

D" = I z'D„(z)dz

(36)

\\;

(lo.luU= J 9{z)[Xz.z']dz (43a,b)

(44)

(4Sa,b)

i|i, 0 0 0 0 (jl. 0 0 0 0 ^1 (j),

aw^/ax aw^/ay

=2NS

= Z N , ' 4

^--

0 0 41,^ 0

0 0 0 0 (49a,b) Thay the bieu thflc (39) va (42a,b,c) vdo bleu thflc (41) bdi todn dao dpng t u do cfla tam FGM cd the v i l t lai nhU sau:

( K - < o ' M ) d - 0 (50) Trong dd ma trdn dp cflng, khdi lupng trong he tpa d d tdng t h i xdc

dmh nhu sau:

M = J

>rA

Tdm FGM hinh vudng cd c h i l u ddy tam h = 0 , 0 1 m dflpc sdn xudt tU vdt lieuAI/AljOj.Thudc tinh vat lieu cuaAl Id:v,„=0.3,E,„=t70GPa, vd p „ = 2707kg/ m ' ,thudctinhvatlieucfla AI^Oj la: v^ = 0.3, E^ = 380GPa v a pj = 3800kg / m ' . Tam s f l d u n g sd lupng diem nflt Id 1 3 x 1 3 . He sd hieu chlnh cSt k = 0.8601. Phuong phap khdng ludi MKG sfl d y n g cdc thdng sd: a = 3,9 = 3 ,

Bdi t o a n 1 : Khdo sdt tan sd dao ddng khdng t h f l nguydn cfla tam FGM vdi t ^ 50 chieu dai vd chieu ddy cua tam a / h , chl sd dp suy gidm n thay ddi. Tdn sd dao ddng dupc khao sdt trong bdi toan la tdn sd dao ddng khdng t h f l nguydn dupc dinh nghia to" = tohjpji^. K i t qud phdn tich tan sd dao ddng t h f l nhat dat duoc tU cdc phuong phdp khdc nhau dUPc t h i hiSn d bdng 1. Cd mdt d i l m chung rd rang t h i hi^n d cac ket qud dat dUPc tU cdc Idi gidi khdc nhau, d d la b i ^ n d d cfla tdn sd dao ddng t f l do t h f l nhat gidm dan khi chi sd n vd t y sd a/h gia tang. Cdc gtd tri tdn sd dao ddng t u do dau tien dat dupc t f l m d hinh d l nghi (S-FSD- MKG) rat phii hap, cd d p chinh xdc cao vdi cac icfi gidi dat dUdc t f l phucmg phdp khac.

1151

(5)

Vh

2

10

20

Phucmg phap

Gidi tich 2D-HOT [27]

S-HSOT [27]

FSDT-IGA [27]

S-FSD-MKG

Gidi tfch 2D410T [27]

S-HSDT [27]

FSDT-IGA [27]

5-FSD-MKG

Gidl tich 2D-HOT [271

FSDT-Ritz [27]

FSDT-IGA [27]

S-FSD-MKG

n = 0 0.9400 0.9297 0.9265 0.9213 0.0578 0.0577 0.0577 0.0565 0.0148 0.0146 0.0148 0.0145

n - 0 . 5 0.8232 0.8110 0.8060 0.8010 0.0492 0.0490 0.0490 0.0480 0.0125 0.0124 0 0125 0 0123

n = l 0.7476 0.7356 0.7330 0.7287 0.0443 0.0442 0.0442 0.0433 0.0113 0.0112 0.0113 0.0111

n = 4 0.5997 0 5924 0.6111 0.6087 0.0381 0.0381 0.0382 0.0375 0.0098 0,0097 0.0098 0.0096

n - 1 0 0 5 4 6 0 0.5412 0.5640 0 J 6 2 1 0.0364 0.0364 0.0366 0.0358 0.0094 0.0093 0.0094 0.0092 Bdng 1. Tan sd dao ddng ddu tien khdng thflnguyen cila tam FGM

Bd) todn 2: Khdo sdt tan sd dao ddng khong t h f l nguyen cfla tam FGM vdi cdc d i l u ki&n b i ^ thay ddi Tdm FGM ty sd a/h = 100. Tan sd dao ddng dudc khdo sdt trong bai todn Id tan so dao ddng khdng thfl nguy&n dflpc dinh nghTaa> = ii»c'f/hJp~E^. K i t qud phdn tfch nam tdn sd dao ddng ddu t i t o dat dupc t f l cdc phUcmg phdp khde nhau dupc t h i hi^n d bdng 2. Hinh 2 the hien nam d^ng dao ddng t f l do eua tam FGM. Dua vao k i t qud cOa bdng 2, ta thdy khi dieu ki^n bidn thay ddi t f l CCCC sang SCSC, SSSS vd SF5F, bien d d cfla Ian sd flng vdi tflng dang dao ddng gidm ddn d i u ddn. Khi chl sd n gia tdng thi bien d d cfla tan sd flng vdi tflng dang dao ddng gidm ddn. K i t qud ve gia t n tdn sd dao ddng t u do dat dupc t f l phUdng phap d l xudt (S-FSD-MKG) rat phfl hop vdi cdc Idi gidl dat dupc t f l phuong phdp khdc

6. K i t luan:

Bdi bdo dd d l xuat mdt m d hinh phdn tfch dao ddng tU do cfla tam FGM sfl dyng md hinh phdn tieh k i t hpp giOa ly thuyet S-FSD vdi phuong phdp khdng lUdi MKG (S-FSD-MKG). Cac vi d u sd v l phdn tfch dao ddng t u do cfla tdm FGM duoe thuc hien va thdo Iudn chi t i l t . Cdc y l u t d dnh hudng d i n dao ddng t u do cua tam FGM chdng han n h u d i l u kidn bifen, chi sd d d suy gidm n cung dupc khdo sat K i t qua cho 6 hinh d l xuat n>di vdi sd I n sd it hem, nhung van c so vdi nhflng "kit qua gidi dUpc t f l cdc phuang

Mode shape 1

Hinh 2. Nam dang dao ddng ddu tien cua tdm FGM vdi dieu kiin biin CCCC

&ngv&n=1

(6)

n Phuang phap Tdn sd dao ddng

M o d e l Mode 2 Mode 3 Mode 4 M o d e s

Sai sd tuang M o d e l Mode 2

ddi cfla S-FSD-MKG (%) Mode 3 Mode 4 Modes Tdm FGM 4 canh bien SFSF

0.5

1

2

FSDT based IGA [27]

S-FSDT based IGA[27) S-FSD-MGK FSDT based IGA[27j S-FSDT based IGA[27]

S-FSD-MGK FSDT based IGA[27]

S-FSDT based IGA[27]

S-FSD-MGK

47.89 47.89 48.85 43.15 43.15 44.08 39.23 39.23 40.14

80.16 80.22 79.28 72.24 72.29 71.63 65.67 65.72 65.32

182 39 182.54 180 88 164.36 164.48 163.90 149.42 149.54 149.90

193.58 193 56 20449 17444 174.42 184.87 158.59 158.57 168 66

232.20 232.26 234.49 209.24 209.29 212.31 190 23 190.28 194.00

2.02 2.01

2.16 2.15

2 31 2.31

- i . n -1.18

-0.84 -0.90

-0.54 -0.61

-0.83 -0.91

-0.28 -0.36

0.32 0.24

5.63 5 £ 4

5.98 5.99

6.34 6.36

0.99 0.96

1.47 1.44

1.98 1.96 Tim FGM 4 canh bien SSSS

0.5

1

2

FSDT based IGA[27]

S-FSDT based IGA[27) S-FSD-MGK

98.14 98 13 95.94 88.43 88.43 86 78 80.40 80.40 79.22

245.25 245.22 245.23 221.00 220.96 222.53 200 92 200 89 203.85

245.25 245.22 245.23 221.00 220.96 222.53 200.92 200.89 203.85

392,20 392.14 382.39 353.42 353.36 350.29 321 30 32125 324.06

490,63 490.10 505.69 442,15 441.63 459.83 401.96 401.50 422.13

-2.24 -2.24

-0.01 0.01

-1.87 -1.87

-1.46 -1,46

0.69 0.71

1.45 1.47

-0.01 0.01

0.69 0,71

1.45 1,47

-2.50 -2.49

-0.89 -0.87

0 ^ 6 0.88

3.07 3.18

4.00 4.12

5.02 5.14

Tim FGM 4 canh bi^n SCSC

OJ

1

2

FSDT based IGA[27]

S-FSD-MGK FSDT based IGA[27 S-FSDT based IGA!27]

S-FSD-MGK

143.87 143.89 145.92 129.64 129.66 132.43 117 87 117 88 121.31

27198 271.99 265.47 245.09 245.09 241.10 222.83 222.82 221.05

344.35 344.31 352.73 310 32 310.27 319.65 282 12 282.08 292 37

469.55 469.59 453.24 423.15 423.16 412.42 384.69 384.70 378.85

508.01 507.54 506.61 457.82 457.36 461.42 416.21 415.80 424.33

1.43 1.41

2.15 2.14

2.93 2.91

-2.40 -2.40

-1,63 - 1 6 3

-0 80 -0.80

2.43 2.45

3.01 3.02

3.63 3.65

-3.47 -3.48

-2.54 -2.54

-1.52 -1.52

-0.28 -0.18

0.79 0.89

1.95 2.05

T i m FGM 4 canh b i e n CCCC

0.5

•

2

FSDT based 1GA[27]

S-FSDT based 1GA[27]

S-FSD-MGK

178,79 178.80 180 95 161 11 161.12 163.62 146.48 146.49 149.32

364.49 364.46 368.78 328 48 328 43 334.20 298.63 298 59 305.70

364.49 364.46 368.78 328.48 328.43 334.20 298.63 298.59 305.70

537 01 537.08 519.85 483.96 483.99 472.61 439.97 440.00 433.75

654.07 652.92 676.29 589.53 588.40 61478 535.92 534.93 564.16

1.21 1.20

1.56 1.55

1.94 1.93

1.18 1.18

1.74 1.76

2.37 2.38

1.18 1.18

1.74 1.76

2.37 2.38

-3.20 -3.21

-2.35 -2.35

-1.41 -1.42

3.40 3.58

4.28 4.48

5.27 5.47

Bdng 2. So sdnh ndm tdn sd dao ddng ddu tldnkhdng t h f l n g u y i n cfla tdm FGM T i b U E U T H A M K H A O

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