# 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 & Optimization, doi: 10.3934/naco.2021060
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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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