Phan tich tan so rieng he khung phang theo phiFO'ng phap khoi \wo>ng p *an bo
Nguyen Duy Hu'ng Nguyen Thanh Dat ABSTRACT
This paper presents mass distribution method for calculating of natural
frequencies of flat frame Euler- Bernoulli. Mass distribution method use form function depends on frequencies which are solved directly from the balance dynamic equations in each element of the frame.
The dynamic stiffness matrix elements, the overall frame structure was established based on this form function and the finite element method.
Eigenvalues equations of the frame are nonlinear and are solved by Wittrick-Williams algorithm. A general computer program to calculate the natural frequencies of the frame is written in MATLAB language.
The results show that the accuracy of mass distribution method is good, especially in the high frequency domain.
KS Nguyen Duy Hu'ng Truong Dai hoc Giao thong van tai TP.HCM
Di dong: 0986315308 Email: nguyenduyhung@
hcmutrans.edu.vn TS Nguyen Thanh Dat Pho Khoa xay dung
Truang Dai hoc Giao thong van tai TRHCM
Didong:0989051723
1. Oat van de
Ket cau khung la mpt trong nhu'ng dang ket cau daoc sir dung rong rai nhat hien nay trong nganh xay diTng. Do do viec nghien CLfu ufng xd chinh xac cua ket cau khung dUdi tac dung cua tai trpng la mpt nhu cau cap thiet, Hau het cac ket cau khung deu co v6 han bac t u d o khi chiu tai trong dpng. Oe giam he co v6 han bac tu"
do thanh mo hinh c6 huu han bac tU do ng\j6\
ta thuong diing mo hinh toan hoc. Dao dong cua khung phang c6 the duoc tap hap tU cac phan tCrdam, nhCrng phan tCrdupc phan tich tu"
nhCng he toa do rieng biet, trong do ma tran do CLfng dUPc thanh lap dUa tren phuong trinh can bang tinh cua phan tu" dam. Ma tran khoi luong CO the CO dang khoi lupng thu gpn tai cac diem ket cau hoac dupc xac dinh tCf ham dang bien dang tinh cua phan t i i dam. Phuang phap phan tCr hUu han la phuang phap so dung de phan tich ket cau. Co hai dang phuang phap phan tCr hUu han dua vao ham chuyen vj trong phan tich dong luc hocThir nhat la phuang phap xap xi, trong do npi suy cac chuyen vi bang each su" dung cac ham dang chuoi da thCrc. ThU hai, duoc xem nhu la phuong phap chinh xac, trong do noi suy cac chuyen vi bang each dung cac ham dang thoa man mot each chinh xac cac phuang trinh can bang tinh. Tuy nhien ca hai phuong phap tren deu bieu losUkem chinh xac bdi vi chung su dung cac ham dang dpc lap vdi tan so. Phuong phap khoi lupng phan bo co the loai trCr sU khong chinh xac nay bang each sd dung cae ham dang phu thupc tan so, la ldi giai chinh xac cua cac phUPng trinh vi phan chuyen dpng, do do chung ta tim duac I6i giai chinh xac eho bai toan dpng cua ket cau. Bdi vi cac ham dang dupe sd dung phu thupc tan so nen ma tran dp cCrng dpng [D] cung phu thuoc vao tan so. Luc nay bai toan tri rieng co dang nhU sau
[D]{qH01 (!) Trong phUPng phap khoi lUpng phan bo, so bac t u do nho hon rat nhieu so vdi khi sir dung phuong phap phan tCr hUu han. Sd bat Ipi cua phuong phap nay nam d sd sieu viet cua ma tran do cCrng dong luc va viec giai bai toan tri rieng phi tuyen.
2. Ctf sof ly thuyet
2.1. Dao dpng dpc true cua thanh Xet mot thanh thang (h'mh 1) co mat do
khoi la p(x), tiet dien la A(x}, modun dan hoi E.
Thanh thuc hien dao dong u(x,t) doc true x.
q{x,t)
p, E, A
Hinh 1: Dam co dac tri/ng thay doi
Khi dam la thanh thang dong chat tiet dien khong doi, phuong trinh dao dong doc tU do
^ - c ' ^ = 0 (2) Ta tim nghiem cua (2) dudi dang sau
u ( x , t ) = v ( x ) T { t ) (3) The (3) vao phuong trinh (2) ta co
v ( x ) T ( t ) - c \ ( x ) T ( t ) = 0
Do ve trai cua phuong trinh phu thuoc vao X, con ve phai phu thuoc vao t, nen hai ve do phai bang hang so. Goi hang so nay la -UJ2
^ - ( X ) T ( t )
Ta nhan duoc hai phuong trinh vi phan v ( x ) . v(x)=0
T ( t ) + a i ' T ( t ) = 0 Viet lai (4) dirdi dang
' + (.) pAv(x)=0
(4) (5)
EA
dx' (6)
2.2. Dao dgng uon cua thanh
Bo qua dao dpng xoSn va doc true, dam chi thuc hien dao dong uon theo phuong y.
f [•"
1 , i , Tiir:
i ElU),nm|
Hinh 2: Dao dong uon ciia thanh dam
Quan he giua do vong va goc xoay co dang r W ( x , t )
tgcp = - = o ( x , t ) (7) Tach mot phan to nho cua dam nhu hinh 3
09.2012 »E^nill?lV 8 5
P(x,t) n i l M l
M(x,t)
CI
V(x,t)mu ) dx
dx Hinh 3: Cac liic tren 1 doan phan to dam
Ap dung nguyen ly d'Alembert, xet can bang cac luc theo phuang dUng, ta co
^ Y = 0
V + p ( x , t ) d x - V dx - f d x = 0 (8) Luc quan tinh phan bo ^dx=|Adx
r t The vao phuong trinh (8) ta duoc;
~ = p - H — ^ (9) rx dt
TCr dieu l<ien can b^ng momen ta duoc
Bo qua VCB bac cao cua p va fi, ta co (10) (11)
(12) ex rx
Thay phuang trinh (9) vaophUOng trinh (12)
^ ^ . ^ = P (13) M +
hay Dao P'lVl
Vdx-
rx
M + — d x V ham rien
rV
= 0
g 2 ve vdi x dan tdi
Hay ^ E l — - + | . i ^ = p 14 ax' l^ ax' j t V
Xet trudng hop dam co tiet dien khong doi El—J- + M—r- = p(x,t) (15) Ta cd phuong
(X r\
trinh dao dong uon tU do
hay vi''^(x,t)+ — w ( x , t ) = 0
(16) (17) Nghiem cua (17) dudi dang phan ly bien so w " ( x , t ) + — w ( x , t ) = 0
The (18) vao (17) ta duoc n t K ( > ' ) - ^ T ( t ) v ( x ) = 0
(18)
(19)
(20) Ttrdd suy ra
_ T ( 0 ^ £ V > )
D J O ^ ptei^cfefj (20) la ham chi phu thuoc vao x, con ve trai la ham chi phu thuoc vao t, cho nen ca hai ve b5ng mpt hang so, ta dat hang sd nay la o)-. TC/dd suy ra
T ( t ) Elv»Xx)^
H v ( x )
> T ( t ) + , . r T ( t ) = 0 (21)
pA(o'v(x)=0(22)
2.3. Ma trdn do cuing dgng luc cua phan tit khung
N\b\ phan tCr l<hung phang c6 hai nut, moi nut cd ba bac tU do la 2 chuyen vi thang va 1 chuyen vi xoay. Phan tCr bi bien dang dpc true bdi thanh phan {u, j j } va bi uon bdi thanh phan {vi 6, «j 0^ }^. Cong tac dung cua hai trudng hdp chiu bien dang doc true va uon ta cd ma tran do cCfng dpng luc cua phan tCr chiu uon + l<eo (nen).
Ma tran do cu'ng dpng iuc cua phan tCr l<hung phang theo ly thuyet dam Euier - Bernoulli
J^ la so nguyen Idn nhat
\'EA (30)
0 0 k;.
Tuong tu, neu ham 1 -coshAsosX = 0 duoc ve ra, nd se cho thay rang J^, sd cac nghiem nho hon gia tri cua X la [31
J„ = i-0,5[1 - ( - l ) ' s g ( l -coshXcosX)] (31) trong dd i la so nguyen Idn nhat < V n va sg(l - coshXcosA) co gia tri +1 hoac -1 tuy thupc vao dau cua (1 - coshXcosX) la dUdng hay am.
NhUvay J^^^ dupc cho bdi
J„, = j"'+ Jb (32) Neu J < R thi F|^ duoc xem ia can dudi F lay
(23)
trong dd:
k*,, = k*jj = (EAi|j/L)cosit)/sinH); k*,, = k'^, - -(EAi|j/L)/sini|j; k^^ = k^^ = (EI/L)F,; k,, = ' k „ = (Ei/
L=)F^; k,, = k,, = - k „ = -k^, = - (EI/L')F^; k,, = k„ - (EI/L=)F^; kjj = k „ = (EI/L)F^; k^, = k^^ = - k „ = - k,,
= - (EI/L')F,
vdi v|(=a) L 1— va F^ (i = 1.. .6) xac dinh bdi;
F, - X ( s i n h A - sin/.)/6;
Fj=X(sinXcoshX-cos/.sinhX)/i); (24) Fj = X'(coshX-cosA)/5
F, =-X'(sinXsinhX)/6 F;=-X^(sinX + sinhA)/5;
Fg =X^(sinXcoshX + cosXsinh?.)/6 5 = l-cosh/.cosX
2.4. Thudt todn Wicttrick-Williams Oau tien ta chpn gia tri tan so thCr uj-F^.
Budc tiep theo ta tinh toan ma tran dp cdng tpng the cua khung va ap dat dieu kien bien, ta thu 6dqc ma tran D(uj). De tim tan so rieng thuf Rcua ket cau ta thuc hien theo cac budc sau. 56 lupng cac tan sd rieng cd gia tri nho hon F^^ la J dupc tinh nhUsau
' j = J„ + s(D(FJl (25) trong dd s{D(F J ) la sd cac phan t d am tren
dudng cheo chinh ciia D»(FJ, vdi D'(F^j la ma tran tam giac tren thu duoc tCf D(FJ bang each dCjng phep khCr Gauss. J„ ia so lupng cac tan sd rieng ed gia tri nhd hdn F^^ neu tat ea cac bae t u do tddng dng vdi q bi ngam lai, va duoc tinh thep eong thde [3]
Jo'X^- (26)
vdi J^^ la sd lupng cae tan so rieng ed gia tri nhd hon F^^ ciia cac phan t d cau thanh he cd 2 dau bi ngam.
Td phudng trinh (24), cae tan so rieng eua mpt phan t d cd hai dau bi ngam se xay ra khi
hoac sinip = 0 va ijj ^ 0 (27) hoac va l-coshXsosX = 0 X * 0 (28) Phuang trinh (27) se duoc thoa man khi
i|j = in 0 = 1,2,3...) (29) Vl vay J^, mdt thanh phan cua J^, phat sinh t d (27) duoc cho boi bieu thde sau
va F^^ duoc nhan ddi ien. Tinh toan lai J vdi gia tri mdi cua F^^. Neu J < R thi ta lap lai budc tinh tren lien tue cho tdi khi J > R. Ngupc lai neu J > R, khi do F^^ trd thanh can tren F^, luc dd F^ = F^.
Moi khi F^ va F, dupe thiet lap, qua trinh lap dupc tiep tue, moi gia tri mdi cua F^^
^' dupe tinh bdi [3]
F„ = (F„ + F,)/2 (33)
va trd thanh gia tri mdi cua F, neu J < R hoac gia tri mdi cua F^ neu J > R. Thu tue ehia ddi khoang dupe thUc hien cho den khi F^^ / (F^^ - F,)
> GCV, vdi GCV ia gia tri hdi tu cho trudc. Gia trj F^^
thda man dieu kien tren chinh la gia tri can tim.
3. Vl du phan ti'ch so
Oe kiem ehdng tinh ehinh xac eua phUdng phap khdi lupng phan bo, mpt vai bai toan phan tieh sd se duoc thuc hien. Tam tan so rieng dau tien cua moi bai toan he khung phang se dddc tinh toan va so sanh vdi ket qua dUde tinh bang 5AP2000 va mpt sd ket qua eua cac tac gia khac.
3.1. Bdi todn 7
Tinh tan so rieng cua khung phang tren hinh 4 IS). Tiet dien cac thanh hinh vudng 0,2m X 0,2m. Cd module dan hoi E = 3.0x10'° N/m', khdi lupng rieng p = 2400 kg/m'.
A A
Hinh 4: Khung phang [51
Sd dung ehuang trinh va 5AP 2000 de tinh toan, ket qua dupc the hien nhu bang 1:
Khi tinh bang PP khoi luang phan bd, khung dupe chia thanh 1, 2 hoac 3 phan t d de kiem ehdng su khdng thay doi cua ket qua. Khi tinh bang 5AP2000, khung dupe chia thanh rat nhieu phan t d d e khao sat su hpi tu cua ket qua. Ket qua thu duac eho thay so luong tan sd khi tinh bang phuang phap khoi lupng phan bo khdng phu thuoc vao viec chia phan td.Trong khi do SAP2000 chi ed the tinh dUdc so lupng tan so bang vdi so phan t d duoc ehia va ed su hdi tu eham. Ben canh dd, sai sd cua tan sd khi tinh bang 5AP2000 vdi ludi chia moi thanh 1 phan t d la rat Idn; khi chia moi thanh thanh hai phan tdthi ket qua kha chinh xac 6 4 tan so dau tien, nhung t d tan sd thd nam thi dd sai ieeh Idn. Tuy nhien, khi chia moi thanh thanh 3 phan t d t r d len thi ket qua se hoi tu nhanh.
Cae ket qua cua SAP2000, vdi dp sai sd cd the chap nhan duoc, deu nhd han cae ket qua cua PP khoi
8 6 ^ ' ^ i ^ ' K i i i ^ ' ^ 0 9 . 2 0 1 2
Bang 1: Cae gia tri tan so rieng (Hz). mdmen quan tinh i ~ lO^m".
Bang 2; Cae gia trj tan sd rieng (Hz)
Idang phan bd va dp chinh xac cd khuynh hudng giam doi vdi cac tan sd cao. Ket qua cua PP bien dp phCtc CO gia tri sai khae khong dang ke so vdi PP khoi lupng phan bd. Dieu nay cCing ehdng minh tinh ehinh xac cua PP khdi luong phan bo, vi ket qua cua PP bien dp phdc cd the xem la ket qua
PhUdng phap KLUdng Phan bd Bien dp phdc [5]
SAP2000 1 elem 2elem 3 elem 4 elem 5 elem 10 elem 20 elem 40 elem Phuong phap KLUOng Phan bo Bien dp phdc [5]
SAP2000 1 elem 2 elem 3 elem 4 elem 5 elem 10 elem 20 elem 40 elem
Tan sd rieng 1
2.071
2 14.232 2.072 1 14.247
3 20.355 20.365
4 31.882 31.898 1.981
2.034 2.042 2.045 2.046 2.047 2.047 2.048
117.47 13.886 14.032 14.051 14.056 14.059 14.059 14.059
117.58 18.574 19.599 19.832 19.925 20.035 20.06 20.067
185.75 30.240 31.260 31.312 31.314 31.310 31.309 31.309 Tan sd rienq
5 59.011 59.041
6 60.519 60.550
7 93.73 93.77
8 122.56
143.2 53.226 56.158 56.961 57.694 57.835 57.869
143.95 54.981 58.115 58.749 59.09 59.118 59.122
262.9 86.97 90.55 91.23 91.27 91.21 91.20
366.25 154.66 108.21 115.31 118.59 118.78 118.8 Chu y: khi tinh bang SAP2000, chung ta tinh vdi cdch chia phdn tdkhdc nhau
Phuong phap KLupng Phan bo
D5M[4]
SAP2000 1 elem 2 elem 3 elem 4 elem 5 elem 10 elem 20 elem 40 elem Phuang phap KLuang Phan bd
DSM[4]
SAP2000 1 elem 2 elem 3 elem 4 elem 5 elem 10 elem 20 elem 40 elem
Tan so rieng 1
0.471
0.456 0.464 0.465 0.466 0.466 0.466 0.466 0.466
2 1.5466
1.3229 1.5078 1.5213 1.5253 1.527 1.5292 1.5297 1.5299
3 3.3988
22.749 3.3244 3.3564 3.3601 3.3609 3.3614 3.3614 3.3614
4 4.8217
22.772 4.5833 4.7434 4.7585 4.7614 4.7625 4.7624 4.7524 Tan sd rieng
5 5.3226
45.51 4.9051 5.1749 5.2196 5.2349 5.2519 5.2556 5.2566
6 7.0027
7.038 55.723 6.0592 6.7998 6.8785 6.8937 6.9012 6.9017 6.9018
7 7.027 7.038 55.73 6.071 6.822 6.902 6.917 6.925 6.926 6.926
8 7.916
55.73 6.6479 7.6103 7.7361 7.7705 7.8016 7.8077 7.8091 chinh xac.
3.2. Bai todn 2
Tim cac tan so dao dpng rieng cua khung 2 tang [4] nhu hinh 5. Modun dan hoi Young E = 10' N/m^; mat cat ngang chU nhat co dien tich A = 10-^ m=; khoi lupng rieng p - 1000 kg/m^;
Hinh 5: Khung phang [4]
SCrdung chuong trinh va SAP2000de tinh toan, ket qua thu dupc nhu bang 2:
Tuong t u nhu bai toan tren, tam tan so rieng dau tien se dUOc tinh toan bang phuang phap khoi lupng phan bo va SAP2000. Ben canh do ta cung so sanh vdi ket qua cua T.H.Chuan [4]. Cac gia tri tan so duoc the hien trong bang 2. Qua vidu nay tacung thay dupc su chinh xac va hieu qua cua phuang phap khoi lupng phan bo.
4. Ket luan
Phan tich dao dpng t u nhien cua ket cau khung phang bang phuong phap khoi lUpng phan bo da dupc trinh bay trong bai bao nay.
Cac ket luan dupc the hien nhu sau:
Ham dang sieu viet cho moi phan tLT dang thanh dUpc suy ra bang each giai phuong trinh chuyen dong du6i dang vi phan dao ham rieng. Cac ma tran dp cufng dpng cua tUng phan tCf va tong the cua khung phang dupe thiet lap. Phuong trinh tinh tan so rieng cua ket cau khung phang ddpc giai bang thuat toan Wittrick - Williams.
- Mpt chuong trinh may tinh de tinh toan tan so rieng cua khung phang dang tong quat vdi so luong nhip va tang bat ky da dUOc xay dung. Ket qua so cung da duoc kiem chCrng vdi phan mem SAP2000.
- Phuong phap nay co dp chinh xac kha tot dac biet la trong mien tan so cao. Vdi lUdi moi mpt thanh dam hoac cpt dupc chia thanh mot phan td, sai so cua tan so rieng la kha nho va nhd hon nhieu so vdi phuang phap phan tCr hUu han dung ham dang da thde Hermit bac 3 cho phan t d thanh.
TAl LIEU THAM KHAO
[1] Chopra A. K., DynamiG of Strudures, Prentice-Hall, New York, 2007.
[2] Williams f.Vi.. Wittrick W.H., An automatic computational procedure for calculating natural frequendes of skeletal strudures. Int. J. Mech. Sci., 12,781-791,1970.
[3] Williams EW., Wittrick W.H., A general algorithm for computing natural frequencies of elastic structures. Quart.
Journ. I^ech. and Applied. )0(IV Pt 3,263- 284,1971.
[4] Tsai Hsiang Chuan, A distnbuted-mass approach for dynamic analysis of Bernoulli-Euler plane frames. Journal of Sound and Vibration, 329,3744-3758,2010.
[5] Nguyen Xuan Hijng, Tinh toan chinh xac ket cau tren may tinh, chUdng trinh ADS - 2001, Ha Noi: Nha Xua't ban Khoa hoc vaKy thuat, 2001.
