基于边界变化的索结构自振频率测试机理 Mechanism of natural frequency measurement of cable structures based on boundary variation 1 first-author 2 corresp 胡迎新，高级工程师，从事道路及桥梁工程设计与工程管理研究；E-mail: 27461561@qq.com 胡迎新，高级工程师，从事道路及桥梁工程设计与工程管理研究；E-mail: 27461561@qq.com 27461561@qq.com 1 1 1 3 1 中南大学 土木工程学院，湖南 长沙，410075 中南大学 土木工程学院，湖南 长沙，410075 School of Civil Engineering, Central South University, Changsha 410075, China School of Civil Engineering, Central South University, Changsha 410075, China 湖南中大设计院有限公司，湖南 长沙，410018 湖南中大设计院有限公司，湖南 长沙，410018 Hunan Zhongda Design Institute Co. Ltd., Changsha 410018, China Hunan Zhongda Design Institute Co. Ltd., Changsha 410018, China 中铁第四勘察设计院集团有限公司，湖北 武汉，430063 中铁第四勘察设计院集团有限公司，湖北 武汉，430063 China Railway Fourth Survey and Design Institute Group Co. Ltd., Wuhan 430063, China China Railway Fourth Survey and Design Institute Group Co. Ltd., Wuhan 430063, China 陈灿华 频率法是现场测试桥梁索力最常用的方法之一，但在测量“短索结构”时无法获得稳定的自振频率，导致索力测试值与真实值存在严重偏差。为探索“短索结构”自振频率测试机理，引入长细比 λ ，推导索结构在不同边界条件下自振频率与长细比的函数关系，提出一种通过长细比判断“短索结构”的方法。为验证长细比判据的有效性，采用通用有限元软件对不同直径的索结构进行仿真分析。采用提出的“短索结构”判定方法，对舟山新城大桥吊索HL1和HL2进行分析，从理论上解释2组吊索在实测中无法获得稳定自振频率的原因，验证长细比判据的工程适用性。研究结果表明：“短索结构”的边界条件受扰动发生的随机变化是频率法无法测得稳定自振频率的主要原因；采用通用有限元软件对不同直径的索结构进行仿真分析，得到的索结构自振频率和频率比的仿真计算结果验证了长细比判据的工程适用性。 The frequency method is one of the most commonly used methods for on-site testing of bridge cable forces, however, when measuring "short cable structures", it is limited by the inability to obtain a stable natural frequency, resulting in significant deviations between the measured force values and the actual values. In order to explore the mechanism of natural frequency testing for "short cable structures", the parameter λ of the slenderness ratio was introduced, the function relationship between the natural frequency and λ of the cable structure under different boundary conditions was derived, and a method for judging "short cable structures" based on λ was proposed. To verify the validity of the slenderness ratio criterion, simulation analyses were conducted on cable structures with different diameters using general finite element software. Furthermore, the "short cable structure" judgment method proposed in this paper was applied to analyze the suspension HL1 and HL2 of Xincheng Bridge in Xincheng, which theoretically explained the reason why stable natural frequencies could not be obtained through measurements and also verified the engineering applicability of the slenderness ratio criterion. The results show that the random variation of the boundary conditions(rotational stiffness) of "short cable structures" due to perturbations is the main reason why the frequency method cannot obtain a stable natural frequency. The calculated natural frequencies and frequency ratios of structures with different diameters are obtained, and these results are in good agreement with those of the theoretical prediction presented in this paper, which verifies the validity of the slenderness ratio criterion. 频率法 短索结构 长细比 有限元模拟 边界条件 frequency method short cable structure slenderness ratio finite element simulation boundary condition 索力调整是悬索桥、斜拉桥和拱桥等含索结构施工过程中的关键工序，对桥梁线形和最终成桥状态具有重要影响。合理的索力调整离不开精准的索力测量，工程中常用测量方法有2种。第一种为直接测量法，如压力表法、磁通量法和压力传感器法 [ 1 1 ] 等，这些方法通过仪器直接读出索力，数据可靠但布设繁琐且造价高昂，多用于特大桥在运营期的健康监测。第二种为间接测量法，即通过测试结构频率、挠度、线形等参数，经理论换算得到索力，所得索力与理论计算方法密切相关，常用计算方法有静力法和动力法2种。其中，静力法常见有“三点弯曲法”，即通过测量桥梁整体或局部线形，建立索力与线形之间的函数关系，进而求得索力，这种方法在实际操作时，常受限于现场索结构具体形式而影响适用性 [ 2 2 - 3 3 2-3 ] 。动力法主要有频率法、地脉动法 [ 4 4 ] 和模态识别法 [ 5 5 ] 等，即通过测量桥梁动力特性，根据索力与动力特性间函数关系求得索力。其中，频率法以操作简单、快捷且经济实用等特点而成为现场索力测试最常用的方法，而地脉动法和模态识别法由于使用条件相对苛刻，在实际测量中运用不多。 频率法是一种通过测试索结构自振频率来确定索力的方法。这种方法由张紧弦理论发展而来，前提是需精准采集索结构自振频率。但对于索结构来说，其自振频率受诸多因素如环境温度 [ 6 6 - 8 8 6-8 ] 、索结构抗弯刚度 [ 9 9 ] 、边界条件与额外支撑条件 [ 10 10 - 11 11 10-11 ] 、阻尼器设置 [ 12 12 ] 和索结构损伤情况 [ 13 13 - 14 14 13-14 ] 等因素影响，其中，索结构抗弯刚度和边界条件的影响最直接，且两者取值难以确定。在实际工程中，索结构的边界条件常常介于固结与铰接之间 [ 15 15 ] ，且与具体施工阶段密切相关，边界条件存在不确定性。目前对索结构边界条件的研究多通过拟合公式再结合实际情况进行解释说明 [ 16 16 ] ，需对“长索结构”与“短索结构”进行分类讨论。分析“短索结构”时，为使拟合值更准确，需考虑更多因素 [ 17 17 ] 。尽管如此，采用拟合公式的方法在实际测量中的效果仍然较差。 在实际运营中，“短索结构”比“长索结构”更易发生如断丝和锈蚀等破损现象 [ 18 18 ] 。对“短索结构”自振频率的精准测量是目前工程中亟待解决的问题。然而，如何在索结构中区分“短索结构”目前还尚无定论。有学者直接按索结构长度来判断 [ 19 19 ] ，也有学者提出采用Irvin参数和抗弯刚度影响系数等 [ 20 20 ] 对“短索结构”进行判定，其不足之处是难以应用于实际测量，且无法解释在测量“短索结构”自振频率时无法得到稳定值的现象。为此，本文在考虑索结构抗弯刚度前提下，对索结构在转动约束和平动约束这2种边界条件下自振频率的变化进行理论分析。引入参数长细比，推导索结构在不同边界条件下自振频率与长细比之间的函数关系，并提出一种通过长细比判断 “短索结构”的方法，对“短索结构”实测频率不稳定的原因进行分析。同时，通过有限元仿真分析对本文提出的计算理论进行验证，以浙江舟山新城大桥为例，验证长细比判定方法在工程中的可行性与适用性。 1 不同边界条件下频率法适应性 理论分析 1 索结构两端边界条件的理想情况为一端固结而另一端铰接，在实际工程中，索结构的边界条件都只是接近理想情况。已有研究表明，在复杂边界条件下，索力与自振频率之间的关系符合超越方程，分析索结构的边界参数变化对自振频率的影响十分困难。为此，考虑索结构两端受转动约束和平动约束2种边界条件，对索结构的自振频率与长细比、抗弯刚度和边界条件等参数的关系进行理论分析。 1.1 　索结构两端受转动约束 2 在对动力特性进行分析时，考虑抗弯刚度的索结构可等效为两端受拉的梁，如 图1 图1 所示。假定索结构采用匀质材料，索结构两端边界处平动刚度无穷大，则索结构的平衡微分方程可表示为 <math display="block" id="M1"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced separators="|"><mrow><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>m</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704544&type= 73.74466705 9.31333351 式中： E 为弹性模量； I 为截面惯性矩； <math id="M2"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704578&type= 8.89000034 3.38666677 为 y 方向位移； <math id="M3"><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704551&type= 3.21733332 3.72533321 为索力； m 为单位长度质量。边界条件为 <math display="block" id="M4"><mi>w</mi><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704565&type= 12.61533356 4.48733330 , <math display="block" id="M5"><mi>w</mi><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704566&type= 12.78466606 4.48733330 <math display="block" id="M6"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mfrac><mrow><mi mathvariant="normal">d</mi><mi>w</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704583&type= 39.36999893 8.29733276 , <math display="block" id="M7"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mfrac><mrow><mi mathvariant="normal">d</mi><mi>w</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>+</mo><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704569&type= 39.53932953 8.29733276 式中： <math id="M8"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704586&type= 3.55599999 3.72533321 为转动刚度； L 为计算长度。 式(1) 式(1) 中索力通常可表示为 <math display="block" id="M9"><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>a</mi><mi>m</mi><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>E</mi><mi>I</mi></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704603&type= 37.76133347 8.38199997 当索结构不考虑自身抗弯刚度时，索力 <math id="M10"><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704604&type= 3.21733332 3.72533321 可表示为 <math display="block" id="M11"><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>m</mi><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704606&type= 19.98133469 8.38199997 式中： a 和 b 为边界转动刚度系数，参考文献[ 20 20 ]， a 的取值范围为[3.5, 4.0]， b 的取值范围为[1.0, 3.0]； n 为自振频率阶次； <math id="M12"><msubsup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi>n</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704621&type= 3.89466691 4.82600021 为索结构考虑抗弯刚度的第 n 阶自振频率； <math id="M13"><msubsup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704592&type= 3.55599999 4.82600021 为不考虑抗弯刚度的第 n 阶自振频率。由 式(4) 式(4) 和 式(5) 式(5) 可得 <math display="block" id="M14"><mi>η</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>L</mi><mroot><mrow><mi>a</mi><mi>m</mi></mrow><mrow/></mroot></mrow></mfrac><mroot><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>E</mi><mi>I</mi></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>L</mi></mrow></mfrac><mroot><mrow><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mfrac></mrow><mrow/></mroot></mrow></mfrac><mo>=</mo><mtext>                        </mtext><mroot><mrow><mfrac><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>b</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>E</mi><mi>I</mi></mrow><mrow><mi>a</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704624&type= 100.66867065 20.99733353 式中： <math id="M15"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 为频率比，反映由刚度变化引起的频率变化程度。为更好地考察边界条件影响，引入索结构长细比 <math id="M16"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 ： <math display="block" id="M17"><mi>λ</mi><mo>=</mo><mfrac><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704598&type= 8.55133343 7.28133297 , <math display="block" id="M18"><mi>i</mi><mo>=</mo><mroot><mrow><mfrac><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704630&type= 11.76866722 8.97466660 式中： A 为面积。由 式(6) 式(6) 和 式(7) 式(7) 可得 <math display="block" id="M19"><mi>η</mi><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>b</mi><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704600&type= 27.85533524 9.90600014 式中： <math id="M20"><mi>ε</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><mi>E</mi><mi>A</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705163&type= 10.75266743 7.36600018 。为考察频率比 <math id="M21"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 对 a 和 b 的灵敏度，令灵敏度系数 <math id="M22"><msub><mrow><mi>η</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>η</mi></mrow><mrow><mo>∂</mo><mi>a</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704635&type= 11.59933376 7.28133297 ， <math id="M23"><msub><mrow><mi>η</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>η</mi></mrow><mrow><mo>∂</mo><mi>b</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704637&type= 11.59933376 7.28133297 。当 b =1， n =1时，由 式(8) 式(8) 可得： <math id="M24"><msub><mrow><mi>η</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mroot><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">4</mn><mi>a</mi><mi>λ</mi><mroot><mrow><mi>a</mi></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704640&type= 22.86000061 11.09133339 。 当 a =4， n =1时，由 式(8) 式(8) 可得： <math id="M25"><msub><mrow><mi>η</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>λ</mi><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mroot><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704669&type= 25.06133461 10.66800022 。 其中： μ= <math id="M26"><mi mathvariant="normal">π</mi><mroot><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>ε</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705175&type= 10.58333302 4.31799984 。在转动约束下， η a 和 η b 随长细比 <math id="M27"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 的变化曲线见 图2 图2 。由 图2 图2 (a)可见：当 a 不变时，随着长细比 <math id="M28"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 增加， η a 绝对值逐渐变小；当 <math id="M29"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 大于 <math id="M30"><mn mathvariant="normal">2</mn><mroot><mrow><mn mathvariant="normal">5</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704676&type= 10.83733368 4.65666676 时， η a 变化趋缓；当长细比不变时，随着 a 增加， η a 绝对值也逐渐减小。由 图2 图2 (b)可见：当 b 不变时， η b 随长细比 <math id="M31"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 的增加而变小，当 <math id="M32"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 达到 <math id="M33"><mroot><mrow><mn mathvariant="normal">10</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704959&type= 10.24466610 4.65666676 时， η b 均小于0.05；当 <math id="M34"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 较小时， η b 随着 b 增加而减小，当 <math id="M35"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705134&type= 1.77800000 2.62466669 大于 <math id="M36"><mroot><mrow><mn mathvariant="normal">10</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704959&type= 10.24466610 4.65666676 时， η b 随着 b 增加而基本保持不变。 (a) 系数 b =1， n =1；(b) 系数 a =4， n =1 上述分析表明：随着长细比 <math id="M39"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 增大，灵敏度系数 η a 和 η b 均下降；当 <math id="M40"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 足够大时( <math id="M41"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 大于 <math id="M42"><mroot><mrow><mn mathvariant="normal">10</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704959&type= 10.24466610 4.65666676 )，灵敏度系数 η b 趋于0。在工程实际中，长索结构的长细比较大，因此，采用频率法可测得其稳定的自振频率。当长细比 <math id="M43"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 较小时(如 <math id="M44"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 < <math id="M45"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704787&type= 2.11666679 3.38666677 )，灵敏度系数 η a 和 η b 随系数 a 和 b 的变化均较大。在工程实际中，短索结构的长细比 <math id="M46"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 较小，采用测量锤敲击“短索结构”时，会造成短索结构的边界条件发生随机变化，这正是无法获得“短索结构”稳定自振频率的原因。 1.2 　索结构两端受平动约束 2 考虑 图3 图3 所示索结构，令索结构上、下两端的平动约束刚度分别为 <math id="M47"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704802&type= 3.89466691 3.72533321 和 <math id="M48"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704805&type= 3.89466691 3.72533321 ，平衡微分方程同样可由 式(1) 式(1) 表示。 此时，索结构两端的位移边界条件为 <math display="block" id="M49"><mi>w</mi><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=44704797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704796&type= 17.52599907 4.48733330 , <math display="block" id="M50"><mi>w</mi><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></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=44704819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704811&type= 17.69533348 4.48733330 为消除索结构两端边界刚体位移影响，令 <math display="block" id="M51"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>L</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=44704844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704821&type= 55.03333664 7.28133297 式中： <math id="M52"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704882&type= 8.89000034 3.38666677 为索结构消除刚体位移后的 y 方向位移。将 式(10) 式(10) 代入 式(1) 式(1) 得 <math display="block" id="M53"><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>m</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mi>m</mi><mo stretchy="false">[</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>L</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">'</mi><mi mathvariant="normal">'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704834&type= 99.82199860 8.29733276 新的边界条件为 <math display="block" id="M54"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704836&type= 12.61533356 4.48733330 , <math display="block" id="M55"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704838&type= 12.78466606 4.48733330 <math display="block" id="M56"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mover accent="true"><mi>w</mi><mo>¯</mo></mover><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mfrac><mrow><mo>∂</mo><mover accent="true"><mi>w</mi><mo>¯</mo></mover></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704841&type= 53.84799957 8.29733276 , <math display="block" id="M57"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mover accent="true"><mi>w</mi><mo>¯</mo></mover><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mfrac><mrow><mo>∂</mo><mover accent="true"><mi>w</mi><mo>¯</mo></mover></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>-</mo><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704852&type= 54.27133560 8.29733276 将 <math id="M58"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704882&type= 8.89000034 3.38666677 表示为各阶主振型的线性叠加， <math display="block" id="M59"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mover><mrow><mi>w</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704867&type= 33.18933105 8.72066593 式中： <math id="M60"><msub><mrow><mover><mrow><mi>w</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704879&type= 8.38199997 4.57200003 为索结构的第 n 阶振动模态； <math id="M61"><msub><mrow><mi>Z</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=44704881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704871&type= 7.19666624 4.48733330 为第 n 阶模态坐标。 用 <math id="M62"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704882&type= 8.89000034 3.38666677 表示的振动方程为 <math display="block" id="M63"><mi>E</mi><mi>I</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>m</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704884&type= 64.51599884 8.29733276 式(15) 式(15) 中，第 n 阶自由振动响应为 <math display="block" id="M64"><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover><mrow><mi>w</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704894&type= 26.07733345 5.16466665 <math display="block" id="M65"><msub><mrow><mover><mrow><mi>w</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</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><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704905&type= 22.77533340 7.28133297 式(16) 式(16) 中： <math id="M66"><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704887&type= 3.97933316 3.72533321 为第 n 阶自振频率。联立式( 11 11 )、( 14 14 )、( 16 16 )和(17)得 <math display="block" id="M67"><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>Z</mi></mrow><mi>¨</mi></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msub><mrow><mi>P</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=44704919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704898&type= 48.51399994 5.58799982 其中： <math display="block" id="M68"><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>L</mi></mrow></msubsup><mrow><msubsup><mrow><mover><mrow><mi>w</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><mi>m</mi><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704921&type= 36.57600021 7.95866632 <math display="block" id="M69"><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>L</mi></mrow></msubsup><mrow><mi>m</mi><mo stretchy="false">[</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>'</mi><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>L</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>L</mi></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>'</mi><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><msub><mrow><mover><mrow><mi>w</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704902&type= 62.90732956 7.95866632 为简化计算，假定 <math id="M70"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704934&type= 16.93333435 3.72533321 ，联立式( 10 10 )、( 13 13 )和(18)求解，可得频率比 <math id="M71"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 为 <math display="block" id="M72"><mi>η</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac></mrow><mrow><mn mathvariant="normal">6</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">4</mn><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704938&type= 45.55066681 18.96533203 , n 为偶数 <math display="block" id="M73"><mi>η</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>=</mo><mroot><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac></mrow><mrow><mn mathvariant="normal">8</mn><mo>-</mo><mn mathvariant="normal">8</mn><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">6</mn><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">4</mn><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704950&type= 80.34867096 18.96533203 , n 为奇数 定义无量纲参数 <math id="M74"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>K</mi><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mi>E</mi><mi>I</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704940&type= 14.56266594 7.78933382 ，其取值范围为[0.1，10.0]。为保证 式(21) 式(21) 和 式(22) 式(22) 的解有物理意义，需满足 <math id="M75"><mn mathvariant="normal">0</mn><mo>≤</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ε</mi></mrow></mfrac><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704943&type= 19.04999924 8.29733276 ，此时， <math id="M76"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 相应取值范围为[ <math id="M77"><mroot><mrow><mn mathvariant="normal">10</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704959&type= 10.24466610 4.65666676 , <math id="M78"><mn mathvariant="normal">10</mn><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704970&type= 5.58799982 3.38666677 ]， <math id="M79"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705086&type= 1.86266661 3.38666677 与 <math id="M80"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704994&type= 4.14866638 3.72533321 和 <math id="M81"><mi>λ</mi><mroot><mrow><mi>ε</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705008&type= 7.53533268 4.31799984 的关系如 图4 图4 所示。选取6组不同长细比，进一步绘制 <math id="M82"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 与 <math id="M83"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44704999&type= 4.14866638 3.72533321 的关系曲线，如 图5 图5 所示。 (a) 自振频率阶次为偶数；(b) 自振频率阶次为奇数 (a) 自振频率阶次为偶数；(b) 自振频率阶次为奇数 由 图4 图4 (a)可知：当自振频率阶次为偶数时，若 <math id="M94"><mi>λ</mi><mroot><mrow><mi>ε</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705070&type= 7.53533268 4.31799984 为定值，则频率比 <math id="M95"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705086&type= 1.86266661 3.38666677 随 <math id="M96"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705132&type= 4.14866638 3.72533321 的变化较平缓，表明改变 <math id="M97"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705132&type= 4.14866638 3.72533321 对索结构自振频率影响较小。 图5 图5 (a)中6条曲线都趋近于水平直线，进一步说明自振频率阶次为偶数时频率比 <math id="M98"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 对 <math id="M99"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705132&type= 4.14866638 3.72533321 不敏感。 图4 图4 (b)与 图5 图5 (b)表明：当自振频率阶次为奇数时，频率比 <math id="M100"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 在 <math id="M101"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705132&type= 4.14866638 3.72533321 小于1时变化较大；随着长细比 <math id="M102"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 增大， <math id="M103"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 变化趋于平缓。 在实际工程中，索结构 <math id="M104"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705116&type= 4.14866638 3.72533321 一般都大于1。当 <math id="M105"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705132&type= 4.14866638 3.72533321 为[1，10]时，频率比 <math id="M106"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 对 <math id="M107"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705132&type= 4.14866638 3.72533321 并不敏感。可见，平动约束不是“短索结构”自振频率测不准的主因。 综合平动约束与转动约束的分析结果可知：当索结构长细比 <math id="M108"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705134&type= 1.77800000 2.62466669 小于 <math id="M109"><mn mathvariant="normal">2</mn><mroot><mrow><mn mathvariant="normal">5</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705147&type= 10.83733368 4.65666676 、采用频率法测试索结构自振频率时，不能忽略边界条件尤其是边界转动刚度变化带来的影响，因此，可将长细比 <math id="M110"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705127&type= 1.77800000 2.62466669 作为“短索结构”的一种判据。 2 有限元模拟验证 1 采用通用有限元软件ABAQUS进行仿真分析。根据前述分析可知，索结构两端边界的转动刚度变化是其自振频率测不准的主因，因此，在模拟边界条件时，只考虑改变边界的转动刚度，忽略其平动刚度。 图6 图6 所示为有限元模型示意图，材料属性见 表1 表1 。考虑5种不同直径(分别为10、20、30、40和50 mm，为方便起见，相应索结构名称分别记为S 10 、S 20 、S 30 、S 40 和S 50 )的索结构，索结构两端的边界条件通过空间梁单元模拟。 长度/mm 弹性 模量/GPa 两端 拉力/kN 截面 形式 泊松比 密度/ (kg·m -3 ) 1 000 206 1 500 圆截面 0.2 7 890 进行有限元分析时，通过控制索结构的单元数量确保计算的收敛性，收敛性验证计算结果见 表2 表2 。从 表2 表2 可见：随着单元数量增加，第1阶自振频率计算结果趋于稳定；当单元数量大于100个时，计算结果收敛性较好，模型满足分析要求。 单元/个 10 50 100 200 300 第1阶自振频率/Hz 403.44 409.21 409.44 409.50 409.51 根据本文提出的长细比 <math id="M111"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">2</mn><mroot><mrow><mn mathvariant="normal">5</mn></mrow><mrow/></mroot><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705162&type= 15.83266735 4.65666676 为“短索结构”的判据，结合 式(7) 式(7) 以及 μ= <math id="M112"><mi mathvariant="normal">π</mi><mroot><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>ε</mi></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705175&type= 10.58333302 4.31799984 和 <math id="M113"><mi>ε</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><mi>E</mi><mi>A</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705163&type= 10.75266743 7.36600018 等关系式，对于圆截面索结构，在索结构计算长度保持不变情况下，长细比判据可转换为直径判据，即 <math display="block" id="M114"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">4</mn><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow><mrow/></mroot><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">4</mn><mi>L</mi></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><mroot><mrow><mfrac><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">20</mn><mi>E</mi><mi mathvariant="normal">π</mi></mrow></mfrac></mrow><mrow/></mroot></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705178&type= 42.50266647 10.66800022 式中： <math id="M115"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705168&type= 3.97933316 3.72533321 为直径判据，直径大于 <math id="M116"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705168&type= 3.97933316 3.72533321 的索结构为“短索结构”。 将 表1 表1 中材料属性数据代入 式(23) 式(23) 可得 D c =29.5 mm，即直径大于29.5 mm的索可认定为“短索结构”。 在实际工程中，索结构的边界条件通常难以确定，本文采用2种方式来处理。第一种是以S 20 和S 50 为分析对象，边界处转动刚度取索结构抗弯刚度0.1倍到10倍。第二种是以所有索结构为分析对象，边界处转动刚度取S 50 抗弯刚度的0.1倍到10倍。 第一种方式的计算结果见图 7 7 (a)和 7 7 (b)。从图 7 7 (a)和 7 7 (b)可见：对于“长索结构”S 20 与“短索结构”S 50 ，当边界处转动刚度从索结构抗弯刚度0.1倍增加到1倍时，2种索的第1阶自振频率均有较大幅度增加(S 20 和S 50 的第1阶自振频率分别增加11.59 Hz和49.19 Hz)；当边界处转动刚度从索结构抗弯刚度的1倍增加到10倍时，第1阶自振频率增幅变小(S 20 和S 50 第1阶自振频率分别增加9.23 Hz和15.43 Hz)。 (a) 第一种方式，索S 20 ；(b) 第一种方式，索S 50 ；(c) 第二种方式，索S 50 第二种方式的计算结果见 图7 图7 (c)。从 图7 图7 (c)可见：对于“长索结构”S 10 和S 20 ，频率比 <math id="M117"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 变化极小，增幅均不超过0.01，说明“长索结构”的自振频率对边界处转动刚度的变化不敏感；S 30 的频率比 <math id="M118"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 增幅约0.05；“短索结构”S 40 和S 50 的频率比 <math id="M119"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44705172&type= 1.86266661 3.38666677 变化较大，增幅分别为0.15和0.30，说明“短索结构”的自振频率对边界处转动刚度的改变较敏感。 在上述2种方式下，不同直径索结构自振频率和频率比计算结果验证了本文“短索结构”判定依据的可行性和有效性。 3 工程实例验证 1 舟山新城大桥改扩建工程主桥采用(36+148+36) m飞燕式钢箱系杆提篮拱结构形式，主桥全长220 m。主跨拱肋矢高37 m，矢跨比为1/4，全桥设13对吊索，吊索沿顺桥向垂直布置，标准间距为8 m，吊索编号见 图8 图8 ，吊索型号除HL1为镀锌钢丝拉索PES(C)7-91外(其中，PES指镀锌钢丝吊索，C指黑色内层、彩色外层的双层结构，7-91指吊索含7股钢丝束，每股钢丝束中包含91根细钢丝线，其余依此类推)，其余吊索均采用镀锌钢丝拉索PES(C)7-73。本文对吊索(HL1至HL7)进行现场实测和分析。 采用基于频率法的索力检测仪对吊索自振频率进行测试，结果见 表3 表3 。在测试过程中，吊索HL1与吊索HL2无法通过索力检测仪采集到稳定的自振频率，2组吊索的索力通过千斤顶张拉反算得到。 2 参数 7 图8中左吊杆编号 HL1 HL2 HL3 HL4 HL5 HL6 HL7 设计索力/kN 1 176 1 058 982 994 1 002 999 979 计算长度/m 3.173 8.087 12.104 15.223 17.451 18.78 19.232 弹性模量/MPa 190 000 190 000 190 000 190 000 190 000 190 000 190 000 等效直径/cm 9.3 8.2 8.2 8.2 8.2 8.2 8.2 实际长细比 136.5 394.5 590.4 742.6 851.3 916.1 938.1 短索结构对应长细比 451.1 434.0 448.7 446.0 444.3 444.9 449.4 实测频率/Hz — — 9.60 7.59 6.62 5.95 5.74 实测索力/kN 1 124 1 106 1 030 1 019 1 026 960 932 根据本文提出的“短索结构”判据，吊索HL1和HL2均为“短索结构”，理论上，这2组吊索无法获得稳定的自振频率。这与 表3 表3 所示结果一致，从而验证了长细比判据的工程适用性。 4 结论 1 1) 引入长细比作为“短索结构”的判据，操作性强，并具备通用性。 2) “短索结构”的边界条件(尤其是转动刚度)受扰动影响发生随机变化，这是频率法无法测得稳定自振频率的主要原因。 3) 本文提出了“短索结构”自振频率测试机理，从理论上验证了“短索结构”无法获得稳定自振频率，且理论结果与仿真结果和现场实测结果均较吻合，验证了长细比判据的工程适用性。 参考文献 YAO Yadong . Measurement of cable forces for automated monitoring of engineering structures using fiber optic sensors: a review [J]. Automation in Construction , 2021 , 126 : 103687 . Measurement of cable forces for automated monitoring of engineering structures using fiber optic sensors: a review 魏建东 , 刘山洪 . 基于拉索静态线形的索力测定 [J]. 工程力学 , 2003 , 20 ( 3 ): 104 - 107 . 基于拉索静态线形的索力测定 WEI Jiandong , LIU Shanhong . Tension calculation of cable by its static profile [J]. Engineering Mechanics , 2003 , 20 ( 3 ): 104 - 107 . Tension calculation of cable by its static profile 葛俊颖 , 苏木标 , 李文平 . 一种测量斜拉桥拉索索力新方法: 垂度法 [J]. 中国铁道科学 , 2018 , 39 ( 4 ): 63 - 70 . 一种测量斜拉桥拉索索力新方法: 垂度法 GE Junying , SU Mubiao , LI Wenping . A new method for measuring cable tension of cable-stayed bridge-cable sag method [J]. China Railway Science , 2018 , 39 ( 4 ): 63 - 70 . A new method for measuring cable tension of cable-stayed bridge-cable sag method 沈之容 , 廖自来 , 虞德群 , 等 . 基于初位移加载法和脉动法的单管通信塔模态测试与数值分析 [J]. 特种结构 , 2020 , 37 ( 5 ): 48 - 51 . 基于初位移加载法和脉动法的单管通信塔模态测试与数值分析 SHEN Zhirong , LIAO Zilai , YU Dequn , et al . Modal test and numerical analysis of single tube communication tower based on pulsating method and initial displacement loading method [J]. Special Structures , 2020 , 37 ( 5 ): 48 - 51 . Modal test and numerical analysis of single tube communication tower based on pulsating method and initial displacement loading method 刘晨凯 , 郑万山 . 基于模态识别的短索索力识别方法 [J]. 公路交通科技 , 2020 , 37 ( 1 ): 74 - 84 . 基于模态识别的短索索力识别方法 LIU Chenkai , ZHENG Wanshan . An estimation method of short cable force based on modal identification [J]. Journal of Highway and Transportation Research and Development , 2020 , 37 ( 1 ): 74 - 84 . An estimation method of short cable force based on modal identification 曹发源 ， 虞庐松 ， 翟启远 . 振动频率法测量斜拉桥索力的影响因素研究 [J]. 铁道建筑 , 2017 , 57 ( 5 ): 27 - 30 . 振动频率法测量斜拉桥索力的影响因素研究 CAO Fayuan , YU Lusong , ZHAI Qiyuan . Research on affecting factors of cable force measurement of cable-stayed bridge by vibration frequency method [J]. Railway Engineering , 2017 , 57 ( 5 ): 27 - 30 . Research on affecting factors of cable force measurement of cable-stayed bridge by vibration frequency method MA Lin . An accurate frequency-based method for identifying cable tension while considering environmental temperature variation [J]. Journal of Sound and Vibration , 2021 , 490 : 115693 . An accurate frequency-based method for identifying cable tension while considering environmental temperature variation 赵珧冰 , 孙测世 , 彭剑 , 等 . 不同初拉力拉索对温度变化的敏感性分析 [J]. 中南大学学报(自然科学版) , 2014 , 45 ( 5 ): 1680 - 1685 . 不同初拉力拉索对温度变化的敏感性分析 ZHAO Yaobing , SUN Ceshi , PENG Jian , et al . Sensitivity analysis of different initial tension forces of suspended cable to temperature changes [J]. Journal of Central South University (Science and Technology) , 2014 , 45 ( 5 ): 1680 - 1685 . Sensitivity analysis of different initial tension forces of suspended cable to temperature changes 任伟新 , 陈刚 . 由基频计算拉索拉力的实用公式 [J]. 土木工程学报 , 2005 , 38 ( 11 ): 26 - 31 . 由基频计算拉索拉力的实用公式 REN Weixin , CHEN Gang . Practical formulas to determine cable tension by using cable fundamental frequency [J]. China Civil Engineering Journal , 2005 , 38 ( 11 ): 26 - 31 . Practical formulas to determine cable tension by using cable fundamental frequency 李红 , 黎晨 . 短吊杆内力识别方法 [J]. 南京理工大学学报 , 2015 , 39 ( 4 ): 467 - 470 . 短吊杆内力识别方法 LI Hong , LI Chen . Methods of recognizing internal force of short suspender [J]. Journal of Nanjing University of Science and Technology , 2015 , 39 ( 4 ): 467 - 470 . Methods of recognizing internal force of short suspender MA L . A highly precise frequency-based method for estimating the tension of an inclined cable with unknown boundary conditions [J]. Journal of Sound and Vibration , 2017 , 409 : 65 - 80 . A highly precise frequency-based method for estimating the tension of an inclined cable with unknown boundary conditions 吉伯海 , 程苗 , 傅中秋 , 等 . 基于振动频率法的斜拉桥索力测试影响因素 [J]. 中南大学学报(自然科学版) , 2015 , 46 ( 7 ): 2620 - 2625 . 基于振动频率法的斜拉桥索力测试影响因素 JI Bohai , CHENG Miao , FU Zhongqiu , et al . Influential factors in cable force measurement of cable-stayed bridges based on vibration frequency method [J]. Journal of Central South University(Science and Technology) , 2015 , 46 ( 7 ): 2620 - 2625 . Influential factors in cable force measurement of cable-stayed bridges based on vibration frequency method 谭冬梅 , 罗素珍 , 瞿伟廉 , 等 . 斜拉索在随机风-车-覆冰联合作用下的疲劳可靠度分析 [J]. 长安大学学报(自然科学版) , 2019 , 39 ( 2 ): 91 - 99 . 斜拉索在随机风-车-覆冰联合作用下的疲劳可靠度分析 TAN Dongmei , LUO Suzhen , QU Weilian , et al . Fatigue reliability analysis of stay cables under combined effect of wind-vehicle load and iced accretion [J]. Journal of Chang’an University(Natural Science Edition) , 2019 , 39 ( 2 ): 91 - 99 . Fatigue reliability analysis of stay cables under combined effect of wind-vehicle load and iced accretion 颜东煌 , 郭鑫 . 斜拉索损伤对在役斜拉桥体系可靠度的影响 [J]. 中南大学学报(自然科学版) , 2020 , 51 ( 1 ): 213 - 220 . 斜拉索损伤对在役斜拉桥体系可靠度的影响 YAN Donghuang , GUO Xin . Influence of damage of stay cables on system reliability of in-service cable-stayed bridges [J]. Journal of Central South University(Science and Technology) , 2020 , 51 ( 1 ): 213 - 220 . Influence of damage of stay cables on system reliability of in-service cable-stayed bridges FANG Zhi , WANG Jianqun . Practical formula for cable tension estimation by vibration method [J]. Journal of Bridge Engineering , 2012 , 17 ( 1 ): 161 - 164 . Practical formula for cable tension estimation by vibration method 燕启清 , 刘世忠 , 朱占龙 , 等 . 基于频率法的短吊杆索力测试 [J]. 铁道科学与工程学报 , 2020 , 17 ( 10 ): 2577 - 2585 . 基于频率法的短吊杆索力测试 YAN Qiqing , LIU Shizhong , ZHU Zhanlong , et al . Determing short suspender tension based on frequency method [J]. Journal of Railway Science and Engineering , 2020 , 17 ( 10 ): 2577 - 2585 . Determing short suspender tension based on frequency method 刘红义 , 和海芳 , 谢进财 , 等 . 基于频率法的短索索力计算公式误差分析 [J]. 铁道建筑 , 2019 , 59 ( 10 ): 15 - 18 . 基于频率法的短索索力计算公式误差分析 LIU Hongyi , HE Haifang , XIE Jincai , et al . Error analysis of short cable force calculation formulas based on frequency method [J]. Railway Engineering , 2019 , 59 ( 10 ): 15 - 18 . Error analysis of short cable force calculation formulas based on frequency method 姚彻 , 康厚军 . 索—梁结构的模态正交性和非线性振动研究 [J]. 中南大学学报(自然科学版) , 2021 , 52 ( 10 ): 3605 - 3612 . 索—梁结构的模态正交性和非线性振动研究 YAO Che , KANG Houjun . Research of orthogonalities of modes and nonlinear vibration in cable-beam models [J]. Journal of Central South University(Science and Technology) , 2021 , 52 ( 10 ): 3605 - 3612 . Research of orthogonalities of modes and nonlinear vibration in cable-beam models 程路超 , 朱妍 , 吴定俊 . 短吊杆索力-频率关系试验研究 [J]. 结构工程师 , 2020 , 36 ( 2 ): 193 - 198 . 短吊杆索力-频率关系试验研究 CHENG Luchao , ZHU Yan , WU Dingjun . Cable tension-frequency test of short suspenders [J]. Structural Engineers , 2020 , 36 ( 2 ): 193 - 198 . Cable tension-frequency test of short suspenders 李国强 , 顾明 , 孙利民 . 拉索振动、动力检测与振动控制理论 [M]. 北京 : 科学出版社 , 2014 : 16 - 20 . 拉索振动、动力检测与振动控制理论 LI Guoqiang , GU Ming , SUN Limin . Theory of cable vibration, dynamic detection and vibration control [M]. Beijing : Science Press , 2014 : 16 - 20 . Theory of cable vibration, dynamic detection and vibration control 10.11817/j.issn.1672-7207.2023.05.016 TU371.3 A 1672-7207（2023）05-1814-09 申成庆, 胡迎新, 黄方林, 等. 基于边界变化的索结构自振频率测试机理[J]. 中南大学学报(自然科学版), 2023, 54(5): 1814-1822. 申成庆, 胡迎新, 黄方林, 等. 基于边界变化的索结构自振频率测试机理[J]. 中南大学学报(自然科学版), 2023, 54(5): 1814-1822. SHEN Chengqing, HU Yingxin, HUANG Fanglin, et al. Mechanism of natural frequency measurement of cable structures based on boundary variation[J]. Journal of Central South University(Science and Technology), 2023, 54(5): 1814-1822. SHEN Chengqing, HU Yingxin, HUANG Fanglin, et al. Mechanism of natural frequency measurement of cable structures based on boundary variation[J]. Journal of Central South University(Science and Technology), 2023, 54(5): 1814-1822. 国家自然科学基金资助项目(12072375) Project(12072375) supported by the National Natural Science Foundation of China 2022-08-10 2022-10-25 2023-05-26 2023-06-20 38909994 中南大学学报(自然科学版) 5 54