The Generalized Variational Principles For The Natural Frequencies Of Elastic Cylindrical Shells

The elastic cylindrical shells are widely used in many kinds of structures of engineering and electric equipments. The variational principles of the natural frequencies of elastic cylindrical shells are the theoretical basis and effective means for solving the vibration problems of shells. Due to the mathematical complexity, it is very difficult to find the natural frequencies of vibrating elastic cylindrical shells in accord with all their equations and boundary conditions. The generalized variational principles of the natural frequencies will overcome these difficulties and provide a more general approximate method for calculating the natural frequencies of elastic cylindrical shells. Therefore, they have significant values in both theory and practice.Firstly, the generalized variational principles of the natural frequencies for the exact Flugge's theory is established in this paper. Secondly, the generalized variational principles of the natural frequencies for the approximate Donnell's theory is proposed. Thirdly, for the convenience of calculation, the frequency variational principle of the Donnell's theory is transformed into that only based on the two variables of stress function and deflection. In order to demonstrate one kind of applications of these principles, the approximate natural frequencies of elastic cylindrical shells are obtained by using the generalized variational principles of the natural frequencies for the approximate Donnell's theory. The results are compared with the ones by using the exact Flugge's theory and FEM. Meanwhile, the advantages and disadvantages of the approximate method are analyzed further.