This paper presents new advances in the arbitrary Lagrangian-Eulerian modal method (ALEM) recently developed for the systematic simulation of the dynamics of general reeving systems. These advances are related to a more convenient model of the sheaves dynamics and the use of axial deformation modes to account for non-constant axial forces within the finite elements. Regarding the sheaves dynamics, the original formulation uses kinematic constraints to account for the torque transmission at the sheaves by neglecting the rotary inertia. One of the advances described in this paper is the use of the rotation angles of the sheaves as generalized coordinates together with the rope-to-sheave no-slip assumption as linear constraint equations. This modeling option guarantees the exact torque balance the sheave without including any non-linear kinematic constraint. Numerical results show the influence in the system dynamics of the sheave rotary inertia.
Regarding the axial forces within the finite elements, the original formulation uses a combination of absolute position coordinates and transverse local modal coordinates to account for the rope absolute position and deformation shape. The axial force, which only depends on the absolute position coordinates, is constant along the element because linear shape functions are assumed to describe the axial displacements. For reeving systems with very long rope spans, as the elevators of high buildings, the constant axial force is inaccurate because the weight of the ropes becomes important and the axial force varies approximately linearly within the rope free span. To account for space-varying axial forces, this paper also introduces modal coordinates in the axial direction. Numerical results show that a set of three modal coordinates in the axial direction is enough to simulate linearly varying axial forces.