Type of Document Master's Thesis Author Subramanian, Jayaram Author's Email Address email@example.com URN etd-11152007-173114 Title TRANSCENDENTAL EIGENVALUE PROBLEMS ASSOCIATED WITH VIBRATION, BUCKLING AND CONTROL Degree Master of Science in Mechanical Engineering (M.S.M.E.) Department Mechanical Engineering Advisory Committee
Advisor Name Title Yitshak M. Ram Committee Chair Michael M. Khonsari Committee Co-Chair Muhammad Wahab Committee Member Su-Seng Pang Committee Member Keywords
- continuous structures
- transcendental eigenvalue problems
- vibration analysis
- discrete model methods
- buckling of plates
Date of Defense 2007-11-13 Availability unrestricted AbstractThe static and dynamic analysis of structures requires us to obtain solutions of their espective governing differential equations subject to appropriate boundary conditions. The dynamic analysis of non-uniform continuous structures is of primary interest, as most traditional methods take the help of discrete models to analyze them. Well established discrete model methods lead to an algebraic eigenvalue problem, the characteristic equation associated with which is a polynomial. The spectral characteristics of a continuous system nevertheless are represented by transcendental functions and cannot be approximated by polynomials efficiently. Hence finite dimensional discrete models are not capable of predicting the response of continuous systems irrespective 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. The idea here is to replace a non-uniform continuous system by a set of continuous system with piecewise constant physical properties. Such approximations lead to a transcendental eigenvalue problem, i.e. a problem with transcendental characteristic equation. A numerical method has been developed to solve such eigenvalue problems. The spectrum of non-uniform rods and beams are approximated with fair accuracy by solving the corresponding transcendental eigenvalue problem. This mathematical model is extended to reconstruct non-uniform rods and beams using a linear polynomial approximation of piecewise area. A piecewise tapered approximation of the physical parameters in non-uniform rods and beams leads to better accuracy in the solution. The ability to use higher order area functions as basic building blocks profoundly reduces the model order when using the mathematical model to analyze complex geometries.
To further study the impact of this method in various problems of engineering the buckling of thin rectangular plates with stepped thickness has been analyzed and compared with the finite element solution. The transcendental eigenvalue method leads to the reduction in matrix sizes when compared with discrete model methods, thus making the solution computationally viable. Finally the transcendental eigenvalue problem associated with the active control of vibration in discrete mass-spring-damper systems has been developed and the proposed mathematical method has been applied.
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