Modal analysis of multi-degree-of-freedom systems with repeated frequencies – analytical approach

The paper deals with the modal analysis of mechanical systems consisting of $ n$ identical masses connected with springs in such a way that the stiffness matrix has the form of a multiband symmetric matrix. The eigenvalue problem formulated for such systems is characterised by repeated eigenvalues to which linearly independent eigenvectors correspond. The solution to the eigenvalue problem has been found for an arbitrary, finite number of degrees of freedom for the fully coupled systems and the systems in which the masses are exclusively connected with the nearest neighbours. Depending on the property of the eigenvectors, two forms of the solution to the initial value problem have been derived. The general deliberations are illustrated with an example of the system with 10 degrees of freedom for 5 different degrees of coupling.