# Optimal Design Formula for Tuned Mass Damper Based on an Analytical Solution of Interaction between Soil and Structure with Rigid Foundation Subjected to Plane SH-Waves

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## Abstract

**:**

_{2}-norm of the system transfer function is introduced to quantify the performances of TMDs. The TMD design problem is then formulated and solved by optimizing the performances. Considering that aspects other than response mitigation, e.g., strokes, damper device costs, etc., may be critical to TMD damping ratios, a design framework is developed by firstly making an informed selection on TMD damping ratios, and subsequently tuning TMD frequency ratios through calibrated formulae. In addition, TMD strokes versus TMD damping ratios are investigated to facilitate the determination of TMD damping ratios. A case study based on a real-existing building system is carried out to illustrate the application of the proposed design framework. The framework has proven to be highly efficient and effective and suitable to for use in practical engineering.

## 1. Introduction

_{2}norm of the system transfer function for displacement and acceleration is introduced for performance evaluation purposes. Optimization problems are formulated as minimizing the performance indices defined through H

_{2}norm objectives for given TMD damping ratios and solved through numerical searches for the multiple representative cases expected in practical engineering. Subsequently, design formulae for the optimal frequency ratio given the TMD damping ratio are calibrated based on the optimal solutions obtained. In addition, a thorough parametric study was carried out on the TMD strokes so to facilitate the determination of TMD damping ratios. The TMD design framework was illustrated through a real existing building structure, showing the given design to have excellent performance and be suitable for practical use.

## 2. Methodology

#### 2.1. The Model

#### 2.2. Analytical Solutions for Soil-Structure-TMD Seismic Interaction

#### 2.3. Performance Evaluation Index

#### 2.4. Dimensionless Calculation Parameters

_{b}is defined as ${\eta}_{b}$

_{=}ω

_{1}a/(πβ). For a typical building, it is very common for the equivalent shear wave velocity of the site to be β = 200 m/s, and it is also typical for the equivalent radius of the foundation to be a = 10 m. Therefore, the first natural frequency of this typical building is ω

_{1}= πβη

_{b}/a = 200π/(6 × 10) = 10π/3, that is, the first-order natural period is T

_{1}= 2π/ω = 0.6 s, which is equivalent to a common reinforced concrete building with 6~12 stories based on the empirical formula ${T}_{1}=\left(0.05~0.10\right)n$ ($n$ is the total number of floors of the building) provided by the “Load Code for the Design of Building Structures (GB50009-2012)” of China.

## 3. The Proposed Design Framework

#### 3.1. Dynamic Behaviors of the TMD System

#### 3.2. Design Formulae

- (1)
- Based on $\mathsf{T}\mathsf{M}\mathsf{D}$ stroke and installation budget limits, design the damping system for the $\mathsf{T}\mathsf{M}\mathsf{D}$.
- (2)
- Estimate the $\xi $ of the damping system and obtain $\widehat{f}\left(\xi \right)$.
- (3)
- Check the mitigation performance as well as practicability of the $\mathsf{T}\mathsf{M}\mathsf{D}$ system.

## 4. Case Study

^{3}kg/m

^{3}[39]. In addition, the parameter η

_{b}takes 1/6, which fits typical engineering cases.

^{3}, as in [38,39].

## 5. Conclusions

- (1)
- The proposed design formulae are easy-to-use while being highly effective and robust to loads of various frequency contents, compared with the traditional design formulae.
- (2)
- For the problem setup in this research, with the consideration of SSI, a global optimal design may not exist. Therefore, for the related optimization problems, extra constraints, e.g., selected damping ratios, are suggested to be considered.
- (3)
- Similar to those observed in cases without SSI, TMD systems with larger mass ratios are more robust to the changes in damping ratio and frequency ratio, while larger TMD damping would lead to a lower TMD stroke.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The soil-structure-TMD seismic interaction model under the excitation of plane SH seismic waves.

**Figure 2.**Seismic mitigation performance in structural displacement and acceleration, as well as damper relative displacement for cases with the mass ratio of (

**a**) 0.01; (

**b**) 0.02; (

**c**) 0.05; (

**d**) 0.10.

**Figure 3.**Data and calibrated formulae for optimal frequency ratios and damping ratios as well as traditional design formulae for the mass ratio of (

**a**) 0.01; (

**b**) 0.02; (

**c**) 0.05; (

**d**) 0.10.

**Figure 4.**TMD stroke (damper displacement relative to the main structure) with respect to optimal frequency ratios given damping ratios for the mass ratio of (

**a**) 0.01; (

**b**) 0.02; (

**c**) 0.05; (

**d**) 0.10. The horizontal axes f is the ratio between the TMD natural frequency and structure natural frequency; the horizontal axes ξ is the ratio between TMD damping ratio and structure damping ratio.

**Figure 6.**The transfer function amplitudes of the relative displacement between the roof center and the basement center during the Landers earthquake, Northridge earthquake, and Borrego Mountain earthquake for mass ratios of (

**a**) 0.01; (

**b**) 0.02; (

**c**) 0.05; (

**d**) 0.10, with damping ratios of 2 (column 1), 4 (column 2), and 6 (column 3).

Optimal Minimizing | $\mathit{\mu}$ | ${\mathit{\psi}}_{6}$ | ${\mathit{\psi}}_{5}$ | ${\mathit{\psi}}_{4}$ | ${\mathit{\psi}}_{3}$ | ${\mathit{\psi}}_{2}$ | ${\mathit{\psi}}_{1}$ | ${\mathit{\psi}}_{0}$ |
---|---|---|---|---|---|---|---|---|

Displacement | 0.01 | 0 | 0 | 2.39 × 10^{−5} | −4.02 × 10^{−4} | 3.31 × 10^{−3} | −8.08 × 10^{−3} | 0.9551 |

0.02 | 0 | 0 | 8.23 × 10^{−6} | −6.73 × 10^{−5} | 1.14 × 10^{−3} | −5.16 × 10^{−3} | 0.9554 | |

0.05 | 0 | 0 | 5.58 × 10^{−5} | −1.17 × 10^{−3} | 0.0101 | −0.0363 | 0.9751 | |

0.10 | 0 | 0 | 3.64 × 10^{−4} | −6.65 × 10^{−3} | 0.0434 | −0.1188 | 1.0260 | |

Acceleration | 0.01 | 3.21 × 10^{−4} | −6.19 × 10^{−3} | 0.0472 | −0.1800 | 0.3599 | −0.3372 | 1.1330 |

0.02 | 2.39 × 10^{−3} | −0.0392 | 0.2543 | −0.8227 | 1.3895 | −1.1436 | 1.4011 | |

0.05 | 0.0102 | −0.1378 | 0.7580 | −2.1706 | 3.4322 | −2.8714 | 2.1358 | |

0.10 | - | - | - | - | - | - | - |

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**MDPI and ACS Style**

Jin, L.; Li, B.; Lin, S.; Li, G.
Optimal Design Formula for Tuned Mass Damper Based on an Analytical Solution of Interaction between Soil and Structure with Rigid Foundation Subjected to Plane SH-Waves. *Buildings* **2023**, *13*, 17.
https://doi.org/10.3390/buildings13010017

**AMA Style**

Jin L, Li B, Lin S, Li G.
Optimal Design Formula for Tuned Mass Damper Based on an Analytical Solution of Interaction between Soil and Structure with Rigid Foundation Subjected to Plane SH-Waves. *Buildings*. 2023; 13(1):17.
https://doi.org/10.3390/buildings13010017

**Chicago/Turabian Style**

Jin, Liguo, Bowei Li, Siqi Lin, and Guangning Li.
2023. "Optimal Design Formula for Tuned Mass Damper Based on an Analytical Solution of Interaction between Soil and Structure with Rigid Foundation Subjected to Plane SH-Waves" *Buildings* 13, no. 1: 17.
https://doi.org/10.3390/buildings13010017