基于结构振动定性理论的轴向承力杆件的模态反问题研究

The evaluation of multi-parameters of axial loaded bars in real-time is significant for the safety of structures, and this can be done by solving modal inverse problems based on ambient vibration data. However, this approach usually suffers from the low confidence of measured data, and thus may give ridiculous results. Besides, the well-posedness of the problems are still not clear. Thus it is proposed that the inverse problem approach can be validated and optimized by employing the qualitative theories of structural vibration. It is expected that by proving the qualitative properties and putting them in consideration, the invalid raw data can be largely filtered out, the modal identification process will be guided and accelerated, the properties of the singular points of the differential equations encountered can be determined, and the well-posedness of different inverse vibration problems of these bars will be discussed. This study can help optimize measuring-processing-solving strategy of the inverse ambient vibration problem approach to evaluate the multi –parameters of axial loaded bars, enhance the quality control of the data used, and finally increase the evaluation accuracy. Furthermore, as an important complement to the qualitative theory of structural vibration, this study can help us have a further understanding on the properties of the vibration of components with both bending and geometric stiffness, and help figure out the characteristics of a group of singular differential equations.

轴向承力杆件是结构工程中的一类重要构件，对它的多参数实时监控对结构的安全有着重要的意义。针对以模态反分析方法监控杆件参数时，噪音大、无效数据多、容易误报、适定性不清楚的问题，提出基于结构振动的定性理论对此类构件的反分析方法进行论证优化。研究预期通过从理论上证明该类构件的一系列定性性质，实现利用定性性质对无效监测数据进行筛除，对随机子空间模态识别过程进行引导和加速，对此类系统的反问题对应的奇异微分方程的奇点性质进行判别，以及对此类系统的反问题的提法的适定性进行论证。研究结果可用于优化环境随机振动反分析方法识别轴向承力杆件参数的测量—处理—反演方案，加强其数据质量控制，提高其识别准确度。同时，本研究是结构振动定性理论的重要补充，对进一步了解同时具有弯曲刚度和几何刚度的构件的固有振动的特点，以及分析一类奇异微分方程的系数和解的特性有着理论意义。

本项目在经典应用数学与力学的框架下研究了欧拉伯努利梁模型弹性变形、弹塑性变形和线性振动的一般定性性质及其反问题，并将其结论推广到有限元离散模型中。研究加强了对经典力学模型、微分方程性质的重新认识，也为模型参数识别、加工工艺设计等实际工程需求提供了理论支持。通过引入一个伴随系统并指出其解的正值性和边条件特性，统一地证明了轴向承力的杆件，只要受压大小不超过某二阶系统的最小特征值，其横向变形满足静力振荡性质，横向振动则满足振荡性质。通过构造法给出离散系统具有振荡性质的充分条件是对任意受结点力情形能找到与之匹配的连续系统，并指出了有限元离散系统并不总是继承对应连续系统的定性性质，提出各种连续梁满足振荡性质的充分条件。利用定性性质指出模态反问题对应微分方程的奇异性，采用Adomian分解法进行求解并对解的适定性进行分析。结合型材弯曲加工工艺探索梁模型的弹塑性反问题，证明了三辊弯曲工艺中变目标曲率梁可逐点确定下压力的充要条件是成型所需弯矩随无量纲弧长的相对变化率小于2，并给出了下压量的设计方案。提出基于欧拉观点的梁模型用于求解大变形弹性弯曲，弯曲塑性加工问题，从而避免使用可能造成计算效率低的接触算法。利用上述模型通过蒙特卡洛模拟分析了梁截面参数随机非均质性的影响，发现随着非均质性的增强塑性弯曲变形量将明显增大。