R E S P O N S E OF A M E C H A N I C A L S Y S T E M TO A

R A N D O M D I S T U R B A N C E

E . T . G r i g o r ' e v a n d V. I . G r o n s k i i UDC 620.178.53

The fundamental equation for the analysis of l inear sys tems under random disturbances is the re la - tion [3, 4]

Sy (o~) = la) (o)) I~ G(o)) (1)

between the spect ra l density Sx(w) of the random stat ionary input disturbance x(t), the frequency response �9 (w) of the sys tem, and the spect ra l density Sy(w) of the response y(t) of the sys tem to that disturbance.

The given relation is valid in the limit as the spect ra l densit ies are defined for infinitely long p ro - cesses , i .e . , when the analytical resolving power tends to zero . The relat ion fails, however, in tests with a cant i levered beam [2]. The spect ra l density of the response of an e lectr ic beam to variants of a na r row- band s ta t ionary random disturbance such that not only the spec t ra l densit ies but also the amplitude d i s t r i -

bu t ions are prac t ica l ly identical turns out to be strongly dependent on the phase distribution of the d is tur -

bance.

It is necessary , therefore , to verify the fulfillment and co r rec tness of Eq. (1) with allowance for the prac t ica l engineering e r r o r s of the spect ra l density es t imates and the finite resolving power of the spectra l

analysis .

We the r e fo re determine the response of a l inear mechanical sys tem to an external periodic d is tur- bance consist ing of segments of sine waves having the same frequency and amplitude, but initial phase angles that differ by 7r (Fig. la). The external disturbance x(t) can be written in the form

b ( ~ t

where T O is the time interval between phase changes and is assumed to be equal to an integral number of per iods of the harmonic with frequency w 0, and b( t /T 0) is the pos i t ive- in tegra l par t of the number t /T 0.

Inasmuch as 2T 0 is the per iod of the external dis turbance, it has a fine spectrum s imi lar to a na r row- band spec t rum for a large number N. For the dispers ions D(kN) of the harmonies with frequencies w k = k27r/2T 0 we readily obtain the express ions

32N~P2 n(n~ ~..,~ = 0 . O~!x = n~ 14N 2 _ (2k -- l)'l ;

Fo r any value of N, roughly 807o of the dispers ion of the total disturbance is associated with two ha r - monics at f requencies differing f rom w0 by the amount Aw = 27r/2T 0 = w0/2N. The form of the normal ized

Dk,%

r - . . . . t

-t7 . . . . , OJ

a b ~a,

Fig. 1

Trans la ted f rom Prikladnaya Mekhanika, Voh 9, No. 1, pp. 105-109, January, 1973. Original

ar t ic le submitted July 19, 1972.

�9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o/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.

86

line spec t rum of the input disturbance for N = 20 is given in Fig. lb. This spec t rum becomes more and more narrowband as the number of per iods in the sine-wave segments is increased. For large enough N the frequency band outside which the total d ispers ion of the external disturbance harmonics is negligibly small , becoming less than the spec t r a l - ana ly t i ca l resolving power, which is p resumed to be finite. The spec t rum of the analyzed external dis turbance in this ease is indistinguishable f rom the spect rum of a sine wave of constant frequency and amplitude.

The response y(t) of the l inear sys tem is considered to be the solution of the differential equation

where we assume, to simplify the calculations, that the natural f requency of the sys tem coincides with the frequency w0 of the sine-wave segments , i .e . ,

C

% = ~/-m" The solution of Eq. (2) in the k-th interval between phase changes of the external disturbance has the

form

gk = %~ e-nt COS (mot + ak) + qo~---~

Here n = q/2m, and the paramete i ' s Xkand a k are determined by the initial conditions for each interval.

We adopt zero-valued initial conditions y(0) = 5;(0) = 0 for the f i rs t interval. The initial conditions for each subsequent interval are charac te r i zed by the displacement y and velocity !~ at the end of the p r e - ceding interval.

Determining the success ive values of the p a r a m e t e r s kkand a k for each interval beginning with the f i rs t , we obtain

%~ = (-- 1) ~+1 ~ V-~ + n ~ [2 - - 2e -~r~ + 2e -2~r" - . . . q~o

�9 .. + (~ I)~-22e -~k-2)~r0 + (~ l)~-'e-~k-unr,];

n ~- ~0 sin % = - - V ~ ' cos % = V ~ "

The process ultimately stabilized after a sufficiently long time (i. e., a sufficiently large number k of intervals). The limiting value of the parameter k k can be found by replacing the bracketed expression in Eq. (3) by the sum of the terms of an infinite geometric progression in which the first term is 2 and the denominator is -e -nT0, i.e., by the expression

Z 2 ( - - e-~r~ ~-' 2 1 + e --nr~

Retaining the index k only to designate whether the interval number is odd or even, we obtain for the l imiting values of the pa r ame te r X k

kk = (-- 1)~+

The response of the sys tem assumes the form

2P V/ g- + . ' qag (t + e -"r")

I I Y k = ( - - 1) k+l 2 P l / ~ 0 + n 2e -'~ P qo)g (1 + e -r'r") cos (%t + a) ........ q% cos % t .

Inasmuch as only the sign of the function differs in adjacent intervals , the dispers ion D of the p ro - cess y can be determined by averaging over just one interval:

To

D = vldt.

0

(3)

(4)

87

d8

aa : I \

a41 42 4ca4 aa a~ a 4 5 6 8 ~a ea Ja ao ~

Fig. 2

Substituting the value of Yk f rom Eq. (4) into the integrand and per forming the integration, we find

p2 [ 2 (I - - e -"r~ (4o~- 5o02n ~ + 3n') ]

D = 2 ~ o, 1 - - nTo(l + e-~ro)((o~ + n%~-~ ~2 ~-/)j"

The damping coefficient n in this express ion is conveniently replaced by the logari thmic decrement of the osci l lat ions e = 7r2n/co 0' and the time T O can be expressed in t e rms of the number N of periods of the sine wave between phase changes of the external dis turbance: T O = N2~v/w 0. Then

D -= 2q~r I -- l__ eN (4n~' + e~) (I 6u ~ + e2) " 2

(5)

For mechanical s t ruc tu res the decrement e usually has a value between 0.01 and 0.30, so that t e rms containing the smal l quantities e 2 and e 4 can be neglected in Eq. (5). We finally obtain

D = D o 1 (6)

Here D O = p2/2q2r is the dispers ion of the sys tem response to a sirlusoida[ disturbance of the same ampli- tude without phase changes.

It is apparent f rom Eq. (6) that the dispers ion D of the response to a disturbance compris ing sine- wave segments of opposite phases is less than the dispers ion D o of the response to a pure sinusoidal dis- turbance. The relat ive reduction of the response dispers ion is given by the equation

TI = - - D O - - 1 " eN

A graph of the relat ive reduction 7/ of the response dispers ion as a function of the product (1/2)eN is given in Fig. 2, in which a logar i thmic scale is used for the independent variable. This graph can be used to easi ly find the relat ive reduction of the d ispers ion for any combination of the decrement e of a s t ructure and the number N of per iods between phase changes of the sine wave.

We see that even for fa i r ly la rge numbers N, i . e . , for the case in which the p roces s has a fair ly nar row-banded line spect rum, the relat ive reduction in the dispers ion may amount to a very considerable value, especial ly for smal l dec rements of the oscil lat ions. For smal l decrements the quantity 7/ is close to unity. Hence the dispers ion D of the react ion to the external action in the form of segments of a sine wave with varying phase may be many t imes sma l l e r than the react ion of the sys tem to a purely sinusoidal

action of the same amplitude.

The foregoing resul t is attributable to the fact that the s teady-s ta te response of the sys tem to sinu- soidal dis turbances does not contain free oscil lat ions, whereas the lat ter are always contained in the r e - sponse of the sys tem to a disturbance in the form of sine-wave segments of different phases (in the steady state). The energy admitted into the sys tem from the external disturbance at the beginning of each new interval af ter a phase change is spent in abating the oscil lat ions that exist in the sys tem at that time out of

88

a b Fig. 3

phase with the driving oscil lat ions. Only af ter the extinction of those oscil lat ions do the amplitudes of the sys tem oscil lat ions in phase with the driving disturbance grow. Multiple repeti t ions of the indicated cycle yields a sizable reduction in the response dispersion. The s teady-s ta te responses of a sys tem to a sinu- soida[ disturbance (Fig. 3a) and to a disturbance in the form of sine-wave segments of different phases (Fig. 3b) are shown schematical ly for i l lustrat ion.

The variat ion of the dispers ions exhibits an analogous pat tern when the phases of the sine-wave seg- ments differ f rom one another by an amount not equal to ~r and when the driving frequency does not coincide with the natural f requency of the sys tem.

It may be inferred from the foregoing discussion that the problem of determining the spectral ,density (or dispersion) of the response of a mechanical sys tem from the spectra l density (or dispersion) of the in- put disturbance is i ncor rec t in the sense that a slight variat ion of the spectral density of the input d is tur - bance within the e r r o r l imits of its es t imate can be matched by a large variat ion of the spectral den,~ity and total dispers ion of the sys tem response .

Abrupt phase changes of the narrowband constituents take place in any random process , including "white noise." An example is afforded by the sudden phase changes of a narrowband p rocess produced by a noise genera tor with noisy diodes in conjunction with the heterodyne fi l ter [1]. It is important in this light to proceed with caution in implementing methods whereby the response of a sys tem to "white noise" or some other random excitation is used to obtain the dynamical cha rac te r i s t i c s and to determine the f r e - quencies, waveforms, and damping decrements of the oscil lat ions of mechanical s t ruc tures .

i,

2.

3,

4.

LITERATURE CITED

E. T. Grigor'ev and V. I. Gronskii, "Phase distribution of a random narrowband process," Trudy Dnepropetrovsk. Inst. Inzh. Zhel.-Dor. Transport, No. 114 (1970). E. T. Grigor'ev, V. I. Gronskii, I. N. Nikolaichuk, and G. L. Demishenko, "Response of a sys- tem to mechanical disturbances with different phase distributions," Prikl. Mekhan., 7, No. 6 (1971).

V. S. Pugachev, Theory of Random Functions and Its Application to Automatic Control Problems [in Russian], Fizmatgiz, Moscow (1960). V. V. Solodovnikov (editor), Automatic Control Theory, Book 2: Analysis and Synthesis of Linear Continuous and Sampled-Data Automatic Control Systems [in Russian], Mashinostroenie, Moscow (1967).

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