Coupled Vertical–Horizontal Vibration Characteristics of a Cold Rolling Mill Based on an Optimized Karman Differential Equation

In the process of strip rolling, mill vibration induces vertical and horizontal displacements in the rolls and rolled parts, affecting the accuracy of the rolling analysis model. Constant changes occurred in the rolling zone between the upper and lower working rolls due to mill vibration, resulting in slight vertical and horizontal displacements of the rolled pieces. These displacements, subsequently, affected the precision and accuracy of the rolling analysis model. A dynamic rolling force optimization model was established in this paper based on the Karman differential equation, metal flow equation, and mixed lubrication friction model. This model took into account the small displacements in both vertical and horizontal directions of the rolled parts, effectively addressing the issue of rolling area variation. A vertical–horizontal coupling vibration model for the cold rolling mill was developed, employing the dynamic rolling force model and lumped mass method. The accuracy of the dynamic rolling force model was validated, and a comprehensive examination of the vibration mechanism of the rolling mill, including exploration of suppression methods, was conducted. The amplitude-frequency response of the coupled vibration system was determined using the multiple scales method, and the effects of external excitation and mill structure parameters on the coupled vibration characteristics were analyzed. The results indicated that the dynamic rolling force optimization model had considered the variations in rolling parameters and could explore the complex vibration patterns of the rolling mill itself from the perspective of varying rolling parameters, effectively addressing the issue of rolling region changes. Furthermore, it exhibited high precision in analyzing vertical and horizontal coupling vibrations in the rolling mill. The simulation results indicated that the primary cause of distance vibrations between the rollers was internal resonance triggered by similar external excitation frequencies and derived frequencies coupled in both vertical and horizontal directions. This was subsequently followed by the movement displacement of the hydraulic cylinder piston and changes in coupling parameters, which had a significant impact on the amplitude and resonance region of the vibration system.