Collisions between a moving mass and an anti-collision device increase structural responses and threaten structural safety. An active mass damper (AMD) with stroke limitations is often used to avoid collisions. However, a stroke-limited AMD control system with a fixed limited area shortens the available AMD stroke and leads to significant control power. To solve this problem, the design approach with variable gain and limited area (VGLA) is proposed in this study. First, the boundary of variable-limited areas is calculated based on the real-time status of the moving mass. The variable gain (VG) expression at the variable limited area is deduced by considering the saturation of AMD stroke. Then, numerical simulations of a stroke-limited AMD control system with VGLA are conducted on a high-rise building structure. These numerical simulations show that the proposed approach has superior stroke-limitation performance compared with a stroke-limited AMD control system with a fixed limited area. Finally, the proposed approach is validated through experiments on a four-story steel frame.

Tuned mass dampers (TMDs) [

A vibration control system should exhibit robust performance. For continuous systems, a robust H_{∞} control method based on linear matrix inequalities (LIM) was proposed [_{∞} and H_{2} control, the active control system of a single-story frame structure is established by considering the interaction between the irregularly controlled structure and foundation. The simulation results illustrate that the control system exhibits robust performance [

To reduce the influence of stroke limitations on the performance of AMD control systems, a stroke-limited AMD control system with variable gain and limited area (VGLA) was proposed in this study. The running states of the VGLA control system are divided into unlimited and variable-limited areas, and the control gain is calculated using an algorithm with H_{∞}/H_{2} and pole assignment constraints. The deceleration mechanism of the moving mass is determined based on the control force and maximum stroke. Then, based on the real-time states of the moving mass, the gain expression in the variable limited area was deduced considering the saturation of the AMD stroke. Furthermore, numerical simulations were conducted using a stroke-limited AMD control system. Finally, the effectiveness of the proposed methodology was verified by conducting an experiment with a four-story steel frame structure.

The main objective of structural control is to reduce the acceleration response of buildings to a comfortable level. An AMD device is commonly installed on the structure, as shown in

The running states of AMD are divided into eight states, as shown in

The force equilibrium of an AMD model (

_{s} and _{w} represent the position matrices of the control forces and excitations, respectively; and

Matrices ∆_{s}, and _{w} are described as follows:

Matrices

Assuming that the state vectors are

Matrices _{1}, _{2}, _{11}, _{12}, _{21}, and _{22} are described as follows:

Based on reference [

_{i}

It is assumed that the matrix of uncertain parameters of structural parameters is

Substituting

Based on reference [_{2} control law with state feedback can be expressed as follows:

Based on reference [_{∞} control law with state feedback can be expressed as follows:

Based on reference [

This problem can be solved by finding a common Lyapunov matrix, denoted as X, that satisfies a set of linear matrix inequalities

To solve this multi-objective control problem, the following convex optimization problem is required:

The state feedback controller design method has H_{2}/H_{∞} performance requirements and closed-loop region pole constraints, which are solved using LMI techniques. Let

The gain of the AMD control system is expressed as follows:

_{a}, _{n}, and _{n+1} denote the gains based on the displacement of the floor in which the AMD is installed, the top floor, and AMD, respectively.

When AMD runs into a variable-limited area, as shown in _{a} denotes the displacement of the floor where the AMD is installed and _{n+1} denotes the displacement of the AMD.

Assuming that _{a} = −_{n+1}, then

Based on

From _{n+1} > 0, the control force at the variable-limited area has the opposite sign to _{n+1} > 0.

Based on

When the AMD runs into the variable limited area (the shadow area shown in _{d}. If AMD is not in a variable-limited area, the gain of the AMD control system is calculated using the regional pole-assignment algorithm.

In the variable-limited area, the velocity of the moving mass decreases. When the stroke of the moving mass reaches

_{max} denote the stroke of the moving mass, the maximum allowable stroke, and the control force of the AMD control system, respectively. _{d} denotes the real-time velocity of the moving mass at the boundaries of a variable-limited area. The accelerations can be determined by differentiating the velocity of the moving mass as follows:

The expected control forces are

Substituting

The high-rise building named KingKey Financial Center in Shenzhen, shown in

Vibration mode | Periods (s) | Frequencies (Hz) |
---|---|---|

1 | 7.15 | 0.14 |

2 | 1.95 | 0.51 |

3 | 0.95 | 1.05 |

4 | 0.64 | 1.55 |

Index | AMD |
---|---|

The auxiliary mass (t) | 250 × 2 |

The effective stroke (m) | ±2 |

The peak power (kW) | 480 × 2 |

The maximum driving force (kN) | 500 × 2 |

The Simulink block diagram shown in

To verify and compare the control performance of the VGLA, the method of stroke-limited AMD control system with VG in reference [

Index | Place- |
Without stroke limit | VG | VGLA | |||
---|---|---|---|---|---|---|---|

Displace- |
Accelera- |
Displace- |
Accelera- |
Displace- |
Accelera- |
||

Control effect (%) | The 87^{th} floor |
27.43 | 28.89 | 27.25 | 28.73 | 27.29 | 28.80 |

The 92^{th} floor |
27.45 | 28.22 | 27.28 | 27.96 | 27.31 | 27.96 | |

The 98^{th} floor |
27.47 | 23.52 | 27.30 | 23.25 | 27.33 | 23.31 | |

Peak control forces (kN) | 800.50 | 1000.00 | 873.93 | ||||

Mean-square control forces (kN) | 139.11 | 148.11 | 143.39 | ||||

Peak strokes (m) | 1.84 | 1.73 | 1.65 | ||||

Mean-square strokes (m) | 35.86 | 33.39 | 33.35 | ||||

Peak power (kW) | 625.93 | 1459.23 | 915.45 | ||||

Mean-square power (kW) | 81.06 | 98.10 | 81.94 |

Furthermore, the peak stroke of the VG is 1.73 m, and the peak stroke of VGLA is 1.65 m. The peak stroke of VG is 5.98% lower than that of the AMD system without stroke limitations. The peak stroke of VGLA is 10.33% lower than that of AMD without stroke limitation. However, the peak control force and power of the VGLA are 12.6% and 37.27% lower than those of the VG, respectively. The mean-square control force and power of the VGLA are 3.20% and 16.47% lower than those of the VG, respectively. Therefore, the control performance of VGLA is significantly better than that of VG.

As shown in

The weak axial displacement and acceleration of the four-floor steel frame structure can be measured by applying the measurement system. The force balance acceleration sensor is used in the system, and the micro-epsilon series laser displacement sensor is used in the system. The structural mass matrix can be obtained using the concentrated mass method. That is, the mass of each floor of the structure is obtained by calculating the geometric size, density, and other parameters of the controlled structure as well as the related parameters of the equipment; the mass matrix is finally combined. The structural stiffness matrix is obtained by utilizing FEM software. The aforementioned components of the AMD control system are shown in

Vibration mode | Periods (s) | Frequencies (Hz) |
---|---|---|

1 | 0.856 | 0.85 |

2 | 0.090 | 2.91 |

3 | 0.034 | 5.40 |

4 | 0.020 | 7.99 |

Index | Without stroke limit | VG | VGLA | ||||
---|---|---|---|---|---|---|---|

Displacement | Acceleration | Displacement | Acceleration | Displacement | Acceleration | ||

The second floor | 44.73 | 77.21 | 32.53 | 79.06 | 44.73 | 76.77 | |

Control effect (%) | The third floor | 28.68 | 59.01 | 33.56 | 66.26 | 28.68 | 66.01 |

The fourth floor | 27.09 | 55.69 | 34.25 | 73.36 | 34.06 | 74.93 | |

Peak control forces (N) | 26.51 | 32.93 | 25.34 | ||||

Mean-square control forces (N) | 13.32 | 13.86 | 13.10 | ||||

Peak power (W) | 2.48 | 7.01 | 5.32 | ||||

Mean-square power (W) | 0.33 | 0.77 | 0.69 | ||||

Peak strokes (cm) | 46.25 | 29.51 | 23.69 | ||||

Mean-square strokes (cm) | 9.19 | 5.40 | 5.23 |

The peak stroke of VGLA is 19.72% less than that of VG, but the peak control force and power of VGLA are 23.05% and 24.11% lower than those of VG, respectively. The mean-square stroke of VGLA is 3.15% less than that of VG. However, the mean-square control power of VGLA is 10.40% lower than that of VG. The experimental results agree with the numerical simulation results (

In order to limit the stroke of the moving mass in the AMD control system, the relation between the stroke and the corresponding feedback gain was established, and the design method of a stroke-limitation system with variable limited area and gain was proposed. Finally, the numerical model of an actual building structure and the experimental model of the four-story frame were taken to verify the necessity and effectiveness of the stroke-limitation system with VGLA. The main conclusions are drawn as follows:

(1) The stroke-limited AMD control system of the VGLA exhibits a good control effect. In a situation where no collision of the control system without stroke limitation occurred, the control effect of the VGLA is approximately equal to that of the control system without stroke limitation. In a situation where a collision of the control system without stroke limitation occurred, the control effects of the VGLA are better than those of the control system without stroke limitation.

(2) The maximum force and stroke of the AMD are considered in the stroke-limitation principle, which can limit the control force and power of the VGLA to permissible ranges.

(3) Considering the variable limited area, the stroke limit effect of VGLA is better than that of VG; however, the control force and power of VGLA are lower than those of VG.

(4) The numerical simulation results agree with the experimental results. The stroke-limitation system with VGLA proposed in this study can effectively limit the stroke to a reasonable range, thus ensuring the control effects and safety of the AMD system.

Based on the variable gain feedback control method, the VGLA method proposed in this paper achieves a stroke limitation control method for building structures with variable-limit areas. The application results show that the proposed method has a short stroke and requires low control power, indicating its potential for effectively controlling structural vibrations in high-rise buildings.

The authors would like to thank Houbing Xing for advice on experimental design.

This research was founded by the Funds for Creative Research Groups of National Natural Science Foundation of China (Grant No. 51921006), the National Natural Science Foundations of China (Grant No. 51978224), the National Major Scientific Research Instrument Development Program of China (Grant No. 51827811), the National Natural Science Foundation of China, (Grant No. 52008141) and the Shenzhen Technology Innovation Program (Grant Nos. JCYJ20170811160003571, JCYJ20180508152238111 and JCYJ20200109112803851).

Zuohua Li and Qinggui Wu contributed equally to this work. Qinggui Wu wrote the paper and summarized the results. Zuohua Li and Jun Teng participated in the data analysis and conceived the study. Chaojun Chen reviewed the study plan.

All data generated and analyzed during this study are included in this published article.

The authors declare that they have no conflicts of interest to report regarding the present study.

_{∞}state feedback control for linear systems with state delay and parameter uncertainty

_{2}control design with regional pole constraints for damping power system oscillations

_{∞}design with pole placement constraints: An LMI approach