## 6 Conclusions  
An optimal TMD design for the response reduction of a cantilever beam is examined in this study within the frequency domain. In addition to the stiffness and damping coefficients of the TMD, there is also optimization of the mass coefficients. The DE algorithm is used in this optimization by considering the constraints to determine the ideal parameters. The performance of the optimum designed TMD can be tested by exposing the cantilever beam-TMD system to a sinusoidal unit harmonic base excitation at the first, second, and third mode natural frequencies of the beam and white noise excitation. The conclusions derived from this study are listed below:  
The properly calculated TMD dampers have been found to be quite successful in decreasing the transfer function of acceleration, displacement, and rotation reduction.  
Even though TMD parameters are very different from each other, all results for the \(f_{1}\) and \(f_{2}\) objective functions provide an effective performance in reducing transfer functions, harmonic and white noise excitation responses.  
The TMD is able to effectively decrease the displacement and acceleration reactions of the cantilever beam in a translational as well as rotational manner with the harmonic base excitations and the random unit white noise excitation. However, it is seen that TMD is not very effective in decreasing the white noise excitation for the acceleration response. It is also observed that TMD is more effective with respect to the reduction of harmonic  
Figure 17: Contour plot of the 6 th node rotation absolute transfer function \(|\mathbf{H}_{R6}(\omega)|\), with respect to frequency and (**a**) TMD mass coefficient, (**b**) TMD stiffness coefficient, and (**c**) TMD damping coefficient considering \(f_{1}\) and \(f_{2}\) minimization for the first mode control.  
load responses than white noise excitation, especially for the acceleration response. The translational and rotational responses of cantilever beams can be effectively decreased by optimally developed TMDs with harmonic base excitations and their peak values.  
The objective function that this study has employed is successful in appropriately developing the TMDs for response reduction.  
The objective function in the frequency domain has been successfully minimized using the DE method.  
When the TMD mass, stiffness, and damping coefficients reach an optimum value in the resonance zone, the transfer function convergences to minimum values.  
The aim of future investigations is to study optimal design and distribution of multiple tuned mass dampers for cantilever beam response reduction.  
Conceptualization, B.O., H.C., M.D., E.A. and E.N.F.; Formal analysis, B.O., H.C., M.D. and E.A.; Funding acquisition, B.O.; Investigation, B.O., H.C. and E.A.; Methodology, B.O., H.C., M.D. and E.A.; Supervision, B.O., E.A.; Validation, B.O., H.C., M.D., E.A. and E.N.F.; Visualization, B.O.; Writing-review & editing, B.O., H.C., M.D., E.A. and E.N.F. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data generated in this study are available from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.