参数自适应VMD在高速列车轴箱轴承故障诊断中的应用

参数自适应 VMD 在高速列车轴箱轴承故障诊断中的应用 Application of parameter adaptive VMD in fault diagnosis of axle box bearing of high-speed train 1 first-author 1 corresp 宋冬利，博士，高级实验师，从事高速列车智能运维研究；E-mail：sdl.cds@163.com 宋冬利，博士，高级实验师，从事高速列车智能运维研究；E-mail：sdl.cds@163.com sdl.cds@163.com 2 2 1 1 西南交通大学 牵引动力国家重点实验室，四川 成都，610031 西南交通大学 牵引动力国家重点实验室，四川 成都，610031 State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China 株洲国创轨道科技有限公司，湖南 株洲，412000 株洲国创轨道科技有限公司，湖南 株洲，412000 Zhuzhou Guochuang Track Technology Co. Ltd., Zhuzhou 412000, China Zhuzhou Guochuang Track Technology Co. Ltd., Zhuzhou 412000, China 伍锦花 针对高速列车轴箱轴承故障特征提取困难和变分模态分解(VMD)参数的人为设置影响分解效果的问题，提出参数自适应VMD轴箱轴承故障诊断方法。首先，以平均包络谱熵为适应度函数，利用麻雀搜索算法自适应地寻找不同工况下的最优模态数 K 和惩罚因子 <math id="M1"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683135&type= 1.77800000 2.37066650 ；其次，对原始轴承时域信号进行VMD分解，利用快速谱峭度图分析最小包络熵的IMF分量，并根据分析结果对该IMF分量进行带通滤波以增强故障特征；最后，对滤波后信号进行希尔伯特包络解调分析，并将分析结果与理论计算所得特征频率进行对比，对轴箱轴承故障进行分类辨识。研究结果表明：与经验模态分解(EMD)、局域均值分解(LMD)、集合经验模态分解(EEMD)等自适应信号分解方法相比，本文所提方法能更有效地降低噪声的影响，提取复杂耦合工况下轴箱轴承振动信号中的故障特征。 Aiming at the difficulty of extracting fault features of axle box bearing of high-speed train and the problem that artificial setting of variational modal decomposition(VMD) parameters will affect the decomposition effect, a parameter adaptive VMD axle box bearing fault diagnosis method was proposed. Firstly, the average envelope spectral entropy was taken as the fitness function, and the sparrow search algorithm was used to adaptively find the optimal mode number K and penalty factor <math id="M2"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683125&type= 1.77800000 2.37066650 under different working conditions. Secondly, the original bearing time domain signal was decomposed by VMD, and the IMF component with the smallest envelope entropy was analyzed by fast spectral kurtosis diagram. According to the analysis results, the IMF component was band-pass filtered to enhance the fault characteristics. Finally, the filtered signal was analyzed by Hilbert envelope demodulation, and the fault classification and identification of axle box bearing were carried out by comparing the analysis result with the calculated characteristic frequency. The results show that the proposed method can reduce the influence of noise more effectively than the adaptive signal decomposition methods such as empirical mode decomposition(EMD), local mean decomposition(LMD) and ensemble empirical mode decomposition(EEMD), and the weak fault features in the vibration signal of axle box bearing under complex coupling conditions are extracted successfully . 高速列车 轴箱轴承 变分模态分解 麻雀搜索算法 快速谱峭度 high-speed train axle box bearing variational modal decomposition sparrow search algorithm fast spectral kurtosis http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683629&type= 14.95016479 14.95016479 轴箱轴承是高速列车走行部传动系统的核心部件 [ 1 1 ] ，起传动和承载作用。根据中国铁道科学研究院 [ 2 2 ] 的研究，传动系统中轴承故障主要集中在轴箱轴承，占比达48%，严重影响了列车行车安全，同时，也造成了一定经济损失。因此，对轴箱轴承的健康进行监测具有重要意义。 我国目前多基于温度和振动对高速列车轴箱轴承进行监测。在温度领域，LIU等 [ 3 3 ] 基于长短期记忆人工神经网络(LSTM)方法提出了轴箱轴承实时故障预警方法，并在CRH3型车的应用中取得了良好的效果。但温度对早期故障不敏感，往往报警时已发生严重故障。振动监测具有可重复检测、诊断结果可靠、可实现故障分类辨识等优点，已成为近年来机械设备领域故障诊断的主要研究手段。针对滚动轴承故障等非平稳、非线性振动信号，比较常用的方法有经验模态分解(EMD) [ 4 4 ] 、小波变换等 [ 5 5 ] 。EMD具有自适应分解特性且在处理非平稳非线性信号方面具有良好的表现，被广泛应用于轴承故障诊断中 [ 6 6 ] ，但是EMD在处理某些信号时存在模态混叠、端点效应较大等缺陷 [ 7 7 ] 。作为另一种非平稳信号分析手段，小波变换在轴承故障诊断得到了广泛应用 [ 8 8 - 9 9 8-9 ] ，但小波基的选择往往依赖人工经验，对部分工况恶劣的非平稳信号分解效果有限。DRAGOMIRETSKIY等 [ 10 10 ] 提出了变分模态分解(VMD)，该方法在处理非平稳非线性信号方面有稳定的表现，逐渐成为用于轴承故障特征提取的热点方法 [ 11 11 - 12 12 11-12 ] 。万书亭等 [ 13 13 ] 的研究表明VMD能有效提取强噪源干扰下的滚动轴承复合故障，且效果比集成经验模态分解的更好。ANEESH等 [ 14 14 ] 的研究表明，与经验小波变换相比，VMD在微弱故障特征提取方面更有优势，但是VMD在分解前需要预先设置分解模态数，参数的设置会极大地影响分解效率和分解效果 [ 15 15 ] 。李华等 [ 16 16 ] 将信息熵用于VMD参数的设置中，但该方法先将 K 和 <math id="M3"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 中的某一个参数固定，再对另一个参数进行优化，容易陷入局部最优。蒋宇涵等 [ 17 17 - 18 18 17-18 ] 分别使用遗传算法和粒子群算法对 K 和 <math id="M4"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 同时寻优，在一定程度上减少了固定单参数进行优化时易陷入局部最优的情况，但研究者发现此类算法迭代时间过长，运算效率不高。 基于上述现状，本文以最小平均包络熵为适应度函数，利用新型生物智能寻优算法即麻雀搜索算法对VMD的2个参数进行自适应优化，进而对VMD分解后的IMF分量进行快速谱峭度分析以确定带通滤波参数，增强信噪比；对滤波后的IMF分量进行希尔伯特包络解调分析，识别淹没在噪声中的微弱轴承故障特征，并通过高速列车轴箱轴承专用试验台验证该方法的有效性。 1 基本理论 1 1.1 　变分模态分解 2 变分模态分解方法是根据Wiener滤波和Hibert变换方法改进得到的，该方法旨在将输入信号实值分解为 K 个估计带宽之和的最小的子信号，分解后的子信号可再现输入信号，同时具有特定的带宽稀疏特性。约束变分问题可写成如下表达式： <math display="block" id="M5"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><msub><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>,</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow></msub><mfenced open="{" close="}" separators="|"><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi>k</mi></mrow></munder><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi mathvariant="normal">j</mi></mrow><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">*</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi mathvariant="normal">j</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">t</mi><mo>.</mo><mtext>    </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>k</mi></mrow></munder><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683140&type= 66.12466431 20.74333191 式中： <math id="M6"><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683143&type= 6.85799980 4.48733330 为原信号经变分模态分解后的第 k 个分量； <math id="M7"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683147&type= 1.10066664 2.62466669 为时间； <math id="M8"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683166&type= 5.67266655 3.38666677 为原始信号； <math id="M9"><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683178&type= 3.80999994 3.72533321 为各模态分量的中心频率；*为卷积运算符号； <math id="M10"><mi>δ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683171&type= 4.99533367 3.38666677 为狄拉克函数。为求解 式(1) 式(1) ，DRAGOMIRETSKIY等 [ 10 10 ] 通过引入二次惩罚函数拉格朗日乘子 <math id="M11"><mi>λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683174&type= 4.91066647 3.38666677 将上式转化为无约束优化问题。修正后的具有增广拉格朗日函数的方程如 式(2) 式(2) 所示。 <math display="block" id="M12"><mi>L</mi><mfenced separators="|"><mrow><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>,</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>,</mo><mi>λ</mi></mrow></mfenced><mo>=</mo><mi>α</mi><munder><mo movablelimits="false">∑</mo><mrow><mi>k</mi></mrow></munder><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi mathvariant="normal">j</mi></mrow><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">*</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi mathvariant="normal">j</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683185&type= 81.02600098 11.17599964 <math display="block" id="M13"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>k</mi></mrow></munder><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfenced open="<" close=">" separators="|"><mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>k</mi></mrow></munder><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683197&type= 60.53666687 11.00666618 式中： <math id="M14"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683189&type= 1.94733346 2.62466669 为惩罚因子。 上述变分问题可通过交替方向乘子算法(ADMM)求解，达到停止条件后结束迭代，得到 K 个IMF分量。具体分解过程见文献[ 10 10 ]。 变分模态分解存在的问题如下：VMD分解过程中，分解模态数 K 和惩罚因子 <math id="M15"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 需预先设置，依赖人工经验。当 K 取值过小时，会出现欠分解的情况，导致模态丢失或者模态混叠；当 K 取值过大时，会出现过分解的情况，导致虚假模态的出现，同时也会对分解速度产生较大影响。而惩罚因子 <math id="M16"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683619&type= 1.94733346 2.62466669 对分解效果同样存在一定影响， <math id="M17"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 设置不合理会显著提升算法的运行时长，出现模态混叠和中心频率重叠的现象，影响其分解效果 [ 19 19 ] 。因此，对 K 和 <math id="M18"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 在不同工况下的自适应寻优进行研究，对于提高变分模态的分解效率和分解效果具有重要意义。 1.2 　麻雀搜索算法 2 SSA(sparrow search algorithm)算法是对麻雀种群的觅食和反捕食行为进行抽象化表述的一种算法。为了简单起见，XUE等 [ 20 20 ] 将麻雀总群划分为寻觅者、加入者、预警者，并制定了相应的规则来建立数学模型。 麻雀种群的位置 <math id="M19"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683226&type= 3.04800010 2.62466669 可用 式(3) 式(3) 表示： <math display="block" id="M20"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2,2</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>d</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683238&type= 39.03133011 18.20333290 式中： n 为麻雀的数量， d 为要优化的变量的维度。所有麻雀的适应度 <math id="M21"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683232&type= 4.14866638 3.72533321 可以用 式(4) 式(4) 表示： <math display="block" id="M22"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="bold-italic">X</mi></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1,2</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><mo>⋯</mo></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>d</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2,1</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2,2</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><mo>⋯</mo></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><mi>d</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><mo>⋯</mo></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>d</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683234&type= 54.69466782 20.40466690 式中： f 为个体的适应度。寻觅者负责寻找食物和指导整个群体的流动，可以在更广泛的地方寻找食物。在每次迭代期间，寻觅者的位置更新表达式如下： <math display="block" id="M23"><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>⋅</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mfrac><mrow><mo>-</mo><mi>i</mi></mrow><mrow><mi>β</mi><mo>⋅</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><mi>Q</mi><mo>⋅</mo><mi mathvariant="bold-italic">L</mi><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683248&type= 52.49333191 16.51000023 式中： <math id="M24"><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683252&type= 4.91066647 4.06400013 为在第 j 维中的第 i 个麻雀的位置信息； <math id="M25"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683256&type= 6.09600019 3.72533321 为迭代次数最大值； <math id="M26"><mi>β</mi><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683272&type= 8.29733276 3.38666677 ，1)，是一个随机数； <math id="M27"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683275&type= 9.73666668 4.40266657 ， <math id="M28"><mn mathvariant="normal">1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683279&type= 2.37066650 3.30200005 ， <math id="M29"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683281&type= 13.29266548 4.48733330 ， <math id="M30"><mn mathvariant="normal">1.0</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683285&type= 5.41866684 3.30200005 ，分别表示报警值和安全阈值； Q 为服从正态分布的随机数； <math id="M31"><mi mathvariant="bold-italic">L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683290&type= 2.37066650 2.62466669 为内部每个元素值为1的矩阵。 当 <math id="M32"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683293&type= 10.75266743 4.48733330 时，周围没有捕食者，寻觅者进入广泛搜索模式；当 <math id="M33"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683300&type= 10.75266743 4.48733330 时，部分麻雀已经意识到了捕食者的存在，其会引导所有麻雀飞到其他安全区域寻找食物。 加入者跟随寻觅者寻找食物，其位置更新式如下： <math display="block" id="M34"><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>Q</mi><mo>⋅</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>,</mo><mi>i</mi><mo>></mo><mi>n</mi><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>+</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfenced><mo>⋅</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>⋅</mo><mi mathvariant="bold-italic">L</mi><mo>,</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683296&type= 62.48400116 18.71133423 式中： <math id="M35"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683303&type= 3.97933316 3.72533321 为寻觅者占据的最佳位置； <math id="M36"><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683305&type= 7.78933382 4.06400013 为当前群体中的最差位置； <math id="M37"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683307&type= 2.70933342 2.62466669 为1× d 矩阵，矩阵中的元素随机分配 <math id="M38"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683309&type= 1.94733346 2.62466669 1或1，此外， <math id="M39"><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">A</mi><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683312&type= 23.53733444 5.58799982 。当 <math id="M40"><mi>i</mi><mo>></mo><mi>n</mi><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683316&type= 9.05933285 2.62466669 时，适应性越差的加入者越有可能挨饿，此时该麻雀需飞往其他地方寻找食物。 预警者的初始位置是在种群中随机生成的，其数学模型如 式(7) 式(7) 所示。 <math display="block" id="M41"><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><mi>β</mi><mo>⋅</mo><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow></mfenced><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>+</mo><mi>M</mi><mo>⋅</mo><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi>t</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">w</mi></mrow></msub></mrow></mfenced><mo>+</mo><mi>ε</mi></mrow></mfrac></mrow></mfenced><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683320&type= 58.33533096 21.08200073 式中： <math id="M42"><msubsup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683331&type= 6.43466663 4.06400013 为当前的全局最佳位置； <math id="M43"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683333&type= 3.64066648 2.62466669 为[-1，1]范围内的一个随机数，表示麻雀移动的方向； <math id="M44"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683335&type= 2.45533323 4.48733330 为当前麻雀的适应度； <math id="M45"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683323&type= 2.96333337 4.48733330 和 <math id="M46"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">w</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683338&type= 3.47133350 4.48733330 分别为当前的全局最佳和最差适应度； <math id="M47"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683340&type= 1.52400005 2.62466669 为中间变量，用于避免分母出现零的情况。为简单起见， <math id="M48"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683343&type= 2.45533323 4.48733330 > <math id="M49"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683326&type= 2.96333337 4.48733330 ，表示麻雀处于群的边缘； <math id="M50"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683343&type= 2.45533323 4.48733330 = <math id="M51"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683345&type= 2.96333337 4.48733330 ，表明麻雀在群体中意识到危险，需要更接近其他麻雀以减小风险。 1.3 　快速谱峭度 2 快速谱峭度图法由ANTONI [ 21 21 ] 提出，具有良好的工程适用性和用户友好性。在快速谱峭度图中，可通过颜色来判别不同中心频率及带宽下的谱峭度，依据图像颜色即可得到该信号的中心频率与最优带宽，并以此为依据对该信号进行带通滤波，增强信号的信噪比。 2 基于参数自适应 VMD 的高速列车轴箱轴承故障诊断方法 1 2.1 　 VMD 参数自适应优化过程 2 香农熵是判断信号稀疏性的一个指标，其值越大，说明信号的不确定性越大；反之，其值越小，说明信号越有序，越有可能包含故障信息 [ 16 16 ] 。在此基础上，唐贵基等 [ 22 22 ] 提出了平均包络熵的概念，即先对信号进行解调，计算出包络信号序列 <math id="M52"><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683347&type= 3.04800010 4.48733330 的香农熵并取其均值。相比于直接求原信号的香浓熵，平均包络熵可以更好地反映原始信号的稀疏特性。平均包络熵 <math id="M53"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683349&type= 3.80999994 3.72533321 的表达式如下： <math display="block" id="M54"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub><mo>=</mo><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>/</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mi>a</mi></mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683351&type= 28.44799995 20.99733353 式中： <math id="M55"><mi>a</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683353&type= 5.84200001 3.38666677 为原信号经希尔伯特解调分析后的包络信号； <math id="M56"><msub><mrow><mi>p</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683355&type= 3.04800010 4.48733330 为 <math id="M57"><mi>a</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683357&type= 5.84200001 3.38666677 的归一化表达式； N 为包络信号数。 为搜寻不同工况下的最佳参数设置，本文将以各分量的平均包络熵的最小值为适应度，用麻雀搜索算法优化VMD的2个参数，适应度函数如 式(9) 式(9) 所示。 <math display="block" id="M58"><mo stretchy="false">(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></munder><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683359&type= 48.51399994 9.65200043 VMD算法中的模态数 K 和惩罚因子 <math id="M59"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 的具体寻优步骤如下： 1) 输入原始轴承振动数据； 2) 设定 K 和 <math id="M60"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 的搜寻范围，初始化SSA各项参数； 3) 计算初始适应度，根据预警值更新寻觅者位置； 4) 更新加入者位置； 5) 根据当前最小平均包络熵更新适应度和最优位置； 6) 循环迭代步骤3)至步骤5)，达到最大迭代次数后输出麻雀的位置，确定当前数据条件下 K 和 <math id="M61"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 的最优取值。 2.2 　轴箱轴承故障诊断流程 2 高速列车轴箱轴承微弱故障特征受环境噪声和其他调制成分影响较大，故障特征提取困难。本文首先利用麻雀搜索算法优化VMD参数，进而对包络熵最小的IMF分量进行快速谱峭度图分析以确定最优中心频率和带宽；然后，对IMF进行带通滤波以增强信噪比；最后，对滤波后信号进行包络解调分析来故障诊断。 图1 图1 所示为基于参数自适应VMD的高速列车轴箱轴承故障诊断过程。 3 仿真信号分析 1 为了模拟环境噪声的影响，使得构造的轴承外圈故障仿真信号更接近实际信号，额外添加高斯白噪声作为背景振动噪声。所构造的轴承外圈故障仿真信号 <math id="M62"><mi>s</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683370&type= 6.18066692 3.72533321 表达式 [ 1 1 ] 如下： <math display="block" id="M63"><mi>s</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></munderover></mstyle><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>⋅</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>C</mi><mfenced separators="|"><mrow><mi>t</mi><mo>-</mo><mi>i</mi><mi>T</mi><mo>-</mo><mi>τ</mi></mrow></mfenced></mrow></msup></mrow><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683373&type= 60.95999908 9.65200043 <math display="block" id="M64"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">n</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi><mo>-</mo><mi>i</mi><mi>T</mi><mo>-</mo><mi>τ</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mi>n</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683375&type= 38.43866730 4.65666676 式中： N p 为故障脉冲数； <math id="M65"><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683372&type= 3.80999994 3.72533321 为初始幅值； <math id="M66"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683389&type= 2.96333337 4.48733330 为系统共振频率； <math id="M67"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683378&type= 2.62466669 4.48733330 为轴承转频； C 为冲击衰减系数； <math id="M68"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683407&type= 2.45533323 2.62466669 为两相邻冲击间的时间间隔； <math id="M69"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683385&type= 2.96333337 4.48733330 为故障特征频率； <math id="M70"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683395&type= 1.43933344 2.62466669 为两相邻脉冲间因相对滑动而产生的时间误差，通常 <math id="M71"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683395&type= 1.43933344 2.62466669 的取值为 <math id="M72"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683407&type= 2.45533323 2.62466669 的1%~2%。 轴承故障仿真信号参数见 表1 表1 。 N /个 <math id="M73"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683410&type= 2.79399991 4.06400013 /Hz <math id="M74"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683412&type= 2.45533323 4.06400013 /Hz C <math id="M75"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683422&type= 2.20133328 2.37066650 /s <math id="M76"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683415&type= 1.35466671 2.37066650 /s <math id="M77"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683428&type= 2.79399991 4.06400013 /Hz 140 2 000 30 500 0.006 7 6.7×10 -5 150 设置采样频率为12 kHz，采样时长为1.5 s，轴承外圈故障仿真信号时域图如 图2 图2 所示。 (a) 无噪仿真信号；(b) 含噪仿真信号 由 图2 图2 (a)可看到明显的周期性冲击成分，叠加噪声后，周期性脉冲已淹没在噪声中，无法观测到故障脉冲(见 图2 图2 (b))。 图3 图3 所示为含噪信号经快速傅里叶变换(FFT)后的频域图。由 图3 图3 可发现仿真信号在2 kHz附近有较高的幅值，轴承故障特征频及其倍频处信号幅值并不明显。 下面按照本文所述方法对该含噪信号进行处理。利用SSA算法对当前条件下VMD的最佳模态数 K 和惩罚因子 <math id="M78"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 的组合进行自适应寻优，寻优过程如 图4 图4 所示。 为了验证本文算法(SSA-VMD)性能的优越性，以最小平均包络熵为适应度函数，用同组仿真数据进行10次实验，取10次实验数据均值，分别用粒子群优化算法(PSO-VMD)和遗传算法(GA-VMD)作为对比。不同算法迭代过程对比如 图5 图5 所示。对迭代时长求标准差，结果如 表2 表2 所示。由 表2 表2 可见：本文所提方法最小平均包络熵比粒子群算法的略高，但迭代次数明显比遗传算法的少，运行效率更高。 方法 平均迭代时长/s 迭代数/次 最小平均包络熵 标准差 SSA-VMD 12.8 5 8.520 0.013 8 GA-VMD 23.2 16 8.542 0.059 7 PSO-VMD 21.0 15 8.513 0.076 4 根据 图4 图4 所示结果，将VMD参数 K 和 <math id="M79"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 分别设置为5和1 385，对含噪信号进行分解，各模态分量如 图6 图6 (a)所示。选择包络熵最小的IMF3进行快速谱峭度图分析，结果如 图6 图6 (b)所示。根据快速谱峭度图分析结果可知，中心频率与最优带宽分别为1 kHz和2 kHz， 图6 图6 (c)所示为IMF3滤波后信号。从 图6 图6 (c)可看出，相比于 图2 图2 (b)，噪声得到有效抑制。 (a) VMD分解后IMF1~IMF5分量时域图；(b) IMF3谱峭度图( B w =2 kHz， f c =1 kHz)；(c) 滤波后IMF3时域 对滤波后的IMF3进行希尔伯特包络解调分析，结果如 图7 图7 (a)所示。 图7 图7 (a)中，不仅轴承故障特征频及其倍频清晰可见，而且轴承转频及其调制成分亦非常明显。为验证本文所提方法的优越性，分别采用EEMD及EMD和LMD等经典自适应信号分解方法对外圈故障仿真信号进行处理，并取其最佳包络谱与本文方法所得结果进行对比，如 图7 图7 (b)~(d)所示。在 图7 图7 (b)中，可观测到3倍频；在图 7 7 (c)和 7 7 (d)中可观测到2倍频，其他谐波成分受噪声干扰影响较大，而本文方法可观测到5倍频，说明本文所提方法有效降低了复杂耦合工况下噪声的影响，能有效辨识淹没在噪声中的外圈故障特征，避免出现误诊。 (a) 本文方法；(b) EEMD方法；(c) EMD方法；(d) LMD方法 4 实验数据分析 1 4.1 　实验台结构 2 为更好地验证方法的有效性，本文采用考虑高速列车轴箱轴承实际服役环境的轴承系统试验台，以最大程度地模拟真实高速列车轴箱轴承运行工况，为模型验证提供更可靠的数据支撑。试验台由激振台、电机驱动系统、加载系统等装置组成，实验台结构示意图如 图8 图8 所示。试验轴承设置于主轴中部，径向载荷由2组垂直安装的弹簧施加在被测轴承上，用以模拟实际轴负荷；主轴与电机通过万向联轴器连接，以模拟列车轴运转；轴向载荷通过外圈加载给试验轴承；通过电磁激振台施加垂向激励以模拟轨道不平顺。在径向载荷、轴向载荷、激振台垂向激励的共同作用下，能最大限度地模拟高速列车轴箱轴承实际运行工况。 试验过程中，轴承转速为800 r/min，激振台频率为50 Hz，负载为1 kN，振动加速度传感器以磁吸的方式贴于被试轴承的轴承座上。采样频率为25.6 kHz。试验对象为某型高速列车双列圆锥滚子轴承，通过打磨机、电火花机等设备自制外圈、滚动体等故障，如 图9 图9 所示。轴承参数如 表3 表3 所示。 (a) 轴承结构；(b) 轴箱轴承实物图；(c) 外圈严重擦伤；(d) 滚动体磨损 节圆直径 D p /mm 滚动体直径 D b /mm 单列滚动体数 N b /个 接触角 <math id="M80"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683542&type= 1.77800000 2.37066650 /(°) 185 26.68 17 12.083 4.2 　外圈故障分析 2 在转速为800 r/min、激振台频率为50 Hz、负载为1 kN的工况下采集外圈故障轴承振动信号。根据 表3 表3 ，结合轴承转速及文献[ 23 23 ]，外圈故障特征频表达式如 式(11) 式(11) 所示。由 式(1) 式(1) 计算可得该型高速列车轴箱轴承在800 r/min转速下的外圈故障特征频约为97 Hz。原始外圈故障时域波形及频谱如 图10 图10 所示。 (a) 时域波形；(b) 频谱 <math display="block" id="M81"><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>×</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mrow></mfenced><mo>×</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">60</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683554&type= 46.65133286 10.32933331 式中： r 为轴的转速。 由 图10 图10 (a)可以看出：由于环境噪声和其他干扰成分的调制影响，很难直接观察到轴承外圈故障脉冲。由 图10 图10 (b)可以看出：在高频带产生了较明显的固有频率成分，轴承微弱故障特征频淹没在其中，也无法被直接观测到。 图11 图11 所示为原始外圈故障信号的包络谱，可见轴承故障特征频及其倍频完全淹没在噪声中。因此，有必要对原始轴承振动信号进行处理以提高信噪比。 下面按照本文所述方法对故障轴承振动数据进行处理。首先，利用麻雀搜索算法对当前工况下的VMD最优 K 和 <math id="M82"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 的组合进行寻优，得到分解参数 K 和 <math id="M83"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683575&type= 1.94733346 2.62466669 的取值分别为6和2200。其次，对原始时域信号进行VMD分解，如 图12 图12 (a)所示。然后，选择包络熵最小值所在分量IMF2进行快速谱峭度图分析，如 图12 图12 (b)所示。根据分析结果可知，中心频率与最优带宽分别为10 666 Hz和4 266 Hz。对IMF2进行带通滤波，滤波后IMF2时域图见 图12 图12 (c)。 (a) VMD分解后IMF1~IMF6分量时域图；(b) IMF2谱峭度图( B w =4.266 kHz， f c =10.666 kHz)；(c) 滤波后IMF2时域图 在外圈故障工况下，将本文方法(SSA-VMD)分别与粒子群优化算法(PSO-VMD)和遗传算法(GA-VMD)进行对比。分别采用不同方法对同一外圈故障振动数据样本进行10次实验，所得实验结果取均值。外圈故障工况下不同优化算法性能对比如 表4 表4 所示。由 表4 表4 可见，本文所提方法的运算效率大于遗传算法和粒子群算法的运算效率。 方法 平均迭代时长/s 迭代数/次 最小平均包络熵 标准差 SSA-VMD 11.3 8 7.648 0.016 9 GA-VMD 17.2 13 7.720 0.048 2 PSO-VMD 22.0 14 8.142 0.054 3 正交性指标 F 可作为信号分解效果的评价标准，其表达式如 式(12) 式(12) 所示 [ 4 4 ] 。 F 越大，说明各子信号相关性越大，越可能出现模态混叠的现象； F 越小，说明各子信号间正交性越强，完全正交的两个信号间的 F 为0。 <math display="block" id="M84"><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>T</mi></mrow></munderover></mstyle><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow></mfrac></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683579&type= 36.15266418 13.71599960 能量保存度 E IEC 是一种表征信号在分解前后能量的对比度的指标，其表达式如 式(13) 式(13) 所示。 E IEC 越高，说明分解过程中能量泄露越少，完全理想的分解状态下 E IEC 为 [ 24 24 ] <math display="block" id="M85"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>T</mi></mrow></munderover></mstyle><mrow><mfenced open="{" close="}" separators="|"><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683581&type= 53.17066193 9.65200043 式中： <math id="M86"><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683565&type= 6.34999990 4.48733330 为信号经分解后的第 i 个分量； <math id="M87"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683586&type= 4.99533367 3.38666677 为原始信号； <math id="M88"><msub><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683598&type= 6.60400009 4.48733330 为趋势项。 对外圈故障信号分别采用本文方法、EMD、EEMD和LMD进行分解后，各IMF分量间的 F 和 E IEC 如 图13 图13 所示。由 图13 图13 可见：本文所提参数自适应VMD分解方法的 F 远小于其他经典自适应分解方法的 F ， E IEC 最大，说明经本文方法分解后各IMF间具有较好的正交性，同时，能量泄露最小。 对滤波后的IMF2进行希尔伯特包络解调分析，结果如 图14 图14 (a)所示。为验证本文所提方法的优越性，分别采用EEMD及EMD、LMD等经典自适应信号分解方法对原始振动信号进行处理，并取其最佳包络谱与本文方法进行对比。在 图14 图14 (b)中，可发现故障特征频及其2倍频，且3倍频幅值和干扰频率幅值比较接近。在 图14 图14 (c)和 图14 图14 (d)中，仅1倍频较为明显，其他谐波成分受噪声干扰影响较大，容易出现误诊；而在 图14 图14 (a)中，故障特征基频及其6次谐波频率均清晰可见，说明本文所提方法可有效降低复杂耦合工况下噪声的影响，且能有效辨识淹没在噪声中的微弱外圈故障特征，避免误诊。 (a) 本文方法；(b) EEMD方法；(c) EMD方法；(d) LMD方法 4.3 　滚动体故障分析 2 在转速为800 r/min，激振台频率为50 Hz，负载为1 kN的工况下采集滚动体故障轴承振动信号。根据文献[ 25 25 ]，滚动体故障特征频如 式(14) 式(14) 所示。根据 表1 表1 中的参数可得该型高速列车轴承在 800 r/min转速下的滚动体故障特征频约为43 Hz。 <math display="block" id="M89"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfrac><mo>⋅</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>α</mi></mrow></mfenced><mo>⋅</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">60</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683614&type= 57.48867035 12.61533356 图15 图15 所示为滚动体故障时频信号及其包络谱。从 图15 图15 (a)中隐约可见周期性冲击特征，但受环境噪声和其他干扰成分的调制影响，难以将其直接判断为滚动体故障。从 图15 图15 (b)可以发现：与外圈故障类似，滚动体故障的频域图中固有频率占主导成分，滚动体微弱故障淹没在其中。对滚动体时域信号求包络谱，结果如 图15 图15 (c)所示。在 图 15 图 15 (c)中同样无法直接观测到轴承故障特征频及其倍频。 (a) 滚动体故障频域信号；(b) 滚动体故障时域信号；(c) 滚动体故障包络谱 下面按本文方法对滚动体故障信号进行处理。利用麻雀搜索算法对当前工况下VMD参数进行自适应寻优，得到参数 K 和 <math id="M90"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43683619&type= 1.94733346 2.62466669 分别为5和1 895。对原始信号进行VMD分解，结果如 图16 图16 (a)所示。选择包络熵最小值所在的IMF3分量进行快速谱峭度图分析，结果如 图16 图16 (b)所示。由 图16 图16 (b)可见中心频率与最优带宽分别为10 400 Hz和1 600 Hz。根据分析结果对IMF3分量进行带通滤波，滤波后IMF3分量的时域信号见 图16 图16 (c)。由 图16 图16 (c)可见IMF3分量已经出现了较为明显的冲击性特征。 (a) VMD分解后IMF1~IMF5分量时域图；(b) IMF3谱峭度图( B w =1.6 kHz， f c =10.4 kHz)；(c) 滤波后IMF3时域图 在滚动体故障工况下，将本文方法(SSA-VMD)分别与粒子群优化算法(PSO-VMD)和遗传算法(GA-VMD)进行对比。分别采用不同方法对同一滚动体故障振动数据样本进行10次实验，所得实验结果取均值。滚动体故障工况下不同优化算法性能对比结果如 表5 表5 所示。由 表5 表5 可见本文所提方法的运算效率大于遗传算法和粒子群算法的运算效率。 方法 平均迭代时长/s 迭代数/次 最小平均包络熵 标准差 SSA-VMD 9.8 9 7.865 0.023 6 GA-VMD 16.6 17 7.880 0.028 4 PSO-VMD 19.4 16 7.964 0.030 5 分别采用本文方法、EMD、EEMD和LMD对滚动体故障信号进行分解，然后对各IMF分量的 F 和 E IEC 进行分析，结果如 图17 图17 所示。由 图17 图17 可见：与外圈故障信号分析结果类似，本文所提参数自适应VMD分解方法的 F 最小， E IEC 最大，说明经本文方法分解后各IMF间具有较好的正交性，同时能量泄露最小。 对滤波后的IMF3分量进行希尔伯特包络解调分析，结果如 图18 图18 (a)所示。由 图18 图18 (a)可见：淹没在噪声中的滚动体故障特征基频及其高次谐波成分成功被提取出来。分别采用EEMD及EMD、LMD方等经典方法对IMF3分量进行分析，并将其结果与本文方法所得结果进行对比。由 图18 图18 (b)和 图18 图18 (c)可见：由于受复杂耦合工况下噪声的影响，仅1倍频较为明显。 图18 图18 (d)中仅可观测到2倍频，可见LMD方法对环境噪声的抑制效果比本文方法的抑制效果差。 (a) 本文方法；(b) EEMD方法；(c) EMD方法； (d) LMD方法 5 结论 1 1) 提出了一种参数自适应变分模态分解和快速谱峭度图结合的信号分析方法。该方法以平均包络谱熵为适应度函数，利用麻雀搜索算法优化VMD的两个分解参数 K 和 α 。本文方法相比于 GA-VMD和PSO-VMD有着更高的运算效率，有效避免了人为选择VMD参数影响分解效果的问题。 2) EMD、EEMD、LMD等经典自适应信号分析方法的对比分析结果表明：经本文方法分解后各IMF分量具有较好的正交性且能量泄露最小。 3) 本文所提方法能有效降低高速列车轴箱轴承振动信号中环境噪声及其他调制频率成分的影响，且可用于提取复杂耦合工况下的微弱轴承故障特征，具有良好的工程适用性。 参考文献 ZHENG Zejun , SONG Dongli , XU Xiao , et al . A fault diagnosis method of bogie axle box bearing based on spectrum whitening demodulation [J]. Sensors(Basel, Switzerland) , 2020 , 20 ( 24 ): 7155 . A fault diagnosis method of bogie axle box bearing based on spectrum whitening demodulation 宋宏智 , 李力 , 杨兴宽 , 等 . 高速机车轴承故障诊断与剩余寿命预测的发展及展望 [J]. 轴承 , 2020 ( 3 ): 61 - 67 . 高速机车轴承故障诊断与剩余寿命预测的发展及展望 SONG Hongzhi , LI Li , YANG Xingkuan , et al . Development and prospect for fault diagnosis and remain life prediction of high speed locomotive bearings [J]. Bearing , 2020 ( 3 ): 61 - 67 . Development and prospect for fault diagnosis and remain life prediction of high speed locomotive bearings LIU Lei , SONG Dongli , ZHANG Weihua . Fault pattern recognition of axle box bearings for high-speed EMU based on onboard real-time temperature data [C]// 2019 Prognostics and System Health Management Conference (PHM-Qingdao) . Qingdao, China : IEEE , 2019 : 1 - 5 . Fault pattern recognition of axle box bearings for high-speed EMU based on onboard real-time temperature data HUANG E , SHEN Z , LONG S , et al . The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis [J]. Proceedings Mathematical Physical & Engineering Sciences , 1998 , 454 ( 1971 ): 903 - 995 . The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis MORLET J , ARENS G , FOURGEAU E , et al . Wave propagation and sampling theory—Part II: sampling theory and complex waves [J]. Geophysics , 1982 , 47 ( 2 ): 222 - 236 . Wave propagation and sampling theory—Part II: sampling theory and complex waves SUN Yongjian , LI Shaohui , WANG Xiaohong . Bearing fault diagnosis based on EMD and improved Chebyshev distance in SDP image [J]. Measurement , 2021 , 176 : 109100 . Bearing fault diagnosis based on EMD and improved Chebyshev distance in SDP image ZHENG Jinde , PAN Haiyang . Mean-optimized mode decomposition: an improved EMD approach for non-stationary signal processing [J]. ISA Transactions , 2020 , 106 : 392 - 401 . Mean-optimized mode decomposition: an improved EMD approach for non-stationary signal processing 李霖 , 张西宁 , 刘书语 , 等 . 基于二维经验小波纹理域特征自适应提取的轴承故障诊断方法 [J]. 西安交通大学学报 , 2021 , 55 ( 12 ): 79 - 86 . 基于二维经验小波纹理域特征自适应提取的轴承故障诊断方法 LI Lin , ZHANG Xining , LIU Shuyu , et al . Bearing fault diagnosis method based on 2D empirical wavelet transform texture domain feature adaptive extraction [J]. Journal of Xi'an Jiaotong University , 2021 , 55 ( 12 ): 79 - 86 . Bearing fault diagnosis method based on 2D empirical wavelet transform texture domain feature adaptive extraction HU Mantang , WANG Guofeng , MA Kaile , et al . Bearing performance degradation assessment based on optimized EWT and CNN [J]. Measurement , 2021 , 172 : 108868 . Bearing performance degradation assessment based on optimized EWT and CNN DRAGOMIRETSKIY K , ZOSSO D . Variational mode decomposition [J]. IEEE Transactions on Signal Processing , 2014 , 62 ( 3 ): 531 - 544 . Variational mode decomposition 陈东宁 , 张运东 , 姚成玉 , 等 . 基于FVMD多尺度排列熵和GK模糊聚类的故障诊断 [J]. 机械工程学报 , 2018 , 54 ( 14 ): 16 - 27 . 基于FVMD多尺度排列熵和GK模糊聚类的故障诊断 CHEN Dongning , ZHANG Yundong , YAO Chengyu , et al . Fault diagnosis based on FVMD multi-scale permutation entropy and GK fuzzy clustering [J]. Journal of Mechanical Engineering , 2018 , 54 ( 14 ): 16 - 27 . Fault diagnosis based on FVMD multi-scale permutation entropy and GK fuzzy clustering 王斐 , 房立清 , 赵玉龙 , 等 . 基于VMD和SVDD的滚动轴承早期微弱故障检测和性能退化评估研究 [J]. 振动与冲击 , 2019 , 38 ( 22 ): 224 - 230 . 基于VMD和SVDD的滚动轴承早期微弱故障检测和性能退化评估研究 WANG Fei , FANG Liqing , ZHAO Yulong , et al . Rolling bearing early weak fault detection and performance degradation assessment based on VMD and SVDD [J]. Journal of Vibration and Shock , 2019 , 38 ( 22 ): 224 - 230 . Rolling bearing early weak fault detection and performance degradation assessment based on VMD and SVDD 万书亭 , 张雄 , 豆龙江 . 强噪源干扰下的滚动轴承复合故障分离方法研究 [J]. 中南大学学报(自然科学版) , 2018 , 49 ( 8 ): 1950 - 1959 . 强噪源干扰下的滚动轴承复合故障分离方法研究 WAN Shuting , ZHANG Xiong , DOU Longjiang . Separation of composite rolling bearings fault features with strong noise interference [J]. Journal of Central South University(Science and Technology) , 2018 , 49 ( 8 ): 1950 - 1959 . Separation of composite rolling bearings fault features with strong noise interference ANEESH C , KUMAR S , HISHAM P M , et al . Performance comparison of variational mode decomposition over empirical wavelet transform for the classification of power quality disturbances using support vector machine [J]. Procedia Computer Science , 2015 , 46 : 372 - 380 . Performance comparison of variational mode decomposition over empirical wavelet transform for the classification of power quality disturbances using support vector machine MIAO Yonghao , ZHAO Ming , LIN Jing . Identification of mechanical compound-fault based on the improved parameter-adaptive variational mode decomposition [J]. ISA Transactions , 2019 , 84 : 82 - 95 . Identification of mechanical compound-fault based on the improved parameter-adaptive variational mode decomposition 李华 , 伍星 , 刘韬 , 等 . 基于信息熵优化变分模态分解的滚动轴承故障特征提取 [J]. 振动与冲击 , 2018 , 37 ( 23 ): 219 - 225 . 基于信息熵优化变分模态分解的滚动轴承故障特征提取 LI Hua , WU Xing , LIU Tao , et al . Bearing fault feature extraction based on VMD optimized with information entropy [J]. Journal of Vibration and Shock , 2018 , 37 ( 23 ): 219 - 225 . Bearing fault feature extraction based on VMD optimized with information entropy 蒋宇涵 , 华春蓉 , 董大伟 , 等 . 基于优化VMD的车轴裂纹和车轮扁疤故障诊断 [J]. 噪声与振动控制 , 2021 , 41 ( 6 ): 71 - 76 . 基于优化VMD的车轴裂纹和车轮扁疤故障诊断 JIANG Yuhan , HUA Chunrong , DONG Dawei , et al . Fault diagnosis of axle cracks and wheel flats based on optimized VMD [J]. Noise and Vibration Control , 2021 , 41 ( 6 ): 71 - 76 . Fault diagnosis of axle cracks and wheel flats based on optimized VMD 司呈鑫 , 章翔峰 , 姜宏 , 等 . 自适应VMD-FSK对低速重载下的转轴裂纹诊断 [J]. 组合机床与自动化加工技术 , 2021 ( 10 ): 111 - 114 . 自适应VMD-FSK对低速重载下的转轴裂纹诊断 SI Chengxin , ZHANG Xiangfeng , JIANG Hong , et al . Fault diagnosis of rotating shaft crack under low speed and heavy load by adaptive VMD-FSK [J]. Modular Machine Tool & Automatic Manufacturing Technique , 2021 ( 10 ): 111 - 114 . Fault diagnosis of rotating shaft crack under low speed and heavy load by adaptive VMD-FSK 周全 . 汽车发动机异响诊断算法及产生机理的研究 [D]. 杭州 : 浙江大学 , 2021 : 35 - 53 . 汽车发动机异响诊断算法及产生机理的研究 ZHOU Quan . Research on diagnosis algorithm and generation mechanism of abnormal noise for automobile engine [D]. Hangzhou : Zhejiang University , 2021 : 35 - 53 . Research on diagnosis algorithm and generation mechanism of abnormal noise for automobile engine XUE Jiankai , SHEN Bo . A novel swarm intelligence optimization approach: sparrow search algorithm [J]. Systems Science & Control Engineering , 2020 , 8 ( 1 ): 22 - 34 . A novel swarm intelligence optimization approach: sparrow search algorithm ANTONI J . Fast computation of the kurtogram for the detection of transient faults [J]. Mechanical Systems and Signal Processing , 2007 , 21 ( 1 ): 108 - 124 . Fast computation of the kurtogram for the detection of transient faults 唐贵基 , 王晓龙 . 自适应最大相关峭度解卷积方法及其在轴承早期故障诊断中的应用 [J]. 中国电机工程学报 , 2015 , 35 ( 6 ): 1436 - 1444 . 自适应最大相关峭度解卷积方法及其在轴承早期故障诊断中的应用 TANG Guiji , WANG Xiaolong . Adaptive maximum correlated kurtosis deconvolution method and its application on incipient fault diagnosis of bearing [J]. Proceedings of the CSEE , 2015 , 35 ( 6 ): 1436 - 1444 . Adaptive maximum correlated kurtosis deconvolution method and its application on incipient fault diagnosis of bearing 胡爱军 , 许莎 , 向玲 , 等 . 滚动轴承外圈多点故障特征分析 [J]. 机械工程学报 , 2020 , 56 ( 21 ): 110 - 120 . 滚动轴承外圈多点故障特征分析 HU Aijun , XU Sha , XIANG Ling , et al . Characteristic analysis of multi-point faults on the outer race of rolling element bearing [J]. Journal of Mechanical Engineering , 2020 , 56 ( 21 ): 110 - 120 . Characteristic analysis of multi-point faults on the outer race of rolling element bearing CHEN Qiuhui , HUANG N , RIEMENSCHNEIDER S , et al . A B-spline approach for empirical mode decompositions [J]. Advances in Computational Mathematics , 2006 , 24 ( 1 ): 171 - 195 . A B-spline approach for empirical mode decompositions 姚徳臣 . 轨道交通轴承故障诊断技术 [M]. 北京 : 中国铁道出版社 , 2017 : 23 - 33 . 轨道交通轴承故障诊断技术 YAO Dechen . Fault diagnosis technology of rail transit bearing [M]. Beijing : China Railway Publishing House , 2017 : 23 - 33 . Fault diagnosis technology of rail transit bearing 10.11817/j.issn.1672-7207.2023.04.013 TH17 A 1672-7207（2023）04-1344-14 董俭雄, 宋冬利, 李林, 等. 参数自适应VMD在高速列车轴箱轴承故障诊断中的应用[J].中南大学学报(自然科学版), 2023, 54(4): 1344-1357. 董俭雄, 宋冬利, 李林, 等. 参数自适应VMD在高速列车轴箱轴承故障诊断中的应用[J].中南大学学报(自然科学版), 2023, 54(4): 1344-1357. DONG Jianxiong, SONG Dongli, LI Lin, et al. Application of parameter adaptive VMD in fault diagnosis of axle box bearing of high-speed train[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1344-1357. DONG Jianxiong, SONG Dongli, LI Lin, et al. Application of parameter adaptive VMD in fault diagnosis of axle box bearing of high-speed train[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1344-1357. 国家重点研发计划项目(2019YFB1405401)；西南交通大学学科交叉基础研究项目(2682021ZTPY099) Project(51925402) supported by the National Key Research and Development Program of China; Project(2682021ZTPY099) supported by the Interdisciplinary Basic Research Project of Southwest Jiaotong University 2022-06-09 2022-08-12 2023-04-26 2023-05-22 37692788 中南大学学报(自然科学版) 4 54