## 1 Introduction
The overall input energy applied to a single-degree-of-freedom system (SDOF) without TMD as well as an SDOF system with TMD can be seen in Figure 1[21]. Figure 1a shows that the input energies brought about by output disruptions like wind and earthquakes are stored in the main frame. Hence, harmful destruction can possibly take place within the structure. Nevertheless, it can be seen in Figure 1b that TMD decreases the dissipated energy by frame. Hence, it is possible to avoid the significant damage to the structure. This study examines the optimum design of a single TMD that is kept at the end node of a cantilever beam so that the translational and rotational dynamic energies can be regulated. Using random vibration theory, this research examines the best configuration for a tuned mass damper located at the end node of the cantilever beam. The cantilever-TMD system is exposed to first, second, and third mode sinusoidal harmonic and random white noise excitation. Using a differential evolution optimization approach that takes into consideration the provided restrictions, the mean-square end node of the beam acceleration and top displacement are minimized based on transfer functions. The optimization of a single-TMD (SDOF) system is less difficult than the optimization of a multi-TMD (MDOF) system. MDOF systems are often decoupled in order to become SDOF systems in conventional solutions. Only one form of control may be used, since common procedures are restricted. A more practical method would be to use TMDs to control multiple frequencies or additional control modes. It should be stated that the mass of the dampers is defined. However, in a traditional design, the mass is often set by the designer in order to determine the best stiffness and damping coefficients. In the proposed method, TMD mass quantity is optimized together with stiffness and damping. A differential evolution (DE) optimization algorithm is used to compute the mass coefficient of TMDs, taking into account the constraints determined, while the equation of motion is included in the frequency domain. The power special density (PSD) function of external disturbance is defined in the form of white light in a given road frequency range, so that there is no restriction on the design by a given frequency. When random vibration theory is considered, the mean square of the acceleration of the end node of the cantilever beam is taken to be an objective function, and the goal is to decrease it. The rotational and translational vibration of the cantilever beam is regulated by the designed TMD. A harmonic base excitation considering the first three natural frequencies that resonate with the fundamental mode of a beam and white noise excitation, which is randomly produced, are used to test it. The results are compared to a few other frequently used methods in the literature to corroborate the findings. The findings of this study show that an appropriately designed TMD is effective in decreasing the cantilever beam's rotational as well as its translational vibrational responses. Section 2 of the remaining work contains information regarding the tuned mass dampers and the definition of the cantilever beam model is included. Section 4 presents a numerical example to show how successful the suggested methods are, and Section 3 describes the optimization issue and the method for the TMD design. Section 3 also provides a full analysis and interpretation of the results. Furthermore, Section 5 describes the findings of this investigation. Figure 1: Total input energy applied to: (**a**) an SDF system without TMD and (**b**) an SDF system with TMD.