# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
## 4 Results
### Inhomogeneous Timoshenko Beams with Uniform Cross-Sections
The beams are composed of zirconia and aluminum with axial gradation according to law (5), where the ceramic volume fraction is an exponential function: \(V_{c}=\frac{e^{x3/L}-1}{e^{2}-1}\). The material and geometrical properties of the beams are described in [11] and detailed in Table 4 as follows:  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n^{\star}\)} & \multicolumn{4}{c}{**0**} & \multicolumn{4}{c}{**1**} & \multicolumn{4}{c}{**2**} \\ \cline{2-13}  & **[14]** & **DTM** & \(\mathbf{\Delta^{\ast\ast}}\)**,\%** & **DTM** \({}^{\ast}\) & **[14]** & **DTM** & \(\mathbf{\Delta_{\text{\%}}}\) & **DTM** \({}^{\ast}\) & **[14]** & **DTM** & \(\mathbf{\Delta_{\text{\%}}}\) & **DTM** \({}^{\ast}\) \\ \hline \multirow{10}{*}{0} & 17.92823 & 17.9268 & 0.01 & 17.78719 & 21.05590 & 21.0543 & 0.01 & 20.89837 & 24.29448 & 24.29265 & 0.01 & 24.11362 \\  & 112.3543 & 112.2918 & 0.06 & 106.6048 & 140.9285 & 140.8521 & 0.05 & 133.8822 & 162.7971 & 162.7083 & 0.05 & 154.6099 \\  & 314.5953 & 314.1788 & 0.13 & 280.3868 & 403.4795 & 402.9577 & 0.13 & 360.512 & 462.7162 & 462.1126 & 0.13 & 413.0607 \\  & 616.4812 & 614.9754 & 0.25 & 508.6127 & 795.2794 & 793.3707 & 0.24 & 658.1122 & 909.6774 & 907.4782 & 0.24 & 751.8375 \\  & 1019.088 & 1015.124 & 0.39 & 774.2471 & 1317.894 & 1312.838 & 0.39 & 1004.356 & 1505.584 & 1499.772 & 0.39 & 1145.767 \\ \hline \multirow{10}{*}{1} & 18.89966 & 18.89824 & 0.01 & 18.75921 & 22.29867 & 22.29709 & 0.01 & 22.14241 & 25.73713 & 25.73531 & 0.01 & 25.55769 \\  & 124.9620 & 124.8901 & 0.06 & 118.3494 & 158.3713 & 158.2830 & 0.06 & 150.2293 & 182.9067 & 182.8039 & 0.06 & 173.4348 \\  & 355.0148 & 354.5324 & 0.14 & 315.5107 & 460.6318 & 460.0258 & 0.13 & 410.8001 & 527.8895 & 527.1870 & 0.13 & 470.1954 \\  & 698.2161 & 696.4732 & 0.25 & 573.9377 & 911.2419 & 909.0284 & 0.25 & 752.4501 & 1041.633 & 1039.076 & 0.25 & 858.5914 \\  & 1155.920 & 1151.340 & 0.40 & 874.6708 & 1512.257 & 1506.403 & 0.39 & 1150.092 & 1726.572 & 1719.828 & 0.39 & 1310.355 \\ \hline \multirow{10}{*}{2} & 18.99432 & 18.99291 & 0.01 & 18.85473 & 22.45531 & 22.45374 & 0.01 & 22.3005 & 25.92123 & 25.91943 & 0.01 & 25.73438 \\  & 127.1780 & 127.1041 & 0.06 & 120.3839 & 162.3684 & 162.2768 & 0.06 & 153.9285 & 187.5542 & 187.4475 & 0.06 & 177.7289 \\  & 363.0379 & 362.5411 & 0.14 & 323.2891 & 475.3808 & 474.7514 & 0.13 & 423.6509 & 544.8226 & 544.0922 & 0.13 & 484.8789 \\  & 714.8085 & 713.0133 & 0.25 & 586.9809 & 941.7464 & 939.4477 & 0.25 & 776.9921 & 1076.597 & 1073.939 & 0.25 & 886.5256 \\  & 1183.930 & 1179.213 & 0.40 & 894.8416 & 1563.728 & 1557.652 & 0.39 & 1188.231 & 1785.535 & 1778.528 & 0.39 & 1353.679 \\ \hline \multirow{10}{*}{3} & 18.97105 & 18.96963 & 0.01 & 18.83099 & 22.45202 & 22.45045 & 0.01 & 22.29715 & 25.91884 & 25.91704 & 0.01 & 25.74104 \\  & 126.3443 & 126.2712 & 0.06 & 119.6185 & 162.2392 & 162.1477 & 0.06 & 153.8091 & 187.4615 & 187.3548 & 0.06 & 177.6432 \\  & 359.5311 & 359.0406 & 0.14 & 319.3864 & 474.8301 & 474.2016 & 0.13 & 423.1715 & 544.4325 & 543.7027 & 0.13 & 484.5410 \\  & 707.2811 & 705.5100 & 0.25 & 581.0888 & 940.5708 & 938.2755 & 0.25 & 776.0493 & 1075.765 & 1073.110 & 0.25 & 885.8633 \\  & 1171.009 & 1166.357 & 0.40 & 885.5952 & 1561.717 & 1555.650 & 0.39 & 1186.747 & 1784.113 & 1777.111 & 0.39 & 1352.638 \\ \hline \multirow{10}{*}{4} & 18.97367 & 18.97225 & 0.01 & 18.83369 & 22.44933 & 22.44776 & 0.01 & 22.29437 & 25.92410 & 25.9223 & 0.01 & 25.74647 \\  & 126.4718 & 126.3986 & 0.06 & 119.7366 & 162.0943 & 162.0029 & 0.06 & 153.6739 From Table 5, it is obvious that the present results for all first four frequencies closely align with those obtained in the referenced study. Finally, uniform aluminum-zirconia Timoshenko beams with axial variation of the material characteristics according to the power law function \(V_{c}=\left(\frac{\pi}{L}\right)^{\beta}\) of the ceramic volume fraction in the rule of mixtures (5) are tested. The natural frequencies of these beams are benchmarked against the findings in [15].