Estimation of cable tension force using the frequency-based system identification method

Abstract

This work proposes a new technique to estimate cable tension force from measured natural frequencies. The proposed method is able to simultaneously identify tension force, flexural rigidity, and axial rigidity of a cable system. Firstly, a finite element model that can consider both sag-extensibility and flexural rigidity is constructed for a target cable system. Next, a frequency-based sensitivity-updating algorithm is applied to identify the model. The proposed approach is applicable to a wide range of a cable system that is beyond the applicable limits of the existing methods. From the experimental works, it is seen that the tension force is determined with an accuracy of 3% by the proposed approach. Furthermore, it is observed that the flexural rigidity of cable with high bending stiffness is proportional to the applied tension force.

Introduction

Modern advances in material, analysis, and construction technology have resulted in increasing number of a long-span cable bridge. Since cables are a crucial element for overall structural safety of the structure, the accurate measurement of cable tension force has practical importance to not only a construction stage but also a maintenance stage. Currently available techniques to estimate the cable tension include the static methods directly measuring the tension by a load cell or a hydraulic jack, and the vibration methods indirectly estimating the tension from measured natural frequencies. In practice, the vibration methods have received increasing attention because of its simplicity and speediness.

Depending on whether the sag-extensibility and bending stiffness are taken into account or not, the existing vibration methods may be classified as the following four categories. The first category utilizes the flat taut string theory that neglects both sag-extensibility and bending stiffness:

$$T=4m{L}^2{\left(\frac{f_n}{n}\right)}^2\text{,}$$

where f_n denotes the nth natural frequency in Hz. The terms T, m, and L denote tension force, mass density, and length of cable, respectively. Given the measured frequency and the mode number, the computation of tension force is straightforward. However, the application of this formula is strictly limited to a flat long slender cable. The second category makes use of the modern cable theory [1], [2], [3] that takes account of the sag-extensibility without bending stiffness. This approach requires additional information of the unstrained length of cable and involves solving a nonlinear characteristic equation by trial-and-error [4]. However, such additional information is often not available in practice. The third category utilizes the following frequency formula of an axially loaded beam that considers the bending stiffness but neglects the sag-extensibility:

$${\left(\frac{f_n}{n}\right)}^2=\left(\frac{1}{4m{L}^2}\right)T+\left(\frac{n^2{\pi }^2}{4m{L}^4}\right)\mathit{EI}\text{,}$$

where EI denotes the flexural rigidity of a cable. Given the measured frequency and the mode number, the linear regression procedures are applied to identify unknown tension force and flexural rigidity simultaneously. This approach is often used by the field engineers because of its simplicity and speediness. The last category takes account of both sag-extensibility and bending stiffness using a practical formula [5], [6], [7]. For the proper use of the proposed formula, a priori knowledge of the axial rigidity and flexural rigidity of the target cable system is required. However, in practice, the flexural rigidity of cable is often neither available nor valid since the shear and bending mechanisms of a cross section of a cable could be different from those of a beam.

The aforementioned vibration methods have at least the following three shortcomings related to their applicable limits. First, the existing vibration methods are based on a closed form relationship between cable tension force and the natural frequencies for a simple mathematical model. Hence, it is not surprising to get accurate estimation of tension force if the target cable system is well represented by the model. However, the estimation result may be significantly distorted if the model could not accurately describe the behavior of the target cable system. For instance, the application of the taut string theory in Eq. (1) to a cable with high sag and high bending stiffness does not guarantee the good results of the estimated cable tension force. Neither does the application of the modern cable theory to a short thick cable such as the tie-rod of an arch bridge. In general, the influence of cable extensibility is negligible for short thick cables but could lead to important errors for long sagged cables [8]. Here, the short thick cables belong to a class of cables that is not slender or not sufficiently tensioned. For such short thick cables, the higher natural frequencies are greater than predicted by the taut string theory, as shown in Tsing Ma Bridge [9]. Second, the existing vibration methods may not be applicable to the cable system whose analytical solution is not known. For instance, the existing vibration methods may not be applicable to the inclined double short hanger of a suspended bridge that consists of two independent cables tied by a clamp and a spacer. Although an application of Eq. (2) to cables with a transverse connector is found [10], such an application is strictly limited to relatively long cables of which the tied-effects are negligible. Third, some vibration methods require not only the measured frequencies but also additional information such as cable effective length and material parameters. However, such additional information that affects accuracy of the resulting tension force is often not available in practice. Therefore, there remains a need to resolve these deficiencies of the existing vibration methods.

The objective of this paper is to introduce a new technique that can estimate cable tension force from measured natural frequencies. The proposed approach resolves the aforementioned deficiencies of the earlier approaches by using a finite element analysis technique and applying a system identification technique [11]. Here, the finite element analysis is adopted in order to extend the applicable limits on geometric complexity of a target cable system, and the state-of-the-art system identification technique is applied in order to overcome the required knowledge of material properties and static shapes of cables. To achieve the objective, the following three tasks are performed. First, the approach to estimate cable tension force from the measured natural frequencies is outlined. Second, a set of numerical comparative study is conducted to examine the accuracy of the proposed approach. Third, the feasibility and practicability of the proposed approach are examined by the laboratory experiments and a field application.