A Highly Stable And Accurate Computational Method For Eigensolutions In Structural Dynamics

Since the natural frequencies and modes of vibration take an essential role in structural dynamics, it is an important aim to have reliable and accurate methods for solving the eigenvalue problem of structure. It is significant to found the natural frequencies and modes of vibration precisely and efficiently both to avoid resonance or prevent buffeting and to analyze the dynamical response of elastic system. It is one of the important tasks of dynamics to find precise and efficient method to solve the eigenproblem.It is stated mathematically as being to solve generalized eigenvalue problem to get the natural frequencies and modes of vibration. There are many methods available to solve linear eigenvalue problem. For the condensed dynamic stiffness matrix in finite models which have finite number of freedom and any dynamic stifmess matrix of infinite system, the element of them is a transcendental function of eigenfrequency. And therefore transcendental eigenvalue problem is needed to be solved in order to get the natural frequency and modes of vibration. In contrast to the many well-established methods for solving linear eigenproblems, there are only a few published methods available for solving transcendental eigenproblems. Further work is needed in this area for the precision and efficiency of these methods available is not satisfactory. The thesis focuses on the research of the transcendental eigenproblems.Base on the Sturm sequence property, Wittrick and Williams has developed an algorithm to solve transcendental eigenvalue problem. It enables one to count the number of frequencies exceeded by a given trial frequency. On the basis of the method. Professor Qi has got a new criterion for detecting the eigenfrequency by using the physical meanings of the matrix used in the congruent transformation of the dynamic stifmess matrix. And by using the property of derivatives of energy norms, the eigenproblem is transformed safely into a specific initial value problem of an ordinary differential equation.The thesis focuses on the new method proposed by professor Qi, and by studying and fully utilizing the characteristics and advantages of the method, we try to solve difficult problems in transcendental eigenvalue problem for the new method.The characteristics of structure which have equal eigenfrequency or close eigenfrequency are analyzed. The characteristics of the energy norm-frequency curves of structure which have equal eigenvalue are studied and compared with that of structure which have scattered eigenvalue only. Combined with the new criterion for detecting eigenfrequency. a new method is found to solve equal eigenfrequency problem. And furthermore, by studying the physicalmeanings of the matrix corresponding to the modes of vibration and force, and considering the physical of eigenfrequency itself, a new criterion is found to distinguish equal eigenfrequency and close eigenfrequency. And finally, the method is developed to find the multi-modes of vibration corresponding to the equal eigenfrequency.As showed in our research work, the new method has unique advantages, which other approaches do not have for transcendental eigenvalue problem and is a valuable research topic.