## 1 Introduction
For the nonproportional structural damping, the imaginary part of a complex stiffness matrix [\(K\)]\({}_{c}\) can be defined with the structural damping matrix [\(H\)] [9]. The magnitude of energy loss depends upon material type and modes of vibration. Experiments demonstrate that the energy loss per cycle due to internal friction is independent on frequency but is roughly related to the square of amplitude. The structural damping constant is acceptable for forced harmonic motion. So, the dynamic analysis of the bar with viscoelastic material is performed with the complex modulus in the frequency domain [11]. [PERSON] [12] suggested a solution using the constant hysteretic damping model for the free vibration response problem. Structural damping always dissipates energy from the system. The loss coefficient \(\eta\) equals the ratio of specific damping capacity per radian to the strain energy. [PERSON] et al. [13] presented a material-specific damping parameter based on the Bernoulli-Euler beam theory as an addition to the proportional damping approach. The modal damping ratio of each mode was calculated using the material damping coefficient. They assumed that the damping energy is proportional to the local modal strain energy. The results were validated experimentally. The authors proposed a material-specific damping parameter that enables the calculation of structure-specific damping parameters based on theoretical or numerical mode forms. Hysterical damping is generally significant for steady-state vibrations, but there are studies for other forms of forced vibrations. [PERSON] et al. [14] proposed a new approach to overcome the instability of the solution. In addition to the real boundary conditions, by creating virtual boundary conditions, they eliminated divergence terms of the complementary solution for the direct integration solution and converged to the exact solution. The following studies are available if we consider the internal viscous damping approaches in a general framework. The developed formulation of variable damping along the bar is based on this approach in the study. [PERSON] et al. [15] analysed the vibration of a Timoshenko beam with partially and fully distributed internal viscous damping. Theyshowed that the natural frequencies decreased with increasing damping, with internal damping fully distributed. Locally distributed internal damping was more effective when the damped segment was at the position with the maximum bending moment. The dissipation function of the beam was written considering the directional velocity, rotational velocity, bending and shearing internal damping coefficients. [PERSON] [16] studied the vibration of the axially loaded Timoshenko beam with locally distributed internal damping of Kelvin-Voigt type. The Eigen frequencies of the damped beams were determined. The effects of the internal damping considering sizes and location parameters on the damped natural frequency were investigated. [PERSON] and [PERSON] [17] studied the vibration of a viscoelastic beam under the consideration of the Euler-Bernoulli beam theory. For the viscoelasticity of the beam, the Kelvin-Voigt model was used. The damping of the beam is assumed to be proportional to the mass and stiffness of the beam as internal. [PERSON] and [PERSON] [3] investigated the vibrations of the beam exposed to the moving harmonic load for the general boundary conditions. For the material, the Kelvin-Voigt model was used. The dissipation function of the beam was written in the form of internal damping of the beam and proportionality constant. [PERSON] [18] studied the transient vibration of the Bernoulli-Euler cantilever beam under base motion with a rigid mass attached at the end. A fractional Kelvin-Voigt model is used to describe the viscoelastic properties of the beam material. A method of calculating the damped natural frequencies of the beam was introduced. He derived the solution for the forced vibration of the beam by using the mode superposition method.