E R R O R S IN T H E R E P R O D U C T I O N OF V I B R A T I O N A L

L O A D S B A S E D ON T H E I R S P E C T R A L D E N S I T Y

E. T . G r i g o r ' e v , V. I , G r o n s k i i , a n d M. G. K o p o r u l i n a

UDC 621.031

Lately, s tatements concerning the organizat ion of tests on vibration stability for the reproduction of a random stat ionary excitation with given spect ra l density have been more frequently expressed. These tendencies were reflected in norms, s tandards , and guiding technical data of various departments and en terpr i ses . The excitation is reproduced using a sys tem of narrow-band f i l ters [2, 4]. A s imi lar f i l ter system also se rves for the analysis of the spect rum of the excitation obtained.

The accuracy of the reproduct ion of the random loads by the indicated method depends on the p a r a m - e te r s of the excitation apparatus and the analyzing apparatus. Fo r e lectrodynamic and electrohydraul ic force exc i te rs , the excitation apparatus is constructed ei ther on one-third octave fi l ters or on f i l ters with constant pass -band width [2]. For a one- thi rd octave band with limit frequencies w k and CCk+ l, the follow- ing relat ions are valid:

1

~ = 2 ~ ~ 1.26; r --~k--.~ 0.206. (-0 h ( (}h+l

As a resul t of this, the relative values of the bands of one-third octave f i l ters (with respect to the maximum threshold frequency), if we begin with the band of maximum frequencies, form a decreas ing geometr ic p rogress ion with f i rs t te rm 0.206 and deonominator 1/1.26. In a two-octave limit, the sequence of te rms of the given p rogress ion has the form: 0.206, 0.164, 0.130; 0.103, 0.082, 0.065. Quantiti[es of approximately the same o rde r are also given by applicable constant-width f i l ters .

If S(~') is a given spect ra l density, then the excitation apparatus constructed on band-pass f i l ters in the state ensures (with unavoidable e r ro r s ) that the d ispers ion

~~ ! ~k~ I

~ok ~o k

will be obtained in each band with boundary frequencies r k and COk+ t. The nature of the variat ion of the spec t ra l density Sx(~') of the excitation that is obtained remains undetermined. The actual spectrmn of excitation, which has a given dispers ion in a given narrow band, can be continuous or d iscre te , and also can represen t a sum of continuous and d iscre te spectra. In this case there will be no constra ints for the var ia t ions of the shape of the continuous spect rum or for the distribution of the d ispers ion of the band between separate lines of discrete (line) spectrum.

In o rder to determine to what extent the shape of the excitation spectrum affects the response of the sys tem, we assume that the excitation has a continuous spect rum with spectra l density descr ibed by the following express ion, which depends on the p a r a m e t e r 5(-1 _< 5 <- 1):

i S h (1 - - 8) + 48h6 co ~_Zn (c% .~ co <~ c00);

S~ (o~) = (I)

~Sk (1 + ~) - - 4 S ~ 6 ~ - - ~ (co0 < o, < ~ ~+,).

Transla ted from Prikladnaya Mekhanika, Vol. 10, No. 6, pp. 129-133, June, 1974. Original ar t icle submitted January 18, 1973.

�9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. IO011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

675

&

15

1,0

~5

0 r162 b 2 o ,..d(.: § l r

a

[,5

/,0

0 u4r ca o w~ v tz C

&

Fig. I

u 5K ~5

/,0 ~5

0 a" ,~ '~,~ .,-I o.,

b

/.5 /,o o,5

0 d

Here S k = D k / A O : is the mean spec t r a l density; w 0 = (w k + 0:k+l)/2 is the mean frequency; and Aw= ~'k+l -Wk is the f requency band-width.

This dependence is shown graphica l ly for 6 = 0.5 in Fig. la ; for 6 = 0 the spec t r a l density is constant in the given band and equals the mean value of S k (Fig. lb); for 6 = - 1 and 6 = 1 we obtain the l imiting case , shown in Fig. lc and d.

In p r o b l e m s dealingwith v ibra t ion s tabi l i ty , the invest igat ion of frequency regions adjacent to the m a x i m a of the ampli tude - f r e q u e n c y c h a r a c t e r i s t i c s of the s t ruc tu re being tested is of g r ea t e s t in teres t . Near resonance , when there a re no nearby natura l v ibra t ions , the a m p l i t u d e - f r e q u e n c y c h a r a c t e r i s t i c s of a sy s t em with many deg rees of f reedom can be approx imated , with sufficient accu racy , by the ampli tude - f r e q u e n c y c h a r a c t e r i s t i c s of a s y s t em having a single degree of f reedom. We l imit ou r se lves to the inves - t igation of the s y s t e m desc r ibed by the different ia l equation

re'b; + q~ + ~y = x (0,

where the externa l exci ta t ion x(t) has spec t r a l densi ty Sx(r de te rmined by Eq. (1).

The modulus of the t r a n s f e r function cor responding to this equation can be wri t ten in the fo rm

l

@ ( ~ ) = 1 [ 1 - f - ( ~ - - 2 / 0 2 + 0 ' ] - ~ , ~ ] (2)

where 0 = w / ~ c is the d imens ion less frequency; cc c = ( c / m is the natura l frequency of the sys tem; and e = ~q/x cr6-~m is the dec remen t of the natura l v ibrat ions .

The spec t r a l densi ty Sy(~) of the response is de te rmined by the equation [3]

Sy (o~) = l r (r (%

and the d i spe r s ion of the response in the invest igated f requency band is de te rmined by the equation

cob+ ]

D~ (6) = f ] �9 (to) 1%x (to) do). oa k

Substituting Eqs. (1) and (2) into this equation, we obtain, a f t e r e l emen ta ry t r ans fo rma t ions ,

S~t%[( 480,, , 48 ( A 2 _ A . ) / 4800' ] Du(8)= - - ~ 1 - -6 - - - - h -6 - . )A x -r---SO +tl +6+-~-~-)A,J.

(3)

Here we have introduced the notation:

Oo Oo O= 02

A~= -b-; A s = - - ; A3= - - ; A~= --b- ; o;

676

-O'b O 0O4 0.08 0.]2 036 e2o d8

q

8

-0,2

O,92 025 I

Fig. 2 Fig. 3

--,j t \ x

.I , !

l, g4 1,03 oo

x

b = 1 + ~ 2 +

- - _ _ - - (DO , O~ c~ Oo--~ 0 o - ~ - , AO----O~--O x. Oc " JOe ' c "

I t is adv i sab le to c o m p a r e the d i s p e r s i o n of the r e sponse Dy(5) for va r ious f o r m s of input exc i ta t ion with the d i s p e r s i o n Dy(0) of the r eac t ion for exc i ta t ion with constant s p e c t r a l dens i ty S k. The resu l t ing

devia t ions wi l l be e s t i m a t e d by the r e l a t i ve e r r o r

Dy(O) - -Dv(6 )=6140~ . (A~- -Aa)+4(A4- -A~) ] ~1 = D v (0) A0 (At + A~) - - 1 �9

To ca lcu la te the e r r o r ,?, we va r i ed the following quant i t ies : the magni tude A0 of the r e l a t ive f r e - quency band-width , the devia t ion 0 o of the mean f requency ~0 of the exc i ta t ion f rom the na tu ra l f requency ~'e of the s y s t e m , and the d e c r e m e n t e of the na tu ra l v ib ra t ions of the sys t em. The p a r a m e t e r 5, d e t e r - mining the shape of the curve of the s p e c t r a l dens i ty of exc i ta t ion was a s s u m e d equal to uni ty, s ince fo r o ther va lues , the e r r o r i s de t e rmined by mul t ip ly ing the given quant i t ies by 6 (since ~ and 5 a r e p r o p o r - t ional) . The functions obtained a r e shown in the f igures .

In Fig. 2 we p r e s e n t the dependence of ~ on A0. F o r cu rves 1, 2, and 3, we have, r e s p e c t i v e l y , 00 = 1, 0.9875, 0.975, and the d e c r e m e n t ma in ta ins the value e = 0.05. F o r cu rves 3, 4, and 5, the d e c r e - ment e = 0.05, 0.1, 0.2, r e s p e c t i v e l y , and 0 o ma in ta ins the value 0.975.

The dependence of 77 on 00 for A0 = 0.1 and e = 0.05 (curve 1) and A0 = 0.05 and e = 0.05, e = 0.1, e = 0.2 (curves 2, 3, and 4, r e spec t ive ly ) a r e shown in Fig . 3.

In Fig. 4 we give the dependence of ~ on ~ for A0 = 0.05; for cu rves 1 arid 2, the p a r a m e t e r 00 = 1 and 00 = 0.975, r e s p e c t i v e l y .

F r o m the poss ib le v a r i a n t s of exc i ta t ions with a n a r r o w - b a n d line spec t rum we c o n s i d e r [1] an e x c i - ta t ion composed of segments of sine c u r v e s of f requeney cc 0 and ampl i tude a = f2T)-k, containing N p e r i o d s and having ze ro o rd ina t e s at the in i t i a l and f inal points . The phase of the s ine curve at the end of the p r e - ceding segment d i f fe r s by ~r f rom the phase at the beginning of the following segment .

We wi l l c o m p a r e the d i s p e r s i o n Dy of the r e s p o n s e of the s y s t e m for such an exc i ta t ion with the d i s - p e r s i o n Dy 0 of i ts r e s p o n s e for a s inuso ida l exc i ta t ion with f requency o: 0 and ampl i tude a. The dependence of the r e l a t i ve e r r o r ~ = ( D y 0 - D y ) / D y 0 of the d i s p e r s i o n of the r e sponse on the number N of p e r i o d s in the s egmen t s of the sine c u r v e s is given in Fig . 5: for c u r v e s 1, 2, and 3, the p a r a m e t e r 00 has the va lues 1, 0.9875, 0.975, r e s p e c t i v e l y , and the d e c r e m e n t ma in ta ins the value e = 0.05; for cu rves 3 , 4, and 5, the d e c r e m e n t e = 0.05, e = 0.1, e = 0.2, r e s p e c t i v e l y , and 00 ma in t a in s the value 0.975.

The dependences p r e s e n t e d he re give a r e p r e s e n t a t i o n of the o r d e r of pos s ib l e e r r o r s in the d i s p e r - s ion of the r e sponse for the reproduc t ion of the loading of s t r u c t u r e s based on the s p e c t r a l dens i ty using a sy s t em of n a r r o w - b a n d f i l t e r s . The r e l a t i ve e r r o r can r each s e v e r a l tens and even hundreds of pe rcen t , with i t being equal ly p robab le that the r e q u i r e d loading wi l l be too high o r too low. F o r o ther condi t ions being equal , the p robab le e r r o r s sha rp ly i n c r e a s e with d e c r e a s i n g d e c r e m e n t s of the na tu ra l v ib ra t ions of the ob jec t s being tes ted .

677

'\ 0.2

0

-0.2 ~'~

-0.40 0./ 0.2 0,3

Fig.

~5

0

- ~

-1,0

-15

-2.~

-35 0 20 40 60 8g 100 N

Fig. 5

In all c a se s , la rge e r r o r s a re probable if the natural frequency of the s t ruc ture is in the excitation frequency band; the e r r o r s will be compara t ive ly small if the natural frequency of the s t ructure is suf- ficiently distant from the excitation frequency band.

The dependence of the e r r o r in d ispers ion of the response on the rat io of the mean frequency of ex- citation to the natural frequency of the s t ruc ture is not a monotonic dependence. The dependence of the e r r o r on the relative excitation band-width is also not always monotonic. An increase (decrease) of the e r r o r occurs for an increase (decrease) of the spect ra l density in the region of the maximum of the ampl i - t u d e - f r e q u e n c y cha rac t e r i s t i c s of the s t ruc ture being tested.

In our analysis , we have postulated a s y m m e t r y in the graph of the spect ra l density of the excitation in the band being considered with respec t to its mean frequency (see Fig. 1). In cases of nonsymmetr ic spec t ra l density it is possible to obtain even m o r e sizable e r r o r s in the dispers ion of the response.

1o

2.

3.

4.

L I T E R A T U R E C I T E D

E. T. Grigor'ev and V. I. Gronskii, "Response of a mechanical system to a random disturbance," Prikl. Mekh., 9, No. 1 (1973). V. T. Koval', "Excitation apparatus for reproducing random vibrations in the investigation of re- liability," Vestnik Mashinostroeniya, No. 6 (1970). V. S. Pugachev, Theory of Random Functions and Its Application to Automatic-Control Problems [in Russian], Fizmatgiz, Moscow (1960). S. H. Crandall (editor), Random Vibration, John Wiley (1958).

678