UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
FACULTA FACULTAD D DE D E INGENIERÍA INGE NIERÍA QUÍMICA QUÍMI CA
ECUACIONES DIFERENCIALES TEMA
APLICACIÓN DE ECUACIONES DIFERENCIALES DIFERENCIALES A LAS OSCILACIONES OSCIL ACIONES LIBRES DE DOS RESORTES ACOPLADOS ACOPLADOS CATEDRÁTICO: ING. WILDER EUFRACIO ARIAS
INTEGRANTES: GARCÍA LAPA RONAL MONTALVO MONTALVO CERRÓN JUAN MUÑOZ SALOMÉ MILAGROS TABOADA SINCHE HILLARY VALLEJOS CHAVEZ BANI
SEMESTRE:
III
HUANCAYO 22 DE JULIO - 20!
RESUMEN
En e!e "n#$%&e' () A*+",-,"n /e e,0-,"$ne /"#e%en,"-+e - +- $,"+-,"$ne +"1%e /e /$ %e$%!e -,$*+-/$))2 -,$*+-/$))2 /e!e%&"n-&$ e+ &$3"&"en!$ /e $,"+-,"n /e /$ %e$%!e -,$*+-/$ /e #$%&- 3e%!",-+ &e/"-n!e e,0-,"$ne +"ne-+e /e e40n/$ 4%-/$ 5 !%-n#$%&-/- /e L-*+-,e2 ,$n"/e%-n/$ !-&1"6n ,$n,e*!$ 17",$ /e +- +e5 /e H$$8e 9:0e e!-1+e,e :0e +- &-4n"!0/ /e +- #0e%;- ne,e-%"- *-%- *%$/0,"% 0n- ,"e%!- e+$n4-,"n en 0n &0e++e e /"%e,!-&en!e *%$*$%,"$n-+ - +- e+$n4-,"n< 5 +- +e5 /e Ne=!$n9 +- #0e%;- $n "n#+0en,"- e>!e%n- :0e -,e+e%-n +$ ,0e%*$ en 0n "!e&- /e %e#e%en,"- "ne%,"-+<. P-%- e+ /e-%%$++ /e-%%$++$ $ /e e!e e>*e%"&e e>*e%"&en!$ n!$ 0!"+";-%e 0!"+";-%e&$ &$ ? &--2 @ %e$%!e %e$%!e 5 $*$%!e &e!7+",$. E&*+e-&$ +- ? &-- *-%- /e!e%&"n-% +- ,$n!-n!e /e+ %e$%!e2 5 3"&$ :0e e+ %e$%!e %e$%!e e ,$&*%"&e ,$&*%"&e $ e!"%- 6!- !%-!- /e %e4%e-% %e4%e-% - 0 +$n4"!0/ +$n4"!0/ n-!0%-+. n-!0%-+. Me/"&$ Me/"&$ +e+$n4-,"n /e+ %e$%!e ,$n ,-/- &--2 en 0n /e!e%&"n-/$ !"e&*$ #"n-+&en!e ,$n +$ /-!$ $1!en"/$ en e+ e>*e%"&en!$ /e!e%&"n-&$ +- ,$n!-n!e /e +$ %e$%!e :0e /e,%"1e e+ &$3"&"en!$ /e 6!$.
OBJETIVOS
OBJETIVO GENERAL:
De!e%&"n-% +- ,$n!-n!e /e +$ %e$%!e &e/"-n!e e,0-,"$ne /"#e%en,"-+e.
OBJETIVOS ESPECÍFICOS:
De-%%$++-% +- e,0-,"$ne +"ne-+e /e e40n/$ 4%-/$ 5 +- !%-n#$%&-/- /e L-*+-,e ,$n +$ /-!$ $1!en"/$ /e #$%&- e>*e%"&en!-+. De!e%&"n-% +- ,$n!-n!e /e +$ %e$%!e &e/"-n!e e%"e /e F$0%"e%.
I"
MARCO TEÓRICO
I" LEY DE HOO#E: L- Le5 /e H$$8e /e,%"1e #en&en$ e+7!",$ ,$&$ +$ :0e e>"1en +$ %e$%!e. E!- +e5 -#"%&- :0e +- /e#$%&-,"n e+7!",- :0e 0#%e 0n ,0e%*$ e *%$*$%,"$n-+ +- #0e%;- :0e *%$/0,e !-+ /e#$%&-,"n2 "e&*%e 5 ,0-n/$ n$ e $1%e*-e e+ +&"!e /e e+-!","/-/. L- #$%&- &7 ,$&n /e %e*%een!-% &-!e&7!",-&en!e +- Le5 /e H$$8e e &e/"-n!e +- e,0-,"n /e+ &0e++e $ %e$%!e2 /$n/e e %e+-,"$n- +- #0e%;- ee%,"/*$% e+ %e$%!e ,$n +- e+$n4-,"n $ -+-%4-&"en!$ *%$3$,-/$ *$% +- #0e%;- e>!e%n-*+",-/- -+ e>!%e&$ /e+ &"&$'
D$n/e e ++-&- ,$n!-n!e e+7!",- /e+ %e$%!e 5 e 0 e+$n4-,"n $ 3-%"-,"n :0e e>*e%"&en!- 0 +$n4"!0/.
[1]
P$% +- Le5 /e H$$8e2 e+ %e$%!e &"&$ ee%,e 0n- #0e%;- /e %e!"!0,"n F $*0e!- +- /"%e,,"n /e+ -+-%4-&"en!$ 5 *%$*$%,"$n-+ - 0 &-4n"!0/ . D",$ en !6%&"n$ "&*+e2 F 82 en /$n/e 8 e 0n- ,$n!-n!e /e *%$*$%,"$n-+"/-/. A0n:0e ,0e%*$ /e /"!"n!$ *e$ *%$/0,en /"!"n!$ -+-%4-&"en!$ /e+ %e$%!e2 !-+ e+e&en!$ e+7!",$ e!- een,"-+&en!e ,-%-,!e%";-/$ *$% 6+ n0&e%$ 8. P$% ee&*+$2 " 0n ,0e%*$ :0e *e- +1. A+-%4- e+ %e$%!e en @ *"e2 en!$n,e2 8 9@< "&*+",- :0e 8 @ +1.*"e.
I"2 LEY DE NE$TON : De*06 :0e 0n- &-- M e 0e!- - 0n %e$%!e2 -:0e++- +$ -+-%4-%- en 0n&-4n"!0/ 5 -+,-n;-%- +- *$","n /e e:0"+"1%"$ en +- ,0-+ 0 *e$ W e e:0"+"1%-/$ *$% +- #0e%;- /e %e!"!0,"n 8. E+ *e$ e /e#"n"/$ *$%' W &. 4
I"% LA TRANSFORMADA LAPLACE"
L- !%-#$%&-/- /e L-*+-,e e 0- ,$n!"n0-&en!e *-%- %e$+3e% e,0-,"$ne /"#e%en,"-+e /e #0n,"$ne ,$n!"n0- - !%-&$' De1"/$ - :0e +- !%-#$%&-/- /e L-*+-,e e 0n- "n!e4%-+2 e!- ,0&*+e ,$n +- *%$*"e/-/e /e +"ne-+"/-/ :0e !"enen +- "n!e4%-+e Un- 3e; :0e e - e!0/"-/$ e+ ,$&*$%!-&"en!$ /e +$ "!e&- /"n7&",$2 e *0e/e *%$,e/e% - /"e-% 5 -n-+";-% +$ "!e&- /e ,$n!%$+ /e &-ne%- "&*+e.
Se- f 0n- #0n,"n /e#"n"/- *-%/e#"ne
2 +- !%-n#$%&-/- /e L-*+-,e /e f (t) e ,$&$
L- +e!%- s %e*%een!- 0n- n0e3- 3-%"-1+e2 *-%- e+ *%$,e$ /e "n!e4%-,"n e ,$n"/e%- ,$n!-n!e.
L- !%-n#$%&-/- /e L-*+-,e ,$n3"e%!e 0n- #0n,"n en t en 0n- #0n,"n en +3-%"-1+e s.
"%" PROPIEDADES DE LA TRANSFORMADA LAPLACE" En +- *%$*"e/-/e e -0&e :0e +- #0n,"$ne # 9!< 5 4 9!< ,$n #0n,"$ne :0e *$een. T%-n#$%&-/- /e L-*+-,e $n' Linealidad: L- !%-n#$%&-/- /e L-*+-,e e /"!%"105e $1%e +- 0&- $
%e!- 5 -,- ,$n!-n!e :0e &0+!"*+",-n.
Primer Teorema de Traslación: L- !%-n#$%&-/- /e L-*+-,e e convierte 0n #-,!$% e>*$nen,"-+ en 0n- traslación en +- 3-%"-1+e s.
Dn/e'
Teorema de la transformada de la derivada: L- !%-n#$%&-/- /e L-*+-,e ,-n,e+- +- /e%"3-/- &0+!"*+",-n/$ *$% +- 3-%"-1+e .
Teorema de la Transformada de la Integral.
Teorema de la Integral de la transformada
S"e&*%e ,0-n/$ e>"!-'
T&'(&)* +& ,* +&(.*+* +& ,* T(*/1'()*+*
I" ECUACIÓN LINEAL:
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O 0-n/$ $!%- n$!-,"n #%e,0en!e'
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2
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p ( p + 1) 1
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L- 0&-!$%"- /e #0e%;- e "40-+- - ,e%$ 5- :0e n$ e !"ene 0n- #0e%;- e>!e%n- $ en!%-/- ,$&$ e 0*0$ ,$n +- *%"&e%- &--. N$!e :0e /e "40-+ #$%&-2 e+ /e*+-;-&"en!$ /e +- e40n/- &-- e 3e%7 -#e,!-/- *$% +- *%"&e%- &--2 5- :0e e+ e40n/$ %e$%!e -,$*+-&1- &--. L- !%-n#$%&-/- /e L-*+-,e /e -&1- e,0-,"$ne /"#e%en,"-+e ,$n"/e%-n/$ ,$n/","$ne "n","-+e n0+- e!7 /-/- *$%'
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II"
MATERIALES: P&* R&'(5& R&6,* 4*,*/;*
PARTE EPERIMENTAL:
M*+&(*
PROCEDIMIENTO: " U/( ,* +' <&* 9'/ ,' (&'(5& = +& )*/&(* /5&(9*,*+*= /9*/+' 9'/ &, (&'(5&"
2" C','9*( &/ ,* <*(5& 7<&('( +&, (&'(5& &/ 7/* )*+&(* <*(* >7& ,'6(*( 7/* )*?'( &5*4,+*+"
%" U*( &, *(9' +& )*+&(* <*(* '5&/&( &, )'+7,'"
DATOS' ELEMENTOS MASA 9&< MASA @ 9&@<
VALORES .Q 4 .Q 4
CONSTANTE 98< CONSTANTE 9@< ELONGACION9< ELONGACION9@<
Q2Q ?2 ?2@ 2@
Un "!e&- ,$n"!en!e /e +$ Re$%!e A 5 B 5 +$ $1e!$ C 5 D -,$*+-/$ en #$%&3e%!",-+ e+ e>!%e&$ /e+ %e$%!e A e!- #"$ en e+ *0n!$ O. C-/- %e$%!e !"enen 0n- ,$n!-n!e /e %e$%!e 5 @ %e*e,!"3-&en!e. L$ $1e!$ !"enen &-- & 5 &@ %e*en!"3-&en!e e+ "!e&- e *$ne - 3"1%-% $!en"en/$ D 5 en +04-% &$3"en/$e C -,"- -%%"1- - 0n- /"!-n,"- - 5 +0e4$ $+!-n/$ -&1$ $1e!$ SOLUCION' P-%- /e!e%&"n-% +- e,0-,"$ne /"#e%en,"-+e /e+ &$3"&"en!$ !$&-&$ en 0n "n!-n!e ! 2 +$ $1e!$ C 5 D e!7n +$,-+";-/$ - +- /"!-n,"- > 5 >@ en 0 %e*e,!"3- *$","$ne /e e:0"+"1%"$. A0&"en/$ :0e +- /"%e,,"$ne -,"- -%%"1- $n *$"!"3-.
C C
D D
SOBRE C' E+ %e$%!e A e!- ee%,"en/$ 0n- #0e%;- $1%e C -,"- -%%"1- 5 +- &-4n"!0/ E+ %e$%!e D e!- ee%,"en/$ 0n- #0e%;- $1%e C -,"- -1-$ 5 +- &-4n"!0/ 3- e% 9>@><.
CALCULOS: H*,,*/+' X 2 y X 1 :
X 1= X f 1 − X 01 X 1= 6.5−3.3 X 1=3.2 X 2= X f 2 − X 02 X 2=11−5.8 X 2=5.3
H*,,*/+' k 1 y k 2 :
F 1= k 1 X 1
k 1=
k 1=
m1 g X 1 0.0579
∗9.8
0.032
k 1=17.73
F 2 =k 2 ( X 2− X 1 )
k 2=
m2 g
( X − X ) 2
k 2=
0.0749
1
∗9.8
0.021
k 2= 36.49
DADAS LAS ECUACIONES DIFERENCIALES: m1 x 1=−k 1 X 1 + k 2 ( X 2− X 1 ) … … … ( 1 ) ,
m2 x 2=−k 2 ( X 2− X 1 ) @@"23 ,
2
O<&(*/+' ,* &97*9'/& +&
m2 D + k 2 ? 5&/&/+' &/ 97&/5* 2 :
ECUACIONES LINEALES DE SEGUNDO ORDEN:
( ( D
1
(
4
x 1∗ D + x 1∗
4
D +
(
17.47
∗¿
4
+
)
)
k 1 + k 2 k 2 k k + D2 + 1 2 x 1=0 m1 m2 m1 m2
k 1 k 2 m1 m2
)
k 1 + k 2 k 2 + ∗ D 2=− x¿ m1 m2
+ 36.49
0.057
4
+
36.49 0.0744
)
2
D =
−17.47∗36.49 0.057 ∗0.0744
2
D + 1437.12 D =−150320.76 raiz = Y c = D ( D + 1437.12 ) = 0 2
m 1 =0
m 2= 0
2
m3=+¿−√ 1437.12 i
m4=+ ¿− √ 1437.12 i
Y c =C 1+ C 2 + C 3 cos 37.91 t + C 4 sin 37.91 t
Y p= x
2
∗a=52.3∗ x
2
'
Y p =2 x∗a ' '
Y p =2∗a ' ''
Y p =0 ;v
Y p = 0
Y p=1437.12 ∗( 2∗a )=−150320.76
a=
−150320.76 2∗1437.12
a = 52.3 2
Y =C 1+ C 2 + C 3 cos 37.91 t + C 4 sin 37.91 t + 52.3 t t =0
y =0
t =0
v =0
0
=C + C + C
0
=−37.91 C sin 37.91t + 37.91 C cos 37.91 t + 2∗52.3∗t
1
2
3
3
DATOS DE UN SOLO RESORTE m 1=00057
4
k 1=17.47 x = 3.2
2
d y dy m 2 + FA + K ∗ y =0 dt dt
2
d y m 2 + K ∗ y =0 dt 0.057 D
2
+ 17.47 Y =0
yc =( 0.057 D + 17.47 ) y =0 2
0.057 D
2
=−17.47
D=+¿−
√
17.47 0.059
i
−¿ 17.51 i +¿ ¿ D =¿ Y =C 1 cos 17.51 t + C 2 sin 17.51t
R&',.&/+' <'( 7)*5'(*: 2
d y m 2 + K ∗ y =0 dt
∞
y =
C x ∑ =
n
n
n
0
C n∗n∗¿ x
n−1
∞
'
y =
¿ ∑ = n
0
∞
y
=∑ C n∗n∗( n −1 ) x n−
' '
2
n= 0
∞
C ∗n∗( n − 1 ) x ∑ =
m∗
n− 2
n
n
∞
+ k ∗∑ C n x n= 0 n= 0
2
∞
∞
C + ∗n + 2∗( n + 1 ) x + k ∗∑ C x =0 ∑ = = n
m∗
n
n
n
n
2
n
0
0
∞
( m∗C + ∗n + 2∗( n +1 ) + k ∗C ) x =0 ∑ = n
n
n
n
2
0
m ( n + 2 ) ( n + 1 ) C n+2 + k C n=0 C n +2=−k C n / m ( n + 2 ) ( n + 1 )
R&&)<,*;*/+' C n +2=
−17.47 C n −306.5 C n = m ( n + 2 ) ( n + 1 ) ( n + 2)( n + 1)
n = 0 … … … … … … . C 2 =
−306.5 C
0
2
n = 1 … … … … … … . C 3 =
−306.5 C 3∗2
n =2 … … … … … … . C 4 =
−306.5 C 4∗3
2
3
y =C 0 + C 1 t + C 2 t + C 3 t
1
2
L- ,$n/","$ne "n","-+e e!7 /-/- *$%'
@ ) ))
• • • •
C0-n/$ ! e "40-+ - ,e%$ 9!<. Re$+3"en/$ *$% L-*+-,e'
[
m 1 s L [ x 1 ] − sx 1 ( 0 ) − x
'
2
( ) ]=k 1 L [ x 2 ] −2 k 1 L [ x 1 ]
1 0
( m 1 s + 2 k 1 ) L ( x 1 ) −k 1 L [ x 2 ] =−m 1 as 2
[
]
m 1 s L ( x 2 ) − sx 2 ( 0 ) − x ' 2 ( 0 )¿=−k 1 L ( x 2)+ k 1 L ( x 1 ) 2
−k 1 L ( x 1 ) + ( m 1 s + k 1 ) L ( x 2 )=0 2
( m s +3 mk s +k ) y ( x 1 )=−ams ( m s + k ) 2
4
2
2
2
(−mas ( m s + ! ) ) = L ( X 1 ) = 2
2
m s
4
2
1 4
[( 2 m s + 3 k ) −5 k ]
L ( X 1 ) =
X 1 =
1
1
2
2
2
L ( X 1 ) =4
3
−m a s + maks 2
3
+ maks ❑ [ ( 2 m s + 3 k ) −√ 5 k ]( ( 2 m s + 3 k ) −√ 5 k ❑) m as
❑
2
−√ 5 √ 5
−√ 5
2 √ 5
ma
a cos
√
❑
2
s
− ❑
❑
1
( 2 m s + 3 k ) −√ 5 k 2
3
−√ 5 2m
. kt −
1
−√ 5
2 √ 5
a cos
−√ 5 √ 5
√
3
ma
+ √ 5
2m
s
( 2 m s + 3 k )❑+ √ 5 k ❑
. kt
2
De "40-+ &$/$ X 2 =
1
√ 5
a cos
√
3
− √ 5 2m
.kt −
1
√ 5
a cos
√
3
+ √ 5
2m
. kt
PRUEBAS EPERIMENTALES PARA X 1 : &
00057
! 2 &6 8
17.47
X 1 =
1
−√ 5
2 √ 5
a cos
√
3
−√ 5 2m
. kt −
1
−√ 5
2 √ 5
a cos
√
3
+ √ 5
2m
. kt
REEMPLAZANDO DATOS X 1 =
1
−√ 5
2 √ 5
a cos
√
−√ 5 1− √ 5 . 17.47∗2− a cos 2∗ 00057 2 √ 5 3
.X@X .Q .X@ M-%4en /e e%%$%' ........................................
√∗
3
+ √ 5
2 00057
. 17.47∗2
PARA X 2 : &
000
! 2 &6 8
%8"
X 2 =
1
√ 5
a cos
√
3
− √ 5 2m
.kt −
1
√ 5
a cos
√
3
+ √ 5
2m
. kt
REEMPLAZANDO DATOS
X 2 =
1
√ 5
a cos
√
−√ 5 1 . 36.49∗2 − a cos 2∗00074 √ 5 3
@?Q.? ?@.? @.X M-%4en /e e%%$% .? ....
√∗ 3
2
+ √ 5
00074
. 36.49∗2
CONCLUSIONES Se /e!e%&"n e+ &$3"&"en!$ /e $,"+-,"n /e+ %e$%!e &e/"-n!e +- e,0-,"$ne +"ne-+e /e e40n/$ 4%-/$ 5 +- !%-n#$%&-/- /e L-*+-,e -,"en/$ 0$ /e +$ /-!$ e>*e%"&en!-+e. Se +$4% ,$n$,e% +- "&*$%!-n,"- /e +- !6,n",- /e !%-n#$%&-/- /e L-*+-,e en +- %e$+0,"n 5 -n7+"" /e *%$1+e&- ,$!"/"-n$ ,$&$ &-- 5 %e$%!e
DISCUSIONES E%%$% /e &e/","n /e +$ /-!$ en e+ &$&en!$ /e +- $,"+-,"n /e+ %e$%!e.
BIBLIOGRAFIA N CARDIELLO, “Elementos de Físi! " de #$ími!% & Edito'i!l (!)el$*
FRAN AYRES2 E,0-,"$ne D"#e%en,"-+e CESAR SAAL R.2 FELI CARRILLO C.2 E,0-,"$ne D"#e%en,"-+e
