An analysis method for rope-driven multibody systems with pulley blocks

For rope-driven systems containing pulley blocks, there are many difficulties in solving problems for such systems due to the high degree of coupling between rope and pulley, unclear boundary conditions for driving drums, and the non-independence of the system's constraint equations. This paper proposes a rope element between pulleys with dynamic boundary properties based on spatial description. Parameters such as arc length coordinates, rope angle, out-of-plane swing angle, and tensile strain of the rope's entry and exit points (i.e. the endpoints of the rope section in contact with a pulley) are introduced to characterize the motion of the contact points between pulley and rope in detail. Later, the deformation energy of the rope section in contact with the pulley is considered, and the boundary conditions between rope and pulley, as well as between rope and drum, are derived. A new method is proposed for addressing the non-independence of the constraint Jacobian matrix in the system's dynamic equations. The proposed method avoids the difficulties that will arise in traditional methods, and allows for efficiently solving for the motion of the pulley blocks and its internal pulleys, as well as the changes in the tension in each rope section within the system and the positions of the contact boundary points. Finally, three classic numerical examples are presented and the results demonstrate that the proposed method can be used to establish complete dynamic equations of a rope-driven system containing pulley blocks and efficiently solve them.