On a comparative study of an accurate spatial discretization method for one-dimensional continuous systems

Abstract

This work provides an in-depth investigation on advantages of a recently developed, new global spatial discretization method over the assumed modes method, and a clear description of the procedure and validity of the new method and its feasibility for arbitrary boundary conditions. A general formulation of the new spatial discretization method is given for second- and fourth-order continuous systems, whose displacements are divided into internal terms and boundary-induced terms, and two examples that consider the longitudinal vibration of a rod and the transverse vibration of a tensioned Euler-Bernoulli beam are used to demonstrate the new spatial discretization method. In the two examples, natural frequencies, mode shapes, harmonic steady-state responses, and transient responses of the systems are calculated using the new spatial discretization method and the assumed modes method, and results are compared with those from exact analyses. Convergence of the new spatial discretization method is investigated using different sets of trial functions for internal and boundary-induced terms. While the new spatial discretization method has additional degrees of freedom at boundaries of a continuous system compared with other global spatial discretization methods, it has the following advantages: (1) compared with the assumed modes method, the new method gives better results in calculating eigensolutions and dynamic responses of the system, and allows more terms to be retained in a trial function expansion due to the slowly growing condition number of the mass matrix of the system; and (2) compared with the exact eigenfunction expansion method, the new method can use sinusoidal functions as trial functions for the internal term rather than complicated eigenfunctions of the system in the expansion solution.