Method Of The Rotating Structure, The Elastic Vibration Of The Hamiltonian System

The research background is based on coupling vibrations in engineering structures. This paper is focused on coupling vibrations of rotating elastic structures. A fundamental theory system and a symplectic method are established by introducing dual system that is based on dual variables. Firstly, kinetic energy and potential energy expressions are obtained by a series of derivation, considering the extrinsic momentum. In the symplectic system, control equations of longitude and transverse coupling vibrations of rotating elastic structures, or Hamilton canonical equations are described. So the solution of the problem is boiled down to the eigenvalues and eigensolutions in the symplectic space. Secondly, considering concrete problems, it obtains actual Hamilton canonical equations of rotating beam and rotating board. Through solving this equation regular formula of the eigensolution is gotten in the simplectic space. So completed eigensolution space is constructed. Lastly, depending on boundary conditions, the relations and characters, among eigenvalues and angular velocities of the rotating structures, are obtained. Some vibrations rules of rotating structures are also revealed. Especially, a physical phenomenon is found that transverse vibration mode shapes and longitudinal vibration mode shapes can transform mutually. Compared with other theses' results, the numerical results of this paper show that symplectic method is simple, directly and efficient. This method also gives a new idea and approach to resolve similar problems.