On point mass identification in rods and beams from minimal frequency measurements

This paper deals with the identification of a small mass point in a vibrating rod based on the knowledge of the variations induced in a pair of natural axial frequencies. The analysis is based on an explicit expression of the frequency sensitivity to point mass variations and allows consideration of non-uniform bars under general boundary conditions. The inverse problem is generally ill-posed, that is, even if the system is not symmetrical, mass points in different locations can still produce identical changes in a pair of natural frequencies. Despite this ill-posedeness, it is found that there are certain situations concerning uniform rods in which the effects of the non-uniqueness of the solution may be considerably reduced by means of a careful choice of the data. Some of the results are also valid for beams in bending and the identification technique can be extended to include the case of two equal point masses. The theoretical results are confirmed by a comparison with dynamic measurements on steel beams with one and two point masses.