Stochastic quadratic eigenanalysis of dynamical and gyroscopic systems

Stochastic quadratic eigenvalue analysis (SQEA) is essential in structural dynamical and gyroscopic systems. In structural dynamics, variability in material properties and geometric imperfections introduce uncertainties in mass and stiffness matrices. SQEA quantifies how these uncertainties influence natural frequencies, damping ratios and mode shapes. Since structural resonances are highly sensitive to parameter changes, SQEA predicts probabilistic frequency shifts to avoid resonance under uncertain conditions. For safety-critical structures, failure modes such as buckling and instability depend on eigenvalues that vary with parameters. SQEA evaluates the probability of these failures and supports reliability-based design. Gyroscopic systems such as rotating machinery and spacecraft exhibit unique dynamics due to gyroscopic effects and are inherently governed by quadratic eigenanalysis, with the gyroscopic term introducing a skew-symmetric matrix. Such systems are prone to flutter, divergence and other instabilities driven by eigenvalue shifts. Uncertainties in rotational speed, damping, stiffness or unbalance may move eigenvalues toward instability. SQEA quantifies these risks. Despite its importance, SQEA is computationally demanding, especially for large-scale systems with high-dimensional random inputs. Balancing accuracy and efficiency is therefore crucial. We introduce two numerical techniques: one transforms the stochastic quadratic eigenvalue problem (SQEP) into a generalized stochastic linear eigenvalue problem, while the other solves SQEPs directly. The methods scale well for large systems and are insensitive to stochastic dimension, allowing hundreds or thousands of random inputs. Applications to civil structures and rolling tires demonstrate their effectiveness, and the approach also extends to stochastic responses of dynamical and gyroscopic systems through stochastic modal decomposition.