Disordered Dynamic Systems Resulting From Approximate Modeling and Analysis

2000 ◽  
Author(s):  
R. B. Bhat ◽  
I. Stiharu
Keyword(s):  
Dynamic Systems ◽  
Disordered Systems ◽  
Ritz Method ◽  
Actual System ◽  
Approximate Methods ◽  
Modeling And Analysis ◽  
Mode Localization ◽  
Exact Analysis ◽  
Curve Veering ◽  
Curve Crossing

Abstract Some dynamic systems exhibit curve veering behavior or avoided crossings when their natural frequencies are plotted against a system parameter, while some other show a curve crossing behavior or frequency coalescence. The curve veering behavior is also observed in disordered systems where the symmetry of the system is slightly perturbed and a mode localization takes place. In some systems while the exact analysis shows a curve crossing trend, approximate analyses show a curve veering behavior. Earlier studies have shown that there is a common pattern in curve veering systems and disordered systems. In the present study the exact analysis is recognized as representing the actual system while the approximate analysis of the same system renders it a disordered system by perturbing the eigenvalues and eigenfunctions from their true values. Since the responses of disordered systems can sometimes show violent changes for small perturbations in the system parameters, the response of a simply supported plate has been obtained both exactly and approximately using the Rayleigh Ritz method, and compared. The conclusions have far reaching implications from the point of the accuracy of the response quantities obtained by approximate methods such as finite element method, the Rayleigh Ritz and Galerkin methods.

scholarly journals Curve Veering: Inherent Behaviour of Some Vibrating Systems

2000 ◽  
Vol 7 (4) ◽  
pp. 241-249 ◽  
Author(s):  
B.B. Bhat
Keyword(s):  
System Parameter ◽  
Ritz Method ◽  
Approximate Solutions ◽  
System Model ◽  
Exact Form ◽  
Actual System ◽  
Vibrating Structures ◽  
Curve Veering ◽  
The Mathematical Model ◽  
Vibrating Systems

The plots of variation of eigenvalues of vibrating structures with a system parameter often cross each other or abruptly veer away avoiding the crossing. The phenomenon is termed as curve veering and has been observed both in approximate solutions as well as in exact solutions associated with vibration of some vibrating systems. An explanation to such behavior is provided and illustrated by solving a simple example. The curve veering behavior is induced into a membrane by introducing a parameter that can change the mathematical model from an exact form to an approximate one. Approximate deflection functions such as those used in Galerkin's method or the Rayleigh Ritz method invariably create an approximate or a ficticious system model in lieu of the actual system. The ficticious system may exhibit curve veering while the corresponding real system has no such behaviour. When the ficticious nature of the system is minimized by using large number of terms in the approximate techniques or by discretisation of the domain as in finite difference or by assuming spline type deflection functions, the curve veering behavior subsides and in some cases almost vanishes.

Mode Localization and Delocalization in Constrained Strings and Beams

1997 ◽  
Author(s):  
Chris H. Riedel ◽  
Chin An Tan
Keyword(s):  
Bernoulli Beam ◽  
Constrained System ◽  
Reflection And Transmission ◽  
Mode Localization ◽  
High Frequencies ◽  
Transmission Coefficients ◽  
Curve Veering ◽  
Elastic Constraints ◽  
Euler Bernoulli Beam ◽  
Free Vibration Response

Abstract The free vibration response of a string and a Euler-Bernoulli beam supported by intermediate elastic constraints is studied and analyzed by the transfer function method. The constrained system consists of three subsystems coupled by constraints imposed at the subsystem interfaces. For both the string and beam systems, curve veering and mode localization are observed in the lower modes when the distance between the elastic constraints is varied. As the mode number increases, the modes of the system become extended indicating that the coupling springs have little effect on the system at higher modes. A wave analysis is employed to further investigate the behavior of the systems at high frequencies. Reflection and transmission coefficients are formulated to show the effects of the constraints on the coupling of the subsystems. The weakly bi-coupled beam produces an interesting phenomena where a particular mode experiences no localization while neighboring modes are localized. The frequency at which this occurs is termed the delocalization frequency. Only one delocalization frequency exists and it occurs where the reflection coefficient of the propagating wave becomes zero.

Wave Analysis of Mode Localization and Delocalization in Elastically Constrained Strings and Beams

1999 ◽  
Vol 121 (2) ◽  
pp. 169-173 ◽  
Author(s):  
C. A. Tan ◽  
C. H. Riedel
Keyword(s):  
Wave Analysis ◽  
Bernoulli Beam ◽  
Transmission Resonance ◽  
Mode Localization ◽  
Curve Veering ◽  
Frequency Behavior ◽  
Elastic Constraints ◽  
And Behavior ◽  
Euler Bernoulli Beam ◽  
Free Vibration Response

The free vibration response of both a string and a Euler-Bernoulli beam supported by intermediate elastic constraints is studied and analyzed. For both the string and beam systems, curve veering and mode localization are observed in the lower modes when the distance between the elastic constraints is varied. As the mode number increases, the modes of the system become extended indicating that the coupling springs have little effect on the systems at higher modes. A wave analysis is employed to show the effects of the constraints on the coupling of the subsystems and high frequency behavior. The beam may exhibit a delocalization phenomenon where a particular mode experiences no localization while other neighboring modes may be localized. The frequency (termed the delocalization frequency) at which this occurs corresponds to a transmission resonance. The delocalization frequency is predicted well by the vibration ratio (Langley, 1995). The existence and behavior of the delocalization are explained analytically by the wave approach.

Approximate methods of identifying dynamic systems

1992 ◽  
Vol 35 (7) ◽  
pp. 767-770
Author(s):  
I. V. Boikov ◽  
T. V. Cherusheva
Keyword(s):  
Dynamic Systems ◽  
Approximate Methods

An Augmented State Formulation for Modeling and Analysis of Multibody Distributed Dynamic Systems

2014 ◽  
Vol 81 (5) ◽  
Author(s):  
K. Noh ◽  
B. Yang
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
Dynamic Systems ◽  
Analytical Method ◽  
Dynamic Problems ◽  
Modeling And Analysis ◽  
Closed Form Solutions ◽  
Time Domains ◽  
Augmented State ◽  
Distributed Transfer Function

Multibody distributed dynamic systems are seen in many engineering applications. Developed in this investigation is a new analytical method for a class of branched multibody distributed systems, which is called the augmented distributed transfer function (DTFM). This method adopts an augmented state formulation to describe the interactions among multiple distributed and lumped bodies, which resolves the problems with conventional transfer function methods in modeling and analysis of multibody distributed systems. As can be seen, the augmented DTFM, without the need for orthogonal system eigenfunctions, produces exact and closed-form solutions of various dynamic problems, in both frequency and time domains.

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