工程科学与技术   2021, Vol. 53 Issue (4): 118-127

1. 青岛理工大学 土木工程学院，山东 青岛 266033;
2. 山东省高等学校蓝色经济区工程建设与安全协同创新中心，山东 青岛 266033

Research on Neural Network Generalization of Cable Force Vibration Measurement
GAI Tongtong1,2, ZENG Sen1,2, YU Dehu1,2, YANG Shujuan1,2, SUN Baodi1,2
1. School of Civil Eng., Qingdao Univ. of Technol., Qingdao 266033, China;
2. Cooperative Innovation Center of Eng. Construction and Safety in Shandong Blue Economic Zone, Qingdao 266033, China
Abstract: The stress state of the cable is related to the safety of the cable system bridge, and the cable force value is an important index to measure the mechanical states of the cable. At present, the difficulty of determining the cable boundary conditions is an important factor affecting the accuracy of the cable force identification results. The ANSYS was used to numerically simulate the cable vibration, and the reliability of the modeling method was verified by the existing cable force calculation formula and the simulation data was generated. Then taken cable length, line density, bending stiffness, first-order frequency, second-order frequency, and third-order frequency as the input parameters, and used cable force as output parameter combined with vibration simulation data to establish BP neural network and generalized regression neural network cable force prediction model. Two neural network cable force prediction models and the existing cable force calculation formula were applied to actual projects for comparison and verification. The results showed that the neural network structure of the BP neural network cable force prediction model was 6–13–13–1, the activation functions between the input layer and the hidden layer 1, the hidden layer 1 and the hidden layer 2, the hidden layer 2 and the output layer were tansig, tansig, purelin, the training algorithm was the L–M optimization algorithm trainlm, the learning rate was 0.1, the number of network iterations was 1 000, the display interval was 100, the mean square error was 0.001, the prediction effect of the cable force prediction model was good, but there was room for further optimization. The best spread value of the generalized regression neural network cable force prediction model was 0.002 15, the prediction effect of the cable force prediction model was better than that of the BP neural network and the existing cable force calculation formula, and the forecast error was basically controlled within 5%. Utilizing the generalized regression neural network to predict the cable force of the bridge can avoid the influence of the judgment error of the cable boundary condition on the accuracy of the cable force recognition result, and improve the accuracy of the cable force recognition, which has a good engineering application value.
Key words: cable force    vibration method    BP neural network    generalized regression neural network

1 拉索振动的数值模拟

1.1 模型的建立、分析与验证

 图1 两端铰接条件下索的振动模态 Fig. 1 Vibration mode of the cable under the condition of hinged support at both ends

 $\qquad {\text{两端铰接：}}T=4m{l}^{2}{f}_{1}^{2}-\frac{\text{π}^{2}}{{l}^{2}}EI$ (1)
 \qquad \begin{aligned}[b] {\text{两端固结：}}&T=4m{l}^{2}{\left({f}_{1}^{2}-2.3{f}_{1}\sqrt{\frac{EI}{m{l}^{4}}}-0.575\frac{EI}{m{l}^{4}}\right)}^{},\\ &\xi =\sqrt{\frac{T}{EI}}l>7.1\\[-18pt]\end{aligned} (2)
 \qquad \begin{aligned}[b] {\text{一固一铰：}}&T\!=\!4m{l}^{2}\left(\!{f}_{1}^{2}\!\!-\!\!1.43{f}_{1}\!\!\sqrt{\frac{EI}{m{l}^{4}}}\!-\!{0.357}\;5\frac{EI}{m{l}^{4}}\right),\\ &\xi =\sqrt{\frac{T}{EI}}l>4.9\\[-18pt]\end{aligned} (3)

1.2 数据模拟

2 BP神经网络索力预测模型 2.1 BP神经网络原理

BP神经网络[16]以其自学习、自适应等能力，可以逼近任意多元非线性函数，网络结构由输入、输出及隐含层构成，其工作原理如图2所示。

 图2 神经元节点工作示意图 Fig. 2 Schematic diagram of neuron node work

 $y\left( x \right) = f\left( {\sum\limits_{i = 1}^n {{w_i}{x_i} - \theta } } \right)$ (4)

 ${\;\;\;\;\;\;\;\;\;\;\;\;\;w_n^{(m)}(i,j)} = w_n^{(m - 1)}(i,j) + \eta \delta _n^{(m)}x_{n - 1}^{(m)}(j)$ (5)

2.2 BP神经网络索力预测模型的建立、训练与预测 2.2.1 BP神经网络索力预测模型结构的确定

 $h = \sqrt {a + b} + c$ (6)

 图3 神经网络结构 Fig. 3 Neural network structure

2.2.2 索力预测模型的训练与预测

 图4 模型的训练、验证及预测曲线 Fig. 4 Model training，verification and prediction curves

 图5 BP神经网络预测误差 Fig. 5 BP neural network prediction errors

 ${\rm{MAPE}} = \frac{{100}}{n}\sum\limits_{t = 1}^n {\left| {\frac{{(y'_t - {y_t})}}{{{y_t}}}} \right|}$ (7)

3 广义回归神经网络索力预测模型 3.1 广义回归神经网络原理

 图6 GRNN 网络拓扑结构[21] Fig. 6 GRNN network topology[21]

1）输入层

2）模式层

 ${p}_{i}=\exp{\left[-\frac{{\left({{X}}-{{{X}}}_{i}\right)}^{{\rm{T}}}\left({{X}}-{{{X}}}_{i}\right)}{2{\sigma }^{2}}\right]}^{},i=1,2,\cdots ,n$ (8)

3）求和层

 ${S\!_{\rm{D}}} = \sum\limits_{i = 1}^n {{p_i}}$ (9)

 ${S\!_{{\rm{N}}j}} = \sum\limits_{i = 1}^n {{y_{ij}}} {p_i},\;j = 1,2, \cdots ,k$ (10)

4）输出层

 ${y}_{j}={\frac{{S}\!_{{\rm{N}}j}}{{S}\!_{\rm{D}}}}_{}{,}_{}j=1,2,\cdots ,k$ (11)

3.2 广义回归神经网络索力预测模型的训练和预测

 图7 广义回归神经网络部分预测值与实际值对比 Fig. 7 Comparison of some predicted values and actual values of GRNN

 图8 广义回归神经网络预测误差 Fig. 8 GRNN prediction errors

 图9 标准化残差直方图及残差正态概率图 Fig. 9 Standardized residual histogram and residual normal probability plot

4 工程案例验证 4.1 工程概况

 图10 开源桥现场测试图 Fig. 10 Kaiyuan bridge field test diagram

4.2 两种神经网络索力预测模型预测值与实测值对比分析

 图11 不同神经网络索力预测模型预测结果对比 Fig. 11 Comparison of prediction results of different neural network cable force prediction models

4.3 不同参考文献中索力识别方法与本文广义回归神经网络识别方法对比

Zui[2]、任伟新[17]、陈淮[4]、王建飞[10]等均对两端固结条件下拉索的索力计算公式进行了推导，其计算公式分别如式（12）、（13）、（14）、（15）所示。

 $T=4m{({f}_{1}l)}^{2}\left[1-2.20\frac{C}{{f}_{1}}-0.550{\left(\frac{C}{{f}_{1}}\right)}^{2}\right],\;\xi\ge 17$ (12)
 $\left\{\!\!\!\!\begin{array}{l}T=m{\left[2l{f}_{1}-\dfrac{2.363}{l}\sqrt{\dfrac{EI}{m}}\right]}^{2},\;18\le \xi \le 210;\\ T=4m{l}^{2}{f}_{1}^{2},\;\xi \ge 210\end{array}\right.$ (13)
 $T=m{({f}_{1}l)}^{2}{\left[1+\sqrt{1-4.314\frac{C}{{f}_{1}}}\right]}^{2}\begin{array}{cc},\xi >20\end{array}$ (14)
 $T=4m{l}^{2}\left({f}_{1}^{2}-2.3{f}_{1}\sqrt{\frac{EI}{m{l}^{4}}}-0.575\frac{EI}{m{l}^{4}}\right),\;\xi >7.1$ (15)

 图12 不同索力识别方法识别误差对比 Fig. 12 Comparison of identification errors of different cable force identification methods

 图13 不同索力识别方法平均绝对误差对比 Fig. 13 Comparison of average absolute errors of different cable force identification methods

5 结　论

1）利用ANSYS中的BEAM188单元，通过释放端头的轴向位移并施加轴向拉力的方式，对不同边界条件下桥梁拉索的振动进行模拟，并将模拟结果利用已有索力计算公式进行验证，结果表明本文建模方式可靠。

2）利用BP神经网络结合模拟数据构建结构为6–13–13–1的索力预测模型，输入索长、线密度、抗弯刚度、1阶频率、2阶频率和3阶频率可直接预测出索力值，且预测效果良好，但还有进一步优化的空间。