Transient mean square response of randomly damped linear systems

Journal of Sound and Vibration (1987) 113(1), 71-79

TRANSIENT MEAN SQUARE RESPONSE OF RANDOMLY DAMPED LINEAR SYSTEMS T.

FANGt AND

E. H.

DOWELL

Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27706, U.S.A. (Received 17 December 1985, and in revised/arm 11 March 1986) Transient responses of a randomly damped one-degree-of-freedom linear system subjected to white noise excitation is discussed. The time varying damping coefficient is considered either as Gaussian white noise or filtered white noise. Perturbation approximations for mean square responses are obtained. The approximation for the white noise damping case is compared with the exact solution obtained by the moment method. It is found that they agree well within the requirement of engineering analysis. Transient responses for both the random damping cases are compared through an example.

1. INTRODUCTION

Some physical considerations lead us to a mathematical model of a randomly damped system, in which the damping coefficient per se is a random time-function. Examples include the damping which results from ambient turbulent flow or the damping in beams and plates due to slipping at the support boundaries, when the normal load between the slipping surfaces is a random function of time, [1]. Such models belong to a wide class of systems, known as parametric systems. There have been many investigations of random parametric vibrations [2,3]. In most of them systems with randomly varying stiffness were considered. Some recent papers [4-7] have been concerned with the general case with varying stiffness terms, damping terms, or even inertia terms, through the moment method and the stochastic averaging method. In this paper the perturbation approximation is applied to obtain the time-dependent mean square response. The varying damping coefficient is supposed to be white noise or filtered white noise. The approximate solution for the white noise damping case is checked with the exact one, obtained by the moment method. It is found that the two solutions agree well within the requirement of engineering analysis. Transient responses for both white noise and filtered white noise damping cases are compared through a numerical example.

2. ANALYSIS

Consider a single-degree-of-freedom linear system with a random time-dependent damping coefficient under stationary white noise excitation. The governing differential equation may be expressed as x+2g(t)woX+ w6x

=

w(t),

(1)

t Visiting Professor.

71 0022-460X/87/040071 +09 $03.00/0

© 1987 Academic Press Inc. (London) Limited

72

T. FANG AND E. H. DOWELL

where Wo is the natural frequency of the system; the time-dependent damping coefficient (( t) can be expressed as (2)

((t) = (0+ j-t(1 (t)

with j-t as a small quantity and ~l (r) is stationary white noise with zero mean and (g\ (t) (\ (t + r» = 21TgS(r). The excitation, w( t), is also stationary white noise with zero mean and (w(t) wet + r» = 21Tj8( r). Both wet) and g\(t) are assumed to be Gaussian and mutually independent. Suppose that the solution of equation (1) can be expressed as a power series of u:

x(t)=xo(t)+j-tXt(t)+j-t2 X2(t)+···.

(3)

Then the mean square value of x( t) has the form (x 2( t» = (x6) + j-t (2(xox\» + j-t 2((xi) + 2(xox2» + O(j-t 2), (3a) where Ot u.2) means a small quantity of order higher than 2. Substituting equations (2) and (3) into equation (1), and equating the terms of the same power of j-t on either side of the equation gives a series of equations of the generating system: xo+2(owoxo+w~xo= wet), ~\ + 2gowox\ + W~Xl = -2wo(.xo,

x2 + 2(OWOX2+ W~X2 = -2WO~tXl' (4) The impulse response of the displacement is

h(t) = (ljj27)owo)(e Pt - e Pt ) , where p

= -(owo + j 77oWo

and

7)0

(0<1,

(5)

=.j 1 - g~. The impulse response of the velocity is

h(t) = (1/j277oWo)(P ePt - p e fll ) ,

(0< 1.

(6)

In what follows the perturbation method is used to look for the non-stationary response under quiescent initial conditions: i.e.,

xo(O) = x\(O)

= X2(0) = ... = 0,

xo(O) = xl(O) == iiO) = ... = O.

Beginning with the first of equations (4), one has

xo(t) =

J:

h(u)w(t-u)du,

xo(t) =

fa' h(u)w(t-u)du.

The time-dependent variance of xo( t) is

(x~(t»= =

J: fa' 21Tj

h(u)(w(t- u)w(t-v»h(v) dv dn

fa'

L

h(u)S(u - v)h(v) dv du = 21Tj

= -( 1Tfl47)~w~)[(e

2p r

fa' h 2(u) du

-1)/ p + (e 2p l - 1)/fi + 2( e -2~o"'ot -1)/ gowo].

From the second of equations (4), one obtains

fa' h(v)Mt-v)xo(t-v)dv -2wo fa' fa'-v h(v)h(r)g\(t - v)w(t - v - r) dr dv x\(t) = -2wo fa' fa'-v l1(v)l1(r)(\(t - v)w(t - v - r) dr dv. x\(t)=-2wo =

(7)

73

RANDOMLY DAMPED SYSTEM TRANSIENT RESPONSE

The time-dependent variance of xl(t) is

(xi(t»)=4w~

LL

L-v L-u h(v)h(u)h(r)h(s)

x (gl (t - v) gl (t - u »)(w(t - v - r) w( t - u - s») ds dr d u dv

= 81TW~f

LL

L-v h(v)h(u)h(r)h(v- u + r)(g,(t- v)gl(t - u») dr du dv.

Noting that

f-v h(r)h(v-u+r)dr =

-(1/ 41J~w~){(p/2) eP(V-U l[e 2p(l-vl -1] - [ppj (p + p)] eP(v-Ul[e(p+p)(l-v l -1] - [ppj (p

+ p)] eP(v-U l[e(P+/l)(I-vl -1]( pj2) eP(v-u)[ e2p(l- v)-

1]} = R

one has

(xi(t»)=161T 2w6Jg = 1T

2fg{(

1Jriw~

L

h 2(v)R(u=v)dv

(.E.

2pp _P+P)(_2__ P + P) + e2p1 + 4pp e(p+Pl'+£e2JlI)t p+p 2 p+p 2pp 2 p+p 2

2pp p+p) 1 3p-p + [( p+ P --2- 2p + 4(p-p)

2pp ] e2pt (p+p)(p-p)

2pp p+p) 1 3p-p + [( p+p--2- 2p+ 4(p-p)

2pp ) 2Jlt (p+p)(p_p) e

+

[2

4PP ] e(p+p)I}. (p+p)2

(8)

The time-dependent covariance of xo( t) and XI (r) is

(xo(t)x,(t») =

-

2wo

fa' fa' L-v h(u)h(v)h(r),

(gl(t - v) wet - u) w( t - v - r») dr dv du = O.

(9)

From the third of equations (4), one obtains

x 2(t) =

-

2wo

L

h(v)g,(t - v)xl(t- v) dv

=4W 6{ L-v f-

v- r h(v)Ji(r)h(s)Mt-v)gt(t-v-r)w(t-v-r-s)dsdrdv.

The time-dependent covariance of xo(t) and X2(t) is v- r (XO(t)X2(t») = 4w~ fh(u)h(v)h(r)h(s)

I L{-V

x (g,(t - v)gl (t - v - r»)(w(t- u)w(t - v - r- s») ds dr dv duo

74

T. FANG AND E. H . DOWELL

Noting that

L-v-r

Ji(S)(W(t- U)W(t- v - r- s) ds = 27TjJi(U - v- r),

f-V

h(r)h(u - v - r)(q(t - V)g( t - v - r» dr

one has (XoXz)

= 167T2w~jgJj(0) = 71'4~~(p-p) 1]oWo

f

= 271'gh(0)h( u -

v)

h( U) faU h( v )Ji(U - V) dv du

{I

1

-[eZp'(2pt-l)+1]-----=[eZjll(2pt-l)+1] 4p 4p

-

+-

+ p-~ [e(P+il)'«p+p)t-l)+l]+ p _ P (eZpt_l) (p+p)2 2p(p-p)

- f

-I)}.

+ P_ (e Zilt -1) _~(e(p+p)t

2p(p-p)

p-p

(10)

Thus, substituting equations (7)-(10) into equation (Ja), gives the second approximation of (x Z( t), with an error of O(p, Z). Then letting t ~ tXJ in it gives the stationary response as 2 (x ) = (71'j/2qow6) + (3p, 27T2fg/ £~w~) + O(p, 2). (11) Therefore, with an error of O(p, 2), (x Z ) may be written as (x

2

)

= (x~)(l + a),

(Ll a)

where (x~) = 7Tj/2£ow6 and a -:;; p, Z67Tgwo/ go. Noting that (x~) is the stationary response of the time-invariant generating system under white noise excitation with a spectral density f, one can recognize (x 2 ) as the stationary response of the same generating system to white noise excitation with a magnified spectral density, (1 + a)f The moment method can be used to check the above result. It is assumed that ql(t) and w( r) are naturally independent white noise processes in the sense of Stratonovich [3]. By letting YI = x and Y2 = x the stochastic differential equation for the Markov vector (YI, yz) for system (1) can be written as

,

dy,

= yz dt,

dyz -:;; (-w~YI - 2gowoyz) dt - p, 2 woY2 d WI + d W z,

(12)

where WI and Wz are Wiener processes, with (d WI) = (d Wz) = 0, (d WI d Wz) = 0, (d Wi) = 271'g dt, and (d W~) = 271'fdt. Then the Fokker-Planck equation for the joint transition probability density, P, can be written as iJp/ at = - Y2 iJp/aYI + w~YI iJp/iJyz 2

, a [( 2qowo-47Tgp, z WOz)Y2P] +-1 -, a [(871'gp,·W 2 2 +oY2+ 27Tf)p].

aY2

2 ilYi

(13 )

The underlined term in equation (13) results from Stratonovich's interpretation. By introducing the notation mL L = (yi), ml2 = (y,h) and m 22 = (y~) , the moment equation for the second moments can be obtained as mil

=2m I2,

ml2 = -w~mll - (2qowo- 471'gj.L 2w~)mI2 + m Z2, mZ2 = -2w~mI2 - (4g owo- 167Tgp, 2w~)m22+ 27Tf

(14)

RANDOMLY DAMPED SYSTEM TRANSIENT RESPONSE

It is easy to obtain the stationary second moment mil =(x

2)=

mIl

75

as

71'1/(2~ow~-87Tgf.L2W6).

(15)

Alternatively, if one assumes that ~l(t) is white noise in Ito's sense, then the result will be mil = (x 2 ) == 71'1/ (2~ow~ - 471'g).L 2 ( 6). It is well known that the random parameter excitation affects the mean square response stability. It is easy to see, from equation (15), that if ~o < 471'g).L 2 W O the stationary mean square response no longer exists. In fact, equation (15) predicts a second moment instability when ~o < 471'g).L 2 wo. Although the perturbation solution (11) does not give any information concerning the mean square stability, within a certain range of stable mean square solutions the perturbation approximations agree well with the exact ones. For example, for ~o = 0'25, Wo = 2, f = 2, g = 1, and f.L = 0-01 ~ 0,05, the numerical results are compared in Table 1. The transient solutions for (x 2 ( t) by the two methods can also be compared for a numerical example. For go = 0'1, Wo = 8, 1= g == 2 and f.L == 0,01, the numerical results are plotted in Figure 1. The approximate solution coincides with the exact one within the expected accuracy. The exact stationary (x 2 ) is equal to 0,0768, while the approximate one is 0·0798. In this case IX = 0·3016. Now we apply the perturbation method to the case when the varying damping is narrow-band limited, like the damping due to slippage when the normal load is a narrow band random one. It is known that narrow band random noise may be approximated by TABLE

I

Comparison of stationary MS responses

0·01 0·02 0·03 0·04 0·05

Perturbation

Exact

Error

solution

solution

(Ufo)

1·5945 1'6656 1'7840 1·9498 2·1630

1·5868 \·6366 1·7271 1'8719 2·0982

0·49 1·77 3·29 4'16 3·09

Moment method 0·06

002-

0'96

4·80

1·92 I (s)

Figure l. Transient response (white noise damping).

76

T. FANG AND E. H. DOWELL

filtered white noise, which is the output of a second order linear filter due to white noise input. The characteristic value of the filter determines the center frequency and the bandwidth of the filtered white noise. We consider here the most severe case when the characteristic value of the filter coincides with that of the generating system, i.e., ~t(t) is assumed to be a stationary zero mean random process with

(~t (t)£t (t + T») = 27T(g e pH + g e fiH ), where g is a complex number and p is the characteristic value. The excitation, wet), and the other parameters of the system remain the same as before. Again, both w( t) and ~t (r) are assumed to be Gaussian and mutually independent. The approximate solution is obtained in the same way as illustrated before. The quantities (x~(t») and (xo(t)xt(t») remain the same as in equations (7) and (9), while (xi(t)) = (7Tzf/21JrilJ)~)(N + N), when

N

3pt at(e P' -1)/ P + Q2(eZP' -1)/2p - a2(e -1)/3p + Q3(e(P+P)1 -1)/ (p + ft) - a3(e(2pt p)t -1)/ (2p + ft) + a4(e(ji-P)t -1)/ (ft - p)

= at t -

- Q4(ejit -1)/ ft + Qs(e(p-Plt -1)/ (p - ft) - as(e(Zp-ji)t -I )/(2p - ft) + Q6t - Q6(ept -1)/ p + Q7( e (p-ji)' -1)/ (p - ft) - a7(e(Zp-p)r -1)/ (2p - ft) + QH( e 2(p-illt -1)/2(p - ft) - as(e pt -1)/ p + a g ( e Zpr -l)/2p - Q9(e (pt2ji)t -1)/ (p + 2ft) + QIO( e(p-ji)t -1)/ (p - ft) - alO(ePt -1)/ ft + at (e" -1)/ p - ii t (e(P-ill l -1)/ (p - ft) + iiz( e( p+Zji)t -1)/ (p + 2ft) - iiz(e(P+PlI -1)/(p + P)

+ ii3(eCzP+p)1 -1)/ (2p + ft) -

ii3(e 2p1-l)/2p + ii4(eCZp-p)t -1)/ (2p - ft)

- iiie 2(p-ji)C -l)/2(p - ft) + iis(e P1-1)/ ft - iist+ ii6(e pt -1)/ p - ii6(eCp-jill -1)/(p - ft)

+ ii7(e PI - 1)/ ft -

ii7t+ ii s(e P'-1)/p- ii s(e(p-p)I_1)/(ft - p) + iig (e 3P' -1)/3p - ag(e CP+ fi)'-1)/(p + ft) + iilO(e(ZP-P)' -1)/(2p - ft) - iiwl + b, (e zPC-l)/2p - iii (e(P+P)t -l)/(p + ft) - bz(e- Pl -1)/ p - bie(P-2 Pl l -1)/ (p - 2ft) ft) - b3t + b4(e Pt -1)/ ft - bie(ZP-Plt -1)/ (2p - ft) + bs(e P' -1)/ P - bs(e P' -1)/ P - b6(e- P1-1)/ p + b6(e- Pt -1)/ ft + b7(e 2(P-p)r -l)/2(p - p) - b7(e(ji- plt -1)/ (ft - p) + bst - bs(e(P-p)1 -1)/ (p - ft) + c t(e 2pt -1)/2p - Ct(ecp+p)t -1)/(p + ft)+ C2t- c2(e(P- plt -1)/(p-ft)

+ b3(e(p-P)1 -1)/(p -

+ c3t -

C3( eCp-jilt -1)/ (p - ft) - c4 ( e -pi -1)/ P -

c4 ( e(p-2plc-

1)/(p - 2ft)

+ cs(e(2 P-f/lc -1)/(2p - ft) -cs(e PI-1)/ ft- c6(e-jit -I)/ft+ c6(e-ji'-1)/ft + c7(e(p-P)' -1)/ (ft - p) - c7(e2(P-fil l -1)/2(p - ft) + cs(e PI -1)/ p - cs(eP' -1)/ p,

77

RAN D OMLY DA MPED SYSTEM TRANSIENT R ESPONSE

where

al = - g e2P ' ,

a2= g[1 -2P!(p + p)],

Q4 = g e

2p

a s = g(P! p ) e

"

2P

a3 = [2p/(p+ ft)](g+ g) -

Cg+ gPl p ),

a6 = -[2P1Cp + ft)](g +g) e (P+P)' , a s = [p g/ (p - 2ft)] e2P',

',

Q7 = [2pg/ Cp+ p )] e(p+jl) t,

= [pgl ( p - 2p) ]{[2p/ ( p + p)] -1}, a lO = -[2ppg/ CP - 2p )(p + p )] e(P+Pl', a9

2ji P 2PP ) bl = g ( 1- p+ p+ 2p-ji- Cp +ji )(2p-p) b3=g 2P_(1 +~) e(P+filt, p +p 2p-p b

=

5

C1

g( 2p p + ji

-1)

2 P P ) 3 Pt ( b2= g 1- Cp + ft)(2p - p ) e ,

I

b4=g~(

p 2 _ 2p - p p+p

b

-1)

e(2p- plt,

g(-L_ 2ft ) e(2p+p)t 2p _ P p + P , 2P 2P b, = -g[pI(2p - ji)] e ' , bs = -g e ', epr

= g(~+;- 2) ,

=

6

I

C2

= -g(~) e

2pt,

C3

= -2g e(P+P )',

c = _ g(l!._ 2ji ) e(2p+Pl' c = _g-(l!._ ~) e pt 4 P p +ji ,5 P p+p , c = g-(J!._ 2P

6

ji p+p

) e(2p+pJ'

C ,7

= g(J!.) e 2p,

P

,

c = 8

g (J!.P _ ~) e pt p+p ,

and

with M = ( r./ 4p 2 )[ e 2P' (2p t -1) + 1] - [;',/ ( p + p )2][e ( P+PJ'«p + ji) t -1 ) + 1]

+~(_~_ p +p

p-p

'1 _+ Pr2 _~+ "':) ce(P+ t - 1) p -2p P PJ

p-p

1 (-_-+--_rl ' 1 --:r2+-'3-_- -r3 ) +-

2p P - P P - P P

P - 2p

+~(~_.!.) (e(2P+PJt 2p+p p P

+~(.!.+ 3p p

P

(e 2I' / -1)

1) _ ~ ( ~ + _1__) (e(2fi+Plt -1)

2p+p P p-2p

1_) (e3Pt - 1), p- 2p

where

p) p

p - -- , r 1-- pg-( 2 -p-p

-2

2

P g- - pg - -pg, r2 = -P g + --=-

p

p

r3=pg(1-~) . 2p - p

For the same numerical ex am ple, mentioned before, the non-stationary responses were ca lculated for different values of g: i.e., g = 4, 40, and 100. The results are plotted in Figure 2. The transient pro cesses for both kinds of var ying damping are compared in Figure 3, where g is cho sen to give the same station ary mean square response , namely g = 100 for filtered white noise damping and g = 4·8 for white noise damping.

78

T. FANG AND E. H. DOWELL

g=100

0·096

9=40

0·072 g=4

"" ~ 0·048 <;»

0·96

1·92

2·88

3·84

4·80

I (5)

Figure 2. Transient response (filtered white noise damping). Filtered white noise (g=100)

0·096

0·072

""

~ 0·048

0,024

1·92

288

3·84

4·80

I(s)

Figure 3. Transient response (white noise and filtered white noise damping).

3. CONCLUSIONS

1. Non-stationary mean square responses of a randomly damped linear system have 'been found by perturbation approximations, within an error of order O(fJ-2). By comparing with the exact solution, it is found the perturbation approximation is an overestimated one. The perturbation method may be more intuitive, while the moment method is more powerful. 2. It is known that for random parameter excitation problems the white noise excitation in Stratonovich's sense is more reasonable than that in Ito's [8]. That is the case in our problem. 3. The time-dependent mean square response experiences larger fluctuations for the filtered white noise damping case than for the white noise one. ACKNOWLEDGMENT

The authors are grateful to the reviewers for their valuable comments.

RANDOMLY DAMPED SYSTEM TRANSIENT RESPONSE

79

REFERENCES 1. E. H. DOWELL 1986 Journal of Sound and Vibration 105, 243-253. Damping in beams and plates due to slipping at the support boundaries. 2. R. A. IBRAHIM 1981 Shock and Vibration Digest 13, 23-35. Parametric vibrations, Part 6, Stochastic problems. 3. V. V. BOLOTIN 1984 Random Vibrations of Elastic Systems. Netherlands: Nijhoff, The Hague. English translation. 4. M. F. DIMENTBERG 1983 in Random Vibrations and Reliability (K. Hennig, editor). Berlin: Akademie-Verlag. Response of systems with randomly varying parameters to external excitation. 5. WEI-QIU ZHU 1983 in Random Vibrations and Reliability (K. Hennig, editor). Berlin: AkademieVerlag. Stochastic averaging of the energy envelope of nearly Lyapunov systems. 6. W. F. Wu and Y. K. LIN 1984 International Journal of Non-Linear Mechanics 19, 349-362. Curnulant-neglect closure for nonlinear oscillations under random parametric and external excitations. 7. R. A. IBRAHIM and A. SOUNDARAJAN 1985 International Journal of Non-Linear Mechanics 20. An improved approach for random parametric response of dynamic systems with nonlinear intertia. 8. R. E. MORTENSEN 1968 NASA CR-1l68. Mathematical problems of modeling stochastic nonlinear dynamic systems.