ROBUST DESIGN OF TUNED LIQUID COLUMN DAMPERS …congress.cimne.com/eccomas/proceedings/compdyn2011/... · Computational Methods in Structural Dynamics and Earthquake ... Tuned liquid

COMPDYN 2011 III ECCOMAS Thematic Conference on

Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, M. Fragiadakis, V. Plevris (eds.)

Corfu, Greece, 25–28 May 2011

ROBUST DESIGN OF TUNED LIQUID COLUMN DAMPERS UNDER STOCHASTIC GROUND MOTION CONSIDERING FUZZY

UNCERTANTIES

G. Quaranta1, S. Chakraborty2, and G. C. Marano3

1 Dept. of Civil and Environmental Engineering, University of California Davis One Shields Avenue, Davis, CA 95616, U.S.A.

[email protected]

2 Dept. of Civil Engineering, Bengal Engineering and Science University Shibpur, Howrah, 711103, India

[email protected]

3 Dept. of Environmental Engineering and Sustainable Development, Technical University of Bari viale del Turismo 10, 74100, Taranto, Italy

[email protected]

Keywords: Credibility Theory, Fuzzy Uncertainty, Optimization, Robust Design, Stochastic Ground Motion, Tuned Liquid Column Damper.

Abstract. The tuned liquid column dampers (TLCDs) have been shown to be effective vibra-tion control devices for flexible structures subjected to long-duration, periodic or harmonic excitations. Their potential applications for seismic protection and retrofitting were recently explored. The optimum TLCD parameters are normally obtained based on the implicit as-sumption that the involved variables are deterministic. However, it is well known that the effi-ciency of TLCDs may be jeopardized if its parameters are not properly tuned to the vibrating mode of interest, for instance as consequence of the unavoidable presence of uncertain va-riables. Thus, the optimization of damper parameters considering model uncertainties has attracted a great deal of interest. The robust design of TLCDs for the passive control of me-chanical systems under random ground motion is investigated in the present paper by coupl-ing random vibration analysis and credibility theory in order to take into account fuzzy uncertainties. In doing so, two antithetical objective functions are considered, and they are the expected value and the variance of an appropriate displacement-based fuzzy structural index. Specifically, this latter one is introduced to characterize the performance variability due to the existence of fuzzy uncertainties affecting, both the system and random loading pa-rameters. A numerical study is performed to demonstrate the applicability of the developed approach.

G. Quaranta, S. Chakraborty and G. C. Marano

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1 INTRODUCTION

Base isolation, passive energy dissipation and active control are three current philosophies for enhancing structural performances and safety against natural and manmade hazards. Among the most popular passive control devices, the tuned mass dampers and the tuned liq-uid dampers are commonly used. A tuned mass damper (TMD) consists of a large mass at-tached to the structure through a spring and a dashpot, and dissipates energy when it is tuned to the frequency of the structure. The tuned liquid damper (TLD) consists of a container which is partially filled with liquid which dissipates energy through the sloshing action of a liquid in the container. For optimum results, the sloshing frequency is tuned to the frequency of the structure. Tuned liquid column dampers (TLCDs) are a class of tuned liquid dampers that impart indirect damping to the primary structure through oscillations of the liquid column in a U-shaped container which includes at least one orifice. It has been also investigated a type of TLCD that have different cross-sections in the horizontal and vertical columns, termed liquid column vibration absorbers (LCVAs). These devices are summarized in Fig. 1.

Main system

TLCD

Main system

TLD

Basemotion

Main system

LCVA

Orifice

a) b) c)

Orifice

Basemotion

Basemotion

Figure 1: Tuned liquid damper (a), Tuned liquid column damper (b) and Liquid column vibration absorber (c).

Potential advantages of TLCDs over the conventional inertial devices (e.g., TMDs) are that the system is inexpensive and requires minimum maintenance and its vibration characteristics (e.g., tuning and damping) are well defined and can be easily modified. For instance, its natu-ral frequency is only determined by the length of the liquid column, and its damping can be overall accounted for by the so-called head loss coefficient that depends on the orifice size in the horizontal column. Additionally, water storage already available in tall buildings can be utilized without introducing excessively extra large mass. The effectiveness of TLCD as pro-tection strategy was primarily addressed for controlling the wind-induced vibration of towers [1] and tall buildings [2]. In doing this, the most of the studies consider a single-degree-of-freedom (SDOF) model of primary system equipped with a TLCD and subject to white noise type of wind loading [3]. On the contrary, the assessment of the TLCD performances for pro-tecting structural systems subject to earthquake load is comparatively less. In this field, the effectiveness of the TLCD performances was investigated in [4] with regard to non-stationary random vibrations simulating an earthquake loading.

All mechanical parameters governing the adopted passive protection strategy have to be properly optimized in order to meet the maximum possible protection level. In doing so, the analysis of all sources of uncertainty is a crucial step of the design process. In this regard, it may be noted that earlier studies were mainly conducted by assuming deterministic protected mechanical systems subject to uncertain dynamic loads, i.e. [3][4][5]. However, this simplifi-cation may jeopardize the effectiveness of the adopted protection strategy. In fact, it was re-ported that the uncertainty in mechanical system parameters might have equal or sometimes even greater influence on the response than the uncertainty in the dynamic excitations [6]. Since passive protection systems do not have any feedback from the structural response, im-proved design strategies should be preferred for taking into account all possible sources of


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uncertainty and their effects on the expected performances. For instance, Marano and co-authors [7][8] investigated the effects due to uncertain mechanical systems parameters for ro-bust optimum design strategies of TMDs. In doing this, the authors considered SDOF systems subject to stationary filtered white noise process and the sensitivity of a performance index against the fluctuations of the uncertain structural parameters is considered via direct pertur-bation method. The application of low-order perturbation methods allow a probabilistic analy-sis by considering few statistical moments of the uncertain parameters (e.g., mean and variance) and do not require full statistical descriptions (e.g., the joint probability density functions). Although researchers and practitioners are familiar with stochastic calculus, a pure probabilistic standpoint in designing these devices may present several limitations.

Since fuzziness is frequently involved in modeling random processes and variables, it is not so reasonable to take into account randomness while ignoring the existence of fuzziness aprioristically [9]. On the contrary, more consistent analysis may be carried out by relaxing conventional probabilistic-based methods in order to include non-probabilistic forms of un-certainty appropriately [10][11]. For instance, the reliability based optimum design of TMDs in seismic vibration control of structures with bounded uncertain parameters was presented in [12]. The optimum design of TLCDs under stochastic earthquake considering uncertain bounded system parameters was discussed in [13]. A procedure for the analysis of structural systems subject to random vibrations considering fuzzy variables was proposed in [14]. A ro-bust-based optimization problem for TMDs was illustrated in [15] which take into account stochastic ground motion and fuzzy variables.

This paper addresses the robust design of TLCDs under stochastic ground motion consider-ing fuzzy uncertainties. In detail, the dynamic loading is modeled as a stationary filtered white noise process whereas both load and system parameters are considered as fuzzy variables. As a consequence, a general performance index is affected by fuzziness itself. Therefore, we pro-pose the application of the credibility measure in order to extract two representative crisp measures. The first one is based on the expected operator for fuzzy variables which needs to be minimized in order to improve the effectiveness of the TLCD. The second one, based on the variance operator for fuzzy variables, must be minimized in order to increase the robust-ness against fuzzy uncertainties. As these measures have an opposite behavior within typical design solution spaces, a multi-objective optimization problem can be formulated in order to define a collection of solutions (in Pareto sense) which are capable of ensuring a satisfactory trade-off between mechanical performances and robustness against the variability due to the uncertain (fuzzy) parameters. Numerical examples at the end will demonstrate the feasibility of the proposed approach of designing TLCDs.

2 DYNAMIC RESPONSE OF TLCD-STRUCTURE SYSTEM

The description of a SDOF system equipped with a TLCD under stochastic ground motion is provided in this section. The stochastic dynamic response analysis of the combined TLCD-structure is evaluated as in [13], by using the state space formulation in time domain and by solving the final Lyapunov equation.

2.1 TLCD system and liquid motion equation

The TLCD considered in the present study is a U-shaped pipe attached to the main system which is modeled as a SDOF system with properties in accordance with the specified mode of vibration. A schematic diagram of a SDOF system equipped with a TLCD under base excita-tion is shown in Fig. 2.


4

ks

cs

zb

..

x

y

ms

h

Bh

Figure 2: Layout of a SDOF system equipped with a TLCD.

The main geometrical and physical data for the TLCD are: cross-sectional area A, the length of the horizontal portion of the tube Bh, the vertical height of the liquid inside the tube h and the density of the liquid ρ. The main system to be protected has a mass equal to ms, a linear stiffness k and a viscous damping cs. The total length of the liquid column is Le=2h+Bh. The mass of the liquid ml=ρALe can be considered as part of ms. An orifice is installed in the horizontal portion of the pipe and ξ denotes the coefficient of head loss which depends on its opening ratio. The orifice is typically placed at the centre of the horizontal portion of the tube. The combined TLCD-structure is subject to base acceleration (see Fig. 2), for instance due to an earthquake. The relative horizontal displacement of the SDOF system is x whereas the dis-placement of the liquid surface (with reference to the unperturbed original configuration) is denoted as y (the time variable is omitted throughout the paper to improve the readability of various equations). In order to derive the motion equation of the liquid within the U-shaped pipe, the following hypotheses are used: (i) the sloshing behavior on the liquid surface is neg-ligible; (ii) the flow is incompressible (i.e., the flow rate is constant); (iii) the dimension of the column cross-section is much smaller than the horizontal length of a TLCD. The equation of motion of the liquid column is [16]:

( )12

2e h bAL y A y y gAy AB x zρ ρ ξ ρ ρ+ + = − +ɺɺ ɺ ɺ ɺɺ ɺɺ (1)

In the above, g is the gravitational acceleration. It is evident in the second term of Eq. (1) that the TLCD response is non-linear as a result of the drag-type forces induced by the orifice. Within the framework of random vibration theory, the motion equation in Eq. (1) is usually linearized via stochastic equivalent linearization technique. First, a linear system is chosen to represent the original non-linear system:

( )2 2e p h bAL y Ac y gAy AB x zρ ρ ρ ρ+ + = − +ɺɺ ɺ ɺɺ ɺɺ (2)

in which cp is the equivalent damping coefficient for the TLCD after linearization. By mini-mizing the mean-square error between Eq. (1) and Eq.(2), it can be found:

24

p

y yc

y

ξ=

ɺ ɺ

ɺ (3)

where i is the operator of expectation. If x is a zero-mean stationary Gaussian process, cp is:


5

( ):2

yp yc standard deviation of the liquid velocity

ξσσ

π= ɺ

ɺ (4)

In Eq. (4), the standard deviation of the liquid velocity is not known a priori: therefore, the dynamic analysis is performed by assuming an initial value for cp (i.e., cp=0) and is updated once the motion equation is solved. This implies an iterative procedure that will be stopped once a convergence criterion is fulfilled (the exact value of cp is achieved after few iterations). The error due to this linearization depends on the head loss coefficient (the error increases as ξ grows): however, for the most part of engineering applications, Eq. (4) provides very good results. By normalizing Eq. (2) with respect to the mass of the liquid in the container, one ob-tains:

2 2p

be e

c gy y y px pz

L L+ + + = −ɺɺ ɺ ɺɺ ɺɺ (5)

where p=Bh/Le is the length ratio.

2.2 Motion equation of the SDOF system equipped with a TLCD

The TLCD is used to protect a SDOF linear system as shown in Fig. 2. The equation of motion of the main system equipped with TLCD is:

( ) ( )2 2s h s s s h b hm AB hA x c x k x m AB hA z AB yρ ρ ρ ρ ρ+ + + + = − + + −ɺɺ ɺ ɺɺɺɺ (6)

If the variable ml=ρABh+ρAh is considered, then above equation can be expressed as:

( ) ( )s l s s s l b em m x c x k x m m z ApL yρ+ + + = − + −ɺɺ ɺ ɺɺɺɺ (7)

The normalization of Eq. (7) with respect to ms yields:

( ) ( )1 1s sb

s s

c kx x x z py

m mµ µ µ+ + + = − + −ɺɺ ɺ ɺɺɺɺ , (8)

in which µ=(ρALe)/ms is the so-called mass ratio. By introducing the notations of the natural frequency ωs=√(ks/ms) and the damping coefficient ξs=cs/(2msωs) for the main system, Eq. (8) can be rewritten as follows:

( ) ( )21 2 1s s s bx x x py zµ ξ ω ω µ µ+ + + + = − +ɺɺ ɺ ɺɺ ɺɺ (9)

The tuning ratio γ=ωl/ωs will be also used, where ωl=√(2g/Le) is the frequency of the liquid. Eq. (5) and Eq. (9) can be also expressed in matrix form:

( ) ( )2

2 201 0

1 100 2

p

e be

ss s

c gp py y y

L zLp x x xµ µ µ

ωξ ω

+ + = − + +

ɺɺ ɺɺɺ

ɺɺ ɺ (10)

If the state space vector u is introduced:

{ }Ty x y x=u ɺ ɺ (11)

then Eq. (10) is reduced to a standard first-order form:

* bz= +u A u rɺ ɺɺ (12)


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where

2 2 2 2* 1 1

× ×− −

=

0 IA

M K M C (13)

and r={0 0 1 1}T. In Eq. (13), 0 and I are the null and the unit matrix, respectively (the sub-scripts denote their size).

2.3 Stochastic model for the ground motion

For base random accelerations, a usual model is obtained via second-order linear filtering of the white noise process. As in several previous studies – see for instance [7] and [13] – the stationary Kanai-Tajimi model is considered in this paper [17][18]. So doing, the excitation at the base is described as follows:

2

2

2

2

f f f f f f

b f f f f f f

x x x w

z x w x x

ξ ω ω

ξ ω ω

+ + = −

= + = +

ɺɺ ɺ

ɺɺ ɺɺɺ (14)

In which w is a stationary Gaussian zero mean white noise process whose power spectral den-sity (PSD) is S0 which can be evaluated as follows

( )2

0 2

2

1 4bf z

f f

Sξ σ

π ξ ω=

+ɺɺ (15)

The standard deviation of the base motion is related to the peak ground acceleration (PGA):

PGA 3bzσ= ɺɺ (16)

Therefore, the dynamic loading model in Eq. (14) depends on two parameters, named the filter frequency ωf and the filter damping ξf.

2.4 Response covariance analysis

The following algebraic matrix equation of order six – the so-called Lyapunov equation – is obtained by taking into account both Eq. (12) and Eq. (14):

6 6T

×+ + =AR RA B 0 (17)

In Eq. (17) the state space matrix A and B are, respectively:

211 12 12 12

2 221 22 21 22

2

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

2 0 2 2 0

2 2 2 2

0 0 0 0 2

e s p e s s

e s f p e s s f f

f f f

m g L m m c L m

m g L m m c L m

ω ξ ωω ω ξ ω ξ ω

ω ξ ω

= − − − −

− −

A (18)


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0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 2 Sπ

=

B , (19)

with

( ) 211 21 12 22

1 11e

e e e e

p pd p m m m m

d d d d

µ µµ µ += + − = = = = (20)

And the global state space vector is:

{ }T

f fy x x y x x=z ɺ ɺ ɺ (21)

The space state covariance matrix R in Eq. (17) takes the form:

=

zz zz

zz zz

R RR

R Rɺ

ɺ ɺ ɺ

, (22)

and each sub-matrix has a size 2×2. Our main interest is focused in the following quantities:

( )2,2xσ = zzR (23)

( )1,1yσ = zzRɺ ɺ ɺ , (24)

that are the standard deviation of the main system displacement equipped with a TLCD and the standard deviation of the liquid velocity, respectively. Eq. (24) is needed to evaluate cp in Eq. (4).

3 ROBUST DESIGN OPTIMIZATION CONSIDERING FUZZY UNCERTANTIES

Once the covariance response analysis is completed, the structural performance index un-der investigation is presented. Thus, the adopted uncertain parameters’ model is introduced. Consequently, the robust design optimization problem of the TLCD is formalized within the framework of the credibility theory.

3.1 Structural performance index

In order to assess the effectiveness of the TLCD, the response covariance analysis is per-formed for the unprotected main system. In this case, the covariance matrix R0 has size 4×4 and can be evaluated as follows:

0 0 0 0 0 4 4T

×+ + =A R R A B 0 , (25)

where


8

2 20

2

0 0 1 0

0 0 0 1

2 2

0 0 2s f s s f f

f f f

ω ω ξ ω ξ ωω ξ ω

= − − − −

A (26)

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 2 Sπ

=

B , (27)

And the global state space vector is:

{ }0 0 0

T

f fx x x x=z ɺ ɺ (28)

Finally, the standard deviation of the unprotected main system displacement is:

( )0 0 0

1,1xσ = z zR (29)

Given a set of TLCD parameters, i.e. d={ γ, ξ}, a dimensionless performance index to as-sess the efficiency of the TLCD is [7]:

( ) ( )0

x

x

σψ

σ=

dd (30)

This function represents a direct stochastic displacement-based performance index: in fact, the designed TLCD is effective in seismic protection as ψ(d) tends to zero.

3.2 Consideration of non-probabilistic uncertainties

The performance index in Eq. (30) is carried out under the assumption that the dynamic loading is the only source of uncertainty, and it is assumed as a pure random process. The ef-fects of uncertain structural parameters are generally considered via direct perturbation me-thod in a probabilistic format by considering few statistical moments of the uncertain parameters. Unfortunately, a pure probabilistic standpoint may be an important limitation for some practical applications. This is because few experimental data are available in many cas-es and, sometimes, literature or expert opinions are the only supports for handling uncertain-ties. In this perspective, alternative (non-probabilistic) uncertain models should be included.

Several studies – see for instance [12][13][14][15] – demonstrated that non-probabilistic sources of uncertainty can play a significant role in the analysis of structural systems subject to random vibration. For instance, the natural frequency of the main system ωs is quite diffi-cult to predict accurately (its real value is usually defined by a full-scale field dynamic testing) and is subject to non-structural induced variations (i.e., due to the temperature). Moreover, different numerical techniques for extracting the natural frequencies from dynamic records may lead to dissimilar numerical values. However, the most relevant source of uncertainty in the main system is usually due to the way by which it dissipates energy when subject to dy-namic loading and, in turn, in the numerical evaluation of ξs. The most important sources of indeterminateness can be detected in the assessment of the filter parameters [15] for which probabilistic models (e.g., probability density functions) are not typically available to design-ers. On the contrary, expert opinions and existing literature data base are often adopted to se-


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lect appropriate values for ωf and ξs. Unfortunately, the variability of the filter parameters does not observe a specific statistical model because of the lack of accurate and consistent in-formation for a statistical treatment. Their numerical values generally depend on the soil type, which is defined using a linguistic-based criterion (stiff, medium or soft soil). Since fuzziness is usually involved in random variables modeling, it is not so reasonable to take into account randomness while ignoring the existence of fuzziness aprioristically [19]. Therefore, an ap-propriate design of TLCDs needs a more complicated analysis regarding the (different) sources of uncertainty. In this paper ωs, ξs, ωf and ξs are considered as fuzzy variables. The use of fuzzy variables for structures subject to random vibrations was proposed in [14] and [15]. Since each interval is an equally possible fuzzy variable, this approach is a generalized version of that in [12] and [13] based on uncertain bounded system parameters.

3.3 Credibility measure, expected value and variance of fuzzy variables

In order to introduce the robust design optimization problem considering fuzzy variables, some definitions from the credibility theory are required. The credibility space is mathemati-cally defined by the triplet (Θ, P(Θ),Cr), where Θ is the nonempty set representing the sample space, P(Θ) the power set of Θ and Cr the credibility measure. In order to define the credibili-ty measure, we consider an event Q, its possibility Pos{Q} (a crisp number indicating the pos-sibility that this event will occur) and its necessity Nec{Q} (another crisp number denoting the impossibility of the opposite set Qopp). The credibility of the fuzzy event Q (denoted as Cr{Q}) is defined as the average between its possibility and necessity of Q:

{ } { }Nec 1 Pos oppQ Q= − (31)

{ } { } { }( )1Cr Pos Nec

2Q Q Q= + (32)

A fuzzy variable bɶ is a measurable function from a credibility space (Θ, P(Θ),Cr) to the set of real numbers. For each fuzzy variable, it is possible to define its membership function (MF) π(b) as follows [20]:

( ) { }{ }π min 2Cr ,1 b b b b= = ∀ ∈ɶ ℝ (33)

Throughout the paper we will suppose that it is possible to define a fuzzy variable by means of its α-cuts. Specifically, we assume that the generic α-cut of a fuzzy variable is the following closed interval:

( ){ } ( ], π 0,1b b b b bααα α α− + = = ∈ ≥ ∀ ∈ ℝ (34)

and the fuzzy variable bɶ is:

( ]0,1

,b b bαα

αα − +

∈

= ɶ ∪ (35)

The representation of the expected value for a fuzzy variable is identical to that for a ran-dom variable [21]. Based on the Choquet integral, the definition of the expected value of a fuzzy variable proposed in [21] is the following:

{ } { }0

0

E Cr d Cr dEb b b s s b s s+∞

−∞

= = ≥ − ≤ ∫ ∫ɶ ɶ ɶ (36)


10

and

{ } ( ){ } ( ){ }( )1Cr sup π 1 sup π

2 v b v b

b b v v b≤ >

≤ = + − ∀ ∈ɶ ℝ (37)

{ } ( ){ } ( ){ }( )1Cr sup π 1 sup π

2 v b v b

b b v v b≥ <

≥ = + − ∀ ∈ɶ ℝ (38)

The variance of a fuzzy variable measures the spread of the distribution around its ex-pected value. In perfect analogy with the probabilistic case, the credibility-based definition of the variance of a fuzzy variable is [20]:

( )2

V E Eb b b = − ɶ ɶ (39)

In this paper, the expected value operator in Eq. (36) will be performed to obtain the like-lihood of an event from fuzzy-type information. Since fuzzy-type outputs can have a different spread around the expected value, the variance operator in Eq. (39) will be adopted to discri-minate their degree of indeterminateness. Both information will be important to formalize the robust design of TLCDs. Some closed form expressions of mean and variance for common MFs are available in [20].

3.4 Robust design optimization of TLCD

The following vector of fuzzy variables (that is, a fuzzy vector) is introduced in order to take into account non-probabilistic uncertainties in the optimum design of the TLCD:

{ } { }1, ,4i s s f fb ω ξ ω ξ== =b…

ɶ ɶ ɶɶ ɶ ɶ (40)

As a consequence, Eq. (30) is a deterministic function involving fuzzy variables (that is, a type of fuzzy function) and whose output is fuzzy itself:

( ) ( )0

x

x

σψ

σ=

dd

ɶɶ

ɶ (41)

with

( ) ( ) ( ) ( ) ( )0 0

; ;x x x xψ ψ σ σ σ σ= = =d d b d d b bɶ ɶ ɶɶ ɶ ɶ (42)

Since fuzzy uncertainties are involved in the proposed analysis, the original displacement-based performance index in Eq. (30) is uncertain (fuzzy) itself. Therefore, instead of aiming to find a single ‘‘best’’ design solution, it may be more interesting to carry out a set of ‘‘good’’ compromise between TLCD performance and sensitivity against fuzziness. This is the key-idea in robust-based design strategies and would be a more consistent way for supporting de-signers and decision makers. Because “performance” and “sensitivity” are often conflicting objectives, a multi-objective optimization problem has to be solved. In this paper, the robust design optimization of the TLCD is performed by solving the following multi-objective opti-mization problem

( ) ( ){ }1 2min ; , ;

. . max min

f f

s t ≤ ≤d

d b d b

d d d

ɶ ɶ (43)

in which


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( ) ( )1 ; Ef ψ= d b dɶ ɶ (44)

and

( ) ( )2 ; Vf ψ= d b dɶ ɶ (45)

The minimization of the first objective function f1 in Eq. (44) aims at improving the effec-tiveness of the TLCD in seismic protection. On the other hand, the robustness of the protec-tion system against fuzziness is improved by minimizing f2 in Eq. (45). Moreover, dmin and dmax in Eq. (43) are the lower and upper bound of the candidate design solutions, respectively.

4 COMPUTATIONAL ISSUES

The proposed strategy of the fuzzy-based robust design of TLCDs needs the analysis of two important computational issues. The first one deals with the resolution strategy for solv-ing the multi-objective optimization in Eq. (43). The second aspect is related to the evaluation of the fuzzy performance index in Eq. (41).

4.1 Solving the multi-objective optimization problem

Since Eq. (44) and Eq. (45) involve crisp outputs only, all existing algorithms for solving multi-objective optimization problems can be used. Therefore, a standard genetic algorithm based multi-objective optimization algorithm [22] is used in this paper. Having so done, all solutions within the Pareto set are equally optimal: it is up to the designer to select a solution in this set depending on the application. The initial population is uniformly generated, the population size is 200, the maximum number of iterations is 500. The multi-objective genetic algorithm is stopped once the average change in the spread of Pareto solutions is less than 1.0 × 10-4. The fitness scaling rank strategy has been chosen as fitness scaling function and the tournament selection (with tournament size equal to 4) as selection operator. The arithmetic crossover with a crossover fraction equal to 80% is selected as crossover operator. An adap-tive feasible strategy is performed as mutation operator.

4.2 Calculation of the fuzzy performance index

An approximation of the output MF in Eq. (41) for a given design vector d can be obtained by repeating a interval analysis at a number of cuts, see for instance [23]. So doing, the inter-val analysis at each α-cut becomes the central point for computing the MF of the fuzzy per-formance index in Eq. (41). In this paper, the Taylor series expansion based approach is chosen [12][13] because of its competitiveness from a computational standpoint.

Let be an α-cut of the ith fuzzy variable in the following form:

, ,c c

i i i i i i ib b b b b b bααα − + = = − ∆ + ∆ (46)

with

2

c i ii

b bb

α α+ −+= (47)

and bc={bic} for i=1,…,4. By making use of interval extension in interval mathematics and

assuming a monotonic behavior as in [12] and [13], the α-cut of the fuzzy performance index is:


12

, αααψ ψ ψ− + = (48)

where

4 4

1 1i i

c c c c

b i b ii i

b bα αψ ψ ψ ψ ψ ψ− −

= == − ∆ = + ∆∑ ∑ (49)

in which the superscript c denotes that the corresponding quantity is evaluated at b=bc

( ) ( )( ) 00

c cxc c x

ccxx

σ σψ ψσσ

= = =b

bb

(50)

( )i i

c

c cb b

ib

ψψ ψ=

∂= =∂

b b

b (51)

Taking into account the above definitions, the computational procedure is based on the fol-lowing steps that must be repeated as many times as the number of α-cuts.

At the beginning, the following Lyapunov equations have to be solved:

( ) 6 6

Tc c c c c×+ + =A R R A B 0 (52)

( )0 0 0 0 0 4 4

Tc c c c c×+ + =A R R A B 0 (53)

from which

( ) ( )0 0 0

2,2 1,1c c c cx xσ σ= =zz z zR R (54)

The central point of the α-cut of fuzzy performance index ψc which appears in Eq. (49) is obtained by introducing the results of Eq. (54) in Eq. (50).

Now, Eq. (49) needs the calculation of the following quantity:

( )

0 0

0

2

, ,i i

i

c c c c

x b x x x bc

bc

x

σ σ σ σψ

σ

−= (55)

where

( )( )

( )( )

0 0

0 0

,,

, ,

1,12,21 1

2 22,2 1,1ii

i i

ccbbc c

x b x bc cσ σ= =

z zzz

zz z z

RR

R R (56)

The quantities in Eq. (56) are evaluated by solving the first-order Lyapunov equations:

( ) ( )( ) 6 6i i i i i

TTc c c c c c c c cb b b b b ×+ + + + =A R R A A R R A B 0 (57)

( ) ( )( )0 0, 0, 0 0, 0 0 0, 0, 4 4i i i i i

TTc c c c c c c c cb b b b b ×+ + + + =A R R A A R R A B 0 (58)

The calculation of the derivates for the matrices in Eq. (57) and Eq. (58) is straightforward and thus omitted for sake of conciseness. The procedure is summarized in Fig. 3.


13

Figure 3: Flow-chart of the resolution strategy to compute fuzzy performance index and objective functions.


14

The monotonic response assumption over the range of uncertain parameter values ensures that no turning points exist on the performance index surface in the parameter space. As a consequence, the upper and lower bounds of the considered α-cut exist at some combination of the end points of the α-cuts of the uncertain parameters. Some preliminary numerical inves-tigations verified that the performance index in Eq. (30) has a monotonic behavior over a range of numerical values of practical interest, as those in the following applications.

5 NUMERICAL APPLICATIONS

This section provides results of numerical study in order to demonstrate the applicability of the proposed design strategy of TLCDs.

5.1 Example

Since the design variable is a 2D vector, a graphical visualization of the objective functions can be obtained, thus showing the conflict between TLCD performance – as evaluated by means of f1 in Eq. (44) and robustness against fuzzy uncertainties, as evaluated by means of f2 in Eq. (45). A list of the adopted numerical values is given in Tab. 1. Fuzzy variables in Tab. 1 have a triangular MF (e1/e2/e3) in which e2 denotes the core value (µ(e2)=1) and the closed interval [e1,e3] the support (µ(e1)→0+ and µ(e3)→0+).

Problem data Numerical value

sωɶ (1.0π/1.5π/2.0π) rad/s

sξɶ (0.02/0.03/0.04)

µ 0.01

p 0.70

fωɶ (4.5π/9π/13.5π) rad/s

fξɶ (0.40/0.60/0.80)

Table 1: Numerical data.

The considered search space is dmin={0.85,0.01} and dmax={1.15,1.20}. The surfaces representing all values of the objective functions over the design space are shown in Fig. 4 and Fig. 5 (the mesh grid size is 50×50). Looking at the performances only (Fig. 4), accepta-ble results are obtained by selecting the tuning ratio close to 1 – i.e. γ within [0.96,1.00] – and by taking the coefficient of head loss values ξ within [0.40,0.90]. Therefore, it seems that de-signers can choose the final solution which best fit their (practical, economical, etc.) needs within a quite large number of alternatives. However, looking at the robustness of the final performances (Fig. 5), the designers should note some issues to be addressed in selecting the candidate solutions space. In fact, a very large variability of the final performances is ob-served in Fig. 5 for ξ within [0.40,0.90] and γ within [0.95,1.02]. In ξ approaches 0.50 and γ→1.03, then the final standard deviation up to one order of magnitude. Therefore, it is ex-pected that the performance variability strongly influences the design strategy for ξ. Where the best values for the tuning ratio occur, if ξ is chosen to met the best performance – that is, within [0.40,0.90] – then a very small variation for γ may lead to a non-proportional and very


15

large worsening of the robustness because the gradient in that zone of the search space is very high. On the other hand, if ξ falls between 0.12 and 0.30, then unexpected variations for γ (i.e. manufacturing errors, bad estimation of the MF for ωs) will not cause an excessive worsening of the TLCD’s robustness.

Figure 4: Objective function f1 over the design space.

Figure 5: Objective function f2 over the design space.

The Pareto front and the set of non-dominated solutions in Fig. 6 and Fig. 7 confirm the above analysis. A linear trade-off exists between f1 and f2 (Fig. 6, on the left). The most of the non-dominated solutions are the couples( ) [ ] [ ], 0.960.0.981 0.025,0.30γ ξ ∈ ⊗ .


16

Although better performances can be obtained by increasing the coefficient of head loss (γ close to 0.985 and ξ between 0.60 and 0.70), the final performance will be more susceptible to large variations due to unexpected errors affecting γ.

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

f1

f 2

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.990

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

γ

ξ

Figure 6: Pareto front and optimal solutions.

Figure 7: Distribution of the Pareto optimal solutions over the search space.

5.2 Sensitivity analyses

Two parameters – the mass ratio µ and the length ratio p – were considered as deterministic input data in Tab. 1. A sensitivity analysis may be carried out for them in order to understand their effects on the selection of the best TLCD parameters.

Different Pareto fronts and non-dominated solutions sets for varying mass ratio are de-picted in Fig. 8. As expected, the optimal tuning ratio decreases as µ grows. Moreover, better performances are achieved when µ increases. For medium and high performances, the robust-


17

ness decreases as µ grows. The trade-off between performance and robustness is still linear. The three Pareto front in Fig. 8 have some common points, which indicates that it is possible to obtain the same performance as well as robustness under different mass ratios scenarios by properly tuning the TLCD parameters.

The effects of the length ratio p on the final set of the non-dominated solutions are less evident. As the length ratio increases, the optimal value for γ decreases. It seems that the Pare-to fronts are stretched and rotate on the worst performance point (f1→1) as p increases.

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 10

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02

0.0225

0.025

0.0275

f1

f 2

0.85 0.865 0.88 0.895 0.91 0.925 0.94 0.955 0.97 0.985 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

γ

ξ

µ=0.0050µ=0.0125µ=0.0200

µ=0.0050µ=0.0125µ=0.0200

Figure 8: Pareto fronts and optimal solutions with varying mass ratio µ.

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 10

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02

0.0225

0.025

f1

f 2

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.990

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

γ

ξ

p=0.3p=0.6p=0.9

p=0.3p=0.6p=0.9

Figure 9: Pareto fronts and optimal solutions with varying length ratio p.


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6 CONCLUSIONS

The paper presented an approach for incorporating fuzziness in the optimum design of TLCDs subject to stochastic ground motion. The main features of the proposed approach are: (i) the use of the Taylor series expansion for computing the final fuzzy performance index and (ii) the application of the credibility theory to define the objective functions of the final multi-objective optimization problem. The main advantage of the proposed procedure deals with its versatility in considering different sources of uncertainties in the robust design of TLCDs.

Numerical applications confirm the important role of non-probabilistic sources of uncer-tainty in random vibrations problems and the effectiveness of our approach in supporting the design of TLCDs. From a practical standpoint, numerical results show that the robustness against (fuzzy) uncertainties influences typical parameters for TLCDs whit respect to the case when they are designed taking into account random dynamic loadings only. In fact, we found that smaller values for the coefficient of head loss may be preferred to avoid an excessive va-riability of the final performances as consequence of unexpected fluctuations of the tuning ratio.

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