# Establishment and Numerical Analysis of Rolling Force Model Based on Dynamic Roll Gap
## 2 Theoretical Analysis of the Rolling Force Modeling  
Accurately calculating a robust rolling force model is essential for constructing cold rolling schedules, establishing rolling mill operating parameters, and developing industrial production process monitoring and management systems. The rolling force model has been extensively studied by researchers both domestically and internationally, providing valuable insights for its application in industrial production. However, to enhance the applicability and robustness of rolling force models, it is crucial to deepen our understanding of the scientific logic behind model establishment. This will enable us to clarify the contextual relevance of various models used in different working conditions [17].  
Cold rolling production is a complex process characterized by multiple parameters with unsteady coupling effects and numerous uncertain factors. To develop a rolling force model that accurately represents the technological process, it is common to simplify and appropriately assume the conditions and control parameters of the industrial production process from the perspective of physics and mathematics. Currently, mathematical and physical model theories exist that can precisely describe the behavior of rolling force, such as the Karman unit pressure differential equation, the Orowan unit pressure equilibrium differential equation, the Stone rolling force calculation formula, and the Bland-Ford-Hill rolling force model.  
The Karman unit pressure differential equation is applicable under the following conditions: the rolling process involves pure plane deformation without significant spreading, there is no shear stress along the height direction of each cross-section in the deformation zone, and the horizontal stress is uniformly distributed. It assumes that the principal stress directions align with the rolled piece's longitudinal, transverse, and vertical directions. The equation considers the element body as the focus of the study. It assumes a constant friction coefficient for the contact arc, neglecting the elastic deformation of both the roll and the rolled piece. Furthermore, it takes equal force exerted by the element body on the width of the rolled piece. Establishing the Orowan unit pressure equilibrium differential equation primarily considers pure plane deformation and negligible spreading of the strip. It replaces the friction stress on the contact surface between the roll and the rolled piece with shear stress. [PERSON]'s rolling force calculation formula is applicable when the thickness of the rolled piece is significantly smaller than the radius of the roll, and the rolling pressure is substantial. It approximates the cold rolling process of the rolled piece as plate upsetting, without accounting for elastic deformation or the elastic contact arc of the rolled pieces. The Bland-Ford rolling force model, derived from the traditional Bland-Ford-Hill cold rolling force formula, is rigorously derived in theory and extensively utilized in practical cold rolling manufacturing. This calculation method is applied in the current study.