A novel approach to determine the frequency equations of combined dynamical systems

A novel approach is presented which can be used to reduce the generalized eigenvalue problem associated with the free vibration of a linear elastic structure carrying lumped elements at s distinct locations. Using N component modes in the assumed-modes method, the free vibration of such a combined dynamical system is governed by the solution of a generalized eigenvalue problem of order N x N, whose stiffness and mass matrices consist of diagonal matrices modified by a total of s rank one matrices, where s corresponds to the number of attachment points. This generalized eigenvalue problem can be manipulated such that the natural frequencies governing free vibration can be calculated instead by solving a much smaller characteristic determinant of order sx s. Interestingly enough, this smaller and simpler characteristic determinant can also be obtained by using the Lagrange multipliers formalism in conjunction with Lagrange's equations. (C) 1999 Academic Press.