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Research Papers

An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable-Car Systems—Part II: Application

[+] Author and Article Information
H. Ren

e-mail: hui.ren@mscsoftware.com

W. D. Zhu

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

1Currently a development engineer at the MSC Software Corporation.

2Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received October 8, 2011; final manuscript received May 6, 2013; published online July 23, 2013. Assoc. Editor: Jean Zu.

J. Vib. Acoust 135(5), 051037 (Jul 23, 2013) (21 pages) Paper No: VIB-11-1233; doi: 10.1115/1.4024558 History: Received October 08, 2011; Revised May 06, 2013

This paper uses the methodology developed in Part I of this work to study the longitudinal, transverse, and their coupled vibrations of moving elevator cable-car systems. A suspension cable is a one-dimensional length-variant distributed-parameter component. When there is only one suspension cable connected to the car, the car is modeled as a point mass. When there are multiple suspension cables, the car is modeled as a rigid body, and the rotation of the car is considered. There are complicated matching conditions between the cable and car, which cannot be satisfied in the classical assumed modes method but can be satisfied in the current method. Hence, not only the longitudinal and transverse displacements but also the internal forces/moment, such as the axial force, the bending moment, and the shear force, which are related to the spatial derivatives of the longitudinal and transverse displacements, are accurately calculated. The results from different choices of boundary motions and trial functions are essentially the same, and the convergence is much faster than that of the assumed modes method. The longitudinal-transverse coupled vibrations of a moving cable-car system are also studied using the current method, and the results are compared with those from the linear models. While the result from the linear model for the transverse vibration agrees well with that from the nonlinear coupled model, the axial force from the linear model can significantly differ from that from the nonlinear model when the car approaches the top of the hoistway.

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References

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Zhu, W. D., and Xu, G. Y., 2003, “Vibration of Elevator Cables With Small Bending Stiffness,” J. Sound Vib., 263(3), pp. 679–699. [CrossRef]
Chi, R. M., and Shu, H. T., 1991, “Longitudinal Vibration of a Hoist Rope Coupled With the Vertical Vibration of an Elevator Car,” J. Sound Vib., 148(1), pp. 154–159. [CrossRef]
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Figures

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Fig. 1

Schematic of an elevator system

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Fig. 2

Models of moving cable-car systems for studying (a) the longitudinal vibration, (b) the transverse vibration, and (c) the longitudinal-transverse coupled vibrations

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Fig. 3

Coordinate systems for a translating cable with a variant length: (a) the initial undeformed configuration and (b) the deformed configuration

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Fig. 5

The longitudinal displacements of (a) the car and (b) the reference particle, the vibratory displacements of (c) the car and (d) the reference particle, and (e) the axial force in the cable, calculated using the fixed-free rod model with ψ0(ξ)=ξ (dashed lines) and ψ0(ξ)=ξ3-ξ2 (dash-dotted lines), and the fixed-fixed rod model (dotted lines). The force balance relation at the lower end of the cable is verified in (f); T↑ and T↓ are shown in dashed and dotted lines, respectively. The quasi-static solutions are shown in solid lines.

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Fig. 6

The longitudinal displacements of (a) the car and (b) the reference particle, the vibratory displacements of (c) the car and (d) the reference particle, and (e) the axial force in the cable, calculated using the fixed-free rod model (dashed lines) and the AMM (dotted lines); T↑ (dashed line) and T↓ (dotted line) calculated using the AMM in (f) are not the same

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Fig. 7

The free transverse displacements of (a) the car and (b) the reference particle calculated using the clamped-pinned tensioned beam model (solid lines), the pinned-pinned beam model (dotted lines), and the string model (dash-dotted lines), and those of (c) the car and (d) the reference particle calculated using the clamped-pinned tensioned beam model (solid lines) and the clamped-pinned beam model (dashed lines)

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Fig. 8

(a) The bending moments and (b) shear forces associated with the free transverse vibration, calculated using the clamped-pinned tensioned beam model (solid lines), the pinned-pinned beam model (dotted lines), and the string model (dash-dotted lines), and (c) the bending moments and (d) shear forces calculated using the clamped-pinned tensioned beam model (solid lines) and the clamped-pinned beam model (dashed lines)

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Fig. 9

The forced transverse displacements of (a) the car and (b) the reference particle, and the corresponding (c) bending moments and (d) shear forces, calculated using the clamped-pinned tensioned beam model (solid lines), the pinned-pinned beam model (dotted lines), and the string model (dash-dotted lines)

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Fig. 10

The transverse displacements of (a) the car and (b) the reference particle, and the longitudinal displacements of (c) the car and (d) the reference particle, calculated using the clamped-pinned tensioned beam and fixed-fixed rod models (dashed lines) and the pinned-pinned beam and fixed-free rod models (dotted lines). The results from the linear models are shown in solid lines.

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Fig. 11

(a) The bending moment, (b) the shear force, and (c) the axial force or tension calculated using the clamped-pinned tensioned beam and fixed-fixed rod models (dashed lines) and the pinned-pinned beam and fixed-free rod models (dotted lines). The results from the linear models are shown in solid lines.

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Fig. 12

Schematic of a moving two-cable one-rigid-body-car system (not to scale)

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Fig. 13

Movement profiles of the (a) left and (b) right cables in Fig. 12

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Fig. 14

(a) The longitudinal displacement of the car, (b) the rotation of the car, the longitudinal displacements of the reference particles on the (c) left and (d) right cables, and the axial forces in the (e) left and (f) right cables, calculated using the fixed-free rod model (dashed lines) and the fixed-fixed rod model (dotted lines). The quasi-static solutions are shown in solid lines.

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Fig. 15

The transverse displacements of the reference particles on the (a) left and (b) right cables and (c) the car, calculated using the clamped-pinned tensioned beam model (dashed lines) and the pinned-pinned beam model (dotted lines)

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Fig. 16

(a) The bending moment and (b) shear force in the left cable, and (c) the bending moment and (d) shear force in the right cable, calculated using the clamped-pinned tensioned beam model (dashed lines) and the pinned-pinned beam model (dotted lines)

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