# Establishment and Numerical Analysis of Rolling Force Model Based on Dynamic Roll Gap
## 1 Introduction  
Cold tandem rolling is widely employed in modern industries as a significant metal forming process. Its primary objective is to achieve precise control over metal strips by using continuous rolling, thereby obtaining the desired thickness, surface quality, and mechanical properties, among other attributes. Rolling force is pivotal in this process as it directly affects stability, material deformation, and product quality. The rolling process involves dynamic changes in the roll gap due to non-steady-state friction lubrication, tension fluctuations, and roll-flattening deformation. The dynamic roll gap model is crucial in formulating process regulations, enhancing product quality, and maximizing production efficiency. The rolling force model forms the foundation for high-precision control of the rolling process and is the core component of the dynamic roll gap model. Therefore, researching a rolling force model which accurately reflects industrial production is an inherently valuable task [1, 2]. The cold tandem rolling process is a complex multi-parameter coupling process with many unstable factors. Numerous studies have been conducted from the perspectives of physics and mathematics on rolling-force modeling of the cold rolling process, domestically and overseas. Many scholars have studied the influencing mechanism of rolling force on lubricating friction and stress factors. [PERSON] et al. [3] developed the total friction stress model with mixed lubrication, and the unit pressure differential equation of thelinearized rolling parts proposed the rolling force model of a cold-rolled strip under mixed lubrication friction. Based on proper mathematical and physical techniques and the unstable hydrodynamic lubrication theory, [PERSON] et al. [4] established a fundamental dynamic roll gap rolling force model. [PERSON] et al. [5] quantitatively analyzed the interaction of unsteady lubrication film thickness, compressive stress, and friction stress in the working area of strip rolling process. [PERSON] et al. [6] used the principle of equivalent interface layers to synchronize the surface roughness deformation with the lubrication effect, obtained the shape change of the micro-working interface of rolls and strips, and came up with a new multi-scale approach to study the pressure and friction of the working interface in the three-dimensional model of cold rolling. [PERSON] et al. [7] proposed that a transition zone between the intake area and the work area is necessary for the strip cold rolling process, established a geometric model of the rolling strip process with the transition zone, and discussed the rolling force by combining the lubrication friction of the intake area, transition area, and work area. Some scholars have also studied the influencing mechanism of rolling force on lubrication friction and tension factors. [PERSON] et al. [8] proposed a connection between rolling force and tension variation, and analyzed the interpretation of the thinning amount and strain hardening, unit friction force, circumferential speed of the work roll, tension, and rolling force. [PERSON] et al. [9] discussed the area where the friction force in the opposite direction is formed in the rolling process, determined that the changes in the circumferential speed of the work roll, the tension at the end of the rolled piece, and other rolling parameters are related to the length of the area, and gave the functional form of the length coefficient of the area, thus improving the influencing factors of the rolling force model. Some scholars have used finite element analysis and experimental means to study the rolling force model. [PERSON] et al.