Eigenvalue bounds of structures with uncertain-but-bounded parameters

Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving uncertain material or geometric parameters, specified as bounds on these parameters. This produces interval stiffness and mass matrices, and the problem is transformed into a generalized interval eigenvalue problem in interval mathematics. However tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem. Often the stiffness and mass matrices can be formed as a non-negative decomposition in the uncertain structural parameters. In this case the eigenvalue bounds may be obtained from the parameter vertex solutions. Even more efficiently, using interval extension from interval mathematics, the generalized interval eigenvalue problem may be divided into two generalized eigenvalue problems for real symmetric matrix pairs. The parameter vertex solution algorithm is compared with Deif's solution, the eigenvalue inclusion principle and the interval perturbation method in numerical examples.