# Structural Nonlinear Damage Identification Method Based on the Kullback–Leibler Distance of Time Domain Model Residuals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic theory of AR model

#### 2.1. AR Model

_{t}is the time series data at time t, y

_{t-I}is the time series data at time t-i, c is a constant, p is the order, ${\phi}_{i}$ is the i-th parameters of the AR model, and ${\epsilon}_{t}$ is the residual of the AR model.

#### 2.2. Order Determination of AR Model

#### 2.3. Parameter Estimation of AR Model

## 3. Damage Identification Based on the KL Distance of Time Series Model Residual

#### 3.1. Nonlinear Damage in Time Domain

_{i}is the stiffness when a breathing crack was closed, and $\alpha $ is the damaged factor when a breathing crack was opened. x

_{i}(t) and x

_{i}(t)(i ≠ 1) represent the displacement of i-th DOF and (i − 1)-th DOF at time t, respectively. In particular, when i = 1, x

_{i−1}(t) = x

_{0}(t) = 0.

#### 3.2. Nonlinear Damage Identification Based on the SOVI

#### 3.3. Nonlinear Damage Identification Method Based on the KL Distance of the AR Model Residual

_{t}}

_{ref}, and the test time series is denoted as {y

_{t}}

_{test}. The AR model established by {y

_{t}}

_{ref}is denoted as {AR}

_{ref}, and the residual of the AR model is denoted as ${\epsilon}_{t}^{ref}$. The AR model established by {y

_{t}}

_{test}is denoted as {AR}

_{test}, and the residual of the AR model is denoted as ${\epsilon}_{t}^{test}$. Therefore, a new time series {y

_{t}}

_{RT}can be established by substituting {y

_{t}}

_{test}into {AR}

_{ref}model, and the residual of the model is denoted as ${\epsilon}_{t}^{RT}$. The information feature difference between the baseline time series {y

_{t}}

_{ref}and the test time series {y

_{t}}

_{test}can be determined by calculating the KL distance:

_{ref}model and ${\left\{{y}_{t}\right\}}_{test}$, ${\epsilon}_{t}^{test}$ is the model residual calculated from {AR}

_{test}model and ${\left\{{y}_{t}\right\}}_{test}$, ${\epsilon}_{t}^{ref}$ is the model residual calculated from {AR}

_{ref}model and ${\left\{{y}_{t}\right\}}_{ref}$, and Var{} is the variance calculation.

#### 3.4. Process of Damage Identification Using the KL Distance of the AR Model Residual

## 4. Numerical Example

#### 4.1. Simulation of Eight-Story Shear Building Model

_{t}} is the acceleration response with measurement noise, $\left\{{{y}^{\prime}}_{t}\right\}$ is the acceleration response without measurement noise, H is the level of measurement noise, and $\left\{{\omega}_{t}\right\}$ is a random amount of white noise at the same level as the magnitude of $\left\{{{y}^{\prime}}_{t}\right\}$.

#### 4.2. Establishment of AR Model

#### 4.3. Nonlinear Damage Identification Results

## 5. Experimental Study on the Stand Structure Model

#### 5.1. Introduction of the Stand Structure Model Experiment

#### 5.2. Modeling Analysis of AR Model

#### 5.3. Damage Identification Results of the Stand Structure Model Experiment

## 6. Discussion

## 7. Conclusions

- (1)
- The proposed method is a damage identification method based on the substructure. The damage indicator of the damaged story is larger than the other stories, which can determine the structural damage location accurately.
- (2)
- The proposed method can accurately distinguish nonlinear damage of a multi-degree-of-freedom shear structure caused by bilinear stiffness changes. This method is robust enough to analyze environmental noise and small damage.
- (3)
- The proposed method can effectively find nonlinear damage in a multi-story and multi-span complex structure caused by bolt looseness, which is beneficial in practical applications.
- (4)
- In this paper, only a multi-degrees-of-freedom shear structure and a stand structure were used to verify the proposed method, and the nonlinear damage identification problem of more structural types should be considered in subsequent research.
- (5)
- This paper only considers the damage identification results of white noise conducted from the ground to the structure, and the damage identification results of excitation at different locations should be considered in subsequent research.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Damage Scenarios | Damage Story | Damage Level | Excitation Amplitude/(kn) |
---|---|---|---|

1 | 1 | 10% | 250 |

2 | 2 | 10% | 250 |

3 | 3 | 10% | 250 |

4 | 4 | 10% | 250 |

5 | 5 | 10% | 250 |

6 | 6 | 10% | 250 |

7 | 7 | 10% | 250 |

8 | 8 | 10% | 250 |

9 | 1 | 30% | 250 |

10 | 2 | 30% | 250 |

11 | 3 | 30% | 250 |

12 | 4 | 30% | 250 |

13 | 5 | 30% | 250 |

14 | 6 | 30% | 250 |

15 | 7 | 30% | 250 |

16 | 8 | 30% | 250 |

17 | 1 | 30% | 100 |

18 | 2 | 30% | 100 |

19 | 3 | 30% | 100 |

20 | 4 | 30% | 100 |

21 | 5 | 30% | 100 |

22 | 6 | 30% | 100 |

23 | 7 | 30% | 100 |

24 | 8 | 30% | 100 |

Damage Scenario | Damaged Region | Bolt-Loosened Braces Number |
---|---|---|

1 | 1 | B2 |

2 | 1 | B1, B2, B3 |

3 | 2 | B5 |

4 | 2 | B4, B5, B6 |

5 | 3 | B8 |

6 | 3 | B7, B8, B9 |

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**MDPI and ACS Style**

Zuo, H.; Guo, H.
Structural Nonlinear Damage Identification Method Based on the Kullback–Leibler Distance of Time Domain Model Residuals. *Remote Sens.* **2023**, *15*, 1135.
https://doi.org/10.3390/rs15041135

**AMA Style**

Zuo H, Guo H.
Structural Nonlinear Damage Identification Method Based on the Kullback–Leibler Distance of Time Domain Model Residuals. *Remote Sensing*. 2023; 15(4):1135.
https://doi.org/10.3390/rs15041135

**Chicago/Turabian Style**

Zuo, Heng, and Huiyong Guo.
2023. "Structural Nonlinear Damage Identification Method Based on the Kullback–Leibler Distance of Time Domain Model Residuals" *Remote Sensing* 15, no. 4: 1135.
https://doi.org/10.3390/rs15041135