Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 161 (2016) 67 – 72
World Multidisciplinary Civil Engineering-Architecture-Urban Planning Symposium 2016, WMCAUS 2016
Reducing Response of Structures by Using Optimum Composite Tuned Mass Dampers Yoyong Arfiadia,* a
Atma Jaya Yogyakarta University, Jalan Babarsari 44, Yogyakarta 55281, Indonesia
Abstract In this paper, the optimization of composite tuned mass dampers in reducing the response of structures subject to earthquake are discussed. Composite tuned mass dampers are mass dampers that consist of two mass dampers connected in series. The mass of the auxiliary dampers is in general relatively smaller than the one of the first damper. However, in this paper the mass ratio of the auxiliary damper to total mass of dampers is varied from 0.1 up to 0.9; and the optimum stiffness and damping of the first and auxiliary dampers are obtained using real coded genetic algorithm (RCGA). From the result of optimization, it is found that the mass ratio of the auxiliary dampers does not significantly affect the response reduction of structures. It is also found that for a certain mass ratio, the resulting stiffness and damping are not unique for achieving the same performance. © Published by Elsevier Ltd. This ©2016 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of WMCAUS 2016. Peer-review under responsibility of the organizing committee of WMCAUS 2016 Keywords: composite tuned mass dampers; vibration control; optimization; real coded genetic algorithms;
1. Introduction The use of tuned mass dampers (TMDs) to reduce the response of structures has been proposed by researchers in the past. These include the classical Den Hartog [6] and Warburton [13] methods. In Den Hartog [6] method, the reduction of response of undamped structures subject to harmonic loading is considered by the addition of a spring mass damper. The extension of analysis was carried out by Warburton [13], where a general mass including spring
* Corresponding author. Tel.: +62-274-487711; fax: +62-274-487748. E-mail address:
[email protected]
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of WMCAUS 2016
doi:10.1016/j.proeng.2016.08.499
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Yoyong Arfiadi / Procedia Engineering 161 (2016) 67 – 72
and damping was considered with the addition of a spring mass damper. The loading is also not necessarily a harmonic loading and might be applied either on the main mass or at the support. In addition to the analytic methods, several numerical methods have also been proposed. Hadi and Arfiadi [8] proposed optimization method by using binary coded genetic algorithms. Bekdas and Nigdeli [3] estimated optimum parameters of TMD by using harmony search. Several discussions on this method are also available ([10], [4], [11], [5]). Leung et al. [9] proposed particle swarm optimization method to optimize the TMD. In this paper, a composite TMD composed of two dampers in series on structure, similar to Nishimura et al. [12], is considered. The optimization method is done by using a modification of real coded genetic algorithms proposed in Arfiadi and Hadi [1][2] and Frans and Arfiadi [7]. However, different from [12] the mass ratio of the dampers is investigated to see this effect on the response reduction of structures. 2. Composite tuned mass dampers formulation A single degree of freedom structure equipped with a mass damper and an additional mass damper is considered in this paper, as shown in Fig. 1. The equation of motions of the structure can be written as:
C U Ms U s s s K s Us
where M s
ª ms «0 « «¬ 0
0 md 0
0 º 0 »» , Cs ma »¼
M s 1s ug
0 º cd ª( cs cd ) « c ( cd ca ) ca »» , K s d « «¬ 0 ca »¼ ca
T T u a @T , U s >u s u d ua @ , U s >us ud ua @ , 1s structure, the first damper, and the auxiliary damper, respectively; c s cd
Us
>u s
(1)
0 º kd ª( k s k d ) « k ( kd k a ) ka »» , d « «¬ 0 ka »¼ ka
>1
ud
,
1 1@T , ms , md , ma = mass of the ca = damping of structure, the first damper,
and the auxiliary damper, respectively; k s , k d , k a = stiffness of the structure, the first damper, and the auxiliary damper, respectively; us , u d , u a = displacement of the main structure, the first damper, and the auxiliary damper, respectively; the over dot (.) is a derivative with respect to time, and ug = ground acceleration.
Fig. 1. Composite tuned mass damper.
In this problem the properties of composite mass damper system are optimized for various mass ratios of auxiliary to the total mass of dampers. The equation of motions can be converted to a state space equation as: AZ E w Z (2) where Z
U s ½ ® ¾, A ¯U s ¿
0 ª « 1 ¬ Ms Ks
I
º
, E M s1C s »¼
0 ½ ® ¾ , and w ¯ 1s ¿
ug
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Yoyong Arfiadi / Procedia Engineering 161 (2016) 67 – 72
3. Performance objective The performance objective is the H2 norm of the transfer function from the ground acceleration to the regulated output, where the regulated output is the displacement of the structure, which can be written as: z
where C z
>1
0 0 0 0 0@ .
Cz Z
(3)
4. Optimization In this case the real coded genetic algorithm (RCGA) is used to optimize the damper properties. The parameters of RCGA to be used in this paper are: population size = 60, maximum generation =1500, probability of mutation = 0.1, probability of crossover = 0.8, percentage of new individual to replace old individual in each generation = 20%. The mutation to be used is a simple mutation, similar to the one in [1]. For the crossover we used asymmetric crossover in order to explore the domain of interest. Note that because the domain of interest is always positive, asymmetric crossover always results in a positive number for positive initial domain, and approaches domain of interest in an asymmetric way as: G1n G2n
r G1 ( 1 r )2 G2
(4a)
r 2 G2 r G1
(4b)
where r = random variable (0-1). The design variables for optimization are cd, kd, ca and ka, for every mass ratio of md/(ma + md). During the process of optimization, the constraints are considered by penalizing each individual so that the fitness of individual that violates the constraint is set to a minimum value that can be accepted by the computer. The constraints are: [ d 1 d 0 and [ a 1 d 0 (5a) and (5b) cd = damping ratio of the first damper, and [ a 2md Zd
ca = damping ratio of auxiliary damper, 2ma Za
k d md = circular frequency of the first damper, and Za
k a ma = circular frequency of the auxiliary
where [ d Zd
damper. The constraints in (5a) and (5b) are taken to make sure that resulting dampers produce underdamped systems. The constraint on negative value of design variables are not necessarily enforced in this case, because of the asymmetric crossover that is used. If the initial values of design variable are assigned as positive numbers, the resulting new individuals are also always positive. Therefore, this type of crossover is useful when the design variable must always be positive. 5. Parametric studies The structural properties are taken from [12], i.e., ms =16.274 t, cs = 5.4 kN-s/m, ks = 1.05 x 103 kN/m, the total mass damper ratio, i.e., (md+ma)/ms = 0.4264/16.274 = 0.0262. In this problem we optimize the damper properties for various ratio of md/(ma +md) from 0.1 to 0.9. In this case the effect of auxiliary mass is investigated with constant total mass damper ratio. The RCGA is then used to optimize the damper properties. The objective is to minimize the transfer function from the ground excitation to the displacement of the structure.
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Yoyong Arfiadi / Procedia Engineering 161 (2016) 67 – 72
6. Results and discussions The results of the optimization are shown in Table 1. The structure is then simulated to the recorded earthquake ground acceleration. The earthquakes to be used are El Centro 1940, Hachinohe 1968, Kobe 1995, and Northridge 1994 ground accelerations. The root mean square (RMS) of displacement of the structure is shown in Table 2. In Table 2 it is shown that the mass ratio md/(ma+md) does not significantly affect the response reduction of the structure. The term TMD in Table 2 means that we have one damper only. The transfer function for the case of uncontrolled, with one TMD only, and with dampers for mass ratio md/(ma+md) = 0.1 and 0.9, is depicted in Fig. 2 (a); while the response of structure subject to Northridge 1994 ground excitation is shown in Fig. 2(b) for md/(ma+md) = 0.3. Note also that the resulting properties of dampers are not unique for the particular performance objective. For example in the case of md/(ma+md) = 0.2, besides the result as shown in Table 1, other resulting parameters are cd = 0.82089 kNs/m, kd = 35.033 kN/m, ca = 0.63075 kNs/m, and ka = 66.78 kN/m. Table 1. Results of optimization. md/( ma +md)
cd (kN-s/m)
kd (kN/m)
ca (kN-s/m)
ka (kN/m)
0.1
0.27804
29.652
9.3276
98.716
0.2
0.080762
51.087
1.6808
34.92
0.3
1.4406
49.144
0.50162
31.774
0.4
0.46393
39.407
1.1623
30.084
0.5
0.42126
29.3
5.0239
47.992
0.6
0.44761
27.992
3.9175
42.275
0.7
0.57888
27.548
0.58644
28.532
0.8
0.51997
28.015
0.68508
17.317
0.9
0.66153
24.305
1.1694
34.927
TMD
0.52376
25.326
-
-
The resulting response for both cases is almost identical as can be seen in Fig. 3.
Fig.2. (a) Transfer function from ground excitation to displacement of structure, (b) Displacement of structure due to Northridge earthquake md/( ma +md) = 0.3.
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Yoyong Arfiadi / Procedia Engineering 161 (2016) 67 – 72
Table 2. RMS of displacement of structure.
md/( ma +md)
RMS displacement due
RMS displacement due
RMS displacement due
RMS displacement due
to El Centro (m) w/o with TMD
to Kobe w/o TMD
to Hachinohe (m) w/o TMD with TMD
to Northridge (m) w/o TMD with TMD
TMD
(m) with TMD
0.1
0.0399
0.0300
0.0685
0.0565
0.0207
0.0141
0.0607
0.0283
0.2
0.0399
0.0311
0.0685
0.0576
0.0207
0.0144
0.0607
0.0300
0.3
0.0399
0.0314
0.0685
0.0585
0.0207
0.0144
0.0607
0.0294
0.4
0.0399
0.0306
0.0685
0.0571
0.0207
0.0143
0.0607
0.0293
0.5
0.0399
0.0316
0.0685
0.0583
0.0207
0.0144
0.0607
0.0291
0.6
0.0399
0.0311
0.0685
0.0576
0.0207
0.0143
0.0607
0.0286
0.7
0.0399
0.0301
0.0685
0.0567
0.0207
0.0141
0.0607
0.0285
0.8
0.0399
0.0317
0.0685
0.0586
0.0207
0.0144
0.0607
0.0283
0.9
0.0399
0.0291
0.0685
0.0553
0.0207
0.0140
0.0607
0.0302
TMD
0.0399
0.0307
0.0685
0.0573
0.0207
0.0142
0.0607
0.0281
Fig. 3. Case of ma/(ma + md) = 0.2 subject to Kobe.
7. Results and discussions The optimization of composite tuned mass dampers has been discussed in this paper. From the simulation, it is found that the ratio of auxiliary mass to the total mass of damper does not affect significantly the response of the structure. It is also possible that the optimum parameters of the dampers are not unique for a certain mass ratio of dampers. References [1] Y. Arfiadi, M.N.S. Hadi, Optimal direct (static) output feedback controller using genetic algorithms. Computers and Structures (2001) 79(17): 1625-34. [2] Y. Arfiadi, M.N.S. Hadi, Optimum placement and properties of tuned mass dampers using hybrid genetic algorithms. International Journal of Optimization in Civil Engineering (2011) 1:167-187 [3] G. Bekdas, S.M. Nigdeli, Estimating optimum parameters of tuned mass dampers using harmony search. Engineering Structures (2011) 33: 2716-2723 [4] G. Bekdas, S.M. Nigdeli, Response of discussion: estimating optimum parameters of tuned mass dampers using harmony search. Engineering Structures (2013) 54: 265-267
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Yoyong Arfiadi / Procedia Engineering 161 (2016) 67 – 72
[5] G. Bekdas, S.M. Nigdeli, Response of discussion: estimating optimum parameters of tuned mass dampers using harmony search. Engineering Structures (2014) 58: 105-106 [6] J.P. Den Hartog, Mechanical vibrations, McGraw-Hill, NY., 1947. [7] R. Frans, Y. Arfiadi, Designing optimum locations and properties of MTMD systems. Procedia Engineering (2015) 125: 892 – 898 [8] M.N.S. Hadi, Y. Arfiadi, Optimum design of absorber for MDOF structures. Journal of Structural Engineering, ASCE (1998) 124: 1272-1280. [9] A.Y.T. Leung, H. Zhang, C.C. Cheng, Y.Y. Lee, Particle swarm optimization lf TMD by non-stationary base excitation during earthquake. Earthquake Engineering and Structural Dynamics (2008) 37: 1223-1246. [10] L.F.F. Miguel, R.H. Lopez, L.F.F. Miguel, Discussion of paper: estimating optimum parameters of tuned mass dampers using harmony search. Engineering Structures (2013) 54: 262-264. [11] H. Saberi, R.P. Hosseini, H. Saberi, Discussion of paper: estimating optimum parameters of tuned mass dampers using harmony search. Engineering Structures (2014) 58: 107-109. [12] I. Nishimura, K. Sasaki, T. Kobori, M. Sakamoto, An intelligent tuned mass damper: an experimental study of an active-passive composite tuned mass damper. AIAA 34th Structures, Structural Dynamics and Materials Conference, Structures, Structural Dynamics, and Materials and Co-located Conferences, La Jolla, CA, USA , 3561-3569, 1993. [13] G.B. Warburton, Optimum absorber parameters for various combination of response and excitation parameters. Earthquake Engineering and Structural Dynamics (1982)10: 381-401.
