non-gaussian equivalent linearization for non-stationary random vibration of hysteretic system

Probabilistic Engineering Mechanics 9 (1994) 15-22

Non-Gaussian equivalent linearization for non- stationary random vibration of hysteretic system

K. Kimura, H. Yasumuro & M. Sakata Department of Mechanical Science, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152, Japan

An approximate analytical technique is developed for obtaining the non- stationary response of a single-degree-of-freedom mechanical system with bilinear hysteresis subjected to amplitude modulated non-white random excitations. The hysteretic behavior is described by introducing an additional state variable and non-linear functions. The equivalent linear coefficients and also the moment equations of the equivalent linear systems are derived using a non- Gaussian probability density function which is composed of a truncated Gaussian probability density function and a couple of delta functions. The mean square responses and the mean dissipated hysteretic energy are calculated by solving the moment equations and are compared with the corresponding digital simulation results.

1 INTRODUCTION

The study of the dynamic response of a hysteretic system subjected to random excitation is important in safety analysis and reliability assessment of structures since hysteretic behavior is frequently encountered in structures under strong seismic excitation. Extensive studies have been carried out on this problem. 1-s In previous papers, 6'7 the authors have reported a computational method by combining the equivalent linearization and moment equation techniques and found that in some situations agreement between the computed and simulated results was not satisfactory. This was thought to be attributable to the assumption of Gaussian response employed in the analysis. In the present paper, the previously proposed technique is modified by assuming a non-Gaussian probability density function in the form of a truncated Gaussian density function and delta functions.

2 MECHANICAL MODEL AND EQUATION OF MOTION

Consider a single-degree-of-freedom hysteretic system as shown in Fig. 1, and described by the equation of motion:

mS~ + cYc + kG(x) = f0(t) (i)

15

where an overdot denotes differentiation with respect to time t; x is the displacement of the system; m, c and k are the mass, damping coefficient and spring constant corresponding to the linear system, respectively; kG(x) is the restoring force of the hysteretic spring, andf0(t ) is a random forcing function. It is assumed that the system is initially at rest and that G(x) describes the characterisic of bilinear hysteretic spring, being composed of straight lines with slopes of unity and (1 - e) as shown in Fig. 2, in which x* denotes a yielding displacement.

By introducing the relationships:

w~ =k/m, ¢= c/2v"-m-k,

eqn (1) can be written as:

T = w 0 t, y = x / x *

(2)

~+2¢~+g(y)=f (r ) (3)

where an overdot hereafter denotes differentiation with respect to the dimensionless time variable r; f ( r ) = fo(-C/OJo)/mx*~og; and g(y) = G(yx*)/x*, where g(y) is the bilinear hysteretic characteristic non-dimension- alized by the yielding displacemnt x*.

The random focing function is assumed to be an amplitude modulated non-white random excitation 8,9 in the form:

f('r) = e('r)n('r), E[n('r)] = 0, E[n('r)n('r + v)] = R(v)

(4)

16 K. Kimura, H. Yasumuro, M. Sakata

X

m i • hysteretic

spring

/ / / / /

Fig. 1. Single-degree-of-freedom hysteretic system.

/ /

z

1

where e(r) is a deterministic envelope function; and n(r) is a Gaussian stationary non-white noise with zero mean and correlation function R(v). The magnitude of the input noise R(0) is given in terms of the stationary standard deviation displacement response of the linear system.

The bilinear characteristic g(y) can be divided into a linear spring and an elasto-plastic type hysteretic characteristic z(y), as illustrated in Fig. 3 and expressed a s ;

g(y) = (1 - e)y + ez(y) (5)

where the coefficient e is a non-linearity parameter. The system with c = 0 represents a linear system. By substituting eqn (5) into eqn (3), the equation of motion of the system can be expressed as:

fi + 2ff~ + (1 - e)y + ez(y) = f ( r ) (6)

3 EQUIVALENT LINEAR COEFFICIENTS UNDER NON-GAUSSIAN A S S U M P T I O N OF RESPONSES

3.1 Equivalent linear system

Since the restoring force of a hysteretic system depends on the time history as well as the instantaneous displacement, it cannot be expressed as a simple function of displacement and velocity. The hysteretic part of the restoring force is assumed to be expressed by using an additional state variable z, and thus three state variables are introduced together with the displacement

G(x)

Fig. 3. Elasto-plastic type hysteretic characteristic.

and the velocity:

Yl = Y, Y2 = .)), Y3 = z (7)

It is apparent from Fig. 3 that the following relation holds:

[Y3I < 1 (8)

The time derivative process, denoted by Y3 =q(Y2,Y3), of the elasto-plastic type hysteretic characteristic is introduced in order to be able to take the time history of the system into consideration:

.Y3 = q(Y2, Y3) =

fY2 f o r l y 3 ] < l ; y 2 < 0 , y 3 = l ; y 2 > 0 , y 3 = - - I

fory2 > 0,y 3 = 1; Y2 < 0, Y3 = -1

(9) The domain of the function q(Y2, Y3) is illustrated in Fig. 4, in which q(Y2, Y3) = Y2 holds in the shaded area and on the solid lines while q(Y2, Y3) = 0 on the broken lines.

The equation of motion of a single-degree-of-freedom system is expressed in terms of the state variable as:

Yl = Y2

.1) 2 ~- - (1 - e)yl - 2~y2 - eY3 +f ( ' r ) (10)

J23 = q(Y2, Y3)

The authors have developed an approximate analytical method, consisting of modification of the equivalent linearization technique and the use of the moment equations of the equivalent linear system, to calculate

Y~

1

Fig. 2. Bilinear hysteretic characteristic. Fig. 4. Domain of the function q(Y2, Y3).

0.80

0.60

0.4-0

0.20

Non-Gaussian equivalent linearization for non-stationary random vibration of hysteretic system

, [ ' ' I

58(~+/) SStY'j-I)

-2.0 - I .0 0 1.0 Y3

2.0

Fig. 5. Marginal probability density function of Y3.

the non-stationary response of a non-linear system

17

functions as shown in Fig. 5 since Pr3 (Y3)=0 for lY3[ > 1 and Y3 can take on the values +1 for a finite time duration. The shaded areas below the Gaussian probability density curve are replaced with a couple of delta functions. The other variables, Yl and Y2, can take on arbitrary values and the Gaussian assumption is thought to be appropriate.

In order to express the truncated Gaussian probability density function with a couple of delta functions we introduce three Gaussian distributed random variables {sl, S2, 6'3} with the covariances given by:

0 - I = = = =

0"23 = E[6"2s31 , o~31 = E[s3$11 (14)

It should be noted that these are time-varying in non-

subjected to non-white excitation. 6'l°'n The non-linear system will be analyzed by this method.

The equivalent linearized equation is written as:

Yl = Y2

3~2 = - (1 - e)yl + 2(y2 - cy3 +f ( r ) (11)

))3 = C l ( r )Y2 + C2( ' r )y3

where Cl('r) and C2(r) are the equivalent linear coefficients and are functions of time variable r. These coefficients are determined in such a way that the expectation of mean square error between eqns (10) and (1 1), i.e.:

E [ e2] = E[{q(Y2, Y3) - C1 ('r)y2 - C2(7")Y3} 2] (12)

is minimized. 12 These are given by:

Cl ( r ) = E[y2]E[Y2q(Y2, Y3)] -E[Y2Y3]E[Y3q(Y2, Y3)] E [ ~ 2 ] E [ y 2] - {E[y2Y3]} 2

C2(r ) = E[y2]E[Y3q(Y2, Y3)]- E[Y2Y3]E[Y2q(Y2, Y3)] E[y2]E[y 2] - ( E[y2Y3]} 2

(13)

3.2 Non-Gaussian probability density function of response

In order to determine the equivalent linear coefficients we need the joint probability density function of Yl,Y2 and Y3, which is usually assumed to be Gaussian in conventional equivalent linearization method. 6 How- ever, with a system with bilinear hysteretic spring the state variable Y3 takes on the values [Y3 [ < 1 as shown in eqn (8), so that the Gaussian assumption is thought to be inappropriate for Y3- For elasto-plastic hysteresis variable, Y3, we may assume a truncated Gaussian density function combined with a couple of delta

stationary behavior. The marginal probability density function for Y3

shown in Fig. 5 is represented as a function of these covariances:

PY3(Y3) -- V ~ 0 - 3 exp \ 2o~3 ] rect(y3)

+ (6(y 3 + 1) + 6(y 3 -- 1)}S (15)

where:

S = "--~ erfc ( ~ 3 ) , erfc (x) = I ;

rect(y3) = u(y3 + 1) - u(y3 - 1)

exp (_~2) d~,

(16)

where 6( ) is Dirac's delta function and u( ) is a unit step function.

The joint probability density functions, Pr2r3(Y2, Y3) and Pr3rl(Y3, Yl), can be derived from the relationship that Pr3 (Y3) has to be a marginal density function of the variables Y2,Y3 or Y3,Yl. These are given as:

1 Pr2r3(Y2, Y3) - 27rD23 exp [ 2D23 ~ °~2

,.~ 0"23 -z--T-~Y2Y30-2o~ + 4 ] rect(y3)

( + 6(Y3 - 1) ~ 1 exp - v~Tr0- 2 2 ~ }

x erfc x/~D23 x/~D230- 2 Y2

+ 6(y3 + 1) ~ - ~ 2 exp ~,-~2]

erfc ( g2 + x \V2D23 v~D23 0-2 Y2 /

(17)


,q, ..re_ v ] where D22 = 0-~0~22 -- 0"42. The expectations in eqn (13) can be evaluated in each time step by using eqns (15)-(20).

0.~

1

Fig. 6. Joint probability density function of Y2 and Ya-

Pr3r,(Y3, Y,) - 27rD31 exp ~ 2D2 k d

~2 Y3Yl ~ rect ~Y3~

1 (__ y21~ + 6(y 3 - 1 ) ~ - - ~ l exp \ 2 4 )

x erfc v ~ 3 1 v~D310"1 Yl

1 (__y2~ + 6(y 3 + 1) ~ - - ~ 1 exp \ 2o~1 )

x erfc ( g l 4 0"~1 ) \v'2D31 vF2Dal 0-1 Yl

(18) /

where D23 ~_ 020-322 _ 0-43, O21 = 0-30"12 2 - 0"~1. The shape of Pr2r3 (Y2, Y3) is schematically shown in Fig. 6.

The probability density functions with respect to Yl and Y2 are given as:

1 (_y2"~ P r , ( Y , ) - V/~0"1 exp k 20"21)'

Pr2(Y2) -V'~0"2 exp \ 2 4 ] (19)

{ 1 e 10" 2 Pr~r2(Yl, Y2) -- 2rD12 exp 2D22

( y 2 _ 0-212 y 2 ~ } x k0-~ l 2~---1-1-1~yly 2 + 0~2, ] (20)

3.3 Non-linear transformation

E[y2Y2] =

E[y2Y3] =

E[y3Yl] =

where:

erf(x) = ix exp (_~2) dE jo

The second moments of the state variables Yl,Y2 and Y3 are given by:

ely2] = 0-2, E[y2] = 0-2,

2 0-~erf - 0-3 exp - E[y2] =

+ ~ erfc

0"22,

2 2 / 1 "x 0"23 erf 1 ~ 1

Vrc \X/20-3,/'

" ~ 0"21 erf (21)

These moments are functions of the covariances of the previously introduced random variables {sl,s2,53} and this relation is symbolically represented in a form of a non-linear transformation T:

E[ymYn] = T(0-;k;j,k = 1,2,3);m,n = 1,2,3 (22)

For given E[y,nyn], ~ k can be determined numerically by the inverse relation:

0-2 k = T- l (E[ymYn];m,n = 1,2,3); j ,k = 1,2,3

(23)

4 MOMENT EQUATIONS AND COMPUTATIONAL PROCEDURES

The moment equations derived from the linearized eqn (1 1) are given by:

~.[y2] _- 2E[ylY2]

/~[y2] = _ aCE[y2] _ 2(1 - e)E[ylY2]

- 2eE[y2Y3] + 2E[y2f]

/~[y2] = 2C2E[y2] + 2CiE[y2y3]

E[YlY2] = - (1 - e)E[y 2] + ELY22] - 2(E[yly2]

- eE[y3yl] + E [ y l f ]

Non-Gaussian equivalent linearization for non-stationary random vibration of hysteretic system 19

I i=1 I

l J E[9..y.I (m, ~ =1, 2, 3) at ~=~,_, I

r" . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . -1

Non-l inear m o m e n t transformation

E[ymy.] = T( a~k; j,k = 1 ,2 ,3 )

{ sty.y.]} ==~ {o-~,~} T-1 non-Gaussian Gaussian

1 0]k (j, k = 1, 2, 3) at r = r~-i

1 I Non-Gauas ian probabi l i ty dens i ty [ I funct ion for {Yx,Y2,Y3}

t.. . . . . . . . . . . . . . . . t . . . . . . . . . . . . . . . _1

Equivalent l inear coefficients

C1, C~ at r = [ r i - l , ri)

1 Second order moment equat ions

J E[9.y.] (m,. = 1,2,~) ~t ~- = ~, J

I Fig. 7. Flow chart for response computation.

~ ' [Y293] = CIE[y 21 - eE[y 2] + (C2 - 2()E[y2Y31

- (1 - e)E[y3Yl] + E[y3f ]

L'[Y3Yl] = CIE[yly2] + E[Y2Y3] + C2E[Y3Yl] (24)

The details o f the computational procedure for solving the moment equations have been reported in previous papers 6J°'ll and a brief description will be given here. The flow chart is given in Fig. 7. We take the equispaced time sequence r0,Tl, T2, . . . , and assume that the values o f the moments E[ymYn] at r = Ti-l are given. The covariances a]k at T = Ti-l are computed through the inverse transformation T-l; thus, the probability density functions (15)-(20) and the equivalent linear coefficients (13) are determined. We further assume that the equivalent coefficients take on the constant values during each small interval, say ['r i_ 1, "ri)- Then, we can obtain a dosed form solution o f the linearized equation using impulse response functions and initial conditions

1.0

~,o.s

/ ~ 1 ' I * I L \ ::::::5

I0 20 30 T

4-0

Fig. 8. Exponential envelope function; a = 0.125, b = 0.25.

at Ti- 1, and we can solve the moment equations (11) by R u n g e - K u t t a - G i l l scheme and thus finally obtain the moments E[ymy,] at r = T i. In comparison with conventional equivalent linearization techniques, the key factor o f the present method lies in the procedure o f non-Gaussian transformation boxed by the broken line in Fig. 7.

5 D I S S I P A T E D ENERGY

The energy dissipated by hysteretic spring during the time interval [0, T] is approximately given by:

W = I : g ( y ) ) ~ d ~ - - 1 { g ( y ) } 2 (25)

where the second term represents the amount of potential energy to be recovered by the linear response at the last unloading stage. The expectation o f the dissipated energy is represented as:

E[W] = [rE[g(y))]dr-½E[{g(y)}2] (26) J O

101 I I

8=2

I

10-I

& r~ 10_2

10-3

10-t i i 0 1.0 2.0 3.

o)

Fig. 9. Power spectra of input noise; ~ = 0-1, A = 1, a0 = 1.

20 3 . 0

2 . 0

1 . 0

' I ' I

K. Kimura, H. Yasumuro, M. Sakata ( L 20

ao=2 -

I ~ I I I i 0 10 20 30 4-0

7- ~ '=0 .1 , E = 0 . 5 , A = I , B = 2

- - Present method(Non-Gaussian); . . . . Previous method(Gaussian); o Simulation

Fig. 1O. Transient rms displacement response to unit step envelope.

Substitution of eqn (5) leads to:

I: E[W] = e E[y2Y3] dr - e(1 - e)E[y3yl]

- ~ E [ y 21 + e(1 2 e)E[y 21 (27)

6 NUMERICAL RESULTS

6.1 Excitation

A random forcing function has been assumed to take the form of a product of a deterministic envelope function and a Gaussian stationary non-white noise with an arbitrary correlation function. In the following compu- tations, a unit step function U(T) or an exponential function is used as an envelope function; these are respectively:

eCT) = U(T)

e(r) = (e -"~ -- e-b~)/max (e -"~ -- e -b,) (28)

The exponential function may be used 13 to simulate

l . O ' I L t ~ I

:+

a0=l

0 10 20 30 $0

T

~o.5 ~.~

( = 0 . 1 , ~ = 0 . 5 , A = I , B = 2 - - Present method(Non-Gau~.sian); . . . . Previous meflmd(Gaussian); o Simulation

Fig. 11. Transient rms response of Y3, hysteretic restoring force, to unit step envelope.

~ o

0 - - 10 20 30 ¢0 T

( = 0 . l , ~ = 0 . 5 , A = I , B = 2 - - Prcsalt method(Non-Gau&sian); . . . . Previous method(Gaussian); o Simulation

Fig. 12. Time variation of hysteretic energy dissipation to unit step envelope.

the envelope characteristics of different sorts of earthquake excitation with appropriate selection of constants a and b, and the function's shape is shown in Fig. 8. Noise with an exponentially decaying harmonic correlation function 14 in the form of:

RCv) = R 0 e -~lvl cos pv (29)

is chosen as a typical example of non-white noise. Constants a and p represent the bandwidth and the dominant frequency of the input power spectrum, respectively. The magnitude of the input noise, R0, is represented by the stationary standard deviation displacement response, a0, of the corresponding linear system. Two parameters, A = a l l and B = p/V/1 - ¢2, are introduced for associating the characteristic of frequency response of the corresponding linear system with that of the input power spectrum.

The relationship between R0 and cr 0 is given by: t5

~°2 = ¢2(1 + A) 2 + (1 - ¢2)(1 + B) 2

A - B + 2 } -I ¢2(1 + A)++-(-i-_- ~)(1 _ B) 2

2 . 0

(30)

J I ' i t + J

r I r I I I i 0 10 20 30 4-0

7" ~ = 0 . 1 , , ¢ = 0 . 5 , A = I , B = 2

- - Present method(Non-Gaussian); . . . . Previous method(Gaussian); o Simulation

F|g. 13. Transient rms displacement response to exponential envelope.

Non-Gaussian equivalent linearization for non-stationary random vibration of hysteretic system 21 1.0 ~ I t [ i I t 2.0 I I I l

aO=2

~ ; ~ 0 . 5

0 I0 20 30 4.0

4 = 0 . 1 , E = 0 . 5 , A = I , B = 2 T - - Pr~ent method(Non-G,'tussian); . . . . Previous method(Gnussian); o Simulation

Fig. 14. Transient rms response of Y3, hysteretic restoring force, to exponential envelope.

~ l . o --1

The shape of the input power spectra is shown in Fig. 9.

6.2 Simulation

Non-stationary response is computed by digital simulation. Noise with a correlation function in the form of eqn (29) was generated as a series of cosine functions with weighted amplitudes, evenly spaced frequencies and random phase angles 16 and 2000 runs were performed to estimate the expectations.

6.3 Numerical results

The computed results for the transient rms displacement and rms restoring force responses and also the transient mean dissipated energy to the unit step envelope function are respectively shown in Figs 10, 11 and 12, while those to the exponential envelope function are shown in Figs 13, 14 and 15, respectively. It is seen that the results for the present computational method are in good agreement with the simulated results and the computation with the assumption of the Gaussian

6 . 0 ~ I J I ~ I h

00=2 |HIH,H H H'''''''H °''"q

¢.0

2 . 0

¢ o ° ~0=1 _

o ........... ,..,, . . . . . , , . , ,~ . . . . . . . . ,.., . . . . . . . . . . / ~ _ " _ " _ " _ " " '_"_''"'_' '_"-'"-' '-"-"-"'-'i

oO ___- .....................

0 -- 10 20 30 40

"7" ( = 0 . 1 , s = 0 . 5 , , 4 = 1 , B = 2

- - P~esent method(Non-Gausstan); . . . . Previous method(Gaussian)" o Simulat on

Fig. 15. Time variation of hysteretie energy dissipation to exponential envelope.

0 -1.5

I I

-!.0 - 0 . 5 Ya 4 = 0 . 1 , E = 0 . 5 , o 0 = l A = I ,

0.5 I

I

.0 1.5

B = 2 , z = 1 0

- - (box) Present method; . . . . (box) Simulation; . . . . (line) Gau~ian

Fig. 16. Shapes of probability density function of Y3.

distribution 6 (broken lines) is significantly improved by the present method (solid lines). Figure 16 shows the comparison of the shapes of the probability density functions of y3 in the case of the unit step envelope input obtained by the present method, simulation and Gaussian response assumption. It is observed that the shape of the non-Gaussian probability density function using a truncated Gaussian function and delta functions agrees well with the simulated results.

7 CONCLUSIONS

Non-stationary response of a single-degree-of-freedom system with bilinear hysteresis subjected to amplitude modulated non-white random excitation is analyzed using the equivalent linearization and moment equation techniques which employ a non-Gaussian probability density function consisting of a truncated Gaussian probability density function and a couple of delta functions. The computed results for the mean square responses, the mean dissipated energies and the shapes of probability density functions agree well with the simulated results.

REFERENCES

1. Roberts, J.B. & Spanos, P.D., Random Vibration and Statistical Linearization. John Wiley, Chichester, UK, 1990.

2. Wen, Y.K., Methods of random vibration for inelastic structures. Appl. Mech. Rev. 42 (1989) 39-52.

3. Iwan, W.D. & Paparizos, G., The stochastic response of strongly yielding systems. Probabilistic Engineering Mechanics, 3 (1988) 75-82.

4. Caughey, T.K., Random excitation of a system with bilinear hysteresis. J. Appl. Mech., 27 (1983) 649-52.

5. Crandall, S.H. & Zhu, W.Q., Random vibration: A survey of recent developments. J. Appl. Mech., 5ll (1983) 953-62.

6. Kimura, K., Yagasaki, K. & Sakata, M., Non-stationary response of a system with bilinear hysteresis subjected to non-white random excitation. J. Sound & Vib., 91 (1983) 181-94.


7. Kimura, K., Yagasaki, K. & Sakata, M., Random vibration of a beam structure with elastoplastic hysteresis. In Proc. 9th World Conf. Earthquake Eng., Tokyo, Vol. V, 1988, pp. 241-6.

8. Shinozuka, M. & Henry, L.R., Random vibration of a beam-column. J. Eng. Mech. Div. ASCE, 91 (1969) 123-43.

9. Bolotin, V.V., Statistical Methods in Structural Mechanics. Holden-Day, 1969.

10. Sakata, M. & Kimura, K., Calculation of the non- stationary mean square response of a nonlinear system subjected to nonwhite excitation. J. Sound & Vib., 73 (1980) 333-43.

11. Kimura, K. & Sakata, M., Moment equations approach to nonstationary responses of a nonlinear system subjected to nonwhite random excitation. In Computational Stochastic

Mechanics, ed. P.D. Spanos & C.A. Brebbia. Elsevier, 1991, pp. 267-78.

12. Lin, Y.K., Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York, 1967.

13. Shinozuka, M. & Sato, Y., Simulation of nonstationary random process. J. Eng. Mech. Div. ASCE, 93 (1967) 11-40.

14. Barnosky, R.L. & Maurer, L.R., Mean square response of simple mechanical system to nonstationary random excitation. J. AppL Mech. ASME, 36 (1969) 221-7.

15. Sakata, M. & Kimura, K., The use of moment equations for calculating the mean square response of a linear system to nonstationary random excitation. J. Sound & Vib., 67 (1979) 383-93.

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non-gaussian equivalent linearization for non-stationary random vibration of hysteretic system

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