scholarly journals Asymptotic Distribution of Eigenvalues of a Constrained Translating String

1997 ◽  
Vol 64 (3) ◽  
pp. 613-619 ◽  
Author(s):  
W. D. Zhu ◽  
C. D. Mote ◽  
B. Z. Guo
Keyword(s):  
Spectral Analysis ◽  
Asymptotic Distribution ◽  
Characteristic Equation ◽  
Infinite Number ◽  
Imaginary Axis ◽  
Irrational Number ◽  
Coupled System ◽  
Numerical Analyses ◽  
Vibration Energy ◽  
Distribution Of Eigenvalues

A new spectral analysis for the asymptotic locations of eigenvalues of a constrained translating string is presented. The constraint modeled by a spring-mass-dashpot is located at any position along the string. Asymptotic solutions for the eigenvalues are determined from the characteristic equation of the coupled system of constraint and string for all constraint parameters. Damping in the constraint dissipates vibration energy in all modes whenever its dimensionless location along the string is an irrational number. It is shown that although all eigenvalues have strictly negative real parts, an infinite number of them approach the imaginary axis. The analytical predictions for the distribution of eigenvalues are validated by numerical analyses.

On the asymptotic distribution of eigenvalues of banded matrices

1988 ◽  
Vol 4 (1) ◽  
pp. 403-417 ◽  
Author(s):  
Jeffrey S. Geronimo ◽  
Evans M. Harrell ◽  
Walter Van Assche
Keyword(s):  
Asymptotic Distribution ◽  
Banded Matrices ◽  
Distribution Of Eigenvalues

Successive Synthesis of Substructure Modes

1991 ◽  
Vol 58 (3) ◽  
pp. 759-765 ◽  
Author(s):  
Luis E. Suarez ◽  
Mahendra P. Singh
Keyword(s):  
Closed Form ◽  
Characteristic Equation ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Closed Form Expression ◽  
Element Discretization ◽  
Root Finding ◽  
Complete Set ◽  
Conventional Solution ◽  
Second Eigenvalue

A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.

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