MODAL ANALYSIS OF NONLINEAR SYSTEMS WITH NONCLASSICAL DAMPING (original) (raw)
Related papers
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.
Soil Dynamics and Earthquake Engineering, 1994
The paper deals with the use of complex modal analysis for computing the response of linear systems having non-proportional damping. The problem of the approximate evaluation of complex modes is first addressed: a second-order perturbation technique, proposed by other researchers, is adopted and modified in view of the application to the analysis of systems having a large number of degrees of freedom. Frequency domain algorithms for the computation of modal response are then tested and a technique for reducing the computational effort due to the performance of inverse Fourier Transforms is proposed. Two examples of application are finally given.
Iterative methods for non-classically damped dynamic systems
Earthquake Engineering & Structural Dynamics, 1994
Non-classically damped structural systems do not easily lend themselves to the modal superposition method because these systems yield coupled second-order differential equations. In this paper, a variety of new computationally efficient iterative methods for determining the response of such systems are developed. The iterative approaches presented here differ from those presented earlier in that they are computationally superior and/or are applicable to the determination of the responses of broader classes of structural systems. Numerical examples, which are designed to evaluate the efficacy of these schemes, show the vastly improved rates of convergence when compared to earlier iterative schemes.
Dynamic Analysis of Nonproportional Damping Structural Systems Time and Frequency-Domain Methods
2001
Structural systems composed of structural elements with different characteristics as NPP and soil and fluid-structure interaction systems present considerable nonproportional damping. The assumption of uncoupled modal equations with assumed modal damping ratios can lead to substantial errors in the dynamic analysis results of those systems. Therefore that assumption is not anymore accepted by the nuclear industry. Appropriate and more rigorous methods for the dynamic analysis of structural systems with nonproportional damping should be developed. In this paper pseudo-force mode superposition methods are developed which consider the nonproportional damping effect through a pseudo-force term in the RHS of the modal equations. These equations are then iteratively solved. The methods presented are considered in time and frequency-domain. The frequency-domain version is very suitable for the analysis of systems with hysteretic damping and frequency-dependent properties. Examples of time ...
Computers & Structures, 1989
For linear structural systems it is possible to use the undamped eigenvectors or load-dependent Ritz vectors to produce a set of modal response equations. When arbitrary viscous damping exists the modal equations are coupled with the modal damping matrix. A robust and efficient numerical algorithm is presented, which solves the coupled modal equations by iteration. It is shown that the numerical integration algorithm always converges. The method produces an exact solution for proportional damping and for loading that varies linearly within an arbitrary time interval. In addition, the algorithm has been modified to incorporate automatically the mode acceleration method and periodic loading. Two numerical examples are presented to illustrate the practical application of the algorithm. A FORTRAN listing of a subroutine is given to facilitate easy implementation of the method in existing computer programs for dynamic response analysis.
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.
A Modal Analysis Method for Structural Models with Non-Modal Damping
2014
Abstract. A general method for the modal decomposition of the equations of motion of damped multi-degree-of-freedom-systems is presented. Two variants of the method are presented, both based on the corresponding eigenvalue problem of the damped structure with symmetric but non-modal damping matrix. The first variant operates with the complex right eigenvectors, normalized relative to the general mass matrix. The second presented variant includes the complex left and right eigenvectors, orthonormal relative to the general stiffness matrix. After initial partitioning of the equations of motion a real modal transformation matrix is built by a combination of two complex transformations, developed analytically be the aid of computer algebra software. For the general case of damped structures with non-diagonalisable symmetric damping matrix a modal analysis can be performed in real arithmetic. Modal damping as a special case is also considered. Two numerical examples with 3 and 10 DOF’s d...
Damping models for structural vibration
Cambridge University, 2000
This dissertation reports a systematic study on analysis and identification of multiple parameter damped mechanical systems. The attention is focused on viscously and non-viscously damped multiple degree-offreedom linear vibrating systems. The non-viscous damping model is such that the damping forces depend on the past history of motion via convolution integrals over some kernel functions. The familiar viscous damping model is a special case of this general linear damping model when the kernel functions have no memory.
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.
Nonclassically Damped Dynamic Systems: An Iterative Approach
This paper presents a new, computationally efficient, iterative technique for determining the dynamic response of nonclassically damped, linear systems. Such systems often arise in structural and mechanical engineering applications. The technique proposed in this paper is heuristically motivated and iteratively obtains the solution of a coupled set of second-order differential equations in terms of the solution to an uncoupled set. Rigorous results regarding sufficient conditions for the convergence of the iterative technique have been provided. These conditions encompass a broad variety of situations which are commonly met in structural dynamics, thereby making the proposed iterative scheme widely applicable. The method also provides new physical insights concerning the decoupling procedure and shows why previous approximate approaches for uncoupling nonclassically damped systems have led to large inaccuracies. Numerical examples are presented to indicate that, even under perhaps the least ideal conditions, the technique converges rapidly to provide the exact time histories of response.