【作者】 赵占山；

【导师】 王德明；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2006， 硕士

【摘要】 本文将微分方程反问题理论应用于结构故障诊断。从微分方程反问题的角度,本文给出了关于原始刚度的微分方程(正问题)的求解方法,和关于由裂纹等故障引起的附加刚度的积分方程(反问题)的一组识别模型,由此来识别由疲劳裂纹等故障引起的结构的微小变化,从而可对裂纹的发生、发展进行在线检测。本文通过Fourier变换,将求解微分方程参数的问题从时域转换到频域上,然后利用小扰动理论将这一反问题进行求解,转化为求解第一类Fredholm积分方程的问题。首先将第一类积分方程进行离散化,然后对Landweber方法进行改进,因为Landweber方法虽然是比较稳定的求解方法,但是计算量大,本文利用矩阵方面的知识对离散得到的矩阵进行处理使得Landweber方法的计算量大大减少,另外,本文模仿Landweber方法构造了一种适用于具有核对称的第一类积分方程的正则化策略并对其进行了有效的改进,使其计算效率大大增加。最后本文利用正则化同伦方法对第一类积分方程进行求解,同伦方法最主要的优点就是求解算子方程时是一种大范围收敛的方法,但是对于不适定的问题,它又克服不了算子方程的不适定性,而在处理反问题的不适定性方面,Tikhonov正则化方法无疑是最著名、最有效的一种方法,用正则化同伦方法求解第一类积分方程可以结合两者的优点。更多还原

【Abstract】 The theory of inverse problems of a partial differential equation is used for the structure fault diagnosis in this paper. A set of identification models of a differential stiffness and of an integral equation on the additional stiffness are given to identify the small stiffness of a variational structure which is caused by the fault in the structure, thus may to the crack of occurrence, development carry on the online examination.This paper through the Fourier transformation, transforms the differential equation of parameter question from the time domain to the frequency domain, then utilizes the small perturbance theory to solve inverse problem. The question is transformed to solve the first kind of Fredholm integral equation. At first, this paper takes the first kind of integral equation to discrete, and takes the Landweber method to improve. Because Landweber method is a well stable solution method, but the computational burden is big. This paper utilizes the knowledge of matrix to reduce greatly the computational burden of the Landweber method. Secondly, this paper imitates Landweber method to structure one kind regularization strategy which is suit in having the symmetric Kernel about the first kind of integral equation and has made the effective improvement to it, which improve computational efficiency obviously. Finally, this paper uses the regularization homotopy method to solve the first kind of integral equation. The most main merit of homotopy method is a wide range convergence method, but regarding to the ill-posed question, it can not overcome ill-posedness of the operator equation. In conquering ill-posed aspect of inverse problem, Tikhonov regularization method is without doubt most famous and the most effective method. The first kind of integral equation with the regularization homotopy method to solve is possible to unify both the merits.更多还原