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ABSTRACT: Tuned mass dampers (TMDs) are control devices consisting of a mass-spring-dashpot system connected to the structure in a way to vibrate out of phase with the main structure, if tuned to the appropriated frequency. In this way the energy is transferred to the TMD reducing amplitude vibration on the main system. The use of TMD to reduce floor vibration has been studied by many researchers lately, however there are still many topics to further development. Through preliminary numerical analysis this work aims to design a TMD to be installed on a platform for dynamic tests located in the University of Brasilia Structure Lab. This platform behaves like reinforced concrete buildings floors subjected to conventional human activity that can cause critical excessive vibrations. Reducing vibration amplitudes using TMD improve human comfort and guarantee structural safety. Numerical modeling of the structure with TMD installed is performed using the finite element ANSYS computational package. The modal analysis of the platform defines natural frequencies and vibration modes, this information allows tuning the TMD properly. The TMD design is based in the study of different parameters such as geometry, material, amount of mass, TMD positioning and the number of devices needed. This study will contribute to vibration control of building floors.

KEY WORDS: Structural dynamics, vibration control, tuned mass damper TMD, slabs, finite element.

1 INTRODUCTION

Nowadays there are more and more flexible and slender civil structures. A high structural flexibility leads to major vibration amplitudes that can be transmitted to people at buildings, causing discomfort, loss of work efficiency due to weariness and even serious health changes. To solve this kind of problem it can be used tuned mass dampers (TMDs) that are spring-mass-dashpot systems designed to have it fundamental frequency next to some structure frequency, in a way that part of the main system energy be transmitted to the TMD. This type of damper has been used successfully also on concrete slabs ([1],[2],[3],[4],[5],[6]). There are several practical cases of structures that present excessive vibrations to pedestrians, among them is the Millenium Footbridge in London, that excessive vibrated laterally right after its opening. The main cause of the vibrations was the lateral force component caused by people walking. The solution for this problem was the placing of passive dampers like showed on Figure 1 ([7], [8]).

Figure 1. Millennium Bridge, London [7] The luxury hotel located in Singapore, presented at the slab cantilevered slab free end large oscillations caused by strong winds, rain and pedestrians. To reduce undesirable vibrations it was designed a TMD with a

5000 kg mass, resonance frequency of 1.0472 Hz with maximum damping of 3978 Ns/m, which the location and the structure can be observed on Figure 2.

Localization TMD

Figure 2. Bay Sands Hotel, Singapore 9] Another case that deserves to be mentioned is the Rio-Niteroi bridge in Rio de Janeiro, that presented high oscillations between 25 and 60 cm in the three mid spans due to wind gusts that very between 90 and 100 Km/h. The solution proposed was to install 32 TMDs with a total mass of 2,2 tons, that reduced more than 80% the vibration amplitudes giving more security to drivers. The dampers were tuned to the bridge first mode. On Figure 3 it is show a drawn on perspective of the TMD placed inside of one of the cellular beams. [10] This Work proposes a design and construction of a TMD to control excessive vibrations on an experimental dynamic platform at the Structures Laboratory of University of Brasilia, designed to better understand concrete floors vibrations.

Study, design and construction of a tuned mass damper (TMD) For concrete building floors

Jorge Eliécer Campuzano Carmona1, Graciela N. Doz2, Suzana Moreira Avila3

1Department of Civil Eng., Faculty of Engineering, University of Brasília, Student PhD, Brazil 2Department of Civil Eng., Faculty of Engineering, University of Brasília, Dr., Brazil 3Department of Civil Eng., Faculty of Engineering, University of Brasília, Dr., Brazil

email: [email protected], [email protected], [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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Study, design and construction of tuned mass damper (TMD) for concrete building floors.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

Figure 3. TMD installed at the Rio-Niterói bridge. [10]

2 COMPUTATIONAL MODELLING

Platform slab discretization is performed using ANSYS finite element library [11]. The elements chosen are those that better simulate the behavior of each structural member: SHELL63 and COMBIN40. That are show on Figure 4 and Figure 5 respectively.

3 12

6

3

3

752

4

8L k

x

6

5Z

YX

y yIJ

zZIJ

XIJ

K,L

JITriangular option

JI1

Figure 4. SHELL63 Geometry

M or M/2

FSLIDEK1

K2

C

GAP M or M/2

J

Operates in nodal coordinate systemY

X

Z

Figure 5. COMBIN40 Geometry SHELL63 element has six degrees of freedom in each one of its four nodes, X, Y and Z translations and torsion around the same axis, it is recommended to simulate membranes behavior and allow loading on its own plane and on the orthogonal one. COMBIN40 is a element that combines springs (one of them sliding) and a parallel dashpot. It is defined by two nodes and has only one degree of freedom at each node, translation, rotation or even pressure and temperature. This element can be used in any type of analysis. The transient and harmonic

dynamics solution [11]. The structure equations of motion is

[ ]{ } [ ]{ } [ ]{ } { }FuKuCuM =++ (1)

The damping matrix adopted is the proportional or Raylegh matrix.

[ ] [ ] [ ]KMC += (2)

First it were performed modal and harmonic analysis using ANSYS [11] for several TMD configurations. The harmonic analysis results obtained were discussed and evaluated to choose the better solution to be designed. Only for the best model it was performed transient analysis.

3 PLATFORM DESCRIPTION

The dynamics platform considered in this paper consist on a concrete slab with two semi-rigid edges and other two free edges. Constructed on Structures Laboratory of University of Brasilia, it is shown non Figure 6 with its corresponding measures. The platform has length 6.1 m, width 4.9 m thickness of 0.1 m, with the concrete compressive strength of 25 MPa and elastic modulus of 29 GPa. Experimental measuring of natural frequencies of the platform were performed exciting the structure by impacts made with a sledgehammer knocking at specific points placed n the middle of the free edges. Another tests were performed exciting the structure by impact of people jumping and knocking they heel at the center of a the slab. A piezoelectric accelerometer 4366 type of Brüel & kjaer (mass = 10 g and 4.80 pC/ms-2 sensitivity) was used attached to ADS2000 acquisition system.

Figure 6. Dynamic platform to test TMD performance. Data acquisition system was configured to acquire from the records of a channel (Channel 0) corresponding to the accelerometer fixed in the platform slab at the time 5x10-3 s resulting in a Nyquist sampling rate of 100 Hz and the duration was 15 s. Figure 7 shows data acquirement through the accelerometer and Figure 8 and Figure 9 present the way the structure was excited by impact loads of a sledgehammer and jumping, respectively.

Figure 7. Accelerometer and data acquisition system.

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Campuzano. J.E, Avilas. S, Doz. G.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

Figure 8. Sledgehammer impact to excite the platform.

Heel drop excitation

Figure 9. Exciting the platform through jumping.

Experimental results are shown on Figure 10. Time domain results are from the impulse generated by jumping, it is analyzed through 15 seconds. Frequency domain results were obtained processing data on MATLAB [12] through Fast Fourier Transform (FFT) algorithm. Three structure harmonics associated to its corresponding natural frequencies can be observed.

4 NUMERICAL ANALYSIS

Aiming to design a vibration control system based on TMD a detailed study was perform. The first case considered was to place a single TMD at the center of the slab. A second analysis combine more than one TMD and also varies it positioning.

0 2 4 6 8 10 12 14 16-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Time (s)

Am

plitu

de (

mV

)

Analysis in time domain - Channel 0

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

Frequency (Hz)

Mag

nitu

de

Analysis in frequency domain - Channel 0

3,64

12,44

16,10

Figure 10. Experimental results (time and frequency domains)

4.1 TMD design

From the modal experimental analysis of the platform it was designed a TMD in a way to reduce platform vibrations. The TMD mass corresponds to various steel plate masses placed one at the top of the other placed vertically over a steel shaft

that is hanging under the slab. TMD stiffness represents the stiffness of steel spirals, in a spring shape, that are positioned between the steel shafts, in a way to generate a resetting force to the steel shaft system and still generate damping through Coulomb forces that create friction between guide shafts and the steel shaft junction. On Figure 11 and Figure 12 it is presented the TMD model proposed to be constructed.

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Study, design and construction of tuned mass damper (TMD) for concrete building floors.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

4.2 Numerical results

The platform slab and the steel profiles that constitute the columns and beams were modeled using SHELL63 element and the damper was modeled with COMBIN40 and MASS21

Figure 11. TMD in perspective view.

Figure 12. TMD in frontal view. To validate the TMD model it is considered only the first vibration mode associated to the fundamental frequency. On Figure 13 it is present the first platform modal shape.

Figure 13. First platform modal shape [13]

From the two proposed studies varying parameters, number and positioning a total of 20 cases of TMD configurations were studied through harmonic and transient ANSYS analysis [14]. The best result obtained is shown on Figure 14 and Figure 15 corresponds to the case of a single TMD positioned on the slab center with the following properties: m=50 kg ; k=25000 N/m ; c=350 Ns/m.

Figure 14. Response frequency of both controlled and uncontrolled cases.

Figure 15. Time history displacement center node: controlled and uncontrolled.

5 BUILDING OF TMD

Based on numerical results and the above TMD drawings it was constructed the TMD that will be future installed and tested at the dynamic platform. From numerical simulations it was obtained optimal values to a good tuning of the platform fundamental frequency. To obtain the value of 50 kg for TMD mass it was needed 10 steel plates of 12.5 mm thickness shown on Figure 16.

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Campuzano. J.E, Avilas. S, Doz. G.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

290.0

182.0

51.4 51.4 51.4 51.4 51.4 16.516.5

35.5

55.6

55.6

35.5

Ø14

Figure 16. Dimensions in mm of the steel plates of the TMD mass. To obtain the stiffness of 25000 N/m, it is necessary 10 spiral springs in compression working in parallel with the following parameters: Ø diameter of the spirals of 2.4 mm, external diameter of spring of 27.9 mm, 6 active spirals. To obtain the theoretical value of stiffness the above parameters are replaced on the following equation of spring stiffness:

364

4

=nR

GdK (3)

where: G = shear modulus d= diameter of the wire section n= number of active spirals R= spring radius On Table 1 it is shown the material and geometrical parameters of the spring. Table 1. Material and geometrical parameters of the TMD spring.

Parameters

E 2E+11 N/m² v 0.3 G 8E+10 N/m² d 0.0024 m R 0.01395 m n 6 K 2509.4 N/m

To find the damping coefficient for the TMD it is needed to calculate an equivalent viscous damping coefficient, that equalize the energy lost per cycle in friction case to the lost energy due to viscous effect. The equivalent damping is calculated through the following expression [15].

0

4=

XN

Ceq (4)

Where: = Friction coefficient

N= Normal force = Resonance frequency

X0= Displacement amplitude From equation (4) it is possible to obtain the normal force that must be done on the bolt (friction Coulomb force) of Figure 11. Table 2. Normal force on the bold in function of equivalent damping coefficient.

Ceq 350 Ns/m 3.1416 3.8 Hz

X0 0.026 m k 0.57

Normal 47.648 N In order to set the torque needed on the screw to achieve the corresponding normal force, it is used the following equation [16].

fLdLdfWd

Tnm

nmm

-coscos+

2= (5)

On Table 3 it is shown values to be used to find needed torque for a screw of ½ inches and screw pitch of 13 thread/inch.

Table 3. Torque needed on the screw (Coulomb force).

Advance of screw 1/13 0.0769 Screw external diameter 0.0127 m Root diameter of the screw 0.0103 m Average diameter of the screw thread, dm 0.0115 m Coefficient of friction in the threads, f 0.12 W = Normal force 47.6475 N 64.8421

n= 13.7901 Torque, T 0.8375 N m

The torque de 0.8375 can be obtained with a torque wrench. As it can be seen on Figure 11 there are two screws, so the torque applied at each one will be of 0.4188 Nm.

5.1 Experimental steps for building the TMD

To build the TMD it was necessary the following procedures. The steel plates were cut and drill the holes to prepare the vibration mass as it was shown on Figure 16. Plates were cut with torch, for this reason the dimensions of the pieces were not calculated according to the exact measure in millimeters and as a consequence the previous weights were slightly below the expected value. The plates were drilled in blocks of five plates with a template to ensure the same distance

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Study, design and construction of tuned mass damper (TMD) for concrete building floors.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

between holes on all parts. The process of making the radial drill holes is shown in Figure 17.

A B

Figure 17. A) drilling of steel plates that compose the vibrating mass, B) drilling of the steel plate serving as a support TMD. Finished the drilling process it was obtained the weight of all pieces of the TMD. The electronic scale in which the weights were obtained has an accuracy of 0.1 grams and is illustrated in Figure 18.

Figure 18. Weights obtained in electronic scale of the different steel pieces of TMD. Table 4 presents all the weights of the different pieces of the TMD set. Weight found experimentally Figure 18. Weights obtained in electronic scale, the different pieces of steel TMD. Has a very close value of the calculated numerically, the difference between the two values is about 288 g. The following step was to measure experimentally the stiffness of the springs bought in a shop and compare to the values obtained through equation (3). Table 4. Weight of the TMD mass pieces.

Piece No Weight [gr]

1 4871.7 2 4864.5 3 4834.8 4 4776.7 5 4798.4 6 4769.6 7 4823 8 4863.1 9 4791.1

10 4862.9 Friction piece 1 513.2 Friction piece 2 527.6

Screw 1 e 2 415.4 Total vibrating mass 49712

To find stiffness experimentally of each spring it was necessary to measure the compression deformation made by a weight hanging at the bottom. The deformation was measured by a clock depth gauge with precession of 0.01 mm, fabrication of Huggenberger Zurich, that recorded the initial and final length of the spring. Masses of 4889.0 g e 2443.0 g were hanging the spring and several times in a way to obtain the mean of the spring stiffness. The mounting of the test to measure spring stiffness is shown on Figure 19.

Figure 19. Mounting of the test to measure spring stiffness Table 5 present the mean values of spring stiffness, adding these values it can be obtained the total stiffness of the TMD parallel springs of 24742.1 N/m, that has a value next to the one found numerically.

Table 5. Mean value of experimental spring stiffness

Spring #

Average [N/m]

1 2410.17 2 2432.51 3 2441.33 4 2499.86 5 2501.27 6 2422.47 7 2381.84 8 2499.43 9 2466.48

10 2522.85 11 2488.22 12 2467.65

Figure 20. Show all the pieces for TMD construction:

1. Guide shaft (friction damping) 2. Support shaft 3. Screw to join the mass plates 4. Piece to friction guide shaft. 5. Springs 6. Supporting plate of springs 7. Supporting plate 8. Supporting plate of the masses

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Campuzano. J.E, Avilas. S, Doz. G.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

12

4

6 7

3

5

8

Figure 20. Pieces of the TMD.

A summary table comparing experimental and numerical values of mass and stiffness of the TMD is show on Table 6. Table 6. Comparative summary of numerical and experimental values of TMD mass and stiffness

Numeric Experimental Mass [M] 50.000 kg 49.712 kg Stiffness [k] 25000 N/m 24742 N/m

The TMD with all pieces assembled is shown on Figure 21. After construction it was hanging under the platform, placed on the point of the greater first modal displacement, the center of the slab, as it can be observed on Figure 13.

12 3

45

6

7

8

Figure 21. TMD constructed with all pieces assembled. Finally on Figure 22 it is shown the TMD placement on the platform bottom center aiming to reduce vertical vibrations.

Figure 22. TMD hanging at the bottom center of the platform.

6 CONCLUSÕES

From the numerical simulations performed the optimal calibration parameters TMD were found. Construction and installation of the TMD in the dynamic test platform was done. Next research steps are perform experimental studies with pedestrian exciting the structure with human activities. These experimental tests will prove the performance of TMD in reducing vibrations in floors.

ACKNOWLEDGMENTS

The authors are deeply grateful to the Brazilian Agency CAPES (National Council for the Improvement of Higher Education) for financially supporting this research.

[1] Varela, W.D., Modelo Teórico Experimental para Análise de Vibrações Induzidas por Pessoas Caminhando Sobre Lajes de Edifícios. Tese Doutorado, COPPE/UFRJ, Rio de Janeiro, 2004. [2] Lima, D.V.F., Controle de Vibrações Induzidas em uma Laje de Academia de Ginástica com a Utilização de Amortecedores de Massa Sintonizados. Dissertação de mestrado, Universidade de Brasília, 2007. [3] Setareh, M., Ritchey, K., Baxter, A. J., Murray, T. M., Pendulum Tuned Mass Dampers for Floor Vibration Control. Journal of Performance of constructed Facilities, ASCE, Vol. 20, No 1, 2006, pp.64-73. [4] Setareh, M., Ritchey, J. K., Murray, T. M., Koo, J. H., Ahmadian, M. Semiactive Tuned Mass Dampers for Floor Vibration Control. Journal of Structural Engineering, ASCE, Vol 133,2007, pp. 242-250. [5] Varela, W.D., Battista, R.C. Control of Vibration Induced by People Walking on large Span Composite Floor Decks. Engineering Structures. Vol 33, No 9, 2011, pp. 2485-2494. [6] Santos, M. D, Lima, D.V.F., Campuzano, J. E., Avila, S.M., Doz, G.N. Vibration control of a Gym Floor Using Tuned Mass Dampers: A Numerical Analysis. Journal Modern Mechanical Engineering, Vol 3, 2013, pp. 9-16. [7] GERB Schwingungsisolierungen GmbH & Co. KG. 2001. Vibration Protection for Structures, Buildings, Machinery and other Equipment with Tuned Mass Dampers. [8] Dallard, P., Fitzpatrick, A. I, Flint, S., Le Bourva, A., Low, R., Smith and M. R. Willford, The London Millennium Footbridge. The Structural Engineer. Vol. 79, No 22,2001, pp,17-33. [9] Maurer Sönhe, Tuned Mass and Viscous Dampers. Technical Information and Products, 2011. pp 13. [10] Battista, R.C. Múltiplos Atenuadores Dinamicos Sincronizados para controle das oscilações induzidas pelo vento na ponte Rio-Niterói.

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Study, design and construction of tuned mass damper (TMD) for concrete building floors.

EURODYN 2014 IX International Conference on Structural Dynamics. 30th June- 2nd July, Porto, Portugal.

[11] ANSYS, Swanson Analysis System, Version 10.8.0.7, 2007. [12] MATLAB, The Language of Technical Computing, Version 7.8.0.347 (R2009a), February, 2009. [13] Campuzano, J. E., Plataforma de ensaios dinâmicos: Estudos Preliminares, Projeto e Construção. Dissertação de Mestrado. Universidade de Brasília. 2011. [14] Campuzano, J. E., Doz. G. N., Avila. S. M. Estudo Numérico Via Elementos Finitos de um Amortecedor de Massa Sintonizado (AMS) para uma Plataforma de Concreto. Proceedings of the XXXIV Iberian Latin-American Congress on Computational Methods in Engineering. CILAMCE 2013. [15] Salamanca, H. E., Vibraciones mecánicas. ME4701, Facultad de ciencias físicas y matemáticas Universidad de Chile, Capitulo 1, 2010. [16] Juvinall, R. C., Marshek, K. Fundamentals of Machine Component Design. Fifth Edition. John Wiley & Sons, INC. 2012.

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