###### Abstract  
The research paper presents a novel approach toward constructing motion equations for structures with attached MTMDs (multiple tuned mass dampers). A primary system with MDOF (multiple dynamic degrees of freedom) was reduced to an equivalent system with a SDOF (single degree of freedom) through the modal approach, and equations from additional MTMDs were added to a thus-created system. Optimization based on \(9\zeta_{2}\) and \(9\zeta_{\infty}\) for the transfer function associated with the generalized displacement of an SDOF system was applied. The research work utilized GA (genetic algorithms) and SA (simulated annealing method) optimization algorithms to determine the stiffness and damping parameters for individual TMDs. The effect of damping and stiffness (MTMD tuning) distribution depending on the number of TMDs was also analyzed. The paper also reviews the impact of primary system mass change on the efficiency of optimized MTMDs, as well as confirms the results of other authors involving greater MTMD effectiveness relative to a single TMD. FEM analysis; vibration control; \(9\zeta_{2}\) and \(9\zeta_{\infty}\) optimization; MTMD; parameter optimization +
Footnote †: journal: Applied Sciences  
with viscous damping. Another new approach to a well-known TMD attached to an SDOF system was reviewed in the work by [PERSON] and [PERSON] [8]. The authors proposed a tuned mass damper inerter (TMDI), which combines a classic tuned mass damper (TMD) with a grounded inerter, which counteracts relative acceleration through additional inertia. The work utilized a modified \"two point\" approach proposed by [PERSON] [1], which enabled determining optimal TMDI parameters. TMD analyses are conducted in various fields of science. The study by [PERSON] et al. [9] involved modeling the vibrations of a drilling mandrel with a vibration damper linked with the mandrel via a viscoelastic coupling. A system with three degrees of freedom with an additionally installed TMD was modeled and the efficiency of such a solution demonstrated. Multiples TMDs (MTMDs) can be introduced to a system in order to improve structural parameters. The efficiency of a spatially distributed MTMD was originally discussed by [PERSON] et al. [10]. The presented results indicated that structural behavior could be improved when an MTMD is tuned for appropriate structural frequencies, maintaining the total mass ratio at the same level as in the case of a TMD. It was demonstrated that an increased number of dampers resulted in the flattening of the response curve over a wider frequency range ([PERSON] and [PERSON] [11]). The performance of TMD and MTMD was compared, and it was shown that the optimal damping values of an MTMD damper were lower than in the case of TMD. The early stages of studies concerning MTMD configurations under simplified and constrained conditions primarily involved to reducing the number of associated design variables. For example, in [12], [PERSON] and [PERSON] studied MTMD with evenly spaced natural frequencies and an equal constant damping, and based on the asymptotic analysis, they demonstrated that such an MTMD was efficient in reducing primary system response. For a finite number of MTMDs with similar constraints, [PERSON] and [PERSON] [13] and [PERSON] [14] presented optimal MTMD parameters for an undamped and damped primary system, respectively. MTMDs with equal damping factors and evenly spaced natural frequencies were also studied by such authors as [PERSON] and [PERSON] [15]. A number of algorithms for selecting optimal MTMD parameter values have been proposed over the years (see [PERSON] and [PERSON] [16]). The study focused on developing optimization algorithms based on the \(\mathfrak{H}_{2}\) and \(\mathfrak{H}_{\infty}\) norms and aimed at minimizing the response to random and harmonic excitations. \(\mathfrak{H}_{\infty}\) optimization was also utilized in the work by [PERSON] [17], for determining optimal ATID (active tuned inerter damper) parameters.