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The free vibration of beams and rods carrying concentrated (lumped) or distributed mass has been extensively investigated in detail for the last three decades. To have an idea about the subject studied in the relevant literature one can refer to the papers listed at the end of this work. Among the vast number of papers, the following can be mentioned. Chen [1] and Goel [2] studied the eigenfrequencies of beams carrying a concentrated mass. Bhat and Wagner [3], and Bhat and Kulkarni [4] obtained the natural frequencies of a cantilevered beam with a slender tip mass. They showed that the gyroscopic elect of the tip mass on the frequencies must be taken into account when its dimensions are considerable compared with those of the carrying beam. Recently, Chan and Zhang [5] studied the free vibration of a cantilever tube partially"filled with liquid, considering it as a beam with distributed mass.Chanet al. [6], and Chan and Wang [7] investigated the free vibration of simply supported and cantilever beams with distributed mass, using Euler}Bernoulli and Timoshenko beam models respectively. Chanet al. [8] treated the free vibration of a beam with two distributed masses in-span. Low [9] derived the frequency equations of a beam with a concentrated mass in-span under classical boundary conditions. Cutchins [10], Batan and Gurgöze [11]dealt with the longitudinal vibrations of rods carrying a concentrated mass. Gürgöze and Inceoglu [12] studied the axial vibration of an elastic rod with external distributed mass. However, it is observed in these works that a system consisting of a mass carried by two different beam segments has not been treated yet. In this paper, a method is presented to obtain the natural frequencies of such a system as shown in Figure 1, due to its practical importance. The general frequency equation derived in the context of this method can also be used to find the eigenfrequencies of the beams either carrying or not carrying a concentrated mass, and non-uniform, two part beams. However, one should remember that the concept of rigidity is an idealization and a theoretical assumption which will not be valid at higher frequencies any more. Therefore, in order to establish a more realistic model for such a system, the intermediate mass must be considered a highly stiff portion of the entire system instead of assuming it ideally rigid.

In this paper, an analytical method is presented to find the eigenfrequencies of a rectangular plate carrying a uniformly distributed mass. Using the standard Galerkin procedure, the equation of motion is reduced to a set of ordinary differential equations. From this set, the frequency equation is obtained. This polynomial equation is solved numerically. Due to the significance of the fundamental frequency of the system, its variation with respect to the non-dimensional parameters associated with the location, the area density and the distribution area of the mass attached to the plate, is investigated. Furthermore, it is shown by a numerical example that the method can be used to study plates with concentrated mass as a special case. Finally, an analysis to obtain the modal surfaces and the related nodal lines is carried out. It is demonstrated that the location of the attachment significantly affects the nodal lines, and modal interchanges may occur.