A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application to Elevator Traveling and Compensation Cables

摘要

A nonlinear, planar model of a slack cable with bending stiffness and arbitrarily movingnends is developed. The model uses the slope angle of the centroid line of the cable tondescribe the motion of the cable, and the resulting integropartial differential equationnwith constraints is derived using Hamilton’s principle. A new method is developed tonobtain the spatially discretized equations, and the Baumgarte stabilization procedure isnused to solve the resulting differential-algebraic equations. The model can be used toncalculate the equilibria and corresponding free vibration characteristics of the cable, asnwell as the dynamic response of the cable under arbitrarily moving ends. The results fornan equilibrium and free vibration characteristics around the equilibrium are experimentallynvalidated on a laboratory steel band. The methodology is applied to elevator travelingnand compensation cables. It is found that a vertical motion of the car can introducena horizontal vibration of a traveling or compensation cable. The results presented arenverified by a commercial finite element software. The current method is shown to be morenefficient than the finite element method as it uses a much smaller number of elements tonreach the same accuracy. Some other interesting features include the condition for antraveling or compensation cable equilibrium to be closest to a natural loop and a directnproof that the catenary solution is unique.