NONSTATIONARY RANDOM

MULTIPLE-MASS SYSTEMS

L. M. Re znikov

V I B R A T I O N S O F

UDC 534.011

The analysis of s t ruc tures subjected to the action of wind, earthquakes, and explosion waves entails the investigation of both s teady-s ta te and t ransient motions elicited by the sudden application of a s ta t ionary dis tur- bance [4]. Nonstat ionari ty of the load, as is typical, e.g., of se ismic effects, explosions, or dis turbances acting on t ranspor ta t ion vehicles upon contact with diverse roadbed features , is also taken into consideration. In the present ar t ic le , nonstat ionary random vibrations of mul t ip le-mass sys tem subjected to kinematic disturbances applied simultaneously or with delay are investigated by the method of monents [3] which is finding ever - in - c reas ing applications in present -day mechanics problems [1, 2].

~_1. N o n s t a t i o n a r y W h i t e N o i s e

Consider the following sys tem of l inear differential equations with variable coefficients, writ ten in Cauchy form:

(t) = B (t)y (t) + OF (t) ~p (t). (1.1)

Here y(t) is a vector of coordinates of order 2n, B(t) is a 2n • square matrix, D is a 2n xm rectangular ma- tr ix, F(t) is an m x m square mat r ix of determinis t ic envelope functions, r is a vector of centered white-noise functions with corre la t ion matr ix ~(t)~'(T)= Q6(t-~) , Q is an m xm square matrix, 6(.) is the Dirac delta func- tion, the overbar denotes averaging over the ensemble of real izat ions, and the prime is the t ransposi t ion sym- bol.

Let I(t) = y(t)y'(t) be the matr ix of c r o s s - c o r r e l a t i o n moments of the sys tem coordinates , and W(t, r) a matr ix of t rans ient response functions. F rom (1.1) we deduce the equation

i (t) = B (t) I (t) + I (t) B" (t) -4- DF (t) ~P (t) y'(t) + y(t) ~P" (t) F" (t) D'; t

y (t) = W (t, ~) y (0) -{- I W (t, "0 OF (~) ~p (~) dr, 0

and for uncorre la ted vec tors y(0) and ~'(t) we have

y (t) ~p" (t) = 0,5 DF (t) Q; ~p (t) y* (t) = O,5 QF" (t) D'.

The matr ix I(t) can be found by integrating the following matr ix differential equation with specified initial con- ditions I(0):

"I (t) = B (t) I (t) + I (t) B" (t) + DF (0 QF" (t) D'. (1.2)

The vibrations of a sys tem with viscous frict ion under the action of nonstat ionary white noise are de- scr ibed by the equation

d [ F ( t ) , ( t ) l + $ F ( t ) r 7, + nx + cx = s l -~ (1.3)

in which the mat r ix of inert ial coefficients is reduced to the unit matrix, H and C are n xn square matr ices , and S 1 and S are n • m rectangular mat r ices . Introducing the extended vector of coordinates y'(t) = ix', {~.- StF(t)@(t)}], we t r ans fo rm Eq. (1.3) to the form (1.1), where

$1 .=[_; Inasmuch as the values of x(O) and k(O) are ss independent of ~(t), in the integration of Eq. (1.2) the initial conditions take the fo rm

Dnepropetrovsk Branch of the Institute of Mechanics, Academy of Sciences of the Ukrainian SSR. T rans - lated f rom Prikladnaya Mekhanika, Vol. 15, No. 7, pp. 88-94, July, 1979. Original ar t ic le submitted March 29, 1977.

0038-5298/79/1507- 0633 $07.50 �9 1980 Plenum Publishing Corporat ion 633

X

= 1 (.=[.]=,+o..,,,,,o) t "xx" xx" lt=o

F r o m this r e s u l t we in fe r I z z ' (t) = Iyy, (t) + 0.5D F(t) Q F' (t) D' 1 + 0.5D 1F(t) Q F' (t) D' + D 1F(t) Q F' (t) D' 16(0). The ad-

di t ional t e r m s in the m a t r i x Izz,( t) not in Iyy,(t) a re equal to z e r o for xx ' , a r e finite quant i t ies fo r x:x' and ~x' ,

and b e c o m e infinite fo r ~b~', as is e n t i r e l y na tura l .

The m a t r i x equat ion (1.2) is equivalent to a s y s t e m of n(2n + 1) o r d i n a r y d i f fe ren t ia l equat ions and is so lvable on a c o m p u t e r .

Fo r l a rge values o f n, it is adv i sab le to use the e i g e n v e c t o r s of the m a t r i x B, p rov ided that the la t ter is independent of t ime . Le t V be the m a t r i x of e igenvec to r s , and A the d iagonal m a t r i x of e igenva lues kj of B.

* * *

Then B = VAV -1, B' = V-1AV, and f r o m Eq. (1.2) we deduce

(0 = a r (t) + r ( t) i + L (t), (1.4)

where

r (0 = v- ' z (O L (0 = V - ' O F (t) QF" (t) D ' V - ' ;

The a s t e r i s k denotes t r ans i t i on to the Hermi t i an -con juga te ma t r ix .

The m a t r i x equat ion (1.4) r e p r e s e n t s a s y s t e m of uncoupled d i f fe ren t ia l equat ions of the f o r m ~ij(t) = (~i +

Xj)Yij (t) + lij (t).

In the spec ia l p r o b l e m of sudden appl ica t ion of s t a t i ona ry r a n d o m p r o c e s s e s , L(t) = cons t for t > O. T h e r e - fore,

o r

At ~ t * r (t) ---- e iF (0) - - 6] e + G; I (t) ---- V {e At iV-11 (0) V - I - - G| e ~t + G}

I (0 = W (t)[I (0) - - I (oo)]W" (t) + I (oo).

Here G---- - - lO n the s t a t i o n a r y solut ion in I(oo) = VGV, and the m a t r i x of t r a n s i e n t r e s p o n s e funct ions is

W(t) = v e h t v -1.

w N o n s t a t i o n a r y W h i t e N o i s e w i t h D e l a y

Fo r d i s t u rbanc e s of the type S F ( t - r ) $ ( t - r ) , the v ibra t ion equat ion for a m u l t i p l e - m a s s s y s t e m is wr i t t en in the f o r m

"9(0 = B y (t) + DF ( t - - T) , (t - - "O.

where

r ( t -T ) = [qj (t--rj}]~=i is a v e c t o r of c e n t e r e d White-noise packe ts ac t ing with d i f fe ren t de lay t i m e s v j ; and

F ( t - r ) = [ f i j ( t - r j ) ] ~ j =l is a m a t r i x of d e t e r m i n i s t i c funct ions of the de lay a r g u m e n t rj.

As in w we obta in

"I (t) = BI (t) + I (t) B" + DF (t - - ~) ~ (t - - ~) y" (t) + y (t) 4" ( t - - r ) F" (t - - ~) D" ;

t

y (t) , " (t - - T) F" (t - - ~) D" = f W (t - - ta) DF (t - - ~) 0

• ~2 (ix - - *) *" (t - - x) dtxF" (t - - , ) D' .

(2.1)

634

We now in t roduce T as the sum of the l a s t two t e r m s in (2.1) and el , fi, di as the i - t h co lumns of the r e - spec t i ve m a t r i c e s E, F, D; Q = [qij]l,,j =l- We wr i t e the c o r r e l a t i o n m a t r i x of the w h i t e - n o i s e funct ions in the

f o r m m m

i = l :=l

Then, mak ing use of the f i l t e r ing p r o p e r t y of the del ta function, we obtain

T -- ~ ~ qu [117 (~I - - ~) Dr, (t - - vj) ti (t - - ~s) D' + 0f l (t - - v:)/~ (t - - T~) D'W (%, - - xt)]. j=l l=l

Thus , the m a t r i x I(t) is ca l cu la t ed by i n t eg ra t i ng the equa t ion I(t) = BI(t) +I( t ) B' +T sub jec t to the s p e c i - f ied in i t ia l condi t ions . Fo r th is ope ra t ion , the va lues of W ( r j - r i ) D fo r v a r i o u s c o m b i n a t i o n s of ind ices i, j a r e d e t e r m i n e d f r o m t h e equa t ion d/dt[W(t)D] = B[W(t)D] + D 6 ( t - r j ) ; W(0)D = 0. F o r the c a s e of z e r o de lay [r = 0, W(0) = 0.5E], the e x p r e s s i o n

T = 0,50 2 ~-~ [fi (t) q~if~ (t) + fj (t) quf~ (t)] 0 ' = OF(t) QF' (t) D' l = l i ~ l

exa c t l y c o r r e s p o n d s to the f r e e t e r m of Eq. (1.2). I f the m a t r i x F ( t - r ) is d iagonal , then

T = ~.~ qufu ( t - - ~) fn (t - - T,) [V, (*j) d~ + dig; (,j)], 1 = 1 i = l

w h e r e the vec to r Yiffj) = W ( r ] - r i ) d i is t he so lu t ion fo r t = r j of the equa t ion ~i(t) = BYi( t )+di /~( t -vi) , Yi(0) = 0. Fo r iden t i ca l funct ions f ( t - r i) with d i f f e ren t de lay t i m e s ri, we have

in Y/$

T = ~ t 2 (t --~j) Tj, r~e r j = Y (~1) d~. - F d y ' (~1); Y (vi)-= ~ Y, (vJ) qu. 1 = 1 / = 1

Final ly , for c o m p l e t e l y c o r r e l a t e d w h i t e - n o i s e funct ions (qi] = 1; i, j = 1, 2, .. . , n), the v e c t o r Y(rj) is the so lu- t ion of the equa t ion Y(t) = BY(t) + D S ( t - r ) sub jec t to z e r o - v a l u e d ini t ia l condi t ions . H e r e 5(t--T) = [6 ( t - -~ i ) ]ml

is the v e c t o r of de layed D i r ac de l ta funct ions .

The g iven e x p r e s s i o n s a r e a l so app l icab le to s y s t e m s with v a r i a b l e p a r a m e t e r s [B = B(t)], and in this c a s e the m a t r i x W(t, tl) depends on the two a r g u m e n t s , but not on t he i r d i f f e r ence .

w N o n s t a t i o n a r y K i n e m a t i c D i s t u r b a n c e D i s t i n c t

f r o m W h i t e N o i s e

Le t a m u l t i p l e - m a s s s y s t e m be ac ted upon by d i s t u r b a n c e s Ui(d/dt)[F(t)~p~ +UF{t)q~~ where U, U 1 a r e nx m r e c t a n g u l a r m a t r i c e s , q~~ is an m - d i m e n s i o n a l v e c t o r of s t a t i o n a r y r a n d o m funct ions with r a t i o n a l - f r a c - t ion s p e c t r a l d e n s i t i e s , and the e l e m e n t s of the m a t r i x ~{t) a r e i n t e g r a b l e p i e c e w i s e - c o n t i n u o u s funct ions . The shap ing f i l t e r t r a n s f o r m i n g the white no ise @(t) into s t a t i o n a r y funct ion q~~ m a y be d e s c r i b e d by the equat ions

~ + Hi+ + C1~ = Sl;~ (t) + S~ (0; ~o (t) = s,~ (0. The d i m e n s i o n s of the m a t r i c e s a r e H1, C 1 (k xk) ; $1, S (k x / ) ; S 2 (m xk) .

The s y s t e m of equa t ions ex tended with r e g a r d for the shap ing f i l t e r is t r a n s f o r m e d to the Cauchy f o r m

~t(t) = B (t) y (t) + D (t) • (t) (3.1)

by se t t ing

t -~ I- x (t) ; D (t) ---- UiF (t) S~SI

Y(O= 1 ,~(t) s~ /' _ ,~ 0 - - s , ~ (t) _ s - - u ~ s i _

635

0 0 011 B (t) = - - C - - H [U~'F (t)q-UF (t)l S~. U~F (t) S~ �9 0 0 0 E "

l o o - c l - n l

The matr ix I(t) = y(t}y'(t) is calculated f rom the equation

"I (t) = B (t) l (t) -~ l (t) S ' (t) q- D (t) QD" (t), (3.2)

where the initial values of the corre la t ion moments of the coordinates x(t), A(t) of the mechanical sys tem cor - respond to the given conditions at t = 0, the c r o s s - c o r r e l a t i o n moments of the coordinates of the sys tem and fil ter are equal to zero, and the corre la t ion moments of the finite coordinates r ~(t)-Sl~(t) a re evaluated for the steady state ( t~ ~o). The lat ter moments a re determined f rom the sys tem of equations obtained f rom (3.2) by putting I(oo) = 0 and retaining in the mat r ices I(t), B(t), and D(t) only elements pertaining to the coordi- nates of the shaping fi l ter.

w V i b r a t i o n s o f t h e S y s t e m in M o t i o n a l o n g a P a t h

w i t h R a n d o m I n h o m o g e n e i t i e s

Let us suppose that kinematic disturbances a re crea ted by the path (track or roadbed) without delay and that the dependence of the path traveled on the t ime r(t) and of the velocity on the time v(t) and path v(r) a re known. Let the sys tem move along a path with i r regular i t i es F(r)~~ where Fir) is an m • matr ix of de- terminis t ic functions with piecewise-continuous f i rs t derivatives and ~p~ is a vector of s tat ionary random functions, for which a shaping fi l ter can be synthesized.

The vibration equations for the extended sys tem

x "t- Hx d- Cx = U1F(r) ~~ d- [U1F(r) d- UF(r)] ep ~ (r); (4.1)

d ~ --- S d~ (r) + S~ (r); ,~o (0 = S~,p (r), dr =

correspond to the moment equation

d l B I (r) + I (r) B" (r) + D (r) QD' (r), (r) (r) (4

which can be integrated to find the corre la t ion moments of the displacements and velocities of the sys tem. Here I(r) = y(r )y ' ( r ) ;

] x (0 [ u~F (0 s ,s~ y (r) == [ qD (r) �9 [; D (r) ----- S ,

0 vC0 0 0 - U~(r) - | ~ I - - Z ; - + -v (Ty] ~,

C H U=(r) Ua ) ; B (r) = v (r) v (r)

0 0 0 o o - - c , - - H, d U3 (r) = U1F (r) S=.

To find the corre la t ion moments of the accelera t ions , it is necessa ry to put S 1 = 0 [otherwise ~(t) x'(t-)-- ~]. Differentiating Eq. (4.1) with respec t to t and ca r ry ing out s t ra ightforward t ransformat ions , we ar r ive at (4.2), in which now

y" (r) ----- x ' (t), x ' (t), r (r), ; D ' (r) = [0 , { U f (r) o (r) S~S}', O, S'] '

V I[ dF (r) dv (r) U~fr)= 11L, -ar d~

d2F (r) H dF (r) .b ~ o(r)]S~-- F (r)v(r) S,Ci } -k - ~ S,;

636

G 5 10 /5 t, sec

Fig . I

O~/OZ - z I

~ F ' ~ - - " " 5 I0 15 20 t, see

Q6

' \

5 10 15 20 25 t, sec

Fig . 2

0 5 10 15 t, sec

Fig . 3 F ig . 4

4- dv (r) S=] + 2 dP (r) Us(r)=Ul{F(r)[--v(r)S~l-Iz-- ~ _ ~ v ( r ) S ~ } +UF(r) sr

E x a m p l e s

F i g u r e 1 g i v e s the v a r i a n c e s of the d i s p l a c e m e n t s in m 2 ( so l id c u r v e s ) and v e l o c i t i e s in m2sec -1 (dashed c u r v e s ) of po in t s x, y, z of a c a b l e [5] s u b j e c t e d to a n o n s t a t i o n a r y g u s t of wind, which c r e a t e s a load a p p r o x i - m a t e d by the func t ion t e - eta(t) , w h e r e ~0(t) i s whi te n o i s e and e = 0.4 s e e - t

C u r v e s r e p r e s e n t i n g the v a r i a t i o n of the v a r i a n c e s of the d i s p l a c e m e n t s and v e l o c i t i e s (in s e e -2) of both m a s s e s of the t w o - m a s s s y s t e m a n a l y z e d in [5] a r e g iven in F i g s . 2 and 3. In the f i r s t e a s e , the b a s e of the s y s - t e m m o v e s a c c o r d i n g to the law t e - e t c ( t ) , w h e r e r i s a s t a t i o n a r y r a n d o m funct ion with unit v a r i a n c e and s p e c t r a l dens i t yS~(c~ ) = 2 a ( ~ 2 + f 1 2 ) ~ t l - w 2 + 2 a i o ~ + ~ 2 + f 1 2 ~ ' 2 ; E=0.3 s e e - t ; a = 7 s e e - t ; fl = 1 8 s e e -1. T h e d o t - d a s h c u r v e in F ig . 2 r e p r e s e n t s the d e t e r m i n i s t i c enve lope of the d r i v i n g funct ion . The s e c o n d e a s e p e r t a i n s to the p r o c e s s of d a m p i n g of t he s y s t e m a f t e r c e s s a t i o n of the s t e a d y - s t a t e mo t ion of the b a s e , fo r a = 0.1 s e c -1 and /3 = 18 s e e -1. I t i s ev iden t f r o m Fig . 3 tha t the v a r i a n c e s of the c o o r d i n a t e s e x c e e d the s t a t i o n a r y v a l u e s a t the beg inn ing of the t r a n s i e n t r e g i m e (t = 0), a s can a l s o happen in the t r a n s i e n t r e g i m e fo l lowing r e m o v a l of the d e t e r m i n i s t i c d r i v e r .

The t i m e d e p e n d e n c e of the v a r i a n c e a . 2 of the v e l o c i t y of one of the m a s s e s of the s a m e s y s t e m is g iven X "

in F ig . 4 for s e v e r a l v a r i a n t s of the r a n d o m func t ion (p(t) [the enve lope F(t) is g iven in F ig . 2]. The c u r v e s a r e n u m b e r e d a c c o r d i n g to the v a l u e s of the p a r a m e t e r s (in s e c -1) of the e x p r e s s i o n g iven above fo r Sqp(~): 1) a = 7, f l = 1 8 ; 2) ~ = 0 . 1 , fl = 1 8 ; 3) ~ : 0 , / ~ = 1 8 ; 4) c~=0 , f l = 3 ; 5 ) ~ = 0 , f l = 1 . C u r v e 6 i s p lo t t ed for the c a s e tSJco) = 1/2:r (white no i s e ) . The n u m b e r s in p a r e n t h e s e s a l o n g s i d e the c u r v e s i n d i c a t e the s c a l e f a c t o r by which he o r d i n a t e s of the c u r v e s m u s t be m u l t i p l i e d . We note tha t for ~ = 0 the s p e c t r a l d e n s i t y Sq~(c0) = (1 /2 )~ (c0 - f l )+

(1/2)5(w+fl), i . e . , ~p (t) = cos(f i t + 0) i s a h a r m o n i c p r o c e s s wi th p h a s e 0 d i s t r i b u t e d u n i f o r m l y in the i n t e r v a l [0, 21r]. A s i s i n c r e a s e d , the p r o c e s s i0 ( t ) b e c o m e s m o r e wideband , and the s u p e r p o s i t i o n of h a r m o n i e s with d i f f e r e n t f r e q u e n -

c i e s t e n d s to s m o o t h the v a r i a n c e of the s y s t e m r e s p o n s e in c o m p a r i s o n wi th the r e s u l t s for ~ = 0 (fl 31) . F o r fi = 1, the s y s t e m e n t e r s a s i t u a t i o n c l o s e to r e s o n a n c e , so tha t the p e a k s of the v a r i a n c e s a r e c o n s i d e r a b l y h i g h e r than in o t h e r c a s e s .

637

4

5 ~ 8 ,'2 ~ sec

Fig. 5

We have invest igated the v ibra t ions of a f l a tca r moving with var iab le veloci ty over a t r a c k with s ta t ionary random i r r egu l a r i t i e s . Ze ro -va lued initial conditions a r e adopted, and the k inemat ic d i s tu rbances a re sup- p re s sed synchronous ly on al l wheel pa i r s . The solid cu rves in Fig. 5 r e p r e s e n t the va r i ances of the a c c e l e r a -

ff2 tions (in m 2 - sec -4) of a wheel pa i r ( ~ ) , of a point of suppor t of the ca r body on the c a r r i a g e (~ . ) , and of the

of the wheelbase (a~.) during motion of a f la tcar over a t r a c k with i r r egu l a r i t i e s for which d2r 2 is midpoint

white noise with a s pec t r a l densi ty ordinate equal to 9.35 �9 10 -6 m 2. sec -3. The dashed curves r e p r e s e n t the s ta t ionary value of the va r i ances of the acce le ra t ion of the same points during motion of the f la tcar with a co r - responding constant veloci ty; the var ia t ion of the veloci ty (m/sec) is r ep re sen t ed by the d o t - dash curve . The s ta t ionary and nonsta t ionary values of the va r i ances of the acce le ra t ions of the wheel pa i r (~.) coincide in the

graph. Calculat ions for t h r e e - and s i x - m a s s hal f -vehic le models have yielded c lose resu l t s , which differ by no more than 6%.

1,

2.

3.

4.

5.

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

F. Y. M. Wan, "Nonsta t ionary response of l inear t i m e - v a r y i n g dynamical s y s t e m s to r andom excitat ion," T rans . ASME, Ser . E: Appl. Mech., 95, No. 2, 422-428 (1973). E. G. Goloskokov, S. I. Detistov, and N. E. Isikov, "Response of mechanica l s y s t e m s to nonsta t ionary se i smic d is turbance ," in: Dynamics and Strength of Machines [in Russian] , No. 22, Vishcha Shkola, Kharkov (1975), pp. 3-5. I. E. Kazakov, Sta t i s t ica l Theory of Control Sys tems in State Space [in Russian] , Nauka, Moscow (1975), p. 432. N. A. Nikolaenko, Probabi l i s t i c Methods for the Dynamic Analysis of Mechanical Engineer ing s t r u c t u r e s [in Russian], Mashinos t roenie , Moscow (1967). L. M. Reznikov, "Calculat ion of the va r i ances of the coordinates of m u l t i p l e - m a s s s y s t e m s in t r ans ien t motion under the action of s t a t ionary random d is tu rbances , " Pr ik l . Mekh., 12, No. 5, 109-115 (1976).

6"~8