02 / 19 / 2007 Maca eno 1
FEM SUBSYSTEM REPLACEMENT TECHNIQUES FOR
STRENGTH PROBLEMS IN VARIABLE GEOMETRY TRUSSES.
Luis M. Maca eno*, Josu Agi ebei ia, Ca los Angulo, Ra ael A ilés
Depa men o Mechanical Enginee ing, Uni e si y o he Basque Coun y
Alameda U quijo s/n, 48013 Bilbao, Spain
* Co esponding au ho :
Luis M. Maca eno
Depa men o Mechanical Enginee ing, Uni e si y o he Basque Coun y UPV/EHU
Escuela Técnica Supe io de Ingenie ía, ETSI
Alameda de U quijo s/n,
48013 Bilbao
SPAIN
Tel: +34 946017399, Fax: +34 946014215
E-mail: [email p o ec ed]
Numbe o wo ds: 5267
Numbe o igu es: 15
Tables: 2
Abs ac
This wo k p esen s he applica ion o a p ocedu e o eplacing FEM subsys ems wi h a
high numbe o do s (possibili y o including mobile in e nal pa s) using Equi alen
Pa ame ic Mac oelemen s (EPMs) wi h a much educed numbe o elemen s o
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Fini e Elemen s in Analysis and Design 44(6/7) : 346-357
(2008),which has been published in inal o m a h ps://doi.o g/10.1016/j. inel.2007.12.003. © 2007 Else ie unde CC BY-NC-ND
license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
FEM subsys em eplacemen echniques o s eng h p oblems in a iable geome y usses
02 / 19 / 2007 Maca eno 2
dec ease signi ican ly he analysis ime wi h an accep able e o . This p ocedu e is
applied o he eplacemen o VGT mobile join s. The equi alence c i e ion p oposed is
based on elas ic s ain ene gies abso bed by bo h bodies. Said eplacemen in ol es
esolu ion o a edundan non-linea equa ion sys em, i e a i ely ocused ia
linea iza ion and subsequen esolu ion ia he Leas Squa es Me hod. The sea ch o
ini ial app oxima ion is suppo ed by Gene ic Algo i hm echniques.
Key wo ds: Va iable Geome y T uss (VGT), Fini e Elemen , Equi alen Pa ame ic
Mac oelemen (EPM), Ene gy Me hod, Op imiza ion.
1. In oduc ion.
Va iable geome y s uc u es a e hose capable o modi ying hei geome y o adap o
di e en loads and wo king condi ions. This is possible because some o he elemen s
comp ising hem can a y hei leng h. These elemen s a e called ac ua o s. Ano he
cha ac e is ic making hem in e es ing is hei high s i ness o weigh a io, which has
con ibu ed o he applica ion o a iable geome y s uc u es in he spa ial esea ch
ield. The s udy o hese s uc u es da es back o he 1980s [1]. A speci ic ype wi hin
his g oup is based on spa ial uss ype s uc u es known as Va iable Geome y T uss
(VGT). I s mos common applica ion is as manipula o s [2, 3 and 4]. These usses a e
o med ia epe i ion o he main module whose opology can be highly a ied [5,
6].The mos widely known VGT is he Double Oc ahed al, made up o wo main
oc ahed al modules [7, 8, 9].
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The MBAD
1
(Mul i Body Analysis and Design) wo kg oup o he Uni e si y o he
Basque Coun y has de eloped a i e-module VGT p o o ype, whe e he geome y o
he main module is also es ablished upon he oc ahed al shape. The eal s uc u e is
shown in Fig. 1. Each module is a pa allel kinema ic mechanism in i sel wi h ac ua o s
se on he ho izon al planes. These planes a e joined among hem using ixed leng h
ba s called longe ons. The join be ween hese ba s and he ac ua o s is done by special
join s. These join s ha e also been de eloped by he wo kg oup and a e pa en ed. In Fig.
2 he e is an exploded iew o he join whe e i can be app ecia ed he di e en
elemen s comp ising he same.
Fig. 1. Fi e-Module VGT P o o ype Fig. 2. Exploded iew o he join .
1
h p://www.ehu.es/mbad
Special piece o bea ings
Join body
Double ing
Simple ing
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To pe o m he analysis on he s uc u e a ini e elemen model was c ea ed in
MSC/Nas an, p eceded by ano he de ailed cinema ic model c ea ed using
MSC/Adams. Fig. 3 shows bo h models indica ing he di e en elemen s comp ising he
i e-module VGT.
Fig. 3. VGT kinema ic and FEM model
In hese s uc u es i is necessa y o pe o m he analysis on a ious con igu a ions. A
p og am has been de eloped in PCL (Pa an Command Language) which au oma ically
c ea es a FEM model in each posi ion o enable di e se s udies. Once he FEM model o
he a iable geome y s uc u e has been c ea ed, i s beha io is s udied unde di e en
load cases and in di e en posi ions. As his is a de ailed model, bo h c ea ion and
analysis ime a e excessi e, so he need o educe he model a ises. The e a e di e en
s a ic model educ ion echniques, whe e he mos widely known a e educ ion ia s a ic
Longe on
Join
Ba en
Ac ua o
Base
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condensa ion [10, 11], i e a i e echniques [12, 13], mac oelemen s [14, 15] and
subs uc u ing echniques [16].
This pape p esen s he applica ion o a s a ic educ ion echnique o FEM models ia
eplacemen o ce ain submodels by equi alen pa ame ic mac oelemen s (EPMs).
These mac oelemen s will ha e a much lowe numbe o elemen s han he submodels
hey eplace, and consequen ly compu a ion cos is d as ically educed. The echnique is
based on elas ic s ain ene gy equi alence be ween he submodel and i s co esponding
mac oelemen . This c i e ion has led o good esul s in o he applica ions [17, 18].
The model o be analyzed is he a iable geome y s uc u e shown in Fig. 1, and he
submodels o be eplaced a e he in e media e join s o he s uc u e. These submodels
consis o 3D e ahed al and hexahed al ype elemen s, likewise igid and con ac
elemen s. Al oge he he e a e app oxima ely 2000 elemen s pe submodel. Fig. 4 shows
he FEM model o one o he join s.
Fig. 4. Join FEM Model
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The ollowing sec ion de ines he ma hema ical de elopmen o he mac oelemen in
de ail. Once i s unc ion has been unde s ood, he me hodology o he model educ ion
echnique is explained; de ining all he a iables likewise he equa ions a ising om
ene gy equi alence. Below a e he esul s ob ained ia applica ion o his echnique o
he join o he i e-module VGT. Finally, a b ie commen a y on hese esul s and he
conclusions a e shown.
2. Ma hema ical de ini ion o he join mac oelemen .
The mac oelemen as unde s ood in his pape is a se ies o elemen s, which join
oge he o ming a new body. I his se o elemen s is in ended o eplace ano he
model, i migh sa is y ce ain es ain s: i s ly i mus be cinema ically equi alen o
he eplaced model and secondly equi alen om a s uc u al iewpoin . The choice o
he ype o elemen s is ee as a as he connec ions wi h he es o he model and he
in e nal mobili y a e p ese ed. In his case, he de ined mac oelemen o eplace he
ini e elemen submodel o he join is o med by eigh 3D beam elemen s wi h ci cula
sec ion and nine nodes, as shown in Fig. 5. Each o hese nodes has six do s, h ee
ansla ions (ux, uy, uz) and h ee o a ions (θx, θy, θz). The beams a e a anged so nodes
1, 3, 6 and 7 co espond o he cen al poin s o he join sphe ical bea ings and nodes 8
and 9 o he link poin s o he ings wi h ac ua o ba s. I can be app ecia ed node 9
eally co espond o he poin midway be ween he wo link poin s o he double ing.
Nodes 2, 4 and 5 a e in e nal and ha e no physical co espondence wi h any o he
join . Node 4 is loca ed on he cen al poin belonging o he o a ion axis o he join
ings. In Fig. 6 i can be app ecia ed he co espondence o he mac oelemen nodes
wi h he join submodel.
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Fig. 5. Mac oelemen Fig. 6. Submodel and mac oelemen
co espondence.
2.1 S i ness ma ix.
The mac oelemen s i ness ma ix is ob ained ia he sum o expanded ma ices o each
elemen , once exp essed in o he global sys em. The axes o his global sys em a e XYZ
shown in Fig. 5. As he mac oelemen consis s o 9 nodes each wi h 6 do s, he inal
ma ix is 54 x 54. The elemen ma ices a e 12 x 12, since 3D beam ype elemen s we e
used.
As said be o e, he mac oelemen mus be cinema ically equi alen o he submodel.
The ings ound he join body can o a e in ela ion o he cen al axis. These o a ion
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angles shown in Fig. 5 ha e been de ined as θ1 and θ2. The o a ion condi ion mus be
included in he mac oelemen . This has been done eleasing he o a ion do ega ding
he cen al axis on he nodes co esponding o elemen s 7 and 8. These elemen s would
co espond o he ings in he submodel. The ac o eleasing do s on a node is equal o
including he null o ce ansmission condi ion in he same di ec ion. The node
unde going his si ua ion is numbe 4, whe e ou elemen s concu numbe ed 3, 4, 7 and
8. Elemen s 7 and 8, as men ioned be o e, would co espond o he ings and elemen s 3
and 4 would ep esen he cen al axis o he eal join . The e o e, con ibu ion o he
momen in di ec ion Z on node 4 o elemen s 7 and 8 mus be cancelled. Fig. 7 shows
eleased do s, likewise he local sys ems o elemen s 7 and 8. Anyway on his node,
e o con inues being ansmi ed in ha di ec ion, al hough only be ween elemen s 4
and 5. In oducing his condi ion means he s i ness ma ix o elemen s 7 and 8 su e
ce ain modi ica ions di ec ly in luencing he mac oelemen s i ness ma ix. Below he
ma hema ical de elopmen is shown in de ail.
Fig. 7. Released do s
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Momen acco ding o he eleased do is no ansmi ed, so i is null. In elemen 7 i
co esponds o Mz on he i s node o he elemen . Thus, using i s equilib ium equa ion
(1) and equaling o ze o i can be ob ained he eleased do which now becomes
dependen on he es .
22
2
11
2
1
2646
0z
z
y
z
z
z
y
z
zL
IE
u
L
IE
L
IE
u
L
IE
M
(1)
2
)(
2
32
121 z
yyz uu
L
(2)
Eq. (2) is eplaced in all he ma ix ows whe e e m θz1 appea s and he new ma ix is
ob ained. This ma ix has ze os in all he ow and column co esponding o he eleased
do . This de elopmen is analogous o elemen 8. All ele an ma ices a e shown on
he Appendix.
Once he p e ious s eps ha e been execu ed o achie e cinema ic equi alence, he
mac oelemen global s i ness ma ix is ob ained ia coupling o all he ma ices o each
elemen in global coo dina es.
A his poin , he mac oelemen has been comple ely de ined, which depends on he
physical p ope ies (E, υ, G) and dimensions o he elemen s comp ising he same, such
as he sec ion o adii. Any o hese a iables may be selec ed as a pa ame e .
2.2 Mac oelemen s i ness ma ix condensa ion.
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Applying Eq. (20) o each displacemen and each pa ame e , he sensi i i y ma ix o
dimensions n x p can be ob ained.
p
nnn
p
p
A
V
A
V
A
V
A
V
A
V
A
V
A
V
A
V
A
V
A
V
...
............
...
...
21
2
2
2
1
2
1
2
1
1
1
(21)
Taking in o accoun all he abo e, he ollowing linea ized equa ion sys em is
o mula ed o k i e a ion, whe e app oxima e elas ic s ain ene gy is equaled o he
submodel ene gy o each displacemen case i.
ni
AA
A
V
VU p
j
k
j
k
j
k
j
k
i
k
ii
...,1
)(
1
1
(22)
This sys em mus be sol ed o each i e a ion k ia he Leas Squa es Me hod, o which
he R e o unc ion is de ined as ollows:
2
1 1
1
1)()(
n
i
p
j
k
j
k
j
k
j
k
i
k
ii
kAA
A
V
VUAR
(23)
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To ob ain alues which minimize he e o unc ion he de i a i es mus be equaled o
ze o o each pa ame e . Thus, he ollowing de e mined compa ible equa ion sys em is
eached o p equa ions and p a iables.
pj
A
AR
k
j
k
...,1
0
)(
1
1
(24)
The solu ion o his sys em is he {A}k+1 pa ame e ec o . Se ou below a e he s op
c i e ia aken in o accoun in he i e a i e p ocess.
3.2 S op c i e ia.
To assess he deg ee o equi alence o he mac oelemen a alue compa ing he ec o s
{U} and {V} mus be de ined. The ollowing o mula is highly app op ia e o his case:
n
VU
n
i
k
ii
1
2
1
(25)
Wi h Eq. (25) i can be e alua ed he e o commi ed in each i e a ion. In he gene al
case, as his is a Leas Squa es me hod, i will no ge a ze o e o . The e o e, he mos
in e es ing s op c i e ion is ha which alues s abiliza ion o he e o commi ed. When
he e o pe cen age inc ease in wo consecu i e i e a ions is less han a ce ain q alue,
he i e a i e p ocess may be deemed inished.
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q
k
kk
1
(26)
In Eq. (26) he absolu e alue is used because he e o is expec ed o be less in each
i e a ion i he me hod con e ges. As an addi ional c i e ion i can also be used he one
based on he op imiza ion pa ame e s a ia ion no m (27).
'
2
1
1
1qAAAA p
j
k
j
k
j
kk
(27)
Howe e , i mus be bo ne in mind, ha i is highly likely he e will be a ce ain numbe
o pa ame e s whose a ia ion has no decisi e impac on he objec i e unc ion alue.
The e o e, p io o assessing he applica ion o Eq. (27), a de ailed s udy mus be
pe o med on pa ame e sensi i i y, as explained in he pa ame e con ol sec ion.
3.3 Ini ial app oxima ion.
The selec ion o ini ial app oxima ion asks equi es special ca e, since i has been
e i ied he me hod used is highly sensi i e o said choice, bo h in p ocess du a ion and
s abili y. He e he use o gene ic algo i hm echniques has been chosen. A simple
gene ic code in eal numbe s wi h 5 digi s in decimal base indica ing he alue o
op imiza ion a iables has been used o his. The a iables chosen o he gene ic
algo i hm will be de ailed in he esul s sec ion.
3.4 Pa ame e con ol.
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Among he op imiza ion me hods, his one belong o hose o s eepes descend o
g adien . The di e ence be ween he ec o ob ained as a solu ion and he p e ious in
each i e a ion p o ides a di ec ion ec o o maximum g adien . This means ha a ying
he pa ame e s in his di ec ion, a maximum a ia ion o he objec i e unc ion is
achie ed. This ec o may be dis o ed in some cases. This is because he a ia ion o
some pa ame e s has almos a null impac on he objec i e unc ion; whe eby i s alue
may inc ease disp opo iona ely wi hou app eciable imp o emen in he solu ion. To
p e en si ua ions o his kind an inc ease con ol has been applied o he pa ame e s.
This con ol consis s o applying a maximum alue allowed. Once he maximum
g adien ec o has been ob ained, a ia ion o each pa ame e is calcula ed. I all alues
a e lowe han he pe cen age pe mi ed, he p ocess con inues unchanged. Should said
alue be exceeded, he ec o is scaled aking i o he allowed limi . This ac ion
p e en s any pa ame e om g owing exagge a edly ye con inues o main ain he
maximum a ia ion di ec ion. The maximum a ia ion alue allowed is a pa ame e
which may be ei he a iable o ixed. Should i be a iable, he mos logical would be
o i o dec ease as he solu ion app oaches op imum, o p e en leaping o ano he
solu ion. In his s udy a ixed alue was chosen.
Apa om delimi ing he pa ame e maximum pe cen age a ia ion, some imes a
comp omise decision mus be aken in ela ion o he alue o some o hem. Applicable
o he ex eme cases commen ed abo e o pa ame e s a ying disp opo iona ely,
wi hou ha dly any imp o emen in he solu ion. This beha io may be quan i ied using
he in o ma ion p o ided by he sensi i i y ma ix. Each ma ix e m indica es he
a ia ion in elas ic s ain ene gy Vi o a uni alue inc ease in he co esponding
pa ame e . When his alue is close o ze o, i means he ene gy ha dly a ies. This
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02 / 19 / 2007 Maca eno 20
alue may be e y low o one displacemen case and high o ano he . The e o e, a
alue which conside s all cases mus be aken as a e e ence. This alue has been
de ined as Sj, o each pa ame e . I mus be calcula ed in each i e a ion since he
sensi i i y ma ix also a ies.
n
A
V
S
n
ij
i
j
1
2
(28)
The alue ob ained in Eq. (28) includes he objec i e unc ion sensi i i y o all
displacemen cases in ela ion o pa ame e Aj. As p e iously indica ed, his sensi i i y
enables unc ion in a iance e alua ion in ela ion o he pa ame e in ques ion.
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3.5 Flow diag am o he op imiza ion p ocess and e m glossa y.
Fig. 8. Flow diag am o he op imiza ion p ocess
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Te m glossa y.
n o al numbe o displacemen cases
i subindex o cu en displacemen case
h numbe o displacemen ec o do s
{δ}i displacemen ec o o case i
[δ] ec o displacemen ma ix
Ui elas ic s ain ene gy o he model o case i
{U} ec o o elas ic s ain ene gies o he model
Vi mac oelemen elas ic s ain ene gy o case i
{V} ec o o mac oelemen elas ic s ain ene gies
p o al numbe o pa ame e s
j cu en pa ame e subindex
Aj pa ame e j
{A} pa ame e ec o
m inc easing pe cen age o pa ame e s in nume ical de i a ion
sensi i i y ma ix
k i e a ion numbe
R e o o objec i e unc ion
ε absolu e e o
q ela i e e o
maximum a ia ion in pa ame e s con ol
Sj pa ame e j sensi i i y
A
V
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4. Resul s.
This sec ion se s ou he op imiza ion p ocess esul s applied o he mac oelemen
desc ibed in sec ion 2. The ma e ial used is aluminum wi h he ollowing
cha ac e is ics:
Young module: E = 70 GPa
Shea module: G = 27 GPa
Poisson Coe icien : υ = 0.33
Me hod applica ion condi ions:
Numbe o mac oelemen pa ame e s: p = 8
Numbe o mas e do s: h = 18
Numbe o displacemen cases: n = 60
Inc eased pe cen age in he sensi i i y ma ix calcula ion: m = 0.01
Maximum Numbe o i e a ions: k_max = 15
Value o admissible q: q = 0.5 %
Maximum a ia ion in pa ame e con ol: = 10 %
S ep one is o c ea e he 60 displacemen cases, o which ec o s {δ} a e c ea ed
andomly. The magni udes o hese displacemen s we e chosen so hey we e highe han
expec ed o he usual wo king condi ions o he eal join . Thus, i is gua an eed co ec
mac oelemen beha io unde no mal condi ions. The andom unc ion (29) is cen e ed
a ze o wi h alues be ween -10-5 and 10-5 me e s.
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)12(10 5 Random
(29)
Once c ea ed he ma ix [δ] wi h he 60 ec o s {δ} each o 18 componen s, ec o {U}
is calcula ed. This ec o can be ob ained in di e en ways, he mos common being
calcula ion o he model elas ic s ain ene gy o each displacemen case as pe he
ollowing o mula:
i
T
i
iFU
2
1
(30)
{F} being he ec o comp ising componen s o eac ion o ces associa ed o mas e
do s whe e displacemen s a e applied. Repea ing Eq. (30) o all displacemen cases, he
ec o {U} is ob ained. As men ioned p e iously his ec o is only calcula ed once.
The sea ch o ini ial app oxima ion {A}0 was ca ied ou execu ing he ee Gene ic
Algo i hm code “Pikaia” [19]. The alues adop ed o he algo i hm alues a e as
ollows:
Numbe o indi iduals in he popula ion: 150
Numbe o gene a ions in he e olu ion: 400
Encoding: decimal
Numbe o digi s o encode e geno ype: 5
C osso e p obabili y: 0.85
Mu a ion mode: adjus able a e based on i ness
Ini ial mu a ion a e: 0.005
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Minimum mu a ion a e: 0.0005
Maximum mu a ion a e: 0.25
Rep oduc ion plan: ull gene a ional eplacemen
Eli ism: ac i e
In his wo k i was decided o limi he pa ame e alue bea ing in mind he physical
ea u es o he submodel. On he o he hand, he sea ch space was no ably educed
aking in o accoun he symme y ex an in he submodel, which implies a ce ain
ela ionship “a p io i” among some pa ame e s.
A e 400 gene a ions he alues ob ained o ini ial app oxima ion (exp essed in
millime e s) we e he ollowing:
T
A810.2971.4822.10917.10000.25000.25999.8989.8
0
These a e he alues ini ially de ining he mac oelemen ; i.e. he alues o he 8 adii o
he ci cula sec ion beams o ming he same. Now he elas ic s ain ene gy Vi can be
calcula ed o each displacemen case. By calcula ing hese alues o he 60 cases,
ec o {V} is o med. Now, i can be assessed he deg ee o equi alence be ween he
mac oelemen and he submodel, using he Eq. (25), which ob ained a alue o ε =
0.0292.
Fig. 9 ep esen s he quad a ic di e ence alues o he U and V ene gies pe
displacemen case i. This igu e con i ms he V alues a e conside ably simila o hose
o U.
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02 / 19 / 2007 Maca eno 32
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FEM subsys em eplacemen echniques o s eng h p oblems in a iable geome y usses
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Appendix. Elemen s i ness ma ices.
Fig. 14. S i ness ma ix o elemen s 1,2,3,4,5 and 6 in local coo dina es
Fig 15. S i ness ma ix o elemen s 7 and 8 in local coo dina es
The ow and column co esponding o he eleased do θz1 all con ain ze os. In his
componen , hese elemen s do no con ibu e s i ness o he mac oelemen .
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02 / 19 / 2007 Maca eno 36
Figu e Legends:
Fig. 1. Fi e-Module VGT P o o ype.
Fig. 2. Exploded iew o he join .
Fig. 3. VGT kinema ic and FEM model.
Fig. 4. Join FEM Model.
Fig. 5. Mac oelemen .
Fig. 6. Real model and mac oelemen co espondence.
Fig. 7. Released do s.
Fig. 8. Flow diag am o he op imiza ion p ocess.
Fig. 9. Di e ence be ween U and V ene gies o he 60 displacemen cases.
Fig. 10. E o e olu ion.
Fig. 11. Pa ame e e olu ion.
Fig. 12. Va ia ion pe cen ages o pa ame e s.
Fig. 13. E o compa ison o se e al “displacemen case g oups”.
Fig. 14. S i ness ma ix o elemen s 1,2,3,4,5 and 6 in local coo dina es.
Fig. 15. S i ness ma ix o elemen s 7 and 8 in local coo dina es.
Tables:
Table 1. Values o pa ame e s and hei ela i e a ia ion.
Table 2. Pa ame e sensi i i y alues.
