## 4 Method Validation
### A Uniform Beam Resting on a Homogeneous Viscoelastic Foundation  
The first example is a uniform beam on a homogeneous viscoelastic foundation. The parameters of the beam include a length \(L=25\) m, a cross-sectional area \(A=1\) m\({}^{2}\), and a second moment of area \(I=1\) m\({}^{4}\). Its material density \(\rho=2700\) kg/m\({}^{3}\), the [PERSON]'s modulus \(E=70\) GPa, the shear correction factor \(\kappa=2/3\), the [PERSON]'s ration \(\mu=1/3\), and the shear modulus \(G=E/2(1+\mu)\). The two ends of the beam are simply supported. Define the dimensionless natural frequency and foundation parameters as follows:  
\[\bar{\omega}=\sqrt{\omega L^{2}\sqrt{\rho A/(EI)}},\quad\bar{k}_{f}=k_{f}L^{4 }/(EI),\quad\bar{c}_{f}=c_{f}L^{2}/\sqrt{\rho A/(EI)} \tag{42}\]where \(\bar{k}_{f}\) and \(\bar{c}_{f}\) are the dimensionless foundation stiffness and damping coefficient, respectively. The convergence of the proposed method is first considered. Two foundation stiffness coefficients are considered, i.e., \(\bar{k}_{f}=1\) and \(\bar{k}_{f}=10^{4}\). The damping coefficient \(\bar{c}_{f}\) is set to zero. The results are shown in Figure 3, where 1 st, 2 nd, and 3 rd denote the first three modes. It can be seen that a rapid convergence is observed, and the values stabilize in a very small range when more than eight Chebyshev polynomials are used. Then, the accuracy of the proposed method is validated by comparing the results with those found in the literature or obtained by the FEM. When \(\bar{c}_{f}=0\), the viscoelastic foundation reduces to a Winkler foundation, Table 1 compares the first three frequency parameters calculated here with the exact solutions computed from the frequency equations reported in reference [33] and the results presented by [PERSON] [34]. The exact solutions were obtained from the coupling differential equations for the transverse vibrations of uniform Timoshenko beams on constant elastic foundations. [PERSON] presented approximate solutions obtained from FEM, where five cubic elements were used. It should be noted that, although the three kinds of results agree well, the present results approach the exact results more closely in mode 3 when compared with the FEM results. When \(\bar{c}_{f}\
eq 0\), complex mode analysis is performed to investigate the transverse vibration of the beam. Assume the solutions of the system's characteristic equation are as follows:  
\[\lambda_{n}=-\delta_{n}+v_{n}i \tag{43}\]  
then, the damped natural frequency \(\omega_{n}=\sqrt{\delta_{n}^{2}+v_{n}^{2}}\). Define the dimensionless parameters  
\[\bar{\omega}_{n}=\sqrt{\omega_{n}L^{2}\sqrt{\rho A/(EI)}},\quad\bar{\delta}_ {n}=\delta_{n}L^{2}\sqrt{\rho A/(EI)}. \tag{44}\]  
Table 2 tabulates the results computed using the Chebyshev spectral method, along with those obtained by the FEM. The finite element analysis is carried out by the COMSOL Multiphysics 6.0 simulation software, using 100 Timoshenko beam elements. Excellent agreement is observed between the two sets of results, which demonstrates the validity of the proposed method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\bar{k}_{f}\)} & \multirow{2}{*}{**Mode**} & \multirow{2}{*}{**FEM [34]**} & \multirow{2}{*}{**Exact [33]**} & \multicolumn{3}{c}{**Chebyshev**} \\ \cline{4-7}  & & & & \(N=8\) & \(N=12\) & \(N=16\) \\ \hline \multirow{4}{*}{\(1\)} & 1 & 3.092 & 3.092 & 3.092 & 3.092 & 3.092 \\  & 2 & 5.882 & 5.881 & 5.881 & 5.881 & 5.881 \\  & 3 & 8.305 & 8.301 & 8.300 & 8.301 & 8.301 \\  & 1 & 9.984 & 9.984 & 9.984 & 9.984 & 9.984 \\ \(10^{4}\) & 2 & 10.187 & 10.187 & 10.187 & 10.187 & 10.187 \\  & 3 & 10.905 & 10.903 & 10.901 & 10.903 & 10.903 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless natural frequency \(\bar{\omega}\) of the uniform beam on a Winkler foundation. Figure 3: Convergence of the first three dimensionless natural frequencies. Finally, the dynamic response of the beam under a distributed step load is also investigated. \[q(x,t)=