## 2 Methods
### Determination of a Transfer Matrix Used to Determine the Objective Function in Optimization Issues  
For a model described with the general motion equation, which is a system with \(N\) degrees of freedom, the transfer matrix \(\mathbf{H}(s)\), the single element of which \(H_{ik}(s)\) is called a transfer function, is a Laplace transform of the impulse response \(h_{ik}(t)\). \[H_{ik}(s)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-st}dt, \tag{30}\]  
where impulse response \(h_{ik}(t)\) is the \(i\)-th response of the system to the \(k\)-th excitation in the form of a unit impulse function \(\delta_{k}\) (Dirac delta) applied at the initial moment \(t=0\), \(j\) means an imaginary unit, and \(s\) is complex variable \(s=\sigma+j\omega\). Therefore, in order to determine impulse responses in the \(i\) points of the system, the excitation shall be adopted in the following form:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]  
In the specific case in which \(\sigma=0\), the input is a complex integral \(e^{-j\omega t}\) at frequency \(\omega\) and \(H_{ik}(j\omega)\), viewed as function of \(\omega\), is known as the frequency response function (FRF) of the system and is given by the Fourier transform. The below Equations (32) and (33) show a pair of Fourier transforms, which can be found in the work by [PERSON] [30]:  
\[H_{ik}(j\omega)=\int_{-\infty}^{\infty}h_{ik}(t)e^{-j\omega t}dt, \tag{32}\]  
\[h_{ik}(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}H_{ik}(j\omega)e^{j\omega t}d\omega. \tag{33}\]  
By substituting the excitation force vector (31) to the Equation (1) and after applying the Fourier transform we get:  
\[(\mathbf{K}-\omega^{2}\mathbf{M}+j\omega\mathbf{C})\mathbf{H}(j\omega)= \mathbf{I}. \tag{34}\]  
In general, \(H_{ik}(j\omega)\) is the complex quantity and can be presented as:  
\[H_{ik}(j\omega)=|H_{ik}|e^{j\Theta_{ik}}=\text{Re}H_{ik}+j\text{Im}H_{ik}, \tag{35}\]  
\[|H_{ik}(j\omega)|=\sqrt{\left(\text{Re}H_{ik}\right)^{2}+\left(\text{Im}H_{ik }\right)^{2}},\text{ arg}H_{ik}=\Theta_{ik}=\text{arctg}\frac{\text{Re}H_{ik}}{\text{Im}H_{ik}}, \tag{36}\]  
where \(|H_{ik}(j\omega)|,\)\(\Theta_{ik}\) are the frequency response function module and argument, respectively. If we assume the denotation of the dynamic stiffness matrix in the form:  
\[\mathbf{G}(j\omega)=\mathbf{K}-\omega^{2}\mathbf{M}+j\omega\mathbf{C}=\text {Re}\mathbf{G}+j\text{Im}\mathbf{G}, \tag{37}\]we get a complex equation:  
\[\text{Re}\mathbf{G}\text{Re}\mathbf{H}-\text{Im}\mathbf{G}\text{Im}\mathbf{H}+j[ \text{Im}\mathbf{G}\text{Re}\mathbf{H}+\text{Re}\mathbf{G}\text{Im}\mathbf{H}]= \mathbf{I}, \tag{38}\]  
and after breaking down, two real equations:  
\[\text{Re}\mathbf{G}\text{Re}\mathbf{H}-\text{Im}\mathbf{G}\text{Im}\mathbf{H} =\mathbf{I}, \tag{39}\]  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]0  
Ultimately, we get a matrix of the real and imaginary section of the full frequency response matrix \(\mathbf{H}(j\omega)\):  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]1  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]2  
Of course, with an appropriate computational procedure, we can directly determine a complex frequency response matrix from the formula:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]3  
Knowing the frequency response matrix, it is possible to determine the structural response within the frequency domain in the form of:  
\[\mathbf{p}(t)=[0,0,\ldots,0,\delta_{k}(t),0,\ldots,0]^{T}. \tag{31}\]4  
where \(\mathbf{q}(j\omega)\), \(\mathbf{p}(j\omega)\) are the Fournier transforms of the displacement and vector of excitation forces.