﻿ 基于时间序列模型的结构损伤识别方法
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (12): 1691-1700, 1755.  DOI: 10.11908/j.issn.0253-374x.2019.12.001 0

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ZHANG Yujian, LUO Yongfeng, GUO Xiaonong, LIU Jun, ZHU Zhaochen. Structural Damage Identification Method Based on Time Series Model[J]. Journal of Tongji University (Natural Science), 2019, 47(12): 1691-1700, 1755.   DOI: 10.11908/j.issn.0253-374x.2019.12.001

文章历史

Structural Damage Identification Method Based on Time Series Model
ZHANG Yujian , LUO Yongfeng , GUO Xiaonong , LIU Jun , ZHU Zhaochen
College of Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: Aiming at the problem of missing damage information caused by the unclear physical meaning of model parameters in the existing structural damage identification method, a damage identification method based on time series model is proposed. Firstly, the general expression of auto-regressive model with eXogenous input (ARX) is derived. The ARX model considering structural dynamic characteristics is established by the simultaneous multi-degree-of-freedom system motion equation. Subsequently, the model is used to predict the nodal acceleration under undamaged conditions, and the parameters of the structural damage, i.e. the damage factor, are constructed according to the difference between the measured data and the predicted data. Finally, the structural damage state is evaluated based on the magnitude and distribution of the damage factor. The numerical results show that the proposed method can perfectly identify single-position and multi-position damage under less measurement dates, and can make a more accurate judgment on the damage degree. At the same time, the influence of the excitation position and measurement noise is small.
Key words: damage identification    time series model    equation of motion    damage factor    measurement noise

1 ARX模型基本原理 1.1 ARX模型一般表达式

 \begin{aligned} y(t) &+a_{1} y(t-\Delta t)+\cdots+a_{n_{a}} y\left(t-n_{a} \Delta t\right)=\\ & b_{1} u(t)+b_{2} u(t-\Delta t)+\cdots+b_{n_{b}} u\left(t-n_{b} \Delta t\right)+\\ & e(t)+c_{1} e(t-\Delta t)+\cdots+c_{n_{c}} e\left(t-n_{c} \Delta t\right) \end{aligned} (1)

 $A(q) y(t)=B(q) u(t)+C(q) e(t)$ (2)

 $A(q)=1+a_{1} q^{-1}+\cdots+a_{n_{a}} q^{-n_{a}}$ (3)
 $B(q)=b_{1}+b_{2} q^{-1}+\cdots+b_{n_{b}} q^{-n_{b}}$ (4)
 $C(q)=1+c_{1} q^{-1}+c_{2} q^{-2}+\cdots+c_{n_{c}} q^{-n_{c}}$ (5)

 $x_{i j}(t) q^{k}=x_{i j}(t-k)$ (6)

 \begin{aligned} y(t) &+a_{1} y(t-\Delta t)+\cdots+a_{n_{a}} y\left(t-n_{a} \Delta t\right)=\\ & b_{1} u(t)+\cdots+b_{n_{b}} u\left(t-n_{b} \Delta t\right)+e(t) \end{aligned} (7)

 \begin{aligned} y(t)+a_{1} y(t-\Delta t)+\cdots+a_{n_{a}} y\left(t-n_{a} \Delta t\right) &=\\ b_{11} u_{1}(t)+\cdots+b_{1 n_{b 1}} u_{1}\left(t-n_{b 1} \Delta t\right)+& \\ b_{21} u_{2}(t)+\cdots+b_{2 n_{b 2}} u_{2}\left(t-n_{b 2} \Delta t\right)+e(t) \end{aligned} (8)

1.2 考虑结构动力特性的ARX模型

 $\boldsymbol{M} \ddot{\boldsymbol{x}}+\boldsymbol{C} \dot{\boldsymbol{x}}+\boldsymbol{K} \boldsymbol{x}=\boldsymbol{F}$ (9)

 $\sum\limits_{j=1}^{N} m_{i j} \ddot{x}_{j}(t)+\sum\limits_{j=1}^{N} c_{i j} \dot{x}_{j}(t)+\sum\limits_{j=1}^{N} k_{i j} x_{j}(t)=f_{i}(t)$ (10)

 $\sum\limits_{j=1}^{N} m_{i j} x_{j}^{(4)}(t)+\sum\limits_{j=1}^{N} c_{i j} x_{j}^{(3)}(t)+\sum\limits_{j=1}^{N} k_{i j} \ddot{x}_{j}(t)=\ddot{f}_{i}(t)$ (11)

 $\sum\limits_{j=1}^{N} m_{i j} x_{j}^{(4)}(t)+\sum\limits_{j=1}^{N} c_{i j} x_{j}^{(3)}(t)+\sum\limits_{j=1}^{N} k_{i j} \ddot{x}_{j}(t)=0$ (12)

 $x_{i}^{(3)}=\left(\ddot{x}_{i}(t+\Delta t)-\ddot{x}_{i}(t-\Delta t)\right) / 2 \Delta t$ (13)
 $x_{i}^{(4)}=\left(\ddot{x}_{i}(t+\Delta t)+\ddot{x}_{i}(t-\Delta t)-2 \ddot{x}_{i}(t)\right) /(\Delta t)^{2}$ (14)

 \begin{aligned} \sum\limits_{j=1}^{N}(&\left.\frac{m_{i j}}{(\Delta t)^{2}}+\frac{c_{i j}}{2 \Delta t}\right) \ddot{x}_{j}(t+\Delta t)=\\ & \sum\limits_{j=1}^{N}\left(\frac{2 m_{i j}}{(\Delta t)^{2}}-k_{i j}\right) \ddot{x}_{j}(t)+\\ & \sum\limits_{j=1}^{N}\left(\frac{c_{i j}}{2 \Delta t}-\frac{m_{i j}}{(\Delta t)^{2}}\right) \ddot{x}_{j}(t-\Delta t) \end{aligned} (15)

 $A_{1} \ddot{x}_{i}(t)+A_{2} \ddot{x}_{i}(t-\Delta t)+A_{3} \ddot{x}_{i}(t-2 \Delta t) =\\~~~ B_{1} \ddot{x}_{i-1}(t)+B_{2} \ddot{x}_{i-1}(t-\Delta t)+B_{3} \ddot{x}_{i-1}(t-2 \Delta t)+ \\~~~ C_{1} \ddot{x}_{i+1}(t)+C_{2} \ddot{x}_{i+1}(t-\Delta t)+C_{3} \ddot{x}_{i+1}(t-2 \Delta t)$ (16)

 $\left\{ {\begin{array}{*{20}{l}} {{A_1} = \frac{{{m_{ii}}}}{{{{(\Delta t)}^2}}} + \frac{{{c_{ii}}}}{{2\Delta t}}, {A_2} = {k_{ii}} - \frac{{2{m_{ij}}}}{{{{(\Delta t)}^2}}}, }\\ {{A_3} = \frac{{{m_{ii}}}}{{{{(\Delta t)}^2}}} - \frac{{{c_{ii}}}}{{2\Delta t}}}\\ {{B_1} = - \frac{{{m_{i, i - 1}}}}{{{{(\Delta t)}^2}}} - \frac{{{C_{i, i - 1}}}}{{2\Delta t}}, {B_2} = \frac{{2{m_{i, i - 1}}}}{{{{(\Delta t)}^2}}} - {k_{i, i - 1}}, }\\ {{B_3} = \frac{{{c_{i, i - 1}}}}{{2\Delta t}} - \frac{{{m_{i, i - 1}}}}{{{{(\Delta t)}^2}}}}\\ {{C_1} = - \frac{{{m_{i, i + 1}}}}{{{{(\Delta t)}^2}}} - \frac{{{c_{i, i + 1}}}}{{2\Delta t}}, {C_2} = \frac{{2{m_{i, i + 1}}}}{{{{(\Delta t)}^2}}} - {k_{i, i + 1}}, }\\ {{C_3} = \frac{{{c_{i, i + 1}}}}{{2\Delta t}} - \frac{{{m_{i, i + 1}}}}{{{{(\Delta t)}^2}}}} \end{array}} \right.$ (17)

2 基于ARX模型的损伤识别方法 2.1 损伤识别机理

 图 1 弹簧质点系统 Fig.1 Spring mass subsystem

 $\ddot{x}_{2}(t)+a_{1} \ddot{x}_{2}(t-\Delta t)+a_{2} \ddot{x}_{2}(t-2 \Delta t) =\\~~~ b_{11} \ddot{x}_{1}(t)+b_{12} \ddot{x}_{1}(t-\Delta t)+b_{13} \ddot{x}_{1}(t-2 \Delta t)+\\~~~ b_{21} \ddot{x}_{3}(t)+b_{22} \ddot{x}_{3}(t-\Delta t)+b_{23} \ddot{x}_{3}(t-2 \Delta t)+e(t)$ (18)

2.2 损伤特征因子

 $\xi=\frac{1}{100 \times\left(Z^{1 / p}\right)+1}$ (19)

 $Z=\frac{\sum\limits_{i=1}^{n}\left(\left|y_{i}-Y_{i}\right|^{p}\right)}{\sum\limits_{i=1}^{n}\left(\left|Y_{i}-\bar{Y}\right|^{p}\right)}$ (20)

2.3 损伤识别流程

(1) 针对损伤识别对象的结构形式，根据式(7)、(15)，确定所采用ARX模型的各多项式阶数；

(2) 通过有限元数值模拟，获取无损伤结构在某瞬时荷载作用后的自由振动加速度响应{Y1, Y2, …, Yi, …}；

(3) 根据{Y1, Y2, …, Yi, …}建立对应于各节点的ARX模型；

(4) 针对待测结构制定测量方案，布置加速度传感器，施加外界激励(如锤击)，获取结构各节点实测加速度时程数据{Y1, Y2, …, Yi, …}；

(5) 将Yi-1Yi+1作为输入，代入对应于节点i的ARX模型，得到未损伤结构中节点i的加速度响应yi

(6) 代入yiYi至式(19)、(20)，计算节点i的损伤因子DF；

(7) 改变i，按步骤(5)~步骤(6)重复计算，最终得到各节点的损伤因子DF；

(8) 根据DF数值对结构损伤状态进行评估.

3 数值算例

3.1 算例一 3.1.1 数值模型

 图 2 简支梁模型 Fig.2 Simple-supported beam model

 图 3 节点加速度时程数据 Fig.3 Acceleration time history data of nodes
3.1.2 损伤工况

3.1.3 测量噪声

 $a_{t}=a_{t, 0}+r_{\mathrm{SN}} \cdot R \cdot n / 100$ (21)

3.1.4 结果与分析

(1) 损伤位置识别

 图 4 单位置损伤识别结果(工况1~工况4，①号激振点，无噪声) Fig.4 Single-position damage identification results (condition 1~4, excitation point 1, without noise)
 图 5 多位置损伤识别结果(工况5~工况8，①号激振点，无噪声) Fig.5 Multi-position damage identification results (condition 5~8, excitation point 1, without noise)
 图 6 识别结果对比(工况6，①号激振点，无噪声) Fig.6 Comparison of identification result (condition 6, excitation point 1, without noise)

(2) 损伤程度识别

 图 7 不同损伤程度下的损伤识别结果(工况2、工况6，①号激振点，无噪声) Fig.7 Identification results of single-position damage under different degree of damage (condition 2 & 6, excitation point 1, without noise)
 图 8 损伤单元相邻节点损伤因子-损伤程度曲线(工况2、工况6，①号激振点，无噪声) Fig.8 Damage factor-damage degree curve of nodes adjacent to damage element (condition 2 & 6, excitation point 1, without noise)

(3) 激振位置对损伤识别结果的影响

 图 9 不同激励位置下损伤识别结果(工况2、工况6，无噪声) Fig.9 Single-position damage identification results under different excitation positions (condition 2 & 6, without noise)

(4) 测量噪声对损伤识别结果的影响

 图 10 不同噪声水平下多位置损伤识别结果(工况2、工况6，①号激振点) Fig.10 Multi-position damage identification results under different noise levels(condition 2 & 6, excitation point 1)

3.2 算例二 3.2.1 数值模型

 图 11 桁架结构模型 Fig.11 Truss structure model

3.2.2 损伤工况

3.2.3 结果与分析

 图 12 损伤位置识别结果 Fig.12 Damage location identification results
 图 13 不同损伤程度下的单位置损伤识别结果 Fig.13 Identification results of single-position damage under different degree of damage
 图 14 损伤因子-损伤程度曲线 Fig.14 Damage factor-damage degree curve

3.2.4 方法对比

 图 15 与现有损伤识别方法的对比 Fig.15 Comparisons with existing damage identification methods

4 结论

(1) 应用本文所提方法建立的ARX模型符合动态数据的实际变化规律，模型拟合程度较高，拟合模型所需的样本数量较少；

(2) 可仅针对疑似损伤区域或重点检测区域进行损伤识别，无需对整体结构进行模态参数识别，所需传感器数量较少；

(3) 实际工程应用中，可通过有限元数值模拟计算基准损伤工况的节点DF数值，并将实测结构所得DF与其进行对比，即可较为准确地判断损伤程度；

(4) 多位置损伤工况下仍具有较好的识别效果.

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