Dynamic response and sliding mode control of a cold rolling mill subjected to harmonic and Gaussian colored noise excitations

During operation, cold rolling mills are susceptible to the coupled effects of random excitations and structural nonlinearities, which can induce complex dynamic behaviors that adversely affect rolling quality and equipment safety. This paper studies the structural dynamic characteristics and vibration suppression for a two-degree-of-freedom cold rolling mill vertical structure model under combined harmonic and random excitation for the first time. Firstly, an averaging method and a stochastic method are extended to derive the amplitude-frequency and steady-state response equations, respectively. Secondly, the response shows the mill exhibits nonlinear hard spring characteristics and bistability in the resonance region. The coexistence and evolution of low- and high-amplitude attractors are further elucidated via the equivalent potential energy diagram and basin of attraction. Additionally, random excitation is a key factor inducing chaotic behavior in the rolling mill. Finally, Gaussian colored noise induces stochastic switching, stochastic P- and D-bifurcations. This can lead to defects in the rolled products, and in severe cases, it may even threaten the safe operation of the rolling mill. To suppress this catastrophic switching, this paper innovatively introduces the improved double power exponential reaching law to design sliding mode control, achieving faster convergence, suppressing chattering and reducing energy consumption. The proposed control has been rigorously proven to be stable and has been effectively verified through numerical simulations. The research findings provide essential theoretical foundations and technical support for the safe design and manufacture of vertical structural models for cold rolling mills in engineering practice.