# Dynamics Analysis of Rotating Cantilever Beams with Free End Mass
## 1 Introduction  
Rotating cantilever beams are widely used in the engineering field. Related issues in dynamic modeling and analyses have always been the research concern of scholars. Reliable dynamic models for accurate operation and control are necessary. The main concern of the modeling process and method is to provide the description of the dynamic stiffening effect or the so-called geometric stiffening effect. There are two common modeling methods to deal with rotating cantilever beams. The first kind of method introduces the axial centrifugal force into the dynamic equations of motion directly to provide a positive stiffening effect. It is direct and convenient for dynamics analysis and simulation. [PERSON] et al. [1] developed this kind of method. [PERSON] [2] used this kind of method to study the effect of hub radius ratios on free vibration. [PERSON] et al. [3] analyzed the coupling effect between bending and torsion motions on frequencies. [PERSON] et al. [4] derived the equations of flapwise and torsional motion for a rotating beam. There are also scholars who consider the foreshortening effect to account for the dynamic stiffening. The foreshortening effect relies on the axial shortening caused by bending deformation, leading to the change in axial position. As a result, the stiffening effect of the rotating beam is noticeable. This kind of method based on the foreshortening effect in the axial direction is also widely used. [PERSON] et al. [5] and [PERSON] et al. [6] introduced the deformation variables instead of Cartesian variables to derive the dynamic equations of motions. The foreshortening effect was included in the deformation variable of stretching. [PERSON] et al. [7] introduced a modeling method called the first-order approximate coupling(FOAC) model that contains the foreshortening effect and [PERSON] strain theory. The foreshortening term was considered to provide a coupling effect and played a positive role in the stiffness of the chordwise motion. [PERSON] et al. [8] also considered the coupling effect on frequency analysis of a rotating functionally graded material beam. This kind of modeling method was used to perform dynamics analysis of rotating beams by different discrete methods, such as the finite element method [7; 9; 10], the assumed mode method [5; 6; 8], and so on. Among a huge number of modeling methods, the axial deformations of rotating cantilever beams were mentioned and verified. Eventually, the axial deformations were used to capture the dynamic stiffening effect in vibration analysis. [PERSON] et al. [11] derived the equation of flapwise motion by obtaining the axial deformation. By using the expression of axial deformation solution of [PERSON], the free vibration analysis was carried out by [PERSON] et al. [12]. Another form of the axial deformation solution was provided by [PERSON] et al. [13]. [PERSON] et al. [14] used this form of the axial deformation expression to study the frequencies of the axial, flapwise, and chordwise vibrations. Based on the FOAC model, [PERSON] et al. [15; 16] investigated the effect of axial deformation on the bending frequency of rotating cantilever beams. In particular applications, rotating cantilever beams do not exist alone. Rotating cantilever beams are used to capture or carry something; thus, the dynamic structures are changed accordingly, and the additional mass will have an important influence on the whole system. These structures can be simplified as rotating beams with free end mass to study the dynamics. The two kinds of methods mentioned above are usually used for rotating beams with free end mass. [PERSON] et al. [17] analyzed the frequency of a rotating beam with tip mass subjected to torsional-bending vibrations. [PERSON] et al. [18] introduced the model of rotating blades carrying tip mass to study dynamic behaviors. [PERSON] [19] studied the random transverse vibrations of a rotating beam with a tip mass.