Generalized Damping Matrix Research Articles

An engineering interpretation of the complex modal mass in structural dynamics

In this paper the modal mass is considered in the case of general damping where it is well-known that the modal mass becomes a complex valued quantity. The aim of this paper is to present a definition of the complex modal mass, which leads to an easier physical understanding of this modal parameter. It is demonstrated that based on the structural modification theory, the complex modal mass can be expressed as a linear combination of the un-damped modal masses, where a complex coefficient matrix can be defined that depends on the general damping matrix and on the mode shapes of the damped and the un-damped system. An important finding is that an apparent mass can be defined so that the modal mass is always equal to the product between the apparent mass and the length squared. The paper includes a 2-DOF example that illustrates the influence of the amount of general damping on modal quantities that are normally considered non-sensitive to damping.

Damped DQE Model Updating of a Three-Story Frame Using Experimental Data

In this paper, following a two-stage methodology, the differential quadrature element (DQE) model of a three-story frame structure is updated for the vibration analysis. In the first stage, the mass and stiffness matrices are updated using the experimental natural frequencies. Then, having the updated mass and stiffness matrices, the structural damping matrix is updated to minimize the error between the experimental and numerical damping ratios. Note that two different damping models are used, including a diagonal matrix with unknown diagonal elements and a general damping model. Since the structural joints of the frames are not completely rigid in practice, several parameters are used to model the flexibility of these joints. The optimum values of the material and geometrical design parameters are obtained by updating the DQE model using the experimental modal parameters obtained through modal testing. Considering the robustness of the evolutionary optimization algorithms in the model updating practice, a combination of particle swarm optimization and artificial bee colony algorithm, that benefits from the advantages of both approaches, is utilized. By updating the DQE model, the effectiveness of the evolutionary optimization algorithms, especially in a high-dimensional optimization problem, e.g., finding the optimum general damping matrix, is studied. The results show that, considering the geometrical lengths of the frame as the design parameters, the natural frequencies of the updated model match better with the experimental ones. In addition, using the general damping matrix, the errors of the damping ratios significantly are decreased.

Uncertainty quantification in computational linear structural dynamics for viscoelastic composite structures

This paper deals with the analysis of a stochastic reduced-order computational model in computational linear dynamics for linear viscoelastic composite structures in the presence of uncertainties. The computational framework proposed is based on a recent theoretical work that allows for constructing the stochastic reduced-order model using the nonparametric probabilistic approach. In the frequency domain, the generalized damping matrix and the generalized stiffness matrix of the stochastic computational reduced-order model are random matrices. Due to the causality of the dynamical system, these two frequency-dependent random matrices are statistically dependent and are linked by a compatibility equation induced by the causality of the system, involving a Hilbert transform. The computational aspects related to the nonparametric stochastic modeling of the reduced stiffness matrix and the reduced damping matrix that are frequency-dependent random matrices are presented. A dedicated numerical approach is developed for obtaining an efficient computation of the Cauchy principal value integrals involved in those equations for which an integration over a broad frequency domain is required. A computational analysis of the propagation of uncertainties is conducted for a composite viscoelastic structure in the frequency range. It is shown that the uncertainties on the damping matrix have a strong influence on the observed statistical dispersion of the stiffness matrix.

AN Identification Method of Damping Matrix for Frame Structures

Nowadays, damping is still the important dynamic characteristic that is the least understood and the most difficult to quantify. The advantage of the proposed damping matrix identification method is that several low order complex modes achieved by Experimental Modal Analysis (EMA) are capable of calculation precision. In order to identify the damping matrix, this paper also proposes a general damping matrix decompositon technique, which can be used in the situation that different parts of the structure have different damping. Then a five-story frame structure is applied to demonstrate the proposed method ,and the results obtained show the suitability of this approach for damping matrix identification in frame structures.

Damping of near-adiabatic magnetization dynamics by excitations of electron-hole pairs

Excitation of electron hole pairs and subsequent relaxation is the most important mechanism of intrinsic damping in near-adiabatic magnetization dynamics in metallic ferromagnets. Intraband scattering dominates in the low temperature regime and is described by the breathing Fermi surface model. A Gilbert-like equation of motion is derived within this model where the constant damping scalar is replaced by a magnetization dependent and in general nonlocal damping matrix. In collinear systems the damping term shows two types of anisotropy, and in a general noncollinear situation nonlocal damping matrices enter the equation of motion.

A Note on Nonproportional Damping

This note deals with three aspects of nonproportional damping in linear damped vibrating systems in which the stiffness and damping matrices are not restricted to being symmetric and positive definite. First, we give results on approximating a general damping matrix by one that commutes with the stiffness matrix when the stiffness matrix is a general diagonalizable matrix, and the damping and stiffness matrices do not commute. The criterion we use for carrying out this approximation is closeness in Euclidean norm between the actual damping matrix and its approximant. When the eigenvalues of the stiffness matrix are all distinct, the best approximant provides justification for the usual practice in structural analysis of disregarding the off-diagonal terms in the transformed damping matrix. However, when the eigenvalues of the stiffness matrix are not distinct, the best approximant to a general damping matrix turns out to be related to a block diagonal matrix, and the aforementioned approximation cannot be justified on the basis of the criterion used here. In this case, even when the damping and stiffness matrices commute, decoupling of the modes is not guaranteed. We show that for general matrices, even for symmetric ones, the response of the approximate system and the actual system can be widely different, in fact qualitatively so. Examples illustrating our results are provided. Second, we present some results related to the difficulty in handling general, nonproportionally damped systems, in which the damping matrix may be indefinite, by considering a simple example of a two degrees-of-freedom system. Last, we use this example to point out the nonintuitive response behavior of general nonproportionally damped systems when the damping matrix is indefinite. Our results point to the need for great caution in approximating nonproportionally damped systems by damping matrices that commute with the stiffness matrix, especially when considering general damping matrices. Such approximations could lead to qualitatively differing responses between the actual system and its proportionally damped approximation.

A new design approach of PD controllers

A PD controller design problem for the system described by a second order dynamic model is investigated in this paper. A new PD type controller is proposed by means of Kharitonov theorem and Bilharz criteria. It is proved that using only two parameters is good enough to stabilize general second order dynamic system despite of its dimensions and coupling characteristics. Unlike the classical PD controller design techniques, this approach does not need to tune the design parameters frequently, so the design procedure is greatly simplified. Especially for the system with asymmetric parties in generalized damping matrix and generalized stiffness matrix, this method transfers the controller design for multivariable system to the choice of a scalar parameter μ. This simple and easy to implement approach can be used for the PD controller design of linear coupled multivariable systems. The attitude control of a spacecraft is used as an example in this study.

Transforming Complex Eigenmodes into Real Ones Based on an Appropriation Technique

A new method allowing the determination of normal modes corresponding to the complex ones identified on a structure is proposed. It is based on an appropriation technique applied to forced responses computed from the identified eigensolutions. The set of applied forces is optimized with respect to an appropriation quality criterion. The knowledge of both complex and normal modes allows the full generalized damping matrix to be determined. The complex modes can then be recalculated using the normal modes and the full generalized damping matrix which provides a test of validity by comparison with the original complex modes. Simulated and experimental tests demonstrate the efficiency of this new method.

A precise time-step integration method by step-response and impulsive-response matrices for dynamic problems

In this paper, a precise time-step integration method for dynamic problems is presented. The second-order differential equations for dynamic problems are manipulated directly. A general damping matrix is considered. The transient responses are expressed in terms of the steady-state responses, the given initial conditions and the step-response and impulsive-response matrices. The steady-state responses for various types of excitations are readily obtainable. The computation of the step-response and impulsive-response matrices and their time derivatives are studied in this paper. A direct computation of these matrices using the Taylor series solutions is not efficient when the time-step size Δt is not small. In this paper, the recurrence formulae relating the response matrices at t=Δt to those at t=Δt/2 are constructed. A recursive procedure is proposed to evaluate these matrices at t=Δt from the matrices at t=Δt/2m. The matrices at t=Δt/2m are obtained from the Taylor series solutions. To improve the computational efficiency, the relations between the response matrices and their time derivatives are investigated. In addition, these matrices are expressed in terms of two symmetric matrices that can also be evaluated recursively. Besides, from the physical point of view, these matrices should be banded for small Δt. Both the stability and accuracy characteristics of the present algorithm are studied. Three numerical examples are used to illustrate the highly precise and stable algorithm. © 1997 John Wiley & Sons, Ltd.

Vibration damping in low-frequency range due to structural complexity. A model based on the theory of fuzzy structures and model parameters estimation

For some large structures with internal structural complexity, the modeling of the “secondary” mechanical subsystems attached to the “master” structure allows the vibration predictions to be improved in the low frequency range for which the master structure has a modal response. Within this context, we present a model of the vibration damping due to the vibrations of mechanical subsystems constituting the internal structural complexity. This vibration-damping model is deduced from the theory of fuzzy structures which was previously developed by the author. Presently, this model uses only the mean part of the probabilistic fuzzy law of the fuzzy substructure. A model of the generalized damping matrix which then includes a modeling of the internal structural complexity is explicitly constructed. This generalized damping matrix depends on parameters related to the fuzzy substructure. Problems related to the model parameters estimation are studied. Finally, an example is presented and allows the theory to be validated.

Parameter estimation for fuzzy structures. A result concerning vibrations in the low-frequency range

In the field of structural vibrations, the structural complexity can be induced by ‘‘secondary’’ mechanical subsystems attached to the ‘‘master’’ structure or by ‘‘local eigenmodes’’ of some continuum elastic subelements of the master structure; these local eigenmodes induce a structural complexity when the model of these subelements can only restitute the elastostatic behavior but not its elastodynamic response. Within this context, a model is presented of the apparent vibration damping of the master structure due to the vibrations of the structural complexity. This vibration-damping model is deduced from the theory of fuzzy structures that was previously developed by the author. Presently, this model uses only the mean part of the probabilistic fuzzy law of the fuzzy substructure. A model of the generalized damping matrix deduced from the model of the structural complexity, is explicitly constructed. This generalized damping matrix depends on parameters related to the fuzzy substructure. Problems related to the model parameters estimation are studied. Finally, an example is presented and allows the theory to be validated.

Response of non‐classically damped structures in the modal subspace

AbstractThe evaluation of the dynamic response of non‐classically damped linear structures requires the solution of an eigenproblem with complex eigenvalues and modal shapes. Since in practice only a small number of complex modes are needed, the complex eigenvalue problem is solved in the modal subspace in which the generalized damping matrix is not uncoupled by classical real modes. It follows that the evaluation of the structural response requires in both cases the determination of complex modes by numerical techniques, which are not as robust as techniques currently used for the solution of the real eigenvalue problem, and the use of complex algebra. In the present paper an unconditionally stable step‐by‐step procedure is presented for the response of non‐classically damped structures in the modal subspace without using complex quantities. The method is based on the evaluation of the fundamental operator in approximated form of the numerical procedure. In addition, the method can be easily modified to incorporate the modal superposition pseudo‐static correction terms.

Second-order structural identification procedure via state-space-based system identification

A theory is presented for transforming the system-theory-based realization models into the corresponding physical-coordinate-based structural models. The theory has been implemented into a computational procedure and applied to several example problems. The results show that the present transformation theory yields an objective model basis possessing a unique set of structural parameters from an infinite set of equivalent system realization models. For proportionally damped systems, the transformation directly and systematically yields the normal modes and modal damping. When nonproportional damping is present, the relative magnitude and phase of the damped mode shapes are separately characterized, and a corrective transformation is then employed to capture the undamped normal modes and nondiagonal modal damping matrix. namics equations has hampered the application (and acceptance) of the system-theory-based structural identification techniques and can eventually curtail the progress of hybrid experimental/analyti- cal modeling and design efforts. The present paper offers a theory for transforming the system- theory-based realization models into the corresponding physical- coordinate-based structural models. Since a key idea employed in the development of the present theory is an objective common basis normalization, it is designated as a common basis-normal- ized structural identification (CBSI) procedure. The resultant model is unique for a given sequence of Markov parameters, which in turn are uniquely determined for a linear structure with given inputs and outputs. Therefore, the realized model, after transformation, has a one-to-one correspondence with the physical parameters of the system and can either yield data for finite ele- ment model correlation or alternatively for direct calculation of mass, stiffness, and damping matrices of the original structure. Specifically, we begin with the so-called McMillan transformation employed by Longman and Juang9 and show that the McMillan transformation does not in general yield the desired structural nor- mal modes. To arrive at an objective transformation basis of gener- alized coordinates, we invoke two invariance properties. The first is the output invariance property, viz., the outputs are invariant with respect to any choice of generalized coordinates. The second is the normal mode identity, viz., for proportionally damped cases the mode shapes for the displacement, velocity, and acceleration vectors are the same. There are several byproducts that the present theory provides, primarily due to the common basis normalization employed in the theory. First, the present transformation theory allows the integra- tion of different realized models with varying actuator and sensor locations if they arise from the same structure. Second, each sensor and actuator or groups of sensors and actuators can be processed in parallel and combined concurrently or sequentially for the con- struction of a global model. Third, it can facilitate a decentralized real-time control implementation. From a structural dynamics point of view, the transformation to an objective basis provides a state- space model coinciding with the canonical form of the second- order equations of motion, thus extracting the classical real-valued parameters of interest to modal testing, mass-normalized normal modes, and the general modal damping matrix, while maintaining the system equivalence properties of the state-space form.

Perturbation Method For The Eigenvalue Problem Of Lightly Damped Systems

An efficient method for the determination of the eigenvalues and eigenvectors of lightly damped systems is developed by means of a perturbation technique. The second order matrix differential equation containing mass, stiffness and damping matrices is normally transformed into a first-order state equation to deal with the general damping matrix. However, the method described in this paper enables us to predict the complex eigenvalues and eigenvectors with relative ease without forming the state equation. The light damping implies that the eigensolution of the damped system differs slightly from the eigensolution of the undamped system, which also implies that we can express the eigensolution of the lightly damped system in terms of a power series expanded from the eigensolution of the undamped system. Once the eigenvalue problem of the undamped system is solved, the higher order terms which reflect the effect of damping can be obtained from the matrix equations, which reduce to simple algebraic equations. A numerical example illustrates the effectiveness of the new method.

Finite element analysis of viscoplastic soils with [formula omitted

A finite element viscoplastic computation is proposed where the strain dependent nonlinear stiffness matrix represents plasticity effects and a general nonlinear strain-rate dependent damping matrix accounts for the overall viscous loss. This model can assimilate any conventional plasticity data measured in the laboratory, where the loss coefficient is characterized by Q −1 . In field testing the same is estimated from seismological observations, usually stated as a strain, strain-rate (or frequency) independent loss factor. It is demonstrated herein that the solution of any auxiliary differential equation even for the constant Q model can be avoided when a Laurent series expansion is sought where the coefficients are calculated by a least square fit of the experimental Q- data . Therein the causality condition is satisfied exactly. Since the procedure yields an integro-differential equation the required time steps are considerably small as compared with those in standard explicit and implicit schemes.

Estimation of the Generalized Damping Matrix from Identified Complex Eigensolutions

ABSTRACT ABSTRACT A direct analytical method is proposed for estimating generalized damping factors and real normal modes of the associated conservative structure by a treatment of the identified complex eigensolutions. The method is based on exploitation of orthonormal relations satisfied by both the identified complex modes and modes of the associated conservative structure. The generalized damping matrix is estimated in matrix blocks, of order equal to the number of eigenmodes at neighboring frequencies. This method exploits generalized masses and implies that they are known with satisfactory precision.

Stochastic Response of Prototype Offshore Structure

The dynamic responses of a prototype space-frame offshore structure to both measured and simulated stochastic wave forces are evaluated. A linearized, nondeterministic frequency domain analysis is made on a two-dimensional model of a prototype offshore production platform located in the Gulf of Mexico. Linear ocean waves are modeled as a zero-mean, stationary, ergodic Gaussian process represented by a one-dimensional wave energy density spectrum. Simulated stochastic wave forces are computed via the Morison equation which is modified to account for both the effects of wave-structure interaction and for the equivalent linearization of the drag force in the time-average, mean-square sense. The dynamic equations of motion for the wave-structure system are solved by the normal mode-superposition technique. The coupled generalized damping matrix is diagonalized through an optimization procedure. Iterative procedures are used to determine the equivalently linearized drag term and to uncouple the generalized damping matrix. Dynamic responses of the prototype production platform are compared using both simulated and measured wave pressure force spectra.

Another Look at Sine-Dwell Mode Testing

A simple and systematic method is described for the measurement of natural modes of vibration through the application of single frequency sinusoidal forces using a minimum number of shakers. It is an explicit procedure which can be prescribed for execution by technical personnel with no previous mode test experience. Anticipated advantages in the application of this selective orthogonal excitation (SOREX) methodology include 1) the achievement of improved mode isolation, 2) elimination of the need for many shakers, 3) shortened test duration, and 4) freedom from dependence upon special insight of the test conductor. The method derives from a rejection of the classical use of phase coherence of response as a test criterion in favor of a more rational and practicable test logic. b / F g_ [G] / [/] [K] m [M] n q, q Q x,x 7 X ¥ Q ',0 Nomenclature = scalar coefficient = number of shakers = vector of applied forces = hysteretic damping coefficient = generalized damping matrix unit matrix generalized stiffness matrix number of modes mass matrix number of local coordinates generalized coordinate vectors vector of generalized forces local coordinate vectors orthogonality par ameter, Eq. (11) square of frequency ratio contaminated mode vector, Eq. (12) real, imaginary components of admittance, Eq.(9) circular frequency