## 1 Introduction  
A tuned mass damper (TMD) is a vibration absorber that includes a mass, a damper, and a spring. The spring and damper are linked to the TMD mass in this device in series, and they are connected in parallel with one another. In practice, the resonance frequency is very important for design [1]. In order to prevent resonance issues in the structures, TMD is a very useful design tool. Inertial forces are involved in the dynamic energy absorption technique of TMD when the primary system is shifted by dynamic excitation forces. This energy absorption signifies the kinetic energy absorption that attains the highest value when the phase angle between the primary structure and the TMD movements is \(\pi/2\) radians. This suggests that the highest kinetic energy absorption takes place when TMD works in the direction opposite to the central structure. Here, there is an increase in the equivalent effective damping quantity due to energy dissipation when the value of the fundamental natural frequency of the main structure is close to that of the TMD frequency value [2, 3]. [PERSON] carried out the first study of TMDs in 1909 to decrease the impact of the hazardous vibration of ship engines on the keel of ships [3]. The theoretical research pertinent to TMDs was first carried out by [PERSON] and [PERSON] [4]. The system regulated by TMDs was examined by [PERSON] and [PERSON], while maintaining the internal damping of the structure [5]. The major mass TMD system was optimized by [PERSON] [6], considering the internal damping of the structure. A few correction factors were included by [PERSON] and [PERSON] [7] in the damping parameters of TMDs. A technique was proposed by [PERSON] and [PERSON] [8] to identify the optimal mass ratio and damping parameter of TMDs. Optimal damping of TMDs was computed by [PERSON] et al. [9] by taking into account the main mass with a natural damping ratio of 5%. [PERSON] and [PERSON] [10] examined the effectiveness of a TMD in a ten-story building in the context of the Mexico City earthquake. [PERSON] and [PERSON] [11] examined the effectiveness of a TMD for earthquake response reduction on a twenty-two-story nonlinear shear building. They reported that a TMD is effective for the response reduction in this nonlinear shear building under low and moderate intensity earthquakes. [PERSON] and [PERSON] [12] used the metaheuristic harmony search algorithm to find the optimal TMD parameters for seismically excited structures. In their study, the mass ratio factor of TMDs was also investigated [13]. [PERSON] et al. [14] proposed a method related to optimal viscous damper distribution in shear buildings controlled with TMDs. An efficient optimal TMD design technique was put forward by [PERSON] and [PERSON] [15] on the basis of the transfer function by employing a differential evolution (DE) algorithm. [PERSON] [16] investigated optimal damper locations in a cantilever beam for dynamic compliance. A rotational and translational vibration absorber was suggested by [PERSON] et al. [17] in order to isolate beam vibration, which is exposed to point or distributed excitations. [PERSON] et al. [18] aimed to decrease the vibration in a cantilever beam by using magnetic TMD. [PERSON] [19] proposed a tuned mass damper for response mitigation in wind-exposed structures. Moreover, it is observed that fuzzy logic and linear quadratic Gaussian controllers improve TMD performance. [PERSON] and [PERSON] [20] proposed that the goal of the optimization process is to identify the minimal TMD mass required to permit the nonlinear objective. The design ramifications of tuned mass dampers under dynamic load and uncertainty in the model are crucial. As shown in Figure 1[21], the input energy in the frame can be significantly decreased if the input energy criterion caused by dynamic excitation holds roughly even, regardless of the existence of inherent damping of the system, and the included tuned mass damper can absorb the dynamic input energy as significantly as possible.