OPTIMUM DESIGN OF TUNED MASS DAMPERS BY DISPLACEMENT AND ENERGY APPROACH

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1 Meccanica dei Materiali e delle trutture Vol. 3 (1), no.1, pp IN: X Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, Dei Materiali DICAM OPIMUM DEIGN OF UNED MA DAMPER BY DIPLACEMEN AND ENERGY APPROACH R. Greco *, G.C. Marano * DICAECH, Department o Civil Engineering, Environmental, erritory, Building and Chemical, echnical University o Bari, via Orabona 4-715, Bari -Italy. el r.greco@poliba.it DICAR, Department o civil Engineering and Architecture echnical University o Bari, via Orabona 4-715, Bari -Italy. el , g.marano@poliba.it. (Ricevuto 1 Giugno 1, Accettato 1 Ottobre 1) Key words: uned Mass Damper, Optimum design, Dissipated energy, uned Mass Damper perormance. Parole chiave: Masse Attonate, Progetto ottimo, Dissipazione energetica. Abstract.. he present work is ocused on the analysis o the eectiveness o uned Mass Dampers: more in detail, the goal is to test the uned Mass Damper perormance both rom the energy and displacement prospective. For this aim a simple one degree o reedom system subject to a dynamic action and equipped with a uned Mass Damper is analyzed. he device perormance is estimated by means o two indicators related to maximum displacement and dissipated energy reduction into the primary system. hese indices will be adopted as objective unction in order to perorm an optimization procedure based on two dierent criteria, aimed at obtaining uned Mass Damper parameters which urnish the better perormance. ommario. Questa memoria concerne il progetto ottimo degli smorzatori a Masse Attonate. Lo studio viene svolto considerando un sistema ad un grado di libertà soggetto ad un sisma rappresentato da una orzante stocastica. Le prestazioni dello smorzatore sono stimate tramite due indicatori: la riduzione dello spostamento dell'oscillatore e la riduzione dell'energia incanalata nella struttura principale. ali indicatori sono pertanto utilizzati come unzioni obiettivo nel progetto ottimo che ornisce i parametri ottimi dello smorzatore a Masse Attonate. 1 INRODUCION he MD technology uses a weight which oscillates at the same period as the main system does, and an additional damper that connects two relatively moving points when the system oscillates. ince the natural requency o MD is tuned in resonance with the undamental Meccanica dei Materiali e delle trutture 3 (1), 1, PP

2 mode o the primary system, a large amount o the main system vibrating energy is transerred to the MD and then dissipated by the damping when the primary system is subjected to external disturbances. his additional system absorbs the vibrations automatically and, in this way, the saety and unctionality o the main system can be greatly enhanced. Many studies have been carried out to demonstrate that MD is very eective in protecting dynamic systems rom vibrations. In [1] the authors discussed the state o the implementation o MD or buildings protection, and the eects were clariied based on various recorded values in actual buildings against both wind and earthquake actions. In [] the environmental loading to be a long-term non-stationary stochastic process characterized by a probabilistic power spectral density unction was considered. In [3] and [4] the MD protection eiciency or a simple system excited at the support has been investigated, considering also dierent environmental conditions with the aim to urnish some promising indications or complete MD optimization. he use o MD has also been demonstrated to be very eective rom the energy point o view. For example, in [5] the authors studied the energy balance and the energy transer among the elastic structure and ound that MD can be very excellent in protecting structures, by increasing the damping energy dissipation. In [6] the author analyzed the transer process o using MD in improving the ability o inelastic structures to dissipate earthquake input energy. Perormance o MD in vibration reduction rom an energy point o view has also been studied in [7]. In [8] the authors analyzed seismic energy dissipation o inelastic structures with multiple MDs. his paper is aimed at evaluating the double perormance o MD both rom an energy point o view and rom the more standard one based on displacement. For this purpose the perormance o MD is estimated by means o two indicators, the maximum displacement reduction and the primary system dissipated energy reduction. hese indices are then adopted as objective unction or perorming an optimization procedure based on two dierent criteria. he analysis is developed in stochastic meaning, considering a generic stationary random action exciting the system here modelled as a simple one degree o reedom system and equipped with a single linear MD. DYNAMICAL YEM AND BAIC EQUAION In the present study MD is considered or the control o the speciic vibration mode o a generic system, representing or example a structural system excited by external actions, such as wind and earthquake. he analytical model o this combined system is shown in Figure 1, where the main system is modeled as a single degree o reedom system (i.e. the main system or the mode-generalized system in the mode being controlled). he dynamic response o this combined system is governed by the dynamic equilibrium equation: My ɺɺ + Cyɺ + Ky=-Mrɺɺ y g (1) where y= ( y,y) is the relative base displacement vector, and M, C and K are mass, damping and stiness matrices, respectively. In equation (1 ) ɺɺ y g(t) is the acceleration that excites the system at its support and it is modeled by means o the Kanai-ajimi process [9] by the ollowing equations: ɺɺ y +ξωy ɺ +ω y =-w () ɺɺ y g=y ɺɺ +w=- ( ξωy ɺ +ω y ) Meccanica dei Materiali e delle trutture 3 (1), 1, PP

3 In equation () w is a stationary Gaussian zero-mean white noise process, whose intensity is, representing the excitation at the bed rock. Moreover, y is the response o the Kanai- ajimi ilter having a requency excitation process at the base o the structure. ω and a damping coeicient ξ. Finally, yɺɺ g is the Figure 1: Mechanical model o a linear MD system. Introducing this seismic model the new state space vector becomes: ( ) z= y y y yɺ yɺ yɺ. (3) R E zz zz, where the symbol [ ] he space state covariance matrix = E denotes the mathematical expectation, is obtained as the solution o a Lyapunov equation [1]: ARzz + RzzA + B = (4) where: A = -ω +ω +ω -ξω +ξω +ξω + δω -( δω +ω ) +ω + δ ξω -( δξω +ξω ) +ξω -ω -ξω (5) he 6x6 matrix B has all null elements except one: [ ] 6,6 (6) he mechanical parameters o the combined system are: k ω =, m k ω s=, s m c c m ξ =, ξ =, δ = (7) m k m k m 3 HE ENERGY EQUAION In recent years eorts have been undertaken to develop the concept o energy dissipation or supplemental damping into a workable technology and a number o these devices have been installed in structures throughout the world [11]. In general, they are characterized by the capability to enhance energy dissipation in the systems in which they are installed. his may Meccanica dei Materiali e delle trutture 3 (1), 1, PP

4 be achieved either by conversion o kinetic energy to heat, or by transerring o energy among vibrating modes. When the simple oscillator considered in this study is subjected to an external excitation, an amount o energy is introduced into it inducing deormation in components: inally, at the end o the event, when the quite is reached, the total energy imparted to system is completely dissipated by damping properties and by inelastic deormations i components show hysteretic behavior. It can be assumed that the damage measure resulting rom inelastic behavior depends on the capacity to dissipate the input energy. From energy prospective, in order to reduce the vibration level and the induced damage one must attempt to minimize the amount o energy dissipated by the primary system. For the analyzed system in igure 1 the relative energy balance equation is: E k (t) + E D (t)+ E (t)=e I (t) (8) where: t t t t E = yɺ My ɺɺ dt, E = yɺ Cyɺ dt, E = yɺ Kydt, E =- yɺ Mrɺɺ y dt (9) k D From energy balance equation it is evident that energy-dissipating devices, such as MD, provide supplemental energy storage and damping energy dissipation, to reducing the structural response caused by seismic input energy. In this sense it is oreseeable that the use o MD will enhance the energy capability o the entire structure and reduce the primary system dynamic responses. Because in this study the eectiveness o MD is considered under a stationary input, the power balance can be considered instead o energy balance: yɺ My ɺɺ + yɺ Cyɺ + yɺ Ky=-yɺ Mrɺɺ x g (1) Introducing the power terms, equation (1) becomes: P k (t) + P D (t)+ P (t)=p I (t) (11) where terms are in order: the kinetic power at the time, the dissipated power, the elastic power, the input power, evaluated at the time t. Ater some rearranging and introducing the mass ratio δ =m /m s one obtains: δ yɺ ɺɺ y + yɺɺɺ y +ξ ω y ɺ +δ ξ ω (yɺ -y ɺ )y ɺ + δ ω (yɺ -y ɺ )y +ω yɺ y =-δ yɺ ɺɺ y -yɺɺɺ y g g In stationary condition the power balance can be written: ξωy ɺ +δ ξ ω (yɺ -y ɺ )yɺ -δ ω yɺ y =-δ yɺ ɺɺ y -yɺɺɺ y s s g and considering the power low in mean value one obtains: δ ξ ω σ +ξωσ -δ ξ ω γ -δ ω γ =-δ γ -γ In equation (14) yy y I g g (1) (13) yɺ s s yɺ yɺ yɺ yɺ y yɺ ɺɺ yb yɺ ɺɺ yb (14) σ and ɺ σ ɺ are the variances o yɺ and yɺ respectively, whereas γ and ɺ ɺ y γ ɺ are the cross-correlations. In the ollowing the attention will be ocused on the dissipated terms: P Dsyst and P Ddevic i.e. the power dissipated by the main system and the power dissipated by the MD device. As well known, the dissipative term o the main system is strictly related to its damage, whereas the amount o the power dissipated by the device points out on its perormance in the protection o the system. I dissipative terms are evaluated in mean value then they assume the orm: µ P =ξωσ Dsyst s y ɺ s (15) µ P =δ Ddevict ξωσ yɺ -δ ξωγyɺ yɺ (16) yy Meccanica dei Materiali e delle trutture 3 (1), 1, PP

5 4. OPIMUM DEIGN OF DM DEVICE In this section the optimum design o MD is developed. For this purpose two dierent perormance indices are introduced in stochastic meanings: these indices will be adopted as objective unctions in the optimum design strategy, aimed at obtaining the best uned Mass Damper mechanical parameters which urnish the better perormance rom the energy prospective and rom the displacement prospective. his means that the main goal o optimization is deining the MD mechanical characteristics that reduce the amount o the energy dissipated by the primary system to be protected or, alternatively, that reduce primary system displacement. hereore, in order to evaluate this double perormance o MD, both rom the energy point o view and rom the deormation approach, the ollowing stochastic indices are introduced in stochastic meaning. µ PDsyst First, the Factor Reduction o Dissipated Power (FRDP) is introduced: FRDP= µ being µ the mean o the power dissipated by the main system when it is equipped with PDsyst the MD, and µ P the same quantity or uncontrolled system. In addition, also the power Dsyst low dissipated by the device µ P will be is considered in the ollowing. Ddevic At the same time the Factor o Reduction o main system Displacement (FRD) is introduced: σ ys FRD=, being σ y the root mean square o the main system displacement when it is σ s ys equipped with the MD, and being σ y the same quantity or the uncontrolled system. s Ater the two stochastic indices are introduced, optimum MD design parameter are evaluated by perorming the solution o the optimum problem. In detail, or the MD the optimum design is ormulated as the evaluation o Design Vector b = (ω,ξ ), which is able to minimize the Objective Function (OF) represented by the MD perormance index (FRDP or the FRD). he ollowing optimum design problems are ormulated: Displacement approach Find b = (ω,ξ ) (17) Which Minimizes σy s OF = FRD= σ Energy approach y s Find b = (ω,ξ ) (16) Which Minimizes µ PDsyst OF = FRDP= µ PDsyst he optimization has been developed by means o Genetic Algorithm strategy. PDsyst 4.1 Numerical analysis In this section the results o optimum design o MD are presented and discussed. Meccanica dei Materiali e delle trutture 3 (1), 1, PP

6 he study considers a simple structural system subject to a iltered stochastic seismic motion and equipped with a MD. he system damping has been assumed ξ =%, the requency ω =π rad/sec, and the mass ratio δ =.1. Input is characterized by the ollowing parameters: ω =rad/sec and ξ =.5. s Figure : Comparison between the two optimization criteria in design variables space. Let - Displacement criterion, Right- Energy criterion. he marked point is the optimum solution. Figure shows the results obtained by two dierent perormance criteria in the space o design variables. First o all, one can point out that the parameters which maximize MD perormance (minimum o the OF) are almost the same in both the adopted criteria, and only a little dierence in optimum parameters can be appreciate in this igure. In detail, ρ is MD larger in the case o energy criterion, whereas ξopt is larger i the displacement criterion is adopted. With regard the MD perormance attained by optimum parameters, i the energy criterion is adopted the OF is almost equal to. (that represents a very good perormance ) whereas in case o displacement approach the OF is almost equal to.45. ω In igure 3 the optimized perormance indices are plotted versus ψ= both in the case o ωs displacement criterion and energy criterion. From igure 3 one can observe that in case o optimum design developed by the energy approach the FRDP assumes approximately a constant value, equal to., over the entire range o requency investigated, which means a reduction o the 8% o the energy dissipated by the main system with respect to the uncontrolled situation. MD opt Figure 3: Comparison between the two optimization criteria. Plots represent minimized OFs and optimum design variables versus. he mass ratio is is.1, and ξ =%. Meccanica dei Materiali e delle trutture 3 (1), 1, PP

7 his result is o great relevance, especially when a hysteretic behaviour takes place into the structure, because as well known, the amount o the energy dissipated is strictly correlated to global and local damage level o the vibrating system. However, exactly a minimum o FRDP is observed aroundψ = 1, i.e. i the dominant requency o the excitation matches the resonance requency o the system to which the MD requency is tuned. his means, obvious, that the maximum eectiveness o MD is reached when the main system is in resonance with ground motion requency. I the displacement criterion is considered the MD perormances are more markedly earthquake dependant, as a variability versus ψ is noticeable. he minimum o the OF, equal almost to.45, is reached or ψ = 1; then, or ψ > 1 the perormance is almost constant, whereas or ψ <1 the MD eectiveness decreases and becomes ineicient when ψ decreases. With regard the optimum parameters one can notice that the energy criterion gives greater ρ MD opt values. Moreover ρ approaches to unit near to resonance condition, even i in the MD opt analyzed case this happens not exactly at ψ = 1 On the contrary, ξ is larger i the displacement criterion is adopted. It is interesting to examine also the amount o power terms when MD parameters are chosen by developing the optimum criteria here proposed. Figures 4 and 5 show the dierent power terms vs. ψ in case o displacement and energy approach, respectively. More precisely, the power terms are shown both in case o uncontrolled system and or system equipped with MD: the MD parameter adopted have been obtain by optimum design. In the upper graph the power input is plotted in both the analyzed cases: µ P is the power input or the uncontrolled system, whereas PMD MD opt µ is the power input or the system equipped with MD. his coincides with the sum o our terms plotted in the lower graph in igure 4, where terms µ or i=1,...4 are whose o the let side in equation (39), considered PMDi in the same order. First o all one can observe that the power introduced into the system by the excitation at the support is increased by the use o MD. It is important to note that the amount o damping power dissipated by the MD is the bigger between various terms; it varies with ψ and reaches the maximum or ψ near to resonance value. Figure 4: Power low in stationary condition or the system versus or the system optimized by displacement criterion. he mass ratio is is.1 and ξ =% On the contrary, the amount o the damping energy dissipated by the main system is lesser with respect to damping power dissipated by the MD and it is approximately insensitive to ψ. In addition, it is smaller i compared with the amount o dissipated energy in the uncontrolled structure. hese results mean that the reduction o the damping energy dissipated by the primary system is balanced by the increase o the damping energy dissipated by MD. Globally, the amount o the energy dissipated by the entire system is larger with respect to the Meccanica dei Materiali e delle trutture 3 (1), 1, PP

8 structure without MD. his suggests that MD has the ability to store a large amount o strain energy and dissipates it in the orm o damping energy, thereby protecting the primary system rom excessive local and global damages. he other two terms in igure 4 and 5 can be considered as power exchange terms, which are responsible o the transerring o the energy rom the primary system to the MD. Figure 5: Power low in stationary condition or the system optimized by energy criterion. he mass ratio is is.1 and ξ =%. REFERENCE [1]. himazu, H. Araki, urvey o actual eectiveness o mass damper systems installed in buildings, Eleventh world conerence on Earthquake Engineering, 1996, Paper no.89. [] H.J. Li, H..L. James, uned Mass Damper Design or Optimally Minimizing Fatigue Damage, Journal o Engineering Mechanics, 18 (8) () [3] G.C. Marano, R. Greco, B. Chiaia, A comparison between dierent optimization criteria or tuned mass dampers design, Journal O ound And Vibration, 39 (11) (1) [4] G.C. Marano,R. Greco, Optimization criteria or tuned mass dampers or structural vibration control under stochastic excitation, Journal o vibration and control, 17 (5) (11) [5] K.K.F Wong, Y.L. Chee, Energy Dissipation o uned Mass Dampers during Earthquake Excitations, he tructural Design o all and pecial Buildings, 13 (4) [6] K.K.F Wong, eismic Energy Dissipation o Inelastic tructures with uned Mass Dampers Journal o. Eng. Mech., 134 (163) (8) [7] F. Pellicano, F.. amani, Vibration reduction on beams subjected to moving loads using linear and nonlinear dynamic absorbers, Journal o ound and Vibration, 35 (9) [8] K.K.F Wong, J. Johnson, eismic energy dissipation o inelastic structures with multiple tuned mass dampers, Journal o Eng. Mech. Asce, 135 (4) (9) [9] ajimi, H., 196, A statistical method o determining the maximum response o a building during earthquake, Proc. o nd World Con. on Earthquake Engineering, okyo, Japan. [1] Lin, Y. K., 1976, Probabilistic theory o structural dynamics, N.Y., Krieger Publishing, Co., Inc... [11] oong, G.F. Dargush, Passive Energy Dissipation ystems in tructural Engineering, John Wiley & ons Ltd, Meccanica dei Materiali e delle trutture 3 (1), 1, PP