Abstract
In this paper, a mathematical model of a controlled system is investigated, created on the basis of a fourth-order differential equation widely used in various fields of science and technology. The problem of managing the behavior of structural elements has been solved. The mechanism of transition from one system to another is considered using the analysis of natural frequencies. The rod can be fixed in different ways (termination, hinge locking, elastic termination, floating termination, free end) [1]. If the ends of the rod are fixed so that resonant vibration frequencies are generated, then the question arises: is it possible to change the fastening of the rod so as to indicate a safe range for controlling the natural frequencies. The question posed by us gives rise to many others, more specific. For example, is it possible to determine how the ends of the bar are fixed by the natural vibration frequencies of the bar? Are they springs, sealed or loose? Such applications are very important especially when the first natural frequency generates a resonance situation. It is necessary to control the natural frequencies so that the system does not receive the first natural frequency for safe operation. The main result is formulated as a theorem. The stress-strain state (SSS) control has been developed for rods with elastic fastening on the left and hinged on the right. The uniqueness theorem for boundary conditions is proved using the analysis of natural frequencies.