# Dynamics Analysis of Rotating Cantilever Beams with Free End Mass
## 4 Frequency of Free Vibration  
For analysis and comparison of normalization, several dimensionless parameters are introduced as follows:  
\[\delta=\frac{a}{L},\quad\alpha=\sqrt{\frac{AL^{2}}{I}},\quad\varphi=\frac{m}{ \rho AL},\quad\gamma=\Omega L^{2}\sqrt{\frac{\rho A}{EI}},\quad f=\omega L^{2} \sqrt{\frac{\rho A}{EI}} \tag{41}\]  
where \(\delta\) is the center body radius ratio, \(\alpha\) is the slenderness ratio of the beam, \(\varphi\) is the mass ratio of the free end mass, \(\gamma\) is the dimensionless rotating angular speed, \(f\) is the dimensionless frequency, and \(\omega\) is the natural frequency. First, to validate the proposed model, the convergence characteristics of the frequency of the free vibration are studied. Considering the case when \(\delta=0\), \(\alpha=70\), \(\varphi=1\), and \(\gamma=10\), the first five-order dimensionless frequencies of axial and chordwise vibrations are listed in Tables 1 and 2, respectively. It can be seen that the first five-order dimensionless frequencies converge as the basis function number \(N\) increases. The chordwise vibration frequencies converge faster than the axial vibration frequencies, and the low-order frequencies converge faster than the high-order frequencies. Reliable converged results can be calculated when \(N=20\), as shown in Tables 1 and 2. Thus, the number of basis functions will be selected as 20 for further computations in this study. The effect of the increasing dimensionless angular speed on the free vibration frequencies of the rotating cantilever beam with free end mass is compared between the proposed model and those from previous models when \(\delta=0\), \(\alpha=70\), and \(\varphi=0.1\). The models \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(N\) & **First Mode** & **Second Mode** & **Third Mode** & **Fourth Mode** & **Fifth Mode** \\ \hline
2 & 61.9539 & 265.6167 & - & - & - \\
3 & 61.3809 & 257.3359 & 490.3962 & - & - \\
4 & 61.0932 & 253.5322 & 480.2201 & 716.2866 & - \\
5 & 60.9205 & 251.3164 & 474.9937 & 705.6272 & 941.8007 \\
6 & 60.8055 & 249.8600 & 471.7183 & 699.8341 & 931.0727 \\
8 & 60.6619 & 248.0580 & 467.7836 & 693.3376 & 920.9901 \\
10 & 60.5758 & 246.9847 & 465.4841 & 689.6878 & 915.7445 \\
12 & 60.5185 & 246.2720 & 463.9700 & 687.3244 & 912.4413 \\
14 & 60.4776 & 245.7641 & 462.8959 & 685.6624 & 910.1503 \\
16 & 60.4469 & 245.3837 & 462.0937 & 684.4277 & 908.4617 \\
18 & 60.4231 & 245.0882 & 461.4716 & 683.4732 & 907.1629 \\
20 & 60.4040 & 244.8519 & 460.9748 & 682.7128 & 906.1318 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Convergence characteristics of the dimensionless axial vibration frequencies when \(\delta=0\), \(\alpha=70\), \(\varphi=1\), and \(\gamma=10\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(N\) & **First Mode** & **Second Mode** & **Third Mode** & **Fourth Mode** & **Fifth Mode** \\ \hline
2 & 3.5560 & 42.8067 & - & - & - \\
3 & 3.3049 & 42.7915 & 92.0261 & - & - \\
4 & 3.2209 & 42.3971 & 91.7619 & 155.8913 & - \\
5 & 3.1877 & 42.3769 & 90.9833 & 154.9753 & 236.5447 \\
6 & 3.1736 & 42.3190 & 90.9766 & 153.8505 & 234.9091 \\
8 & 3.1636 & 42.3002 & 90.8286 & 153.4911 & 233.2298 \\
10 & 3.1609 & 42.2948 & 90.7856 & 153.3735 & 232.7474 \\
12 & 3.1594 & 42.2943 & 90.7703 & 153.3392 & 232.5747 \\
14 & 3.1567 & 42.2934 & 90.7582 & 153.2802 & 232.4191 \\
16 & 3.1551 & 42.2929 & 90.7410 & 153.2054 & 232.3796 \\
18 & 3.1540 & 42.2925 & 90.7248 & 153.1925 & 232.3580 \\
20 & 3.1534 & 42.2918 & 90.7154 & 153.1888 & 232.3059 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Convergence characteristics of the dimensionless chordwise vibration frequencies when \(\delta=0\), \(\alpha=70\), \(\varphi=1\), and \(\gamma=10\). used for comparison are the proposed model, the previous model from [PERSON] et al. [31], and the model without stiffening effect. Figure 6 presents the dimensionless first-order frequencies of chordwise and axial free vibration.