do an tot nghiep ok in_verson new_3

8/21/2019 Do an Tot Nghiep OK In_verson New_3

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MC LC

LI NI U.

CHNG 1. O HM V TCH PHN CP PHN S

1.1 Kh|i nim v{ nh ngha m u o h{m v{ tch ph}ncp nguyn....

1.2 Biu thc hp nht gia o h{m v{ tch ph}n cp nguyn ...

1.3 nh ngha o h{m v{ tch ph}n cp ph}n s ...

1.4 C|c tnh cht ca o h{m v{ tch ph}n cp ph}n s..

1.5 Mt s v d v o h{m, v{ tch ph}n cp ph}n s.

1.6 Php bin i Fourier v{ Laplace ca o h{m cp ph}n s. .............................

CHNG 2. CC PHNG PHP S TNH TON DAO NG C H C O HM CP PHN S

2.1 Xp x th{nh phn o h{m cp ph}n s.

2.2 Phng ph|p s gii phng trnh vi ph}n cp ph}n s..

2.2.1 Phng ph|p Newmark

2.2.2 Phng ph|p Runge-Kutta-Nystrm...

2.2.3 Phng ph|p Runge-Kutta.

2.2.4 Phng ph|p Gauss.

2.2.5 So s|nh chnh x|c v{ thi gian tnh gia c|c phng ph|p tnh.

2.3 Kt lun..

CHNG 3. TNH TON DAO NG CA MNG MY TRN NN N NHT CP PHN S..

3.1 Cc m hnh h {n nht..

3.2 H dao ng mt bc t do.

3.3 H dao ng hai bc t do...

3.4 H dao ng nhiu bc t do.

3.5 So snh kt qu tnh to|n gia c|c m hnh l thuyt v{ vi kt qu thc nghim

3.6 Kt lun.

CHNG 4. CHNG TRNH FDESOLVER TNH TON DAO NG ....

4.1 Tng quan v chng trnh

4.2 S dng chng trnh

KT LUN..

TI LIU THAM KHO..

PH LC..

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GVHD: GS.TSKH. Nguyn Vn Khang

SVTH: Dng Vn Lc KSTNCT- K54 1

LI NI UV{i thp k gn }y, nhiu ng dng ca o h{m cp ph}n s trong c|c lnh vc vt l, ha hc,

c kh, giao thng vn ti, x}y dng, t{i chnh v{ c|c ng{nh khoa hc kh|c ~ c quan t}m nghin

cu. Oldham v{ Spanier ng dng o h{m cp ph}n s v{o qu| trnh khuych t|n; Kempfle m t c

h tt dn; Bagley v{ Torvik, Caputo nghin cu v tnh cht ca c|c vt liu mi

C|c nh{ c hc cng bt u nghin cu vic |p dng o h{m cp ph}n s v{o c|c h dao

ng, ng lc hc nh h {n nht v{ nht do. Nutting (1921, 1943) l{ mt trong nhng nh{ nghin

cu u tin ngh rng hin tng chng ng sut c th c m hnh thng qua thi gian bc ph}n

s. Shimizu (1995) nghin cu dao ng v{ c tnh xung ca b dao ng vi m hnh Kelvin Voigt

ph}n s ca vt liu {n nht da trn gel silicone v{ chng minh mt s tnh cht kh|c bit gia kh

nng gim chn ca vt liu n{y so vi vt liu o h{m cp nguyn. Zhang v{ Shimizu (1999) nghin

cu mt v{i phng din quan trng v trng th|i tt dn ca b dao ng {n nht c m hnh biquy lut kt cu Kelvin Voigt ph}n s. H ~ tho lun s nh hng ca iu kin u ti trng th|i

tt dn

Ta ~ bit quan h gia lc v{ bin dng ca c|c b gim chn {n nht c dng

, 0 1.pf t D x t p

Trong f t l{ lc t|c dng, x t l{ dch chuyn, l{ h s cn khng tuyn tnh,p

p

p

dD

dx

l{ to|n t o h{m cp ph}n s.Thc t rng i vi nhng bin dng ln, |p ng phi tuyn xut hin. Mt s m hnh c

xut gii thch s |p ng phi tuyn. Mt m hnh c th c a ra l{ mt l xo phi tuyn

c thm v{o v phi ca phng trnh trn. Mt m hnh kh|c c a ra biZhimizu v Nasuno

yu cu i vi mt s vt liu {n nht, tnh phi tuyn xut ph|t t o h{m cp ph}n s ca bin

dng nn.

|n tt nghipn{y trnh b{y c|c m hnh c bn ca o h{m cp ph}n s, v{ c|c phng

ph|p s gii phng trnh vi ph}n cha o h{mcp ph}n s. p dng tnh to|n mt v{i m hnh dao

ng phi tuyn ca c|c h {n nht c cha o h{m cp ph}n s.

Qua }y em xin gi li cm n ch}n th{nh nthy GS.TSKH.Nguyn Vn Khang, thy ~ tn

tnh gip em trong sut thi gian thc hin |n tt nghip n{y.

H Ni, Ngy 10 thng 05nm 2014

Sinh vin thc hin: Dng Vn Lc

Email: [email protected]
mailto:[email protected]:[email protected]:[email protected]


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CHNG 1Equation Chapter (Next) Section 1

O HM V TCH PHN CP PHN S1.1 Kh|i nim v{ nh ngha m u o h{m v{ tch ph}n cp nguyn

Chng ta s dng n v N l{ nhng s nguyn dng, , , , ,p q r v Q l{ nhng s bt k.

Cho mt h{m s f x . Ta k hiu o h{m cp 1, cp 2,... cp n ,ca h{m f x nh sau

2

2, ,..., ,...

n

n

df x d f x d f x

dx dx dx (1.1)

Ngo{i ra ta cng c c|c k hiu o h{m tng t

2

2, ,..., ,...

n

n

df x d f x d f x

dx dx dx (1.2)

o h{m ca h{m f x theo x a bng o h{m theo x ca n

2 2

2 2, ,..., ,...

n n

n n

df x d f x d f x d f x d f xdf

dxd x a dx dxd x a d x a

(1.3)

Do tch ph}n l{ s nghch o ca o h{m nn ta vit

1

0 01

0

xd f x

f x dxdx

(1.4)

C|c tch ph}n nhiu lp c k hiu

12

1 0 02

0 0

xxd f x

dx f x dxdx

(1.5)

1 2 1

1 2 1 0 0

0 0 0 0

.nx x xxn

n nn

d f xdx dx dx f x dx

dx

(1.6)

Khi gii hn di kh|c 0, c|c tch ph}n s c vit

1

0 01

x

a

d f xf x dx

d x a

(1.7)

1 2 1

1 2 1 0 0.nx x xxn

n nn

a a a a

d f xdx dx dx f x dx

d x a

(1.8)

Lu phng trnh sau ng vi o h{m nhng khng ng vi tch ph}n

n n

n n

d f x d f x

dxd x a

(1.9)

Tc l{


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.

n n

n n

d f x d f x

dxd x a

(1.10)

o h{m cp n thng c vit nf x

T ta s s dng i vi tch ph}n

1 2 1

1 2 1 0 0.nx x xx

n

n n

a a a a

f x dx dx dx f x dx

(1.11)

Vi p l{ s bt k

.p p p

p

p p p

d f x d f x d f xf x

dxdxd x a

(1.12)

.

p p

p p

x b

d f x d fb

d x a d x a

(1.13)

Mt s k hiu sau thng c s dng

.

ppa x

p pa

D f xd f x

D f xd x a

(1.14)

1.2 Biu thc hp nht gia o h{m v{ tch ph}n cp nguyn

1.2.1 o h{m cp n

Trc khi gii thiu o h{m cp ph}n s, ta s rt ra biu thc hp nht cho o h{m v{ tch

ph}n cp nguyn. u tin, ta c nh ngha o h{m cp 1 ca h{m f x

11

1 0 0lim lim .x x

d f x df x f x f x xx f x f x x

dx xdx

(1.15)

o h{m cp 2 ca h{m f x

20 0

2 0 0

2

0

2lim lim

lim lim

lim 2 2

x x

x x

x

f x f x x f x x f x xd f x f x f x x x x

x xdx

x f x f x x f x x

(1.16)

Tng t ta c o h{m cp 3

33

30

lim 3 3 2 3 .x

d f xx f x f x x f x x f x x

dx

(1.17)

Bi c|c h s trong nhng phng trnh trn gn ging vi h s nh thc Newton, ta c th

vit o h{m cp n


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00

lim 1 .

n nn j

nx

j

nd f xx f x j x

jdx

(1.18)

Gi thit rng tt c c|c o h{m u tn ti v{ x tin ti 0 lin tc, ngha l{ tt c nhng gi|

tr ca n u tin ti 0. i vi s biu din hp nht vi tch ph}n, ta s cn c mt gii hn cht. l{m c iu n{y, chia khong x a thnh N on bng nhau

, 1,2,3...Nx x a N N (1.19)Thay v{o phng trnh (1.35)

00

lim 1 .N

n nn j

N Nnx

j

nd f xx f x j x

jdx

(1.20)

Ch rng h s nh thcn

j

= 0 nu j n , (1.20) c vit li nh sau

1

00

lim 1 .N

n Nn j

N Nn xj

nd f xx f x j x

jdx

(1.21)

T (1.19) v{ (1.21) suy ra

1

0

lim 1 .

nn Nj

n Nj

nd f x x a x af x j

jN Ndx

(1.22)

1.2.2 Tch ph}n nhiu lp ca mt h{m s

B}y gi ta s ch v{o biu thc tch ph}n n lp ca f x . V mt tch ph}n cp nguyn c

nh ngha qua din tch, ta biu din n vi tng Riemann

11 1

0 01

0

1

0

0

lim 2

lim .

N

N

x

a x a x

a

N N N Nx

N

N Nx

j

d f xI f x D f x f x dx

d x a

x f x f x x f x x f a x

x f x j x

(1.23)

Tch ph}n 2 lp:

122 2

1 0 02

2

0

12

00

lim 2 3 2

lim 1 .

N

N

xx

a x a x

a a

N N N Nx

N

N Nx

j

d f xI f x D f x dx f x dx

d x a

x f x f x x f x x Nf a x

x j f x j x

(1.24)

i vi tch ph}n 3 lp


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2 133 3

2 1 0 03

13

0 0

1 2lim .

2N

x xx

a x a x

a a a

N

N N

x j

d f xI f x D f x dx dx f x dx

d x a

j jx f x j x

(1.25)

Tng t vi tch ph}n n lp vit nh sau

1 1

1 2 0 0

1

00

1lim .

n

N

x xxnn n

a x a x n nn

a a a

Nn

N Nx

j

d f xI f x D f x dx dx f x dx

d x a

j nx f x j x

j

(1.26)

1

0

1lim .

nn N

n N j

j nd f x x a x af x j

jN Nd x a

(1.27)

1.2.3 Shp nht gia to|n t o h{m cp n v{ tch ph}n n lp

B}y gi ta thay n n vi n nhn gi| tr }m th phng trnh (1.27) c dng

1

0

1lim .

nn N

n Nj

j nd f x x a x af x j

jN Nd x a

(1.28)

So s|nh phng trnh (1.22) v{ (1.28) ta thy

1

1 j n j n

j j

(1.29)

Tht vy ta s chng minh cng thc (1.29)

Theo nh ngha

1 2 1 1 21 1

! !

11 !

! 1 !

j jn n n n n j j n j n n

j j j

j nj n

jj n

(1.30)

Vi 1 !, 1n n n n n

1 2 1 1!

! ! ! 1 1

m m m m m k mm

k k k m k k m k

Thay1m j n

k j

ta c

1 ,

1 1 ,

1 .

m j n

k j

m k n

Mt kh|c

11

1

j n j n j n

j j n j

(1.31)

Do c th vit biu thc (1.22) v{ (1.28) di mt dng chung


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1

0

lim .1

nn Nn

a x nN

j

d f x j nx a x aD f x f x j

N n j Nd x a

(1.32)

Trong n nhn c gi| tr nguyn }m v{ dng.

1.3 nhngha o h{m v{ tch ph}n cp ph}n s1.3.1 nhngha o h{m v{ tch ph}n cp ph}n s theo Grnwald-Letnikov.

Cng thc (1.32) ng vi mi ty , ta t c nh ngha c bn v{ tng qu|t nht theo - Letnikov

1

0

lim .1

pp Np

a x p Nj

d f x j px a x aD f x f x j

N p j Nd x a

(1.33)

Vi p l{ s thc ty .

C|ch nh ngha theo Grunwald - Letnikov nh trn c u im l{ o h{m, tch ph}n cp ph}n

s c tm thng qua gi| tr ca h{m, khng cn c|c php tnh tch ph}n v{ o h{m ca n.

Mt kh|c ngi ta ~ chng minh c rng h{m 0p p c th khng hu hn nhng

t s

j p

p

hu hn.

H s:

11

1 j

j p j pA

j j p

(1.34)

c gi l{ h s Grnwald

1.3.2 nh ngha o h{m v{ tch ph}n cp ph}n s theo Riemann Liouville.

Vi 0p o h{m, tch ph}n cp ph}n s c dng

11

, 0 .

xpp

a x

a

D f x x y f y dy pp

(1.35)

Vi 0p

11

, 0, 1 .( )

xnn pp

a x n

a

dD f x x y f y dy p n p n

n p dx

(1.36)

nh ngha theo Riemann Liouville c ng dng rt ph bin. Tch ph}n trong phng trnh(1.35) ch hi t vi 0p . Tuy nhin,vi 0p b{i to|n c bin i bng vic |p t iu kin

n p trong phng trnh (1.36).

1.3.3 nh ngha o h{m v{ tch ph}n cp ph}n s theo Caputo.

Ta c

11

, 1

tnn pp

a t n

a

dD f t t f d n p n

n p dt

(1.37)

Vi c|c gi| tr u


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11

22

lim ,

lim ,

lim .

pa t

t a

pa t

t a

p na t nt a

D f t b

D f t b

D f t b

(1.38)

B{i to|n gi| tr u n{y v mt to|n hc ho{n to{n hp l nhng v mt ng dng, ngha vt

l ca nhng iu kin u rt kh l gii. gii quyt iu n{y, Caputo a ra mt nh ngha kh|c

ca o h{m v{ tch ph}n cp ph}n s nh sau

11

, 0 1 .x

n p nC p

a x

a

D f x x y f y dy n p nn p

(1.39)

1lim .x

n n nC p

a xp n a

D f x f a f y dy f x

(1.40)

1.3.4 Mt s nh ngha o h{m v{ tch ph}n cp ph}n s kh|c

1.3.4.1 o hm cp phn sdng dy (dng Miller Ross)

nh ngha

0 1 ;

,

p

np p p px x x x

n

dD p

dx

D f x D D D f x

(1.41)

Phng trnh trn c gi l{ o h{m cp ph}n s dng d~y. Trong pxD c nh ngha

dng Riemann Liouville (hoc dng Grunwald - Letnikov). Ta c nh ngha o h{m cp ph}n s

dng d~y

1 2

1 2

,

,

, 1 .

npp ppx x x x

n

n ppa x a x

n

D f x D D D f x

p p p p

d d dD f x D f x n p n

dx dx dx

(1.42)

Nu s dng dng Caputo

, 1 .n pC p C

a x a x

n

d d dD f x D f x n p n

dx dx dx

(1.43)

1.3.4.2 nh ngha dng Weyl (tch phn Weyl)

Xut ph|t t nh ngha Riemann Liouville, cho a ta c nh ngha dng Weyl

11.

xpW p

a x

D f x x y f y dyp

(1.44)

1.3.4.3 nh ngha Davison Essex


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1

1

0

,1

x kn kp

x n k k

x y d f ydD f x dy

dx dx

(1.45)

Vi , 0 1, 0 1,p n k n n l{ s nguyn.

Khi 0k nh ngha Davision Essex tr v nh ngha Riemann- Liouville vi 0,a p n

1

( )

1

0

1.

1

xnp np

x n

dD f x x y f y dy

n pdx

(1.70)

1.4 C|c tnh cht ca o h{m v{ tch ph}n cp ph}n s

1.4.1 Tnh cht tuyn tnh

o h{m cp p ca tng

1 1 2 2 1 21 2

p p p

p p pd c f x c f x d f x d f xc c

d x a d x a d x a

(1.46)

1.4.2 Quy tc Leibniz

o h{m v{ tch ph}n cp p ca tch hai h{m f v g

0.

p p j j

p p j jj

d f x g x p d f x d g x

jd x a d x a d x a

(1.47)

Trong h s nh thc c x|c nh bng vic thay th giai tha vi h{m Gamma tng ng.

0

.jp p j

a x a x

j

pD f x g x D f x g x

j

(1.48)

1.4.3 Tnh cht bin i thang bc

Php bin i thang bc ca mt h{m s i vi gii hn di a

,f x f x a a (1.49)Vi l{ h s thang bc khng i. Nu gii hn di l{ 0, (1.49) tr th{nh

,f x f x (1.50)Khi 0a , ta c s thay i thang bc

, .

p pp

p p

d f X d f X X x a a

d x a d X a

(1.51)

Khi 0a

.

p pp

p p

d f x d f x

dx d x

(1.52)

1.4.4 o h{m v{ tch ph}n cp ph}n s ca mt chui


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Cho 1 chui h{m hi t u 0

j

j

f x

. o h{m v{ tch ph}n cp ph}n s ca chui

0 0

.

qqj

jq q

j j

d fdf x

d x a d x a

(1.53)

Vi h{m c khai trin th{nh chui ly tha 0

j

j

j

f x a x a

, |p dng cng thc

Riemann

1, 1

1

p qq p

q

p xd xp

p qd x a

(1.54)

0 0

,q j j

p p qn n

j jq

j j

pn j n

d na x a a x a

pn qn j nd x an

(1.55)

Trong q ly gi| tr bt k nhng 01, 0,jp a nn 1.4.5 Tnh cht hp th{nh

1.4.5.1 Khi m, n nguyn dng

,

m n m nn m

n m n m m n

d f x d f x d f xd d

d x a d x a d x a d x a d x a

(1.56)

.

m n m nn m

n m n m m n

d f x d f x d f xd d

d x a d x a d x a d x a d x a

(1.57)

.

n m nm

m n m n

d f x d f xd

dx d x a d x a

(1.58)

1

.!

km m nn nm k n

n m m nk n m

d f x d f x x adf a

kd x a d x a d x a

(1.59)

V d 1.1Cho hm 3xf x e . Hy tm

3 3

3 3, .

d f x df xd d

dx dxdx dx

p dng (1.58) v (1.59) ta c

3 2

2 2 1

3 2 0 0 .

d f x d f xdf x f x f

dx dx dx


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1 13 3

0

1 10 ,

3 3

xx x

f x e dx e f

2 23 3

0

1 1 10 ,

3 9 9

x

x xf x e dx e f

Do

33

3

1 1.

9 3 9

xd f xd xedx dx

T (1.59) ta tm c

23 2 23

3 2

1 10 .

2 9 3 9 2

xdf x d f xd x x xf edxdx dx

1.4.5.2 Khip, q l cc s bt k

Cng thc sau ch ng khi c iu kin x|c nh n{o

,

p q pq

q p q p

d f x d f xd

d x a d x a d x a

(1.60)

Gi s f x c khai trin th{nh chui ly tha

0

, 1, khng nguyn.p j

j

j

f x a x a p j p

(1.61)

Khi (1.60) ch ng khi f x tha m~n iu kin di }y

0.

pp

p p

d f xdf x

d x a d x a

(1.62)

Tng qu|t, quy tc hp th{nh i vi ,p q

.

pq

q p

q p pq p p

q p q p p p

d f xd

d x a d x a

d f x d f xd df x

d x a d x a d x a d x a

(1.63)

Ch rng khi p n ta c


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1

0

.1

nq

q n

q n nq n n

q n q n n n

k q n kq n n

q nk

d f xd

d x a d x a

d f x d f xd d

f xd x a d x a d x a d x a

d f x x a f a

k q nd x a

(1.64)

V d 1.2 Cho hm 1 2f x x .

Vi 0, 1 2, 1 2a p q .

a. Kim tra iu kin (1.62)

1 2 1 2 1 2 1 21 2 1 2 1 2

1 2 1 2 1 2

1 20 0

0

d d x d x x x

dx dx dx

Vy iu kin (1.62) khng c tha m~n.

b.Tnh

1 2 1 2 1 2

1 2 1 2

d d x

dx dx

Theo (1.63) ta c

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 21 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 .d d x d x d d d xx

dx dx dx dx dx dx

Do

1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 20 0

d x d d x

dx dx dx

Dn n

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 0 0.

d d x d x d x x x

dx dx dx dx

1.5 Mt s v d v o h{m, v{ tch ph}n cp ph}n s

1.5.1 o h{m v{ tch ph}n cpph}n s ca mt hng s

Trc tin ta s dng nh ngha Grunwald , ta c o h{m v{ tch ph}n cp ph}n s ca 1C

1

0

1lim .

1

pp N

p Nj

d j pN

x a j pd x a

(1.65)

Theo tnh cht ca h{m Gamma


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1

0

, lim 1.1 1

1.

1

pN

Nj

pp

p

j p N p N N p

p j p N N

d x a

pd x a

(1.66)

Khi Cl{ mt hng s bt k ta c

1.

1

pp p

p p

d C d x aC C

pd x a d x a

(1.67)

1.5.2 o h{m v{ tch ph}n cp ph}n s ca h{m f x x a

S dng nh ngha Grunwald - Letnikov i vi h{m f x x a

1

0

1 11 1

0 0

lim1

lim lim .1 1

pp N

p Nj

N Np p p

N Nj j

d x a j p N x a j x aN

x a j p Nd x a

j p j j px a N N

p j p j

(1.68)

Kt hp vi tnh cht ca h{m Gamma, s lin h gia c|c h{m Gamma

1

0

, 2 1 12 1

N

j

j p p N pp p p

p j p N

ta c

1 1

,1 2

pp

p

d x a px a

p pd x a

(1.69)

1

1 1.

2 2 2

ppp

p

d x a x ap px a

p p pd x a

(1.70)

1.5.3 o h{m v{tch ph}n cp ph}n s ca 1p

f x x a p

Xut ph|t t nh ngha Riemann Liouville, mi lin h gia c|c h{m Bta v{ h{m Gamma, ta

c o h{m v{ tch ph}n cp q ca h{m f x

1, 1.

1

p p qq

q

d x a p x ap

p qd x a

(1.103)

1.5.4 o h{m v{ tch ph}n cp ph}n s ca h{m 1 p

f x x

x}y dng cng thc cho tt c gi| tr ,p q ta vit 1 1x a x a

p dng cng thc nh thc cho h{m f x


14/78



0

11 1 1 .

1 1

p j p j j

j

px a x a

j p j

(1.71)

Tcng thc o h{m v{ tch ph}n cp ph}n s ca mt chui s, cng thc Riemann cng vi

tnh cht ca h{m Gamma v{ h{m Bta, ta c

1 1

, .

p p qq

xq

d x xq q p

qd x a

(1.72)

vix l{ h{m bta khng y .

1.5.5 o h{m v{ tch ph}n cp ph}n s ca h{m bc nhy n v v{ h{m Delta Dirac

1.5.5.1 Hm Heaviside

Ta c hm Heaviside

000

0 khi.

1 khi

x xH x x

x x

Khi 00, p a x x

0

0

0

0 1

0

1 1

1

1 0 1 1.

1

xp

p p

a

px x

p p

a x

H x xdH x x dy

p x yd x a

x xdy dy

p p px y x y

(1.73)

M rng vi h{m 0f x H x x

0

0

0

0 khi

khi

x xf x H x x

f x x x

.

0 0 0

0

.

pp

p p

d f xdf x H x x H x x a x x

d x a d x x

(1.74)

1.5.5.2 Hm Delta Dirac 0x x

Ta c hm Delta Dirac 0 x x

0 0x

a

x x f y dy f x nu 0a x x .

Chn

11 001

0 ,x

pp

p

a

x xf x y dy x x p

x y

1

0 00 1

1.

pxp

p p

a

x x x xdx x dy

p px yd x a

(1.75)


15/78



T (1.74) v{ (1.75) ta c mi lin h

10 0

1

0 0

,

.

p p

p p

d H x x d x x

d x a d x a

d H x x x xdx

(1.76)

1.5.6 o h{m v{ tch ph}n cp ph}n s ca h{m atf t e

Hm atf t e c o h{m cp ph}n s

1

0

1,

tpp a

tD f t t e dp

(1.77)

t u t ta c

1

0

1 t

a u tp ptD f t u e dup

(1.78)

1 1,10

.

tatp p au pt p

eD f t u e du t E at

p

Trong 1,1 pE at l hm Mittag Leffler hai tham s.

Vy :

1,1 .p pt pD f t t E at

(1.79)

1.5.7 o h{m v{ tch ph}n cp ph}n s ca h{m sin , cosf t t f t t

Hm cxf x e c o h{m cp ph}n s

.p cx p cxxD e c e

Vi c i ta c . pp i x i x

xD e i e

2 2cos sin .2 2

i i ppi e i i e

2 cos sin .2 2

i x pp i x p p p

xD e e x p i x p

(1.80)

Mt kh|c ta c cos sini xe x i x

cos sin cos sinp i x p p px x x xD e D x i x D x iD x

sin sin2

.

cos cos2

p p

x

p p

x

D x x p

D x x p

(1.81)


16/78



1.6 Php bin i Fourier v{ Laplace ca o h{m cp ph}n s

1.6.1 Php bin i Laplace

Php bin i Laplace l{ mt php bin i tch ph}n bin mt h{m f t trong min thi gian

sang Lf s trong mt phng phc

0

, .stL Lf t f s f s e f t dt f t

L (1.82)

Laplace ngc ca nh Lf s

1 1 .Lf s f t f t f t L L L (1.83)Nu f t l{ mt h{m gc v{ Lf s l{ nh ca n th ti mi im lin tc ca f t ta c

1 .2

ist

L

i

f t e f s dsi

(1.84)

Trong tch ph}n c tnh dc theo ng thng ng Re s .

1.6.2 S tn ti ca php bin i Laplace

Hm f t c gi l{ h{m cp m khi t nu tn ti c|c hng s C, K, T sao cho

,CTf t Ke t T (1.85)Khi Laplace ca f t s tn ti vi . s C

1.6.3 Tnh cht

a. nh l vi ph}n

Nu f t c c|c o h{m ti cp n v Lf t f sL ta s c c|c o h{m ca n

2

2 11 2

0 ;

0 0 ;

0 0 0 0 .

L

L

n n nn n n

L

f t s f s f

f t s f s sf f

f t s f s s f s f sf f

L

L

L

(1.86)

b. Tch chp ca 2 h{m s

Cho 2 h{m s f t v g t .Tch chp ca 2 h{m s l{ mt h{m s ca t

0

,t

f t g t f t g d (1.87)

Nu Lf t f sL , Lg t g sL , nh ca tch chp bng tch c|c nh

. .L Lf g f g f s g s L L L. (1.88)

1.6.4 Php bin i Laplace ca o h{m v{ tch ph}n cp ph}n s


17/78



1.6.4.1 Php bin i Laplace ca o hm v tch phn cp phn s Riemann - Liouville

Khi 0a ta c

1

0

0

0

1,

.

tn p p n

t

np

tn

g t t f d D f t

n pd

g t D f tdt

(1.89)

Hay

0 0 ,n p

t tD g t D f t (1.90)

Php bin i Laplace

1

1

0

0

0 .n

n kn n k

t L

k

D g t s g s s g

L (1.91)

1

1

0

1 ,

t n p

n p tg t t f d H t f tn p n p

11

,n p p nL Lg s g t t f t s f sn p

L L L

1 1

0

11

0 0

0

0 0 ,

0 .

n k p k

t

np p k p k

t L t

k

g D f

D f t s f s s D f

L (1.92)

1.6.4.2 Php bin i Laplace ca o hm v tch phn cp phn s Caputo

1

0

0

1,

tn p nC p

tD f t t f dn p

(1.93)

t ng f

1

0 0

0

11

0

1, 1

0 .

tn pC p p n

t t

np kC p p k

o t L

k

D f t t g d D g t n p nn p

D f t s f s s f

L (1.94)

1.6.5 Php bin i Fourier ca o h{m v{ tch ph}n cp ph}n s

1.6.5.1 Php bin i Fourier

nh ngha 1: Php bin i thun

.i tF f t e f t dt

F (1.95)

Ch : Php bin i Fourier ca h{m f t tn ti khi f t l{ h{m kh tch tuyt i tcl f t

kh tch , ). Ni c|ch kh|c nu f t l{ h{m kh tch lin tc tng khc v{tch ph}n ca h{m


18/78



f t dt

th s tn ti php bin i Fourier ca f t .

nh ngha 2: Pho bin i Fourier ngc cho ta h{m gc f t

1 1 .2

i tf t F e F d

F (1.96)

1.6.5.2 Php bin i Fourier ca o hm cp phn s

Gi s:

1

2

i t

f t F

f t F e d

F

(1.97)

Do vy:

1 1

2 2

p p i t p i tD f t D F e d F D e d

(1.98)

Mt kh|c:

qq i t i t

D e i e

(1.99)

Thay (1.99) vo (1.98) ta c:

1 1 1

2 2 2

p pp p i t i t i tD x t F D e d F i e d i F e d

(1.100)

Do vy ta c cp bin i Fourier i vi o h{m cp ph}n s l{:

1

ppD x t i X

F

F

(1.101)

Trong : F l{ nh ca f t qua php bin i Fourier:

1.6.5.3 Tnh chtca php bin i Fourier

a. Bin i Fourier ca o h{m

.d

f t F f t i F

dt

F, (1.102)

.n

n

n

df t i F

dt

F (1.103)

b. Tch chp ca 2 h{m s

Cho 2 h{m s f t v g t .Tch chp ca 2 h{m s l{ mt h{m s ca t

,f t g t f t g d

(1.104)

Nu f t F F , g t G F , nh ca tch chp bng tch c|c nh

. .f g f g F G F F F. (1.105)


19/78



CHNG 2 Equation Section 2CC PHNG PHP S TNH TON DAO NG CA C H

CHA O HM CP PHN S

2.1 Xp x th{nh phn o h{m cp ph}n s2.1.1 nh ngha o h{m cp ph}n s

Theo nh ngha o h{m cp ph}n s p bi Riemann-Liouville vi ,x a b .

11

tnn pp

RL n

a

dD x t x x d

n p d

(2.1)

Trong : . l{ h{m Gamma, n l{ s nguyn tha m~n 1n p n , hay 1n p . Mt c|ch kh|c

o h{m cp ph}n s c xp x bi Grnwald-Letnikov nh sau:

0

0

1lim 1

t pt

kp

GL ptk

pD x t x t k t

kt

(2.2)

Trong cng thc n{y tng gii hn trn hi t tuyt i vi mi p>0. Tuy nhin chnh s di dctrong cng thc m{ s dng cng thc n{y c mt s hn ch. V{ vic s dng cng thc n{y tnhto|n th thng khng n nh, cng nh chnh x|c khng cao.

nh ngha o h{m cp ph}n s ca Caputo:

11

t nn pp

C n

a

d xD x t x d

n p d

(2.3)

Trong : . l{ h{m Gamma, n l{ s nguyn tha m~n 1n p n , hay 1n p . Theo nh

ngha ca Caputo c mt s thun li, nh trong vic bin i Laplace v{ tr|nh c phi x l iukin u.

Mnh lin h gia 2 nh ngha ca Riemann-Liouville v Caputo:

1

0 1

p kknp p

RL C kk

x ad xD x t D x t a

dt p k

(2.4)

Trong chng n{y s xut vic xp x vi 0 1p v{ vi 1 2p i vi nhng trng hp siu

khuch t|n chn.2.1.2 Xp x th{nh phn cp ph}n s vi 0


20/78



Th{nh phn o h{m cp ph}n s c xp x nh sau:

1

1

00 1

0

( ) 1 ( ) ( ) 1

(1 ) (1 ) (1 )

n n

n

t t

p p

RL n n n np p

tn n

x t x d x d D x t t I I I

p p pt t

(2.7)

Trong :

1 20 1

1

10

( ) ( )2

2 ( )

nt nn

n p p p pin nn

x xx d h x ihI

t h t iht

(2.8)

1

1 2 2

1 1

( )(1 )

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

n

n

t p p p

n n n np

t n

x d t t tI x x x

p p p p pt

(2.9)

Thay (2.5) vo (2.9) ta c:

1

1

1 1 1

( ) 2 1(1 )(2 ) 2

n

n

t p

n n n n np

t n

x d tI x x p x txp p tt

(2.10)

Do vy t (2.7) ta c:

0 1 0 1 2 1 1 11

, , , , ,... , , ,(1 )

p

n n n p n n n nD x t I I I x x x x x x xp

(2.11)

2.1.2.2Sdng o hm cp hai.

Th{nh phn o h{m cp ph}n s c xp x nh sau:

0

0

0

1 10

00

1 100

0

1 1

1 1 1

( ) 1

(1 ) (2 )

( ) 1( ) ( )( )

(1 ) (2 )

1

(1 )

n n

nn

n

t t

p p

RL n nq p

t n n

tt

p pp

n n n

t

p p p

n n n

x x t xdD x t d t dp dt p pt t

x tt x t x t d

p p

x tt x t t x t d

p p

Ip

0 0

0 0 0 1 2 1

1( )

(2 )

, , , , ...

n

p n

J J tp

x x x x x x

(2.12)

Tong :1

0 0

1

0 0

( ) ,

( ) ( )( ) ( )n n

n

p

n

t t

p

n n t

J x t t

J t x t d y d

(2.13)

Cng thc (2.13) c xp x bng cng thc hnh thang nh sau:


21/78



Hnh 2.1.Xp x tch ph}n bng cng thc hnh thang

Ta tnh c:

0

1 2

1 1 1

0 0

1

/2 /2 1 /2

0

( ) 0

( ) ( ) ( ) ( ) ( ), ( 1)2 2 2

( ) ( ) ( ), ( 0)2 2 4

n n n n n

n n n

n n

n t j t j t j t j t n

j j

n

n t h j t h j t h n

j

J t

h h hJ t y y y y y t n

h h hJ t y y y t n

(2.14)

2.1.2.3 S dng sai phn

Theo t{i liu [31] th{nh phn o h{m cp ph}n s c xp x nh sau :

0

0

0

0 0

0

1 1

1 1

1 1

1 1

n n

n

t t

p p

RL n nq p

t n n

t

np p

n n np

x xdD x t d x t t d

p dt pt t

x tx t t d x t t I t

p p

(2.15)

Ta c:

11

00

n j ht nn n

n p p

j jh

x t x tI t d d

(2.16)

S dng sai ph}n:

1n j n jnx v

x th

(2.17)

Thay (2.17) v{o (2.16) ta c:

1 11 11

1

0 0

1 11 11 1

1 10 0

1 1

11 1

j h j h pn nn j n j

n n j n jpj jjh jh

n nj h pp p

n j n j n j n jpjhj j

x vI t d x v d

hh

x v x v j jh p h p

(2.18)

Thay (2.18) v{o (2.15) ta c:

0 0


22/78



11 1

0 1

0

0 1 2 1

11 11

2 1

, , ,... ,

npp p

n n j n jp p pj

p n n

pD x t x t x v j j

h p n h p

x x x x x

(2.19)

2.1.2.4 S dng tch phn sVic xp x theo s im Gauss c li th l{ thi gian tnh to|n c n tuy nhin vic xp x n{y l{

km chnh x|c hn c|c phng ph|p kh|c.

Xp x th{nh phn o h{m cp ph}n s

0

1

p p p

RL C

x tD x t t D x t

p

(2.20)

0 1

pt

p

C

tD x t Dx d

p

(2.21)

Trong h{m Gamma c nh ngha:

1

0

z pp e z dz

(2.22)

ng thi h{m Gamma c tnh cht sau:

1sin( )

p pp

(2.23)

Thay (2.22) v (2.23) vo (2.21) ta c:

0 0

2sin pt

p z

C

p z dzD x t e Dx d

t z

(2.24)

t:

2z t y (2.25)

Do vy t (2.24) v (2.25) ta nhn c:

22 1

0 0

2sin t t yp pC

pD x t y e Dx d dy

(2.26)

t:

22 1

0

2sin, ,

tt yp p

y t y e Dx d

(2.27)

T (2.26) ta c:

2 1 2, ,pD y t y Dx t y y t (2.28)

S dng phng ph|p Gauss n im xp x o h{m cp ph}n s:

0

, w ,in

yp

C i i

i n

D x t y t dy e y t

(2.29)

Trong :n l s im Gauss.


23/78



w :i

l{ h{m trng lng

:i

l{ im Gauss ph thuc v{o s im Gauss dng xp x

,iy t c x|c nh bng c|ch gii h phng trnh vi ph}n sau:

2 1 2, , , 1..pi i i iy t y x t y y t i n (2.30)

2.1.3 Xp x th{nh phn cp ph}n s vi 1


24/78



111

0

212 1,

,1 ,20

1

, 2 1

0

1

2

21

2 2

23

k

k

xjp

p

C j j

k x

pjk k kp t

C j j k

k

p j

j k k k k

k

D x t x x dp

x t x t x t tD x t d

p t p

td x t x t x t

p

(2.36)

Trong :

2 2

, 1p p

j kd j k j k

V d 2.1:

Xt hm: 4x t t . Vi bc ca o h{m 1 2p theo [32] ta c:

424

5

p p p p

C RL GLD x t D x t D x t tp

(2.37)

Kt qu thu c bng c|c c|ch xp x kh|c nhau:

- Kt qu s dng sai ph}n vi o h{m cp mt, v{ o h{m cp hai

- S dng o h{m cp mt 1.51st d

D x

, o h{m cp hai 1.52nd d

D x

, v{ sai s tuyt i tng ng

1 2

,st d nd d

Bng 2.1 So s|nh kt qu xp x vi p=1.5.

Time 1.5 exactD x 1.5 1st dD x 1st d 1.5 2nd dD x 2nd d

0.1 0.0228 0.02283512 -1.6622e-006 0.02280769 -2.9094e-005

0.2 0.1292 0.12921109 2.6702e-005 0.12910254 -8.1846e-005

0.4 0.7308 0.73097847 2.0123e-004 0.730546647 -2.3059e-004

0.5 1.2766 1.276967347 3.5205e-004 1.276293360 -3.2194e-004

Nhn xt :

- i vi b{i to|n phi tnh to|n khi th{nh phn o h{m cp s xpD t vi 1 2p :

, x, , , 1 2px f t D x x p

Ta a th{nh phn o h{m cp ph}n s v dngD vi 0 1 , tht vy t:

1 ,p y x t

Khi h phng trnh thu c l:

1, x, , , x, ,

, y, y,

pyx

x f t D x x f t D x x

f t D xy

(2.38)

Vi iu kin u cn tha m~n l{: 0 0, 0 0x x


25/78



2.2 Phng ph|p s gii phng trnh vi ph}n cp ph}n s

Xt phng trnh dao ng c dng tng qu|t nh sau:

( ) ( )pmx bx c x D x kx f t (2.39)

iu kin u:0 0(0) , (0)x x x x (2.40)

Chia li thi gian, vi bc h.

0 0 0,1,2,3...nt t nh t n t n (2.41)

Chuyn (2.39) v dng ch s.

( ) ( ), 0pn n n n n nmx bx c x D x kx f t n (2.42)

2.2.1 Phng ph|p Newmark

Xp x ,n nx x theo cng thc Newmark

Theo cng thc xp x Newmark.

1 1 1 2 1 1 121 1 1

1 , , , ,2

n n n n n n n n nx x x x x x x x xt t

(2.43)

1 1 1 1 1

1 1 1 1

(1 ) 1 12

, , , , ,

n n n n n n n n

n n n n

x x tx tx x t x x xt

x x x x

(2.44)

i vi th{nh phn cp ph}n s ta c th c xp x theo o h{m cp mt, o hm cp haihoc sdng sai ph}n. Cui cng ta a (2 .42) v phng trnh i s sau:

2 1 1 1 1 1 1 1, , , , , , , , , ( ) ( )n n n n n n n n n p n nm x x x x b x x x x c x kx f t (2.45)

Phng trnh trn l{ phng trnh i s vi n l{ nx , c gii bng phng ph|p s.

Tnh c nx , ri ta tnh c ,n nx x nh sau:

1 1

1 1 12

(1 )

1 1 11

2

n n n n

n n n n n

x x tx tx

x x x x xt t

(2.46)

2.2.2 Phng ph|p Runge-Kutta-Nystrm

Ho{n to{n tng t, gii phng trnh vi ph}n cha o h{m cp ph}n s th mu cht l{ vic xpx th{nh phn o h{m cp ph}n s. s dng cng thc Runge-Kutta-Nystrm ta c th xp x th{nhphn o h{m cp ph}n s theo o h{m cp mt, o h{m cp hai hoc s dng sai ph}n. Bng c|ch ta a c (2.42) v dng phng trnh vi ph}n cp 2 dng ch s nh sau:

( ) ( )

1( ) ( ) ( , , )

( , , )

n n n p n n

n n n n p n n n n

n n n n

mx bx c x kx f t

x f t bx c x kx g t x xm

x g t x x

(2.47)

iu kin u:


26/78



0 0 0 0,x t x x t x (2.48)

p dng cng thc Runge-Kutta-Nystrm:

1 1 2 3

1 1 2 3 4

1 1 1

x x x k k k ,3

1x x k 2k 2k k ,

3

x ( , x , x ).

n n n

n n

n n n n

hh

g t h

(2.49)

Trong :

1

2 1 1

3 1 2

4 3 3

k , x , x ;2

k , x x k , x k ;2 2 2 4

k , x x k , x k ;2 2 2 4

k , x x k , x 2k .2

n n n

n n n n

n n n n

n n n n

hg t

h h h hg t

h h h hg t

hg t h h h

(2.50)

2.2.3 Phng ph|p Runge-Kutta

i vi c|c h c hc dao ng cp ph}n s nhiu bc t do ta c th a c v dng h phngtrnh vi ph}n cp mt nh sau:

( ) 1, , , ... , 0 1

T

n jt p p p px f x x p (2.51)

iu kin u :

0 x 0

Thnh phn o h{m cp ph}n s c th c xp x theo o h{m cp mt, o h{m cp haihoc sdng sai ph}n. a (2.51) v phng trnh vi ph}n dng ch snh sau:

, ,pn n n

tx f x (2.52)

p dng cng thc Runge-Kutta cho h phng trnh vi ph}n cp mt (2.52) nh sau:

1 1 2 3 41

2 26

i i h x x k k k k (2.53)

Trong :

1

2 1

3 2

2 3

,

,2 2

,2 2

,

p

p

p

p

i i

i i

i i

i i

t

h ht

h ht

t h h

k f x

k f x k

k f x k

k f x k

(2.54)

2.2.4 Phng ph|p Gaussi vi phngph|p n{y phng trnh (2.39) c a vh phngtrnh sau:


27/78



1

2 1 2

1w ,

, , , 1..

i

ny

i i

i

p

i i i i

x t v t

c xkv t f t x t e y t

m m m

y t y v t y y t i n

(2.55)

Trong :n l s im Gauss ~ c trnh by c th mc 2.1 trn.

- w :i

l{ h{m trng lng

- :i

l{ im Gauss ph thuc v{o s im Gauss dng xp x

V{ vicgii h phng trnh vi ph}n (2 .55) c th c thc hin bng phng php Runge-Kutta nhmc 2.2.3

2.2.5 So s|nh chnh x|c v{ thi gian tnh gia c|c phng ph|p tnh

2.2.5.1So snh chnh xc so s|nh chnh x|c ta xt v d sau:

V d 2.2

0.5 30.8Dx t x t x f t (2.56)

Trong :

2

3

7 5 3 2

9 7 7 92 4 4 2

10 10 10 10

8 128 128 42 9 7

10 0.5 35 25 25 10 10

f t t t t t t t t

t t t t t t

(2.57)

iu kin u: 0 0, 0 0x x (2.58)

Nghim chnh x|c ca (2.56) l:

2

ex

9 7

10 10actx t t t

(2.59)

th nghim ca (2.56) c biu din nh Hnh 2.2

Hnh 2.2.Nghim chnh x|c ca VD 2.2


28/78



Bng 2.1So s|nh nghim chnh x|c v{ kt qu tnh to|n ca c|c phng ph|p (s im li: n=1000)

TimeExactx anackovicAtx Rayx Zhang Shimizux Newmarkx

2ndderivative

RKNx

2ndderivative

0.25 0.01828125 0.018253191 0.0182813 0.018507464 0.018508641 0.018281311

0.5 0.02 0.019851524 0.0200026 0.020643625 0.020647787 0.020000016

0.75 -0.00421875 -0.004492587-

0.00419593-0.003348193 -0.003343554 -0.004218918

1 0.03 0.029380011 0.0300995 0.030526185 0.030530651 0.029999869

Bng 2.2So s|nh sai s % tng i ca c|c phng ph|p (s im li: n=1000)

TimeanackovicAtx Rayx Zhang Shimizux Newmarkx

2nd

derivative

RKNx

2nd

derivative0.25 0.153486% 0.000274% 1.237407% 1.243849% 0.000332%

0.5 0.742382% 0.013000% 3.218126% 3.238937% 0.000080%

0.75 6.490951% 0.540919% 20.635433% 20.745378% 0.003975%

1 2.066632% 0.331667% 1.753949% 1.768835% 0.000437%

2.2.5.2So snh thi gian tnh ton

V d 2.3 Xt mt h mng m|y c phng trnh dao ng nh sau:

2 0,390,17 616 1043 0;1 342 ( )

x tD x t D x tx t

(2.60)

iu kin u: 0 0; 0 0.64x x (2.61)

Kt qu tnh to|n theo c|c phng ph|p:

Hnh 2.3.Kt qu gii, v{ thi gian tnh to|n ca c|c phng ph|p s


29/78



Bng 2.3Thi gian tnh to|n ca c|c phng ph|p kh|c nhautheo s im li

n Zhang Shimizut

1st derivative

Newmarkt

2ndderivative

RKNt

2ndderivative

100 6.972 2.289 2.872200 12.081 3.067 4.179

400 24.268 6.261 8.441

600 37.302 9.524 12.912

800 51.042 13.158 17.348

1000 64.862 16.58 21.608


,nmx A kx mg (2.62)

Trong :

2

1 12

2

1( ), 1 / ; 1,08;

4

0.005( ); 0.277( ); 0.020( ); 4620( / ); 5020( / ); 60

(0) 0; (0) 2 1.08 /

q

n a

q

D c x H c A

H m m kg m k N m Ns m h mm

x x gh m s

(2.63)

Kt qu thu c

Hnh 2.4.Kt qu gii, v{ thi gian tnh to|n ca c|c phng ph|p sVD2.4

Bng 2.4Thi gian tnh to|n ca c|c phng ph|p kh|c nhau theo s im li

n Zhang Shimizut

1st derivative

Newmarkt

2ndderivative

RKNt

2ndderivative

100 7.193 1.896 2.201

200 13.27 3.813 4.424

400 26.912 7.072 8.339

600 41.478 9.898 12.606

800 55.552 14.147 17.415

1000 70.057 18.267 21.984


30/78




,nmx A kx mg (2.64)

Trong :

23/2 1/2 2

1 12

2

2 1 1 1; 1 / ; 1,095;

3 3 4

0.005( ); 0.277( ); 0.020( ); 4620( / ); 5020( / ); 60

(0) 0; (0) 2 1.08 /

q q

n a a

q

D D c x H c A

H m m kg m k N m Ns m h mm

x x gh m s

(2.65)

Kt qu thu c:

Hnh 2.5.Kt qu gii, v{ thi gian tnh to|n ca c|c phng ph|p s VD2.5

Bng 2.5.Thi gian tnh to|n ca c|c phng ph|p kh|c nhau theo s im li

n Zhang Shimizut Newmarkt

2ndderivative

RKNt

2ndderivative

100 21.692 5.599 2.222

200 42.504 11.449 4.507

400 88.373 24.146 9.608

600 131.26 34.325 14.978

800 181.616 48.706 21.569

1000 224.114 59.445 27.942

2.3 Kt lun

- Mc 2.1.2 ~ a ra 3phng ph|p xp x th{nh phn o h{m cp ph}n s : theo o h{m cp 1, oh{m cp 2, v{ s dng sai ph}n. ng thi mc 2.2 a ra 3 phng ph|p gii l{ Newmark, Runge-Kutta-Nystrmv Runge-Kutta do vy m{ ta c th c 9cch gii kh|c nhau.im chung ca c|cphng ph|p n{y l{ amtphng trnh vi ph}nthng thngv dngphng trnh vi ph}n hocphng trnh i sdng ch s.

- Vic s dng phng ph|p Runge-Kutta-Nystrm tnh to|n dao ng phi tuyn cha o h{m cpph}n s cho nhiu u im vt tri v chnh x|c, thi gian tnh, cng nh vic lp trnh thut to|ntrn my tnh.


31/78



CHNG 3Equation Section 3

TNH TON DAO NG CA MNG MY TRN

NN N NHT CP PHN S

3.1 Cc m hnh h {n nhtVi s ph|t trin khoa hc k thut, ng{y c{ng c nhiu c|c vt liu mi ra i (nh vt liu silicone,

vt liu cao su), nhng m hnh {n nht c in vi o h{m cp nguyn khng th hin c y tnh cht ca vt liu. Do gii quyt vn n{y, o h{m cp ph}n s c |p dng bi c|cnh{ nghin cu trong thi gian gn }y.

Hnh 1.M hnh c in Hnh 2.M hnh mi

Trong lc |p ng ca h {n nht c in c dng nh sau:

nF kx cx (3.1)

Vi c|c h {n nht mi, lc |p ngc dng cha th{nh phn cp ph}n s nh sau: pvF kx c x D xb x (3.2)

Trong :

- C|c h s: , ,k c l{ c|c h s ca vt liu.

- Cc hm iu chnh ,c x b x .

Cc hm ( )c x v ( )b x l{ h{m ca x vi (0) (0) 1c b . Hm ( )b x gii thch t|c ng ca lc cn nhttrong trng hp bin dng ln.Cc m hnh lc, theo t{i liu [5] ~ a ra 5 m hnh ( )c x v ( )b x nh sau:

- M hnh 1:4( ) 1 3 , ( ) 1c x x b x

- M hnh 2:2

1 2( ) 1 , ( ) 1c x c x c x b x

- M hnh 3:2

1 2

2

1( ) , ( ) 1

1c x c x b x

c x

- M hnh 4:1

1( ) , ( ) 1(1 )

c x b xc x

- M hnh 5:


32/78



1

1( ) 1, ( )

1c x b x

c x

Ngoi ra ta c th vit (3.2) di dng ng sutg}y ra bn trong vt liu {n nht nh sau:

v x nF kx f kx A (3.3)

Vi A l{ din tch mt ct ngang ca vt mu, v{ trong trng hp n{y c xem nh vt mu l{ mtvt tr.

Theo t{i liu[4] a ra 4m hnh ng sutn c dng nh sau:

- M hnh IIa v IVa: 2

1qn ax D

(3.4)

- M hnh IIIb: 2 2

11 1q qn a ax D D

(3.5)

- M hnh IIIc: 33 2

2 1 11 1

3 3

q q

n a ax D D

(3.6)

- M hnh IVc: 3/2 1/2 2

2

2 1 1 1

3 3

q q

n a ax D D

(3.7)

Trong : 11 /c x H Vi:H l{ chiu cao ca vt mu v{

1c l{ h s iu chnh m hnh.

3.2 H dao ng mt bc t do

3.2.1 M hnh Kelvin Voigh

Hnh 3.M hnh Kelvin Voigh

lp phng trnh dao ng, ta s dng nh l chuyn ng khi tm:

e vmx F f f F mg (3.8)

Trong :

m

F

x

k , c(x)


33/78



-ef : Th{nh phn lc tuyn tnh

ef kx (3.9)

-vf : Th{nh phn lc cn phi truyn cp ph}n s

pvf c x D x (3.10)

- F : Th{nh phn ngoi lc

Do vy (3.8) c vit li nh sau:

pmx c x D x kx F mg (3.11)

Vit di dng ng sut:

nmx A x kx F mg (3.12)

Nhn xt :

- i vi m hnh n{y ta c th |p dng cho c|c trng hp c th sau

+ H dao ng chu kch ng va p, phng trnh c dng nh sau.

pmx c x D x kx mg (3.13)

Vi :

- m l{ khi lng ca vt dao ng v{ vt kch ng :0 1m m m

- iu kin u: 00 ; 0 2m

x x ghm

+ H chu kch ng lch tm (iu ha), phng trnh c dng nh sau: 2 2 sin( )p emx c x D x kx mg m e t (3.14)

Vi:

- m l{ khi lng ca vt, v{ vt lch t}m :1 e

m m m

- iu kin u: 0 ; 0x x


34/78



V d m phng s:

V d 3.1: Cho h mng my chu kch ng va p c phng trnh dao ng nh sau:

2

0

4

0 1 0

4 0.5 0

0 1

0,5; 0,5; 1; 2; 1 3 ; 1,8

0.52 1 3 1; 0 0, 0 2 2.10.1,8 3

1

pmD x t c x D x t b x t kx t mg

m m k c x x hm

x x D x x x x ghm m

(3.15)

Kt qu thu c:

Hnh 3.1.Dch chuyn v{ gia tc ca VD 3.1

V d 3.2: Cho h mng m|y chu kch ng va p c phng trnh dao ng nh sau:

20 1 2

20 1 0

2 1/2 0

0 1

1

0,5; 0,5; 1; 2; 1 ; 1,8

2 1 1; 0 0, 0 2 3

pmx t c x c x D x t kx t mg

m m k c x x x h

mx t x x D x t x t x x gh

m m

(3.16)

Kt qu thu c:



35/78




20 1 2

2

0 1 0

2 0

20 1

1

1

0, 5; 0, 5; 1; 2; 1,8

12 1; 0 0, 0 2 3

1

p

p

mx t c x D x t kx t mg c x

m m k h

mx t x D x t x t x x gh

m mx

(3.17)

Kt qu thu c:



0

1

0 1 0

0

0 1

1,

1

0,5; 0,5; 1; 2; 1,8

12 1; 0 0, 0 2 3

1

p

p

mx t D x t kx t mg c x

m m k h

mx t D x t x t x x gh

x m m

(3.18)

Kt qu thu c:



36/78




0 2

1

0 1 0

1/2 0

20 1

1,

1

0,5; 0,5; 1; 2; 1,8

12 1; 0 0, 0 2 3

1

pmx t D x t kx t mg

c x

m m k h

mx t D x t x t x x gh

m mx

(3.19)

Kt qu thu c:



2

3

1

, (0) 0; (0) 1.08( / )

1

0.277( ); 0.020( ); 4620( / );

0.005( ); 5020( / ); 1.08

x

q

x a

q

mx f kx mg x x m s

f A D

m kg m k N m

H m Ns m c

(3.20)

Kt qu thu c:



37/78




2 2

3

1

, (0) 0, (0) 1.08( / )

11 1

0.277( ); 0.020( ); 4620( / );

0.005( ); 5020( / ); 1.10

x

q q

x a a

q

mx f kx mg x x m s

f A D A D

m kg m k N m

H m Ns m c

(3.21)

Kt qu thu c:



33 2

3

1

, (0) 0, (0) 1.08( / )

2 1 11 1

3 3

0.277( ); 0.020( ); 4620( / );

0.005( ); 5020( / ); 1.08

x

q q

x a a

q

mx f kx mg x x m s

f A D A D

m kg m k N m

H m Ns m c

(3.22)

Kt qu thu c:



38/78




3/2 1/2 2

2

3

1

, (0) 0, (0) 1.08( / )

2 1 1 1

3 3

0.277( ); 0.020( ); 4620( / );

0.005( ); 5020( / ); 1.148

x

q q

x a a

q

mx f kx mg x x m s

f A D A D

m kg m k N m

H m Ns m c

(3.23)

Kt qu thu c:


V d 3.10: Cho h mng m|y chu kch ng lch t}m (iu ha) c phng trnh dao ng nh sau:

2 2

0 0

4 0.5

sin( ),

1 3 sin(2 ), 0 0, 0 3

pmD x t c x t D x t b x t kx t m e t

x x D x x t x x

(3.24)

Kt qu thu c:



39/78



V d 3.11: Cho h mng m|y chu kch ng iu ha c phng trnh dao ng nh sau:

2 20 1 2 0

2 1/2

1 sin( ),

2 1 sin(2 ); 0 0, 0 0

pmx t c x c x D x t kx t m e t

x t x x D x t x t t x x

(3.25)

Kt qu thu c:



2 20 1 02

2

2

2

1 sin( ),1

12 sin(2 ), 0 0, 0 0,

1

p

p

mx t c x D x t kx t m e t c x

x t x D x t x t t x xx

(3.26)

Kt qu thu c:



40/78




20 0

1

1sin( ),

1

12 sin(2 ), 0 0, 0 0

1

p

p

mx t D x t kx t m e t c x

x t D x t x t t x xx

(3.27)

Kt qu thu c:



20 02

1

1/2

2

1sin( ),

1

12 sin(2 ), 0 0, 0 0

1

pmx t D x t kx t m e t c x

x t D x t x t t x xx

(3.28)

Kt qu thu c:



41/78



2.2.2 M hnh Maxwell

Hnh 4.M hnh Maxwell


e

e

mx F f F mg

f mx F mg

(3.29)

Trong :


1 1e

e

ff kx x

k (3.30)

- vf : Th{nh phn lc cn phi truyn cp ph}n s 2 2

p

vf c x D x (3.31)

-1 2,x x : Dch chuyn ca th{nh phn tuyntnh v{ phi tuyn cp ph}n s

1 2 2 1x x x x x x (3.32)

n gin, th h{m dch chuyn c(x) trong trng hp n{y cchn l{: 1c x .

Thay (3.32) vo (3.31) ta c:

2 2 1 1 1p p p

vf c x D x c x x D x x D x x (3.33)

Thay (3.30) vo (3.33) ta c:

1p p p pe

v e e

p p

e e

ff f D x x D x D x D f

k k

f D x D fk

(3.34)

Thay (3.39) vo (3.34) ta c phng trnh dao ng ca h nh sau:

2

p p

p p p

mx F mg D x D mx F mg

kD x mx D x mg F D F

k k

(3.35)

m

F

x

k

, c(x)=1


42/78



2.2.3 M hnh dao ng cp 3

Hnh 5.M hnh dao ng cp 3


e v

e v

mx F f f F mg

f mx f F mg

(3.36)

Trong :

-vf : Th{nh phn lc cn phi truyn cp ph}n s

pvf c x D x (3.37)

n gin, th h{m dch chuyn c(x) trong trng hp n{y c chn l: 1c x .


1 1e

e

ff kx x

k (3.38)

-cf : Th{nh phn lc cn nht.

2cf cx (3.39)

-1 2,x x : Dch chuyn ca th{nh phn tuyn tnh v{ phi tuyn cp ph}n s

1 2 2 1x x x x x x (3.40)

Thay (3.40) vo (3.39) ta c:

2 1cf cx c x x (3.41)

Thay (3.28) v (3.41) ta c:

2 1e

c e

e e

f cf cx c x x c x cx f

k k

cf cx f

k

(3.42)

Thay (3.36) vo (3.42) ta c phng trnh dao ng ca h nh sau:

m

F

x

, c(x) =1k

c


43/78



1

0v v

p p

cmx f F mg cx mx f F

k

c c cmx mx D x cx D x F F mg

k k k

(3.43)

V d m phng s:

V d 3.15: Cho h dao ng bc ba nh sau:

1,5 sin ,

0 0, 0 0, 0 0, 2

x x D x x x t

x x x

(3.44)

gii phng trnh trn ta a v dng h phng trnh vi ph}n bc mt sau:

0.5

, 0 0, 0 0, 0 0

sin

x yy x

y z x y z

z x y z t z D y y x

(3.45)

Kt qu thu c:



44/78



3.3 H dao ng hai bc t do

Trong phn n{y ta xt m hnh ba rn trn nn {n nht cp ph}n s chu kch ng va p. Trong khi lng

0m l{ u ba, 1m l{ e, 2m l{ mng. Vi gi thit 0 1 0 2,m m m m nn ln tnh v{

nh hng ng lc do0m g}y ra sau khi va chm l{ khng |ng k nn b qua khi kho s|t.C hai m

hnh c kho s|t nh sau:

Hnh 6. M hnh 1 Hnh 7.M hnh 2

( n gin cho vic lp h phng trnh vi phn th hm iu chnh c(x), v b(x) ca b cn nht cpphn s c chn bng 1)

Thit lp h phng trnh dao ng i vi m hnh 1:

- Phng php s dng phng trnh Lagrange loi II:Biu thc ng nng ca h c dng:

2 2

1 1 2 2

1 1

2 2T m x m x (3.46)

Biu thc th nng, h{m hao t|n ca h:

2 2 2

1 2 1 2 2 1 2 1

1 1 1( ) ; ( )

2 2 2k x x k x c x x (3.47)

Lc sinh ra do b {n nht:

2 2pf D x (3.48)

Biu thc lc suy rng:

1 1 1 1 2 1 1 2 1

1 1

2 2 2 2 2 1 1 2 1 1 2 2 2

2 2

( ) ( )

( ) ( ) p

Q F k x x c x x F x x

Q f F k x k x x c x x D x F x x

(3.49)

Th v{o phng trnh Lagrange loi II:

m

F1

k1 , p

m2

F2

k2 , p

m1

F1

k2 , p

m2

F2

k1 c1


45/78



i

i i

d T TQ

dt x x

(3.50)

Ta nhn c h phng trnh vi ph}n dao ng l:

1 1 1 1 2 1 1 2 1

2 2 2 1 1 2 1 1 2 2 2 2

( ) ( )( ) ( ) ( )p

m x c x x k x x F m x D x k x x c x x k x F

(3.51)

- Phng php tch vt, s dng nh l chuyn ng khi tm

Hnh 8. Tch vt vi m hnh 1

Lc |p ng g}y nn bi h {n nht th nht:

1 1 1 1 2 1 1 2( )upF k l c l k x x c x x (3.52)

Lc |p ng g}y nn bi h {n nht th hai: w 2 2 2 2

p pdoF k l D l kx D x (3.53)

p dng nh l chuyn nh khi t}m vi vt th nht:

1 1 1 1 1 2 1 1 2

1 1 1 1 2 1 1 2 1

( )

( ) ( )

upm x F F F k x x c x x

m x c x x k x x F

(3.54)

p dng nh l chuyn nhkhi t}m vi vt th hai:

m1

F1

m2

F2

x2


46/78



2 2 2 w 2 1 2 1 1 2 2 2

2 2 2 2 2 1 2 1 1 2 2

( )

( )

p

up do

p

m x F F F F k x x c x x kx D x

m x c x D x kx k x x c x x F

(3.55)

Vy h phng trnh vi ph}n thu c l:

1 1 1 1 2 1 1 2 1

2 2 2 1 1 2 1 1 2 2 2

( ) ( )

( )pm x c x x k x x F

m x D x k x x c x x kx F

(3.56)

Vit di dng ma trn:

1 1 1 1 1 1 1 1 1

2 22 2 1 1 2 1 1 2 2 2

00

0 p

m x c c x k k x F

D xm x c c x k k k x F

(3.57)

Nu vit h phng trnh vi ph}n trn di dng ng sut ca b {n nht cp ph}n s, th h phngtrnh c dng nh sau:

1 1 1 1 1 1 1 1 1

22 2 1 1 2 1 1 2 2 2

000 n

m x c c x k k x F A xm x c c x k k k x F

(3.58)

Thit lp h phng trnh dao ng i vi m hnh 2:

- Phng php s dng phng trnh Lagrange loi II:Biu thc ng nng ca h c dng:

2 2

1 1 2 2

1 1

2 2T m x m x (3.59)

Biu thc th nng, h{m hao t|n ca h:

2 2

1 2 1 2 2

1 1( ) ; 0

2 2k x x k x (3.60)

Lc sinh ra do b {n nht:

1 1 1 2 2 2 2;p pf D x x f D x (3.61)

Biu thc lc suy rng

1 1 1 1 1 2 1 1 2 1

1 1

2 2 1 2 2 2 1 1 2 2 1 1 2 2

2 2

( )

( )

p

p p

Q f F k x x D x x F x x

Q f f F k x k x x D x D x x F x x

Th v{o phng trnh Lagrange loi II:

i

i i

d T TQ

dt x x

Ta nhn c h phng trnh vi ph}n dao ng l:

1 1 1 1 2 1 1 2 1

2 2 2 2 1 1 2 1 1 2 2 2 2

( )

( ) ( )

p

p p

m x D x x k x x F

m x D x D x x k x x k x F

(3.62)


47/78



- Phng php tch vt, s dng nh l chuyn ng khi tm

Hnh 9. T|ch vt vi m hnh 2

Lc |p ng g}y nn bi h {n nht th nht: 1 1 1 1 1 2 1 1 2

p pupF k l D l k x x D x x (3.63)

Lc |p ng g}y nn bi h {n nht th hai: w 2 2 2 2 2 2 2 2

p pdoF k l D l k x D x (3.64)

p dng nh l chuyn nh khi t}m vi vt th nht:

1 1 1 1 1 1 2 1 1 2

1 1 1 1 2 1 1 2 1( )

pup

p

m x F F F k x x D x x

m x c x x D x x F

(3.65)

p dng nh l chuyn nh khi t}m vi vt th hai:

2 2 2 w 2 1 1 2 1 1 2 2 2 2 2

2 2 2 2 2 2 1 1 2 1 1 2 2

p p

up do

p p

m x F F F F k x x D x x k x D x

m x D x k x k x x D x x F

(3.66)

Vy h phng trnh vi ph}n thu c l{:

1 1 1 1 2 1 1 2 1

2 2 2 2 2 1 1 2 1 1 2 2

( ) p

p p

m x c x x D x x F

m x D x kx k x x D x x F

(3.67)

Vit di dng ma trn:

m1

F1

m2

F2

x2


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1 1 1 1 1 11 1 2

2 2 1 1 2 2 22 2 1 1 2

0

0

p

p p

m x k k x F D x x

m x k k k x F D x D x x

(3.68)

Nu vit h phng trnh vi ph}n trn di dng ng sut ca b {n nht cp ph}n s, th h phngtrnh c dng nh sau:

1

2 1

1 1 21 1 1 1 1 1

2 2 1 1 2 2 22 2 1 1 2

0

0

n

n n

A x xm x k k x F

m x k k k x F A x A x x

(3.69)

Nhnxt:

i vi b{i to|n h chu kch ng va p, iu kin u c tha m~n l:

1 2 1 20; 0; 0; ; 0;t x x x u x (3.70)

Vn tc ban u u c x|c nh theo s va chm ca u ba v{ e. Gi thit qu| trnh va chm xy ral{ rt ngn nn c|c khi lng

0 1,m m tha m~n nh lut xung lng

0 0 1 1 0 0 1 1v vm u m u m u m u (3.71)

Vi:0 1,v vu u - vn tc trc va chm; 0 1,u u - vn tc sau va chm;

Trong trng hp tng qu|t, tnh c|c gi| tr0 1,u u ngi ta s dng gi thit ca Newton:

0 1

0 1v v

u uk

u u

(3.72)

Trong c|c trng hp t ra ta c:1 00; 2v vu u gh . T c|c phng trnh ta c :

0

0 1

(1 )

2

k m

u ghm m

(3.73)

V d m phng s:

V d 3.16: Cho h c phng trnh dao ng nh sau:

1 1 1

4 0.5

2 22 2 2

01 0 1 1 1 1 0,

2(1 3 )0 1 0 0 1 2 0

x x x

x D xx x x

(3.74)

iu kin u:

T T

0 0 0 ; 0 5 0 x x

Kt qu thu c:


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Hnh 3.16.Dch chuyn v{ giatc ca VD 3.16

V d 3.17: Cho h phng trnh dao ng nh sau:

4 0.5

1 1 1 2 1 2

4 0.5

2 2 2 2

1 0 1 1 02(1 3( ) )

0 1 1 2 02(1 3 )

x x x x D x x

x x x D x

(3.75)

iu kin u:

T T

0 0 0 ; 0 5 0 x x

Kt qu thu c:



1 1 1

4 0.5

2 22 2 2

01 0 1 1 1 1 0

2(1 3 )0 1 1 1 1 2 0

x x x

x D xx x x

(3.76)

iu kin u:

T T

0 0 0 ; 0 3 0 x x

Kt qu thu c:


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4 0.5

1 1 1 2 1 2

4 0.5 4 0.5

2 2 2 2 1 2 1 2

1 0 1 1 02(1 3( ) )

0 1 1 2 02(1 3 ) 2(1 3( ) )

x x x x D x x

x x x D x x x D x x

(3.77)

iu kin u:

T T

0 0 0 ; 0 3 0 x x

Kt qu thu c:



1 1 1

22 2 2

1 12

01 0 1 1 1 1 0

0 1 1 1 1 2 0

1; 1 / ; 1; 2; c 1.1; 3

n

q

n a

x x x

A xx x x

x D c x H A H

(3.78)

iu kin u:

T T

0 0 ; 3 0 x x


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Kt qu thu c:



1 1 21 1

2 2 1 1 22 2

1 1 2 12

1 0 1 1 0

0 1 1 2 0

1; 1 / ; 1; 2; c 1.1; 3

n

n n

q

n a

A x xx x

A x A x xx x

x D c x H A A H

(3.79)

iu kin u:

T T

0 0 0 ; 0 3 0 x x

Kt qu thu c:



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3.4 H dao ng nhiu bc t do

i vi c|c h c nhiu bc t do ta c th a v dng sau:

( ) t pMx Cx Dx Kx F (3.80)

Trong :- , , ,M C K D l{ c|c ma trn khi lng, cn nht, cng, cn {n nhtcp ph}n s.- (p )x c dng nh sau :

1

( ) ... ,

n

p

p

D

D

p

x

x

x

(3.81)

- D c dng nh sau :11 12 1

21 22

1

...

... ...

... ... ... ...

... ...

n

n nn

D (3.82)

Ta d{ng a (3.80) v dng phng trnh cp 2 nh sau:

( ), , ,f t px x x x (3.83)

ng thi ta cngd d{nga h (3.83) c v dng cp mt, tht vy :

t:

y x (3.84)

H (3.83) a c v dng sau:

( ), , ,f t

p

yx

x y xy (3.85)

Hay (3.85) c dng sau:

( ), ,f t pz z z (3.86)

Trong :

xz

y (3.87)


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3.5 So s|nh kt qu tnh to|n gia c|c m hnh l thuyt v{ vi kt qu thc nghim

3.5.1 So s|nh m hnh mng m|y mtbc tdo

M hnh {n nht c in:

2mD x t cx t kx t F (3.88)

M hnh {n nht mi:

- Dng m hnh lc: 2 0

pmD x t c x t D x t b x t kx t F (3.89)

- Dng m hnh ng sut: 2 nmD x t A x kx t F (3.90)

iu kin u:

0 ; 0x x (3.91)

V d m phng s:


- M hnh {n nht c in:2 1x x x (3.92)

- M hnh {n nht midng m hnh lc: 4 0.52 1 3 1x x D x x (3.93)

iu kin u:

0 0, 0 3x x

Kt quthu c:

Hnh 3.22.So snh dch chuyn v{ gia tc ca VD 3.22


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- M hnh {n nht c in:2 1x x x (3.94)

- M hnh {n nht midng ng sut:

1 12

1

1; 1 / ; 1; 2; c 1.1; 3

n

q

n a

x A x x

x D c x H A H

(3.95)

iu kin u:

0 0, 0 0x x

Kt qu thu c:

Hnh 3.23.So s|nh dch chuyn v{ gia tc ca VD 3.23

V d 3.24: Cho h c phng trnh dao ngnh sau:

- M hnh {n nht dng m hnh lc: 4 0.52 1 3 1x x D x x (3.96)

- M hnh {n nht mi dng ng sut:

1 12

1

1; 1 / ; 1; 2; c 1.1; 3

n

q

n a

x A x x

x D c x H A H

(3.97)

iu kin u: 0 0, 0 0x x

Kt qu thu c:


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Hnh 3.24.So s|nh dch chuyn v{ gia tc ca VD 3.24

3.5.2 So snh m hnh mng my hai bc tdo

M hnh {n nht c in:

1 1 1 1 1 1 1 1 1

2 2 1 1 2 2 1 1 2 2 2

0

0

m x c c x k k x F

m x c c c x k k k x F

(3.98)

M hnh {n nht mi:

+ M hnh 1:

-

Dng m hnh lc:

1 1 1 1 1 1 1 1 1

2 22 2 1 1 2 1 1 2 2 2

00

0 p

m x c c x k k x F

D xm x c c x k k k x F

(3.99)

- Dng m hnh ng sut:

1 1 1 1 1 1 1 1 1

22 2 1 1 2 1 1 2 2 2

00

0 n

m x c c x k k x F

A xm x c c x k k k x F

(3.100)

+ M hnh 2:

- Dng m hnh lc:

1 1 1 1 1 11 1 2

2 2 1 1 2 2 22 2 1 1 2

0

0

p

p p

m x k k x F D x x

m x k k k x F D x D x x

(3.101)

- Dngm hnh ng sut:

1

2 1

1 1 21 1 1 1 1 1

2 2 1 1 2 2 22 2 1 1 2

0

0

n

n n

A x xm x k k x F

m x k k k x F A x A x x

(3.102)

iu kin u: 0 ; 0x x (3.103)


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V d m phng s:


- M hnh {n nht c in:1 1 1

2 2 2

1 0 1 1 1 1 0

0 1 1 3 1 2 0

x x x

x x x

(3.104)

- M hnh {n nht midng m hnh lc:

1 1 1

4 0.5

2 22 2 2

01 0 1 1 1 1 0

2(1 3 )0 1 1 1 1 2 0

x x x

x D xx x x

(3.105)

iu kin u:

T T

0 0 0 ; 0 3 0 x x

Kt qu thu c:

Hnh 3.25.aDch chuyn v{ gia tc ca m hnh {n nht c in VD 3.25

Hnh 3.25.bDch chuyn v{ gia tc ca ca m hnh {n nht mi VD 3.25


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2 2 2

1 0 2 2 1 1 0

0 1 2 4 1 2 0

x x x

x x x

(3.106)


0.5

1 1 1 2

0.5 0.5

2 2 2 1 2

1 0 1 1 02

0 1 1 2 02 2

x x D x x

x x D x D x x

(3.107)

iu kin u:

T T

0 0 0 ; 0 3 0 x x

Kt qu thu c:




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2 2 2

1 0 1 1 1 1 0

0 1 1 3 1 2 0

x x x

x x x

(3.108)


1 1 1

22 2 2

1 12

01 0 1 1 1 1 0

0 1 1 1 1 2 0

1; 1 / ; 1; 2; c 1.1; 3

n

q

n a

x x x

A xx x x

x D c x H A H

(3.109)

iu kin u:

T T0 0 0 ; 0 3 0 x x

Kt qu thu c:




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2 2 2

1 0 2 2 1 1 0

0 1 2 4 1 2 0

x x x

x x x

(3.110)


1 1 21 1

2 2 1 1 22 2

1 1 2 12

1 0 1 1 0

0 1 1 2 0

1; 1 / ; 1; 2; c 1.1; 3

n

n n

q

n a

A x xx x

A x A x xx x

x D c x H A A H

(3.111)

iu kin u:

T T0 0 ; 3 0 x x

Kt qu thu c:



Ghi ch: c|c v d trn, k hiu t c gi| tr ngc du vi gia tc, hay t a t


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3.5.3 So s|nh kt qu tnh to|n gia c|c m hnh l thuyt vi kt qu thc nghim

Cho h dao ng mng my nh sau:

Hnh 3.29.H dao ng chu kch ng va p(a)Trc va chm; (b)Btu va chm; (c) Sau va chm.

Phng trnhdao ng ca h l{:

xmx f kx mg (3.112)iu kin u:

(0) 0; (0) 2x x gh (3.113)

Vi h l{ cao ca vt nng ri.Trong :

- m khi lng ca 2 vt. 1 2m m m - x nf A ( n c tnh theo cc m hnh [3.4-3.7])

S liu:

2

0.277( ); 0.020( ); 4620( / );

0.005( ); 5020( );

(0) 0; (0) 2

q

m kg m k N m

H m Ns m

x x gh

Kt qu thu c:

(Nt lin l kt qu tnh ton cc m hnh l thuyt, v ng chm chm l kt qu thc nghim)


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- M hnh IIa v IVa vi kt qu thc nghim

Hnh 3.30.Kt qu so s|nh m hnh l thuyt IIa v{ thc nghim vi h=30mm




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- M hnh IIb vi kt qu thc nghim

Hnh 3.33.Kt qu so s|nh m hnh l thuyt IIbv{ thc nghim vi h=30mm




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- M hnh IIIc vi kt qu thc nghim

Hnh 3.36.Kt qu so s|nh m hnh l thuyt IIIc v{ thc nghim vi h=30mm




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- M hnh IVc vi kt qu thc nghim

Hnh 3.39.Kt qu so s|nh m hnh l thuyt IVc v{ thc nghim vi h=30mm

Hnh 3.40.Kt qu so s|nh m hnh l thuyt IVc v{ thc nghim vi h=60mm

Hnh 3.41.Kt qu so s|nh m hnh l thuyt IVcv{ thc nghim vi h=100mm


65/78



3.6 Kt lun

- i vi c|c m hnh {n nht dng lc, v{ dng ng sutth cho kt qu l xp x nh nhau.

- i vi c|c m hnh {n nht mi cho kt qu tnh to|n trn th bng c|c ng phc tp hn, dovy m{ ta c th s dng c|c m hnh n{y m hnh c|c vt liu mi, vi c|c tnh cht c hc phc

tphn.- Qua kt qu so s|nh kt qu so s|nh gia c|cm hnh l thuyt v{ thc nghim, ta thy rng m hnhIVc l{ m hnh ph hp nht vi kt qu thc nghim.


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CHNG 4 Equation Section 4

CHNG TRNHFDE SOLVER TNH TON DAO NG

C H CO HM CP PHN S

4.1 Tng quan v chng trnhFDE Solver

Chng trnh FDE Solver (Fractional Derivative Equation Solver) phin bn u tin (v1.0a) clp trnh gii c|c phng trnh dao ng cp ph}n s, dng nh sau:

( ) ( )pmx bx c x D x kx f t (4.1)

iu kin u:

00 0, 0x x v (4.2)

Giao din chnh ca chng trnh nh sau:

Hnh 4.1.Giao din chng trnh FDE Solver.

Cc thng s cn nhp:

- Khi lng ca vt:m

- H s cn nht: b

- H s cn {n nht:


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- Bc o h{m cp ph}n s: p

- H{m dch chuyn: c(x)

- cng ca l xo: k

- iu kin u: 0

0x v

La chn phng php trnh ton:

- Phng ph|p xp x o h{m cp ph}n s (bng o h{m cp mt/ o h{m cp hai)

- Phng ph|p gii (Newmark/ Runge-kutta-Nystrm)

Phn ha hin th th ca kt qu:

Hnh 4.2.Phn ha hin th kt qu tnh to|n


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4.2 S dng chng trnh FDE Solver

s dng chng trnh, ta nhp c|c thng s ca phngtrnh (4.1), v{ la chn phng ph|p gii

cho b{i to|n, sau nhn nt Calculate thc hin vic tnh to|n.

V d vi b{i to|n sau:

4 0.51 3 1, 0 0, 0 3x x D x x x x (4.3)

Kt qu thu c nh sau:

Hnh 4.3.Kt qu tnh to|n s dng chng trnhFDE Solver


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KT LUNVi s ph|t trin khoa hc k thut, ng{y c{ng c nhiu c|c vt liu mi ra i (nh vt liu

silicone, vt liu cao su), nhng m hnh {n nht c in vi o h{m cp nguyn khng th hin

c y tnh cht ca vt liu. Do gii quyt vn n{y, o h{m cp ph}n s c |p dng

nghin cu trong thi gian gn }y. |n ~ bc u nghin cu tnh to|n dao ng ca mt s h

c hc c cha o h{m cp ph}n s. |n gm 4chng vi ni dung nh sau:

Chng mt trnh b{y c|c vn c bn ca o h{m cp ph}n s da trn t{i liu [1, 5, 7, 30]

Chng hai trnh b{y v c|c phng ph|p s kh|c nhau xp x th{nh phn o h{m cp ph}n s ,

cng nh c|c phng ph|p s tnh to|n dao ng ca c h cha th{nh phn o h{m cp ph}n s

da trn nhngt{i liu [3, 20, 28, 29, 30, 31, 32, 33]

Chng ba trnh b{y v tnh to|n dao ng ca mng m|y trn nn {n nht cp ph}n s.

Chng bn gii thiu v chng trnh PDE Solver tnh to|n dao ng .

|n a ra c|c phng ph|p s gii quyt c|c b{i to|n dao ng phi tuyn cp ph}n s.

Hng nghin cu sp ti l{ m rng tnh to|n dao ng cho nhiu m hnh phc tp hn, nghin cu

v iu kin u ca phng trnh vi ph}n cp ph}n s v{ s n nh ca c|c phng ph|p s, cng

nhx dng c|c kt qu thc nghim tm ra c|c quy lut phi tuyn cp ph}n s mi.

Hy vng nhng kt qu ca |nn{y c th l{m c s nghin cu cho c|c b{i to|n trong lnh

vc dao ng phi tuyn cp ph}n s. Tuy nhin dokh nng v{ trnh cn hn ch cng nh thi

gian c hn nn b|o c|o n{y ca em chc chn cn nhiu thiu st, em rt mong s nhn c nhng

kin ng gp ph bnh v{ b sung ca c|c thy c gi|o trong b mn em c th ho{n thin v{ b

sung thm kin thc.

Mt ln na em xin ch}n th{nh cm n Thy GS.TSKH. Nguyn Vn Khang ~ tn tnh gip em

trong thi giannghin cu v{ ho{n th{nh |nny!


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TI LIU THAM KHO[1] Nguyn Vn Khang (2009), Bi ging ng lc hc h c o hm cpphn s, Trng i hc

Bch Khoa.

[2] Nguyn Vn Khang (2004), Dao ng k thut, NXB Khoa hc k thut, H{ Ni.

[3] Nguyen Van Khang, N. Shimizu, M. Fukunaga, Duong Van Lac, Bui Thi Thuy, Calculation of

responses of a nonlinear fractional derivative model of impulse motion for viscoelastic

materials using Runge-Kutta-Nystrm method, Tuyn tp hi ngh c hc to{n quc, H{ Ni

(2014).

[4] M. Fukunaga and N. Shimizu. Comparison of fractional derivative models for finite

deformation with experiments of impulse response. Journal of Vibration and Control July

10, (2013).

[5] M. Fukunaga, N. Shimizu, H. Nasuno, A nonlinear fractional derivative model of impulse

motion for viscoelastic materials, Physica Scripta T136 (2009) 014010 (6pp).

[6] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, Boston, New York (1974).

[7] I. Podluny, Fractional Differential Equations, Academic Press, Boston, New York (1999).

[8] R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to

viscoelasticity, Journal of Rheology, 27 (3), 201-210, (1983).

[9] R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped

structures, AIAA Journal, 23 (6), 918-925, (1985).

[10] K. Diethelm (1997), An Algorithm for the Numerical Solution of Differential Equations of

Fractional Order, IMA J. Numer. Anal., Vol.5, pp. 1-6.

[11] K. Diethelm (2003), Fractional Differential Equations, Vorlesunysskrifit der TU Braunschweig.

[12] L. E. Suarez, A. Shokooh (1995), Response of Systems with Damping Materials Modeled using

Fractional Calculus, ASME J. Appl. Mech, Vol.48, No.11, pp. 1-9.

[13] L. E. Suarez, A. Shokooh (1997), An Eigenvector Expansion Method for the Solution of Motion

Containing Fractional Derivatives, ASME J. Appl. Mech, Vol.64, pp. 629-635.

[14] M. Fukunaga, N. Shimizu (2003), Initial Condition Problems of Fractional Viscoelastic

Equations, Proceedings of the 2003 ASME Design Engineering Technical Conferences,

September 2-6, 2003, Chicago Illinois, VSA.

[15] M. Fukunaga, N. Shimizu (2004), Role of Prehistories in the Initial Value Problems of

Fractional Viscoelastic Equations, International Journal of Nonlinear Dynamics, Vol.38, No.1-2,

pp. 207-220.

[16] M. Fukunaga, N. Shimizu (2009), Analysis of Impulse Response of Gel by Nonlinear Fractional

Derivative Models, Proceedings of the ASME 2009 International Design Engineering Technical

Conferences, September 2, 2009, San Diego, California USA.

[17] Neville J. Ford and A. Charles Simpson (2003), Numerical and Analytical Treatment of

Differential Equations of Fractional Order, Numerical Analysis Report 387, Manchester Centrefor Computational Mathematics, Manchester.


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[18] N. Gil-Negrete, J. Vinolas, L. Kari (2009), A Nonlinear Rubber Material Model Combing

Fractional Order Viscoelasticity and Amplitude Dependent Effects, ASME J. Appl. Mech, Vol.76,

pp. 110091-110099.

[19] N. Shimizu (1995), Dynamic Characteristics of a Viscoelastic Oscillator, Trans. Jps. Soc. Mech.

Eng., Vol.61, No.583, C, p. 166-170.

[20] N. Shimizu, W. Zhang (1999), Fractional Calculus Approach to Dynamic Problems of

Viscoelastic Materials, International Journal of JSME, Series C, Vol.42, No.4, pp. 825 -837.

[21] N. Shimizu, H. Nasuno (2007), Modeling and Analysis of Nonlinear Viscoelastic Systems by

means of Fractional Calculus Numerical Integration Algorithms, International Conference on

Material Theory and Nonlinear Dynamics, Hanoi.

[22] P. G. Nutting (1921), A New General Law of Deformation, J. of the Frankin Inst, 191, pp. 679-

685.

[23] P.G. Nutting (1943), A General Stress-Strain-Time Formula, J. of the Frankin Inst, 235, pp.

do an tot nghiep ok in_verson new_3

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