The paper presents an extension of the subspace iteration method for application in systems with viscoelastic damping elements, which are described by both classical and fractional models. The presented method enables determination of only a certain number of eigenvalues and associated eigenvectors. This is very useful, especially when the considered problem has many degrees of freedom, in which case it would be very time-consuming or even impossible to determine all the eigenvalues. At the same time, from the engineering point of view, it is not necessary to know all the eigenvalues. In this paper, the subspace iteration method is used for the first time to solve the nonlinear eigenproblems which appear in the dynamic analysis of systems with viscoelastic damping elements. The proposed solution consists of assuming the number of eigenvalues to be determined, taking the initial point of iteration and then solving the nonlinear reduced eigenproblem with Hermitian matrices in each iterative loop. The solution to the reduced eigenproblem is obtained using the continuation method, and it is an additional novelty in the paper. The correctness and effectiveness of the proposed approach are illustrated with numerical examples.

本文提出了子空间迭代方法的扩展，用于在具有粘弹性阻尼元件的系统中的应用，该系统由经典模型和分数模型描述。所提出的方法能够仅确定一定数量的特征值和相关的特征向量。这非常有用，尤其是当所考虑的问题具有很多自由度时，在这种情况下，确定所有特征值将非常耗时甚至不可能。同时，从工程的角度来看，没有必要知道所有的特征值。本文首次采用子空间迭代法求解具有粘弹性阻尼单元的系统动力学分析中出现的非线性特征问题。建议的解决方案包括假设要确定的特征值的数量，取迭代的初始点，然后在每个迭代循环中使用 Hermitian 矩阵求解非线性简化特征问题。使用延拓法得到了约简本征问题的解，是本文的又一个新颖之处。数值例子说明了所提出方法的正确性和有效性。