Steady State Dynamics Of Systems With Fractional Order Derivative Damping Models

Abstract

Rubber like materials find wide applications in damping treatment of structures, vibration isolations and they appear prominently in the form of hoses in many structures such as aircraft engines. The study reported in this thesis addresses a few issues in computational modeling of vibration of structures with some of its components made up of rubber like materials. Specifically, the study explores the use of fractional derivatives in representing the constitutive laws of such material and focuses its attention on problems of parameter identification in linear time invariant systems with fractional order damping models. The thesis is divided into four chapters and two annexures. A review of literature related to mathematical modeling of damping with emphasis on fractional order derivative models is presented in chapter 1. The review covers lternatives available for modeling energy dissipation that include viscous, structural and hybrid damping models. The advantages of using fractional order derivative models in this context is pointed out and papers dealing with solution of differential equations with fractional order derivatives are reviewed. Issues related to finite element modeling and random vibration analysis of systems with fractional order damping models are also covered. The review recognizes the problems of system parameter identification based on inverse eigensensitivity and inverse FRF sensitivity as problems requiring further research. The problem of determination of derivatives of eigensolutions and FRF-s with respect to system parameters of linear time invariant systems with fractional order damping models is considered in chapter 2. The eigensolutions here are obtained as solutions of a generalized asymmetric eigenvalue problem. The order of system matrices here depends upon the mechanical degrees of freedom and also somewhat artificially on the fractional order of the derivative terms. The formulary for first and second order eigenderivatives are developed taking account of these features. This derivation also takes into account the various orthogonality relations satisfied by the complex valued eigenvectors. The system FRF-s are obtained by a straight forward inversion of the system dynamic stiffness matrix and also by using a series solution in terms of system eigensolutions. As might be expected, the two solutions lead to identical results. The first and the second order derivatives of FRF-s are obtained based on system dynamic matrix and without taking recourse to modal summation. Numerical examples that bring out various facets of eigensolutions, FRF-s and their sensitivities are presented with reference to single and multi degree freedom systems. The application sensitivity analysis developed in chapter 2 to problems of system parameter identification is considered in chapter 3. Methods based on inverse eigensensitivity and inverse FRF sensitivity are outlined. The scope of these methods cover first and second order analyses and applications to single and multi degree freedom systems. While most illustrations are based on synthetic measurement data, limited efforts are also made to implement the identification methods using laboratory measurement data. The experimental work has involved the measurement of FRF-s on a system consisting of two steel tubes connected by a rubber hose. The two system identification methods are shown to perform well especially when information on second order sensitivity are included in the analysis. The method based on inverse eigensolution is shown to become increasingly unwieldy to apply as the order of the system matrices increases while the FRF based method does not suffer from this drawback. The FRF based method also has the advantage that the prior knowledge of order of fractional order derivative terms is not needed in its implementation while such knowledge is assumed in the method based on eigensolutions. While the methods are shown to perform satisfactorily when synthetic measurement data is used, their success is not uniformly good when laboratory measurement data are employed. Chapter 4 presents a summary of contributions made in the thesis and also enlists a few suggestions for further research. Annexure I provides a précis of elementary notion of fractional order derivatives and integrals. A case study on finite element analysis of aircraft engine component made up of metallic and rubber materials is outlined in Annexure II and the study points towards possible advantages of using fractional order damping models in the study of such structures.