## 4 Method Validation
### A Non-Uniform AFG Beam on an Inhomogeneous Viscoelastic Foundation  
The second example is a non-uniform AFG beam resting on a variable viscoelastic foundation, as shown in Figure 1. Its material, cross-section, foundation, and load are all non-uniform or inhomogeneous. The beam is of a length \(L=2\text{ m}\), and its cross-section varies along the longitudinal direction. \[b(x)=b_{0}e^{-(x-L/2)^{2}/2},\quad h(x)=h_{0}e^{-(x-L/2)^{2}/2} \tag{46}\]  
where the width and the height \(b_{0}=h_{0}=0.05\text{ m}\). The material parameters also vary axially, following the laws  
\[\rho(x)=\rho_{1}+(\rho_{2}-\rho_{1})(x/L)^{n},\quad E(x)=E_{1}+(E_{2}-E_{1})( x/L)^{n} \tag{47}\]  
where the material gradient index \(n=2\). The material properties of the Zirconia component are \(E_{1}=200\text{ GPa}\), \(\rho_{1}=5700\text{ kg}/\text{m}^{3}\), \(\mu_{1}=1/3\), and those of the Aluminum component are \(E_{2}=70\text{ GPa}\), \(\rho_{2}=2702\text{ kg}/\text{m}^{3}\), \(\mu_{2}=1/3\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**B.C. Mode**} & \multicolumn{5}{c}{\(\bar{k}_{f}=1\), \(\bar{c}_{f}=1\)} & \multicolumn{5}{c}{\(\bar{k}_{f}=10^{4}\), \(\bar{c}_{f}=1\)} \\ \cline{2-10}  & \multicolumn{2}{c}{\(\omega_{n}\)} & \multicolumn{2}{c}{\(\delta_{n}\)} & \multicolumn{2}{c}{\(\omega_{n}\)} & \multicolumn{2}{c}{\(\delta_{n}\)} \\ \cline{3-10}  & & **Present** & **FEM** & **Present** & **FEM** & **Present** & **FEM** & **Present** & **FEM** \\ \hline \multirow{3}{*}{PP} & 1 & 3.0915 & 3.0915 & 0.4931 & 0.4931 & 9.9845 & 9.9845 & 0.4916 & 0.4916 \\  & 2 & 5.8815 & 5.8816 & 0.4803 & 0.4803 & 10.1869 & 10.1869 & 0.4769 & 0.4769 \\  & 3 & 8.3011 & 8.3016 & 0.4710 & 0.4710 & 10.9029 & 10.9029 & 0.4672 & 0.4672 \\  & 1 & 4.4046 & 4.4047 & 0.4944 & 0.4944 & 10.0619 & 10.0619 & 0.4930 & 0.4930 \\ \multirow{3}{*}{CC} & 2 & 6.8241 & 6.8243 & 0.4847 & 0.4847 & 10.4307 & 10.4307 & 0.4821 & 0.4821 \\  & 3 & 8.9414 & 8.9422 & 0.4776 & 0.4776 & 11.2323 & 11.2327 & 0.4750 & 0.4750 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensionless natural frequency \(\bar{\omega}_{n}\) and decrement coefficient \(\bar{k}_{n}\) of the uniform beam on a viscoelastic foundation. Figure 4: Vibration of the uniform beam on a viscoelastic foundation under a distributed step load. (**a**) \(x=L/4\); (**b**) \(x=L/2\). The varying foundation stiffness and damping coefficient can be given by:  
\[k_{f}(x)=k_{f1}+\Big{(}k_{f2}-k_{f1}\Big{)}(x/L),\quad c_{f}(x)=c_{f1}+\Big{(}c _{f2}-c_{f1}\Big{)}(x/L) \tag{48}\]  
where  
\[k_{f1}=\bar{k}_{f}E_{1}I_{0}/L^{4},\quad k_{f2}=2\bar{k}_{f}E_{1}I_{0}/L^{4}, \tag{49}\]  
\[c_{f1}=\bar{c}_{f}\sqrt{\rho_{1}A_{0}E_{1}I_{0}}/L^{2},\quad c_{f2}=2\bar{c}_{f }\sqrt{\rho_{1}A_{0}E_{1}I_{0}}/L^{2} \tag{50}\]  
where \(\bar{k}_{f}=0.6\pi^{4}\) is the dimensionless foundation stiffness, \(\bar{c}_{f}=1\) is the dimensionless foundation damping coefficient, and \(A_{0}=b_{0}h_{0}\), \(I_{0}=b_{0}{h_{0}}^{3}/12\). Define the dimensionless natural frequency and decrement coefficient as follows:  
\[\tilde{\omega}_{n}=\sqrt{\omega_{n}L^{2}\sqrt{\rho_{1}A_{0}/(E_{1}I_{0})}}, \quad\tilde{\delta}_{n}=\delta_{n}L^{2}\sqrt{\rho_{1}A_{0}/(E_{1}I_{0})}, \tag{51}\]  
A total of 16 Chebyshev polynomials are used in the Chebyshev spectral method, and 100 Timoshenko beam elements are used in the FEM. Both of their results are shown in Table 3. It can be found that, under all kinds of boundary conditions, the complex mode results are in good agreement with the finite element results, and the difference between the two sets of results is rarely small. The dynamic response of the non-uniform AFG beam under three types of uniformly distributed loads is also studied.