A Symmetric Inverse Vibration Problem

1992 ◽  
Vol 114 (4) ◽  
pp. 564-568 ◽  
Author(s):  
L. Starek ◽  
D. J. Inman ◽  
A. Kress
Keyword(s):  
Natural Frequencies ◽  
Computer Code ◽  
Inverse Eigenvalue Problem ◽  
Repeated Eigenvalues ◽  
Stiffness Matrices ◽  
Rigid Body Modes ◽  
Inverse Eigenvalue ◽  
Inverse Vibration ◽  
Vector Differential Equation ◽  
Vibrating Systems

This paper considers the 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 coefficient matrices, assumed to represent the mass, damping, and stiffness matrices, given the natural frequencies and damping ratios of the structure (i.e., the system eigenvalues). The approach presented here allows for repeated eigenvalues, whether simple or not, and for rigid body modes. The method is algorithmic and results in a computer code for determining mass normalized damping, and stiffness matrices for the case that each mode of the system is underdamped.

A Symmetric Positive Definite Inverse Vibration Problem With Underdamped Modes

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.

A Constrained Convex Approach to Modal Design Optimization of Vibrating Systems

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Dario Richiedei ◽  
Alberto Trevisani ◽  
Gabriele Zanardo
Keyword(s):  
Mechanical Design ◽  
Design Method ◽  
Inverse Eigenvalue Problem ◽  
Test Cases ◽  
Vibration Modes ◽  
Flexible Design ◽  
Stiffness Matrices ◽  
Inverse Eigenvalue ◽  
Design Requirements ◽  
Vibrating Systems

This paper introduces a general and flexible design method for the inverse modal optimization of undamped vibrating systems, i.e., for the computation of mass and stiffness linear modifications ensuring the desired system eigenstructure. The technique is suitable for the design of new systems or the optimization of the existing ones and can handle several design requirements and constraints. Paramount strengths of the method are its capability to modify an arbitrary number of parameters and assigned vibration modes, as well as the possibility of dealing with mass and stiffness matrices with arbitrary topologies. To this purpose, the modification problem is formulated as a constrained inverse eigenvalue problem and then solved within the frame of convex optimization. The effectiveness of the method is assessed by applying it to two different test cases. In particular, the second investigation deals with a meaningful mechanical design application: the optimization of a system recalling an industrial vibratory feeder. The results highlight the generality of the method and its capability to ensure the achievement of the prescribed eigenstructure.

A Symmetric Inverse Vibration Problem for Nonproportional Underdamped Systems

1997 ◽  
Vol 64 (3) ◽  
pp. 601-605 ◽  
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.

scholarly journals Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Roberto Caracciolo ◽  
Dario Richiedei ◽  
Iacopo Tamellin
Keyword(s):  
Structural Modification ◽  
Natural Frequencies ◽  
Adjoint System ◽  
Global Optimum ◽  
Inverse Eigenvalue Problem ◽  
Partial Assignment ◽  
Inverse Eigenvalue ◽  
Novel Method ◽  
Vibrating Systems ◽  
Homotopy Optimization

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick’s constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples.

scholarly journals A Structured Approach to Solve the Inverse Eigenvalue Problem for a Beam with Added Mass

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Farhad Mir Hosseini ◽  
Natalie Baddour
Keyword(s):  
Natural Frequencies ◽  
Point Of View ◽  
Inverse Eigenvalue Problem ◽  
Bernoulli Beam ◽  
Combined System ◽  
Vibrational System ◽  
Single Mass ◽  
Inverse Eigenvalue ◽  
Euler Bernoulli Beam ◽  
Structured Approach

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this paper, a method to impose two natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam and carrying a single mass attachment is evaluated.

Design of Nonproportional Damped Systems via Symmetric Positive Inverse Problems

2004 ◽  
Vol 126 (2) ◽  
pp. 212-219 ◽  
Author(s):  
L. Starek ◽  
D. J. Inman
Keyword(s):  
Eigenvalue Problems ◽  
Positive Definite ◽  
Inverse Eigenvalue Problem ◽  
Eigenvalues And Eigenvectors ◽  
Inverse Eigenvalue Problems ◽  
Complex Eigenvalues ◽  
Inverse Eigenvalue ◽  
Nonproportional Damping ◽  
Damped Systems ◽  
Vibrating Systems

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.

scholarly journals An Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Yongxin Yuan
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Inverse Eigenvalue Problem ◽  
Stiffness Matrices ◽  
Inverse Eigenvalue ◽  
Second Order Systems ◽  
Necessary And Sufficient ◽  
Semidefinite Matrix

The inverse eigenvalue problem of constructing symmetric positive semidefinite matrix (written as ) and real-valued skew-symmetric matrix (i.e., ) of order for the quadratic pencil , where , are given analytical mass and stiffness matrices, so that has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are specified.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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