Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) through the Modal Approach system. Optimization based on Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) through the Modal Approach Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) through the Modal Approach Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) through the Modal Approach Optimization of Multiple Tuned Mass Damper (MTMD) Parameters for a Primary System Reduced to a Single Degree of Freedom (SDOF) through the Modal Approach

: 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 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. the analysis of linear MTMDs with numerous practical conﬁgurations, attached to an SDOF structure, subjected to white noise input. Six practical conﬁgurations were developed and analyzed comparatively, each of which was linearly restricted with distributed tuning factors, mass ratios, damping factors, and their combinations. In [28], Stanikzai et al. analyzed TMDs and MTMDs distributed at various degrees of freedom of an MDOF system in a building with base isolation (BI). The analysis involved 40 earthquake ground motions for the adopted pattern of an MDOF structure with attached TMD and MTMDs and indicated the efﬁciency of the latter. In [29], Yin et al. analyzed a new type of TMD system named pounding tuned mass damper (PTMD). The coupled equations were created by combining the equations of motion of both the bridge and moving vehicles. In order to compare the damping performance, a parametric study of the various numbers and locations, mass ratio, and stiffness of the MPTMDs were investigated. The article presents an alternative method for developing motion equations for a MTMD structure. The method enables adding single TMDs or MTMD groups to completely different degrees of freedom of the primary system. The system of equations allows for easy MTMD tuning for complex vibrations modes, with MTMDs located in local maxima of these vibrations modes, while still analyzing the SDOF system with attached MTMDs. MTMD parameter optimization based on 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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 ℌ ℌ ∞ MTMD; 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. Keywords: FEM analysis; vibration control; ℌ 2 and ℌ ∞ optimization; MTMD; parameter optimization 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 ℌ 2 and ℌ ∞ 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. The calculation process involved using the aforementioned GA and SA optimization methods and the objective functions based on 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. kind of TMD, where the damping element is not connected with the primary mass but with the substrate (other motion equations); it is impossible to directly apply the optimal parameter formulas proposed in this study as parameters for a standard TDOF system. please note that optimal TMD parameters based on the 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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 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 ℌ 2 and ℌ ∞ 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. 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 ℌ 2 and ℌ ∞ 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.


Introduction
The first general formulation of the research problem involving the parameter optimization in a TMD attached to a SDOF structure was suggested by Den Hartog [1]. He determined generally known formulas for TMD tuning and a critical damping ratio in consideration of a TDOF (two degrees of freedom) system with harmonic excitation. Recommendations in terms of the optimal TMD parameter set can also be found in the work by Warburton [2] for a deterministic case, that is, when the main mass of a SDOF system undergoes harmonic excitation. The response of a linear system to broadband response is different to the case of a harmonic excitation, because the first one occurs at system's natural frequencies. In cases where broadband excitation has an almost constant spectrum within the natural frequency range, it is convenient to replace it with white noise. In his work [3] Warburton provided formulas for optimal TMD parameters at exactly such excitation. Similar discussions were conducted by Bakre and Jangid in [4] and [5]. A different approach to the issue of determining TMD parameter values was proposed by Krenk [6], who suggested aligning the ordinates of three points A, B (just like Den Hartog) and one central, between the previous two. He derived a new optimal damping factor value, which is 15% higher than the classic result.
The most recent work that need mentioning include the one by Batou and Adhikari [7], which focused on an SDOF system with an attached viscoelastic damper. A standard rheological model as a TMD model was used to obtain analytically optimal TMD parameters. Classic results for a TMD with viscous damping can be obtained as a specific case for this damper type. It was also of which was linearly restricted with distributed tuning factors, mass ratios, damping factors and their combinations. In [27], Stanikzai et al. analyzed TMDs and MTMDs distributed at various degrees of freedom of an MDOF system in a building with base isolation (BI). The analysis involved 40 earthquake ground motions for the adopted pattern of an MDOF structure with attached TMD and MTMDs, and indicated the efficiency of the latter. In [28], Yin et al. analyzed new type of TMD system named pounding tuned mass damper (PTMD). The coupled equations were created by combining the equations of motion of both the bridge and moving vehicles. In order to compare the damping performance, a parametric study of the various numbers and locations, mass ratio and stiffness of the MPTMDs were investigated.
The article presents a new method for developing motion equations for a MTMD structure. The method enables adding single TMDs or MTMD groups to completely different degrees of freedom of the primary system. The system of equations allows for easy MTMD tuning for complex vibrations modes, with MTMDs located in local maxima of these vibrations modes, while still analyzing the SDOF system with attached MTMDs. MTMD parameter optimization based on H2 and H∞ norms for the transfer function associated with the generalized displacement of an SDOF system was proposed. 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 comparison and discussion were based on previously developed models of a primary system with an attached MTMD.

Original proposal of a motion equations for a primary system with attached MTMDs, reduced to an SDOF system through the modal approach
In order to be able to model an equivalent system with a single degree of freedom (SDOF) for a system with multiple degrees of freedom using the modal approach, several conditions need to be satisfied, such as the spatial load distribution must be relatively uniform, and in the case of concentrated load, it should be located near the highest ordinate of the first vibration mode, and natural frequencies higher than i  are not close to this frequency.
Using this relationship enables obtaining an equation, which describes an equivalent motion of a SDOF system, corresponding to a system with N degrees of freedom. The primary system is reduced through the modal approach. The main assumption is the orthogonality of the main structure C damping matrix. In general, for N degrees of primary system, the motion equation solution: expands into a series of eigenvectors where () t ψ is the primary coordinate vector, i a is i-th eigenvector, while W is the matrix corresponding mode shapes in its columns. If we include the Ni mode of the vibrations, which have the greatest share in the vibration system, the above equation can be expressed as: If we normalize the eigenvectors in the manner shown below, we get the following motion equations: where: Designations as in Figure 2. were adopted for the analyzed situation. The diagram shows a beam structure with eliminated degrees of rotation and degrees of freedom along the X-axis direction of the global coordinate system.

Figure 1.
The primary system, adopted first and i-th mode shape, together with an SDOF equivalent system.
The following designations for the diagram above were adopted: j=1,2,…N -system degrees of freedom; i=1,2,…Ni-number of vibration modes taken into account (further vibration eigenmodes not necessarily included in the analysis)i index means any subsequent mode number included; Assuming that natural frequencies corresponding to the modes, which are decisive for structural vibrations, are separated from each other, the dynamic response in the case of excitations with a selected vibration mode, can be determined using the formula: where: () i t  -primary coordinate associated with the i-th vibration eigenmode, ji a -coordinate of the i-th vibration mode of the j-th degree of freedom, Ni -number of vibration eigenmodes taken into account.
The S point to which the system is reduced, is called the equivalent system mass reduction point. It is adopted at the point of the highest coordinate of the ai mode shape or the point of application of concentrated load within the system, including, e.g. force originating from a mechanical vibration damper.
After bilaterally dividing the modal motion equation (4) by the coordinate of the i-th mode shape of the equivalent system mass reduction point ji a and after normalizing eigenvectors in the form of 1 we get an equivalent system motion equation: where: 1,2,.., A new tuned mass damper (TMD) or multiple TMDs (MTMDs) can be attached to the new system. In general, attached TMDs do not have to be located at the S system reduction point. If the S system reduction point is an attachment point for a single TMD, we get a 2DOF system, which is a well-known issue addressed by Den Hartog in [1], in the case of no damping of the primary system. Of course, TMD is tuned to a frequency near i  , for which the equivalent system was determined.
The case of using MTMDs located in various degrees of freedom of the primary system requires a separate discussion. MTMD arrangement diagram is shown in Figure 2. Of course, each TMD shall be tuned to near frequency i  . If we introduce location vectors for each MTMD, with the value 1 present on the degree of freedom to which the tuned mass damper is attached to, we can write the following structural displacement equation, for a degree of freedom to which the k-th TMD is attached to: where T 0,0,1,...,0 k =    e is the location vector of the k-th damper. The value 1 is present on the degree of freedom to which the k-th TMD is attached to, T ki k i a = ea is the ordinate of the i-th eigenvector at the TMD attachment point.
We also have to introduce the value of the displacements, velocity and relative acceleration, which describe the motion of additional MTMDs. Generalized displacements, velocities and relative accelerations for the dampers can be expressed with the following formulas: A system of motion equations for such a case, with the number of degrees of freedom of 1 k N + is shown by the formula below:

q t c q t k q t p t T m q c q q k q q
The force from each damper, which needs to the attached to the degree of freedom associated with the k-th TMD, taking into account absolute values, can be expressed with the following formulas: whereas while using relative values: Changing to generalized force, for absolute values we get: and for relative quantities: 11 By substituting the additional damping force values to motion equations, and by using the relationships for absolute values (17), (18), we get: Whereas when using relative values (19), (20), we get: Motion equations in matrix form, for the most general case, with absolute values, are shown by the relationship below: where: From the generalized displacement and load vector let us separate vector blocks associated with the primary ordinate and actual TMD displacements: where: 11 , , , Similarly, the blocks associated with the discussed degrees of freedom shall also be separated from the matrix ,, M C K : 11 The size of blocks  12 21 0 0 0 0 The new system suggested by the authors, significantly limits the time consumption of the calculations, and also enables the optimization of MTMD parameters discussed below. In general, i N equivalent systems can be created (as many as natural frequencies are taken into account). Each of these systems, with the number i, is a system with 1 k N + degrees of freedom. Achieving full displacement at the S reduction point, requires determining the response for each of the created systems, and then applying the formula: This approach is obviously not convenient. However, the method of reducing the system to an SDOF is very useful for structures with simple static diagrams (beams, cantilever structures such as chimneys, masts), when the lowest vibration natural frequency is taken into account. Next, let us determine the parameters of an SDOF equivalent system. For such a system, using the aforementioned equations, we can create new motion equations with attached single TMDs or MTMDs tuned to a distinguished natural frequency. of Fourier transforms, which can be found in the work by Bendat [29]:

Determination of a transfer matrix used to determine the objective function in optimization issues
where: impulse response () ik ht is the i-th response of the system to the k-th excitation in the form of a unit impulse function k  (Dirac delta) applied at the initial moment 0 t = , j means an imaginary unit, () means the frequency domain . Therefore, in order to determine impulse responses in the i points of the system, the excitation shall be adopted in the following form: and by substituting the aforementioned load vector mode to the equation (1), and after applying the Fourier transform, we get: where: , ik ik H transfer function module and argument, respectively.
If we assume the denotation of the dynamic stiffness matrix in the form: we get a complex equation:

Re Re
Im Im

Im Re
Re Im Of course, with an appropriate computational procedure at our disposal, we can directly determine a complex transfer matrix from the formula: Knowing the transfer matrix, it is possible to determine the structural response within the frequency domain, in the form of: where ( ), ( )  qp are the Fournier transforms of the displacement and load vector. The structure's load Fourier transform is expressed by the formula: If we determine the Fourier retransform of the structure's load vector, we get the following relationship: ( ) H . This corresponds to strengthening the response after passing a system at the j degree of freedom after an excitation at the same j degree of freedom. The transfer function argument, which is the phase shift angle between the load and the response, is omitted in objective functions, since it does not determine the MTMD operational effectiveness. This proposal is very similar to the objective functions known in the source literature as the H2 norm, which correspond to the objective functions based on the spectral response of a structure to white noise input. An objective function form similar to the H∞ norm known from the source literature was also proposed. In this case, the optimization criterion is the minimization of the maximum values of the transfer function () jj H  (see Figure 3), which can be described by the formula:

The issue of optimization
The analysis used two optimization methods in the form of a standard genetic algorithm (GA), and a simulated annealing (SA) algorithm. Ready libraries for the DELPHI software environment were used.
The procedure for the numerical determination of the optimal parameters can be summed up in the following steps: 1. Calculations of the eigenvalues and eigenvectors in FEM software, adopting baseline frequency i  , to which MTMDs will be tuned; 2. Adopting a point for reducing a system to an equivalent SDOF system, normalization of eigenvectors and determination of the parameters for the equation (7); 3. Adopting a TMD number, and initial parameters , , t t t k k k m c k for the TMDs; 4. Adopting variables, which will be subject to the optimization process, e.g. , (56) at each optimization step. 6. After reaching the desired calculation accuracy (change of the J1 or J2) value, saving the obtained MTMD parameters and calculations of the equivalent system transfer function () jj H  associated with the j degree of freedom, which was adopted as the equivalent system reduction point.

Input data
The analysis adopted a cantilever structure in the form of a chimney made of S235 steel, with a height of h=160m. The chimney parameters were selected so as to obtain the first circular natural frequency equal to 1 The chimney diameter and its wall thickness were adopted as a constant, along its entire height. These values equal to, respectively: D=4010 mm, g=25.5 mm. The cantilever structure was divided into 40 beam elements with the following parameters: Young modulus E=210 GPa, Poisson coefficient =0.3 and density =7850 kg/m 3 . First 3 natural vibration modes, and their corresponding eigenvalues were calculated. FEA software was used to import a normalized eigenvector for the first mode of the natural vibrations, which was then used to construct motion equations for the structure reduced to SDOF system with attached TMD. Figure 4 shows a FEM model with the numberings of the bars and modes, and a presentation of the first three mode shapes and normalized eigenvectors.   The calculation process involved using the aforementioned GA and SA optimization methods, and the objective functions based on H2 and H∞ norms in the form of J1 and J2. The frequency range when determining the objective function was (0.0 rad/s: rad/s).

Numerical optimization resultscomparison of the GA and SA heuristic methods
The efficiency of the GA and SA optimization was checked based on the example of 10 TMDs. Both heuristic methods do not guarantee finding an exact solution, which is the consequence of the very specificity of the methods. Figure 5 shows the optimal TMD parameter values obtained using both methods. Compared to SA, the GA method does not provide results convergent with the results presented in the work by Zuo and Nayfeh [19]. The obtained results are discussed in greater detail in Chapter 4. Only the SA optimization method was applied in the further part of the optimization calculations.

Numerical optimization results -SA method
Optimal MTMD parameters, which are discussed below, were obtained as a result of computations. Table 2 shows the end values of the J1 and J2 objective functions, after the completion of the SA optimization process, using the J1 and J2 objective functions. In addition, it also specifies the percentage change of the objective function value, relative to a 1 TMD system. Computation accuracy was adopted at a level of 1E-10.  Table 3 and Table 4 show optimal MTMD parameters obtained with optimized J1 and J2 objective functions. A tabular presentation of the 20 TMDs was omitted due to the high number of the results. Results for these cases are shown as drawings in the next chapter. Figure 5a shows a graph of the transfer function module for an equivalent system

Impact of primary system mistuning
The analysis also covered the impact by the changing mass of the equivalent structure 41,1 m , on the value of the equivalent structure transfer function 41,41 H for a structure with attached MTMDs with optimal parameters, pre-determined based on the J1 and J2 objective function. A 10% reduction and increase in the equivalent structure mass was assumed, which results in the change of the natural circular frequencies to a value of

Discussion
The analyses presented in chapter 3 were used as a base to compare the application of the SA optimization method and the J1 and J2 objective function with known analytical solutions in terms of a single TMD. Table 5 shows known formulas for the determination of the optimal parameters for TMD usually determined from a TDOF system, with stated values of the optimal parameters. The table also presents original results for 1 TMD, taking into account natural damping of the primary system, as well as without it. Figure 8 shows an equivalent system transfer function module graph for optimal TMD parameters.  When analyzing the obtained solution (see Table 6 and Figure 8a), in the case of no damping of the primary system 1 0  = and in comparison with the Den Hartog solution [1], which has been known for years, on can state very good conformity of the optimal tuning parameters 1  and the TMD damping factor 1 t  . Whereas optimal TMD parameters proposed by Warburton [2], upon a harmonic excitation are close to the parameters obtained through calculations of the J2 objective function optimization. The solution proposed by Ren [30] suits a different kind of TMDs, where the damping element is not connected with the primary mass but with the substrate (other motion equations), hence, it is impossible to directly apply the optimal parameter formulas proposed in this study as parameters for a standard TDOF system. Furthermore, please note that optimal TMD parameters based on the H2 norm (white noise input), i.e., determined from the J1 objective function, provide similar TMD efficiency as suggested by Warburton in [3].
In the case of an analysis covering a structure including natural damping of the primary system, we can observe that transfer function module graphs, developed based on optimal Den Hartog and Warburton parameters, are no longer consistent with the graphs based on the parameters obtained through the H2 and H∞ methods, that is, the J1 and J2 objective functions. This, of course, stems from the fact that these formulas do not include natural damping 1  . Whereas, when comparing to the Matsuhis solution [31], where the optimal parameters were obtained based on the stability criterion, taking into account 1  , it should be concluded that his solution was optimal neither for harmonic excitations, as well as white noise input.
In [19], Zuo     In the course of analyzing the results shown in chapter 3, which concern the optimization of MTMDs located at the tip of a h=160m high chimney, higher effectiveness of MTMDs relative to a single TMD can be confirmed (see Table 2 and Figure 5). This is particularly clear to optimal parameters determined with the J2 objective function, where the location of the flat section of graph 41,41 H in the area of 1  can be seen increasingly lower (Figure 5b). Another issue that needs to be stressed are the graphs 41,41 H (Figure 5a) for TMD tuning in the form of white noise (H2 optimization of the J1 function) exhibit a broader damper impact range on the side of higher input frequencies.
Furthermore, Figure 11 shows a function variability graph depending on the number of TMDs. Of course, higher MTMD efficiency relative to TMDs can be observed for both objective functions, J1 and J2, applied within the numerical optimization. Furthermore, Figure 11 shows the values of individual objective functions for the optimal parameters obtained using both optimization methods, namely, H2 (J1 function) and H∞ (J2 function). It can be seen (see Figure 11a) that in the case of random excitation, which corresponds to the value of the J1 function and TMDs tuned to harmonic excitation (optical parameters based on the optimization of function J2), we get a lower MTMD efficiency. Figure 11b indicates an inverse relationship. When analyzing the results concerning the primary system mass change (see Figure 6, 7), for an approximately 5% difference in the value of 1  ("structural mistuning"), MTMDs are more effective than a single TMD. A lower effectiveness with a higher number of TMDs tuned to harmonic excitations can be observed only in the case of TMDs tuned based on J2 optimization and reducing the primary system mass (increase in frequency In the event of such a primary system mistuning, better effectiveness is exhibited by MTMDs tuned based on J1, which results from the aforementioned broadband excitation on the side of frequencies higher than 1  . It can also be concluded that a reduction in the primary system mass has a more adverse influence on the structural response and MTMD operation than mass increase. These relationships are shown in Figure 12. It can also be assumed that with a low difference (below 5%) between the calculated natural frequencies of the structure and the ones determined for a real structure, it is advisable to use MTMDs instead of a single TMD. The aforementioned analyses confirm the general conclusions included in the works by Zuo and Nayfeh [19], as well as Li and Ni [20] concerning the higher effectiveness of MTMDs relative to TMDs. Additionally, for analysis purposes, it was decided to present the results of optimal MTMD parameters in the form of optimal tuning ratio k  and optimal damping ratio TMDs, conclusions can be drawn regarding the application of the J1 objective function based on the H2 norm, and the J2 function based on the H∞ norm. As far as the tuning distribution is concerned, we can observe increased factor k  for TMDs tuned above and below the 1  frequency. The area of TMDs tuned to values beyond 1  exhibits a significantly higher growth of the value k  , especially for the parameters obtained from the J1 objective function, which is particularly well illustrated by the graphs 1 kk  − − .
If we consider the optimal tuning ratio graphs for individual t k  , we can observe a derivative characteristic, but only for the J1 function, although the increments

Conclusions
The adopted new method for developing motion equations enables adding single TMDs or MTMDs to completely different degrees of freedom of the primary system. The system of equations allows for easy MTMD tuning for complex vibrations modes, with MTMDs located in local maxima of these vibrations modes, while still analyzing the SDOF system with attached MTMDs. Locating MTMDs at the tip of the chimney provided a possibility to compare the results with studies on MTMD tuning. The next stage of the research will involve the parameter optimization in MTMDs unevenly distributed within the primary system, but still analyzed as an SDOF equivalent system.