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Eigensolution of nonclassically damped structures by complex subspace iteration

机译：非经典阻尼结构通过复杂子空间迭代的特征解

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A nonclassically damped structure includes the consistent damping matrix. All structural matrices are symmetric, the stiffness matrix has to be positive definite. This leads to a quadratic eigenproblem with complex eigenvalues and eigenvectors. The common solution technique is the transformation into a linear equation system in state space form, which doubles the order of the system, and to solve it e.g. by the Lanczos method. The presented approach solves the quadratic eigenproblem directly, i.e. the order is not in- creased and the sparseness or bandedness of the system matrices is retained. The solution of the eigenproblem is obtained by the principle of subspace iteration. In comparison to undamped modes, the complex subspace iteration operates on a modal subspace of double size. The iteration sequence is based on a filter principle. This enables to utilise prefactorised system equations and starting information of eigenvectors from a previous solution, even when the structural matrices are updated. Shift techniques can be applied, which makes the algorithm useful for numerically tough problems including a large number of overdamped eigensolutions.
机译：非经典阻尼结构包括一致的阻尼矩阵。所有结构矩阵都是对称的，刚度矩阵必须是正定的。这导致具有复杂特征值和特征向量的二次特征问题。常见的解决方法是将状态空间形式的线性方程组转换为系统的阶次并对其求解，例如用Lanczos方法。提出的方法直接解决了二次特征问题，即不增加阶数，并且保留了系统矩阵的稀疏性或带状性。本征问题的解决方案是通过子空间迭代原理获得的。与无阻尼模式相比，复杂子空间迭代在两倍大小的模态子空间上运行。迭代序列基于过滤原理。这样即使在更新结构矩阵时也可以利用预先求解的系统方程式和来自先前解决方案的特征向量的起始信息。可以应用平移技术，这使得该算法对于包括大量过阻尼本征解在内的数值难题非常有用。

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