INTERVAL-ANALYSIS OF VIBRATING SYSTEMS

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 p arameters, initial conditions or forcing functions defined as interval s rather than single-valued quantities have been modelled using interv al calculus. The eigenvalue problem for determining natural frequencie s and vibrating modes of a system in the case of system matrix element s given as a range of values cannot be solved by exhaustion, due to th e prohibitively large number of solutions of the point-number eigenval ue problem. Eigenvalue solution with interval evaluation of the common ly used numerical techniques is not feasible, because if they are appl ied directly the solution intervals diverge. An optimization technique was used to obtain the minimum-radius intervals of the solution for t he eigenvalue sensitivity problem. To assure monotonicity and absolute inclusion, necessary for convergence to the exact interval, a converg ing interval halving sequence was developed for finite width interval matrices. For numerical tests of the method, a Monte Carlo solution wa s 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 syste m parameters cannot be found with interval evaluation of the commonly used numerical techniques, because if they are applied directly the so lution intervals diverge. Interval modal analysis and the interval sol ution of the eigenvalue problem were developed. An application is pres ented for the dynamic response of a rotor with interval bearing proper ties.