The paper presents the results of tests on dynamic stability of Bernoulli-Euler beam with damages. Damages (cracks) were modeled using three rotational springs. An analysis of the influence of crack depth and their position relative to the beam ends on dynamic stability of the beam was carried out. The problem of dynamic stability was solved by applying the mode summation method. Applying an orthogonal condition of eigenfunctions, the dynamic of the system was described with the use of the Mathieu equation. The obtained equation allowed the dynamic stability of the tested system to be analyzed. Stable and unstable solutions were analyzed using the Strutt card.

The study analyzed the influence of periodic and aperiodic stiffness distribution for the four-element Bernoulli-Euler beam on the first two eigenfrequencies and the dynamic stability of the system. The influence of increasing the ratio of cross-sections of the analyzed elements was also analyzed. Significant differences were found in eigenfrequencies and dynamic stability. Using the variational Hamilton principle, the equation of motion was derived, on the basis of which the values of the eigenfrequencies were determined, and the transformation into the form of the Mathieu equation made it possible to determine the dynamic stability for the analyzed structures.

The study analyzed the influence of materials and different types of damping on the dynamic stability of the Bernoulli-Euler beam. Using the mode summation method and applying an orthogonal condition of eigenfunctions and describing the analyzed system with the Mathieu equation, the problem of dynamic stability was solved. By examining the influence of internal and external damping and damping in the beam supports, their influence on the regions of stability and instability of the solution to the Mathieu equation was determined.