A review of methods for constrained eigenvalue problems

J.G.M. Kerstens
Technical University Eindhoven, The Netherlands

Numerous studies are concerned with vibrations or buckling of constrained systems realising that complex systems can be analysed when starting from unconstrained or known problems. Over the years various constrained eigenvalue formulations have been published and put into practical use. The most important ones are the Lagrangian multiplier method (modal synthesis method or component modes synthesis method), the receptance method and the modal constraint method. In this paper the similarities and merits of the various methods are discussed. It is striking that so far the similarities of the various constrained eigenvalue expressions have not been reported nor has the similarity of the eigenvalue expressions of these methods with Weinstein's determinant for intermediate problems of the first type been noticed. The eigenvalue formulations of the Lagrangian multiplier method and that of the receptance appear to be similar to Weinstein's determinant for intermediate problems of the first type. The modal constraint method is based on an extension of Weinstein’s method for intermediate problems of the first type and offers some significant advantages, i.e. the resulting eigenvalue formulation of the modal constraint method has a standard form in contrast with that of the other mentioned methods in that they have the known and unknown eigenvalues in the denominator of the eigenvalue formulations. Further, zero modal displacements, persistent and multiple eigenvalues do require special attention using these methods whereas this is not the case for the modal constraint method. Based on the similarities of the various constrained eigenvalue expressions a number of interesting conclusions are drawn.

Key words: Vibration, buckling, constraints, component mode synthesis, modal synthesis method, receptance method, modal constraint method