12

Pendulo Lagrange

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Page 1: Pendulo Lagrange
Page 2: Pendulo Lagrange
Page 3: Pendulo Lagrange

(1)

(2)

( ) ( ) ( )

( ) ( )

x ( ) ( ) () ( ) (

y t l t( ) () (

p p( )(( )

( ) (3)U M gl t(cosM glM gl

(4)1 ( ) ( )K Mx1 (r

J Kx(

2 2( ) 1

cKKmprrm gJ KJ2

2

22

+)Mx1 (2

( )Mx( 2

(5)1 ( )K M x1 (2

cKK cMM 2

Page 4: Pendulo Lagrange

(6)( ( ) 2 ( ) ( ) ( ) ( ) ) 1 ( )cosK M x t l x t t t l t I t21

2p p p p p

2 2 2 2a a a a= - + +

(7)( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )cosK M M x t M l x t t t t M l I t21

21

c p p p p p p2 2 2 2a a a a= + - + + +

(8)( ) ( )cos cosL M M x M l x M l I M gl21

21

c p p p p p p p p2 2 2 2a a a a a= + - + + + -

( )

dtd

xL

xL F B x

dtd L L B 9

eq

p

22

22

22

22

a aa

- = -

- =-

oo

oo

`

`

j

j

( )

( ) ( )

( )

sin

sin cos

sin cosx

M M I M M l M l

I M l B x M l I M l M l B

M l g I M l F2

c p p c p p p p

p p p eq p p p p p p p p

p p p p p

2 2 2 2

2 2 3 2

2 2

a

aa a a

a a=

+ + +

- + - + -

+ + +p

o o o= G

( )( )

( ) ( )

cos cos

M M I M M l M l sin

M M M l gsin M M B M l sin cos

M l B x M l F10

c p p c p p p p

c p p p c p p p p

p p eq p p

2 2 2 2

2 2 2

aa

a a a a a

a a- +=

+ + +

+ - + -

p

o o

o= G

( ) ( )x

( )M M I M M l

M l g I M l B x M l B I M l F2 2 2 2

c p p c p p

p p p p p eq p p p p p p

2

a a- + - + +=

+ +p

o o6 @

Page 5: Pendulo Lagrange

( )( )

( ) ( )

I M M l( )(

M M M l g M l B x M M B M L F( )( ( )

c p p c p p) M M lM M

) ) M LM L2

a =I) p) I)

M l g M l B x M M BM l B x M M B) ( )(M lM l) M l BM l B B)p

o o6 @

( )

Fr p

n K T

T n K I

IR

V K

rrg g mK TK T

m m t mT n K IT n K I

mIIm

m mKK ~

=

=

( )Fr R

n K n KV

r R

n K n K Kx

2

2

mp mr Rrg g m tK n KK n K

mp mr Rrg g m t mK n K KK n K

= -

1 3

2 4

Page 6: Pendulo Lagrange

( )( )

( )

( )

zI M M l)

M l gz I M l B( )r R

n K n K Kz M l B z

r Rn K n K

V

2c p p c p p) M M lM M

M lM l eq)mp mr Rr

g g m t mK n K KK n K Kp p pM l BM l B

p p p

mp mr Rrg g m tK n KK n K

3

2 2ll 2

2

2

3 4

2

=I) p) I)

+BeqBI M l( ))

+(

cc mR

T

SRR

SSS

SSSSS

TTSSTTTT

SSSS

V

X

WVV

WWW

WWWWW

XXWWXXXX

WWWW

( )

( ))

( )

zI M M l)

M M M l gz M l B)r R

n K n K Kz M M B z( )

M lr R

n K n KV

c p p c p p) M M lM M

) eq

mp mr Rrg g m t mK n K KK n K K

c p p)B)

p pM lMmp mr Rr

g g m tK n KK n K

4 2

2

2

3 4

=I) p) I)

+BeqBM l gz M l)M lM l) M lM l (

+

cc mR

T

SRR

SSS

SSSSS

TTSSTTTT

SSSS

V

X

WVV

WWW

WWWWW

XXWWXXXX

WWWW

( )( )X X u(

0

0

0

0

0

0

2 1972.

27 6669.

1

0

16 797.

38 7034.

1

1

0 0071.

0 0891.

0

0

2 206.

5 083.

=-

-

-+

R

T

SRR

SSS

SSS

SSSSTT

SS

R

T

SRR

SSS

SSS

SSSSTT

SS

V

X

WVV

WWW

WWW

WWWWXX

WW

V

X

WVV

WWW

WWW

WWWWXX

WW

X AX BU

Page 7: Pendulo Lagrange

16.8861 26.4438 379.6803 ( )P S 16.8861CPP4 3 2

SS 26 443826 4438

, , ,0 4 8338 4 583638 17 136288. , . , .. , . ,1 ,,, 4 83384 8338

( )M B AB A B A B2 3B AB A6 @

(1 )

0

0

2

5

2

5

36

84

36

84

588

1479

588

1479

9539

23982

=-

-

-

-

-

-

R

T

SRR

SSS

SSS

SSSSTT

SS

V

X

WVV

WWW

WWW

WWWWXX

WW

( )

356 0866.

26 6641.

15 9762.

1

26 6641.

15 9762.

1

0

15 9762.

1

0

0

1

0

0

0

=

-

-

-R

T

SRR

SSS

SSS

SSSSTT

SS

V

X

WVV

WWW

WWW

WWWWXX

WW

( )

51 4716.

0

0

0

0 1657.

0 0015.

51 4716.

0

2 2772.

5 247.

0 1657.

0 0015.

0

0

2 2772.

5 247.

=

-

-

R

T

SRR

SSS

SSS

SSSSTT

SS

V

X

WVV

WWW

WWW

WWWWXX

WW

. . . . ( )23. . . . (. . . (= 6 @

( )

. . . . ( )

K

K

(23 0667 166 125 402 4901 479 1006. . . .. . T

(9. . . .. . .

1=

= -

-66

@@@

Page 8: Pendulo Lagrange

( )e e u 24eC U= +

A

C

B

0

0

0

C

U

=-

=

=

;

G

E

1 0 0 0= 6 @

K1 ( )kiK 25= -6 @

(26). . . . .K 90 635 124 292 53 5441 29 5463 96 08121 = - - -6 @

Page 9: Pendulo Lagrange
Page 10: Pendulo Lagrange
Page 11: Pendulo Lagrange

(2 )( )x X u yB L)X u yu y1 = -( ) uu)X uuXX

( )

. .221 1.

19 2.

1497 2.

4553 3.

1 2.

226 2.

175 6.

5222 9.

=-

-

-

R

T

SRR

SSS

SSS

SSSSTT

SS

V

X

WVV

WWW

WWW

WWWWXX

WW

Page 12: Pendulo Lagrange