裂纹齿轮时变啮合刚度与传动系统振动机理 Time varying meshing stiffness of cracked gears and fault vibration for transmission system 1 2 3 4 corresp first-author 莫帅，博士，博士生导师，从事传动系统动力学与故障诊断研究；E-mail：moshuai2010@163.com. 莫帅，博士，博士生导师，从事传动系统动力学与故障诊断研究；E-mail：moshuai2010@163.com. moshuai2010@163.com 1 1 1 5 6 天津工业大学 机械工程学院，天津，300387 天津工业大学 机械工程学院，天津，300387 School of Mechanical Engineering; Tiangong University, Tianjin 300387, China School of Mechanical Engineering; Tiangong University, Tianjin 300387, China 广西大学 机械工程学院，广西 南宁，530004 广西大学 机械工程学院，广西 南宁，530004 School of Mechanical Engineering; Guangxi University, Nanning 530004, China School of Mechanical Engineering; Guangxi University, Nanning 530004, China 华中科技大学 数字制造装备与技术国家重点实验室，湖北 武汉，430074 华中科技大学 数字制造装备与技术国家重点实验室，湖北 武汉，430074 State Key Lab of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China State Key Lab of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China 江苏万基传动科技有限公司，江苏 泰州，225400 江苏万基传动科技有限公司，江苏 泰州，225400 Jiangsu Wanji Transmission Technology Co. Ltd., Taizhou 225400, China Jiangsu Wanji Transmission Technology Co. Ltd., Taizhou 225400, China 北京航空航天大学 虚拟现实技术与系统国家重点实验室，北京，100191 北京航空航天大学 虚拟现实技术与系统国家重点实验室，北京，100191 State Key Laboratory of Virtual Reality Technology and System, Beihang University, Beijing 100191, China State Key Laboratory of Virtual Reality Technology and System, Beihang University, Beijing 100191, China 宁波中大力德智能传动股份有限公司，浙江 宁波，315301 宁波中大力德智能传动股份有限公司，浙江 宁波，315301 Ningbo Zhongda Leader Transmission Equipment Co. Ltd, Ningbo 315301, China Ningbo Zhongda Leader Transmission Equipment Co. Ltd, Ningbo 315301, China 赵俊 研究齿轮裂纹对时变啮合刚度和振动特性的影响机理。首先，将齿轮齿廓分为过渡曲线、渐开线非啮合区和渐开线啮合区3个部分建立精确的齿廓模型，再结合势能法改进时变啮合刚度计算方法。其次，建立齿轮裂纹分析模型，将裂纹扩展路径和有效厚度的限制线分别假设为直线和抛物线，根据几何法和裂纹终止点的位置，改进有效截面积和截面惯性矩的计算方法，求解不同裂纹状态下的时变啮合刚度曲线。最后，建立六自由度裂纹齿轮故障动力学模型，采用龙格-库塔法求解不同裂纹下齿轮传动系统的振动特性和幅频特性，通过小波变换对振动特征进行时频分析。同时，采用统计指标的方法，研究齿轮裂纹对传动系统振动响应的敏感度。研究结果表明：裂纹齿轮在啮合过程中会产生冲击特征，随着裂纹情况的加剧，冲击加剧；峭度对振动响应最为敏感。 The influence mechanism of gear crack on time-varying meshing stiffness and vibration characteristics was studied. Firstly, the gear tooth profile was divided into three parts, i. e., transition curve, involute non-meshing area and involute meshing area, to establish an accurate tooth profile model, and then the calculation method of time-varying meshing stiffness was improved by combining the potential energy method. Secondly, the gear crack analysis model was established. The crack propagation path and the limiting line of effective thickness were assumed to be straight lines and parabolas, respectively. According to the geometric method and the position of crack termination point, the calculation methods of effective cross-sectional area and cross-sectional inertia moment were improved, and the time-varying meshing stiffness curves in different crack states were solved. Finally, the six free degree failure dynamics of cracked gear was established, and the vibration and amplitude-frequency characteristics of gear transmission system with different cracks were solved by Runge-Kutta method. The time-frequency analysis of the vibration characteristics was carried out through wavelet transform. At the same time, the statistical index method was used to study the sensitivity of the gear crack to the vibration response of the transmission system. The results show that the cracked gear will produce impact characteristics in the meshing process. With the aggravation of the crack, the impact will be more obvious. The kurtosis is the most sensitive to the vibration response. 齿轮裂纹 时变啮合刚度 齿轮动力学模型 振动响应 故障机理 gear crack time varying meshing stiffness gear dynamics model vibration response failure mechanism http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704523&type= 14.66796875 14.66793823 对齿轮系统进行健康监测对于防止系统突然失效至关重要。裂纹是齿轮主要故障之一，裂纹齿轮的时变啮合刚度会受到裂纹变化的影响 [ 1 1 - 3 3 1-3 ] 。许德涛等 [ 4 4 ] 基于有限元研究了裂纹的扩展路径对时变啮合刚度的影响；WU等 [ 5 5 ] 建立了齿轮裂纹分析模型，计算了不同齿根裂纹深度下的时变啮合刚度，并研究了裂纹扩展路径对齿轮传动系统的振动响应；CHEN等 [ 6 6 - 7 7 6-7 ] 提出了不同齿轮裂纹长度下时变啮合刚度的计算模型，研究了沿齿宽方向的裂纹对齿轮传动系统的动态响应及统计指标的影响；MA等 [ 8 8 ] 考虑了真实的过渡曲线，对能量法求解时变啮合刚度的方法进行修正，同时将齿轮裂纹的扩展路径和限制线假设为不同形式的曲线，并用有限元求解时变啮合刚度以验证裂纹扩展路径和有效厚度的限制线分别假设为直线和抛物线的合理性；REZAEI等 [ 9 9 ] 对斜齿轮中裂纹位置的检测原理进行了研究，提出了一种检测方法；孟宗等 [ 10 10 ] 研究了沿齿厚方向扩展的齿轮裂纹对时变啮合刚度影响；YU等 [ 11 11 ] 对齿轮齿根裂纹的系统动力学特性进行了分析。 如上所述，研究者致力于裂纹齿轮时变啮合刚度建模和动力学特性研究，本文在已有的研究基础上，建立精确齿轮齿廓模型和构建包含裂纹位置、裂纹深度和裂纹长度的裂纹齿轮时变啮合刚度计算模型，定量研究裂纹齿轮的时变啮合刚度变化。最后,根据六自由度齿轮动力学振动模型，仿真求解得到裂纹齿轮的振动特征和幅频特性，通过小波变换分析法研究裂纹故障下的齿轮振动特征，采用统计指标的方法对齿轮传动系统故障振动的敏感度进行研究，为齿轮传动系统的裂纹故障检测提供理论基础。 1 裂纹齿轮副时变啮合刚度计算建模 1 1.1 　健康齿轮副时变啮合刚度计算建模 2 本文将轮齿考虑为悬臂梁结构，通过对齿轮齿廓的精确建模和受力变形情况的分析，修正时变化啮合刚度的计算模型。齿轮齿廓模型如 图1 图1 所示。齿轮齿廓为标准渐开线齿廓，包括过渡圆角曲线 AB 段和渐开线齿廓 BD 段，其中， BC 段为渐开线齿廓的非啮合区， α 为压力角， α p 为啮合角； α 1 和 α 2 分别为点 A 和 B 的齿基角； F 为啮合力； h x 为啮合点有效厚度； x A 、 x B 、 x C 和 x p 分别为点 A 、 B 、 C 、 p 的横坐标。 根据文献[ 12 12 ]，渐开线部分可以用齿轮转角 θ 来表达，过渡曲线 AB 段由齿轮加工制造过程中刀具确定， AB 段的方程如 式(1) 式(1) 所示： <math display="block" id="M1"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mi>R</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>φ</mi><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>ϕ</mi><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo>-</mo><mi>φ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mi>R</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>φ</mi><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>ϕ</mi><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo>-</mo><mi>φ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msup><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mi>m</mi><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>φ</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>A</mi><mo>/</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>m</mi><mo stretchy="false">(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>+</mo><msup><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">π</mi><mi>m</mi><mo>/</mo><mn mathvariant="normal">4</mn><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mi>m</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>α</mi><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704332&type= 57.15000153 32.34266663 式中， <math id="M2"><mi>ϕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704364&type= 1.94733346 3.38666677 为过渡曲线的参变量， <math id="M3"><mi>α</mi><mo>≤</mo><mi>ϕ</mi><mo>≤</mo><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704357&type= 15.23999977 3.38666677 ； R 为分度圆半径； <math id="M4"><msup><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704343&type= 2.87866688 2.96333337 为顶隙系数； <math id="M5"><msup><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704369&type= 3.04800010 2.96333337 为齿顶高系数； m 为齿轮模数； <math id="M6"><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=44704371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704360&type= 3.80999994 3.72533321 为刀具圆角圆心到轮齿中线的距离； <math id="M7"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704347&type= 3.64066648 3.72533321 为刀具圆角圆心到齿槽中线的距离； R 1 为刀具圆角半径； φ 为刀具圆角圆心与刀具圆心的连线和中心距延长线的夹角。 由变截面弹性悬臂梁理论 [ 13 13 ] 可知，将齿轮考虑为悬臂梁结构，则齿轮的赫兹刚度 k h 、轴向压缩刚度 k a 、弯曲刚度 k b 和剪切刚度 k s 的计算公式如 式(2) 式(2) 所示。 <math display="block" id="M8"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfrac><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>E</mi><mi>I</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>+</mo></mtd></mtr><mtr><mtd><mtext>         </mtext><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>E</mi><mi>I</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mtext>         </mtext><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>E</mi><mi>I</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></mfrac><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow/><mrow><mi>B</mi></mrow></msub></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1.2</mn><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><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mi>G</mi><mi>h</mi><mi>L</mi></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1.2</mn><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><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mi>G</mi><mi>h</mi><mi>L</mi></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo></mtd></mtr><mtr><mtd><mtext>         </mtext><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1.2</mn><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><msub><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mi>G</mi><mi>h</mi><mi>L</mi></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></mrow></mfrac><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>α</mi></mrow><mrow><mi>E</mi><mi>h</mi><mi>L</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>+</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>α</mi></mrow><mrow><mi>E</mi><mi>h</mi><mi>L</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>+</mo></mtd></mtr><mtr><mtd><mtext>         </mtext><msubsup><mo>∫</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>α</mi></mrow><mrow><mi>E</mi><mi>h</mi><mi>L</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">π</mi><mi>E</mi><mi>L</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704372&type= 65.36266327 85.76733398 式中： E 为弹性模量； L 为齿轮齿宽； G 为剪切模量； v 为泊松比； y p 为 p 点纵坐标； I 为齿廓任意处的截面惯性矩 [ 14 14 ] 。 <math display="block" id="M9"><mi>I</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">12</mn></mrow></mfrac><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704395&type= 16.08666801 7.19666624 式中： h 为有效厚度。 在齿轮啮合过程中，一般假设其受到的力为线性变化，此时，基体刚度 k f 计算公式如 式(4) 式(4) 所示。 <math display="block" id="M10"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">[</mo><msup><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>α</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">]</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><msub><mrow><mi>α</mi></mrow><mrow><mi mathvariant="normal">P</mi></mrow></msub></mrow><mrow><mi>E</mi><mi>L</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704397&type= 77.04666901 12.86933327 式中： u f =R b /cos α - R f ； S f 为齿根圆弧长 S f =2 θ f R f ； θ f 为齿根圆弧对应圆心角的一半， h f = r f / r int ； r int 为轮毂半径； r f 为齿根圆半径； P * 、 Q * 、 M * 和 L * 为计算参数，可由 式(5) 式(5) 计算得到。 <math display="block" id="M11"><msup><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><msubsup><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>/</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>+</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>+</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704383&type= 76.03067017 4.82600021 式中： <math id="M12"><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704384&type= 3.30200005 3.72533321 ~ <math id="M13"><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704377&type= 3.21733332 3.72533321 为拟合系数； <math id="M14"><msup><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704409&type= 4.31799984 2.96333337 代表基体刚度计算参数 P * ， Q * ， M * 和 L * 。 基体刚度计算参数的拟合系数如 表1 表1 所示。综上所述，时变啮合刚度的计算公式如 式(6) 式(6) 所示。 <math display="block" id="M15"><mi>k</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>i</mi></mrow></munderover></mstyle><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mfenced close="" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>h</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704388&type= 64.76999664 10.66800022 <math display="block" id="M16"><mfenced open="" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704405&type= 39.62400055 10.66800022 ； i =1，2 2 参数 6 拟合系数 A i B i C i D i E i F i L * -50.752 0×10 -5 185.500 0×10 -3 0.053 8×10 -4 53.300 0×10 -3 0.289 5 0.923 6 Q * -6.204 2×10 -5 9.088 9×10 -3 -4.096 0×10 -4 7.829 7×10 -3 -0.147 2 0.690 4 M * 60.111 1×10 -5 28.100 0×10 -3 -83.431 0×10 -4 -9.925 6×10 -3 0.162 4 0.908 6 P * -5.574 0×10 -5 -1.998 6×10 -3 -2.301 5×10 -4 4.770 2×10 -3 0.027 1 6.804 5 1.2 　齿轮裂纹沿齿厚分布计算建模 2 齿轮在啮合过程中受弯曲应力作用，导致齿轮裂纹会沿2个不同方向扩展，分别为沿齿厚扩展和沿齿宽扩展。当齿轮裂纹沿齿厚扩展时，通常将齿轮裂纹的扩展路径假设为直线，有效厚度的限制线假设为抛物线，通过有效厚度来计算有效截面积和截面惯性矩。 沿齿厚的裂纹几何模型如 图2 图2 所示，其中， α c 为裂纹扩展角， q 为裂纹深度， q max 为裂纹起始点与裂纹扩展最左端点之间的距离， S 为裂纹终止点。过 S 点与齿廓齿顶点 D 的抛物线作为有效厚度的限制线，此时，轮齿上任意位置有效厚度的计算公式如 式(7) 式(7) 所示。 (a) q < q max ；(b) q > q max <math display="block" id="M17"><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">k</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">2</mn><mi>y</mi><mo>,</mo><mi>x</mi><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>+</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>></mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704416&type= 60.62133789 14.90133286 式中： h crack 为裂纹齿轮的有效厚度； <math id="M18"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704426&type= 1.94733346 3.38666677 为齿轮齿廓上任意一点的纵坐标； x S 和 y S 分别为裂纹终止点 S 的横坐标和纵坐标。 裂纹沿齿厚通过齿中线后，裂纹深度 q=q max +q S ， q S 的最大值等于 q max 。而实际上，在裂纹扩展到整个齿厚之前，轮齿就会发生断裂，当裂纹通过齿中线以后，轮齿上任意位置有效厚度的计算公式如 式(8) 式(8) 所示。 <math display="block" id="M19"><msub><mrow><mi>h</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">k</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">2</mn><mi>y</mi><mo>,</mo><mi>x</mi><mo><</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mi>y</mi><mo>-</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</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>x</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>≤</mo><mi>x</mi><mo>≤</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>+</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>></mo><msub><mrow><mi>x</mi></mrow><mrow><mi>S</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704442&type= 60.62133789 26.67000008 1.3 　齿轮裂纹沿齿宽分布计算建模 2 为了研究裂纹沿齿宽不均匀扩展对时变啮合刚度的影响，用微元法将整个齿宽分成许多独立的薄片，薄片的裂纹深度为 q x ，且薄片沿齿宽方向的裂纹为贯穿状态，故可将薄片假设为裂纹沿齿厚扩展模型进行分析。沿齿宽方向的裂纹几何模型如 图3 图3 所示。 (a) 裂纹齿轮；(b) 齿宽切片；(c) 裂纹模型 当沿齿宽的裂纹没有贯穿时，沿齿宽裂纹的长度 q ( x )计算公式如 式(9) 式(9) 所示。 <math display="block" id="M20"><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mroot><mrow><mfrac><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mrow><mrow/></mroot><mo>,</mo><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi>b</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>L</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704453&type= 40.13199997 14.73200035 当沿齿宽的裂纹贯穿时，沿齿宽裂纹的长度 q ( x )计算公式如 式(10) 式(10) 所示。 <math display="block" id="M21"><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mfrac><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>L</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>L</mi><mo>-</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704431&type= 40.30133438 9.56733322 式中： <math id="M22"><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704448&type= 3.21733332 4.48733330 为初始裂纹深度； <math id="M23"><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704450&type= 3.21733332 4.48733330 为齿轮非工作侧贯穿裂纹深度。 2 齿轮裂纹对时变啮合刚度和载荷分配的影响机理 1 为了研究不同裂纹深度、裂纹长度和裂纹扩展角对时变啮合刚度的影响，采用如 表2 表2 所示的齿轮传动系统基本参数。 齿轮 齿数/个 模数/mm 压力角 α /(°) 齿宽 b /mm 弹性模量 E / (N·m -2 ) 泊松比 ν 质量/kg 转动惯量/(kg·mm -2 ) 转速 n / (rad·min -1 ) 主动轮 24 3 20 18 2.06×10 11 0.3 0.81 1 180 2 000 从动轮 40 3 20 18 2.06×10 11 0.3 1.20 5 720 1 200 2.1 　齿轮裂纹对时变啮合刚度的影响机理 2 初始裂纹深度 q 0 =2.4 mm，不同裂纹扩展角下的时变啮合刚度如 图4 图4 (a)所示。假设裂纹沿齿宽方向贯穿，随着裂纹扩展角 α c 的增大，时变啮合刚度出现减小的现象。 (a) 不同裂纹扩展角；(b) 不同裂纹深度； (c) 不同裂纹长度 裂纹扩展角为 α c =60°，裂纹深度 q =0~5.6 mm，间隔为0.8 mm，不同裂纹深度下的时变啮合刚度如 图4 图4 (b)所示。假设裂纹沿齿宽方向贯穿，当齿轮出现裂纹时，刚度就会减小，当齿轮裂纹较小时，刚度的减小量不大；随着裂纹深度的增大，刚度的减小程度加剧，特别是当裂纹接近4.8 mm时，啮合刚度大幅降低，因为裂纹穿过齿中线轮齿变形会突然加剧；随着 q 的增大，轮齿有断裂的趋势，啮合刚度大幅下降。 初始裂纹深度 q 0 =4.0 mm，裂纹扩展角 α c =60°，不同裂纹长度下的时变啮合刚度如 图4 图4 (c)所示。从 图4 图4 (c)可见：随着裂纹的产生和加剧，会出现与 图4 图4 (b)中一样的变化趋势，但刚度减小的幅度与初始裂纹深度有关。 2.2 　齿轮裂纹对载荷分配的影响机理 2 在齿轮啮合过程中，裂纹齿轮受到的接触载荷会随着啮合位置的改变而改变，通过计算载荷分配比来研究齿轮受力情况，载荷分配系数 <math id="M24"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">r</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704461&type= 4.57200003 3.72533321 计算公式如 式(11) 式(11) 所示。 <math display="block" id="M25"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">r</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>R</mi></mrow><mrow><mi>b</mi><mi>p</mi></mrow></msub><mo>/</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704471&type= 20.74333191 3.72533321 式中： F i 为第 i 对齿的接触力； R bp 为主动轮的基圆半径； T 0 为主动轮转矩。 裂纹齿轮接触载荷的变化如 图5 图5 所示。从 图5 图5 可见：初始裂纹 q 0 =2 mm；在单齿啮合区，只有裂纹齿轮参与啮合，其载荷比为1；在双齿啮合区，裂纹齿轮副接触载荷减少，而其相邻的健康齿轮副接触载荷增大，导致裂纹齿副的承载能力下降。 (a) 裂纹齿轮副；(b) 健康齿轮副 3 含裂纹的齿轮传动系统故障振动响应 1 3.1 　含裂纹的齿轮传动系统故障动力学建模 2 齿轮系统动力学模型如 图6 图6 所示。 图6 图6 中， m p 主动轮质量， m g 从动轮质量， K m 为时变啮合刚度， C m 为齿间啮合阻尼， r 1 和 r 2 为分度圆半径， J 1 和 J 2 轮转动惯量， T 1 和 T 2 为施加的扭矩， K p1 和 K p2 为径向支撑刚度， C p1 和 C p2 为径向支撑阻尼，下标1表示主动轮，下标2表示从动轮。该模型考虑 x 方向、 y 方向和 θ 方向的自由度。根据 图6 图6 建立的动力学方程如 式(12) 式(12) 所示。 <math display="block" id="M26"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mi>¨</mi></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">f</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>y</mi></mrow><mi>¨</mi></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>R</mi><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>m</mi></mrow><mrow><mi>g</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mi>¨</mi></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>m</mi></mrow><mrow><mi>g</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>y</mi></mrow><mi>¨</mi></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mi>¨</mi></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi>R</mi><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>θ</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704488&type= 76.70800018 70.61199951 裂纹齿轮的动态特性计算流程图如 图7 图7 所示。 3.2 　齿轮裂纹对传动系统故障振动响应的影响 2 为了研究裂纹对齿轮传动系统振动的影响，将裂纹齿轮的时变啮合刚度代入动力学方程，得到不同裂纹深度和长度下的振动位移和副频特性。裂纹参数分为4种情况，即case1， q 0 =0 mm， q 1 = 0 mm；case2， q 0 =2.4 mm， q 1 =0 mm；case3， q 0 =2.4 mm， q 1 =2.4 mm；case4， q 0 =4.0 mm， q 1 = 4.0 mm， 图8 图8 所示为不同情况下沿齿轮接触线 y 方向的振动位移和频域响应图。从 图8 图8 可见：健康齿轮的振动位移比较平稳，而裂纹齿轮会产生振动冲击，当裂纹较小时，这种振动冲击信号并不明显，而随着裂纹的加深，振动冲击会逐渐加剧。 (a) case1振动位移；(b) case1幅频特性；(c) case2振动位移；(d) case2幅频特性； (e) case3振动位移；(f) case3幅频特性；(g) case4振动位移；(h) case4幅频特性 采用小波变换的方法对振动位移进行时频分析。动态特性的时频分析结果如 图9 图9 所示。从 图9 图9 可见：在裂纹处，颜色明显变亮，其振动幅值变大；随着裂纹深度的增加，其裂纹的振动位移越明显，振动幅值越大。 (a) case1；(b) case2；(c) case3；(d) case4 3.3 　齿轮裂纹对传动系统振动响应灵敏性分析 2 许多基于时域波形的简单信号度量在机械故障诊断中广泛使用，统计指标均方根、峭度和Sr函数常用于总体振动水平评估，其中，Sr函数表达式为 <math display="block" id="M27"><mi>S</mi><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>/</mo><mi>N</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><mi>x</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>-</mo><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow><mo>/</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>/</mo><mi>N</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mi>x</mi><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>-</mo><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704497&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704517&type= 78.40132904 11.93799973 式中： N 为振动位移数据点总数； <math id="M28"><mover accent="true"><mi>x</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704528&type= 1.60866666 2.37066650 为振动位移的平均值； x ( n )为振动位移中第 n 个幅值； n =1，2，3，…， N 。 每个统计指标的数量级存在差异，为了综合评价各统计指标的敏感性，对统计指标按 式(14) 式(14) 进行转换。 <math display="block" id="M29"><msub><mrow><mi>Y</mi></mrow><mrow><mi mathvariant="normal">S</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><mn mathvariant="normal">100</mn><mi mathvariant="normal">%</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704519&type= 35.13666534 4.48733330 式中： Y S 为转化后的裂纹齿轮振动位移统计指标； Y j 为转化前的裂纹齿轮振动位移统计指标， j =2，3，4，分别对应不同裂纹扩展角、裂纹长度和裂纹深度的裂纹； Y 1 为转化后的健康齿轮振动位移统计指标。振动位移的统计指标如 图10 图10 所示。从 图10 图10 可见：随着裂纹深度的增大，3种统计指标都表现出增大的趋势，但峭度对齿轮裂纹的振动信号更敏感。 (a) 裂纹扩展角；(b) 裂纹深度；(c) 裂纹长度 4 结论 1 1) 对齿廓过渡曲线部分进行精确建模，考虑裂纹沿齿厚和齿宽2个方向扩展，将裂纹扩展路径和有效厚度限制线分别假设为直线和抛物线，根据建立刚度计算模型，求解裂纹齿轮的时变啮合刚度曲线，发现沿齿厚方向和齿宽方向上有着相同的变化趋势，随着裂纹深度或长度的增加，时变啮合刚度减小，且时变啮合刚度的减小量随着裂纹程度的加剧而增大。对齿轮啮合过程中的接触载荷进行分析，当裂纹增大时，裂纹齿上的载荷减小，相邻齿轮副的接触载荷增大，即裂纹齿轮副承载能力下降，相邻齿轮副易产生裂纹故障。 2) 为了研究齿轮裂纹对齿轮动力学系统的影响，建立六自由度裂纹齿轮故障动力学模型，同时对其振动特性和幅频特性采用小波变换原理进行分析。当裂纹程度加剧时，时变啮合刚度减小，振动幅值增大，会引起齿轮传动系统出现更大的噪声和冲击。 3) 随着裂纹深度的增大，振动位移的均方根、峭度和Sr函数都增大，但峭度对齿轮裂纹的振动信号最敏感。 参考文献 CHEN Yuan , ZHU Rupeng , JIN Guanghu , et al . Influence of crack depth on dynamic characteristics of spur gear system [J]. Journal of Vibration Engineering & Technologies , 2019 , 7 ( 3 ): 227 - 233 . Influence of crack depth on dynamic characteristics of spur gear system YANG Lantao , WANG Liming , YU Wennian , et al . Investigation of tooth crack opening state on time varying meshing stiffness and dynamic response of spur gear pair [J]. Engineering Failure Analysis , 2021 , 121 : 105181 . Investigation of tooth crack opening state on time varying meshing stiffness and dynamic response of spur gear pair RAGHUWANSHI N K , PAREY A . Experimental measurement of gear mesh stiffness of cracked spur gear by strain gauge technique [J]. Measurement , 2016 , 86 : 266 - 275 . Experimental measurement of gear mesh stiffness of cracked spur gear by strain gauge technique 许德涛 , 唐进元 , 周炜 . 基于扩展有限元法的齿根裂纹扩展规律 [J]. 中南大学学报(自然科学版) , 2016 , 47 ( 8 ): 2668 - 2675 . 基于扩展有限元法的齿根裂纹扩展规律 XU Detao , TANG Jinyuan , ZHOU Wei . Tooth root crack propagation regularity based on extended finite element method [J]. Journal of Central South University(Science and Technology) , 2016 , 47 ( 8 ): 2668 - 2675 . Tooth root crack propagation regularity based on extended finite element method WU Siyan , ZUO M J , PAREY A . Simulation of spur gear dynamics and estimation of fault growth [J]. Journal of Sound and Vibration , 2008 , 317 ( 3/4/5 ): 608 - 624 . Simulation of spur gear dynamics and estimation of fault growth CHEN Zaigang , ZHANG Jie , ZHAI Wanming , et al . Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack [J]. Engineering Failure Analysis , 2017 , 82 : 72 - 81 . Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack CHEN Zaigang , SHAO Yimin . Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth [J]. Engineering Failure Analysis , 2011 , 18 ( 8 ): 2149 - 2164 . Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth MA Hui , SONG Rongze , PANG Xu , et al . Time-varying mesh stiffness calculation of cracked spur gears [J]. Engineering Failure Analysis , 2014 , 44 : 179 - 194 . Time-varying mesh stiffness calculation of cracked spur gears REZAEI M , POURSINA M , JAZI S H , et al . Multi crack detection in helical gear teeth using transmission error ratio [J]. Journal of Mechanical Science and Technology , 2019 , 33 ( 3 ): 1115 - 1121 . Multi crack detection in helical gear teeth using transmission error ratio 孟宗 , 石桂霞 , 王福林 , 等 . 基于时变啮合刚度的裂纹故障齿轮振动特征分析 [J]. 机械工程学报 , 2020 , 56 ( 17 ): 108 - 115 . 基于时变啮合刚度的裂纹故障齿轮振动特征分析 MENG Zong , SHI Guixia , WANG Fulin , et al . Vibration characteristic analysis of cracked gear based on time-varying meshing stiffness [J]. Journal of Mechanical Engineering , 2020 , 56 ( 17 ): 108 - 115 . Vibration characteristic analysis of cracked gear based on time-varying meshing stiffness YU Wennian , MECHEFSKE C K , TIMUSK M . Effects of tooth plastic inclination deformation due to spatial cracks on the dynamic features of a gear system [J]. Nonlinear Dynamics , 2017 , 87 ( 4 ): 2643 - 2659 . Effects of tooth plastic inclination deformation due to spatial cracks on the dynamic features of a gear system PATEL A , SHAKYA P . Spur gear crack modelling and analysis under variable speed conditions using variational mode decomposition [J]. Mechanism and Machine Theory , 2021 , 164 : 104357 . Spur gear crack modelling and analysis under variable speed conditions using variational mode decomposition VERMA J G , KUMAR S , KANKAR P K . Crack growth modeling in spur gear tooth and its effect on mesh stiffness using extended finite element method [J]. Engineering Failure Analysis , 2018 , 94 : 109 - 120 . Crack growth modeling in spur gear tooth and its effect on mesh stiffness using extended finite element method 莫帅 , 岳宗享 , 冯志友 , 等 . 面齿轮分汇流系统动力学均载特性研究 [J]. 华中科技大学学报(自然科学版) , 2020 , 48 ( 2 ): 23 - 28 . 面齿轮分汇流系统动力学均载特性研究 MO Shuai , YUE Zongxiang , FENG Zhiyou , et al . Analytical investigation on load sharing characteristics for face gear split flow system [J]. Journal of Huazhong University of Science and Technology(Natural Science Edition) , 2020 , 48 ( 2 ): 23 - 28 . Analytical investigation on load sharing characteristics for face gear split flow system 10.11817/j.issn.1672-7207.2023.05.012 TH132 A 1672-7207（2023）05-1769-10 莫帅, 王檑, 胡庆森, 等. 裂纹齿轮时变啮合刚度与传动系统振动机理[J]. 中南大学学报(自然科学版), 2023, 54(5): 1769-1778. 莫帅, 王檑, 胡庆森, 等. 裂纹齿轮时变啮合刚度与传动系统振动机理[J]. 中南大学学报(自然科学版), 2023, 54(5): 1769-1778. MO Shuai, WANG Lei, HU Qingsen, et al. Time varying meshing stiffness of cracked gears and fault vibration for transmission system[J]. Journal of Central South University(Science and Technology), 2023, 54(5): 1769-1778. MO Shuai, WANG Lei, HU Qingsen, et al. Time varying meshing stiffness of cracked gears and fault vibration for transmission system[J]. Journal of Central South University(Science and Technology), 2023, 54(5): 1769-1778. 国家自然科学基金资助项目(51805368)；中国科协青年人才托举工程(2018QNRC001)；数字制造装备与技术国家重点实验室开放基金资助项目(DMETKF2021017)；江苏省泰州市双创计划项目(20212333) Project(51805368) supported by the National Natural Science Foundation of China; Project(2018QNRC001) supported by the Young Elite Scientists Sponsorship Program by CAST; Project(DMETKF2021017) supported by the Open Fund Project of State Key Laboratory of Digital Manufacturing Equipment And Technology; Project(20212333) supported by the Jiangsu Taizhou Mass Entrepreneurship and Innovation Program 2022-05-19 2022-06-21 2023-05-26 2023-06-20 38909993 中南大学学报(自然科学版) 5 54