Modified ODE-solver for vibration of tube-in-tube structures Original Research Article

For some complex boundary value problems, such as ordinary differential equations (ODEs) with eigenvalue problems, finding a closed form of solution is usually very difficult, sometimes even impossible. The main purpose of this paper is to present a numerical solution of eigenvalues for free vibration of tube-in-tube tall buildings by using ODE solver modified in the paper. In order that an existing ODE solver may be compatible with solving the eigenvalue problem, a solution strategy for the ODE solver is modified. That is, the strategy organically puts the following techniques together: ‘Standard’ transform for non-linear problem; Inverse iteration; Orthogonalization and eigenvalue shift, etc. In comparison to the results calculated by other analytical methods and the finite element method shows that the present ODE solver can be conveniently taken for accurate and reliable solutions of the eigenvalue problem of the vibration.