DESIGN OF TLCD UNDER RANDOM LOADS: A NEW FORMULATION

Meccanica dei Materiali e delle Strutture Vol. 3 (0), no.3, pp. -8 ISSN: 035-679X Dipartimento di Ingegneria Civile Ambientale e dei Materiali - DICAM DESIGN OF TLCD UNDER RANDOM LOADS: A NEW FORMULATION Alberto Di Matteo *, Francesco Lo Iacono, Giacomo Navarra, Antonina Pirrotta * * Dipartimento di Ingegneria Civile, Ambientale e dei Materiali (DICAM) Università degli Studi di Palermo, Viale delle Scienze, 908 Palermo, Italy e-mail: antonina.pirrotta@unipa.it Facoltà di Ingegneria ed Architettura Università degli Studi di Enna Kore, Cittadella Universitaria, 9400 Enna, Italy e-mail: giacomo.navarra@unikore.it; francesco.loiacono@unikore.it (Ricevuto 0 Giugno 0, Accettato 0 Ottobre 0) Key words: passive control, tuned liquid column damper, stochastic linearization. Abstract. Several types of passive control devices have been proposed in recent years in order to reduce the dynamic responses of different kind of structural systems. Among them, the Tuned Liquid Column Damper (TLCD) proved to be very effective in reducing vibration of structures. This paper aims at developing a pre-design simplified formulation, by means of a stochastic linearization technique, able to predict the effectiveness of TLCD when subjected to random agencies. The numerical result are then validated through an experimental campaign on a small scale SDOF shear-type model built in the Experimental Dynamic Laboratory of University of Palermo and equipped with a TLCD excited at the base with random noises through a shaking table. INTRODUCTION Tuned Liquid Column Damper (TLCD) systems can be considered as a particular type of passive mass dampers and represent an effective alternative to Tuned Mass Damper (TMD) systems [] to control the vibration level of structures. TLCDs dissipate vibration energy by a combined action of the movement of the liquid in a U-shaped container, the restoring force on the liquid due to the gravity and the damping effect due to the passage of the liquid through the orifice with inherent head loss characteristics. The TLCD is generally modelled as a single degree of freedom (SDOF) oscillator which is rigidly attached to a vibrating structure [] and, like TMDs, the effectiveness of a TLCD depends on proper tuning and damping value. However, unlike traditional TMDs, the TLCD response is non-linear and the optimal parameters cannot be established a-priori and Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8

numerical optimization methods are needed [3]-[5]. Several investigations on TLCDs controlled structures under stochastic loads have been recently conducted by many authors and some useful design formulas for TLCD have been derived [6]. Recently, the authors of this paper presented a modified mathematical formulation of the TLCD equation of motion which has been proved to be very effective in the prediction of the experimental system vibrations when the input is deterministic and periodic [7]-[8]. Due to the randomness of environmental forces a deeper investigation of the behaviour of TLCD controlled systems is needed. In this paper the stochastic linearization technique is used in order to develop a pre-design simplified formula for the optimal TLCD parameters on lightly damped single-degree-of-freedom structures loaded by random agencies. The ready-touse and straightforward proposed formulation has been verified by comparison with the numerical Monte Carlo simulation based on the non-linear complete system and, finally, the theoretical and numerical results are proved to be in very good agreement with the results obtained by an experimental campaign on a small scale SDOF shear-type model built in the Experimental Dynamic Laboratory of University of Palermo and equipped with a TLCD excited at the base with random noises through a shaking table. PROBLEM FORMULATION Let it be assumed that a main structure (Figure -a), characterized by a light-damped sheartype single-degree-of-freedom (SDOF) system, is subjected to base excitation and is mitigated by using a tuned liquid column damper (Figure -b). x (t) 5 mm b d 500 mm y (t) h x (t) 50 mm upper plate a) b) Figure. a) SDOF shear-type model; b) SDOF system with TLCD. While the equation of motion of the uncontrolled system may be expressed as in equation (), the classical formulation of the equations of motion, widely used in literature [], for the TLCD controlled system can be represented as reported in (). ( ) + ( ) + ( ) = ( ) Mx ɺɺ t Cxɺ t Kx t Mx ɺɺ t () In equations () and (), M, C and K are the mass, damping and stiffness parameters of the SDOF system, respectively, xɺɺ represents the ground acceleration, x and y are the g g Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8

displacement of the SDOF with reference to the ground, and the displacement of the liquid in vertical columns, respectively, mtlcd = ρ AL is the total liquid mass of the TLCD, mh = ρ Ab equals to the liquid mass of the only horizontal portion, being ρ the liquid mass density, A the cross sectional area, and L the total liquid length inside the tube (L=b+h). The head loss coefficient and the gravitational constant are denoted by ξ and g, respectively. The upper dots mean time derivatives. ( M m ) ɺɺ x ( t) mɺɺ y ( t) Cxɺ ( t) Kx( t) ( M m ) ɺɺ x ( t) + TLCD + h + + = + TLCD g mɺɺ hx t m ɺɺ TLCDy t A yɺ t yɺ t Agy t mɺɺ hxg t ( ) + ( ) + ρ ξ ( ) ( ) + ρ ( ) = ( ) In order to reduce the number of the parameters involved, let us express the equations of motion in a more convenient dimensionless form. This can be achieved by dividing the equations () by M and by m, respectively: TLCD ( µ ) ɺɺ x( t) αµ ɺɺ y ( t) ζ ω xɺ ( t) ω x( t) ( µ ) ɺɺ xg ( t) αɺɺ x( t) + ɺɺ y ( t) + c yɺ ( t) yɺ ( t) + ω y ( t) = αɺɺ x ( t) + + + + = + g where µ = mtlcd M, α = mh mtlcd, ω = K M, ω = g L, ζ = C Mω ; c = ξ L. Since the damping term in the second of equations (3) is nonlinear, even supposing that the uncontrolled structure behaves linearly, the whole damper-structure system has inherent nonlinear properties. It follows that simple design techniques as Response Spectra (RS) method may not be pursued. In order to overcome this difficulty, this paper aims at defining an equivalent system in which a linear viscous damping coefficient is chosen through a simplified procedure by using the Statistical Linearization Technique (SLT) [9]. () (3) 3 STOCHASTIC LINEARIZATION OF TLCD Let us consider that the structure-damper system depicted in Figure -b is excited by random forces at base that can be modelled as zero mean Gaussian processes. It follows that the displacements and their derivatives are stochastic processes too and, as customary, are denoted with capital letter. Due to the presence of the nonlinear damping term, the response processes are non-gaussian. According to the SLT, the original nonlinear system (3) is replaced by a linear equivalent one as follows: ( µ ) Xɺɺ ( t) αµ Yɺɺ ( t) ζ ω Xɺ ( t) ω X ( t) ( µ ) Xɺɺ g ( t) α Xɺɺ ( t) + Yɺɺ ( t) + ζ ω Yɺ ( t) + ω Y ( t) = α Xɺɺ ( t) + + + + = + g where ζ is the equivalent damping ratio that is to be chosen in order to minimize the mean square error made in passing from (3) to (4). Omitting for clarity s sake the dependence from the time, we have: ( c Y Y ζ ωy ) = ζ E min ɺ ɺ ɺ (5) where E[ ] means ensemble average. Once the minimum is performed and the responses Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8 3 (4)

processes have been considered as Gaussian ones, the equivalent damping is obtained as: c ζ = σ Y ω π (6) ɺ where σy ɺ is the standard deviation of the velocity of the fluid. The use of equation (6) for design purposes is not straightforward since σy ɺ is still unknown and it implicitly depends on ζ, then, as usually an iterative procedure is necessary [9]. Since the main goal of the present paper is to readily find the equivalent damping ratio of TLCDs in a design framework, the iterative procedure cannot be pursued. Equation (6) provides a relationship between the equivalent damping ratio ζ and the standard deviation of the fluid velocity σy ɺ. In order to find a direct, even simplified, relationship between the input characterization GXɺɺ g ( ω) and the estimated value for ζ, a closed-form solution in terms of steady state response statistic σy ɺ is needed. In order to do so, let us suppose that the input can be modelled as a zero-mean stationary Gaussian white noise process. In this framework, for the linear system (4) the Lyapunov equation of the evolution of the covariance matrix can be used and the steady state response statistics could be easily evaluated. After some algebra the exact solution for the steady state variance of the fluid velocity can be expressed as: π S σɺ = (7) 0 y zω in which the factor z, has a very cumbersome expression which cannot be used for practical design purposes. In order to obtain a design formula we have to introduce the following approximations. We can assume that ω is practically coincident with ω, since the damper has to be tuned with the main system; moreover, if the main system is lightly damped, the higher powers of ζ can be neglected. Introducing these assumptions the following expression of z is obtained: ( ) z = µ ζ + γζ (8) where the dimensionless parameter γ = µ + µ α depends only on the geometry of the TLCD. In such a way the variance of the fluid velocity σy ɺ can be directly related with the white noise strength S 0 and by eliminating σy ɺ, a direct expression that provides the equivalent damping ratio ζ as a function of the white noise strength S 0 is obtained: S c ζ ζ + γζ = (9) 0 ( ) 3 µω The nonlinear algebraic equation (9) can be easily numerically solved in order to obtain a good estimate of ζ to be used for design purposes. In the following sections the proposed procedure will be validated through numerical and experimental tests. 4 NUMERICAL VALIDATION In the present section the simplified formulation is validated by means of numerical simulations. In order to define a realistic analytical system on which numerical integration of Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8 4

motion equations can be performed, we will now refer to the analytical model of TLCD () whose parameters, as reported in Table, have been computed by identifying the dynamical properties of the experimental model that will be presented in the next section. The steady state response statistics of the TLCD-controlled system are obtained through a 000 samples Monte Carlo simulation directly integrating the equations of motion by means of a 4 th -order Runge-Kutta scheme. The response statistics of the uncontrolled system () and of the equivalent linear system (4) may be evaluated either by a Monte Carlo simulation, or by means of a stochastic analysis, since these are linear systems. In a similar way the steady state Power Spectral Density (PSD) functions have been computed. Obviously, in the definition of the equivalent linear system (4) the value for ζ is determined by means of the simplified proposed procedure (9). E[X ] [m ] 5x0-3 4x0-3 3x0-3 x0-3 0-3 0x0 0 uncontrolled nonlinear equivalent linear 0 0.00 0.004 0.006 0.008 0.0 S 0 [(m/s ) /(rad/s)] E[Y ] [m ] x0-3 x0-3 8x0-4 4x0-4 0x0 0 nonlinear equivalent linear 0 0.00 0.004 0.006 0.008 0.0 S 0 [(m/s ) /(rad/s)] Figure. Variance of displacements versus input strength: a) uncontrolled system (green line), main displacement of non linear system (red line) and equivalent linear system (orange dashed line); b) fluid displacement of non linear system (blue line) and equivalent linear system (blue dashed line) Figure shows the variances of the main system displacement E X ( t) and of the fluid displacement E Y ( t) for different values of the input strength S 0, respectively. These variances are computed for the uncontrolled system and compared with those obtained for the TLCD-controlled system by using both nonlinear equations (3), and equivalent linear equations (4). It is evident that the control system is effective and that the equivalent linear system fits the nonlinear one even for large values of the input strength. G(f) [m /(rad/s)] 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0-0 Output PSD G xx,unc G xx,nl G xx_lin G yy,nl G yy_lin a) 0 3 4 5 b) 0 3 4 5 f [Hz] f [Hz] Figure 3. Steady state PSD functions: a) for a low value of input strength; b) for an high value of input strength; uncontrolled system displacement (green solid line), main system TLCD-controlled displacement (red lines), fluid displacement (blue lines); solid lines for nonlinear system, dashed lines for equivalent linear system. Figure 3-a reports steady state PSD functions for a low value of the input strength 4 S 0 = 0 (m/s ) /(rad/s). From this picture is clearly visible the effect of the TLCD that generates two peaks in the controlled system (solid red line) instead of the higher single peak G(f) [m /(rad/s)] 0-0 - 0-3 0-4 0-5 0-6 0-7 0-8 Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8 5

of the uncontrolled system (green solid line). Furthermore, it is possible to note that dashed lines (equivalent linear system) cannot be distinguished by the solid lines (nonlinear system) for this level of input strength. In Figure 3-b steady state PSD functions for an high value of the input strength S 0 = 0 (m/s ) /(rad/s) are reported. From this picture it is evident the higher value of damping associated to the current value of input strength. Moreover, now it is possible to distinguish PSDs obtained by the equivalent linear system (dashed lines) from those computed by from nonlinear system (solid lines), which also exhibit a superharmonic frequency at 4.35 Hz due to the nonlinearity. 5 EXPERIMENTAL VALIDATION In the present section the numerical results obtained by using the proposed formulation for the equivalent linear damping estimation are validated by means of an experimental campaign. In order to do so, a small-scale shear-type SDOF model equipped with a TLCD has been built in the Laboratory of Experimental Dynamic at University of Palermo. Figure 4-a reports a picture of the uncontrolled model over a shake table, while in Figure 4-b the picture of the TLCD-controlled system is reported. The main uncontrolled system is a small scale single degree of freedom (SDOF) shear-type frame composed by two steel columns (length 500 mm, width 50 mm and thickness mm) and two nylon rigid plates (length 50 mm, width 50 mm and thickness 5 mm) as base and floor respectively. The total mass model is 4.507 kg, of which 0.408 kg takes into account the dead weight of the tube mass of TLCD. a) b) Figure 4. Experimental one storey shear-type model setup; a) uncontrolled system; b) TLCD-controlled system. The TLCD device, is a U shaped cylinder tube, with constant cross section 3 A =.9 0 m, rigidly connected to the upper plate of the main system, to create a simple TLCD-controlled system. The centerlines of the vertical branches are at a distance of 05 mm 3 each other. The tube has been filled with water ( ρ = 000 kg/m ), up to a level of 40 mm from the centerline of the base tube in order to be tuned with the uncontrolled system. The water that has been poured has a total liquid mass m TLCD = 0. 48 kg. and total liquid length of L = 85 mm, which corresponds to a mass ratio µ = m M TLCD = 0.098, close to 0%. The acceleration responses, at the base and at the storey of both uncontrolled and Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8 6

controlled systems, have been acquired using miniaturized piezoelectric accelerometers and then processed using a self-developed signal processing software in LabView and MATLAB environment. Both uncontrolled and controlled models have been excited at the base through a shaking table that provides the displacement controlled ground motion. Table. Parameters for the definition of a structure-tlcd system - eq. () and eq. (3). parameter value parameter value M 4.503 kg µ 0.098 unitless m TLCD 0.48 kg α 0.8794 unitless m h 0.368 kg ω 0.07 rad/s C 0.53 Ns/m ω 0.366 rad/s K 45.77 N/m ζ 0.008 unitless ρ 000 kg/m 3 c 5.479 m - A.9 0-3 m ξ unitless As a first stage, the dynamic parameters of the uncontrolled system and of the TLCDcontrolled system were identified by exciting the structure with broadband noises in the range 0.5 0 Hz. The dynamic parameters were identified by using some well-known parameter extraction techniques [0] such as Rational Fractional Polynomial method, genetic algorithm and particle-swarm optimization method that provide similar estimations of the parameters that are reported in Table. 0.6 0.4 0. uncontrolled exp. uncontrolled num. nonlinear exp. nonlinear num. equivalent linear num. 0 0 0.4 0.8..6 Figure 5: Experimental validation in term of variances. Variances of uncontrolled system (green) and TLCDcontrolled systems (red). Solid lines indicates numerical results (eq. () for uncontrolled system, eq. (3) for TLCD-controlled system), dashed line stands for equivalent linear system (eq. (4)), dots stand for experimental results. The effectiveness of the control has been validated by computing the response statistics in terms of variances from 50 samples of ground acceleration for several levels of input strength and then comparing them with the variances obtained by solving numerically the equation (3) and (4). This comparison is reported in Figure 5 showing that the experimental results are in a good agreement with the numerical ones and the effectiveness of the control in reducing structural responses is clearly visible. Moreover, the equivalent linear system follows very closely the trends of both experimental statistics and numerical results by using the nonlinear equation, thus proving the reliability of the proposed formulation in predicting structural responses, even in experimental field. Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8 7

CONCLUSIONS In the present paper the effectiveness of a simplified pre-design formulation for TLCD passive control system subjected to random loads has been investigated. The proposed formulation has been validated through some numerical simulations and an experimental campaign on a small scale SDOF shear-type model in the Laboratory of Experimental Dynamic at University of Palermo. Results showed that stochastic linearization technique works very well both in frequency and in time domain. Moreover, the response statistics obtained by classical nonlinear formulation, by the equivalent linear system obtained by using the proposed formulation and by the experimental results are very close to each others, thus proving the reliability of the proposed approach. The design formulation outlined in the paper can easily be applied also to random processes which have Response Spectra coherent with building codes. REFERENCE [] Warburton G.B., 98, Optimum Absorber Parameters for Various Combinations of Response and Excitation Parameters, Earthquake Engineering and Structural Dynamics, 0, 38-40. [] Sakai F., Takeda S., Tamaki, T., 989, Tuned liquid column damper- new type device for suppression of building vibrations, Proceedings of the international conference on highrise buildings, 96-93. [3] Hochrainer M.J., 005, Tuned liquid column damper for structural control, Acta Mechanica, 75, 57 76. [4] Farshidianfar A., Oliazadeh P., 009, Closed form optimal solution of a tuned liquid column damper responding to earthquake, World Academy of Science, Engineering and Technology, 59, 59-64. [5] Yalla S.K., Kareem A., 000, Optimum absorber parameters for Tuned Liquid Column Dampers, Journal of Structural Engineering, 6(8), 906-95. [6] Chang C.C., 999, Mass dampers and their optimal designs for building vibration control, Engineering Structures,, 454-463. [7] Di Matteo A., Lo Iacono F., Navarra G., Pirrotta A., 0, The control performance of TLCD and TMD: experimental investigation, 5th European Conference on Structural Control, 8 0 June, Genoa, Italy. [8] Di Matteo A., Lo Iacono F., Navarra G., Pirrotta A., 0, The TLCD Passive Control: Numerical Investigations Vs Experimental Results, Proceedings of the ASME 0 - International Mechanical Engineering Congress & Exposition IMECE0, 9 5 November, 0, Houston, Texas, USA. [9] Roberts J.B., Spanos P.D., 990, Random Vibration and Statistical Linearization, Wiley, New York, USA. [0] Ewins D. J., 984, Modal Testing: Theory and Practice. Research Studies Press, Taunton, Somerset, England. Meccanica dei Materiali e delle Strutture 3 (0), 3, PP. -8 8