# The transcendental eigenvalue problem and its application in system identification

## Identifier

etd-0401103-155847

## Degree

Doctor of Philosophy (PhD)

## Department

Mechanical Engineering

Dissertation

## Abstract

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.

2003

## Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Yitshak M. Ram