
Type of Document Dissertation Author Singh, Kumar Vikram Author's Email Address ksingh2@lsu.edu URN etd0401103155847 Title The Transcendental Eigenvalue Problem and Its Application in System Identification Degree Doctor of Philosophy (Ph.D.) Department Mechanical Engineering Advisory Committee
Advisor Name Title Yitshak M. Ram Committee Chair George Z. Voyiadjis Committee Member Michael C. Murphy Committee Member Michael M. Khonsari Committee Member SuSeng Pang Committee Member Joseph A. Giaime Dean's Representative Keywords
 system identification
 parameter estimation
 continuous system
 beams
 free vibration of nonuniform rods
 eigenvlaue problem
 algorithm
 modal analysis
 mathematical modeling
 inverse problem
 vibration
Date of Defense 20030303 Availability unrestricted Abstract An accurate mathematical model is needed to solve direct and inverse problems related toengineering analysis and design. Inverse problems of identifying the physical parameters
of a nonuniform continuous system based on the spectral data are still unsolved.
Traditional methods, for the system identification purpose, describe the continuous
structure by a certain discrete model. In dynamic analysis, finite element or finite
difference approximation methods are frequently used and they lead to an algebraic
eigenvalue problem. The characteristic equation associated with the algebraic eigenvalue
problem is a polynomial. Whereas, the spectral characteristic of a continuous system is
represented by certain transcendental function, thus it cannot be approximated by the
polynomials efficiently. Hence, finite dimensional discrete models are not capable of
identifying the physical parameters accurately regardless of the model order used.
In this research, a new low order analytical model is developed, which approximates the
dynamic behavior of the continuous system accurately and solves the associated inverse
problem. The main idea here is to replace the continuous system with variable physical
parameters by another continuous system with piecewise uniform physical properties.
Such approximations lead to transcendental eigenvalue problems with transcendental
matrix elements. Numerical methods are developed to solve such eigenvalue problems.
The spectrum of nonuniform rods and beams are approximated with fair accuracy by
solving associated transcendental eigenvalue problems. This mathematical model is
extended to reconstruct the physical parameters of the nonuniform rods and beams.
There is no unique solution for the inverse problem associated with the continuous
system. However, based on several observations a conjecture is established by which the
solution, that satisfies the given data by its lowest spectrum, is considered the unique
solution. Physical parameters of nonuniform rods and beams were identified using the
appropriate spectral data. Modal analysis experiments are conducted to obtain the
spectrum of the realistic structure. The parameter estimation technique is validated by
using the experimental data of a piecewise beam. Besides the applications in system
identification of rods and beams, this mathematical model can be used in other areas of
engineering such as vibration control and damage detection.
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