## 1 Introduction
[36] revealed that the stability of layered rock mass was controlled by the external force, the modulus of the elasticity, the severity of rock mass, the inter layer shear stress, and the geometry based on the cusp catastrophe theory. [PERSON] et al. [37; 38] established the cusp catastrophe model on the basis of the energy theory, pointing out that the instability of the support system was caused by the incompatibility of energy release. The results showed that the catastrophe theory can explain the nonlinear dynamic characteristics of the instability of a coal-rock system. The deformation of the immediate roof accumulates elastic energy under the load of the overlying rock layer, and when the energy accumulates to a certain extent, it will be released instantly, resulting in the step sinking of the main roof. The crushing of the support and the ribbing of the coal wall have obvious instantaneity and abruptness. Therefore, the catastrophe theory is introduced into the stability analysis and parameter calculation during the first weighting of the working face to solve the problem of sudden disaster in coal mining. In the system of main roof-immediate roof-support, the deformation of the main roof is generally given, and the influence degree of the support on the main roof and the effect degree of the main roof on the support depend on the mechanical properties (stiffness) of the immediate roof medium and the yield ability of the support. The key point of roof control is the control of the immediate roof, and the role of immediate roof in the relationship between the support and surrounding rock is of vital importance [21; 22; 39]. According to the unstable collapse process and pressure characteristics of the main roof structure of large mining height, the main roof will break and rotate, compressing the immediate roof and support when the immediate roof and support system are not enough to support the main roof. When the deformation of the immediate roof rock exceeds its deformation critical value, the immediate roof will be crushed, and the main roof will slide along the coal wall, resulting in a wide range of roof cutting, support crushing, and step sinking (Figure 1) [40]. Therefore, the immediate roof rock mass controls the instability of the main roof, and the key point to control the large-scale pressure of the main roof is to control the instability of the immediate roof under the abutment pressure. Therefore, the main objective of this study was to explore the general process and weighting characteristics of the main roof along the coal wall during the first weighting of large mining height in shallow coal seam, to analyze the mechanical mechanism of the main roof instability, to reveal the disaster-causing factors, and to explore the determination method of support resistance to ensure the stability of working face. In this study, according to the structural characteristics of the main roof cutting and instability along the coal wall during the first weighting, the system mechanical model consists of the main roof, immediate roof and support is established. This model is based on catastrophe theory and the energy conservation principle analysis method, as well as through considering the nonlinear characteristics of the failure and instability of the immediate roof. A new analysis method was proposed to study the mechanical mechanism of roof cutting along a coal wall during the first weighting. The instability process was analyzed theoretically, and the sufficient and necessary conditions for the sudden instability of the system were successfully obtained. The main influencing factors of the system catastrophe, such as stiffness ratio \(K\), rock material parameters, external load \(Q\), and weighting interval \(l\) are discussed. We also compared and analyzed the effects of elastic modulus \(E\), support stiffness \(k_{1}\), cross-sectional area \(a\), and immediate roof thickness \(H\) on the stiffness ratio \(K\).