## 2 Methods of Analysis
### Dynamic Stability Analysis  
In the general case, the study of the structural instability problems leads to non-linear issues, which are solved by iterative or incremental-iterative analysis of large displacement gradients. However, in the case of dynamic instability analysis, the nature of motion is studied. A quasi-linear approach is sufficient for determining the conditions under which the motion is of an unsteady nature, with solutions that increase indefinitely with time. Admittedly, the determination of the magnitude of the amplitudes of these oscillations can only be obtained from non-linear equations of vibration, no less; without knowing the magnitude of the amplitudes, quasi-linear theory gives a sufficiently complete and accurate view of the issue of instability. The equation of motion with time-varying coefficients with the inclusion of a periodic load (1) takes the following form:  
\[\mathbf{M}\ddot{\mathbf{q}}(t)+[\mathbf{K}_{L}+\mathbf{K}_{G}(\mathbf{S})+P(t )\mathbf{K}_{GN}(\mathbf{N})]\mathbf{q}(t)=0, \tag{12}\]  
where \(\mathbf{M}\in\mathbb{R}^{m\times\mathbf{m}}\) is a consistent mass matrix, and \(\ddot{\mathbf{q}}(t)\in\mathbb{R}^{m\times 1}\) is an acceleration vector.  
The boundaries of the stable and unstable regions (Ince-Strutt maps) are determined using the periodic solutions of the equation of motion (12). The problem of dynamic instability analysis leads to determining the conditions under which Equation (12) has nonzero solutions. In this paper, the dynamic instability analysis is carried out using the harmonic balance method (for a detailed description, see [23; 25]). This method leads to the boundaries of the first, second, third, etc., resonance regions. The focus of this consideration is the first (main) resonance region. The equation sufficient to do this takes the following form:  
\[\text{det}\bigg{\{}\mathbf{K}_{L}+\mathbf{K}_{G}(\mathbf{S})+\bigg{(}1\pm \frac{1}{2}\frac{P_{t}}{P}\bigg{)}\mathbf{K}_{GN}(\mathbf{N})-\frac{\theta^{ 2}}{4}\mathbf{M}\bigg{\}}=0. \tag{13}\]  
Figure 2: Process for determining the stiffness of Geiger domes.  
Based on Equation (13), the main unstable regions \(A_{\eta}(S_{i})\) for the \(i\)-th initial prestress level \(S_{i}\) are determined in the plane of a pulsatility index \(v\) and a resonance frequency \(\eta\):  
\[v=\frac{P_{t}}{P},\ \ \eta=\frac{\theta}{2\pi}. \tag{14}\]  
For quantitative assessment, the nondimensional parameter  
\[\lambda=\frac{A_{\eta}(S_{i})}{A_{\eta}(S_{min})} \tag{15}\]  
is introduced, which expresses the ratio of the area of the unstable region at the \(i\)-th level of initial prestress \(S_{i}\) to the area of the unstable region at the minimum initial prestress level \(S_{min}\). The range of parameter \(\lambda\) is defined as a value between 1 and 0. In the case of \(\lambda=1\), there is the potential for the excitation of unstable motion, whereas in the case of \(\lambda=0\), such a risk is absent. Additionally, from the perspective of structural durability, the characteristics of the changes in the parameter with growth in the initial prestress level are important. The best solution is obtained when the parameter \(\lambda\) decreases exponentially as the initial prestress level increases.