Monte Carlo

damping

ISBN905«J9,B80

hysteretic

Biot

witl1

system

&Spanos (eds)~ 200' 88JI8tn8. ~.

nonlinear

a

of

simulations

Monte

Carlo

Ab1feCaItJs.",.IkNJ. ~

P.D.Spanos

I.B.Ryon C.llairin Engineering,Rice University,Houston.Tex..UM Spyro

Tsavachidis

MechanicalEngineering.Rice University,Houston.Ta. USA

ABSTRACT:

In the paper the Biot hysteretic model involving an infinite collection of elementswhosedynamic characteristicsare specified through a probability density-like function is re-examined;in the limit case,the Biot model yields a dynamic systemwith ideal hystereticdampingwhich is known to be problematic for random vibration analysis. It is shown that bona fide Monte Carlo simulations can be conductedfor the numerical

The statisticallineari-

a

of

dynamics.

system

accuracy

the

the

of assess

to

used

representation

the

in

further

are

studies

encountered Carlo

integrals

Monte

the

to

of pertain

calculations

which

double

by treating,cautiously,the integrodifferential equationwhichis involved;this is basedon recur-

model

results

sive

Biot

zation procedureadoptedin detennining the responseof the hystereticallydampedsystem to white noise.

1 INTRODUcnON

m

for where

(5)

is

model

this ;(t)],

for f-

where x(t) is the Hilbert Transformof x(t), that is, . I (6)

-ft.

K_t-t

~

domain,

frequency

the

in

written,

be

can

(5)

(7)

F(m)=k(I+f1isgn(m)]X(m),

any

at

exci-

the physically

is

of

output

the future

is,

model

the

that

the

that

shows

on

system, depends clearly

instant

non-causal

is

a

time

This

tation.

given

it

where the functions of frequencyF(ro) and X(ro) are the Fourier transformsof F(t) and x(t), and the function sgn(ro)is the signum function. This model yields a loss modulus independentof frequency.However,

into

model

damping

hysteretic

ideal

unrealizable. There have been some attemptsto incorporate

where A is a constant,and (I) is the circular frequency of oscillation. That is, the energy dissipation per cycle is proportional to the frequency of oscillation. Thus, the viscous damper is unable to accountfor the phenomenonof hystereticdamping. Early linear hysteretic damping models have been, the complex stiffness model (Soroka 1949), frequency dependentdamping (Bishop 1955), and the so-called 'ideal hysteretic damping' (Crandall 1991). The restoring force for the complex stiffnessmodel is P(x)- k(1+ i1\)x, (3)

as

Equation

(2)

,

Am

=

E.,...

If sinusoidal motion is considered, the energy dissipation per cycle is given by

11

force F(t)

(1)

dt

a(t)=--

P(x)=c~.

rorepresents thefrequency of a si-

the symbol

nusoidal input. The models given by equations(3) and (4) are well suited for steady-stateharmonicvibrations. For an arbitrary input, however,the use of frequency dependentparametersin the time domain can lead to dubious results suchas a complex-valued responseto a real-valuedinput. This problemmay be addressed by the ideal hystereticdampingmodel.The +

that

x(t)

showed

the

1950)

k(x(t)

(Lazan

=

1935),

many engineering materials the internal damping is such that the energy dissipation per cycle of deformation is almost independent of the frequency at which the stressis applied. This discovery has necessitated a mathematical model that accounts for this phenomenon.Note that the commonly used viscous damper, introduced by Rayleigh (Rayleigh 1877), producesa force that is proportional to the velocity of motion. That is,

restoring

Early

experiments(Kimball &. Lovell 1927), (Wegel

&. Walther

where k is a stiffness parameter, i = .J=I, and '1 is the loss factor of the element. Alternatively, the restoring force can be expressedas F(x)=k(1+i..!l)x, (4)

(J = IJJrJfe-r,C'-t)x(t)dt.

rithm describedin (Inaudi & Kelly 1995) introduces bounds for the values of the loss factor fl. Specifiin

(SDOF)

with

ideal

x(t)dt.

e-r,c'-t)

r.bjrJf

kx+

(1)=

Jol '. Equation (13) can be seen as a weighted sum with weights b rJ. If the number of spring-dashpotelements is allowed to tend to infinity, the summation is replaced by integration. In this casethe weights are replacedby a weighting function g(r) .That is,

studies of a single-

system

hys-

teretic damping (Crandall 1963). Spanos and Zeldin

proved that a quiescentSDOF systemconsisting of a mass appended to an ideal hysteretic element is un-

(I) = kx+ Jim~ lJiri

initial

quiescent

with

system

the

Therefore,

sense.

N

stablein the bounded-input-bounded-output (DIDO)

I

been made for random vibration degree-of-freedom

resultsclearly show the inappropriatenessof the ideal

g(r)

(14)

;(t)d'Mr.

e-r(I-t)

as £

>

r

g(r)

I. defined

2kT1

(IS)

T1is the lossfactorof the Biot modelandEis Theresponseof the Biot elementcan be

where

Biot introduced a linear visco-elastic model for hysteretic damping that satisfies the causality requirement (Biot 1958).Biot's hystereticdamperconsistsof an infinite number of spring - viscousdampercombinationsconnectedin parallel, seeFigure I.

r