## 1 Introduction  
Beams are important structural elements both alone or as system elements. Their dynamic conditions have especially been the subject of study for many researchers. The boundary conditions and their damping properties affecting the dynamic state are significant in vibration assessment [1, 2, 3, 4, 5]. Damping has an important place in the vibration analyses of beam and plate elements that form the basis of engineering applications. Many mathematical models are utilised in the damped vibration analysis of continuous systems to represent the dissipation and damping mechanisms. Mathematical models that characterise the mechanical characteristics of materials are an ongoing topic in engineering study. Due to their linearity, the Kelvin-Voigt and Maxwell models are simpler and more suitable for modelling. Damping applications in boundary supports are an alternative to surface damping applications using viscoelastic materials. Although the damping effect is obtained by the layer effect on the surface in practice, it is also very common to apply it as a boundary condition [1, 2, 3]. [PERSON] et al. [1] used a complex normal mode analysis approach to study the forced vibration of a beam with viscoelastic boundary supports. The viscoelastic support regions were defined in terms of equivalent complex stiffness coefficients. They separated the equations of motion using complex modes with complex stiffness and investigated the impact of viscoelastic supports on forced vibration response. The Young's modulus of the viscoelastic material is a complex number that consist of \(E_{r}\), the storage modulus and \(\eta\), the loss factor of the material. In addition to the investigation of the vibration characteristics of the beam with viscoelastic boundary conditions, the response of the beam under various boundary conditions has been the focus of interest for researchers [3; 4]. Studies have been carried out according to the characteristics of the external force. The damping effect was investigated by applying the force randomly and harmonically from the centre or end points of the beam [2; 3; 4; 5]. The addition of mass and damping elements to the beam and the effects of their locations on the vibration response have been the subject of research on the optimum vibration values [5; 6]. [PERSON] and [PERSON] [4] conducted a study to determine the best position for viscoelastic supports in reducing vibration levels in a beam that is exposed to harmonic force. Their analysis was based on the Rayleigh damping model, which combines stiffness and mass matrices. Damping has a complicated and unpredictable nature. The dissipating energy is a physical phenomenon and arises via damping in vibrating systems. Considering velocity, displacement and strain parameters, the researchers tried to model the viscoelasticity and dissipation behaviour of solid metallic material. Studies are carried out for hysteretic and viscous damping and equivalent damping in the literature. Viscous and hysteretic damping models are linear damping models. Hysteresis losses in materials due to slippage between micro-surfaces cause damping in typical structures without considering specific viscous, Coulomb and viscoelastic damping elements. It is common to consider viscous damping as either proportional or Rayleigh-type. This is a result of the damping matrix being capable of diagonalization using the same real eigenvectors employed to diagonalize the mass and stiffness matrices [7; 8; 9; 10]. [PERSON] and [PERSON] [9] described structural damping with two real positive constants like the proportional viscous damping case. Therefore, the damping matrix [\(H\)] can be written proportional to the stiffness [\(K\)] and mass [\(M\)] matrices ([\(H\)] = \(\
u\)[\(K\)] + \(\mu\)[\(M\)]). Each mode has its damping loss factor for a system with proportional structural damping. \(\
u\) and \(\mu\) are damping constants. When \(\mu\) = 0, the damping matrix only depends on the stiffness matrix, meaning the damping loss factor for all modes is constant \(\
u\).