scholarly journals A Modal Perturbation Method for Eigenvalue Problem of Non-Proportionally Damped System

2020 ◽  
Vol 10 (1) ◽  
pp. 341
Author(s):  
Danguang Pan ◽  
Xiangqiu Fu ◽  
Qingjun Chen ◽  
Pan Lu ◽  
Jinpeng Tan
Keyword(s):  
Perturbation Method ◽  
State Space ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Algebraic Equations ◽  
Engineering Structures ◽  
Damped System ◽  
Real Mode ◽  
Nonlinear Algebraic Equations ◽  
Practical Engineering

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 eigenvectors monotonically converge to exact solutions; (2) the accuracy of the proposed method is significantly higher than the first-order perturbation method and proportional damping method. The calculation time of the proposed method is shorter than the state space method; (3) the method is particularly suitable for finding a few individual orders of frequency and mode of a system with highly non-proportional damping.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

Compressed sensing based on dictionary learning for underdetermined modal identification

2020 ◽  
Vol 64 (1-4) ◽  
pp. 129-136
Author(s):  
Wei Guan ◽  
Longlei Dong ◽  
Jinxiong Zhou
Keyword(s):  
Compressed Sensing ◽  
Dictionary Learning ◽  
Engineering Application ◽  
Modal Identification ◽  
Mode Shapes ◽  
Engineering Structures ◽  
New Approach ◽  
Learned Dictionary ◽  
Complete Measurement ◽  
Practical Engineering

With the engineering structures becoming more complicated, it is difficult to obtain complete measurement responses with limited sensors. Thus, carrying out the underdetermined modal identification will have practical engineering application values. In this paper, a new approach for underdetermined blind modal identification based on dictionary learning in the framework of compressed sensing (CS) is proposed. The principal idea is to estimate modal shapes using a clustering technique, and recover modal responses combing the estimated mode shapes matrix and the learned dictionary. The experiment results on a typical cantilever beam structure illustrate that the proposed method can perform accurate dynamic parameters identification whether in underdetermined case or determined case.

Asymptotic Treatment of Non-Classically Damped Linear Systems

1995 ◽  
Vol 48 (11S) ◽  
pp. S111-S117
Author(s):  
W. Fang ◽  
J.-G. Tseng ◽  
J. A. Wickert
Keyword(s):  
Eigenvalue Problem ◽  
Dynamic System ◽  
Perturbation Method ◽  
Quadratic Forms ◽  
Vibration Modes ◽  
Discrete Dynamic System ◽  
Damped System ◽  
Base Solution ◽  
The Stability ◽  
Dissipative Terms

The presence of non-classical dissipation in a general discrete dynamic system is investigated through a perturbation method for the eigenvalues and vectors. Results accurate to second-order are obtained, with corrections to the base solution being expressed in terms of readily-calculated quadratic forms. Exact solutions, and the derived asymptotic ones, are compared with the predictions of the so-called method of approximate decoupling, in which certain non-classical dissipative terms are omitted from calculations in the eigenvalue problem. The perturbation method is discussed through its application in several examples, indicating circumstances in which a non-classically damped system can be well-approximated by an “equivalent” classically damped one. Somewhat surprisingly, the addition of non-classical damping does not necessarily increase the stability of all vibration modes, and the perturbation method is shown to be useful in identifying those critical modes.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

The Use of Modan 3D in Experimental Modal Analysis

2013 ◽  
Vol 486 ◽  
pp. 36-41 ◽  
Author(s):  
Róbert Huňady ◽  
František Trebuňa ◽  
Martin Hagara ◽  
Martin Schrötter
Keyword(s):  
Modal Analysis ◽  
Vibration Analysis ◽  
High Speed ◽  
Mechanical Vibration ◽  
Steel Sample ◽  
Experimental Modal Analysis ◽  
Image Correlation ◽  
Optical Methods ◽  
Mode Shapes ◽  
Vibration Modes

Experimental modal analysis is a relatively young part of dynamics, which deals with the vibration modes identification of machines or their parts. Its development has started since the beginning of the eighties, when the computers hardware equipment has improved and the fast Fourier transform (FFT) could be used for the results determination. Nowadays it provides an uncountable set of vibration analysis possibilities starting with conventional contact transducers of acceleration and ending with modern noncontact optical methods. In this contribution we mention the use of high-speed digital image correlation by experimental determination of mode shapes and modal frequencies. The aim of our work is to create a program application called Modan 3D enabling the performing of experimental modal analysis and operational modal analysis. In this paper the experimental modal analysis of a thin steel sample performed with Q-450 Dantec Dynamics is described. In Modan 3D the experiment data were processed and the vibration modes were determined. The reached results were verified by PULSE modulus specialized for mechanical vibration analysis.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

Identifying VIV Vibration Modes by Use of the Empirical Orthogonal Functions Technique

2002 ◽  
Author(s):  
Gudmund Kleiven
Keyword(s):  
Model Test ◽  
Vibration Mode ◽  
Multivariate Data ◽  
Empirical Orthogonal Functions ◽  
Mode Shapes ◽  
Data Sets ◽  
Vibration Modes ◽  
Ocean Engineering ◽  
Orthogonal Functions ◽  
Related Technique

The Empirical Orthogonal Functions (EOF) technique has widely being used by oceanographers and meteorologists, while the Singular Value Decomposition (SVD being a related technique is frequently used in the statistics community. Another related technique called Principal Component Analysis (PCA) is observed being used for instance in pattern recognition. The predominant applications of these techniques are data compression of multivariate data sets which also facilitates subsequent statistical analysis of such data sets. Within Ocean Engineering the EOF technique is not yet widely in use, although there are several areas where multivariate data sets occur and where the EOF technique could represent a supplementary analysis technique. Examples are oceanographic data, in particular current data. Furthermore data sets of model- or full-scale data of loads and responses of slender bodies, such as pipelines and risers are relevant examples. One attractive property of the EOF technique is that it does not require any a priori information on the physical system by which the data is generated. In the present paper a description of the EOF technique is given. Thereafter an example on use of the EOF technique is presented. The example is analysis of response data from a model test of a pipeline in a long free span exposed to current. The model test program was carried out in order to identify the occurrence of multi-mode vibrations and vibration mode amplitudes. In the present example the EOF technique demonstrates the capability of identifying predominant vibration modes of inline as well as cross-flow vibrations. Vibration mode shapes together with mode amplitudes and frequencies are also estimated. Although the present example is not sufficient for concluding on the applicability of the EOF technique on a general basis, the results of the present example demonstrate some of the potential of the technique.

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