In this paper, a canonical transformation is proposed to solve the eigenvalue problem related to the dynamics of rotor-bearing systems. In this problem, all matrices are real, but they may not be symmetric, which leads to the appearance of complex eigenvalues and eigenvectors. The bi-iteration method is selected to solve the original eigenproblem whereas the QR algorithm is adopted to solve the reduced or projected problem. A new canonical transformation of the global eigenproblem which reduces the quadratic eigenproblem to a linear eigenproblem, maintaining numerical stability since all that is required is that the stiffness matrix is well-conditioned, which is always true when it comes to applications in dynamic problems. The proposed technique is good for obtaining dominant eigenvalues and corresponding eigenvectors of real nonsymmetric matrices and it possesses the following properties: (i) the matrix is not transformed, therefore sparsity is maintained, (ii) partial eigensolutions can be obtained and (iii) use may be made of good eigenvectors predictions.

中文翻译：

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一种求解涡轮机械动力学非对称本征系统的数值方法

在本文中，提出了一个典型的变换来解决与转子轴承系统动力学相关的特征值问题。在这个问题中，所有矩阵都是实数，但它们可能不是对称的，这导致出现了复杂的特征值和特征向量。选择双迭代方法解决原始特征问题，而采用QR算法解决简化或投影问题。全局特征问题的新规范变换，将二次特征问题简化为线性特征问题，保持数值稳定性，因为所需要的只是刚度矩阵是良好条件的，这在动态问题的应用中总是正确的。