DETERMINATION OF THE DAMPING CHARACTERISTICS OF STRUCTURES BY TRANSIENT TESTING USING ZOOM-FFT

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DETERMINATION OF THE DAMPING CHARACTERISTICS OF STRUCTURES BY

TRANSIENT TESTING USING ZOOM-FFT

R. Adams, D. Lin

To cite this version:

R. Adams, D. Lin. DETERMINATION OF THE DAMPING CHARACTERISTICS OF STRUC-

TURES BY TRANSIENT TESTING USING ZOOM-FFT. Journal de Physique Colloques, 1983, 44

(C9), pp.C9-363-C9-369. �10.1051/jphyscol:1983953�. �jpa-00223403�

JOURNAL DE PHYSIQUE

Colloque C9, supplgment au n012, Tome 44, dgcernbre 1983 page C9-363

DETERMINATION OF THE DAMPING CHARACTERISTICS OF STRUCTURES BY TRANSIENT TESTING USING ZOOM-FFT

R . D . Adams and D . X . Linr

Reader i n Mechanical Engineering, University of B r i s t o l , B r i s t o l , BS8 ITR,

U . K .

haad ad Mechanical Engineering I n s t i t u t e , Sian, China, and a t Department o f Mechanical Engineering, University of B r i s t o l , B r i s t o l , BS8

I

TR,

U . K.

Re'sum6 - Nous d i s c u t o n s une n o u v e l l e t e c h n i q u e pour d e t e r m i n e r l ' a m o r t i s s e m e n t des s t r u c t u r e s . C e t t e t e c h n i q u e e s t fonder s u r l ' e s s a i t r a n s i t o i r e 'Zoom-FFT'.

Nous de'crivons l e p r i n c i p e de c e t t e e s s a i e t nous donnons l e s r e s u l t a t s e x p e r i m e n t a l s pour quelques mate'riaux.

A b s t r a c t

-

A new t e c h n i q u e of determining t h e damping p r o p e r t i e s of s t r u c t u r e s i s d i s c u s s e d . The t e c h n i q u e i s based on t h e t r a n s i e n t t e s t t e c h n i q u e u s i n g 'Zoom1-FFT. The p r i n c i p l e o f t h i s t e c h n i q u e is d e s c r i b e d and t h e e x p e r i m e n t a l r e s u l t s a r e given f o r s e v e r a l m a t e r i a l s .

I . INTRODUCTION

For many y e a r s , e x p e r i m e n t a l d e t e r m i n a t i o n of t h e dynamic c h a r a c t e r i s t i c s of s t r u c - t u r e s has been based on s t e a d y - s t a t e d i s c r e t e frequency methods, i n which t h e n a t u r a l f r e q u e n c i e s and damping r a t i o s a r e d e r i v e d from v e c t o r diagrams /I/. This method, l i k e a l l s t e a d y - s t a t e methods, i s t e d i o u s and time-consuming and cannot be r e a d i l y a p p l i e d o u t s i d e t h e l a b o r a t o r y ; a l s o , c o n s i d e r a b l e expense may be i n c u r r e d i f a l e n g t h y t e s t programme is r e q u i r e d i n o r d e r t o i n v e s t i g a t e t h e c h a r a c t e r i s t i c s of a b u i l t - u p s t r u c t u r e . The q u a s i - s t e a d y - s t a t e t e s t method may reduce some t e s t t i m e ; h e r e , t h e e x c i t a t i o n frequency i s c o n t i n u o u s l y v a r i e d through t h e frequency r a n g e of i n t e r e s t . However, e x c e p t w i t h very slow sweep r a t e s , t h e measured v a l u e s of n a t u r a l f r e q u e n c i e s and damping r a t i o s d e r i v e d from such a t e s t a r e i n a c c u r a t e because t h e assumption t h a t t h e system response a t t a i n s t r u e s t e a d y - s t a t e l e v e l s i s n o t u s u a l l y v a l i d .

I t i s w e l l known t h a t t h e frequency r e s p o n s e of a s t r u c t u r e may b e d e r i v e d from t h e response t o a u n i t impulse. Even though t h e u n i t impulse i s u n a t t a i n a b l e i n p r a c t i c e , t h e concept o f impulsive e x c i t a t i o n l o g i c a l l y l e a d s toward e x c i t a t i o n by a s i n g l e p u l s e o f simple geometric shape and s h o r t d u r a t i o n . I n t h i s way, t h e p u l s e c l o s e l y approximates t o an impulse. White i n d i c a t e d t h a t i f t h e p u l s e i s of s h o r t d u r a t i o n i n comparison t o t h e s h o r t e s t p e r i o d of t h e system under t e s t , t h e n only t h e response of t h e system need by a n a l y s e d / 2 / . I n view of t h i s , t h e method of s i n g l e p u l s e e x c i t a t i o n o r t h e s o - c a l l e d "Hammer t e s t t ' i s a u s e f u l technique.

However, t h e r e a r e some r e s t r i c t i o n s when u s i n g t h e hammer t e s t , s i n c e l i t t l e con- t r o l can b e e x e r c i s e d over t h e range o f f r e q u e n c i e s e x c i t e d . On t h e o t h e r hand, when t h e energy l e v e l of t h e hammer i s r e l a t i v e l y low, o t h e r e x c i t a t i o n methods

/3,4/ may b e used.

The advent o f a c c u r a t e and f a s t a n a l o g u e - t o - d i g i t a l c o n v e r t e r s h a s p e r m i t t e d t h e use of t r a n s i e n t t e s t methods i n which t h e time-domain r e s p o n s e t o a n impulse i s con- v e r t e d t o t h e frequency domain by u s i n g t h e F a s t F o u r i e r Transform t e c h n i q u e . During t h e l a s t twenty y e a r s , t r a n s i e n t t e s t i n g t e c h n i q u e s have been d e v e l o p e d ' i n a wide v a r i e t y of methods f o r handling r e s p o n s e d a t a (2-6/. Most of t h e s e r e q u i r e a g r e a t d e a l o f mathematical p r o c e s s i n g and s t i l l i n c u r g r e a t expense i n computing, w h i l e t h e r e s u l t s may n o t b e o b t a i n e d o u t s i d e a l a b o r a t o r y . However, t h e advent of Zoom-FFT p r o v i d e s a method by which i n c r e a s e d r e s o l u t i o n can be o b t a i n e d w i t h i n a s m a l l e r p a r t of t h e frequency range. The Zoom s p e c t r a , a s shown i n F i g . 1, a r e of h i g h r e s o l u t i o n s o t h a t t h e r e s o n a n t f r e q u e n c i e s and damping r a t i o s may b e o b t a i n e d d i r e c t l y and c l e a r l y w i t h o u t t h e need f o r a c c e s s t o l a r g e computers. T h e r e f o r e , Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983953

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LIN X 9.18E-2

F i g . I The Zoom spectrum of an off-axis laminated beam of E-Glass

(e -

^45')

t h e method p r e s e n t e d i n t h i s paper i s p r a c t i c a b l e f o r wide a p p l i c a t i o n , such a s i n monitoring and diagnosing machine f a u l t s / 7 / and i n n o n - d e s t r u c t i v e t e s t i n g /8/.

I n t h i s p a p e r , t h e a p p l i c a t i o n of t h e Zoom-FFT method and t h e hammer t e s t t o d e t e r - mine t h e damping c h a r a c t e r i s t i c s o f s t r u c t u r e s i s d i s c u s s e d , and some r e s u l t s a r e

given f o r Glass F i b r e Reinforced P l a s t i c s (GFRP) p l a t e s and beams. F i n a l l y , a comparison i s made w i t h r e s u l t s o b t a i n e d by more c o n v e n t i o n a l methods.

11.1 L i n e a r , lightly-damped m u l t i degree-of-freedom system

Considar t h e c a s e of a l i n e a r , lightly-damped s i n g l e degree-of-freedom system e x c i t e d by a s i n g l e p u l s e

Y ( t ) = H(w) F ( t ) where

The F o u r i e r Transform of Eqn ( 1 ) i s Y(w)

=

H(w) F ( o )

where Y(w) and F(w) a r e t h e spectrum of t h e system and o f t h e f o r c e r e s p e c t i v e l y . I n t h e c a s e o f an impulse, F(w) may be c o n s i d e r e d t o b e e s s e n t i a l l y c o n s t a n t o v e r a broad frequency range.

Now i f

I F ( w ) ~

=

C

t h e n

IY(u)

1 ⁼

C I H ( W ) I

Owing t o t h e low damping, t h e frequency r e s p o n s e f u n c t i o n may b e made up of i n d i v i - d u a l modes which a r e w e l l s e p a r a t e d i n frequency, s o t h a t each mode can b e analyzed s e p a r a t e l y . Hence, i n t h e c a s e of m u l t i degree-of-freedom s y s t e m s , t h e spectrum of each mode would b e a s f o l l o w s :

Yi(w)

=

Hi(w) Fi(w) s o t h a t

Iyi(w)I

=

cIHi(w)I

where ^I -1

and

me,

is t h e ith e q u i v a l e n t mass which r e l a t e s t o t h e system parameters and i s dependent on t h e n a t u r a l frequency w ni.

ci

is t h e modal v i s c o u s damping r a t i o . For t h e a c c e l e r a t i o n r e s p o n s e , we have

where

I f t h e curve of (ai(w)

I

i n t h e r e s o n a n t r a n g e were determined e x p e r i m e n t a l l y , t h e modal v i s c o u s damping r a t i o S i and n a t u r a l frequency f n i would b e o b t a i n e d e i t h e r by u s i n g curve f i t t i n g o r by applying t h e " h a l f power p o i n t " t e c h n i q u e . Thus, t h e damping r a t i o s of each mode would be:

s p e c i f i c damping c a p a c i t y (SDC) Jli

=

4 a Si

l o g a r i t h m i c decrement 6 i

=

2 a Si

I

^{( 1 0 )}

l o s s f a c t o r q i

=

2 Si

The r e s u l t o f a normal FFT a n a l y s i s shows a d i s t r i b u t i o n of frequency from z e r o up t o t h e Nyquist frequency f N , while t h e frequency r e s o l u t i o n i s determined by t h e number of frequency l i n e s up t o f~ (normally h a l f t h e number o f t h e o r i g i n a l d a t a sample /9/.

The d e f i n i t i o n of t h e D i s c r e t e F o u r i e r Transform (DFT) i s

and

where

fk

=

k A f , and tn

= n

^~t

I f t h e r e c o r d i n g time i s T

=

N A t and t h e sampling number is N , t h e n t h e sampling

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frequency i s f s = N/T. Because of t h e u n c e r t a i n t y p r i n c i p l e /9/, t h e sampling r a t e should be g r e a t e r t h a n twice ( u s u a l l y 2.56 t i m e s ) f n . So, f N

=

fs/2.56 and t h e r e s o l u t i o n is B

=

fs/N. For example, when equipment with a 1 K t r a n s f o r m i s used ( i . e . N

=

1024 p o i n t s ) , i f f N

=

400 Hz, t h e r e c o r d i n g time w i l l be 1 s e c and t h e r e s o l u t i o n B

=

1 Hz. This IS i n s u f f i c i e n t f o r determining t h e resonance c u r v e , because t h e Af of t h e h a l f power p o i n t s i s about 1 Hz f o r a system having 5% SDC and f n

=

125 Hz. For t h i s r e a s o n , it i s d e s i r a b l e t o o b t a i n a c o n s i d e r a b l y f i n e r r e s o l u t i o n over a l i m i t e d p o r t i o n o f t h e spectrum. But by t h e u s e of t h e "Zoom-FFT"

procedure, t h i s c o n d i t i o n can now b e d e a l t w i t h adequately.

The frequency range used i n t h e Zoom-FFT method depends on t h e damping o f t h e specimen o r s t r u c t u r e and t h e c o r r e c t range can only b e found by e x p e r i e n c e . For g l a s s f i b r e r e i n f o r c e d p l a s t i c s , a frequency range of 1 3 1 t o 1 5 1 Hz was used f o r one mode ( s e e Fig. 1 ) w h i l e , f o r a carbon s t e e l beam, it was n e c e s s a r y t o reduce t h e frequency range t o 113.5 t o 114.5 Hz.

11.4 T e s t i n g arrangement

The t e s t i n g arrangements a r e a s i l l u s t r a t e d i n Fig. 2. I n Fig. 2 ( a ) t h e specimen i s suspended v e r t i c a l l y by two f l e x i b l e s t r i n g s t h a t e n a b l e s t h e specimen t o be c o n s i - dered a s b e i n g " f r e e - f r e e " . I n t h i s way, t h e n a t u r a l f r e q u e n c i e s and damping r a t i o s can b e measured mode by mode without a l t e r i n g t h e arrangement. However, owing t o t h e u s e of an a c c e l e r o m e t e r a t t a c h e d t o t h e specimen a s a pick-up, t h e e f f e c t of added mass has t o be t a k e n i n t o account. I f t h e r a t i o of attachment mass/specimen mass is l e s s t h a n 0.1%, t h i s e f f e c t can b e n e g l e c t e d . I n F i g . 2 ( b ) , t h e specimen is shown p l a c e d on sponge s u p p o r t s s o t h a t a non-contacting pick-up, such a s a capaci- t a n c e t r a n s d u c e r , i s used. I n o r d e r t o reduce e x t e r n a l e f f e c t s from t h e s u p p o r t s , t h e n o d a l l y s u p p o r t e d arrangement ( F i g . 2 ( b ) ) i s recommended. However, i n u s i n g t h i s arrangement, t h e n o d a l l i n e o f t h e specimen must be determined f i r s t , and t h e s u p p o r t s have t o b e moved f o r measuring any o t h e r mode.

Accelerometer

Charge

L

Amplifier

Fl'T Andlyser

microcomputer

Microphone

Ampltfior ^co FPT Analyaer

Sponge nodal supports

F i g . 2 Apparatus

Having s e t up t h e specimen a s i n F i g . 2 ( a ) o r ( b ) , t h e t e s t i n g procedure i s a s f o l l o w s :

The specimen i s tapped by t h e hammer, and t h e t r a n s i e n t response i s processed by t h e Zoom-FFT a n a l y s e r , s o producing t h e zoom spectrum. The microcomputer (HP-85) which is i n t e r f a c e d t o t h e FFT a n a l y z e r r e c o r d s t h e c a l c u l a t e s t h e spectrum t o o b t a i n t h e n a t u r a l frequency and damping r a t i o immediately.

111. DISCUSSION

To v e r i f y t h e t r a n s i e n t t e s t a n a l y s e d by Zoom-FFT, u n i d i r e c t i o n a l and o f f - a x i s lami- n a t e d E-glass p l a t e s and beams have been t e s t e d . The r e s u l t s a r e shown i n Fig. 3 and Table 1.

9

0 10 20 30 LO SO 60 70 80 90 pi< angle (eO)

Fig. 3 Comparison t o t h e r e s u l t s o f o f f - a x i s laminated beams of E-Glass

- - - , -

T h e o r e t i c a l v a l u e s of E and SDC r e s p e c t i v e l y .

8 O E and SDC experimental d a t a o b t a i n e d by t e c h n i q u e a s Ref. 11.

x A E and SDC e x p e r i m e n t a l d a t a o b t a i n e d by t r a n s i e n t (zoom) method Good agreement i s shown i n F i g . 3 f o r t h e r e s u l t s o f o f f - a x i s l a m i n a t e s i n E-glass beams a t a s e r i e s of a n g l e s u s i n g Zoom-FFT from t h e t r a n s i e n t t e s t and from t h e technique i l l u s t r a t e d i n Ref. 11. The o t h e r e x p e r i m e n t a l r e s u l t s (method of r e f e r e n c e 1 1 ) were obtained by s t e a d y - s t a t e methods and t h e continuous l i n e i s a t h e o r e t i c a l p r e d i c t i o n .

T e s t s were t h e n made on p l a t e s u s i n g both accelerometer and c a p a c i t a n c e t r a n s d u c e r s (Table 1 ) . I t i s seen t h a t t h e damping r e s u l t s o b t a i n e d u s i n g t h e accelerometer a r e h i g h e r t h a n t h o s e o b t a i n e d u s i n g t h e c a p a c i t a n c e t r a n s d u c e r , owing t o t h e s l i g h t a d d i t i o n a l damping of t h e accelerometer l e a d . The h i g h e r t h e mode, t h e more s i g - n i f i c a n t i s t h e e f f e c t of t h e attachment of t h e a c c e l e r o m e t e r . Also, t h e l i g h t e r t h e specimen, t h e g r e a t e r i s t h e e f f e c t of t h e attachment, which e v e n t u a l l y l e a d s t o very s e r i o u s e r r o r s . When l i g h t , beam specimens a r e t e s t e d , it i s suggested t h a t t h e a c c e l e r o m e t e r should not be used. But t h i s d i f f e r e n c e i n damping i s n o t due t o any e r r o r i n t h e Zoom-FFT method b u t simply t o t h e use of an i n a p p r o p r i a t e t r a n s - d u c e r , t h e a c c e l e r o m e t e r . When a f r e e decay method was used f o r t h e same p l a t e and

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modes of v i b r a t i o n , t h e damping v a l u e s o b t a i n e d were e x a c t l y t h e same a s when u s i n g t h e same t r a n s d u c e r and t h e Zoom-FFT method.

Furthermore, i f t h e Young's Modulus o f t h e p l a t e i s r e q u i r e d , t h i s can be obtained from t h e v a l u e s o f t h e n a t u r a l frequency p a r a m e t e r , X , which is g i v e n i n Ref. 1 0 ,

f a 2

where X

= - 6

^{( a}

⁼

p l a t e s i d e l e n g t h , t

=

p l a t e t h i c k n e s s , 0

=

d e n s i t y , t

ET

=

t r a n s v e r s e Young's modulus). The modulus E can be determined from f n . T

Table 1 The comparison of average SDC values JI and n a t u r a l f r e q u e n c i e s f n o b t a i n e d by u s i n g a c c e l e r o m e t e r and c a p a c i t a n c e t r a n s d u c e r s . Specimen is u n i d i r e c t i o n a l E-glass l a m i n a t e .

o: pick-up s i t e .

The c a s e i n which t h e t r a n s i e n t response i s l o n g e r t h a n t h e r e c o r d i n g time h a s a l s o been i n v e s t i g a t e d . Using Zoom-FFT t h e s p e c i f i c damping c a p a c i t i e s of carbon s t e e l beams are about 0.07%

-

0.14% which i s i n agreement with t h e r e s u l t s shown i n Ref.

I V . CONCLUSIONS

error of SDC

E=U

^{x l O O X}

$c

9 . 3

1 8 . 9 6

2 0 . 5 6 Mode Shape

I Fl

f i b r e d i r e c t i o n

r]

^{I '}

I n t h i s paper a method i s d e s c r i b e d of d e t e r m i n i n g t h e dynamic c h a r a c t e r i s t i c s of s t r u c t u r e s and specimens u s i n g t r a n s i e n t t e s t i n g t e c h n i q u e s and a p p l y i n g Zoom-FFT a n a l y s i s . Although i n t h i s p a p e r we have o n l y d i s c u s s e d r e s u l t s from specimens of a q u i t e s i m p l e s h a p e , such as p l a t e s and beams, t h i s method can a l s o b e a p p l i e d t o more complex s t r u c t u r e s .

REFERENCES

1. ^KENEDY

,

^{C. C.} and PANCU, C. D. P

. ^,

J o u r n a l o f A e r o n a u t i c a l S c i e n c e ,

E,

^{( 1947} )

,

603.

Using a c c e l e r o m e t e r

2. WHITE, R. G., J , Roy. Aero. Soc., (19691,

73,

1047.

3. KANDIANIS, F., J o u r n a l o f Sound and V i b r a t i o n

2,

(1971), 203.

Using c a p a c i t a n c e transducer f n c

(hz)

5 6 . 8

8 9 . 9 6

1 4 8 . 2 5 a c c . mass

p l a t e Inass

-

^{500 1}

-

^{500 1}

1

500

fna (hz)

5 6 . 1

8 9 . 6

1 4 7 . 3

$c

( X )

8 . 0 3

5 . 5 9

5 . 5 1 (%)

8 . 7 8

6 . 6 5

6 . 6 4 7

WHITE, R. 5

.,

J. Sound a n d V i b r a t i o n ,

15,

( 1 9 7 1 ) , 1 4 7 .

CLARKSON,

';.

L. a n d MERCER, C . A . , AIAA J o u r n a l ,

3,

( 1 9 6 5 ) , 2287.

CAWLEY, P. a n d ADAMS, R. D . , J . Sound a n d V i b r a t i o n ,

2,

( 1 9 7 9 ) , 1 2 3 .

HERLUFSEN, H., "Order a n a l y s i s u s i n g Zoom FFT", B. & K A p p l i c a t i o n Notes 012-81.

AL-AGHA, H. a n d ADAMS, R. D . , " D e t e r m i n a t i o n o f m u l t i p l e damage s i t e s b y m e a s u r e m e n t s o f s t r u c t u r a l n a t u r a l f r e q u e n c i e s " . l o t h World Conf. on NDT, Moscow, 1 9 8 2 .

PAPOULIS, A . , " S i g n a l a n a l y s i s " . McGraw-Hill Book Company, 1 9 7 7 , 273-276.

CAWLEY, P . a n d ADAMS, R.D., J . Composite M a t e r i a l s , 1 2 , ( 1 9 7 8 ) , 336. - ADAMS, R.D. and BACON, D . G . C . , J. P h y s . D : Appl. P h y s . ,

5,

( 1 9 7 3 ) , 27.

ADAMS, R.D., J . Sound a n d V i b r a t i o n , 2 3 , ( 1 9 7 2 ) , 1 9 9 .

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