C S THY A C HC

CHNG V. VN NG N NH CA ND TRONG

CC LP KHNG NG NHT

Nu tng cha nc c cng thnh phn thch hc v hs thm th gi l tng cha nc ng nht. Tng chanc ng nht c th ng hng hoc d hng.

Khng nn nhm khi nim d hng vi khi nim khngng nht. Tnh d hng do c im kin trc v cu toca t gy ra, cn tnh khng ng nht do s khcnhau v thnh phn thch hc ca t gy ra. Trong tnhin thng gp cc tng cha nc khng ng nht.

Cc tng cha nc c th khng ng nht trong mt ctv c trn bnh din. Kamenxki G.N. v Guxeinzade M.A c nhiu ng gp vo vic nghin cu l thuyt vnng ca nc di t trong cc tng cha nc khngng nht. Guxeinzade dng l thuyt hm s bin sphc nghin cu s vn ng ca cht lng trong cctng cha nc c h s thm thay i trn mt bng.

Nghin cu vn ng ca nc trong cc tngcha nc khng ng nht trong mt ct, Kamenxki G.N chia thnh ba dng ch yu sau.

1. Tng cha nc gm nhiu lp, cc lp c hs thm khc nhau sp xp theo mt th t nhtnh;

2. Tng cha nc gm hai lp, thng gp nhtkhi lp trn c h s thm nh hn so vi lpdi;

3. Tng cha nc c h s thm thay i t thoc thay i t ngt theo hng vn ng.

Sau y chng ta s nghin cu mt s trnghp thng gp.

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

1) Khi nc thm song song vi mtlp:

V d vn ng ca nc song song vimt lp trong tng cha nc c cu tonhiu lp (hnh IV.1a). Trong trng hpny, tr s gradien p lc tit din btk l khng i v lu lng n v cadng chy mi mt lp c th biu dintheo nh lut acxi nh sau

lp th 1 q1 = k1h1I lp th 2 q2 = k2h2I lp th n qn = knhnI y, k1, k2; h1, h2, l h s thm v

chiu dy ca cc lp 1,2

Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. a- vn ng song song

vi mt lp;

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

1) Khi nc thm song song vi mtlp:

Cng v vi v cc biu thc trnchng ta nhn c lu lng n vca dng nc ngm trong tng chanc

(IV-1) Cch khc, thay tng cha nc

khng ng nht bng tng chanc tng ng ng nht c hs thm ktb, lu lng n v catng cha nc tng ng l:

Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. a- vn ng song song

vi mt lp;

( )=

++==n

ii Ihkhkqq

12211 ...

hIkq tb= (IV-2)

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

1) Khi nc thm song song vi mtlp:

y, h - chiu dy ca tng chanc tng ng, bng tng chiudy tng lp ring ca tng chanc khng ng nht.

T (IV-1) v (IV-2) chng ta nhnc cng thc tnh h s thmtrung bnh ca tng cha nc khinc vn ng song song vi mt lp

(IV-3)

Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. a- vn ng song song

vi mt lp; ......

21

2211

++++=

hhhkhkktb

Palubarinva Kotsina P.Ya (1952) gi ktb l h s thm tng ng hoc h s thmdn dng: N l gi tr trung bnh cn bng ca tng cha nc gm nhiu lp, c tnhn chiu dy ca mi lp, v vy, ktb cn gi l h s thm trung bnh cn bng

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

2) Khi nc vn ng vung gc vimt lp:

Khi nc di t vn ng vunggc vi mt lp (xem hnh IV-1b), theo nh lut acxi tc thm trongmi mt lp s bng y, H1, H2, l tn tht p lc trong milp; h1, h2, v k1, k2 l chiu dyv h s thm ca cc lp.

lp th 1

lp th 2 (IV-4)

lp th n

Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. b- vn ng vung gc

vi mt lp;

1

11111 h

HkIkv ==

2

22222 h

HkIkv ==

n

nnnnn h

HkIkv ==

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

2) Khi nc vn ng vung gc vimt lp:

V tnh lin tc ca dng thm nntc thm qua mi mt lp s bngnhau. T phng trnh (IV-4) chngta c tng tn tht p lc khi ncvn ng qua tng cha nc:

H = H1 + H2 + Hn = (IV-5) Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. b- vn ng vung gc

vi mt lp;

++= ...

2

2

1

1

kh

khv

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

2) Khi nc vn ng vung gc vimt lp:

Tng tn tht p lc khi nc, vnng vung gc vi mt lp c thxc nh bng cch khc: thay tngcha nc ktb v chiu dy h bngton b chiu dy ca tng chanc khng ng nht (hnh IV-1b), y, v - tc thm ca lp tngng (bng tc thm trong lpthc t).

(IV-6)

Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. b- vn ng vung gc

vi mt lp;

tbkhvH .=

1. Xc nh h s thm trung bnh cacc tng cha nc khng ng nht

2) Khi nc vn ng vung gc vimt lp:

So snh cc phng trnh (VI-5) v(IV-6), sau khi rt gn chng ta tmc gi tr h s thm trung bnh khinc thm vung gc vi mt lp.

(IV-7)Hnh IV.1. Vn ng canc trong tng cha

nc khng ng nht. b- vn ng vung gc

vi mt lp;

n

n

ntb

kh

kh

kh

hhhk++++++=

...

...

2

2

1

1

21

Chng ta hon ton c th chng minh c h s thmtrung bnh khi nc vn ng song song vi mt lp l cci, cn h s thm trung bnh khi nc vn ng vung gcvi mt lp l cc tiu Trong trng hp nc vn ngnghing vi mt lp h s thm trung bnh s c gi tr trunggian.

2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht

1) Tng cha nc khng ng nht c cu to phctp:

Nghin cu vn ng ca nc ngm trong tng chanc nm nghing c cu to phc tp, ngm ncthay i theo c phng ngang ln phng thng ng. Lu lng n v ca dng ngm xc nh theo cngthc gn ng Kamenski. Chng minh cng thc nhsau:

Phng trnh Duypuy i vi lu lng ca dng ngmtrong trng hp trn c dng

(IV-8)

y K - h s thm trung bnh mt ct bt kdxdHKhq =

2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht

1) Tng cha nc khng ng nht c cu to phctp:

Tch phn biu thc trn trong khong t mt ct 1 n2; p dng nh l v gi tr trung bnh ca tch phnhm s Kh=f(h) s c mang ra nmgoi du tch phn dng gi tr trung bnh f(Htb):

(IV-9)

Sau tch bin s v tch phn biu thc (IV-8), tmc cng thc gn ng tnh lu lng n v cadng ngm

(IV-10) K1 v K2 - h s thm trung bnh p83 mt ct 1 v 2; h1

v h2 - b dy dng ngm mt ct ; H1, H2 caotrnh mc nc; L - khong cch gia 2 mt ct.

2)( 2211 hKhKHf tb

+=

LHHhKhKq 212211

2+=

2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht

2) Tng cha nc gm hai lp : Chng ta nghin cu tng cha nc

gm hai lp v lp di c h s thmln hn lp trn nhiu ln. Cu to catng cha nc nh vy rt ph bintrong nhng tam gic chu ca ccsng

Xt tng cha nc gm hai lp cy cch nc nm ngang. Trongtrng hp ny c th coi nh ncdi t vn ng c chung mt mt

Hnh IV.2. Vn ngca nc di t

trong tng cha ncgm hai lp

p lc ng gch t trn (hnh IV.2). Khi , gi thitrng phn trn nc di t vn ng vi b mt tdo, cn phn di nc di t vn ng nh ncp lc.

2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht

2) Tng cha nc gm hai lp : Phng trnh lu lng n v ca

dng nc ngm trong tng chanc c dng sau

(IV-11)

Sau khi bin i biu thc (IV-11) vtch phn t mt ct 1 n mt ct 2, chng ta nhn c

(IV-12)

Hnh IV.2. Vn ngca nc di t

trong tng cha ncgm hai lp

+=dxdhmk

dxdhhkq 21

Lhhmk

Lhhkq 212

22

21

1 2+=

2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht

2) Tng cha nc c h sthm thay i t ngt theophng vn ng :

Trng hp nghin cu c dngnh hnh IV.3.

gii bi ton trn chng tathng dng phng php phnon.

Hnh IV.3. Tng cha nc ch s thm thay i t ngt

theo phng vn ngLy mt ct x trng vi ranh gii tip xc gia hai lp chanc c dn nc khc nhau (xem hnh IV.3). Trongphm vi t mt ct 1 n mt ct x v t mt ct x n mtct 2 tng cha nc l ng nht.Lu lng n v ca dng nc ngm gia mt ct 1 vmt ct x s bng

(IV-13) 1

221

1 2Lhhkq x=

2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht

2) Tng cha nc c h sthm thay i t ngt theophng vn ng :

Cn lu lng n v ca dngngm gia mt ct x v mt ct2 c xc nh nh sau

(IV-14) Hnh IV.3. Tng cha nc ch s thm thay i t ngttheo phng vn ng

2

22

2

2 2Lhhkq x =

T hai phng trnh trn chng ta nhn c cng thclu lng n v ca dng nc ngm trong tng chanc c h s thm thay i t ngt theo phng vnng:

(IV-15)

+

=2

2

1

1

22

21

2kL

kL

hhq

3. Dng chy qua lp thm nc yu Chng ta s nghin cu vn ng

ca nc di t trong tng chanc gm nhiu lp, c nhng lpthm nc tt c ngn cchbng nhng lp thm nc yu(xem hnh IV.4).

Theo Girinxki Miatiev trong tngcha nc tn ti hai dng vnng: 1) nc vn ng trong lptheo hng t min cung cp nmin thot v 2) nc chy xuynqua lp thm nc yu.

Hnh IV.4. S vn ngca nc trong hai lptheo Girinxki - Miatiev

Tng hp hai dng vn ng trn to thnh dng phc tp lm sng tc dim vn ng ca dng tng hp chng ta ly ra mt nguyn t dngthm dx (hnh IV.4). Dng cc k hiu sau: k1, m1 - h s thm v chiu dyca lp thm nc yu; k2, m2 - h s thm v chiu dy lp cha nc vib mt p lc; H p lc ca lp trn; Hx p lc ca lp cha ncnghin cu.

3. Dng chy qua lp thm nc yu Tham gia vo s cn bng ca

nc trong nguyn t dng thm ccc i lng sau: 1) dng ncchy n nguyn t q; 2) dngxuyn tng q* chy qua lp thmnc yu; 3) dng nc chy ra tnguyn t q + dq.

Khi vn ng n nh phng trnhcn bng ca nc trong nguyn tdng thm c dng sau

q + q* = q + dq (IV-16) do q* = dq (IV-17)

Hnh IV.4. S vn ngca nc trong hai lptheo Girinxki - Miatiev

Theo nh lut acxi lu lng n v ca dng nc dit trong tng cha nc c biu din bng cng thc

dxdHmkq x.22= (IV-18)

3. Dng chy qua lp thm nc yu Vi phn phng trnh (IV-18) theo

bin s x chng ta nhn c (IV-19)

bng cch khc, chng ta xc nhdq theo cng thc lu lng cadng chy xuyn qua lp thmnc yu

(IV-20)

T hai phng trnh (IV-19) v (IV-20) chng ta tm c phng trnhcn bng trong nguyn t dngthm:

Hnh IV.4. S vn ngca nc trong hai lptheo Girinxki - Miatiev

2

2

22 . dxHdmkdq x=

dxmHHkqdq x1

1* . ==

2221

1. dxHdmkdx

mHHk xx =

2(IV-21)

3. Dng chy qua lp thm nc yu t H Hx = H, phng trnh trn

vit li c dng n gin hn (IV-22)

y, .

H s b gi l h s chy xuyntng.

Phng trnh (IV-22) l phngtrnh vi phn tuyn tinh ng nht. T l thuyt phng trnh vi phn

Hnh IV.4. S vn ngca nc trong hai lptheo Girinxki - Miatiev

221

12

kmmkb =

0222

= Hbdx

Hd

chng ta bit li gii ca phng trnh (IV-22) c dngH = C1exp(bx) + C2 exp (-bx) (IV-23)

y C1 v C2 l cc bng s c xc nh theo iu kin bin gii.Ty tng trng hp c th chng ta s c nhng li gii khc nhau. Da vo li gii tm c chng ta nghin cu c im vn ng ca trc di t khi c dng xuyn qua lp thm nc yu cung cp cho tng cha nc.

HT CHNG V

C S THY A C HC1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 1. Xc nh h s thm trung bnh ca cc tng cha nc khng ng nht 2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht 2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht 2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht 2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht 2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht 2. Cc phng trnh vn ng ca ND trong cc lp khng ng nht 3. Dng chy qua lp thm nc yu3. Dng chy qua lp thm nc yu3. Dng chy qua lp thm nc yu3. Dng chy qua lp thm nc yu