Analysis of Multiple Tuned Mass Damper System For Controlling Torsional Response K

Analysis of Multiple Tuned Mass Damper System For Controlling Torsional Response
K. Majumdar1*and R.Debbarma2
1M.Tech Scholar, Dept. of Structural Engineering, NIT Agartala, [email protected]
2 Associate Professor, Dept. of Structural Engineering, NIT Agartala, [email protected] gmail.com

Abstract
In the present study, the performance of Multiple Tuned Mass Dampers (MTMDs) to control the earthquake induced response of a torsionally coupled system is investigated. MTMDs with uniformly distributed frequencies are taken for this study. The arrangement of MTMDs is in a row covering the width of the system. A parametric study involving parameters like mass ratio, damping ratio, frequency ratio, normalized eccentricity ratio and the ratio of uncoupled torsional to lateral frequency is carried out for different seismic time history data, in time domain method. Numerical example is taken to find the effectiveness of MTMDs on reducing both torsional and translational response of the coupled structure. A comparative study is made between torsionally coupled and uncoupled structures. It is shown that the effectiveness of MTMDs in controlling lateral response of torsionally coupled system increases with increase in mass ratio, damping ratio but decreases with an increase in degree of asymmetry.

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KEYWORDS: vibration control; torsionally coupled system; real earthquake; MTMDs
Introduction
The structural vibrations are effectively reduced by the passive control device, the TMD. The tuning of the damper is done to a particular structural frequency so that when that frequency is excited, the damper will resonate out of phase with the structural motion and the energy is exhausted by the damper inertia force acting on the structure. The effectiveness of TMD in reducing structural response is decreased by leaps and bounds by mis-tuning of a TMD. So by using more than one TMD, the effectiveness can be improved. It was shown by Iwanami and Seto4 in their research that two- TMDs are much more effective in reducing the structural response than a single one. Then, Multiple-Tuned-Mass-Damper(MTMDs) with distributed natural frequencies were suggested by Xu and Igusa8 and also worked upon by Yamaguchi and Harnpornchai5, Abe and Fujino1, Jangid6 and Abe and Igusa. From their research it was shown that the MTMDs have several advantages over a single TMD. This shows that a considerable work is carried out on the use of TMD and MTMDs in suppressing the seismic response of buildings idealized as a planar model. Even Jangid and Dutta7 have worked on the effect of MTMDS in controlling the torsional response of a system in frequency domain, but very few work has been done to investigate the effect of MTMDs in controlling the torsional response of the system under real earthquake excitation. The present study throws some light in this problem. The objective of the study are : (i) To show the difference between the behaviours of torsionally coupled and uncoupled models with MTMDs.(ii)To study how the frequency bandwidth for torsional and translational responses of a torsionally coupled system varies (iii)To investigate the variation of the eccentricity of the main system, the uncoupled to lateral frequency ratio etc. with the frequency bandwidth of the system model.
Structural Configuration
The system model consists of a main system on which n number of TMDs with different dynamic characteristics are placed as depicted in Figure 1. The main structural system which is shown is essentially a torsionally coupled system, where the Centre of Resistance (CR) of the main system does not coincide with the Centre of Mass(CM). For this reason, torsional effects are shown by the main system when it is excited in the lateral direction by various earthquakes. The TMDs with uniformly distributed frequencies are evenly placed about the CM of the main system in a row covering the width, b. The stiffness and damping ratios are kept constant. As a result, the CR of TMDs coincides with the CM of the main system. The total mass of the TMDs is the sum of the masses of all TMDs. The system is subjected to a lateral force at the CM of the main system. As a result, the main system and each TMD vibrate in the lateral direction. Also, due to torsional coupling the main system undergoes torsional vibration. Thus, the total degrees of freedom(DOF) of the combined system is n+2.Two uncoupled frequency parameters of the main system are defined as
(3.1)
(3.2)

Plan view
Figure 1. Model of a torsionally coupled system with multiple tuned mass dampers
Where ms is the mass of the main system, ks =(kS1+ks2) is the lateral stiffness of the main system , about the CM, and rs is the radius of gyration of the main system about the CM;ks1 and ks2 are the stiffness and ys1 and ys2 are the distance from the CM of the resisting elements, respectively (refer Figure 1).The eccentricity between the CR and the CM of the main system is given by
(3.3)
The frequencies and may be interpreted as the natural frequencies of the main system if it were torsionally uncoupled, i.e. a system with eS=0; but ms the mass of the main system ks and k? are the same as in the coupled system. The parameters ks1 , ks2 and ys1(=ys2) are adjusted to provide the desired values of ?s , ?? , es.. Let wt be the average frequency of MTMDs (i.e. )and n be the total number of MTMDs then the natural frequency of the jth TMD is expressed as
(3.4)
where the parameter ? is the non-dimensional frequency bandwidth of MTMDs defined as
(3.5)
If kT and ?T are the constant stiffness and damping ratio of each TMD, respectively, then the mass and the damping constant of the jth TMD are expressed as
(3.6)
(3.7)
The ratio of the total mass, mT of MTMDs to the mass of the main system,(ms) is defined as the mass ratio
(3.8)
The constant stiffness required for each TMDs can be expressed as
(3.9) Since the torsionally coupled system with one way eccentricity is characterised by two natural frequencies, it is difficult to define a tuning ratio for MTMDs as it is done for an uncoupled single degree of freedom(SDOF) system. So two different tuning frequency ratios are considered in the study, namely
and (3.10)
Where is the uncoupled natural frequency of lateral vibration defined earlier(Equation(3.1))andis the first natural frequency of the torsionally coupled main system.

Equations of Motions
The n+2 equations of motion of the system in Figure 1 are expressed
(3.11)
Where is the displacement vector of the system model, and are the displacement and rotation of the main system , is the displacement of the jth tuned mass damper, f(t) is the lateral excitation force acting at the CM of the main system, , , , are expressed as
(3.12)
(3.13)
(3.14)
where , and are the elements of the damping matrix of the main system without MTMDs which are obtained by assuming a modal damping , ?s ,in both modes of vibration , is the coupling term between the translational and the torsional degrees of freedom for the main system ,and yj is the distance of the jth TMD from the CM of the main system.

Numerical Study
A building is modeled as a SDOF structure and TMDs are mounted on top of the primary structure as shown in Figure 1, used for the analysis. The properties of primary structure are ms=2.0×105 kg, ks=7.89×106 N/m and damping ratio of structures, ?s =2% .The structure has a natural Time Period T=1sec, which is tuned by frequency of the TMDs. Some parameters are held constant. These are:
?=1%; b/rs=1;The structure is also an asymmetric structure with es/rs not being equal to zero .Here four values of normalized eccentricity ratio(es/rs=0,0.1,0.2,0.3) and three values of ??/?s (0.5,1,2)are considered which indicates a torsionally flexible system. To know the impact of seismic excitation on the performance of TMDs, past earthquake ground motion record is selected for the analysis. The response quantity of interest are the frequency bandwidth of MTMDs. The result is first obtained for MTMDs tuned to the uncoupled lateral frequency of the main system and their damping ratio ?T is kept at 1%. The ground acceleration time histories of earthquakes taken for this present study are given below
Table-1: Details of selected earthquake time history records
Earthquake Peak Ground acceleration
(PGA) Duration
(sec) Time Interval between
data prints(sec)
Parkfield earthquake (1966) 0.4339g 43.96 0.02
Colinga earthquake (1983) 0.1246g 39.96 0.005
The variation of translational displacements and accelerations of structures with time considering time history data of previously selected earthquakes data of Park field earthquake using TMDs and without TMD have been shown in Figure 2. to Figure 5., which shows the effectiveness of the damper in reducing peak structural responses
The results show that the effectiveness of both STMD and MTMDs to reduce the structural displacement significant for some earthquake records and for some records the reductions are not that much effective, i.e. for Parkfield Earthquake the reductions in displacements and accelerations for STMD and MTMDs are significant.
The peak linear structural displacement reduction for STMD here is 49.3% whereas for MTMDs (3 nos) and MTMDs (5 nos) it is 54.37% and 54.75% respectively. The results are obtained for es/rs=0.1 and w?/ws=0.5(i.e. a torsionally flexible system).The results also shows that an increase in number of dampers increases the reduction percentage. In other words MTMDs are more effective in controlling the translational response of a structure than a single TMD.

(a) (b) (c)
Figure 2. Variations of translational displacement of structures using (a) Single TMD (b) MTMDs(3 Nos.) and (c) MTMDs (5 Nos) under Parkfield Earthquake.

(a) (b) (c)
Figure 3. Variations of translational accelerations of structures using (a) Single TMD (b) MTMDs (3 Nos.) and (c) MTMDs (5 Nos) under Parkfield Earthquake.

(a) STMD (b)MTMD(3 Nos) (c) MTMD(5 Nos)
Figure 4. Variations of torsional displacement of structures using (a) STMD (b) MTMDs(3 Nos.) and (c) MTMDs (5 Nos) under Parkfield earthquake
(a) STMD (b)MTMD(3 Nos) (c) MTMD(5 Nos)
Figure 5. Variations of torsional acceleration of structures using (a) STMD (b) MTMDs(3 Nos.) and (c) MTMDs (5 Nos) under Parkfield earthquake
Parametric Study
In Figure 6. the variation of the peak displacement for translation and torsion considering three TMD is plotted against the frequency bandwidth ? for ??/ ?s=1.The responses increases with the increase of eccentricity. This shows that if the MTMDs are designed for asymmetric structures without taking notice into the effect of torsional coupling, then the effectiveness of MTMDs are overestimated. It is seen that with the increase of ? the peak responses increases under real earthquake excitation. Further it is seen that the effectiveness of MTMDs in suppressing both translational and torsional responses decreases with the increase of eccentricity ratio.

Figure 6. Variations of peak response structures with the frequency bandwidth for ??/?s=1,?T=1% and f=1 (a)for translation and (b) for torsion under Parkfield earthquake
The variation of ? with the corresponding eccentricity ratio is given in the above table and also the peak translational and torsional responses are shown. Figure 7 depicts the same variation for torsionally flexible structure i.e. for ??/?s=0.5. It is shown in the graphs that the MTMDs fares well for torsionally flexible structures than that of the structure with intermediate torsional stiffness. With the increase of eccentricity the response decreases and the effectiveness of MTMDs increases. For torsionally flexible structures the trend is opposite to the trend observed for the case of the structure with intermediate torsional stiffness(es/rs=1).One very obvious reason for this may be attributed to the spacing of natural frequencies.

Table 2. Frequency bandwidth and effectiveness of MTMDs(??/?s=1,?T=1%,tuning ratio f=1)
? 0 0.1 0.2 0.3 0.4 0.5 0.6
es/rs0 xs(m) 0.049 0.0491 0.0493 0.0514 0.0537 0.0551 0.0558
?s (rad) – – – – – – –
0.1 xs0.0468 0.0468 0.047 0.0472 0.0474 0.0475 0.0477
?s0.0042 0.0042 0.0049 0.005 0.0052 0.0053 0.0053
0.2 xs0.0545 0.0568 0.0537 0.0495 0.0505 0.0518 0.0525
?s0.0069 0.006 0.006 0.0054 0.0056 0.0056 0.0057
0.3 xs0.0721 0.0714 0.0681 0.0639 0.0658 0.0682 0.0689
?s0.0094 0.0095 0.009 0.0087 0.0093 0.0097 0.0098
For smaller es/rs the natural frequencies are closely spaced and hence the contribution of the mode other than the one which is being suppressed, significantly influences the response. The variation of ? with the corresponding eccentricity ratio is given in the Table 3 and also the peak translational and torsional responses are shown . It is seen that the reduction of torsional response is good and as the MTMDs are tuned to the first mode of vibration which has significant torsion and so the reduction of torsion due to the MTMDs is also significant.

Figure 7. Variations of peak response structures with the frequency bandwidth for ??/?s=0.5,?T=1% and f=1 (a)for translation and (b) for torsion under Parkfield earthquake
Furthermore, Table-2 and Table-3 shows that the reduction of translational response due to MTMDs is more for ??/?s=0.5 than it is for ??/?s=1.The reason for this is that for a given value of es/rs, torsional coupling is more for the latter and so the effectiveness of MTMDs become less for ??/?s=1.
Table 3. Frequency bandwidth and effectiveness of MTMDs(??/?s=0.5,?T=1%,tuning ratio f=1)
? 0 0.1 0.2 0.3 0.4 0.5 0.6
es/rs0 xs0.049 0.0491 0.0493 0.0517 0.0539 0.0552 0.0558
?s- – – – – – –
0.1 xs0.0486 0.0487 0.0490 0.0491 0.0516 0.053 0.0536
?s0.000801 0.000803 0.000864 0.000902 0.000948 0.000945 0.000926
0.2 xs0.0482 0.0455 0.0457 0.0469 0.0461 0.0463 0.0465
?s0.0042 0.004 0.0042 0.0044 0.0043 0.0043 0.0043
0.3 xs0.0413 0.0414 0.0416 0.0418 0.0420 0.0422 0.0423
?s0.0056 0.0057 0.006 0.0059 0.0058 0.0058 0.0058

Figure 8. Variations of peak response structures with the frequency bandwidth for ??/?s=2, ?T=1% and f=1 (a)for translation and (b) for torsion under Parkfield earthquake.

Figure 8. shows the same variations for ??/?s=2, i.e. torsionally stiff system with lateral and torsional frequencies well separated.The variation of peak displacement for the translational and torsional responses with the frequency bandwidth are small for all values of es/rs, and it follows the same trend as that of ??/?s=1.The effectiveness of MTMDs does not change by a significant amount for all values of es/rs,
Effect of mass ratio
Figure 9 shows the variation of mass ratio with peak displacement of structure. It is evident from Figure 9 that TMD with higher mass ratio is more effective in suppressing the structural displacement. Here in the above figure (Fig 14) we can see that the peak response of the structure decreases with the increase of mass ratio of structure for both STMD and MTMD.

Figure 9. The influence of mass ratio on the maximum translational displacement of structure and liquid for 1%, 3% and 5% damping ratio considering real time history data of Colinga earthquake (1966) for (a) MTMD(3 nos).

Effect of tuning ratio
Figure 10 shows the variation of tuning ratio with peak displacements for different structures. The study is conducted for three different structures namely torsionally flexible structures, torsionally intermediate structures and torsionally stiff structures. From this figure it can be be deduced that there exists no unique value for the tuning frequency for achieving maximum suppression of response for asymmetric systems. It depends upon the parameters of ??/?s and es/rs. But it is possible to obtain a tuning frequency for a given combination of ??/?s and es/rs for which the response of the asymmetric structure would be maximum.

Figure 10: The effect of tuning ratio on the maximum translational displacement of structure for 1%, 3% and 5% damping ratio considering real time history data of Colinga earthquake (1966)for (a) MTMD(3 nos).

This could be achieved by trial and error or by running an optimization program. . Doing this, the effective design of MTMDs can be realised for a meaningful reduction of response of asymmetric systems having realistic amount of degree of asymmetry.

Conclusions
Performance of MTMDs for controlling the response of a torsionally coupled system for real earthquakes is studied. A simple one-way eccentric model having two degrees of freedom is provided with MTMDs. The results lead to the following conclusions:
Effectiveness of MTMDs in controlling the translational response is slightly less than that of the corresponding symmetric systems though in some cases(??/?s=0.5) its effectiveness increases than that of the corresponding symmetric systems.
The frequency bandwidth of MTMDs corresponding to maximum reduction in response of a system changes with the change in the eccentricity of the asymmetric system. So if that frequency of MTMDs is computed by ignoring the effect of torsional coupling may not effectively control the response
MTMDs are more effective than a single TMD even for torsionally coupled structures. But the effectiveness of MTMDs decreases with the increase in eccentricity of the system. However, the case is just opposite for torsionally flexible structures.

The translational response of an asymmetric system considering torsional coupling decreases with the increase in mass ratio of the MTMDs
Tuning frequency of MTMDs for which the maximum response is achieved for asymmetric system depends upon the degree of asymmetry (??/?sand es/rs)
References
Abe, M., and Fujino, Y.(1994),”Dynamic characterisation of multiple tuned mass dampers and some design formulas”, Earthq. Engng. Struct. Dyn., Vol 23, No.8, pp. 813-835
Bekdas, G., and Nigdeli, S.M.(2013),”Mass ratio factor for optimum tuned Mass Damper strategies”, International Journal of Mechanical Sciences, Vol 71, pp.68-84
J.E Brock, “A note on the damped vibration absorber”,J. Appl. Mech ASME , Vol 13,A-284(1946)
Iwanami and Seto, K. (1984),” Optimum design of dual tuned mass dampers and their effectiveness”, Proc. JSME(C), Vol 50, No 449, pp.44-52
H.Yamaguchi and N.Harnpornchai. (1993)”Fundamental characteristics if multiple tuned mass dampers for suppressing harmonically forced oscillations”, Earthquake.eng.struct.dyn, Vol. 22, No. 1, pp.51-62
Jangid, R. S. (1995)” Dynamic characteristic of structures with multiple tuned mass dampers”, Structural Engineering Mechanics, Vol. 3, No. 5, pp. 497–509.

Jangid, R.S and Datta, T.K.(1997),”Performance of Multiple Tuned Mass Dampers For Torsionally Coupled System”, Earthquake eng.struct.dyn., Vol. 26, No. 3, pp. 307-317(1997)
Igusa,T., and Xu , K. (1994)” Vibration control using multiple tuned mass damper”, Journal of Sound and Vibration, Vol. 175, No. 4, pp. 491– 503.

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