## 1 Introduction
[PERSON] [19] analysed the bending vibration of an axially loaded twisted beam with locally distributed internal damping of the Kelvin-Voigt type. They used [PERSON]'s beam theory and [PERSON]'s principle. The Eigen frequencies are studied concerning the internal damping, the sizes and positions of the damped segment, twist angle and the axial load. He showed how different damped beam parameters affect the vibration characteristics of the beam. Due to the integration of smart materials and passive controllers in present-day technologies, the research on vibration in elastic systems with viscoelastic damping has increased. It is also possible to manufacture components with specific material distribution coefficients and different properties like Functionally Graded Materials (FGM). This allows combining solidness, thermal conductivity and friction in exact proportions. The transition between different materials through the thickness or length of the beam can be described as an exponential or sigmoid function. This change is only taken into account in the direction of thickness or length, or thickness and length (bi-directional functionally graded material, BDFGM). Generally, material properties are considered as varying along the thickness direction in the studies on FGM beams. Despite the constraints of the manufacturing process, researchers are highly interested in this technology because of its numerous benefits. However, despite the extensive research on beams and plates, little has been conducted on beams and plates that are supported by springs and dampers, considering their viscoelastic properties. Furthermore, as far as I am aware, the dynamic response characteristics of a beam supported by axially FG viscoelastic bars have not been investigated yet. In the studies of beams with Functionally Graded Materials, the damping was applied discretely or defined as the material damping for the FGM beam. [PERSON] et al. [20] explored the vibration response of beams made of functionally graded (FG) and viscoelastic materials installed on the Pasternak foundation in their study. They aimed to investigate the effectiveness of vibration suppression in a simply supported FG beam subjected to a point harmonic load. They introduced fractionally damped absorbers to reduce the beam deflection at natural frequencies to achieve this. A new method was developed to obtain analytical solutions using Laplace transformations. [PERSON] and [PERSON] [21] utilised the finite element method to investigate the resonance frequency response of a beam using the Euler-Bernoulli beam theory with functionally graded material under viscoelastic boundary conditions. The material characteristics of the beam varied with thickness according to a power law distribution. Different stiffness and damping coefficients are employed in viscoelastic support components to achieve various boundary conditions. [PERSON] et al. [22] conducted a study on the dynamic stiffness matrix of a double-functionally graded Timoshenko beam system on Winkler-Pasternak under axial stress. They also took into account the damping of the connecting layer. The research compared the effect of gradient parameters, foundation parameters, axial loading and connection stiffness on the frequency and buckling load. Furthermore, they studied the effects of the damping factor on the system. FG beam with an elastic foundation and spring supports was investigated by [PERSON] et al. [23] for its free vibration response. The exponential law dictated that Young's modulus, mass density and beam width should vary in thickness and axial direction. [PERSON] et al. [24] conducted a study on the vibration response of a thick beam made of functionally graded porous material subjected to a dynamic sine pulse load with damping effects. They used an accepted finite element model and incorporated the damping effect into the model using the Kelvin-Voigt viscoelastic constitutive model.