# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems

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## Abstract

**:**

## 1. Introduction

`hinfnorm`function of the Matlab Robust Control Toolbox [29,30]. The ATOP solutions are obtained with the genetic algorithm (GA) solver provided by the Matlab Global Optimization Toolbox, which allows using hybrid sets of discrete and continuous optimization variables, permits defining a sufficiently wide variety of optimization constraints and facilitates an easy implementation of parallel computing [31]. To demonstrate the flexibility of the proposed methodology, three different DDSs are designed for the seismic protection of a MBS formed by $m=5$ adjacent five-story buildings. After that, a proper set of numerical simulations are conducted using the full-scale 180-component of the El Centro 1940 seismic record as ground acceleration disturbance. The obtained results corroborate the effectiveness of the proposed design procedure and confirm its computational efficiency for large-scale problems.

**Remark**

**1.**

**Remark**

**2.**

## 2. Mathematical Model

#### 2.1. Plain Building Model

#### 2.2. Interstory and Interbuilding Damping Models

**Remark**

**3.**

#### 2.3. Overall Multibuilding Model

#### 2.4. State-Space Model and Output Variables

## 3. Optimization Procedure

#### 3.1. Allowed Damper Positions, Dampers Allocation and Damping Coefficients

#### 3.2. Optimization Variables and Associated Multibuilding Model

#### 3.3. Objective Function and Optimization Constraints

**Remark**

**4.**

`ss()`function of the Matlab Control System Toolbox [34], a state-space representation of the linear time-invariant model in Equation (60) can be created with

`sys=ss(A,B,Cz,0)`. After that, the corresponding ${H}_{\infty}$ system-norm in Equation (61) can be readily computed with the function

`hinfnorm`of the Matlab Robust Control Toolbox [30] in the form

`gamma=hinfnorm(sys)`.

**Remark**

**5.**

## 4. DDS Designs

**Remark**

**6.**

`PopulationSize`, 0.9 for the

`CrossoverFraction`, 20 for the

`EliteCount`and 500 for the

`MaxGenerations`parameters. To take advantage of the CPU multi-core architecture, the GA solver has been enforced to run in parallel mode by enabling the option

`UseParallel`. Also, to improve the relative accuracy in the computation of the ${H}_{\infty}$-norm, the tolerance in the function

`hinfnorm`has been decreased to ${10}^{-3}$ [30].

**Remark**

**7.**

`rng(125)`has been used to set a common random seed for all the computed DDS configurations. That random seed has been arbitrarily chosen, which confirms the effectiveness of the proposed design methodology and indicates that improved results could be possibly obtained by exploring a wider set of random seeds [29].

**Remark**

**8.**

## 5. Seismic Responses

**Remark**

**9.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ATOP | allocation-tuning optimization problem |

DDS | distributed damping system |

FVD | fluid viscous damper |

GA | genetic algorithm |

MBS | multibuilding system |

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**Figure 1.**System of $m=5$ adjacent buildings equipped with a distributed set of interstory and interbuilding dampers.

**Figure 2.**Schematic mechanical model of a five-story building equipped with a distributed system of ${\widehat{n}}_{j}=3$ supplemental interstory dampers implemented at the story levels ${\widehat{p}}_{1}^{j}=1$, ${\widehat{p}}_{2}^{j}=3$ and ${\widehat{p}}_{3}^{j}=4$.

**Figure 3.**Interbuilding separation of the stories ${s}_{i}^{j}$ and ${s}_{i}^{j+1}$ corresponding to the interbuilding approaching ${\tilde{r}}_{i}^{j}\left(t\right)=-\{{q}_{i}^{j+1}\left(t\right)-{q}_{i}^{j}\left(t\right)\}=2\delta $ for adjacent buildings ${\mathcal{B}}^{\left(j\right)}$ and ${\mathcal{B}}^{(j+1)}$ with an interbuilding gap ${\Delta}_{j}$.

**Figure 4.**Schemes of allowed damper positions for a three-building system. Interstory scheme $\widehat{\sigma}=[{\widehat{\sigma}}^{\left(1\right)},{\widehat{\sigma}}^{\left(2\right)},{\widehat{\sigma}}^{\left(3\right)}]$ with ${\widehat{\eta}}_{1}=3$, ${\widehat{\eta}}_{2}=0$ and ${\widehat{\eta}}_{3}=3$ (blue dashed rectangles). Interbuilding scheme $\tilde{\sigma}=[{\tilde{\sigma}}^{\left(1\right)},{\tilde{\sigma}}^{\left(2\right)}]$ with ${\tilde{\eta}}_{1}=2$ and ${\tilde{\eta}}_{2}=2$ (red dotted rectangles).

**Figure 5.**Damping configuration DC1. Full-linked distributed damping system (DDS) with three actuated buildings, $\widehat{n}=8$ interstory dampers and $\tilde{n}=4$ interbuilding dampers.

**Figure 6.**Damping configuration DC2. Full-linked DDS with two actuated buildings, $\widehat{n}=7$ interstory dampers and $\tilde{n}=5$ interbuilding dampers.

**Figure 7.**Damping configuration DC3. Full-linked DDS with a single actuated building, $\widehat{n}=4$ interstory dampers and $\tilde{n}=8$ interbuilding dampers.

**Figure 8.**Full-scale 180-component of El Centro 1940 ground-acceleration seismic record with an absolute acceleration-peak of 3.417 m/s${}^{2}$. Data available at Strong-Motion Virtual Data Center (VDC) (ftp://strongmotioncenter.org/vdc/smdb/1940/c/139u37el.c0a).

**Figure 9.**Maximum absolute interstory drifts corresponding to the nonactuated multibuilding system (plain configuration) and the damping configurations DC1, DC2 and DC3.

**Figure 10.**Maximum absolute story total-accelerations attained by the nonactuated multibuilding system (plain configuration) and the damping configurations DC1, DC2 and DC3.

**Figure 11.**Maximum interbuilding approachings produced by the nonactuated multibuilding system (plain configuration) and the damping configurations DC1, DC2 and DC3.

**Figure 12.**Fully unlinked damping configuration DC4 obtained by suppressing the interbuilding dampers in the linked configuration DC2.

**Figure 13.**Maximum absolute interstory drifts corresponding to the nonactuated multibuilding system (plain configuration), the linked configuration DC2 and the unlinked configuration DC4.

**Figure 14.**Maximum absolute story total-accelerations attained by the nonactuated multibuilding system (plain configuration), the linked configuration DC2 and the unlinked configuration DC4.

**Figure 15.**Maximum interbuilding approachings produced by the nonactuated multibuilding system (plain configuration), the linked configuration DC2 and the unlinked configuration DC4.

**Table 1.**Values of the damping coefficients corresponding to the linked damping configurations DC1, DC2 and DC3 ($\times {10}^{7}$ Ns/m).

Conf. | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{4}$ | ${\mathit{d}}_{5}$ | ${\mathit{d}}_{6}$ | ${\mathit{d}}_{7}$ | ${\mathit{d}}_{8}$ | ${\mathit{d}}_{9}$ | ${\mathit{d}}_{10}$ | ${\mathit{d}}_{11}$ | ${\mathit{d}}_{12}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

DC1 | 1.4480 | 1.5137 | 1.3414 | 2.0098 | 2.1014 | 1.9725 | 1.5460 | 1.3074 | 0.4334 | 0.3683 | 0.8023 | 0.1528 |

DC2 | 1.8626 | 1.9882 | 1.7986 | 1.6611 | 2.1895 | 1.9489 | 1.6484 | 0.6311 | 0.4362 | 0.1519 | 0.0006 | 0.6828 |

DC3 | 2.6792 | 2.7151 | 2.5843 | 2.1407 | 0.2790 | 0.9873 | 0.7835 | 0.0009 | 1.3963 | 0.9310 | 0.0007 | 0.5018 |

Conf. | Act. Build. | ${\mathit{H}}_{\mathit{\infty}}$ Norm | opt. vars. | Generations | funct. aval. | Time (s) | Parallel Time (s) |
---|---|---|---|---|---|---|---|

Plain conf. | — | 0.8090 | — | — | — | — | — |

DC1 | 1, 3, 5 | 0.0897 | 47 | 206 | 41,400 | 273.5 | 167.9 |

DC2 | 2, 4 | 0.0970 | 34 | 161 | 32,400 | 242.8 | 157.5 |

DC3 | 1 | 0.1457 | 37 | 288 | 57,800 | 409.0 | 103.8 |

**Table 3.**${H}_{\infty}$ norm and overall maximum peak-values corresponding to the nonactuated multibuilding system (plain configuration), the linked configurations DC1, DC2 and DC3, and the unlinked configuration DC4.

Conf. | Act. Build. | ${\mathit{H}}_{\mathit{\infty}}$ Norm | Max. drift (cm) | Max. accel. (m/s${}^{2}$) | Max. Approach. (cm) |
---|---|---|---|---|---|

Plain | — | 0.8090 | 5.38 | 9.62 | 20.00 |

DC1 | 1, 3, 5 | 0.0897 | 2.56 | 5.71 | 22.70 |

DC2 | 2, 4 | 0.0970 | 2.76 | 5.62 | 23.15 |

DC3 | 1 | 0.1457 | 3.25 | 7.02 | 21.92 |

DC4 | 2, 4 | 0.6272 | 5.38 | 9.62 | 20.05 |

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**MDPI and ACS Style**

Palacios-Quiñonero, F.; Rubió-Massegú, J.; Rossell, J.M.; Karimi, H.R.
Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems. *Appl. Sci.* **2020**, *10*, 2383.
https://doi.org/10.3390/app10072383

**AMA Style**

Palacios-Quiñonero F, Rubió-Massegú J, Rossell JM, Karimi HR.
Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems. *Applied Sciences*. 2020; 10(7):2383.
https://doi.org/10.3390/app10072383

**Chicago/Turabian Style**

Palacios-Quiñonero, Francisco, Josep Rubió-Massegú, Josep M. Rossell, and Hamid Reza Karimi.
2020. "Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems" *Applied Sciences* 10, no. 7: 2383.
https://doi.org/10.3390/app10072383