## 1 Introduction
The study investigated the impact of layer stacking sequence, material gradation index and porosity parameter on the dynamic response of the beam. The researchers used the Kelvin-Voigt material model due to its linearity. \(\eta\) is the damping ratio defined as the ratio of the damping constant to [PERSON]'s module. The damping ratio is selected as \(\eta=0.0001\) to represent the damping matrix [C] as follows. [C] = \(\eta[K]\), where [\(K\)] is the stiffness matrix. The transition between different materials through the length of the bar can be described as an exponential function. The axial bar approach is used for the system elements that are subjected to a coercive force in the axial direction. Therefore, the axial bar theory is used in the formulations with the assumption that the displacements in the direction of other axes are small [25; 26; 27]. Vibration damping studies using axial FG bars are more recent and very few in number. [PERSON] and [PERSON] [28] designed a beam with an Axially Functionally Graded design, considering the deflection mode. They used the material properties and geometric parameters to realise the vibration control of specific regions on the beam structures. [PERSON] [29] considered an Axially Functionally Graded (AFG) bar as a support element. He investigated the frequency parameters of the AFG bar for the viscoelastic and point mass boundary conditions. He interpreted the structural behaviour of the AFG bar for the power law exponent \(n\). The vibration isolation and the force transmitted to the ground have been considered in the context of isotropic and FG beams in the literature. These studies used internal material damping and discrete damping elements for vibration suppression. In this study, internal damping coefficients are defined by using material longitudinal damping values and [PERSON]'s moduli from the literature. A functionally graded material approach is used to formulate the exponential variation of the internal damping coefficients between the two ends of the bar. This approach is based on the dissipation behaviour of solid metallic material losses due to slippage between micro-surfaces. Bars with varying materials and internal damping properties in the axial direction are attached to the beam at the ends under the effect of harmonic force. It is impossible to minimise both the vibration displacements and the force transmitted to the ground. Either the vibration amplitudes or the force transmitted to the ground remain large. Smaller displacements can be achieved with such structures according to spring support. By controlling the material and damping distribution, it will be possible to provide the desired frequency value and isolation. When we evaluate the system as application areas, the following observations are made:  
* The system consists of beam and AFG viscoelastic bars. The system or only AFG viscoelastic bars can be used as an alternative to the spring and damping system to minimise the force transmitted to the ground in the foundations of large mass machines and to minimise the vibration amplitudes of machines. Variable internal damping and stiffness values along the axis of the structure at the support provide advantages in terms of stress and amplitude compared to other structures;
* Passive dampers are used in buildings against stimuli such as earthquakes, wind, etc. Such energy dissipation devices are called ADAS (Added Damping and Stiffness). The ADAS can dissipate a part of the seismic energy input to a structure by hysteretic contribution. The structure considered in the paper absorbs large displacements,especially between floors, by axially varying internal damping. It is also safe in terms of structural stress;
* In anti-terrorist structures such as mushroom barriers, it will be possible to make more successful anti-terrorist equipment by using the advantage of the varying stiffness and material damping of the rods in the axial direction (collision direction) in the steel pipe.