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M ult iple Force Identi ficat ion for om plex Struc turesb y R o
b e r t A d a m s a n d J a m e s F D o y leA B S T R A C T - - T h
i s p a p e r p r e s e n t s a m e t h o d fo r d e t e r m i n in
gf o rc e h i s t o r i es us i ng e x p e r i m e n t a l ly m e a
s u r e d r e s p o n s e s . I ti s bas ed on a rec u rs i v e re
f o rm u l a t i on o f t he gov e rn i ng equa -t i ons i n c on j
unc t i on w i t h a gene ra l f i n it e e l em en t p rog ram , t
h i sla t ter a s p e c t m a k i n g i t a p p l i c a b l e t o c
om p l ex s t ruc t u res , I t c ande t e rm i ne m u l t i p le i
s o l a t ed unc o r re l a t ed ) f o r c e h is t o r i es asw e
l l a s d i s tr i b u te d p r e s s u r e s a n d t r a ct io n s
a n d a l l o w s f o r t h ed a t a c o l l e c t e d t o b e o f
d i s s im i l ar t y pe . A s a dem ons t ra t i on o ft h e m e t
h o d a n d o f i ts scalabil i ty f o r c e rec ons t ruc t i ons
f o r ani m p a c t e d s h e l l a n d a n i m p a c t e d p l a t
e a r e d e t e r m i n e d us i nga c c e l e r o m e t e r a n d
s t r a i n g a g e d a t a .K E Y W O R D S - - F i n i t e e l e
m e n t m e t h o d , f o rc e i d e n ti fi ca t io n ,regu l a r
i z a ti on , s t r uc t u resI n t r o d u c t i o nT h e r e s p
o n s e u x , t ) o f a g en e ra l l i n ea r s t ru c tu ra l sy
s t em toa s in g l e ex c i t a t i o n l o ad P t ) can b e w r i
t t en a s
u x , t ) = / G x , t - ~ ) P ~ ) d ~I0
w h e r e G x , t ) i s t h e sy s t em resp o n se fu n c t io
n . T h e fo rcei d e n ti f ic a t io n p r o b l e m i s u s u a
l l y p o s e d a s : g i v e n s o m e m e a -s u r e m e n t s u
( t ) ( p e r h a p s i m p e r f e c t l y ) , a n d k n o w l e d
g e o f t h es y s t e m f u n c t io n G x , t ) ( p e r h a p s i
m p e r f e c t l y ) , d e t e r m i n e t h el o a d P t ) . T h
i s p ro b l em i s d i f f i cu l t f o r tw o q u i t e s ep a ra
t er easo n s . T h e f i r s t is t h a t i t i s a h ig h ly i l
l- co n d i t i o n ed p r o b -l e m w h i c h m e a n s t h a t s
m a l l e r r o r s i n t h e m e a s u r e m e n t s o rth e m o d
e l in g can c au se v e ry l a rg e v a r i an ces i n t h e id en
t i f i edfo rces . S eco n d , t h e n eed to k n o w th e sy s t
em fu n c t io n i n an a -l y t ic a l f o r m h a s m e a n t t h
a t m o s t s t r u c t ur a l s y s t e m s a n a l y z e dh av e
b een r e l a t i v e ly s im p le .R e f e r e n c e 1 g iv e s a
n e x c e l l e n t s u m m a r y o f t h e l i t e ra t u reo n fo
rce i d en t i f i ca t i o n a s w e l l a s an o v e rv i ew o f
t h e su b j ec ti t s e lf ; m o r e r ecen t c i t a t i o n s
can b e fo u n d in R ef . 2 . A v a r i e tyo f m e t h o d s h a
v e b e e n u s e d , b u t th e m o s t c o m m o n s o l u t io
ns c h e m e i s s o m e f o r m o f d e c o n v o lu t i on . R e
f e r e n c e 3 s u m -m a r i z e s s o m e o f t h e m a i n d e
c o n v o l u ti o n m e t h o d s , a l t h o u g hth ese a r e n
o t d i r ec t l y ap p l i ed t o t h e fo rce i d en t i f ica t
i o n p ro b -l em . R efe ren ces 2 , 4 , 5 , an d 6 u sed F o u r
i e r m e th o d s b e cau seo f t h e r e l a t i v e s im p l i c
i t y o f t h e i n v e r s io n an d b ecau se i t i ss u i t a b
ly m a t c h e d t o t h e s p e c t ra l e l e m e n t a p p r o a
c h . D e r i v -in g an a ly t i ca l r e sp o n se fu n c t io n
s i s a fo rm id ab le ch a l l en g e ;R ef . 7 an d R ef s . 8 an
d 9 l o o k ed a t h a l f - sp a ce an d h a l f -p l an e
R o b e r t A d a m s i s a P r o f e s s o r a n d J a m e s E
D o y l e i s a G r a d u a t e S t u d e n t ,S c h o o l o f A e
r o n a u t i c s & A s t r o n a u t i c s , P u r d u e U n i
v e rs i t y, W e s t L a f a y e t te ,I n d i a n a 4 7 9 0 7
.Orig in a l ma nusc r ip t submit ted . Jan uary 7 , 2000 .Final
manusc r ip t received . Oc to ber 15, 2001 .
p ro b lem s , r e sp ec t iv e ly ; g en e ra l f r am e s t ru
c tu res w ere so lv edb y th e i n t ro d u c t io n o f sp ec t r
a l e l em en t s i n R ef . 2 ; f o ld edp la t e s t ru c tu res
w ere so lv ed u s in g p l a t e sp ec t r a l e l em en t s . 10E
ach o f t h ese a r e q u i t e l im i t ed , i n t h a t t h ey a
r e r e s t r i c t ed t or e l a t iv e l y s i m p l e g e o m e
t r i e s nd d id n o t ad d res s t h e q u es t i o no f m u l t
i p l e o r d i s t r i b u t ed fo rces .T h e b e g i n n i n g s
o f a d e c o n v o l u t i o n m e t h o d c o u p l e d w i t ha
g en e ra l f i n it e e l em en t ap p ro ach w as g iv en in R ef
. 1 1 w h ichu s e d a n F E M p r o g r a m t o g e n e r a t e w
a v e l e t s o l ut i on s w h i c hco u ld b e sy n th es i ze d
i n t h e i n v e r se so lu t i o n . A q u i t e d i f f e r en
ta p p r o a c h w a s p i o n e e r e d i n R e f s . 1 2 - 1 4 f
o r h e a t c o n d u c ti o np ro b le m s an d R e f s . 1 5 an d
1 6 fo r fo rce i d en ti f i ca t i o n p ro b -l e m . T h i s a
p p r o a c h us e s id e a s f r o m d y n a m i c p r o g r a m m
i n g ,b u t w h a t s e t s i t ap a r t f ro m th e ab o v e r e
f e r en ces i s t h a t i t d ea l sw i th t h e d i sc r e t i
zed fo rm o f th e g o v e rn in g eq u a t io n s . T h i sm ak es
i t su i tab l e fo r co u p l in g w i th a f in i t e e l em en t
m o d e l in go f t h e s t ru c tu re . H o w ev e r , a s p o in
t ed o u t i n R ef . 1 5, o n e o ft h e d i s a d v a n t a g e s
o f t h e m e t h o d i s t h a t t h e a m o u n t o f c o m -p u
ta t i o n s i n c reases d ram a t i ca l l y a s t h e o rd e r o
f t h e m o d e lin c reases . I n d eed , th i s n ecess i t a t
ed t h e au th o r s to i n t ro d u cea m o d a l r ed u c t io n
sch e m e su ch th a t th e f i n a l s i ze o f th e p ro b -l em
s tu d i ed i n R ef s . 1 5 an d 1 6 w as n in e d eg rees o f f r
eed o m .M o re d e t a i l s an d ap p l i ca t i o n s can b e fo
u n d in R ef . 1 7.
I t i s t h e g o a l o f t h i s p ap e r t o d ev e lo p a m e
th o d fo r fo rceid en t i f i ca t i o n t h a t i s ro b u s t
en o u g h to b e ap p l i ed t o co m -p lex s t ru c tu res , b u
t w h o se co m p u ta t i o n a l co s t s s ca l e a s fo ra fo
rw ard f i n i t e e l em en t an a ly s i s . F u r th e rm o re ,
w e w i sh t ob e ab l e t o d e t e rm in e m u l t i p l e i so l
a t ed fo rces a s w e l l a s d i s -t r i b u t ed p res su res
an d t r ac t io n s . A f t e r d ev e lo p in g th e m a inin g
red i en t s o f t h e m e th o d , w e i l l u s t r a t e i t s a
t t r i b u t e s u s in ge x p e r i m e n t a l d a t a f r o m t
h e i m p a c t o f a 3 - D s h e l l a n d f o r th eim p ac t o f
a p l a t e w i th a h o l e .T h e E s s e n t i a l D i f f i c u
l t y
B efo re d ev e lo p in g th e sp ec i f i c s o f t h e ac tu a
l m e th o d , w ef i r s t sk e t ch t h e o u t l i n es o f a n
a iv e m e th o d o f fo rce i d en t i f i-ca t i o n . T h i s w
i l l a l so se rv e t o i n t ro d u ce so m e o f t h e m a inno
tat ions .
U s in g th e d i sc r e t i za t i o n a f fo rd ed b y t h e f
i n i t e e l em en tm e th o d , 1 8 ' 19 t h e g o v e rn in g d
y n am ic eq u a t io n s fo r a g en e ra lco m p lex (b u t l in
ea r ) s t ru c tu re can b e d i sc re t i zed i n sp ace a s
[M]{//} + [C]{ti} + [K]{ u} = {P} = [Bg]{g}. (1)In t h i s, {u }
i s t h e v ec to r o f a l l th e f r ee d eg ree s o f f r ee d o
m ndi s o f s ize {m, x 1}, [K ] i s the [mu x mu ] s t i f t '
ness matr ix ,[C] i s the [mu x m u ] d am p in g m a t r ix , {P }
is t h e {mu x 1}v ec to r o f a l l ap p l i ed l o ad s ( so m e
o f w h ich a r e ze ro ) , {g } i st h e {m e x 1 } su b se t o f
{P } v ec to r o f n o n -ze ro ap p l i ed l o ad s ,a n d [Bg] i
s t h e [m u m g ] m a t r ix t h a t a s so c i a t e s t h ese l
o ad s
E x p e r i m e n t a l M e c h a n i c s 9 5
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w i th t h e d eg rees o f f r eed o m . L e t t h ese g o v e
rn in g eq u a t io n sb e d i sc re t i zed i n t im e a s t h e r
ecu r r en ce r e l a t i o n s
{u}n+i = [A]{u }n + [B]{g}n. 2 )In t h i s , n i s t h e su b sc
r ip t o v e r th e d i sc r e t i zed t im e ; {u} n o wco n ta in
s t h e s t a t e v ec to r s ( a t l ea s t d i sp l ace m en t an
d v e lo c i t y )an d i s o f s i ze m > 2 m , ; t h e v ec to
r o f fo rc in g t e rm s , {g } , i s o fs i ze {rag x 1}; and [B]
i s the [m x mg] m at r ix t h a t a s so c i a t e sth e fo rces w
i th t h e D o E S o m e sp ec i f i c fo rm s fo r [A ] an d [B ]a
re g iv en in A p p en d ix I .In t h e fo rw ard p ro b l em , g
iv en th e i n i t i a l co n d i t i o n s a s{U }l = k n o w n ,
an d th e ap p l i ed l o ad h i s to r i e s {g } , w e ca nso lv
e fo r th e r e sp o n se r ecu r s iv e ly f ro m E q (2 ) . I n t
h e in v e r sep ro b lem o f i n t e r e s t h e re , t h e ap p l
i ed l o ad s a r e u n k n o w n b u tw e k n o w s o m e i n f o
r m a t i o n a b o u t t h e r e sp o n s e s ; w e w i s h t ou
se t h i s i n fo rm a t io n t o d e t e rm in e t h e ap p l i ed
l o ad s . In p a r -t i cul a r , a s su m e w e h av e a v ec to
r o f m e asu re m e n t s {d } o f s i ze{m d x 1} w h ich a r e r
e l a t ed t o th e s t ru c tu ra l D o F acco rd in g t o
{d}n r [Q]{u}n .N o te t h a t t h e [m d mu] m at r ix [Q ] co
u ld b e a d i f f e r en cere l a t i o n a s w o u ld b e t h e
case fo r s t r a in s . W e w an t t o f in d t h ef o r c e s {
g} t h at m a k e t h e s y s t e m b e s t m a t c h t h e m e a s
u r e m e n t s .C o n s id e r t h e g en e ra l l ea s t sq u a
res e r ro r g iv en b y
N
n = l{u}n+l = [A]{u)n + [B]{g}n,
w h e r e r W j c o u l d b e a g e n e ra l w e i g h t i n g a
r r a y o n t h e d a taa l t h o u g h w e w i l l t ak e i t a s
d i ag o n a l . O u r o b j ec t i v e i s to f i n dth e se t o f
fo rces {g} t h a t m in im ize t h i s e r ro r fu n c t io n a l
.
I t is p o ss ib l e t o e s t ab l i sh a g lo b a l sy s t em
o f s im u l t an e o u seq u a t io n s b y t h e u su a l p ro
ced u res fo r m in im iz in g t h e l eas tsq u a res . T h a t i
s , w e a r r an g e E q (2 ) t o fo rm
[ f i, ] { ~} = [ / Y ] { ~ }, { ~ } = { { U } l , " - , { U} N}
,{ g } = { { g } l , , { g } N } T .
S o lv e fo r {~ } in t e rm s o f {~ } . T h a t i s , f i r s
t d eco m p o se [ e{ ]an d th en so lv e [G ] a s t h e se r i e s
o f b ack - su b s t i t u t i o n s
[ A ] [ G ] = [LrDU][G] = [/}].T h e m a t r ix [G ] i s t h e
co l l ec t i o n o f fo rw ard so lu t i o n s fo r u n i tl o a
ds a p p l i e d f o r e a c h o f t h e u n k n o w n f o r c e s.
W e t h e r e f o r ecan w r i t e t h e ac tu a l fo rw a rd so lu
t i o n a s
{~} = [G ]{~ } .S u b s t i t u t e t h is i n to t h e e r ro r
eq u a t io n an d m in im ize w i th r e -sp ec t t o {g } . T h e
r e su l t i n g sy s t e m i s
[GT QTW QG ]{~} = [QG ]r[W]{d}.T h i s n a iv e sch em e i s n o
t p r ac t i ca l fo r tw o v e ry im p o r t an tr easo n s .C o n
s i d e r a d y n a m i c p r o b l e m w h e r e t h e r e a r e
mu d e-g rees o f f r ee d o m a t N t im e s t ep s ; t h i s w o
u ld l ead t oth e v e ry l a rg e sy s t em a r r ay [A ] o f s i
ze [ mu x N ) x
mu N ) ] . A sy s t em w i th 1 0 , 00 0 D o F o v e r 2 0 0 0 t
im es t ep s_ h as n ea r ly 1 0 0 1 06 u n k n o w n s an d a sy s
t em s i zeo f [A ] w h ich is t h e sq u a re o f t h a t . T h i
s w o u ld l ead t oa R A M m e m o r y r e q u ir e m e n t o f 3
x 1 09 M B . F u r t h e r-m o r e , s o l v i n g th e s y s t e m
o f e q u a t io n s w o u l d c o s t a p p r o x -i m a t e l y N
3 o p e ra t i o n s, w h i c h o n a 1 G F l o p s u p e r c o m
-p u t e r w o u l d t a k e 1 0 1 3 seco n d s o r 3 0 0 0 0 0 y
ea r s ! C lea r ly , t h i sa p p r o a c h t o s o l v i n g t he
p r o b l e m d o e s n o t s c a l e v e r y w e l l a n dth e p
ro ced u re i s r e s t r ic t ed t o v e ry sm a l l p ro b l e m
s o n ly . E v eni f ad v an tag e i s t ak en o f t h e sp ec i a
l n a tu re o f [A ] a s g iv en inA p p en d ix I , t h e n u m b
er s w o u ld s t i ll b e o u t r ag eo u s ly l a rg e .T h e r e
i s a n o t h e r r e a s o n w h y t h e a b o v e s c h e m e w i
l l n o tw o rk ev e n i f t h e co m p u ta t i o n a l co s t co
u ld b e a f fo rd ed : t h e e s -t ab l i sh ed s y s t em o f eq
u a t io n s i s n o to r io u s ly i l l - co n d i ti o n ed .T h
a t i s , an y sm a l l p e r tu rb a t io n s i n t h e i n p u t
d a t a o r sy s t emp aram ete r s w i l l cau se s ig n i f i can
t ch an g es i n t h e e s t im a tedp a r a m e t e r s .T h e t w
o k e y f a c t o r s t h a t m u s t b e a d d r e s s e d a r e t
h e r e f o r es i ze an d il l - co n d i t io n in g . W i th t h
i s in m in d , so m e o f t h ea t t r i b u te s t h e m e th o d
sh o u ld h av e a r e :
9 B e ab l e t o an a ly ze s t ru c tu res w i th co m p lex i
t i e s a s u su -a l l y f o u n d i n f o r w a r d p r o b l e m
s a n d h a n d l e d b y t h ef in it e e l em e n t m e th o d
.
9 B e a b l e t o d e t e r m i n e m a n y u n k n o w n f o r
c e h i s t o r i e sfo rm in g u n k n o w n sh ap e d i s t r i b
u tio n s a s w e l l a s ac t i n gin d ep en d en t o f each o th
e r .
9 B e ab l e t o u t i l i ze m an y sen so r s , o f d i f f e
r en t t y p es , d is -t r i b u t ed t h ro u g h -o u t t h e s
t ru c tu re an d n o t p l aced i n an yp ar t i cu l a r ly o p t
im ized p o s i t i o n s .
9 F r o m a c o m p u t a t i o n a l c o s t a s p e c t, b e s
c a l a b l e s i m i l a rto fo rw ard p ro b l em s u s in g t h
e f i n i t e e l em en t m e th o d .
In t h e fo l l o w in g , t h e i l l - co n d i t i o n in g i
s h an d led b y t h e i n t ro -d u c t io n o f r eg u la r i za
t i o n t e rm s . 3 W e co p e w i th t h e p ro b l emo f s i z e
b y us i n g a n a l g o r i t h m a d a p t e d f r o m B e l l m
a n ' s i d e a sin d y n am ic p ro g ram m in g ; 2 ~ th i s s ch
em e es sen t i a l l y a l l o w s ar ecu r s iv e ( i n t im e)
so lu t i o n o f t h e p ro b l em .R e c u r s i v e o r m u l a
t i o n o f t h e P r o b l e m
C o n s id e r t h e g en e ra l l ea s t sq u a res e r ro r g
iv en b yN VE u, g = Z ] { d - Qu}ffFWJ{d- Qu}n
Ln = ] 3 )+ k{g),r~[Hl{g}~]
{ U } n + l : [A]{u}n } [B]{g}n (4)w h ere t h e su m m at io n
i s o v e r a l l t h e t im e s t ep s a s d i scu ssedb e fo re
an d th e seco n d t e r m o n th e r i g h t o f E q (3 ) is t h e
r eg -n l a r i za t i o n t e rm , a s d i scu ssed i n A p p en d
ix I I . T h e m a in i d eat o b e d e v e l o p e d i s t o p e r
f o r m t h e m i n i m i z a t i o n r e c u r s i v e lyra th e r
t h an g lo b a l ly , w o rk in g f ro m th e l a s t t im e s t
ep . T h eo r ig in a l fo rm o f t h i s a lg o r i t h m w as g
iv en in R ef s . 2 0 an d 2 1an d h as m u ch in co m m o n w i th
t h e K a lm an f i lt er . 2 2 -2 4Ricatti Equation
W e w i l l n o t g iv e t h e d e t a i l ed d e r iv a t i o n
b u t j u s t h in t a t i tsessen t ia ls ; a fu l l der ivat ion
is g iven in Ref . 17 .
2 6 9 V oL 4 2 No . 1 M a r ch 2 0 0 2
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C o n s id e r t h e a rb i t r a ry t im e s t ep n ; t h e o p
t im ized p a r t i a le r r o r s u m i s
E u, g) - G u, g)N W
= Z [ [ d - Q u } ; F W J { d - Q u } j_
+ k { g } ; [ H ] { g } j ]w h e r e E n m e a n s t h e s u m m
a t i o n b e g i n s a t j = n . T h e k e yto t h e a lg o r i t
h m i s t o r ew r i t e t h e p a r t i a l su m in t e rm s o f t
h eso lu t i o n t o a p rev io u s ly o p t im ized p ro b l em .
T h u s , a t t h eea r l i e r t im e
m i n [( u , g ) = { Q u - d } _ l F w J { Q u - d } n - 1n - 1
~ 5 ~+ k{g}n_ 1[ H ] { g } n - 1
1n ( u n = A u n - I + B g n - 1 )w h e r e t h e p a r e n t h
e s is o f E * m e a n s f u n c t i o n o f . I t c a n b esh o w
n th a t E * i s q u ad ra t i c i n {u}n fo r an y n an d th e re
fo recan b e w r i t t en a s t h e r ecu r s io n r e l a t i o
n
E ? , u ) = { u } V . + Gw h ere [R n ] , {Sn} an d C n a re co
e f f i c i en t s . I t is f o r t h ese co -e f f i c i en t s t
h a t a r ecu r r en c e r e l a t i o n i s e s t ab l i sh ed . D
ef in in gth e i n v e r se t e rm
[ D , ] = [ 2 k [ H ] + 2 [ B ] T [ R , ] [ B ] ] - It h e n ( a
f t e r s o m e m a n i p u l a t i o n ) w e g e t t h e r e c u r
r e n c ere l a t i o n s
[Rn-1] ---- [ Q ] T F W j [ Q ]+ [ A ] T[ R n - 2 R n B D n B T
R n ] [ A ]
{ S n 1 } ~ - 2 [ Q ] T [W J { d } , - i+ [ A ] T [ I -- 2 R n B
D n B T ] { s n } . (5)
T h i s i s a fo r m o f t h e R ica t t i eq u a t io n . 25 T
h ere i s n o n ee dto co m p u te t h e co e f f i c i en t C n s
in ce i t d i sap p ea r s o n d i f f e r -en t i a t io n . T h e
s t a r t e r v a lu es fo r th e r ecu r s io n a r e o b t a in
edb y lo o k in g a t t h e o p t im ized e r ro r fu n c t io n a
t t h e en d p o in tn - - N . T h i s g iv es
[ R N ] = [ Q ] r r W J [ Q ] , { S N } - ~ - - 2 [ Q ] T F W J
{ d } N 9 (6 )In t h i s w ay , s t a r t i n g w i th t h e en d v
a lu es , w e r ecu r s iv e ly d e -t e rm in e an d s to re ( e i
t h e r i n t e rn a ll y o r t o d i sk ) t h e q u an t i t i e
s[Dn ] [2 Rn B ] T [A] a nd [Dn ] [B] T { Sn } . T h i s i s ca l l
ed t h e b ack -w a r d s w e e p . D u r i n g t h e f o r w a r d
s w e e p , w e c o m p u t e
{g*}n = - [ D n + I ] [ 2R B ] T + I [A ]{u}n- - [ D n + I ] [ B
] T { s n + I }
{U}n+l = [A]{u}n + [B]{g}n . (7 )
A s d i scu ssed i n A p p en d ix I , th e m an ip u la t i o n
s w i th t h e l a rg ea r r ay [A ] a r e d o n e i n an e f f i c
i en t m an n e r .
T h es e a r e t h e fo rm o f th e eq u a t io n s u sed i n R
ef s . 1 5 an d1 6 fo r fo rce i d en t if i ca t i o n an d R ef s
. 1 2 -1 4 fo r h ea t co n d u c-t i o n p ro b l em s . A s d i
scu ssed n ex t , t h ese eq u a t io n s a r e s t i l lu n su i t
ed fo r s ca l i n g t o l a rg e s i zed f i n it e e l em e n t p
ro b l em s .
T h e a r r a y s i n t ro d u c e d w i t h t h i s f o r m o f
t h e i n v e r s e p r o b -l e m a r e o f s i z e{d} = [md
1]
[W J = { m d 1} d iagonal[ Q ] = r ind x m]
[R ~] = [m x m ] sy m m e t r i c{ } = { m x }
[D n 1 ] = r ing x mg] s y m m e t r i c[Dn] = r ing x mg] s y m
m e t r i c .
T h e s e a r r a y s s h o w a h u g e i m p r o v e m e n t i
n s t o r a g e r eq u i r e -m e n t s a s c o m p a r e d t o t
he n a i v e m e t h o d . H o w e v e r , t h e y a res t i ll n o
t i n a g o o d fo rm fo r s ca l i n g t o l a rg e s i ze p ro b
l em s . Inp a r t i cu l a r, t h e sq u a re , f u l l y p o p u
la t ed a r r a y [R n ] i s o f s i ze[m x m ] an d r eca l l th a
t m > 2 m u . F o r o u r t a r g e t s y s t e m o f1 0 0 0 0 D
o F th i s i s o n t h e o rd e r o f 1 0 0 0 M b y tes (w h en s
to redin d o u b le p rec i s io n ) . H en ce , an y m an ip u la
t i o n s i n v o lv in g itw i l l b e co m p u ta t i o n a l ly
i n t en s iv e a s w e l l a s r eq u i r i n g l a rg es to rag e
. F u r th e rm o re , t h e sy s t em req u i r em en t s a r e t
h en o r -d e r s o f m a g n i t u d e g r e a t e r t h a n n e e
d e d f o r t h e c o r r e s p o n d i n gf o r w a r d a n a l y
s i s v i a F E M .
T h ere i s an o th e r r easo n fo r av o id in g t h e u se o
f t h e R ica t t ifo rm o f t h e eq u a t io n s fo r la rg e sy
s t em s : a s p o in t ed o u t inR ef . 2 6 , t h ey a r e p ro n
e t o a n u m er i ca l i n s t ab i l i t y u n l e s s sp e -c i
a l p r ecau t io n s a r e t ak en to p rese rv e t h e p o s i t
i v e d e f in it en esso f [ R , ] . F o r s m a l l s y s t e m s
, e n f o rc i n g t h e s y m m e t r y o f [ R ~]by set t ing
[Rn] = 1 TRn + R, ] i s su ff ic ien t , bu t i t i s no t c le
arth a t t h i s i s su f f i c i en t fo r l a rg e s y s t em s
.
T h e eq u a t io n s w i l l n o w b e m o d i f i ed w i th t
h e i n t en t io n o fa v o i d i n g h a v i n g t o f o r m t h
e m a t r i x [ R n ] a l to g e th e r an d th u sav o id an y d
raw b a ck s a s so c i a t ed w i th i t.Time Invar ian t S ys
tems
F o r m o s t s t ru c tu ra l p ro b l e m s o f i n t e r e s
t , th e sy s t em i s t im ein v a r i an t ( t h e m a t r ix [A
] is co n s tan t ) . T h e e s sen ce o f t h eap p ro a ch to t
ak e ad v an tag e o f t hi s i s to r ea l i ze t h a t w h en th
es y s t e m d o e s n o t c h a n g e m a n y o f t h e i n t e r
m e d i a t e m a t r i c e sd o n o t c h a n g e b e t w e e n t
i m e s t e p s a n d t h a t t h e r e c u r r e n c ere l a t i o
n s can b e w r i t ten i n t e rm s o f th e i r d i f f e r en
ces . T h i sw as f i r s t in t ro d u ced in t h e se r i e s o f
p ap e r s b y K a i l a th 2 6'2 7w h o d i scu sses a n u m b er
o f v a r i a t i o n s o f t h e b as i c a lg o r i t h m ;d e r
iv a t i o n s can a l so b e fo u n d in R ef s . 1 7 an d 2
8.
C o n s i d e r t h e s y m m e t r i c a r r a y [ R n ] a t tw
o t im es i n t h er e c u r s i o n g i v e n b y[R n -1 ] = [ Q T
W Q ] + [ a r ] [R n ] [ I - 2 B D n B T ] [ R n ] [ A ][ R n -2 ]
= [ Q T W Q ] + [ A T ] [ R n - 1 ] [ I - 2 B D n - I B T ]
[ R n - 1 ] [ A ] .T h ere i s a g o o d d ea l o f d u p l i ca
t i o n i n g o in g f ro m o n e l ev e l t oth e n ex t ; t h e o
n ly r ecu r s iv e t e rm n eed ed a t t h e seco n d l ev e li s
[D n ] w h ich i s a l so o b t a in ed f ro m [R n ] a s t h e i n
v e r se o f
[ D n 1] = 2 k [ H ] + 2 [ B T R n B ] .
ExperimentalMechanics 9 7
-
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4/12
T h i s m o t i v a t e s e x p r e s s i n g t h e q u a n t i
t i e s i n t e r m s o fd i f f e r e n c e s
[ / x R n ] - - [ R n ] - [ R n - l ] .T h e r e c u r s i o n r
e l a t i o n s a r e w r i t t e n f o r t h e d i f f e r e n c e
s .
T h e f u l l s e q u e n c e o f r e c u r s i o n r e l a t i
o n s i s[D~-11 ] = [D~ 11 - 2 [ B T y , ] [ L n l [ Y [ B ][ 0 n -
1 ] = [ 0 2 1 ] - 1[ L n - 1 ] = [Ln] + 2 [LnyT B][On-1][BTynLn][ Y
, - I ] = [Ar][Y, , - 2K nB r Yn1
[ K ~ - , ] = [ K , , ] - [ Y n - 2 K , , B r Y n ] [ L n Y f B
] [ D , , - , ]{ S n - i ' = - 2 [ O T W ] { d } n -I + [ A T ] {{
S n } - - 2 [ K n B r ]{ Sn }.
I t i s c l e a r t h a t t h e s e c o m p u t a t i o n s c a
n b e s e q u e n c e d s o t h a tm a n y o f t h e p a r ti a l p
r o d u c t s c an b e r e - u se d . W h e n d o i n gt h i s , t
h e c a r r y - o v e r a r r a y s s h o u l d b e c h o s e n t o
b e t h e s m a l lo n e s . O n e s u c h p o s s i b l e s c h e
m e i s g i v e n i n R e f . 2 8. T h ei n i ti a l c o n d i t i
o n s f o r t h e r e c u r s i v e v a r i a b l e s a r e
[ D ~ 1] = 2 X [ H ] + 2 [ B T Q T ] [ W J [ Q B ][DN] = [ 0 N 1
] - 1[YN] = [AT][Q T]
[KN] = [QTW][QB][DN][ L N ] = - [ W J + 2 [ W J [ Q B ] [ D N ]
[ B r Q ] T F W j{ S N } --- - 2 [ Q T w ] { d } N .
D u r i n g t h e b a c k w a r d p h a s e o f th e c o m p u t
a t i o n s , t h eq u a n t i t i e s
[DnBTI{Sn}, [ K : ]a r e s t o r e d e i th e r t o d i s k o r
i n - c o r e . T h e s e c a n t h e n b e u s e d i nt h e f o r
w a r d c a l c u la t i on s . T h e f o r w a r d s t a g e o f
th e c o m p u t a -t i o n s i s g i v e n b y
{g}n = - [ K T + I ] [ A I { u } n - [ D n + l B r ] { & + l
}{ u} n+ l = [ d ] { u } n + [ B ] { g } n . ( 8)
T h i s r e a r r a n g e m e n t o f t h e e q u a ti o n s s h
o u l d g i v e e x a c t ly t h es a m e r e s u l ts a s w h e n
u s i n g t h e R i c a t t i f o r m . T h e p r i m a r yr e s t
r i c ti o n i n c u r r e d i s t h a t i t i s a s s u m e d t h
a t t h e s y s t e m m a t r i x[ A ] d o e s n o t c h a n g e b
e t w e e n t i m e i n c r e m e n t s .
T h i s n e w f o r m o f th e r e c u r s i v e r e l a t io n
s d e a l s w i t h a r r a y so f s i z e
[ D n 1 ] = [rag x ra g] s y m m e t r i c[ D n ] = [mg x mg] s
y m m e t r i c[Ln] = [md x rod] s y m m e t r i c[Y~I = [m
rod]
[ K ~ I = [ m x mgl{Sn} = [m x 1].
T h e l a r g e s t a r r a y s a re o f si z e [ m x m d ] a n
d [ m x rag], a n ds i n c e b o t h m d a n d rng ( n u m b e r o
f se n s o r s a n d n u m b e r o f
f o r c e s , r e s p e c t i v e l y ) a r e s i g n i f i c a
n t l y l e s s t h a n m , t h e r e i s ah u g e r e d u c t i o
n i n t h e s t o r a g e r e q u i r e m e n t s .
T h e d e r i v e d e q u a t i o n s h a v e t h e f u r t h e
r a d v a n t a g e t h a t t h e ya r e m o r e s t a b l e t h a
n t h e R i c a t t i f o r m . 2 6'2 9T h e C o m p u t e r P ro g
r a m n v e r s e
B o t h t h e R i c a t t i a n d t i m e i n v a r i a n t e q
u a t i o n s a r e i n c o r -p o r a t e d in t o a c o m p u t e
r p r o g r a m c a l l e d I n v e r s e . I m p l i c i ta n d e
x p l i c i t v e r s i o n s o f t h e t i m e i n t e g r a t i o
n s a r e s e l e c t a b le .
E x i s t i n g f i n i t e e l e m e n t c o d e t o p r o d u
c e t h e s t i f f n e s s ,m a s s , a n d d a m p i n g a r r a
y s o f 3 - D t h i n - w a l l e d s h e l l s t ru c -t u re s w
i t h r e i n fo r c e m e n t s w a s m o d i f i e d a n d i n c
o r p o r a t e d i n t oZ n v e r s e ; c o n s e q u e n t l y i
t c a n h a n d l e th e s a m e p r o b l e m s a st h e o r i g i
n a l f i n it e e l e m e n t p r o g r a m . T h e m o d e l o f
t h e s t r u c -t u r e i s c r e a t e d u s i n g t h e m e s h
g e n e r a t i n g p r o g r a m a s s o c i a t e dw i t h t h e
f i n i t e e l e m e n t p r o g r a m .
T h r o u g h t h e m a tr i x [ Q ] , a n y n u m b e r o f m i
x e d s e n s o rs c a nb e u s e d . I n t h e c a s e o f t h e i
m p l i c i t s c h e m e , t h e a c c e l e r o m e t e rd a t a
c a n b e u s e d d i r e c t l y ; i n t h e e x p l i c i t s c h
e m e i t n e e d s t ob e i n t e g r a t e d o n c e t o b e p u
t i n th e f o r m o f v e l o c i t y .
T h r o u g h t h e m a t r i x [Bg], a n y n u m b e r o f f o
rc e s a n d m o -m e n t s c a n b e d e t e rm i n e d . A s s e
e n i n th e e q u a t io n s o f A p -p e n d i x I , w e a c t u
a l l y d e t e r m i n e t h e r a t e o f f o r c e a n d n o t t
h ef o r c e i t se l f . T h i s h a s t h e e f f e c t o f a d d
i n g r e g u l a r i z a t i o n i nt h e t i m e d i r e c t i o
n ; t h i s is t h e o n l y t i m e r e g u l a r i z a t i o n i
m p l e -m e n t e d . T h e c h o i c e o f z e r o , fi rs t , a
n d s e c o n d o r d e r s p a c er e g u l a r i z a t i o n i s
i m p l e m e n t e d .
N o t e t h a t i f w e s t o r e i n - c o r e t h e n w e n e
e d [ ( m x m x) x N]m e m o r y , w h e r e N i s t h e n u m b e
r o f ti m e s t e ps . F o r e x a m p l e ,i f t h e s y s t e m
s i z e is 1 0 0 0 0 , w i t h 2 0 u n k n o w n f o r c e s , a n
d 2 0 0 0t i m e s t e p s , t hi s c a n l e a d t o o v e r 3 G B
o f m e m o r y . H e n c e i t i so n l y f o r s i m p l e p r o
b l e m s t h a t i n - c o r e s t o r a g e i s p e r f o r m e d
.M o r e d e t a i l s l i k e t h i s , a n d o n t h e p r o g r
a m i n g e n e r a l , c a n b ef o u n d i n R e f . 2 8 .
I n o r d e r t o e x er c i se t h e p r o g r a m a n d d e t
e r m i n e s o m e o f t h em e t r i cs , w e n o w a p p l y i t
t o a c y l i n d e r p r o b l e m u s i n g s y n t h e ti cd a t
a . T h i s w i l l b e u s e d t o d e m o n s t r a t e t h e a b
i l i t y o f t h ep r o g r a m t o h a n d l e l a r g e p r o b
le m s . I t w i l l a ls o d e m o n s t r a t et h e n e c e s s
i t y o f r e g u l a r i z a ti o n .
T h i s e x a m p l e p r o b l e m , s h o w n i n F i g . 1 ,
p r e s e n t s a c o m -p l e x p r o b l e m , i n t h a t i t i
s t h r e e d i m e n s i o n a l w i t h a l a r g en u m b e r o
f d e g r ee s o f f r e e d o m . T h e e l e m e n t s u s e d a
r e s u c ht h a t th e r e a r e s i x d e g r e e s o f f r e e d
o m a t e a c h n o d e : t h r e e d i s -p l a c e m e n t s , a
n d t h r e e r o ta t io n s . T h e t o t al n u m b e r o f D o
F f o rt h e m e s h s h o w n i s 3 4 5 6 a n d t h i s t r a n s
l a t e s i n t o a t o t a l s y s -t e m s i z e ( s t a t e v e
c t o r p l u s u n k n o w n f o r c e s ) f o r t h e i n v e r s
ep r o b l e m a simpl ic i t : S ize = 10375 , A t = 5 .0 b~s, N =
400e x p l i c it : S i z e = 6 3 7 5 , A t = 0 .4 t xs , N = 2 0 0
0 .
T h e s e n u m b e r s a r e o n t h e o r d e r o f th o s e p
r e s e n t e d i n t h ei n t r o d u c t o r y d i s c u s s i o
n .
T h e d i s t r i b u t e d l o a d a l o n g t h e w e s t e d
g e i s c o m p r i s e d o fs e v e n u n k n o w n f o r c e s w
i t h a p u l s e - l i k e h i st o r y . T h e s e n -s o r s a r
e n o t d i s t r i b u te d i n a n y p a r t i c u l a r o r o p
t i m i z e d m a n n e rf o r c a lc u l a t in g t h e s e f o r
c e s - - i n d e e d w e p u r p o s e l y , f o r th er e s u l t
s t o b e s h o w n , d i d n o t u s e a n y s e n s o r s i n t h
e i m m e d i -a t e l o c a t i o n o f t h e u n k n o w n f o r
c e s . A l s o , t h e s y n t h e t i c d a t aw e r e c o n t a
m i n a t e d w i t h G a u s s i a n n o i s e h a v i n g a s t a
n d a r dd e v i a t i o n o f 1 0 % o f th e a v e r a g e v e l o
c i t y .
T w o s e t s o f r e s u l t s a r e s h o w n i n F i g . 2 u
s i n g d i f f e r e n ta m o u n t s o f r e g u l a r i z a t i
o n . R e g u l a r i z a t i o n i s s e e n t o b e a ne s s e n
t i a l i n g r e d i e n t. T h e t i m i n g s f o r t h e s o l
u t i o n s a r e :
2 8 9 VoL 42 No. 1 March2002
-
8/12/2019 Souleymane BF02411048
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F ig . 1 Sh e l l m e s h w it h p a ra m e t e r s
a m : 8 9 m m 3 . 5 i n .): n g t h : 5 n m 2 , 0 i n . )l i c k
: 3 m m . 1 2 5 i n . )u m i n u m
im p l i c i t : 5 3 m in u tesexp l ic i t : 10 .6 hours .
T h e r e l a t i v e c o s t b e t w e e n t h e b a c k w a r
d a n d f o r w a r d p o r -t i o n s o f t h e so lu t io n i s
ap p ro x im a te ly 8 :1 fo r b o th i n t eg ra t io nsch em es .
B o th se t s o f t im in g s a re s ig n i f ican t ly sm a l l e
r t h anth e n a iv e sch em e p resen ted in t h e in t ro d u c t
io n . T h e d i f f e r -e n c e b e t w e e n t h e t w o s c h e
m e s i s t h a t t h e i m p l i c i t m e t h o dcan u se a t im
e s t ep r e l a t ed to t h e f r eq u en c y co n ten t o f t h
eex c i t a t i o n s , w h ereas t h e s t ep s i ze fo r t h e ex
p l i c i t m e th o d i sd i c t a t ed b y th e e l em en t s i
ze . In th i s i n s t an ce , t h e ad v an tag el i e s w i th t
h e im p l i c i t m e th o d .E x p e r i m e n t I : 3 D C y l i
n d r i c a l S h e l l
T h e f i r s t ex p e r im e n t is o n an im p ac ted sh e l l
. W h i l e o n ly as in g le fo rce i s id en t i f ied , i t d em
o n s t r a t e s so m e o f t h e p rac t i ca li s su es a s so c
i a t ed w i th u s in g ex p e r im en ta l d a t a .Ex p e r im e
n t a l Se t u p
T h e se t -u p fo r t h e ex p e r im en t is p re sen ted in F
ig. 3 . A5 1 m m (2 in ) l o n g cy l in d e r w i th 3 m m (1 /8 i
n ) w a l l t h i ck -n e s s w a s c o n s tr u c t e d f r o m a
l u m i n u m . A c c e l e r o m e t e r s w e r ep lace d o n th
e to p (9 0 ~ an d 1 8 0 ~ f ro m th e im p ac t s i t e , a ssh o
w n in F ig . 3, an d a t t ach ed u s in g b ees w ax . T h e sp
ec i -m e n w a s s u s p e n d e d b y s t r in g s a n d i m p a
c t e d m i d w a y 2 5 m m(1 in ) a long the leng th .D a t a f o
r t h e e x p e r i m e n t s w e r e c o l l e c t e d b y m e a n
s o f a nO m eg a In s t ru m en t s D A S -5 8 d a t a acq u i s i
t i o n ca rd ; t h i s i s a1 2 - bi t c a r d c a p a b l e o f 1
M H z s a m p l i n g r a t e a n d s t o r i n g u pto 1 0 6 d a t
a p o in t s in t h e o n -b o a rd m em o ry . A to t a l o f 8c h
a n n e l s c a n b e s a m p l e d ; h o w e v e r , i n c r e a s
i n g t h e n u m b e ro f ch an n e l s d ec rease s t h e f a s t
e s t p o ss ib l e sam p l in g r a te . F o rt h is e x p e r i m
e n t , f o u r c h a n n e l s w e r e s a m p l e d a t 2 5 0 k H
z , o ra 4 ~ s t im e s t ep .A c c e l e r o m e t e r s ( P C B 3
0 9 A ) a n d a m o d i f i e d f o r c e t ra n s -d u c e r ( P C
B 2 0 0 A 0 5 ) w e r e u s e d t o c o l le c t th e d a ta . T h
e c o n -n e c t io n s b e t w e e n t h e a c c e l e r o m e t e
r s a n d t h e c o m p u t e r w e r et h r o u g h t h e u se o f
a n O m e g a I n s t r u m e n t s B N C - 5 8 m u l t i p l e x
-ing un i t .
T h e a c c e l e r o m e t e r o u tp u t s w e r e i n t e g
ra t e d a c c o r d i n g t oth e t r ap ezo id a l ru l e
A = .0. O 1 6
tim eF ig . 2 Seven fo rce recons t ruc t ions fo r the she l l
us ingd i f fe ren t amounts o f regu la r iza t ion
v n + l = V n + 8 9 + a n + l ) A tt o d e t e rm in e th e v e
lo c i t i e s . B ec au se o f s l ig h t o f f se t s i n th
eacce l e ra t i o n v o l t ag es , a t r en d i s so m et im es o
b se rv ed in t h ev e lo c i ty. T h i s t r en d i s e s t im a
ted an d su b t r ac t ed ; w e r e fe r t oth e d e t r en d ed d
a t a a s t h e r aw d a t a . T h e ex p l i c i t i n t eg ra t
io no p t io n in n v e r s e r eq u i r e s a sm a l l A t , an d
so th e v a lu es o fth e i n p u t d a t a a r e o b ta in ed f ro
m th e m easu red d a t a b y l i n ea rin t e rp o la t io n . T h
e im p l i c i t i n t eg ra t io n o p t io n d o es n o t r e -q
u i r e i n t e rp o la t io n an d can u s e t h e acce l e ra t i
o n d a t a d i r ec t ly .
T h e r e c o r d e d f o r c e a n d ( p r o c e s s e d ) v e
l o c i t y d a t a a r esh o w n in F ig . 4 . N o te t h a t t h
e im p ac t i s a d o u b le im p ac to n a v e ry sm a l l t im e
sca l e .F o r wa r d P r o b le m
A g o o d in v e r se an a ly s i s i s p red i ca t ed o n a h
ig h f id e l i t yf in i te e l em e n t m o d e l o f t h e s t
ru c tu re b e in g an a ly zed . T h a t i s ,t h is m o d e l m u
s t c o r r e s p o n d a c c u r a t e l y t o t h e r e a l e x p
e r i m e n -t a l s i t u a tio n in t e rm s o f m a te r i a l p
ro p e r t i e s (Y o u n g s m o d u lu s ,m ass d en s i ty ) , d
im en s io n s (d i am e te r , t h i ck n ess ) , an d sen so rlo
ca t io n s . T h e re fo re , b e fo re an y so lu t io n to an in
v e r se p ro b -l em i s a t t em p ted , t h e m o d e l in g sh
o u ld b e v e r i f i ed b y d o in g af o r w a r d p r o b l e m
.
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F ig . 3 - - Ex p e r im e n t a l s e t - u p
1.0
n
N = ' (501b)
I I ~ I i I I
0 4 0 0 8 0 0 1 2 0 0 1 6 0 0T i m e [gs]o Measured-- Model Nf/~
~ ' ~ 9 0 d e g p ~
.5 M
.0180 deg
i r i r i i i i0 4 0 0 8 0 0 1 2 0 0 1 6 0 0T i m e [gs]
F ig . 4 - -Recorded fo rce and ve loc i t y t races fo r the
she l l . A lsoshow n is a com par ison w i th the fo rward p rob
lem resu lts
A c o n v e r g e n c e s t u d y w a s p e r f o r m e d t o d
e t e r m i n e t h ep r o p e r e l e m e n t s i z e f o r th e F
E M m o d e l ; t h e d a t a f r o m t h efo rce t r an sd u ce r
p ro v id ed th e i n p u t fo rce h i s to ry fo r t h i ss tu d
y. F ro m th e r e su l t s , i t w as d e t e rm in ed th a t 3 m
m(1 /8 i n ) e l em en t m o d u les w o u ld b e an accep t ab l e
s i ze fo r t h e
I n v e r s e an a ly s i s . A m o d e l w as t h en c r ea t
ed s im i l a r t oth a t o f F ig . 1 u s in g 1 4 08 fo ld ed p l
a t e e l em e n t s . T h i s c r ea t e sa p r o b l e m w i t h
a s y s t e m s i z e o f 4 4 8 6 d e g r e e s o f f r e e d o m
.T h e r e su l t s fo r t h e fo rw ard p ro b l em can b e seen i
n F ig . 4 .T h e tw o d a t a s e t s a r e a lm o s t i n p h ase
w i th each o the r . T oach iev e t h ese r e su l t s , t h e e f
f ec t i v e d i am e te r o f t h e cy l in d e rw as t ak en a t
t h e av e rag e r ad iu s .I n v e r s e S t u d y
T h e f o r c e r e c o n st r u c ti o n s f r o m t h e e x p
e r i m e n t a l v e l o c i tyin p u t s a r e sh o w n in F ig .
5 . I t is p l eas in g t o n o t e t h a t th ed o u b le im p a c
t i s d e t ec t ed . N o w , a w o rd ab o u t t h e ch o iceo f r
eg u la r i za t i o n is i n o rd e r . F o r t r an s i en t p ro
b l em s , i t cana lw ay s b e a r r an g ed th a t t h e re i s
an i n i t i a l ze ro fo rce h ead e r ,o r an i n i t ia l p e r
io d o f q u i e scen ce . O n r eco n s t ru c t i o n w i thp e r
f ec t d a t a , t h i s w o u ld b e ze ro ; w i th n o i sy d a t
a t h i s w o u ldo sc i l l a t e ab o u t ze ro ; an d w i th p o
o r p a ram ete r s t h i s w i l l sh o wa t ren d . W e tw eak th
e r eg u la r i za t i o n t o g iv e t h e b es t l o o k in
gheader .
A c c e l e r o m e t e r 2 o n i ts o w n d o e s a g o o d j o
b b u t th e s a m ei s n o t t r u e fo r A c ce l e ro m e te r 1
. A t f ir s t s i g h t , t h i s m ig h tap p ea r t o i n d i ca
t e t h a t so m eh o w th ese d a t a a r e co r ru p t ed .H o w
e v e r , t h e f a c t t h a t u s i n g b o t h a c c e l e r o m
e t e r s t o g e t h e rg iv es so m ew h a t im p ro v ed r e su
l ts d o es n o t b ea r t h i s ou t . I fa c c e l e r o m e t e
r 1 w a s i n d e e d c o n t a m i n a t e d , t h e n m i x i n g
b o t ha c c e l e r o m e t e r d a t a w o u l d a l so s h o w t
h e c o n ta m i n a t i o n . Am o re r easo n ab le co n jec tu
re i s t h a t p o s i t i o n 1 i s v e ry sen s i -t ive , in an
i l l -cond i t ion ing sense, to s l igh t er ro rs in the data
.
O u r co n jec tu re h as im p o r t an t im p l i ca t i o n s
fo r i n v e r sem eth o d s , so w e d ec id ed to fu r th e r i n
v es t i g a t e t h e n a tu re o fth e d i sc r ep an cy b y m a
n ip u la t i n g sy n th e t i c d a t a i n a co u p le o fw a y
s a n d c o m p a r i n g t he p e r f o r m a n c e o f p o s i ti
o n s 1 a n d 2 .F i r s t, t h e sen s i t i v i t y d u e t o t h
e p o s i t i o n in g o f t h e ac -c e l e r o m e t e r i n r e
l a t io n t o t h e n o d e u s e d i n th e I n v e r s ean a ly
s i s w as ex am in ed . T h e se r e su lt s a r e sh o w n in F
ig . 6an d d e m o n s t r a t e t h a t , f o r a g iv en m i sa l
i g n m en t , t h e 9 0 ~ p o -s i t i o n i s m o re se n s i t i
v e t h an t h e 1 8 0 ~ p o s i t i o n . H o w e v er ,t h e s y
m p t o m o f t he e r r o r ( a c o n s ta n t f r e q u e n c y s
u p e r p o s i-t i o n ) d o es n o t r e f l ec t t h e sy m p to
m sh o w n in th e ex p e r im en ta lresu l ts .
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F ig . 5- -Compar ison of reconst ructed forces f rom I n v e r
s eand force transducer data for the she l l experiment
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j \ ~ . ~ ~ . ~ 1 80 ,+6 mm
/ ~ . . . . ~ 9 0 ,+ 3 m m
.0I I ~ I ~ I r I
0 400 800 1200 1600Time [gs]F ig . 6- -Ef fe c t on fo rce
reconstructions of sensor m is-posit ioning in the hoop direct
ion
A s a s e c o n d s t u d y, a p p r o x i m a t e l y t he s a
m e a m o u n t o f d r i f ts een i n t h e ex p e r im en ta l v
e lo c i t i e s w as ad d ed to t h e v e lo c i -t i e s g e n e
r a te d s y n t h e ti c a ll y . T h i s a m o u n t w a s d e t
e r m i n e d b ye x a m i n i n g t h e r a w d a t a f r o m t h
e a c c e l e r o m e t e r s a n d e s t i m a t -in g t h e d ig
i t iz in g l ev e l o f th e v o l t ag es r eco rd ed . T h i s v
a lu ew as fo u n d to b e a ro u n d 1 5 m / s (6 0 0 in / s ) . T
h ese ad ju s t ed v e -l o c it i e s w e r e u s e d t o r e c o
n s t r u c t t h e f o r c e f r o m t h e r e s p o n s ea t t h
e tw o n o m in a l 9 0 ~ an d 1 8 0 ~ p o s i t i o n s an d th e
r e su l t s a r ep resen t ed i n F ig . 7 . A g a in , t h e 9 0
~ p o s i t i o n i s m o re sen s i t i v eto s l ig h t d ev i a
t i o n s i n th e i n p u t d a ta . T h i s t im e th e sy m p to
mis s im i l a r t o t h a t o f t h e ex p e r im en ta l r e su
lt s . I t is n o t ed t h a t
1
0I
0
-- Inverse~ , ..........Transducer, ' : k . , , . / , ' . , ,
,
8Odeo
400 800 1200 1600T i m e [gs]Fig. 7--E ffec t on force
reconstruct ions of adding dr i f t tovelocity inputs
u s i n g b o t h a c c e l e r o m e t e r s d o e s n o t g i
v e a s i m p l e a v e r a g e b u tg iv es a r e su l t t h a t i
s o f t en b e t t e r t h an t h e i n d iv id u a l r e su l t s
.T h ese s tu d i e s i n d i ca t e a co n ce rn t h a t i s q u i
t e c ru c i a l i n
in v e r se s tu d i e s an d i s d i f f e r en t f ro m th e
ex p e r im en ta l i s su eso f p o s i t io n accu racy an d s ig
n a l f id e l it y . W h en d o in g an i n -v e r se an a ly s i
s , t he l o ca t i o n o f t h e sen so r ( s ) an d n o t j u s t
t h e i raccu ra cy is v e ry im p o r t an t . T h i s i s n o t
so m eth in g t h a t cana l w a y s b e d e c i d e d i n a d v a
n c e ; f o r e x a m p l e , i f t h e i m p a c tw as a t t h e 2
7 0 ~ l o ca t i o n t h en t h e ro l e s o f t h e acce l e ro m
ete r sw o u ld b e r ev e r sed . T h e im p l i ca t i o n i s t
h a t m u l t i p l e s en so r sm u s t b e u s e d , a n d t h a
t t h e y b e c o m b i n e d t h r o u g h t h e u s e o fr eg u
la r i za t i o n . B o th o f t h ese a sp ec t s a r e p a r t o
f th e p ro g ra mI n v e r s e
E x p e r i m e n t I1 : I n P l a n e P r o b l e mT h e s e c
o n d e x p e r i m e n t d e m o n s t r a t e s t h e a b i l i
ty o f t h e
m eth o d to d e t e rm in e m u l t i p l e u n co r r e l a t
ed fo rces . I t i s n o tco n v en ien t t o a r r an g e m u l t
i p l e im p ac t s o f a s tru c tu re ; i n s t ead ,w e u s e m
u l t i p l e f o r c e s a s s o c ia t e d w i t h t h e d e c o
u p l i n g o f aco m p lex s t ru c tu re . T h i s a l so a l l o
w s a d em o n s t r a t i o n o f th eo n e - s i d e d H o p k i
n s o n b a r .One-Sided Hopkinson Bar
In ex p e r im en ta l v e r i f i ca t i o n s tu d ie s , i t
o f t en o ccu r s t h a t w en eed to ap p ly a k n o w n fo rce h
i s to ry to a s t ruc tu re . I n so m ein s t an ces , w h ere t
h e f r eq u en cy co n ten t o f t h e ex c i t a t i o n i sl o w
, w e c a n u s e a n i n s t r u m e n t e d h a m m e r a s i s c
o m m o n i nm o d a l an a ly s i s , 3 ~ an d a s d o n e i n t h
e p rev io u s ex p e r im en t . I no th e r i n s t an ces , h o
w ev er , w e m ay n eed a h ig h en e rg y , h ig hf r e q u e n c
y i n p u t a n d h e r e th e c o m m o n f o r c e t r a n sd u c
e r s d on o t h av e t h e r eq u i r ed sp ec i f i ca t i o n s
. T h e o n e - s id ed H o p k in -so n b a r w as d es ig n ed fo
r t h i s p u rp o se .
A w i d e l y u s e d m e t h o d f o r d y n a m i c m a t e r
i a l a n a l y s e s i sb y m e a n s o f t h e s p l i t H o p k
i n s o n p r e s s u r e b a r ( S H P B ) } ]T h e s t an d a rd
c o n f ig u ra t i o n h as ev o lv ed in to o n e w h ich h astw
o lo n g b a r s o f i d en t i ca l m a te r i a l , i n s t ru m
en ted w i th s t r a ing a g e s , a n d w i t h a s m a l l s p e
c i m e n p l a c e d b e tw e e n t h e m . T h ef i r s t r o d ,
o r i n p u t b a r , i s im p ac t ed b y f i r i n g a p ro j ec
t i l e an d a
Experimental Mechanics 9 31
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s t re s s w a v e i s ge n e r a t e d . T h i s w a v e c a u
s e s t h e s p e c i m e n i nb e t w e e n t h e t w o b a r s t
o b e d y n a m i c a l l y l o a d e d . T y p i c a l ly ,b e c a
u s e t h e s p e c i m e n i s s m a l l , t h e w a v e p r o p a
g a t i o n e f f e c t sw i t h i n t h e s p e c i m e n a r e n
e g l i g i b le , a n d i t q u i c k l y r e a c h e s au n i f o
r m s t r e s s s t a te . U n d e r t h i s c i r c u m s t a n c
e , b o t h t h e r e -f l e c t e d a n d t r a n s m i t t e d s
i g n a l s i n t h e l o n g b a r s h a v e a s i m p l er e l a
t i o n s h i p t o t h e s p e c i m e n s t r e s s ( a n d s t r
a i n ) s t a t e . U s i n gt h e s tr a in g a g e s t o m e a s
u r e t h e w a v e s p r o p a g a t e d i n b o t h t h ei n p u
t a n d o u t p u t b a r s, t h e s t r e s s a n d s t r a in h i
s t o r i e s o f t h es p e c i m e n c a n b e d e t e r m i n e
d . S i n c e t h e b a r s a re s u c h t h a t t h ec r o s s - s
e c t i o n a l r a d i u s i s n o t v e r y l a r g e , d i s p e
r s i o n e f f e c t s a r em i n i m a l a n d th e r e f o r e
th e a n a l y s i s c a n b e d o n e u s i n g 1 - D r o dtheory
.
I n th e o n e - s i d e d H o p k i n s o n b a r c o n f i g u
r a t io n a s s h o w n i nF i g . 8 , t h e s p e c i m e n i s t
h e c o m p l e x s tr u c tu r e f o r w h i c h w ew i s h t o a
p p l y t h e k n o w n f o r c e . A p u l s e o f s h o r t d u r
a t i o n i sg e n e r a t e d a t o n e e n d o f t h e ba r , u s
u a l l y b y i m p a c t i n g i t. T h ei m p o r t a n t d i f f
e r e n c e i n c o m p a r i s o n t o t h e S H P B i s th a t w
a v ee f f e c t s i n t h e s p e c i m e n a r e s ig n i f ic a
n t a n d c o n s e q u e n t l y t h er e f l e c t e d s i g n a
l d o e s n o t h a v e a s i m p l e r e l a t i o n t o t h e t r
a n s -m i t t e d f o r c e . T h a t i s, s i n c e t h e d u r a
t i o n o f t h e e v e n t c o v e r ss e v e r a l p a s s a g e
s o f t h e o r i g i n a l p u l s e u p a n d d o w n t h e i n
-c i d e n t b a r , s i m p l e t i m e s h i f t i n g o f t h e
r e c o r d e d s i g n a l i s n ol o n g e r a d e q u a t e . B
o t h t h e i m p a c t i n g f o r c e a n d t h e t ra n s m i t
-t e d f o r c e m u s t , i n g e n e r a l , b e d e t e r m i n
e d s i m u l t a n e o u s l y .
T h e o n e - s i d e d H o p k i n s o n b ar w a s d e s i g n
e d w i t h t h e f o l -l o w i n g p a r a m e t e r s : I t i s
2 5 4 m m ( 1 0 0 i n ) l o n g w i t h B L HS R - 4 s e m i c o n
d u c t o r g a g e s ( m o d e l S N B 1 - 1 6 - 3 5 - 6 ) p l a c
eda t th e m i d - a n d t h r ee - q u a rt e r s p o in t s f r o
m t h e i m p a c t e d e n d .T h e r o d i s m a d e o f s t e e
l w i t h a c r o s s - s e c t i o n a l d i a m e t e r o f6 . 3
m m ( 0 . 2 5 i n ). T h e r a t h e r l o n g l e n g t h o f t h
e r o d a l l o w sf o r p u l s e s o f l o n g d u r a t io n t o
b e g e n e r a t e d .I n v e r s e P r o b l e m
T h e f i n i t e e l e m e n t c o n v e r g e n c e s t u d y
i n d i c a t e d a m a x i -m u m e l e m e n t s iz e o f 6 .3 m
m ( 0 .2 5 i n ) m o d u l e s c o u l d b e u s e di n th e a n a
l y s is . T h e m o d e l a n d m e s h a r e s h o w n i n F i g
. 9 , As y m m e t r i c m o d e l w a s u s e d t o a l l o w f o
r m o r e e l e m e n t s o f as m a l l e r s i z e ; t h e n u m
b e r o f e l e m e n t s f o r t h e r e s u lt i n g m o d e lw a
s 2 1 5 9 p l a t e e l e m e n t s , w h i c h c r e a t e s a p r
o b l e m w i t h as y s t e m s i z e o f 3 4 0 7 d e g r e e s o
f f r e e d o m .
T h e s p e c i m e n w a s m a c h i n e d f r o m a 6 .3 m m (
1 /4 i n ) t h ic ks h e e t o f a l u m i n u m , c u t t o b e a
3 0 5 m m ( 1 2 i n ) s q u a r e w i t h
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V I / I / V V l tA , 4 A / I / I / I / V V ~1 ]A I I I V/ V V I / V
[ /A I V I / I / V U/ I / I / I / I / u/ v v v v vv v v w l / w ~ /
v w wI , / V V V V I A A A / q A / V I / V 1 /V V b d / V ~ A / V V
V ~~ V I A A / 1 / I . d A A / 1 / V 1 /W V L . V V L / V V V W W
WW V V I ~ / W V [ /W V V I ~ A / V I / VW V V I / V V I / V V I A
/ V I A /I / V V V I / I / V I / I / ~ / q A / I / ] /. I / I / I A
/ V V I A / I / I /I / I l l ~ I l l /V I l lA / I /
/ / I / I A / V I / I / I / ] /[ / I A /V I / I I V I /
/ I / V V V V I f V V I .A / V V V V I / V V I .A / V V V V V I
N V I ,/ I / I / I / q / V V V l / I ,I A A / I , / I / I / I I V V
W I ,/ I / I / I / I/ V V I / I / I/ V I / I / V V I / V I ,/ I / I
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/ V V I / V V V ] / bI / V V I A / I / W V W I ,V V ~ A A A / V V V
V I ,I / V V l / 1 / V l / W W bW V V V I / V / / V I / V I
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A / V V 1 / V LV I I I / V I I I / LA A / V V W V W L~ / V V t A A
/ I . a / I A ,V] ] I I ] VI V1 L~ A A / I / ] / V V V V LV V V V V
I / I / W V I ,V I m / v v v l . . q / , / V bI ~ / I / I / I / W V
~AAAA /WVCAAAA/I / I /V~AAAA/VV~A / V I / I / W b/ ] / V I / I / V
V b/ ] A / I / I / I / V b/ ] A / I / I / I I V b~ q A / V I / I I
V b
I / V V I AI A / W V IV I / I A / V IU 1 / I A / V IW V V I / IV
V I / V I / IVVI /VVIIAAA/1AI A / V V V II A / I / I I I / 1I A A A
A / IV I A / I / V IV I A / I / I / II / I / I / V I / Iv i / i / 1
/ i / ii / 1 / i / I / i / ivv t~ v IVI /1AAAVVIAAAVI /V~A
F i g . 9 - - S q u a r e p l a t e w i t h c e n t r a l h o l
e ( d i m e n s i o n s i n m m ) .T h e H o p k i n s o n b a r i
s a t t a c h e d t o t h e l e f t s i d e
a 7 6 m m 4 i n ) d i a m e t e r h o l e b o r e d in it s c e
n te r . A s m a l lh o l e w a s d r i l l e d a n d ta p p e d a
t t h e m i d - p o i n t o f o n e o f t h ee d g e s f o r t h e
c o n n e c t i o n o f t h e o n e - s i d e d H o p k i n s o n b
ar .A c c e l e r o m e t e r s w e r e p l a c e d o n t h e e d g
e o f th e f o r c e in p u ta n d o n t h e o p p o s i t e e d g
e o f t h e p l a t e at 5 0 . 8 m m 2 i n ) o f ft h e f o r c e i
n p u t l i n e , a s s h o w n i n F ig . 9 . T h e a c c e l e r
o m e t e ra n d s t r ai n g a g e d a t a w e r e c o l l e c t e
d u s i n g t h e d a t a a c q u i s it i o ns y s t e m o f F i g
. 3 . T h e d a t a w e r e c o l l e c t e d a t a 4 t z s r a te
o v e r4 0 0 0 l ~ S .
T h e a c c e l e r a t i o n d a ta w e r e c o n v e r t e d t
o v e l o c i t ie s a n d d e -t r e n d e d , a n d t h e s t r a
i n g a g e d a t a w e r e m o d i f i e d t o a c c o u n tf o r
t h e s li g h t n o n l in e a r i t y o f th e W h e a t s t o n
e b r i d g e s i n c et h e r e s is t a n c e c h a n g e o f t h
e s e m i c o n d u c t o r g a g e s i s r e la t i v e lyl a r g
e ) . T h e s e d a t a a r e s h o w n i n F i g . 1 0 .
T h e i n v e r s e p r o b l e m w a s d i v i d e d i n t o t
w o s e p a r a t e p r o b -l e m s c o r r e s p o n d i n g t o
t h e b a r a n d t h e p l a te . T h e d a t a f r o mt h e t w o
s t r ai n g a g e s o n t h e H o p k i n s o n b a r w e r e u s
e d s i m u l -t a n e o u s l y t o r e c o n s t r u c t t h e i
m p a c t f o r c e a n d t h e f o r c e i n p u tt o t h e p la t
e . T h e s e f o r c e r e c o n s t r u c t i o n s a r e s h o w
n l a b e l l e da s P a n d R s r e s p e c t i v e l y , i n F i
g . 1 1 . T h e i m p a c t f o r c e i sa s e x p e c t e d , i n
th a t i t i s e s s e n t i a l l y a s i n g l e p u l s e i n p
u t . T h er e c o n s t r u c t i o n s o f t h e f o r c e a t t
h e c o n n e c t i o n p o i n t , o n t h eo t h e r h a n d , a
r e q u i te c o m p l i c a t e d p e r s i s ti n g w i t h s i g
n i f ic a n ta m p l i t u d e f o r th e f u l l d u r a ti o n o
f t h e re c o r d i n g . N o t e t h a tt h e c o n n e c t i o n
f o r c e i s l a r g e r t h a n t h e im p a c t f o r c e b e c
a u s eo f t h e h i g h i m p e d a n c e o f t h e p l a te .
T h e R s f o r c e f r o m t h e H o p k i n s o n b a r w a s
u s e d a s i n -p u t t o t h e f o r w a r d p r o b l e m f o r
t h e p l a t e a n d t h e c o m p u t e dv e l o c i t i e s a r
e c o m p a r e d t o t h e m e a s u r e d o n e s i n F ig . 1 0
.T h e c o m p a r i s o n i s q u i te g o o d c o n s i d e r i n
g t h e e x te n d e d t i m e
3 2 9 V o L 4 2 N o 1 M a r c h 2 0 0 2
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o
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-100 r - ~ ~ 4
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0T ime [~s ]F i g. 1 0 - - Ex p
e r im e n t a l l y r e c o r d e d da t a. T o p , s t r a in s
fr o mHopk inson bar . Bo t tom, ve loc i t ies computed f romacce
le ra t ions
p e r i o d o f th e c o m p a r i s o n . T h e d a t a f r o m
t h e a c c e l e r o m e t e r sw ere t h en u sed a s sep a ra t
e s in g l e sen so r in p u t s t o r eco n s t ru c tth e fo rce
a t t h e co n n ec t io n p o in t . T h ese fo rce r eco n s t ru
c t io n sa re sh o w n l ab e l l ed a s R e an d R ~ in F ig . 1
1 . T h e c o m p ar i so no f t h e r eco n s t ru c t ed fo rces
a t t h e co n n ec t io n p o in t ag ree q u i t ec lo se ly b o
th i n ch a rac t e r an d m ag n i tu d e . T h i s i s s ig n if
i can tb ecau se th e fo rce t r an sm i t t ed ac ro ss t h e b o
u n d ary i s q u i t ec o m p l e x a n d h a s b e e n r e c o n
s t r u c t e d f r o m d i f f e r e n t s e n s o rty p es p l
ace d o n q u i t e d i f f e r en t s t ru c tu ra l t y p es .D i
s c u s s i o n
T h i s p a p e r b r i n g s t o g e t h e r a n u m b e r o f
t e c h n o l o g i e s t oso lv e th e fo rce i d en t i fi ca t
io n p ro b lem o n co m p lex s t ru c tu res .
1 ~ o e l I 9l uminum
\ Im pa ct n bar Pref 2 2 0 0 N 5 0 0 ] b )
Acc. #2........ Acce lerome ters- - Hopkinson Bar
1 -1 , 0 . 0 _ J / 1 ~~ J ~ \Acc. #------~ -'
P(8R~(t),R~(t)
N(8 , R~(t)
0 0 0 1 3 0 1 0 0 1. 1 2 0 0 0 . 4 0 0 0 .Ti m e [1as]
F ig . 11- -Fo rce recons t ruc t ions fo r the p la te w i th a
ho le
A t t h e c o r e i s t he e m b e d d i n g o f t h e f i n it
e e l e m e n t m o d e l i n g o fc o m p l e x s t ru c t u re s
t o d e t e r m i n e t h e s y s t e m m a t r i x . C o n s e -q
u en t ly , s t ru c tu res a s co m p lex a s i s u su a l ly m o
d e le d w i th t h ef in it e e l em en t m e th o d ca n b e h an
d led co n v en ien t ly . S p ec i a le f fo r t w as m ad e to i
n su re t h a t t h e co m p u ta t io n a l co s t s s ca l ein a
m an n er s im i l a r t o t h e fo rw ard p ro b lem u s in g th e
f i n i t ee l e m e n t m e t h o d . T h e p r o b l e m o f i l
l- c o n d it i o ni n g i s h a n d l e dth ro u g h r eg u la r i
za t io n . T h i s a l l ev i a t e s t h e b u rd en o f p ro p e
rsen so r p l acem en t an d , a s a r e su l t , t h e d i f f i
cu l t p ro b lem o ft r ac t io n an d p res su re d i s t ri b u
t io n s can b e so lv ed . T h i s o p e n su p t h e e x c i t i
n g p o s s i b il i t y o f s o l v i ng p r o b l e m s w i t h u
n k n o w nb o u n d ary co n d i t i o n s o r u s in g su b d o m
ain s fo r t h e m o d e l in g .A p re l im in a ry in v es t ig a
t io n o f t h ese a r e g iv en in R ef s . 2 8and 32.
F u r th e r e f f i c i en c i e s can s t i l l b e ach iev ed
in t h e p ro g ram -m i n g ; f o r e x a m p l e , o p e r a t io
n s o n t h e b a n d e d r e p r e s e n t a ti o no f t h e s t i
f fn ess m a t r ix can b e r ep l aced w i th sk y l in e o p e ra
-t i o n s . T h i s w i l l d ec rease t h e s to rag e r eq u i r
em en t s a s w e l l a si n c r e a s e t h e s p ee d . A n a s p
e c t o f th e p r o g r a m m i n g t h a t c o u l dh av e a p ro
fo u n d e f f ec t i n t h e fu tu re i s t h a t t h e b ack w
ard so -lu t io n s t ag e ( i n p a r ti cu l a r , t h e m a t r
i ces [ n o r [K n ] ) ac tu a l lyso lv es fo r an y fo rce h i s
to ry a t t h e s t a t ed fo rce l o ca t io n s i n -d e p e n d
e n t o f t h e m e a s u r e d d a t a . T h u s i n v e r s e p r
o b l e m s t h a tr eq u i r e i t e r a t i o n ( e . g . , f o
rce l o ca t io n ) o r u se m u l t i p l e d a t a se t s( e . g
. , r ea l - t im e h ea l th m o n i to r in g ) n eed so lv e th
e b ack w ards t ag e o n ly o n ce (w h ich i s t h e ex p en s iv
e s t ag e ) an d u se t h efo rw a rd s t ag e fo r m an ip u la t
in g th e ac tu a l i n p u t d a ta . T h eim p l i ca t io n s o
f t h i s a r e s ig n i f i can t .
T h e p o w e r o f t h e p r e s e n t e d m e t h o d i s t h
a t i t e m b e d s ag en e ra l p u rp o s e F E M p ro g ram in i
t. I ro n i ca l ly , t h i s can a l sob e p e rce iv ed a s a d
raw b ack , i n t h a t a s t h e s t ru c tu ra l co m -p lex i ty
i n c reases ( so t h a t i t r eq u i r e s a n ew e l em en t t y
p e , s ay )so m u s t t h e co m p lex i ty o f t h e i n v e r se
co d in g a l so i n c rease .T h u s t h e p o w e r o f c o m m e
r c i a l F E M p r o g r a m s c a n n o t b e u t i-l i zed . In
th i s r eg a rd , t h e m e th o d s p re sen ted in R ef s . 1 1
an d3 3 o f f e r a s ig n i f ican t ad v an tag e b ec au se th e
F E M an a ly s i s i s
E x p e r i m e n t a l M e c h a n i c s 9 33
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8/12/2019 Souleymane BF02411048
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r u n a s a n e x t e r n a l p r o c e s s t o t h e i n v e r
s e p r o g r a m a n d t h e r e -f o r e th e p o w e r o f c o m
m e r c i a l F E M c o d e s c a n b e le v e r a g eT h e m e t h
o d s d o n o t s c a l e a s w e l l a s t h e o n e p r e s e n t
e d i n t h i sp a p e r , s o i t w i l l b e i n t e r e s t i n
g t o s e e w h i c h o f t h e a t t r i b u t e s - -s c a la b i
li t y o r g e n e r a l i t y - - t u r n s o u t t o b e m o r e
i m p o r t a n t .A p p e n d i x I S o m e F o r m s f o r [ ] a
n d [B]
W e h a v e i m p l e m e n t e d t w o s c h e m e s f o r t h
e ti m e d i s c r e t iz a -t i o n o f t h e e q u a t i o n s o
f m o t i o n . O n e i s t h e c o n d i t i o n a l l y s t a -b
l e e x p l i c i t c e n t r a l d i f f e r e n c e m e t h o d ,
1 8 '1 9 a n d t h e o t h e r i st h e u n c o n d i t i o n a l l
y s t a b l e i m p l i c i t N e w a r k o r c o n s t a n t a c c
e l -e ra t ion method .18 ,19
U s i n g t h e c e n t r a l d i f f e r e n c e f i n i t e d
i f f e r e n c e s c h e m e , t h eg o v e r n i n g s y s t e m
o f e q u a t i o n s c a n b e w r i tt e n a s
i =g n+l
[ ( I - M l l K A t 2 ) ( I - M I I C A t ) A t M l l B g A t 2
]
] oit + 0 Atg n g n
w i t h [ M 1 ] - ~ [ M + 1 C A t ] a n d [ I ] a s t h e u n i
t m a t r ix . N o t etha t the ra te o f fo rce , {g}n , is t aken
as the ac tua l ap p l ied lo ad ;t h i s w i l l g i v e t h e e q
u i v a l e n t o f f ir s t o r d e r r e g n l a r i z a t i o n
i n t h et i m e d i r e c t i o n . T h i s r e l a t i o n i s a
b b r e v i a t e d a s
{u}n+~ = [A]{u }n -t- [B]{g}n ( A 1 )w h e r e n i s t h e s u b
s c r i p t o v e r t h e d i s c r e t iz e d t i m e . I n t h is
,t h e d e g r e e s o f f r e e d o m v e c t o r , { u} , is o f
s i z e { ( m = 2 m u +mg) x 1} .T h e s y s t e m a r r a y [ A ]
i s v e r y l a r g e a n d t h e r e f o r e i t i s t oo u r a d
v a n t a g e t o g i v e s p e c i a l t r e a t m e n t t o i t s
m a n i p u l a t i o n s .T h e s t r u c t u r e o f t h i s m a
t r i x i s s h o w n i n F i g . A 1 . B e c a u s e o ft h e n a
t u r e o f t h e s tr u c t u r a l s y s t e m s , t h e s t i f
f n e ss m a t r i x [ K ]i s b a n d e d a n d s y m m e t r i c ;
c o n s e q u e n t l y , t h e t w o u p p e r l e f tp o r t i on
s c a n b e m a d e b a n d e d ( a l th o u g h n o t s y m m e t
r i c ) . A si s u s u a l w i t h e x p l i c i t d i s c r e t i
z a t i o n s c h e m e s , b o t h t h e m a s sa n d d a m p i n
g m a t r i c e s a r e t a k e n a s d i a g o n a l a n d h e n c
e t h et w o u p p e r m i d d l e p o r t i o n s a r e d i a g o
n a l o n l y . T h e f i r s t t w ob o t t o m m a t r i c e s a
r e z e r o a n d t h e t h i r d i s d i a g o n a l w i t h u n i
t y .T h e r e m a i n i n g f a r le f t m a t ri c e s a r e m o
s t l y z e r o s b u t b e c a u s em g
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Ill--1 =Z I
2 z l - / )4~- t2 Z l - - I
AtZ(2Z 2 - - I )
4A 7 (Z2 - - I )
At Z32 A t Z 3(4Z 3 - 1)
Z4
2~ z 44~tSt2 z4
w i t h
Z4g2~Z4g
4 Z4g A t
Z1 - -- - [K - 1] F 2 C A t + 4 M J / A t 2,Z 2 = [ K - 1 ] [ C
A t + 4 M J / A t 2,Z 3 = [ K - 1 ] [ M I / A t 2 ,Z 4 = [ K - 1 ]
[ B g ]
a n d [ K 1 ] = [ K + 2 C / A t + 4 M / A t 2 ] . T h i s c a n
a l s o b ea b b r e v i a t e d a s E q ( A 1 ) .
A s i n t h e e x p l i c i t c a s e , w e n e v e r d o a c t
u a l l y a s s e m b l et h e [ A ] m a t r i x , b u t i n s t e
a d r e t a i n t h e i n d i v i d u a l c o m p o n e n tm a t r
i c e s , a n d t h e m a t r i x p r o d u c t s a r e p e r f o r
m e d i n a n e f f i -c i e n t m a n n e r . T h e e x p e n s i
v e c o m p u t a t i o n s a r e t h e p ro d u c t s[ K 1 ] l [ M
i J { v j } . W e d o n o t f o r m t h e i n v e r s e o f th e s
t i f f-n e s s m a t r i x ( s i n c e t h i s w o u l d b e t o o
e x p e n s i v e a n d r e q u i r et o o m u c h m e m o r y ) ;
r a t h e r , t h e p r o d u c t i s a c c o m p l i s h e d i nb
a n d e d f o r m a s f o l lo w s :
{ z } = [ K ~ I ] F M i ] { v j } o r [ K = U D U I { z } = { M
i v j } .T h e [ U D U ] d e c o m p o s i t i o n i s p e r f o r
m e d o n l y o n c e , th u st h e c o m p u t a t i o n a l c o s
t is e s s e n t i a l l y t h a t o f t h e b a c k -s u b s t i t
u ti o n s . T h i s i s t h e s a m e c o s t as o c c u r s i n t
h e f o r -w a r d i m p l i c i t p r o b l e m . T h e a r r a y
s a s s o c i a te d w i t h d e s c r i b in gt h e i m p l i c i
t s y s t e m a r e s i m i l a r i n s i z e to t h a t o f t h e
e x p l i c i ts c h e m e e x c e p t t h a t s i z e { u} = { ( m
= 2 m , + rag x 1}.A p pen d i x I : M i n i m i z i ng P r i nc i
p l e w it hR e g u l a r i z a t i o n
T h e u s u a l l e a s t s q u a r es p r o c e d u r e i s p o
s e d i n t h e f o r m o ff i n d i n g t h e s e t o f d i s c r
e t i z e d u n k n o w n s { u} g i v e n th e s e t o fd a t a {
d } s o t h a t t h e p o s i t i v e f u n c t i o n a l & = X
2 is a m i n i m u m ;tha t i s , '
m i n i m i z e : & = [d - A u ] T [ d - A u ] .
cOt~N. mc6
0 3r e
d a t a f i t
r r or n o r m
F i g . A 2 T r a d e o f f c u r v e
T h e n u m b e r o f d a t ap o i n t s i s ty p i c a l l y l
a rg e r t h a n t h e n u m -b e r o f u n k n o w n s a n d t h e
r e fo r e m i n i m i z i n g w i t h r e s p e c t tot h e u n k
n o w n { u} g i v es a d e t e rm i n a t e s y s t e m o f e q u
at i o n s .I t i s a c t u a l l y v e r y d i f f i c u l t t o g
e t r o b u s t a n s w e r s f o r t h e s ep r o b l e m s ( e v
e n w h e n t h e r e i s a n ex c e s s o f d a t a) b e c a u s e
t h e ya r e h i g h l y i l l - c o n d i t i o n e d . R e g u l
a r i z a t i o n i s u s e d t o m a k ei l l - c o n d i t i o n
e d p r o b l e m s b e t t e r c o n d i t i o n e d .
W e n o w p a r a p h r a s e s o m e o f t h e d is c u s s io
n o f re g u l a r iz a -t i o n a s g i v e n i n R e f . 3 ; a m
o r e m a t h e m a t i c a l t r e a t m e n t ( p l u sa n e x t
e n s i v e b i b l i o g r a p h y ) i s g i v e n i n R e f . 3 4
. T h e e s s e n t i a li d e a i n o u r i n v e r s e t h e o r
y i s t h e o b j e c t i v e
m i n i m i z e : & + k ~ = { d - A u } T { d - A u }+ ? , {
u } r [ H ] { u }
f o r v a r i o u s v a l u e s o f 0 < k < o c a l o n g
t h e s o - c a l l e d t r a d e -o f f c u r v e o f F ig . A 2 .
T h e r e a r e t w o p o s i t i v e f u n c t i o n a l s ,
&a n d 2 . T h e f i r st m e a s u r e s t h e a g r e e m e n
t o f th e m o d e l t o t h ed a t a; w h e n & b y i t s e l
f i s m i n i m i z e d , t h e a g r e e m e n t w i t h t h ed a
t a b e c o m e s v e r y g o o d b u t t h e s o l u t io n b e c
o m e s u n s t a b l e.T h i s r e f l e c t s t h a t & a l o
n e t y p i c a l l y d e f i n e s a n i l l - c o n d i t i o n e
dm i n i m i z a t i o n p r o b l e m . T h e s e c o n d t er m 2
i s i n t r o d u c e d t oo v e r c o m e t h is p r o b l e m ; i
t m e a s u r e s t h e s m o o t h n e s s o f th ed e s i r e d s
o l u t i o n o r s o m e t i m e s a q u a n t i t y r e f l e c t
i n g a p r i o r ij u d g m e n t s a b o u t th e so l u t i o n
. T h i s f u n c t i o n is c a l l e d t h er e g u l a r i z i n
g o p e r a t o r . M i n i m i z i n g ~ b y i t s e l f g i v e s
a s o l u t i o nt h a t i s w e l l - b e h a v e d b u t c l e a
r l y d o e s n o t r e f l e c t t h e d a t a .
I n c r e a s i n g X p u l l s th e s o l u t io n a w a y f r
o m m i n i m i z i n g X 2i n f a v o r o f m i n i m i z i n g {
u } r [ H ] { u } . W e s e t t le o n a b e s tv a l u e o f k b y
s o m e c r i te r i o n r a n g i n g f r o m f a i rl y o b j e c
t iv et o e n t i r e l y s u b j e c t i v e ; t h i s i s d i s c
u s s e d a l i t t l e f u r t h e r i n t h em a i n t e x t
.
T h e r e g u l a r i z a t i o n m e t h o d w e u s e i s g e
n e r a l l y c a l l e dT i k h o n o v 3 5 - 37 r e g u l a r i z
a t i o n . T y p i c a l l y , th e f u n c t i o n a li n v o l v
e s s o m e m e a s u r e s o f s m o o t h n e s s t h a t d e ri
v e f r o m f i rs to r h i g h e r d e r i v at i v e s a n d g i
v e n b y [ H I = [ D ] r [ D ] w h e r e[ D ] i s t h e d i f f e
r e n c e m a t r i x . D e t a i l e d d i s c u s s i o n o f th
e d i f -f e r e n t c h o i c e s o f r e g u l a r i z a t i o n
i s g i v e n i n R e f s . 2 8 a n d 3 2a n d t h e r e f o r e w
i l l n o t b e e l a b o r a t e d o n h e r e .
T h e a r r a y s a s s o c i a t e d w i t h d e s c r i b i n
g t h e r e g u l a r i z a t i o na r e o f s i z e
E x p e r i m e n t a l M e c h a n i c s 9 35
-
8/12/2019 Souleymane BF02411048
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[ D ] = [ ( ra g - 1 m g ][ H I = [ m g x r a g ]
B o t h a r r a ys a r e r e l a t i v e l y s m a l l s i n c e
u s u a l l y mg ~ m uc k n ow l e d gmen t s
T h i s w o r k w a s s u p p o r t e d i n p a r t b y a U . S
. A r m y M u l t i -D i s c i p l i n a r y U n i v e r s i t y R
e s e a r c h I n it i a t i v e U n i t e d S t a t e sA r m y G r
a n t N o . D A A H 0 4 - 9 6 - 1 0 3 3 1 ) a w a r d e d t o P u r
d u eU n i v e r s i t y .
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Press, W.H., Flannery, B.P., Teukolsky, S.A., a nd Vetterling, W
..T, Nu-mer ical Recipes, 2r id ed., Cambridge University Press,
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3 6 9 Vo L 4 2 N o . 1 Ma r ch 2 0 0 2
