Ứng dụng phương pháp xoáy rời rạc để xác định và khảo sát các đặc tính khí động của cánh khí cụ bay trong dòng khí dưới âm

1
DANH MUC CAC K ^ HIEU
0

a

Toe do am thanh trong dong khong nhieu, (m/s).

b

D6 dai dac tnmg, (m).

b^.3^

Day eung khi dong trung binh c u a canh, ( m ) .

bjjj

D a y cung ddu mtit canh, ( m ) .

b

Day eung g6'c cua canh, ( m ) .

I

Sai canh, (m).

S

Di6n tieh cua canh khi cu bay, (m^).

p

Mat d 6 cua khong khi, (kg/m^).

p

A p sua't dong khi, (N/m^).

q = ^ ^

D o n g ap, (N/m^).

Y

Luc nang eiia khi cu bay, (N).

X

L u c can chinh dien eiia khi cu bay, (N).

Y
c = —

H6 sd lire nang eiia canh khi cu bay.

^^

Cy

D a o h a m h6 s6' luc nang theo goe t^n a

Cy"

Dao ham he so luc nang theo goe quay co^

m"

Dao ham he so momen doc theo goe tan a

mf

Dao ham he so momen doc theo goe quay co^

m^'

Dao ham he so momen ngang theo goe quay co^

M

So Mach cua dong khong nhilu.

Re

So Reynolds.

UQ

van toe eiia dong khong nhieu dong, (m/s).

YV

Toe do cam umg khong thu" nguyen.

X, y, z

Toa do cua mot di^m, (m).

^ = _ ; y] = i-; ^ zz — Toa do khong thur nguyen cua mot diem,
b
b
b
' *
a

Goe t&i, (do).


p

2
Goe trucrt canh, (d6).

X

Goe mui tSn canh, (do).

r.
r=

Cudng do eiia xoay, (mVs).
Cudng do khong thii nguyer

o

The van toe, (mVs).

m

J'
's

A=

Do gian dai eiia canh.

Tl =

D6 that eiia canh.

K


3
MUCLUC

DANH MUC CAC KY HIEU

1

MUC LUC

3

M6DAU

6

CHUONG I: TONG QUAN CAC PHUONG PHAP XAC DEMH DAC
TINH KHI DONG CUA

CANH

8

TRONG DONG DUCtt AM

1.1 He true toa d6

8

1.2 Canh eiia khi cu bay, cac tham so hinh hoc

9

1.3 Cac dac tinh khi dong eiia canh khi cu bay

12

1.4 T6ng quan cac phuofng phap xac dinh dac tinh khi dong

cua canh khi cu bay trong dong khi dudri am

14

1.4.1 Phudng phap tinh toan ly thuyd't

14

1.4.2 Phudng phap thirc nghiem

16

1.4.3 Phuong phap vat ly khi dong

17

CHUtJNG H: T R U O N G VAN TOC CAM UNG Bdl CAC HE XOAY

18

TRONG DONG KHI DUCil AM
2.1 Trudng van toe cam ling boi doan xoay

18

2.2 van toe cam utig bcri he xoay xien hinh mong ngUa

21

2.3 van toe cam ling bcri xoay xien hinh mong ngua trong

cac trudng hdp rieng

27

2.3.1 van toe cam utig bcri he xoay xien hinh mong ngua

27

khi goe x=0
2.3.2 Van toe cam utig bdi he xoay xien hinh mong ngua
khi goe X'^O , y=0

28


•

4

2.4 Van tdc cam ung bcri mat phang xoay
CHlTONG m: P H U O N G PHAP XOAY RClI RAC XAC DINH DAC

29
31

TINH KHI DONG CUA CANH KHI CU BAY TRONG DONG KHI
f.

A

DUOIAM
3.1 Dinh ly Giukovsky cho phSn tijf canh eo sai hihi han

31

3.2 Bai toan xac dinh cac dac tinh khi dong cua canh

34

3.3 Di^u ki6n bi6n

35

3.4 M6 hinh xoay

38

3.5 H^ phudng trinh xac dinh cudng d6 eiia eae xoay

44

3.6 Xac dinh eae dac tinh khi dong cua canh

46

3.7 Cac dac tinh khi dong eiia canh trong dong khi chiu nen

47

dudfi am

CHUONG IV: KET QUA TINH TOAN VA KHAO SAT CAC DAC

50

TINH KHI DONG CUA CANH 6 TOC Dp DU6l AM
4.1 Gidri thieu ehuong trinh xac dinh cac dao ham khi dong

50

eiia canh
4.2 Ki^m nghiSm do hoi tu va do chinh xac

51

4.2.1 D o h o i t u

51

4.2.2 Do chinh xac

57

4.3 So sanh vdi ket qua t h i nghiSm trong 6'ng thdi khi dong
duciamOT-l

59

4.3.1 M6 ta thi nghiem

59

4.3.2 Che' d6 thdi va cac ket qua do

60

4.4 Xac dinh va khao sat cac dac tinh khi dong cua canh

64

4.4.1 Su phan bo he so dao ham C "^ theo sai canh

^^

4.4.2 Su phan bo ap sua't theo day cung cua canh

66


5
4.4.3 Su phu thu6c cac dao ham khi d6ng vao hinh dang
^

69

cua canh
4.4.4 Su phu thuoc eae dao ham khi dong vao so M

•K^TLUAN

TAI LIEU THAM KHAO

80
. 8 3

85


6
MCJDAU
Su ra d5i va nhip do phat tri^n manh me cua nganh ky thuat Hang khong
lu6n gan liin vdi nhflng thanh tuu eiia Imh vue khi dong hoc, chuydn nghien curu
cac qui luat ehuy^n dong cua eh^t khi va su tac dung tudng tac giua dong khi vdfi
* vat chay bao noi chung va vol cac phSn tii cua khi cu bay noi rieng.
Cac luc tijr dong khi tac dung len b^ mat cua eae ph^n tijf khi cu bay nhu:
canh, duoi, than khong nhihig ehi phu thuoc vao che do bay dac trung bcri cac
tham so nhu: van t6e, do cao bay va cac goe xac dinh vi tri eiia khi cu bay so vcri
dong khi ma eon phu thuoc vao hinh dang ben ngoai, kich thude cua timg phan
tir cung nhu sir phoi tri chinh trong so do e^u thanh khi cu bay.
Canh cua khi eu bay bao gom canh nang va cac canh dieu khi^n. Giong
nhu 6 khi cu bay, hinh dang eiia canh tren binh do cung la mot trong cac yeu to
anh hudng cd ban den cac dac tinh khi d6ng cua canh 6 cac che d6 bay. Chinh
vi thd', xu hudtig hoan thien va cai tien cac dac tinh khi dong cua canh a mot dai
r6ng thu6e cac ehS' do bay, thucfng xua't phat tir nhung ket qua nghien curu ve sir
thay d6i hinh dang ben ngoai cua canh.
Lich six phat tri^n eiia nganh hang khong cho tha'y rang eiing vofi su ra dori
cua cac the he dong ecf hang khong tien tien, su thay ddi hinh dang ben ngoai
cua canh tir canh diip den canh dofn c6 do gian dai Idn, tien den canh c6 goe mui
ten Icfn, do gian dai nho va cuoi cung canh eo hinh dang phiic tap, thay ddi hinh
hoc trong khi bay da lam thay ddi tiTng budfc ve chat eae dac tinh khi dong cua
canh noi rieng va cua khi cu bay noi chung.

Nhihig thanh tuu eiia llnh vuc nghien eun khi dong ly thuyet va thuc
nghidm, cu the sir sang tao cua cac nha bac hoc ve cac phuong phap tinh toan va
phuong phap xijf ly cac so lieu thuc nghiem da giai quyet thanh cong nhieu bai
toan ve hop ly va toi uu hoa hinh dang ben ngoai cua canh va cac phan tu khac
eiia khi cu bay.
Ngay nay, trong linh vue nghien curu khi dong cac khi cu bay, mot hudng
di mdi da va dang hinh thanh, do la thuc nghiem tinh toan so tren eo so cac mo


7
hinh toan hoc vdi su trcr giup hieu qua cua cac thiet bi cong nghe thong tin.
Hudng nghidn curu nay cho phep trong mot khoang thcri gian ngan eo th^ tinh
toan m6t s6 ludng lorn cac phucmg an thig't ke khi dong khi cu bay. Cac phucfng
•phap dUde sijf dung phd bien trong thuc nghiem so hien nay do la:
- Doi vcfi mo hinh khi cu bay c6 th^ tieh, thu&ng diing phucfng phap panen.
Theo phudng phap nay, be mat khi cu bay dUde thay the bang nhieu eae panen
phing, hinh chiJ nhat. Viee tinh toan dude tien hanh doi vdri timg panen sau do
t6ng hofp lai. PhUdng phap nay thuat toan phlJc tap , do chinh xac khong cao ma
khS'i lucmg tinh toan lai qua 1cm.
- Di don gian hoa trong qua trinh tinh toan, mo hinh tinh toan doi vofi khi
cu bay cd thi tieh duoe thay the bang mo hinh mat nang mong. Doi vdi loai mo
hinh nay ton tai ph6 bien eo eae phucfng phap nhu: Phuong phap phSn tu huu han
[16], phuong phap sai phan hiJu han [6], phuong phap xoay rcri rac [10], [11],
[12], [13], [14]. Trong cac phuomg phap neu tren , phucfng phap xoay r5i rac la
phuofng phap duoe suf dung rong rai, c6 hieu qua va do chinh xac cao. De tai cua
luan van ufng dung phuofng phap xoay rcri rac va sijf dung may tinh di xac dinh va
khao sat eae dac tinh khi dong cua canh trong moi quan he phu thuoc vofi hinh
dang ben ngoai eiia no d cac toe do duori am.

•

Luan van gom:
Chuofng I: Tdng quan cac phucfng phap xac dinh dac tinh khi dong cua canh
trong dong khi dudfi am.
Chucmg II: Trudfng van toe cam utig bcri cac he xoay trong dong khi dudi am.
Chuc^g III: Phucfng phap xoay rcri rac xac dinh cac dinh dac tinh khi dong cua
canh trong dong khi dudi am.
Chucmg IV: Ket qua tinh toan va khao sat cac dac tinh khi dong cua canh khi cu
bay.
Tac gia luan van xin chan thanh bay to long biet ofn sau sac den cac thay
va cac dong nghiep da tan tinh giup dd tac gia hoan thanh cac noi dung cua luan
van nay.


8
CHirONG I
T 6 N G QUAN CAC

PHUONG PHAP XAC DINH DAC TINH

KHI DONG CUA CANH TRONG DONG DUOl AM
I

1.1. He true toa dp:
Nghien cihi cac loai canh cua khi cu bay, thudng su dung he true toa do
lien ket hinh 1.1. He true toa do OXYZ cd true OX hudng theo chieu ehuye'n
dong eiia canh, true OY nlm trong mat phang doi xiing cua canh, true OZ hudng
theo nira canh phai.

My>0

Hinh 1.1: He true toa do xac dinh cac dac tinh khi dong canh khi cu
bay.
Ky hieu: U^ - Vec tcf van toe tuyet doi, goe toa do O, Q - Vec tcr van toe cua
canh quay quanh cac true toa do. Cac thong so chuyen dong tuyet doi cua canh
tren cac true eiia he true toa do dong OXYZ:
Uo = iU„,+jU„^ + kU„,
Q = i Q, + jQy + kQ,

(11)


Vi tri eiia canh ddi vdi dong chay bao dac tnmg bang eae goe: Goe ta'n a
va goe trucrt p. Cac thanh ph^n van toe U^ lien he vdi goe tifn a va goe trugt p:
Uox = U^, cos a cos p
U„y = - U„ sin a cos P

(1.2)

U,, = -U„sinp
Khi xet bai toan chay bao canh eae khi cu bay trong khuon kh6 tuye'n tinh thi
mdi lien he giiia eae thanh ph^n van toe cua U^ vdi cac goe a va P cd dang:

u„, = u.
(1.3)

a = u,
^^

P = u.

1.2. Canh cua khf cu bay, cac tham so hinh hoc:
Canh cua khi cu bay phd bien la canh doi xiing tren binh do c6 mep canh
tnrdc va sau la nhihig doan thSng vdi goe mui ten khong d6i (hinh 1.2a) hoac la
nhihig dudng thang gay khiic vdi goe miii ten thay d6i (hinh 1.2 b), hoac la
nhiftig dudng cong (hinh 1.2 c).
X

Hinh 1.2a


10

Hinh 1.2b

Hinh 1.2c

Hinh 1.2. Cac dang canh tren binh do.
Ky hieu: 1 - sai canh, bg- Day cung goe canh, b^,^ - Day cung miit canh,
Xi - Gdc mui ten mep canh sau canh, S - Dien tieh canh. Dang canh tren binh do
xac dinh bang cac tham so dac trUng: Do gian dai X, do that r\ va goe mui ten
mep canh trude Xo(1.4)

S

-

^

Doi vdi canh cd mep canh trude la doan thang, khi thay d6i cac tham so X,
T], Xo s^ nhan duoc nhieu dang canh tren binh do khac nhau.

Ky hieu b' - Day cung canh cua tie't dien Z theo sai canh va Xe gdc mui ten
eiia ducmg thang chia day cung theo ti le 9 tren hinh 1.2a Dai lucfng b' va tgXo
xac dinh bang cac bieu thiic:
(1.5)

b' = b^ • Z(tgXo-tgXi)
Hoac:

b'

(1.5')

= i-z(i-n)

tgXe=tgXo-2-^(1—)9

Hoac:

tgXe=tg)Co- — ( ^ )
X

r\-\-\

-

;z =

2z

(1.6)

(1.6)


11
Canh eiia khi cu bay cd mep trude va sau la dudng thing gay khiic. Cac diem gay
eiia mep canh chia canh thanh cac viing 8 = 1 , 2 ...n (tren hinh 1.2b). Gia su cac
tham s6 dac trung trong moi vung Sj da biet:
Tis= ^ ; t g X o e ; i > ^

(1.7)

6 day: U - Sai cua vung canh e^, b'^ - Day cung canh a diem gay. Khi dd:

S = ^ Z ( — + —K)

(1.8)

'-

(1.9)

'

(—-^)(^-l)

tgXec=tgXoe- 2 ^ ( - L - ± ) l

(1.10)

Canh cua khi cu bay cd mep trude canh la dudng cong xac dinh bang
phucfng trinh:

thi:

Xo(z>^ ;b(i).^

(1.11)

tgxe=^(^-e-^

(1.12)

1

d(z)

d(z)

*

Ngoai nhihig tham so hinh hoc neu tren con cd cac tham so dac trung khac
phue vu cho qua trinh khao sat cac dac tinh khi dong cua canh nhu:
- Tam ap sua't ky hieu toa do theo true OX la: x^^ - diem dat eiia t6ng hcfp
lire khi dong len canh.
- Tieu cu khi dong ky hieu toa do theo true OX la: Xp - diem ma momen
doc mjj khong phu thuoc vao gdc ta'n a khi van toe U^ khong d6i.
- Day cung khi dong trung binh ky hieu (b^a^) la day cung eiia canh thuc
hien phep trung binh hoa theo dien tieh canh:
b,,,=A|b=dz

(1.13)

L2

Xcax=^ j x b dz
S-.:

(1.14)


12
d day:

X(au^ - Toa do theo true OX di^m ddu eiia day cung khi dong

trung binh.
D6i vdi dang canh dd'i xihig tren binh do vdi cac mep canh la dudng thing
•c6:
^
=^
b,
3'

— 1 _
ri(ri+l)'

^cax ^ ^ t g X o ( Y ] + 2 )

b_

12

n

(1.15)
(1.16)

L3. Cac dac tinh khi dong ciia canh khi cu bay.
Cac tham so dac trung cho chuydn dong diimg (khong phj thuoc vao thdi
gian) cua canh vdi tdc do U^ dudi am nhu trong 1.1 la: q^ = (a,p,(Ox, cOy, (o^)
1=1,2...n
Dac tinh khi d6ng cua canh khi cu bay la tap hop cac dai ludng xac dinh
sir tac dung tucfng h6 giira canh vdi khong khi trong cac ehuye'n dong cu the' cua
canh. Dac tinh khi dong cho phep xac dinh cac luc va momen khi tac dung len
canh eung nhu cac moi quan he phu thuoc giua chiing vdi nhau, giua chiing vdi
hinh dang cac tham so hinh hoc, cac tham so chuyen dong cua canh. Dac tinh
khi dong eiia canh th^ hien qua:
- Cac luc va momen khi dong: lire can X, luc nang Y, luc canh Z va
momen lieng M^^, momen hudng My va momen chiic ngdc M^.
- Cac he so khi dong tucfng ling khong thii nguyen : c^^, Cy, c^, m^^, my, m^.
Moi lien he giua cac luc, momen khi dong vdi cac he so khong thii nguyen
tucfng utig:
X=c,^S;M,=m,^S.b
Y =c § - ^ S •
Y

Cy

^

^

M , = m y ^ ^ S.b

,

Z = c^S;

M,=m,^^S.b

(1.13)


13
Cac he so khi dong khong thiJ nguyen cd the bieu thi qua cac he so dao
ham khi d6ng nhu sau:
(1.14)
i=I
n

(1.14')
i=l

6 day: c?' =5c.

5qi

;m?' =

9m

dq\

- Quan he giffa cac he so khi dong khong thu* nguyen nhu: chat lugng khi
d6ng va tieu cu khi dong Xp:
K=

^
c

;(1.15)

Xp =

m

(1.16)

X

Khac vdi cac luc va momen, cac he so khi dong khong thur nguyen khong
phu thuoc vao d6ng ap (-—- ) va kich thude hinh hoe eiia canh (S, b, 1) ma lai
phu thu6c vao hinh dang eiia canh tren binh do (x, A,, r\ ). Cac tham s6' chuyen
U.

dong (a, P, 0)^, co , co^) sd mach M

va so Reynol Re = Uo.b

00

Luc nang Y, momen doc M^ va momen ngang M^ sinh ra do cd do chenh
ap giiJa mat dudi va mat tren canh (AP = P^ - Px^). Cac bieu thiic d^ tinh luc nang
va cac momen tren hinh 1.3 la:
Y


14
Hinh 1,3: Sa do tinh luc va cac momen canh khi cu bay,
Y=

JJ APdxdz

(1.17)

h
M^=

Jj APxdxdz

(1.18)

M^=

JJ APzdxdz

(1.19)

S t6ng dien tieh eiia canh.

Sijf dung do chenh ap va eae toa do khdng thiJ nguyen:
AP - ^' 'f
2

^=b^

^=^

^

b'

- 2z
z= —
I

Khi dd he so luc nang va eae he so momen doc va ngang ducfe xac dinh:
b

2 l/2b5|

S = 2 ^ J jAPd?d!:
S

(1.18)

0.

b

2 l/2b?

m, = - 2 ^ J jAP?d?d!:
S

b

(1.19)

0.

2 l'2bi:

m, = - 2 ^ J JAPCdgdC

(1.20)

O day: ^o, ^1 toa do khong thu" nguyen eiia diem thuoc mep trude va sau
c^nh.
1.4. Tong quan cac phir0ng phap xac dinh dac tinh khi dong cua canh khi
cu bay trong dong khi dirdi am.
Xac dinh, khao sat cac dac tinh khi dong cua khi cu bay noi chung va ciia
canh noi rieng cd cac phucfng phap nghien cihi chinh sau:
1.4.1 Phirong phap tinh toan ly thuyet


15
Bao gom phucfng phap giai tieh va phucfng phap so. Rieng phucfng phap
s6 trong nhung nam gin day vdi su phat tri^n nhanh va manh cua Imh vue cong
nghe thong tin, ket hop vdi may tinh da hinh thanh phucfng phap thix nghiem so

« [15] trong nghien euti khao sat eae dac tinh khi dong cua canh va khi cu bay.
Bai toan chay bao cac vat trong moi trudng khi thuc (khi cd do nhdt) tren
cd sd cac mo hinh toan hoe ve Idp bien roi, cac phucfng phap so va ket hcfp vdi
may tinh cd th^ giai quyet ducfe hdu het cac van de cua bai toan chay bao dat ra.
Tuy nhien, doi vdi bai toan chay bao cac vat cd hinh dang khong gian phiic tap,
con nhi^u v^n de chua giai quyet. Vi du d so' Reynoil Idn tUcfng umg vdi dieu
kien dong khi ciia canh va cac khi cu bay khi chuyen dong, d^ xac dinh cac dac
tinh khi dong, thuc te' cho tha'y khong e^n doi hoi phai giai bai toan chay bao vat
trong dieu kien khi thuc ma chi edn giai bai toan chay bao vat (canh) tren cof sd
m6 hinh ciia chSii khi hoac cha^t long ly tudng va Idp bien. Bai toan chay bao vat
(canh) CO hinh dang khong gian phiJc tap, eiing vdi viec lira chon cac hinh dang
t6'i uu, cac phuong phap tinh toan ly thuyet cd xu hudng hoan thien doi vdi cac
mo hinh toan hoe.
Gia thuyet ve dong chay the' va ve sir thay the* vat chay bao bang cac dac
trung thuy khi dong (xoay, nguon, hut, ludng cue...) la nhirng giai phap hieu qua
de' hoan thien mo hinh dong chay bao chat khi hoac chat long ly tudng. Mo hinh
chay bao cd luu sd van tdc ciia Giukovsky, Traplugin lam ro co che tao ra lire
nang ciia canh va khi cu bay.
Phuong phap bien d6i bao giae cua Giukovsky da giai quye't bai toan chay
bao cac profil canh dcfn gian, cac bai toan luong phut trong chat khi ly tudng
khdng chiu nen. Hudng chung va ph6 bien de giai cac bai toan chay bao vat
trong dong chat khf khong chiu nen la quan niem thay the' dudng thang hoac mat
phang bao quanh vat bang cac dac trUng trong thuy khi dong hoc, khi dd bai toan
chay bao dua ve giai cac phucfng trinh tieh phan.
Ddi vdi bai toan chay bao canh cd sai huu han, do gian dai canh X nhd.
Hien tucfng tach dong va chay tran tren cac mep canh anh hudng ro ret den dac
tinh ciia dong chay bao. Mo hinh dong chay bao ddi vdi trudng hop nay vln la


•

16

dong the ne'u mat canh ducfe tiep tue la mot mat phang xoay tu do (eae dai xoay
xu^t phat tir mep sau va miit canh tien v^ v6 cue).
Phucfng phap sd d^ giai bai toan chay bao cac loai canh ciia khi cu bay cd
hinh dang phufc tap va trong eae t6 hcfp giiJa canh vdi cac phin tiJf khac eiia khi cu
bay la phucfng phap xoay rdi rac [10 ] ,[11],[12],[16].
Ph^n Idn mo hinh toan hoc trong cac phucfng phap tinh toan ly thuyet
thudng dUde tuye'n tinh hoa d^ cd nhutig ket qua giai tieh. Ngoai ra eon nhan
tha^y rang tuyen tinh hoa cac bai toan chay bao khi cd nhirng nhieu dong nhd
trong dong khi eon tao ra cd sd d^ xac dinh cac dac tinh khi dong ciia canh cd
sai hffu han trong dong khi vudt am.
1.4.2 Phuong phap thuc nghiem
Phudng phap thuc nghiem cd vai trd quan trong trong qua trinh tie'p can,
nhan biet v^ ban cha't cua hien tUdng va cac dac trung cua dac tinh khi dong cua
khi cu bay. Mae du phucfng phap tinh toan giai tieh va phudng phap sd da dat
dude nhi^u thanh tuu, nhung phUdng phap thuc nghiem d^ nghien cihi, khao sat
eae dac tinh khi dong cua canh va khi cu bay vln la nhu cSu cSn thiet nhlm cung
ca'p nhiJng ket qua de tham dinh cac phudng phap tinh toan ly thuyet eung nhu
lam cd sd di so sanh va lira chon cac ke't qua nhan dudc trong qua trinh nghiep
cihi khao sat.
Phudng phap thuc nghiem phSn Idn dude tie'n hanh tren cac mo hinh trong
cac dng th6i khi dong cd cac kha nang dieu chinh dUdc mot sd tham sd cua dong
khi nhu van tdc, mat do . . .
Nghien cuti khao sat cac dac tinh khi dong ciia canh va khi cu bay tren cac
mo hinh eiia vat thuc, ddi hoi phai dap utig cac tieu chuan cua ly thuyet dong
dang nhu: ve hinh hoe, dong hoc va dong lire hoc. Thuc te cho tha'y rang dam
bao tieu chu^n dong lire hoc mot each tuyet ddi la viec ra't khd khan. Chinh vi
vay, tuy thuoc vao ban chat ciia hien tucfng vat ly, phucfng phap thuc nghiem tren

cac md hinh chi dam bao mot each cd ban hoac mot phan nao dd cua tieu chuan
dong dang ve dong lire hoe ddi vdi vat chay bao kich thude thuc ma thoi.
Nhin chung xac dinh, khao sat cac dac tinh khi dong cua canh, cung nhu
cua khi cu bay bang phucmg phap thuc nghiem ddi hdi phai dau tu nhieu.


•

17

1.4.3 Phuong phap vat ly khi dong:
Phudng phap vat 1^ khi ddng canh va cac khi cu bay nham nghien cihi,
khao sat cSiu tnic dong chay bao canh va khi cu bay, cac trudng van tdc, ap suat,
nhiet do va mat do eiia dong khi. Phudng phap quan sat cac ph6 ciia dong chay
bao canh va khi cu bay cho phep tim hi^u cac qua trinh vat ly xay ra, ly giai cac
d^u hieu dac biet, dong thdi eung c^p cac sd lieu dSu vao cho viec xay dung cac
mo hinh toan hoe eiia bai toan dat ra.
Hien nay ton tai rat nhi^u phudng phap d^ quan sat phd dong chay bao
canh va khi cu bay nhu: Phucfng phap khdi, phUdng phap mang chat long,
phudng phap quang hoc ...


18
CHl/ONG n
TRl/CJNG V^N T6C CAM IDNG B 6 I CAC HE XOAY
TRONG DONG KHI Dl/Cfl AM
• -

Phucfng phap ap dung di xac dinh va khao sat cac dac tinh khi ddng cua
canh cac khi cu bay dude xay dung tren cd sd ly thuyet xoay trong ddng khi dudi

am.
Trong cac cong tnnh [10], [11], [12], [13]... tren cd sd cong thirc xac dinh
van tdc cam dug cua Bioxavara, cac tac gia da nghien cihi ve trudng van tdc cam
umg bdi cac he xoay khac nhau trong ddng khi dudi am. Dudi day tien hanh khao
sat, he thong nhutig ket qua nghien curu ve trudng van tdc cam ling bdi cac he
xoay trong ddng khi dudi am, dac biet dua ra nhirng hiiu thiie tdng quat d^ xac
dinh cac thanh ph^n van toe cam umg eiia he xoay xien hinh mdng ngua.
2.1 Trudng van tdc cam umg bdi doan xoay:
Trong th^ tieh khong khi gidi han, cd mot doan xoay A^ A2 bat ky cd
cudng do r + khong ddi tren chieu dai ciia doan xoay (xem hinh 2.1).
«

Trong he true toa do de cac OXYZ, diim Aj cd toa do tUdng umg x,, y,, z,,
di^m A2 cd toa do tUdng umg X2, y2, ^- Chon di^m M(xo, yo, zo) bat ky trong
khong gian, la di^m d^ tinh van tdc W cam umg bdi doan xoay A^Aj.
Dung mat phang di qua doan thang A1A2 va diem M. Ndi A^, A2 vdi diem
M. Gdc d cac dinh Ai va A2 ky hieu la cp^ va (p2 ' khoang each tir diem M den
doan thang A1A2 ky hieu la r. Theo cong thiic cua Bioxavara [20], van tdc cam
umg bdi doan xoay dude xac dinh.
W=^
(Cos^, +Cos^2)
4m

(2.1)

CJ day van tdc cam umg W cd phUdng vuong gdc vdi mat phang A1MA2,
hudng theo chieu tucfng umg vdi chieu quay cua cudng do xoay r+.


19

Ky hieu cac thanh phin van tdc cam ling W theo cac true toa do OX, OY,
OZ la Wx, Wy va Wz. Xay dung cac bi^u thiic di xac dinh Wx, Wy va Wz.

Hinh 2.1 Xac dinh tdc do do doan xoay cd hudng
bat ky trong khong gian, tao ra tai cac di^m Ian can.

Ky hieu phap tuyen cua mat phang A^ A2 M la OK vdi vec td ddn vi la n,
chieu dudng ciia phap tuyen la'y theo chieu dudng eiia xoay 7~^. Gdc giira phap
tuyen n vdi cac true toa do la P^ P2 va P3. Khi dd hinh chieu ciia vec td van tdc
cam umg W tren cac true toa do Ox, Oy, Oz xac dinh bang cac bieu thufc:
W , = W cospi; Wy= W C0SP2; W,= W cosp3.

.

(2.2)

Viet phudng trinh cac canh eiia tam giae A^ A2 M khi biet toa do cac dinh
x -X

y-Yi

x^-x,

Y2-YI

X -x,

y-Yi

>

z-z,
Z2-Z, '

_

z-z,
9

XQ

— X|

Yo-Yi

X - X Q

^2 ~ ^0

ZQ

>

— Z]

(2.3)

Y-Yo _ z - z .
Y2-Y0

Z2-Z0

y

Tiir (2.3) xac dinh cac goe A:
COSrp, =

r.r,:

(2.4)


20
(2.5)

COS'f

VA*'+B*^+C''

(2.6)

r=
Vdi;
A;
A;

= (Xi- X2) (xi- Xo) + (Yi- y^) (Yi- Yo) + (z,- Zj) (z,- ZQ);
= (X2- Xi) (Xj- Xo) + (y2- Yi) (Y2- yo) + (Z2- Zi) (Z2- Zo);

(2.7)
(2-8)

Fi = V(x, - X o ) ' +(y, - Y o ) ' +(z, - Z o ) ' ;

(2.9)

^2= V(x2 -Xo)' +(y2 - y o ) ' +(z2 - Z Q ) ' ;

(2.10)

ri2 = V ( x 2 - x , ) ' + ( y 2 - y , ) ' + ( z 2 - z , ) ' ;

(2.11)

A*= (Xo- X,)(Y2-Y,)- (YO-YI)(X2- XI);

(2.12)

B*= (yo-yi)(z2-Zi) -(vzi)(Y2-yi);

(2.13)

C*= (ZO-Z,)(X2-XI)- (XO-X,)(Z2-ZI).

(2.14)

D^ tinh cac thanh ph^n W^ ,Wy va W^ cua vec to cam iJng W can thiet xac dinh
gia tri cac cosin chi phuong ciia phap tuyen vdi mat phang A,A2 M. Phuong trinh
ciia mat phSng AjA2 Mc6 dang:

A X + BY + C Z + D = 0.

(2.15)

Khi do:
A

cospi=
VA^
cosp2 =

+ C^
B

(2.16)

VA^ + B^ +c'
C

COSP3 =

VA'+B'

+C'

Cac he sd A,B,C trong phucfng trinh (2.15) diroe xac dinh bang phucfng trinh
mat phang AjA^ M di qua 3 di^m Ai(Xj,yi,Zi); A2(x2,y2,Z2) ^^ M(^o'yo'Zo) ^6
dang sau:
x-x,

^0

y-y,

^i Yo

Yi

X: - X , y . - y ,

z-z,
^
z

= 0.

(2.17)


21
Dong nhlft thlic cac he so nhan duoc:
A = (Y2- yi)(Zi - Zo)- (Zj - Zi)(Y,-Yo); "^
B = (Z2- Z,)(X,- Xo)- (X2- Xi)(Zi- Zo);
«

y

(2.18)

C = (Xj -X,)(Y, -YO) -(Y2-YI)(XI-XO).

Theo bi^u thiic (2.19) va bi^u thiic (2.12),(2.13) va (2.14) nhan tha'y
A*=A;B*=B;C*=C
Suy ra:

VA'+B'+C'

= VA''+B*'+C*'

(2.19)

Thay (2.3),(2.4),(2.5) vao (2.1) va ket hop vdd cac bi^u thiJc (2.18),(2.19) ta co:
B*
•2
An (A''+B'^+C*')
r.

W_=

)

(2.20)

- ( ^ + ^ )
47r(A'^+B*^+r^)^ r

(2.21)

A*
4n (A'^+B'^+C*') r

(2.22)

r
'
W=

c*

*2

•)

Trong d6' A*,B*,C*,ri,r2 duoc xac dinh theo cac bi^u thiJc til (2.7) den (2.14).
2.2 V$n toe cam umg bdi he xoay xien hinh mong ngua.
Xet mot he xoay gom 1 doan xoay lien ket AjAj va 2 soi xoay tu do song
song vdi van toe Uo, xua't phat tir diem d^u va cuoi cua xoay lien ket AjAj. Xoay
lien ke't lech 1 goe % so vdi true OZ. Khoang each giiJa 2 xoay: 1^. Hai xoay tu do
va xoay lien ke't nam tren eiing mat phang XOZ. (Xem hinh ve 2.2.) Cac xoay
trong he xoay cucmg do khong d6i duoc the hien: r+=Uo IQ F , f eucfng do xoay
khong thii nguyen.

. M(Xo,yo,Zo)

r.

00

U,

Ai(Xi,0,Zi)

F7 (^

Hmh 2.2: He xoay xien hinh mong ngua

00


22

Xac dinh van toe cam umg bdi he xoay tai diem M trong khong gian.
Di^m M CO toa do (x^, yo, z^). Van tdc cam ihig do he xoay gay ra la tdng cac
van tdc cam ihig xoay lien ket AjAj va 2 xoay tu do A,oo va A2X •
W = U+V

(2.23)

Trong dd U la van tdc cam dug do xoay lien ke't A,A2. Van tdc nay
duoc xac dinh theo hiiu ihxic (2.20); (2.21); (2.22) vdi luu y cac toa do yj = y,
= 0.
Xet van tdc cam thig do 2 xoay tu do gay ra.
V=V,+V2

(2.24)

Ap dung cong thiie Bioxavara ddi vdi sdi xoay A2ao ta cd:
V2= ^ ( l + cos^3)

(2.25)

471 r

6 day r + : cudng do xoay va dung bang cudng do xoay lien ke't AjA^.
r: Khoang each tir M tdi true soi xoay.
(Py Gdc tao bdi giffa MA2 va AjooBie't toa do di^m M(\^, yo^^o), ta cd:
r = VYO+(22-^0)'
Cos (p3 = .

(2.25)
'''"''

(2.26)

V(x2-Xo)'+(z2-Zo)'+yo
Tijf day suy ra:
V.x= 0;
^2y= - . y^- ^ " ^ ° \ . (1 + ,
\^=

^""^

,

, )

(2.27)

— ^ ; ^'y°^ ^ (1+ .
^°'^^

)
47c(y„+(z,-z„)
^-x^y+z^-z^f+yl

(2.28)

Bang each bien ddi tuong tu doi vdi soi xoay tu do Ajoo ta co van tdc
cam umg.
Vu= 0;
V =
-^ (^1 "^°^
(1 +
^" "'^i
)
"
47i(y-+(z,-z„)=
4{x,-xS-^z,-z„f+yl

4;t(y„+(z,-zJ
Cuoi Cling cd:

^{x.-xJ'Hz.-zJ'+yl

(2 29)


23

w, = u, + v,,+v2x
(2.31)

Khai tri^n cu thi hiiu thurc (2.31) ta co:

B

W =-^(

4;r \ 4 * ' + B * ' + C * '

(2.32)

(^ + ^ ) )
r,1

r*2

c
w, = - ^ (__^____(AL+AL)
+ ^ —
4;r ^^*^+B*^+C*^^ r,

*il +

r,

*

y:+(z,-z„y

hZ^).^IlZl^ (, + ^iili^)

(2.33)

y^+(z,-zj
r
\V

-

^

>i*

^

A*

(

/ ^ l

A*
• ^ 2

4;r^^*^+B-+C*^^,

^(1 +

X o - X i

-)+ , '?

7 - 7
X

•

^1

r, ^

^o

»

y^+(z,-zj^

' ° ,, (1 + ^^^^^)

(2.34)

y:+(z2-Zo)"

Bidu thi cac gia tri Xj = - —tgx, Zj = -\J2
ma

X2=YtgX.

(2.35)

Z2 = lo/2

Tinh duoc:
A*=

(2.36)

-l„yotgx

B*= yo-l„

(2.37)

C* = (z, + 1^2) 1„. tg X - (Xo + l„/2 tg X). lo

(2.38)

= Zo • lo t g X - Xo • lo

r,= j(^tgx+xJ^+y^+(V2 + zJ^

(2.39)

r.= , & x - x j ' + y „ ^ + ( L / 2 - z J ^

(2.40)

A,* = lo. tgx (lo/2 tg X + x„) + 1„ (lo/2 + Zo)=
=%
T T ! ^y ^ + lo x„ tgx +loZo=
2 tg^x + lo Xo tgx +2Y + 'o^o =2.Cos

L

, lo

.

+ x^sinx +z„cosx )
2.Cosy ( Cosy

A,* = lo. tgx (lo/2 tg X -Xo) + lo (lo/2 - z„)=

(2.41)


24
1.

1

2.Cos X

Cosx

(2.42)

x„sinx -z„cosx )

Thay th^ cac hiiu thiic tir (2.36) den (2.42) vao (2.32) nhan duoc
loY,

W. =

47r l„(y: + y„Vx+(z„tgx-xJ/^

"\

"

COSY

ft

(TTT—+
2COSY

x,Sinx+z,Cosx)

-i^(-A___x^Sinx-ZoCosx)
I COSY

2COSY
>

L2

,

..2

'(^tgx-xJ^+(^-zJ^+y;

tgX+x„)^+(^+zJ^+y^

vay:
Yo

w. =

4;r yo+yotgx+(Zntgx-xJ'
r

(—-— + x^Sinx + ZoCosy)
Cosx__2Cosx

i-Z^r
Cosx 2Cosx

(~iBx + ^y + (k+^of + yl

x,Sinx-z„Cosx)
K2.43)

( Yjt g X - x J ^ + ( ^ - z J ^ + y „ ^

>^

W =- i l r
^

^otg;ir-Xo

4;r Jy„' + y^'tg'x + (z„tgx - x J '

COSY

( 2COSY
— i — + x„Sinx +z„Cosx)

J ( Y t g x + x J ^ + ( ^ + zJ^+y;

'

(

'

Cosy ZCosy

x^Sinx -z^Cosy)

^(^tgy-xJ^+(^-zJ^+y;

x„+-rtgx

+ z.

0+

(1 +
yl+i^

+ ^oY

J ( ^ t g x + x J ^ + ( - ^ + z J ^ + y^
• \

x„-:rt8X

-z.
(1 +
.2

.1.

yo+(^-Zo)

•)

'(^tgx-xj=+(|-zj=+y^

(2.44)


25

w, =

- Yo tgZ

4;r

X

.2*_2

yo + yotgx +(z„tgx-xj'
(r:r^—+ x„Sinx+z„Cosx)

COSY

2COSY

'

I COSY

(:'

ZCOSY

XoSinx - z^Cosx)

'(^tgx-xJ^+(^-zJ^+y;

( f t g x + x „ ) ^ + ( ^ + zJ^ + y„^

x„+ytgx

Yo

0+

(1+

y\+
( | t g x + x J ^ + ( ^ + zJ^+y^

x„+^tgx
(1 +

•)

(2.45)

(^tgx-xJ^+(^-zJ^+y^

y^+(^-0^

Chuyen qua toa do khong thii nguyen

(2.46)

Datr+ = u „ b r ; ^ 0 = ^ ; Tio=^; Co=^;
b

b

D

d day r la hang sd khong ddi nao dd, b: do dai dac tnmg.
Khi dd bidu thiic ( 2.43) viet lai thanh:

W

'

Cosy

(:

=

'"

2b.CosY

Uor

llo

4K

(^o+^;tgx+(^otgx-?o)"

+? ,Sinx +C oCosy)

>

(.

lo

j COSY 2b.Cosy

j(^.gx*«.)=*(^H.)"-.;

^^

- ? „Sinx -^ oCosx)
>2.47)

J(5*-l.)'+(^-c.)".,;
^J

B\i\x thiic (2.44) trd thanh:
? o tgx - ?
^

1
COSY

471

(_

lo

( i l ' + i l ' t g ' x +CotgX - ? o )
+? „Sinx +C „Cosx)

2b.COSY

^(^tgx.?J^.(^HJ-.^

' (-^^
I COSY zbCosy

? „Sinx-? „Cosx)

j( ^ t g x - ? J ^ + ( ^ - ? J ^ + r , |