# American Institute of Mathematical Sciences

doi: 10.3934/naco.2021060
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## Robust optimum design of tuned mass dampers for high-rise buildings subject to wind-induced vibration

 1 Federal University of Rio Grande do Sul (UFRGS), Postgraduate Program in Civil Engineering, Av. Osvaldo Aranha 99, 90035-190 Porto Alegre, RS, Brazil 2 Federal University of Rio Grande do Sul (UFRGS), Department of Mechanical Engineering, Av. Sarmento Leite 425, 90050-170 Porto Alegre, RS, Brazil

*Corresponding author: Bibiana Bertolin Rossato

Received  June 2021 Revised  November 2021 Early access December 2021

Fund Project: The first author is supported by CNPq grant 130511/2018–8

Control devices are commonly applied to suppress structural displacement due to dynamic loads. In this work, a study of the optimum tuned mass dampers (TMDs) design was carried out, installed in a high-rise building subject to wind-induced vibration. Tuned mass dampers are the most known passive energy device and their design are an important area of study. A mathematical model to consider the wind force in the time domain was introduced. A procedure to obtain the robust design of tuned mass dampers was proposed through the optimization under uncertainties, which considers the uncertainties present in the structural properties of the building and also in the dynamic excitation. This method led to the robust design of TMDs, whereas the device performance became insensitive to the randomness of the input variables of the optimization problem. The robust design was compared with a design obtained by a deterministic optimization and the advantages of using an optimization under uncertainties are shown. In addition, the proposed methodology was compared with traditional TMD design methods, showing again the superiority of the proposed methodology.

Citation: Bibiana Bertolin Rossato, Letícia Fleck Fadel Miguel. Robust optimum design of tuned mass dampers for high-rise buildings subject to wind-induced vibration. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021060
##### References:
 [1] ABNT - Brazilian Association of Technical Standards, NBR 6123: Foras devidas ao vento em edificaes, Rio de Janeiro, Brazil, 1988. [2] V. Boonyapinyo, A. Aksorn and P. Lukkunaprasit, Suppression of aerodynamic response of suspension bridges during erection and after completion by using tuned mass dampers, Wind and Structures, 10 (2007), 1-22.  doi: 10.12989/was.2007.10.1.001. [3] F. S. Brando and L. F. F. Miguel, Vibration control in buildings under seismic excitation using optimized tuned mass dampers, Frattura ed Integrità Strutturale, 54 (2020), 66-87.  doi: 10.3221/IGF-ESIS.54.05. [4] G. Chen and J. Wu, Experimental study on multiple tuned mass dampers to reduce seismic responses of a three-storey building structure, Earthquake Engineering and Structural Dynamics, 32 (2003), 793-810. [5] J. P. Den Hartog, in Mechanical Vibration (4th ed), McGraw-Hill, New York, (1956). [6] S. Elias and V. Matsagar, Optimum tuned mass damper for wind and earthquake response control of high-rise building, In: Advances in Structural Engineering (eds. V. Matsagar), Springer, New Delhi [7] M. S. Gonalves, R. H. Lopez and L. F. F. Miguel, Search group algorithm: A new metaheuristic method for the optimization of truss structures, Computers and Structures, 153 (2015), 165-184.  doi: 10.1016/j.compstruc.2015.03.003. [8] L. Ierimonti, I. Venanzi and L. Caracoglia, Life-cycle damage-based cost analysis of tall buildings equipped with tuned mass dampers, Journal of Wind Engineering and Industrial Aerodynamics, 14 (2018), 54-64. [9] T. Igusa and K. Xu, Vibration control using multiple tuned mass dampers, Journal of Sound and Vibration, 175 (1994), 491-503. [10] J. C. Kaimal, Spectral characteristics of surface-layer turbulence, Quart J Roy Meteorol Soc, 98 (1972), 563-589.  doi: 10.1002/qj.49709841707. [11] A. Kareem, T. Kijewski and Y. Tamura, Mitigation of motions of tall buildings with specific examples of recent applications, Wind and Structures, 2 (1999), 201-251.  doi: 10.12989/was.1999.2.3.201. [12] A. Kawaguchi, A. Teramura and Y. Omote, Time history response of a tall building with a tuned mass damper under wind force, Journal of Wind Engineering and Industrial Aerodynamics, 43 (1992), 1946-1960.  doi: 10.1016/0167-6105(92)90625-k. [13] C-L. Lee, Optimal design theories and applications of tuned mass dampers, Engineering Structures, 28 (2006), 43-53.  doi: 10.1016/j.engstruct.2005.06.023. [14] M. Y. Liu, Analytical and experimental research on wind-induced vibration in high-rise buildings with tuned liquid column dampers, Wind and Structures, 6 (2003), 71-90.  doi: 10.12989/was.2003.6.1.071. [15] M. Y. Liu, Wind-induced vibration of high-rise building with tuned mass damper including soil-structure interaction, Journal of Wind Engineering and Industrial Aerodynamics, 96 (2008), 1092-1102.  doi: 10.1016/j.jweia.2007.06.034. [16] Q. Liu, Optimal design of wind-induced vibration control of tall buildings and high-rise structures, Wind and Structures, 2 (1999), 69-83.  doi: 10.12989/was.1999.2.1.069. [17] G. C. Marano, Robust optimum design of tuned mass dampers devices in random vibrations mitigation, Journal of Sound and Vibration, 313 (2008), 472-492.  doi: 10.1016/j.jsv.2007.12.020. [18] L. F. F. Miguel, A novel approach to the optimum design of MTMDs under seismic excitations, Structural Control and Health Monitoring, 23 (2016), 1290-1313.  doi: 10.1002/stc.1845. [19] L. F. F. Miguel, Assessment of code recommendations through simulation of EPS wind loads along a segment of a transmission line, Engineering Structures, 43 (2012), 1-11.  doi: 10.1016/j.engstruct.2012.05.004. [20] M. Mohebbi, Designing optimal multiple tuned mass dampers using genetic algorithms (GA) for mitigating the seismic response of structures, Journal of Vibration and Control, 19 (2013), 605-625.  doi: 10.1177/1077546311434520. [21] V. B. Patil and R. S. Jangid, Optimum multiple tuned mass dampers for wind excited benchmark building, Journal of Civil Engineering and Management, 17 (2011), 540-557. [22] R. Rana and T. T. Soong, Parametric study and simplified design of tuned mass dampers", Engineering Structures, Engineering Structures, 20(3) (1998), 193-204.  doi: 10.1016/S0141-0296(97)00078-3. [23] M. Shinozuka and C. M. Jan, Digital simulation of random processes and its applications, Journal of Sound and Vibration, 25 (1972), 111-128.  doi: 10.1016/0022-460x(72)90600-1. [24] H. Tanaka and C. Y. Mak, Effect of tuned mass dampers on wind induced response of tall buildings, Journal of Wind Engineering and Industrial Aerodynamics, 14 (1983), 357-368. [25] L. S. Vellar, Robust optimum design of multiple tuned mass dampers for vibration control in buildings subjected to seismic excitation, Shock and Vibration, 1 (2019), 1-9.  doi: 10.1007/s00158-014-1129-4. [26] I. Venanzi, Robust optimal design of tuned mass dampers for tall buildings with uncertain parameters, Industrial Structural and Multidisciplinary Optimization, 51 (2015), 239-250.  doi: 10.1007/s00158-014-1129-4. [27] G. B. Warburton, Optimum absorbers parameters for various combinations of response and excitation, Earthquake Engineering and Structural Dynamics, 10(3) (1982), 381-401.  doi: 10.1002/eqe.4290100304. [28] C. Yang, Benchmark problem for response control of wind-excited tall buildings, Journal of Engineering Mechanics, 130 (2004), 437-446.  doi: 10.1061/(asce)0733-9399(2004)130:4(437). [29] C. Zang, A review of robust optimal design and its application in dynamics, Computers and Structures, 83 (2005), 315-326.  doi: 10.1016/j.compstruc.2004.10.007.

show all references

##### References:
 [1] ABNT - Brazilian Association of Technical Standards, NBR 6123: Foras devidas ao vento em edificaes, Rio de Janeiro, Brazil, 1988. [2] V. Boonyapinyo, A. Aksorn and P. Lukkunaprasit, Suppression of aerodynamic response of suspension bridges during erection and after completion by using tuned mass dampers, Wind and Structures, 10 (2007), 1-22.  doi: 10.12989/was.2007.10.1.001. [3] F. S. Brando and L. F. F. Miguel, Vibration control in buildings under seismic excitation using optimized tuned mass dampers, Frattura ed Integrità Strutturale, 54 (2020), 66-87.  doi: 10.3221/IGF-ESIS.54.05. [4] G. Chen and J. Wu, Experimental study on multiple tuned mass dampers to reduce seismic responses of a three-storey building structure, Earthquake Engineering and Structural Dynamics, 32 (2003), 793-810. [5] J. P. Den Hartog, in Mechanical Vibration (4th ed), McGraw-Hill, New York, (1956). [6] S. Elias and V. Matsagar, Optimum tuned mass damper for wind and earthquake response control of high-rise building, In: Advances in Structural Engineering (eds. V. Matsagar), Springer, New Delhi [7] M. S. Gonalves, R. H. Lopez and L. F. F. Miguel, Search group algorithm: A new metaheuristic method for the optimization of truss structures, Computers and Structures, 153 (2015), 165-184.  doi: 10.1016/j.compstruc.2015.03.003. [8] L. Ierimonti, I. Venanzi and L. Caracoglia, Life-cycle damage-based cost analysis of tall buildings equipped with tuned mass dampers, Journal of Wind Engineering and Industrial Aerodynamics, 14 (2018), 54-64. [9] T. Igusa and K. Xu, Vibration control using multiple tuned mass dampers, Journal of Sound and Vibration, 175 (1994), 491-503. [10] J. C. Kaimal, Spectral characteristics of surface-layer turbulence, Quart J Roy Meteorol Soc, 98 (1972), 563-589.  doi: 10.1002/qj.49709841707. [11] A. Kareem, T. Kijewski and Y. Tamura, Mitigation of motions of tall buildings with specific examples of recent applications, Wind and Structures, 2 (1999), 201-251.  doi: 10.12989/was.1999.2.3.201. [12] A. Kawaguchi, A. Teramura and Y. Omote, Time history response of a tall building with a tuned mass damper under wind force, Journal of Wind Engineering and Industrial Aerodynamics, 43 (1992), 1946-1960.  doi: 10.1016/0167-6105(92)90625-k. [13] C-L. Lee, Optimal design theories and applications of tuned mass dampers, Engineering Structures, 28 (2006), 43-53.  doi: 10.1016/j.engstruct.2005.06.023. [14] M. Y. Liu, Analytical and experimental research on wind-induced vibration in high-rise buildings with tuned liquid column dampers, Wind and Structures, 6 (2003), 71-90.  doi: 10.12989/was.2003.6.1.071. [15] M. Y. Liu, Wind-induced vibration of high-rise building with tuned mass damper including soil-structure interaction, Journal of Wind Engineering and Industrial Aerodynamics, 96 (2008), 1092-1102.  doi: 10.1016/j.jweia.2007.06.034. [16] Q. Liu, Optimal design of wind-induced vibration control of tall buildings and high-rise structures, Wind and Structures, 2 (1999), 69-83.  doi: 10.12989/was.1999.2.1.069. [17] G. C. Marano, Robust optimum design of tuned mass dampers devices in random vibrations mitigation, Journal of Sound and Vibration, 313 (2008), 472-492.  doi: 10.1016/j.jsv.2007.12.020. [18] L. F. F. Miguel, A novel approach to the optimum design of MTMDs under seismic excitations, Structural Control and Health Monitoring, 23 (2016), 1290-1313.  doi: 10.1002/stc.1845. [19] L. F. F. Miguel, Assessment of code recommendations through simulation of EPS wind loads along a segment of a transmission line, Engineering Structures, 43 (2012), 1-11.  doi: 10.1016/j.engstruct.2012.05.004. [20] M. Mohebbi, Designing optimal multiple tuned mass dampers using genetic algorithms (GA) for mitigating the seismic response of structures, Journal of Vibration and Control, 19 (2013), 605-625.  doi: 10.1177/1077546311434520. [21] V. B. Patil and R. S. Jangid, Optimum multiple tuned mass dampers for wind excited benchmark building, Journal of Civil Engineering and Management, 17 (2011), 540-557. [22] R. Rana and T. T. Soong, Parametric study and simplified design of tuned mass dampers", Engineering Structures, Engineering Structures, 20(3) (1998), 193-204.  doi: 10.1016/S0141-0296(97)00078-3. [23] M. Shinozuka and C. M. Jan, Digital simulation of random processes and its applications, Journal of Sound and Vibration, 25 (1972), 111-128.  doi: 10.1016/0022-460x(72)90600-1. [24] H. Tanaka and C. Y. Mak, Effect of tuned mass dampers on wind induced response of tall buildings, Journal of Wind Engineering and Industrial Aerodynamics, 14 (1983), 357-368. [25] L. S. Vellar, Robust optimum design of multiple tuned mass dampers for vibration control in buildings subjected to seismic excitation, Shock and Vibration, 1 (2019), 1-9.  doi: 10.1007/s00158-014-1129-4. [26] I. Venanzi, Robust optimal design of tuned mass dampers for tall buildings with uncertain parameters, Industrial Structural and Multidisciplinary Optimization, 51 (2015), 239-250.  doi: 10.1007/s00158-014-1129-4. [27] G. B. Warburton, Optimum absorbers parameters for various combinations of response and excitation, Earthquake Engineering and Structural Dynamics, 10(3) (1982), 381-401.  doi: 10.1002/eqe.4290100304. [28] C. Yang, Benchmark problem for response control of wind-excited tall buildings, Journal of Engineering Mechanics, 130 (2004), 437-446.  doi: 10.1061/(asce)0733-9399(2004)130:4(437). [29] C. Zang, A review of robust optimal design and its application in dynamics, Computers and Structures, 83 (2005), 315-326.  doi: 10.1016/j.compstruc.2004.10.007.
Total wind velocity for the Z = 160m
Convergence by the number of samples. (a) Convergence of the mean of $u_{max}$, (b) Convergence of the standard derivation of $u_{max}$
Probability density function of maximum displacement
Maximum displacement per building height
Displacement at the top of the building for the cases: without TMD (yellow curve), with 1 TMD designed by Den Hartog's method (blue curve), with 1 TMD designed by Warburton's method (red curve), and with 1 TMD optimized by the proposed optimization method (green curve), for coefficients of variation equal zero for all random variables
Input random variables of the building and of the TMDs
 Random Variable Mean Value CV Story mass $m_{i}$ 9.8$\times$105kg 5$\%$ Story stiffness $k_{1}$ 2.13$\times$109N/m 10$\%$ Story stiffness $k_{40}$ 9.98$\times$108N/m 10$\%$ Damping ratio $\zeta$ 0.016 10$\%$ Spring constant of each TMD $k_{tmd}$ Design variable 5$\%$ Damping constant of each TMD $c_{tmd}$ Design variable 5$\%$
 Random Variable Mean Value CV Story mass $m_{i}$ 9.8$\times$105kg 5$\%$ Story stiffness $k_{1}$ 2.13$\times$109N/m 10$\%$ Story stiffness $k_{40}$ 9.98$\times$108N/m 10$\%$ Damping ratio $\zeta$ 0.016 10$\%$ Spring constant of each TMD $k_{tmd}$ Design variable 5$\%$ Damping constant of each TMD $c_{tmd}$ Design variable 5$\%$
Input random variable of the wind excitation
 $\textbf{Random Variable}$ $\textbf{Mean Value}$ $\textbf{CV}$ Mean wind velocity $V_{0}$ 43 m/s 15%
 $\textbf{Random Variable}$ $\textbf{Mean Value}$ $\textbf{CV}$ Mean wind velocity $V_{0}$ 43 m/s 15%
Resulting TMD robust design (scenario with 1 TMD on the top)
 $\textbf{Run}$ $N_{TMDs}$ $E(k_{tmd}$)[kN/m] $E(c_{tmd}$)[kNs/m] $E(u_{max}$)[m] - Without dampers 0.628 1 1 1907.808 269.153 0.411 2 1 1964.466 250.027 0.416 3 1 1916.927 281.217 0.414
 $\textbf{Run}$ $N_{TMDs}$ $E(k_{tmd}$)[kN/m] $E(c_{tmd}$)[kNs/m] $E(u_{max}$)[m] - Without dampers 0.628 1 1 1907.808 269.153 0.411 2 1 1964.466 250.027 0.416 3 1 1916.927 281.217 0.414
Resulting TMD robust design
 $\textbf{Parameter}$ $\textbf{Without TMD}$ With TMD $\textbf{Reduction}$ $E(u_{max})[m]$ 0.6283 0.411 34.6% $Var(u_{max})[m^2]$ 0.0458 0.0180 60.7% $Std(u_{max})[m]$ 0.214 0.1344 37.2% $E(a_{max})[m/s^2]$ 1.417 0.8669 38.8%
 $\textbf{Parameter}$ $\textbf{Without TMD}$ With TMD $\textbf{Reduction}$ $E(u_{max})[m]$ 0.6283 0.411 34.6% $Var(u_{max})[m^2]$ 0.0458 0.0180 60.7% $Std(u_{max})[m]$ 0.214 0.1344 37.2% $E(a_{max})[m/s^2]$ 1.417 0.8669 38.8%
Resulting TMD robust design (scenario with 5 TMDs on the top)
 $\textbf{TMD Number}$ $\textbf{1}$ $\textbf{2}$ $\textbf{3}$ $\textbf{4}$ $\textbf{5}$ $E(k_{tmd})[kN/m]$ 515.37 315.48 386.02 429.65 372.79 $E(c_{tmd})[kN/m]$ 27.953 16.103 35.951 26.562 30.231 $E(u_{max})[m]$ 0.415
 $\textbf{TMD Number}$ $\textbf{1}$ $\textbf{2}$ $\textbf{3}$ $\textbf{4}$ $\textbf{5}$ $E(k_{tmd})[kN/m]$ 515.37 315.48 386.02 429.65 372.79 $E(c_{tmd})[kN/m]$ 27.953 16.103 35.951 26.562 30.231 $E(u_{max})[m]$ 0.415
Comparing the robust optimization with a deterministic optimization
 $\textbf{Parameter}$ $\textbf{Robust Optimization}$ $\textbf{Deterministic Optimization}$ Number of TMDs 1 1 $k_{tmd}[kN/m]$ 1907.78 1727.77 $c_{tmd}[kN/m]$ 269.11 147.47 $E(u_{max})[m]$ 0.411 0.435 $Var(u_{max})[m^2]$ 0.018 0.0224
 $\textbf{Parameter}$ $\textbf{Robust Optimization}$ $\textbf{Deterministic Optimization}$ Number of TMDs 1 1 $k_{tmd}[kN/m]$ 1907.78 1727.77 $c_{tmd}[kN/m]$ 269.11 147.47 $E(u_{max})[m]$ 0.411 0.435 $Var(u_{max})[m^2]$ 0.018 0.0224
Comparing the robust optimization with SGA and GA
 $\textbf{Parameter}$ Robust Optimization with SGA Robust Optimization with GA Number of TMDs 1 1 $k_{tmd}[kN/m]$ 1907.78 2057.38 $c_{tmd}[kN/m]$ 269.11 761.65 $E(u_{max})[m]$ 0.411 0.449
 $\textbf{Parameter}$ Robust Optimization with SGA Robust Optimization with GA Number of TMDs 1 1 $k_{tmd}[kN/m]$ 1907.78 2057.38 $c_{tmd}[kN/m]$ 269.11 761.65 $E(u_{max})[m]$ 0.411 0.449
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