EACWE 5Florence, Italy

19th – 23rd July 2009

Flying Sphere image © Museo Ideale L. Da Vinci

Keywords: wind excitation, non linear viscous devices, stochastic dynamics, equivalent linearization,stochastic averaging.

ABSTRACT

A SDoF oscillator with a non linear viscous device and subjected to random wind excitation isconsidered. The damping force of the non linear viscous device is proportional to the velocitymodulus powered to a, being a a positive real number smaller than one. As an exact solution does notexist, the response statistics can be found by means of Monte Carlo simulation or by using someapproximate methods of stochastic dynamics. Among these the stochastic equivalent linearization isvery popular, but it yields a Gaussian response to a Gaussian excitation when the response of a nonlinear oscillator is not Gaussian. In this paper equivalent linearization and stochastic averaging arecompared with simulation. The former proves to be adequate and better than the latter as regards theinstantaneous statistics. On the other hand, it causes a light overevaluation of the largest valueaverage, which is better estimated by the stochastic averaging method. However, stochastic lineariza-tion is preferable as it requires very short times of computation.

Contact person: Dept. of Structural Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano;phone 390223994382; fax 390223994220. E-mail floris@stru.polimi.it

Stochastic wind response of SDoF structures with non linearviscous dampers

Claudio FlorisDept. of Structural Engineering, Politecnico di Milano, Milano, Italy, floris@stru.polimi.it

1. INTRODUCTION

Nowadays, the use of damping devices for reducing or suppressing the vibrations caused by wind hasbecome very frequent (e.g. see Xu et al., 1992, Kareem, 1997). To quote a notable case, theMillenium bridge in London (UK), exhibited unacceptable flexural and torsional vibrations, whichhave been reduced by adding viscous dampers at the supports.

In engineering practice it is common to account for these devices by increasing the damping ratioin the linear equations of motion. However, many devices for suppressing vibrations have apronounced non linear behavior. Among these, there are the friction devices and the devices thatprofit from the movement of a semifluid material, which goes from a cell to another. Attention is herefocused on the latter.

The force-displacement and stress-strain relationships for a solid material in the uniaxial state ofstress and for a Newtonian fluid are, respectively

11111111 or)(,or eh=s×h=e=s×= &&dtFEdkF d (1, 2). (1).

Some materials that are used in the damping devices are neither a perfect solid nor a perfect fluidso that the suitable force-displacement relationship is

)sgn()( ddCtF d&& a

×= (3),

where sgn denotes the sign function [sgn(a) = -1, 0, +1 when a is negative, zero, or positive]. Theexponent a falls in the interval ( )1,0 . When a = 0 we have the pure (Coulombian) friction.

The exact determination of the dynamics of an oscillator with a damping device governed by Eq.(3) subjected to a stochastic force is not possible. There are two alternative ways: (1) Monte Carlosimulation (MC), which is accurate but computationally onerous; (2) approximate methods of thestochastic dynamics.

In fact, in stochastic dynamics the exact solutions are few, and are restricted to oscillators directlyexcited by a Gaussian or Poisson white noise. In these cases, the steady state probability densityfunction (PDF) of the response is given by a partial differential equation, the so called Fokker-Planck-Kolmogorov (FPK) equation. The FPK equation is analytically solvable in the presence of lineardamping and non linear restoring force deriving from a potential function. When the damping is nonlinear, the conditions under which the FPK equation is solvable are very cumbersome if the excitationis parametric. If the excitation is external, substantially the condition of solvability requires that thedamping is a function of the mechanical energy of the oscillator. For this wide and non simple matterthe reader is referred to Chap. 5 of the book of Lin and Cai (1995), to the papers by Caughey et al.(1963), Liu (1969), Caughey and Ma (1982), Zhu and his collaborators ( Zhu et al. 1990, Zhu andYang 1996, Huang and Zhu 2000), Soize (1991), Wang and his collaborators (1997, 1998) forGaussian white noise excitation, and to Vasta (1995) and to Proppe (2003) for Poisson white noiseexcitation. Unfortunately, no exact solutions exist for coloured excitations even when these aredescribed by filter equations having a white noise as primary excitation, and the damping devices areused in structures that are stressed by strong motion or by wind turbulence that are coloured stochasticprocesses. Thus, the analyst must resort to approximate methods, two of which will be presented innext section.

2. ANALYTICAL METHODS OF SOLUTION

With reference to wind excitation, the motion of a SDoF oscillator with a non linear viscous deviceis governed by the following equation

)()()sgn()()(2)( 22000 tbUUtXXtXtXtX d +g=w+x+wz+ a &&&&& (4),

where U is mean wind speed blowing in x direction, U(t) is wind turbulence in the same direction.The other symbols mean: UbMACMC pdd g=r=g=x 2,2/1, , being M the structural mass, r

the air density, and Cp the aerodynamic coefficient. Moreover, w0 is the structural pulsation, and z0the ratio of critical damping. For simplicity’s sake, the square of U(t) is neglected.

For general consent U(t) is assumed to be a zero mean stationary coloured Gaussian randomprocess. As previously advanced, notwithstanding the Gaussianity of the input, exact methods forfinding the response of (4) in a statistical sense do not exist. The simplest approximate method is thestochastic equivalent linearization (EL), which replaces the non linear system (4) by an equivalentlinear one so that the response is easily found. This method was used by Di Paola and Navarra (2005)for earthquake excitation, and by Rüdinger (2006) for white noise excitation. The EL method gives aGaussian response to a Gaussian excitation, while the true response of a non linear system is notGaussian. Thus, it is important to ascertain the validity of the EL method.

Analytical method that seize the response non-Gaussianity are the moment equation approach ofItô’s stochastic calculus (ME), the stochastic averaging (STA), and the path integral solution (PIS),and the equivalent non linearization (NEL). Both ME and STA methods require that the excitation isa Gaussian broad band process, which can be approximated by a white noise. In the followingsubsections brief presentations of the EL, ME, and STA methods will be given, while for PIS methodsee Barone et al. (2007).

The fundamental principles of the NEL method can be found in the Chap. 5 of the book by To(2003). The NEL method is not applicable to the present case as the excitation is a coloured process,when NEL requires that it is a white noise Gaussian process. Rüdinger and Krenk (2003) applied itsuccessfully to an oscillator of the type (4) excited by a Gaussian white noise with a > 1, but practicaldamping devices are in general characterized by a < 1 (Terenzi 1999; Di Paola et al. 2007).

2.1 Equivalent linearization (EL)Papers and books on the EL method are very numerous: however, in the book by Roberts and

Spanos (1990) and in the paper by Socha (2005) one finds all he needs. In this method the originalnon linear system is replaced by a linear system equivalent to the former in some statistical sense:

)()()(2)( 2200 tbUUtXtXtX e +g=w+wz+ &&& (5).

If Eq. (5) is used instead of Eq. (4), the error one makes is

E XXXX ed &&&&000 2)sgn(2 wz-x+wz=

a(6).

In the classic EL method the error is minimized in mean square, that is

0][ 2=

z¶

¶

e

E E (7).

From Eq. (7) it is obtained

[ ]2

1

0XE

XEde

&

&úûù

êëé

x+z=z

+a

(8).

Eq. (8) contains expectations of the type

( ) ò¥+

¥-

g-g

÷÷

ø

ö

çç

è

æ

s-sp=úû

ùêëé xxxXE

XX &

&&&

&& dexp2

21(9).

Since the standard deviation X&s is a priori unknown, the EL method must be applied in an iterativefashion till convergence is achieved.

2.2 Moment equation approach (ME)In order to apply the ME, a narrow band process may be approximated by the output of a linear

filter excited by a Gaussian white noise. The coloured process U(t) is represented by the output U (or

by U& ) of a second order linear filter such as (Benfratello et al. 1996, Floris et al. 2001)

)()(ˆ)(ˆ2)(ˆ 2 tWwtUtUtU ffff p=w+wz+ &&& (10),

where W(t) is a zero mean stationary Gaussian white noise with unit PSD. The filter parameters zf, wf,wf must be selected in such a way that the PSD of )(ˆ tU as given by Eq. (10), that is

( ) ,4 222222úûù

êëé wwz+w-wp ffffw matches the actual turbulence PSD.

Then, the augmented system of Eqs. (4, 10) is recast in incremental form by introducing the four

state variables ,ˆ,ˆ,, 4321 UzUzXzXz && ==== and in this way four Itô's stochastic differentialequations are obtained (for the principles of Itô's stochastic calculus the reader may refer to Di Paola,1993, and to Soong and Grigoriu, 1993). By applying Itô's differential rule to the non-anticipatingfunction ,4321

srqp zzzz=f the differential equations ruling the evolution of the response moments arewritten down. In symbolic form they read as:

îíì ×g+w-úû

ùêëéx-b-+= -+

-a+-

21,1

20243

121201,1 )sgn( UmzzzzzzEmqmpm rsqp

srqpdpqrsrsqppqrs&

} ( )+w-b-+++ -++-+-- 1,12

1,1,,,11,1, srpqfrspqfsrqpsrqprsqp mmsmrmbm 22 -srpqf mwp (11),

where .2,2 000 fff wz=bwz=b Eq. (11) contains the expectation úûù

êëé -a )sgn( 243

1212 zzzzzzE srqp ,

whose evaluation requires the knowledge of the joint PDF of the response. As was done by Fatica andFloris (2002) for the case of the oscillator with pure friction, this can be expressed by a modifiedtruncated Gram-Charlier series, that is

=)( 4321 uuuup

[ ]l

l

k

k

j

j

i

iNlkji

i j k l

lkji

u

up

u

up

u

up

u

uplkji

uHuHuHuHE

4

40

3

30

2

20

1

10

0 0 0 0

4321

d

)(d

d

)(d

d

)(d

d

)(d!!!!

)()()()(å å å å

=+++

= = = == (12)

where Hi(x) is the Hermite polynomial of order i in the variable x, the numerator of the fraction can benamed Hermite moment, ,

izii zu s= and p0 is the standard Gaussian univariate PDF. N = 4, that is a

closure at the fourth order, should be enough for the present problem. Since EL and STA performwell, this method of solution is not pursued further here.

2.3 Stochastic averaging (STA)The classic (first order) stochastic averaging method is considered here (see the Chap. 6 of the

book by Soong and Grigoriu, 2003, Roberts and Spanos, 1986, Zhu, 1988). STA is a suitable methodof analysis of oscillators with non linear damping (Roberts 1977, 1978; Spanos 1978). The stochasticaveraging was applied to wind response probably for the first time by Lin and Holmes (1978) in theirpioneer work. More recently, it has been applied to a wind problem by Krenk and Nielsen (1999).

In order to apply the stochastic averaging method to the present problem, Eq. (4) is recast as

)()()()sgn()()(2)()( 02

0020 tFXhtbUUXtXtXtXtX d +z-=+úû

ùêëé g+x+wz-=w+

a &&&&&& (13),

where ).()(,, 00 tUbtFdd === zggzxx With the variable transformation ),(cos)()( ttAtX j=

)(sin)()( 0 ttAtX jw-=& , in which )()( 0 ttt Y+w=j , the averaged equations of the method are

)()(

2)()()( 1

0

0020

020

0

20 tW

SA

SAhtA FFFFs w

wpz+

wwpz

+wz

=& )()(

)()( 20

00

0

20 tW

SAh

At FF

c wwpz

+wz

=Y&

(14, 15)

W1 and W2 are uncorrelated unit strength Gaussian white noises, the PSD of F(t) is calculated in thestructural frequency w0, being ).()( 2 w=w UUFF SbS The functions hs(A) and hs(A) are

( ) ( )òòpp

jjjw-jp

=jjjw-jp

=2

00

2

00 dsinsin,cos

21)(dcossin,cos

21)( AAhAhAAhAh cs

(16, 17).

Inserting the expression of h(·) in Eq. (16) and expanding, it is obtained

( )

òò

p+aaa

p

júûù

êëé jg+jjwx-jw-

p=

=jjjw-jp

=

2

0

210

220

2

00

dsin)sinsgn(sinsin221

dcossin,cos21)(

UAAA

AAhAh

d

s

(18).

It must be stressed that: (1) the second addendum in the integral (18) must be numerically evaluatedas a + 1 is not an integer number; (2) clearly, the integral of the third addendum vanishes. As regardsthe point (2), this fact is a feature of the first order stochastic averaging method, which is not capableof accounting for an excitation with non zero mean. However, in the present case the non-linearity isconfined to the damping so that the effect of Ug 2 can be considered apart.

The FPK equations associated with Eqs. (14, 15) respectively are easily solvable giving that thePDF of Y is uniform between 0 and 2p, and

)(2

1),(d)()(

2exp)(00

0 apaxxpuuhSaqap AXXa

sFF

A p=úû

ùêë

éwp

w= ò && (19, 20).

where q is a normalization constant, and ( ) 2120

22 w+= xxa & . Eqs. (19, 20) allow the computation ofthe second order statistics of the response. Since Ug 2 is dropped in Eq. (13), these statistics refer tothe oscillations around the mean value. Once the second order statistics are known, it is necessary todetermine the largest value statistics, which can be obtained only approximately.

3. ANALYSES

The oscillator of Eq. (4) is analyzed by means of MC, EL, and STA. Two series of analyses areperformed that are distinguished by the coefficient xd in (4) -0.10 or 0.30-, while the exponent a isworth always 0.30. The other parameters take the following values:

· z0 = 0.02, w0 = 2p rad/s;· A = 20 m2, Cp = 1, r = 1.25 kg/m3, M = 2161 kg.

As previously stated, the average wind speed U is not considered in the analyses as it causes astatic drift only that can be computed apart. On the other hand, it determines the wind turbulencePSD. The Kaimal’s modified equation (Simiu and Scanlan 1996) has been chosen for this function:

53251

2200),(

2*

-

÷÷ø

öççè

æ

p

w+

p=w

Uz

UzuzSUU (21).

U takes the values 10, 20, 30, 40 m/s: with z = 20 m and adopting a logarithmic wind profile with z0=1 m, the shear velocity u* can be computed in Eq. (21).

In applying MC simulation samples with 100,000 motion histories are constructed: in each historyEq. (4) without the constant term is solved by means of fourth order Runge-Kutta method; theresponse statistical moments are computed by ensemble averaging. Each simulation requires 24 hoursabout of computation on a standard desk top computer. Specific programs in Fortran language havebeen written for both EL and STA method. The computing time of the EL method are negligible,while STA requires from 30 minutes to 2 hours of computation as U increases since some doubleintegrals are to be evaluated. The largest value averages are computed by a program written inMAPLE language profiting in this way from the ability of MAPLE of performing symbolicoperations.

As regards the instantaneous statistics, it has been chosen to show the standard deviations ,Xs

X&s . The former quantity is commonly used to define the design value:

[ ] XXd gTXx s+m=@ max (22).

where the peak factor g is implicitly affected by the standard deviation of the velocity.Figures 1, 2 regard the case a = 0.30, xd = 0.10, while figures 3, 4 the case a = 0.30, xd = 0.30. In

all the figures the curves for the non controlled oscillator (xd = 0) are also traced to ascertain theefficacy of the damping device. However, the determination of the optimum values of the damper isbeyond the aim of this paper.

As regards sX (figures 1, 3), the EL method is in perfect agreement with the simulation: the twocurves coalesce, and are indistinguishable in the plots. On the contrary, STA underestimates the re-sponse displacement standard deviation: thus, the use of this method more complicated than the EL

Figure 1: case a = 0.30, xd = 0.10, plot of sX against U . non controlledoscillator; controlled oscillator: simulation, EL, STA.

Figure 2: case a = 0.30, xd = 0.10: plot ofX&

s against U . non controlled

oscillator; controlled oscillator: simulation, EL, STA.

is not justified, and EL is preferable. However, it is notable that the reduction of sX is not importanteven for the larger value of xd (it is recalled that this parameter implicitly contains the damper mass, whichcan be a non large fraction only of the structural mass).

Figures 2, 4 reveal that both the analytical methods seize the velocity standard deviationX&s exactly: the respective curves coalesce with the simulative one. Even this quantity is modestly

reduced by the damper.Knowkedge of the largest value statistics is necessary in order to perform checks for the ultimate

limit states: a definition of a peak factor to be used in Eq. (22) is not attempted here, and the largestvalue average E[max X½T] is directly given. In stochastic process theory exact solutions for thelargest value statistics do not exist even for the simple case of a Gaussian process. For this there aretwo approximate expressions of largest value CDF, the former of which is based on the hypothesis

Figure 3: case a = 0.30, xd = 0.30, plot of sX against U . non controlledoscillator; controlled oscillator: simulation, EL, STA.

Figure 4: case a = 0.30, xd = 0.30: plot ofX&

s against U . non controlled

oscillator; controlled oscillator: simulation, EL, STA.

that the upcrossings of a given level constitute a Poisson process, while the latter is due to Vanmarcke

(1975). They are, respectively:

[ ] ( )úú

û

ù

êê

ë

é

-

-×n-×÷

øö

çèæ -==<=

-

--+-n- +

2

22

5.0

5.00,

)(,

1

1exp1)(max)(u

kuuu

TXTx

TXe

eeexFeTxXPxF X (1

(23, 24).The nomenclature in Eqs. (23, 24) is:

Figure 5: plot of the largest value average against U , T = 40 s, case a = 0.30, xd =0.10 . simulation, EL, STA.

Figure 6: plot of the largest value average against U , T = 40 s, case a = 0.30,xd = 0.30 . simulation, EL, STA.

==n ò¥+

¥-+ xxxpxx XXX &&& & d),()(

222 )(21)(21)(21

2d

21 XXX x

X

Xxx

XXexeex s-

¥+

¥-

s-s-

ps

s=

sps ò &&

&

&& &

(25)

21

202.1

0 122 l×l

l-=p=

s=

ps

s=n+ qqkxu

XX

X& (26 - 29).

Eqs. (23 - 28) are valid for a stationary Gaussian process, being Eq. 28 empirical (Vanmarcke 1975).The parameter q in Eq. (29) is a measure of the bandwidth of the process X(t) (Vanmarcke 1975). It

can be defined for a process whichever provided that the unilateral PSD SXX(w) is known, from whichthe spectral moments lj are computed as

ò¥+

www=l0

d)(XXj

j S (30).

In stochastic dynamics the PSD SXX(w) of a response process is known in general for Gaussian re-sponses only, while for non Gaussian responses exact solutions are very rare (Dimentberg et al.1995), and approximate solutions may be difficult to obtain (Bouc 1994; Krenk and Roberts 1999;Failla et al. 2003). Hence, particularly in applicative computations one resigns to include the band-width effect in the largest value CDF, for which in general it is assumed a form like exp [-aX(x)],being aX(x) the decaying rate (Lutes et al. 1980). In this paper, the expression suggested by Winter-stein (1987) has been adopted

( ) ( )( )( ) 831

12142432612

exp)(4

244

33

20

-a+

+--a+-a+÷÷ø

öççè

æ-n=a + uuuuuTxX (31),

where 43 and, aas= Xxu are the skewness and the kurtosis of the process XXU s= , respec-tively (in this case 3a is zero as the response X is symmetric). The mean upcrossing rate of the zeroline with positive velocity is computed as

ò¥++ =n

00 d),0()( xxpxx XX &&& & (32).

where the PDF XXp & is given by Eq. (20).The plots of the largest value averages referring to T = 40 s, which is the duration of the simula-

tions, are in the figures 5, 6 for the cases a = 0.30, xd = 0.10 and a = 0.30, xd = 0.30, respectively. Asregards the EL method, the curves deriving from the Poissonian CDF (23) are not reported as theyevidently overestimate the largest value averages. The curves obtained by using Vanmarcke’s CDFare quite adequate even if they are on the safe side with respect to the simulation. Clearly, as regardsthe largest value average estimates the results by STA and Eq. (31) are better but not to a such extentto justify the larger amount of computations. For comparison’s sake the largest value averagespresented herein refer to a duration of 40 s, the same as that of the simulations: for design purposes aduration of 10 - 60 minutes should be considered as usual in the standards, but in this way the simul-ations would become unaffordable.

4. CONCLUSIONS

In this paper the response to wind excitation of an SDoF oscillator controlled by a passive damperwith non linear power law is considered. The principal aim is to ascertain whether the stochasticequivalent linearization is adequate for determining the response statistics. In fact, this method givesa Gaussian response to a Gaussian excitation when the response of a non linear system is not such.

Besides the linearization method, the response statistics are computed by Monte Carlo simulationand stochastic averaging. The equivalent linearization proves to be very suitable as it requires a shorttime for computation, gives very accurate estimates of the instantaneous standard deviations andacceptable estimates of the largest value averages. Stochastic averaging is worse in estimating theformer and better in estimating the latter, which is not enough to compensate for longer times ofcomputation, which on the other hand remain much smaller than those required by Monte Carlosimulation.

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