﻿ 基于加速度频响函数小波分解的模型修正方法
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Model updating method based on wavelet decomposition of acceleration frequency response function
PENG Zhenrui, CAO Mingming, LIU Mandong
School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
Abstract: To improve the efficiency of model updating and satisfy its robustness for the measured environmental noise, the Kriging model and wavelet decomposition are introduced into the model updating of the acceleration frequency response function. Firstly, the acceleration frequency response function is decomposed by wavelets, and the obtained wavelet coefficients with large amplitudes in the first layer are used to represent the original frequency response function. Secondly, the Latin hypercube sampling is utilized to design the primary parameters to be updated, and the sensitivity analysis of each parameter carried out according to the design results to determine the parameters to be modified, which are then used as the inputs of the Kriging model, while the corresponding wavelet coefficients as the outputs of the model. The optimal correlation coefficients of the Kriging model are found through the mixed grey wolf algorithm, with which an accurate and effective Kriging model is established. Finally, with the error between the wavelet coefficients calculated from the Kriging model and those from tests as the objective function, a minimization problem is solved by the water cycle algorithm for parameter updating. Numerical examples show the effectiveness of the proposed model updating method. When Gaussian white noise with a signal-to-noise ratio of 5 dB is added to the acceleration frequency response function, the updating error is smaller than 4%, proving the robustness of the method against random noise.
Keywords: model updating    Kriging model    acceleration frequency response function    wavelet decomposition    correlation coefficients

1 加速度频响函数小波分解 1.1 加速度频响函数

 $( - {\omega ^2}\mathit{\boldsymbol{M}} + {\rm{i}}\omega \mathit{\boldsymbol{C}} + \mathit{\boldsymbol{K}})\mathit{\boldsymbol{X}}(\omega ) = \mathit{\boldsymbol{F}}(\omega )$ （1）

 $\mathit{\boldsymbol{X}}(\omega ) = \mathit{\boldsymbol{H}}(\omega )\mathit{\boldsymbol{F}}(\omega )$ （2）

 $\mathit{\boldsymbol{H}}(\omega ) = {\left( {\mathit{\boldsymbol{M}} - \frac{{{\rm{i}}\mathit{\boldsymbol{C}}}}{\omega } - \frac{\mathit{\boldsymbol{K}}}{{{\omega ^2}}}} \right)^{ - 1}}$ （3）
1.2 小波变换

 $y(t) = s(t) + o(t)\;\;\;{\kern 1pt} t = 0,1, \cdots ,T - 1$ （4）

 ${\lambda _{jk}} = {2^{ - \frac{j}{2}}}\sum\limits_{n = 0}^{T - 1} y (n)\psi ({2^{ - j}} - k)\;\;\;{\kern 1pt} j,k \in {\rm{Z}}$ （5）

 ${\lambda _{jk}} = {\lambda _{jk}}(s) + {\lambda _{jk}}(o)$ （6）

2 基于Kriging模型的模型修正理论 2.1 Kriging模型的建立

Kriging模型是一个基于随机过程的代理模型，最初在地质统计学中提出。模型包含了线性回归部分和非参数部分：

 $y(\mathit{\boldsymbol{x}}) = \sum\limits_{i = 1}^p {{\beta _i}} {f_i}(\mathit{\boldsymbol{x}}) + z(\mathit{\boldsymbol{x}}) = {\mathit{\boldsymbol{f}}^{\rm{T}}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{\beta }} + z(\mathit{\boldsymbol{x}})$ （7）

 ${\rm{ Cov}} [z({x_i}),z({x_j})] = {\sigma ^2}R({x_i},{x_j})$ （8）

 $R({x_i},{x_j}) = {\rm{exp}}\left( { - \sum\limits_{k = 1}^m {{\theta _k}} |x_i^k - x_j^k{|^2}} \right)$ （9）

 $\mathit{\boldsymbol{\hat \beta }} = {({\mathit{\boldsymbol{\hat F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{\hat F}})^{ - 1}}{\mathit{\boldsymbol{\hat F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{Y}}$ （10）

 ${\hat \sigma ^2} = \frac{1}{n}{(\mathit{\boldsymbol{Y}} - \mathit{\boldsymbol{\hat F\beta }})^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}(\mathit{\boldsymbol{Y}} - \mathit{\boldsymbol{\hat F\beta }})$ （11）

Kriging模型构造完成后，下一步就是预测待测点的响应值。对于任意待测点x0，其响应值${\hat y}$(x0)为

 $\hat y({x_0}) = {\mathit{\boldsymbol{f}}^{\rm{T}}}({x_0})\mathit{\boldsymbol{\hat \beta }} + {\mathit{\boldsymbol{r}}^{\rm{T}}}({x_0}){\mathit{\boldsymbol{R}}^{ - 1}}(\mathit{\boldsymbol{Y}} - \mathit{\boldsymbol{\hat F\hat \beta }})$ （12）

2.2 混合灰狼算法

1) 参数初始化。设置种群规模Npop、最大迭代次数MaxIt、自变量维数nVar、缩放因子下上界beta_min和beta_max，交叉概率pCR，参数下上界lb和ub。

2) 对种群个体进行DE变异操作，产生中间体；并进行竞争选择操作形成父代种群、子代种群和变异种群个体。

3) 社会等级。计算种群中每个灰狼个体的适应度值，并依据适应度值的大小进行排序，确定灰狼种群中的社会等级，最优解Xα为头狼，第2和第3最优解XβXδ为狼群中第2和第3等级狼，其余的候选解X为狼群中最低等级狼。

4) 搜索包围猎物。灰狼搜索猎物过程中，灰狼接近并包围猎物行为可以表示为

 ${\mathit{\boldsymbol{D}} = |\mathit{\boldsymbol{C}} \cdot {\mathit{\boldsymbol{X}}_p}(t) - \mathit{\boldsymbol{X}}(t)|}$ （13）
 ${\mathit{\boldsymbol{X}}(t + 1) = {\mathit{\boldsymbol{X}}_p}(t) - \mathit{\boldsymbol{A}} \cdot \mathit{\boldsymbol{D}}}$ （14）

5) 追捕猎物。当灰狼搜索到猎物所在位置时，首先，头狼Xα带领狼群对猎物进行包围；然后，头狼Xα带领Xβ狼和Xδ狼对猎物进行攻击捕捉。在灰狼群体中，XαXβXδ狼距离猎物位置最近，因此可以通过这三者的位置来计算灰狼个体向猎物移动的位置。并根据式(15)计算出种群中最优个体XαXβXδ与其他灰狼个体之间的距离，再根据式(16)计算出其余灰狼的移动方向。最后由式(17)更新灰狼位置。

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{D}}_\alpha } = |{\mathit{\boldsymbol{C}}_1} \cdot {\mathit{\boldsymbol{X}}_\alpha }(t) - \mathit{\boldsymbol{X}}(t)|\\ {\mathit{\boldsymbol{D}}_\beta } = |{\mathit{\boldsymbol{C}}_2} \cdot {\mathit{\boldsymbol{X}}_\beta }(t) - \mathit{\boldsymbol{X}}(t)|\\ {\mathit{\boldsymbol{D}}_\delta } = |{\mathit{\boldsymbol{C}}_3} \cdot {\mathit{\boldsymbol{X}}_\delta }(t) - \mathit{\boldsymbol{X}}(t)| \end{array} \right.$ （15）
 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{X}}_1}(t) = {\mathit{\boldsymbol{X}}_\alpha }(t) - {\mathit{\boldsymbol{A}}_1} \cdot {\mathit{\boldsymbol{D}}_\alpha }\\ {\mathit{\boldsymbol{X}}_2}(t) = {\mathit{\boldsymbol{X}}_\beta }(t) - {\mathit{\boldsymbol{A}}_2} \cdot {\mathit{\boldsymbol{D}}_\beta }\\ {\mathit{\boldsymbol{X}}_3}(t) = {\mathit{\boldsymbol{X}}_\delta }(t) - {\mathit{\boldsymbol{A}}_3} \cdot {\mathit{\boldsymbol{D}}_\delta } \end{array} \right.$ （16）
 $\mathit{\boldsymbol{X}}(t + 1) = \frac{{{\mathit{\boldsymbol{X}}_1}(t) + {\mathit{\boldsymbol{X}}_2}(t) + {\mathit{\boldsymbol{X}}_3}(t)}}{3}$ （17）

6) 对种群个体进行交叉、选择操作保留优良成分并产生新子代个体，计算个体的适应度值。

7) 更新灰狼XαXβXδ及其他对应的位置信息。

8) 判断是否达到最大迭代次数MaxIt，如果是则停止并输出当前最优解，否则返回3)继续执行。

2.3 混合灰狼算法优化Kriging模型

Kriging模型中的相关系数θk决定着预测响应值精度，只有当所建立Kriging模型精度足够高时，代理模型对修正结果的误差影响才会最小。所以相关系数的确定对于构建代理模型至关重要。本文采用拉丁超立方抽样方法抽取一定数量的样本点，然后把抽取的样本点分成训练集和测试集。以训练集作为Kriging模型的输入变量，以加速度频响函数小波分解得到的第1层小波系数作为响应值来构建Kriging模型，以测试集Kriging模型的均方误差(MSE)均值平均值作为混合灰狼算法的目标函数，寻得最优相关系数。最后以训练集作为样本点建立Kriging模型。

2.4 模型修正

 ${\rm{obj}} = \sum\limits_{i = 1}^K {{{({{\hat \lambda }_i} - {\lambda _i})}^2}}$ （18）

 图 1 模型修正流程图 Fig. 1 Flow chart of model updating
3 数值算例

 图 2 三维桁架模型结构图 Fig. 2 Structure drawing of three-dimensional truss model

 杆类型 弹性模量/GPa 横截面积/mm2 密度/(kg·m-3) 上弦杆 190 85.5 7 800 下弦杆 190 85.5 7 800 直腹杆 190 141.0 7 800 斜腹杆 190 45.0 7 800

3.1 待修正参数选取与试验设计

 参数 试验值 初始误差/% 有限元值 E/GPa 190 10 209 ρ/(kg·m-3) 7 800 -10 7 020 A1/mm2 85. 5 10 95 A2/mm2 85. 5 -10 76 A3/mm2 141 10 156 A4/mm2 45 -10 40

 图 3 各参数灵敏度 Fig. 3 Sensitivity of each parameter

3.2 Kriging模型的构造及评估

 图 4 混合灰狼算法迭代曲线 Fig. 4 Iteration curve of DE-GWO

 图 5 训练集样本第5个小波系数预测值 Fig. 5 Predicted value of the fifth wavelet coefficient of training set sample
 图 6 测试集样本第10个小波系数预测值 Fig. 6 Predicted value of the 10th wavelet coefficient of test set sample
 ${\rm{RMSE}} = \frac{1}{{k\bar \lambda }}\sqrt {\sum\limits_{i = 1}^k {{{({\lambda _i} - {{\hat \lambda }_i})}^2}} }$ （19）

3.3 模型修正

 修正参数 试验值 有限元值 初始误差/% 修正值 平均误差/% E/GPa 190 209 10 190.011 856 6.24×10-3 ρ/(kg·m-3) 7 800 7 020 -10 7 802.893 8 3.71×10-2 A1/mm2 85.5 95 10 85.510 858 5 1.27×10-2
 图 7 水循环迭代曲线 Fig. 7 Iteration curve of WCA

 图 8 Kriging模型、试验模型及有限元模型频响函数曲线 Fig. 8 FRF curves of Kriging model, test model and finite element model

 图 9 修正前后频响函数曲线 Fig. 9 FRF curves before and after updating

 图 10 20 dB和5 dB信噪比下加速度频响函数 Fig. 10 FRF under 20 dB and 5 dB SNRs

 图 11 不同信噪比下小波系数曲线和局部放大图 Fig. 11 Wavelet coefficient curves and local enlarged curves under different SNRs

 修正参数 初始误差/% 无噪声误差/% 平均误差/% 信噪比30 dB 信噪比20 dB 信噪比10 dB 信噪比5 dB E/GPa 10 6.24×10-3 0.286 0.533 1.430 3.416 ρ/(kg·m-3) -10 3.71×10-2 0.485 0.453 1.718 3.718 A1/mm2 10 1.27×10-2 0.326 0.595 1.067 2.079
4 结 论

1) 将小波分解引入频响函数模型修正中，既保留了利用频响函数进行模型修正无需进行模态识别的优点，同时也避开了频响函数中频率点的选择难题。

2) 在构造Kriging模型时，用混合灰狼算法对Kriging模型相关系数进行寻优，使所构造的Kriging模型具有良好的拟合精度和预测能力，能代替有限元模型进行迭代计算，提高了模型修正效率。

3) 将加速度频响函数进行小波分解，用幅值较大的小波系数来表征频响函数进行模型修正，修正效果较好。算例表明，在不同噪声水平下，使用本文所提方法仍然可以得出较满意的修正效果，即使当信噪比低至5 dB时，各参数修正误差仍然低于4%，证明了本文所提方法对于强噪声的鲁棒性。

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http://dx.doi.org/10.7527/S1000-6893.2020.23548

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#### 文章信息

PENG Zhenrui, CAO Mingming, LIU Mandong

Model updating method based on wavelet decomposition of acceleration frequency response function

Acta Aeronautica et Astronautica Sinica, 2020, 41(7): 223548.
http://dx.doi.org/10.7527/S1000-6893.2020.23548