## 2 Methods  
Original Proposal of 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 excitation force 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 \(\omega_{i}\) 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 \(\mathbf{C}\) damping matrix and a main structure is treated as a linear time-invariant system (LTI system). It is a system that produces an output from any input signal subject to linearity and time constraints. In general, for \(N\) degrees of primary system, the motion equation solution:  
\[\mathbf{M}\mathbf{\tilde{q}}(t)+\mathbf{C}\mathbf{\tilde{q}}(t)+\mathbf{K} \mathbf{q}(t)=\mathbf{p}(t) \tag{1}\]  
expands into a series of eigenvectors  
\[\mathbf{q}(t)=\sum_{i=1}^{N}\mathbf{a}_{i}\psi_{i}(t)=\mathbf{W}\psi(t), \tag{2}\]  
where \(\psi(t)\) is the modal coordinate vector and \(\mathbf{a}_{i}\) is \(i\)-th eigenvector, while \(\mathbf{W}\) is the matrix corresponding mode shapes in its columns. If we include the \(N_{i}\) mode of the vibrations, which have the greatest share in the vibration system, the above equation can be expressed as:  
\[\mathbf{q}(t)=\sum_{i=1}^{N_{i}}\mathbf{a}_{i}\psi_{i}(t). \tag{3}\]  
If we normalize the eigenvectors in the manner shown below, we get the following motion equations:  
\[\widetilde{m}_{i}\ddot{\psi}_{i}(t)+\widetilde{c}_{i}\dot{\psi}_{i}(t)+ \widetilde{k}_{i}\psi_{i}(t)=\widetilde{p}_{i}(t), \tag{4}\]  
where \(i=1,2,\dots N_{i}\), \(\widetilde{m}_{i}=\mathbf{a}_{i}^{T}\mathbf{Ma}_{i}=1\), \(\widetilde{c}_{i}=\mathbf{a}_{i}^{T}\mathbf{Ca}_{i}=2\widetilde{c}_{i}\omega_ {i}\), \(\widetilde{k}_{i}=\mathbf{a}_{i}^{T}\mathbf{Ka}_{i}=\omega_{i}^{2}\), and \(\widetilde{p}_{i}(t)=\mathbf{a}_{i}^{T}\mathbf{p}(t)\). Designations as in Figure 1 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. The following designations for the diagram above were adopted: \(j\) = 1,2 \(\dots\)\(N\)--system degrees of freedom; \(I\) = 1,2 \(\dots\)\(N_{i}\)- number of vibration modes taken into account (further vibration eigenmodes not necessarily included in the analysis)--\(i\) index means any subsequent mode number included; and \(q_{i}(t)\)--displacement of \(j\)-th degree of freedom. 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:  
\[q_{j}(t)=\sum_{i=1}^{N_{i}}a_{ji}\psi_{i}(t), \tag{5}\]  
where \(\psi_{i}(t)\) is the modal coordinate associated with the \(i\)-th vibration eigenmode, \(a_{ji}\) is the coordinate of the \(i\)-th vibration mode of the \(j\)-th degree of freedom, and \(N_{i}\) is the 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 \(\mathbf{a}_{i}\) mode shape or the point of application of concentrated load within the system, including, e.g., force originating from a mechanical vibration damper.