Applying mathematical models and numerical methods is crucial for describing and simulating the metal cold-rolling process, wherein the accurate prediction of rolling force is an effective way to improve the quality of rolled sheets. This paper considers key influencing parameters such as friction lubrication, stress, tension, and roll-flattening radius during the rolling process and establishes a calculation model for the friction coefficient and roll-flattening radius. By considering the coupling effect of the dynamic roll gap on rolling force, a rolling force model for non-steady-state friction lubrication during the rolling process is obtained. The correctness of the proposed model is verified by comparing it with industrial measurement results. The influences of the friction coefficient, stress, tension before and after rolling, and roll-flattening radius on rolling force are quantitatively studied. The results show that the rolling force increases with an increase in the friction coefficient. When the friction coefficient exceeds 0.2, the rate of increase slows down, approaching dry friction conditions. The rolling force increases linearly with stress but decreases with increasing tension before and after rolling. The rolling force model, considering the roll-flattening radius, provides numerical calculation results that are closer to an industrial measured rolling force. This work contributes to a better understanding of the mechanism behind the improvement of the cold rolling process.

A Smith predictor‐Taylor series‐based LQG (STLQG) control to compensate time delay of a semiactive suspension system is newly presented. This control consists of a Taylor series‐based LQG (TLQG) control and a Smith predictor based on the TLQG. The TLQG control compensates one half of time delay to decrease magnification from whole time delay compensation. The Smith predictor based on the TLQG compensates the other half to decrease horizontal shift from whole time delay compensation using the Smith predictor‐based LQG. Finally, a practical case illustrates advantages of the STLQG control.

To reveal the influence of surface morphology characteristics in mixed lubrication on the contact characteristics of the rolling interface, a random three-dimensional rough surface model based on Gaussian distribution theory was established. The model utilizes the finite element method (FEM) to simulate the regular contact and tangential sliding behavior of micro-asperities at the rolling interface in mixed lubrication conditions. The connection bearing capacity of models with varied roughness in mixed lubrication was studied. Furthermore, the effect of various sliding and normal indentation amounts on the normal and friction stress was investigated. The simulation result reveals that the roughness of the surface influences the distribution of the lubricating oil film. The lubricating oil layer between the interfaces with a lower roughness has a higher bearing capacity due to its more uniform distribution of peaks and valleys. An increase in the normal indentation amount raises the friction stress and normal stress. In contrast, an increase in sliding lowers the normal pressure, substantially impacting the fluctuation of the friction coefficient dramatically. Finally, the random three-dimensional rough surface model is verified by comparing it with the experimental data in the related literature.

Ubiquitiform is a new theory of finite-order self-similar physical structure and it is more reasonable to describe real engineering surfaces by ubiquitiform rather than fractal. In this paper, by introducing the frequency truncation criterion, a new analytical expression of the two-dimensional W–M function based on the ubiquitiform theory is firstly derived and constructed and the two-dimensional ubiquitiformal curve characterization under different contact characteristic parameters is achieved. On this basis, the anisotropic three-dimensional surface W–M function with ubiquitiformal features is constructed, and the evolution law of the anisotropic three-dimensional surface morphology under the regulation of the ubiquitiformal complexity is investigated. Then, an improved adaptive box counting algorithm is proposed, and the lower limit of the metric scale in the self-similarity region of the asperities on the rough surface is determined and then the computation method of the ubiquitiformal complexity is established. At last, the validity and accuracy of the method are confirmed by the Koch curves. Key findings include: (1) higher ubiquitiformal complexity D corresponds to increased surface irregularity and complexity; (2) the characteristic scale factor G affects surface height only; (3) reducing the lower limit of metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ increases surface undulation frequency, revealing finer details. This research provides a rationale and quantitative guidance for the matching design of critical joint interfaces in modern precision machinery.

To address the vertical nonlinear vibration of a rolling mill under dynamic rolling force, a vibration model coupling piecewise nonlinear stiffness constraints and dynamic rolling force is constructed. The mathematical expression of the dynamic rolling force is derived based on the Bland-Ford-Hill rolling theory. The harmonic balance method and numerical integration of the nonlinear equation of motion were employed to analyze the three-dimensional amplitude–frequency characteristics of the key parameters. The Melnikov method was adopted to determine the critical conditions for chaos onset in the sense of the Smale horseshoe, and the theoretical threshold was validated using phase trajectories, Poincaré sections, bifurcation diagrams, and basins of attraction. The results show that the linear stiffness of the press-down cylinder dominates the resonance characteristics, whereas the increased nonlinear stiffness amplifies both the response amplitude and offset. A higher damping ratio effectively suppresses resonance, whereas larger rolling force fluctuations reduce the system stability. The Melnikov threshold closely matches the dissipation threshold from numerical simulations, confirming that the system enters chaos through period-doubling bifurcations. Bi-parameter response analysis indicates that optimizing the linear stiffness of the press-down cylinder is a priority for enhancing the system stability, and moderate nonlinear stiffness further aids in vibration control. Additionally, the increased rolling force fluctuations induce potential well transitions and limit cycle splitting, thereby intensifying the nonlinear vibration behavior.

Friction at the cold rolling interface is affected jointly by the surface roughness, lubrication state, local pressure, and relative sliding. A constant friction coefficient is therefore insufficient to describe its non-uniform distribution along the contact arc. Accordingly, this study proposes a macro–micro two-scale mixed-lubrication and dynamic friction model based on the measured roll roughness. First, the measured roll roughness profile was represented within a finite effective scale interval by a scaled and truncated Weierstrass–Mandelbrot (W–M) function. The parameters D and G were obtained as finite-scale W–M roughness parameters and were introduced into a mixed-lubrication load-sharing model to calculate the local mixed-lubrication friction coefficient. The pressure distribution along the contact arc was calculated using the Karman equation, and the local macroscopic pressure was mapped to a representative microscopic contact load. Finally, the mixed-lubrication friction coefficient was used to calibrate the dynamic friction factor separately in the forward-slip and backward-slip zones, and the friction stress distribution along the contact arc was calculated. For the selected effective scale interval and preprocessing procedure, the fitted W–M roughness parameters were D = 1.528 and G = 9.15 × 10⁻⁸ m. The W–M parameter D had a more significant influence on the mixed-lubrication friction coefficient and load-sharing behavior than the scale parameter G. Increasing the rolling speed strengthened the oil-film load-carrying effect and reduced the equivalent interfacial friction coefficient. The friction stress was positive in the backward-slip zone and negative in the forward-slip zone, with a direction reversal near the neutral point. Field forward-slip inversion showed that both the simulated and measured equivalent friction coefficients decreased with increasing rolling speed, with a difference of approximately 0.009~0.017. The proposed model can capture the main trend of cold rolling interfacial friction with variations in the rolling speed and contact state.