Optimum design of tuned mass damper systems for seismic structures

Transcription

1 Earthquake Resistant Engineering Structures VII 175 Optimum esign of tune mass amper systems for seismic structures I. Abulsalam, M. Al-Janabi & M. G. Al-Taweel Department of Civil Engineering, Faculty of Engineering, Altahai University, Sirte, Libya Abstract Tune mass ampers are well known evices for the passive control of vibrations in builings subjecte to earthquake loaings. Various methos have been propose for the esign of tune mass amper (TMD) systems. In the present work, a metho is suggeste for obtaining the values of the parameters require for esigning an efficient TMD system when attache to a SDOF system. In this metho, the values of the optimum frequency ratio an optimum amping ratio for the TMD system are efine as the values that will reuce the maximum isplacement of the structure to a minimum value when subjecte to a specific earthquake time-acceleration history. For this purpose, a MATLAB computer program is evelope. The program consists of a ynamic analysis subroutine embee in a nonlinear constraine optimization program. The suggeste metho is use in selecte case stuies showing its efficiency when compare to other methos for esigning TMD systems attache to SDOF systems. Keywors: tune mass amper, control of structures, optimum esign, earthquake time-history, optimum esign. 1 Introuction The tune mass amper (TMD) system represents an important type of passive control evices of structures subjecte to ynamic loas. It can be installe to new or existing structures to improve their resistance to earthquakes an wins. A TMD system consists of a mass, a spring an a amper. If these properties are properly esigne an selecte, then the TMD evice can be effective in suppressing unesirable vibrations inuce by earthquake or win loas. oi: /eres090161

2 176 Earthquake Resistant Engineering Structures VII Obtaining the optimal esign for a TMD system has been the goal of many researchers for many ecaes. In these researches ifferent assumptions have been mae regaring the simulation of the acting ynamic force, its location an the criteria use in efining the optimal esign parameters. Den Hartog [1] has erive the formula for the optimum values of the TMD parameters for a SDOF structure when subjecte to a harmonic loa. An extension has been mae by Warburton an Ayorine [2] an Tsai an Lin [3], where amping in the main mass was consiere an several types of harmonic excitations were examine. Extensive research was also mae by Warburton [4] an Rana an Soong [5], where formulas for several types of excitations were evelope. In this case the harmonic an ranom excitations were applie either on the main system or at the base of the structure. Saek et al. [6] suggeste a metho for estimating the esign parameters of TMDs for seismic applications, the criterion use to obtain the optimum parameters was to select, for a given mass ratio, the frequency an amping ratios that woul result in equal an large moal amping in the first two moes of vibration. As can be notice from the brief literature review, various assumptions have been mae regaring the earthquake loaing (Harmonic or Ranom), an about the location of the acting force (on the structure or at its base). In the present work, a metho is suggeste to obtain the optimum TMD parameters. In this metho, earthquake recors are use to obtain the optimal TMD properties when subjecte to the earthquake forces at its base. 2 Description of the suggeste metho Consier the TMD system with mass m, spring stiffness k an amping coefficient c shown in Figure 1 attache to a SDOF structure with a stiffness K, mass M an a structure amping coefficient C. The resulting overall structure will have two egrees of freeom. The basic features of the metho suggeste for obtaining the optimum esign parameters of a TMD system are as follows. Figure 1: SDOF structures with the TMD system.

3 Earthquake Resistant Engineering Structures VII Excitation force In orer to obtain the optimum esign parameters of the TMD system certain assumptions regaring the excitation force shoul be mae. To simulate actual behavior, it is assume in this stuy that the structure is subjecte to base excitation. The excitation force vector is compute from time acceleration history for a given earthquake. In the present work all time acceleration histories are taken from actual earthquake recors. It shoul be note that research is eveloping about generating groun earthquake time histories for esign (Fengxin et al. [7], Aballa an Hag-Elhassam [8] an Varpasuo an Geler [9]). 2.2 Optimization criterion an optimization parameters In the present work, the optimization criterion an parameters use by many authors is aopte (Den Hartog [1], Rana an Soong [5] an Tsai an Lin [3]). In this criterion, the optimum esign parameters k, c for a given m are efine as those values that minimize the maximum relative isplacement of the structure when subjecte to an excitation. The maximum relative isplacement of a regular SDOF shear builing frame usually occurs at the top. 2.3 Statement of the problem as a constraine nonlinear optimization problem Figure 2 shows a shear builing structure provie with TMD system at the top floor. When the structure is subjecte to a given earthquake excitation (acceleration-time & x& g ), then u t is the relative isplacement occurring at the top of the frame. Defining: u tmax : Maximum u t occurring uring the earthquake uration. u t max Figure 2: Shear builing uner earthquake excitation with the TMD system. For a given total structure properties ([K], [C], [M]), TMD mass m an earthquake excitation & x&, u tmax will be function of c, k only. This problem can g

4 178 Earthquake Resistant Engineering Structures VII be put in the following optimization problem: Fin c, k that minimizes the following objective function: u tmax = f(c, k ) subjecte to the inequality constrains c > 0 an k > 0. This problem can be classifie as multivariable, nonlinear constraine minimization problem. For the treatment of such problem, one of the functions available in the MATLAB [10] optimization toolbox is use. 2.4 Develope computer program for optimum esign of a TMD system As a first stage, a MATLAB computer program is evelope for the analysis of SDOF structure with TMD system when subjecte to earthquake excitation. This program is base on Newmark s metho. As a secon stage, the above mentione program is embee in another nonlinear constraine optimization MATLAB program. This last program is use to obtain the optimum esign parameters of the TMD system as mentione in previous section. Details of the MATLAB software are given in [11]. 2.5 Convergence of the propose metho to the optimum solution To emonstrate the capability of the propose metho to catch the optimum esign parameters for a TMD system when attache to structures, many problems are examine (Al-Taweel [11]). One of these verification problems is iscusse herein. A single story shear builing with properties shown in Figure 3 is consiere. A TMD is attache to the top with mass m = 1.5 ton equal to 3% of the total mass of the structure. u t max C = 6.283kN s / m K = kN / m M = 50ton m = 1. ton 5 Figure 3: Structure stuie as verification problem. The objective is to etermine the optimum value of TMD stiffness k an amping c that will minimize isplacement u tmax at the top when the structure is subjecte to El-Centro earthquake excitation. To unerstan the variation of u tmax with various values of k an c the first stage software is use to compute u tmax for k (40 to 70 kn/m with steps of.2) an c (0 to 1.9 kn-s/m with steps of 0.1). The results are plotte as three-imensional function surface [u tmax = f(c, k )] in Figure 4 an as contour lines as shown in Figure 5. Next, the secon stage software is use to obtain the optimum esign parameters through

5 Earthquake Resistant Engineering Structures VII 179 minimization process, as escribe in previous sections. For the minimization process it is given that for this problem, the upper boun an the lower boun value of the stiffness k are 0 an 1000 kn/m respectively, while the upper boun an lower boun of the amping are 0 an 100 kn-s/m respectively. After running the problem, it is foun that the optimum values of stiffness an amping of the TMD are k = kn/m an c =0.643 kn-s/m. The corresponing value of u tmax is m.When projecting these results on the contour plot in Figure 5 it can clearly notice that the solution given by the secon stage optimization software represents the minimum value for the surface or contour plot shown in Figures 5 an 6. This proves the capability of the software to catch the minimum value of u tmax an the corresponing optimum values of k an c u t max k Figure 4: Variation of u t max with k an c for the example in Figure 3 as a three imensional surface. 3 Case stuy one: efficiency of the present stuy metho in reucing SDOF structure vibrations uner earthquake excitations In this case, a SDOF structures with M=30 Ton, K= 2700kN/m (f=1.5 Hz) is stuie. TMD systems with 3% mass ratio are tune to control vibrations in the structure when subjecte to 18 earthquake recors. These earthquake recors cover a wie range of earthquake ominant frequency (from 0.3 to 4.82 Hz.). The properties of TMD systems k, c are compute using three methos (as shown in Figure 6) an the present stuy metho using the MATLAB program c

6 Earthquake Resistant Engineering Structures VII evelope in this work. This was repeate for each of the 18 earthquakes excitations. Figure 6 shows the variation of percentage reuction in maximum isplacement u tmax with earthquake ominant frequency for the investigate structure. The results showe that for all 18 Earthquake recors the u k c t max 0.13 = m = kn / m = kn s / m k Figure 5: c Variation of contour line. u t max with k an c for the example in Figure 3 as a Den Hartog % Reuction in utmax pacoima2(4.82) Figure 6: kern1(3.05) pacoima1(2.36) elcent2(2.16) parkf2(2.07) oakwh1(1.67) elcent1(1.48) kern2(1.38) parkf1(0.98) nrige3(0.89) sanfern1(0.59) oakwh2(0.59) mexcit1(0.49) mexcit2(0.49) nrige1(0.39) Earthquakes Name & Dom. frequency (Hz) Percentage of reuction in u f = 1.5 Hz. t max nrige2(0.39) santam2(0.39) sanfern2(0.30) Fahim Saek Warburton Present stuy for various earthquake recors for

7 Earthquake Resistant Engineering Structures VII 181 metho propose in this stuy gave the maximum reuction in u tmax. On the contrary to other methos, the present stuy metho gave ifferent values for the optimums k, c because it consiers the earthquake characteristic in obtaining these optimum values. The figure also shows that the efficiency of the TMD system for all methos may vary with the characteristic of earthquakes. The maximum reuction in isplacement observe is about 80% in this case stuy. 4 Case stuy two: effect of TMD mass ratio on the present stuy results The main purpose here is to investigate the effect of the variation of TMD mass ratio on the optimum values of TMD frequency ratio f opt an TMD amping ratio ξ opt when compute accoring to the present stuy metho. Figure 7 shows the SDOF structures use in the present investigation. A TMD system with mass ratio ranging from 0 to 0.1 is attache to the structure. Also, three earthquake recors are use in the stuy. The evelope MATLAB software is use to compute the optimum ξ opt an f opt. ω = K M C ξ = 2ωM µ = m M ω = f = k m ω ω c ξ = 2ω m Figure 7: µ = (0 0.1) m = (0 4) ton ξ = 0 K = (9870) kn / m f = 2.5Hz Structure stuie in the case stuy. Figure 8 shows the effect of TMD mass ratio variation on the optimum TMD frequency ratio f opt for three earthquake excitations. The figures also show the variation of f opt with µ as compute by the Den Hartog metho. The results show that this relation is greatly affecte by the type earthquake excitation to the extent that there is no unique shape for this relation as given by the Den Hartog Metho. Similar behavior is also notice for the relation between ξ opt an the mass ratio µ as shown in Figure 9. The important conclusion from the above mentione figures is that the relation of f opt or ξ opt with µ is highly affecte by the earthquake type an cannot preicte by simple equations as in the Den Hartog or other similar methos. Figure 10 shows the variation of maximum isplacement u tmax of the structure (when controlle with ξ opt an f opt ) with mass ratio µ. The figure shows that for some earthquakes, the maximum isplacement ecreases as the mass ratio µ increases, however there are other cases which o not show the same tren [11], an in same cases the isplacement may increase with the increase of µ.

8 182 Earthquake Resistant Engineering Structures VII Damping ratio xopt Kern Earth. Elcentro Earth. Sanfern. Earth. TMD by Den Hartog Mass ratio m Figure 8: Variation of ξ opt with µ for various earthquakes Frequency ratio f opt Kern Earth. Elcentro. Earth. Sanfern. Earth. TMD by Den Hartog Mass ratio m Figure 9: Variation of f opt with µ for various earthquakes.

9 Earthquake Resistant Engineering Structures VII Max. Struct. Displ. u tmax (m) Kern Earth. Sanfern. Earth Mass ratio m Figure 10: Figure 10: Variation of u tmax with µ for various earthquakes. The conclusion here is that the relation between maximum isplacement u tmax an mass ratio µ is also affecte by the earthquake type an there is no general tren for this relation as etermine by Den Hartog or other similar methos. 5 Conclusions The following main conclusions can be rawn from the present stuy: For a given SDOF structure, earthquake excitation an TMD mass, the present stuy metho an software evelope are capable to trace an compute the optimum values for k an c. The TMD system esigne accoring to the present stuy metho is more effective in reucing maximum structure isplacement than other methos, such as Deng Hartog. This is also foun true for a wie range of earthquake excitations an structure frequencies. The efficiency of the TMD system esigne accoring to the present stuy metho an other methos are generally affecte by the earthquake excitation. This means that the earthquake characteristics have an important effect on the TMD behavior an shoul be consiere in the esign process.

10 184 Earthquake Resistant Engineering Structures VII With the increase of research about preicting earthquake timeacceleration history, the importance of the present stuy will increase as an efficient metho for esigning TMD systems. For SDOF structures with TMD system esigne accoring to the present stuy metho, the relationship between f opt (optimum tuning frequency) or ξ opt (optimum TMD amping ratio) with µ (TMD mass ratio) is affecte by the earthquake type an cannot be preicte by simple equation as in the Den Hartog or similar methos as shown. For some earthquakes, the maximum structure isplacements ecrease with the increase of µ (as in the Den Hartog metho). However, there are cases for other earthquakes that showe ifferent trens. This implies that the earthquake type may affect the tren of the relationship between the maximum structure isplacement an mass ratio µ. References [1] Den Hartog, J. P. Mechanical Vibration, McGraw-Hill, New York. N.Y., [2] Warburton, G. B. an Ayorine, E. O., Optimum absorber parameters for simple systems, Earthq. Engng Struct. Dynam, 8, pp , [3] Tsai, H. C. an Lin, G. C., Optimum tune mass ampers for minimizing steay-state response of support excite an ampe systems, Earthq. Engng Struct. Dynam., 22, pp , [4] Warburton, G. B., Optimal absorber parameters for various combinations of response an excitation parameters, Earthq. Engng Struct. Dynam, 10, pp , [5] Rahul Rana an T. T. Soong, Parametric, stuy an simplifie esign of tune mass ampers, Engineering Structures, 20(3), pp , [6] Saek, F., Mohraz, B., Taylor, A. an Chung, R., A Metho of Estimating The Parameters Of Tune Mass Dampers For Seismic Applications, Earthq. Engng Struct. Dynam, 26, pp , [7] Fengxin, Z., Yushan, Z. an Hongshan, L., Artificial groun motion compatible with specifie groun shaking peaks an target response spectrum, Journal of Earthquake Engineering an Engineering Vibration, 1, pp 41-48, (2006). [8] Aballa, J. A. & Y. M. Hag-Elhassam, Simulation of Earthquake Groun Motion for Generation of Artificial Accelerograms, ERES V, [9] Varpasuo, P. H. A. & Geler, J. M. V., Generation of Design Groun Motion Time Histories for Lianyungang Nuclear Power Plant, ICOSSAR 01, New port Beach, pp.17-22, [10] MATLAB Version 7.10, The Language of Technical Computing, The Math Works, Inc. (2005). [11] Al-Taweel, M. G., Optimum Design of Tune Mass Damper Systems for Seismic Structures, MSc Thesis Submitte To The Faculty of Engineering, Al-Tahai University, Sirte, Libya, 2007.