Design of Nonproportional Damped Systems via Symmetric Positive Inverse Problems

This paper summarizes the authors’ previous efforts on solving inverse eigenvalue problems for linear vibrating systems described by a vector differential equations with constant coefficient matrices and nonproportional damping. The inverse problem of interest here is that of determining symmetric, real, positive definite coefficient matrices assumed to represent mass normalized velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. Two previous solutions to the symmetric inverse eigenvalue problem, presented by Starek and Inman, are reviewed and then extended to the design of underdamped vibrating systems with nonproportional damping.

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A Symmetric Positive Definite Inverse Vibration Problem With Underdamped Modes

Volume 3C: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration Control, Analysis, and Identification ◽

10.1115/detc1995-0682 ◽

1995 ◽

Author(s):

Ladislav Starek ◽

Daniel J. Inman ◽

Deborah F. Pilkev

Keyword(s):

Positive Definite ◽

Inverse Eigenvalue Problem ◽

Eigenvalues And Eigenvectors ◽

Symmetric Positive Definite ◽

Vibration Problem ◽

Damped System ◽

Stiffness Matrices ◽

Inverse Eigenvalue ◽

Inverse Vibration ◽

Vector Differential Equation

Abstract This manuscript considers a symmetric positive definite inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices. The inverse problem of interest here is that of determining real symmetric, positive definite coefficient matrices assumed to represent the mass normalized velocity and position coefficient matrices, given a set of specified eigenvalues and eigenvectors. The approach presented here gives an alternative solution to a symmetric inverse vibration problem presented by Starek and Inman (1992) and extends these results to include the definiteness of the coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of the system is underdamped. The result provides an algorithm for determining a non-proportional damped system which will have symmetric positive definite coefficient matrices.

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Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede

Special Matrices ◽

10.1515/spma-2018-0008 ◽

2018 ◽

Vol 6 (1) ◽

pp. 77-92 ◽

Cited By ~ 1

Author(s):

Debashish Sharma ◽

Mausumi Sen

Keyword(s):

Eigenvalue Problem ◽

Recurrence Relations ◽

Eigenvalue Problems ◽

Inverse Eigenvalue Problem ◽

Characteristic Polynomials ◽

Inverse Eigenvalue Problems ◽

Inverse Eigenvalue ◽

Principal Submatrices ◽

Arrow Matrix ◽

New System

Abstract The reconstruction of a matrix having a pre-defined structure from given spectral data is known as an inverse eigenvalue problem (IEP). In this paper, we consider two IEPs involving the reconstruction of matrices whose graph is a special type of tree called a centipede. We introduce a special type of centipede called dense centipede.We study two IEPs concerning the reconstruction of matrices whose graph is a dense centipede from given partial eigen data. In order to solve these IEPs, a new system of nomenclature of dense centipedes is developed and a new scheme is adopted for labelling the vertices of a dense centipede as per this nomenclature . Using this scheme of labelling, any matrix of a dense centipede can be represented in a special form which we define as a connected arrow matrix. For such a matrix, we derive the recurrence relations among the characteristic polynomials of the leading principal submatrices and use them to solve the above problems. Some numerical results are also provided to illustrate the applicability of the solutions obtained in the paper.

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Inverse eigenvalue problems for damped vibrating systems

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(87)90306-4 ◽

1987 ◽

Vol 123 (1) ◽

pp. 238-261 ◽

Cited By ~ 39

Author(s):

P Lancaster ◽

J Maroulas

Keyword(s):

Eigenvalue Problems ◽

Inverse Eigenvalue Problems ◽

Inverse Eigenvalue ◽

Vibrating Systems

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A Symmetric Inverse Vibration Problem for Nonproportional Underdamped Systems

Journal of Applied Mechanics ◽

10.1115/1.2788935 ◽

1997 ◽

Vol 64 (3) ◽

pp. 601-605 ◽

Cited By ~ 24

Author(s):

L. Starek ◽

D. J. Inman

Keyword(s):

Complex Conjugate ◽

Eigenvalues And Eigenvectors ◽

Vibration Problem ◽

Damped System ◽

Stiffness Matrices ◽

Complex Eigenvalues ◽

Nonproportional Damping ◽

Inverse Vibration ◽

Vector Differential Equation ◽

Vibrating Systems

This paper considers a symmetric inverse vibration problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices and nonproportional damping. The inverse problem of interest here is that of determining real symmetric, coefficient matrices assumed to represent the mass normalized velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. The approach presented here gives an alternative solution to a symmetric inverse vibration problem presented by Starek and Inman (1992) and extends these results to include noncommuting (or commuting) coefficient matrices which preserve eigenvalues, eigenvectors, and definiteness. Furthermore, if the eigenvalues are all complex conjugate pairs (underdamped case) with negative real parts, the inverse procedure described here results in symmetric positive definite coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of the system is underdamped. The result provides an algorithm for determining a nonproportional (or proportional) damped system which will have symmetric coefficient matrices and the specified spectral and modal data.

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Inverse Eigenvalue Problems for Singular Rank One Perturbations of a Sturm-Liouville Operator

Advances in Mathematical Physics ◽

10.1155/2021/1681873 ◽

2021 ◽

Vol 2021 ◽

pp. 1-5

Author(s):

Xuewen Wu

Keyword(s):

Eigenvalue Problem ◽

Potential Function ◽

Liouville Operator ◽

Eigenvalue Problems ◽

Inverse Eigenvalue Problem ◽

Inverse Eigenvalue Problems ◽

Rank One ◽

Inverse Eigenvalue ◽

Sturm Liouville

This paper is concerned with the inverse eigenvalue problem for singular rank one perturbations of a Sturm-Liouville operator. We determine uniquely the potential function from the spectra of the Sturm-Liouville operator and its rank one perturbations.

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A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

Abstract and Applied Analysis ◽

10.1155/2014/721346 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Weiping Shen

Keyword(s):

Newton Method ◽

Quadratic Convergence ◽

Eigenvalue Problems ◽

Generalized Newton Method ◽

Inexact Newton Method ◽

Numerical Tests ◽

Inverse Eigenvalue Problems ◽

Inverse Eigenvalue ◽

Generalized Newton ◽

Special Case

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

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