Synchronization and frequency capture in a system of two coupled oscillators of reversible cold rolling mills

Selfoscillation may occur at high speeds in multicellular cold rolling mills. As a rule, this happens on the last two squares, the third fourth or the fourth fifth. The rolling speed should be greater than 1000 m/min. The occurrence of selfoscillations is accompanied by phase synchronization of vibration displacements of the mill's crates. The source of the oscillations are nonconservative forces in the contact zone of the mill's working rolls. A system consisting of two oscillators connected by an elastic coupling is chosen as the model. This is a classic problem in which the coupling is usually considered small. Synchronization of two oscillators can occur both at the fundamental harmonic frequency and at a multiple of three. The digital twin of the mill was obtained using the MathCAD package. The conditions for the occurrence of selfoscillations, the bifurcation of the birth of the limit cycle, are considered in a wide range of values of the elastic coupling between the oscillators. The oscillators selfoscillate under the influence of nonzero initial conditions or external influences. The forces generated by the working rolls when rolling a strip of metal can act as an external influence. Regardless of the frequency of the disturbance, the frequency of selfoscillations corresponds to the natural frequency of the oscillator. The influence of the rolled strip on the conditions of selfoscillation was carried out in a wide range of variations in tension forces. In the entire tension range of the band connecting the oscillators, selfoscillations occur at the same frequency. As the tension increases from zero to a certain critical value, the amplitude increases smoothly on the decoiler and decreases on the mill crate. With a further increase in tension, the oscillators, the mill and the uncoiler, oscillate with the same amplitude. Under the external influence of the harmonic force generated by the mill cage, selfoscillations occur with a steady limiting cycle. When the critical tension level is reached, the system of two oscillators connected by an elastic bond turns into an aperiodic system. During the operation of the rolling mill, the actual tension can vary from 100 to 170 kN. The maximum tension level does not exceed 220 kN. By changing the tension level, the vibration state of the rolling mill can be stabilized.