DERIVATION OF THE BOUNDARY-CONDITIONS FOR THE EQUATION OF THE VIBRATIONS OF A THIN-PLATE BY THE METHOD OF GENERALIZED-FUNCTIONS

The boundary conditions on the free boundary of a thin vibrating plate of variable thickness, when the thickness of the plate is changed abr uptly, are derived. The left-hand side of the fourth-order differentia l equation describing the vibrations of the plate has a singularity of the delta-function type and its derivative. Since the right-hand side of this equation has no singularity, it is natural to equate the coef ficients of the generalized functions to zero. These equations also re present the boundary conditions. This procedure for finding the bounda ry conditions is not new [1, 2], but its application to a fourth-order equation involves the need to take partial derivatives of the delta-f unction distributed over the contour. A formula for calculating such d erivatives is derived and is used to obtain the boundary conditions.