## 6 Conclusions  
In this study, we investigated the transverse vibration of a free-free beam with unequal end masses and obtained an exact frequency equation. In this equation, the rotary inertia of the end masses was also taken into account. The first five natural frequencies and mode shapes of the beam were obtained. The influences of tip masses and rotary inertias on the natural frequencies and mode shapes were presented by obtaining the analytical results of the frequency equation. In order to verify the natural frequency values obtained analytically, an experimental study was carried out and the first five natural frequencies of the free-free beam with unequal end masses were obtained. Some of the results obtained from this study and the main novelties and contributions of this paper can be summarized:  
* The natural frequencies and mode shapes of the free-free beam with unequal end masses were obtained by including the rotary inertia of the end masses. * The first five nondimensional frequency parameters were obtained and presented in a table for different values of \(\Omega_{1},\alpha_{1},\ \Omega_{2},\ \alpha_{2}\). * The first and second mode shapes of the free-free beam with unequal end masses were obtained for different values of \(\alpha_{2}\). * It was concluded that the increase in the rotational inertia of the end mass was more effective at low frequencies and mode shapes of the structure. * The increase in the distance between the gravity center of the end mass and the beam connection point decreases the beam's natural frequencies. * Experimental testing was performed to verify the frequency equation. The results obtained were in good agreement with analytical results. Due to undesirable interference effects in the experimental study, the difference between the experimental results and the analytical results could be accepted by up to 15%. In our study, the highest difference was 1.86%. This showed that the frequency equation obtained analytically represented the real operating conditions as well as possible. The results obtained from this research are helpful for designing engineering structures, such as energy harvesters and Stockbridge-type dynamic vibration absorbers. Moreover, when designing Stockbridge dampers or dynamic vibration absorbers, our results will be helpful in determining tip masses, the distances between the gravity center of the end mass and the beam connection point, and tip/beam mass ratios. Taking advantage of this work, the distance \(d_{2}\) can be increased in the \(x\)-axis to obtain suitable frequency and mode shapes in the dynamic absorber design. However, since this may cause some difficulties in terms of structural design, the position of \(G_{2}\) can also be designed in cases when \(\xi<1\), as shown in Figure 1b. This means that the end mass center of gravity is also slidable to the left of the attachment point \(B\). This will afford convenience to the designer in terms of construction. Conceptualization, H.B. and O.K.M. ; Methodology, H.B. and O.K.M. ; Software, H.B. ; Validation, H.B. and O.K.M. ; Formal analysis, H.B. and O.K.M. ; Investigation, H.B. and O.K.M. ; Resources, H.B. ; Data curation, H.B. and O.K.M. ; Writing--original draft, H.B. and O.K.M. ; Visualization, H.B. All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Natural Frequencies** & **Analytical (Hz)** & **Experimental (Hz)** & **Error (\%)** \\ \hline
1 st & 30.086 & 29.68 & 1.35 \\
2 nd & 100.012 & 98.15 & 1.86 \\
3 rd & 191.337 & 188.6 & 1.43 \\
4 th & 293.348 & 298 & \(-1.59\) \\
5 th & 437.994 & 442 & \(-0.91\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The first five natural frequencies of the beam-end mass system. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data used in this study are available from the corresponding author upon request.