Interval analysis of vibrating systems

Abstract

Interval calculus is a tool to evaluate a mathematical expression for ranges of values of its parameters. The basic mathematical operations are defined in the interval algebra. Vibrating systems having system parameters, initial conditions or forcing functions defined as intervals rather than single-valued quantities have been modelled using interval calculus. The eigenvalue problem for determining natural frequencies and vibrating modes of a system in the case of system matrix elements given as a range of values cannot be solved by exhaustion, due to the prohibitively large number of solutions of the point-number eigenvalue problem. Eigenvalue solution with interval evaluation of the commonly used numerical techniques is not feasible, because if they are applied directly the solution intervals diverge. An optimization technique was used to obtain the minimum-radius intervals of the solution for the eigenvalue sensitivity problem. To assure monotonicity and absolute inclusion, necessary for convergence to the exact interval, a converging interval halving sequence was developed for finite width interval matrices. For numerical tests of the method, a Monte Carlo solution was developed. The results showed that interval analysis can predict the range of the eigenvalues with sufficient accuracy. The response of a linear system to general excitation for interval matrices of the system parameters cannot be found with interval evaluation of the commonly used numerical techniques, because if they are applied directly the solution intervals diverge. Interval modal analysis and the interval solution of the eigenvalue problem were developed. An application is presented for the dynamic response of a rotor with interval bearing properties.