Cha P.

A new approach is presented to discretize an arbitrarily supported linear structure carrying various lumped attachments. Specifically, the exact modes of vibration, i.e., the exact natural frequencies and mode shapes, of the linear structure are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration. This is achieved by identifying a set of minimum changes in the finite element mass and stiffness matrices and enforcing certain constraint conditions. Once the updated matrices for the linear structure are found, the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. Numerical experiments show that using only a few elements, the proposed scheme returns higher natural frequencies that are nearly identical to those obtained by using a finite element model with a very fine mesh. The new method is easy to apply and efficient to use. It remains applicable for any combination of attachments, and is valid for a combined system that is either undamped or damped.