In structural dynamics, schemes such as the assumed-modes method [1] or the Lagrange multipliers formalism [2,3] are often used to obtain the approximate modes of vibration of complex dynamical systems consisting of a continuous structure combined with various spring}mass attachments. The assumed-modes method [1] is a procedure for discretizing an arbitrary structure prior to obtaining the governing equations of motion. This method consists of assuming a solution of the free vibration problem in the form of a series composed of a linear combination of N spatial functions multiplied by the time-varying generalized co-ordinates. The spatial functions must satisfy the boundary conditions of the unconstrained system, de"ned here as the arbitrary structure without the constraints. This series is then substituted into the expressions for the kinetic and potential energies, thus reducing them to discrete form, and the equations of motion are derived by means of Lagrange's equations. Assuming R spring}mass systems are attached to the unconstrained system at distinct locations, then the mass and sti!ness matrices of the combined system can be expressed as the sum of diagonal matrices and R rank-one matrices. The modes of vibration of the combined system correspond to the eigensolutions of an N;N generalized eigenvalue problem.
In reference [4], an approach to reduce the aforementioned generalized eigenvalue problem was presented. Speci"cally, for a system with R spring}mass attachments at distinct locations, the N;N generalized eigenvalue problem was manipulated so that its characteristic determinant is equivalent to that of a smaller R;R matrix (where it was assumed RN, since in practice, a large number of component modes, N, is generally used to ensure convergence and su$cient accuracy), each element of which involves a sum of N terms.
Interestingly, this reduced characteristic determinant can also be obtained by using the Lagrange multipliers formalism [2,3]. This method is based on using the spatial functions of the unconstrained structure in a Rayleigh}Ritz analysis with the constraint conditions enforced by means of Lagrange multipliers. Using this particular approach, R Lagrange multipliers and R constraint variables are introduced in the analysis. Manipulating the equations of motion, the eigenvalues must satisfy the zeros of the constraint equations in matrix form. Under certain conditions, the R;R characteristic determinant that needs to be solved is shown to be identical to that obtained by mathematically manipulating the N;N generalized eigenvalue problem as obtained from the assumed-modes method [4].