Analytical/numerical matching applied to a finite beam with multiple nonperiodically spaced constraints

Abstract

Analytical/numerical matching (ANM) is used to solve for structural vibrations of a finite beam constrained by multiple, nonperiodic supports. The ANM solution decomposes the stated problem into global, local, and matching problems. The global problem addresses large scale effects, with structural discontinuities replaced by smooth distributed forces. The local problem models the rapidly changing region around a structural discontinuity. These constituent problems are solved independently by the most efficient method available. Here, the local problem is solved numerically using finite element analysis (FEA) and the matching problem is solved analytically. The global problem is modeled using FEA with third-order Bernoulli–Euler beam elements. However, the solution procedure of the FEA model must be modified, in order to be compatible with ANM. Two-dimensional plane elements are required only in the local solution, to resolve the discontinuity. This finite-element order reduction provides significant savings in computation time. Comparison with traditional solution methods shows that the ANM approach accurately solves for local effects at each support, in addition to the overall response of the vibrating structure. The framework of applying ANM to more complex problems is also presented. [Work sponsored by ONR.]