Modal Analysis Procedure Using Complex Left and Right Eigenvectors of Non-Proportionally Damped Structures (original) (raw)
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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...
Journal of Sound and Vibration, 2006
This paper presents an efficient numerical method that approximates the complex eigenvalues and eigenvectors in structural systems with viscoelastic damping materials, characterised by the complex structural damping matrix. This new method begins from the solution of the undamped system and approximates the complex eigenpair by finite increments using the eigenvector derivatives and the Rayleigh quotient. It is implemented in three different approaches: single-step, incremental and iterative schemes. The single-step technique is presented for systems with low and medium damping. From numerical examples, it can be verified that the errors committed by the approximate single-step technique with respect to the exact solutions, obtained by the IRAM method, are less than 0.2% when the loss factor of the material damping is lower than 1; this is a considerable improvement on other approximate methods. For higher damped systems an improvement to the previous approach is proposed by an incremental technique that keeps the accuracy without significantly increasing the computational time. The complex eigenproblem for materials whose mechanical properties are dependent on frequency is solved by a fast iterative approach, whose validity is proved using a four-parameter fractional derivative model. r (M.J. Elejabarrieta).
Memoria Investigaciones en Ingeniería N23, 2022
The use of finite element method software allows the modal and spectral analysis of complex structures with an accessible computational cost. In contrast, with the theory of damped systems of a degree of freedom, solutions can be obtained analytically that describe with greater generality the vibrations of structures subjected to variable forces over time. However, with analytical methods, only very simple structures can be studied. This paper presents a method that allows to calculate the rigidity and effective mass of a structure from the values of the angular frequencies of the structure, with their corresponding inertial loads. Next, the structure can be analyzed as a cushioned system of a degree of freedom. In this way, it is possible to calculate the displacements and accelerations that the structure will suffer when it is excited by an external force variable over time. Subsequently, using D'Alembert's principle and a finite element program, the stresses can be calculated by a static analysis.
Modal Decomposition Procedures for Fe-Based Stuctural Models with Non-Proportional Damping
2018
This paper presents two variants of a general method (Cramer, Stanoev 2008), (Stanoev 2013, 2014, 2016, 2017) for modal transformation of the equations of motion for multi-degree-offreedom-systems (MDOFS) with non-modal symmetric damping matrix. The first variant is described in comparison with a similar method, presented in earlier publications (Chu M. T., Buono N. T. 2008), (Ma Fai F., Morzfeld M., Imam A., 2010). The equations of motion are stated in state-space formulation. The final modal decomposition is performed by a purely real-space transformation matrix, which is derived by a combination of two complex transformations using the complex left and right eigenvectors of the associated special eigenvalue problem. The eigenvector normalization is performed in two different ways. Analytical expressions for all presented variants of the modal transformation basis are developed by the aid of computer algebra software. The proposed modal procedures retain all common advantages of t...
An Engineering Interpretation of the Complex Eigensolution of Linear Dynamic Systems
2005
In traditional finite element based modal analysis of linear non-conservative structures, the modal shapes are determined solely based on stiffness and mass. Damping effects are included by implicitly assuming that the damping matrix can be diagonalized by the undamped modes. The approach gives real valued mode shapes and modal coordinates. While this framework is suitable for analysis of most lightly damped structural systems, it is insufficient for interpretation of the free vibration and resonant response of structures with e.g. significant nonclassical damping, gyroscopic or other effects resulting in a complex eigensolution. In this paper, the more general approach based on complex eigenvalues and eigenvectors is employed. We give an interpretation of the complex eigensolution that describes free and resonant vibrations of a generally damped linear structure. The interpretation show how the different parts of the complex eigensolutions; i.e. the complex left and right eigenvect...
Computer Methods in Applied Mechanics and Engineering, 2006
In this paper efficient numerical methods to approximate the complex eigenvalues and eigenvectors in non-proportional and non-viscous systems are presented. These methods are specially conceived for practical engineering applications making use of the finite element analysis to determinate the effect that potential damping treatments have on the natural frequencies and mode shapes of structural systems. Considering the solution of the undamped problem, the complex eigenpair is estimated by finite increments using the eigenvector derivatives. For non-proportional viscous systems with low and medium damping, a simple single-step technique is presented whose rapidity and accuracy is verified by means of numerical applications. For higher damped systems an incremental approach is proposed that keeps the accuracy without significantly increasing the computational time. For non-viscously damped systems a fast iterative modality is suggested, which allows to approximate, in an efficient and simple way, the complex eigenpair. As numerical applications, the study of a metallic beam with free layer damping treatment is completed using finite element procedures, where the damping material is modelized by an exponential model whose parameters are obtained from curve fitting to experimental data.
A Modal Perturbation Method for Eigenvalue Problem of Non-Proportionally Damped System
Applied Sciences, 2020
The non-proportionally damped system is very common in practical engineering structures. The dynamic equations for these systems, in which the damping matrices are coupled, are very time consuming to solve. In this paper, a modal perturbation method is proposed, which only requires the first few lower real mode shapes of a corresponding undamped system to obtain the complex mode shapes of non-proportionally damped system. In this method, an equivalent proportionally damped system is constructed by taking the real mode shapes of a corresponding undamped system and then transforming the characteristic equation of state space into a set of nonlinear algebraic equations by using the vibration modes of an equivalent proportionally damped system. Two numerical examples are used to illustrate the validity and accuracy of the proposed modal perturbation method. The numerical results show that: (1) with the increase of vibration modes of the corresponding undamped system, the eigenvalues and...
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.
MODAL ANALYSIS OF NONLINEAR SYSTEMS WITH NONCLASSICAL DAMPING
This paper presents a mode-superposition procedure for the analysis of nonlinear problems in structural dynamics where damping cannot be assumed proportional. The procedure consists of treating the nonlinearity as a pseudo force and using a complex eigenvalue solution to decouple the equations of motion. The response time history of a twenty-degree-of-freedom system with nonproportional damping to a base excitation is obtained using the proposed procedure and compared with that from a direct integration of the equations of motion. The comparison indicates excellent agreement. Few studies have used the mode-superposition procedure to solve nonlinear problems in structural dynamics (Molnar e t al. ; Riead ' ; Shah e t al. ; Stricklin and Haisler '). Such procedures have been limited to classically damped systems where proportional damping is assumed to uncouple the equa--tions of motion.