## 4 Verification Studies  
No study has been found in the literature on the vibration of viscoelastic FG bar-supported beams subjected to a harmonic force at the centre point. Therefore, the frequency parameters of the simply supported beam [11] are used to verify the simulation results. The simply supported boundary condition is obtained by making the bar supports rigid. The value of Young's modulus \(E\) of beam is 2.1 \(\times\) 10\({}^{11}\) N/m\({}^{2}\) for the steel beam. Infinite axial stiffness can be simulated by setting the values of \(E_{La}\) and \(E_{R}\) equal to 70 \(\times\) 10\({}^{80}\) N/m\({}^{2}\) and \(na\) = 1 for the axial bars. Figure 5 shows the force transmissibility of a steel beam supported with rigid bars, obtained in a MATLAB environment. The scanning interval is taken as 10\({}^{-8}\) Hz when calculating the frequency parameters in all the tables for the harmonic analysis of the system in MATLAB. Considering the simulation time in graphical representations, the scanning interval for harmonic analysis in MATLAB is assumed to be 0.01 Hz. Consequently, the calculated results are compared with the exact results of the simply supported isotropic steel beam in Table 2. The exact results are obtained using [PERSON]'s analytical values [11]. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(E\) (GPa) & \(\rho\) (kg/m\({}^{3}\)) & \multicolumn{2}{c}{**Flexural Loss Factor**} & \multicolumn{2}{c}{**Longitudinal Loss**} & \\  & & \(\eta\) & **Factor \({}^{*}\)\(\eta\)** & \(c_{A}\)=\(E\eta\) (GPa)** \\ \hline Al/Aluminium & 72 & 2700 & (0.3–10) \(\times\) 10\({}^{-5}\) & 1 \(\times\) 10\({}^{-4}\) & 72 \(\times\) 10\({}^{5}\) \\ St/Steel & 210 & 7800 & (0.2–3) \(\times\) 10\({}^{-4}\) & 3 \(\times\) 10\({}^{-4}\) & 630 \(\times\) 10\({}^{5}\) \\ Br/Brass & 95 & 8500 & (0.2–1) \(\times\) 10\({}^{-3}\) & 1 \(\times\) 10\({}^{-3}\) & 950 \(\times\) 10\({}^{5}\) \\ Ag/Silver & 80 & 10,500 & 4 \(\times\) 10\({}^{-4}\) & 3 \(\times\) 10\({}^{-3}\) & 2400 \(\times\) 10\({}^{5}\) \\ Cu/Copper (polycrystalline) & 125 & 8900 & 2 \(\times\) 10\({}^{-3}\) & 2 \(\times\) 10\({}^{-3}\) & 2500 \(\times\) 10\({}^{5}\) \\ Pb/Lead(pure) & 17 & 11,300 & (5–30) \(\times\) 10\({}^{-2}\) & 2 \(\times\) 10\({}^{-2}\) & 3400 \(\times\) 10\({}^{5}\) \\ \hline \end{tabular} \({}^{*}\)The values are used in the study  
\end{table} 
Table 1: Material properties [8]. Figure 5: The force transmissibility of the isotropic steel beam supported with rigid bars. The finite element model is also created with 3-D truss and beam elements in ANSYS to compare and visualise the results. To better understand the effect of the bar and beam on the vibration response, simulations are performed in the ANSYS environment. Three-dimensional truss and beam elements are connected with revolute joints. The system is fixed at the ends of aluminium bars. In addition, out-plane nodal rotation and displacements of the system are fixed to obtain in-plane motion (Figure 6). In the Ansys model, 64 elements are used for each of the beams and bars as in the Matlab model. The bars and beam have a rectangular cross-section with a width of 100 mm and a height of 10 mm. Their length is 1000 mm. Harmonic analysis is performed between 0 and 2000 Hz. The frequency response and vibration motions obtained are shown in Figure 7. Table 2 compares the results of this study with the results of the Ansys harmonic and modal analysis under the same boundary conditions. Since there is a harmonic force acting on the centre point of the