# Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End
## 6 Conclusions  
A high-rate experimental testbed is studied. The testbed is characterized as being a clamped-pinned-free beam with a mass at the free end. Euler-Bernoulli beam theory is applied to derive the transcendental equation for a general case applicable to the system pinned at an arbitrary location and with an arbitrary mass. The eigenvalues and mode shapes were presented under various test conditions. Experimental tests were conducted and results compared with the theoretical calculations of the first five natural frequencies. The comparison of results exhibited a good match in frequency values for the first three modes. The errors increase with the higher modes. The difference in higher modes can be attributed to the electromagnet vibrating separate to the beam, the beam vibrating within the rollers, and the rotational inertia of the mass not taken into consideration. Nevertheless, the percent difference of all modes between the theoretical and experimental values fell below 20% except for one case. These results confirm that within reason, the theory matches the experimental results.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Proposed Model** & **Experiment** & **Difference** \\
**Mode** & **(Hz)** & **(Hz)** & **(\%)** \\ \hline
1 & 23.09 & 25.87 & 12.0 \\
2 & 173.89 & 195.62 & 12.5 \\
3 & 526.12 & 614.09 & 16.7 \\
4 & 1015.61 & 1088.84 & 7.2 \\
5 & 1292.51 & 1421.86 & 10.0 \\ \hline \end{tabular}
\end{table}
Table 7: Pinned at \(a=150\) mm and \(\frac{m_{\text{ethanol}}}{m_{\text{basem}}}=0.2\).  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Proposed Model** & **Experiment** & **Difference** \\
**Mode** & **(Hz)** & **(Hz)** & **(\%)** \\ \hline
1 & 28.02 & 31.39 & 12.0 \\
2 & 221.20 & 259.65 & 17.4 \\
3 & 595.97 & 646.99 & 8.6 \\
4 & 798.79 & 950.15 & 18.9 \\
5 & 1479.27 & 1741.16 & 17.7 \\ \hline \end{tabular}
\end{table}
Table 8: Pinned at \(a=200\) mm and \(\frac{m_{\text{ethanol}}}{m_{\text{basem}}}=0.2\).  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Proposed Model** & **Experiment** & **Difference** \\
**Mode** & **(Hz)** & **(Hz)** & **(\%)** \\ \hline
1 & 16.73 & 17.75 & 6.1 \\
2 & 118.28 & 128.88 & 9.0 \\
3 & 350.24 & 378.68 & 8.1 \\
4 & 710.71 & 872.01 & 22.7 \\
5 & 1202.46 & 1400.19 & 16.4 \\ \hline \end{tabular}
\end{table}
Table 5: Pinned at \(a=50\) mm and \(\frac{m_{\text{ethanol}}}{m_{\text{basem}}}=0.2\).  
The analytical model developed here can be useful in the design and numerical assessment of structural health monitoring solutions designed for systems operating in high-rate dynamic environments. Furthermore, the experimental quantification of the error in the higher modes validates and defines the bounds for which the high-rate experimental testbed is best utilized. Future work will include the evaluation of algorithms and methodologies for the SHM of structures experiences high-rate dynamic events.  
[PERSON] performed the analytical work, numerical simulations, and experimental activities, and led the write-up; [PERSON] and [PERSON] co-led the investigation; [PERSON] assisted in the collection of experimental data and validation of the analytical work. All authors contributed to preparing and proofreading the manuscript.  
The material in this paper is based upon work supported by the Air Force Office of Scientific Research (AFOSR) award numbers FA9550-17-1-0131 and FA9550-17 RWCOR503, and AFRL/RWK contract number FA8651-17-D-0002. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Air Force. Additionally, this work was partially supported by the National Science Foundation Grant No. 1850012. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.  
The authors declares no conflict of interest.