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.

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 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.

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.

Identification of a Nonproportional Damping Matrix from Incomplete Modal Information

1991 ◽  
Vol 113 (2) ◽  
pp. 219-224 ◽  
Author(s):  
C. Minas ◽  
D. J. Inman
Keyword(s):  
Experimental Data ◽  
Weighted Least Squares ◽  
Positive Semidefinite ◽  
Eigenvalues And Eigenvectors ◽  
Damping Matrix ◽  
Reduction Techniques ◽  
Stiffness Matrices ◽  
Hysteretic Damping ◽  
Nonproportional Damping ◽  
Pseudo Inverse

In modeling structures the damping matrix is the most difficult to represent. This is even more difficult in complicated structures that are not lightly damped. The work presented here provides a method of modeling the damping matrix of a structure from incomplete experimental data combined with a reasonable representation of the mass and stiffness matrices developed by finite element methods and reduced by standard model reduction techniques. The proposed technique uses the reduced mass and stiffness matrices and the experimentally obtained eigenvalues and eigenvectors in a weighted least squares or a pseudo-inverse scheme (depending on the number of the equations that are available) to solve for the damping matrix. The results are illustrated through several examples. As an indication of the accuracy of the method, fictitious examples where the damping matrix is originally known are considered. The proposed method identifies the exact viscous or hysteretic damping matrix by only using a partial set, half of the system’s eigenvalues and eigenvectors. The damping matrix is assumed to be real, symmetric, and positive semidefinite.

scholarly journals On the residual motion in damped vibrating systems

2013 ◽  
Vol 40 (1) ◽  
pp. 5-15
Author(s):  
Ranislav Bulatovic
Keyword(s):  
Positive Definite ◽  
Symmetric Positive Definite ◽  
Damping Matrix ◽  
Stiffness Matrices ◽  
Residual Motion ◽  
Vibrating Systems ◽  
Symmetric Systems

In this paper, linear vibrating systems, in which the inertia and stiffness matrices are symmetric positive definite and the damping matrix is symmetric positive semi-definite, are studied. Such a system may possess undamped modes, in which case the system is said to have residual motion. Several formulae for the number of independent undamped modes, associated with purely imaginary eigenvalues of the system, are derived. The main results formulated for symmetric systems are then generalized to asymmetric and symmetrizable systems. Several examples are used to illustrate the validity and application of the present results.

An Out-of-Core Eigen-Solver with OpenMP Parallel Scheme for Large Spare Damped System

2019 ◽  
Vol 16 (07) ◽  
pp. 1950038 ◽  
Author(s):  
S. H. Ju ◽  
H. H. Hsu
Keyword(s):  
Large Scale ◽  
Block Size ◽  
Lanczos Method ◽  
Preconditioned Conjugate Gradient ◽  
Damped System ◽  
Parallel Scheme ◽  
Complex Eigenvalues ◽  
Lanczos Iteration ◽  
Block Lanczos ◽  
Block Lanczos Method

An out-of-core block Lanczos method with the OpenMP parallel scheme was developed to solve large spare damped eigenproblems. The symmetric generalized eigenproblem is first solved using the block Lanczos method with the preconditioned conjugate gradient (PCG) method, and the condensed damped eigenproblem is then solved to obtain the complex eigenvalues. Since the PCG solvers and out-of-core schemes are used, a large-scale eigenproblem can be solved using minimal computer memory. The out-of-core arrays only need to be read once in each Lanczos iteration, so the proposed method requires little extra CPU time. In addition, the second-level OpenMP parallel computation in the PCG solver is suggested to avoid using a large block size that often increases the number of iterations needed to achieve convergence.

scholarly journals The inverse vibration problem for fixed beam submerged in fluid

2019 ◽  
Vol 405 ◽  
pp. 012018
Author(s):  
M Havlásek ◽  
V Habán ◽  
M Hudec ◽  
F Pochylý
Keyword(s):  
Vibration Problem ◽  
Inverse Vibration ◽  
Fixed Beam

Inverse Vibration Problem Used for the Characterization of the Damping Added by a Trim Foam on a Plate

2020 ◽  
Author(s):  
Meryem Le Deunf ◽  
Charles Pezerat ◽  
Frédéric Ablitzer ◽  
Nicolas Merlette
Keyword(s):  
Vibration Problem ◽  
Inverse Vibration

A variational approach for solving an inverse vibration problem

2006 ◽  
Vol 14 (5) ◽  
pp. 557-577 ◽  
Author(s):  
Leonardo D. Chiwiacowsky ◽  
Haroldo F. De Campos Velho ◽  
Paolo Gasbarri¶
Keyword(s):  
Variational Approach ◽  
Vibration Problem ◽  
Inverse Vibration
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