On the identification of the axial force and bending stiffness of stay cables anchored to flexible supports

Abstract

The generic model of a cable with small bending stiffness and anchored to flexible supports in rotation and translation is considered. An asymptotic analysis of the natural frequencies of this generic model is derived and shows that, for small bending stiffness, the first few natural frequencies can be expressed as a function of the cable axial force, the small bending stiffness and a single dimensionless group collecting the information of all other problem parameters. This formulation is used to develop an identification procedure of the cable axial force. Two formulations are proposed, one numerical and one semi-analytical based on a simple linear regression model. Both methods do not attempt at separately identifying the problem parameters since the observability analysis has revealed that only the cable axial force, the bending stiffness and the dimensionless group can be identified. In particular, the second method is very simple to implement and provides estimates of the cable axial force which account for the flexibility of the support. The proposed method can therefore be seen as an extension of usual identification techniques based on linear regressions of natural frequencies vs. mode number relations, by considering at the same time the bending stiffness and the deformability of supports. Being simple and robust as shown by means of an uncertainty quantification analysis, the proposed method can be conveniently embedded in the framework of a continuous monitoring strategy.

Minimal requirements for the vibration-based identification of the axial force, the bending stiffness and the flexural boundary conditions in cables

“…They take values in the left-bounded interval [0, ∞[ with hinged and clamped boundary conditions corresponding respectively to the zero lower bound value and to the infinite upper limit value. For modeling and identification purposes [28], the rotational degree-of-fixity parameters…”

“…In general, the zeros of Eq. ( 18) can be evaluated through any suitable root finding algorithm and, in this paper, the bisection method was successively applied to obtain them numerically, by using the dichotomy algorithm that has already been validated against a finite element model in [28]. However, in order to get a deeper understanding of the influence of each parameter on the dynamics of the cable, analytical formulas for the natural frequencies and the mode shapes are derived as well in the appendix.…”

“…that has been developed to approximate the first few frequencies in [28] and generalizes the formula derived by…”

“…First, methods that do not include any measurement of the mode shapes are only able to identify two independent parameters; and this is also conditioned upon small noise on measured natural frequencies. For instance, in [8], the axial force and a global rotational stiffness (equivalent of ρ r ) have been correctly identified because the bending stiffness was known in advance while, in [28], ρ r was fixed to a pragmatic value in order to determine the fundamental frequency and the bending stiffness parameter. These observations are corroborated in this paper as well by looking at Figure 4 and Figure 5, which illustrate the evolution of the objective function close to a given set of target parameters when modal amplitudes are included in the measurements or are not considered at all, respectively.…”

“…On the sole basis of measured natural frequencies, which are not modified by the swapping of the two rotational end restraints, existing methods have shown that it is only possible to determine a single dimensionless group, accounting for both of them all at once [28], when the measurements are corrupted by external noise or when the model contains epistemic uncertainties. However, by complementing the natural frequencies with synchronous measurements of the dynamic deflections of the element at several locations along its length, its rotational end restraints [29] can be identified after its axial force [30] when its bending stiffness is known in advance, by contrast with the recent attempts to make the simultaneous identification of both the axial force and the bending stiffness regardless of the boundary conditions [31,17,32].…”