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Frontiers of Structural and Civil Engineering

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Front. Struct. Civ. Eng.    2017, Vol. 11 Issue (1) : 66-89    https://doi.org/10.1007/s11709-016-0356-8
RESEARCH ARTICLE
Control efficiency optimization and Sobol’s sensitivity indices of MTMDs design parameters for buffeting and flutter vibrations in a cable stayed bridge
Nazim Abdul NARIMAN()
Institute of Structural Mechanics, School of Civil Engineering, Bauhaus University Weimar, Weimar 99423, Germany
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Abstract

This paper studies optimization of three design parameters (mass ratio, frequency ratio and damping ratio) of multiple tuned mass dampers MTMDs that are applied in a cable stayed bridge excited by a strong wind using minimax optimization technique. ABAQUS finite element program is utilized to run numerical simulations with the support of MATLAB codes and Fast Fourier Transform FFT technique. The optimum values of these three parameters are validated with two benchmarks from the literature, first with Wang and coauthors and then with Lin and coauthors. The validation procedure detected a good agreement between the results. Box-Behnken experimental method is dedicated to formulate the surrogate models to represent the control efficiency of the vertical and torsional vibrations. Sobol’s sensitivity indices are calculated for the design parameters in addition to their interaction orders. The optimization results revealed better performance of the MTMDs in controlling the vertical and the torsional vibrations for higher mode shapes. Furthermore, the calculated rational effects of each design parameter facilitate to increase the control efficiency of the MTMDs in conjunction with the support of the surrogate models.
Keywords MTMDs      power spectral density      fast Fourier transform      minimax optimization technique      Sobol’s sensitivity indices      Box-Behnken method     
Corresponding Author(s): Nazim Abdul NARIMAN   
Online First Date: 09 November 2016    Issue Date: 27 February 2017
 Cite this article:   
Nazim Abdul NARIMAN. Control efficiency optimization and Sobol’s sensitivity indices of MTMDs design parameters for buffeting and flutter vibrations in a cable stayed bridge[J]. Front. Struct. Civ. Eng., 2017, 11(1): 66-89.
 URL:  
https://academic.hep.com.cn/fsce/EN/10.1007/s11709-016-0356-8
https://academic.hep.com.cn/fsce/EN/Y2017/V11/I1/66
Fig.1  TMD attached to the primary mass
Fig.2  Location of TMDs for suppressing (a) vertical and (b) torsional vibrations
Fig.3  Layout of the TMDs
Fig.4  Wind speed fluctuation profile
mode shape type of vibration frequency (Hz)
1 vertical 0.242
3 vertical 0.509
6 vertical–torsional 0.776
7 torsional 0.789
9 vertical 0.883
11 lateral–torsional 0.986
16 vertical–torsional 1.208
20 vertical–torsional 1.631
Tab.1  Vibration mode shapes data
Fig.5  Eight mode shapes of vibrations. (a) Mode shape 1; (b) mode shape 3; (c) mode shape 6; (d) mode shape 7; (e) mode shape 9; (f) mode shape 11; (g) mode shape 16; (h) mode shape 20
Fig.6  Power spectral densities of displacements at the mid span. (a) Vertical displacement; (b) torsional displacement
mass ratio m frequency ratio f TMD stiffness Kd (N/m) TMD damping coefficient Cd(N.s/m)
0.25% 1.0 54183 6982
0.75% 1.0 162549 20947
1.25% 1.0 270915 34912
1.75% 1.0 379281 48876
2.25% 1.0 487647 62841
Tab.2  Parameters of TMDs for multiple mass ratios
mass ratio m vertical vibration control efficiency %
0.25% -33.20
0.75% 12.77
1.25% -9.24
1.75% 15.12
2.25% 34.76
Tab.3  Vertical response control efficiency
Fig.7  TMD mass ratio effect on the vertical response
mass ratio μ torsional vibration control efficiency (%)
0.25% 13.72
0.75% 33.80
1.25% 39.59
1.75% 50.81
2.25% 58.47
Tab.4  Torsional response control efficiency
Fig.8  TMD mass ratio effect on torsional response
Fig.9  TMDs mass ratio effect on vertical and torsional vibrations. (a) No TMDs; (b) mass ratio= 0.25%; (c) mass ratio= 0.75%; (d) mass ratio= 1.25%; (e) mass ratio= 1.75%; (f) mass ratio= 2.25%
frequency ratio
f		 mass ratio
μ		 TMD stiffness Kd
N/m		 TMD damping coefficient Cd
N.s/m
0.8 2.25% 312094 50273
0.9 2.25% 394994 56557
1.0 2.25% 487647 62841
1.1 2.25% 590052 69125
1.2 2.25% 702211 75409
Tab.5  Parameters of the TMD for multiple frequency ratios
frequency ratio f vertical vibration control efficiency (%)
0.8 -21.76
0.9 6.16
1.0 34.76
1.1 46.18
1.2 50.47
Tab.6  Vertical response control efficiency
Fig.10  TMD frequency ratio effect on the vertical response
frequency ratio f torsional vibration control efficiency (%)
0.8 58.74
0.9 58.25
1.0 58.47
1.1 58.36
1.2 58.12
Tab.7  Torsional response control efficiency
Fig.11  TMD frequency ratio effect on torsional response
damping ratio ξ mass ratio m TMD stiffness Kd
(N/m)		 TMD damping coefficient Cd
(N.s/m)
0.01 2.25% 487647 12568
0.05 2.25% 487647 62841
0.10 2.25% 487647 125682
0.15 2.25% 487647 188523
0.2 2.25% 487647 251364
Tab.8  Parameters of the TMD for multiple Damping ratios
damping ratio ξ vertical vibration control efficiency %
0.01 33.03
0.05 34.76
0.10 38.93
0.15 43.74
0.2 46.00
Tab.9  vertical response control efficiency
Fig.12  TMD Damping ratio effect on the vertical response
damping ratio ξ torsional vibration control efficiency (%)
0.01 41.52
0.05 41.54
0.10 41.52
0.15 41.53
0.2 41.56
Tab.10  Torsional response control efficiency
Fig.13  TMD Damping ratio effect on the torsional response
Fig.14  Validation for TMDs mass ratio effect on the vertical vibration of the deck
Fig.15  Validation for TMDs damping ratio effect on the vertical vibration of the deck
Fig.16  Box–Behnken experimental design. (a) The design, as derived from a cube; (b) interlocking 22 factorial experiments
variable symbol coded variable level
low center high
−1 0 + 1
mass ratio X1 0.0025 0.0125 0.0225
frequency ratio X2 0.8 1.0 1.2
damping ratio X3 0.01 0.105 0.20
Tab.11  The level of variables chosen for the Box-Behnken design
run No. coded level of variables actual level of variables
X1 X2 X3 mass
ratio		 frequency
ratio		 damping
ratio
1 -1 -1 0 0.0025 0.8 0.105
2 + 1 -1 0 0.0225 0.8 0.105
3 -1 + 1 0 0.0025 1.2 0.105
4 + 1 + 1 0 0.0225 1.2 0.105
5 -1 0 -1 0.0025 1.0 0.01
6 + 1 0 -1 0.0225 1.0 0.01
7 -1 0 + 1 0.0025 1.0 0.2
8 + 1 0 + 1 0.0225 1.0 0.2
9 0 -1 -1 0.0125 0.8 0.01
10 0 + 1 -1 0.0125 1.2 0.01
11 0 -1 + 1 0.0125 0.8 0.2
12 0 + 1 + 1 0.0125 1.2 0.2
13 0 0 0 0.0125 1.0 0.105
14 0 0 0 0.0125 1.0 0.105
15 0 0 0 0.0125 1.0 0.105
Tab.12  Box-Behnken design with coded and actual values for three size fractions
Fig.17  Coefficient of regression-vertical vibration control efficiency
Fig.18  Coefficient of regression-torsional vibration control efficiency
sensitivity indices vertical vibration control efficiency torsional vibration control efficiency
first order X1 0.6851 0.9997
first order X2 0.2176 0.0000
first order X3 0.0840 0.0001
sum of first orders 0.999329 0.999961
interaction between X1 and X2 0.0091 0.0001
interaction between X1 and X3 0.0030 0.0001
interaction between X2 and X3 0.0006 0.0000
total order of X1 0.6979 0.9999
total order of X2 0.2279 0.0002
total order of X3 0.0883 0.0002
sum of total orders 1.0141 1.0003
Tab.13  Sensitivity indices
Fig.19  Convergence of sensitivity indices –vertical vibration control efficiency
Fig.20  Convergence of sensitivity indices –Torsional vibration control efficiency
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