# Dynamic Characteristic and Parameter Analysis of a Modular Building with Suspended Floors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Configuration of Module Building with Suspended Floors

#### 2.1. Configuration of Connection Nodes and Modules

#### 2.2. Structural Calculation Diagram

_{i}and m

_{si}are the concentrated masses of the main structure and substructure of the ith story, respectively. K

_{i}, k

_{si}, and k

_{vi}are the stiffness of the ith story’s main structure, the sum of the stiffness of the springs connecting the ith story’s suspended floors and main structure and the equivalent stiffness of the horizontal component of the force of the ith story’s sling, respectively. C

_{i}and c

_{si}are the damping coefficient of the material of the ith story’s main structure and the sum of the damping coefficient of dampers connecting the ith story’s substructure and main structure, respectively. l

_{i}is the length of the ith story’s sling.

## 3. Motion Equation

_{si}and v

_{mi}represent the displacement response of the substructure and the main structure of the ith story, respectively.

#### 3.1. Structure with the Swing-Suspended Floors

_{S}

_{,i}, f

_{D}

_{,I}and f

_{I}

_{,i}are the spring force, damping force and inertia force of ith story’s main structure, respectively. f

_{S}

_{,i,s}, f

_{D}

_{,i,s}and f

_{I}

_{,i,s}are the spring force, damping force and inertia force of ith story’s suspended floor, respectively. k

_{vi}can be obtained from the simple pendulum model and be written as:

_{di}is the difference between the displacement of the ith story substructure and of the jth story main structure and can be written as (·)

_{di}=(·)

_{si}−(·)

_{i}

_{−1}. The ith story’s suspended floor is connected to the jth story’s main structure by springs and dampers, and connected to the ith story’s main structure by slings. Therefore, its stiffness matrix will have an additional coupling phenomenon compared with the ordinary FIS system. A defined matrix (3 × 3), [k]

_{I}, representing the stiffness matrix of the ith suspended floor is given by:

_{I}, can be obtained similarly:

_{c}]

_{i}is defined as the stiffness of a module with suspended floor, which represents the connecting process of the ith suspended floor and main structure and is written as:

_{j}is related to the jth story and ith story structure vibration at the same time, only the processing of the coefficients before v

_{j}is considered in matrix concatenation. v

_{j}in Equation (3) is extracted from the dynamic balance equation of the ith story main structure. Its coefficient is the sum of the elements of row 3 and column 3 of [K

_{c}]

_{j}and the elements of row 1 and column 1 of [K

_{c}]

_{i}. This paper uses the symbol “

^{~}” to define such matrix concatenation operations,

_{a}]

_{j}

_{,i}is defined to represent the stiffness matrix of the assembled structure formed by connecting the jth and ith story:

_{a}]

_{j}

_{,i}can be written as:

_{I}and [c]

_{i}represent the damping matrix of the main and substructure of the ith story. [C

_{c}]

_{i}represents the damping matrix of the structure formed by connecting the substructure and the main structure of ith story, and [C] represents the damping matrix of the total structure formed by connecting all stories. These matrices can be written as:

_{c}]

_{i}represents the mass matrix of the ith story which combines the substructure and main structure, and [M] represents the mass matrix of the total structure. These matrices can be written as:

_{c}]

_{1}and [C

_{c}]

_{1}, representing the stiffness and damping matrices of the underlying structure, are given by:

#### 3.2. Structure with the Locked, Suspended Floors

## 4. Effect of Increasing Modules

_{r}represents the lateral stiffness provided by one column in the module and the four columns in one module provide the same lateral stiffness. Finite element analysis can be used to determine K

_{x}

^{*}and K

_{y}

^{*}, which stand for the lateral stiffness of a single module along the x and y directions, respectively.

_{xr}and k

_{yr}, provided by each module column along the x and y directions can be written as:

_{r}′. It is concluded that there is always a linear relationship between the lateral stiffness of the composed modular column, k

_{r}′, and the lateral stiffness of a single modular column, k

_{r}.

_{x}

^{*}= 4447 N/mm and K

_{y}

^{*}= 4612 N/mm were obtained according to the finite element analysis. Values of k

_{xr}= 1112 N/mm and k

_{yr}= 1153 N/mm were obtained according to Equations (26) and (27). The horizontal assembly scheme 3 is shown in Figure 3c, and its finite element modeling is shown in Figure 4c,d. It is concluded that K

_{x}

^{*}= 8771 N/mm and K

_{y}

^{*}= 13,699 N/mm.

_{xr}′ = 1080 N/mm and k

_{yr}′ = 2271.75 N/mm were obtained according to Equations (28) and (29), that is, k

_{xr}′ = 1.0 k

_{xr}and k

_{yr}′ = 2.0 k

_{yr}.

_{x}

_{1}′ = 2.0 k

_{x}

_{1}and k

_{x}

_{2}′ = 2.0 k

_{x}

_{2}. Thus, columns 1 and 2 can be called the stiffness enhancement columns along the x direction.

_{x}

_{3}′ = 1.0 k

_{x}

_{3}. Therefore, the total lateral stiffness provided by columns 1, 2 and 3 in the x direction is k

_{x1}′ + k

_{x}

_{2}′ + k

_{x}

_{3}′ = 2.0 k

_{x}

_{1}+ 2.0 k

_{x}

_{2}+ 1.0 k

_{x}

_{3}. Similarly, the total lateral stiffness provided by the three columns in the y direction is k

_{y}

_{1}′ + k

_{y}

_{2}′ + k

_{y}

_{3}′ = 2.0 k

_{x}

_{1}+ 1.0 k

_{x}

_{2}+ 2.0 k

_{x}

_{3}. According to the rule and the certain assembly scheme, the lateral stiffness, K, of the story can be calculated through Equations (30) and (31), as long as the number of stiffness enhancement columns along a certain direction is calculated:

_{r}is taken as k

_{0}, r = 1, 2, …, q. Then, Equations (30) and (31) can be simplified to:

## 5. Optimal Frequency and Damping Ratios

#### 5.1. Structural Parameters and Objective Functions

_{T}, are two design parameters that can be used to frame the optimization problem [16]. The parameters of the dynamic characteristics of the modular building can be written as:

^{opt}is the optimal natural frequency of the substructure. ω

_{1}is the frequency of the first mode of the main structure. m

_{si}is the mass of the ith story substructure. k

_{si}

^{opt}and c

_{si}

^{opt}are the optimal spring stiffness and damper damping coefficient of the ith story substructure, respectively. The same spring and damper are utilized in each module since the modular building demands that the equipment requirements be as uniform as possible. Therefore, only a set of optimal ν

^{opt}and ζ

_{T}

^{opt}are required for each certain condition. Several researchers have reported ν

^{opt}= 0.7 and ζ

_{T}

^{opt}= 0.4 for a concrete structure [16]. Thus, ν

^{opt}= 0.7 and ζ

_{T}

^{opt}= 0.4 were used as the midpoints of their respective discussions in this paper.

_{m}and v

_{s}, of the main structure and the substructure, as well as their variances, ${\sigma}_{m}^{2}$ and ${\sigma}_{s}^{2}$. However, the displacement response variance is related to the power, S

_{0}, of white noise excitation. This paper uses the dimensionless displacement variance indexes, R

_{m}and R

_{s}, as the objective functions of structural optimization [24].

_{m}and G

_{s}are the displacement response transfer functions of the main structure and the secondary structure, respectively. Dimensionless variables λ = ω/ω

_{m}and υ = ω

_{s}/ω

_{m}are introduced so that Equations (36) and (37) can be further written as:

_{0}, so the functions F

_{m}and F

_{s}are not derived in detail. The specific derivation process can be found in the literature [24].

_{m}and R

_{s}, which are independent of S

_{0}, are obtained:

_{m}is the natural frequency of the main structure. R

_{m}and R

_{s}are the non-dimensional displacement variances of the standardized main structure and substructure according to the different white noise excitation intensities and the natural frequency of the main structure.

#### 5.2. Optimal Parameters

_{i}

^{*}= 9168.35 kg, i = 1, 2,…, n−1, and M

_{n}

^{*}= 18,005.50 kg. The mass of the ith story substructure of a single module is taken as m

_{si}

^{*}= 8837.15 kg, i = 1, 2,…, n. The lateral stiffness of the ith story’s main structure of a single module is taken as K

_{i}

^{*}= 4447 kN/m. The damping ratio of the steel structure is taken as ξ = 0.02. The damping coefficient of the ith story main structure C

_{i}can be written as:

_{T}∈ [0.0, 0.8]. The objective functions are Equations (40) and (41).

_{s}

_{1}and that of the main structure R

_{1}are given in Figure 8.

_{T}= 0, the response of the substructures is obviously larger than those of the substructures equipped with dampers. When the damping ratio was taken as ζ

_{T}> 0.1, the boundary benefit of the response of the substructures decreased gradually. When the damping ratio was taken as ζ

_{T}= 0.5, the response of the substructures was smallest. When the modular building had only one story, the response of the main structure did not change with the tuning ratio and damping ratio. The dynamic response of a single-story building was not significantly different when modular buildings had different plane layouts and assembly schemes. A tuning ratio of ν = 0.1 and a damping ratio of ζ

_{T}= 0.5 are recommended. When the number of modules was twenty with a damping ratio of ζ

_{T}= 0.5, the sum of the damping coefficients of all the dampers was 0.7 × 10

^{6}N·s/m. The damping coefficients of dampers commonly used in TMD are 0.5~4.0 × 10

^{6}N·s/m, which can meet the requirements.

_{T}> 0.5, there was no significant change.

_{T}= 0.5 for building a suspended floor module.

_{si}and that of the main structure R

_{i}were investigated and are shown in Figure 9 and Figure 10, respectively.

_{T}= 0. The boundary advantage of the substructure response steadily declined when the damping ratio was ζ

_{T}> 0.1. When the damping ratio was ζ

_{T}= 0.5, the substructures’ responses were at their smallest. The main structural responses were likewise substantial, with a damping ratio of ζ

_{T}= 0. When the tuning ratio was v > 0.7, the responses of the main structure were not noticeably affected with increasing v. The response of the main structures was the least when ν = 0.8~1.0 and ζ

_{T}= 0.5, respectively.

_{T}of 0.5 is still recommended.

## 6. Conclusions and Discussions

_{T}.

- The simplified lateral stiffness calculation method has good calculation accuracy, and the lateral stiffness of modular buildings increases linearly with the increase in the number of modules;
- Modular buildings with suspended floors are recommended to have different tuning frequency ratios depending on the vibration control objects;
- To control the vibration of the substructure, the tuning ratio is recommended to be ν = 0.1~0.2. To minimize the response of the main structure, a ν = 0.8~1.0 is recommended;
- It is not necessary for the dampers’ initial stiffness to be very high for a modular structure with suspended floors. As a result, it is advised to utilize magnetorheological or viscous dampers of the velocity-dependent type, with a damping ratio of ζ
_{T}= 0.5; - However, there are some limitations to this study: (a) Although FIS vibration technology is used in this study, it is not modularized with the building, and only considers the case that the tuning frequency of each floor is the same. (b) The emphasis of this paper is only on the analysis of the structure. To make this system practical, a scheme concerning acoustic insulation, thermal insulation and other properties related to the building function needs to be researched. (c) This paper puts forward the concept of a modular building introduced by FIS, which is poorly considered in terms of economic benefits and practicability. (d) This paper only discusses the response of the structure under the excitation of white noise and needs to include cases experiencing small, medium and large earthquakes.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Finite element analysis: (

**a**) calculation diagram and model of (

**b**) scheme 1, (

**c**) scheme 3 and (

**d**) connecting plate.

**Figure 8.**Responses of (

**a**,

**c**,

**e**,

**g**) the substructures and (

**b**,

**d**,

**f**,

**h**) the main structures formed by 3, 6, 10, and 20 modules.

**Figure 9.**Responses of the (

**a**,

**c**,

**e**) 1st, 2nd and 3rd story of the substructure and (

**b**,

**d**,

**f**) main structure of a three-story building.

**Figure 10.**Responses of the (

**a**,

**c**,

**e**,

**g**,

**i**) 1st, 2nd, 3rd, 4th and 5th story of the substructure and (

**b**,

**d**,

**f**,

**h**,

**j**) main structure of a five-story building.

**Table 1.**Comparison between lateral stiffness of the story calculated by finite element analysis and Equation (32).

Schemes | Directions | Number of the Stiffness Enhancement Columns | Lateral Stiffness (N/mm) | |
---|---|---|---|---|

Finite Element Method | Equation (32) | |||

Scheme 1 | x | 0 | 4447 | 4447 |

y | 0 | 4612 | 4612 | |

Scheme 2 | x | 4 | 13,409 | 13,341 |

y | 0 | 9098 | 9224 | |

Scheme 3 | x | 0 | 8771 | 8894 |

y | 4 | 13,699 | 13,836 | |

Scheme 4 | x | 0 | 13,097 | 13,341 |

y | 8 | 22,716 | 23,060 | |

Scheme 5 | x | 4 | 18,400 | 17,788 |

y | 4 | 18,847 | 18,448 | |

Scheme 6 | x | 4 | 17,880 | 17,788 |

y | 4 | 18,058 | 18,448 | |

Scheme 7 | x | 8 | 22,809 | 22,235 |

y | 0 | 13,587 | 13,836 |

**Table 2.**Comparison between the recommended values of this study and the optimal values reported in the literature [16].

Tuning Frequency Ratio, Ν | Tuning Damping Ratio, Ζ_{t} | |||
---|---|---|---|---|

For the Main Structure | For the Substructure | For the Main Structure | For the Substructure | |

This study | 0.8 to 1.0 | 0.1 to 0.2 | 0.5 | 0.5 |

The literature [16] | 0.7 | — | 0.4 | — |

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## Share and Cite

**MDPI and ACS Style**

He, Q.; Zhang, S.; Shang, J.
Dynamic Characteristic and Parameter Analysis of a Modular Building with Suspended Floors. *Buildings* **2023**, *13*, 7.
https://doi.org/10.3390/buildings13010007

**AMA Style**

He Q, Zhang S, Shang J.
Dynamic Characteristic and Parameter Analysis of a Modular Building with Suspended Floors. *Buildings*. 2023; 13(1):7.
https://doi.org/10.3390/buildings13010007

**Chicago/Turabian Style**

He, Qingguang, Shiquan Zhang, and Jiying Shang.
2023. "Dynamic Characteristic and Parameter Analysis of a Modular Building with Suspended Floors" *Buildings* 13, no. 1: 7.
https://doi.org/10.3390/buildings13010007