Numerical presentation of eigenvalue curve veering and mode localization in coupled pendulums

AIAA Journal ◽  
2001 ◽  
Vol 39 ◽  
pp. 2001-1816
Author(s):  
X. L. Liu
Keyword(s):  
Mode Localization ◽  
Curve Veering

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.

Transfer Function Formulation of Constrained Distributed Systems

1991 ◽  
Author(s):  
C. A. Tan ◽  
C. H. Chung
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
Constrained Systems ◽  
Natural Boundary ◽  
Mode Localization ◽  
Function Formulation ◽  
General Displacement ◽  
Curve Veering ◽  
Natural Boundary Conditions ◽  
Pointwise Constraints

Abstract The transfer function formulation of constrained distributed systems is presented. The methodology is illustrated for subsystems under pointwise constraints and distributed systems consisting of multiple subsystems. A general displacement method (GDM) is used to determine the eigensolution of the constrained systems. It is shown that GDM requires only the natural boundary conditions (force constraints) be imposed at the subsystem interface. The methodology is applied to two examples. Curve veering and mode localization phenomena are found in an elastic structure on an elastic foundation.

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.

Mathematical Insights of Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

2013 ◽  
Author(s):  
Y. F. Chen ◽  
I. Y. Shen
Keyword(s):  
Natural Frequencies ◽  
Necessary Condition ◽  
Localized Modes ◽  
Mode Localization ◽  
Visual Method ◽  
Mean Frequency ◽  
Cyclic Symmetric ◽  
Complete Set ◽  
Curve Veering ◽  
Cyclic Symmetric Rotors

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors with mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be easily reoriented when the mistune is present to form localized modes. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh-Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we can develop an effective visual method — through use of the deviatoric component and the rotor mistune — to precisely identify those modes needed to form localized modes. Finally, we show that curve veering with respect to engine orders is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. A numerical example confirms the findings above.

scholarly journals Curve veering and mode localization in a buckling problem

1989 ◽  
Vol 40 (5) ◽  
pp. 758-761 ◽  
Author(s):  
Christophe Pierre ◽  
Raymond H. Plaut
Keyword(s):  
Mode Localization ◽  
Curve Veering

Mathematical Insights Into Linear Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Y. F. Chen ◽  
I. Y. Shen
Keyword(s):  
Natural Frequencies ◽  
Necessary Condition ◽  
Localized Modes ◽  
Mode Localization ◽  
Linear Mode ◽  
Blade System ◽  
Cyclic Symmetric ◽  
Complete Set ◽  
Curve Veering ◽  
Cyclic Symmetric Rotors

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors that contain slight mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be linearly combined to form localized modes when the mistune is present. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh–Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we develop an effective visual method—through use of the deviatoric component and the rotor mistune—to precisely identify those modes needed to form localized modes. Finally, we show that curve veering is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. Numerical examples on a disk–blade system with mistune confirm all the findings above.

Sign in / Sign up

Export Citation Format

Share Document