Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spill-over. This paper studies the spill-over phenomenon in the updating of quadratic pencils.

Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a mass-spring system, are serially linked. The discussion recasts both undamped or damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described. AMS subject classifications. 65F18, 15A22, 93B55

The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping, and the stiffness matrices, respectively, such that they are closest to the given analytical matrices and satisfy the measured data. The difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite. Based on an equivalent dual optimization version of the IQEP, we present a quadratically convergent Newton-type method. Our numerical experiments confirm the high efficiency of the proposed method.

Abstract. Quadratic pencils arise in many areas of important applications. The underlying physical systems often impose inherent structures, which include the predetermined inner-connectivity among elements within the physical system and the mandatory nonnegativity of physical parameters, on the pencils. In the inverse problem of reconstructing a quadratic pencil from prescribed eigeninformation, respecting the desirable structure becomes important but more challenging both theoretically and practically. The issue of whether a structured inverse eigenvalue problem is solvable or not is problem dependent and has to be addressed structure by structure. In an earlier work, physical systems that can be modeled under the paradigm of a serially linked mass-spring system have been considered via specifically formulated inequality systems. In this paper, the framework is generalized to arbitrary generally linked systems. In particular, given any configuration of inner-connectivity in a mass-spring system, this paper presents a mechanism that systematically and automatically generates a corresponding inequality system. A numerical approach is proposed to determine whether the inverse problem is solvable and, if yes, computes the coefficient matrices while providing an estimate of the residual error. The most important feature of this approach is that it is problem independent, that is, the approach is general and robust for any kind of physical configuration. The ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported.

Abstract. Many natural phenomena can be modeled by a second-order dynamical system M ¨y+C ˙y+Ky = f(t), where y(t) stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that, given merely the desirable structure of the spectrum, namely, given the cardinalities of real or complex eigenvalues but not the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed. AMS subject classifications. 65F18, 15A22, 93B55

Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. This paper studies the spillover phenomenon in the updating of quadratic pencils. In particular, it is shown that an updating with no spillover is always possible for undamped quadratic pencils, whereas spillover for damped quadratic pencils is generally unpreventable. Nomenclature A, B = parameter matrices; Eq. (37) C = damping matrix in a pencil C0 = initial damping matrix in a pencil D = diagonal matrix; Eq. (25) f t = external force H, ^H = intermediate matrices K = stiffness matrix in a pencil

In this paper we concern the inverse problem of constructing the n-by-n real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(λ): = λ2I + λC + K (where I is the identity matrix) possesses the given partial eigendata. We first provide the sufficient and necessary conditions for the existence of an exact solution to the inverse problem from the self-conjugate set of prescribed four eigenpairs. To find a physical solution for the inverse problem where the matrices C and K are weakly diagonally dominant and have positive diagonal elements and negative off-diagonal elements, we consider the inverse problem from the partial measured noisy eigendata. We propose a regularized smoothing Newton method for solving the inverse problem. The global and quadratic convergence of our approach is established under some mild assumptions. Some numerical examples and a practical engineering application in vibrations show the efficiency of our method.