Modal Damping Matrix Research Articles

Wind-tunnel experimental study on identification of modal aerodynamic damping matrix for operating wind turbines

Wind-tunnel experimental study on identification of modal aerodynamic damping matrix for operating wind turbines

Model Updating for Systems with General Proportional Damping Using Complex FRFs

In this paper, we propose a model updating method for systems with nonviscous proportional damping. In comparison to the traditional viscous damping model, the introduction of nonviscous damping will not only reduce the vibration of the system but also change the resonance frequencies. Therefore, most of the existing updating methods cannot be directly applied to systems with nonviscous damping. In many works, for simplicity, the Rayleigh damping model has been applied in the model updating procedure. However, the assumption of Rayleigh damping may result in large errors of damping for higher modes. To capture the variation of modal damping ratio with frequency in a more general way, the diagonal elements of the modal damping matrix and relaxation parameter are updated to characterize the damping energy dissipation of the structure by the proposed method. Spatial and modal incompleteness are both discussed for the updating procedure. Numerical simulations and experimental examples are adopted to validate the effectiveness of the proposed method. The results show that the systems with general proportional damping can be predicted more accurately by the proposed updating method.

Rheological-dynamical analogy for analysis of low cycle fatigue in seismically isolated regular bridges

This paper examines low cycle fatigue in seismically isolated regular (SIR) bridges, taking into account the steady-state vibrations due to the lateral force of vehicles or any other lateral impacts of a permanent character. Bridges are modeled as two-degree-of-freedom (TDOF) systems in which the force presents sinusoidal impact on the centre of gravity of the bridge superstructure. Inelastic stiffnesses and modal damping ratios for the column and isolation bearing are obtained using the rheological-dynamical analogy (RDA). The background of this inelastic theory is presented in the framework of a mathematical-physical analogy between the rheological model of viscoelastoplastic (VEP) material and dynamical model with viscous damping. To estimate quantitatively the influence of isolation material characteristics on damping, corresponding frequency response functions for bent column and isolation bearings in bridges are analyzed. In addition, the paper develops the real damping matrix, which includes effects of damages due to fatigue, using the modal damping matrix and taking account of mode orthogonality.

Modal damping variations in nonlinear dynamical systems

For any linear dynamical system coupled with one or more nonlinear dynamical attachments, the effect of nonlinear energy content on modal damping variations of the entire system has not been clearly addressed in the literature. Accordingly, a novel method is employed here to formulate the amplitude-dependent modal damping matrix for such nonlinear dynamical systems using an amplitude-dependent stiffness approach. The proposed method is directly applied into the equations of motion where numerical and analytical solutions are not required to be known a priori. This advantage is highly desirable to study the dynamical behavior of nonlinear dynamical systems by direct application of methods into equations of motion. Accordingly, the modal damping content variations under the effect of amplitude-dependent stiffness are investigated here. The method is based on linearizing the nonlinear coupling stiffness where a scaled amplitude-dependent stiffness has been obtained to replace the original nonlinear coupling stiffness in the system. Accordingly, the amplitude-dependent modal damping matrix in modal coordinates is obtained and investigated. Consequently, new significant findings regarding modal damping content variations under the effect of the change in nonlinear energy during oscillation are achieved through this study. Furthermore, the nonlinear amplitude-dependent modal damping matrix of the equivalent system is found to be satisfying all matrix similarity conditions with the linear amplitude-independent modal damping matrix of the original system. These findings are expected to be of significant impact on passive nonlinear targeted energy transfer for shock mitigation and energy-harvesting fields.

Closed-form eigensolutions of nonviscously, nonproportionally damped systems based on continuous damping sensitivity

In this paper, nonviscous, nonproportional, vibrating structures are considered. Nonviscously damped systems are characterized by dissipative mechanisms which depend on the history of the response velocities via hereditary kernel functions. Solutions of the free motion equation lead to a nonlinear eigenvalue problem involving mass, stiffness and damping matrices. Viscoelasticity leads to a frequency dependence of this latter. In this work, a novel closed-form expression to estimate complex eigenvalues is derived. The key point is to consider the damping model as perturbed by a continuous fictitious parameter. Assuming then the eigensolutions as function of this parameter, the computation of the eigenvalues sensitivity leads to an ordinary differential equation, from whose solution arises the proposed analytical formula. The resulting expression explicitly depends on the viscoelasticity (frequency derivatives of the damping function), the nonproportionality (influence of the modal damping matrix off-diagonal terms). Eigenvectors are obtained using existing methods requiring only the corresponding eigenvalue. The method is validated using a numerical example which compares proposed with exact ones and with those determined from the linear first order approximation in terms of the damping matrix. Frequency response functions are also plotted showing that the proposed approach is valid even for moderately or highly damped systems.

Eigensolutions of non-proportionally damped systems based on continuous damping sensitivity

The viscous damping model has been widely used to represent dissipative forces in structures under mechanical vibrations. In multiple degree of freedom systems, such behavior is mathematically modeled by a damping matrix, which in general presents non-proportionality, that is, it does not become diagonal in the modal space of the undamped problem. Eigensolutions of non-proportional systems are usually estimated assuming that the modal damping matrix is diagonally dominant (neglecting the off-diagonal terms) or, in the general case, using the state–space approach. In this paper, a new closed-form expression for the complex eigenvalues of non-proportionally damped system is proposed. The approach is derived assuming small damping and involves not only the diagonal terms of the modal damping matrix, but also the off-diagonal terms, which appear under higher order. The validity of the proposed approach is illustrated through a numerical example.

A Mechanical Approach to the Design of Independent Modal Space Control for Vibration Suppression

Many systems have, by their nature, a small damping and therefore they are potentially subjected to dangerous vibration phenomena. The aim of active vibration control is to contain this phenomenon, improve the dynamic performance of the system, and increase its fatigue strength. A way to reach this goal is to increase the system damping, preferably without changing its natural frequencies and vibration modes. In the past decades this has been achieved by developing the well-known independent modal space control (IMSC) technique. The paper describes a new approach to the synthesis of a modal controller to suppress vibrations in structures. It turns from the traditional formulation of the problem and it demonstrates how the performance of the controller can be evaluated through the analysis of the modal damping matrix of the controlled system. The ability to easily manage this information allows us to synthesize an efficient modal controller. Furthermore, it enables us to easily evaluate the stability of the control, the effects of spillover, and the consequent effectiveness in reducing vibration. Theoretical aspects are supported by experimental applications on a large flexible system.

Efficient uncoupled stochastic analysis with non-proportional damping

The use of normal modes of vibration in the analysis of structures with non-proportional damping reduces the size of the resulting set of governing equations, but does not decouple them. A common practice consists in decoupling the equations by disregarding the off-diagonal elements in the modal damping matrix. Recently, an approximation based on an asymptotic expansion of the modal transfer matrix has been proposed in a deterministic framework to partially account for off-diagonal terms, but still with a set of uncoupled equations. This paper aims at extending this method in a stochastic context. First the mathematical background is introduced and the method is illustrated with a simple example. Then its relevance is demonstrated within the context of the structural analysis of a large and realistic structure.

A new approach to the synthesis of modal control laws in active structural vibration control

Many systems have, by their nature, a small damping and they are therefore potentially subjected to dangerous vibratory phenomena. The aim of active vibration control is to contain these phenomena, increasing the natural damping of the system and its fatigue life, preferably without changing its natural frequencies and vibration modes. The modal control technique is the favorite amongst mechanical and structural engineers because of its representation in modal coordinates. However, especially in the field of active vibration control, the traditional design of the controller developed in state coordinates makes it difficult to understand how the mechanical parameters of the system are modified. The present paper introduces a new approach to the synthesis of a modal controller to suppress vibrations in structures. Itturns away from the traditional formulation of the problem showing how the performance of the designed controller can be evaluated through the analysis of the resulting modal damping matrix of the controlled system. The approach is based on the relationship existing between the shape of the matrix and the possibility that the control forces are dissipative (and, thus, ensuring the system stability). This analysis allows us to easily evaluate spillover effects, due to the presence of unmodeled modes, the stability of the control and the consequent effectiveness in reducing vibration. The opportunity to easily manage this information allows the synthesis of an efficient modal controller. Theoretical aspects are supported by numerical simulations and experimental tests.

Study on Equivalent Damping Ratio of Shenzhen Baoan International Airport T3 Terminal

Today, the structures constituted by different materials, increase more and more, especially the lower part is concrete, the upper is steel. For this type of structural system, modal damping matrix is non-diagonal matrix, the earthquake response equation is coupled on the modal damping matrix, the modal coupling of the non-proportional damping system leads to the traditional real modal analysis methods not be directly applied. For such structure, the changes of the damping ratio are analyzed in this article. Finally, the equivalent damping ratio of Shenzhen Airport is obtained using the energy theory.

Study on Equivalent Damping Ratio of Shenzhen Baoan International Airport T3 Terminal

Today, the structures constituted by different materials, increase more and more, especially the lower part is concrete, the upper is steel. For this type of structural system, modal damping matrix is non-diagonal matrix, the earthquake response equation is coupled on the modal damping matrix, the modal coupling of the non-proportional damping system leads to the traditional real modal analysis methods not be directly applied. For such structure, the changes of the damping ratio are analyzed in this article. Finally, the equivalent damping ratio of Shenzhen Airport is obtained using the energy theory.

Perturbation spectrum method for seismic analysis of non-classically damped systems

Fundamental principles from structural dynamics, random theory and perturbation methods are adopted to develop a new response spectrum combination rule for the seismic analysis of non-classically damped systems, such as structure-damper systems. The approach, which is named the perturbation spectrum method, can provide a more accurate evaluation of a non-classically damped system’s mean peak response in terms of the ground response spectrum. To account for the effect of non-classical damping, all elements are included in the proposed method for seismic analysis of structure, which is usually approximated by ignoring the off-diagonal elements of the modal damping matrix. Moreover, as has been adopted in the traditional Complete Quadratic Combination (CQC) method, the white noise model is also used to simplify the expressions of perturbation correlation coefficients. Finally, numerical work is performed to examine the accuracy of the proposed method by comparing the approximate results with exact ones and to demonstrate the importance of the neglected off-diagonal elements of the modal damping matrix. In the examined cases, the proposed method shows good agreement with direct time-history integration. Also, the perturbation spectrum method leads to a more efficient and economical calculation by avoiding the integral and complex operation.

Asymptotic expansion of slightly coupled modal dynamic transfer functions

In case of non-diagonal modal damping, normal modes of vibration do not decouple modal equations. The usual way to handle such a non-diagonal modal damping matrix is to neglect its off-diagonal elements. In this paper, we propose an approximate method based on an asymptotic expansion of the transfer function. It is intermediate between the classical decoupling approximation and the formal solution requiring a full matrix inversion. Indeed, on the one hand, it allows to partially account for modal coupling and, on the other hand, still allows the modal equations to be solved independently from each other. We first provide the mathematical background necessary to canvass the proposed method, then consider a benchmark against which the benefits of the method are measured.

On The Decoupling Approximation in Damped Linear Systems

The principal coordinates of a non-classically damped linear system are coupled by non-zero off-diagonal elements of the modal damping matrix. In the analysis of non-classically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is widely believed that if the modal damping matrix is diagonally dominant, then errors due to the decoupling approximation must be small. In addition, it is intuitively accepted that the more diagonal the modal damping matrix, the smaller will be the errors due to the decoupling approximation. Two numerical indices are proposed in this paper to measure quantitatively the degree of being diagonal in modal damping. It is demonstrated that, over a finite range, errors due to the decoupling approximation can continuously increase while the modal damping matrix becomes more and more diagonal with its off-diagonal elements decreasing in magnitude continuously. An explanation for this unexpected behavior is offered. Within a practical range of engineering applications, diagonal dominance of the modal damping matrix may not be sufficient for neglecting modal coupling in a damped system.

Diagonal dominance of damping and the decoupling approximation in linear vibratory systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant. In addition, the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant. It is shown numerically in this paper that, over a finite range, errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off-diagonal elements decreasing continuously in magnitude. An explanation for these unexpected drifts of decoupling errors is provided with the use of complex coupling coefficients. Small off-diagonal elements in the modal damping matrix are not sufficient to ensure small errors due to the decoupling approximation. Any error-criterion based solely upon the diagonal dominance of the modal damping matrix would not be accurate.

Rayleigh Quotient and Dissipative 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.

A remark about the decoupling approximation of damped linear systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. Contrary to widely accepted beliefs, it is shown numerically that over a finite range, errors due to the decoupling approximation can continuously increase at any specified rate while the modal damping matrix becomes more and more diagonal.

Modal control of vibration in rotating machines and other generally damped systems

Second-order matrix equations arise in the description of real dynamical systems. Traditional modal control approaches utilise the eigenvectors of the undamped system to diagonalise the system matrices. A regrettable consequence of this approach is the discarding of residual off-diagonal terms in the modal damping matrix. This has particular importance for systems containing skew-symmetry in the damping matrix which is entirely discarded in the modal damping matrix. In this paper, a method to utilise modal control using the decoupled second-order matrix equations involving non-classical damping is proposed. An example of modal control successfully applied to a rotating system is presented in which the system damping matrix contains skew-symmetric components.

Observability of Self-Sensing System Using Extended Kalman Filter

Nomenclature Bp = input matrix (l m) Cp = diagonal piezoelectric capacitance matrix (m m) In = unit matrix (n n) K = constant-charge stiffness matrix (l l) l = number of DOF of structure M = mass matrix (l l) m = number of piezoelectric actuators Ok n = zero matrix (k n) Q = charge vector (m 1) V = voltage vector (m 1) w = external force vector (l 1) x = displacement vector (l 1) z = state vector in Eq. (7) (2l 1) ze = extended state vector in Eq. (10) [ 2l m 1] e = observer gain matrix [ 2l m m] = diagonal modal damping matrix (l l) = modal displacement vector (l 1) = eigen matrix (l l) = diagonal constant-charge modal stiffness matrix (l l)

Fast frequency response analysis of partially damped structures with non-proportional viscous damping

This paper presents a new method for the modal frequency response analysis of partially damped structural system with non-proportional viscous damping. The basic idea is to handle the modal viscous damping matrix by noting that the rank of the viscous damping matrix is typically very low for problems of interest. Then, the Sherman–Morrison–Woodbury formula provides a convenient expression for the inverse of equation which includes the low-rank matrix. The new method, fast frequency response analysis (FFRA) algorithm, dramatically improves the performance of the modal frequency response analysis compared to conventional methods in industry with the same accuracy.