Dynamic Analysis of Nonproportional Damping Structural Systems Time and Frequency-Domain Methods (original) (raw)
Related papers
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.
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.
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.
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...
SOME OBSERVATIONS ON THE CHARACTERIZATION OF STRUCTURAL DAMPING
Journal of Sound and Vibration, 2002
This paper deals with the characterization of damping in dynamical structural systems. In particular, the problem of how the modal damping ratios change with di!erent boundary conditions is addressed. It is shown that only Rayleigh-type damping is actually independent of boundary conditions and modal damping ratios can be easily converted from one boundary condition to another. This condition applies independently to continuous, discrete and discretized systems.
This paper considers the dynamics of complex structures with dry friction dampers attached, such as turbomachinery bladed disks. Two extensions of the Hybrid Frequency/Time domain (HFT) method are introduced to calculate efficiently the steady-state forced response of (1) realistic, tuned assemblies with cyclic properties; and (2), structures represented by reduced-order models which feature -for sake of accuracy and convenience -a high ratio of linear degrees of freedom to nonlinear frictional degrees of freedom. It is shown in particular that for cyclic systems, considering the disk as a rigid support is no longer required to carry out the nonlinear analysis, and that the exact dynamics of a flexible bladed-disk structure can be entirely deduced from the dynamics of one of its sectors. Three numerical examples are also presented. They show that the proposed modifications of the HFT method allow one to study with great efficiency the dynamics of complex, tuned or mistuned flexible assemblies, without sacrificing (1) the fidelity of the modeling of the structure, (2) the realism of the modeling of the frictional interfaces, which can possibly involve variable normal load and lost of contact; and (3), the accuracy of the nonlinear analysis, that is, the number of Fourier coefficients retained to approximate the periodic, steady-state response of the structure. domain method, and confirmed that large scale, frictiondamped systems can be efficiently studied with this approach.
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.
The dynamic characterization is important in making accurate predictions of the seismic response of the hybrid structures dominated by different damping mechanisms. Different damping characteristics arise from the construction of hybrid structure with different materials: steel for the upper part; reinforced concrete for the lower main part and interaction with supporting soil. The process of modeling damping matrices and experimental verification is challenging because damping cannot be determined via static tests as can mass and stiffness. The assumption of classical damping is not appropriate if the system to be analyzed consists of two or more parts with significantly different levels of damping. The dynamic response of structures is critically determined by the damping mechanisms, and its value is very important for the design and analysis of vibrating structures. A numerical algorithm capable of evaluating the equivalent modal damping ratio from structural components is desirable for improving seismic design. Two approaches are considered to explore the dynamic response of hybrid tower of cable-stayed bridges: The first approach makes use of a simplified model of 2 coupled lumped masses to investigate the effects of subsystems different damping, mass ratio, frequency ratio on dynamic characteristics and equivalent modal damping; the second approach employs a detailed numerical step-by step integration procedure.