In this paper, a new system of damage detection equations for damage identification of cylindrical shells is developed based on classic finite element method, incomplete noisy modal data and model updating. Classic finite element technique in conjunction with Sander's thin shell theory is deployed in order to construct the finite element model of cylindrical shells and compute vibration data. To overcome the difficulty of incomplete data, “system equivalent reduction expansion process” (SEREP) method is applied to find a proper complete approximation of incomplete modal data. The proposed system is usually ill-posed due to noisy data and ill-conditioned system derived from the model updating process. To tackle this issue, a constraint as a regularization parameter is employed in model updating problem to differentiate potentially damaged elements and undamaged elements. This constraint is imposed on the model updating process by defining a Reduction Index (RI) and forcing severities of undamaged elements to tend to zero. Furthermore, the presented system is non-square and large depending on the number of structural unknowns. For these reasons, Biconjugate gradient (BCG) method along with an appropriate preconditioner are employed to solve the system of equations. Not only does the preconditioner stabilize the solution against numerical errors and inaccuracies caused by noisy data, but also it alters the system to a square one. Numerical studies are performed to demonstrate the efficiency of the presented method. Moreover, a statistical analysis is carried out to evaluate the effect of different sequences of random noises on the damage detection results. Findings show that the proposed method is capable of detecting the severities and sites of damages for cylindrical shells in the presence of noisy data.

本文基于经典有限元法，不完全噪声模态数据和模型更新，开发了一种新的圆柱壳损伤识别方程。运用经典的有限元技术结合桑德的薄壳理论，以建立圆柱壳的有限元模型并计算振动数据。为了克服不完整数据的困难，采用“系统等效缩减扩展过程”（SEREP）方法来找到不完整模态数据的适当完全近似值。所提出的系统通常由于来自模型更新过程的嘈杂数据和病态系统而处于不适状态。为了解决这个问题，在模型更新问题中使用约束作为正则化参数来区分潜在损坏的元素和未损坏的元素。通过定义缩减指数（RI）并强制未损坏元素的严重性趋于零，此约束被施加到模型更新过程中。此外，所提出的系统是非正方形的，并且很大，取决于结构未知数的数量。由于这些原因，采用双共轭梯度（BCG）方法以及适当的预处理器来求解方程组。预处理器不仅可以使解决方案稳定，以防止由于噪声数据导致的数值错误和不准确性，而且还可以将系统更改为正方形。进行数值研究以证明所提出方法的效率。此外，进行统计分析以评估不同随机噪声序列对损坏检测结果的影响。结果表明，所提出的方法能够在存在噪声数据的情况下检测圆柱壳的严重程度和损坏部位。