Static equilibrium model and analysis of rope rescue systems

In rope technique training or practical rescue operations, accurately calculating the equilibrium positions of rope systems and the loads borne by the system is of significant importance for assessing safety risks in emergency rescue missions and optimizing the configuration of fall protection equipment. Addressing the static equilibrium problem of rope systems under typical conditions, this study establishes a static equilibrium model for rope systems based on the three-force concurrent principle, obtaining high-precision analytical solutions for equilibrium intervals and equipment forces. Research indicates that whether a rope system achieves equilibrium and the range of equilibrium positions are directly related to the pulley friction coefficient, anchor point coordinates, and pulley positions. Under specific conditions, simultaneous satisfaction of the subsystem equilibrium test function inequalities and constraints between subsystems ensures the system reaches equilibrium, and the forces on each piece of equipment can be determined accordingly. Additionally, the pulley friction coefficient affects the solution interval: a higher friction coefficient widens the solvable interval, making equilibrium easier to achieve.