Natural Frequencies of a Structure with Bounded Uncertainty

Abstract

Frequency analysis of a structural system with bounded uncertainty is presented. An interval (set-theoretic) formulation is used to quantify the uncertainty present in the structure’s parameters such as material properties. Independent variations for each element are considered. Using the developed interval finite-element method, it is proven that, in the presence of any physically allowable uncertainty in the structural stiffness, the solutions to two deterministic problems are sufficient to obtain the exact bounds on the system’s fundamental frequencies. Therefore, calculating the bounds on frequencies does not require a combinatorial solution procedure. Several example problems that illustrate the developed algorithm with comparison to existing interval eigenvalue solution are presented. The solutions show no overestimation in the bounds of calculated frequencies which has been a difficulty with other interval procedures.