## Formulation of the Problem  
### Optimization of the Selected Dynamic Parameters of Laminate Structures  
The dynamic behavior of structures whose numerical model is known can be described by the following equation:  
\[\mathbf{M}\dot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}= \mathbf{P}, \tag{1}\]  
where \begin{tabular}{l l}
**M** & is a mass matrix, \\
**C** & is a damping matrix, \\
**K** & is a stiffness matrix, \\
**x** & is a vector of nodal displacements, and \\
**P** & is a vector of external forces at nodes, \\ \end{tabular}   
while \(\ddot{\mathbf{x}}\) and \(\dot{\mathbf{x}}\) indicate the second and the first derivative of \(\mathbf{x}\) after time \(t\), respectively. For linear models, such as those considered in this paper, matrices \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are constant over time \(t\), while the vector of external forces (excitations) \(\mathbf{P}\) and, obviously, the vector of nodal displacements \(\mathbf{x}\) are variable. If excitation forces do not occur, i.e., \(\mathbf{P}\equiv(0)\), and the damping is neglected, i.e., \(\mathbf{C}\equiv(0)\), then Equation (1) is simplified to  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{0}. \tag{2}\]  
This equation leads to the generalized eigenproblem [30],  
\[\mathbf{K}\mathbf{\Phi}=\mathbf{M}\mathbf{\Phi}\mathbf{\Omega}^{2}, \tag{3}\]  
where matrix \(\mathbf{\Phi}\) consists of eigenvectors \(\phi_{i}\) (in columns), and \(\mathbf{\Omega}\) is a diagonal matrix with natural frequencies \(f_{i}\) adequate for the vectors \(\phi_{i}\). Knowing the values of natural frequencies \(f_{i}\) is crucial to avoiding the _resonance_ phenomenon, which occurs when, for any \(i\), \(f_{i}\approx p\) (\(p\) is the frequency of an excitation load). Resonance is an extremely dangerous phenomenon for engineering structures since it leads to structure oscillations with increasing amplitude (theoretically, amplitude can increase to infinity where undamped systems are concerned). In turn, these oscillations cause serious structural failures. One of the most popular methods used to avoid resonance is to change the structure's geometry, stiffness, and/or mass (see Equation (3)) in order to offset the natural frequencies from the excitation force frequency \(p\). For engineering structures made of laminates, there is a unique approach that cannot be applied to isotropic materials. The stiffness \(\mathbf{K}\) can be altered without any change to the structure's geometry or mass, and the stiffness can be changed by changing the lamination fiber angles. As described by [PERSON] and [PERSON] in [31], changes in the lamination angles have a serious impact on natural frequencies. Although some natural frequencies increase and others decrease, in both cases, the changes are relatively big [31]. The avoidance of resonance for a known frequency of the excitation force should, therefore, be possible only by changing the stacking sequence. However, although it would be relatively easy to choose a lamination angle for one or two layers, it is not a trivial matter for three, four, or more layers. This paper presents a novel method for selecting lamination angles to prevent the appearance of the resonance phenomenon using the optimization approach. The optimization task described above can be written as  
\[\boldsymbol{\Lambda}^{*}=\operatorname*{arg\,min}_{\boldsymbol{\Lambda}\in \mathbb{L}^{n}}\left\{\mathrm{g}(\boldsymbol{\Lambda})\right\}, \tag{4}\]  
where \(\boldsymbol{\Lambda}\) is a vector of lamination angles, \(\boldsymbol{\Lambda}=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}\); \(\mathbb{L}^{n}\) is an \(n\)-dimensional space of possible values of \(\boldsymbol{\Lambda}\) (herein defined by \(\lambda^{\text{min}}\) and \(\lambda^{\text{max}}\), the lower and upper limits of \(\lambda_{i}\)); and \(\mathrm{g}(\boldsymbol{\Lambda})\) is an objective function to be minimized over the possible values of lamination angles \(\lambda_{i}\).