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An Introduction to the measurement and interpretation of dynamic ice
loads on compliant structures
Frederking, R. M. W.
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An Introduction to the Measurement and
Interpretation of Dynamic Ice Loads on
Compliant Structures
by R. Frederking
Reprinted from
IAHR Special Report 845, February 7989
Working Group on Ice Forces: 4th Stateof-th+Afl Report
p. 167-187
(IRC Paper No. 1600) Reprinted with permission
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AN INTRODUCTION TO THE MEASUREKNT AND INTERPRETATION OF DYNAMIC ICE LOADS ON COMPLIANT STRUCTURES '
R. F r e d e r k i n g S e n i o r Research O f f i c e r Geotechni ca 1 Sect i o n I n s t i t u t e f o r Research i n Construct i o n N a t i o n a l Research C o u n c i l o f Canada CANADA Abstract
Some o f t h e fundamental f a c t o r s i n t h e behaviour of f l e x i b l e s t r u c t u r e s s u b j e c t t o v a r i o u s t y p e s o f i c e 1 oading a r e . introduced. Terminology i s presented and v a r i o u s aspects o f t h e b a s i c equations o f m o t i o n o f a compliant s t r u c t u r e a r e discussed. Measurement of dynamic response o f s t r u c t u r e s , a n a l y s i s u s i n g t r a n s f o r m a t i o n between t h e t i m e domain and frequency domain, and i n t e r p r e t a t i o n o f r e s u l t s a r e t r e a t e d . Recommendat io n s a r e made on measurement methods. The r e s u l t s o f 1 aboratory t e s t s w i l l be used t o i l l u s t r a t e t h e s e methods.
I n t roduct i on
Many s t r u c t u r e s are s u b j e c t t o dynamic i c e loads. These can be slender. s t r u c t u r e s such as l i g h t p i e r s o r b r i d g e p i e r s . I n t h i s case t h e whole s t r u c t u r e may v i b r a t e , e i t h e r as a r e s u l t o f t h e compliance of t h e s t r u c t u r e i t s e l f o r i t s foundation. F o r very l a r g e s t r u c t u r e s such as off-shore s t r u c t u r e s o r ships, t h e whole s t r u c t u r e may v i b r a t e , as above and a d d i t i o n a l l y , components o f t h e s t r u c t u r e may v i b r a t e . The dynamic n a t u r e o f t h e i c e load i s s t i l l s u b j e c t t o considerable d i s p u t e , as witnessed by t h e two o t h e r s t a t e - o f - t h e - a r t r e p o r t s t o t h i s Symposium on t h e s u b j e c t (MZih'ttanen, 1988 and Sodhi, 1988). The o b j e c t i v e of t h i s review i s t o introduce some o f t h e fundamentals o f dynamic s i g n a l a c q u i s i t i o n and analysis i n t h e c o n t e x t o f i c e load measurements i n t h e f i e l d and laboratory. It i s aimed a t t h e person e n t e r i n g t h e f i e l d of dynamic s i g n a l a c q u i s i t i o n and a n a l y s i s , not t h e expert. By removing some of t h e ambiguity i n t h e basic data, t h e e f f o r t s i n i n t e r p r e t i n g r e s u l t s can be made more e f f e c t i v e .
Background
A l l s t r u c t u r e s are f l e x i b l e t o some e x t e n t and consequently, when subjected t o a load, they w i l l deform. Depending on t h e nature of t h e loading, o s c i l l a t o r y motions o f t h e s t r u c t u r e can be set up. There a r e numerous reference works on v i b r a t i o n s , eg. Thomson, 1981. The f o l lowing w i l l r e v f ew selected fundamentals o f v i b r a t i o n analysis. Free v i b r a t i o n describes a system moving under t h e a c t i o n o f f o r c e s i n t e r n a l t o i t . Such a system w i l l v i b r a t e a t one o r more n a t u r a l frequencies determined by t h e mass and s t i f f n e s s p r o p e r t i e s o f t h e system. I f v i b r a t i o n s take place as a r e s u l t o f e x t e r n a l forces i t i s termed f o r c e d v i b r a t i o n . For o s c i l l a t o r y e x t e r n a l forces, t h e system i s f o r c e d t o o s c i l l a t e a t t h e f o r c i n g frequency. I f t h e f o r c i n g frequency corresponds t o one of t h e n a t u r a l frequencies o f t h e system, resonance occurs and e x c e s s i v e l y l a r g e o s c i 1 l a t ions may resu 1 t.
There a r e a number o f types o f p e r i o d i c motion, t h e s i m p l e s t being p e r i o d i c motion; i.e. motion having a constarrt frequency. One of t h e
x = A s i n 2 n t / - r ( 1
where x i s p o s i t i o n a t t i m e t, t extends f r o m - - t o +-, A i s t h e a m p l i t u d e and 7 i s t h e p e r i o d o f t h e o s c i l l a t i o n . P e r i o d i c m o t i o n can r e s u l t i n complex wave forms which a r e represented by t h e F o u r i e r s e r i e s . Methods a r e now a v a i l a b l e t o d e t e r m i n e t h e F o u r i e r s p e c t r u m ( f r e q u e n c y c h a r a c t e r i s t i c s ) o f a complex wave form. T h i s w i l l b e discussed i n more d e t a i l 1 ater. Another t y p e o f o s c i l l a t o r y mot i o n i s t r a n s i e n t motion, w h i c h has a f i n i t e s t a r t and end, ti c t c tf. T r a n s i e n t m o t i o n i s n o t
p e r i o d i c , i n t h a t i t repeats i n d e f i n i t e l y , b u t i t can be analysed f o r i t s frequency content. Random motion i s n o w d e t e r m i n i s t i c ; i .e. a f u n c t i o n cannot be w r i t t e n which can p r e d i c t f u t u r e motions. It i s p o s s i b l e , however, t o determi ne frequency, phase, and amp1 i t u d e c h a r a c t e r i s t i c s o f random motions and d e s c r i b e them i n s t a t i s t i c a l t e r n .
M o t i o n i s d e s c r i b e d as b e i n g a s i n g l e degree o f freedom when p o s i t i o n i s s p e c i f i e d by a s i n g l e s p a c i a l coordinate. A s i n g l e p a r t i c l e moving i n space would have t h r e e degrees o f freedom ( p o s i t i o n s p e c i f i e d by c o o r d i n a t e s o f t h e p o i n t ) . A r i g i d body would have t h r e e p o s i t i o n a l c o o r d i n a t e s and t h r e e r o t a t i o n a l c o o r d i n a t e s , s i x degrees of freedom A
continuous e l a s t i c body would have an i n f i n i t e nunber o f degrees o f freedom, a l t h o u g h i n p r a c t i c e i t i s p o s s i b l e t o d e s c r i b e t h e system w i t h a f i n i t e number o f degrees o f freedom.
The m o t i o n o f most s y s t e m , a t l e a s t f o r some a m p l i t u d e range, can be d e s c r i b e d as 1 inear. A simple s i n g l e degree o f freedom system w i t h damping
i s d e s c r i b e d by t h e e q u a t i o n where x ( t ) = t i m e s e r i e s o f displacement ;(t) = f i r s t d e r i v a t i v e o f x ( t ) w i t h respect t o t i m e ? ( t ) = second d e r i v a t i v e o f x ( t ) w i t h r e s p e c t t o t i m e n = on/Zn =
E=
m n a t u r a l frequency k = s t i f f n e s s o f t h e s t r u c t u r e m = mass o f t h e s t r u c t u r eF
= c / 4 m f n = c r i t i c a l damping r a t i o c = damping constantf ( t ) = t i m e s e r i e s o f i n p u t f o r c e
See F i g u r e 1 f o r a schematic r e p r e s e n t a t i o n o f such a system. Mass, m, damping constant, c, and s t i f f n e s s , k , a r e f i x e d constants o f t h e system. Note t h a t t h e power on each o f t h e s e t e r n i s one. E q u a t i o n ( 2 ) i s t h e f a m i l i a r l i n e a r second o r d e r d i f f e r e n t i a l e q u a t i o n t o w h i c h t h e r e a r e c l o s e d form s o l u t i o n s f o r many cases o f f ( t ) . I f f ( t ) = 0, t h e r e s u l t i n g m o t i o n i s f r e e damped v i b r a t i o n . I f f ( t ) t 0, t h e n i t i s f o r c e d v i b r a t i o n .
I n a l i n e a r system as d e s c r i b e d above, t h e r e i s a 1 i n e a r r e l a t i o n between cause and e f f e c t .
F i g u r e
1
Schematic o f a s i n g l e degree o f freedom system w i t h damping s u b j e c t t o f o r c e d v i b r a t i o n , f ( t ) .Some s y s t e m , however, a r e nonl inear. The d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g a n o n l i n e a r o s c i l l a t o r y system has t h e general form
S o l u t i o n t o t h i s e q u a t i o n i s much more d i f f i c u l t and numerical methods g e n e r a l l y have t o be used. A s p e c i a l f o r m o f a n o n l i n e a r system i s t h e autonomous system, one i n w h i c h time, t, does n o t appear e x p l i c i t l y . The e q u a t i o n f o r an autonomous system i s o f t h e form
h
reduced t o a system o f two f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s and a
s o l u t i o n d e t e r m i n e d i n t h e phase plane (x,; p l a n e ) . The f o l l o r i n g methods
w h i c h w i l l b e d i s c u s s e d a p p l y o n l y t o l i n e a r systems, and cannot be a p p l i e d
t o a non-1 i n e a r system. Sel f - e x c i t e d o s c i l l a t i o n s come a b o u t as a r e s u l t o f t h e m o t i o n i t s e l f . F l u t t e r o f a n a i r c r a f t w i n g i s a c l a s s i c a l example. I n t h e s i m p l e s t form t h e m o t i o n i s i n d u c e d by an e x t e r n a l e x c i t a t i o n f o r c e w h i c h i s some f u n c t i o n o f t h e v e l o c i t y , f(;); i.e. f ( i ) may be 1 in e a r o r n o n l i n e a r i n
i.
R e a r r a n g i n g e q u a t i o n ( 5 ) i t c a n b e s e e n t h a t t h e a p p a r e n t damping, ( ( ( i ) = c i-
f c o u l d b e n e g a t i v e , depending o n t h e n a t u r e o f f ( i ) . W i t h n e g a t i v e a p p a r e n t damping t h e ampl i t u d e o f v i b r a t i o n becomes p r o g r e s s i v e l y l a r g e r , p o s s i b l y 1 e a d i n g t o s t r u c t u r a l f a i l u r e . Note t h a t a l t h o u g h t h e s t r u c t u r a l consequences a r es
i m i l a r , s e l f - e x c i t e d o s c i l l a t i o n s a r e n o t a resonant phenomenon. Somep o s s i b i l i t i e s f o r t h e f o r m o f apparent damping a r e p r e s e n t e d i n F i g u r e 2 .
F o r f(;) =
0
( d a s h e d l i n e ) t h e u s u a l c a s e o f p o s i t i v e v i s c o u s dampingo b t a i n s ( c c o n s t a n t ) . The s o l i d l i n e p r e s e n t s a case o f n e g a t i v e
dampinq. F o r s m a l l v e l o c i t i e s , a p p a r e n t damping I$(;) i s n e g a t i v e and t h e
ampl i t u d e o f o s c i l l a t i o n w i l l increase. F o r l a r g e r v e l o c i t i e s damping
becomes p o s i t i v e and a m p l i t u d e o s c i l l a t i o n s w i l l t e n d t o a l i m i t . A t h i r d case i s shown by t h e d o t t e d l i n e , one where damping i s i n i t i t a l l y p o s i t i v e
f o r s m a l l v e l o c i t i e s , b u t becomes n e g a t i v e f o r l a r g e r v e l o c i t i e s . It .
s h o u l d be remembered t h a t n e g a t i v e damping does n o t imply t h a t energy i s b e i n g c r e a t e d i n t h e system, b u t t h a t i t i s b e i n g b r o u g h t i n t o i t from some o u t s i d e source.
The o b j e c t i v e i n any s e t o f e x p e r i m e n t a l measurements i s t o e x t r a c t
i n f o r m a t i o n f r o m measured data. Q u a n t i t i e s o f i n t e r e s t i n s t r u c t u r e
response i n c l u d e 1 oads, deformat i o n s , a c c e l e r a t i o n s , s t ra i ns, etc. The
q u a l i t y o f t h e a c q u i r e d d a t a s i g n i f i c a n t l y a f f e c t s t h e u s e f u l n e s s o f t h e
F i g u r e 2 I d e a l i z e d r e p r e s e n t a t i o n o f two examples o f " n e g a t i v e " apparent damping, s o l i d l i n e (
-
) and dotted l i n e (.. ...); and normal viscous damping, dashed 1 i ne (---).poor data. It i s most d e s i r a b l e t o have an i n t e g r a t e d approach f o r a l l phases of a v i b r a t i o n a n a l y s i s task; i .e. measurement ( i n s t r u m e n t a t i o n ) , d a t a acquisi'tion, s i g n a l processing and analysis. F i g u r e 3 il l u s t r a t e s schemat i c a l l y t h e elements t o an o v e r a l l approach t o measuri ng. V i b r a t ion o r s i g n a l a n a l y s i s i s t h e t o o l which can be a p p l i e d t o b e t t e r understand dynamic l o a d i n g o f a s t r u c t u r e . F i n a l l y i t should be remembered t h a t a l l measurement programs a r e i t e r a t i v e . Something must be known about a s i g n a l before i t can be s u c c e s s f u l l y measured. Once i n i t i a l l y measured, i t can be improved upon. I
)
I
EXTRACTION PROCESSING INTERPRETATIONr
1
4 Measuring Techniques
A1 1 measurements i n v o l v e some s o r t of t r a n s d u c e r which converts response of t h e s t r u c t u r e i n t o an e l e c t r o n i c s i g n a l which can be measured and recorded. Transducers may be d i v i d e d i n t o two basic categories; p a s s i v e ones such as s t r a i n gauges ( l o a d c e l l s , pressure c e l l s ) , displacement transducers (LVDTs, DCDTs) which r e q u i r e e x t e r n a l e x c i t a t i o n , and a c t i v e ones which s e l f-generate a s i g n a l ; i .e. electromagnetic
(seismometers o r geophones f o r v e l o c i t y ) o r p i ezo-el e c t r i c (acceleraneters ) which generate a charge. Each category o f t r a n s d u c e r and type w i t h i n
c a t e g o r i e s has i t s own c h a r a c t e r i s t i c s . The t r a n s d u c e r has t o be selected i n r e l a t i o n t o t h e v a r i a b l e which i s t o be measured. Frequency response, s e n s i t i v i t y , etc. have t o be t a i l o r e d t o t h e requiremerrt. To measure a h i g h v i b r a t i o n frequency, t h e n a t u r a l frequency o f t h e sensor must be even l a r g e r . This means i t s mass has t o be small and/or i t s s t i f f n e s s large. On t h e o t h e r hand, t o o b t a i n h i g h s e n s i t i v i t y t h e s t i f f n e s s has t o be small. A consequence o f t h i s i s t h a t h i g h frequency response sensors w i l l tend t o have l o w s e n s i t i v i t y , and v i c e versa.
A t y p i c a l frequency response curve i s shown i n F i g u r e 4 t o i l l u s t r a t e sensor behaviour. The s o l i d curve shows t h e response o f a strain-gauged a c c e l e r a n e t e r o r l o a d c e l l , t h e dashed curve t h a t o f a p i e z o - e l ' e c t r i c acceleraneter. The strain-gauged device has response down t o a zero frequency, somet imes r e f e r r e d t o as DC, whereas t h e p i ezo-el e c t r i c device i s s u b j e c t t o u n d e r s h o o t when t h e low frequency c u t - o f f , fL, becanes t o o h i g h r e l a t i v e t o t h e frequency range o f t h e s i g n a l , fa t o f b i n F i g u r e 4. On t h e o t h e r hand i f t h e h i g h frequency l i m i t , fH, i s exceeded b o t h types o f transducers are s u b j e c t t o r i n g i n s , i.e. h i g h amplitude resonant response. P i ezo-el e c t r i c sensors have almost no damping so t h e resonant amplitudes can become very large.
D i g i t a l Processing
The f i r s t stage o f d i g i t a l processing i s sampling. A continuous analogue s i g n a l i s sampled a t a p a r t i c u l a r t i m e i n t e r v a l , ~ t , and t h e amplitude a t t h a t t i m e i n s t a n t determined. The e r r o r i n amplitude d i g i t i z i n g i s a f u n c t i o n o f t h e nunber o f
-
b i t s ( b i n a r y d i g i t s ) i n t h e analogue t o d i g i t a l conversion system. For a s i n g l e sample, represented asFREQUENCY
F i g u r e 4 Frequency response f o r various types o f transducers, s o l i d l i n e (
1
f o r strain-gauged a c c e l e r m e t e r , dashed l i n e (---) p i e z o - e l e c t r i c acceleraneter.a b i n a r y nunber w i t h N b i t s , t h e e r r o r , e, i s
z - ~ / z .
Normal l y , w i t h 10 o r 12 b i t systems t h i s r e s o l u t i o n e r r o r i s i n s i g n i f i c a n t (1/2048 o r 1/8192), however i f t h e f u l l range i s not being used i t can become q u i t e apparent. The f r e q u e n c y o f sampling, fS = l / h t , i s t h e n e x t c o n s i d e r a t i o n . I n o r d e r t o preserve t h e frequency content o f t h e s i g n a l i t should be sampled a t l e a s t a t t w i c e t h e maximum s i g n i f i c a n t frequency o f t h e s i g n a l , fmax. This i s known as t h e Nyquist c r i t e n o n . I n p r a c t i c e a s l i g h t l y h i g h e r sampling frequency i s usedI f sampling i s not done a t a s u f f i c i e n t l y h i g h r a t e , an a l i a s i n g e r r o r occurs, t h a t i s a s i g n a l appears t o have a frequency which i s lower t h a n i t r e a l l y i s . An example o f a l i a s i n g i s shown i n F i g u r e 5. fmax here i s 10 Hz, b u t by s a m p l i n g a t f S = 8 Hz, b t = 0 . 1 2 5 ~ ~ t h e a p p a r e n t frequency becomes 2 Hz. It would appear t h a t sampling a t a h i g h e r r a t e would e l i m i n a t e a l i a s i n g ; however, i n p r i n c i p l e , a l l t h i s does i s move up t h e frequency t h r e s h o l d f o r which i t might occur. The s o l u t i o n t o c o n t r o l l i n g a l i a s i n g e r r o r s i s t o i n t r o d u c e an a n t i a l i a s i n g f i l t e r between t h e s i g n a l source and t h e d i g i t i z e r . This i s a sharp low pass f i l t e r which r e j e c t s s i g n a l components w i t h f r e q u e n c y components g r e a t e r t h a n f m a X . It i s almost impossible t o c o r r e c t f o r t h e e f f e c t o f a l i a s i n g once i t has occurred.
0 0.2 0.4 0.6 0.8 1 .O TIME, sec
I
At2 = 0.125 SeC At ,= 0.025 SeCFREQUENCY
F i g u r e 5 R e p r e s e n t a t i o n o f a l i a s i n g i n t h e t i m e domain and frequency doma i n , sampl i ng f r e q u e n c y
,
f = 8 Hz; a c t u a l frequency 10 H z ,s a1 i a s e d frequency, 2 Hz.
Transform
Operatiom
An i d e a l i z a t i o n o f t h e l o a d i n g s i t u a t i o n f o r dynamic i c e / s t r u c t u r e i n t e r a c t i o n i s shown s c h e m a t i c a l l y i n F i g u r e 6. It i s p o s s i b l e t o measure a r e a c t i o n o r response f o r c e on t h e foundation, r ( t ) , o r response a c c e l e r a t i o n ? ( t ) a t some p o i n t i n t h e s t r u c t u r e remote from t h e i c e i n p u t f o r c e f ( t ) . T h i s i s what t h e s t r u c t u r e d e s i g n e r requires. I n i c e mechanics, however, we a r e i n t e r e s t e d i n knowing t h e i c e i n p u t force, f ( t ) , and t h e f a c t o r s which i n f l u e n c e it. This f o r c e i s o f t e n awkward o r
d i f f i c u l t t o measure. The n a t u r e o f t h e response, r ( t ) , i s dependent upon t h e dynamic c h a r a c t e r i s t i c s o f t h e s t r u c t u r e and can be r e l a t e d t o t h e f o r c i n g f u n c t i o n , f ( t ) , through t h e c o n v o l u t i o n i n t e g r a l o f t h e impulse response f u n c t i o n o f t h e s t r u c t u r e , h ( t )
,
FOU NOATION F i g u r e 6 Schematic o f i c e l o a d i n g o f a p i e r .
To e s t a b l i s h t h e r e l a t i o n between t h e f o r c e and t h e response, i t i s more convenient t o t r a n s f o r m t h e above r e l a t i o n i n t o t h e frequency domain:
where w i s t h e c i r c u l a r frequency; R, F and H a r e a l l complex f u n c t i o n s i n t h e frequency domain f r o m w h i c h expressions f o r ampl i t u d e and phase can be extracted. The frequency response f u n c t i o n , H(w), can be d e r i v e d from measured c h a r a c t e r i s t i c s o f t h e system. T h i s i s done by performing a " p l u c k i n g " o r s t e p u n l o a d i n g experiment (e.g. Haynes, 1986) f r o m w h i c h t h e n a t u r a l frequency, f and damping r a t i o ,
c,
can be d e t e n n i ned. Once t h e s en
'
f a c t o r s a r e k n w n f o r t h e s i n g l e degree o f freedom system d e s c r i b e d by E q u a t i o n (2), t h e ampl i t u d e o f t h e frequency response f u n c t i o n , a l s o r e f e r r e d t o as t h e " t r a n s f e r function", can be c a l c u l a t e d from t h e f o l l w i n g expression:
Note t h a t t h e r e i s a l s o a f u n c t i o n f o r t h e phase angle. 176
The F o u r i e r J t r a n s f o r m method p r o v i d e s a means f o r c o n v e r t i n g f u n c t i o n s from t h e t i m e domain t o t h e frequency domain and v i c e versa. When s i g n a l s a r e recorded i n d i g i t a l f o r m a t equal t i m e i n t e r v a l s , At, t h e d i s c r e t e F o u r i e r t r a n s f o r m o f a t i m e s e r i e s x ( n A t ) o f N p o i n t s i n t o a f u n c t i o n X(nAf) i n t h e frequency domain i s
f o r n = 0,1,2,... N-1. I n a s i m i l a r manner, t h e i n v e r s e d i s c r e t e F o u r i e r t r a n s f o r m i s
I n a s e t o f experiments t h e t i m e s e r i e s o f t h e response, i n t h i s case t h e f o r c e , r ( t ) , i s measured. The measured response f u n c t i o n c o u l d a l s o be a t i m e s e r i e s o f t h e a c c e l e r a t i o n , ? ( t ) . The t i m e s e r i e s response f u n c t i o n , r ( t ) , i s converted i n t o t h e frequency domain, R(w), by a p p l y i n g t h e d i s c r e t e F o u r i e r t r a n s f o r m t o i t ( E q u a t i o n 11). Knowing t h e amp1 i t u d e o f t h e t r a n s f e r f u n c t i o n H(w) f r o m E q u a t i o n (10) and t h e phase, t h e i c e f o r c i n g f u n c t i o n i n t h e frequency domain, F(w), i s c a l c u l a t e d by t r a n s p o s i n g E q u a t i o n ( 4 )
a g a i n remembering t h a t F, R and H a r e complex functions. The i n v e r s e d i s c r e t e F o u r i e r t r a n s f o r m ( E q u a t i o n 12) i s t h e n a p p l i e d t o F(w) t o o b t a i n t h e t i m e s e r i e s f o r t h e i n p u t i c e f o r c e f ( t ) .
The b a s i c methods f o r c a r r y i n g o u t t h i s a n a l y s i s a r e r e l a t i v e l y s t r a i g h t f o r w a r d , p a r t i c u l a r l y w i t h t h e ready a v a i l a b i l i t y o f hardware t o d i g i t i z e analogue s i g n a l s and s o f t w a r e t o p e r f o r m d i s c r e t e F o u r i e r transforms. It m s t be k e p t i n mind, however, t h a t t h e r e a r e c e r t a i n 1 im i t a t i o n s i n a c t u a l appl i c a t i o n s (Ra i n e r , 1986). These i n c l u d e sampl i ng frequency, t r a n s f o r m o f l o n g records and wrap-around o r leakage. To a v o i d " a l i a s i n g " o f a t i m e s e r i e s s i g n a l , i t must be sampled a t o r above t h e N y q u i s t frequency, as discussed e a r l i e r . To adequately d e f i n e t h e t i m e r e c o r d from a p p l i c a t i o n o f t h e t r a n s f e r f u n c t i o n , t h e sampling frequency
should be a t l e a s t s i x times, and p r e f e r a b l y t e n times, t h e h i g h e s t s i g n i f i c a n t frequency component i n t h e o r i g i n a l s i g n a l .
The d i s c r e t e F o u r i e r a n a l y s i s i s performed on samples of l e n g t h N. To e s t a b l i s h t h e sample l e n g t h N, some p r a c t i c a l considerations have t o be t a k e n . The maximum s i g n i f i c a n t frequency, fmax. has t o be selected, which e s t a b l i s h e s f S from E q u a t i o n ( 7 ) . N e x t t h e r e s o l u t i o n i n t h e frequency domain, ~ f , has t o be s p e c i f i e d . F i n a l l y t h e sample l e n g t h N i s c a l c u l a t e d f r a n
N i s then adjusted upwards o r downwards t o s a t i s f y t h e c o n d i t i o n t h a t N i s e q u a l t o 2", m b e i n g a n i n t e g e r . I f t h e N a r r i v e d a t i s t o o l a r g e , computational problems c o u l d be encountered. Some judgments have t o be made on t h e frequency r e s o l u t i o n , ~ f . Where t h e a v a i l a b l e s i g n a l l e n g t h i s s e v e r a l t i m e s t h a t r e q u i r e d f o r a n a l y s i s , Tt, i t may we1 1 be advantageous t o break t h e record i n t o a number o f s m a l l e r , o v e r l a p p i n g segments of t h e r e q u i r e d length. The r e s u l t s from t h e segments a r e averaged t o o b t a i n t h e frequency c h a r a c t e r i s t i ' c s o f t h e whole record. The d i s c r e t e F o u r i e r t r a n s f o r m i s a c i r c u l a r f u n c t i o n which can be viewed as r e p e a t i n g i t s e l f each N set o f points. I f t h e end and s t a r t values o f t h e record a r e not t h e same, a f i c t i t i o u s impulse i s imposed on t h e s i g n a l which r e s u l t s i n
i t s reappearance a t t h e beginning, a phenomenon known as wrap-around. The e x p l a n a t i o n f o r t h i s phenanenon i s t h e leakage o f energy a t frequencies t o e i t h e r s i d e of t h e a c t u a l frequency. Windowing i s a method o f s e l e c t i v e f i l t e r i n g which reduces t h e e f f e c t o f leakage. See f o r example Braun
(1986). The minimum value o f ~f i s c o n t r o l l e d by t h e t o t a l s i g n a l l e n g t h a v a i l a b l e f o r a n a l y s i s , Tt, according t o t h e r e l a t i o n
If t h e s i g n a l l e n g t h i s t o o s h o r t , ~f becomes l a r g e and r e s o l u t i o n i s l o s t i n t h e frequency domain. I n d i v i d u a l frequency components m i g h t not be d i s t i n g u i s h e d . Also, t h e amplitudes o f peaks i n t h e frequency domain a r e attenuated. This i s r e f e r r e d t o as a b i a s e r r o r .
-
Bias e r r o r s can be recognized by decreasing t h e bandwidth bf; i.e. i n c r e a s i n g N and r e p e a t i n g t h e a n a l y s i s .The e f f e c t o f some o f t h e above noted e r r o n p l u s o t h e r problems which can be encountered i n t h e measurement and i n t e r p r e t a t i o n o f dynamic loads
I w i l l now be i l l u s t r a t e d w i t h an example. The case i n p o i n t i s a s e r i e s of
p h y s i c a l model t e s t s c a r r i e d o u t by F r e d e r k i n g and Timco (1988) t o measure i c e loads on a r i g i d s t r u c t u r e on a compliant foundat ion. The experimental s e t up a1 lowed t h e s t i f f n e s s o f t h e foundation, k, t o be s e l e c t e d a t -200
1
I I I I II
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 TIME,s
0 5 10 15 2 0 2 5 3 0 F R E Q U E N C Y , f , H zF i g u r e 7 Time s e r i e s o f measured response force, r ( t ) , f r u n a step unloading t e s t and i t s t r a n s f o r m i n t o t h e frequency domain, ~f =
0.3 Hz.
predetermined values. Damping r a t i o , 5 , v a r i ed w i t h s t i f fnes s, b u t c o u l d n o t be c o n t r o l l e d . F i g u r e 6 approximates t h e t e s t arrangements. To o b t a i n t h e n a t u r a l f r e q u e n c y , fn, o f t h e s t r u c t u r e , a s t e p unloading t e s t was c a r r i e d out. The t i m e s e r i e s o f t h e response l o a d and i t s conversion i n t o t h e frequency domain f o r one p a r t i c u l a r case i s shown i n F i g u r e 7. Here t h e frequency i n t e r v a l ~f was 0.3 Hz. A n a t u r a l frequency, fn, of 17.6 Hz and a r e l a t i v e damping, C, o f 0.025 was c a l c u l a t e d . Determining t h e n a t u r a l frequency and r e l a t i v e damping from step unloading t e s t s , and assuming a l i n e a r system, an i d e a l i z e d t r a n s f e r f u n c t i o n can be determined f o r each case u s i n g Equation (10). The i d e a l i z e d t r a n s f e r f u n c t i o n s f o r f i v e d i f f e r e n t f o u n d a t i o n s t i f f n e s s e s a r e i l l u s t r a t e d i n F i g u r e 8. Note t h a t t h e value o f t h e s i n g l e degree o f freedom t r a n s f e r f u n c t i o n becomes l e s s t h a n one once a p a r t i c u l a r value o f frequency i s exceeded. Since t h e t r a n s f e r f u n c t i o n i s deconvolved w i t h t h e frequency domain response f u n c t i o n (Equation 13), i t can be seen t h a t t h e amplitude o f h i g h e r frequency components would be arnplif i
ed.
P r a c t i c a l c o n s i d e r a t ions Suggest t h i s i s not t h e case. A s o l u t i o n i s t o a p p l y a c u t - o f f frequency beyond which t h e amplitude o f t h e t r a n s f e r f u n c t i o n i s taken t o be one. This p o i n t i s n o r m a l l y about 4/3 o f t h e n a t u r a l frequency, see F i g u r e 8. An5
10
15
20
25
3 0
35
4 0
45
5 0
F R E Q U E N C Y .
H z
F i g u r e 8 Amplitude o f t r a n s f e r f u n c t i o n , H(u), f o r f i v e d i f f e r e n t foundat i o n s t i f fnes ses
,
k.1
a l t e r n a t i v e c u t o f f frequency o f 36 Hz was applied t o t h e two h i g h e r s t i f f n e s s foundat ions since i n f l u e n c e o f t h e c a r r i age t o w i n g system may s t a r t t o i n t r u d e a t t h i s p o i n t .
TO examine t h e e f f e c t o f t h e sampling frequency, f S , a 6 second l o n g r e c o r d o f a t i m e s e r i es o f measured response, r ( t ) , was selected. The minimum band width, hfmin, which can be d e r i v e d from t h e sample i s 0.17 Hz. The o r i g i n a l sampling was done a t 100 Hz which, f r a n Equation (7), i n d i c a t e s a maximum s i g n i f i c a n t frequency o f 40 Hz. I n t h i s case t h e n a t u r a l frequency o f t h e s t r u c t u r e was 9 Hz and a c u t - o f f frequency of 12.5 Hz (see F i g u r e 8 ) was used i n t h e a p p l i c a t i o n o f t h e t r a n s f e r f u n c t i o n i n a r r i v i n g a t t h e i c e i n p u t o r "compensated" force, f ( t ) . F i g u r e 9 ( a ) i s t h e measured response f o r c e t i m e s e r i e s . F i g u r e 9 ( c ) i s i t s t r a n s f o r m i n t o t h e frequency domain. The peak frequency i s 3.3 Hz. A p p l i c a t i o n of t h e t r a n s f e r f u n c t i o n f o r t h e
9
Hz s t r u c t u r e (Equation 13) y i e l d s t h e "compensatedn frequency spectrum o f F i g u r e 9(d). Note t h a t t h e energy a t t h e 9 Hz n a t u r a l frequency has been removed and frequency of t h e peak energy has been reduced t o 2.5 Hz. The r e s u l t i n g "compensated" i c e i n p u t f o r c e , f ( t ) , i s presented i n F i g u r e 9(b). Although not presented here, t h e a n a l y s i s was a l s o a p p l i e d t o a 33 second l o n g sample f r a n t h e same t e s t . I n t h i s case t h e minimun frequency band w i d t h i s 0.03 Hz and t h e peak frequency was 3.8 Hz. The frequency spectrum had much more r e s o l u t i o n t h a n t h a t o f Figures 9(c,d) and had h i g h e r peaks by about 20%. This i s evidence o f some bias e r r o r due t o u s i n g a s h o r t record. F i g u r e 10 i s t h e same s i g n a l as t h e one presented i n F i g u r e 9, b u t i n t h i s case sampled a t a frequency o f 20 Hz. Figure 10(a) i s t h e measured response f o r c e sampled a t 20 Hz. F i g u r e 1 0 ( c ) i s i t s t r a n s f o r m i n t o t h e frequency domain. It can be seen t h a t t h e peak frequency i s now about 4 Hz, an i n c r e a s e f r a n t h e 3.3 Hz determined a t t h e h i g h e r sampling frequency. Also t h e magnitude of t h e power s p e c t r a l d e n s i t y has increased. This c o u l d be due t o a l i a s i n g of t h e frequency components g r e a t e r t h a n 20 Hz, which would i n c r e a s e t h e power spectra1 d e n s i t y a t lower frequenci es. A p p l i c a t i o n of t h e t r a n s f e r f u n c t i o n f o r t h e 9 Hz s t r u c t u r e y i e l ds t h e "compensated" frequency spectrum of F i g u r e 10(d). The r e s u l t i n g i c e i n p u t f o r c e t i m e s e r i e s , f ( t ) , i s presented i n F i g u r e lO(b).F i g u r e 9 Time and frequency domain records of measured response f o r c e ,
r ( t ) , and i c e input f o r c e , x ( t ) , f o r sampling a t 100 Hz; 9 Hz
F i g u r e 10 Time and frequency domain r e c o r d s o f measured response f o r c e , r ( t ) , and i c e i n p u t f o r c e , x ( t ) , f o r sampling a t 20 Hz; 9
Hz
TIME. t .
s
a ) TIME INTERVAL 7 2 . 5 TO 9 0 . 5s
TIME, 1 ,s
b ) TIME INTERVAL 70.5 TO 8 8 . 5s
F i g u r e 11 Example of wrap-around e f f e c t r e s u l t i n g f r a n a s l i g h t s h i f t i n t h e t i m e i n t e r v a l , Tt, s e l e c t e d f o r a n a l y s i s .One o f t h e problems discussed e a r l i e r was t h a t o f wrap-around. This i s i l l u s t r a t e d i n F i g u r e 11, which shows t h e e f f e c t o f t a k i n g a t i m e s l i c e , f o r a n a l y s i s purposes, f r a n d i f f e r e n t i n t e r v a l s i n t h e t i m e s e r i e s . I n F i g u r e l l ( a ) , i n which t h e s t a r t and end values of t h e response force, r ( t ) , a r e very s i m i l a r , t h e i c e i n p u t f o r c e t i m e s e r i e s , f ( t ) , i s v i r t u a l l y
2
i d e n t i c a l t o r ( t ) . I n F i g u r e l l ( b ) t h e s t a r t and end v a l u e s o f r ( t ) a r e v e r y d i f f e r e n t and t h i s shows up i n t h e magnitude of t h e s t a r t and end o f t h e i c e i n p u t f o r c e t i m e s e r i e s , f ( t ) . A p p l i c a t i o n of windowing would have reduced t h i s e f f e c t
.
An i n t r o d u c t i o n t o t h e t o p i c o f dynamic s i g n a l measurement and a n a l y s i s has been made i n t h e c o n t e x t o f t h e behaviour o f f l e x i b l e s t r u c t u r e s s u b j e c t t o i c e loading. The b a s i c equations d e s c r i b i ng t h e f o r c e s and motions o f a v i b r a t o r y system have been presented and t h e associ ated terminology defined. Many commerci a1 programs a r e a v a i l a b l e f o r d o i n g d i s c r e t e F o u r i e r transforms. I n u s i n g them i t i s important t o understand t h e i r l i m i t a t i o n s . The techniques o f d i g i t a l s i g n a l a c q u i s i t i o n and processing a r e considered, and some of t h e e r r o r s and p i t f a l l s which m i g h t be encountered i n p r a c t i s e have been i 1 lu s t r a t e d .
Recomwndat ions
Since most i c e l o a d i n g events a r e o f a random nature, t h e r e s u l t s f r u n them s h o u l d be t r e a t e d i n a p r o b a b i l i s t i c fashion. This i s t h e approach which i s being taken i n new codes f o r environmental loads on offshore s t r u c t u r e s , eg. t h a t o f t h e Canadian Standards Associ a t ion. I n examining p u b l i s h e d d a t a where dynamics might be a f a c t o r , c a u t i o n should be e x e r c i sed i n i n t e r p r e t a t ion, u n l e s s t h e measurement and a n a l y s i s methods a r e explained.
The f o l l o w i n g check l i s t i s suggested i n p l a n n i n g and executing a f i e l d o r l a b o r a t o r y measurement program:
i ) Ensure t h a t a l l transducers, s i g n a l c o n d i t i o n e r s and analogue t o d i g i t a l converters have s a t i s fa c t o r y response c h a r a c t e r i s t i c s over t h e expected frequency range.
i
i )
P r e s e l e c t t h e maximum s i g n i f i c a n t s i g n a l frequency, fmaX. This e s t a b l i s h e s t h e sampling frequency, fS,i i i ) I n t r o d u c e an a n t i - a l i a s i n g f i l t e r b e f o r e t h e i n p u t t o t h e analogue t o d i g i t a l c o n v e r t e r t o r e j e c t a l l s i g n a l components w i t h frequencies g r e a t e r t h a n fmax.
i v ) E s t a b l i s h t h e m i n i m m number o f samples, N, r e q u i r e d i n t h e record,
where hf i s t h e r e q u i r e d frequency r e s o l u t i o n i n t h e frequency domain. Note t h a t N must meet t h e requirement t h a t i t i s some p w e r o f 2; i.e. 2, 4, 8, 16 o r 32...
v ) V e r i f y t h a t t h e s i g n a l l e n g t h a v a i l a b l e f o r a n a l y s i s , Tt, meets t h e requirement f o r t h e frequency r e s o l u t i o n
v i ) Run a t e s t w i t h t h e above f a c t o r s and examine t h e r e s u l t s t o see i f t h e y a r e reasonable. Re-adjust t h e sampl i n g frequency and r e p e a t t h e a n a l y s i s t o see if t h e r e i s any change i n t h e r e s u l t s . Remember t h a t i t i s an i t e r a t i v e process.
The above steps a r e n o t e x h a u s t i v e b u t d o p r o v i d e some guidance i n dyanamic measuring and a n a l y s i s program.
References
Braun, S. ed., 1986. M e c h a n i c a l S i g n a t u r e A n a l y s i s , t h e o r y and a p p l i cation, Academic Press, London, p. 385.
Frederking, R. and Timco. G.W. 1988. I c e Loads on a R i g i d S t r u c t u r e w i t h a Compl i a n t Foundation, Proc. POAC '87, Fairbanks, v. 3, pp. 393-402.
Haynes, F.D. 1986. V i b r a t i o n a n a l y s i s o f t h e Yamachiche L i g h t p i e r . I n t . J . of A n a l y t i c a l and Experimental Model A n a l y s i s , A p r i l 1986, Vol. 1, Flo. 2 , pp. 9-18.
Maattanen, M. 1988. ice-lnduced V i b r a t i o n s o f Structures--Sel f Exci t a t i o n ,
t o appear i n proceeding o f IAHR I c e Symposium 1988, Sapporo.
Rainer, J.H. (1986). A p p l i c a t i o n s o f t h e F o u r i e r transform t o t h e processing o f v i b r a t i o n s i g n a l s. National Research Counci 1 Canada, Ottawa, BRN 233, p. 24.
Sodhi, 0. 1988. Ice-Induced V i b r a t i o n s o f S t r u c t u r e s , t o appear i n proceeding of IAHR I c e Symposium 1988, Sapporo.
Thomson, W.T. (1981). Theory o f v i b r a t i o n s w i t h a p p l i c a t i o n s . London, George A1 l e n and Unwin, p. 467.
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