Uncertainty Analysis on Dynamics of Mechanical Systems

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. Using this method, it is shown that calculating the bounds on the dynamic response does not require a combinatorial solution procedure. Exemplar problem that illustrates the behavior of the method and comparison with combinatorial and Monte-Carlo simulation results is presented.