## 1 Introduction  
A vibrating beam-mass system can be found both in classic and modern engineering applications, such as micromechanical systems, atomic force microscope (AFM) probes, nanosensors, robotic manipulators, turbine blades, Stockbridge dampers, etc. In order to operate safely for these structures, it is very important to determine their dynamic characteristics during the design progress. The modeling of these structures is as close as possible to actual operating conditions; therefore, obtaining the equation of motion, solving the frequency equation, and determining the change of the natural frequencies depending on the end mass have great importance. Many researchers have investigated free transverse vibration of beam-mass systems and a considerable amount of research outcomes have been brought together in some reference books [1; 2; 3; 4; 5; 6]. The effects of rotary inertia, mass size, mass positions, springs, beam materials, etc. on the natural frequencies and mode shapes were investigated in the last decades [7; 8; 9; 10; 11; 12; 13; 14; 15]. In one of the earliest studies on beam-mass systems, [PERSON] and [PERSON] [16] studied the free transverse vibration of a Euler-Bernoulli (EB) beam with an end mass rotating at a constant speed. [PERSON] [17] investigated the effect of the end mass on the natural frequencies and mode shapes of the beam using the modified Reissner variation theorem, by taking the beam from the work of [PERSON] and [PERSON] [16]. [PERSON] and [PERSON] [18] studied the natural frequencies of beam-mass systems in transverse vibration. They considered the Euler-Bernoulli type beam with different boundary conditions and the masses in their study were concentrated masses. They compared analytical and numerical results. [PERSON] et al. [19] investigated the transverse vibration of the clamped-pinned-free beam with the end mass. They used the Euler-Bernoulli beam and considered the end mass as a point mass. In addition, they presented the results for varying end mass values. [PERSON] et al. [20] studied the cantilever beam with a large tip mass. They worked numerically and experimentally and emphasized the importance of including inertia when the tip mass is large. [PERSON] et al. [21] investigated the dynamic analysis of rotating cantilever beams with a tip mass. They derived the equations of motion for free vibration analysis of the rotating cantilever beam with a tip mass by using [PERSON]'s principle and a [PERSON] method. Many of these studies about beam-mass systems are concerned with cantilevers, simply supported beams, and pinned beams. Free-free beams are a special case of beams with no end restrictions. Because of these properties, they are frequently used in specific engineering applications. For instance, a free-free beam can be used in the modeling of the mechanical behavior of space structures and rockets. These engineering structures can be represented as free-free beams with one end carrying a trigger load [22; 23]. A beam carrying a mass at both ends is a special class of free-free beam. Because of their importance in engineering fields, a fundamental issue is to determine the natural frequencies and mode shapes of the free-free beam with tip masses. [PERSON] and [PERSON] [24] studied free-free beams with a concentrated mass by using the finite element method (FEM) via the extended Hamilton's principle and their theory was based on the Euler-Bernoulli beam theory. [PERSON] and [PERSON] [25] obtained the natural frequencies and mode shapes of a free-free beam with large end masses connected to the beam by torsion springs. [PERSON] and [PERSON] [26] investigated a spinning unconstrained beam with a concentrated mass. In their study, the beam was subjected to a thrust. [PERSON] [27] offered an approximative formula for the first natural frequencies of the free-free beam with identical concentrated tip masses.