Title page for ETD etd-0401103-155847

Type of Document Dissertation
Author Singh, Kumar Vikram
Author's Email Address ksingh2@lsu.edu
URN etd-0401103-155847
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
Su-Seng Pang Committee Member
Joseph A. Giaime Dean's Representative
  • system identification
  • parameter estimation
  • continuous system
  • beams
  • free vibration of non-uniform rods
  • eigenvlaue problem
  • algorithm
  • modal analysis
  • mathematical modeling
  • inverse problem
  • vibration
Date of Defense 2003-03-03
Availability unrestricted
An accurate mathematical model is needed to solve direct and inverse problems related to

engineering analysis and design. Inverse problems of identifying the physical parameters

of a non-uniform 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 non-uniform 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 non-uniform 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 non-uniform 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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