xv - 02

111
11t1
A
A
PUBLICAÇAO DA ABCM • ASSOCIAÇAO BRASILEIRA DE CIENCIAS MECANICAS
VOL. XV • NQ2 • 1993
ISSN 0100-7386
A
A
REVISTA BRASilEIRA DE CIENCIAS MECANICAS
JOURNAL OF THE BRAZILIAN SOCIEIY OF MECHANICAL SCffiNCES
REViSTA Bf;IASJI..BAA: ~E C~IAS MEcliii~S..
.. JOURN~OFf~e::~~~~~~ç~
. ' , vor r .t4', (~97!i) .ma de Janeiro: AsSQc~o 'l!raslt~fllll[e_ (l!ê~Cias
·
•· ,
1111ed~s
·TtfmeMr\íl
tneiUf 11NrêJfc1<l,S:blbJroglif~s,
i-•Mer;ánlcà ·
. ISSN-01 00.7386
EDITOR:
Leonardo Goldsleln Jr.
UNICAMP- F[M- DETF · CP 6122
13083--970 Campinas· SP
Tel: (0192} 39·3006 Fax: (0192} 39-3722
EDITORES ASSOCIADOS:
Agenor de Toledo Fleury
IPT- lnslitllto de Pesquisas Tecnológicas
Divisão de Mecânica e Eletricidade - Agrupamento de Sistemas de Controle
Cidade Universitária • C.P. 7141
01064-970 S1o Paulo - SP
Tel: (011} 268-2211 R-504 Fax: (011) 869-3353
Carlos Alberto Carrasco Allemani
UNICAMP- FEM- DE · C.P. 6122
13083-970 Campinas - SP
Tel: (0192) 39·8435 Fax: (0192) 39-3722
José Augusto Ramos do Amarai
NUCLEN • NUCLEBRÁS ENGENHARIA. SA
Superinter>dênçia de Esti1J1Uras e Componentes Mecanicos.
R: Visconde de Ouro Preto, 5
2225().180 • Rio de Janeiro · RJ
Tel: (021) 552-2772 R·269 ou 552·1095 Fax: (021) 552·2993
Waller L. Weingaertner
Universidade Federal de Santa Catarina
Dep~ de Eng' Mecanica - lab. Mecânica de Precisão
Compus - Trindade · C.P. 476
88049 Aorianópolis • SC
Tel: (0482) 31-9395/34·52n Fax: (0482) 34-1519
CORPO EDITORIAl:
Alcir de Faro Orlando (PUC • RJ)
Antonio Francisco Fortes (UnB)
Armando Aibertazzi Jr. (UFSC)
Atai r Rios Neto (INPE)
Benedito Moraes Purquerio (EESC • USP)
Caio Mario Coslll (EMBRACO)
Carlos Alberto de Almeida (PUC · RJ)
Carlos Alberto Martin (UFSC)
Clovis Raimundo Maliska (UFSC)
Emanuel Rocha Woiskl (UNESP · FfiS)
Francisco Emnlo Baccaro l'ligro (IPT - SP)
Françisco José Sim~es (UFPb)
Genesio José Menon (EFfl)
Hans lngo Weber (UNICAMP)
Henrique Ro:zenfeld (EESC USP)
Jair Carlos Dut11 (UFSC)
João Alziro He11 de Jornada (UFRGS)
José João de Espindola (UFSC)
Jurandlr ltizo Yanagihara (EP USP)
Urio Schaeter (Uff!GS)
Lou rival Boehs (UFSC)
luis Carlos Sandoval Goes (ITA)
Man:io ZMani (UFMG)
Moyses Zir>deluk (COPPE • UFRJ}
Nisio de Carvalho Lobo Brum (COPPE - UFRJ)
Nivaldo lemos Cupini (UNICAMP)
Paulo Afonso de Oliveira Sovlero (rTA)
Paulo Eigi Míyagi (EP USP)
Rogerio Martins Saldanha da Gama (LNCC)
Valder Sleffen Jr. (UFU)
REVISTA FINANCIADA COM RECURSOS DO
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MCT
a Publicações Cientificas
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RBCM • J. of the Braz. Soe. Meehanleal Seleneee
Vol. XV· n• 2 - 1993 · pp. 112-123
ISSN 0100·7386
Printed in Brazn
Escolha da Ferramenta no Torneamento
de Resina Reforçada com Fibras de
Vidro e Carbono
Choice of Too/ in Turning of Carbon and Glass Fiber
Reinforced Resin
J. R. Ferreira
E FEl - Escola Federal de Engenharia de ltajubá
Departamento de Produção
Cx. Postal 50
37500-{)00 ltajubá · MG
N. L. Cuplnl
UNICAMP · Universidade Estadual de Campinas
Departamento de Engenharia de Fabricação
Cx. Postal 6122
13081-970 Campinas · SP
Resumo
Este trabalho consiste na realização de experimentos para analisar o comportamento de ferramentas de diversos
materiais no torneamento de compósitos de resina fenólica reforçada com fibras de vidro e carbono. Durante os
cn~ios foram observados a variação dos desgastes das ferramenl.'lS, da rugosidade superficial da peça e da força
de avanço. Uma das mais importantes conclusões deste trabalho é que a rugosidade superficial do compósito
aumenta de maneira progressiva com o desgaste da ferramenta, viabilizando somente o emprego da ferramenta
de diamante em torpeameoto de acabamento. Em operação de desbaste o metal duro apresentou-se como a
melhor solução. pois apesar de sofrer desgaste superior ao diamante é mais barato e dispensa cuidados
operacionais especiais.
Palavras-chave: Usinagem - Ferramentas- Materiais Compósitos
Abatract
Experiments were carried ou tio study lhe pcrformaoce of differeot tool materiais. They were tested io turníog of
carbon and gl~ss fiber reinforced fenol ic rosin composite. Durlng the experiments, tool 'M:Br, surfuce roughness,
aod fced force behavior were observed. Experimental results sbowed that, dueto the great iotlueoce of tool wear
on workpiece surface roughness, only diamond tools are suitable for use in fioish tumíng. lo rougb turoiog. the
cemented caJbide tools showed to be the best solution. Altbough it showed larger wear thao diamood, the cosi/
benefit ratio is much lower and special precautions are not oecessary.
Keywords: Machining - Tool · Composite Material
Introdução
O emprego de materiais compósitos à base de resinas fenólicas reforçadas com fibras tem crescido
notavelmente nos últimos anos, especialmente nas indústrias aeronáutica, aeroespacial e
automobilística. Como conseqüência do aumento e da diversificação do emprego dos materiais
compósitos, cresce cada vez mais a necessidade de usiná-los, para satisfazer as exigências com relação
à precisão dimensional e à qualidade superficial dos componentes (Wunsch e Spur, 1988).
Os plásticos reforçados com fibras são moldados e a massa mais importante de sua aplicação até
na;sos dias é na forma de placas, lâminas e chapas. Nelas incide principalmente operações de corte e
furação. Pouco encontra-se em pesquisa e desenvolvimento, relatado ou publicado sobre outras
operações como: torneamento, fresamento, retificação etc. (Reinhart, 1987). Este trabalho tem por
objetivo contribuir para aplicações alternativas dos plásticos reforçados com fibras, na fabricação de
peças e componentes com geometrias diversas, particularmente no caso de torneamento.
Considerando-se estes aspectos, o trabalho apresenta uma experiência de torneamento de um
compósito de resina fenóhca reforçada com fibras de vidro e carbono, confeccionados nas formas de
placas laminadas e tubos booinados (Fig. 1). No torneamento do compósito de tecido bobinado serão
Manuscript receiYed: June 1993. Associate Technical Editor: Walter L 'Neingaertner
J. R. Ferreira e N. L Qlpinl
113
avaliados a evolução do desgaste das ferramentas, da força de avanço e da rugosidade da peça, em
função do comprimento de corte usinado. Na operação de torneamento do compósi to de tecido
laminado serão estudados a innuencia da velocidade de corte no desgaste da ferramenta e da
orienmção das fibras na rugosidade da peça.
De posse dos resultados obtidos, apresenta-se neste trabalho a escolha da melhcr ferramenm para o
processo de torneamento do compósito, em operações de desbaste e acabamento.
Procedimentos Experimentais
Material, Ferramenta e Equipamentos. O material compósito ensaiado apresenta as seguintes
propriedades: Tecido bibrido de fibra de carbono (70%) e fibra de vidro (30%); resina fenólica (35 a
40% em massa) e densidade de 1.4 a 1,5 g!cm3 .
As ferramentas utilizadas foram: metal duro classe ISO KlO, cerâmica pura (A1203+Zr02),
cer âmi ca mista (AI 20 3 + Ti C), cerâmica reforçada com whiskers (AI203+SiC), SIALON
(AJ~3 +Si 3 N 4), nitreto de boro cúbico (CBN) e diamante policristalino (PCO). As fe rramentas de
metal duro, CBN e diamante tinham geometria ISO SNUN 120412 e as cerâmicas tinham geometria
ISO SNUN 120712.
DlREÇÃO
A
115
DAS
CORTE
FIBRAS
A-A
ESC.: 1: 2
Fig. 1 a)
Corpoa da prova da material com~lto - tacldo bobinado
Nomenclatura
ap ~ profundidade de corte
a = ângulo de posição das
fibras relativa À direçáo
de corte
c., = custo de afiação da
femsmen1a
C, .. custo da ferramenta
Ctt "' custo da ferramenta por
vida
CP "' custo da ferramenta por
peça
Cpr .. custo elo portaferramenta
f = avanço de corte
= força de avanço
lc "' comprimento de corte
n, = número de afiações da
ferramenta
número de vidas da
ferramenta
número de vidas do
porta-ferramenta
Ra = rugosidade média
sh = salÁrio do operador
Sm ., salário da máquina
T = vida da ferramenta
= tempo de aproximação e
recuo da ferramenta
F,
t.
1.c
t11
~
\>
t.
~
= tempo de corte
= tempo de troca da
ferramenta
= tempos lmprodutivoe
= tempo de preparação
da máquina
= tempos secundários
= tempo total de
confecçio por peça
Vc = velocidade de corte
V8 = desgaste de flanco
Z = número de peças do lote
Esoolha da Ferramenta no Torneamento de Resina Reforçada oom Fibras de Vidro e Carbono
114
OIREÇio DAS FIBRAS
t3~A
ESC.: 1:2
CORTE A·A
Fig. 1 b) Corpos da prova de malerlal eomp6elto -1eeido laminado
Os equipamentos utilizados foram: Torno CNC com potência de 22kW, rugosímetro digital,
microscópio com mesa de duas coordenadas, aspirador de pó industrial e equipamentos de proteção
individual.
Método Experimental. No torneamento do compósito de tecido bobinado fez-se uma comparação
do desempenho de todas ferramentas ensaiadas. Para isto, adotou-se uma condição de corte constante
para todas as ferramentas: velocidade de corte Vc=320 m/mío. avanço f= 0,1 mm/rote profundidade
de corte ap= 1 mm. Para cada condição de ensaio ferramenUI-peça foram medidos o desgaste da
ferramenta, a força de avanço e a rugosidade superficial da peça. Este ensaio serviu para classificar as
ferramentas baseado no desempenho de cada uma delas.
No torneamento do compósito de tecido laminado, trabalhou-se com ferramentas de metal duro e
diamante, as escolhidas do ensaio anterior. Com o metal duro estudou-se a evolução do desgaste da
ferra menU! com as seguintes velocidades de corte: 25, 100, 200, 320 e 450 m/min. Com a ferramenta
de diamante verificou-se a variação da rugosidade superficial da peça em relação ao ângulo de posição
das libras do laminado. 1\S condições de corte utilizadas foram: velocidade de corte V ç=200 m/m.in,
profundidade de corte ap= I mm e três avança; diferentes, f=O,l, 0,2 e 0,3 mm/rot.
Resultados e Discussões
A Fig. 2 mostra a evolução do desgaste das ferramentas em função do comprimento de corte
usinado, no torneamento do compósito de tecido bobinado. As ferramentas de materiais cerâmicos
apresentam um fraco desempenho no torneamento do compósito reforçado com fibras de vidro e
carbono, exemplificado pela ferramenta de SIALON que sofreu o maior desgaste entre todas as
ferramentas ensaiadas. O mecanismo de desgaste provável neste caso foi por abrasão mecânica, pois
não foi observado nenhuma quebra ou avaria da ferramenta no ensaio. O melhor desempenho entre as
ferramentas mecânicas obteve-se com a alumina reforçada com whiskers de SiC, mas devido ao
elevado custo da ferramenta comparado ao metal duro, não justifica seu emprego.
J. R Ferreira e N. L. Cupinl
1111
1 , 6~------------------------------------------~
I
Vc • 320 m /min
f • 0,1 mm/rot
ap • tmm
Mot.: tecido bobinado
1,..
-~ 1,2 g
o
u
5
K10
la.
..,•
CBN
10
12
14
Comprimento de Corte lc
Fig. 2
16
18
( m x 102)
20
22
ESC.:5 :-1
o..g ..te de ftanco d.. fef!'amentae v.rsua eomP'Imento 1M corta
A Tabela I mostra os custos por vida, por aresta e as vidas das ferramentas para um critério de
desgaste de flanco Ve=0,9 mm.
A ferramenta de CBN apresentou um desempenho superior às ferramentas de cerâmicas e metal
duro, mas ficou aquém da ferramenta de diamante. O custo do CBN de US$ 142,00 por aresta, superior
ao do diamante (US$ 93,00) inviabiliza a sua aplicação no torneamento do compósito. Portanto, a
escolha da ferramenta fica emre o metal duro por apresentar o menor custo por aresta e por vida, a
cerâmica pura que apresentou o menor custo por a resta e por vida em relação as cerâmicas, e o
diamante que apresentou o melhor comportamento de todas as ferramentas.
Tl.,.la I. Cuatoa por ar..ta • por vldll daa ferramenta•
Custo/ar.
(CR$)
Custo/ar.
(US$)
Vida lc(m)
VB: 0,9mm
Custo/Vida
(CRS/m)
Cerâmica pura
141 ,60
3,54
750
0,19
Cerâmica mista
180,00
4,50
580
0 ,31
Cerâmica ref.
SIC
900,00
22,50
1280
0,70
SIALON
450,00
11 ,25
450
1,00
Metal duro
29.20
0 ,73
1850
1,57
CBN
5680,00
142,00
lc = 2300 m
VB 0,32 m
2,47
Diamante (PCD)
3720,00
93,00
lc "' 2300 m
VB • 0,32m
1,86
Ferramentas
=
116
Escolha da Ferramenta no Torneamento de Resina Retorc;ada oom Fibras de VidtO e Carbono
Em operação de desbasle deve ser fei1 o um estudo mais delalhado, pois o custo por aresta da
ferromentn de diamante é 127 vezes o cus1o da ferramenta de metal duro. Bm função dis10 deve ser
realizada uma análise de custO/benefício de usinagem.
Levanlando-se alguns dados sobre o 1orneamento do compósito de tecido bobinado
especificamente, pode-se fazer uma comparação da> custos relativos à usinagem entre as ferramenUlS
de metal duro, cerâmica pura e diaman1e. O custo relativo à usinagem por peça (Cp) é dado por
(ferraresi, 1977):
(1)
sendO:
It • Ic•~
+I +I + ~
+ (~-.!_)Ir
z
Tz t
Cr1
Cr
•
n 1 Cpr
+C - + n2
nf n 2
n3
(2)
-
(3)
p/ Inserto reafiávcl
(4)
A Tabela 11 mostra os valores dO> tempa> e custos de usinagem para as ferramentas de metal duro,
cerâmica pura e diamante, no torneamento de um peça de compósito de tecido bobinado.
Os valores obtidas na Tabela H mo5trnm que a utilização de ferramentas de diamante e de cerâmica
pura em operação de 10meamento de desbaste de compósito de 1ecido bobinado apresenta um custo de
usinagem por peça superior ao metal duro. Isto abrange as demais cerâmicas pois a relação custO/vida
desla$ ferramentas foram maiores que a obtida para cerâmica pura. Portanto, para uma relação de
custolheneffcio o metal duro moslrou ser a melhor de tcx:las as ferramenLa$ ell!laiadas o o torneamento
de desbaste do comp6süo.
Em operação de acabamento, a Fig. 3 mostra o comportamento da rugosidade (Ra) da peça em
relação ao comprimento de corte usinado para todas as ferramentas ensaiadas.
A rugosidade aumentou drasticamente nas peças usinadas com ferramentas cerâmicas, devidO aos
elevada> desgastes ocorridos nestas ferramentas. Enquanto que nas peças usinadas com ferramentas de
metal duro, CBN e diamante, a rugosidade apresentou menor taxa de crescimento. A ferramenta que
assegurou o melhor acabamento superficial foi o diamante.
Tabela 11. Tam~ a cuat oajpeça da ualnagam para aa farramantaa
Descrição
MO KJO
VB = O,Smm
Cer. pura
VB = 0,5 mm
Diamante
VB =0,5 mm
Tempo de corte da peça te (min)
5,78
5,78
5,78
Tempos improdutivos li (min)
3,7
3,7
3,7
Tempo de troca ferram. ln (min)
0,5
0,5
0.5
Número de peças do lote Z
10
10
10
J. R. Ferreint e N. L Cupini
117
Cer. pura
MDKlO
VB = 0,5 mm
VB=0,5 mm
Diamante
VB=0,5mm
Tempo total confec. peça ft (mln)
11 ,43
12,60
9,54
Vida das ferramentas T (min)
1,44
0,91
25,73
Descrição
5
Número de afiações n 1
8
8
6
Salário do operador Sh (US$/h)
3,00
3,00
3,00
SaJãrio da máquina Sm (US$/h)
20,00
20,00
20,00
Custo da ferramenta Cf (US$)
5,84
28,32
93,00
Custo ferramenta/Vida Cft (US$)
1,06
3,54
30,55
Número de vidas da ferramenta n2
13,95
Custo afia.ç áo ferramenta Caf (U$$)
Custo porta-ferramenta Cpf (US$)
131,00
131,00
131,00
Custo de usinagem/peça Cp (US$)
9,63
29,40
10,52
Isto acontece devido as de laminações das fibras e os sulcos formados pela presença de resina no material comp6si to. Sendo que estes sulcos e dela mi naçOes aumentam de intensidade com o crescimento ele
desgaste da ferramenta (Takeshita e Webara, 1985). Portanto, quanto mais resistente ao desgaste é a
ferramenta, melhor seu comportamento em relação ao acabamento da peça.
1 6.---------------------------------------------~
12
K10
6
-3o
~
•o
PCO
4
Vc " 320 m/rftin .
t • O. t """' rot
a, • hiM
g' 2
III:
at.~
tecido bobtnodo
ESC.:5 : 1
118
Esoolha da Ferramenta no Torneamento de Resina Reforçada oom Fibras de Vidro e Carbono
Observa-se também que as médias dos valores da rugosidade do compósito apresentaram uma taxa
de crescimento aproximadamente constante oom o comprimento de corte. Explica-se este fato pelo
mecamsmo de desgaste da ferramenta desenvàvido na usinagem de compósito reforçado oom fibras.
O desgaste neste caso é Jredominantcmente abrasivo, ocorrendo uma espécie de polimento somente oa
supcrfTcie de folga da ferramenta (Fig. 4a). Neste caso não são observados os desgastes de cratera e de
entalhe, presentes nas ferramentas de metal duro quando da usioagem de aços dúteis (Fig. 4b), onde há
grande variação na rugosidade da peç.a (Ferroresi, 1977).
SUPERFICIE
DE
SA(DA
FLANCO
1 . Torneamento de meteria! eompóelto ·Metal duro K10
DESGASTE
DE
CRATERA...,
DESGASTE
OE
ENTALHE"
b. Torneernemo de açoe dOtei• VJAE/ASKT 1~)- Metal duro P20
Fig. 4 Meeanl•~ de <IM9aete da
~anta
de metal duro
J. R Ferreira e N. L Cupini
119
A Fig. 5 mO>tra a variação da força de avanço com o comprimento de corte. Para alguns materiais
de ferramentas como as cerâmicas, o crescimento da força é acentuado. Para o diamante e o CBN, a
força de avanço permaneceu quase constante. Justifica-se este fato pela intensidade do desgaste sofrido
pelas ferramentas, confirmando os resultados obtidos na Fig. 2
O valor da força de avanço medida no torneamento do compOOito com ferramenta de diamante foi
de 5 kgf. Este valor de força de avanço é da mesma ordem de grandeza das obtidas no corte de aços
(Ferraresi, 1977). O material compósito apesar de apresentar baixa densidade requer grandes forças de
usinagem, provavelmente devido às suas boas propriedades mecânicas.
3~
_30
'ti.
N
...
w
u
cn
~
-2 5
La..
~2 0
c
o
>
4(
15
-3
..
o
11..
5
o
2
10
1
14
CompriiMnto de Corte lc ("'
16
1t
1 o2l
1
20
ESC.:5: 1
Fig. 5 Força de avanço ver.u. c omprimento de corte
No torneamento do compósito de tecido laminado observoo-se a influência da velocidade de cone
no desgaste da ferramenta de metal duro, como também a variação da rugosidade com a direção das
fibras oo laminado.
A Fig. 6 mostra a evolução do desgaste da ferramenta de metal duro com a velocidade de corte.
Observa-se que até a velocidade de 320 m/min, o desgaste da ferramenta varioo pouco. De 320 m/min
a 450 m/min houve um rápido crescimento do desgaste, devido ao maior atrito ferramenta-peça e à
menor resistência ao desgaste da ferramenta em temperaturas mais elevadas, onde ocorreu inclusive a
queima da resina fenólica (Hasegawa, 1984)
120
Esoolha da Ferlllmenta no Torneamento de Resina Reforll8da oom Fíb18S de Vidro e Carbono
1,2r----------------------------------------------------,
E
E 1,0
f "' 0,1 mm/ rot
op = 1 mm
....
Ferramenta : Metal Duro
Mot .: tecido laminado
o11)
o
u
c
o
~
~
0,4-L--- - - -- ---c>------
0
01
•
~ 0,2
75
125
17~
2
Velocidade de
+- lc • 300m
FI~
~
3
275
Corte
Vc
~
( m / min)
+ lc:" 600m
375
4 5
ESC .: 1: 5
l
6 o..gut. de nanco ve,..ue velocidade de cort.
A fig. 7 mostra a variação da rugosidade com o ângulo da di reção das fibras (a) em torno de 45 e
225 graus em relação à direção de corte, fK)iS nesta posição a tendência da ferramenta é de dei aminar as
fibras, deixando a superfície do compósito mais áspera. Na Fig. 7, também pode ser observado uma
forte influência do avanço (f) na rugosidade da peça, de maneira similar como na usinagem dos metais
(Wunsch e Spur, 1988).
Para finalizar, observou-se a influência da disposição das fibras do mat.erial compósito no desgaste
da ferramenta . A Fig. 8 apresenta as curvas de desgaste das ferramentas de metal duro em função do
comprimento de corte. O comportamento dos desgastes das ferramentas são comparados entre os
torneumentos dos compósitos de tecidos bobinado e laminado.
J . R. Ferreira e N. L, Cupini
121
v8 • co- o.os , ••
12
~· 200M1Min.
ap• 1111lft
FerraNnta PCO
Mat: tecido laMinado
ln
10
8
-36
E
~
o
a::
••
.,
.,o
••o
2
01
~
a::
45
90
Anoulo •
[Jf • 0,1 "'"' /rot
135
180
225
Poaiçlo do Fibra ex: (Graus)
UJ f •
0,2 "'"'' rot
ESC.: 1:3
Ot • 0,3 "'"' /rot
Fig. 7 Rugoeld•d• R. em ~o do engulo de poelçlo d . . tlbr. .
122
Esoolha da Ferramenta no TorM8lllento de Resina Reforçade oom Fílllas de VIdro e Carbono
1,4
1,2
e
e
otn
III
>
1,0
~
o
cj
(/')
o
0,8
w
11.
Vc =320 mlmin
f =0,1 mm / rot
ap= 1 mm
F e rromenta :
Metal ouro K10
•
-; 0,4
&
•
~o.
o.
8
Comprimento de
-&Tecido Bobinado
Fig. 8
o..g ..u
2
Corte lc ( m
14
x 1cf1
18
ESC.:5:1
+ Tecido Laminado
de ftanco veraue comprimento de cone
A ferramenta de metal duro se desgasta mais rapidamente no corte do compósito de tecido
laminado do que no bobinado. A maior abrasividade do tecido laminado deve-se ao fato da variação da
orientação das fibras em relação à direção de corte no torneamento, onde se tem a cada instante uma
nova pa>içáo da fibra em relação à ferramenta.
Conclusões
• Em função dos resultados obtidos pode-se tirar as seguintes conclusões:
• Conforme as considerações efetuadas nas Thbelas 1 e ll, o uso de ferramentas cerâmicas e de CBN
no torneamento de compósito não é viável do ponto de vista técnico e econômico;
• A rugosidade da peça aumenta de maneira progressiva com o desgaste de flanco da ferramenta
podendo levar à dei aminação das fibras;
• Existe uma grande dispersão na medida da rugosidade do compósito, devido aos defeitos
originada; por delaminação e sulcos na superficie da peça;
• Em operação de acabamento, somente a ferramenta de diamante pode assegurar uma qual idade
superficial com rugosidade R~ na faixa de 2,5 a 5,0 IUll;
• Apesar do compósito ser um material leve mediram-se forças de corte nos patamares alcançados
em corte de aça;;
• Devido à presença do desgaste de flanco e à inexistência de desgaste de cratera na ferramenta, a
força de avanço aumentou notavelmente com a evolução do desgaste;
123
J. R FeiT'eira e N. L Cupinl
• O compósito de tecido laminado é mais abrasivo à ferramenta do que o compósito de tecido
booinado, e
• Em operação de desbaste, a utilização da ferramenta de metal duro apresentou-se como a melhor
solução, pois apesar de sofrer desgaste supedor ao diamante, resulta numa relação custo,tbeneffcio
mais vantajosa e dispensa cuidados operacionais especiais.
Agradecimentos
Ao CTA • Centro Técnico Aeroespacial, pelo fornecimento do material compósito. À Dffer
Diamantes Industriais, Sandvík. SKF Ferramentas, Brasimet e Winter do Brasil, pelo fornecimento de
ferramentas. À FINEP pelo apoio financeiro e ao aluno Frederico Ueda pelo auxílio dado 00$ ensaios.
Referências
Ferraresi, O. ( 1977) "Fundame ntos da Uslnagem dos Metais", Editon. Edgard Blucber, São Pulo.
Hu9wa, Y. ( 1984) "Ciuracteristlcs olTool Weu la CuuiagGFRP", 5th Laternational Conference on Production
Englneering, pp. 185-190, Tokyo.
Koplev , A. e Lystnp, A. (1983) "Tlae CulliDg Procesa, Cblps, and Olttiog Forces ln Macbloing CFRP",
Composites, vol. 14-4, pp. 371-376.
Malhotra, S.K. (1989) " Hlg.ll Speed Steel Tool Wear Stodles in Maclaiolng of Glus Fiber-Reio(orced Plutics",
Wear, vol. 132. pp. 327-336.
Relllhan, TJ., (1988) "Eaglaeeriag Mat.e rials Haoci)oolc", Composiles, \\li. 1, ASM lnterutional, Ohio.
Takeshlta, H. e Wehara, K. (1985) "Cuttlog Mechanism oC Some Composlte Materiais", Second lnternational Metal
Canlng ConCereace, Tokyo UDiversity, Japan.
Wunscb, U.E. e Spur, G. (1988) "Turning of Fiber-Rei nforced Plastics•, Manufacturing Review, \bl. l, pp. 124129.
RBCM - J . of the Braz. Soe. M.chanlcal Sclancu
Voi. XIV - n"2·1993pp. 124-135
ISSN 0100·7386
Printed ln Bnuíl
Mechanical Systems ldentification through
Fourier-Series Tlme-Domain Technique
Identificação de Sistemas Mecânicos Através das Séries de
Fourier- Técnica no Domfnio do Tempo
G. P. Melo
~h. Eng. Oepartment
FEIS/UNESP
15378 Ilha Solteira - SP
Brazll
V. Steffen Jr.
~h.
Eng. Department
Univ. Fed. Ubet1ándia
38400 Uberlãndia • MG
Bnuil
Abetract
A time-domain tecllnique based on Fourier series is used in tllis work for the pa rameters identific.ation of
mechanic.al systems with mulli-degrees of freedom . Tbe metbod is based o n the ortbogon.ality property of lhe
Fourier series which enables lhe integration of the equations of motion. These equations are then converted to a
linear algebraic modellhat is solved to obtain the uoknown parameters. Tbis way tbe method can be summariz.ed
as fo llows:
I . Expansion of the input aod ourput signals in Fourier series;
2. lntegratio n of lhe equations of motioo using ao operatiooal malrix lo integra te the series;
3. Tbe parameten; are calculated through a least-square estimation melhod.
K~ywords: para meter identincation, modal analysis, time-domain techoique, Fourier series.
Resumo
Apresenta -se neste trabalho um m~todo para identificação de Sistemas Mcclnicos com vários graus de liberdade
ope rando no domfnio do tempo. O m~todo baseia-se na expansão das funções de excitação e de resposta do
sistema em termos de séries de Fourier e na transformação das equações diferenciais do movimento em equações
alg~brieas por meio de integrações sucessivas e da utilização de uma matriz operacional para integração das
funções que formam aquelas séries. Desta forma, o m~todo pode ser sumariudo em três etapas fundamentais:
I. E.xpansão da excitação e da resposta em séries de Fourier;
2. Integração das equações de movimento e emprego de uma matriz operacional para integração das s~ries de
Fourier;
3. f.stimativa dos parâmetros pelo m~todo dos mlnimos quadrados.
Palavras-chan: identjficação de parâmetros, análise modal, t~nica oo domlnio do tempo, Séries de Fourier.
lntroduction
l n recenl years a great number of methods for the identification of dynamical systems using
orthogonal functions have been developed. Most of applications are however devoted to control
problems. Walsh (Chen and Hsiac, 1975), Block Pulse (Jiang and Schanfelberger, 1985) and Fourier
(Chun, 1987) functions and Chebyshev (Paraskevopoulos, 1983), Jacobi (Lin and Shih, 1985),
Laguerre ( Hwangand Shih, 1987) and Hermite (Paraskevopoulos aod Kerkeris, 1983) polynomials are
examples of such ortbogonal functions used to identify linear-time-invariant lumped parameters
systems. l n a previous work a metbod based on the expansioo of the input and output time functions in
terms of Fourier series was presented by one of the author5 of lhis paper (Steffen, 1991) focusing
mechanical vibrarions problems.
ln the presented paper the method is tested for different mechanical engineering applications:
dynamical seaJ; three d.o.f. sys1em; external force identification. Numerical simulatlons and
experimentaltests confirm the applicability of the method The influence of noise was analysed.
Manuscript receflled: January 1993. As&oàate Technical Edilor: AQenor de 10iedo Fleury
125
G. P. Melo and V. Stelfen .k.
Baste Governing Equations
Tbe method is now briefly presented for linear time-invariant mechanical systems with N degrees
of freedom. Fc:r lhose systems a widely used model is
[MJ {x(t)} +[C] {x(t)} + [KJ {x(t)}
- {f(a)}
(1)
were [M], [C] and [K] are respeaively the inertia, damping and stiffness N-order matrices;
T
{x(t)} - {x 1 (t) x2 (t) ... x0 (t)}
is
th e
vector
of
displacements
and
{f(t)} T- {f 1 (t) f 2 (t) ... f 0 (t)} isthevectorofinput forces.
This equation can be rewritten in the following state-form equation:
{x (t) }]
[
+ (AhNx2N
[{x (t) }]
{x(t)} 2N><l
{
= (B)2NxN f(t)
}
Nd
(2)
{x(t)} 2Nxt
where
(O)NxN
(I)NxN
J
(A)2NM2N[
[M) -I [K) N ;~
and
[M) -1 (C) N" N
[ B) 2NxN _ [(0) N xNJ
l
(M) Matrix [A) contain ali tbe dynamic cbaracteristics of the system: eigeo-values (natural
frequencies and modal damping factors) and eigen-vectors can be extracted.
After double integration Eq. (2) becomes
f{x(,;)}d-c]-[
t
c
{O} ]-[{x(O)}]t+
{x(O)}
{x(O)}
{X ( t)}
t t
J{x(,;)}d't2
+[A) oo
tr
-[A) [
{O} ]•- (BJ
{x(O)}
{II {f(,;) }d't2}
(3)
CC
I{x ('t)}d't
o
x;(t) and f;(t),l=l, N are now expaoded in Fourier series:
X;(t)- {X;} {0(t)}, Í= l,N
f;(t) -
{F;} {0(t)}, i- l,N
(4)
126
Me<:hank:al Systems ldentit;c.ation Through Fourler-5eries Time-DomaJn Technlque
w here
T
{<I> ( t) } , 2 - [ 1, cas ( 2Jtt/ T) •..., cas ( 2Jtrt/ T). sin (21tt/ T), ...
... , sin21trt/T] . ( r2- 2r + 1)
T
{ X } r2-
•
•
and
[xo,XI' .... xr'x l, ... ,x rl
T
{F}r2- Uo.ft'""'fr,r
l .....
r rl
lt is possible to wríte:
T
{x(t)}Nwt • [X)Nxr2{<J>(t)}r2MI
{ r (I) } N
X
and
1 - [ F] ~ X r2 { <J> ( t) } r2 X I
(S)
where
[X) • [ {X 1} {X 2 }
...
{XN})
and
[ F] • [ { F 1} { F2} ... { FN} )
Substituting Eq. (5) into Eq. (4) gives:
(X)T(P]l
[
-[
{x(O)} {e}T(P)
+A[[X)T[P]~
- (A) [
2
+
{x(O)} {e}T+{x(O)} {e}T[P 2Nxr2
(X]T J2N xr2
[X)T(Pd
J
Nu
2
J
{O}
{e} T[P) • (BJ
{x(O)} 2Nxr2
[F)T[P] ~xr2
(6)
l n thi s equa1ion 1 was written as 1 - {e} T [ P) {0(t)} and [P) is the operational matrix of
integratíon whí ch is determined from tbe following integml property of the basís vector { 0} of tbe
orthogonal scries:
J. .... J { <I> ('t)} (d't)
o
0
(7)
[P]"{<J>(t)}
-
o
n times
and
T/2
(P)r2xr2 •
I
---1{O}
(O]
---
---- ---J --[O] r ><r
- rI - T / 21t [-I r xr
T / 2Jt { e}
I
where
- T
-T/ :n:{e}r
Tl2x{l}rxr
[O]rxr
G. P. Melo and V. Steften .X.
127
o o ... o
o 112 o ... o
1
1
112
{ê}r- l/3•[il,.r1/ r
O O 1/3
... o
O O
O 1/r
O
Fa- a given set of initial cooditions Eq. (6) can be rearranged as
[A] [O)- [B] [G) - [E]
(8)
wbere
[O) -
l
J
[X] T [P] 2
X)T[P) - {x(O)} {e}T[P
[G] -
UFJ
T
[P] ~
'
2Nxr2
and
J
({x(O)}{e} T -[X]){P]
T
[E)[
{x(O)} {e}T+ {x(O)} {e}T[P) -(X)T
2Nxr'2
Then
(9)
wbere
[H] - UA) ·. _ [B'l
lJ2Nx3N
aod
(JJ _
Í[oJl
UGU3N xr'2
A Jeast-square technique is used to obtain the unlcnown parameters (natural frequencies, modal
damping factors).
(H) T - ( [J] (J) T) -1 [J) [E] T
(lO)
Tbe equations above can be easily adapted for lhe identification of the excitation forces.
Applicatlons
Three applications are now presented to demonstrate the performance of the above metbodology.
128
Mechanical Systems ldentification Through Fourier·Series llme·Domain Technique
1. The method was firstl y applied to a two degree of freedom system which represents a dynamic
seal as shown in Fig. 1.
kyy, kyz.kzy,kzz
Fig. 1 Dynamlc auJ
The ~uation of motion which represents the dyoamics of the seal is:
(MJ {x (t)} + (CJ {x (t)} + (KJ {x(t)}
~
{f(t)}
(11)
whcre
[K] ·
{ x
(r) }
~::
=
-:.:]
ÍY (1)l,
Lz(t)j
aoó
{r(,)}~ fr
y(
>l
1
~z(t)J
where Myy=26.200 Kg, Mu=26.200 Kg, Cyy=C:u=l124.000 Ns/m, Cyz=Czy=720.000 Ns/m,
Kyy=Kzz=468430.000 N/m, Kyz=Kzy=42811.000 N /m, fy(t)=fz(t)=y(O)::z(O)=O .OOO,
y-
1000, i - 0,000 and 0,00 s t s 0,08s.
129
G. P. Melo and V. Stelfen .k.
The response { x ( t) } is obtained using a fourtb-order Runge-Kutta technique. {x ( t) } and
{ f ( t) } are expanded on Fourier series to permit the identification of the model ptrameters p:esented
in Thble I.
Table I - Theoretlcal end ldenttled peremete,.
• • percentual magnltudt dHrerencee
Tbeoretical
Values
ldentified
Values
Relative
Differeoce
w 0 (rd/s)
121 .689
120.418
1.04
rd/s)
119,012
118.010
0.84
0,209
0,199
-25.384 +
1119.012
-23.966 +
1118.010
1.04*
8gen y2
Vector& yl
0.000 +
11.000
-0.061 +
i 1.017
1.88*
w 0 ( rd/s)
147.536
147.826
0.20
wd (rd/s)
146.493
146.824
0.26
0.119
0.116
-17.516 +
i 146.493
-17.184 +
i 146.824
0.20'"
0.000.
11.000
-0.039.
i 1.038
3.80*
Parameters
Mode
wd (
~
8gen
Values
2
!;
8gen
Values
Elgen y2
Vectors yl
2. A real three degree-of-freedom system as sbown in Fig. 2 was used for identificatioo purposes.
The equatioffi to represent tbe undamped dynamic !rodei of tbe above system are given by:
-
2
2
lxa + ( 4KL + 2KaL ) a - O
-
2
lytl +4KB tl =O
Mi+ ( 4K + 2K8 ) z - O
wbere
lx, ly and M are tbe lnertia and mass coefficients;
K and Ka are stiffness coefficients;
L and B are lengtbs, and
a., tl, z are angular and linear displacements
(12)
Meçhanical Systems ldenlifi<:ation Through Fourier·Series Tlme·Oomain Techníque
130
1·Ampllfier
2·Signal Analyser
3·Micro computer
4-Piotter
z
y
Ag. 2
a) 3 D.O.F. T. .t rlg acheme b) 3 D.O.F. Model
Three accelerometers were used to measure the system response. The signals were manipulated to
obtain o.(t) , j}(t) and z(t). Three sets of 1024 points-sampling were used. Tbe sampling time was T=0.7
seconds.
Tbese accelerometers were placed at points D, G and F of the structure
Thble 2 shows theoretical and identified paramete.rs.
a. P. Melo end V. Stellen .l-.
131
'nl~ 2
- Theoretcal and ldentfled parameten
• • parcenul magnltudl dlfferancM
Theoretical
ldentified
Rela tive
~lues
'hlues
Difference
w 0 ( rd/s)
175.889
174.671
0.69
(rd/s)
175.899
174.656
0.71
0.000
0.001
0.000 +
1175.899
-0.160 +
1174.620
Mode
Parameters
wd
~
Elgen
Values
2
Eigen
Vector&
yl
yl
1.000 +
10.000
1.00010.000
Eigen
Vectors
y2
yl
0 .000+
10.000
0.037 +
10.017
Eigen
Vector&
y3
yl
0.000+
10.000
-0.03210.053
w 0 ( rd/s)
219.908
221 .056
0.52
wd ( rd/s)
219.908
221.059
0 .52
0 .000
0.002
0 .000 +
i 219.908
-0.650 +
i 221 .060
~
Elgen
Values
3
o.~
0 .52•
Elgen
Vectors
yl
y3
0.000+
10.000
-0.001 +
i0.004
Eigen
Vectors
y2
y3
0.000+
10.000
0 .009 10.008
Elgen
Vectors
y3
y3
1.000 +
10.000
1.000 +
10.000
w 0 ( rd/s)
304.481
290.339
4 .64
w d (rd/S)
304.481
290.338
4.64
0.000
0.003
0 .000 +
i 304.481
-0.092 +
i 290.340
~
Elgen
Values
Elgen
Vector&
yl
y2
0.000 +
10.000
-0.020+
10.006
Elgen
Vectors
y2
y2
1.000 +
10.000
1.00010.000
Elgen
Vectors
y3
y2
0.000 +
iO.OOO
-0.03910.064
4.64•
132
Mechanlcal Systemsldentífication lhrough Fourier.S.ries llme-Oomain Technlque
3. The las! applicarion deals wilh force idenriflcarion of a singie-degree-of-freedom system where
M=lO.O kg, C=40.0 Ns/m and K=lOOO.O N/ m. For simularion purposes the system response for ao
excitati on force f(t)=50.0 N ( x (O) - :X (O) - 0,0) was calculared and is given by:
x(t)- e-
21
(13)
(-0,050cos(9,798t) -O,Ot0sin(9,7898t)) +0,05
Using Eq. (13) 1024 points were generated for 0,00 s t s 1,50 seconds aod the Fourier coefficienrs
w ere calculated.
The results are presented for different siluarions:
A - Parameters idenrification reraining five terms in Fourier expansions (Table 3)
B - Parameters idenlificarion reraining ten terms in Fourier expans!ons (Thble 4)
C- lnflue.nce inrroduced in the identi1ied parameters when considering the introduction of 10% of
random noise in the answer x(t) - (Table 5)
D - ldentification of the exciration force f(t) - Fig. 3
Tabr. 3 - TMofetieal anel ldenfllad para~tare (r•S)
Theoretical
Vcllues
ldentified
Values
Rela tive
Difference (%)
a.
4.000
4.054
1.34
ao
100.000
100.256
0.25
bo
0.100
0.100
0.00
d (m)
0.000
0.007
d 1 (m/s)
0.000
-o.001
M (kg)
10.000
9.980
0.20
C (Ns/m)
40.000
40.455
1 .13
1000.000
1000.555
0.05
0.200
0.202
1.00
w (rd/s)
10.000
10.013
0.12
wd (rd/s)
9.798
9.793
0.05
'
0
K (N/m)
~
0
<l. P. Melo and V. Steften .W.
133
Teblt 4 • Theoretlcel end ldenlfted peremett,. (rw10)
Theoretical
Yalues
ldentified
Yalues
Relative
Difference (%)
a.I
4.000
4.044
1.09
ao
100.000
100.163
0.16
bo
0.100
0.100
0 .00
d0 (m)
0.000
0.004
d 1 (m/s)
0.000
-0.001
M (kg)
10.000
9.990
0 .10
C (Ns/m)
40.000
40.398
0.99
K (N/m)
1000.000
1000.628
0.06
0.200
0.202
1.00
(rdfs)
10.000
10.008
0.08
wd (rd/a)
9.798
9.801
0.3
!;
ó>
0
Tables 3 and 4 permit to observe that increasing the number of retained terms in Fourier series
decrease the relative difference between tbeoretical anel idenlified parameters.
lable 5 • Theoretlcelend ldentlfted peremetet'• (r=10} • 10% of rendom nol••
Theoretical
Values
Identified
VàJues
Re la tive
Difference (%)
a.I
4.000
4.026
0.64
ao
100.000
100.583
0.58
bo
0.100
0.105
5.00
d 0 (m)
0.000
0.050
d 1 (m/s)
0.000
0.016
M (kg)
10.000
9.569
4 .30
C (Ns/m)
40.000
38.228
4.44
K (N/m)
1000.000
952.485
4.75
!;
0.200
0.201
0.50
w 0 (rd/s)
10.000
10.029
0.29
wd (rd/s)
9.798
9.824
0.26
Mecllanical Systems ldentification Through Fourier·Series lime·Oomain Technique
134
Remarlc: The random noise introduced in the answer signal is obtained as follows:
1) The RMS of the sampled data x(t) is obtained and the value NR=lO% RMS is detennined
2) N random numbers are produced each one of them is associated to each element of x(t)
3) The product NR by the random numbers is evaluated, with (0.0) average and (1.0) standard
deviatioo.
Fig. 3 shows the results retaining fíve aod ten tenns in the Fourier series.
60.0
FORÇA(Nl
Fig. 3 Excltatlon forc• ld•ntlf lca11on
Conclusion
This paper presented a time-domain tecbnique for identification based on the Fourier series
expansions of the system excitation and response. The integration of the series is made using an
operational matrix. Simulation and experimental results obtained for different mechanical engineering
systems show that the method is very simple to be implemented and very little sensitive to noise. It was
observ.ed that lhe quality of the results is improved when increasing the number of harlllOnics talcen in
the Fourier expansions.
References
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Cbun, H. Y. (1987) "Sysrem ldentification via Fourier Series", lnternational Journal of Systems Science, 18: 1191·
1194.
Hwang, C. and Sbih, Y. P. (1982) "Paramerer ldenrification via Laguerre Polynomials", lnrernarional Journa'l of
Sysrems Science, 13: 2()9-217.
135
G.
P. Melo and v. Steffen .k.
Jiaag, Z . H. nd Sclanfclbergcr, W. ( 1985) "A Ncw Algorltllm for Siagle-laplt-Siaglc-Oitpal Systcm
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Melo, G. P. (1992) "ldelltificação de Sistemas Mecia.loos aiDV~s do M~todo das ~ries de Folrier- Um Mttodo ao
DomfnJo do Tempo", Tese de Mestrado - UnlveraldJide Federal de UberiJndla, 114p.
ParaslcevopouJos, P. N. and Kerkeris, G. Tii (1983) "Hennite Series Approach to System ldeali.fication, Aaalysis
ud Optfmal Cootrol", Proc. 6th lat. Symp. on Methods and Appl. of Measurement and Cootrol, Acta Press,
Canada, 1: 146-149.
Parask.evopoulos, P. N., Sparis, P. O. aad MourouiSos, S. G. (1985) "The Fourier Serles Opcratiooal Malrlx of
lntegration", Intemational Joumal ofSystems Scieoce, 16: 171-176.
Panskevo poulos, P. N. ( 1983) "Chebychev Serles Approacll to System ldeatlrtcatloa, Analysis and Optimal
Control", Joumal of lhe Fraalcüo l11Stltute, 316: 135-157.
Steffea Jr., V. ud JUde, O. A. ( 1991) "Aa ldeatificatioa Metltod of Malti-Degree-of-Freedom Systems Based oa
Fourler Series", The latematiooal JouraaJ of Aaalyticalaad Experimeatal Modal Aaalysis, pp. 271-278, ocL
RBCM - J . of tt. Bru. Soe. U.Chllnlcal Sc:lenc. .
Vol XV·n'2 · 1993·pp.136·149
ISSN OtG0-7386
Printed in 8ra2il
Updating Rotor-Bearing Finite Element
Models Using Experimental Frequency
Response Functions
Joe6 Roberto F. Arruda
Department ol Compulational Mechanics
State Uníversíty ol Campinas
Campinas 13083-970. Bra.zil
Marcus A. V. Duarte
Department ol Medlanical Engineering
Federal Unlwrsity ol Uberlândíe.
Uberlándia 38400-<lOO, Bra.zil
Abstract
Most Finile Elemen l model updatiog techniques reqwre experimental modal parametel'3, which are obtained
from modal tests. ln tbe case of real rotor-bearing systems ii is frequently unfeasible to perform complete modal
tesls dueto difficulties such as exciting a rotating shafl and dealing with bearing nonlinearlties. ln Lhis paper, the
po!~.~ibility of updaling the Finite Element model of a rolor-bearing syslem by estimating the bearing stiffness and
damping coefficienls from n few measured Frequency Response Functions is investigated. The proposed model
updating procedure consists of applying a nonlinear parameler eslimation techoique to correcta few PE model
parametel'6 cbosen a priori. The parameter estimation problem is presented in its general forro, from wbich the
various metbods can be derived, and tben particularired to a nonlinear least squares problem wbere the objective
function to be minimized is lhe sum of the square differences between the logarithm of lhe theoretical and
experimental Frequency Rcsponse Function (FRF) magnitudes. The importance of using the logarithm of lhe
mngnitudes and its equivalence lo lhe maxlmiution of the correlation between lhe tbeoretical and experimental
functions is pointed out. A closed form expression for the sensitivity matrix of the logarithm of the FRF
magnitudes is dcrived. and ii is shown that tbe finite differences approximation produces sufficieolly accurate
results ai a m uch lower computational cosi. The issues of ideotifiability and parameter estimalion ·errors are
addressed. A numerical example of a flexible rotor supported by two ortho tropic bearings is presented to
ii lustrare the proposed melhod.
lntroduction
Although the tools for theoretical dynamic analysis are increasiogly sophisticated and refined
models can bc treated ato rclatively low oost and high speed with modero computers, dynamic tesling
is still of forcmost importance dueto the difficulties in predicting the dynamic behavior of certai o
structure compooents, e.g., mecbanical joints. Rotor systems always include a nonlinear,
nonconservative kind of joint which influences dramatically its dynamic behavior, namely the
bearings. Fluid/structure interoction is also very difficult to model. ln both cases, linearized models
moy be obtained by lhe experimental estimation of some linearized parameter values whicb describe
the syStem behovior for the operating dynamic lood leveis. This estimation is dane by comparing
theO!'etical model predictions with measured data and updating Lhe former so that tbe predictions fit lhe
experimental data. Different approaches exist to the problem of updating theoretical models of
structures (lmr.egun and Visser, 1991). The ma>t suitable when a small set of physical parameters of a
FE moclel are to be corrected is tbe so-called inverse sensitivity approacb. lnverse sensitivity metbods
are usually applied in the modal domai o (Lallement and Zhang (1988), Natke (1988), and Carneiro and
Arruda ( 1990)), andare usually referred to as indirect method<;.
l n the case of industrial rotor systems, however, it is extremelly awk:ward, wben feasible, to
perform a lhorough modal test aod extract modal parameters. This is rel ated to experimental
difficulties in exci ling and measuring the responscs on a rotating structure as well as to modal
parameter estimation problerns dueto the variation of the modal parameters with the rotating
frequency and to lhe noolinear behavior of bearings and seals. Hence, in tbe case of rotor syst.ems, it is
desirable to use model updating techniques that need only a few measured system respooses, those
Menuscript received: July 1993. Associate Tedlnical Editor: Agenor de lOiedo Aeury
137
J. R. F. Arruda and M. A. V. Duarte
whicb can be easily measured. Using dynamic responses in the updating procedure is usually referred
to as tbe direct approach (Natke, 1986). When· using direct ínverse sensltivity rotor Flnite Elements
{FE} model updating methods, unbalance responses and Frequency Response Functions (FRFs) are
examples of experimental dynamic response functions whicb can be used.ln previous papers (Arruda,
1987 and 1989), one of tbe authors proposed the FE model updating of rotors based on the unbalance
response curve fit. The method was shown to work wilh simulated data, but the problem with its
implementation is lhe experimental difficulty in measuring steady-state unbalance respooses near lhe
rotor criticai speeds.
When updating a FE model, one has to choose the theoretical parameters to be corrected. They are
either k.nown a priori or have to be "localized". Tbe localizatlon problem is usually based on a
sensitivity analysis (Blakely and Walton, 1984). ln the case of rotor-bearing systems, the rotor i s
usually well modeled, while the bearing and seal stiffness and damping coefficients are quite poorly
predicted tbeorerically. Heoce, the parameters whlch must be estimated when updating tbe modelare
k:nown a priori. ln this paper, the passibility of updating FE rotor models by Lhe estimation of bearing
stiffness aod damping coefficients using measured FRFs is investigated. ln practical applications, the
FRFs may be obtained by exciting a few accessible shaft locations with ao impact hammer. The FRF
curve fit is implemented using a nonlinear parameter estimation metbod whicb, in its most general
forro, is a Regressive Maximum a Priori metbod and, in its simplest form, when there is no information
about either the statistics of the estimated parameters or the statistics of the experimental data, is the
Ord.l nary Nonlinear Least Squares metbod. Tbe importance of using the magnitudes of tbe FRFs in
logarithmic scale is pointed out and it is shown that this improves the smoothness of the objective
function, whicb becomes similar in shape to the complement of the correlation coefficienr between
theoretical and experimental FRFs as a functlon of the parameters.
1\vo methods for calculating the sensitivity matrix- finite differences and analytical formulationare implemented and compared. The problems of identifiability and error estimates are adressed and
the techniques used to reduce the computational efford are discussed. A numerical example consisting
of a flexible rotor supported by two ortbotropic bearings illustrates tbe use of tbe proposed
methodology.
Nomenclature
B ( w) ..dynamic stiffness matrix
= viscous damping matrix
c
Cyy
.
•
.
Czz "' bearrng hneanzed
damping coeflicients
D(Q) =generalized damping
matrix including
gyroscopic effect
any dynamlc response
F
function, e.g .. a FRF
predicted dynamic
FI
response function
elemenl of vector F
FI,
1
measured dynamic
Fx
response functlon
scaled measured
fx
functlon ln mtudmum
correlalion method
G
coriolis matrlx
(k) (gyroscoplc effect)
h
= step taken ln the search
direction to minimize
J (p)
H ( w) =frequency response
matrlx
H .. ( w) ,.frequenc~ response
'J
function (F F)
HR = real part of a FRF
=
H! = lr:t-fnary part of a FRF
I =
-1
~ = imaginary part of the
sensltivity
J ( p) = objective function of the
parameters p
J ( p) = objective functlon for the
· estimated parameters p
stíffness matrix
K
kyy
bearing llnearized
k'lZ
stiffness coefficients
mass matrix
M
ne number of elements of
nP
p
p
Po
Pk
p(k)
~~c:r,~~~f
elements of
vector p
vector whose elements
are the parameters to
estimate
= estimate of the
parameters p
= inllial theoretlcal guess
values for parameters p
model parameter, kth
element of vector p
= estimate of lhe
parameters at the kth
=
iteralion step
!R = real part of the sensilivity
s<k> = sensitivity matrix aIlhe
kth iteration slep
S.. = element of the sensitivity
IJ
matrix
weighting matrix for the
WF
experimental data
weighting matrix for lhe
model parameters
v ar [p] =cVariance of lhe
parameter eslimates
scaling factor used ln the
ll maximum
correlation
method
y
relativa confidence in lhe
lheoretlcal model
increment ln the
ôj
parameter values
âpj = increment in the
parameter values for
derivativa computation
âp{k)
wP
search direction
w = frequency ln rad/s
Q
rota11ng frequency ln
Ql
rad/$
variance
138
Updating Rotor-bearing Anile Bement Models Ueíng Experimental Frequency Aesponse Functions
Model Updating Procedure
The estimation of model parameters that update a FE model is done by minimizing the difference
betwcen lhe predicted and the measured respooses (outputs) or excitations (inputs). Depeneling on that
the method may be classified as ao "output error" or an "input error" rnethod, respectively (Eykhoff,
1974 ). Frequency Response Functions (FRFs) are obtained by normalizing tbe systern responses with
respect to the excitation. As a consequence, estimating parameters from the FRFs cbaracterizes ao
output error method. Output error rnethods are not biased by additive Gaussian noise but they are
ilcrative anel hence prooe to convergence problems (Natlce, 1986). Puning the experimentally oblained
FRF magnitudes evaluated at differeot frequencies in a vector called Fx' and accordingly, the values
predicted by the theoretical model with a set of paramcter values, p, in Ft and given an initial set of
guess values for the parameters, pQ! the model parameters are obtained by minimizing the objective
function (Beck and Arnold (19TT), Blakely anel Wcllton (1984), and Natke (1988)):
(1)
ln lhis cxpre-<~Sion, WF is lhe weighting matrix of the experimental data, expresslng the rclative
confidence (statistically the inverse of the covariance matrix) in the experimental FRFs. WP is a
weigh1ing matrix cxpressing the relative coofidence in the theoretical rnodel parameters (of which the
stalistics is usuatly unknown). The scalar y expresses a global relative confidence between the
thcoretical and the experimental models. lf y - O and Wf is the inverse of the covariance ma1rix of
the experimental data, tlte solution leads to the Maximum Ltkelyhood (ML) estimator.lf y - 1, WF is
the inverse of the covariance matrix of the experimental data and W~ is the inverse of the covariance
matrix of lhe theoretical parameters, tbe solutioo is the Maximum a Posteriori (MAP) estimator (Beck:
and Arnold, 1977). Finally, if y--. oo the minimization leads to the trivial solution p - p0, and lhe
experimenml data is not t.aken into account.
The weighting matrices may be chosen empírically in practical situations, giving lhe dynamicist
the possibility of introducing any subjective a priori lcnowledge about the problem. When the statistics
of lhe data and model parameters are used, the approach is sometimes referred to as Bayesian (Natke,
1988). It is interesting to note at this poinl lhat the Kalman Filter formulation does not apply to the
model updating problemas stated,here because the model parameters are oot time-varying by
hypothesis, i.c., the FE model parame1crs are supposed to be constant intime. When ~-(p) is a
noolinear function oftbe p1rameters p,the MAP estimator p, which minimizes hs (f) of cq. 1, may
be shown to be derived from an iterative process where a new set of parameters p( • I) is obtained
from the previous values p(k) by the equations:
f>
(k + 1)
-P
(k)
+h
( k)
ap
(k)
(2)
(3)
where tJ. p (k) is the search direction and h (k) is the step taken in thc search directíon, which is
determined via a cne-dimensional search algoríthm. S(k) is the sensitivity matrix at iterative step k:,
given by tbe Jacobi ao of function f 1 with respect to the parameters p , i.e.:
(4)
139
J. R. F. Amlda and M. A V. Duarte
lntroduciog lhe one-dimeosiooaL search is a modification of the Gauss's alga-ithm. Different onedimensional search methods may ·be used. ln tbis paper the Box-Kaoemasu modíflcation as described
by Beck and Arnold (1977) was used . Algóritbrns for tbe solutioo of the nonllnear least squares
problem are well-known and wiJl not be detailed in this paper. The algorithm may be made recursive
wheo the measurements are made sequentially. ln this case the parameter estimatlon algorithm may be
seen as a filter instead of a curve fit. When the expsimental data are FRFs, vectors with complex
values at different frequencies in a given frequency range are usuaJiy obtaioed simultaneously using
Discrete Fourier Seríes metllods. lf ali the experimental data is available at once tbere is no oeed for
sequential estimation algorítbms. Tbe so-called direct method (Beck and Arnold, 1977) can help
saving memory space if the FRFs are measured sequentiaJiy but no rntermediate estimation of the
parameters is needed, i.e., ooly the finaJ update after aJI the experimental data bas been used is sought.
ln tbe numerlcaJ example treated in this paper 4 FRFs were used in each case and the direct sequential
method was used. The computation of tbe sensitivity matrix is discussed i o anotber section of tbis
paper.
Objective Functlon
ln the case of rotor FE model updating by the estimation of bearing s tiffness and damping
coefficients, tbere is usually a very low confidence in the initial guess vaJues (theoretical predictions)
and it is conveniem to make y .. O in Eq. 1. lt can beshown that tbe discretization of tbe FRF curve in
the case of lightly-damped structures may Iead to severa! problems of coovergence because of tbe
existenoe of many local minima in lhe objective functioo.
ln some of the cases simulated with tbe oumerical example described hereafter, there was no
convergence of the iterative noolinear search. 1t was observed tbat when convergence problems
occurred the first iteration step generally decreased tbe square error objective fuoction but it decreased
the correlation coefficieot between tbe theoretical and experimental FRFs. A maximum correlation
iterative searcb was sought to improve tbe convergence ln those case. A formulation for parameter
estimation by lhe maximization of the co.rrelation cüefficient was presented by Arruda (1992), where it
is shown that the sarne convergence improvement can be achieved using the logarithm of the FRF
magnitudes witb the nonlinear least-squares estimatjon. The objective function for tbe maximum
correlation technique is given by:
[F;F (p)] 2
J
(p) ~ 1- - -- - - (5)
MC
11Fxii 2 11Ft(p) 11 2
where 11 F1 {p) 11 denotes the Eoclidean oa-m ofvector F\(p). Figure 1 shows tbe plots of scaled
1
cross-sections of JL.S and JMC (p) for F in botb linear and ogarithmic scales. ln this case F is the
magnitude of one of tbe rotor system example FRFs and p is one of the bearlng stlffness coefficients
of wbich p0 is the exact value. As it can be observed in Fig. 1 tbe shape of Jl,.S (p) and JMc(p) is
approximately the sarne in tbe neigbborhood of the solution p0 when a logar1tbmic scale F is used.
Arruda (1992) showed tbat the relation between JLS ( p) and J MC ( p) was:
JLS
J
MC
- - -2
11 Fxll
(6)
140
Updating Rotor-bearing Finita Element Models Using Experimental Requency AeflponM Functions
,,
1,00
~
'
~-
r'
\
~-
\
'
I
I
I
I
I
'I
'I
I
I
I
0,50
..... \
-LS lin ·. ·
...... LS log
__ Me lin
-·-MC lo g
~00 +-----------~~----------~
0,50
1,50
1,00
FIIJ. 1 Cron H ctiOM ot tne dlttl,.nt eealed objecllve fun<:tion• in the Mlghbofhood of the •olu11on
p0
ln the implementcd computer code it is possible to cboose among the four different objective
functions. II was observed that it is usually more efficienl to starl searcbing for the maximum
correlation with the FRFs in logarithmic scale and, ofter a high levei of correlation is attained, e.g.,
0 .95. switch to the least squares objective function with the FRFs still in togorithmic scate. The
refinement of the solutioo may be otained taking the FRF in lineor scale in the few last iterative steps.
Arruda ( 1992) showed Lhat, in order to minimize JMC ( p) it is necessary to introduce aoother iterative
scheme besides the nonlinear least·squares loop. A scaled experimental FRF is introduced:
(7)
where.
11
13 -
F,ll2+ 2(p(k+ I)- P<~c.> l Ts~~t/• +
{p (k• t>- P<k>> TsTk>s<~t> {p<"+I)- í'> <"J> s
FTF+ ("' (k•l)_ ;.,(k)) ST F
X
I
1-'
I-'
(k)
()
X
~~~tially, j3 - l and the least·squares search direction is calculated fro m Eq. 3. The solution
6 p
is then pu1 i n1o Eq. 8 10 calculate j3 wbich is used 10 scate F" i n Eq. 7. Wlth tbe scated curve
Fx, the search direction is recatcul;\c~d from Eq . 3. This rrocedure is repeated untit convergence of the
values of the seorcb direction 6 p . ln the one·dimcnsional search (Box·Kanemasu modification)
performed with the calculated search directioo the objective function is also given by Eq. 5.
J. R F. Nnldaand M. A. V. Duarte
141
Sensitivlty Matrix Calculatlon
Many recent papers treat tbe JrOblem á calculaúng tbe sensiúvity matrix S of Eq. 4 (Adelman anel
Haftka (1986), Sbarp and Brooks (1988), Sutter et aJ (1986), and Vcinboacker (1989)). ln tbis section
tbe basjc formulation of tbe derivative of tbe FRF with respect to the structural parameters will be
reviewed. Because of the iterative cbaracter of lhe updating rnethod it is interesting to talre the simple
first order finite dltierences approximation for the deriva tive, whicb has tbe lowest computational cost:
c3Fi,
s,J - -
apJ
F,, (pi+ À pi) - F1; (pi)
. - - - - - -- À
(9)
PJ
Tbe cboice of À pi.may be criticai. lt sbould be os small as possible butthere are limitations dueto
numerical truncation. 1ne Brown and Dennis rule (Beck: and Arnold, 1977) wos used here:
(lO)
wbere,
IPjl < 10- 6
10-31Pjl if IPjl :t 10-6
10-
9
if
(11)
The Frequency Respoose Matrix of a r<Xor system may be expressed as:
H ( ro) - B(ro)-
1
(12)
with,
B(ro)- K -c}M+iroD(Q)
(13)
D (Q) • C+QG
(14)
wbere K, M and C are tbe stiffness, mass, and viscous damping matrices respectiveJy, G is lhe
Ccriolis rnatrix, and O is tbe rotatioo speed. lt is po;sible to calculate tbe sensitivity of any FRF with
respect to a parameter pk using tbe lcnown property for the inverse square rnatrix derivative(Martinez.,
1981):
.
(15)
sothat,
( 16)
For a given FRF HiJ ( ro) , the sensitivity matrix elements are given by:
142
Updating Rotor-bearlng Ante Sement Models Using Experimental Frequency Response Functiona
(17)
The derivalives that appear in Eq. 16 must be obtained analytically, wbich is usually
Straighforward. ln tbe case wbere tbe parameters are lhe bearing stifiness and dampng coefficients, il
is sufficienl to replace tbe parameter value by 1 aod watch for tbe elements of lhe ma1rices K and C
that vary. The derivative of the elemenls lhat do not vary is zero and lhe deriva tive of the elements thal
vary is equal 10 tbe new value of the element. The cl05ed-form solution of lhe sensitiviries has a higber
computaliooal cosi because i1 is necessary to invert matrix B ( w), or, numerically, solve a linear
sys1em of algebraic equations for each frequency w at wbicb tbe FRF has beeo measured (usually
hundreds or tbousands of frequeocy !ines). The fioite differeoces approximation gives fairly precise
results wben compared to lhe exaet, closed-form solutioo. ln lhe numcrical example treated in this
paper the sensi1ivities have been calculated by finile differences and the receptaoce FRFs
(displacement/force) were taken in dB scale witb 1 m/N reference. Jt can be shown that, fo;:
11.I) (w)
(18)
- iJ ll .. (<v) • ffi+i~
iJpk lj
(19)
the sensitivity of the absolute value in dB scale is given by:
(20)
ln the investigated numerical examples il was no1 possible to depict in graphical plots of the
scnsilivity curves (lhe columns of the sensitivity matrix) any difference between lhe sensitivities
obtaincd analytically and by finite differences.
Estlmation Errors
The variances of the generalized no nlinear least-squares eslimates of the paramelers,
additive Gaussian noisc may be shown to be approxima1ed by (Eykhoff, 1974):
T
var{p] • [S S)
·I 2
o
p, for
(21)
2
ln the above equation o is lhe variance of lhe ooise and cao be estimated witb:
2
J (p)
o·--
0
e
-
(22)
np
where ne is the number of elements of vector F and n is the oumber of elemcnts of vector p .
For ( ne- nf') - oo the probability diStribution of 1ends be Gaussian and lhe normal coofidence
intervals for the eslimated parame1ers rnay be determ.ined (Beck and Arnold, 1977).
p
to
143
J . R. F. Arruda and M. A V. Duarte
Unlqueness and ldentlftablllty
A quesdoo that is always present in model updating aod parameter estimadon problems is whether
the models are uni que. Given a set of modal parameters (natural ftequencies, mode sbapes aod modal
dampings), it is only passible to define uniquely tbe structure stlffness, mass and damping matrices if
lhe model is complete, i.e., if ali the modes are known at every degree-of-fteedom. This Is practically
impossible because real structures are continuous and it is too costly to experimentally identify high
orde: modes (Berman, 1988). So, it is meaningless to try to idenrify lhe model of the structure, but,
hopefully, it is possible to identify a model consistent with the a priori knO'Nledge about the structure
and with the measured data. ln tbe case where only a small set ri parameters ri the theoretical model is
estimated from experimental data, this problem may be formulated as an identifiability problem. The
question is whetber the parameters may be independently estlmated from the available data. There
exist different identifiabillty cri teria (Beck and Arnold, 1977). For tbe specificcase of nonlinear leastsquares estimatioo, one of tbem consists of look:ing for the number of nonnegligJble srngular values of
tbe sensitivity matrix, which are related to tbe linear independence of the columns of this matrix. lt
may be proved tbat if tbe columns of the sensitivity matrix for a giveo group of parameters are linearly
dependent, tben tbat group of parameters are linearly dependeot, then that group of parameters cannot
be estimated uniquely from the available data {Beck aod Arnold, 1977), i.e., lhe rank of lhe sensitivity
matrix is equal to lhe number of parameters wbicb can be estimated independeotly.
Computer Code
The proposed metbod was implemented ln a 16 bit personnal computer. A Finite Element code for
rotor systems developed by Benhier, Ferraris and Lalanne (1983) was rnodified to include the model
updating procedure. The calculation of FRFs ( O constanl, w varying) by inversioo of the dynamic
matrix B ( w ) with the equilibrium equations reduced using the first eigen-modes of tbe noorotating
system was implemeoted. Throughout the iterative process and wbeo calculating tbe sensitivities by
finite differences, lhe modal data base of tbe nonrolatiog system is kept constant and only the finite
ele ments which involve tbe parameters whicb are being estimated are re-calculated. When
convergence is attained the modal data base is updated before the final results are given. The
implemented code is híghly interactive so tbat it is possible, at each iteration step, to ch.ange the FRF
being curve-fitted. tbe frequency range and the parameters to be estimated. This alJO'NS tbe use of any
oonalgorithmic rule due to tbe knowledge and experience of the dynamiscist about rotordynamic
structural problems.
Numerical Example
A numerical example is presented io this seclion to illustrate the proposed rotor model updating
me thod . Fig. 2 shows the ortbotroplc flexible rotor system consisting of a single bollow spool
supported by two bearings, which was reported in Chen et ai. (1987). The FE nodes, referred to as
stations, are depicted in tbe figure. The bearings are considered to be orthotropic with two unknown
stiffness coefficieots and two unknown damping coefficients in each bearing. The rotating speed is
constaot at 5000 rpm. The geometry and material properties of the rotor-bearing system are given in
Thbles 1, 2 and 3.
Upd&ting Aotor-bearing Anile Element Modela Uling Elcpertmental Frequency Aeaponae Fuootions
1eomm
....__...
2
3
Fig. 2
4
IS
6
.,
e 9
Scheme of the rotor·b..rlng syst.m numericel example
Table 1 Rator-beartng system example • ahaft dllta
Axiallocation
Inner radius
(mm)
Outer radius
(mm)
0000.0
12.70
25.40
2
0088.9
12.70
25.40
3
0203.2
12.70
25.40
4
0274.3
12.70
34.29
5
0591 .8
20.32
34.29
6
0896.6
20.32
34.29
7
1049.0
20.32
34.29
8
1193.8
15.24
38.10
9
1270.0
15.24
38.10
Station number
Table 2
(mm)
Rotcr-bearlgn sy.tem example • ciiiCl cS.ta
Mass (kg)
Diametral
inertia (kg.m2)
Polar inertia
11 .38
0.0982
0.1953
4
07.88
0.0835
0.1671
7
07.70
0.0880
0.1761
8
21.71
0.2224
0.4448
Station number
(lcg.m2)
144
J . R F. Atruda and M. A. V. Duarte
145
Bearing
81
.350
.438
.350
.440
B2
.525
.613
.200
.610
The influence of tbe excitation and measurement point loc:ations was investigated aod results are
summarized in Tables 4 and S. 1Wo response measurement loc:ations were used in each case and at
Tabla 4 Hlgheat among the beartng paramatar aatlmatlon arrora for dltfarant axcltatlon polnt locatlona;
raaponM maaaurad at atatlona 2 and 8.
Excitation point location
Maxímum relative
estimation error (%)
2
3
4
5
6
7
8
9
station 2 bearing
0.03
1.60
0 .45
0.44
0.44
0.44
0.09
0.36
549.1
station 9 bearing
0.92
256.9
285.6
0.66
0.61
0.61
0.02
0.02
1.22
5 Hlgheat among tha bearlng paramatar aafmation arrora tw dl1'tarant raaponaa m...uramant locationa
(ona ot U'ta maaauramant atatlona la kapt thcad whlla the CJihar ta moYad; excltatlon ta tlxad et etatlon 45).
~bla
Maximum relative
estimation error (%)
Meas urement pont loc:ations at stations 9 and:
1
2
3
4
5
6
7
8
station 2 bearing
0.23
0.44
0.14
0.26
12.32
33.03
36.5
84.86
statlon 9 bearing
5.00
0.61
0.15
0.09
2 .16
3 ..26
1.91
1.62
9
Measurement pont loc:ations at stations 2 and:
Maximum relative
estimation error (% )
2
3
4
5
6
7
8
9
station 2 bearing
5 .90
6.89
34.29
0.50
13.50
0.17
3.66
0 .44
station 9 bearing
81 .11
461 .3
242.1
1.27
35.15
0.03
27.55
0.61
eacb location the FRFs on both di rections, x aod z, were simulated. The 4 FRFs were used
sequentially to find the searcb direction of Eq. 3 at each iteration s1ep. The FRFs were simulated with a
frequency resolution of 11.72 cpm in the frequency range O- 3000 cpm (O- 50Hz). Random Gaussian
error of variance equal to 0 .5% of tbe FRFs rms amplitude was added to lhe FRFs to siroulale
measuremeot ooise.
ln rotordynamic applicalions, the usual measuremen1 poinl locations are the bearings, where
ooncon1actiog sensors are frequently installed. From lhe simulated results they have proved to be
convenient for tbe estimation of tbe bearing parameters. lt can be seen from Thble 4 that the rotor mid-
Updating Rotor~earing Anke 8ement Models Using Elcperimental Frequency Response Functions
146
span is a good location for the excitation if the two bearings are to besimultaneously identified.lf ooly
one bearings is to be ldentified ata time, the excitation and lhe measurement point locations may be
situated at the beariog itself.
The logaritbm of Lhe quotient between the biggest and the smallest singular values of tbe
sensitivity matrix was chosen as an index to quantify the identifiability. Fig. 3 sbows tbe good
correlation that
xlO
~
83
ll>
55
.....
o
.....
.....
"O
ll>
......
o
E
:.;: 28
VI
ll>
.I l
.22
.33
.H
.5e
identifiability index
x
10
2
Fig. 3 Correlatlon .,_tween lhe ldentltlablllty Index and the maxlmum . .ttmatlon error
exists between this index and the estimation errors. Fig. 4 shows the percentage of successful trials
.11
.22
.33
.44
.55
identifiability index
Fig. 4
Pe~entage
of aucce. .N trlata •• a Mctlon oftheldentlftabltlty Index
as a function of the identifiability index. ln tllis context, a successful trial means that the
parameters were estirnated with an error of the sarne magnitude as tbe simulated measurement noise.
J . R. F. Atrudaand M. A. V. Duarte
147
The additive Gaussian noise levei influenoe is shown in Fig. 5. lt can be seen that lhe seosilivity
wilh respect to lhe measurement noise is small for noise leveis under 4% and acceptable for noise
leveis up to 10%.
The amounl of measured data did not show a strong influence on the precision of lhe eslimales.
Thble 6 shows some illustrarive results relating the number of frequency Iines of the FRF used to
.otl
o
o
~
~ .90
'0
'-
o
~ .60"
o
"'~o
.§
......
o
o
.JO
o
o
o o o
o
(I)
«>
• • o o o o
o
o
o
00~----,------r----~~----~-----r----~
.18
.37
.55
.73
.92
lllO
meosurement error ( 0 /o)
Fig. 5 l nftuanc. of tht al mula1ed rnMtUrtmtnt noiH on tht
~111matar ..u matlon ai'I"'rt
T1ble 6 lnftuenca of the number otfrequency llnea
N. of frequency Iines
60
120
256
500
1000
2000
Max. estim. error (%)
1.86
0.70
1.14
0.61
0.94
0.97
update lhe model and the maximum error foWld ln the estimated bearing parametei'S. These results
were obtained witb a 2% additive noise levei. For higher noíse leveis the averaging effect of taking a
greater number of frequency lines woold bave had a more significam effect.
Concluslons
The proposed FE model updating method applies whenever a small number of theoretical
parameters selected a priori need to be corrected based on experimental data. Depending on the extent
of the knowledge of the statistics of the problem the method can be classified as Ordinary Least
Squares, Maximum Likelyhood or Maximum a Posteriori. lts implementatioo can be recwsive or not.
The method is versatile and nny experimentally obtained dynamic response whicb can be predicted
with the FE model can be used. As the dynamic resp:>nses are oonlinear in the structural parame1ers, a
modified Gauss's noolinear iterative approach is used.
The application of the method to the Fin1te Element model updating of rotor-bearing systems usi ng
measured Frequency Response Functions was presented. The method wns osed to estimate bearing
stiffness and damping coefficients of a rotor-bearing nwnerical example.
Updating Rotor-bearing Anb1 Element Models U&ing Experimental fntquency Response Functions
148
1\vo methods for the calculalion ofthe sensitivity matrix- cl~ed form and finite differences- were
compared and the latter was preferred because it is sufficiently accurate and bas a much lower
computational cost than the former.
With respect to the influence of the excillltion and measurement point locatiom, it was shown that
for two-bearings r01ors the bearing locations are adequate for measurlng the respomes wbile the shaft
mid span is a good location for the excitatioo. ln practical cases a previous numerical simulation can be
dooe to indicare the best among the possible excitation and respoose point locations.
The identifiability problem was investigated. lt was shown that an identifiubility index can be used
to indicate if a given set of parameters can be estimated from the available measured data. This index is
the logarithm of the quotient between the highest and the lowest singular values of the sensitivity
matrix. ln the numerical example, a good correlation was fouoo between this index and the estimation
errors and convergence of the model updating method. The arder of magnitude of lhe parameter
estimation crrors was shown 10 be predictable with lhe proposed method.
References
Adelmao, H.M. and Haftll:a, R.T. ( 1986) "S.:ositivity Aoalysis of Discrclc Structural Systems", AIAA Journal,
24(5): 823-832.
Arruda, J.R.F. (1987) "Rotor Fioile Elemenr Model Adjustlog" Proc. of rhe 9th Brazillan Coogress of Mochaoical
Enginecring, Floriaaópolis, Brnil: 741-744.
Arruda, J.R.F. (1989) "Rotor Model Adjusting by Unbalancc Response Curvc-fining", Proc. of lbc 7th lnt. Modal
Analysis Confcrencc, Las Vcgas, USA: 479- 484.
Arruda, J.R.F. ( 1992) "Objective Functions for tbc Curve-Fito C Frequency Response Functloos", AIAA Journal ,
30(3): 855-857.
Beck, J.V. and Arnold, KJ. (1977) "Parametcrs Estimarioo ln Engiocering and Scicnce", John Wilcy, New York.
13emu.n, A. (1988) ''Nonunique Struclural Systcm ldentificatioo", Proc. of lhe 6111 lnt. Modal Analysls Conferencc,
Orlando, USA: 355-358.
Bcrthier. P., Ferraris, G . and Lalanne, M. (1983) "Prediction o CCriticai Spceds, Unbalancc and Nonsynchronous
Forced Rcspoose of Rorors", The Shock aod Vlbration Bulletin, 53: 103- lll.
Blakely, K.D. and Walton, W.B. (1984) "Selection oC Measurement and Para meter Uncertaintíes for FE Model
Revi sion", Proc. o( tbc 2°d lnt. Modal Analysis Conference, Orlando, USA: 82-88.
Cuneiro, S .H.S. and Arruda, J.R.F. ( 1990) "Updating Mcchanical Joiot Properties Based ou Experimentally
Determined Modlll Parameters'', Proc. of rhe gch Int. Modal Analysis Conference, Klssímmce, USA: 1169·
1175.
C hen , W.J . et ai. (1987) "Beari11g Parametcr ldentification of Rotor-Beuing Systems Using Nonlincar
Programming Techoiques•, Mecha nicai and Aerospace Englneering Department Publicatloo, Arizona State
University, Tempe, Arizona, USA.
Eykhoff, P. (1974) ''Systcm ldenlifkalion", John Wiley, London.
lmregun, M. and Visse r, WJ. ( 1991) "A Review o( Model Updating Tecbnlques", Tbe Shock aod Vibration Digesl,
23(1 ): 9-20.
Lallemcnl, G. and Zbang. O. ( 1988) "lnverse sensiriviry based on lbe elgen.wlutions: analysls of some difficulties
encountered io tlte problem of parameter correction of fioite elcmeot models", Proc. of the 13111 lnl. Sem. o o
Modal Analysis, Leuve n, Belg.ium : 19-23.
Martinez, D.R. ( 1981) " Paumelcr Estlmation ln Structunl Dynamics Modcls", Sandia Narional Laboratorles
Report SAND80-0J35, Albuque rque, USA.
Nalke, H.G. (1986) "lmprovemcnt of Analytícal Models Witb Input/Output Measuremenls Contra Experimental
Modal Analysis~ , Proc. o( lhe 4dl lnt. Modal Analysis Conference, Los Allgeles, USA: 409-413.
Natke, H.G. (1 988) "Updating Compurational Models in the Frcquency Domain Based oo Measured Data: A
Survey", Probabilisric Engineering Mechanics, 3(1): 28-35.
Sharp. R.S. and Brooks, P.C. (1988) "Sensitivities of Frequenc y Respoose F~actions of Linear Dynami c Systems to
VJriations in Design Para meter Values", Joomal of Sound ud Vibrarion, 126(1): 167-172.
149
J. R. F. Arruda and M. A V. Duarte
Suner, T.R. et ai. (1988) ''Comparisoo of Severa I Methods for Calculating Vibration Mode Sbape Derlvatives",
AlAAJoumal, 26(12): 1.506-1511.
Vaoboacker, P. (1989) "Seositivity Analysis of Mechanical Structures Basedoo ExperimentaHy Determlned Modal
Parameters", Proc. of the 71h Int. Modal Aoalysis Conference, Las Vegas, USA: 534-541.
RBCM · J. of lhe Braz. Soe. Mechllnlcal Sc..nc••
Voi.XV·n•2· 1993 - pp. 150·164
ISSN 0100·7386
Printed in Bra.zil
On Geometric Nonlinear FE Solution of
Unidimensional Structure Dynamics
C. A. Almeida
J.A. Plrea Alves
Department of Mechanical Engineering
Pontifícia Uníversidade Católica do F110 de Janeiro
Gávea • Rio de Janeiro • Brazll
Abstract
A numerical procedtue for geometric nonlinear analysis of framed structures in presented. The tini te element
model employs 3-D Hnear to cubically varying space dlsplacemen.IS and is capable of transmitting stress only in
the di recrion normal to the cross-seclion. h is assumed that thls normal stress is constant over the cross~ectionaJ
area, and lha! the area itself remains constanl, allowing tbe element for large displacement analysis bul under
small strain condition. Two algorithms for lhe automatic incremenral solution of the resuiLing nonlioear
equilibrium equarions ín slalic analysis are accessed. These include the displacemenr controltechnique aod the
minimizatioo of residual displacemenls, which allow to calcula te rhe patb response of general truss structures,
even of lbose tbat exbibit unstable configurations. The formulation has been implemented and lhe resulls of
various exam pies illustrate tbe elemeot effectiveness in botb static and dynamic analyses.
Ktywo rds: Geometric Nonlinear Analysis, Automatic Incremental Solution, Displacemenr Conrrol,
Minimizatioo of residual displacements.
Resumo
Um procedimento numérico para a análise não linear geométrica de estruluras entreliçadas é apresentado. O
modelo de elementos finitos utiliza deslocamentos tri-dimensionais com interpolação espacial de ordem linear a
cúbica e é capaz de transmitir tensões apenas na direção normal de seção transversal. Considera-se esta tensão
normal constante sobre a área da seção reta que permanece constante, permitindo ao elemento representar
grandes deslocamentos mas sob a condição de pequenas deformações. Dois algorítmos para a solução
incremental automática das equações de equiHbrio não lineares são apresentados. Estes incluem as técnicas de
incremento controlado do deslocamento e de minimização dos deslocamentos residuais que permi tem obter a
resposta de estruluras de barras em geral, mesmo apresentando configurações insráveis. A formulação fo i
implementada e os resultados com vários exemplos ilustram a efetividade do elemento em análises estáticas e
dinâmicas.
Palavras-cha vt: Análise Não Linear Geométrica, Solução Automática Incremental, Controle do Deslocamento,
Minimização dos Deslocamentos Residuais
lntroduction
The important requirements for a reliable nonlinear finite element analysis of structures are, in
general, the use of accurate finite element model for geometry and kinematics representation and lhe
use of an efficient procedure for the solution of the incremental equations of motion. For a given
solutíon accuracy, the procedures for assemblage and solution of the equilibrium equations are
efficient when de computer costs are low and the numerical solution is obtained in a reliable manner
with a minimum amount of analysis effort.
The need of nonlinear analysis of structures in recent years has significantly increased dueto the
adoption of limit static and optimum design concepts, which have resulted in lhe use of slender
members in framed structures. Since the bebavior of these structures near to the limit (pre· and pastbuckling) loads is complex, and difficult to predict accurately by observalion, mucb empbasis is
curreritly being placed on lhe developmcnl of more general and automatic solution schemes. Extensive
research efforts have been devoted in the past decade, to the development of reliable and efficlent
numerical algcrithms (Batoz and Dhat (1979); Rides (1979); Ramm (1980); Crisfeld (19tH); Yang and
MacGuire (1985); Chan ( 1988)) that differ from the classical Newton-Raphson numeric scheme by the
invdvement with a large number of degrees-of-freedom. The standard approach relies on the fact that
a nonlinear solution can be obtained by successive linearized results in an iterativa process ata
Manuscript receíved: July 1993. As6ociate Technical Editor: Agenor de Toledo Reury
C. A Almeida and J . A Pires Alves
151
constant load. lbis condition results in failure of convergence in the analysis of structures displaying
limit points io the force-displacemen.t relationship (snap throughibaclc bebaviors) because the tangent
stlffness matrlx is 111-conditloned or simply because the Joad levei Is beyond the solution path . A
comparatlve study on tbe advantages and disadvantages ln tbe use ri tbe various soludon procedures is
reported in Bath and Ciments (1980), Qarlc and Hancock (1988) and Pires Alves (1992).
The objectives of tbis paper is to present a methodology tbat is simple and effective and predicts
accurately tbe significant deformations in general linear and geometric nonlinear analysis of framed
structures. The element is a two to four-node lsoparametric displacement based finlte element with 3field displacements to represent constant uniaxial stress states normal to tbe cross-section. Based on
the discussions and practica1 applications reported in Pires Alves (1992), concerningseveral numerical
solulion schemes that using surface constraint equations are employed to solve finite element illconditiooed equilibrium equations, two methods are presented in this worlc: a) the constant iocrement
displacement and b) the minimum residual displacement. 8oth methods employ the displacement as a
control parameter. However, in the former, tbe lncrement of a sole displacement compooent is used ln
lhe marching proress to obtain lhe solutlon corresponding loading and displacement components that
satisfies equilibrium wbile in lhe second approach tbe objective is to minimize (or to eliminate) the
dlsplacement vector associated to the unbalaoce load in tbe iterative process.
ln the next section of this paper we briefiy review Lhe formulation of an unidimensional element
with emphasis on lhe geometric nonlinearities incorporated in the finite element equilibrium equations
to be solved for. Metbods for the numerical solution are presented afterwards. The element bas been
implemented in the computer and following, we present the analysls results of some problems that
demonstrate tbe validity of the methcxlology in general noolinear analysis.
The Truss Element Formulation
The tru8s element is a structural model that transmits axial forces only. Normal stresses are
uniform over the cross-sectional area whicb remains constant during deformation. As a consequence,
althougb tbe element can undergo to large displacements, only lhe small strain condition is allowed in
tbe element model.
Considering a truss witb arbitrary orientation in space and described by two to four nodes, as
shown in Fig. 1, the global coordinates of tbe elernent at times O and tare respectively
o
X; (r)
- 1,2,3
(1)
k• I
ln Eq. (1) 0x~ and 1x~ are tbe nodal pant global coordinates ai rimes O and 1 respectively, h (r)
are the Lagrangi<ln interpalation functions defined in Fig. 1, N is the number of element nodes an~·r is
the element local coordlnate. Using equation (1) it follows tbat
N
l
l
0
~
I k
ui(r)- xi(r)- x;(r)- ~ hk(r) ui
(2)
k• I
and
N
u1 (r) -
}: hk (r) u~
(3)
k• I
are the total displacement field and the increment displacement field, at time t, respectively, wbile tuk
1
and u~ are tbese quaotities at tbe element nodes. Thus, with tbe aid of tbe following definitions
I
152
On Geometric Nonlinear FE Solution of Unldlmentionel Structure Dynamk:s
..
®
•
3
2
h (r) =( - 9r + 9r ~ r-1 )/ 16
1
3
2
h 2 ( r ) =(9 r + 9 r - r - 1 ) / 16
3
2
h 3 (r)=(2 7r- 9 r- 27 r .... 9 )/ 16
h 4 ( r )=(- 2 7 ~ - 9 r 2.. 2 7 r .. 9 ) / 16
(4)
(5)
ú -
Íul ui ui .... uN uN uN1
L 1 2 3
1 2 3J
(6)
(7)
equations (l) to (3) can be rewriuen in the matrix form
(8)
Since lhe only stress component considered is normal to the cross-section area we have, for the
corresponding local elementlongitudinal strain, in tbe Thtal Lagrangian formulation (Bathe (1987)),
(9)
153
C. A Almeida anel J. A Pires Alws
where tbe linear compooent is expressed by
(10)
and the nonlinear term is
)j
2
2
1
1
2
3
0
- - -) +
- - ) +(au
-'I 1 1 -[(élu
(au
2
aox1
aox1
o
(11)
Ô x1
botb defined with respect to the cocrdinate system attacbed to tbe element body at time O. Plugging in
equations (8) into the above results in lhe strain-displacement transfoonation matrices; as an example,
for a straight two-noded element, witb length L and space angles o.l' a 2, a 3 , lhese matrices result in
t
o8 L
..
( 1/L) [ -cosa 1
-cosa2
-cosa3
cosa 1 cosa 2
cosa3]
(12)
and
1
o o
1
I
2
O~L- (l!2L)
o o
o o -1 o
1 o o -1
1 o o
1 o
-1
(13)
1
ln static analysis, COffiidering the linearized equation of motion with incremental decompositions
obtained from the principie of virtual displacements, tbe discretizations i o equation (8) yield to the
matrix form equilibrium equation
(14)
wbere tbe stress matrix ~ and the vector /fo are simply the second Plola-Kirchboff tensor with a sole
non-zero (;i 11 component, JC is the linear constitutive matrix and 1 + AtR is the equivalent applied
externa! loading reduced to the nodes degrees-of-freedom ai time t + l!.t. These equations are the
necessary ingredients for the assemblage of the total equilibrium system of equations. 1t should be
pointed out that the elemeot matrices are obtaioed in this formulation directly corresponding to the
global displacement components. However, and alteroative procedure is to use local displacements
referred to tbe coordinate aligned witb the element to obtain the element matrices and thus to use a
transfonnation to tbe global CO<rdinate system.
On Geomelric Nonlinear FE Sc*ltlon of Unidlmensional Structure Dynemlcs
154
Numerlcal Solution Strategies
The essence of an automatic numerical solutioo is to seek the load-displacement relation such the
structure is assumed subjected to a fixed direction varying load. Thus, considering the equilibrium
condition in equation ( 14), at time t + ât , the followiog iterative proce<lure can be derived from the
Newton-Raphson iteration scheme,
(15)
with
(16)
where the s1rucrure displacement increment vector u(i), at iteraúof; (i} and the load parameter 1 + 611..,
at lime t + ât , are tbe problem state variables to be solved for. 1K '- 1 is the coeficient matrix, R is a
vector direction of ex1ernally applied nodal point loads and 1 • 61F< 1- J) is a vector of nodal poinl
forces equivalent (in the virtual work sense) to the internal stresses. When iteraction is performed in
the load-displacemenl space dcscribed in equations (15) and ( 16) the additional equation
(17)
is used to constraint the load step size.
Various constraint equations proposed io the literature for the funclion f are surveyed by Pires
Alves ( 1992). From tbosc, two numerical schemes were found suitable for general applications in
structurc analysis, namely, the constaot increment displacement component method aod the minimum
residual displacement method. ln the first, a method devised by Powell aod Simons (1981), the load
vector is modified a1 each step such the displacement incremeot is prescribed ata certain degree-offreedom. During iteratio~. this displacement increment component is kept constant while the load
incremcnt parameterâi..(•J is adjusted for equilibrium. From combining equations (15) and (16) the
vec1or displacement increment, at iteration (i), can be decomposed as
.(i)
U
•
,,(i)
"U
_.,( 1).(1)
+ Ull.
Ur
(18)
where ü~i) and ú~i) are the unbalance and the residual displacement increments, respectively,
obtained from the following systerns of equations
tK(i-t)
.. (i)
Uu
- l+t.LÀ.(i) R _t+AIF(i-l)
(19)
and
tK (I-1)
• (i)
Ur
R
(20)
For a specified displncemenl compooent increment m to remain coostant (i.e. u~~) • 0), we bave
155
C. A. Almeida and J . A. Pires Alves
b
u(m) •
T. (i)
u
(21)
-O
in which vector b contains zeroes except for unit at m-th component. Thus, equations (18) and (21)
yields the load step coostraint equation
f(AÀ
(i)
(i)
) - u(m) + AÀ
(i)
(i)
(22)
ur(m) • O
wbich gives
(23)
Since R is a constant vector, ur(m) is calculated onde during each step cycte - modified NewtonRapbson procedure -. and softening or stiffening system behavior is represented by a negative or
positive sign of A/...(') , respectively. Also, the numerical procedure requires the displacement
component 1 u~~) being monotonically aescent ln the sotuôon load interval.
Considering a new step, the toad parameter_inaement for tbe first iteration is obtained by simply
imposing in equatjon (18) a prescribed value u (m) for the m-tb component of vector displacement
inaement with u~~) • O, and result.ing in
(24)
ln tbe second metbod the constraint equation is obtained by imposing a minimum value for the
norrn of the increment displacement in each step cycle. ln accordance with the iterative procedure
objectives, the motivation for thls cri teria is to eliminare the increment disptacement vector by
imposing a minimum value of its Euclidian norrn. This condition is expressed by the equatlon
(25)
or, considering the increment displacement decomposition in equation (18), in the following vector
forro
(26)
which gjves
.. ~(i)
uf\.
-
-( • (i)T , (i ) )/(!\(i)T . (i))
u,
. Uu
u,
.
u,
, for i>l
(27)
Equation {26) states the constraint condition for selection of the load parameter: the displacement
inaement vector in each iteration is onhogonal to the residual increment displacement vector. The load
parameter at the flrst iteration of each step shall be arbitrarily chosen but sbould not be too large to
ensure convergence in the iteration process. lts size is related to the degree-of-nonlinearity and, in
On Geometric Nonlneat FE So4ution ol Unidimensiooal Strudure Dynatnics
156
order to characterize the sLructure overall behavior, a nondimension quantity is sug~ed in Bergan et
ai. (1978) as a current stiffness parameter correction factor of the load increment óÃ (1), used in lhe
first iteration of the first step cycle. Thus,
T
(I)
ÓÀ
- ÓÀ
:::(J)]T
R . Ur
-(I)
[
(28)
-T • (I)
R .ur
where ~! t) is the residual displacement vector in thc first SJep anel y is an exponent factor assigned by
the analyst for com rol of the load size acx:ording to the seriousness of the nonlinearity.
Numerical Analysis
The aformentioned numeric algorithms have been implemented with tbe truss element
formulation . the results of some sample analysis that are reported in tbis paper illustrate the poinrs
madc previously and give some insight imo how the solution of a specific kind of nonlinear problem
should be approached.
1\vo Rod Are h Structure
The plane structure shown in Fig. 2ls considered for analysis comparison because it is simple and
E=2. 1xf0 6
L= 10.
•
A=2A
A= 1.
TOL= W- 4
Physlcal Problem
Model Used
-4
Fig. 2 The twa roei arch atructwe conelde,..d (energy tderance for convergenc• uaed • 1 xlO
the analytic solution for static aoalysis is avail ablc in the Jiterature (Bathe, 1987). Because of
symmetry one rwo-node truss element is employed i o lhe numeric represeotation. The aoalysis is
limiled to the material linear behavior and, as shown in Figs. 3, lhe solutions for lhe Joad-displacement
relation furnished by the constant inc@Oleot displacement merhod and the minimum residual
displacement method (with y • 1 and óÃ(I) - 4.22) are both ln good agreement witb anal ytic
results. 1t sbould be noted tbat, for the classical monotonically incremented loading procedure, lhe
displacement path with snap-through from A to Bis likely to be followed, instead tbe stiffness
sofrcni ng path. The sarne structure was also coosidered for dynamic anal ysis witb lhe mass of the bar
lumped at tbe element nodes and the Newmark metbod with constant average acceleration (a s 0.5
C. A Almeida end J. A Plree Alws
157
20.
t),
15.
Analytic Solution
• • • e
10.
Minimum Residual
Displacement - M R D
( w/ load correction factor · "Y=1
5.
o
- 5.
- 10.
20.
t)..
15.
e e e e
Analyti c Solution , Ref. Ct1 J
Constant Displacement
Increment
10.
5.
O.
-5.
-1 0.
Figa. 3 • L..oado(liaplac.ment comperieona
and ~ - 0.25, Batbe (1987) employed to direct integrate tbe equilibrium equation in lhe time domai o.
Solutions for lhe displacements obtained witb different time step integratioo sizes are sbown in Fig. 4
and used for selection of the time integration parameter. The dynamic analysis results with a .01 sec
step and three initial conditionsare shown in Fig. 5, in the pbase plane. These analysis required tbree
158
On Geometric Nonllnear FE Solutlon of Unldlmensional Slru<:nJre [)ynamiC$
2~----~------~----~------r------r------r---~
u
DT = .001
........._ DT= .01
........_ DT= .02
-6
-8+------r-----.~----~-----r-----.------~----~
0.00
0 .05
0 .10
0 .15
0.20
0.25
0.30 t
035
Fig. 4 Tlme etep evelue11on ln lhe l ntegr•tlon of tht equlllbrlum equetlon uelng the Newmerk method,
v
50
DT =.01
JNITIAL CONDITIONS
1.1.1
Cl)
""
o
-50
-6
o
~
uo vo
A 172!50 .300
o.
8 43410 .7(17
o.
c
o.
74818 t.t4
u
Fig. 5 PhiM pl1ne eolutlone fot flrM dltt.rent lnltlll concltlone
iterations, in average, for eacb step cycle anda total of 35 time steps. Employing tbe Runge-Kutta
159
C. A. Almeida and J . A. Plrw Alves
procedure for integration of the equilibrium equation, a step size of .001 sec, smaller than wíth the
Newmark method, was required for convergence and resulting io a solution very close to the obtained
with tbe finite element tecbnique.
Three Rod ln plane Structure
The assemblage of elastic hinged bars in Fig. 6 has been considered for analysls i o tbe literature
E = 3.x 106
A= 1.
~
R =4. X 10
5
3
\\\\\\\
30
5
Fig. 11 The1hree hlnged-ber etruc1ure conaldered
using the displacement control metbod (Powell and Simons, 1981) and the constant work metbod
(Bathe and Dvorkin, 1983). The structure load-displacement responses possess lhe snap-through and
the snap-back bebaviors when subjected to a compresslve load at node 1. Due to the snap-back
behavior of nodes 1 and 3 displacement responses, node 2 displacement sbould be used as controlling
displacement. Numerical solutions employing the Q.W>Jacement contrai algorithm and lhe method of
minimum residual displaoements with a load factor A"A.<t) - 1.542 and y - 1 were obtained and the
responses automatically traced. As sbown ln Fig. 7, the agreement of lhe numerical results with
previously obtained solutions is good.
Ttvee-Dimenelon Rod Structure
Tbe spatialtruss in Fig. 8 was loaded vertically and subjected to two sets of b~ndary conditions at
the structure base. First, the condition of ali nodes at the base fixed is considered and lhe problem
reduces to tbe one degree-of-freedom case addressed in lhe first example. The cross section area of the
bars was adjusted to give the sarne results for lhe vertical displacement of node 7, as shown io Fig. 3.
ln the second case, free bounclary conditions were coosidered for all base nodes of tbe strocture exoept
fo r node 1 kept fixed and for node 6 set freei o x 1 -direction only, Numerical solutions using the
minimum residual displacement and constant increment displacement methods are presente<! in Figs. 9
for the load-displacement relation at node 7, in lhe vertical direction and at node 4, in the xrdirec tion,
respectively.
On Geometric Nonlil'leat FE Solutlon of Unldlmensional Structure Dynamk:s
160
OOOConstont Oisp. lncrement, Ret ( 10J
• • • Coostort Disp. lncrement <Node~
Node®
4.
/
-
MRO ("Y= 1. )
Node 1
2.
6
E=3. x10
A= 1.
~
R=f . xiO
- 2.
Fig. 7
Foree-dleplacement numerlcal reaponeea at the nodea of the atructiM'a
Free Fali of a Chaln
A chain originally fixed between two horizontal points is released atone end. The shape of the
hanging chain m rest is obtained nwnerically by imposing the equilibrium ar each joint, considering the
weight and the internal forces in eacb chain segment. First, tbe chain is beld stand by a solenoid, which
has the electric current shutted off when a stroboscopic ligbt is activated, releasi ng the chain fixture ai
B and allowing ii for motion. The numerical simulation of the cbain falling, shown in Fig. 10, was
obtained using 46 two-node truss elements with Jumped mass. The Newmark time integration
procedure was employed in a step-by-step solution with a time step of .SE-4 sec., for positions ata
time interval of .1464 sec. Due 10 the solenoid lime lag during the chain release, the second numerical
pC6ilion shown at configuration b, corresponds to time .1662sec., made coincidem with the fifth chain
exposure in the experimental counterparl of the phenomena presented in Fig.11. The strobe light
frequency was adjusted to 27.3 Hz. ll is remarkable tbat the conputed positions follow the experiment
cven in fine details as the curling of the free end that takes place while tbe cbain flings back across the
verticalline. The small differences presented are certainly dueto the fricdoo condition O()( induded in
the nwnerical model representation.
161
C. A. Almeida and J . A. Pire& Alves
t
>. R
Cut A-A
~~
~-~~l···
Member t- 7
E= 2. t x t0 6
R = 1000 .
'f= 15 °
A = . t 66
Concluslons
A finite elemeot formulation has been shown applicable for general truss structure analysís
including geometric nonlinearies and dynamícs. It eoables analysis of structure motions possessing
large displacements but under the condition of small stroins. Sampling analysis bave shown the use of
automaric stepping procedure in prcdictiog tbe load-<lefiection respoose of strucrures preseotiog snapthrouglvback behaviors. 1be examples shown illustrate the applícability of the formulation to a certai o
extent and the application of the methodology of analysis to more general elasto-plastic behavlor is
subject of continuing work.
Acknowledgements
The authors wish to aclcnowledge lhe financial support provided by the MCf to CfC-PUC/RJ, and
by CAPES for the supr.ort of J. A. Alves MSc' Studies at the Dept. of Mechaoical Engineering.
References
Bathe, K. J. aod Cimento, A. P. (1980) "Some Practical Procedures for the Solullon llf Nonlloear Fioite Element
Equatioos", J. Comp. Methods Appl. Mech. Engng., vol. 22, pp. 59-85.
Bathe, K. J. and Dvorlcio, E. N. (1983) ~on tbe Automatic Solutlon of Nonlinear Finite Elemenl Equa&ions",
Computers nd Structures, vol. 17. n. 5-6, pp. 871-879.
Batbe, K. J. (1987) "Plnite Elemenl Proeedures in E.ngineering Analysís", Prentice·Hall.
Batoz:, J. L and Oball, G. ( 1979) "lncremenul Displacemenl Algorit.llms for Noalinear Problems", ln li. J. Num.
Met. Eng., vol. 14, pp. 1262-1267.
Bergan, P. G., Horrimore, G., Kralceland, B. and Soreide, T. H. (1978) "Sohuion Techniques for Nonlioear Finite
Element Problelll$", lnL J. for Na.m. Met. in Eng., vol. 12, pp. 1677-1696.
Chao, S. L. (1988) "Geomelric and Material Nonlinear Aoalysis of Ream/Columas and Frames Using tbe
Mlnimum Residual Dispiacemenl Method", lo ti. J. Num. Met. Eng., vol. 26, pp. 2M7-2669.
On Geometric Notllillear FE Solutlon of Unldlmeneional Structure Oynamice
t~
15.
• • • Con st. Incr. Oispl.
I
M.R. D.
10.
-. 3
-5.
20.~--------------------------------------
t~
15.
10.
5.
Constont Increment
• • • • Displocement Control Ctu >
7
MR.D.
!
162
163
C. A. Aknelda and J . A. Pire$ Mies
E =2K 1o5 N/mm2
L=529mm
A=10mm 2
Fixed Point
M= 153,3 g
46 Two-node Truss Elemen1S
S<Menoid
L
Chain at Rest~,."' a
................
''\c
__
- - - FALLING
-RETURN
Fig. 10 Ch•ln conllgur•tlon predlcted bV lhe flnlte element model
Fig . 11
.,
Conftgura11ona of 1 " " talllng chaln (pend!Jum like moton)
On Geometric Nonlinear FE Soartion of Unldlmensional Slructure Oynamics
164
Clark, M. J. aad Hancock, G . J. (1988) ''A Suady of lncrementai-Iterative Strategies for Nonllnear AnalysisM,
Research Repon n. R857, Tbe Unlversily o( Sydney, Scbool of Civiland Mtnlng Eng.
C ri sfeld, M. A. (J 981) "A Fast locrcmen tal/ h eutivc Sohu lon Procedure tbat Handles Snap-Tbrougb", Comp.
Struct., vol. 13, pp. 55-62
Pires Alves, J. A . (1992) uSoluç.io Incremental Autom,tica de Treliças com Não-Linearidade Geomttrica,
Utlliu ndo o M~todo de Elementos Flnitosn, Msc. Tbesls, DEM-PUCIRJ.
Powell G. and Simoos, J. (1981) wlmproved
vol. 17, pp. 1455-1467.
lt~ratioo
Strategy for Nonlinear Structuresn, lati. J. Num. Met. Eng.,
Ramm , E. ( 1980) "Suategles for Traclng lhe Nonlinear Response Near Limit Points'', Proc. of tbe Europe-US
Wo rksbop on Nonllnur Finite Element Analysis in Structural Mecbanics, pp. 63-89.
Riks, E. ( 1979) ''An Incremental Approac b to tbe Sohuion of Suppiog ud Buckllog Problem.s", lntl. J. Solids
Srrucr., voJ. 15, pp. 529-551.
Yang, Y. B. and Mcguire, W. ( 1985) "A Work Cootrol Metbod for Geometrically Nonlioear Aoalysis", Proc. lot.
Conf. on Advao ces i o Num. Mel. ln Eng, UK, Swanse.t, pp. 913-921.
RBCM - J. of the Braz. Soe. Mec:hllnlcel Sc;ienc:H
Vo/. XV- n• 2- 1993 · pp. 165-171
ISSN 0100·7386
Printed in Brazll
Fundamentos Científicos de la lngeniería
Mecánica
Scientific Foundations of Mechanical Engineering
Juan Joaé Scala Eatalella
Departamento de Flsica Aplicada
Universidad Politecnica Madrid
E. T.S. de lngenieros lndustriales
Jose Gutlerrez Abascal. 2-Madrid-Eepaila
Abatract
The paper discusses lhe question of looking for the scientific found.ations of technology ln preference to the
technological found.ations of science.
Key words: Science, Technology, Mechanical Eogineering
El análisis de los fundamentos científicos de la ingeniería mecânica es parte de un estudio más
general, que nos conduciria a la consideración de cuáles soo los fundamentos científicos de la técnica.
En cuantas ocasiones he sido invitado a abordar este tema me he planteado la misma pregunta: <,por
qué siempre se buscan los fundamentos científicos de la técnica y sistemáticamente se olvidan los
fundamentos tecnológicos de la cit:n<-ia?.
Cierto es que en nuestra tradición académica, no companida por otros países y, en especial, los
anglosajones, se suelen presentar los contenidos por la vía lógico-deductiva. Ab ren camino las
matemáticas, seguidas por las Hsicas y las ciencias dei ingeniero, para cerrar el arco coo el entrallUido
de las tecnologías, cuyos métodos se ofrecen firmemente apoyados en aquéllas.
Pero forzoso es reconocer que históricamente las cosas no fueroo así o, ai menos, en bastantes
ocasiones siguieron e l orden inverso. Muchos capítulos de la ma temática nacíeron con el afán de
formular fenómenos ffsicos; e incontables avances técnicos tuvieronlugar antes que la ffsica formulara
las leyes que los rigen.
Cuando Watt patenta su máquina en 1769 faltan 55 aõos para que el iogeniero francés Nicolas
Léonard Sadi Carnot (1796-1832) presente su estudio sobre la potencia que se puede obtener dei calor;
faltan 73 aõos para que Julian Robert Meyer (1824-1878) sugiera la equivalencia y conservación de
todas las formas de energía; faltan 74 afios para que James Prescon Joule publique sus medidas que
dan el equivalente mecánico dei calor; faltan 81 ai'los para que Rudolf Julius Emannuel Clausius
(1822~1888) y Lad Kelvin (1824-1907) formulen la segunda ley de la termodinâmica; faltan 129 aõos
para que Ludwig Boltzmann (1844-1906), después de fundar la física estadística, escrib iera
desesperado, coo una desesperacióo que le llevó al suicídio: •soy consciente de ser só! o una persona
que lucha débil mente contra la corriente de su tiempo". Así se adelantaba la técnica a la ciencia.
Si esto es cierto para la termod inâmica, qué no podemos decir de la mecânica, cuya historia
aparece en los albores de la propia historia de la civilización. LQué sabían dei esfuerzo cortante los
hombres dei paleolftico cuando tallabao sus hachas? lQuién había introducido el coocepto de centro
instantáneo de rotación cuaodo se realizó ese invento inconmensurable de la ingeniería mecânica, que
fue la rueda, y que ignoraroo Impor tan tes civilizaciones, como las dei conti nente americano, tan
avanzadas en otras ramas dei saber, como la astrooanfa? i. Y qué decir de las construcciones de RollUI?
Y la misma Edad de las TLnieblas, no lo fue tanto en artilugios mecánicos, autómatas, relojes y
transmisiones ingeniosas de movimientos en molinas de aire y agua como se desarrollaron en la Edad
Media. Nombres tan ii ustres en el mundo de la física, desde el slracusano Arqufmedes, basta e l hombre
típico dei Renacimicnto, que fue Leonardo da Vinci, bien pueden merecer el título de ingenicros
mecánicos. Seria un interesante trabajo de erudiáon estudiar la presencia de la ingenieria mecánica en
la historia de la civilización, pero noes esto para lo que me habéis llamado en esta sesión inaugural dei
congreso.
Lect\Jre at lhe I Congreso lbefoamericano de lngenleria Mecanica. Madrid-Espaiía- September 1993.
RBCM lnvíted paper
Fundamento Cientiliooe de la lngenieria Mecanica
166
Cuando una rama de la ingenlerfa tiene un nombre que coincide coo el de una parte clásica de la
física, como es la mecánica, parece que el tema está bastante centrado. La mecánica es aqu~lla parte de
la ffsica que estudia et movimlento de tos cuerpos y tas causas que to producen. La llamada
clásicamente mecáoica racional, porque maneja entes de razón, es el capítulo de ta física que requiere
menos hipótesis físicas. Las teyes de Newton y los teoremas de la cantidad de movimiento, el
momento cinético y ta energfa cinética permiten levantar et compacto edificio de la mecánica vectorial
o newtoniana, que da cuenta del comportamiento dei sólido rígido ante las solicitaciones dinámicas.
Casi como casos particulares, si bien es clásico su tratamiento separado y anterior a ta dlnámlca,
están ta cinemática, conjunto de relaciones espacio-temporales, verdadera geometria dei rnovimiento,
y la estática que estudia las condiciones de equitibrio de los cuer~.
Sin embargo, forzoso es reconocer que ta mecánica teórica no nació ai hilo de tos pragmáticos
problemas de ta ingenierfa, sino dei esfuerzo de unos sabios que trataron de descobrir el orden
exiStente en et Universo, ese orden que San Agustín eleva a uno de los atributos que hace que lo creado
por Dios sea bueno. La obsesión por descobrir este orden necesario en el mundo condujo a no pocos
errares científicos, corno aquéllos en que cayó et propio Kepler.
No obstante, 1)rcho Brabe ya apuntaba en otro sentido, evitando construir una cosmografia
apoyada en especutaciones abstractas y recomendando basarla en la observación para elevarse
gradualmente hasta et coocx:imiento de las causas.
Galiteo, como es bien sabido, aplica et método experimental, y Newton introduce un nuevo
método en et estudio de ta mecánlca: et cálculo. A partir dei hecho de que tos planetas describen
elipses atrededor dei Sol, situado en uoo de los focos, ínfiere que la fuerza atractiva debe decrecer con
el cuadrado de ta distancia. Ningúo m~todo experimental hubiera permitido llegar a esta conclusión,
que Halley y Hooke habfan entreviSto por ta misma época.
LI egamos asf a Joseph Louis Lagrange que en 1788 publica su Mécanique Analytique en ta que
aplica et cálculo de variaciones a las coordenadas generalizadas para ofrecer una nueva visión de ta
dinámica: et formalismo tagrangiaoo, tocai mente analítico. Esta rnodelización de ta mecáoica culmina
cuando Laplace extiende la teorfa de ta gravitación de Newton ai conjunto dei sistema solar, to que
COilSiituía un complejo problema. Según tas observaciones realizadas desde et tiempo de JYcbo Brahe,
es decir, durante dos sigtos, ta órbita de Júpiter se estaba reduc:iendo continuamente, mi entras que la de
Saturno se amptiaba. El becho había sorprendido a Newton quien, at no poder explicar
matemáticamente fenánenos tan complejos, supuso una periódica intervención divina para arreglar las
cosas y evitar la catástrofe. Laplace desarrolló tas soluciones en seóe hasta las terceras potencias de tas
excentricidades y de tas inclinaciones, antes despreciadas, concluyendo en ta invariancia dei
movimiento medio planetario. Era el paso más importante en ta mecánica celeste desde Newtoo.
Laplace resumió tos resultados obtenidos por media de cálculos matemáticos y la apticación de la
ley de gravitacióo de Newton eo su gran obra titulada Traité de Mécanique Céleste. En ella demuestra
que ct Universo, como reatización de un magnffico orden, responderá perfectamente a las teyes de la
fTsica .
Todo esto era asf mirando ai cieto, que no es precisamente el trabajo de los íngenieros mecánicos
de hoy. Pero aún en esa máquina marnvillosa que es el Universo, andaba enredando un fantasma que
nunca podemos conjurar. En efecto, Laplace no lo explicó todo, ai menos todo to que boy sabemos. Por
ejemplo, la lierra no es un cuerpo rígido, tos rnovimientos de tos fluidos que afettan ai manto y a ta
corteza, así corno las variaciones dei nivel global de los mares, debidos a la congetación o fusi6n dei
hielo polar alterao el momento de inercia dei planeta y modifican a su velocidad de rotacíón. Bt
sistema no es, además, conservativo, pues tos fenómenos de marca frenan los movimientos de tos
cuerpos celestes y, en el caso de la li erra, SUfOOen un incremento gradual de casi dos mitisegundos por
siglo.
Laplace no se planteó estas reservas y afirmó que et sistema está diseõado para una duración
eterna, mediante los mismos pónei pios que prevatecían tan admirabtemente en la lierra misma para la
conservación de los indivíduos y para ta perpetuidad de las especies.
Et fantasma omnipresente dedicado a perturbar e.l cumptimiento de las leyes en et mundo mecánico
es et rozamiento. En efecto, tos sistemas soo disipativos y ta energía mecánica no se conserva. Las
leyes de conservación son uno de tos pitares de la física. Una de ellas se refiere a ta electricidad y es la
167
Juan José Scala Estalella
conservación de la carga eléctrica, que hasta el momento se presenta inconmovible frente a los
modelos cuántioo y relativista. Otras dos se inscriben en la física de partfculas: la oonservación dei
número de leptones y dei número de bariones. Las otras tres se encuadran en la mecánica: la
oonservación de la cantidad de movimento, dei momento cinético y de la eoergfa. Las tres entran en
quiebra por la enojosa presencia dei rozamiento, pero aún en interacciones instantáneas, en que las
fuerzas intervinientes pueden ser representadas per funciones de Dirac, como choques y percusiooes,
se salvan las dos primeras, pero no la tercera.
En la Mecânica clásica se conservaban dos cantidades: la masa y la energiá. Pero Einstein probó
que la segunda no era más que una manisfestaclón de la primera. Por ello, actualmente se predica só! o
la conservación de la energiá, que puede manifestarse de muy diversas formas : cinética, potencial,
másica, electromagnética, etc. Puesto que los sistemas mecánicos son disipativos, está siempre
presente una transformación de su energiá en calor; el ingeniero mecânico no puede ignorar esta
transformación, que genera un calor con frecuencia iodeseable y que se ve obligado a difundir al media
ambiente o a sistemas refrigerantes y esto le conduce ai mundo de la termodinámica.
Esta degeneración- pennítanme utilizar esta pai abra- de la energiá mecánica debe ser minimi:Uida.
Luego nos ocuparemos de ello. Entretanto, pensemos que el calor es la forma menos noble de la
energía. Esperema; no ofender con ello a nuestros colegas de técnicas energéticas, pero ellos lo saben
mejor que nadie. La energiá calorffica es, en el fondo, energía mecánica; la temperatura está
estrechamente relacionada con la energía cinética media de las moléculas. Emonces, i,dónde está la
diferencia?. Es bien simple. La energfa mecánica es ordenada, ni'ientras que la calorífica es
desordenada. Siempre es fácil, y casi inevitable, pasar dei ordenai desorden, pero no recíprocamente.
Para extraer dei calor energía mecánica hay que encontrar un foco cal iente y uno frfo . Só! o asf se puede
recuperar algo, nunca toda. Si éstos no se encontraran se babrfa llegado ai equilíbrio térmico, único
estado perfectamente estable de un Universo, que sólo podría salir de ahí por una nuctuación
afortunada, como puede ser la que di o origen a la vida.
La disimétrica dificultad de ambas transformaciones (energía mecánica en calor o la recíproca)
queda bien patente meditando el contexto histórico en que tuvieron lugar. Moviendo entre las manos
un palo terminado en punta, coo rotación alternativa, sobre una base de madera, o bien hacieodo saltar
chispas en el golpe entre dos piedras, alguien consiguió geoerar el fuego. Quién fuera la persona o
civili:Uición dei invento ha quebrado ignorado en la noche de los tiempos, pero se sabe que ocurrió en
el paleoHtico media, cuando el hombre de Neanderthal se movía por la 1ierra, varias decenas de miles
de a na; antes de la revolucióo dei Neolftico. Se había pasado de la energfa mecánica ai fuego. El hecho
pasó a la mitologfa como un regalo de Promcteo a Jc:g hombres, después de haber robado el fuego a los
díoses, lo que le valió coroo castigo vivi r encadenado a una roca por toda la etcrnidad.
Pero la transformación dei fuego en energfa mccánica tuvo que esperar bastante más. Fue durante
la Edad Media cuaodo la tecnologra militar conoció un avance ciertamente revolucionaria que alteró
las regias dei arte de la guerra. Parece que en Espaõa lo utilizó por primera vez Alfonso el Sabio en el
sitio de Niebla: nos referimos a la pólvora. Las armas de fuego hacen su aparici6n. No queremc:g hac:er
un juego de pai abras, sino invitar a la refleldón, considerando que oo es el fuego como arma, cooocido
desde antiguo, primero en la lucha prehistórica dei hombre con los animales y, posteriormente, de los
bombres entre sr. Los romanc:g ya arrojaban con sus catapultas objetos ardientes sobre las fortale:Uis
enemigas. Ahora ya no es el fuego el que hiere, sino el que impulsa. La energía dei proyectil es, como
siempre, mecânica. Pera se ha dado el primer paso de la termodinámica: se ha aprendido a converti r la
energia química potencial de la pólvora en energfa calorffica, y ésta en energía mecánica. Falta mucho
para que el ingeniero militar Sadi Carnot (1796-1832) publique su libro "Renexiones sobre la potencia
motriz dei fuego" (1824), que puede, coo razón, ser considerado como la piedra fundamental de la
ciencia termodinámica.
Pera una cosa es lograr una explosión y otra más difícil es consegui r controlar la generación de
energia mccánica con fines útiles, eoteodiendo por tales los no destructivos. Habían de pasar unos
seiscientos anos hasta que Nicolaus August Oito (1832-1891) construyera su motor de explosión de
cuatro tiempa;, dei que él mismo decfa: "Es brutal, pero funciona" . Recordemos que la fusión nuclear
conoció también este retraso de su empleo como fuente de energía coo respec1o a su uso como medi o
de destrucción.
Fundamento Cientific:os de la lngenleria Mecanlca
168
En meclnica leórica es muy fácil cortar el nudo que supone el carácter disipativo de los sistemas,
afirmando, como es cierto, que el problema dei rozamiento es de naturaleza termodinâmica y
refugiándose en el estudio de los sistemas conserva ti vos. Pero ai ingenfero meçánico no Ie es permitido
realizar esas pi ruelas de abstracción inlelectual y se verá conducido, de grado o por fuerza, a
entendérselas con el mundo de la rermodinámica y de. la rransmisión dei calor.
Para disrninuir los efectos dei rozamiento se introducen capas fluidas entre las superflcies de
contacto. Aunque se pretendiera separar la ingeniería hidráulica de la ingenierfa mecánica, no sería
soslayable el uso de las leyes de la mecánica de fluidos en sus aspectos cinemáticos y dinâmicos y,
como recurso matemático ineludible, la teoría general de campos, que permitirá formular la evolución
dei fluido.
No siernpre se pretende minimizar el rozamiento. En muchas ocasiones se encueotra en él un
medio de disipar energía, como en los procesos de frenado, o de transformar la naturaleza dei
rnovimiento, como en el caso de la rodadura. Sea como fuere, se hacen presentes simultaneamente
fenómenos de interacción molecular, rugosidad y efectos ventosa, cuyo estudio está lejos de ser
simple.
La ingenierfa mecánica no se limita a mover objetos, sino que debe conformarlos adecuadamente,
y la idea de conformar, dar forma, nos está evocando esa parte, hoy tan olvidada de la matemática, que
es la geometria. Y no me refiro a la geometria de los hiperespacios curvos de Riemann, tan fructífera
para estudiar la relatividad generalizada, o a las geometrías no euclfdeas. Quiero llevar nuestra
reflexión bacia el peligroso olvido en que se encuentra la geometria métrica, la que se desarolla en el
espacio euclfdeo tridimensional. Esa geomet~fa, que en el siglo XIX fue mucho más aliá de lo q11e
pudi era interesar a un ingeniero, pero que tloreció en su máximo esplendor en la Escuela Politécnica
de París, con oombres tan ilustres como el de Monge.
Pcr uno de esos movimientos pendulares de la vida académica, hoy ha caído sobre ella el velo dei
silencio. Cierto es que eo un ambiente universitario en que tanto se premia la investigación, resulta
poco atractivo para los matemáticos un campo tan agotado, que dio, entre otros frutos, gruesos
vol úrnenes sobre las propriedades dei triângulo. Pero una cosa es su actual esterilidad para la actividad
investigadora (afirmación que aún bago con reservas) y otra bien distinta es que no contenga
herramientas utilfsimas para la iogenierfa. Y las cienclas básicas deben servir a esos fines. Por la
misma razón podríamos estar tentados los físicos de hacer algó muy parecido coo la Mecánica
Racional, prácticamente cerrada desde los tiempos de Poincaré. Pero sus Jeyes siguen siendo válidas y
extremamente útiles.
El ingeniero necesita de La geometrfa. Y no sólo como una ciencia que Ie ofrece unos entes de
razón, sino como unas técnicas de represeotación que le perrnitan Uevar a cabo su trabajo. iOué es, si
no, el dibujo? Hágase sobre un tablero o sobre la pantalla de un computador, siempre quedará el
desafío que supone representar métricameote el R3 en el R2
Definida la forma, es necesacio realizaria y no de cualquier manera, sino sobre materiales
determinados, coo la precisión métrica requerida y dejando las superficies en un estado adecuado. Bl
desarollo en el tiempo compcrta procesos espaciales y la geometrfa, abora convertida en cinemática,
reclama su puesto, tanto para estudiar las fases de mecanizado de una pieza, como para proyectar el
movimiento dei brazo de un robot. Como un escultor, el ingeoiero deberá arrancar la materia a la
materia o darle forma pasándola por estado de fluidez, Uegaodo a medidas muy exactas, que cada vez
más Ie acercan a las dimensio,nes moleculares. Bs impensable que en este JX"oceSO se igncre la õsica de
la materia y la ciencia de los materiales, como fundamentos científicos para llevar a buen término un
trabajo coo exigencias crecientes.
Vale la pena pensar cuáles fueron los medias que se utili1..aron para arrancar la materia de la
mareria. Desde tiempos muy remotos se di o por supuesto que para conformar un cuerpo bacra falta
otro constituído de material más .<Juro: se cortóla madera o La carne de los animales coo bachas de
piedra. Pera hubo que dar forma y filo a las propias piedras y no siempre se encontraron cuerpos más
duros. Golpeando piedra contra piedra se hacían saltar trozos de una de ellas. La primera ídea,
consistente en un proceso de corte, evoluciona bacia un proceso de erosión, en el que el agente es la
energfa.
169
Juan .bsé Scala E&talella
Esta es lo que ocurre en procesos como el cborro de arena o el corte con charro de agua y así se
evoluciona a las técnica de electro-erosióo. Es casi incre1ble, pero la ingeniería corta y perfora
actualmente coo partículas como los fotones, ique no tienen masa! ;.Quién pudo pensar que la luz seóa
una herramienta en mane~ dei ingeniero mecánico? Pera esto es asf, y la nueva herramieota tiene que
ser cooducida con las leyes de la óptica. En cuando a su génesis y en la interaccióo delláser coo la
materia no se pueden ignorar los fundamentos cuánticos de estos fenómenos. Seamos, pues, muy
prudentes ai pasar una página de un libra de física afirmando:"Este no es mi campo y, por consiguiente,
no me interesa".
Una vez constituídos los elementos mecánicos, éstos están llamados a reunirse en sistemas
alta mente complejos, interactuando entre sí de todas las formas posibles (rocladura, pivotamiento,
deslizamiento, tracción y compresión). Esta capacidad de los conjuntos que integran las máquinas y
mecanismos ya entusiasmaron a Leonardo da Vinci (1452-1519), pintor, escultor, pero tambiéo
cientifico e iogeniero. El mismo, en una carta a Ludovico Sforza, se considera ingeniero militar e
hidráulico. Sus cooocimientos geométricos, así como dei calor y la óptica, le llevaron a aplicar con
todo rigor las Jeyes de la perspectiva, creando una verdadera ~ciencia de la pintura" en la que llegó a
coo fundir la visión coo la percepción. Estaba convencido que la fuente dei cooocimiento consistfa en
la observación, la experiencia, "saper vedere''. Esta preocupación, llevada a la práctica, ha quedado
recogida en sus libras de notas, no sólo de arquitectura y pintura, sino también sobre anatomfa y, en
especial, los que sobre mecánica se enoontraron en la Biblioteca Nacional de Madrid.
Leonardo se interesa por todo aquello que, de una forma genérica, podemos designar como
cadenas cinemáticas. La transmlsión de movimiento por media de tornillos roscados, engranajes,
mecanismos giratorios, elevadores hidráulicos y transmisiones de todo tipo, centra su atención y
curiosidad. No solamente se ocupa dei sólido rfgido. como el movimiento de las flechas en torno a su
centro de masa, sino dei mucho más complejo movimiento de los fluidos: dei agua, a la que llama
''vetturnle deUa natura" (portadora de la naturaleza), y dei alre, aJ que observa por la evolucióo de las
nubes y de las masas de humo. Estudia y dibuja con minuciosidad los torbellinos y remolinos que se
originan en los vertederos, y desea llevar esos conocimientos a fines prácticos, disenando compuertas
para los canales de la reglón dei Arno y Lombardfa, asi como imaginando la maoera de remedar las
capacidades de aves y peces, en lo que más tarde serán submarinos y aeroplanos.
Mayor misterio encerraba la dinámica. Leonardo era consciente de la acdón resistente que ejercen
las fuertas de rozamiento, por lo cual tiene que existir un principio motor. AI buscarlo le falia su
capacidad observadora y s itúa a la fuerza en un plano trascendente, llamándola "virtu spirituale",
llegando así ai concepto escolástico dei Primer Motor (Primo Motore). Quizá esto pueda resultar
sorprendente, pero Newton seguirá hablando de las fuerzas ocultas de la Naturaleza. jCuál hubiera sido
el asombro y entusiasmo de aquellos hombres ai contemplar las actuales realizaciones de la ingenleóa
mecánica! .
La integración de los elementos mecánicos cn conjuntos cada vez más complejos plantes nuevos
problemas, sobre todo si se tiene en cuenta que los distintos elementos son producidos en lugares
distintos por empresas distintas. Es necesario tener una perfecta información sobre las caractedsticas
que tendrá la obra dei otro. A las técnicas dei contrai numérico de las máquinas se une el diseõo
asistido por computador, iotegrándose con la propia mecaoización. Las cosas van hoy más aliá,
apuntando ai CIM, en un esfuerw para integrar y hacer transparente la información abundantíslma que
se maneja en el proceso de fabricación.
Nuevas exigencias para el ingeniero mecánico: la informática se introduce en su mundo y el
tratamiento automático de la información se hace necesario desde el diseno hasta el montaje. Y con la
informática, la automática, que, bueno será recordalo, no fue invento de los electrónicos, sino de los
mecánicos. Ah( está, como recuerdo permanente, el símbdo de la especialidad mecánica en el escudo
de la ingcnierfa industrial: el regulador de Watt.
Mucbo hemos hablado basta aquí dei movimiento y presiento el malestar de los colegas cuya
aspiración es lograr sistemas que permaoezcan firmes, inmóviles, resistentes a todas las solicitudes.
Dccir que la estática es el único fundamento cientffico de las ciencias de la comtrucción es simplificar
demasiado las cosas. En efeao, las leyes dei equilíbrio de un cuerpo rígido se redocen a la nulidad de
la resultante y dei momento dei sistema de fuer:zas que actúan sobre él.
Fundamento Cientffioos de la lngenleria Mecanica
170
Pero los elementos resistentes no se consideran como cuerpos rígidos. El ingeniero estudia sus
deformacíones hasta el estado plástico y debe analizar su capacidad de resistencia. Su cálculo se
introduce en el seno de la materia a través de las ciencias de la elastici<;lad y la resistencia de
materiales. El cálculo tensorialle permitirá estudiar el comportamiento dei material en el entorno de un
punto y la teoria de la rotura nos enfrenta nuevameote coo la co~tituc:ióo de la ma teria, que estudia la
física dei estado sólido.
Pa- otra parte la situación se complica bastante por la oaturaleza de las solicitaciooes. Estas no son
estáticas, sino dinámicas: esfuerzos variab.les, periódicos o no, choques y percusiooes, cargas móviles,
etc., hacen que las estructuras puedan p:esentar fenómena. oscilatorios coo Ia; delicados problemas de
resonancia.
Una vez más debcmos recordar que toda la teoria física de las ondas nació inspirada en el estudio
de las oscilaciones mecánicas: cuerda vibrante, tubos sonoros, oscilaciones en placas y membranas,
etc. Se llegó asf a establecer la ecuación de ondas y fue posteriormente cuanclo la aoalogía matemática
que ofreciael comportamiento de los campos eléctricos y magnéticos variables, a partir de las
ecuaciones de Maxwell, hizo pensar en la naturalez.a ondulatoria de la luz y de la radiación
electromagnética en general.
lndependientemente de este hecho, la ingeniería mecánica hace uso extensivo de las ondas para
escrutar et interior de los materiales coo que trabaja, tanto su homogeoeidad e isotropía, como su
· continuidad y el estado tensorial a que los someten las soticitaciones externas. Los rayos X, la
fotoelasticidad, los ultrasonidos, etc., son actualmente herramientas de uso habitual. La misma
metrología ba visto incrementado el nível tecnológico mediante el empleo de sistemas láser.
Hay un principio general , que conviene tener presente: las leyes físicas son siempre
modelizaciones de una realidad más compleja. Los entes que se manejan en la ciencia soo
idealizaciones (el sólido rígido, el sólido homogéneo e isotrópo, el hllo inextensible y perfectamente
flexible, el líquido y el gas ideales, etc). De las variables que iotervienen en el fenómeno se retieoen
unas cuantas que se estiman son significativas.
La ingeniería también modeli.z a sus situaciones. AI fin y ai cabo, un modelo no es más que uo
sistema ideal que se comporta como el sistema real coo un grado de aproximación suficiente, pero que
es más sencillo de calcular, más fácil de construir, más rápido en reaccionar o más seguro de utilizar.
Cualquiera de estas cuatro situaciones pueden conducir ai ingeniero mecânico a modelizar su
problema. Pero los modelos soo cada vez más complejos, es decir, más próximos a la situacióo real.
Soo, sobre todo, los métodos de cálculo los que han propiciado esta complejidad creciente. Los
computadores han permitido Ilevar el cálculo numérico a donde nunca bubiera podido Hegar el análisis
matemático.
Aun así conviene no prescindir de éste. El análisis puede ofrecer una visión de la realidad tis ica
altamente elocuente, cuando un alud de números tienda a ocultar fácil mente la visión global dei
fenómeno. En ningún caso serfa plausible prescindir de las leyes que rigen la naturaleza, para lanzarse
a un empirismo ciego o p:ogresar a tientas entre puro cálculo numérico. Análisis, cálculo y experiencia
deben ser los sólidos apoyos sobre los que se sastenga el trabajo dei ingeniero. Ese tríp<XIe dará a sus
conclusiones una gran seguridad y estabilidad.
Hago un llamarniento a reflexionar eo esta línea a quienes tienen la responsabilidad de definir los
planes de estudio por los que ban de formarse los nuevos ingenieros. La palabra se repite sin cesar,
pero coo frecuencia se la traiciona. La; centros de enseõanza debeo ofrecer una verdadera formación y
no una mera información. La galopante evdución tecnológica exige ser prudente ai elegi r los sectores
en que deba hacerse esa gran inversión intelectual, que es la preparacióo de los futuros técnicos. Y
cuando hablo de sectores, no me refiero a especialidades, sino a esos grandes estratos que son
matemáticas, física, ciencias dei ingeniero y tecnologfas. Como babrán observado be eludido las
pai abras troncal y troncalidad, porque sencillamente las considero de poco gusto. En efeao, quizá en
esta imagen botánica a que nos lleva una nomenclatura legislativa poco afortunada, se descubrirá no
tardando que habrá muchas troncales que se anden por las ramas.
Y antes de terminar esta presentación sobre los fundamentos científicos de la ingenlerfa mecánlca,
debo confesar que he fracasado en mi intento. Mi (Ximera idea fue ofrecerles una inteligente selección .
Creo que esto es lo que se esperaba de mí. Pero ai hilo de mis reflexiones, acabaron desfilando todos
171
Juan Joeé Scala Esaalella
los campos de la física y, de su mano, toda la matemática. AJ darme cuenta de ello, resonaban en mi
memoria de los tiempos de estudiante aquellas pai abras que tantas veces nos repetra el gran maestro O.
Pedro Puig Adam, catedrático de matemáticas de esta Escuela: "Lo único que no se aplica, es lo que no
se corxx:e".
Nada hay más práctico que una buena teoria. Los profesores de las ciencias básicas somos
conscientes de nuestra ignorancia enciclopédica. Por eso nos merece mucho respeto la sabiduría
monográfica de los especialistas. Pero éstos deben convencerse que les es conveniente, de vez eo
cuando, salir de su paro y recibir la brisa refrescante y confortadora dei vasto océaoo de la ciencia.
RBCM • J. ot tlw Bru. Soe. lihchanlcal SC'-nc"
Vol. XV · n• 2 • 1993 • pp. 172·198
ISSN 0100·7386
Printed ln Bra2il
Finita Elements Analysis of Parabolic
Problems: Combined lnfluence of Adaptive
Mesh Refinement and Automatic Time
Step Contrai
Paulo Roberto Maclel
~ra
University College ol swaneea
Oept ol Civil Engineering
SA28PP • Singleton Par1<. Swansea • Wales, U.K.
Summary
An adaptíve noite element method with mesh refinemeot, by mesh enríchmeot, in time aod space and an
automatic time stepping control is described for a cla.s s of parabolic problems. The computatiooal domai o is
represented by ao assembly of 4-noded isoparametric elements. The mesb enrichmenl is achieved by locally
subdividiog each element loto four new elemenls in those regioos where further resolution is required duriog lhe
aoalysis. The Euler backward lime marchlng scheme is used and lhe time step sizes are graduaUy adapted. Botb
lhe space and lhe time discretization error estimatioo adopted involve ooly lhe computed approximate solution
and those are use to dietate lhe adaptatlon. A discussioo of tbe combined influence of time and spatial adaptatioo
in tbe cootext of parabolic problema is al.so presented Sample model applications are included to demonstrate
the efficieocy and accuracy, as well as tbe potentialities for engineeriog analysis, of lhe outlined procedure.
Keyworcls; Finite Element Analysis; Parabolic Problem; Adaptive Mesb Refinement; Automatic Time Step
Control.
lntroduction
The reliability of an approximate solution computed using lhe finite elemenl melhod is direclly
related to lhe capacity of the discrete model to represent the physical problem under consideratioo.
Basically two requirements must be fulfilled: the first is to develop techniques which can adequately
represent the geometry of the computational domain. This task requires the development of suitable
mesh generators and is clearly independeot of the problem solution. Since it is unacceptably inefficient
to construct a fine discretization everywhere, it is necessary to connect the discretization with the
solution. This is the second requirement which cao be provided once some techniques that permit an
assessment of tbe error in the solution aod a full adaptive processare combined. ln this way the
discrete model can detect the main features of the problem solutioo.
Nowadays, there is general agreement that adaptivity will become a standard feature of numerical
metbods software in the near future. A substantial progress was madé in this area in recent years.
However, there are numerous problems that still have to be overcome if adaptivity is to be used with
confidence in the solution of practical problems.
The so-called H-version d an adaptive method has received most attention for elliptic probletm. ln
this paper, one possible adaptive procedure based upon the use of such a method is presented lo the
solutioo of problems govemed by parabolic differential equations.
It is well known that the smoothness of solutions of parabolic problems vary considerably in space
and time, exhibiting, for instaoce, ioitial transients, where highly oscillatory components of the
solution are rapidly decayiog. Therefore, efficient computational metbods for this class of problems
require tbe use of mesh spacing and time steps which are variable, ideally lo both space and time. A
discussion of the combioed influence of adaptively defining the time step and tbe mesh spacing is
presented ln thls paper.
An outline of this paper is as follows: ln the next section a model parabolic partia! dlfferential
equation as well as the discretization methods adopted are described. Then, a brief comment on the
main characteristics of the equivalent discrete equations and formulations are presented. 'lhe following
section describes the errar analysis used. A brief description of the mesh spacing and the time step
control strategies are presented, with the correspmding algorithms. Practical implications associated to
Manuecllpt receMid: April 1993. Technical Editor: Leonardo Goldsteln Jr.
P. R. Madel Lyra
173
tbeir implementation are fully discussed. Some oumerical results illustrating lhe performance of lhe
procedures are reported. Fioally, the maio conclusioos are summarized anda brief comment on
possible extensiom and furtber optimizalions are presented.
The Parabollc Equatlon
According to lhe Fourier taw for conduction in continuous medium, two-dimeosionat transient
heal conduction probtems can be described (Zienkiewicz anel Morgan, 1983) by tbe equation:
in Qxl
(01)
where:
+ = temperature;
Q
2
= spacedomain (x,y) EOC!Jl ;
I = timedcmain IEI(O, T] ;
t)
= rate of heal generation per unir volume;
f(
Kx, KY = lhermal conductivity(Kx - KY • K ln an isotropic medi um);
"- = coeffictenl of convection or radiation; and
C = beat capaci1y coefficient (C - pCP), where p is the density and CP is the specific
heat at coostant pressure.
Boundary conditions:
•Oirichlet boundary condirions- lhe temperature is prescribed ( ;)
(02)
•The Cauchy boundary condirion
1. Heat flow cooditioo (q) • (a -O)
2. Convection or radiation condi tion (a (
+- +al ) , (q- O, a .. O)
(03)
in which nx and n are the components of the direction cosines of tbe unit outward normal 10 tbe
boundary, a is thl convection or radiation coefficient on the boundary, and ; is the ambient
temperature or the temperature of the externa! radlative source.
a
0
To render the inltinllboundary-value problem well pose<!, an initial condition on temperature ( )
must atso be specified at time t0 :
+
(04)
Th obtain the equivalent discrete problem lhe standard fioite element procedure was followed
(Zienldewicz and Morgan, 1983). Firs1, a spatial dlscretization was applied and oext, on lhe resulting
system of firs1 order ordlnary differentiat equations, discretization io time by finite difference.
174
Flnite EHtment Analysis ol Parabok Problema; CQmbined lnfluenoe ol Adaptive Mesh...
Spatlal Dlacretlzatlon
The computational domai o wbicb does not vary witb time is discretized by linear 4-nodes
isoparametric finite elements and an approximation solution is sougbt in lhe form
nn
t•th- ~Ni (x, y)•
-
-
4>j(t) -
!'l+~h
(05)
where nn is lhe IWmber of nodes in tbe mesb, NJ are lhe piecewise linear shape funcdons associated
with node j and q,j {t) is the nodal value of tbe temperature at time 1, kept constant.
Applying a Galerkin weight residual procedure, integrating by parts and ioserting the
approximation q,h in Eq. (01), yields a coupled system of first order ordínary differential equations.
d(~)
C - - +K~b - F
-
dt
-
with
th- ~ob
at
0
(06)
t- t
where:
!< - !<"+!<c is generically called "stiffness" matrix:
!<k is the conductivily matrix;
!<c is lhe convective matrix;
Ç is the consistem capacity rnatrix;
F is lhe independent term vector whicb arises dueto lhe imposition of "tbermaJ load" on tbe domain
ãnd/or tbe boundary conditions.
Time Dlacretlzatlon
Discretization intime by ftnite differences where
~h~ n + 1 • f~ + 1 - 4>~
(07)
tl. t" +-i
dt
in which
tl. t"+ 1 isatimestepattime ln+ I
and
;~ + e.
<1
-O);~+ o;~ • 1
(08)
where 9 is a weigbt parameter (Os e s 1)
lnserting Eqs. (07) and (08) into Eq. (06) tbe problem at time t"+
linear algebric equations, as follows
1
can be expressed as a system of
(09)
P. R Maâel Lyra
175
where:
(10)
(11)
in whicb
~, the independent term vector evaluated at time n, and ;
0
,
the initial coodition, are lcnown.
Eq. (09) has to be solved at each time step. The parameter 6 can vary from O (explicit scheme,
after lumping the capacity matrlx) to 1 (fully implicit scheme).
ln this paper, we will concentrare our attenúon on the Euler/Backward scheme. &j. (09) becomes
in this case:
(12)
A discontinuous Galerkin method (Johnson, 1987) can be applied to discretize &j. (01). lt makes
use of a finite element formulation to discretize in time, where the approximotion functions a~e
0
polynomials with degree at mail g and may be discoodnuoos intime atlhe disaete time leveis t •
Employing constant functions (g • O) on eacb time interval (I"+ ) the following equivalent
discrete problem of &j. (01) is obtaincd:
rLô
t
n+1
J
n
!C+Ç ~hn+1 • Çtb+
J f(t)dt
(13)
This expression can be recognized as a simple variant of the Euler/Ba'ikward scbeme wbere the
inde~ndent 1crm involves the evaluation of an integral of f (t) over 1" + rafher than tbe v alue of
F" • at t" + . ln pracúce tbat integral is replaced by an average of F over I"+ , íf the function f ( t)
varies smoothly in time.
Olscusslon About the Time Dlscrete Formulatlon
ln a matbematical sense, there are basically four requirements to be applied to a numerical scbeme
in arder tbat it be reliable: Consistency, Stability, Convergence and Accuracy.
The primary requirement of any algorithm is tlult ir must converge, i .e., that the numerlcal solution
npproaches the exact solution of the differential equation as tbe discretization is refined. To establish
the convergence one can appeal to the Lax &juiva.lence Theorem, wbich may be stated (Hughes, 1~7)
as: " For a well-pased initial value problem anda coosistent discreti2Jltion, stability is a necessary and
sufficient condition for convergence". Therefore it is sufficient to consider only consistency and
stability.
We will concentrate on the two-level time stepping scheme represented ln Eqs. (08-ll). The
adoption of tlult class of methods in beat conduction is extremely popular because of its cbaracteristics
of easy programrning, little use of memory ond sufficiently precise results.
Hughes (1987) has proved that schemes of tbis class are consistent. Furlhermore, the arder of
accu racy, i.e., the rate at which the discretc formulation goes lo lhe differentlal equation since t:J. t
tends to zero, is "one" for ali e E [O, 1) , except for e - 112 , in wflich case it is "two".
Stability establisbes a relation between the computed and the exact solution of the discretized
equation. It is required tbat any component of any round-off error should not be amplified witbout
bound.
Finite Bement Analyais ol Parabofic Problema: Combine<! lnlluenoe ot Adaptive Meah...
176
For linear heat conduction problems, stabílity criteria can be srudled using modal approach (or
spectral analysis) and two basic classes can be establisbed for tbe two-level time stepping methcds:
CondiJionally stable, in whicb stability imposes a time step restriction, and
Unconc6tiONJ I/y stable, wbere no time step restrictioo is necessary.
Whenever 8
~
1/ 2 on Eqs. (08-11), unconditlonally stable methods arlses; otherwise, for
~~~1, for a one-<limensional problem (Zienkiewicz and
Taylor, 1983), since the high frequency is O (h ) • can be approximately exp-essed as:
8 < 1/2, the time step has to satisfy a limlt
pc h
6
<
I
cr
2
p
2k ( 1 - 20)
(14)
wbere h is the element lengtb. ln two-<limensiooal problems, h is a representative element lenght
anel must be multiplíed ín Eq. ( 14) by 1/2 for a regular quadrilateral mesh ( 6 x - 6 y - b) (Press et
ai., 1988). ln practice a safety factor (0.90-1.00) is a.lso adopted to comider any mesh írregularity.
h is ímportant to stress here tbat even an uncondítionally stable scbeme can exhibit an undesírable
oscillatory behavíor íf no restrictíon is imposed to tbe time step size. lbís is caused by the existence of
very hígh frequencíes in the transíent response. The oscillatory limit can be expressed approximately1
for one/dimensional problems (Zienkiewícz and Morgan, 1983), as
pc h
6 I
<
OIC
p
2
4k(l -6)
(15)
The only scheme oscillatioo-free intime, withio the two-levels class, is the Euler-Baclcward which
satisfies the oscillatory conditioo whatever value is adopted for the time step. Otherwise, some
numerical procedures can be employed to diminisb or eliminate these effects either by introducing
numerical damping (Tezduyar and Liou,l990) or by weighting tbe forcing term (Bettencourt,
Zienk.iewicz and Cantin, 1983), or simple averaging successive time steps solution io some maoner
(Hugbes, 19tH). One of these ideas is to empoy tbe Euler-Backward scheme (ncxmally for lhe tirst or
in severe cases for 2 or 3 time steps) until the effects of the bigber tbermal frequencies on the transient
response have decayed and use otber scheme from this point oowards.
Once consisteocy and stability are verified, convergence is automatic. The accuracy requirement
allied with some particular cbaracteristics of the parabolic problems are considered nexL
Many beat conduclion problems tend eventually to an equilibrium, often over very loog lime
scales. This makes conditionally stable methods loolt unauractive. Moreover, very refined mesh will
nonnally be necessary to represent accumtely the solution bebavior and excessively small time steps
are required for stability reasons, even if h is only moderately small.
The semi-discrete problem (Eq. 06) is an example of a stiff initial value problem, mainly for the
case in which the factor k/ (pc ) (diffusivity parameter) becomes small and when highly distorted
elements are present. The stiffn~ is due to the fact thal the eigenspectrum of the tbermal frequencies
is very large. Higber freque.ncies are dominant in the early stages of a transient response, particularly if
drastic changes i o initlal conditions or thermal load is iovolved. As the solution tends to the sieady
state condition it is governed by the lower frequencies. An algorithm which can deaease tbe amplitude
of the higher frequencies will be desírable, and tberefore oscHiation-free schemes are indicaled.
Very little information is available on the stability of general non-Jinear discretized problems.
Within the framework of tbe \bn Neumman metbod it can be said that tbe stabilily of Jinearized
equations, with frozen coefficients, is necessary for the stability ofthe nonlinear form but it is certainly
not sufficient ( Hirscb, 1989). The generated higb-frequency waves, througb a combination of the
Fourier modes on finite mesh, will reappear as low-frequency waves and tbe solution may deteriorate.
This non-linear phenomenoo is called "aliasing", and can be avoided by providing enough dissipation
in tbe scbeme to damp lhe high frequencies.
1T7
P. R. Madel Lyra
Fa the solution of nonlinear thennal transient problems using the trapezoidal family of metbods at
time integratioo, aU schemes with 6 < 1 becomeonly conditiooally stable, as shown by Hughes(l977)
and only lhe Euler-Baclcward scheme retains its unconditional stability. Furtbermore, tbe EulerBaclcward scheme is less sensitive to the gradual deficiency caused when the reduce inlegration has
been used. This, apart from generally reducing tbe computational COSI, is particularly beneficial for
coupled problem analysis (Owen and Damjanic, 19fH).
From an engineering point of view, besides the mathematicaJ requirements, it is necessary to
consider practical questions related 10 compu ter efficiency and reliability answers beyond pre-assigned
COSI.
Tbe cboice of the so-called implicit methods requires lhe solution of a system of equations at each
time interval. On lhe other hand, the EuJer-Porward (6- O), wben a lumped or diagonal capacity
matrix (C) is used, is a fully explicit method, and no system of equalions must be solved. Tbe
computatÍonaJ advantage of an explicit scheme is accompenied by t:be severe stability restrictioo and,
as was already stressed, tbe time step could be prohibitive, even more if we bave a refined mesh.
Some technics have been proposed (Hughes, 1987; Smolinski, 1991) io order to improve the
efficiency of an expliclt scheme. Tbe basic idea is 10 combine ii wilh a domain decomposi tion metbod
where the mesh is subdivided in some groups of elements and in eacb group a rnaximum allowable
time step is used. Tbis is an interesting approach, mainly wben combined with ao adaptative mesh
refinement (Smolinski, 1991). Another idea (Hughes and Uu, 1978; Thzduyar and Li ou, 1990) attemps
to combine tbe advantages of both methods. The mesb is agai.n panitioned and the "stifr' part of it is
integrated implicitly and the remainder ofthe mesh explicitly. This is a very efficient scheme when the
regions are defined adaptively and coupled problems are being analysed. Tbe maio disadvantage of
these schemes is lhe necessity of an appropriate interchange of information across the boundaries of
the groups, which requires a specific data strucrure.
Tbe extra cost involved at each step of an implicit method is more than compensated once larger
time steps may be taken. And, normally, they are more efficient methods, particularly if an automatic
ti me step adjustment is available.
ln order to satisfy an accuracy requirement very small Lime steps are neccssary during tbe initial
transient. However, outside tbe transientlarger time step may be used. Consequently, ooe would like to
use a method that au10matically adapt the size of tbe time step according to lhe smoothness of solution.
ln tbat way, the number of systems of equations to be solved are reduced aod the efficiency improved.
The strong stability and free-oscillation behavior of the Euler-Backward scheme, together with an
accuracy control and an adaptive time step size definition presents the best choice to deal witb ali
classes (linear, noo-linear, couple, ...) of parabolic problems. Hence, an efficient strategy must be
developed to provi de cost effective numericaJ solution and robust tools.
ln transient analysis it is generally thougbt that the accuracy can be improved, indefinitely, as the
time steps decrease in magnitude. However, in the finite element ap(roximations very small time steps
may cause stability problems, which lead to pbysically unreasonable results. lt is shown by Katz and
Werner (19&3) tbat this is dueto lhe violation of the discrete maximum principie. ln other words, if a
consistent capacity matrix is used, the solution couJd present an oscillatory behavior if the space
discretization and the time step do not satisfy a stability condition which limit the time ~tep size in
tenns of the element size.
This stability condition can be demonstrated for one dimensional heat conduction problem (Rank:
and Werner, 1983). Using a parametric study an extension to two-dimension is presented in Murti,
Valliappan aod Khalili -Naghadeh, 1989, where for an isoparametric element a lower bound for the
time step magnitude was proposed to be
(16)
178
F1111te 8ement Anetysis ot Parabollc Problems: Comblned lnlluence of Adaptive Mesh...
in which y is a rorrection factor introduced in order 10 lake into account other parameters Iike: the
nature of input loading, type of etement, aspect ratio, etc. The parametric study carried out by Murti et
ai., 1989, identifies a realistic value of the rorrection factor. For an isoparametric linear elemenl in an
irregular mesh y s 2.0 was rerommended.
ln a stiff problem oormally the accuracy requirements impose a time step size that make it difficult
or even impossible to choose a space discretization which satisfies the stability rondition (Hughes,
1'177). These difficulties can be overcome if a lumped capadty matrix is used. ln this case no limit on
the minimum size of the time step is imposed, only the accuracy (or physical reasons) will determine
the time step size. Therefore, even using an implicit technique a diagonal "mass" matrix is
recommended.
A Posterior! Error Analysis
A proper errar estimator to deal with a parabolic problem must take into accountthe two main
errors involved in the discretization: the space discretization error and the time discretization errar.
Over tbe past severa! years substantial progress has been made in the development of a posteriori
errar estimators: Babuska and Rheinboldt (1978);Gago et ai. (1983); Zienldewicz and Zhu (1987,
1991); and Johnson, Nie and Thomee (1990). These mostly deal with the spatial discretization error in
elliptic problems. ln this section a brief description of the error estimators adopted in the present paper
is presented.
Using the Disrontinuous Galerkin method, Johnson, Nie and Thomee ( 1990) demonstrated a
suitablc a posteriori error analysis for parabolic problems. Their dem?n~tration was based on the
1
assumptions that: a) The exact solution is bounded; b) The step ratio ó. t ,. I ó. t is also bounded; c)
The ratio ó. t' l t' is sufficient1y small; d) The spatial discretization satisfies tbe regularity rondition; e)
The initial time steps are selected separately. The derived errar estimators for lhe variant of the BulerBackward, given in Eq. (13) can be written as
for n
1, ... , N
m
(17)
sup 11 v (s) li L and 11 11 H, Is tbe norm
in which 11 11~.-: is Lrnorm, llvll.., 1
SE (0
2
corresponding to the Hilbert space H2(Q). ' •
For more details about this expression we rccommend Johnson (1987) and Johnson, Nie and
Thomee (1990). ln Eq. (17) the first term on lhe right hand side bounds the time discretization error
and the scrond term tbe space discrelization error.
Spatial Discrete Error- Elllptlc Problema
Three maio types of a posleriori errar estimarors can be distinguished. First, the residual type of
error estimator, presented by Babuska and Rheinboldt (1978) and !ater by Gago et ai. (1983). Second,
ao interpolatioo rype of estimator (Demkowicz, Devloo and Oden, 1985; and Jobnson, 1987). And
finally, the pC6t processing type of estimator, whicb is cornputed using a rerovered higher arder finite
element solution presenteei by Zhu and Zienkiewicz (1987).
ln lhe evaluation of the space portion of the discretizalion error we adopted the third type
mentioned, since it is easy to be i~plemented, with no additional data strucrure reqoired and fast to
compute, because only the first derivalives have to be evaluated. Furthermore, it gives not onJy global
but also element error estimation.
A local solution errar is glven by
e
-+
- ;
-!I>~
"
(18)
P. R Maàel Lyra
179
<r, in terms of tbe gradient of lhe nodal unlcnowns, by
(19)
Using lhe discrete energy norm, the error over an eJement is defined as
)dO~
1
1
1/2
(20)
- [J <eTo - e )d0+ J (e 1..e
-o- ·o
-+ -+
Qj
Qj
r
where 8 is an operator tb.at relates the gradient of the varlable with itself. The global errc:x is
11·.11,- [~~ (I ~.IIJ
(21)
ln order 10 have a normali7.ed error ii is useful to define the parameter rJ such Lha!:
(22)
which provides a measure of the global error of a given flnire element mesh, and where
ll9hiiE -
[im
J (9~!)-lpb)
J - l QJ
dO+
(9~Àth) ll/2
o,
J
f
dQ
(23)
Other norms may be uscd, like the L 2 norm in wbich case the constitutive matrix (!>) mus1 be
replaccd by the ideotity matrix in tbe above equations.
ln the standard F.E.M. formulation the gradient in a typical element is evaluared by direct
substitution of tbe local variables field i o an appropriate constitutive equation:
(24)
wbere tb - ~ ~b
Since, only tbe continuity of the unknown, and not of its derivatives, is impa>ed in tbe analysis, the
unknown gradients are discoorinuous and unrealistic. lndeed that element gradients a re mapped
completei y out of the subspace of the interpolation fuoctions, i.e., they are not ín the finire-dimensional
subspace of the space comaining the actual unknown function ' ·
Braucbli and Odeo (1971) and, independently, Hinton and Campbell (1974) have s hown a
consistent procedure for calculating stress (gradients) in the F.E.M., wnich is based on the idea of
conjugate approximations. Wlth only a small increase on computation a gradient field consístent witb
tbe approximations can be obtained. Moreover, íf tbe unknown approximations are contiDlJOUS, tbe
consistent gradient will also be contiouous and will be m<re accurate, in a mean square sense.
Finita Bement Analyals ol Parsbolic Problllms: Comblned ln"-'enoe ol Adapti\le Me&h...
180
The procedure presented in Brauchli and Oden (1971), and i o Hioton and Campbell (1974) was
later extended by Zienltiewicz anc:l Zhu (1990 and 1991) as a local recovery technique which is io fact
a particular projection technique to recover lhe derivatives (gradients) of the F.E. solution. A
systematic procedure for computing consistent, cooti nuous gradient distributions in F.E.
approxirnation can be summarlzed as follows:
Com~te the fundamental rnatrix
~
Nclm
_ :L ~j :L r1~!~odOl
Lo,
J
_
j _ 1
Com~te
(25)
j - 1
the discontlnuws gradient field using Eq. (24) for each element and evaluate
Nelm
~- L J <~!~h)do-
Nclm
{L LQJIJ <~!P!J)dol}~b
J
(26)
j- 1
j - L Qj
The consistent gradienl distribution over the entire domain is obtained after solving the following
linear system of equations
(27)
Noting the best quality of the gradient obtained usi ng the projection technique, Zienldewicz and
Zhu (1987) proposed an error estimatos where the exact gradient 9 (not available) of the solution is
replaced by the consistent gradient 9 in Eqs. (19-22). The second term in Eqs. (20) and (23) is
omitted since it is of blgber order.
Reliability of this error estimator has been established mathematically. Also lt has been
successfully used in adaptive analysis for many types of elements in different physical problems.
Remarks:
a. Eq. (27) can be approxirnately solved by using a Jurnped matrix ~L such that:
(28)
A successively better solution can be obtained by iteration using
....,.(n)
9
..... (n-1)
- 9
+M• L
t( P-M9...,.(n-1))
•
•
(29)
ln practice the iteration must be carried out only a fixed number of times (normally 3 or 4), if ít is
judged important.
b. The best and most obvious projection (Brauchli and Oden, 1971) is obtained using
N
- -N •
- o
(30)
Otber projections, even using a simple nodal averaging of the F.E. gradients, give good resuJts
(Brauchli and Oden, 1971; and Hínton and C8mpbell, 1974).
P. R Madel Lyra
181
c. lt is interesting to stress tbat lhe procedure above presented can be used repeatedly in order to
determine an approximation to a higher order derivative, as can be seen in Zienlclewicz and Taylor
(1989).
d. The contiouous gradient field obtained using the projectioo tecbnique does oot satisfy the
overa!l equilibrium conditions. If it is judged important, tbe residual that results can be minlmized by
the application of Loubignac's iteration scheme (Cantln, Loubignac and Touzot, 1978). Also, if the
procedure described above is to be used in a piecewise homogeneous problem it must be carried out
over each material separately, since the gradieot jumps are an iotegrated part of the problem in110lving
multi pie material.
e. Based on rhe well known existence of some exceptional points of the domain, known io advance
for certai o elements, wbere the FE solutlon is superconvergent, Zienlciewicz and Zhu (1990 and 1991)
recently proposed a more accurate way to improve tbe gradient of the solution. They have
demonstrated tbat for even shaped fuoctions tbe oew procedure couJd represeot a big improvement if
compared with the projectjons described before. Nonetbeless, for linear elements, the only advantage
of the new scheme is tbe fact tbat no global system must be solved.
Time Dlacrete Error - Parabollc Problema
To evaJuate lhe time fractioo of lhe discretizatioo error we adopted the procedure proposed by
Johnson (1987), whicb is very suitable and easy to use as an indicator. 1t is also efficient in tbe sense
that almost optimal, and not extremely small, time steps are geoerated wheo this est.imator is used to
adapt lhe time step.
The second part of Eq. (17) can be writteo as
li ~o 11 ~me
1
1
s Cd
t" + 11
~h 11 .. 1 •
• •
c
li
;~ + 1 - f~ li L
(31)
2
where c is a constant, solution independent io linear problems, and 11 ~h li was replaced by the discrete
equivalent in central difference. This expression represents an estimate of tbe time discrete error and
must be limited by a pre-assigned tolerance 6 during tbe time integration.
ln arder to have ao efficient algorithm the error estimator must be kept close to the tolerance. ln
that way we must seek the time step size by solving the following nonlinear problem.
which can be linearized taking the estimare of 11 ~h li by tbe numerical tangent at the end of the time
interval ln , and the next time step may be estimated as
with
n~1
(33)
Tbe error estimator is normalized usiog tbe present solution:
11 ~"IIL
~
1
- ---------------(11~"11 ~2 ll;~li~J
112
(34)
+
Matbematlcal vaJidity of this simple error estimator can be found in Hughes nnd Liu (1978). Also
some numerical results for one-dimensional problems are presented there.
Flnlloe Bement Ana~ of Parabolic Ploblema; Comblned lnlluenoe ol Adapti\le Me.n...
182
The error analysis, in wbich the total error lo a specific oorm is estimated and the relative
contributions of lhe io<ividual elements is indicated, is by itself important to help in lhe interpretation
of the results. Also, it can be c::ombined witb a full adaptive process of refinement to govem tbe
adaptation.
Tranalent Adaptlve Procedure
An adaptive algorithm may be consldered to be a computational procedure for constructing a
discrete model for a given problem such that, ideally, tbe error of the corresponding approxírnate
solution is within a given tolerance, in a given norm and such that the number of degrees of freedom to
be aoalysed is minimal.
The main aim of this paper is to presenta procedure to deal with parabolic problems where both
space and time are adapted automatically, based in an error estimation whicb involves only the
computed solution.
The previous adaptive procedures developed to solve parabolic problems have based their scbemes
in the interpretation of tbe parabolic problemas a succession of elliptic problems, and defined the
adaptive strategies based ooly oo the spatial error. Since a I~ of acx:uracy at ao intermediate step can
n~ be recovered in a future solution it is necessary to cboose a very small time step to guarantee that
tbis component of the discrete error is not important when compared to the spatlal discrete error. ln
such a way there Is no advantage of the progressively smOdh nature of most of tbe parabàic problerns
is used.
There exíst many otber methods in the literature to obtain ao automatic time step control for
parabolic problems. Tbe adaptation is normally based ln certain c::omputation on a local error
ttuncation. The maio disadvantage of lhis procedl.l'e is the cost involved, since they require i o priociple
lhe results of a bigher arder metbod.
As stressed above the parabolic problem (Eq 01) can be ioterpreted as a sucessive solutioo of
elliptic problems. ln that way, the time and space discretization can be decoupled and lhe errar analysis
and c::onsequent adaptation proceed independently. The proposed strategy wbich coordinates the
adaptation can be surnmarized as follows:
A. Spatlal Control
1. Generate the model;
2. Determine the time step and advance tbe solution over (see details ln the algorithm below);
3. Update lhe iodependentvector (p-);
4. Estimate the spatial discretization error (B<p. 20, 21, 22);
5. Refine tbe mesb according to the refinement sttategy;
6 . lf atleast one element was refined in the step 5:
Then:
1
. Reset time over -6 t" + ;
. ai 1n+l -u
,. n+l
. . ld
. Store the soI utlon
1
as .101t1a
ata;
. Go to lhe step 1;
Otberwise:
. lf the time of tbe analysis bas elaJl'õed or tbe solution reacbs tbe steady-state:
. Stop.
Else,
. Carry oo witb the time integration;
. Goto step 2;
Remarlcs:
P. R. Madel Lyra
183
A.1- To start the time integration a time step equal to "lOE-03 times the stability timestep limit for
the explicit method" is adopted. This Is done to reduce the discrete error [See Eq. {17)] only to the
spatial component. ln this way, for the first step this algoritbm falis back into the steady state
algorithm, where the mesh is refined until the error estirnated is below tbe pre-assigned tolerance. Anel
then it works, initially, as a pre-processor in the transient analysis. Moreover, the use of lhis idea
produces an initial solution whicb will be used to evaluate the error intime after the first time step.
A.2- ln parabolic problems, where lhe solution cbanges smoolhly intime, lhe "ojtimal" grid used
at the previous time step should be a very g<XXJ approximation LO lhe desired grid at the advanced time
step. It is sometimes important in tenns of computational efficiency to define a number m of steps,
after which the spatial error is checked. The number of steps, that the analysis can advance in time
before changing lhe mesb, depends on the problem under consideration, and it is important to keep in
mind that if some cbanging feature of the solution is lost at an intermediate step, then this loss may not
be recovered in future solutions.
A.3- The solution of the resultant system of algebricequations is performed by the Preconditioned
Conjugate Gradient algoritbm implemented in an element- by-element procedure. Tbe coarser grid
solution is also used as ao improved initial guess in a similar way as tbe strategy so-called " nested
iteration" (Brigg;, 1987).
A.4 - An initial structured mesh is used, and after local refinements non-conforming nodes are
introduced and an additional data structure must be used. A tree data s tructure to define the
connectivity is used. To ensure that the method is confonning, restrictions are imposed at element levei
(Demkowicz, Dcvloo and Oden, 1985; and Lyra et ai., 1988).
A.5 - There are severa! refinement srrategies based on tbe "Principie of Equidistributioo of error".
Convergence ratio, reliability and cost depend on lhe chosen strategy (Lyra et ai., 1989). A common
problem of adaptive refinement procedures, is that they lend to concentrate an excessive amount of
elements about the singularities. Tbe reason is that, as the strategy tends to equidlstribute tbe error, it
will sacrifice accuracy in lhe regions wbere tbe solution is smooth to diminish the error at tbe point/line
of singularity. A srrategy to provide acceptable results for problems with slrong singularities was
applied. The basic idea consists of eliminating tbe error associated with tbe singularity in order to
capture other features in the solution ( this strategy is even more important wben byperbollc or
advection-dominant parabolic problems are considered). A brief description of the adojted strategy is
given below:
. Evaluate the allowable error per element in tbe whole domain ( Q) :
e
pe
- toll •
[
li~.li~+ 11th 11 ~]
112
(35)
Nelm
. Splitthe computational domain in two parts:
r:t • - {UOe/levei e <Maxlevel
where Q -
o.•
or lle• +llcE <e pe }
(36)
u o· . and n• no· . .. cll
. Evaluate lhe allowable error per element, purging the error thar corresponds to the sub domain
o• :
(37)
1&4
Finlte Bemenl .Analy&is of Parabolic Problems: Combined lnfluenoe of Adaptive Mesh...
Define the elemeots to be refined using epe• • :
lf levei <Max levei and 11 e li c ~e • • then refine element "e".
-+
c
E
pe
Another important measure mentioned durin; the analysis of the numerical resulls is the
normalized error correspondent to the sub domaln O • :
••
TJ
<11~.11~> r;l •
- - - -- -- · 100(%)
(li ~.11~ +li ;bll!) ~
2
(38)
A6- Normally in heat conduction problems tbere are special features, such as beat sources, heat
flux, corners, obstacles, which are assumed at some specified pa>itions. The refinement capabilities are
necessary to deal wilb singularities originated at lhese special points. Nevertheless, ao unrefined
capability can represem or not an advantage since a more complex data structure is needed and ii is
necessary to computation to eli.minate elemeots, nodes and reorganize the data structure. An unrefined
capability must be used at hyperbolic or advection-dominated parabolic partia! differentlal problems
where the solution migrates over the grid. Another scbeme extremely effective is the use of ao adaptive
unstructured remeshing technique, as presented by Peraire et ai. (1987).
B. Tlme Control
1. Compute~";
2. Estimate the time discretization error (Eqs. 31, 34);
3 . lf the normalize<! error is smaller thao 6 :
Then:
. Evaluate t:J. 1° • 1 using Eq. (33);
•
O+
J
. Advance ttme owr t:J. 1
;
Else:
. Resettime over t:J. t\
n +I
6 I
. .Takc: t:J.t
-2
Remarks:
8.1 · Tbe smallest time step obtained ai the beginning of Lhe algorithm is kept in me.mory to be
used, as an initíal guess, in the case of the existen<:e of more than coe transíent oo the problem solution.
8.2 · Owing to the discrete nature of tbe finite element, the idealizatioo has to be in such a way that
the highest frequeocy in the finite spectrum to be predicted accurately can be represente<!. As argued in
Hirsch (1987), the shortest reasonable wavelength (Àmio) represented oo a mesb wii~J!ze h is
Àmi - 2h (The spacing (h) used 10 compute the time step is taken as h - (area)
of Lhe
smaYiest clement). Based on lhis physical argumeot aod on lhe time that this fastest decaying
compooenl is reduced to 1/10 of its initial value, Sampaio showed, for a one dimensional beat
conduction problem, that a time-scale approximately balf of the stability límit gíven by Eq. (14) can be
defined for lhe problem.
This time scale is adopted as the first real time step to start the time integration. Tbis is a very
rigorous criteria and using it one can expect oo (or ooly very little) reduction io lhe time steps at lhe
beginning of the integration in time. Tbe procedure carries on with the time step being adapted
(normally increasing) during the interval of the analysis.
lf the error tolerancc is extremely severe (or ü the space discretization is not in agree.ment with the
time discretization dictated by tbe error) ao excessive reduction in the step when lhe problem presents
a sharp initial condition is expected. ln real applicatioos, it is not necessary to continue this reduction
since it is caused by a singularity in the continuous model that no real problem displays (there is no
instantaneous load or change in tbe boundary conditions), Tberefore, in similarity witb tbe space
P. R. Madel Lyra
185
discretization, ·a I()Y{er limit in the time discretízation is imposed. A factor of 1/5 ofthe stability limit is
adopted. ln that way only in a extremely small interval tbe error cri teria c:an not be satisfied, if so, but
outside this interval it is guaranteed that the solulion is below the pre-assigned tolerance. lf a COil'Sistent
"mass" matrix has been used, lhe lower bound given by Bq. (16) must be imposed.
B.J - A li mil in the relatioo between 6 ln+ 1I ti tn was imposed (normally from "1.5 10 2.0"). lbis
is neccssary becouse of the assumplions on tbe demons tration of the error (Bq. 17), and thus has a
practical interpreuuion relatcd lo the apfl"oximate nature of the error estimation. ln the examples tested
here this restricrion was activated only in the initial transient, when the initial step was very
conservative and when the problem was close to the steady-state solution, where the step started to
grow very steep once the global error decreased fast.
B.4 - There a re severa! ways 10 construct a lumped "mass" matrix. The "Special Lumping
Technique" developed by Hintoo, Rock and Zienkiewic:t (1976) is adopted. The reason for this choice
is that for ao arbitrary element it always produces a positive diagonal matrix. For an isoparametric
linear elernent it is equivalent to the row-surn technique or to the nodal quadrnture rule.
8 .5 - No study was done in order to investigate the coostants invdved in the error estimator given
by Johnson (1987). This could be important if a more sharp error estimation is desired for a special
problem.
Using the described procedure, lhe main objective of bounding lhe discrele errar by a given
tolerance, can be achieved, as will be shown in the sample model problems of lhe next section.
Application
From an efficiency poin1 of view, i 1 will always be beneficial to begin an adaptive approach wi1h a
suitable mesh, provided by a pre.analysis based on the analyst experience. H()I.Vever, in order to stress
the adaptivity ideas, very coorse initial discretizations are adopted in the following examples. The
tolerance adop1ed in Lhe error analysis is not very light and tbe Lrnorm is used in the examples
analyzed.
The first application focus on Lhe time step adapúve procedure. The geometry, material property
and initinl discrete modelare described at Fig. 1.
4m
k
1m
-
o
= 1. O \1/m-C
J o
o=
0
e • oc·
c
1. O Wsm-C
X
I
I
Fig. 1 lnltlal Olacret. Modal and Problam 1 Oeacrlptlon
Thc tolerance adopted in both space (TJ) and time (I') discrete error was 10%. The maximum
1
0
levei of refinemeot adopted was 6 anda bound of 3!2 was used in the relation fl t + I ó 1°:
Finlte Element Analyais ol Parabolc Problema: Comblned lnlluenoe ol Adapdlle MMh ...
186
The final mesh obtained is shown in Fig. 2.
Fig. 2 Final Meeh
lf lhe code has the capability of derefinement, the final mesh expected is tbe initial one, for this
specific problem. But the presented mesb allows the transieol solution, as shown in Fig. 3. The resulls
1.2
1
Q)
Â> .- .- . -à-·- · - . -~:;- . - · -· -A-·- ·- ·-·-·-·"X·)(.......
)( )(.
.... ~.....X
~~!e ..:t..~Ã- - ·Â · - <À · -
············-~............ x.............)(............................
0.8
~
:s
m
~
0.6
·. ·o.
.·o.
~
Q)
o.
0.4
E-4
0.2
eQ)
B.
time = 0 . 01
time = 0.12 --+--·
time = 1.02 . 8 ..
time = 10 . 5 -X···-time = 97.5
• •• -r.t
&.;~'-' • • • • -
o
-0.2
o
0.5
1
1.5
2
X ( y =
2. 5
o. o )
3
3.5
4
Fig. 3 Temperature Dlatrlbulon at Sample llmee
are in agreement witll lhe analytical solution presented in Damjanic anel Owen (1982) aod free of any
space oscillatioo.
P. R. Maciel Lyra
187
The time step adaptation can be seen in Fig. 4 and the time discrete error, both the L2 and
0..
Cl)
+J
(f)
50
45
40
35
30
25
20
15
10
5
o
o
20
40
60
80
100
Time
Fig. 4 • TliM S.p Evolulon
maximum oorm, in Fig. 5. The temperature agaínst time for sorne sample points are presented in Fig.6.
0.8
-o
0. 7
..............~ .............. ~................ :.. !:;2 ··· ~
.........
.
0.6
..
...
..
..
.
..
..
..
.
...
..
...
...
.
................................
. .................
..
..
.. ..........................
....
...
..
.
.
.
..
.
.............................. . ......................................... ........
.
.
..
..
.
....
..
...
..
.
................................................................ ...................
..
..
..
.
..
...
..
.
.
.
.
cJp
0.5
~
0.4
~
~
til
.
• • • • • • • • • • • • ·:-- . . .
•• o . . . . . . . .
0.3
0.2
0.1
o
o
..
-~
: maxiinum
. . . . . . . . . . . . ~- . . . . . o . . . o • • • • • •
•• o . . . . . . . . . .
o-~-
-~- ·
! .. ... .......... - ~ ...... -- .. o .. .
••••••••• • • •• • :
.. .............
-~
o
. . . . . . . . . . . . . ..
.
.
•
•
o
•
•
•
20
40
60
Time
Fig. S - llme DIKretl Error Evolutlon
•
.
.
o
80
100
Finite Bement Analy&is of Parabolc Problema: Combined lnnuenoe ol Adaptive Mesh...
188
1.2
1
0.8
X =
Q)
~
;::l
+J
IÓ
X
X
0.6
1.0 -+-
= 2. 0 -+--·
=
4. 0 ·0· -
~
Q)
o. 0.4
E
Q)
E-4
0.2
o
o
20
40
60
80
100
Time
Fig 6 • ~m,-ratura w. Tlme
Once more no oscillation and an agreement with the analytical solution were obtained.
ln a-der to empbasise tbe impa-tance and tbe efficiency of tbe time step adaptation procedure used,
the sarne problem with the mesh presented in Fig. 2 was computed using a fixed time step equal to lhe
smallest one (~ t- 2.44E-04) obtained when using adaptivity. A detail of the time error at the
beginning of the transient is presented in Fig. 7.11 can be seen that the time error is kept below and
close to the tolerance adopted ( 10%), which is in some sense optimal. The spatial errors present similar
values, the global estimators (TJ) are basically the sarne (5.9%). However, the purged estimators
(,.... •) present some difference, Tl• • - 3.22 with adaptation and Tl• • - 0.64 witb fixed step.
This is an interesting result that demonstrates an interdependence of the two parts of the discretized
error. To cover the time of the analysjs presented here only 18 steps were necessary against 130 wben
no adaptatioo is used. A detail of th.e solution along the dcxnain at tbe end of this interval shows no Jost
of quality in the solution when the adaptation is used (Fig. 8).
It is very important to use a lirniter in the relation between lhe step size in eacb interval to preserve
the quality of the solution. 1t was observed that a compromise between the cost and quality of the
solution is influenced by tbis parameter. If tbe maximum norm 11 11 .. was used one could expect more
freedcxn in tbat choice, but it could be conservative. The use of a technique for elirninating the danaln
wbere tbe solution bas converged (presented by Devloo (1987) in a different context) seems to be
dwbly promising. Firstly, ii reduces tbe number of equations to be computed; and secoodly, only the
region where the transient solution still exists wiU contribute to. the _error evaluation, whicb results in
P. R. Maàel Lyra
189
0.15
fixed
adapted
0.1
~-·
~
o
~
~
0.05
til
o
-0. 0 5
o
0.01
0.03
0 . 02
Time
Fig. 7
Tlme Dl.crete Error Hletory Uelng Flxed and Adap11ve llme Stap
1
ada pted o
fixed
0.8
Q)
~
;::J
0.6
.w
cn
~
0.4
Q)
o..
E
Q)
E-4
'
~
~
0.2
~
o
··· ······~~~~---~- --~---~-------
o
Fig a
0.5
1
1.5
2
2.5
3
3.5
4
Coord. X ( y=O.O )
Oetell ct1 the lHiperatln et.trlbutlon at T1rne 3.1 E-42
reduction i o tbe iocrease of lhe time step. Coosequently, good results are expected for the whole
domai o when tbe solution is close to the steady·state, even if a blgger value of the limi t in ti me step
relatioo is used.
Flnite Ektment Atlalysis ol Parabolic Problem&: Comblned lnlluenoe of Adaptille Melfl...
190
The second application consists of ao "infioite" domain witb a punctual heat source. The finite
discrele model adopted in lhe analysis and lhe physical cbaracteristics are presented in Fig. 9. The
Fig. 9 lnHiel Olecrete Model end Problem 2 Oeecrlptlon
domain was lruncated a1 a distance r - 250.0m, the adopted tolerance in bolh space (TI) and time
( f.l) diseteie error was 20%. The maximum levei of re6nemen1 was agaio 6 and the limil in the time
step rela1ion 2.
Punctual heat source:
limit~(t>O)
2JtrK
Boundary conditions:
~
•
~o
r-. oo ( t >O)
lnüial conditions:
~ - ~o
Vr
01:: O
at
1 -
where: ~o - 0 .0°C
k - 1.0 W/m°C
3
C - l.OE-8 Wsm °C
O
191
P. R. Madel L)'fa
Q- 1.6 W/ m
The final meshes obtained using the transient and also the steady-state strategies are presented in
Fig. 10, andare similar. The global error estimators (Tt) in the steady-state for both strategies are
•>
b)
Fig. 10 FIMI M..ha: a) lta.-lentAnalyala; b) StMdy-Stata Analyala
192
F1111te Bement Analyllis of Patabolic Problema: Comblned lntluenoe of Adaptiw Meeh...
37.33% and 37.59%, and tbe purged erra estimators (rt •) are 6.99% and 4.76%, respectively. The
criteria adopted to deal wilh higher arder singularities (Remark A5) was activated in bolh cases.
An outline d lhe solution distribution for some sampte times is depicted in Fig. 11. The solution at
1.8
1.0E-6
1.0E-5
1.0E-4
time= l.OE-1
steady-state
1.6
1.4
Q)
~
1. 2
~
1
;:J
~
~-·
·B··
X
~o. 8
e
~o. 6
0.4
0.2
o
-60
-40
-20
o
20
Diameter
40
60
Fig. 11 ,..mpera-.re Ole1rlbUllon at Sample nmee.
time l.OE-04 and also ai the Sleady-state are presented in Fig. 12, logether with lhe "analytical"
solution (in fact, an approximate solution using the second class Bessel furr;:rions).
The results are not very good since a crude toleran<:e in lhe errar analysis was adopled and also
because lhe "analylical" solution could not be exact enougb. Tbe space and time discrele errar
evolution are presented in Fig. 13 and 14.
The incompalible limit in the maximum number of refinement (maxlevel- 6) with the
tolerance adopted aod also tbe use of the cri teria to deal with bigher order singularities 'can be observed
in Fig. 14. More refinemenl in space must be allowed in order to reduce lhe global errar to lhe preassigned standard (20%); however, outside tbe neigbbabood of the singularity tbe errar was kept well
below the toleraoce. The errar intime also depicts the behavior of tbe problem, in wbicb a beat so1uce
was suddenly activated at lhe beginning. lf no constraint was adopted the time step obtained by tbe
adaptive strategy was 2.33E-12 Once tbe stability time step limit in tbat problem is equalto 9.54E-09
tbe use of tbe lower limit described in remark 8.2 is justified.
The time step evolution presents a similar bebavior as lhe one presented in Fig. 4. Tbe solution for
some sample points are presented in Fig. 15.
P. R Madet l'tfa
193
1.8
1.8
""'::J 1.5
1.5
<V
~ 1.2
1.2
0.9
0.6
0.3
""'0.. o. 9
<V
e o. 6
<V
~o.
3
o
o
o
100
200
Diameter
o
100
200
Diarneter
Fig. 12 • Tempe111ture Dlttrlbutlon: a) At time 1.0E..O.: b) At St..dy.Stata
70
global -+--purged -+--·
60
--
50
o
30
cJp
40
~
~
~
~
20
10
o
o
0.0005
0.001
Time
Fig. 13 ·a.,_ Diecrete EITor Hlatoty
Using the mesh described in Fig. lOa, with a fixed time step equal to tbe srability limit, the sarne
problem was run. The detail o f the begínning of the discrete time error evolutioo is presented in Fig.
16. rt can be noted how conservative and inefficient the use of fixed time step can be, since the error
becomes mucb smaller than tbe tolerance adopted.
Flnlte Element Analyais ol Parabolc Problema: Combined lnlluenc:e ol Adaptil.<e Meth ...
0.8
0.7
-
0.5
~
0.4
~
0.3
o
~
l:al
L2-+-
maximum -+--·
0.6
dP
1<-+.
'+
' +...... ...... __
0.2
0.1
o
194
--...._._
o
--- ---
0.0005
Time
---- -----0.001
Fig. 14 nme Dlec:,.te Enor Hletory
Q)
~
~
~
ro
~
Q)
~
E
Q)
f-4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
x= 31.25 ~
x= 62.50 -+--·
x= 156.25 ·B··
r:1
q·"
8
• • • • •
.r.'\. ••• •• • ••••• ••
El ... .... .... "l.:T
o
o
0.0005
Time
0.001
Fig. 15 Tempennur. Dletrtbutlon 1t Sim~ Polnte
A reasonable agreement was achieved for the solution using fixed or adapted steps; these results
are presented io Figs. 17 and 18. It must be noted tbat to reacb tbe time levei 0.1, which already
cbaracterizes the steady-state, only 83 steps were necessary witb adaptive time stepping, agalnst
approximately 1E+07 steps that woold be necessary ifthe step was kept fixed uotil tbe sarne time levei.
P. R Maciel Lyra
195
1
I
fixed
adapted
0.8 -
-~-·
!
-
0.6
-
0 .4
-
0.2
0
-
~
,..
~-~~--~---~-----~------~--f- ................................... .. .
I
o
le-06
Se-07
Fig. 16 Tlma Dlacn~ta Error Hlatory Ualng Flxed and Adapha11ma Stap
0.3
fixed
adapted o
<lJ
~
~
0.2
+->
ro
~
<lJ
o.
E
<lJ
0.1
f--4
o
o
Fig. 17 Datall
5
10
15
Diameter
ot tha Tamparatwa Olatrlbu1Jon at Tlma 0.36E-os
20
196
Flníte Elemtmt Analyás of Parabollc Problema: Combined lnlluence of Adapllve Mesh...
0.06
fixed
·a dapted
0 . 05
Q)
~
::l
.w
C'O
1-4
Q)
o..
E
Q)
~
0.04
0.03
0.02
E-t
0.01
o
o
2e-06
le-06
3e-06
Time
Fig. 18 Temperatura at x -
31.25
Conclusions
Adaptive procedures save time in data preparation or mesh input data and provi de more efficient
and reliable resulls, mainly with poor prwictabilily a in which the analysl has no previous experience.
Although the analysis presenled for lhe simple model problems described are nol conclusive, lhey
give some insight in tbe characteristics of the proposed transient mesb refinemenl procedure. It is
important to emphasize lhat lhe 1ecbniques adopled are robusl ín lhe sense lhal they are stable and free
from any lcind of oscillation.
ln order to have a full adaptive scheme for parabolic problem it is neoessary to adapt the slep also
in space. Johnson, Nie and Thomee (1990) have already proposed some ideas in this area. Another
approach consists in subdividing tbe mesh and having an implicit-explicit algorithm wbere in the
implicit part the time step is adapted sim.ilarly to the way shown bere.
To ensure major flexibility and to allow more complex problems, as for example, when lhe beat
source is moving, Dechaumphai and Morgan (1990) the derefinement capabilty is essential, and the
unstruclured remeshing technique presents ilself as lhe besc choice to be combined witb the ideas bere
described.
Acknowledgment
The author would like to acknowledge the financial support of CNPq (Brazilian Research Council)
and to thank his colleague Dr. Mareio M. de Farias for lhe &glish revision.
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Vol. XV· n• 2 · 1993 • pp. f99o207
ISSN 0 100-7386
Printed in &a2íl
Comportamento Mecânico em Compressão
Uniaxial de um Compósito Epoxi-Aiumina.
Uniaxial Compressíon Mechanical Behavior of an
Alumina-Epoxy Composite
Fabio Horta Motta Marquee da Coeta
Joe• Roberto Moraee d'Aimelda
Dep. Ciência dos Materiais e Metalurgia
Pontiflcia Universidade Católica do Rio de Janeiro
22453°900 Rio de Janeiro RJ
o
o
Resumo
Neste trabalho é feita a análise do comportamento mecânico em compressão uniaxial de um compósito epoxio
alumioa e da matriz epoxi sem reforço. A incorporação das partfculas de alumioa, mesmo em fraçóes
volumétricas pequenas, acarreta uma mudança significativa no modo de fal lla do material. A falha explosiva,
característica d o comporramento da matri~ epoxi cm compressão, é alterada pa ra uma falha estável no
compósito. O aumento da energia absorvida no processo de fratu ra do compósito, em relação~ matriz epoxi sem
reforço, foi atribufdo a parcela de energia de propagação. Os mecanismos de aumento da energia de propagação
foram identiucados como sendo ancoragem e/ou desvio de trinca nas interfaces partfculaoma triz.
Palavras-Chave: Compósito, Reforço por Partfculas, Ancoragem de Trinca, Energia na Fratura.
Abetract
Tho: mecbanical bebaviour in uniaxial com pression of ao alumina-epoxy composite and its neat matrix epoxy
resin wal! analyscd. The iocorporation of lhe alumina parti eles produced a marked cbange in tbe fai lure mode of
the composite. Tbe common neal resin burst fallure was chan.g ed lo a stable composile failure, even witb lhe low
volume fractioo of alumio a particles used in tbis work. Tbe composíte bas a higber energy at fracture than lhe
epoxy resin matrix and this difference was attrlbuted 10 tbe propagatíon e nergy term. Crack aochor ing and
deOexion ai lhe resin-alumina interfaces were ldentified as lhe maio energy absorbing mechanisms.
Key~M:>rds: Composite, Particle Reinforcemenl, Crack Ancboring, Fracture Energy.
Introdução
Polímeros termofixos são, na atualidade, largamente usados como adesivos e como matrizes em
compósitos. Entretanto, estes materiais são frágeis, apresentando valores extremamente baixos tanto
para o fator de intensidade de tensões ( K1 ), como para a taxa de liberação de ener~ia ( 0 1 ).
Tipicamente, para as resinas epoxi, os valor~ de K 1 e 0 1 variam entre 0,6-0,8 MPam 1 e 0,1!6,2
k.J m"2, respectivamente (Bandyopadhyay, 1990).
c
c
Com o objetivo de aumentar a tenacidade pode-se incorporar diferentes cargas aos polímeros
termofixos, tais como fibras, elastOmeros e partfculas rígidas (Low c May, 1988; Bandyopadhyay,
Pearce e Mestan, 1985; Kinlocb, Maxwell e Young, 1985) ou mesmo polfmeros termoplásticos
(Bucknall e Oilbert, 1989).
O mecanismo de aumento de tenacidade atuante é função do tipo de interação entre a frente de
trinca e as cargas. Para compósitos de matriz polimérica os mecanismos propostos são: i) ancoragem e/
ou deflexão da trinca (Low e Mai, 1988; Lange e Radford, 1971), ii) manutenção da união entre as
faces da trinca por partículas elastoméricas estiradas ou ptt fibras (Bandyopadhyay, Pearce e Mestan,
1985) e iii) desenvolvimento de deformação plástica localizada nn ponta da tri nca (Kinloch, Maxwell e
Young, 1985). A incorporação de partículas rígidas acarreta, ainda, um aumento da rigidez do
compósito fabricado.
Este trabalho tem por objetivo analisar o efeito da incor poração de partículas rígidas sobre o
comportamento mecânico, cm compressão uniaxial, de um compósito pdimérico com matriz epoxi e
partículas de alumina. Os micromecanismos de fratura atuantes no compósito são analisados e
comparados aqueles da matriz sem reforço.
Manuscript received: July 1993. Aaaoci.ate lecbnlcal Editor: Agenor de Toledo Aeury
Comportamento Mecanlco em Compress4o Unlllldal de um Composito Epo»-Aiumina
200
Material e Procedimento Experimental
Usou-se como matriz um sistema epoxi de cura a frio, formado pela resina epoxi bifuncional
digliddil ~ter do bisfenol A (DGEBA, DER 331), tendo como agente de cura uma poliamina alifática
(trietileno tetraamina, TETA, DEH 24). Panículas rígidas de alumina (Al6SG, ALCOA) com grau de
pureza de 99,9% e tamanho médio de 0,26 J.U1l foram usadas como carga. A razão n•..sina/ agente de
cura/ carga empregada na preparação dos compósitos foi de 100/13/30 panes, em peso.
Os compósitos foram fabricados pela incorporação da alumina à resina já pré-formulada com o
agente de cura. Este procedimento diminui o tempo disponível para a homogeneização da mistura
antes do início da gelatinização, porém impede que se forme um gradiente na distribuição das
partfculas ao longo da espessura da amostra. Antes da incorporação, a alumlna foi seca e peneirada
para diminuir a tendencia a foonação de aglomerados. A mistura homogeneiUida foi vazada em molde
aberto de alumínio com área de 100 x 220 mm2 e 15 mm de espessura. Placas de resina sem reforço
foram obtidas de modo similar.
A densidade do compósito fabricado foi medida segundo a norma ASTM 0792-86. As frações
volumétricas de carga ( VP ) e de vazios ( Yv ) foram cbtidas a partir elas equaçóes da miaornecânica de
compósitos (Schoutens, 1984), a saber:
1
vP - - (p Pp - Pm c
p (1 m
v v)]
(1)
(2)
onde p é a densidade e M a fração em massa. Os fndices p, m e c designam a carga de alumina, a
matriz e o compósito, respectivamente. Para os cálculos de V e V v foram tornados valores padrões
tabelados para as densidades da matriz epoxi e de ai um ina ( p mp - 1160 kglm3 e pP - 2970 kglm\
O cálculo das frações volumétricas a partir das equações 1 e 2 foi adotado porque, conforme
discutido abaixo, o emprego de microscopia quantitativa não forneceu bons resultados, devido as
características da microestrutura do compósito.
0_s e nsaios ~e compressão foram realiUido~fm máquina de ensaios de acionamento mecânico, a
veloctdade nommal constante de V • 1,7 x 10 · m/s. De acordo com a norma AS1M D695M foram
usados corpos de prova prismáticos (ou cilíndricos) com 20 mm de comprimento e 10 x 10 mm de
seção (ou 10 mm de diâmetro). Os valores reportados são a média de pelo menos 05 ensaios.
Os corpos de prova fratllrados foram analisados por microscopia ótica (MO). As superfícies de
fratura foram analisadas por microscopia eletrônica de varredura (MEV) no modo secundário e com
voltagem de aceleração do feixe de elétrons entre 15-20 kV. Para evitar acúmulo de carga, as fraturas
foram recobertas com uma camada condutora de ouro-paládio.
Nomenclatura
Gl c "' taxa de liberação de
energia
Kl c = fator de intensidade de
tensões
Mm
fra~o em massa da
restna matriz
MP a frl!lção em massa de
partfculas
p
carga aplicada
vP
• fraç6o volumétrica de
particulas
vv • fração volumétrica de
vazios
• densidade de energia
total
= densidade de energia
até a carga máxima
lJ p ~ densidade de energia de
propagação
u
ucm
Pc • densidade do compósito
Pm : densidade da matriz
Pp = densidade das
partículas
• deformação
o .. tensão
Or c tensão de ruptura
CJy • tensão de escoamento
.11 c deslocamento
t
F. H. M. M. Coata e J. R. M. d'Almeida
201
Resultados Experimentais
A Fig. 1 mostra urna microestrutura típica do compósito fabricado. Dois aspectos podem ser
Fig. 1 Mlcroutruture do com~lto apaxl4.tumlna. ~ partlculaa da alumlna tormam
agloma111dos com dlimatro variAvat.
destacado.." nesta micrografia: i) observa-se um baixo teor de vazios e ii) as partículas de a lurnina
formam aglomerados com tamanhos variáveis. A presença destes aglomerada; dificultou a medição da
fração volumétrica pelos métodos usuais de microscopia quantitativa (da Silva, Jr et ai., 1991) devido a
dificuldade de se definir. dentro doo aglomerados, a matriz das partículas de alumina. Na Thb. 1 estão
listados os valores das frações volumétricas calculadas pelas equações 1 e 2 e o valor medido para a
densidade do compósito.
Tabela 1 Fraç"-• volurn61rleu da partlculu • vazio. • denaldade do compóelto
p (kg/m )
1.400
Vp (%)
8
Vv (%)
-o
Na rig. 2 está mootrado o aspecto geral das curvas carga, P, vs desiQalmento, âl, obtidas para a
resina sem carga e o compósito. Os resultados dos ensaios estão listados na Thb. 2. Como esperado, a
incorporação de partículas rígidas acarretou um aumento no módulo de elasticidade e na tensão de
escoamento do compósito. Entretanto, a principal diferença entre o compósito e a resina matriz é a
mudança no comportamento à fratura. As amostras de resina sem carga fraturam explosivamente,
enquanto nos compósitos a fratura ocorre de um modo controlado, sem estilhaçamento dos corpos de
prova (Fig. 3).
Comportamento Mecanico em Compressão Unialóal de um Composíto Epold·~mina
202
.."'
500
O L-----L-----L-----L-----L-----L-----L---~L---~----~----~~
o
ó l Cm m )
•
lO
Fig. 2 Forma geMrlca clu curvu exp.tmenta• carga (P) w deelocamento (âJ) obtldM noe enealoe de
compreee6o. a) ,..,,. epad e b) comp6elto
Fig. 3 Mpecto geral doe corpoe de prove do compóel1o antM {a) e depole (b) do enealo.
F. H. M. M. Coeta e J. R. M. d'Aimelda
203
Tabela 2 Proprl.ctadH mdnle... M6duto de eluleldade (E), t.nelo de Nptur. (o r)
e teneio de eaeoamento (oy)
E(GPa)
Matriz
Compósito
2.04+0.1
2.91 +0.09
or(MPa)
oy(MPa)
92+5
Análise dos Resultados
A ooservação macroscópca do modo de falha das amOStras indica que existe uma grande mudança
oo compatamento à fratura entre o compósito e a resina pura. Na superfície de fratura da matriz epoxi
sem reforço existem apenas estrias semelhantes as descritas por Atsuta e Turner (1982) e que são
formadas pela junção, através de mecanismos de deformação plástica localizada, de trincas que se
propagam em planos próximos (Fig. 4).
Fig. 4 Super11de de ntura di ma1rlz epaxl. M mareee topogrifteae aio ea•••• formada• pela junçlo de
trlneaa que ae propagam em planoa dlferentea.
A ocorrência de plasticidade localizada é o principal mecanismo de deformação em polfmeros
termofixos (Smith, Kaiser e Roulin-Moloney, 1988). Assim sendo, o comportamento à compressão
uniaxial da resina epoxi OGEBNTETA é governado pela capacidade deste material sustentar um
proce!'.SO de deformação plástica. Uma vez. exaurido este processo a ruptura ocorre de forma abrupta,
com a propagação instável de trinca, que leva a um estilbaçamento dos corpos de prova. O
estilhaçamento, que é provocado pela ramificação das trincas, indica que estas já haviam atingido sua
máxima velocidade de propagação e que a energia extra disponfvel foi liberada pela criação de oovas
superfícies (Ravi-Chandar e Knauss, 1984; Cantwell, Roulin-Moloney e Kaiser, 1988). Em alguns
casos foi ~ível identll1car na superffcie de fratura regiões de transição mostrando estrias e, também,
uma área com superfície lisa (Fig. 5). A morfologia lisa é característica de fratura instável em regiões
sob compressão (d' Almeida e Graça, 1990).
Comportamento Mecanico em Compressão Uni&lOal de um Composito EpolCi-Aiumlna
F~ . S Ragll o dll hnal~ o na fratura da matriz apaxl.
a) aatrlu a b) ragt6o llaa carac1ar1 ..ca de fratura tnat6vat.
Pnr outro lado, na superffcie de fratura do compósito foram observadas diferentes marcas
lopográfieílo.; superficiais (Fig. 6). Além das marcas de deformação plástica na matriz epoxi, observam-
Fig. 8
S upar11cla da fratura do compóet10 mot~trando mareu carac1ar1aUcu de ancoragam da .-Inca ( --+ )
tn•rfacaa rompidu ( ~) a datormaç6o plbttca toca Iluda na matriz apoxl ( • ).
se ma rcas caracterfsticas de ancoragem de trinca e decoesão na interface partfcula-matríz. Estes do is
últimos aspectos topográficos indicam que o processo de fratura dos compósitos a lumina-epoxi, aqui
estudado, pode ser subdividido em duas etapas distintas, a saber:
i) nos primeiros estágios do carregamento, com a matriz rigidamente ligada às partículas, a resina
epoxi, que tem propriedades elásticas menores que as das partículas, tem a sua deformação restringida
pelas partículas. Esta restrição à deformação da matriz gera um estado de tensão hidrostática na região
da matriz próximo as partículas (Krock., 1967). A falha da interface resulta em uma mudança do estado
de deformação plana para tensão plana. A matriz pode. então, escoar lívremente. Assim, nas regiões
próximas a interfaces rompidas obst rva-se a maior concentração de marcas de deformação plástica na
F. H. M. M. Coa:a e J. R. M. d'Aimeida
matriz. (Fig. 7). Esta restrição imposta pelas partfculas à deformabilidade da matriz resulta no aumento
Fig. 7
Mare~~a
de daformaçlo plutlca locall:zada na matriz apoxl, aclJacentta a Interface. rompidle.
a) Aapecto geral a b) Detalha da 6raa marcada na fig. 7a.
do limite de escoamento e da curva a vs E nominal do compósito em relação à matriz sem reforço,
porém. implica em uma red ução da deformação na tensão nominal máxima do compósito (cf., Fig. 2).
ii) exaurida a capacidade de deformação plástica na matriz, as trincas existentes e nucleadas
preferencialmente nas interfaces são aceleradas, resultando na perda da capacidade do material
suportar carga. Entretanto, a existência de interfaces abertas/íntegras atua no sentido de divergi r/
ancorar as trincas. A ancoragem de trinca pelas partículas de alumina ou o desvio da trinca nas
interfaces rompidas representam mecanismos absorvedores de energia eficientes, que interferem no
processo global de fratura, travando localmente a frente de trinca (Kinloch, Maxwell e Young, 1985;
Nascimento, Abreu e d ' Almeida, 1991) c reduzindo a velocidade de propagação (Kinloch, Maxwell e
Young, 1985) devido a criação de múlliplas trincas secundárias (Bandyopadbay, 1990, Kinloch,
Maxwell e Young, 1985). Deste modo, a falha do compósito ocorre de um modo controlado, sem
ruptura catastrófica.
De fato, valores obtidos a partir de ensaios de tenacidade à fratura para o fator de intensidade de
tensão, K 1• mostram que, para a velocidade de ensaio usada, a incorporação de alumina à matriz epox:i
aumenta a tenacidade do comp6si10 em relação a resina pura (d' Almeida, 1992).
Compor1amento
~nioo
em Compressão Unlllldal de um Compoaito Epolei-Aiumina
Uma avaliação qualitativa da diferença descrita entre o processo de fratura em compressão do
compósito e da resina sem carga, pode ser feita avaliando-se o valor da energia de deformação
acumulada durante o ensaio (i.e., a área sob a curva P vs 61). Na Tab. 3 estão listados os valores
encontrados para a densidade de energia de deformaÇlio, sendo U a densidade de energia total, Uclll a
densidade de energia até a carga máxima e UPa densidade de energia de propaga~o. Pode-se ver que
o aumento da energia de deformação total do compósito em relação ao polimero-matriz sem carga esté
associado, principalmente, ao termo de propagação ( Up)· Esta parcela de energia é maior para o
compósito e pcxle ser atribuída aos meamismos de ancoragem e desvio de trinca pelos aglomerados de
ulumina.
-6
3
Tabela 3 O.naidada da energia dt dtftlrmaçio (Jx 10 I m )
u
Ucm
compósito
60,3
55,3
5,0
matriz.
54,4
-54,0
<1
Conclusão
A Introdução de partículas finas de alumina em uma resina epoxi bifunciona1 de cura a frio, altera
o modo de falha em compressão do compósito em relação à matriz sem reforço. A diferença no
comportumento à fratura foi as..<.OCiuda !l ~reei a de energia de propagaÇlio ele trinca. Os mt:(;aflismos de
ancoragem e desvio de trinca foram identificados como sendo os respoosáveis pela maior resistência à
prorsgação de trinca do compósi to em relaÇlio à matriz.
Agradecimento
Os autores agradecem à Dow Química pelo fornecimento da resina epoxi e ao MCT pelo apoio
financeiro.
Bibliografia
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Rubber Toughened Epoxy Resíns'', Proceeding of lhe 6'b Cburcbill College Cooference oo Deforma tio o, Yield
and Fraaure Polymers, paper 18, Cambridge, Inglaterra.
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Resins", Journal of Marerial.s Science, \bl. 23, pp. 1615-1631.
d'Aimeida , J.R_.M. (1992) ''Epoxy and Alumina -Epoxy Composlte Fracture Mecbanisms Evaluulon", Anais do
Simpósio fbero Americano de Pollmeros, pp. 543-544, Vigo, Espanha.
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Ptllimetilmeucrilaro", Aluis do 11 Micromal, pp. 213-216. São Paulo, Brasil.
da Silva, Jr. 1-I.P., Paciornik, S., Galllcío, E.G. e d'Aimeida, J.R.M. (1991) ''Anilise Metalogrifi~ de Conjugado
Epoxí-Microesferas Ocas de Vidro", AnAis do XIII Col64uio da S.B.M.E., Vol. lll, pp. 49-50, Caxambu,
Brasil.
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Mate riais Science, ~1. 20, pp. 4169--U84.
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R.l·l. eds., Addiso n-Wesley, Reading, Massachusetts, cap. 16.
Lange, F.F. e Radford, K.C. ( 1971) "Fra cture Energy of 111 EpoxJ Composile System", Joumal of Materiais
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Epoxy Resins". Journal of Materiais Scieac~. ~1. 23, pp. 3833-3842.
RBCM - J . of 1he Bl"'z. Soe. Mec:hanlcal Sc:leVol. XV· n• 2 · 1993
ISSN 0100·7386
Printed in Bralll
Abstracts
Ferreira, J . R. and Cuplnl, N. L , 1993, "Cholce of Toolln TUrnlng of Carbon and Gl...
Flber Relnforced Plaatlc" (ln Portugueee), RBCM - J. of the Braz. Soe. Mechanlcal
Sclenc•, Vol15, n.2, pp. 112-123.
Experiments werc carried out to study the performance of differcnt too! matc:rials. Tbey were tcsted in turning of
carbon and glass fi bcr rcinforced fenoli c resin composite. During lhe expcriments, tool wear, surface roughness,
and fced force bchavíor were observcc:l Experimental resulls s howed lha I, dueto the great influence of lool wear
o n workpiece surface roughness, oo ly diamond tools are s uítable for use ln finish turning. lo rough turning, lhe
cemen ted carbide tools showed to be lhe besl solution. Nthougb ii showcd larger wcar lhan diamond, lhe cosi/
benefit ralio is much lower and special precautions are nol oecessary.
Kty words: Machining · Tool - Composite Material
Melo, G. P. and Steffen Jr., V., 1993, " Mechanlcal Syetemaldentlflcatlon Through
Fourler Serleellme Domaln Technlque", RBCM- J. of the Braz. Soe. Mechanlcal
Sclencea, Vol15, n.2, pp. 124-135.
A time-domai n tec bniquc bascd on Fourler series is used in lhis wo rk for lhe parameters ide ntlflcatl oo of
mecbanical systems with multi-<legrces of frecdom. Tbe metbod is based on the orthogonalíty property of lhe
Fourier series which enablcs lhe integratioo of lhe cqualions of motion. Tbcse cquations are then convertcd to a
linear algebraic model that is solvcd 10 obt.ain lhe unkoown parametc:rs. Tbis way lhe method can be su mm ariud
as follows:
l . Cxp!nsio o of the input and output signals in Fourier series;
2 . lnt eg ration of lhe equations of motion using an operati onal matrix to iotc:grate tbe series;
3. The parameters are calc ulated lhrough a leasl·square cstimalioo me thod.
Ktywords: Parameter ldentification - Modal Analys is - lime-Dom aio Technique • Fourier Series
Arruda, J. R. F. and Duarte, M. A. V., 1993, " Updetlng Rotor-Bearlng Flnlte Element
Modela Ualng Experimental Frequency Reaponae Functlona", RBCM - J . of the Braz.
Soe. Mechanical Sclencea, Vo115, n.2, pp. 136-149.
Mosl Finite Element model updati og techniques require experimental modal parameters, wbich are obt.aiocd
from modaltests. ln lhe case of rea l rotor-bearing systems it is frequently unfeasible lo perform complete modal
tests dueto difficullies .such as exciting a rot.ating shaft and dealing wilh bearing nonlinearities. lo this paper, lhe
possibility o f updating lhe Finitc Element model of a rotor-beariog system by estimating lhe bearing stiffness and
damping ooefficients from a few measurcd Frcquency Rc:5ponse Functioos is investigatcd. The proposcd model
updating procedure consists of applying a nonlinear para meter estimation tc:cbnique to correcta few FE model
parameters choseo a priori. The parameter estimation problem is presen tcd in iú general forro, from which lhe
vario us metbods can be derived, and tben particularized to a nonlinear leas t squares problem where the objective
functi on to be minimized is lhe s um of tbe square differences between lhe logarithm of lhe th eoretica l and
experimental Frequency Response Function (FRF) magnitudes. The im portao ce of usiog the logarithm of lhe
magni tudes and its equivaleoce to the maximízation of tbe corrclation between tbe theoretical and experimental
fuo c tíons is pointed o ut. A cl osed forro expressio n for lhe sensitivity matrix of lhe loguitbm of lhe FRF
magn itudes is derived, and it is showo lhal the finíte differenccs approximatíon produces sufficiently accurate
results ata much lower compu tatio nal cosi. The issues of ideotifiability aod parameter estimalioo errors are
addressed. A oumerica l example o f a fle.:<ible roto r s upported by two ortbo tro pic bearings is preseoted to
illustrate lhe proposed method.
Keywords: Rotor-Bearing • Finite Elemeol Model - Frcquency Response Functions
209
RBCM - Journal ol lhe Brazllian Socíety ol Mechanical Sdences
Almeida, C. A. and Plree Alvee, J . A., 1993, " On Geomelrle Nonllnear FE Solutlon of
Unldlmenelonal Struc1ure Dynemlc.", RBCM - J . of the Braz. Soe. Mechanleal
Selencee, Vo115, n.2, pp. 150-164.
A oumerical prooedure for geometric oonlioear analysis of framed structures iD presented. The finite element
mcxlel employs 3-0 linear to cubieally varying space displacements and is capable of transmitting stress only iD
the direction normal to the cross-section. lt is assumed tbat tbis normal slress is constant over lhe cross·sectional
arca, and tbal lbe area itself remains constaol, atlowing tbe element for large di.splacement analysis but under
small strain condition. Two algoritbms for tbe automatic incremental solution of tbe resulling noolinear
equilibrium equations in static analysis are accessed. These iDclude the displacemeot oontroltecbnique aod lhe
mlnimization of residual displacemeots, whicb allow to calcula te tbe path response of general truss structures,
eveo of tbose tbat exbibít UJIStable coniigurations. The formulatioo bas been implemented aod tbe resuhs of
various examples illustrale tbe clement effectiveness in botb static aod dyoamic analyses.
Keywords: Geometric Nonlinear Aoalysis - Automatic Incremental Solution - Displacement Coo1rol Minimization of Residual Displacemenls
Estalella, J. J. S., 1993, "Sclentlfle Foundatlon of Meehanlcal Englneerlng" (ln
Spanleh), RBCM • J. of the Braz. Soe. Meehanlcal Seleneea, Vol15, n.2, pp. 165· 171 .
The paper di.scusses tbe question of looking for tbe scientific foundations of tecbnology in preference lo lhe
lechnological foundalions of science.
Keywords: Science - Tech.nology - Mecba nical Engineering
Lyra, M. R. P., 1993, " Finlte Element Anllllyele of Parabolle Problema: Comblned
lnfluence of Adapttve Mesh Reflnement and Automatlc Time Step Contrai", RBCM- J .
ofthe Braz. Soe. Mechanleat Selencee, Vol 15, n.2, pp. 172-198.
An adaptive fin.ite element metbod with mesh reflnement, by mesb eoricbment, intime and space and ao
automatic time stepping control is described for a class of parabollc problems. The computational domai o is
represeoted by ao assem bly of 4-noded isoparametric elements. The mesb enrícbment is achieved by loeally
subdividing each element into four new elements in tbose regions wbere fur1ber resolution is required during tbe
analysis. The Euler backward lime marchlng sebe me is used aod the time step sizes are gradually adapted. Both
lhe space and lhe tim e discretization error estimation adopled iovolve only lhe oomputed approximate solution
and tbose are use to dictate tbe adaptation. A discussion of the oombined ioOuence of time and spatial adaptation
in tbe oontext of parabolic problems is also presented. Sam pie model applications are included to demonstra te
tbe efficiency and accuracy, as well as lhe potentialities for engineering analysis, of the outlincd procedure.
Keywords: Fioite Element Analysis- Parabolic Problem • Adaptive Mesb Refinemeot • Automa tic lime Step
Control
Marques da Coeta, F. H. M. and d'Aimelda, J . R. M., 1993, "Unlaxlal Compreeelon
Meehanlcal Behavlour of an Alumln•Epoxy Compoelte" (ln Portuguese), RBCM • J . of
the Braz. Soe. Meehanlcal Selencea, Vol15, n.2, pp. 199-207.
The mecbanical bebaviour iD uniaxial compression of an alumina-epoxy composite and iLS oeat matrix epoxy
resio was analysed. The incorporatioo of the alumioa partícles produced a marked change in lhe failure mode of
Lhe composite. The common neat resin burst failure was changed to a stable composite failure, even wíth tbe low
volume fraction of alumina particles used in tbis work. The oomposite bas a bigber cnergy at fracture tban tbe
epoxy resin matrix and tbis differeoce was attributed to tbe propagation energy term. Crack anchoring and
deflexion ai the resin-alumlna interfaces were identified as lhe maio energy absorbing mechaoisms.
Keywords: Compositc • Particle Reinforcement • Crnck Ancboring • Fractme Energy.
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• References should be llsted in alphabetical order, according to lhe tast name of the flrst author. at lhe end ol
paper. Some sample references tollow:
Bordalo. S.N.. Ferzlger, J.H. and Kllne, S.J., t 989, 'lhe Devetopment ot Zonal Models for lurbutence',
Proceedlngs, 10" ABCM • Mechantcal Enotneertno Conference, Vol. I, Aio de Janeiro. Brazll, pp. 41-44.
Clark, J.A., 1986, Prlvate Communication. University of Michigan, Ann Arbor, MI.
Coimbra, A.L.. 1978, 1..essons ol Continuum Mechanlcs', Editora Edgard Blucher Lida. São Paulo, Brazil.
Kanduka.r. S.G. anel Shah, R.K., 1989. "Asymptotk; Eflectiveness • NT\J Formulas for Multiphase Ptate Heal
Exchangers', ASME Journal of Heallransler, Vol 111, pp. 314·32 t
McCormak. RW.. 1988. "Dn lhe Development ot Efficlettt Algorlhrns for Three Dimensional Fluid Flow', Joumal
of lhe Btanllan Society ot Mechamcal Sclences. Vol. 10, pp. 323-346.
Silva. Ul.M.. 1988. "New Integral FormWtlon for Problems ln Mechantcos', (ln portuguesa), Ph.D. ThesiS.
Federal Unlversity of Sanu Caunna. Aor1anópolis, SC, Btwl.
Sparrow. E.M. t980a. 'forced·Convection Heat lransfer in a Duct Having Spanwlse·Periodic Rectangulai
Protuberances·. Numerical Heal lranster. Vol. 3, pp, 149·167.
Sparrow, E.M. 1980b. 'fluid·to-Auld Con)ugate Heal Transler for a Vertical Plpe-lntemat Forced Convection
aod Externa! Naturlll Convectton", ASME Joumal of Heat l ransfer. Vol. 102. pp. 402-407.
ISSN O100- 7386
REVISTA BRASILEIRA DE CI~NCIAS MECÂNICAS
JOURNAL OF THE BRAZIUAN SOCIETY OF MECHANJCAL SCIENCES
VOL. Xv- N° 2-1993
Machlnlng
• Choice of Toai in Tuming of Carbon and Glass
Fiber Reinforced Plastic (ln Portuguesa)
Mechanica/ Systems /dentlflcatlon
• Mechanical Systems ldentification
Through Fourier Serles Time Dornain
Technique
• Updating Rotor-Bearing Finite Element
Models Using Experimental Frequency
Response Functlons
Structure Dynam/cs
• On Geometric·Nonlinear FE SoiU1ion of
Unidimensional Structure Oynamics
Sclence and Technology
• Scientific Foundation of Mechanical Engineering
(ln Spanish)
Rnne Element
• Finite Element Analysis of Parabolíc
Problems: Combined lnfluence of Adaptive Mesh
Refinement and Automalic Time Step Contrai
Fracture Mechanlcs
• Uniaxial Compression Mechanical Behaviour ol an
Alumina-Epoxy Composite
(ln Portuguese)
• Abstracts - Vol. 15- N1 2- 1993
João Roberto Ferreira
and Nlvaldo Lemos Cupini
112
Gilberto Pexoto de Melo
and Valder Steffen Jr.
124
José Roberto França Arruda
and Marcus A. V. Duarte
136
Carlos Alberto Almeida
and J. A. Pires Alves
150
Juan Jose Scala Estalella
166
Paulo Roberto Maciellyra
172
Fábio Horta Motta Marques da
Costa and José Roberto
Moraes d'Aimeida
199
208