# TMD Design by an Entropy Index for Seismic Control of Tall Shear-Bending Buildings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Shear-Bending Lumped Mass Model

_{i}and θ

_{i}. $\mathbf{C}$ is the damping matrix.

^{th}term, ${\mathsf{\omega}}^{2}{\mathrm{m}}_{\mathrm{T}}$ ($\mathsf{\omega}$ is the tuning frequency), construct the recovery force from the relative motion of the TMD to its installed story. At the same time, the ith line of ${\left[{\mathbf{K}}^{\mathrm{y}\mathrm{y}}\right]}^{\mathrm{*}}$ should also consider the retroaction of the TMD.

#### 2.2. HSV-Based Entropy Index

#### 2.2.1. The Entropy Based on Hankel Singular Values (HSVs)

**u**=$\mathbf{M}\left\{1\right\}{\ddot{\mathrm{a}}}_{\mathrm{g}}$ as in Equation (9), the equations of motion and responses of a system can be written in the state-space form of:

**z**is the output, $\mathbf{A}$ is the state matrix, $\mathbf{C}$ is the damping matrix, and $\mathbf{B},{\mathbf{C}}_{out},{\mathbf{D}}_{out}$ are input coefficient, output gain, and output coefficient matrices, respectively.

**B**matrix; while TMD force is essentially “internal”, because it is the direct function of structural responses, and thus, its location impact is on the mass and/or the stiffness matrix, in all, the

**A**matrix.

#### 2.2.2. Vibration Reduction Evaluation of TMDs

## 3. Building Information

^{2}; roof: 1299.97 kN/m

^{2}; other stories: 1579.73 kN/m

^{2}. Pei et al. [11] obtained the stiffness of the frame part and the shear-wall part, respectively, by pushover analysis in Midas, in which the story shear (V)-drift (u) relationships of the frame and the wall were obtained by weakening the walls and the frame manually in turn, and the M-θ relationships were obtained from story shear V and rotation angle θ. The stiffness of the frames and the shear walls are denoted by k

_{0}and k

_{s,}respectively. The values are shown in Table 1.

## 4. HSV Indices of the Structure with and without TMD

#### 4.1. Entropy Index Ratios

- (1)
- Transfer Equation (10) into a modal form, and then calculate grammians ${\mathbf{W}}_{c}$ and ${\mathbf{W}}_{o}$.
- (2)
- Calculate $\mathsf{\gamma}$ by Equation (13), and then the index ${\mathsf{\sigma}}_{u}$ of the uncontrolled building (without TMD) by Equation (18), including inter-story drifts ${\mathsf{\sigma}}_{u}$ and acceleration ${\mathsf{\sigma}}_{u}$.
- (3)
- Calculate ${\mathsf{\sigma}}_{ij}$ of the controlled building with TMD tuning to the first three modes and placing on stories 4–12 in turn, to get a 3 × 12 matrix, as the flow chart is shown in Figure 4.
- (4)
- (5)
- Figure out the TMD location case with maximum index reduction for each tuning.

#### 4.2. Validation by Responses to White Noise Excitation

#### 4.3. Comparison with the Shear Structure Model

_{b}. The first three natural periods of the shear structure are 0.818, 0.324, and 0.207. The modal nodes are at stories 12th, 6th, and 9th & 4th. But by the index σ, the optimal TMDs should be 4th/1st mode, 8th/2nd mode, and 8th/3rd modes, which are different from those in shear-bending structure.

## 5. Earthquake Responses of TMD Controlled Shear-Bending Structure

^{2}, corresponding to the Intensity 7 fortification level in Chinese code.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Elias, S.; Matsagar, V. Seismic response control of steel benchmark building with a tuned mass damper. Asian J. Civ. Eng.
**2020**, 21, 267–280. [Google Scholar] [CrossRef] - Wu, J.N. Seismic Performance, Design and Placement of Multiple Tuned Mass Dampers in Building Applications. Ph.D. Thesis, University of Missouri, Columbia, MO, USA, 2000. [Google Scholar]
- Daniel, Y.; Lavan, O. Allocation and sizing of multiple tuned mass dampers for seismic control of irregular structures. In Seismic Behaviour and Design of Irregular and Complex Civil Structures; Springer: Berlin/Heidelberg, Germany, 2013; pp. 323–338. [Google Scholar]
- Rahmani, H.R.; Könke, C.; Bojórquez, E. Seismic control of tall buildings using distributed multiple tuned mass dampers. Adv. Civ. Eng.
**2019**, 2019, 6480384. [Google Scholar] [CrossRef][Green Version] - Suresh, L.; Mini, K.M. Effect of Multiple tuned mass dampers for vibration control in high-rise buildings. Pract. Period. Struct. Des. Constr.
**2019**, 24, 04019031. [Google Scholar] [CrossRef] - Chey, M.H.; Kim, S.I.; Mun, H.J. Structural control strategy of using optimum tuned mass damper system for reinforced concrete framed structure. J. Eng. Appl. Sci.
**2017**, 12, 7038–7045. [Google Scholar] - Sun, G.-J.; Li, A.-Q. Analysis of mode participation factor and modal contribution of structures. J. Disaster Prev. Mitig. Eng.
**2009**, 29, 485–490. (In Chinese) [Google Scholar] - Kaneko, K. Optimal design method of tuned mass damper effective in reducing overall bending vibration in steel buildings with inter story dampers. J. Struct. Constr. Eng. (Trans. AIJ)
**2017**, 82, 1003–1012. [Google Scholar] [CrossRef][Green Version] - Ping, X.; Akira, N. Optimum design of tuned mass damper story system integrated into bending-shear type building based on H∞, H
_{2}, and stability maximum criteria. Struct. Control. Health Monit.**2015**, 22, 918–938. [Google Scholar] - Wada, A.; Iwata, M.; Shimizu, K.; Abe, S.; Kawai, H. Damage Control-Based Design of Buildings, 1st ed.; China Construction Industry Press: Beijing, China, 2014; pp. 93–122. [Google Scholar]
- Pei, X.-Z.; Zhang, Y.-X. Accuracy research on structural bending model of frame-shear wall structure. Earthq. Resist. Eng. Retrofit.
**2013**, 35, 1–8. (In Chinese) [Google Scholar] - Kang, S.; Shin, S. Determination of optimal accelerometer locations for bridges using frequency-domain Hankel matrix. J. Korea Inst. Struct. Maint. Insp.
**2016**, 20, 27–34. [Google Scholar] - Bigoni, C.; Zhang, Z.-Y.; Hesthaven, J.S. Systematic sensor placement for structural anomaly detection in the absence of damaged states. Comput. Methods Appl. Mech. Eng.
**2020**, 371, 113315. [Google Scholar] [CrossRef] - Fu, J.; Zhang, H.; Sun, Y. Model reduction by minimizing information loss based on cross-Grammian matrix. J. Zhejiang Univ. (Eng. Sci.)
**2009**, 43, 817–826. (In Chinese) [Google Scholar] - Fu, J.; Zhong, C.; Ding, Y. Approach to model reduction based on Hilbert-Schmidt norm and cross-Gramian. Inf. Control.
**2010**, 39, 402–407. (In Chinese) [Google Scholar] - Gawronski, W.K. Dynamics and Control of Structures: A Modal Approach; Springer: New York, NY, USA, 1998. [Google Scholar]
- Fernando, K.V.; Nich, O.H. On the structure of balanced and other principal representations of SISO systems. IEEE Trans. Autom. Control
**1983**, 28, 228–231. [Google Scholar] [CrossRef]

**Figure 1.**Schemes of the shear-bending lumped mass model of the multi-story building; (

**a**) Deformed shape with shear & bending springs. (

**b**) Separating shear and bending deformation between the ith and (i + 1)th lumped masses.

**Figure 2.**The 12-story frame-shear wall building. (

**a**) Plan; (

**b**) The lumped mass model: a column + shear wall (shear-bending coupled).

**Figure 9.**Frequency responses to white noise Δt = 0.01 s, with optimal TMDs (shear–bending structure).

**Figure 13.**Max. responses to white noise, with TMDs’ tuning considering and not considering bending (shear–bending structure, Δt = 0.005 s).

**Figure 14.**Response spectra of the seven earthquake ground motions. (

**a**) Acceleration response spectra; (

**b**) Displacement response spectra.

**Figure 15.**Max. Displacement reduction ratios (controlled/uncontrolled) of the shear-bending structure to seven scaled earthquakes.

Stories | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

frame k_{0} (10^{8}) | 14.3 | 17.2 | 16.9 | 16.1 | 14.9 | 14.1 | 13.7 | 12.6 | 11.3 | 9.85 | 8.15 | 5.20 |

wall k_{s} (10^{8}) | 65.3 | 35.5 | 23.5 | 17.3 | 13.5 | 10.8 | 8.65 | 6.89 | 5.34 | 3.90 | 2.51 | 1.14 |

wall k_{b} (10^{8}) | 5260 | 2250 | 1420 | 967 | 701 | 500 | 353 | 241 | 155 | 90.6 | 43.8 | 13.7 |

ϕ (N.m^{2},deduced) | 46.7 | 57.7 | 55.0 | 50.8 | 47.1 | 41.9 | 36.8 | 31.4 | 25.9 | 20.5 | 15.2 | 1.78 |

_{0}: shear stiffness of the frame; k

_{s}: shear stiffness of the wall; k

_{b}: bending stiffness; ϕ: stiffness ratio.

ChiChi | Kobe | Taft | ElCentro | WenChuan | Sylmar | Newhall | Effective No. | |
---|---|---|---|---|---|---|---|---|

12th/1st mode | 0.6645 | 0.8769 | 0.9950 | 0.9815 | 0.9605 | 1.5264 | 0.9392 | 6 |

4th/1st mode | 0.7316 | 0.9413 | 0.9686 | 1.1364 | 0.8817 | 1.0769 | 1.0461 | 4 |

6th/2nd mode | 1.0118 | 1.1155 | 1.0090 | 1.2117 | 0.9698 | 1.1998 | 1.2233 | 1 |

8th/3rd mode | 0.9830 | 0.9506 | 0.9972 | 1.2009 | 1.1845 | 1.3705 | 1.2800 | 3 |

ChiChi | Kobe | Taft | ElCentro | WenChuan | Sylmar | Newhall | |
---|---|---|---|---|---|---|---|

12th/1st mode | 0.9634 | 0.6987 | 0.6279 | 0.6182 | 0.7860 | 0.7610 | 0.5077 |

4th/1st mode | 0.9542 | 0.7499 | 0.8208 | 0.9863 | 0.8302 | 0.8628 | 0.8835 |

6th/2nd mode | 0.7559 | 0.9880 | 0.6631 | 0.8371 | 0.7158 | 0.7703 | 0.9781 |

8th/3rd mode | 0.6275 | 0.6341 | 0.7082 | 0.8230 | 0.9749 | 0.9061 | 1.0288 |

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**MDPI and ACS Style**

Wang, Y.; Qu, Z.
TMD Design by an Entropy Index for Seismic Control of Tall Shear-Bending Buildings. *Entropy* **2023**, *25*, 1110.
https://doi.org/10.3390/e25081110

**AMA Style**

Wang Y, Qu Z.
TMD Design by an Entropy Index for Seismic Control of Tall Shear-Bending Buildings. *Entropy*. 2023; 25(8):1110.
https://doi.org/10.3390/e25081110

**Chicago/Turabian Style**

Wang, Yumei, and Zhe Qu.
2023. "TMD Design by an Entropy Index for Seismic Control of Tall Shear-Bending Buildings" *Entropy* 25, no. 8: 1110.
https://doi.org/10.3390/e25081110