chuong2 giao trinh ndht

Chng 2: M HNH HA

Hunh Thi Hong B mn iu khin T ng

1

Chng 2

M HNH HA Chng 2: MO HNH HOA 2.1. Gii thieu 2.2. Phan tch chc nang 2.3. Phan tch vat ly 2.4. Phan tch toan hoc 2.5 Mt s th d Tham kho: [Smith, 1994], chng 2, 4. [Johansson, 1993], chng 7. 2.1 GII THIEU Mo hnh hoa la phng phap xay dng mo hnh toan cua he thong bang cach phan tch he thong thanh cac khoi chc nang, trong o mo hnh toan cua cac khoi chc nng a biet hoac co the rut ra c da vao cac qui luat vat ly, sau o cac khoi chc nang c ket noi toan hc e c mo hnh cua he thong. Ba bc mo hnh hoa: Phan tch chc nang Phan tch vat ly Phan tch toan hoc

Chng 2: M HNH HA


2

2.2 PHAN TCH CHC NANG (tham khao chng 2, [Smith, 1994])

2.2.1 Khai niem

Phan tch chc nang la phan tch he thong can mo hnh hoa thanh nhieu he thong con, moi he thong con gom nhieu bo phan chc nang (functional component).

Khi phan tch chc nang can e y lien ket vat ly (connectivity) va quan he nhan qua (causality) gia cac thanh phan ben trong he thong.

Ba bc phan tch chc nang: Co lap he thong Phan tch he thong con Xac nh cac quan he nhan qua 2.2.2 Co lap he thong - Lien ket ngoai Xac nh gii han cua he thong can mo hnh hoa, cat ket noi gia he thong khao sat vi moi trng ngoai, moi ket noi b cat c thay the bang mot cong e mo ta s tng tac gia he thong va moi trng.

Hnh 2.1: He thong co mot cong lien ket vi moi trng

He thong Moi trng U

Y

bien cua he thong

Chng 2: M HNH HA


3

Cong (port) : la mot cap au cuoi ma qua o nang lng hoac cong suat vao hoac ra khoi he thong. Mot he thong co the co nhieu cong (multiport system). Bon loai cong thng gap: c kh (Structural), ien (Electrical), nhiet (thermal), lu chat (fluid) Loai cong

Ten (Ky hieu) S o Co lap

1. C

K

H

a. Tnh tien (Structural Translation - ST)

b. Quay (Structural Rotation - SR)

c. Phc hp (Structural Complex - SC)

3.

IEN

a. ien dan (Electrical Conduction EC)

b. ien bc xa (Electrical Radiation ER)

Chng 2: M HNH HA


4

3. N

HIE

T

a. Dan nhiet (Thermal Conduction TC)

b. oi lu nhiet (Thermal Convention TV)

c. Bc xa nhiet (Thermal Radiation TR)

4. L

U

CH

AT

a. Lu chat trong (Fluid Internal FI)

a. Lu chat ngoai(Fluid External FE)

Th du 2.1: Co lap he canh tay may

Chng 2: M HNH HA


5

Hnh 2.2: S olien ket ngoai cua canh tay robot

Th du 2.2: Co lap he thong lam mat

(a)

(b)

Hnh 2.3: He thong lam mat (a) S o he thong (b) S o trao oi nhiet

Chng 2: M HNH HA


6

(a)

(b)

Hnh 2.4: He thong lam mat (a) S o a cong cua he thong

(b) S o a cong cua bo trao oi nhiet

Hnh 2.5: S o a cong lu chat long trong he thong lam mat

Chng 2: M HNH HA


7

2.2.3 Phan tch he thong con - Lien ket trong Phan tch he thong sau khi co lap thanh cac he thong con (subsystem), sau o tiep tuc phan tch cac he thong con chi tiet en cac bo phan (component), thay the lien ket gia cac bo phan bang cac cong.

Th du 2.3: Phan tch lien ket trong he canh tay robot

Hnh 2.6: S o khoi canh tay may chi tiet en cac he thong con

Hnh 2.7: S o khoi canh tay may chi tiet en cac bo phan

Chng 2: M HNH HA


8

2.2.4 Quan he nhan qua - Cac bien cua he thong V cong la au cuoi ma qua o cong suat (nang lng) truyen vao ra he thong nen quan he nhan qua cua cong c xac nh bi cac bien nh ngha cong suat tai cong.

Th du 2.4: Canh tay may

Hnh 2.8: S o khoi hoan chnh cua canh tay may

Chng 2: M HNH HA


9

2.3 PHAN TCH VAT LY

2.3.1 Phng phap phan tch vat ly

2.3.1.1 Cac qui luat vat ly

a. Quan h c bn gia lng, th v dng

H thng vt l c th chia thnh 4 loi: in (Electrical) C (Machenical) Nhit (Thermal) Lu cht (Fluid)

Mt h thng phc tp c th gm nhiu h thng con thuc 4 loi ni trn.

Mi loi h thng c 3 phn t c bn (basis element): Tr (resistance) Dung (capacitance) Cm (inductance) hay qun tnh (inertia)

Cc phn t c bn ny c nh ngha da trn 3 bin: Lng (quantity) Th (potential) Thi gian (time).

Bng 2.1: Cc bin c s dng nh ngha cc yu t c bn ca cc loi h thng.

Loi H thng

Bin Lng Th Thi gian

in in tch (Charge)

in th (Voltage)

Giy

C kh Khong cch (Distance)

Lc (Force)

Giy

Lu cht (lng)

Th tch (Volume)

p sut (Pressure)

Giy

Nhit Nhit nng (Heat energy)

Nhit (Temperature)

Giy

Chng 2: M HNH HA


10

Cc bin khc c nh ngha da trn 3 bin c bn trn. Cng dng l bin thin lng trong mt n v thi gian

(hay cng dng l tc bin thin lng).

lngdong o cng dtd (2.1) Cong suat

dongocngthesuat cong

nh ngha cc phn t c bn (Quan h gia lng, th v dng)

Tr l s chng li s chuyn ng hay dng vt cht, nng lng. Tr c o bng th cn thit chuyn mt n v lng trong mt n v thi gian (giy).

dong o cng

thetr (2.2)

Dung biu din mi quan h gia lng v th. Dung c o bng lng cn thit l cho th bin thin mt n v.

the

lngdung (2.3)

(2.1) & (2.3) dt dong o cngdungthe 1 (2.4)

Cm hay qun tnh l s chng li s thay i trng thi chuyn ng. Cm c o bng th cn thit lm tc bin thin ca lng thay i mt n v.

dong o cngcamthedtd (2.5)

Chng 2: M HNH HA


11

* Cc phng trnh cn bng

Cc nh lut bo ton khi lng, nng lng, v xung lng l cc nh lut c bn c s dng khi m hnh ha. Phng trnh cn bng c bn c dng tng qut nh sau: radongvaodongluytch dong (2.6) Nu h thng khng c cc phn t tch tr khi lng, nng lng v xung lng th phng trnh (2.6) tr thnh: radongvaodong 0 (2.7) Nu h thng c phn t tch tr khi lng, nng lng hay xung lng th s tch tr ny lm thay i trng thi ca h thng.

ra dongvao dongthai trang bien dtd (2.8)

Cc hin tng t nhin xy ra theo hng lm ti thiu nng

lng, v nhiu bi ton m hnh ha m t iu kin cn bng lin quan n s ti thiu nng lng. 2.3.1.2 L tng ha cc phn t vt l Cc nguyn tc l tng ha:

Nguyn tc thun ha: nhn ra nh hng vt l c bn chi phi hot ng ca i tng v dng cc phn t thun biu din.

Hnh 2.9: Mo hnh tu ien gom cac phan t thuan nhat

ien moi C R

Chng 2: M HNH HA


12

Nguyn tc tp trung ha: cc nh hng vt l thc lun phn b trong mt min hay khng gian nht nh (d nh). Cc nh hng phn b ny c th l tng ha bng cch m hnh ha tp trung.

Hnh 2.10: Mo hnh van nhay

Nguyn tc tuyn tnh ha: tt c cc h thng thc u l h

phi tuyn l tng ha bng cch tuyn tnh ha.

Hnh 2.11: ac tnhphi tuyen cua ien tr

Chng 2: M HNH HA


13

2.3.1.3 S tng ng ca cc quan h vt l

Do cc hin tng vt l c s tng ng nn c th m hnh ha h c bng h in, h nhit bng h in,

Hnh 2.12: S tng ng gia cc phn t c bn ca cc loi h thng vt l

Chng 2: M HNH HA


14

2.3.2 Phan tch vat ly he thong ien

2.3.2.1 Cc phn t in

Cc bin c bn trong h thng in: in lng: q [C] in th: u [V] Cng dng in: i [A]

(2.1) dtdqi

Cc phn t in c bn in tr: S

lR [] (2.2)

iuR

in dung: d

SC [F] (2.3)

uqC

(2.4) idtCu 1

in cm: b

NrL2 [H]

(2.5) dtdiLu

Ngun p l tng: Ngun dng l tng

2.3.2.2 Phng trnh cn bng in nh lut Kirchoff v dng. nh lut Kirchoff v p.

2.3.2.3 Phng php gii tch mch in Phng php dng vng Phng php th nh

Chng 2: M HNH HA


15

2.3.3 Phan tch vat ly he thong c 2.3.3.1 Cc phn t c Chuyn ng thng Cc bin:

Khong cch: x [m] Lc: f [N] Tc : v [m/sec]

(2.1) dtdxv

Cc phn t: Tr: bRM

b: h s ma st nht [N.sec/m]

(2.2) vfb

vfRM

Dung: kCM /1

k: cng l xo [N/m]

(2.3) fx

k1

fxCM

(2.4) kxvdtkf vdtCf M1

x f

b

f

x f

k

f

Chng 2: M HNH HA


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Qun tnh c: m : khi lng [kg]

(2.5) madtdvmf

Chuyn ng quay Lc Moment Khong cch Gc quay Vn tc Vn tc gc Gia tc Gia tc gc Qun tnh Moment qun tnh 2.3.3.2 Phng trnh cn bng c Phng trnh cn bng lc. Phng trnh Euler Lagrange:

qP

qL

qL

dtd

Trong : UTL U: th nng T: ng nng P: nng lng tiu hao q: ta tng qut : ngoi lc (hay moment)

x f

m

Chng 2: M HNH HA


17

2.3.3.3 S tng ong gia he thong c va he thong ien

Hnh 2.13: S tng ong gia cac phan t

Hnh 2.14: S tng ong gia cac nguon

Chng 2: M HNH HA


18

2.3.4 Phan tch vat ly he thong nhiet 2.3.4.1 Cac phan t nhiet Cac bien trong he thong nhiet :

Nhit : [0C] Nhit nng: Q [J] Dng nhit: H [J/sec]

(2.1) dtdQH

Cac phan t nhiet: Nhit tr:

Nhit tr dn nhit: Sk

lRC

T trong : kC: l h s dn nhit ca mi trng truyn nhit l: l chiu di ca mi trng truyn nhit S: l tit din ngang ca mi trng truyn nhit (2.2)

HRT

Nhit tr i lu + Bc x nhit: xem (Smith,1994)

Nhit dung: McCT . trong : c: l nhit dung ring ca mi trng truyn nhit M: l khi lng ca mi trng truyn nhit (2.3)

QCT (2.4) HdtCT

1

Qun tnh nhit: TI (2.5)

dtdHIT

Chng 2: M HNH HA


19

Hnh 2.15: Nhiet tr (a) Nhiet tr truyen nhiet (b) Nhiet tr oi lu (c) Nhiet tr bc xa

Chng 2: M HNH HA


20

Hnh 2.16: Nhiet dung

2.3.4.2 Cac phng trnh can bang nhiet 2.3.5 Phan tch vat ly he lu chat long 2.3.5.1 Cac phan t lu chat Cac bien trong he lu chat :

p sut: p [N/m2] Th tch: V [m3] Lu lng: q [m3/sec]

(2.1) dtdVq

Cac phan t lu chat: Lu tr:

Lu tr ca ng ng: 4128dlRL [N.sec/m]

(2.2) qpRL

Cng thc trn ch ng trong trng hp lu cht chy tng (c hng), v ng ng dn lu cht di (l>20d) Lu tr ca van: phi tuyn

Chng 2: M HNH HA


21

Hnh 2.17: Lu tr

Dung: g

ACL [m5/N]

(2.3) pVCL

(2.4) qdtCp L1

Chng 2: M HNH HA


22

Hnh 2.18: Lu dung

Qun tnh: alIL [Nsec2/m5]

(2.5) dtdqIp L

2.3.5.2 Cac phng trnh can bang trong h lu cht

Chng 2: M HNH HA


23

2.3.6 Mt s th d Th du 2.5: M hnh ha h thng sau: Cch 1: Dng nh lut Newton Chiu ln phng chuyn ng, p dng nh lut II Newton: sin)()( PtFtxm Cch 2: Dng cng thc EulerLagrange: ng nng: 2

21 xmT

Th nng: sinmgxU sin

21 2 mgxxmUTL

p dng cng thc EulerLagrange: F

xL

xL

dtd

Fmgxmdt

d sin

Fmgxm sin Th du 2.6: Mo hnh hoa he thong giam soc cua xe may (phuoc nhung)

Cch 1: p dng nh lut Newton:

)()()()( txbtkxtftxm

)()()()( tftkxtxbtxm

P

N F

x

x

f

b k

m

Chng 2: M HNH HA


24

Cch 2: p dng cng thc Euler-Lagrange: ng nng: 2

21 xmT

Th nng: 221 kxU

2221

21 kxxmUTL

p dng cng thc EulerLagrange: xbf

xL

xL

dtd

xbfkxxmdtd

)()()()( tftkxtxbtxm Cch 3: p dng s tng ng gia h thng in v h thng c: Th: Uf Lng: qx Dung: Ck /1 Tr: Rb Cm: Lm Quan h gia dng v p ca mch in tng trn l: )(1)( sULs

CsRsI

)(1)( 2 sULs

CRs

ssI

)(1)( 2 sULsC

RssQ

)()(1)()( tutqC

tqRtqL Do quan h gia cc i lng ca h c l:

)()()( tfkxtxbtxm

U R C L

i

Chng 2: M HNH HA


25

Th du 2.7: Mo hnh hoa he con lac ngc

Chu thch: M : trong lng xe [Kg] m : trong lng con lac [Kg] l : chieu dai con lac [m] u : lc tac ong vao xe [N] g : gia toc trong trng [m/s2] x : v tr xe [m] : goc gia con lac va phng thang ng [rad] Cach 1: Dung nh luat Newton Goi (xP, yP) la toa o cua vat nang m au con lac, ta co: sinlxxP (1) coslyP (2) Ap dung nh luat II Newton cho chuyen ong theo phng x, ta co:

Fdt

xdmdt

xdM P 22

2

2

(3)

Thay xG bieu thc (1) vao bieu thc (3) suy ra:

Flxdtdm

dtxdM sin2

2

2

2

(4)

Khai trien cac ao ham bieu thc (4) va rut gon ta c: FmlmlxmM )(cos)(sin)( 2 (5)

x

x F M

m

l

mg lcos

lsin

Chng 2: M HNH HA


26

Mat khac, ap dung nh luat II Newton cho chuyen ong quay cua con lac quanh truc ta c:

sinsincos 22

2

2

mglldt

ydmldt

xdm PP (6) Thay (1) va (2) vao (6) suy ra:

sinsin)cos(cos)sin( 22

2

2

mgllldtdmllx

dtdm

(7)

Khai trien cac ao ham bieu thc (7) va rut gon ta c: sincos mgmlxm (8) T (5) va (8), ta co the de dang tnh c:

22

)(cossincos)(sin

mmMmgmlFx

(9)

lmMml

mlgmMF)()(cos

)sin(cos)(sin)(cos2

2

(10) Cach 2: Dung cong thc EulerLagrange: Goi (xP, yP) la toa o cua vat nang m au con lac, ta co: sinlxxP (1) coslyP (2) ong nang cua vat nang au con lac:

2222 sin21cos

21

21

21 lmlxmymxmT PPP

22221cos

21 mlxmlxmTP

ong nang cua xe:

221 xMTC

ong nang cua he thong:

22221cos

21 mlxmlxmMTTT CP

The nang cua he thong = the nang vat nang au con lac: cosmglU

Chng 2: M HNH HA


27

cos21cos

21 222 mglmlxmlxmMUTL (3)

Phng trnh Euler Lagrange:

FxL

xL

dtd

(4)

0

LL

dtd

(5) Thay (4) vao (3): FmlmlxmM 2)(sin)(cos 0sincos mgmlxm Suy ra:

22

)(cossincos)(sin

mmMmgmlFx

lmMml

mlgmMF)()(cos

)sin(cos)(sin)(cos2

2

Th du 2.8: Mo hnh he tay may hai bac t do

l1, l2 : chieu dai cua 2 m1, m2 : khoi lng 1, 2 : goc quay cua cac khp canh tay 1, 2 : moment lam quay cac khp noi

Toa o cua canh tay may trong he toa o De-cac la: 111 sinlx 111 cosly 22112 sinsin llx 22112 coscos lly

1

2

l1

l2

m1

m2

y1

y2

x2 x1

Chng 2: M HNH HA


28

Van toc:

111

111

1

11 sin

cos

ll

yx

v

222111

222111

2

22 sinsin

coscos

llll

yx

v

ong nang:

)(21)(

21 2

2222

21

211 yxmyxmT

222222121221211 21

21

21 lmlmlmT

)sinsincos(cos 212121212 llm The nang: )coscos(cos 22112111 llgmglmU Do o:

22222

21

212

21

211 2

121

21 lmlmlmUTL

)sinsincos(cos 212121212 llm )coscos(cos 22112111 llgmglm (1) Phng trnh EulerLagrange:

111

LL

dtd

(2)

222

LL

dtd

(3)

Thay (1) vao (2) va (3) ta c: 221212121

2121 )sinsincos(cos)( llmlmm

11121222121212 sin)()sincoscos(sin glmmllm (4)

121212122

222 )sinsincos(cos llmlm

2222212121212 sin)sincoscos(sin glmllm (5)

Chng 2: M HNH HA


29

Th du 2.9: Mo hnh toan bon cha chat long (Single Tank) A: tiet dien ngang bon cha a: tiet dien van xa k: he so t le vi cong suat may bm

Phng trnh can bang: )()()( tqtqtAhdtd

outin Dong vao: )()( tkutqin Dong ra: )()/2()( 222 tpCatq Dout trong o: )()( tghtp : ap suat 6.0DC : he so xa

Suy ra: )(2)(1)( tghaCtkuA

th D Th du 2.10: He bon noi tiep (Cascade Tank)

)(2)(21)(

)(2)(1)(

2221112

2

111111

1

tghCatghCaA

th

tghCatukA

th

DD

D

h(t) u(t)

qin

qout

h1(t) u(t)

qin1

qout2 h2(t)

qout1=qin2

Chng 2: M HNH HA


30

Th du 2.11: He bon lien ket (Coupled Tank) Mo hnh toan:

)()(2)()(sgn)(2)(1)(

)()(2)()(sgn)(2)(1)(

21121212222222

2

21122112111111

1

ththgaththCtghCatukA

th

ththgaththCtghCatukA

th

DD

DD

Th du 2.12: S tng ong gia he lu chat va he thong ien:

e 2 mo hnh tren tng ng ta can gia thiet bon cha rat ln, khi he thong van hanh o cao mc chat long trong bon cha thay oi khong ang ke.

Chng 2: M HNH HA


31

Th du 2.13: Mo hnh lo say )(tS : nhiet o nguon nhiet )(t : nhiet o lo say )(tH : dong nhiet Ta co:

Dong nhiet: T

S

RtttH )()()(

Phng trnh can bang: )()( tHdt

tdCT trong o:

AkdRT 2

: nhiet tr d: chieu dai lo say A: tiet dien ngang k: he so dan nhiet TC cM : nhiet dung c: nhiet dung rieng cua moi trng truyen nhiet M: khoi lng moi trng truyen nhiet Suy ra mo hnh toan hoc cua lo say la:

)()()( ttdt

tdCR STT Trng hp lo say dai, co the mo hnh hoa bang cach chia lam nhieu ngan:

(t) S(t) H(t)

2(t) S(t) H1(t) 1(t)

d/4 d/4 d/2

H2(t)

Chng 2: M HNH HA


32

Dong nhiet:

1

11

)()()(T

S

RtttH

2

212

)()()(TR

tttH Phng trnh can bang:

)()()( 2111 tHtHdttdCT

)()( 222 tHdttdCT

Suy ra:

2

2122

2

21

1

111

)()()(

)()()()()(

TT

TT

ST

Rtt

dttdC

Rtt

Rtt

dttdC

2.4 PHAN TCH TOAN HOC (tham khao chng 5 va chng 8, (Smith, 1994)) Phan tch toan hoc:

Ket hp tat ca cac he phng trnh mo ta ac tnh ong cua cac bo phan chc nang e c he phng trnh mo ta he thong.

Tuyen tnh hoa quan he phi tuyen e c mo ta toan hoc

tuyen tnh Xet he phi tuyen bac n co p ngo vao, q ngo ra mo ta bi phng trnh trang thai.

))(),(()())(),(()(

tttttt

uxhyuxfx

trong o nt )(x la vector trang thai, pt )(u la vector tn hieu vao, qt )(y la vector tn hieu r; n(.)f , q(.)h la vector ham mo ta ac tnh ong cua he phi tuyen.

Chng 2: M HNH HA


33

Khai trien Taylor xung quanh iem lam viec tnh ),( ux ta co the mo ta he thong bang phng trnh trang thai tuyen tnh:

)(~)(~)(~)(~)(~)(~

tttttt

uDxCyuBxAx

trong o: ),( ,)()(~

)()(~)()(~

uxhyyyyuuuxxx

tttttt

Cac ma tran trang thai cua he tuyen tnh gan ung c tnh nh sau:

)(2

2

2

2

1

2

1

2

1

1

1

)(

u,x

,xxfA

n

nn

n

n

n

n

u

xf

xf

xf

xf

xf

xf

xf

xf

xf

)(21

2

2

2

1

2

1

2

1

1

1

)(

u,x

u,xufB

p

nnn

p

p

uf

uf

uf

uf

uf

uf

uf

uf

uf

)(21

2

2

2

1

2

1

2

1

1

1

)(

u,x

u,xxhC

n

qqq

n

n

xh

xh

xh

xh

xh

xh

xh

xh

xh

Chng 2: M HNH HA


34

)(21

2

2

2

1

2

1

2

1

1

1

)(

u,x

u,xuhD

p

qqq

p

p

uh

uh

uh

uh

uh

uh

uh

uh

uh

ai so s o khoi Phng phap s o dong tn hieu va cong

thc Mason e tm ham truyen tng ng cua he tuyen tnh. anh gia s phu hp cua mo hnh Dung mo hnh e d bao ap ng cua he thong oi vi tn

hieu vao cho trc.

Th du 2.14: Mo hnh toan he con lac ngc truyen ong dung ong c DC, xet anh hng cua ma sat: * ac tnh ong cua he xecon lac co xet en anh hng cua ma sat: Tng t nh th du 2.7, tuy nhien lc tac ong phai ke them lc ma sat: CfFmlmlxmM )(cos)(sin)( 2 (1) Pfmgmlxm sincos (2) trong o: Cf lc ma sat tac ong len xe Pf lc ma sat tac ong len con lac

Chng 2: M HNH HA


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* ac tnh ma sat: Gia thiet co ca ma sat tnh va ma sat nht tac ong lam can tr chuyen ong cua xe va con lac. Cac lc ma sat nay co the mo ta bang cac phng trnh sau: xBeAxf x

xCxC

x )sgn( (3) BeAf CP )sgn( (4) trong o xA , xB , xC , A , B , 0C . * ac tnh ong c: baaaaa EIRVIL (5) bb KE mlmm BTTJ (6) agim IKKT FrTl rxKg / aL : ien cam phan ng aR : ien tr phan ng aI : dong ien phan ng aV : ien ap phan ng mT : moment ong c lT : moment tai : toc o quay ong c r : ban knh pu-li mJ : moment quan tnh ong c mB : he so ma sat nht iK : he so moment iK : he so moment gK : he so giam toc bE : sc phan ien F : lc tac ong vao xe ac tnh ong c co the bieu dien theo aI , x va F nh sau:

Chng 2: M HNH HA


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abg

aaaa VxrKK

IRIL (7)

FIrKK

xrBK

xr

JKa

gimgmg 22 (8)

Do o mo hnh toan hoc cua he xe con lac vi tn hieu vao la ien ap cap cho ong c nh sau:

abg

aaaa

P

Caigmgmg

VxrKK

IRIL

fmgmlxm

fIrKK

xrBK

mlmlxr

JKmM

sincos

)(cos)(sin)( 22

2

Th du 2.15: Mo hnh tuyen tnh cua he con lac ngc xung quanh v tr thang ng. He phng trnh mo ta ac tnh ong phi tuyen cua he con lac ngc (xem th du 7):

22

)(cossincos)(sin

mmMmgmlux

(1)

lmMml

mlgmMu)()(cos

)sin(cos)(sin)(cos2

2

(2) at Txxt ],,,[)( x : vector trang thai, ta c:

(.))(cos

sincos)(sin)(

)()(cos)sin(cos)(sin)(cos

)(

)()()()(

21

11221

4

21

221111

2

4

3

2

1

fx

xmmMxxmgxxmlu

txlmMxml

xxxmlxgmMxutx

txtxtxtx

Chng 2: M HNH HA


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(.))()(

)()(

)(3

1

h

txtx

txt

ty

iem can bang v tr thang ng: )0,(),( 0ux Tuyen tnh hoa xung quanh iem can bang:

)(~)(~)(~)(~)(~)(~

tttttt

uDxCyuBxAx

trong o:

0001000

000)(0010

)0(4

4

3

4

2

4

1

4

4

3

3

3

2

3

1

3

4

2

3

2

2

2

1

2

4

1

3

1

2

1

1

1

)(

gMm

gMl

mM

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

xf

u

,

,xxfA

0

M

Ml

ufufufuf

u u 10

10

)0(

4

3

2

1

)(

,

,x

fB

0

01000001

)(4

2

3

2

2

2

1

2

4

1

3

1

2

1

1

1

)(

u,x

,xxhC

xh

xh

xh

xh

xh

xh

xh

xh

u

00

)(

2

1

)(

u,x

,x

hD

uhuh

u u

Chng 2: M HNH HA


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chuong2 giao trinh ndht

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