UNIVERSIDADE FEDERAL DE OURO PRETO ESCOLA DE MINAS DEPARTAMENTO DE ENGENHARIA CIVIL MESTRADO EM CONSTRUÇÃO METÁLICA

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UNIVERSIDADE FEDERAL DE OURO PRETO

ESCOLA DE MINAS

DEPARTAMENTO DE ENGENHARIA CIVIL

MESTRADO EM CONSTRUÇÃO METÁLICA

ANÁLISE DINÂMICA DE SISTEMAS

FUNDAÇÃO-ESTRUTURA METÁLICA COM

AMORTECIMENTO NÃO-PROPORCIONAL E

NÃO-LINEARIDADE FÍSICA

por

Andréa Gonçalves Rodrigues

orientada por

Antônio Maria Claret de Gouvêia, D.Sc.

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CONVÊNIO USIMINAS/UFOP/FUNDAÇÃO GORCEIX

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ANÁLISE DINÂMICA DE SISTEMAS FUNDAÇÃO-ESTRUTURA METÁLICA COM AMORTECIMENTO NÃO-PROPORCIONAL E

NÃO-LINEARIDADE FÍSICA

Andréa Gonçalves Rodrigues

TESE SUBMETIDA AO CORPO DOCENTE DO MESTRADO EM

CONSTRUÇÃO METÁLICA DO DEPARTAMENTO DE ENGENHARIA CIVIL DA ESCOLA DE MINAS COMO PARTE DOS

REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE MESTRE EM CIÊNCIAS DE ENGENHARIA CIVIL (M.Sc.)

Aprovada por:

Dutervil Geraldo de Magalhães, Eng. de Minas, Metalurgia e Civil

Fernando Venâncio Filho, D.Sc.

Francisco Célio de Araújo, Dr.-Ing.

José Elias Laier, D.Sc.

Sebastião Arthur Lopes de Andrade, Ph.D.

Antônio Maria Claret de Gouvêia, D.Sc. Presidente

Ouro Preto - MG – Brasil Dezembro 1994 !

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AGRADECIMENTOS !

Ao Prof. Antônio Maria Claret de Gouvêia,

pela orientação e apoio prestados em todas as etapas deste trabalho.

Ao Prof. Fernando Venâncio Filho, pelo apoio científico.

A USIMINAS,

pelo apoio dado ao Mestrado em Construção Metálica.

Aos amigos.

A minha família. !

! M! SUMÁRIO

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1. CONSIDERAÇÕES GERAIS

1.1 Introdução

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1.2 Objetivos! !

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1.3 Descrição Sumária! !

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1.4 Revisão Bibliográfica! !

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2. EQUAÇÕES DE EQUILÍBRIO DINÂMICO! !

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65,-1! 4! 1! -5631;0:5,<1! -0! :0660*! %! 0! ;1,6<0,<5! -5! 0:1.<5;8:5,<1! U86<5./<8;1! 5! 8*! 0! >,8-0-5!8:0T8,7.80L!B5665!;061*!0!5Z>02?1!-1!5Z>83FY.81!-8,9:8;1!/! ! ! QNLOV! ! ! ! . v c

fv =− !

). t ( p kv v c v

m!!+ !+ =

, kv ) D 2 ( i f_e =

(

)

v

! J! !

! ! ! ! ! !

5Z>02?1!Z>5!61:5,<5!D1-5!65.!<.0<0-0!.8T1.160:5,<5!,1!-1:F,81!-0!].5Zxa,;80!L!

(:Y1.0!1!0:1.<5;8:5,<1!,06!56<.><>.06!,?1!65e0!486;161!5*!68:*!U86<5./<8;1*!!,0! D.7<8;0*! 1! :5;0,86:1! -5! 0:1.<5;8:5,<1! .503! /! ;1:>:5,<5! 0D.1_8:0-1! D531! 0:1.<5;8:5,<1! 486;161*! Z>5! 3540! 0! >:0! ]1.:0! ;1,45,85,<5! -0! 5Z>02?1! -5! :148:5,<1!! -0! 56<.><>.0L! B0! :081.80! -16! ;0616*! 1! :1-531! -5! 0:1.<5;8:5,<1! 486;161! 5Z>84035,<5! 0-1<0-1! 5:! 56<.><>.06! .56>3<0! 5:! >:0! 0D.1_8:02?1! .0^17453! -16! .56>3<0-16! 5_D5.8:5,<086L!

B16! D.1Y35:06! -8,9:8;16! Z>5! 3540:! 5:! ;1,68-5.02?1! 16! 5]58<16! -0! 8,<5.02?1! 6131@56<.><>.0*! 0! -8668D02?1! -5! 5,5.T80! Z>5! 56<7! 0661;80-0! w! D.1D0T02?1! -5! 1,-06! 5376<8;06! ,1! 6131*! ;1,68-5.0-1! >:! :581! 5376<8;1*! 65:8@8,]8,8<1*! U1:1Ta,51! 5!!! 861<.`D8;1*!.56>3<0!,1!0:1.<5;8:5,<1!T51:/<.8;1!1>!D1.!.0-802?1L!

C0.0! ;1,;58<>0.! <03! 0:1.<5;8:5,<1*! 65e0! >:0! ]>,-02?1! 5:! D30;0! 48Y.0,-1! 45.<8;03:5,<5! 61Y.5! >:! 65:8@56D021! 5376<8;1*! =8T>.0! NLfL! %0! D30;0*! ]1,<5! -5! 48Y.02?1*! D.1D0T0:@65*! ,1! 65:8@56D021! 5376<8;1*! 1,-06! -5! ;1.D1*! ;1:! >:0! ].5,<5! -5! 1,-0! U5:86]/.8;0*! 5!1,-06!-5! #0{358TU*!;1:!].5,<56!-5!1,-0!;83F,-.8;06L!(6656!-186!<8D16!-5! 1,-06*!w!:5-8-0!Z>5!65!0]06<0:!-0!]1,<5*!D.1D0T0:@65!5:!>:!413>:5!;0-0!45^!:081.! -5!:0<5.803L!%56<5!:1-1*!0!-5,68-0-5!-5!5,5.T80!5*!;1,65Z>5,<5:5,<5*!0! 0:D38<>-5!-5! 48Y.02?1! -5;.56;5:! 5:! ;0-0! 1,-0*! ;1,6<8<>8,-1@65! ,1! 0:1.<5;8:5,<1! T51:/<.8;1! 1>!! D1.!.0-802?1L!

)>Y6<8<>8,-1@65! 1! 65:8@56D021! D1.! >:! 686<5:0! :0660@:130@!!!!!!!!!!!!!!! 0:1.<5;5-1.! 5! 8T>030,-1@65! 06! 0:D38<>-56! 1Y65.40-16! ,1! 686<5:0! :0660@:130@

0:1.<5;5-1.! 5! ,1! 686<5:0! ]>,-02?1! .FT8-0@65:8@56D021! 5376<8;1*! D1-5@65! -5<5.:8,0.! 1! ;15]8;85,<5! -5! 0:1.<5;8:5,<1! 486;161! 5Z>84035,<5! 01! 0:1.<5;8:5,<1! T51:/<.8;1L! +! W0Y530! NLI! :16<.0! 06! 5_D.566[56! -1! 0:1.<5;8:5,<1! 486;161! 5Z>84035,<5! ]1.,5;8-06! D1.! #8;U0.<!QIJPtV!5!B140y!QIJPfVL!

+! ;1,68-5.02?1! -1! 0:1.<5;8:5,<1! U86<5./<8;1! ,0! 0,73865! -8,9:8;0! -5! 686<5:06! 6131@56<.><>.0! <0:Y/:! D1-5! 65.! ]58<0! 0! D0.<8.! -0! ;1,-82?1! -5! .0-802?1*! Z>5! ;1,->^! 0! ;15]8;85,<56!-5!.8T8-5^!-8,9:8;16*!,0!]1.:0!>,8-8:5,681,03*!56;.8<16!D1.!

! !

QNLJV! ! !

! ! ! ! ! ! !

, c ia k

! It! !

! ! ! ! !

1,-5!k1 5 c1 6?1! ]>,2[56! -06! D.1D.85-0-56! ]F68;06! 5! T51:/<.8;06! -16! 686<5:0*!aO! /! 0!

].5Zxa,;80!0-8:5,681,03!5!aOc1!/!1!0:1.<5;8:5,<1!T51:/<.8;1!,1!686<5:0L!

'<838^0,-1! 1! D.8,;FD81! -0! ;1..56D1,-a,;80! 0! D0.<8.! -0! (ZL! QNLJV*! D1-5@65! 6>Y6<8<>8.!1!:`->31!5376<8;1!.503!D531!5Z>84035,<5!;1:D35_1n!

! !

!QNLItV! !

!

1,-5! %! /! 0! ;1,6<0,<5! -5! 0:1.<5;8:5,<1! U86<5./<8;1! 1>! 56<.><>.03! 5! 8*! 0! >,8-0-5! 8:0T8,7.80L!!

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

(

)

! II! !

! ! !

! ! ! ! ! ! ! ! ! ! ! ! !

=8T>.0!NLI!S1:D1,5,<56!Y768;06!-5!>:!686<5:0!-5!>:!T.0>!-5!38Y5.-0-5L! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! IN! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

=8T>.0!NLf!S1,;58<1!-5!0:1.<5;8:5,<1!T51:/<.8;1!Q#8;U0.<*!IJPtV! !

! If! ! ! ! ! ! ! ! ! ! ! ! ! W0Y530!NLI!S15]8;85,<56!-5!0:1.<5;8:5,<1!486;161!5Z>84035,<5!D0.0!!!!!!!!!!!!!!!!!! ]>,-02[56!.FT8-06!;8.;>30.56*!65T>,-1!#8;U0.<!QIJPtV!5!B140y!QIJPfV! ! ! ! ! ! ! A1-1!-5! !M8Y.02?1! ! ! #0^?1!-5!A0660!1>!-5! !",/.;80! ! ! S15]8;85,<5!-5!+:1.<5;8:5,<1! ! M5.<8;03! ! ! !! z1.8^1,<03! ! ! ! ! ! !!! #1<0;81,03! ! ! ! ! ! !!! W1.681,03! ! ! ! ! ! ! ! !

(

)

(

)

(

)

O x r m 1 32 8 7 B ρ ν − ν − = x x B 288 . 0 = ξ

(

)

ψ

(

)

ψ ψ ψ + = ξ B B 1 15 . 0 5 O r I B ρ = θ θ θ

θ = ₊

! IK! ! ! ! ! !

3. MÉTODO DA SUPERPOSIÇÃO MODAL E PSEUDOFORÇAS

3.1 Formulação! !

)5e0! >:! 686<5:0! 56<.><>.03! -86;.5<8^0-1! 5:! 535:5,<16! ]8,8<16*! <1<038^0,-1!N! T.0>6! -5! 38Y5.-0-5*! -1<0-1! -5! 0:1.<5;8:5,<1! 486;161*! 5_D.5661! D1.! >:0! :0<.8^!cL!! (,<?1*!6>0!5Z>02?1!-5!:148:5,<1!/! ! ! QfLIV! ! ! +!613>2?1!-1!D.1Y35:0!-5!0><14031.! ! ! !QfLNV! ! !

;1,->^! w6! ].5Zxa,;806! ,0<>.086! -5! 48Y.02?1! ωωωωi, i = 1,2,...,N, 5! 016! :1-16! -5!!!!!!!!

48Y.02?1! ;1..56D1,-5,<56! φφφφi, i = 1,2,...,N. $6! :1-16! -5! 48Y.02?1! 6?1!!!!!!!!!!!

,1.:038^0-16!-5!0;1.-1!;1:! ! ! !QfLfV! ! !

1,-5!δ8e! /! 1! -53<0! -5! |.1,5;y5.L! +! :0<.8^!ΦΦΦΦ! ;>e06! ;13>,06! 6?1! 16! :1-16! ,1.:086! -5! 48Y.02?1!φ8! /! 0! :0<.8^! :1-03! 5! <5:! 06! D.1D.85-0-56! -5! 1.<1T1,038-0-5! 5:! .5302?1! w6! :0<.8^56!-5!:0660!5!-5!.8T8-5^L!(6606!D.1D.85-0-56!6?1!5_D.56606!D1.! ! ! !QfLKV! ! 5! ! QfLpV! ! ). t ( p kv v c v

m!!+ !+ =

i i

i m

kφφφφ ₌ωωωω2 φφφφ

, m _{j ij}

T

i φφφφ δδδδ

φφφφ =

I m

T _ΦΦΦΦ= Φ Φ Φ Φ , k

T _{ΦΦΦΦ ΛΛΛΛ} Φ

Φ Φ

! Ip! ! ! ! 1,-5!I!/!0!:0<.8^!8-5,<8-0-5!-5!1.-5:!N*!;>e16!535:5,<16!<a:!0!-8:5,6?1!-5!:0660*!5!! Λ ΛΛ

Λ! /! 0! :0<.8^! -80T1,03! ]1.:0-0! D5306! ].5Zxa,;806! ,0<>.086! -5! 48Y.02?1! 53540-06! 01! Z>0-.0-1L!

+<.04/6! -0! :0<.8^! :1-03*! 06! ;11.-5,0-06! ]F68;06! Q-5631;0:5,<16V! v! 6?1! <.0,6]1.:0-06!5:!;11.-5,0-06!T5,5.038^0-06*!86<1!/*!]0^5,-1!,0!(ZL!QfLIV! ! ! !QfLoV! ! !

1,-5!Y! /! 1! 45<1.! -5! ;11.-5,0-06! T5,5.038^0-06*! D./@:>3<8D38;0,-1! 0:Y16! 16! 30-16! -0! (ZL! QfLIV! D1.! ΦΦΦΦT! 5! ;1,68-5.0,-1! 06! (Z6L! QfLKV! 5! QfLpV*! 0! 65T>8,<5! 5Z>02?1! 5:! ;11.-5,0-06!:1-086!/!1Y<8-0! ! ! !QfLPV! ! ! B5660!5Z>02?1*!C!/!0!:0<.8^!-5!0:1.<5;8:5,<1!T5,5.038^0-0!-0-0!D1.! ! ! !QfLOV! ! ! 5!P!/!1!45<1.!-5!]1.206!T5,5.038^0-06!-0-1!D1.! ! ! !QfLJV! ! !

h>0,-1! 1! 0:1.<5;8:5,<1! -1! 686<5:0! /! ,?1@D.1D1.;81,03*!C ,?1! /! -80T1,03L! +! T.0,-5^0! .530<840! -16! 535:5,<16! -5! ]1.0! -0! -80T1,03! -5! C! D0.0! 16! 535:5,<16! -0!! -80T1,03!-5!C!/!5_D.5660!D1.! ! ! QfLItV! ! ! !

$!F,-8;5!-5!0;1D30:5,<1!-1!0:1.<5;8:5,<1!,1!686<5:0*!α*!/!-5]8,8-1!;1:1!1!:7_8:1! 4031.! -5!α8e! 5! /! >:! -16! 8,-8;0-1.56! -1! Z>0,<1! 535! /! ,?1@D.1D1.;81,03*! ;1,]1.:5! D.1D16<1!D1.!M5,9,;81@=83U1!5!S30.5<!QIJJIVL!

! !

, Y v=ΦΦΦΦ

. P Y Y C

Y!!+ ! +ΛΛΛΛ =

Φ ΦΦ Φ Φ Φ Φ Φ C C₌ T

. p P_=ΦΦΦΦT

! Io! ! ! ! ! +!:0<.8^!-5!0:1.<5;8:5,<1!T5,5.038^0-0!Q,?1@-80T1,03V!D1-5!65.!D16<0*!;1:1!0! 61:0!-5!->06!:0<.8^56*!-0!65T>8,<5!]1.:0! ! ! QfLIIV! ! !

1,-5!Cd /! 0! :0<.8^! -80T1,03*! ;>e16! 535:5,<16! 6?1! 16! 535:5,<16! -0! -80T1,03! -5!C*! 5!!!!!!

Cf!<5:!535:5,<16!,>316!,0!-80T1,03!5!16!;1..56D1,-5,<56!535:5,<16!-5!]1.0!-0!-80T1,03!

-5!CL!)>Y6<8<>8,-1!C!-0!(ZL!QfLIIV!,0!(ZL!QfLPV*!1Y</:@65! ! ! !QfLINV! ! !

B5660! 5Z>02?1! 5:! ;11.-5,0-06! :1-086*! 1! 0;1D30:5,<1! -548-1! 01! 0:1.<5;8:5,<1!!!,?1@D.1D1.;81,03!!!/!!;1,68-5.0-1!!;1:1!!>:!!45<1.!!!-5!!!!!D65>-1]1.206! !!!!!!!!!!!!!!,1! 65T>,-1! :5:Y.1L! C1.! 1><.1! 30-1*! 1! D.8:58.1! :5:Y.1! /! -560;1D30-1!!!!!!! ;1:1!1;1..5!,16!686<5:06!-1<0-16!-5!0:1.<5;8:5,<1!D.1D1.;81,03L!

+! (ZL! QfLINV! /! .561348-0! D1.! >:! D.1;5661! 8<5.0<841*! ;>e1! y@/68:1! D0661! /! -0-1! D1.! ! ! QfLIfV! ! !

+!;0.T0!/!0D.1_8:0-0!D1.!>:0!D138T1,03*!;1:1! ]18!]58<1!D1.! "Y.0U8:Y5T148;!5! b8361,! QIJOJVL! (:! ;0-0! D0661*! 1! 8@/68:1! 535:5,<1! -5!Y(k), Yi(k) /! ;03;>30-1! D530!

8,<5T.03n! ! ! !QfLIKV! ! ! !

;1:!;1,-82[56!8,8;8086!8T>086!w6!;1,-82[56!]8,086!,1!D0661!0,<5.81.L!B0!(ZL!QfLIKV*!ωdi!/!

0! ].5Zxa,;80! 0:1.<5;8-0! ,1! 8@/68:1! :1-1! -5! 48Y.02?1! 5! ! ! ! ! ! /! 0! 4531;8-0-5! -0!!!!!!!!!!!!!!!!!!!!! e@/68:0!;11.-5,0-0!T5,5.038^0-0L! ! ! ! ! , C C C= _d+ _f

. Y C P Y Y C

Y!!_{+ d}! _{+ΛΛΛΛ = − f} !

(

)

( ) _{C Y}( ) _Y( ) _{P C Y}( )_.

Y k 1

f k

k d

k ₊ ! _{+ = −} ! −

!

! _ΛΛΛΛ

( )

_∫

_∑

_{( )}

− _{ω −τ τ}

− ω = < t B I e I e 8 -< I y e 8e 8 8 -y

8 ! QC S • V5 68, < - *

I

• Q<V !

! IP! ! ! ! ! !

':! ;.8</.81! -5! ;1,45.Ta,;80! -1! D.1;5661! 8<5.0<841! 56<0Y535;8-1! ,0! (ZL! QfLIfV! D0.0!0!8@/68:0!;11.-5,0-0!/!! ! ! QfLIpV! ! !

1,-5! ! ! ! ! ! /! 6>]8;85,<5:5,<5! D5Z>5,1! 5! <eQe! ‚! I*N*LLL*~V! /! 1! <5:D1! -86;.5<1! ,1! Z>03! 0!! 8,<5T.03!/!;03;>30-0L!

+<.04/6! -5! -845.616! 5_5:D316! ,>:/.8;16*! 1Y65.40@65! Z>5! 0D5,06! /! ,5;5667.81! 45.8]8;0.!0!;1,45.Ta,;80!-16!4031.56!:7_8:16!-06!;11.-5,0-06!T5,5.038^0-06L!+668:*!1! ;.8</.81!-5!;1,45.Ta,;80!-0!(ZL!QfLIpV!D1-5!65.!5_D.5661!D1.! ! ! QfLIoV! ! !

C1.! 1><.1! 30-1*! >:0! D.5;86?1! 0-5Z>0-0! /! 1Y<8-0! ,1! ;73;>31! -0! .56D16<0*! 65! 1! ;.8</.81! /! 60<86]58<1! ,16! :1-16! :086! 68T,8]8;0<8416L! (665! ]0<1! 45:! .5->^8.! 1! 56]1.21! ;1:D><0;81,03! 5_8T8-1! D531! :/<1-1L! (_5:D316! ,>:/.8;16! 65.?1! 0D.565,<0-16! ,1! ;0DF<>31!KL!

! ! !

3.2 Condição de Convergência! !

! !

)5e0!•8θQ<V!1!4031.!5_0<1!-0!8@/68:0!;11.-5,0-0!T5,5.038^0-0!,1!y@/68:1!D0661!5! 65e0! •8αQyVQ<V! 1! ;1..56D1,-5,<5! 4031.! 0D.1_8:0-1! ,1! y@/68:1! D0661L! $6! 5..16! ,0! ;11.-5,0-0! T5,5.038^0-0! 0D.1_8:0-0! 5! 6>06! -5.840-06! ,1! y@/68:1! D0661! -1! D.1;5661! 8<5.0<841!6?1!-5]8,8-16!D1.! ! ! ! ! ! ! ! ! ! ! , ) t ( Y ) t ( Y ) t ( Y j ) k ( i j ) 1 k ( i j ) k (

i − _<ε

! IO! ! ! ! ! !QfLIPV! ! ! QfLIOV! ! ! !QfLIJV! ! !

$! D.1;5661! 8<5.0<841! /! ;1,45.T5,<5! 65*! -0-1!ε! D168<841! 5! 0.Y8<.0.80:5,<5! D5Z>5,1*!!! 5_86<5!k*!<03!Z>5! ! ! !QfLNtV! ! !

+! ;1,-82?1! -5! ;1,45.Ta,;80! -1! D.1;5661! 8<5.0<841! /! 1Y<8-0*! 5:! 65T>8-0*! 5:! ]>,2?1! -06! ].5Zxa,;806! ,0<>.086! -5! 48Y.02?1! 5! -16! 535:5,<16! -06! :0<.8^56! -5! :0660! 5!!! -5! 0:1.<5;8:5,<1! T5,5.038^0-06L! +! D.5:8660! Y768;0! D0.0! 0! 1Y<5,2?1! -0! ;1,-82?1! -5! ;1,45.Ta,;80! ;1,686<5! ,0! -5]8,82?1! -5! >:0! 48^8,U0,20! -5! •8θQ<V! w! Z>03! D5.<5,20!! •8αQyVQ<VL!+!65T>8,<5!-5]8,82?1!/!D.1D16<0!

! !

!QfLNIV! !

!

1,-5! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! /! 1! :`->31! -1! :7_8:1! 5..1! ,0! 8@/68:0! ;11.-5,0-0! T5,5.038^0-0!!!!!!!!!!!!!!!!!!!!!!!! ,1! D0661! y! 5!θy! /! <03! Z>5! ! ! ! ! ! ! ! ! ! ! ! ! .0-80,16! ,1! 8,6<0,<5! 5:! Z>5!ς8QyV! 1;1..5L!!!!!!!!!!!!!!!!!!!!!!!! (660!-5]8,82?1!/!83>6<.0-0!,0!=8T>.0!fLI!1,-5!/!0-:8<8-1!1!68,03!D168<841L!!

S1,68-5.5! ,0! (ZL! QfLIfV! 0! 5_D.566?1! -0! 8@/68:0! ;11.-5,0-0! T5,5.038^0-0*! 1>! 65e0*!! ! ! ! QfLNNV! ! ! ! ! ! ! !

( ) _{Y Y ,}

Y_ik(t)= _iα(t)(t)− _iθ(t) ∆

( ) _{Y Y ,}

Y_ik(t)= _iα(t)(t)− _iθ(t)

∆! ! !

( ) _{Y Y ,}

Y_ik(t)= _iα(t)(t)− _iθ(t)

∆!! !! !!

( ) _.

Y_ik (t) <ε ∆

( ) _Y( ) ( )_sen

₍

₎

Y k _{i k}

i ) t ( i ) t ( k

i = ±ς ω +θ

θ α

( ) ( )k

max ,i k

i = ∆Y

ς

(

)

( ) ( ) ( )

_∑

α _{+ +ω = −} N

1 j1 j ), t ( 1 k i ij i ) t ( k i 2 i ) t ( k i ii ) t ( k

i C Y Y P C Y

! IJ! ! ! ! ! !

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( )_cos

(

)

C Y C P Y Y C Y k j 1 k j j N 1 j 1 j N 1 j1 j ij ) t ( j ij i ) t ( k i 2 i ) t ( k i ii ) t ( k i θ + ω ς ω − = ω + + − ≠ = ≠= θ α α α

∑

∑

! ! !

∑

θ _{+ +ω = −} N

1 j 1 j ). t ( i ij i ) t ( i 2 i ) t ( i ii ) t (

i CY Y P C Y

Y!! ! !

( ) ( ) ( )

( )_cos

(

)

! Nt! !! ! ! ! ! ! ! QfLNoV! ! ! ! !

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( ) D _C ( )_cos

(

)

Y k1 _{j k 1 k}

j j ij N 1 j 1

j 2i

ji )

t ( k

i = _ω ως ω +θ +λ

∆ − − ≠ =

∑

(

)

(

)

[

]

ji i 2 2 ji

ji 1 2

D = −β + ξβ −

( )

_∑

i C |.

D | | Y | _max ( ) ( ) ( )

_∑

i C |.

D | |

Y

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3.3 Processos Total e Parcial! !

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ji _{ω ≤}

ω

∑

i C |

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! ! ! !

3.4 Algoritmo para Solução de um Sistema Linear Usando o Método de Superposição Modal Clássico com Pseudoforças –

Processo Parcial

"L!"BiS"$! !

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!

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k 2 _i

i

i====ωωωω ΦΦΦΦ ==== Φ

ΦΦ Φ

! Nf! ! ! ! ! ! ! ! ! ! !

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. y Y , y Y ), t ( p P ) t ( P 0 T 0 0 T 0 j T j j ! ! _====ΦΦΦΦ

Φ Φ Φ Φ ==== Φ Φ Φ Φ ==== ==== . C C C==== _d ++++ _f

! NK! ! ! ! ! ! !

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{

}

5 V < Q

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= −ξω

( ) ( )

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C !

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( ) ( )

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• • • y 8e I y 8e y

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! NP! 4. EXEMPLOS NUMÉRICOS

4.1 Exemplo 1: Sistema mecânico flexível com amortecedor concentrado! !

!

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!

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! !

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! ! ! !

4.2 Exemplo 2: Sistema Massa-Mola-Amortecedor com 2 Graus de Liberdade !

!

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4.3 Exemplo 3: Modelo de Usina Nuclear Simplificado em um Sistema de Oito Graus de Liberdade

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40 Figura 4.1.1 Sistema dinâmico amortecido (Exemplo 1).

51 Figura 4.1.5-h Deslocamento na direção 8 (m) x tempo (s).

53 Figura 4.2.1 Sistema massa-mola-amortecedor (Exemplo 2).

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! !

5. ANÁLISE NÃO-LINEAR !

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5.1 Introdução! !

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! !

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