(untitled) (Multidisciplinary Journal of Science and Technology, 2026)
In this work, the issue of analytically verifying the priority of vibrations of a mechanical system with elastic dissipative characteristics of the hysteresis type under the influence of random parametric excitations together with a dynamic damper is considered, depending on the design parameters of the system. In this case, solutions are sought in the form of exponents, a system of equations in normal form is formed, and the system of motion differential equations is averaged using the averaging method. The system of motion differential equations of the vibration-protected system is brought to the system of Ito differential equations. Based on the solutions obtained in the form of exponents, characteristic equations and their roots are determined. According to the main theorem of priority theory, analytical expressions of priority conditions are obtained based on the roots of characteristic equations. The boundaries of the priority and unstable regions are determined and analyzed in an analytical form.