Linear dynamic analysis of SDOF systems with random properties subject … · 2014-05-19 · Linear...

Linear dynamic analysis of SDOF systems

with random properties subject to

earthquake excitations

H.U. Koyliioglu*, S.R.K. Nielsen^ A.S. Cakmak*

"Department of Civil Engineering and Operations,

Princeton University, Princeton, NJ 08544, USA

^ Department of Building Technology and

Structural Engineering, University of Aalborg,

DK-9000 Aalborg, Denmark

ABSTRACT

The present paper deals with the dynamic analysis of linear single de-gree of freedom (SDOF) systems with random mass, random stiffness andrandom damping subject to random seismic loading. The importanceof considering random system model is implemented with a detailed ex-amination of the impulse response function of SDOF systems and withseveral exact solutions. Response characteristics are initially determinedon condition of a specific realization of the structural system. Then, totalprobability theorem is applied to evaluate unconditional expectations ofthe response. Conventional zeroth, first and second order perturbationsolutions are compared to the exact solutions for the response statistics.Random damping and random stiffness effects are investigated separatelyconsidering eigenvibrations, stationary response under harmonic loadingsand horizontal nonstationary earthquake excitations. For eigenvibrations,in both cases, perturbation solutions carry divergent secular terms. Whenonly random stiffness is considered, due to the secular divergent terms,perturbation solutions blow up with time, even for very small variabilitiesof the random variables. In the case of only random damping, as the di-vergent secular terms are under the governing control of the exponentialdecay, the existing deviations in the perturbed solutions become neitherobservable nor important. Hence, perturbation solutions don't divergefrom the exact solution and are observed to be very good approximations.Under harmonic loadings, after the dissipation of eigen vibrations, per-turbation solutions don't possess any secular terms and are very good ap-proximations except for the cases in which the system is excited at or closeto its resonance frequency. Finally, the nonstationary random response ofstochastic SDOF systems subject to nonstationary random seismic exci-

Transactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

776 Soil Dynamics and Earthquake Engineering

tations are studied. It is concluded that perturbation solutions are goodapproximations in the case of random damping. However, in the case ofrandom stiffness, significant deviations are observed, especially for shortduration earthquakes. On the other hand, perturbations solutions seemto be admissable and useful in the estimation of the maximum varianceof the response which is expected within the first few natural periods.

INTRODUCTION

Random vibrations theory of deterministic structures, in which only theloading is considered as a random process, has been well developed. Sev-eral papers, review articles and books have been published in this field.The present study applies the principles of random vibrations theory ofdeterministic structures to random vibrations theory of stochastic struc-tures. Only SDOF systems will be covered in this paper. Based on thefundamentals of this study, the extension to multi-degree of freedom sys-tems will be investigated in a forthcoming paper.

All structures possess uncertanties in their material properties and/or ge-ometry which are due to physical imperfections, model inaccuracies andsystem complexities. Such uncertainties are spatially distributed overthe structure and can be mathematically modeled using either randomvariables or random processes which may be functions of time and/orspace. Random vibration analysis of stochastic structures considers ran-dom structural models under random loading. It should be pointed outthat the utilized random model must fulfill all the physical requirementsof the problem with probability 1.

Stochastic structural models have been considered by several researchers.In this respect, eigenvalue analysis of the random systems was attackedabout twenty-five years ago and some reconsiderations or applications ofearlier studies appeared recently. Random eigenvalue problems have beendiscussed by Boyce [5], Fox and Kapoor [24], Collins and Thompson [9],Shinozuka and Astill [40], Hasselman and Hart [29], Grigoriu [27], Na-gashima and Tstusumi [38], and, Zhu and Wu [55].

In 1980's, the analysis of the response variability of stochastic systemsreceived a lot of attention, consequently a new field, " Stochastic FiniteElements " was coined to stochastic mechanics. Although there have beenpapers on Monte Carlo solutions and reliability considerations, most of thestudies done in stochastic finite elements have been on the second momentanalysis of stochastic systems under deterministic loading.

An account for the state of the art in 1986 was given in a review articleby Vanmarcke et al. [50] in which local averaging technique, Monte Carlosimulations and reliability considerations are stressed out. Following de-velopments till 1988 were reviewed by Benaroya and Rehak [4]. In thisreview, available perturbation and linear partial derivative methods for


Soil Dynamics and Earthquake Engineering 777

second moment methods are discussed. Another good review in recentdevelopments in finite elements, in which different discretization methodsand reliability applications are outlined, is given by Der Kiureghian etal. [21]. Recently, an excellent update review has been provided in a re-port by Brenner [7]. Brenner gave a comprehensive overview of the basicresearch in static stochastic finite elements in a survey concerning theoret-ical backgrounds, strengths and weaknesses of various available stochasticfinite element methods. The comparisons in this report are mainly focusedon discretization schemes and future reliability applications.

As noted in these reviews, the evolution of stochastic finite element meth-ods took place in the last decade. Handa and Anderson [28] employed firstorder perturbation method in connection with stochastic finite elements tosolve simple static problems. Hisada and Nakagiri [30] incorporated sec-ond order perturbation method to static stochastic finite elements prob-lems when structures have uncertain shapes. Vanmarcke and Grigoriu[52] used local averaging technique of Vanmarcke [49] , a way of discretelyrepresenting random fields with the finite element method, to evaluatethe second-order statistics of a beam with random rigidity under staticloading. Liu et al. [35, 36, 37] developed an approximate second-orderperturbation technique for linear and nonlinear static and dynamic prob-lems. For dynamic problems, they observed good results in the mean, butsignificant deviations in the variance with time were present even if withvery small variabilities of the random variables. In their example, thecoefficient of variation for stiffness was taken as only 0.05. Lawrence [32]introduced basis random variable method in connection with Galerkin fi-nite elements. The principle in this method is to expand the randomfields as the sum of random variables multiplied by linearly independentand preferably orthogonal deterministic functions. Yamazaki, Shinozukaand Dasgupta [54] used Neumann expansion technique in static stochasticfinite element problems to compare Neumann expansion technique withperturbation methods and Monte Carlo solutions. Spanos and Ghanem[43] proposed employing Karhunen-Loeve expansion for discretization ofthe random fields followed by Neumann expansion method for the solutionof the static random equations. Later, Ghanem and Spanos [26] introducedthe polynomial chaos concept to stochastic finite elements. In this method,the random response is simply expressed as a convergent series along anorthogonal polynomial basis. Der Kiureghian et al. [21] proposed a newdiscretization method named optimum linear estimation which is merelyan interpolation method between nodal points where optimal determinis-tic interpolation functions are selected. Following the fundamentals of Liuet al. [35]'s work, Teigen et al. [47, 48] investigated nonlinear concretestructures under static random loads. Recently, Chang and Yang [8] de-veloped a mean-centered approximate second-order perturbation methodin conjuction with modal expansion to solve nonlinear dynamic problems



with structural uncertainties for short durations. They employed equiv-alent linearization with Gaussian closure to treat nonlinearities and thelocal averaging method of Vanmarcke [49] to discretize random fields.

In all these aforementioned methods, the representation of the randomfields by a small number of random variables is not exact and usuallymesh dependent. The consequent limits on the finite-element size whichdepends on the correlation length of the stochastic field urges the analystto use very fine meshes for shortly correlated fields. In a series of pa-pers, Deodatis, Takada and Shinozuka introduced a new method in whichGalerkin finite elements with deterministic shape functions is applied tostochastic differential equations. This new method named "Weighted In-tegrals" overcame the major drawbacks of the other methods, i.e. theresults are mesh independent and random fields can be accurately repre-sented by a small number of random variables of interest, Deodatis [13,15, 17], Takada [44, 45, 46], Deodatis and Shinozuka [18]. There have alsobeen some simple studies on stochastic shape functions, e.g. Dasguptaand Yip [12], Deodatis [15, 16].

The ultimate goal of stochastic response analysis is reliability consider-ations with limit states ranging from serviceability and applicability re-quirements to total collapse. In this respect, reliability of stochastic sys-tems have been discussed by Der Kiureghian and Liu [19, 20], Liu and DerKiureghian [34], Ghanem and Spanos [26], Lawrence et al. [33], Deodatisand Shinozuka [18], Der Kiureghian et al. [21]. In the mean time, spectral-distribution free bounds for the second-order moments of the response ofstochastic systems are proposed by Shinozuka and Deodatis [42], Deodatisand Shinozuka [14], Deodatis [15, 16].

The motivation of the present study arises from the fact that the presentmethods of analysis for random structures subject to deterministic or ran-dom dynamic loadings are not satisfactory, even in the case of small vari-ability of the random structural properties. As follows from the title ofthe paper, and in order to clearly emphasize the reasons for the shortcom-ings of the previous attempts, only linear SDOF systems are considered.However, the SDOF equation is assumed to be obtained from a modalexpansion of a structural system retaining as an uncoupled term for eachmode.

Mass, damping and stiffness of the SDOF system are assumed to be ran-dom functions of a finite set of basic variables X, defined from the dis-cretization of the random structural field. The probabilistic structure ofX, e.g. in terms of the joint probability density function, /x(x) is assumedto be known.

For a specific realization of X = x, the response of the SDOF system,y(t; x), can be calculated. Then, the unconditional expectations E[y"(t\ X)]



can be obtained exactly using the total probability theorem.

E[y"(t;X)] = ...y"(t;x)/x(x)dx (1)

R."

where /x(x) is the joint probability density function of vector X.

Using second order Taylor's expansion approximation for y(f; X) aboutthe mean value XQ = E[X] provides

; X) = y(t;xo) + yj(f;xo)A + yjt(xo)A%,A%t (2)

where A%, = Xj - E[Xj], y,j(*;x<,) = af-y(*;x<,) and y,,,(f;xo) =

dX*dX- yfa Xo). In equation (2), summation over the dummy indices mustbe performed. Following (2), it is straight forward to obtain the first andsecond order perturbed approximations for the mean value function, / y(£),and the variance function, <7y(Z).

= y((;xo) 4- t Xo) (3)

(4)

where Cjk = E[&Xj A-X"*] signifies the components of the covariance ma-trix. First order perturbation solution for the mean and the varianceconsists of only the first terms of (3) and (4). Thus, for the mean valueszeroth order perturbation solution, the solution for the mean deterministicsystem, is the same with the first order perturbed solution. If the thirdand fourth order terms are neglected in the variance, equation (4), thesolution must be called approximate second order perturbation solution.Because, in this case, although the mean is second order accurate, thevariance becomes first order accurate, e.g. Liu et al.'s work [36]. Hence,the variance of the first order solution and the approximate second ordersolution are the same.

Using Gaussian closure to evaluate the third and fourth order centralmoments simplifies equation (4) to

(*;xo)Cj-* + -y,;&(t; xo)y,fm(*; Xo)Cj,Ctm (5)



Along the text, exact solutions are compared with mean centered zeroth,first and second order perturbation solutions in several problems. The re-sponse of stochastic SDOF systems under eigenvibrations, harmonic load-ings and horizontal earthquake random nonstationary earthquake loadingsare studied in detail.

STOCHASTIC SDOF SYSTEMS UNDER EIGENVIBRATIONS

The equation of motion of a linear SDOF system with viscous dampinghas the following form.

my + cy + ky = /(*) y(0) = yo y(0) = yo (6)

where the combined stochastic variables m = m(X), c = c(X) and k =fc(X) are respectively mass, viscous damping and the stiffness, y denotesthe displacement of the SDOF system, yo and yo are the initial values forthe displacement and velocity, and, f(t) is the time dependent loading.

Omitting the explicit indication of the dependency on the structural sys-tem parameters, X, equation (6) can be rewritten as

m( y + 2(woy +wgy ) = f(t) (7)

where

"I = — C = —— (8)° m ^ ^ '

wo denotes the undamped circular natural frequency and ( is the viscousdamping ratio, characterizing damping as a nondimensional number.

The impulse response function ( Green's function ) for the differentialoperator of equation (7) is

sinh(\/(2 — IwoZ) , (^ > 1, f > 0

h(t; X) = { ~ exp(-wo^) , C = 1, * > 0 (9)

, (<1, f>0

The displacement can always be written in the integral form as

fty(*;X) = [c/i(*) + mA(*)]yo+m/i(*)yo + / h(t - r)f(r)dr (10)

Jo

Consequently, the eigenvibrations, f(r) == 0, can be written as

y(t; X) = [ch(t) -h mA(t)]yo -f mfc(t)yo (11)



Since for most of the civil engineering structures, damping ratio varies from0.01 to 0.10, only the underdamped systems where £ < 1 is considered inthis study. The Green's function for the underdamped SDOF systems interms of the physical parameters of equation (6) is

sin | \l771

(12)

In what follows, the Green's function of equation (12) is studied in detail.Because it is clear from equation (10) that response under any forcingfunction and any initial values for the displacement and velocity can easilybe calculated using the Green's function.

Since ra, c and k depend on the basic random variables of the system,X, the Green's function, A(*;X), as denned by equation (12) becomesa stochastic function which is generated by X. Despite the linearity ofthe equation of motion, Green's function is a highly nonlinear function ofthe random quantities, so does the solution which is evaluated from theDuhamel's integral, equation (10).

In the case of eigenvibrations, the first order derivative of the displacement(11) with respect to Xj, yj(t; X), is proportional to t, the second derivativeyjk(t] X) is proportional to t*, etc., which imply a behaviour identical tosecular terms in singular perturbation methods using the Taylor expansionof the response. This questions the application of all the classical first andsecond order perturbation techniques for the mean and the variance ofthe response , at least in the case of eigenvibration problems and externalexcitations with impulsive character. Actually, such defects have beenobserved by several researchers, see e.g. Frisch [23], Nakagiri et al. [39],Vanmarcke et al. [50], Liu et al. [36].

0.5 1 1.5 2 2.5 3 3.5 4eigen periods

Figure 1. The exact E[h(t,m,c^k)] and its first ( zeroth ) and secondorder perturbed approximations versus eigenperiod nondimensional time.m = /%m = 1, c and k are mutually independent parabolically distributedwith, respectively, PD(b, = 0.10, j = 0.04) and PD(b* = 1.00, - =

0.20).



As an example, consider the case f(r) = 0, yo = 0, t/o = "•• Theny(£, m,c, A;) = /i(i, m,c, A;). The mean value of the stochastic Green'sfunction, E[h(t, m,c, A;)], and the first and second order perturbed ap-proximations of this quantity as a function of eigenperiod nondimensionaltime are plotted in Figure 1. For simplicity, ra is assumed to be determin-istic whereas c and k are modeled as mutually independent parabolicallydistributed random variables. Parabolic distribution as well as other prob-ability distributions refered in this paper are summarized in Appendix 1.It should be noted that all the physical bounds of the problem, i.e. k > 0,c > 0, £ < 1, are fulfilled for all these probability distributions with prob-ability 1.

The numerical values chosen for this example and for all the followingexamples in the text follow the legend under the figures. The computa-tions for expectations at each time is carried out using Mathematica [53].As shown in Figure 1, the first and second order perturbations for themean of the stochastic Green's function differ sooner or later significantlyfrom the exact solution. The second order approximation only postponesthe time of deviation. The deviation of the perturbation solution getsworse in time. Therefore, the application of conventional mean centeredperturbation methods for long durations is not able.

The zeroth order perturbation solution represents the response of the de-terministic SDOF system with mean parameters, /i™, /%c, /i&. The factthat this solution is significantly different than the mean of the stochasticresponse, should convince engineers that stochastic analysis is definitelynot only worthwhile but also essential.

1.5

0.5

Gaussian Closure /

0 0.5 1 1.5 2 2.5 3 3.5 4eigen periods

Figure 2. The exact Var[h(t, m, c, A:)], first order perturbation and secondorder perturbation with Gaussian closure of this quantity versus eigenpe-riod nondimensional time for the example of Figure 1.

For the same problem, exact variance of the response, Var[/i(£, m, c, fc], itsfirst and second order perturbations with Gaussian closure, equation (12),versus eigenperiod nondimensional time are plotted in Figure 2. Althoughgood fits are observed for very short durations, both first and the secondorder perturbations with Gaussian closure blow up with time.



-5 Second order, •'

0 1 2 3 4 5 6eigen periods

Figure 3. Random stifness analysis. The exact E[h(t,m,k)] and pertur-bation approximations versus eigenperiods. m = firn = 1, k is uniformlydistributed in [0.70,1.30] with /** = 1.00 and cr*. = 0.1732.

The difference in the amplitude of the curves in Figure 1 is not governedby the damping random term on the exponential multiplied with time. Itis mainly due to the random stiffness term and the change in phase is dueto the random damped frequency. In order to clearly show these effects, ananalytical solution for the mean value of an undamped stochastic Green'sfunction h(t,m,k) is derived. For simplicity, fc, or equivalently Wg, isconsidered as a random variable uniformly distributed in the interval [/it —\/3<7&, Hk + \/3crjk], where p* = E[k] and v\ = Var[fc]. With m = 1, it canbe shown that

E(h(t,m,k)] =

This exact solution and perturbation solutions versus eigenperiod nondi-mensional time are plotted in Figure 3. For the present undamped case,the zeroth order solution is an undamped sine curve with a period de-termined by the stiffness pk- As shown in (13), the exact solution willdamp out inversely with time. This effect is present for all the consid-ered distributions, see Fig. 5.a, and will be termed stochastic artificialdamping.

10, . , , • 1 'o _~ Gaussian Closure /

e 6

S 4

0 0.5 1 1.5 2 2.5 3 3.5 4eigen periods

Figure 4. Random stiffness analysis. The exact Var[h(t, m, k)] and per-turbation approximations versus eigenperiod for the example of Fig. 3.



For the same example, the variance of the Green's function is plotted as afunction of eigenperiod nondimensional time in Figure 4. Exact variancestabilizes after some oscillations and the perturbations are meaninglessafter a few periods. First order perturbation and Gaussian closure resultsdiverge with second order and fourth order errors, respectively.

Uniform— Parabolic

Triangular

0 32 4eigen periods eigen periods

Figure 5.a ) Only random stiffness analysis. Probability density functiondependency of E[h(t, m, fc)]. Figure 5.b ) Probability density function de-pendency of Var[h(t, m, k)}. m = /%^ = 1, k is a, respectively, uniformly,parabolically and triangularly distributed random variable with fj,k = 1and (7k = 0.1732.

Figures 5.a and 5.b show the corresponding results for Figure 3 and Figure4 in which the mean and the variance of k is kept constant while theprobability density function (pdf) assigned to it is altered. These resultsshow that the mean and the variance of /i(t, ra, k) is not strongly dependenton the pdf of fc beyond the second moment properties. Hence, a betterapproach for the solution of any transient vibration problem than usingsecond moment based perturbation methods is to assign an engineering-wise acceptable pdf to k with mean /%& and standard deviation <%& and tocarry out the analysis accordingly to obtain a closed form solution suchas (13).

First order— Second order

Exact

4 5 6eigen periods

Figure 6. Random damping analysis. Exact £[/i(t,m,c, k)} and its per-turbation approximations versus eigenperiods. m = fj,m = 1, fc = /i* = 1and c is parabolically distributed with PD(fAc = 0.10, = 0.04).

In order to investigate the random damping effect, a similar analysis for



damped impulse response function, h(t, m,c, &), is worked out in Figure6. Stiffness and mass are kept deterministic and only damping is assumedto be a parabolically distributed random variable. In this case, the arisingsecular terms of perturbation solutions are under the governing controlof the exponential decay, thus, very good results are observed for theperturbed solutions. For lightly damped systems, the secular terms of theperturbation approximations only become significant after the dissipationof the eigen vibrations. Hence, perturbation solutions will never divergeand are very good approximations for lightly damped systems. In order toshow the governing control of the exponential decay, respectively an exactsolution and its second order approximation are given analytically in (14)and (15) assuming c to be uniformly distributed in [fie — \/3<Jc, Me + VocrJ.

-"**(e '** - e~ ***) (14)

(15)

In this case, second order perturbation underestimates the exact result.For PC = 2<7c and a<.t — 1, approximately 5.1 percent error is present afterthe eigenvibration has decayed to one fifth, thus, the absolute differencebetween the exact solution and the second order perturbed solution isapproximately 1 percent of the amplitude of the initial eigenvibration.Although the error slowly tends to grow up to 100 percent as t — > oo,because of the exponential decay term, (14) and (15) diminish faster.Therefore, the differences are neither observable nor important. Similarresults are also valid for the second order statistics of the response.

In conclusion, the present analysis has shown that for eigenvi brat ions,stochastic response of the SDOF with random stiffness cannot be deter-mined using perturbation methods due to secular terms in the expansion.Extensions to third and higher order will not improve the solution. On thecontrary, secular terms of the corresponding order will arise, thus the so-lution will blow out faster. When only the damping coefficient is random,perturbation solutions are seemed to be very good approximations.

STOCHASTIC SDOF SYSTEMS UNDER HARMONIC LOADING

The previous section dealt with the response due to initial values andimpulsive type of loadings. In this section, the opposite limit is studied,where the transient effects have decayed and stationary vibrations havebeen attained.

Under the harmonic excitation, f(r)

/(*)== rlcos(Of) (16)



with a deterministic circular frequency fl and a deterministic amplitude A,the stationary response of the SDOF system for each realization becomesharmonic with the same circular frequency ft.

t; m, c, fc) = \H(Sl)\A

where

(17)

(18)

(19)

In this case, the partial derivatives of y(t', m, c, k) with respect to m, cand k of any order will not contain any secular terms. Hence, there is notendency that the first and second order perturbation solutions eventuallydeviate arbitrarily from the exact solution, as in the transient case.

0.4

~ 0.2

^ 0w -0.2

-0.40 0.5 1 1.5 2 2.5

eigen periods

Figure 7. Stationary harmonic vibrations. Exact E[y(t, m, c, k)] and itsperturbation approximations versus eigenperiods. ft = 2, ra = /Xm = 1,c = fj,c = 0.10 and k is uniform in [0.70,1.30] with /x* = 1.00 and cr& =0.1732.

This has been shown in Figure 7 for O = 2<\/E[k]. Only k is taken as arandom variable with uniform distribution. The exact solution, first andthe second order approximations will be harmonic functions with samecircular frequency with slightly different phases and amplitudes. In thepresent case, well away from the expected resonance range, ft ~ E[k\,the random phase is very small for all realizations of the structure, andno visible phase difference between the solutions are observed. However,in the expected resonance range, substantial phase differences may bepresent along with large differences in the amplitudes of the perturbedapproximations.

Any non-impulsive external loading can be expressed in terms of a possibleinfinite sum of harmonic terms through its Fourier transform. For each



of these harmonic terms, the previous analysis is valid. Then, it can bestated that the first and second order perturbation analysis can be appliedwithout significant errors for the stationary response where any responsefrom the initial values has been dissipated.

The presented discussions on stochastic eigenvibrations and stochastic sta-tionary harmonic vibrations clearly explain the reasons for the numericalobservations of Liu et al. [36], Nakagiri et al. [39], Vanmarcke et al. [50].

STOCHASTIC SDOF SYSTEMS UNDER HORIZONTAL EARTH-QUAKE EXCITATIONS

Horizontal ground earthquake accelerations, Xg, are zero-mean nonsta-tionary random processes which are mostly modeled as time modulatedstationary stochastic processes.

X, = A(t)B(t) (20)

where A(t) is a deterministic modulation function and B(t) is a stationaryzero-mean stochastic process.

Next, an analytically tractable example is worked out to calculate the timedependent variance for the displacement of an oscillator subject to a timemodulated white-noise excitation. Let B(t) = W(t) denote unit whitenoise and y(t) signify the horizontal displacement of the mass relativeto the ground surface. Consequently, the excitation force f(t) becomes— mA(t)W(t). Under this loading the conditional variance of the displace-ment as a function of time, for each realization of the oscillator, with zeroinitial conditions can be evaluated from (5) as

t t

<%;(f) = m? f fh(t- r)h(t - t)A(r)A(T)E(W(T)W(f)}dTdt (21)

0 0

For unit white noise

EMT) (f)] = f(r-f) (22)

Substitution of (22) into (21) yields

t

(h(t-T)A(T)]*dT (23)

For many forms of time envelope, -4(t), this integral can be evaluatedanalytically, e.g. a sum of two exponentials fit.

A(t) = a(e~** - e *) (24)



where duration and the strength of the earthquake is defined by the en-velope parameters, a, /? and 7. In what follows, the random stiffness andrandom damping affects are investigated separately using A(t) in the form

ooof (24). The energy content of an earthquake is proportional to / A(t)*dt.

oKeeping this as constant and altering the envelope parameters a, /?, 7is tantamount to studying different types of earthquakes in duration andstrength with the same energy content., Three different envelope functionswith the same energy content are assigned to represent, respectively, veryshort duration, El-Centro and very long duration earthquakes, see theright corner of Figure 8.

A ( El-Centro )

- eigen periods

__ First order- Only c is random

,. Only k is random

3 4 5 6eigen periods

Figure 8. The unconditional variance E(cr*(t)\ for the stochastic SDOFsystem with m = /i^ = 1, c and k uniformly distributed in [0.03,0.07]and [0.7,1.3], respectively, subject to El-Centro Earthquake, A, (a =2.32, /? = 0.09, 7 = 1.49).



20

18

16

14

12

10

Fust order— Only c is random

Only k is random

0 OJ 1 1J 2 2J 3 3Jeigco periods

Figure 9. The unconditional variance E[cTy(t)] of the same stochasticSDOF system of Figure 8 subject to a short duration earthquake, B,(a = 533, 0 = 1.9, 7 = 1.95).

15

0.5

First order— Only c is random

Only k is random

14

eigen periodsFigure 10. The unconditional variance £[<jy(£)] for the same SDOF systemof Figure 8 subject to a longer duration earthquake, C, (a = 0.7109, j3 =0.01, 7 = 2.00).

The corresponding exact unconditional variance for the stochastic oscilla-tor and its first order perturbed approximation are plotted as a functionof eigenperiods in Figure 8, Figure 9 and Figure 10 . It is clear from thefigures that in the case of only random damping, perturbation solutionis very accurate for all three kinds of earthquakes. However, as shown inFigure 9, for only random stiffness, due to the secular terms, perturbationsolutions are not accurate for short duration earthquakes. The effect of



stochastic artificial damping can be seen clearly in Figure 9. Althoughit is not shown, as they carry higher order secular terms, second orderperturbations diverge faster. On the other hand, according to the studiedcases, perturbation solutions seem to be good in estimating the maximumvariance of the SDOF system in any case.

CONCLUSION

The dynamic response of linear SDOF systems with random mass, damp-ing and/or stiffness subject to deterministic and random excitations hasbeen studied. The conventional zeroth, first and second order pertur-bation solutions are compared to the exact solution which is evaluatedapplying the total probability theorem, to the conditional results derivedfor a specific realization of the random variables.

Random damping and random stiffness effects are investigated separatelyconsidering two limit behaviours of oscillators; eigenvibrations and sta-tionary response under harmonic loadings. For eigenvibrations, in bothcases, perturbation solutions carry divergent secular terms. When onlyrandom stiffness is considered, due to the secular divergent terms, pertur-bation solutions blow up with time. In the case of only random damping,since the divergent secular terms are under the governing control of theexponential decay, the existing deviations in the perturbed solutions be-come neither observable nor important. Hence, perturbation solutionsdon't diverge from the exact solution and are seemed to be very goodapproximations. Under harmonic loading, after the dissipation of eigenvibrations, perturbation solutions don't possess any secular terms and arevery good approximations except for the case in which the system is ex-cited at or close to its resonance frequency.

Finally, short duration nonstationary random response of stochastic SDOFsystems subject to random seismic excitations are investigated in detail. Itis observed that perturbation methods cire good approximations in the caseof random damping. When only stiffness is considered random, due to thesecular terms, perturbation solutions are poor approximations for shortduration earthquakes, yet seem to be useful in estimating the maximumvariance of the SDOF system in any case.

In summary, it is concluded that perturbation solutions for dynamic prob-lems can be used in the case of random damping. In the case of randomstiffness, due to the secular terms in eigenvibrations, perturbations solu-tions may significantly diverge from the exact solution with time, even inthe case of small variabilities of random variables.

ACKNOWLEDGEMENT

The present research was supported by the Danish Technical ResearchCouncil under grant no. 16-5043-1.



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APPENDIX

Uniform Distribution, U(a, b)

fx(x) = -r — , o<z<6 M-l)0 — CL

E(X] = i±* (A-2)

7or[%] = j - ( -3)

Symmetric Triangular Distribution, TI>(6, )

, 6-a<x < &

/x(x)

= 6

Var(X] = y (A.6)

Symmetric Parabolic Distribution, PD(b, -j=)

fx(x) = (-2% + 2fex - tf -h a^) , b - a < x < 6 -f a (A.T)

= 6 (A8)


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