A P P L I C A T I O N S

An Error Associated with Graphical Parametric Operations

Object of this paper is to analyze error, suggest means of computing it and means of avoiding it

by V. J. Parks and A. J. Durelli

ABSTRACT-- In grid and moir~ analyses, patterns developed in a deformed body are sometimes superposed to obtain the difference of their parameters, as when it is desired to obtain velocities by superimposing two isothetic patterns. In this operation, an inherent di f f icul ty develops because, as a con- sequence of the deformation, not all points can be matched. An error follows which is analyzed here. Means to avoid or to compute the error are suggested.

Introduction A n i n d i c a t i o n of t he pos s i b i l i t y of u s i n g p a r a m e t r i c o p e r a t i o n s to d e t e r m i n e fields g r a p h i c a l l y can be f o u n d in C o k e r and F i l o n ) T h e y s u g g e s t tha t , if two f ami l i e s of curves , one of i s o c h r o m a t i c s ( ~ -- ,re) a n d t h e o t h e r of i sopach ics (~, + r p r o c e e d by t h e s a m e equa l s teps, a n d a re supe rposed , t h e i s o b a r s al a n d ~., can b e o b t a i n e d g r a p h i c a l l y b y c o n n e c t i n g t h e i n t e r s e c t i o n s a long d i a g o n a l s of the q u a d r i l a t e r a l s f o r m e d b y t h e c u r v e s of t he f ami l i e s of i s o c h r o m a t i c s a n d i sopachics . T h e p a r a m e t e r s of t h e i s oba r s a re g i v e n b y t h e a d d i t i o n a n d s u b t r a c t i o n of t h e p a r a m - e t e r s of t h e two s u p e r p o s e d curves . P i r a r d h a d a s i m i l a r p o i n t of v i ew in h is c o n s i d e r a t i o n on moi r6 . '2 The a p p r o a c h h a s also b e e n fo l l owed b y Oster , W a s - s e r m a n a n d Z w e r l i n g 3 a n d Post.* T h e a u t h o r s of t h i s p a p e r h a v e s u g g e s t e d a n u m b e r of a p p l i c a t i o n s of t h e m e t h o d in t h e i n t e r p r e t a t i o n of m o i r 6 f r inges , 5 i n - c l u d i n g i ts use to d e t e r m i n e ve loc i t y a n d a c c e l e r a t i o n fieldsP, 7 One of t he a u t h o r s h a s also a p p l i e d i t w h e n s p a t i a l f i l t e r i ng t e c h n i q u e s a re used. s T h e t wo f a m i - l ies of cu rves , or con tour s , w h i c h a r e a d d e d or s u b - t r a c t e d ( s u c h as t h e i s o c h r o m a t i c s or i sopachics , a b o v e ) w i l l b e ca l l ed h e r e t he g e n e r a t i n g p a t t e r n s . T h e r e s u l t i n g t w o f a m i l i e s of c u r v e s ( s u c h as t h e i soba r s a b o v e ) wi l l be ca l l ed t he g e n e r a t e d p a t t e r n s .

Es sen t i a l l y , t he m e t h o d cons i s t s in a d d i n g or s u b - t r a c t i n g t h e p a r a m e t e r s of t w o con tour s , one f r o m each g e n e r a t i n g p a t t e r n , t h a t i n t e r s e c t a t a poin t . T h e r e s u l t of t h e a d d i t i o n a n d s u b t r a c t i o n wi l l be t h e

V. ]. Parks and A. 1. Durelli are Associate Professor and Professor, respectively, Civil and Mechanical Engineering Department, The Catholic Unicersit!! o[ America, Washington, D. C.

p a r a m e t e r s of two o t h e r con tour s , one of e ach of t he t w o g e n e r a t e d p a t t e r n s pa s s ing t h r o u g h t h a t s a m e poin t . No di f f icul t ies a p p e a r w h e n t h e m e t h o d is u sed as s u g g e s t e d b y C o k e r a n d F i lon , a l t h o u g h it h a s no t b e e n f o u n d v e r y p r a c t i c a l b e c a u s e it r e q u i r e s o b t a i n i n g g e n e r a t i n g c o n t o u r s s e p a r a t e d b y t h e s a m e p a r a m e t r i c s tep a n d h a n d d r a w i n g of t h e g e n e r a t e d con tour s . T h e m e t h o d b e c a m e m o r e p r a c t i c a l w h e n m o i r 6 effects (or m o i r ~ - o f - m o i r 6 effects) cou ld be used.

W i t h t he e x t e n s i o n of t h e p a r a m e t r i c o p e r a t i o n to t h e c o m p u t a t i o n of t i m e d e r i v a t i v e s such as ve loc i t i e s a n d acce l e r a t i ons , P o s t :~ n o t e d t h a t a c o m p l i c a t i o n a p p e a r e d . S o m e t i m e s , w h e n t he two g e n e r a t i n g p a t - t e r n s f r o m d i f f e ren t load l eve l s a re supe rposed , the two s u p e r p o s e d p o i n t s at t h e i n t e r s e c t i o n of two con - t o u r s a re no t p o i n t s t h a t o r i g i n a l l y h a d t he s a m e pos i t ion . In o t h e r words , t h e y do no t c o r r e s p o n d to t h e s a m e o r i g i n a l p o i n t because , b e t w e e n t he two l eve l s of load, t h e b o d y ha s b e e n d e f o r m e d in the p lane . Th i s s i t u a t i o n m a y occur , in gene ra l , in p a t - t e r n s such as i so t ach ic s ( e q u a l - v e l o c i t y c o n t o u r s ) a n d i soep i t ach i c s ( e q u a l - a c c e l e r a t i o n c o n t o u r s ) ob - t a i n e d f r o m i so the t i c s ( e q u a l - d i s p l a c e m e n t c o n t o u r s ) of d i f f e r e n t load levels . I t m a y also occu r in p a t t e r n s o b t a i n e d f r o m s u b t r a c t i n g ou t a n i n i t i a l m i s m a t c h of a n u n d e f o r m e d b o d y f r o m t h e p a t t e r n of a de - f o r m e d body. A n e r r o r fo l lows t h i s s i t ua t i on . The o b j e c t of th i s p a p e r is to a n a l y z e t h i s e r ro r , sugges t m e a n s of c o m p u t i n g it, a n d m e a n s to avo id it.

Computation of the Error C o n s i d e r two s u p e r p o s e d g e n e r a t i n g p a t t e r n s of

o r d e r e d c o n t o u r s e a c h f r o m a d i f f e r e n t l oad l e v e l of a d e f o r m e d body . T h e o r d e r of t h e c o n t o u r s of t h e g e n e r a t e d p a t t e r n , o b t a i n e d b y m o i r ~ ef fec t or b y g r a p h i c a l l y c o n n e c t i n g t h e i n t e r s e c t i o n s of l ines, in g e n e r a l is g i v e n b y t h e d i f f e r ence of t h e o r d e r s of t h e c o n t o u r s of t h e g e n e r a t i n g p a t t e r n s t h a t i n t e r - sect a t a po in t . ( I n some cases i t m a y be t h e sum. ) C o n s i d e r t h e p o i n t P, in Fig. 1, w h i c h h a s s h i f t e d an a m o u n t • f r o m pos i t ion P ' to P" b e t w e e n t h e t w o

Experime.tal Mecha~dcs [ 429

A P P L I C A T I O N S

levels of load. Say, at the lower level of load i point P belongs to a contour of order m ' and, at the higher level of load, to a contour of m " . The difference in contour values at point P should be ( m " -- m ' ) . But, in the superposition, the point P has two images, P ' and P" . Say that, at P', the h igher - leve l contour order is k" and, at P" , the lower - leve l contour order is q'. The generated pa t te rn wil l have contour param- eters ( k " - - m ' ) at P' and ( m " -- q ' ) at P ' . Then, the error wil l be ( k " -- m " ) at P ' or ( m ' - - q ' ) at P" .

These two values of er ror may be equal, which would mean that the contour of the generated pat- tern would pass through both images P' and P " and indicates that the situation cannot be corrected by using the midpoint of P' and P" .

In general, it can be said that, if the contour of the generated pat tern is re fe r red to the h igher- load- level geometry, the er ror in the order of the gener- ated contour will be equal the number of generat ing contours on the lower - load- leve l pat tern ( m ' - - q') fall ing be tween the two images of the point. And if the contours of the generated pat tern are referred to the lower - load- leve l geometry, the error in the order of the generated contour wi l l be equal to the num- ber of generat ing contours on the h igher - load- leve l pat tern ( k " - - m " ) fal l ing between the two images of the point. Note that, if the motion of the point is along one of the generat ing contours, then the gen- erated contour value is correct with respect to the

GENEF CON'

EVEL 'ATIERN

Fig. 1--Point P shown on patterns corresponding to two load levels. The point moves from P' to P"

other generat ing pat tern geometry (ei ther k" - - m " z O , o r m ' - - q ' = O ) .

Fig. 2--Superposed isothetics of the vertical displacement for two load levels from one quadrant of a ring under diametral compression. The moira pattern obtained is the vertical velocity (isotachics) corrected for four different portions of the ring

430 I September 1971

A P P L I C A T I O N S

Correction of the Error I f t h e r e is some w a y to i d e n t i f y t h e p o s i t i o n of a

p o i n t a t b o t h l eve l s of load, t h e n t h e g e n e r a l m e t h o d to c o r r e c t t h e g e n e r a t e d p a t t e r n is to s h i f t t h e t w o g e n e r a t i n g p a t t e r n s to m a k e t h e t w o i m a g e s of t h e p o i n t coincide. T h i s can b e a c h i e v e d w h e n t h e r e is a g r id on t h e mode l . In w h a t fo l lows, c o r r e c t i o n p r o - c e d u r e s for some p a r t i c u l a r s i t u a t i o n s wi l l b e d e - sc r ibed .

Velocities

W h e n the g e n e r a t e d p a t t e r n is a v e l o c i t y c o m -

p o n e n t p a t t e r n , say u, a c o r r e c t i o n c a n b e m a d e if t h e o t h e r v e l o c i t y c o m p o n e n t p a t t e r n is also k n o w n ,

say v. The m o t i o n of a p o i n t b e t w e e n t he two load l eve l s in a r e g i o n of a g e n e r a t e d c o n t o u r c an be a p - p r o x i m a t e d b y t h e ve loc i t ies , a n d t h e i m a g e s can b e s h i f t e d or a v e c t o r of t h e m o t i o n c a n be d r a w n to sca le on t he image . T h e n u m b e r of g e n e r a t i n g c o n - t o u r s a long t h i s v e c t o r t h e n g ives t h e c o r r e c t i o n to be app l i ed to t he v e l o c i t y c o n t o u r s p a r a m e t e r s . B y r e p e a t i n g t h e p roces s w i t h t h e c o r r e c t e d va lues , m o r e p rec i se v a l u e s of t h e v e l o c i t y c an b e o b t a i n e d .

F i g u r e 2 s h o w s t h e f ield of v e r t i c a l c o m p o n e n t s of ve loc i t i e s o b t a i n e d f r o m m o i r 6 of t w o i so the t i c s co r - r e s p o n d i n g to t w o d i f f e r en t load l eve l s of a r i n g s u b - j e c t e d to d i a m e t r a l com pr e s s i on . T h e i so the t i c s a r e s u p e r p o s e d b y m a t c h i n g a t four d i f f e r e n t po in t s of t h e r ing : (1) t he cen te r , (2) t h e top, (3) t he r i g h t e n d Gf t he h o r i z o n t a l d i a m e t e r a n d (4) a p o i n t on t he i n n e r b o u n d a r y of t h e r ing. In e v e r y case, t h e two i m a g e s of e ach of t h e axes w e r e k e p t p a r a l l e l to each o the r . T h e c e n t e r of t he r i n g is a s s u m e d no t to h a v e m o v e d a n d is t he r e f e r e n c e poin t . T h e f r i n g e s in t h e m a t c h e d a r ea on each p a t t e r n g ive t h e m o s t a c c u r a t e resu l t s . T h e d e v i a t i o n f r o m these in o t h e r a r e a s g ives t h e a m o u n t of e r ro r .

T h e p r o c e d u r e fo r m a t c h i n g t he p o i n t of t h e f o u r t h p a t t e r n is as fo l lows. T h e f r i n g e o r d e r s at t h a t po in t

on t h e v e r t i c a l a n d h o r i z o n t a l v e l o c i t y f ie lds o b - t a i n e d b y m a t c h i n g t h e c e n t e r a n d a l i g n i n g t h e a x e s w e r e - -19 ( see u p p e r l e f t of Fig. 2) a n d 11, r e s p e c - t ive ly . To m a t c h t h i s p o i n t o n t h e i n ~ e r b o u n d a r y , s ince t h e g r a t i n g d e n s i t y is 1000 lpi , t h e l o w e r l e v e l i so the t i c p a t t e r n w a s s h i f t e d 0.011 in. to t h e l e f t a n d 0.019 in. down . H o w e v e r , w i t h t h i s c o r r e c t i o n , t h e f r i n g e s a t t h e p o i n t n o w h a d n e w v a l u e s of - -16 a n d 16. Thus , a n a d d i t i o n a l c o r r e c t i o n w a s r e q u i r e d . S h i f t i n g 0.016 in. to t h e l e f t a n d 0.016 in. d o w n a g a i n f r o m t h e o r i g i n a l pos i t i on g a v e t h e f r i n g e v a l u e 17 at t h e ; p o i n t as Shown in t h e p a t t e r n a t t h e l o w e r r i g h t of Fig. 2. B e c a u s e t h e s e c o n d c o r r e c t i o n w a s o n l y one f r i nge , f u r t h e r c o r r e c t i o n of t h e p a t t e r n w a s d e e m e d u n n e c e s s a r y .

Isothetics Obtained with Grating Mismatch

If a g r a t i n g on a n u n d e f o r m e d s p e c i m e n is no t m a t c h e d w i t h t he m a s t e r g r a t i n g , t h e s u p e r p o s i t i o n of t h e d e f o r m e d s p e c i m e n g r a t i n g a n d t h e m a s t e r g r a t i n g p r o d u c e s a m o i r 6 t h a t g ives a f i c t i t i ous i so- the t ic . Th i s s i t u a t i o n is r e f e r r e d to as g r a t i n g m i s - m a t c h a n d m a y be p r o d u c e d on p u r p o s e or a c c i d e n t - al ly. T h e m i s m a t c h c a n be due to a r o t a t i o n of t he u n d e f o r m e d s p e c i m e n l ines w i t h r e s p e c t to t h e m a s t e r - g r a t i n g l ines, ( r o t a t i o n a l m i s m a t c h ) or due to a d i f f e r e n c e in p i t c h of t h e two sets of l ines (p i t ch m i s m a t c h ) or to a c o m b i n a t i o n of bo th . In a n y case, t h e m o i r 6 p a t t e r n o b t a i n e d by s u p e r p o s i n g t h e m a s t e r g r a t i n g on t h e u n d e f o r m e d g r a t i n g h a s to be s u b - t r a c t e d f r o m t h e m o i r 6 p a t t e r n o b t a i n e d b y s u p e r - pos ing t h e m a s t e r g r a t i n g on t h e d e f o r m e d g r a t i n g to o b t a i n t h e i so the t ics . T h e m o i r 6 on t he u n d e f o r m e d b o d y c a n be c o n s i d e r e d a l o w - l o a d - l e v e l p a t t e r n w i t h a f i c t i t i ous d i s p l a c e m e n t .

T h e s u b t r a c t i o n c a n be d o n e b y p l o t t i n g t h e t w o se ts of f r i n g e s a n d s u b t r a c t i n g one c u r v e f r o m t h e o the r , or b y s u p e r p o s i n g t h e t w o m o i r 6 p a t t e r n s ( g e n e r a t i n g p a t t e r n s ) to p r o d u c e a m o i r 6 - o f - m o i r 6

POINT P' P"

HORIZONTAL x' x" POSITION

MASTER GRATING x' x" ORDER P P

UNDEFOF GRATING ORDER

DEFORMED ROD% GRATING ORDER

UNDEFORMED BODY CONTOUR VALUE

DEFORMED BODY CONTOUR VAtUE

GENERATED CONTOURS

x_, X'L P~ P,

!'_ -k" "~'~*~ p - - - - rll" :not significant} P

m'= x' _ X ' x" x" P Ps q'=

(not significant) P Ps

k" m" not significant)

CONTOURS ON DEFORMED BODY PRODUCED BY THESE IxNO CRATING

k" I" m" MASTER DEFORMED BODY

/NON-CORRECTED ~ ISOTHETICS ]

,, GENERATED CONTOURS P\,~ IISOTHETICS)

(m" q ' l = H " - p ' ) etc.

CONTOURS ON UNDEFORMED BODY DUE TO PITCH MISMATCH OF THE "PNO GRATINGS

IF1TITIOUS ISOTflETICS)

"/" ',,~i- p I ~ p, MASTER UNDEFORMED ORATING BODY GRATING

m' n' o' p'

Fig. 3- -Mot ion of point shown on patterns of a deformed body and undeformed body with pi tch mismatch

Experimental Mechanics 1431

A P P L I C A T I O N S

(generated pa t te rn) . In e i ther case, the displace- ment wil l be subject t o the error ment ioned prev i - ously in regions where the points do not coincide. The error can be corrected in the moir~-of-moir~ pa t te rn by ei ther of the methods described above (shifting or laying off the motion vector) .

If the error is due to a pitch mismatch, it can also be corrected in a simple way as follows. Two gener- ating pat terns (Fig. 3) are produced by (1) a master grat ing and a mismatched specimen grating on the undeformed body, and (2) a master grating and a deformed mismatched grating. The pitch of the master grat ing is called p. The pitch of the mis- matched grating on the undeformed body is called Ps. The difference between p and Ps is what produces the pat tern on the undeformed body and is the "pitch" mismatch. The images of a point P in the undeformed and deformed bodies in Fig. 3 are given the symbols P' and P " as before. Say that point P " has the hori- zontal coordinate va lue x" . The master grating order at P" is x " / p . The order of the deformed-specimen- grat ing line at P " is then ( x " / p ) - - m " , since m " is the order of the contour produced by the master gra t - ing and deformed specimen grating at P". But the original position of this point is at P'. The order of the undeformed-spec imen-gra t ing line at P', which is x ' / p s , must equal the order of the deformed speci- men grat ing at P " since they refer to the same grat- ing line

32' 3C ~

- - m " ( 1 ) Ps P

The motion of the point in the x direction is

o r

u ~ x " -- x ' = x " - - ( x " / p -- m " ) p s

u = Ps Ira" - - x " ( 1 / p - - 1 / p s ) ] .

(2)

Note however that q', the contour line which is on the undeformed body at P", is the difference of grat- ing line orders of the master and undeformed speci- men at x" , then

X " X " q ' - (3)

P Ps

u : Ps ( m " - - q') (4) Therefore

Note that ( m " - - q') is the order of the generated contour line. The mismatched specimen pitch used in eq (4) gives then the correct calibration of the iso- thetic contour lines that were obtained by super- posing a pat tern of contour lines f rom a deformed body on a p i tch-mismatch pat tern from an unde- formed body. As known, the isothetics give displace- ment of the points located on them. Since the iso- thetics are associated with the displacements, these points are at the deformed position of the body, for instance, point P".

If the subtraction of the initial mismatch is done by plott ing curves, the same approach should be fol-

lowed. The specimen pitch should be used to cali- brate the difference of the two fringe orders, and the result obtained is re fe r red to the deformed-body positions at all points.

The "p i tch-mismatch" technique is often used to increase the number of fr inges on a deformed body. This permits a more accurate determinat ion of the displacement derivative. The actual displacement der ivat ive is obtained by subtracting the straight line represent ing the initial fringes from the curve r e p - resenting the fringes in the deformed body. It fol- lows f rom the reasoning above that this difference must be mult ipl ied by the undeformed specimen pitch Ps to obtain the correct displacement derivat ive. This is true despite the fact that both sets at fringes were obtained using a master grat ing of pitch p.

I sopach i c s O b t a i n e d w i t h a W e d g e A n g l e

Another case of pract ical interest develops in the determinat ion of isopachics. In this case, the error sometimes has an analytic expression as when the shape of the undeformed plate is a wedge. By super- posing the contour lines of equal thickness of the deformed plate (deformed-s ta te isobathics) on the contour lines of equal thickness of the undeformed wedge (init ial isobathics), the undeformed-wedge thickness t ---- ax is subtracted-out and the isopachics are obtained, t is the thickness of the wedge, a its angle and x the distance of a point f rom the apex of the wedge. The generated pat tern wil l have an error proport ional to the displacement of points in the direction of the wedge angle equal to (At)fic ~-~ au, where (At)fie is the e r ror in at produced by a dis- p lacement u in the direction of the wedge angle a.

Observation on Isothetics The error discussed above does n o t apply to the

most common moir~ pattern, which is the one giving displacement components (isothetics) obtained wi th- out mismatch. Displacement components obtained by a moir~ of a deformed and undeformed gratings give the exact values of displacement of the deformed body, even for large deformation.

R e f e r e n c e s 1. Coker, E. G. and Filon, L. N., A Treatise on Photoelasticity,

Cambridge University Press, N e w York, 178 (1931). 2. Pirard, A., "'Considdrations sat la mdthode du moir~ en photo-

dlasticitC'" Rev. Univ. des Mines, 9e Serie, tome XVI(4), 177-200 (196o).

3. Oster, G., ~Vasserman, M. and Zwerling, C., "Theoretical Interpretation of Moir~ Patterns," ]nl. Optical Soc, Am., 54 (2), 169-175 (1964).

4. Post, D,, The Generic Nature of the Absolute Retardation Method of Photoelasticlty. EXPERIMENTAL MECHANICS, 7 (6), 233- 241 (tune 1967).

5. Durelli, A. 1. and Parks, V. ]., "'Moir~ Fringes as Parametric Curves," EXPERIr,~ENTAL MECrIANICS,, 7 (3), 97-104 (March 1967).

6. Durelli, A. 1., "'Visual Representation of the Kinematics o[ the Continuum," EXPEI~d[MENTAL MECHANICS, 6 (3), 113-139 (March 1966).

7. Durelli, A. 1, and Parks, V. 1., Moird Analysis of Strain, Pren- tice-Hall, Englewood Cliffs, N. 1. (1970).

8. Clark, ]. A. and Durelli, A. 1., "'Separation of Additive and Subtractive Moird Patterns," lnl. of Appl. Mech., 38 (1), 266-269 (March 1971),

9. Post, D. (Private Communication).

432 [ September 1971