Tuyen tap cdng trinh Hdi nghi cffhgc todn qudc
Ky niem 30 ndm thdnh lap Vien Cff hge vd 30 ndm Tgp chi Cff hge Hd Ndi, ngdy 8-9/4/2009
Dao dong phi tuyen tam composite lop co dang luon song
Khiic Van Phu, Le Kha Hoa
Hoc vien Hciu cdn
Tom tat: Tren ca sd cpian hi chuyin vi - biin dgng do OS. TSKH Dcio Hiiy Bich di xudt vd md rong cdch tiep can ciia Seydel ddi vdi vat lieu composite, bdi bdo thiit lap hi phuang trinh chuyin dong cua tdm composite lap dgng licgn sdng cd xet din yiu td phi tuyen hinh hoc. Sic dung phuang phdp Bubndv-Galerkin vd phicang phdp Runger-Kutta de phdn tich ddng luc phi tuyen tdm composite lap luan sdng, da nhdn dugc quan he giica tdn sd-bien do dao ddng phi tuyin vd cdc ddp icng phi tuyin ciia ldm. Kit qud sd nhdn dugc cho thdy dnh hudng eiia cdc tiidng sd hinh hge vd dieu kiin biin din ddp icng ddng hoc ciia tdm composite lap dgng licgn sdng.
1. M6' diu
Cae ket cau composite lop dang tam va v6 mong eo gan gia cuong hoac eo dang hinh hoc lugn song dugc sir dung nhieu trong thuc tien. Nhieu ket qua nghien ciru cac bai toan tTnh va dgng ciia tam phang composite lop dugc phan anh tap trung trong cae cu6n sach ciia Chia |4|.
T6ng quan ve mgt loat bai bao tinh dao dgng ciia tam phang duge de cap trong cong trinh ciia Sathyamoonthy [5]. Tuy nhien doi vai tam eo dang lugn song thi moi nhan iugc mgt vai ket qua nghien ciru [1, 3]. Trong bai bao nay, tae gia tap trung nghien ciru ve dao dgng eiia tam composite lop lugn song co xet den yeu t6 phi tuyen hinh hge.
2. Thiet lap phuong trinh dao dong phi tuyen ciia tam composite lop luon song
Hinh 1: Mo hinh eiia tam CPS lop lugn song
Xet tim composite lop co dang lugn song (hinh 1), chiu tae dung ciia lue phan b6 deu q vuong goe voi mat phang trung binh eiia tam. Gia thiet tam eo dang lugn song theo phuang true X, duong trung boa dugc bieu dien duoi dang ham sin:
212 Klnic Vdn Pint, Li Khd Hod
. nx z = H sm —
/
trong do: H - la bien do nua song (mui ten H), / - la do dai nira buac song Theo [3] ta eo quan he giira truong bien dang va trirong chuyen vj:
du 1 ( d^v
(2.1)
'' = &ni^|-'^"'
?!• 1
CI' 2 GW
r.vv du_ dv)
^d}' dx
dw 5)1' dx d}'
Xy=-
J.v,.=-2 d'w d'w dy
d'M'
(2.2)
dxdy
trong do: k la do cong ciia duong trung boa trong mat phang (x, z)
k= ^—.r-r^l
( , . . - ) •
- / / ^ s i n — vai H « l , /- /
u, V, w la chuyen vj eiia mgt diem bat ky thuge mat giua eiia tam theo phuong tuong irng vdi eae true x, y, z.
Tir quan he irng suat bien dang, tich phan theo chieu day tam va phat trien each tiep can ciia Seydeldoi vai vat lieu composite, ta nhan dirge bieu thirc lire va in6 men dii voi tim CPS luo'n song d6i xiina;:
N,=A^\.e, + A^je^, Ny = A ,,.£_, + A 22.Sy
M.,=D;^.z, + D;2.Zy,
A/,, = D|Vj, + D*2.j, , (2.3)
trong do Aij [i,J = 1,2,6) la eae hang so do cimg mang. Cac hing s i do cirng uon D* dugc xac djnh theo quan diem ciia Seydel [7] th6ng qua cac hing so d6 cirng uon D,j eiia tim phing, tuong irng theo e6ng thirc:
s I
trong do: D3 = Z),, + 2D( 56 / = h.H' 0,81 l + 2,5.(-)2
2/
;r H
\if^
cos — d x w /. 1 + ~ 2 ^X \ TT' H' ,I V ir-
Theo [6] phuong trinh chuyen dgng eiia tam co dang:
Dao ddng phi tuyin tdm composite lap cd dgng lugn sdng 213
dN,. dN_, dx dy dN^,. dN..
• = . / ,
d u dr
— ^ i
d'v
(2.4)
dx dy dt' d'M, d'M.y d'M d
r i + 2 •^ + r ^ + — 2 ^..^. -,,.2 g dx'
dy
dxdy dy ., dw dw N^,.—+N^ —
•'•' dx ^ dy j
., dw 8w N^ — + N —
^ dx dy J d^w
+ <7 = -^o-rT---^2 dr
d W O W
dx^dt- dy-dt-
N Ih
trong do: -'i = ^ ( P^^^^'dz (i = 0, 1, 2); p ' * ' la Idioi lugng rieng a lap thir k; n la s i lop ciia
tam)
Thay (2.2) vao (2.3) ta nhan d u g c quan he giiia noi lire va chuyen vj N=Ay du
Ny=A,
'V.V. = '^(.e
dx 21, dx ) du 1 ( dw\
— + - -kw dx 2{dx )
du dv dw dw dy dx dx dy
+ A,-,
+ /!,,
9v ]_
dy 2 dv 1
— + — dy 2
dw s'5y ,
9 VI'
M^ = -D 11-
dx 1 M 2 - . J •
dy' dy . Mv=-D^,.—^-D,2.—T-
ox' dy
A/ - i n * 9 " " '
M,y=-2D(,^. dxdy
(2.5)
Thay (2.5) vao phuo'ng trinh chuyen dgng (2.4) ta thu duge he p h u o n g trinh vi phan dao ham rieng d6i voi cac chuyen vj u, v, w:
d'li d'u d'v , dw d'w , dw d'w
dx^ dy- ^ ^^'dxdy " dv dx' dx dy'
I , ^ -.dw d w , ,,je Tix , ,,7r' . nx dw , d'lt + {^2+ A(,)'T--^—+ A^wH — COS--+ A^^H —sm—.-— = Ja-ry,
dy dxdy r I I' I ax dt' d'v , d'v , ^ , , d'u , dw d'w , dw d'w dy dx dxdy dy dy ay dx' / , . sdw d'w , HTT' . nx dw , d^v
+ {^2 + ^ 6 6 ) — T - ^ - + -^12 - ^ ^ S ' " - r i r = -^0 T ^ ' dx dxdy / - / dy dt'
.. 8\v , . d^w 1 , f dwX d'w 1
0-^.2(D:,.2D-J^.O:,^--A,^^J^--A.,
dw ] d'w1 f dwY d'w 1 '2'\'d^) '8^'2 ''
dw
dy j dx' du d'w d'w du d'w du d'w
dy dx dx dx dy dy dxcy - 2 4 dv d^w , H-K' . TLv d^w , dv d^w , Hn' . TUX d
66,_,_,:. -'ii-7r-s'"-r^^rT~^i2
dx dxdy V' I dx' dy dx'd'w
-A ,2 / . -sin — ir dv d'w dw d'u dw d'v _
dy dy ox dt dy at dt 'Ji
I dy' ' d\v d'w \
^dx'dt' dy'dt' ,
(2.6)
214 Khiic Vdn Phu, Li Khd Hod
Day la he phuang trinh vi phan phi tuyen tong quat eiia ly thuyet tam CPS lop dang lugn song. Cae phuong trinh nay diing de khao sat bai toan tTnh va dgng eiia tam CPS lap lugn song.
3. Phuong phap giai
3.1. Dao dgng phi tuyen cua tain composite lop lugn sdng
Khao sat tam composite lugn song hinh ehu' nhat co eae th6ng s6 hinh hge nhu hinh I. Tam chju lien ket ban le bon canh. Khi do truong chuyen vi eo the chgn nhu sau:
-, I -. mnx . nnv
" = '^m„(')cos s m — ^
a b ,, /,s . mnx nnv
a b
„, . N . mnx . nny w = lr„„, (/jsin sm -
(3.1)
a b
Thay (3.1) vao (2.6) sau d6 sir dung phuo'ng phap Bubnov - Galerkin ta thu dugc he phuo'ng trinh:
^I l^„»i (') + ^n'"™, ( 0 + «13'^„m (0 + «14'-'',™ (0 + ^15 ^""' ( 0 = 0'
«2l'™i (0+^22^^™, (') + «2.l'f^™, (') + «24'C (0+«25 '''"" ( 0 = 0 -
«.,"',„„ (')+«32"»„ (0+«.^3»'i(0+«34^,™ (')W™. ( 0 + ^35'^,™ ( 0 " ' " ' . (') +«36'^',„„ {!)Umn {i) + a.JV„^„ {f)V„,„ (t)+0^^ JV„,„ (/) = fl,, q{t), trong do:
ab J I ' " ^ . { nn
AA I +A6 mnn'
{Ai+'^bb)^
^ Hn'mb(a -2m I' ~a~H)(, na
«i3=^ii J-, r 7 \ 1-C0S-- 2la[a'-Am-l-) V /
« 2 4 = '
Aab 9mnn' ah
4 ^bb 4ab 9m nn'
ad
\ "'
I a
a/J
\^ 2 3
2 4 , -{A^,-A,,)^~-
ei j ab' ab
J„
. 1 nn
b
Hn'nrnl
«22 = «I2. "23 = - ^ P — TT
a' -4m I
-cos-
2 3
m nn
ab
2A22\~j-{A2-Ab)^,^
n- I '"^ , o/'n* , Tn* \i'"'n~7T , nn
0,5 =a\s.
«„ =(-l)"'"'a ab 128 3 4
mn
AA~\ +Ai\ — Sabm^l'H
i(m''r'-a'')(9m''r'-a''f"' I "
2 2 4
m n n
+ 2 ( 3 4 2 - 2 4 6 ) ^ ^ ^ ^ + 3421 , a'b' \ b
(3.2)
Dao ddng phi tuyin tdm composite lop cd dgng liegit sdng 215
8a6
9mnn
?,ab -ccp
,3 2 3
., . 1 inn \ ,^ , , s mn n
24.1 I +(242+4<;)-'11
a ab'
"35
9mnn -ap
N3 2 3
-, . \ nn \ . , , \m nn 2 4 2 I — ) + ( 2 4 2 + 4 6 ) - ^ ^ -'36
8fe 9n;r aP,
Sa ab \ fln = -Jn ap, O3S = -—^•'37 JQ + J2
9mn Aab
mn a
mnn'
-aP, voi a = 0 neu n clian
1 neu n le P =
neu m chan neu m le
Theo [1], [3] d6i voi bai toan dao dgng duoi tac dgng ciia lire kieh dgng vuong goc voi mat trung binh ciia tam thi chuyen vj u, v thuong eo gia tri rat nho so vai w. D6ng thai theo Volmir [9], neu kh6ng tinh den sir truyen song trong mat phang tam thi cae s6 hang quan tinh theo huang nay eo the bo qua nen he phuong trinh (3.2) eo the viet:
«l l^,„„ (0 + "vJnm (0 + "BI^^/™ (0 + «I4^/™ (0 = 0,
«21 ^''/m. ( 0 + «22^m« ( 0 + «23^^«„, ( 0 + ^24^™, ( 0 = 0' aiX^V,„n ( 0 + «32fn'« ( 0 + « 3 3 ^ i (/) + «34^™, ( 0 ^ ™ (') + + «35^'™, {}YK,n ( 0 + ^38 «^«"' ( 0 = ^^39 "7(0
(3.3)
Ta thay hai phuo'ng trinh dau ciia (3.3) la hai phuong trinh dai so tuyen tinh d6i vai
^m,XO,y„„Xt), vi vay ta co thi bilu diin U„,„{'), V,„„{t) theo W,„„{t) sau do thay vao phuong trinh thir ba ciia (3.3) ta thu dugc:
W,„„ + col [Km + 7 ^L + ^^L ) = P,n„ ( 0 (3.4)
trong do:
vo'i:
k,=
h =
2 «31
«38
'^2l"l3 - ' ^ I 2 ' ' 2 3
" l l « 2 l - ' » 1 2 " 2 2 fl||fl23 - « 2 2 " l 3
"^11^21 ~ " l 2 " 2 2
034^1 + « 3 5 * 3 + " 3 2
^31
j^ _ « 2 I « 1 4 - « I 2 « 2 4
fl,|fl21 - a i 2 ' 2 2 2 , ai|fl94 - « 2 T f l | 4
^4 -
''ll'^2l - ' ' ' l 2 ' ' 2 2
^«33+«34^2+«35^4_ ^^^^^_ (;) ^ i?3L^^(,-) ( 3 5 )
Tnedng hgp dao ddng tie do tuyin tinh
Voi dao dgng tu do tuyen tinh, tir phuong trinh (3.4) suy ra:
W„,n+C0lW„,„ = 0
Do do Q)Q chinh la tan so dao dgng rieng eiia tam:
.-.-.J3.6)
Dn I I +
2 2 4 r
^ / ,.,• ^^f \m n n ,^* nn
2 ( z ) , 2 + 2 D , , ) - 5 ^ + D , [ - ^
•'38
J0+J2 mn a
nn
216 Klnic Vdn Phu. Li Khd Hod 3.2. Quan he giua tdn s6 vd bien do dao dgng tie do phi tuyen
Thay JK = Acosat vao phuong trinh thuan nhat (phuong trinh dao dgng tu do phi tuyen) cua(3.4)ta dirge:
X = A{ col- (o -\cos CO t + q A ' 0) I COS' w t + AA' CO 'Q COS' (01 = 0
2i!tm
Ti'eh phan tren mgt ehu ky dao dgng: f Xcoscotdt^O din din (voi dieu kien
0
A^o. (o^oy.
, , 2 ^ ^ / l 3 ^ 0 (3.7)
£)at 1/- = . ^ : binh phuong ti so tin s i dao dgng phi tuyin va tuyen tinh eiia tam, ta nhan dugc:
co\ 0 1 ^^ . 2
y = 1 + — A 4
Nhu vay trong dao dgng phi tuyin cua tim composite, d6 thi bieu dien quan he giira binh phuong ti s6 tan so va bien do la duong cong parabol.
3,5
§• 2,5
Hinh 2: Quan he binh phuong ti s6 tan s6 va bien do trong dao dgng phi tuyen truang hgp ban le b6n canh
3.3. Phdn tich dgng lire phi tuyen cua tdm composite lap lugn sdng
Khao sat irng sir ciia tam composite lop lugn song, duoi tac dung ciia lire dgng phan b6 deu /'(') = Po s'n ^ ' ta xuat phat tir phuong trinh (3.4) irng voi in6i cap (m, n) xac djnh.
- Xet khi/« = ;? = ] phuong trinh (3.4) CO dang:
"•' + «^"^('",„„+7"'";™^^"'»;l) = p " ( ' ) (3.8) Liic nay ;(„,„ dong vai tro gia trj chuyen vj (do vong) tai diem giira eiia tam, (n^," la tin si
Dao ddng phi tuyen tdm composite lop cd dgng lugn sdng \\1
dao d6ng rieng thir nhat ciia tain:
cor'=-
D n ^ + 2 ( D , ; + 2 D ; ) ^ + D ; 2 - i
^1 » ' rf- It- H a b
J o + 4 ^ - — + 7T .a- b'
(3.9)
77'", /l''' la cac gia trj bieu thirc 77, X dugc tinh theo c6ng thirc (3.5) irng vai m = n = \ 16pQsinQ/
P (0 =
, 7 ' I 1 '
a' b'
(3.10)
Phuong trinh (3.8) dugc giai theo phuo'ng phap Runger - Kutta.
4. Khiio sat so
4.1. Biin bdn li bdn cgnh
Xet mgt tam CPS hinh ehu nhat xep lap d6i xirng eo dang lugn song, eo lien ket ban le 4 canh, chju lire phan b6 deu c6 tan s6 va bien do luc kich dgng la Q = 40, p^ = 300NI nr Tain c6 th6ng s6 hinh hge va eau tao nhu sau: a = 0,9 m; b=l,5in; Cau triic xep lop: [45"/-45°/-45°/45"].
Chieu day mgt lop h = 1mm, bien do nira song H=4em, d6 dai nira buac song 1=0,3m. Hang so dan h6i eiia vat lieu AS4/3501 graphite/epoxy:
£, = 144,8 C P a , ^ , =9,67 GPfl,G,, =G^. =4,\4 GPa, 1^12=0,3 p = 1 3 8 9 , 2 3 ^
m'
Kliao sat s6 ta thu duoc d6 thi:
Hinh 3: Ddp img phi tuyen ciia tdm CPS lap lugn sdng (p(,=300N/m')
Khi tang s6 diem len ta duoc d6 thi hinh 5.
Hinh -/' Ouan hi dw/dt vd w.
Hinh 3 va hinh 5 bieu dien dap irng phi tuyin ciia tam. Quan he giira dvv/dt va vv dirge bieu dien tren hinh 4. Hinh 6 bieu dien dap irng phi tuyen eiia tam theo thai gian voi cae bien do tai trgng 77, =100V/m- p2=200Nlm' p,=2QQN/m-
218 Khiic Vdn Phu, Le Khd Hod
i P3=300 Wm''. P2=700 N/m
Hinh 5: Ddp img phi tuyin cita tdm composite Hinh 6: Ddp i'mg phi tuyin cita tdm img vdi lap lugn sdng (po=300N/m') cdc biin do tdi trong khde nhau Khao sat anh huong eiia bien do niia song H (miii ten H) den tan s6 dao dgng va he s6 phi tuyen ciia bien do dao d6ng. Ket qua khao sat duge the hien trong bang 1:
Bdng 1: Anh hudng cita miii ten H din binh phuong tdn sd dao ddng vd cdc hi sd phi tuyin ciia phuang trinh vi phdn
H (cm) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
«S7^(S"')
1.2416e+003 1.7793e+003 1.8022e+003 1.8407e+003 1.8950e+003 1.9658e+003 2.0538e+003 2.1596e+003 2.2844e+003 2.4292e+003 2.5952e+003
77 (m-')
0
9.037 le+003 1.7756e+004 2.5947e+004 3.3436e+004 4.0086e+004 4.581 le+004 5.0567e+004 5.4355e+004 5.7210e+004 5.9194e+004
A(m--) -5.81284e+004 -5.6556e+004 -5.5837e+004 -5.467le+004 -5.3103e+004 -5.1190e+004 -4.8998e+004 -4.6596e+004 -4.405 le+004 -4.1426e+004 -3.8776e+004
Til' ket qua khao sat tren, ta thay rang khi mui ten H tang len thi tin s i dao dgng, he s i phi tuyen ciia bien do dao dgng tang len. Dilu do chiing to khi mui ten H tang len thi kha nang chju lire eiia tam tang. Vai H=0, tam tro ve tim phing, khi do ta thay cac c6ng thirc xac dinh tin si dao dgng, he s6 phi tuyen cac d6 cirng D* bang D .
Khao sat anh buong eiia chieu day tain, kit qua khao sat nhan dugc nhu bang 2:
Nhin vao bang 2 ta thay neu chieu day eiia lop tang len (tu'c chieu day tim tang) thi tan si dao dgng cang tang, gia trj he s6 phi tuyen va he s6 lire kinh dgng cang nho. He s6 lire kinh dgng giam dieu do chirng to:
+ Khi chieu day tam tang thi kha nang chiu lire ciia tam tang.
Dao ddng phi tuyin tdm composite lap cd dgng lugn sdng 219
+ Chieu day tam tang den mgt gia trj nhat dinh, thi anh huong eiia ylu to phi tuyin hinh hge la kh6ng dang ke. Liic do bai toan chi can xet den m6 hinh tuyin tinh.
Bdng 2: Anh hudng cita do ddy lap din binh phieong tdn sd dao ddng vd cdc hi sd phi tuyin cita phuang trinh vi phdn
h (mm) 1 1.5 2 2.5 3 3.5 4 4.5 5
(0- (s-')
2.2844e+003 4.4739e+003 7.5391e+003 1.1480e+004 1.6296e+004 2.1987e+004 2.8554e+004 3.5995e+004 4.431 le+004
77 (m -')
5.4355e+004 2.7754e+004 1.6470e+004 1.0816e+004 7.6189e+003 5.6466e+003 4.3478e+003 3.4488e+003 2.8014e+003
X (m --) -4.405le+004 -2.2492e+004 -1.3347e+004 -8.7653e+003 -6.1746e+003 -4.576le+003 -3.5236e+003 -2.7950e+003 -2.2704e+003
fl3(,/cr3S
-0.0292 -0.0194 -0.0146 -0.0117 -0.0097 -0.0083 -0.0073 -0.0065 -0.0058 Cung voi lue kieh dgng po=300 N/m" va Q. = 40, nhung khi bien do nira song H=0 ta thu dugc d6 thj dap irng phi tuyen eiia tam eiia tam phang:
Nhin vao hinh 5 va hinh 7 ta thay dao dgng phi tuyen eiia tam lugn song va tam phang deu CO dang phach dieu hoa. Tuy nhien cimg voi lire kich d6ng nhu nhau nhung voi tain lugn song (hinh 5) thi bien do dao dgng nho hon (.W^^.^ ~1*\^~^m) so voi bien do dao dgng eiia tam phang (J^^ax ~ 3 * 10~ m) (hinh 7). Dieu do chirng t6 tam lugn song co kha nang chiu lire t6t ban tam phang.
^ - 2 1 0
Kinh 7: Ddp irng phi tuyin cita tdm CPS lap phdng
Hinh 8: Quan hi w — dw/dt cua tdm phdng CPS lap
4.2. Ngam bon canh
Hinh 9 in6 ta dap irng phi tuyin eiia tim CPS lap lugn song, hinh 10 mo ta quan he giira bien do dao dgng diim giira tim va dao ham bien do theo thoi gian trong truong hgp dieu ki$n bien la ngam bin canh. Nhin vao hinh 10 ta thiy dao dgng ciia tam trong truang hgp ngam bon canh eung co dang phach dieu hoa. Tir hinh 9 va hinh 5 ta thay ring trong truong hgp ngam b6n
220 Khiic Vdn Phii, Li Khd Hod
canh thi bien do dao dgng ciia tim la nho hon (1)',,,,^ «1,8*10 ' w ) trong truong hgp ban le bin eanh(H' ~ 7*10"''7;7).
C 0 5 .2 -15 .05
Hinh 9: Dap irng phi tuyen tam Composite lop lugn song (N4)
5. Ket loan
Hinh 10: Quan he w - dw/dt (N4)
Tren co' so de xuat ve truang bien dang chuyen vj eiia GS.TSKH Dao Huy Bich d6i vdi 1am CO dang lugn song va mo rgng each tiep can eiia Seydel d6i voi vat lieu CPS, cac tac gia da xay dirng duge he phuong trinh chuyen dgng ciia tam CPS lap dang lugn song co xet den yeu to phi tii>'en hinh bgc.
Phan tich dgng luc phi tuyen ciia tam CPS lop lugn song voi eae dieu kien bien tren. Cac dap ling phi tuyen dugc tinh toan theo phuong phap Runger— Kutta.
Tdc gid xin chdn thdnh cam on OS. TSKH Ddo Huy Bich dd hudng ddn hoan thdnh cdng Irinh nay.
Cdng trinh ndy hoan thdnh vdi sir tdi trg cita de tdi OGTD-08-07.
6. T a i Heu t h a m k h a o
[1], Khiic \'an Plu'i, Le Van Dan (2007), Dao dong tam composite lop co dang lugn song, Turin tap cong Irinh khoa hoc Hdi nghi Ca hoc loan qudc Idn thic S, Nha .xiiSl ban Dai hoc Bach khoa Ha Noi.
Tran Ich Thinh (1994), \'dl lieu Compozil ca hoc vd linh todn kit cdu. Nha xuSl ban Giao due.
Dao Huy Bich, Khue Van Phu (2006), 'Non-linear analysis on stability of corrugated cross-ply laminated composite plates', Vietnam Journal of Mechanics 2S(4). pp. 197-206.
C.Y.Chia. (1980), Non-linear analysis of plates. Mc Grant-Hill. Inc.
M. Sathyamooitly. (1987), Non-linear vibration analysis of plates: a review and survey of current developmenf. Applied Mechanic Review, 40. 1553-1561.
Redch^ .(.N. (2004). Mechanics (f Laminated Composite Plates and Shells: Theory and analvsrs. 2'"' Edition. CRC Press.
'^•-"'l-dell E, (1931). Schuhknickvcr.suche mil Welhlechlafeln. DVL
i-jci i c i i t .
[2].
[4].
[5].
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