Numerical Simulation of Lamb Wave Dispersion Characteristics and Stratified Defect Detection in the Aluminum-based Glass Wool Lamination
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摘要:
针对铝基玻璃棉层合板分层缺陷定位检测困难的问题,发展有限元特征频率法分析该结构中Lamb波的频散特性,得到层合板中对称模态与反对称模态的频散曲线,发现Lamb波在层合板中的传播特性呈类周期性分布,且存在铝基层与玻璃棉层模态混合而成的奇异模态.为进一步分析铝基玻璃棉层合板Lamb波的传播特性及与分层缺陷的交互作用,基于有限元瞬态分析模拟了Lamb波在其中的传播特性及与缺陷的交互作用.结果表明:模拟结果与有限元特征频率法计算结果得到了很好的吻合;A2模态在53 kHz时,得到了理想的回波信号,且波结构能量分布较为均匀,可利用该类频点定位缺陷位置;利用健康信号与缺陷信号的差分计算,通过波包形心法可以较准确地定位缺陷位置,证实了本文方法的可行性.
Abstract:To overcome the difficulty in the detection of layered defects in aluminum-based glass wool laminates, the dispersion characteristics of Lamb wave in the structure were analyzed by the finite element method. The dispersion curve of symmetric and antisymmetric modes in the laminated plate was obtained.It is found that the propagation characteristics of Lamb wave in the laminated plate are periodic distribution and have strange modes which are composed of the aluminum base layer and glass wool layer's mode mixing. In order to further analyze the propagation characteristic of Lamb wave in the aluminum-based glass wool laminatesand the interaction with layered defects, the propagation characteristics of Lamb wave and its interaction with defects were simulated based on the finite element transient analysis.The results show that the simulations are well matched with the calculation of finite element characteristic frequency method. When A2 mode is 53 kHz, the ideal echo signal can be obtained, and the energy distribution of wave-structure is more homogeneous. This type of frequency points can be used to locate the defect location; Using the difference calculation of the health signal and the defect signal, the position of the defect can be accurately positioned by the wave envelope centroid method, which supports the feasibility of the method.
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Keywords:
- aluminum-based glass wool /
- laminates /
- Lamb wave /
- defect detection
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铝基玻璃棉层合板被广泛应用于工农业及生活的各个领域.相较单层板壳结构而言,该类结构具有抗冲击性强、刚度大、隔热、隔声性能好等优点[1].但该类结构长期使用容易出现玻璃棉与铝基分离、脱层等现象[2-3]影响结构的使用性能,故对其脱层缺陷的在线检测就显得尤为重要.
传统板类结构快速检测主要是基于Lamb波,关于Lamb波在板类结构中传播特性的研究,在理论研究与数值计算的发展上已较为成熟[4].铝基玻璃棉层合板由于其基板与芯层的弹性常数具有较大的差异,因而与复合材料多层板和单层板的波动特性差异显著,且由于其频散特性较为复杂,应用常用的理论求解方法存在较多问题,故主要是借鉴Lamb波在普通板中的波动理论或通过实验的方法进行分析研究[5-6].
为究明此类结构中Lamb波的波动特性,近年来众多学者采用不同的方法展开了此方面的研究工作,如利用级数展开法和谱有限元法[7-8]. He等[9]采用勒让德级数展开法研究了各向异性复合材料层合板中Lamb波的传播特性,得到耦合和纤维取向对传播特性的影响. Gopalakrishnan[10]基于二维谱有限元方法得到了不同纤维方向复合材料层合板中Lamb波的频散曲线.刘锋等[11]基于谱有限元对Lamb在复合材料层合板中的传播特性进行了模拟,证实了谱有限元方法在该方面的优越性.此外,Xiao等[12]采用空耦传感器基于时频分析方法试验测试了铝基玻璃纤维层合板中Lamb波的频散曲线. Ma等[13]基于回传射线矩阵法研究了Lamb波在复合材料层合板中的频散特性,与试验结果一致性较好.在分层缺陷检测与数值模拟方面,Tian等[14]采用模态数据采集,结合波数分析方法研究了Lamb波与分层缺陷之间的交互作用. Yelve等[15]基于Lamb波的非线性效应研究了复合材料层合板中的分层缺陷,通过提取基波和高次谐波的信息以定位分层缺陷. Schmidt[16]针对Lamb波在复合材料层合板中应用的多模态特性,设计了一种考虑频散和衰减特性可选择模态的传感器,应用于分层缺陷的检测,取得了较好的效果.刘增华等[17-19]基于RAPID和Chirp激励等方法,对复合材料板中的Lamb波进行成像与信号分析,实现了波中缺陷的准确定位.上述学者近几年的研究工作对于Lamb波在层合板检测中的应用具有很好的指导意义.
有限元特征频率法作为一种适用性较广的方法,已被成功地用于求解复杂波导类结构和预应力波导结构的频散特性,如钢轨与异型波导结构等[20-22].为进一步验证有限元特征频率法应用的普适性,通过求解分析铝基玻璃棉层合板中的振动模态,探究Lamb波在其中的频散特性,并采用有限元数值模拟Lamb波在此类结构中的传播特性及与分层缺陷的交互作用,以期验证有限元特征频率法分析该类结构频散特性的可行性.
1. 频散特性计算
1.1 有限元特征频率法基本理论
有限元特征频率法可通过模型网格划分建立单元节点坐标,再建立形函数、位移场与应变场.在此基础上构建求解模型的刚度矩阵与质量矩阵,其通用方程式表示为[23]
$$ \mathit{\boldsymbol{K}}={{\iiint_{V}{\mathit{\boldsymbol{B}}}}^{\text{T}}}\mathit{\boldsymbol{DB}}\text{d}V $$ (1) $$ \mathit{\boldsymbol{M}}={\iiint_{V}\rho \mathit{\boldsymbol{N}}^{\text{T}}}\mathit{\boldsymbol{N}}\text{d}V $$ (2) 式中:K为刚度矩阵;M为质量矩阵;N为形函数矩阵;B为应变位移关系矩阵(可由形函数矩阵N给出,如式(3));D为弹性矩阵;ρ为材料密度.
$$ {\mathit{\boldsymbol{B}}_i} = \left[{\begin{array}{*{20}{c}} {\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{x}}}}}&0&0\\ 0&{\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{y}}}}}&0\\ 0&0&{\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{z}}}}}\\ 0&{\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{z}}}}}&{\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{y}}}}}\\ {\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{z}}}}}&0&{\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{x}}}}}\\ {\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{y}}}}}&{\frac{{\partial {\mathit{\boldsymbol{N}}_i}}}{{\partial \mathit{\boldsymbol{x}}}}}&0 \end{array}} \right] $$ (3) 在不考虑外部荷载的理想情况下,用有限元法构建结构的自由振动方程式,可化解得到特征方程式
$$ \mathit{\boldsymbol{KU}} - {\omega ^2}\mathit{\boldsymbol{MU}} = 0 $$ (4) 式中:ω为结构的特征频率;U为特征频率对应的振型位移矢量矩阵.
根据边界条件确定模态振型位移矢量矩阵,对方程(4)进行求解即可得到对应振型的特征频率.根据上述计算方法得到的特征频率在频散分析中只属于普通频率点,并不具有特殊意义.边界条件的不同决定了所求的模态类型,如可利用中面的对称简化实现对称模态的求解等.由于该方法求得的特征频率点是离散的,不便于后续分析模态的群速度.为使频散特性分析具有较好的连续性,可采用不同的模型长度分别求解特征频率点,利用波数连续性的原则进行群速度的计算.此外,还可利用端面的对称边界设置使模型长度简化一半,从而节省求解的自由度数.
有限元数值计算得到的各模态相速度与群速度分别由式(5)与式(6)给出.对于任意频率点的相速度与群速度可由相邻特征频率点的线性插值给出.
$$ {c_{\rm{p}}} = \frac{{fL}}{n} $$ (5) $$ {c_{\rm{g}}} = \Delta fL $$ (6) 式中:cp为相速度;f为特征频率;L为模型长度;n为振动周期数;cg为群速度;Δf为同一模态相邻特征频率的差值.
1.2 几何物理参数及边界条件
研究对象中铝基厚度0.6mm,对应拉梅常数λ=51.64GPa,剪切模量μ=26.7GPa,密度ρ=2700kg/m3;玻璃棉厚度15mm,其对应等效拉梅常数λ=0.008GPa,剪切模量μ=0.0125GPa,密度ρ=48kg/m3.
为便于铝基玻璃棉层合板中Lamb波各类模态的识别与统计,求解时对称模态与反对称模态分别设置计算.图 1为求解对称模态与反对称模态的边界条件.
1.3 Lamb波频散特性计算及分析
为使得到的相速度和群速度具有较好的连续性和平滑性,模型长度分别选取500、600、700和800mm,每种模型长度计算特征频率500个,利用Matlab波数统计与模态识别分离程序统计各个模态类型及其对应的振动周期数.根据式(5)(6)计算得到的Lamb波对称模态和反对称模态的频散曲线如图 2和图 3所示.
从相速度与群速度频散曲线发现,层合板中对称模态的传播特性与反对称模态较为类似,对称模态与反对称模态的频散特性呈现周期性分布,每隔约35kHz的频宽出现能量传播速度最大的模态,且每种模态群速度最大值较为接近.在频散较大相邻模态交接处,两大类模态均表现出频散较大的特点,表明此频段的模态不适合在缺陷检测中使用.对称模态与反对称模态中均出现奇异模态,该类模态频散特性较为复杂.通过模态振型分析,发现该类模态主要是铝基层和玻璃棉芯层的模态振型存在较大差异导致的,但2种模态在结合面处恰好满足位移与应力连续的条件,二者模态叠加形成新的模态,在群速度的表现上出现奇异特征.相关研究也表明,层合板中导波的频散特性与各层板特性相关,说明模态表现奇异处可参考各层板的频散特性[24].
由图 2和图 3可知,S1模态在35kHz,S2模态在70kHz,A1模态在18kHz,A2模态在53kHz时,频散较小,且群速度值较大,符合缺陷检测的需求.故在数值模拟分析波的传播特性时,选取上述模态对应的激励频率进行研究,以期得到适合层合板分层缺陷检测的最佳模态与激励频率.
2. 数值模拟与分析
2.1 Lamb波传播特性的数值模拟
数值模拟计算采用COMSOL Multiphysics5.2a结构力学瞬态分析,模型参数见1.2,模型长度选取1000mm.模拟计算参数及选取依据见表 1[25].
表 1 有限元模拟计算参数选取方法Table 1. Parameters for finite element simulation参数名称 选取依据 选取值 分析时长t t≥l/cS≈0.5ms 1ms 分析步长Δt Δt≤ ${\rm{min}}\left\{ \begin{array}{l} {L_{{\rm{min}}}}/{c_{\rm{L}}} \approx 0.6\mu s\\ 1/20f \approx 0.7\mu s \end{array} \right. $ 0.5μs 最大网格单元尺寸Lmax Lmax ≤λS/10≈10mm 8mm 由于18kHz检测频率对缺陷灵敏度较低,数值模拟主要分析35、70和53kHz激励频率下Lamb波的传播特性.激励激励信号采用10个周期经汉宁窗调制的正弦信号,其中35kHz和70kHz采用对称加载方式以激励S1模态和S2模态,53kHz采用反对称加载方式以激励A2模态,加载方式参加文献[26].所有频点均在端面激励并接收回波信号,数值模拟结果如图 4所示.
图 4结果表明激励频率为35kHz和53kHz时,接收到的一次端面回波和二次端面回波均较为独立,信噪比较高.激励频率为70kHz时, 接收到的回波信号存在一定的干扰,且干扰信号频散较大,根据其速度值推测可能是S0或A1模态.考虑到低频对缺陷的敏感度较低,且激励频率为35kHz时,根据其波结构发现该频点的能量主要集中在铝基层,故35kHz和70kHz频点不太适合用于缺陷检测.
采用互相关算法计算上述几个频点数值模拟的群速度值分别为,35kHz时4945m/s,53kHz时4886m/s,70kHz时4892m/s,与特征频率法计算得到值偏差分别为1.6%、0.86%和0.83%,表明数值模拟激励的模态与理论分析的结果相符,证实了模拟的可行性与准确性.
2.2 Lamb波与分层缺陷交互作用的模拟
在有分层缺陷的模型结构中,在板长500mm处,人造长度2mm、深度1mm的分层缺陷.为提高计算精度与速度,将模型分成3段,分层缺陷段采用自由网格划分方式,其他2段采用映射网格划分方式.模型网格及分层缺陷如图 5所示.
根据传播特性数值模拟结果,为使检测频率对缺陷更为敏感,选取53kHz时的A2模态,数值模拟其与分层缺陷的交互作用.由于检测频率波长较大,遇到缺陷时的反射信号较弱,无法直接观察到缺陷的回波信号,故通过对健康信号与缺陷信号进行差分计算[27],通过差分信号定位缺陷位置,结果如图 6所示.
由图 6可知,差分信号中缺陷信号信噪比较高,可以通过其定位缺陷位置.但由于差分信号中的缺陷信号相位发生较大的变化,根据最大幅值法和互相关算法定位时间误差较大.通过波包形心法[28]定位缺陷的位置与实际位置偏差为2.56%,表明通过对健康信号与缺陷信号进行差分计算,以差分信号定位缺陷位置具有较高的准确性.
3. 结论
1) 基于有限元特征频率法,计算了层合板中Lamb波的频散特性,发现对称模态与反对称模态频散特性较为相似,每隔约35kHz的频宽出现频散较小的模态,此时其群速度传播最快,表明可以利用此特性选择适合检测的模态与激励频率.
2) 层合板中存在面板层和中间层模态不同的现象,但2种模态在结合面满足位移与应力连续的条件,混合而形成奇异模态;该类模态频散特性较为复杂,其波动特性还有待于后续研究.
3) 在频散较小群速度传播的极大值点,有限元瞬态模拟了Lamb波的传播特性,数值模拟结果与理论分析的结果相符,证实了模拟的可行性与准确性;当S1模态激励频率为35kHz,A2模态激励频率为53kHz时,回波信号较为理想,可以用于缺陷检测.
4) 通过人造分层缺陷,数值模拟了53kHz的A2模态与分层缺陷的交互作用;针对低频信号对缺陷敏感度较低的问题,通过利用健康信号与缺陷信号的差分计算,以波包形心法计算缺陷回波时间定位缺陷位置,模拟结果与实际偏差较小,表明该方法可以较准确地定位缺陷位置.
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表 1 有限元模拟计算参数选取方法
Table 1 Parameters for finite element simulation
参数名称 选取依据 选取值 分析时长t t≥l/cS≈0.5ms 1ms 分析步长Δt Δt≤ ${\rm{min}}\left\{ \begin{array}{l} {L_{{\rm{min}}}}/{c_{\rm{L}}} \approx 0.6\mu s\\ 1/20f \approx 0.7\mu s \end{array} \right. $ 0.5μs 最大网格单元尺寸Lmax Lmax ≤λS/10≈10mm 8mm -
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