Exact dynamic analysis of multi-segment cable systems

doi:10.1016/j.ymssp.2020.107053

Mechanical Systems and Signal Processing

Volume 146, 1 January 2021, 107053

https://doi.org/10.1016/j.ymssp.2020.107053 Get rights and content

Highlights

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An exact method for dynamic analysis of multi-segment cable systems is proposed.
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Consider multiple effects such as flexural stiffness, sag, and elastic supports simultaneously.
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The closed-formed frequency equation of the system is obtained.
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The accuracy of the method has been verified by the full scaled real cable test.

Abstract

Cable-supported structure is a common system widely used in engineering, with the rapid development of the infrastructure and transportation industry, its structural form has become more and more complicated. Multi-segment cable system, which consists of a suspension cable and several lateral supports, such as the cable-damper system, the main cable of a suspension bridge, etc., is one of the important cable support systems. Due to the low structural rigidity and damping, its dynamic problems have always been the focus of engineering. To obtain a more general conclusion, a unified dynamic model considering the effect of cable sag, additional cable force, flexural stiffness, and the support stiffness of lateral components are considered in this paper for the first time. However, the exact analysis of this model is difficult, in view of this, the dynamic stiffness method is employed in this paper to investigate the dynamic characteristic of the multi-segment cable system. The accuracy of the proposed method is verified by comparing with finite element solutions and experimental results. Results show that the position of lateral supports has a strong coupling effect and influence the lower-order mode more significantly than higher-order modes, and the maximum frequency value can be reached by installing the supports equidistantly.

Introduction

A suspension cable with several lateral supports is a common cable-supported system in engineering, such as the cross ties system, suspension-hanger systems, and cable-damper systems [1], [2], [3], [4]. Divided by lateral supports, each cable segment will follow different dynamic configurations during vibration. To achieve an exact analysis of the dynamic characteristic of the multi-segment cable system, it is necessary to consider the influence of supports on dynamic configuration and additional cable force [5], [6], [7]. However, existing researches can only give the analytical expression of the additional cable force for the one- or two-segment cable system, but cannot generalize it to multiple segments.

The dynamic analysis method of cable can be roughly divided into three categories, that is analytical methods, numerical methods, and semi-analytical semi-numerical methods. Because the analytical method can give the solution of the dynamic characteristics or response in an analytical form, it has a wide application in modal analysis and parameter identification of cable structures due to the conciseness and explicitness of the solution [8]. However, since the analytical method can take only a few design parameters into consideration, the reliability and applicability of the results are limited. To obtain a, some numerical methods including finite element method (FEM) [9], Galerkin method [10], and finite difference method (FDM) [11] are employed to dynamic analysis of cable systems. Among them, the FEM is more commonly used but its calculation accuracy depends on the number of element divisions, therefore, its computational efficiency is limited and not conducive to the parameter analysis of cable structures. In view of this, the semi-analytical semi-numerical method is a better choice because it combines the advantages of both analytical and numerical methods.

The semi-analytical and semi-numerical methods commonly used in cable dynamic analysis are transfer matrix method (TMM) and dynamic stiffness method (DSM). The TMM is a method to solve modal frequencies and mode shapes of a system based on the study of the conversion relationship between the state vectors of each node of the system. Kang etc. [12] reported a systematic investigation on the linear and nonlinear dynamics of a suspended cable by the TMM, taking bending stiffness into account, and find that the bending stiffness plays a considerable role in changing the natural frequencies and mode shapes. Wang and Kang [13] proposed an in-plane dynamic model and transfer method for dynamic analysis of a suspension bridge, and the effect of some key parameters on modal frequencies and mode shapes are investigated. An alternative method is the DSM [14], which transforms structural displacement functions from the time domain to frequency domain and obtains the general solution of the governing differential equation (GDE) in frequency domain by variables separation. Then, combined with the nodal force and displacement boundary conditions, the frequency equation and dynamic equilibrium equation of the system is determined, and the dynamic characteristic and response of the system can then be obtained by solving the equations [15], [25]. Han etc. [16] made a systematic overview of the vertical vibration of the suspension bridge and proposed a unified method using DSM considering the effect of stiffness of hangers and towers.

In this paper, the DSM is employed to derive the frequency equation of the multi-segment cable system considering the effect of the bending stiffness, sag, additional cable force, and lateral supports. Since the frequency equation is generally a complicated transcendental equation, a reliable solution is to use the Wittrick-Williams (W-W) algorithm [17], [18]. By using the W-W algorithm, the modal frequencies and mode shapes of the system is solved. The correctness of results is verified by comparing with the finite element solutions and experimental results. Finally, the effects of parameters on modal frequencies and mode shapes are systematically investigated.

Section snippets

Dynamic model

When the cable system is attached with several lateral components like dampers and hangers, the cable will be divided into several cable segments and each segment will follow different dynamic configurations during the vibration. In view of this, considering a shallow sagged cable shown in Fig. 1, in this model, the boundary condition can be pinned, clamped or elastic support, and the cable is elastically supported by several lateral springs along its length direction. Without loss of

Verification

To verify the accuracy of the proposed method, the verification process will be arranged by two sections. In Section 3.1, the results of this paper are compared with FEM solutions to validate the proposed method. To further illustrate the effectiveness of the method, we did a real cable experiment while the results and conclusions are shown in Section 3.2.

Parameter study

In actual engineering, the number, stiffness, and position of the lateral supports are critical to dynamic performance of multi-segment cable structures. In fact, the study of the installation position of the support is usually to discuss the maximum damping ratio or optimal control efficiency of the specified mode. For example, studying the installation position of the damper in a cable-damper system can make it obtain the optimal damping ratio, besides, for a multi-segment system such as the

Conclusions

In this paper, the proposed dynamic analysis method for multi-segment cable system is verified. Then the dynamic characteristic a shallow sagged cable with several lateral supports is analyzed to investigate the effect of the number, stiffness, and relative installation position of lateral supports on the dynamic characteristic of the multi-segment cable system. Some main results can be concluded as

1)
The proposed dynamic model and analytical method give us a novel way of dynamic analysis of a

CRediT authorship contribution statement

Han Fei: . Deng Zichen: . Dan Danhui: Project administration.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grant No. 51878490); the National key R&D Program of China (2017YFF0205605); Shanghai Urban Construction Design Research Institute Project ‘Bridge Safe Operation Big Data Acquisition Technology and Structure Monitoring System Research’; and the Ministry of Transport Construction Science and Technology Project ‘Medium-Small Span Bridge Structure Network Level Safety Monitoring and Evaluation’.

References (25)

J. Lardies et al.
Modal parameter identification of stay cables from output-only measurements
Mech. Syst. Sig. Process.
(2011)
N. Barbieri et al.
Dynamical analysis of transmission line cables. Part 1—linear theory
Mech. Syst. Sig. Process.
(2004)
N. Barbieri et al.
Dynamical analysis of transmission line cables. Part 2—damping estimation
Mech. Syst. Sig. Process.
(2004)
R. Barbieri et al.
Dynamical analysis of transmission line cables. Part 3—Nonlinear theory
Mech. Syst. Sig. Process.
(2008)
H. Fei et al.
Free vibration of the complex cable system − an exact method using symbolic computation
Mech. Syst. Sig. Process.
(2020)
Han Fei et al.
Experimental and theoretical study on cable-supporting system
Mech. Syst. Sig. Process.
(2020)
Y.Q. Ni et al.
Dynamic analysis of large-diameter sagged cables taking into account flexural rigidity
J. Sound Vib.
(2002)
X. Liu et al.
A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems
Mech. Syst. Sig. Process.
(2017)
F. Han et al.
Exact dynamic characteristic analysis of a double-beam system interconnected by a viscoelastic layer
Compos. B Eng.
(2019)
H. Fei et al.
A novel method for dynamic analysis of complex multi-segment cable systems
Mech. Syst. Sig. Process.
(2020)

F.W. Williams et al.

An automatic computational procedure for calculating natural frequencies of skeletal structures

Int. J. Mech. Sci.

(1970)

Fei Han et al.

Exact dynamic analysis of shallow sagged cable system — theory and experimental verification

Int. J. Struct. Stab. Dyn.

(2019)

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View full text

Exact dynamic analysis of multi-segment cable systems

Highlights

Abstract

Introduction

Section snippets

Dynamic model

Verification

Parameter study

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

J. Sound Vib.

Mech. Syst. Sig. Process.

Compos. B Eng.

Mech. Syst. Sig. Process.

Int. J. Mech. Sci.

Exact dynamic analysis of shallow sagged cable system — theory and experimental verification

Int. J. Struct. Stab. Dyn.