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Bulletin of the Seismological Society of America. Vol. 60~ No. 3, pp. 807-81 7. June, 1970

A P P L IC A T I O N O F T H E B I L I N E A R Z - T R A N S F O R M M E T H O D T O G R O U N DM O T I O N S T U D I E S

B y J. RAY STAGNER AND GARY C. HART: i

ABSTRACTA recursive relationship in the time domain is der ived for the response of a singledegree of freedom oscillator to an arbitrary digitized ground motion. In develop-ing the relationship the oscillator's continuous frequency response characteristicsare expressed in terms of an equivalent sampled function Z-plane representationusing the bilinear Z-transform method. Oscillator response time histories and r e-sponse spectra for prescribed ground motions are compared with exa ct solu-tions. Applicat ion of the presented method to instrument correction studies is dis-cussed.

INTRODUCTIONThe calculation of the response time history for a l inear single degree of freedom

osci l lator to a prescr ibed ground motion t ime his tory has been discussed by otherresearchers (Benioff, 1934; Housn er, 1941). Prior to the i mp act of the digital com-puter , emphasis was placed on methods of calculat ing which considered the groundmotion to be a cont inuous funct ion of t ime. However , now with the increased avai l -ability and speed of digital computers the response of a single degree of freedomosci llator to a n arbi t rary ground motion is usual ly calculated using digi tal techniques.Dif ferent authors have proposed methods for calculat ing digi t ized response t imehistories (Nigam and Jennings, 1969; Brady, 1966). In general, these methods repre-sent both the forcing function and oscillator response as a digitized stepwise linearfunct ion in the t ime domain. The relat ive mer i t of each method is descr ibed in theoriginal papers.

In this technical paper we shall explicitly represent the oscillator 's response char-acter is t ics in the f requency domain using a bi l inear Z- transform. This f requencyrepresentat ion enables us to develop a t ime domain recursive relat ion for calculat ingstructural response. Three basic steps are necessary to develop this relation. First ,the s ta ndard cont inuous Laplace t ransform s-plane representat ion of the groundmotion-oscillator displacement response relationship is derived. Second, this standardcontinouous Laplace s-plane representat ion is expressed in terms of an equivalentsampled funct ion Z-plane f requency representat ion. Third , the paral lel ism betweenthe Z-plane bilinear transform and finite difference operators is util ized in order todevelop a recursive relation in the time domain for calculating oscillator response.

The recursive relationship we derive, using this digitized frequency approach,expresses the oscillator 's relative displacement in terms of the ground acceleration atthe t ime of in terest and the oscil lator displacement and ground accelerat ion at thetwo previous instants in t ime. In addi t ion to our method providing an al ternate wayof calculating oscillator response it also results in approximately a threefold savingsin computer computat ion t ime when compared to Simpson 's rule in tegrat ion ofDuhamel 's in tegral .

Applicat ion of the presented approach to instrument correct ion s tudies fol lowsdirect ly . We discuss br ief ly this problem and formulate the necessary reeursive equa-tion for pre-processing the instrument record prior to response analysis.

8O9

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~10 BULL ETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

THEORETICAL R ESPON SE DEVELOPMENTT h e t h e o r e t i c a l s t e p s t h a t a r e n e c e s s a r y i n o r d e r t o d e v e l o p a r e c u r s i v e e q u a t i o n

for ca l cu l a t i ng t he r e sponse o f a s i ng l e degree o f f r eedom osc i l l a to r t o a p resc r ibedg r o u n d m o t i o n t i m e h i s t o r y a r e p r e s e n t e d i n t h i s s e c t i o n . T h e s e s t e p s a r e b a s e d i np a r t o n t h e b i l i n e a r Z - t r a n s f o r m m e t h o d ( G o l d e n a n d K a i s e r , 1 9 6 4 ) .Th e d i f f e ren t i a l equa t i o n o f m ot ion fo r a si ng le degree o f f r eed om o sc i l la to r sub-j e e r e d t o b a s e m o t i o n i s

2 ( t ) - + - 2BO~n2(t) + ~ n 2 z ( t) = - - a ( t ) ( 1 )wh ere 13 i s t he c r i t i ca l d am ping r a t i o , COn s t he osc i l l a t o r 's u nd am pe d na tu ra l f r equ enc yo f v i b r a t i o n a n d a ( t ) i s t h e g r o u n d a c c e l e r a t i o n . T a k i n g t h e L a p l a c e t r a n s f o r m o feq ua t i on (1 ) fo r ' the spec i a l case o f ze ro in i t i a l cond i t i ons y i e lds

2[ s ~ + 2 ~ co ,~ s + ~ o n ] X ( S ) ---- - - A ( S ) ( 2 )w h e r e s i s t h e L a p l a c e v a r i a b l e a n d A ( s ) a n d X ( s ) a r e t h e L a p l a c e t r a n s f o r m s o fg r o u n d a c c e l e r a t i o n a n d r e l a t i v e d i s p l a c e m e n t , r e s p e c t i v e l y . U s i n g e q u a t i o n ( 2 ) i tf o ll o w s t h a t t h e s -p l a n e t r a n s f e r f u n c t i o n r e l a ti n g t h e g r o u n d a c c e l e r a ti o n a n d r e l a ti v ed i s p l a c e m e n t i s

_ x ( s ) _ 1 / [ s 2 -t- 2/~cons -t- ~on2]. ( 3 )H ( s ) A ( s )E q u a t i o n ( 3 ) i s a s t a n d a r d L a p l a c e f o rc e - re s p o n s e r e la t io n s h i p . W h i le t h i s e q u a t i o n

i s u s e f u l f o r s o m e m a t h e m a t i c a l g r o u n d m o t i o n s , i n p r a c t i c e w e u s u a l l y r e p r e s e n t t h eg r o u n d m o t i o n a n d c o r r e s p o n d i n g o s c i l l a t o r r e s p o n s e a s s a m p l e d d a t a p o i n t s d i g i t i z e da t u n i f o r m t i m e i n c r e m e n t s ( d a t a d i g i t i z e d a t n o n - u n i f o r m t i m e i n t e r v a l s c a n a l w a y sb e t r a n s f o r m e d i n to ~ u n i f o r m l y s a m p l e d f u n c t i o n b y a n i n t e rp o l a t io n p r o c e ss ) .I n o r d e r t o e x p r e s s e q u a t i o n ( 3 ) i n a d i g i t i z e d f o r m i t i s a d v a n t a g e o u s t o i n t r o d u c et h e c o n c e p t o f t h e Z - t r a n s f o r m . H o w e v e r , w h i l e d i r e c t a p p l i c a ti o n o f t h i s t r a n s f o r mm e ~ h o d p r e s e r v e s t h e a m p l i t u d e c h a r a c t e ri s ti c s o f t h e s - p la n e f u n c t i o n g i v e n b y e q u a -t i on (3 ) , i t warps t he f r equency sca l e . I n o rde r t o compensa t e fo r t h i s e f f ec t , i t i s nec -e s s a r y t o p r e - w a r p t h e f r e q u e n c y c h a r a c t e r is t ic s o f e q u a t i o n ( 3). W i t h t h i s i n m i n d w ed e f in e t h e n e w v a r i a b l e

w h e r e

a n d

sl =-- sk ( 4 a )

2~ 1 - - x i t a n ( ~ n ~ t / 2 ) . (4c)

T h e d i g i t iz e d g r o u n d m o t i o n is sa m p l e d a t a u f ii fo r m r a t e a n d t h e t i m e i n c r e m e n ti s d e n o t e d A t . E q u a t i o n ( 4 ) p r e - w a r p s t h e f r e q u e n c y s c a l e a n d w h e n i t i s a p p l i e dt o e q u a t i o n ( 3 ) w e o b t a i n a n e w p r e - w a r p e d s - p l a n e t r a n s f o r m

H ( S l ) = - - ] c 2 / [ 8 1 2 - t - 2 /~ ( .. 0 1 S l - t - o j1 2 ] ( 5 )

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B I L I N E A R Z -T R A N S F O R $~ [ M E T H O D 811Now the Z-transform method can be applied to equation (5) and both the ampli-

tude and phase characteristics of the oscillator's original s-plane transform are pre-served. We first define the Z-plane variable as

Z - -~ e - s l A t

Then we observe that 81 and z are also related identically by the equationsl = 2(1 - z ) / A t ( 1 + z ) (6)

in the limit as At -~ 0. Upon substituting equation (6) into equation (5) it followsdirectly th at the transform of the single degree of freedom oscillator is

whereH ( z ) = -k12(1 -4- 2 z - t- z 2 ) / ( B o + B l z --? B 2 z 2 ) (7)

l c l = k a r l 2k 2 = ~ l A t / 2

B o = 1 + 2 ~ k 2 - ~ k2 2B 1 = 2 ( k 2 2 - - 1 )B2 = 1 -- 2~k2 -+- k 22 .

Note that these coefficients are only a function of natural frequency, sample rate andcritical damping ratio. Equation (7) can be interpreted as the digitized counterpartof equation (2). Just as equation (2) provides a s-plane representation of the oscil-lator, it can be shown that equation (7) provides an equally valuable Z-plane repre-sentation of the oscillator (Robinson and Treitel, 1964).The computational advantages of the Z-transform method and the correspondencebetween it and the finite difference operators has been noted in the li terature (Robinsonand Treitel, 1964). It immediately follows from this reference that equation (7) canbe written in the equivalent recursive time domain representationx[n At] = -kl-~2Bo a [n n t] + 2a[(n -- 1)At] -~ a [ ( n - - 2)At]

J ~ l X [ ( ~, - - 1 ) z~ t ] -~ - X [ ( n - - 2 ) A t ] ( 8 )+ g o go This recursive equation expresses the response at time t = n a t in terms of the ground

acceleration at the time of interest and the ground acceleration and oscillator responseat the two previous incremental times. In a following section we shall apply equation(8) in calculating the response of a single degree of freedom oscillator to mathematicaland actual ground motions.

Discussion OF A N IN S T R U M E N T C O R R E C T IO N P R O C E D U R EThe correction of a recorded ground motion for instrument response characteristics

can also be carried out using the bilinear Z-transform method. In this correction therecording instrument is visualized as an oscillator with an established natural fre-

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812 B U L L E T I N O F T H E S E I S M O L O G I C A L S O C I E TY O F A M E R I C Aq u e n c y a n d c r it ic a l d a m p i n g r a t i o . T h a t i s, e q u a t i o n ( 3 ) c a n b e i n v e r t e d i n o r d e r tor e p r e s e n t t h e i n s t r u m e n t ' s c o n t i n u o u s s - p l a n e t r a n s f e r f u n c t i o n a s

G(s) --- A ( s ) _ _@2 ~_ 2 ~ s - ~ ~ 2) . ( 9 )

T h i s c o n t i n u o u s i n s t r u m e n t t r a n s f e r f u n c t i o n c a n b e r e p r e s e n t e d i n t h e d i g i t i z e df r e q u e n c y d o m a i n u s in g t h e p r e v i o u s l y d is c u s s e d p r e - w a r p i n g a n d Z - t r a n s f o r m m e t h -o d s . I t t h e r e f o r e fo l lo w s t h a t e q u a t i o n ( 9 ) b e c o m e s

G(z) = - - ( B o + B l Z -Jr B 2 z 2 ) / k 1 2 ( l @ 2 z @ z 2) ( l o )w h e r e t h e c o n s t a n t s h a v e b e e n p r e v i o u s l y d e f i n e d .

A r e c u r s iv e r e la t i o n s h ip a n a l o g o u s t o e q u a t i o n ( 8 ) c a n n o w b e d e v e l o p e d f o r e q u a -t i o n ( 1 0 ) a n d i t c a n b e s h o w n t o b e e q u a l t o

u[nht] = k12G(1) { B o x~ A t ] + B l x [ ( n - - 1 ) A ~ d - B 2 x [ ( n - - 2 ) A t ] }d - 2 a [ ( n - 1 ) A ~ d - a [ ( n - 2 )A ~ ( 1 1 )

w h e r e e q u a t i o n ( 1 1 ) h a s b e e n d i v i d e d b y G ( 1 ) i n o r d e r to n o r m a l i z e t h e r e l a t io n s t ot h e r e s p o n s e a t z e r o f r e q u e n c y .

T h i s r e c u r s i v e re l a t io n s h i p h a s t h e i n v e r s e c h a r a c t e r is t ic s o f e q u a t i o n ( 8 ) a n da s s u m e s t h a t t h e o u t p u t o f t h e m e a s u r i n g in s t r u m e n t i s p r o p o r t i o n a l t o t h e r e l a ti v ed i s p l a c e m e n t o f a s i n g l e d e g r e e o f f r e e d o m o s c i l l a t o r . F o r s u c h a m e a s u r e m e n t s y s -t e m , e q u a t i o n ( 1 1 ) c a n b e u s e d a s a p re - p r o c e s s o r t o r e d u c e t h e i n f lu e n c e o f t h ese i smic de t ec to r on acce l e rograms pr io r t o ana lys i s . A l l t ha t i s r equ i r ed i s a knowledgeo f t h e i n s t r u m e n t n a t u r a l f r e q u e n c y a n d c r it ic a l d a m p i n g r a t io .

O S C IL L A T O R R E S P O N S E C A L C U L A TI O N ST h e r e c u r s i v e r e l a ti o n e x p r e s s e d in e q u a t i o n ( 8 ) i s n o w u s e d t o c a l c u l a t e a si n gl e

d e g r e e o f f r e e d o m o s c i l l a t o r ' s r e s p o n s e t i m e h i s t o r y f o r t h r e e m a t h e m a t i c a l l y p r e -s c r i b e d g r o u n d m o t i o n s . A l s o , p l o t s a r e g i v e n o f c a l c u l a t e d r e s p o n s e s p e c t r a f o r t h eN - S c o m p o n e n t o f t h e 1 94 0 E 1 C e n t r o e a r t h q u a k e , u s i n g o u r m e t h o d a n d D u h a m e l ' si n t e g r a l w i t h S i m p s o n ' s r u l e i n t e g r a t i o n .F i r s t , w e a p p l i e d a u n i t i m p u l s e g r o u n d m o t i o n t o t h e o s c i l l a t o r a n d i t s r e l a t i v ed i s p l a c e m e n t t i m e h i s t o r y f o r 0 , 2 0 a n d 5 0 p e r c e n t c r i t i c a l d a m p i n g i s a s s h o w n i nF i g u r e 1 . I n t h i s f i g u r e t h e c r o s s m a r k s r e p r e s e n t o u r r e c u r s i v e l y c a l c u l a t e d r e s p o n s eand the so l id l i nes r epresen t t he exac t con t inuous r esponse o f t he osc i l l a to r . Thesep l o t s s h o w n o n o t i c e a b le d i f fe r e n ce b e t w e e n t h e e x a c t a n d t h e r e e u r s iv e l y c a l c u l a te dr e s p o n s e . T h e o s c i l l a t o r ' s u n d a m p e d n a t u r M f r e q u e n c y a n d s a m p l e r a t e w e r e 1 c p sa n d 1 0 0 s a m p l e s / s e c , r e s p e c t i v e l y .

Second , t he osc i l l a to r r e sponse t o a un i t s t ep func t ion was ca l cu l a t ed fo r 0 , 20a n d 5 0 p e r c e n t c r i ti c a l d a m p i n g , s e e F i g u r e 2 . A s w i t h t h e p r e v i o u s l y d i s c u ss e d u n i ti m p u l s e r e s p o n s e t h e f i g u r e s h o w s a n e x c e l l e n t c o r r e l a t i o n w i t h t h e e x a c t r e s p o n s et i m e h i s t o r y . T h e s a m p l e r a t e a n d o s c i ll a to r n a t u r a l f r e q u e n c y w e r e 1 0 0 / s e t a n d 1 c p s,r e s p e c t i v e l y .

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BILINEAR Z-TRANSFORM METHOD 813The oscillator response as a function of sinusoidal ground motion frequency is

shown in Fig. 3 for 0, 20 and 50 per cent critical damping. This plot shows, as ex-pected, that the peak response occurs when the ground frequency is 1 cycle/sec,(i.e., same as the oscillator's natural frequency). With a sample rate of 100/see theexact and recursively calculated curves again are indistinguishable.

0.2B = 0-(po.~o F , , ~, l , . , ~ , ~ o . ~ o , \ f .

t , i , i-o., kW ~,'- 0 . 2 I I I I 10 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5

S E C O N D SFIG. 1. Impulse response time history.2.0

1.5

om 1.oZ

0.5

o o

i , ' f f ! t 77 , , - ~ . . ~ t , . - + - . t . ,y i i ;\ J , W ,, ,

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5S E C O N D S

FIG. 2. Step respo nse t ime h is tory .

In order to s tudy the effect of decreasing the sample rate we calculated the responseof the same linear oscillator to the three previously studied mathematical groundmotions. However, this time the sample rate was set at 20/see (At = 0.05 see) foran oscillator with a natural frequency of 1 cps. Figures 4 and 5 show the responseto have a very slight amplitude and phase shift, respectively. While this discrepancyhas little engineering significance we have purposely shown the response spectra for asinusoidal ground motion, Figure 6, so as to more completely assess this difference.Note that when the ground frequency is 5 cps the discrepancy between the exact andour response is large (approximately 40 per cent). However, we must realize th at whenthe ground frequency is 5 cps its period is 0.20 sec and therefore we only have four

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I 0 0 . 0 0 I

I 0 . 0 0 ~ -A ,

I.oo ~.~'~---.z

0 .10

O . O J 0

f B = 0~ , ~ 0 . 2 00 . 5 0

I I I I rI 2 3 4FREQUENCY

F I G . 3 . S i n u s o i d a l r e s p o n s e s p e e t r ~ .

0 . 2

~z 1.0

= 1 . 0 H Z

0.1

z :

-O J

-0 .2 I , I t I I0 0 .5 1 ,0 1 .5 2 .0 2 .5SECONDSF i e , 4 " . I m p u l s e r e s p o n s e t i m e h i s t o r y (2 0 s a m p l e s / s e e ) .

2_.01

0 ,5

0 0 0 .5 1.0 i .5SECONDS

8 1 4 B U L L E T I N O F T H E S E I S M O L O G I C A L S O C I E T Y O F , ~M E R IC A

f.un = 1 .00 HZ

2 .0 2 .5F I G . 5 . S t e p r e s p o n s e t i r ae h i s t o r y ( 20 s a m p l e s / s e e ) .

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B I L I N E A R Z - T R A N S F O R M M E T H O D 8 1 5

I00.00

B = OI0.00 , .~. = 1.00 HZ

1 . 00 :

0 . 1 0

0 . 01 I I i I II 2 3 4 5

F R E O U E N C YFIG. 6. Sinusoidal respo nse sp ec tra (20 sam ples/se e)..2

w 0

J - I

- 20

021; n ' O 0 O o 0 s

t = 0 , 0 1 s e c

I I I I f P I I0 . 5 1 .0 1 .5 2 . 0 2 . 5 3 , 0 3 , 5 4 . 0

SECONDSFIG. 7. Re la t ive e r ro r vs t ime .

5 0

o .I

/% 5

r I I I I I I F I f f I I I I I l lhO 10 . 0F i G . 8 . E l C e n t r o ( 1 9 40 ) r e s p o n s e s p e c , tr a ..

d a t a p o i n t s p e r p e r i o d . I f w e r e s t r ic t o u r s e l v e s t o a g r o u n d m o t i o n f r e q u e n c y r a n g ew h i c h h a s a t l e a s t 1 0 d a t a p o i n t s p e r p e r i o d , ( i. e. , e q u a l t o o r l e ss t h a n 2 e p s ) t h e nt h e r e i s n o n o t i c e a b l e d i f f er e n c e b e t w e e n t h e e x a c t r e s p o n s e s p e c t r a a n d o u r r e e u r s i v e l yc a l c u l a t e d r e s p o n s e s p e c t ra .

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816 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMEI~ICA

T h e p r e c e d i n g p l o t s a re m i s l e a d i n g as to t h e t r u e a c c u r a c y o f t h e p r e s e n t e d m e t h o d .T h a t i s , t h e s c a le o f t h e p l o t s i s s u c h a s t o n o t s h o w s m a l l p e r c e n t a g e e r r o r s i n t h e a p -p r o x i m a t e m e t h o d . F i g u r e 7 s h o w s a p l o t o f t h e r e l a t i v e e r r o r , d e f i n e d

R e l a t i v e E r r o r = ( E x a c t R e s p o n s e - R e e u r s i v e l y C a l c u l a te d R e s p o n s e )S t a t i c R e s p o n s ef o r a s t e p g r o u n d m o t i o n w i t h 0 a n d 2 0 p e r c e n t c r i t ic a l d a m p i n g . T h e r e l a t i v e e r r o rc a n b e a c c u r a t e l y e s t i m a t e d u s i n g t h e e q u a t i o n

E ~ l ~ ti v , = - - ( 0 . 5 co,~At)e ~t sin (x ,~t). (1 2)W h i l e t h e r e s p o n s e t i m e h i s t o r y c a n b e c o r r e c t e d f o r t h i s e r r o r i t is n o t u s u a l l y n e c e s -s a r y f o r m o s t g r o u n d m o t i o n p r o b l e m s . T h i s c o n c lu s io n is b a s e d u p o n t h e f a c t t h a t t h em a x i m u m e r r o r o c cu r s w h e n t h e r e s p o n s e is n e a r z er o , w h e r e a s , t h e a c c u r a c y o f t h ep r o p o s e d m e t h o d i s e x c e ll e n t a t o r n e a r p e a k r e s p o n s e t im e s .

F i g u r e 8 s h o w s th e r e s p o n s e s p e c t r a f o r t h e N - S c o m p o n e n t o f t h e 1 94 0 E1 C e n t r ee a r t h q u a k e . T h e c ro s s m a r k s a n d s o li d l in e s c o rr e s p o n d t o t h e r e e u r s iv e a n d S i m p s o n ' sr u l e c a l c u l a t e d r e s p o n s e, r e s p e c t i v e l y . T h e s a m p l e r a t e u s e d f o r t h e s e c a l c u l a t i o n sw a s 1 0 0 /s e e. T h e r e e u r s i v e r e l a t i o n ' s c o m p u t e r c o m p u t a t i o n t i m e w a s a p p r o x i m a t e l yo n e t h i rd t h a t w h e n u s i n g S i m p s o n ' s ru l e.

C O N C L U S I O N

W e h a v e p r e s e n t e d a b i l i n e a r Z - t r a n s f o r m p r o c e d u r e f o r c a l c u l a t i n g t h e d i s p l a c e -m e n t r e s p o n s e t i m e h i s t o r y o f a s i ng l e d e g r e e o f f r e e d o m o s c i ll a to r . T h e d i s p l a c e m e n tt i m e h i s t o r y c a l c u l a t e d u s i n g o u r r e c u r s iv e r e l at i o n s h i p s h o w e d e x c e ll e n t a g r e e m e n tw i t h t h e e x a c t c o n t i n u o u s r e s p o n s e s o lu t io n s f o r t h r e e m a t h e m a t i c a l g r o u n d m o t i o n s .A l so g o od a g r e e m e n t w a s d e m o n s t r a t e d f o r t h e r e s p o n se s p e c t r a o f t h e N - S c o m p o n e n to f t h e 1 94 0 E 1 C e n t r e e a r t h q u a k e . T h e r e c u r s i v e p r o c e d u r e r e s u l t e d i n a t h r e e f o l ds a v in g s in c o m p u t e r c o m p u t a t i o n t i m e . A s w i t h a ll d i g it iz e d r e s p o n s e c a l cu l a t io n sc a r e m u s t b e t a k e n i n e i t h e r p r e p r o e e s s i n g t h e r e c o r d s u s i n g e s t a b l i s h e d d i g i ta l f i lt e r in gt e c h n i q u e s a n d / o r s e l e c ti n g " h i g h " s a m p l e r at e s.

ACKNOWLEDGMENTSThe authors wish to ~hank Mr. Inn Stubbs, Oeotronies , for h is vocal and f inancia l support forpa r t o f th i s re sea rch . Also , we wou ld l ike to aeknowIedge the a ss i s tance o f the R epor t s Group o fthe School o f Eng inee r ing and App l ied Sc ience under the superv is ion o f Es te l le E . D orsey .

REFERENCESBenioff , H. (1934). Calcula t ion of tbe response of an osci l la tor to arbi trary ground motion, Bull.Seism. Soc. Am. 24,398-403.Brady, A. G. (1966). Studies of Response to Earthquake Grou.nd Motion, Ea r th q u a k e En g r . Re s .Lab . , Ca l i f . In s t . o f Teeh . , Pasadena .Golden, IR.. M. and J . F . K aise r (1964). I )es ign of wideb and sam pled d ata f il ters, The Bell SystemTech. J. 43, No. 4 , par t 2 .Housner , G. W. (1941) . Ca lcu la t ion o f the re sponse of an osc i l la to r to a rb i t r a ry g round m ot ion ,Bull. Seism. Soc. Am. 31, 143-149.

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BILINEAR Z-TRANSFORM M ETHOD 81 7N i g a m , N . C . a n d P . C . J e n n i n g s ( 1 9 69 ) . C a l c u l a t i o n o f r e s p o n se s p e c t r a f r o m s t r o n g - m o t i o ne a r t h q u a k e r e c o r d s , Bull. Seism. Soc. Am. 39,909--922.Rob i nson , E . A . and S . T re i t e l (1964). P r i nc i p l e s o f d i g it a l f i l te r i ng , Geophysics. 29.~[ECHANICS AND ~TRUCTURE DEPARTMENTSCHOOL OF ENGINEERING AND APPLIED SCIENCEUNIVERSITY OF CALIFORNIAI~OS ANGELES~ CALIFORNIA

M anusc r i p t r ece i ved Oc t obe r 27 , 1969 .