Frequency Domain Analysis of Continuous Systems with Viscous Generalized Damping (original) (raw)
Related papers
2005
To develop methodologies for free and forced vibration analysis of damped linear MDOF systems in a generalized and unified manner.
Modal analysis of non-classically damped linear systems
Earthquake Engineering & Structural Dynamics, 1986
A critical, textbook-like review of the generalized modal superposition method of evaluating the dynamic response of nonclassically damped linear systems is presented, which it is hoped will increase the attractiveness of the method to structural engineers and its application in structural engineering practice and research. Special attention is given to identifying the physical significance of the various elements of the solution and to simplifying its implementation. It is shown that the displacements of a nonclassically damped n-degree-of-freedom system may be expressed as a linear combination of the displacements and velocities of n similarly excited single-degree-of-freedom systems, and that once the natural frequencies of vibration of the system have been determined, its response to an arbitrary excitation may be computed with only minimal computational effort beyond that required for the analysis of a classically damped system of the same size. The concepts involved are illustrated by a series of exqmples, and comprehensive numerical data for a three-degree-offreedom system are presented which elucidate the effects of several important parameters. The exact solutions for the system are also compared over a wide range of conditions with those computed approximately considering the system to be classically damped, and the interrelationship of two sets of solutions is discussed.
Experimental Identification of Generalized Proportional Viscous Damping Matrix
Journal of Vibration and Acoustics, 2009
A simple and easy-to-implement algorithm to identify a generalized proportional viscous damping matrix is developed in this work. The chief advantage of the proposed technique is that only a single drive-point frequency response function (FRF) measurement is needed. Such FRFs are routinely measured using the standard techniques of an experimental modal analysis, such as impulse test. The practical utility of the proposed identification scheme is illustrated on three representative structures: (1) a free-free beam in flexural vibration, (2) a quasiperiodic three-cantilever structure made by inserting slots in a plate in out-of-plane flexural vibration, and (3) a point-coupled-beam system. The finite element method is used to obtain the mass and stiffness matrices for each system, and the damping matrix is fitted to a measured variation of the damping (modal damping factors) with the natural frequency of vibration. The fitted viscous damping matrix does accommodate for any smooth variation of damping with frequency, as opposed to the conventional proportional damping matrix. It is concluded that a more generalized viscous damping matrix, allowing for a smooth variation of damping as a function of frequency, can be accommodated within the framework of standard finite element modeling and vibration analysis of linear systems.
Time and frequency response of structures with frequency dependent, non-proportional linear damping
Journal of Sound and Vibration, 2014
A method to compute the non-stationary time and frequency response of structures with a frequency-dependent non-proportional linear damping, called the resonance modes method, is presented in this paper. It consists of two main steps. The first step aims at spotting the structure resonance modes, which are the solutions of the matrix non-linear eigenvalue problem obtained using the finite element method in the complex plane. This step requires a complex eigensolver and an iterative scheme, a perturbation technique or a combination of both. The second step uses the computed resonance modes and an analytical expression of the inverse Laplace transform to deduce the time or frequency response of structures to general excitations. The response of an aluminum plate damped with an elastomer treatment to a point-force excitation, computed with the classical modal approach, the direct solution and the presented method shows its precision and efficiency. An acoustic power computation finally validates the implementation of a fast variant, based on the perturbation technique, for vibroacoustic applications.
A Real-Space Modal Analysis Method for Non-Proportional Damped Structures
Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016), 2016
The inclusion of damping in the equations of motion of FEM-based structural models yields a complex (quadratic) eigenvalue problem. In this paper is presented a variant of a general method [4], [5] for real-space modal transformation of damped multi-degree-offreedom-systems (MDOFS) with non-modal (non-proportional) symmetric damping matrix. The method is based on the conjugated complex right eigenvectors of the system, normalized relative to the general mass matrix. After state-space formulation of the equations of motion a real modal transformation matrix is built by a combination of two complex transformations, which is the main advantage of the presented method. Analytically expressions for the modal transformation basis are developed be the aid of computer algebra software (MATLAB). Applying the suggested method to the special case of proportionally damped system, an analytical expression for the constant phase lag of the free vibration modes has been derived. The conversion of the developed general real transformation matrix into the modal matrix of the undamped problem is analytically proved by taking into account the synchronous free oscillations in this special case. The derived formulas for the modal transformation basis contain the real and the imaginary parts of the eigenvectors and the associated eigenvalues. A numerical examplevibration of a rotor blade of a wind turbine-demonstrates the performance of the presented modal decomposition method for the general case of nonproportional damped system. The damping matrix of this example contains structural and aerodynamic damping. The initial computation of the complex eigensolution of the FEM beam model in the presented example and all subsequent computations are done by the aid of the Symbolic Math Toolbox of MATLAB. The suggested procedure can be applied in structural systems containing different damping and energy-loss mechanism in various parts of the structure and also in structure-environment interaction problems, where a non-modal damping matrix is occurring.
THE MODES OF NON-HOMOGENEOUS DAMPED BEAMS
Journal of Sound and Vibration, 2001
This short note is concerned with computing the eigenvalues and eigenfunctions of a continuous beam model with damping, using the separation of variables approach. The beam considered has di!erent sti!ness, damping and mass properties in each of two parts. Pinned boundary conditions are assumed at each end, although other boundary conditions may be applied at the ends quite simply. Although applications are not considered in detail, one possible example is a thin beam partly submerged in a #uid. The #uid would add considerable damping and mass to the beam structure, and possibly some sti!ness. Yang and Zhang [1] calculated these added mass and damping coe$cients for parallel #at plates.
Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), 2017
In the general case of non-proportionally damped structural model the associated quadratic eigenvalue problem leads to complex eigenvalues and eigenvectors. The modal decomposition of the equations of motion is usually to be performed in complex space. In this paper are presented possible variants of a general method [2]-[5] for modal transformation of damped multi-degree-of-freedom-systems (MDOFS) with non-modal symmetric damping matrix. The assembly of a modal transformation matrix in real space is based either on the conjugated complex left eigenvectors, or on the right eigenvectors, or on a combination of the left and right eigenvectors of the system. The eigenvector normalization can be performed with respect to the general mass or to the general stiffness matrix. The equations of motion are stated in state-space formulation. The developed real-space modal transformation matrix is always built by a combination of two complex transformations. Analytically expressions for all presented variants of the modal transformation basis are developed be the aid of computer algebra software. Those formulas operate with the real and the imaginary parts of the eigenvectors and the associated eigenvalues. All variants of the suggested modal procedure retain the common advantages of the classic modal decomposition of the equations of motion. The vibrations of a rotor blade of a wind turbine subjected to wind thrust loads have been calculated in two variants to demonstrate the performance of the presented modal analysis procedures. The initial computation of the complex eigenvalue solution of the FEM beam model and all subsequent computations are done by the aid of computer algebra software. The suggested procedures can be applied in structural systems containing different damping and energy-loss mechanism in various parts of the structure, described by non-proportional damping matrix.
Solution of Eigenvalue Problems for Non-Classically Damped Systems with Multiple Frequencies
Journal of Sound and Vibration, 1999
An efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of non-classically damped structural systems with multiple eigenvalues. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem through matrix augmentation of the quadratic eigenvalue problem. In the iteration methods, such as the inverse iteration method and the subspace iteration method, singularity may occur during the factorizing process when the shift value is close to an eigenvalue of the system. However, even though the shift value is an eigenvalue of the system, the proposed method provides non-singularity, and that is analytically proved. Since the modified Newton-Raphson technique is adapted to the proposed method, initial values are needed. Because the Lanczos method effectively produces better initial values than other methods, the results of the Lanczos method are taken as the initial values of the proposed method. Two numerical examples are presented to demonstrate the effectiveness of the proposed method and the results are compared with those of the well-known subspace iteration method and the Lanczos method.