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A new method for dynamic modal analysis of a system with interval uncertainty is developed that is capable of obtaining the bounds on dynamic response of the system. The method uses an interval formulation to quantify the uncertainty present in the system's parameters such as material properties. The existence of uncertainty is considered as the presence of perturbation in a pseudo-deterministic system at each stage of analysis. Having this consideration, first, an interval eigenvalue problem is performed using the concept of monotonic behavior of eigenvalues which leads to a computationally efficient procedure to determine the bounds on the system's natural frequencies. Then, using the procedures for perturbation of invariant subspaces of matrices, the bounds on directional deviation of each mode shape are obtained. Following this, an interval modal analysis is performed to obtain the bounds on the system's total response.

In order to ensure the safety of a structure, adequate strength for structural elements must be provided. In addition, the catastrophic deformations such as buckling must be prevented. In most buckling analyses, structural properties and applied loads are considered certain. Using the linear finite element method, the deterministic buckling analysis is done in two main steps. First, a static analysis is performed using an arbitrary ordinate of applied load. Using the obtained element axial forces, the geometric stiffness of the structure is assembled. Second, performing an eigenvalue problem between the structure's elastic and geometric stiffness matrices yields the structure's critical buckling loads. However, these deterministic approaches disregard uncertainty in the structure's material and geometric properties. In this work, an interval formulation is used to represent the uncertainty in the structure's parameters such as material characteristics.