Natural frequencies of structures with interval parameters

• Published in: Journal of sound and vibration Vol. 347; pp. 79 - 95
• Main Authors: Sofi, A., Muscolino, G., Elishakoff, I.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 07.07.2015
• Subjects: Eigenvalue bounds; Eigenvalues; Extra unitary interval; Generalized interval eigenvalue problem; Improved interval analysis; Interval natural frequencies; Interval uncertainties; Intervals; Mathematical models; Resonant frequency; Sensitivity analysis; Stiffness matrices; Uncertainty; Vibration; Improved interval analysis; Sensitivity analysis; Extra unitary interval; Eigenvalue bounds; Interval uncertainties; Generalized interval eigenvalue problem; Interval natural frequencies
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2015.02.037

This paper deals with the evaluation of the lower and upper bounds of the natural frequencies of structures with uncertain-but-bounded parameters. The solution of the generalized interval eigenvalue problem is pursued by taking into account the actual variability and dependencies of uncertain structural parameters affecting the mass and stiffness matrices. To this aim, interval uncertainties are handled by applying the improved interval analysis via extra unitary interval (EUI), recently introduced by the first two authors. By associating an EUI to each uncertain-but-bounded parameter, the cases of mass and stiffness matrices affected by fully disjoint, completely or partially coincident uncertainties are considered. Then, based on sensitivity analysis, it is shown that the bounds of the interval eigenvalues can be evaluated as solution of two appropriate deterministic eigenvalue problems without requiring any combinatorial procedure. If the eigenvalues are monotonic functions of the uncertain parameters, then the exact bounds are obtained. The accuracy of the proposed method is demonstrated by numerical results concerning truss and beam structures with material and/or geometrical uncertainties. •Free vibrations of structures with interval stiffness and mass parameters are analyzed.•A method for the solution of the generalized interval eigenvalue problem is proposed.•Uncertainties are handled by the improved interval analysis via extra unitary interval.•The eigenvalue bounds are evaluated solving two deterministic eigenvalue problems.•The bounds are exact if the eigenvalues are monotonic functions of uncertainties.