Repeated eigenvalues and their derivatives of structural vibration systems with general nonproportional viscous damping

Highlights

• Study of lightly damped viscous vibration systems with repeated eigenvalues.

• Establishing that most of such systems do not lead to system defectiveness.

• Discovery that defectiveness can occur when viscous damping is considerably high.

• Case of repeated eigenvalues in general nonproportional viscous damping.

• Development of a new method for computing the eigen derivatives for such a case.

Abstract

Repeated vibration modes often occur in practice in structural vibration systems with general nonproportional viscous damping. However, the topic remains perhaps the least understood in vibration analysis. Some researchers have suggested that the inclusion of nonproportional viscous damping renders a system defective in the case of repeated eigenvalues, while others have assumed that a complete set of linearly independent eigenvectors can always be found regardless of the forms and magnitudes of viscous damping. This paper seeks to first establish that for light non-proportional viscous damping, a system with repeated eigenvalues generally does not become defective based on perturbation theory and realistic numerical examples since rigorous theoretical proof is believed to be difficult. Once non-defectiveness is confirmed, a method for computing the eigen derivatives with repeated eigenvalues in the case of general viscous damping is developed. Effect of mode truncation on numerical accuracy has been discussed. When the magnitude of non-proportional viscous damping becomes considerably high however, it is possible for a damped system to become defective when repeated modes occur. This can have profound practical implications where very high damping is desired as in the cases of vibration suspension and absorber designs, since system defectiveness can lead to major difficulties in the applications of conventional vibration analyses.

Introduction

Eigenvalue and eigenvector derivatives with repeated eigenvalues have attracted intensive research interest over the years. Systematic eigensensitivity analysis of multiple eigenvalues was conducted for a symmetric eigenvalue problem depending on several system parameters [1], [2], [3], [4]. An explicit formula was developed using singular value decomposition to compute required bases for eigenspaces, as well as to keep track of the dimensions of state variables and the conditioning of the state-space equations [5]. The existing method proposed in [1], [2] was further extended to compute the derivatives for the case where both repeated eigenvalues and repeated eigenvalue derivatives were present [6]. Extensions were made to the modal expansion procedure originally formulated by Fox and Kapoor [7] for distinct modes to allow the computation of sensitivity information for mode shapes associated with repeated roots and repeated root derivatives [8]. Further, a modal expansion method was discussed and applied to compute eigenderivatives of repeated eigenvalues [9]. Also, eigenderivatives with repeated eigenvalues of the generalized nondefective eigenproblem was examined [10]. A generalized inverse was employed for determining the particular solution of eigenvector derivatives with and without repeated eigenvalues [11]. The simultaneous iteration method first proposed for distinct eigenvalues was further developed for derivatives of eigenvectors in which the dominant eigenvalue was repeated [12]. For the special case where design variable change affects the stiffness in only one direction but not the mass matrix, a much simplified method was developed for computing eigenderivatives of doubly repeated eigenvalues [13]. A novel extension to Nelson's method [14] was discussed and used to calculate the first order derivatives of eigenvectors when the derivatives of the associated eigenvalues became equal, together with continuity of eigenvectors [15]. A set of nonmodal vectors were obtained from the modes associated with the repeated eigenvalue which were orthogonal to the eigenvectors and were used to compute eigenvector derivatives [16].

The modal mass is important in characterizing the dynamical behavior of a base driven structure and the calculation of the effective mass sensitivities was generalized to the case of repeated eigenvalues [17]. Conditions on the parameterization were derived and formulated as theorems, which ensured the existence of derivatives of eigenvectors with respect to these parameters [18]. A combined method based on the constrained generalized inverse of the frequency-shifted stiffness matrix was formulated which became applicable to all nondefective systems [19]. Without explicit use of the eigenvectors, a novel algorithm was designed to calculate derivatives of eigenvalues with respect to system model parameters for both distinct and repeated eigenvalues [20]. To further improve accuracy, a higher-precision dynamic flexibility expression was proposed based on a geometrical series expansion which possesses excellent convergence [21]. Also, by employing symmetry properties of cyclic structures in order to reduce computational effort, a method was presented for eigenvector derivatives which was suitable for parallel implementation [22].

To avoid the use of second order differentiations, an adjoint method was proposed to compute adjoint variables from the simultaneous linear system equations [23]. Nelson’s method was extended to the case of repeated eigenvalues for symmetric real eigen systems with improved condition number of the coefficient matrix [24]. The governing equations for the particular solutions of eigenvector derivatives were augmented by requiring the solution to be mass orthogonal with respect to the repeated modes and adjusting the corresponding coefficients so that the coefficient matrix of the augmented system became non-singular and had improved condition numbers [25]. A method of eigenvector-sensitivity was proposed for repeated eigenvalues and eigenvalue derivatives in which an extended system with a nonsingular coefficient matrix was constructed to derive the needed particular solutions [26]. Similarly, for asymmetric quadratic eigenvalue problems with repeated eigenvalues, a numerical algorithm was developed for eigenvector derivatives by introducing additional normalization conditions to calculate the required particular solution [27]. Also, a preconditioned conjugate gradient method was proposed for repeated eigenvalues by leveraging on the iterative eigen solutions such as the Lanczos or the subspace iteration methods [28]. A general framework for computing eigenvector sensitivity was discussed which was capable of tracking specific mode shapes selected beforehand in the case of multiple eigenvalues [29].

The eigenvalue problem of vibration systems with viscous damping can in general be transformed into standard general complex eigenvalue problem with symmetric system matrices using augmented state space formulation [30], though non-symmetric formulation [31] has also be attempted. The classical modal method and Nelson’s method were extended and applied to viscously damped systems to compute eigen derivatives in the case of distinct eigenvalues [32], [33]. A computationally more efficient method was developed to deal with mode truncation considering in practice when only the partial eigen solution is usually carried out [34]. Based on the Kronecker algebra and matrix calculus, the derivatives of eigenvalues with respect to the model parameters for linear damped systems was proposed [35]. Eigen sensitivity analysis was carried out to predict eigenvectors and eigenvalues of the non-proportionally damped structure due to the changes in structural mass, damping and stiffness through iterations [36]. First-order derivatives of complex eigenvectors of general non-defective eigen systems were examined by introducing new normalization conditions [37]. The sensitivity and stability of the linearization of transforming a nonlinear quadratic eigenvalue problem to a general linear complex eigenvalue problem were examined [38]. The derivatives of semi-simple eigenvalues and the corresponding eigenvectors of the quadratic matrix polynomial were discussed, together with proposed measures to improve the condition of the coefficient matrix [39].

For eigen derivatives with repeated eigenvalues of viscously damped systems, different methods have also been developed. Derivatives of a repeated eigenvalue of viscously damped vibrating systems with respect to system parameters were successfully implemented on the subspace spanned by the eigenvectors [40]. A simplified method for the simultaneous computation of first-, second- and higher-order derivatives of eigenvalues and eigenvectors associated with repeated eigenvalues was presented and was found to be numerically stable and efficient [41]. A procedure was formulated to compute the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems by avoiding the rather undesirable state space representation [42]. For quadratic eigenvalue problems, new algorithms, which were valid and applicable much more generally, were proposed, analyzed, and tested [43]. The computation of eigen solution sensitivity of viscously damped eigen systems with repeated eigenvalues was examined and an efficient algorithm was proposed which worked within the physical N-space without resorting to state-space equations [44]. Nevertheless, a fundamental assumption behind the development of these methods is that the damped system with repeated eigenvalues under consideration remains non-defective so that complete set of eigenvectors can be found which form a complete linearly independent vector base from which the desired eigenvector derivatives can be found.

However, recent studies have shown that a general dynamic system with non-proportional viscous damping possessing repeated eigenvalues may become defective [45], [46]. A defective system does not have a complete set of linearly independent eigenvectors, and is therefore not diagonalizable. In dynamic problems of structure and fluid interactions such as flutter of airplane and missile wings or long blades of turbines, defective systems often arise and special analytical methods are required for the proper vibration analysis of these systems [47]. Other examples of defective systems include the eigenvalue problem of the zeros, or anti-resonances of frequency response functions [48]. From the outset, it is somewhat puzzling for a viscously damped system to become defective in the presence of repeated eigenvalues, since in the particular case when damping is proportional, the system is perfectly non-defective with its complete eigenvectors forming a complete linearly independent vector base [49]. Due to the very general form a proportional damping can assume [49], in the case of lightly damped systems which include most engineering structural systems [50], a non-proportional damping can perhaps be considered mathematically as a perturbation to a proportional damping case. Such damping perturbation will lead to perturbations in classical normal modes and based on the theory of perturbations of eigenvalue problems [51], provided the damping perturbation is small relative to stiffness of the system in Euclidean norms, complete linearly independent eigenvectors can still be found and the system remains non-defective. However, when the damping of a system becomes high such as those in vibration suspension and vibration absorber [52], then the damping perturbation can no longer be considered small, leading drastic changes in system characteristics and resulting in a subsequent defective system in the presence of repeated eigenvalues. Nonetheless, such a conjecture cannot be neatly and analytically proven and we will resort to perturbation theory together with numerical case studies to support our argument.

This paper examines eigenvalue and eigenvector derivatives for vibration systems with general non-proportional viscous damping in the case of repeated eigenvalues. It first seeks to establish, based on both perturbation theory and numerical case examples, that for most lightly damped pracrical vibration systems, the existence of repeated eigenvalues does not generally lead to defective systems and as a result, complete set of linearly independent eigenvectors can be found. This is important since most vibration analyses developed to date are only applicable to non-defective systems. Once non-defectiveness is confirmed, a new method is then developed for computing the eigenderivatives with repeated eigenvalues in the case of general non-proportional viscous damping. Further, the effect of mode truncation on numerical accuracy of eigenvector derivatives is also discussed. When the non-proportional viscous damping becomes considerably high however, it is possible for a system to become defective when repeated modes occur. This can have profound practical implications where very high damping is desired as in the cases of vibration suspension and absorber designs, since system defectiveness can lead to major difficulties in the applications of conventional vibration analyses.