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.

Updating the Damping Matrix Using Frequency Response Data

1995 ◽  
Author(s):  
Cécile Reix ◽  
Alain Gerard ◽  
Christian Tombini
Keyword(s):  
Frequency Response ◽  
Weighted Least Squares ◽  
Element Model ◽  
Mode Shapes ◽  
Minor Effect ◽  
Sensitivity Matrix ◽  
Damping Matrix ◽  
Stiffness Matrices ◽  
Weighted Least Squares Estimation ◽  
A Minor

Abstract This paper presents a method for the updating of the damping matrix of a linear dynamic system. For this dynamic study, it is presumed that the characteristic mass and stiffness matrices are perfectly known thanks to the updating of the experimental and calculed frequencies and mode shapes as from a finit element model. Furthermore, it is accepted that damping has only a minor effect on the frequencies and mode shapes of a structure (a hypothesis that has been verified for structures with low damping). It is proposed to adjuste the coefficients of the [D] hysteretic damping matrix as from the superposition of the experimental and analytical Frequency Response Functions (FRF). The frequencies and mode shapes are extracted from the solutions of the caracteristic equation (3) resulting from the classic dynamic equation. An analytical FRF is calculed and then used to establish the sensitivity matrix, translating the influence of the updating parameters on the FRF. To update the [D] matrix, we use a non-linear weighted least squares estimation.

Identification of Multi-Degree of Freedom Systems With Nonproportional Damping Using the Resonant Decay Method

2004 ◽  
Vol 126 (2) ◽  
pp. 298-306 ◽  
Author(s):  
Steven Naylor ◽  
Michael F. Platten ◽  
Jan R. Wright ◽  
Jonathan E. Cooper
Keyword(s):  
Measurement Noise ◽  
Mode Shapes ◽  
Damping Matrix ◽  
Modal Damping ◽  
Stiffness Matrices ◽  
Rigid Body Modes ◽  
Nonproportional Damping ◽  
Modal Mass ◽  
Damped Systems ◽  
Modal Space

This paper describes an extension of the force appropriation approach which permits the identification of the modal mass, damping and stiffness matrices of nonproportionally damped systems using multiple exciters. Appropriated excitation bursts are applied to the system at each natural frequency, followed by a regression analysis in modal space. The approach is illustrated on a simulated model of a plate with discrete dampers positioned to introduce significant damping nonproportionality. The influence of out-of-band flexible and rigid body modes, imperfect appropriation, measurement noise and impure mode shapes is considered. The method is shown to provide adequate estimates of the modal damping matrix.

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.

scholarly journals Modeling and simulation of the enzymatic degradation of 2,4,6-trichlorophenol using soybean peroxidase

2021 ◽  
Author(s):  
Alexandre Santuchi da Cunha ◽  
Ardson dos Santos Vianna Junior ◽  
Enzo Laurenti
Keyword(s):  
Experimental Data ◽  
Enzymatic Degradation ◽  
Weighted Least Squares ◽  
Industrial Effluents ◽  
Complex Model ◽  
Enzyme Inactivation ◽  
Coefficient Of Determination ◽  
Soybean Peroxidase ◽  
Data Set ◽  
Function Coefficient

Abstract The enzymatic degradation of organic pollutants is a promising and ecological method for the remediation of industrial effluents. 2,4,6-Trichlorophenol is a major pollutant in many residual waters, and its consumption has been linked to lymphomas, leukemia, and liver cancer. The goal of the present work is to comprehend the enzymatic degradation of 2,4,6-trichlorophenol using soybean peroxidase. Different assumptions for the kinetic model were evaluated, and the simulations were compared to experimental data, which was obtained in a microreactor. The literature pointed out that the bi-bi ping-pong model represents well the kinetics of soybean peroxidase degradation. Since it is a complex model, some reactions can be considered or not. Six different possibilities for the model were considered, regarding different combinations of the generated enzyme forms that depend on the hypotheses for simplifying the model. The adjustment of the models was compared based on different metrics, such as the value of the objective function, coefficient of determination and root-mean-square error. The process modeling was obtained by the mass balance of all the reaction components, and all the simulations were performed in MATLAB® R2015a. Reaction parameters were estimated based on the weighted least squares between the experimental data set and the values predicted by the model. The results showed that the data were better adjusted by the model that considers all the enzyme forms, including enzyme inactivation. Therefore, a better comprehension of the reaction mechanism was achieved, which allows a more precise reactor project and process simulation.

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.

Generalizing of Numerically Solving Methods of Eigenvalue Problems to Asymmetrical, Damping Included Case

2001 ◽  
Author(s):  
Shahram Rezaei
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Lanczos Method ◽  
Subspace Method ◽  
Damping Matrix ◽  
Stiffness Matrices

Abstract In this paper, “Subspace” method is generalized to asymmetrical case. In the new algorithm described here, “Lanczos” method is used to find the first subspace and to solve the eigenvalue problem resulted in generalized subspace method. To solve the standard eigenvalue problem developed by “Lanczos” method “Jacoby” method is used. If eigenvalue problem includes damping matrix, that will be imported in new defined mass and stiffness matrices.

scholarly journals Approximation of stability radii for large-scale dissipative Hamiltonian systems

2020 ◽  
Vol 46 (1) ◽  
Author(s):  
Nicat Aliyev ◽  
Volker Mehrmann ◽  
Emre Mengi
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Order Reduction ◽  
Interpolation Property ◽  
Element Model ◽  
Positive Semidefinite ◽  
Stability Radius ◽  
Linear Time Invariant ◽  
Reduction Techniques ◽  
Stability Radii

Abstract A linear time-invariant dissipative Hamiltonian (DH) system $\dot x = (J-R)Q x$ẋ=(J−R)Qx, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625–1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + BΔCH for given matrices B, C, and another with respect to Hermitian perturbations in the form R + BΔBH,Δ = ΔH. We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.

Identification of the Damping Matrix

2001 ◽  
Author(s):  
Menahem Baruch
Keyword(s):  
Frequency Response Function ◽  
Degrees Of Freedom ◽  
Linear Structure ◽  
Dynamic Structure ◽  
Viscous Damping ◽  
Structural System ◽  
Linear Dynamic ◽  
Damping Matrix ◽  
Stiffness Matrices ◽  
Mass Matrices

Abstract Only experiments can provide the data necessary to obtain the damping matrix of a dynamic structural system. In the method proposed here the damping matrix can be separated from the mass and stiffness matrices and obtained in an independent of them way. Two methods are presented. In the first method it is assumed that all the degrees of freedom can be loaded and measured. Several methods for calculation of the damping, mass and stiffness matrices, using the experimental data are presented. In the second method the load is employed only in some chosen points. However, it is assumed again that all the degrees of freedom are measured. In order to identify the damping, stiffness and mass matrices of the structure the measured quantities are forced to comply with the general laws for a linear structure. The structure is idealized to be a linear dynamic structure with viscous damping. The measured quantities are measured during the tests at discrete points of the Frequency Response Function.

Rayleigh Quotient and Dissipative Systems

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
A. Srikantha Phani ◽  
S. Adhikari
Keyword(s):  
Perturbation Theory ◽  
Rayleigh Quotient ◽  
Dissipative Systems ◽  
Diagonal Dominance ◽  
Dissipative Forces ◽  
Damping Matrix ◽  
Modal Damping ◽  
Nonproportional Damping ◽  
Vibrating Systems

Rayleigh quotients in the context of linear, nonconservative vibrating systems with viscous and nonviscous dissipative forces are studied in this paper. Of particular interest is the stationarity property of Rayleigh-like quotients for dissipative systems. Stationarity properties are examined based on the perturbation theory. It is shown that Rayleigh quotients with stationary properties exist for systems with proportional viscous and nonviscous damping forces. It is also shown that the stationarity property of Rayleigh quotients in the case of nonproportional damping (viscous and nonviscous) is conditional upon the diagonal dominance of the modal damping matrix.

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