On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε . The density is of order O ( ε –m ) in the small region, the concentrated mass , and it is of order O (1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε →O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m : Low‐ and high‐frequency vibrations are studied for m >2. We also consider the case where the thickness plate h depends on ε ; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.