UNIVERSIDAD UNIVERS IDAD CA CATOLICA TOLICA DE SANT SA NTA A MARIA
FACULTAD DE CIENCIAS E INGENIERIASFISICASY FORMALES
PROGRAMA PROFECIONAL DE INGENIERIA MECANICA ELECTRICA Y MECATRONICA
MECANICA COMPUTACIONAL II INFORME DE FASE N°1 NOMBRE CODIGO SECCI ON MEDINA VILLEGAS ARNULFO 200820168 A ANDRE 1 SALINAS BARREDA EDISON 200880236 A ERICK 1 ARENAS OVIEDO ALVARO 200820391 A ALONSO 1 MONZ ONZÓN AR ARU DIE DIEGO GO YA YAIR 2008201 016 A 1
ING! "UAN CARLOS CUADROS ARE#UIPA$PER% 2009$10$8
TRABAJO DE FASE N°1
TRABAJO DE FASE N°1
1! S& '( )*(+, )*(+, (- ,./, )(')( )(')( /+ ,+/( ,+/( &-+)*& &-+)*&4 4 5(*&,5(*&,- 4* 7()&4 7()&4 )( ,*&* ,*&* /+, 5-5/-, (+ -, ,'(: (- -&/&)4 ;-/&* *&),7(+( /,+)4 (- ,+/( ('( --(+4 < )(',&4 4+;4*7( '( )*(+(! L, ,', , -, /( (- +&5(- )(- ,./, )&'7&+/<( ('= dy y =−k √ y dt
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
Resolución de 2.1.1 y 2.1.2 adjuntado en el archivo de Excel 2!1!3 Grafica de Euler 3
2.5
2
1.5
1
0.5
0
-0.5 0
10
20
30
40
50
60
Grafica de RK2 3
2.5
2
1.5
1
0.5
0 0
10
20
30
40
50
60
VALOR REAL DE LA INTEGRACION= 1 −0.06
0
∫ 3
t
dy = dt √ y 0
∫
1 ( 2 √ 3 )= t 0.06
t =57,7
2! I7-(7(+( /+ *4.*,7, (+ MATLAB /( '4-/&4+( (- *4-(7, ,+(*&4*! E*4.*,7, )((* '4-&&,* /,- )( -4' )4' 74)4' '( (7-(,* (+ -, '4-/&?+: < /+, 5( (, -, (-(&?+ 4* ,*( )(- /'/,*&4 )((* 74'*,* -, *('/(', ,*4-(7, (+ ;4*7, ,/-,* < .*;&,! A)>/+,* '/ )&,.*,7, )( ;-/>4! Código: clc, clear all q=menu('seleccione el metodo','metodo de Euler','Metodo de RK2'); switch q case 1 F=inline('-!"#sqrt($)'); h=!%; $(1)=&; t(1)=; i=1; while $(i) $(i1)=$(i)F($(i))#h; t(i1)=t(i)h;
*rint('+%!% +%!% +%!% +%!%n',t(i),$(i),t(i1),$(i1)) i=i1;
*rint('+%!% +%!% +%!% +%!%n',t(i),$(i),t(i1),$(i1)) i=i1;
end rid on *lot(t,$) rid on case 2 F=inline('-!"#sqrt($)'); h=!%; $(1)=&; t(1)=; i=1; while $(i) .1=F($(i)); $1(i1)=$(i)F($(i))#h; i $1(i1)/ 0rea. end t(i1)=t(i)h; .2=F($1(i1)); $n(i1)=$(i)(h2)#(.1.2); *rint('+%!% +%!% +%!% +%!%n',t(i),$(i),.1,t(i1),$1(i1),.2,$n(i1)) $(i1)=$n(i1); i=i1; end rid on *lot(t,$) rid on end
Algorito:
&! &&!
&&&!
E-(.&* (- 74)4 , *(,-&,* S& '( (-&.( (- 74)4 )( E/-(* &! I+.*(',* -, ;/+&?+: : < 0: 0: &1 &&! M&(+*,' <&H0 1! <&1<&;<&J 2! &1& 3! &&1 S& '( (-&.( (- 74)4 )( RK2 &! I+.*(',* -, ;/+&?+: : < 0: 0: &1 &&! M&(+*,' <&H0 1! K1;<& 2! <&1<&F<&J 3! &1& ! @2F<1&1 ! <+&1<&2J@1@2 6! <&1<+&1 ! &&1
<&H0 y ( i + 1 ) = y ( i ) +
dy ( y ( i ) )∗h dt
+%!%
+%!%
+%!%
<&H0 y ( i + 1 ) = y ( i ) +
dy ( y ( i ) )∗h dt
t ( i +1 ) =t ( i ) + h
&&1 &: <&: &1: <&1 <&H0 K1=
dy ( y ( i ) ) dt
y1 (i + 1 )= y (i )+
dy ( y ( i ) )∗h dt
t ( i + 1 ) =t ( i ) + h K2=
dy ( y ( i + 1 ) ) dt
y n (i + 1 )= y (i )+
h 2
( K1 + K2 )
y ( i + 1 ) = y n ( i + 1 )
&&1 &:<&:K1:&1:<1&1:@2:<+&1 INICION h=!%;$(1)=&;t(1)=;i=1;q
1 2 FIN !iagraa de flujo:
3! E' ;*(/(+( /( (+ -4' ,+-&'&' ,5,+,)4' )( &+.(+&(*, '/*>,+ ;/+&4+(' )( B(''(-: 474 (+ (- ('/)&4 )( -4' ,74' (-*&4'! D&,' ;/+&4+(' 4* -4 .(+(*,- +4 '4+ '/'(&-(' )( (5,-/,*'( (+ ;4*7, )&*(, <: 4* (--4: +4 (' *,*4 /( ('+ 47&-,),' (+ ,-,' 7,(7&,' ('+),*! P4* (>(7-4: x "1#x$
1!8 2 2!2 2! 2!6
0!81 0!6 0!6 0!202 0!08
E'&7( J 1(2.1) 4+ (- /'4 )( /+ 4-&+47&4 )( &+(*4-,&?+ )( L,.*,+.( )( .*,)4 7&74 )( ,/(*)4 , -4' ),4' *44*&4+,)4'! S& (- 5,-4* 5(*),)(*4 (' 0!68292: ,., (- -/-4 )(- (**4* ,'4-/4 < (- (**4* *(-,&54 x = 2.1
f ( x n )= 0.568292
∣ Ea∣=¿ ∣ Er ∣=¿ F ( x
)=
(
x−2
)(
x − 2.2
)(
x − 2.4
)(
x − 2.6
)(
0.5815 ) +¿
3! E' ;*(/(+( /( (+ -4' ,+-&'&' ,5,+,)4' )( &+.(+&(*, '/*>,+ ;/+&4+(' )( B(''(-: 474 (+ (- ('/)&4 )( -4' ,74' (-*&4'! D&,' ;/+&4+(' 4* -4 .(+(*,- +4 '4+ '/'(&-(' )( (5,-/,*'( (+ ;4*7, )&*(, <: 4* (--4: +4 (' *,*4 /( ('+ 47&-,),' (+ ,-,' 7,(7&,' ('+),*! P4* (>(7-4: x "1#x$
1!8 2 2!2 2! 2!6
0!81 0!6 0!6 0!202 0!08
E'&7( J 1(2.1) 4+ (- /'4 )( /+ 4-&+47&4 )( &+(*4-,&?+ )( L,.*,+.( )( .*,)4 7&74 )( ,/(*)4 , -4' ),4' *44*&4+,)4'! S& (- 5,-4* 5(*),)(*4 (' 0!68292: ,., (- -/-4 )(- (**4* ,'4-/4 < (- (**4* *(-,&54 x = 2.1
f ( x n )= 0.568292
∣ Ea∣=¿ ∣ Er ∣=¿ F ( x 2,1 )=
(
− )(
x−2 1.8
2
x − 2.2 1.8 − 2.2
)(
x − 2.4 1.8− 2.4
( − )( − )( − )( − )( ( ( (
x −1.8 x −2.2 x −2.4
x −2.6
2
2
1.8
x −1.8 2.2− 1.8
x −1.8 2.4 −1.8
x −1.8 2.6 −1.8
2
2.2
2
)(
x −2
)(
)( )(
2.2− 2
− )(
x −2 2.4
2
x −2 2.6 −2
)(
2.4
x − 2.4 2.2 −2.4
x − 2.2 2.4− 2.2
x − 2.2 2.6− 2.2
)( )( )(
2.6
)(
x − 2.6 1.8− 2.6
)(
0.5815 ) +¿
0.5767 )+¿
x −2.6 2.2− 2.6
x − 2.6 2.4 −2.6
x −2.4 2.6 −2.4
)(
0.5560 ) +¿
)(
0.5202 )+¿
)(
0.4708 ) +¿
x%2.1
( −− )( −− )( −− )( −− ) ( )( −− )( −− )( −− )( )+¿ )( −− )( −− )( −− ) ( ) +¿ )( −− )( −− )( −− )( )+¿
F ( x 2,1 )=
( ( (
2.1 2 1.8 2
2.1 2.2 2.1 2.4 1.8 2.2 1.8 2.4
2.1 2.6 0.5815 ) +¿ 1.8 2.6
2.1−1.8 2−1.8
2.1 2.2 2 2.2
2.1−1.8 2.2− 1.8
2.1 2.2
2 2
2.1 2.2
2.4 2.4
2.1 2.2
2.6 0.5560 2.6
2.1− 1.8 2.4 −1.8
2.1 2.4
2 2
2.1 2.4
2.2 2.2
2.1 2.4
2.6 0.5202 2.6
2.1 2.4 2.1 2.6 0.5767 2 2.4 2 2.6
(
2.1−1.8 2.6 −1.8
)(
2.1− 2 2.6 −2
)(
2.1 −2.2 2.6− 2.2
)(
)
2.1 −2.4 ( 0.4708 ) +¿ 2.6 −2.4
F ( x 2,1 )=0,571147
•
∣
E a =
∣
E a =
∣
F ( x 2,1 ) − F ( x n ) F x 2,1
∗100
0,571147 −0,568292 0,571147
∣∗
100
E a =0,499871
•
Er =∣0.0182 ( x − 1.8 )( x −2 )( x − 2.2 )( x − 2.4 )( x − 2.6)∣
x
&#x$
1!8 2 2!2 2! 2!6
0!81 0!6 0!6 0!202 0!08
$ $0!02 $0!103 $0!19 $0!2
Er =8.1900∗10
$ $ $ $ $0!198 $ $0!188 0!016 $0!01 0!0312
−6
! ,)4' -4' ),4' )( -, '&./&(+( ,-,= x
f#x$
1 2 3 8
3 6 19 99 291
$ $ $ $ '.'1(2
1! C,-/-( f(4) 4+ (- /'4 )( 4-&+47&4' )( &+(*4-,&?+ )( N(Q4+ )( ?*)(+(' )( 1 , ! E-&>, -4' /+4' ,'( ,*, 4(+(* /+, /(+, (,&/)! 2! E'&7( (- (**4* ,*, ,), *()&&?+ '(.+ -, (/,&?+ )( (**4* Rn
F(4)%))) x % *
, Grado + 1 : F ( 4 )= F ( x 0 ) + F ( x 1 , x 0 ) ( x − x 0 )
3
F 19 99
$ 0
F ( 4 )=19 + 40 ( 4−3 ) F ( 4 )=59
•
E**4* R+ = 3
F 19 99
$ 0
,
2-1
-
Rn =14∗( x − x 0 )∗( x − x 1 ) Rn =∣−14∣=14
Grado + 2 : F ( 4 )= F ( x 0 ) + F ( x 1 , x 0 ) ( x − x 0 ) + F ( x 2, x 1 , x 0 ) ( x − x 0 ) ( x − x 1)
2 3
F 6 19 99
$ 13 0
$ $ 9
1*
F ( 4 )=6 + 13 ( 4− 2 ) + 9 ( 4 −2 ) ( 4− 3) F ( 4 )=50
•
E**4* R+ =
2 3
F 6 19 99
$ 13 0
$ $ 9
,
2-1
-
1*
1
Rn =1∗( x − x 0 )∗( x − x 1)∗( x − x 2) Rn =∣−2∣=2
Grado + / : F ( 4 )= F ( x 0 ) + F ( x 1 , x 0 ) ( x − x 0 ) + F ( x 2, x 1 , x 0 ) ( x − x 0 )( x − x 1 ) + F ( x 3 , x 2 x 1 , x 0 )
( x − x ) ( x − x ) ( x − x ) 0
1 2 3
F 3 6 19 99
$ 3 13 0
1
2
$ $ 9
$ $ $ 1
F ( 4 )=3 + 3 ( 4−1 ) + 5 ( 4 −1 ) ( 4− 2 ) + 1 ( 4 −1 ) ( 4 − 2 )( 4− 3) F ( 4 )= 48
•
E**4* R+ =
1 2 3
F 3 6 19 99
$ 3 13 0
$ $ 9
$ $ $ 1
,
2-1
-
1*
1
'
Rn =0
) Grado + * : x 4 , x 3 , x 2 x 1 , x0 F ( 4 )= F ( x 0 ) + F ( x 1 , x 0 ) ( x − x 0 ) + F ( x 2, x 1 , x 0 ) ( x − x 0 )( x − x 1 ) + F ( x 3 , x 2 x 1 , x 0 ) ( x − x 0 ) ( x − x 1) ( x − x 2 ) + F ¿
1 2 3
( x − x 0 ) ( x − x1 ) ( x − x 2 ) ( x − x 3 ) F 3 6 19 99 291
$ 3 13 0 96
$ $ 9 1
$ $ $ 1 1
$ $ $ $ 0
F ( 4 )=3 + 3 ( 4−1 ) + 5 ( 4 −1 ) ( 4− 2 ) + 1 ( 4 −1 ) ( 4 − 2 ) ( 4 −3 ) + 0 ( 4 − 1)( 4− 2 )( 4 − 3 )( 4 −5 ) F ( 4 )= 48
•
E**4* R+ =
1 2 3
F 3 6 19 99 291
$ 3 13 0 96
$ $ 9 1
$ $ $ 1 1
$ $ $ $ 0
(
***
10/
1-
1
'
Rn =0
'
! S/4+., /( (', )&'(,+)4 /+ ,+/( (';*&4 ,*, ,-7,(+,* ,./, ,*, /+ 4-,)4 (/(4 )(- ,'! E- 54-/7(+ )( -/&)4 /( /()( 4+(+(* (- ,+/( '( ,-/-, 4+=
V = π h
2
( 3R − h ) 3
)4+)( V54-/7(+73: *4;/+)&),) )(- ,./, (+ (- ,+/( 7: < R*,)&4 )(- ,+/( 7 R('/(-5, 4* (- 74)4 )( -, F,-', P4'&&?+ ,', /( (- (**4* *(-,&54 '( 7(+4* 4 &./,- /( 0!($ ! ntervalo: 1:/3
a
f#a$
4
f#4$
xr
f#xr$
e
1!00000 1!893 2!0169 2!0261 2!0268 2!02690 2!02691
$ 21!6222 $3!211 $0!26 $0!0181 $0!00136 $0!00010 $0!00001
3!0000 3!0000 3!0000 3!0000 3!0000 3!0000 3!0000
26!86 26!86 26!86 26!86 26!86 26!86 26!86
1!893 2!0169 2!0261 2!0268 2!0269 2!02691 2!02691
$3!211 $0!26 $0!0181 $0!00136 $0!0001 $0!00001 0
100 !9031 0!628 0!0338 0!0026 0!00018 0!00001
R('/(-5, 4* (- 74)4 )( N(Q4+ R,'4+ )( 2° O*)(+: ,', /( (- (**4* *(-,&54 '( 7(+4* 4 &./,- /( 0!($!
xr
e
1!9 2!02682 2!02691 2!02691
100!0000 0 6!21 0!001 0!00000
6! I7-(7(+( /+ *4.*,7, (+ MATLAB /( '4-/&4+( (- *4-(7, ,+(*&4*! E- *4.*,7, )((* '4-&&,* /,- )( -4' )4' 74)4' '( (7-(,* (+ -, '4-/&?+: < /+, 5( (, -, (-(&?+ 4* ,*( )(- /'/,*&4 )((* 74'*,* -, *('/(', ,- *4-(7, (+ ;4*7, ,/-,* < .*;&,! A)>/+,* '/ )&,.*,7, )( ;-/>4! Código: clc,clear all q=menu('elia el metodo' ,'Falsa 3osicion','4ewton Ra*hson 2do orden' ); switch q case 1 a=1; 0=&; e=!%#15-6; 7=; er=1; F=inline('(*i#752#(8-7)&)-&'); while F(a)#F(0) a=in*ut('inrese un nue9o *rimer 9alor: ' ); 0=in*ut('inrese un nue9o seundo 9alor: ' ); end while e/er 7r=a-((F(a)#(0-a))(F(0)-F(a))); er=a0s((7r-7)7r)#1; 7=7r; *rint('+%!% +%!% +%!% +%!% +%!% +%!% +%!%n',a,F(a),0,F(0),7r,F(7r),er) i F(7r)#F(a)/ 0=7r; elsei F(7r)#F(a) a=7r; elsei F(7r)#F(a)== 0rea. end end case 2 F=inline('(*i#752#(8-7)&)-&'); F=inline('*i#("#7-752)' ); 2F=inline('*i#("-2#7)'); er=1; e=!%#15-6; con=in; while con1 7=1!8; con=a0s(F(7)#2F(7)(F(7)52)); end while e/er 71=7-(F(7)2F(7))((sqrt((F(7)52)(2#2F(7)#F(7))))2F(7)); 72=7-(F(7)2F(7))-((sqrt((F(7)52)(2#2F(7)#F(7))))2F(7)); e1=a0s((71-7)71)#1; e2=a0s((72-7)72)#1; i e1/e2 7=71; er=e1; else 7=72; er=e2; end *rint('+%!% +%!%n',7,er) end end
Algorito:
A-.4*&74 )( -, ;,-', 4'&&?+=
&! &&!
I+.*(',* ;: ,: : (': 00: (*100 C47,*,* 1! S& ;,J;0 ,! M&(+*,' ('(* f ( a )∗( b −a ) x r =a − &! f ( b )− f ( a )
∣ ∣
e r =
&&! &&&!
x r − x 0 x r
∗100
C47,*,* 1! S& ;*J;,0 ,! * 2! S& ;*J;,H0 ,! ,* 3! S& ;J;,0 ,! *0 S& ;,J;H0 V4-5(* , ()&* /+ &+(*5,-4
2! ,!
A-.4*&74 )( N(Q4+ R,'4+ )( '(./+)4 4*)(+
&! I+.*(',* ;: ;: ;: (': 4+&+;: (*&+; &&! M&(+*,' 4+H1 1! I+.*(',* 0 ' ' ' f ( x 0 )∗ f ( x 0 ) 2 ' 2! con = f ( x )
∣
0
∣
&&&! M&(+*,' ('(* ' '' f ( x ) −2∗ f ( x )∗ f ( x ) √ ' ' 1! x = x − f ' ' ( x ) + f ( x )
f ( x 0 )
2
'
1
0
0
0
0
0
0
' ' ' f ( x ) − 2∗ f ( x )∗ f ( x ) √ ' ' 2! x = x − f ' ' ( x ) − f ( x )
f ( x 0 )
2
'
1
0
0
0
0
∣ ∣ ∣ ∣
3!
e 1=
!
e 2=
x r − x 0 x r
x r − x 0 x r
∗100
∗100
! C47,*,* ,! S& (1(2 &! 01 &&! (*(1 ! S&+4 &! 02 &&! (*(2
0
0
!iagraa de flujo
D&,.*,7, )( -, ;,-', 4'&&?+=
INICIO ;: , :: (': (*100: 00 ;,J;H0 ,: ('(* f ( a )∗( b −a ) x r = a − f ( b )− f ( a )
∣ ∣
e r =
;,J;*H0 ;,J;*0 A: ;,: : ;: *: ;* :(*
,* * *0 FIN
x r − x 0 x r
∗100
D&,.*,7, )( ;-/>4 )( N(Q4+ R,'4+ )( S(./+)4 4*)(+
I+&&4 ;: ;: ;: 0: (': (*100: 4+&+;
4+H1 0
∣
con =
'
' '
∣
f ( x 0 )∗ f ( x 0 ) '
2
f ( x 0 )
('(*
' '' f ( x ) −2∗ f ( x )∗ f ( x ) √ + x = x − ' ' ' ' f ( x ) f ( x )
f ( x 0 )
2
'
1
0
0
0
0
0
0
' ' ' f ( x ) − 2∗ f ( x )∗ f ( x ) √ − x = x − ' ' ' ' f ( x ) f ( x )
f ( x 0 )
2
'
1
∣ ∣
e 1=
x r − x 0 x r
∗100
∣ ∣
e 2=
(1(2 01 (*(2 02
FIN
0
0
0
(*(1
0
x r − x 0 x r
∗100
0
0
D&,.*,7, )(- *4.*,7, )( '(-(&?+ )( M4)4 INICIO
1 R(,-&,* -4' *4()&7&(+4' )( -, ;,-', 4'&&?+ R(,-&,* -4' *4()&7&(+4' )( N(Q4+$R,+4'4+ )( '(./+)4 4*)(+ 2 F&+
! S/4+., /( -, ;/(*, ,&, ,**&, )( -, *('&'(+&, )(- ,&*( '4*( /+ 4>(4 /( ,( (' *44*&4+,- ,- /,)*,)4 )( -, 5(-4&),)! P,*, ('( ,'4: -, 5(-4&),) '( ,-/-, 4+= )4+)( 4(;&&(+( )( ,**,'*( )( '(./+)4 4*)(+!
v ( t ) =
√
gm tanh Ca
(√ ) gCa t m
9.8 m /¿ '2 : 768!1 @.: < 0!2 ,
S&
Kg m : /'( &+(.*,&?+ ,+,-&, ,*, )((*7&+,* /
,+ -(>4' ,( (- 4>(4 (+ 10 '(./+)4'! ,., -4 7&'74: (*4 (5,-/ -, &+(.*,- (7-(,+)4 -, *(.-, )( S&7'4+ 13 47/(',! P*/(( 4+ )&;(*(+(' ,', 4(+(* *(' ).&4' '&.+&;&,&54' )( (,&/)!
I+(.*,&?+ A+,-&,= 10
∫ 0
√
9.8∗68.1 ∗tanh 0.25
(√
)
9.8∗ 0.25 ∗t dt =333.9262 68.1
I+(.*,&?+ 4* S&7'4+ 13 C/,*, )(*&5,), F6228231880J1$ ,+1362J1362W12JW2W2J,+1362J1362W12JJ1362W12$ 3361231880J,+1362J1362W12JW3J1$ ,+1362J1362W12JW2J1362W12 M4 =max x ∈ [0,10 ]∣ f ∣ 4
P,*, 5,-4* 7&74 )( -, /,*, )(*&5,), 2!2 F 0!232 ,0
10 b− a 4 h M4 180
h=
b− a
h
∫ f ( t ) dt " 3
! =
0
[
h ≅1
n =10
2n
0!00000 1!00000 2!00000 3!00000 !00000 !00000 6!00000 !00000 8!00000 9!00000 10!00000
x' x1 x2 x/ x* x0 x x, x( xx1'
10
h 0.4356
0!000
2n − 1
# f ( x$ )+ 4
2n − 2
f ( xi ) + 2 ∑ f ( xi ) + f ( x ∑ = = i
1
0 9!681 18!11 26!902 33!0832 38!186 2!06 !883 6!9266 8!3 9!3918
f#x'$ f#x1$ f#x2$ f#x/$ f#x*$ f#x0$ f#x$ f#x,$ f#x($ f#x-$ f#x1'$
i
2
2n
)
]
10
∫ f ( x ) dx " 13 [ 0 + 4 ( 9,1841 + 26,5002 +38.1846 + 44,8830 + 48,3755 ) + 2 ( 18,711 +33,0832 + 42.0446 +46,
! =
0
SX33:06
CONCLUSIONES:
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•
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