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Research Papers

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

[+] Author and Article Information
W. D. Zhu1

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250wzhu@umbc.edu

H. Ren

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250renhui1@umbc.edu

C. Xiao

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250chuangx@umbc.edu

1

Corresponding author.

J. Appl. Mech 78(4), 041017 (Apr 15, 2011) (13 pages) doi:10.1115/1.4003348 History: Received August 26, 2009; Revised December 27, 2010; Posted January 04, 2011; Published April 15, 2011; Online April 15, 2011

A nonlinear, planar model of a slack cable with bending stiffness and arbitrarily moving ends is developed. The model uses the slope angle of the centroid line of the cable to describe the motion of the cable, and the resulting integropartial differential equation with constraints is derived using Hamilton’s principle. A new method is developed to obtain the spatially discretized equations, and the Baumgarte stabilization procedure is used to solve the resulting differential-algebraic equations. The model can be used to calculate the equilibria and corresponding free vibration characteristics of the cable, as well as the dynamic response of the cable under arbitrarily moving ends. The results for an equilibrium and free vibration characteristics around the equilibrium are experimentally validated on a laboratory steel band. The methodology is applied to elevator traveling and compensation cables. It is found that a vertical motion of the car can introduce a horizontal vibration of a traveling or compensation cable. The results presented are verified by a commercial finite element software. The current method is shown to be more efficient than the finite element method as it uses a much smaller number of elements to reach the same accuracy. Some other interesting features include the condition for a traveling or compensation cable equilibrium to be closest to a natural loop and a direct proof that the catenary solution is unique.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of an elevator system

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Figure 2

A cable with moving ends and its coordinates

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Figure 3

The kth (a) undeformed and (b) deformed cable elements

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Figure 5

(a) Schematic of the experimental setup, and (b) the roving laser method to measure the responses of a point on the lower loop of the band in the x and y directions

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Figure 6

Equilibrium solutions of the band with fixed (solid lines) and pinned (dashed lines) boundaries. For fixed boundaries, θ¯0=−π/2; θ¯N=π/2 in (a), θ¯N=−3π/2 in (b), θ¯N=5π/2 in (c), θ¯N=−7π/2 in (d), and θ¯N=9π/2 in (e).

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Figure 7

Comparison of (a) the equilibrium of the band corresponding to the solid line in Fig. 6, and (b) its first 40 natural frequencies of vibration around the equilibrium: solid lines, experimental results; dashed lines, results from the current method; and dotted lines, FEM results using ABAQUS 6.7

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Figure 8

The measured FRFs with the excitation and measurement points at Points 18 and 22, respectively (solid), and the excitation and measurement points reversed (dashed)

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Figure 9

(a) Equilibrium of an elevator traveling cable with the car at the initial height (dashed line, the current method; dotted line, the FEM using ABAQUS 6.7 ), and (b) the first ten natural frequencies of the cable with the car at different positions (solid line, the current method; ●, the FEM using ABAQUS 6.7 ). Three particles on the cable with s=3 m, 10 m, and 17 m are indicated in (a).

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Figure 10

Dynamic responses of the three particles on the cable, as indicated in Fig. 9 (solid lines, the current method; dashed lines, the FEM using ABAQUS 6.7 ): ((a) and (b)) horizontal and vertical positions of Particle 1, respectively; ((c) and (d)) horizontal and vertical positions of Particle 2, respectively; and ((e) and (f)) horizontal and vertical positions of Particle 3, respectively

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Figure 11

Movement profile of the car: (a) position, (b) velocity, and (c) acceleration

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Figure 12

Dynamic responses of the three particles on the cable, as indicated in Fig. 9, with the movement profile of the car shown in Fig. 1 (solid lines, the current method; dashed lines, the FEM using ABAQUS 6.7 ): ((a) and (b)) horizontal and vertical positions of Particle 1, respectively; ((c) and (d)) horizontal and vertical positions of Particle 2, respectively; and ((e) and (f)) horizontal and vertical positions of Particle 3, respectively

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Figure 13

Dynamic responses of three particles on the cable with s=4 m (Particle 1), 15 m (Particle 2), and 26 m (Particle 3) (solid lines, the current method; dashed lines, the FEM using ABAQUS 6.7 ): ((a) and (b)) horizontal and vertical positions of Particle 1, respectively; ((c) and (d)) horizontal and vertical positions of Particle 2, respectively; and ((e) and (f)) horizontal and vertical positions of Particle 3, respectively

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Figure 4

Equilibria of the cable or the lower loop of the cable for different σ and H: (a) H=0, and σ=0.1 (dashed line), 0.05 (solid line), and 0.01 (dotted line); (b) σ=0.05, and H=0 (solid line), 30 m (dashed line), 60 m (dotted line), and 90 m (dashed-dotted line); (c) σ=0.1, with H the same as those in (b); and (d) σ=0.01, with H the same as those in (b)

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