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 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 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.

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.

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.

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.

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