# Frequency-Based Cable Tension Identification Using a Nonlinear Model with Complex Boundary Constraints

Authors: Wen-ming Zhang; Zhi-wei Wang
Journal: Shock and Vibration
 (2023)

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## Abstract

Accurate estimation of the cable force affects the bridges’ long-term integrity and serviceability directly. The frequency method is often used for cable tension testing in bridge engineering. However, the generally used cable tension calculation formulae are based on “ideal hinge” or “ideal fixed” boundary conditions. The inclination, bending stiffness, and sag-extensibility of the cable are not properly considered, which results in non-negligible errors. A frequency-based method for precisely determining the tensile force of a cable with unknown rotational and support constraint stiffnesses at the boundary was proposed. A nonlinear mathematical model of the vibration of the cable was established. In addition to parameters such as inclination, sag, and bending stiffness, the effects of unknown rotational and support constraint stiffnesses at both ends of the cable were also considered. The finite difference method was employed to discretize and solve the mode equation of the cable vibration. A frequency-based sensitivity-updating algorithm was applied that can identify simultaneously several system parameters using multiorder measured natural frequencies. Calculation of the matrix eigenvalue derivatives was the key to obtaining the system sensitivity matrix. Numerical examples indicated that the algorithm can be used efficiently and precisely to identify multiple system parameters of the cable, including its tension, bending stiffness, and boundary constraint stiffnesses.


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## Body

## 1. Introduction

Cables are widely used in bridge engineering, for applications such as stay cables for cable-stayed bridges, flexible hangers for tied-arch bridges, and hangers for suspension bridges. The accuracy of the cable force testing affects the bridge construction monitoring, the structural stress state assessment, and the maintenance strategy formulation. Based on the basic principle that there is specific correspondence between the natural vibration frequencies and the cable tension, the cable tension can be determined by the measurement of the vibration frequencies. The frequency-based method is simple in operation and low in cost during bridge construction and service. In consequence, the approach is widely used for tension testing. However, practical applications have shown that the frequency-based method may produce non-negligible errors, mainly because of the inclination, sag-extensibility, bending stiffness, and boundary conditions of the cable have not been properly considered. Especially for applications with short and thick (high linear stiffness) cables, e.g., the short hangers of suspension bridges, this method has been observed to have low accuracy and may even lead to completely incorrect results. The main reason is that the vibration frequencies of the short hangers are particularly affected by boundary constraint stiffnesses.In the earlier research, there were mainly three classical calculation theories of the frequency-based method. (1) The flat taut string theory neglects the influence of bending stiffness and sag-extensibility of the cable. The cable tension is obtained directly according to the measured frequency and frequency order. It applies only to flat and slender cables with low bending stiffness. (2) The axial-loaded slender beam theory considers the influence of bending stiffness but ignores the sag-extensibility of the cable. It also requires the boundary condition to be ideally hinged or fixed. Several researchers studied the cable with pinned end conditions at the two ends [1, 2] and the cable with fixed end conditions at the two ends [3–5]. However, in engineering, the boundary conditions of the cable mostly show unknown rotational and support constraint stiffnesses. For example, one end of the anchorage cable strand of a suspension bridge is connected to the cantilever linkage of the front anchor facet. The opposite end is stacked with the other strands in the saddle groove and the “bearings” at both the ends of the cable vibrate in pace with the vibration of the cable. (3) The last approach is the application of practical calculation formulae that consider the bending stiffness and sag effect of the cable. However, to use these formulae, the bending stiffness and axial stiffness of the cable must be known. In practical engineering, the deformation mechanism of a beam and that of a cable are not exactly the same. Therefore, the use of the formulae for a beam to calculate the axial stiffness and bending stiffness of a cable is not always trustworthy.Many researchers have conducted in-depth research on the vibration method for the estimation of cable tension. Irvine et al. [6] systematically studied the out-of-plane and in-plane linear vibration of the cable with a sag-to-span ratio of within 1 : 8 and verified the theoretical correctness through experiments. Shimada and Nishimura [7] explored the influence of cable bending stiffness on cable tension estimation through experimental research. They found that a larger error resulted when using the vibration method to estimate the tensile force of a short cable, as the method ignores bending stiffness. Zui et al. [3] expressed the physical properties of the cable in the form of dimensionless parameters. They derived practical formulae for cable tensile force estimation which were presented by piece-wise functions and suitable for any cable regardless of the tensile force and the length of the cable. Mehrabi and Tabatabai [8] introduced a finite difference method for discretizing the vibration mode equation of the horizontal cable. A lot of important parameters such as the sag, bending stiffness, and the intermediate dampers or springs of the cable were taken into account. Based on the energy method and the curve fitting method, Ren et al. [4] gave empirical formulae adopting only the fundamental frequency of the cable.Many difficulties in cable tension estimation will be imposed by the unknown complex boundary conditions when adopting the traditional cable force identification methods. Yan et al. [9] developed an innovative methodology independent of the cable boundary conditions. The problem of identifying the cable force was transformed into searching the zero-amplitude points from the mode shapes of the cable in their research. Chen et al. [10] put forward a new approach for cable tension identification using mode shape functions to eliminate the effects of complex boundary conditions. They considered that the cable with complex boundary conditions can be generally predicted by adopting an explicit formula of an equivalent cable model with pinned end conditions at the two ends. The abovementioned research attempted to give practical formulae for estimating the cable tension based on the vibration method but did not consider systematically the effects of cable inclination, bending stiffness, sag-extensibility, and the unknown boundary constraint stiffnesses on the dynamic characteristics of the vibration of the cable. Therefore, their applications have relatively large limitations.In recent years, some numerical iterative algorithms which can estimate cable tensile force, bending stiffness, and some other parameters of the cable have been put forward. Kim and Park [11] established a finite element model taking into account the bending stiffness and sag effect of the cable. A frequency-based sensitivity-updating algorithm was adopted to determine the parameters of the cable model from the measured multiorder frequencies. Liao et al. [12] eliminated the errors between the frequencies calculated by an accurate finite element model of the cable and the measured frequencies by adopting a least squares optimization technique. This method can also determine the cable tension and other system parameters simultaneously. Li et al. [13] put forward an extended Kalman filter algorithm that can identify the cable tension varying with time in real-time. Zarbaf et al. [14] introduced an error function that corresponds to the differences between the calculated frequencies and the measured frequencies of the stay cable. Then, the particle swarm optimization algorithm and genetic algorithm were employed to minimize the error function and obtain the cable tension. However, the models established in these studies still did not consider systematically cable inclination, sag effect, bending stiffness, and the unknown boundary constraint stiffnesses at the cable end.The technique developed in the present paper solves the disadvantages of classical theories. Compared with some modern cable theories, this paper established a nonlinear analytical model of the cable vibration which accurately accounted not only for the cable’s inclination, bending stiffness, and sag effect but also for the influence of unknown rotational constraint stiffness and lateral support stiffness at the cable ends. The finite difference method was utilized to discretize the vibration mode equation of the cable, which transformed the complex analytical solution process into a simple numerical solution process. On this basis, the frequency-based sensitivity-updating algorithm could be used to identify simultaneously and accurately multiple system parameters of the cable according to multiple measured frequencies, including the cable tensile force, axial stiffness, bending stiffness, and unknown rotational and lateral stiffnesses at the cable end. Finally, the effectiveness and feasibility of the method developed in the present paper were verified by numerical examples.


## 2. Overall Idea

The overall idea of the method put forward in the present paper is shown in Figure1. First, an analysis of the free vibration of the cable was conducted to establish the free vibration equation. The equation considers the inclination, bending stiffness, and sag-extensibility of the cable, which then was converted into a vibration mode equation by the mode separation method. Second, the boundary constraint equilibrium equations at the cable ends were established. Then, the finite difference method was used to discretize the vibration mode equation and the boundary constraint equilibrium equations to obtain the overall stiffness matrix of the cable. After the discretized vibration mode equation was established, the modal frequencies and mode shapes could be determined by eigenvalue analysis. Finally, a frequency-based sensitivity-updating algorithm was applied to identify simultaneously several system parameters using multiorder measured natural frequencies, including cable tension, axial stiffness, bending stiffness, and cable-end constraint stiffnesses.Figure 1
Overall idea flowchart.The frequency-based sensitivity-updating algorithm for cable parameter identification is based on the concept of the Newton–Raphson method. Its derivative-based root-finding feature gives the algorithm a satisfactory convergence speed. Herein, the “sensitivity” refers to the variation of the cable’s vibration eigenvalues (frequencies) along with the cable parameters; and the “updating” denotes the automatic update of the cable parameters with the help of the eigenvalue sensitivity matrixα in each iteration. The detailed process is described as follows. First, proper initial iterative values of the system parameters were set to gain the initial overall stiffness matrix of the cable. Second, the eigenvalue analysis was conducted to obtain the eigenvalue difference vector δ λ which represents the differences between the calculated and measured vibration eigenvalues. Third, the derivatives of the vibration eigenvalues with respect to the system parameters were calculated, and then, the eigenvalue sensitivity matrix α was obtained. Essentially, α is the Jacobian matrix representing the fastest decreasing direction of the eigenvalue difference δ λ (or say the fastest convergent direction) in the solution space during the iteration. Finally, the correction coefficient of each cable parameter (denoted by F) could be determined based on α and δ λ. The initial iterative values of the cable parameters for the second iteration could be gained using F obtained from the first iteration. The abovementioned process was repeated until convergence, and then, the accurate values of the system parameters were obtained. This algorithm could run automatically and efficiently when the appropriate initial iterative values were first determined.


## 3. Methodology

### 3.1. Free Vibration Analysis

#### 3.1.1. Free Vibration Equation

Figure2 shows the schematic of an inclined cable model along with the coordinate system. Point A is taken as the coordinate origin. Compared with the previous study [6–8], the proposed model gives systematic consideration to the inclination angle, bending stiffness, sag-extensibility, and geometric nonlinearity of the cable. Especially, the complex boundary constraints at the cable ends were first comprehensively modeled, which would be introduced in detail in Section 3.3.Figure 2
Schematic of an inclined cable.Only the in-plane vibration of the cable is considered. Assume that it is uncoupled between the axial vibration and the transverse in-plane vibration of the cable. According to the D’Alembert principle, the equations of the chord-wise and transverse in-plane motions of the inclined cable are derived from the following equations:(1)∂∂sT+τdxds+∂u∂s+mgsinθ=m∂2u∂t2+cu′∂u∂t,(2)∂∂sT+τdyds+∂v∂s−∂2∂s2EI∂2η∂x2+mgcosθ=m∂2v∂t2+cv′∂v∂t,where T(x) denotes the initial cable tension; τx,t denotes the additional tension caused by vibration; s is the arc-length coordinate; yx denotes the static equilibrium profile; ux,t and vx,t represent the displacement in the x-direction and y-direction caused by vibration, respectively; ηx,t=y+v denotes the sum of the displacements in the y-direction; g denotes the gravitational acceleration; EI(x) denotes the bending stiffness; m(x) denotes the mass per unit length; θ denotes the inclination angle; cu′x and cv′x represent the axial and transverse in-plane viscous damping coefficient per unit length, respectively. In addition, Hm denotes the mean value of the chord-wise component of T(x); EA(x) denotes the axial stiffness; and L denotes the cable length.It is generally known thatT+τ=H+hds/dx, where H and h denote the chord-wise component of T and τ, respectively. Furthermore, the axial vibration of the cable can be neglected. Then, equations (1) and (2) become(3)∂∂sH+h+mgsinθ=0.(4)∂∂sH+hdydx+∂v∂x−∂2∂s2EI∂2η∂x2+mgcosθ=m∂2v∂t2+cv′∂v∂t.If the sag-to-span ratio of the cable is no more than 1/10, ∂s can be replaced by ∂x. In this case, we can find that ∂h/∂x = 0 according to equation (3), therefore, h(t) becomes a variable varying with time alone. Substituting the static equilibrium differential equation into equation (4) and neglecting the second-order terms results in(5)Hx∂2v∂x2+H′x∂v∂x+hd2ydx2−∂2∂x2EI∂2v∂x2=m∂2v∂t2+cv′∂v∂t.Equation (5) is the free vibration equation of an inclined cable, and the following steps are taken to transform it into an eigenvalue calculation.


#### 3.1.2. Vibration Mode Equation

The vibration of a cable is the superposition of several modes, by using the mode separation method; the variablevx,t can be expressed as follows:(6)vx,t=wxqt,where wx denotes the mode shapes independent of time, and q(t) denotes the generalized coordinates dependent on time alone. Considering the damped free vibration of a cable, q is generally written as qt=ept, where p=−ζω±iωD, in which ζ denotes the damping ratio; ω and ωD denote the nondamping and damped circular frequencies of the free vibration of the cable, respectively.With the separation of the variablevx,t, equation (5) becomes(7)d2dx2EId2wdx2q−Hd2wdx2q−H′dwdxq−hd2ydx2+cv′pwq+mp2wq=0.Expressing the additional cable tension componenth as a linear correlation expression of the generalized coordinates q, then, the generalized coordinates q can be factored out as a common factor. Based on the relationship between the additional cable tension (τ) and the cable elongation, Irvine [1] derived the following expression:(8)∫0Lhds/dx3EAdx=∫0L∂u∂xdx+∫0Ldydx∂v∂xdx.Due to the assumption that chord-wise displacementu equals zero, the first term of the right side of equation (8) is zero. As h is a variable varying with time alone, h can be written as follows:(9)h=∫0Ldy/dx∂v/∂xdx∫0Lds/dx3/EAdx=−∫0Ld2y/dx2wdx∫0Ldy/dx2+13/2/EAdxq.Substituting equation (9) into (7), eliminating the common factor q, and expanding the first term of equation (7) results in(10)d2EIdx2d2wdx2+2dEIdxd3wdx3+EId4wdx4−Hd2wdx2−H′dwdx+∫0Ld2y/dx2wdx∫0Ldy/dx2+13/2/EAdxd2ydx2+cv′pw+mp2w=0.Equation (10) is the nonlinear vibration mode equation of an inclined cable. The first five terms of the left side are the linear stiffness part of the cable while the sixth term denotes the nonlinear stiffness part. By using the finite difference method to discretize this fourth-order nonlinear differential equation, the eigenvalue problem will be solved by a numerical technique below.


### 3.2. Finite Difference Method for Discretizing the Mode Equation

The differential schemes of individual terms in equation (10) are transformed into difference schemes after using the finite difference method for discretization. Figure 3 shows a schematic of a discretized inclined cable with n internal nodes and n + 1 segments. The projection length on the chord line of each cable segment is expressed as a = L/(n + 1).Figure 3
Schematic of a discretized inclined cable.Defining the difference schemes of mode shape functionswx in the following forms:(11)dwxidx=wi+1−wi−12a,d2wxidx2=wi+1−2wi+wi−1a2d3wxidx3=wi+2−2wi+1+2wi−1−wi−22a3,d4wxidx4=wi+2−4wi+1+6wi−4wi−1+wi−2a4.Furthermore, the difference schemes of EI(x), EA(x), and y(x) are the same as wx.Converting the differential scheme into the difference scheme at nodei (i = 1, 2, …, n) by adopting the abovementioned formulae, which then gives the matrix form of the discretized mode equation:(12)Kw+Cpw+Mp2w=0.In which,(13)K=K1+K2,wT=w1,w2,…,wn;M=diagm1,m2,…,mn;C=diagcv1′,cv2′,…,cvn′,where K1 and K2 are the linear stiffness matrix and the nonlinear stiffness matrix, respectively; n represents the number of internal nodes; wi and mi denote the mode shape displacement and mass per unit length at node i (i = 1, 2, …, n), respectively; cvi′=cvi/a, and cvi denotes the transverse in-plane viscous damping coefficient of the damper at node i.The linear stiffness matrixK1 is expressed as follows:(14)K1=QU1W10RS2U2W2V3D3S3U3W3−−−−−ViDiSiUiWi−−−−−Vn−2Dn−2Sn−2Un−2Wn−2Vn−1Dn−1Sn−1P0VnDnTn×n.Each internal node of the discretization model of the cable corresponds to one row ofK1, in which:(15)Vi=−12a4EIi+1−2EIi−EIi−1Wi=12a4EIi+1+2EIi−EIi−1Si=1a4−2EIi+1+10EIi−2EIi−1+2Hia2Di=1a42EIi+1−6EIi−Hia2+Hi+1−Hi−14a2Ui=1a4−6EIi+2EIi−1−Hia2−Hi+1−Hi−14a2,where EIi and Hi are the bending stiffness and the chord-wise component of the cable tensile force at the internal node i, respectively. D and U incorporate the effect associated with the inclined angle of the cable, compared with previous research on horizontal cables [8]. Meanwhile, the effect of the boundary conditions is incorporated into Q, T, R, and P. The expressions for them need to be determined based on the boundary equilibrium equations described below.The nonlinear stiffness matrixK2 is expressed as follows:(16)K2=rsT;rT=r1,r2,…,rn;sT=s1,s2,…,sn,in which,(17)ri=si∑i=1nti3/EAi;si=yi+1−2yi+yi−1a2;ti=yi+1−yi−1/2a2+11/2.It is necessary to solve the static equilibrium profile due to the self-weight of the cable because some of the terms used depend ony as is reflected in the expressions for K2.When the damping ratioζ is very small, the effect of damping can be neglected. At this time, the damped circular frequencies ωD is equal to the undamped circular frequencies ω of the free vibration of the cable, and p2 = –ω2. Then, equation (12) becomes(18)Kw−Mω2w=0,Equation (18) indicates that the modal frequencies and the mode shapes of a cable in free vibration are the generalized eigenvalues and eigenvectors of the mass matrix and stiffness matrix of a cable, respectively.


### 3.3. Boundary Constraint Equilibrium Equations

In order to calculate the elementsQ, T, R, and P in the matrix K1, the boundary conditions of the cable should be taken into account. In practical engineering, the constraints at the cable ends are usually neither simply pinned nor completely fixed. Herein, it is assumed that the boundary conditions of the cable are elastic support and elastic embedding at both ends. Figure 4 shows an inclined cable model under the boundary conditions of elastic support and elastic embedding, in which Kr1 and Kr2 are the rotational stiffness at the cable ends, and Ks1 and Ks2 are the lateral support stiffness at the cable ends.Figure 4
Schematic of a cable under the boundary of elastic embedding and elastic support.According to the equilibrium of the moment and shear force at the two ends of the cable, the boundary constraint equilibrium equations of the cable are derived as follows:(19)Kr1dwdxx=0−EId2wdx2x=0=0,Ks1wx=0−Hdwdxx=0+EId3wdx3x=0=0,(20)Kr2dwdxx=l+EId2wdx2x=l=0,Ks2wx=l+Hdwdxx=l−EId3wdx3x=l=0.After discretizing the calculation model of the cable, equations (19) and (20) become(21)Kr1w1−w−12a−EI0w1−2w0+w−1a2=0,(22)Ks1w0−H0w1−w−12a+Kr1w1−2w0+w−1a2=0,(23)Kr2wn+2−wn2a+EIn+1wn+2−2wn+1+wna2=0,(24)Ks2wn+1+Hn+1wn+2−wn2a+Kr2wn+2−2wn+1+wna2=0.To determine the elementsQ and T, one should replace w−1 and w0 with a factor containing w1 and replace wn+1 and wn+2 with a factor containing wn. Analogously, to determine the elements R and P, w0 should be expressed by a factor containing w1, and wn+1 should be expressed by a factor containing wn. Based on equations (23)–(26), one can obtain the following equations:(25)w0=2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12w1w−1=Kr1a−2EI0Kr1a+2EI0w1+4EI0Kr1a+2EI0⋅2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12w1wn+1=2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22wnwn+2=Kr2a−2EIn+1Kr2a+2EIn+1wn+4EIn+1Kr2a+2EIn+1⋅2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22wn.Then, the elementsQ, T, R, and P in the matrix K1 can be determined as follows:(26)Q=S1+Kr1a−2EI0Kr1a+2EI0V1+2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12D1+4EI0Kr1a+2EI0V1,(27)T=Sn+Kr2a−2EIn+1Kr2a+2EIn+1Wn+2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22Un+4EIn+1Kr2a+2EIn+1Wn,(28)R=D2+2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12V2,(29)P=Un−1+2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22Wn−1,where EI0 and EIn+1 denote the bending stiffness at the two ends, and H0 and Hn+1 denote the chord-wise component of the cable tensile force at the two ends.


### 3.4. Static Profile of the Cable

One of the premises of finding the value of each element in the stiffness matrixK2 is to obtain the static displacement yi (i = 1, 2, …, n) of each node of the cable under the static equilibrium state. That is, it is necessary to calculate the static equilibrium profile due to the self-weight of the cable. The static profile is often presumed to be a parabola [1]. This assumption is appropriate when the cable is an equal-section horizontal cable that does not consider the bending stiffness. However, when the bending stiffness of the cable cannot be neglected, or the sectional characteristic varies along the cable, the true static profile of the cable will deviate significantly from the parabola. Therefore, the finite difference method is employed to calculate the static profile of the cable in this study.The static equilibrium differential equation of an inclined cable subjected to tension is as follows:(30)d2dx2EId2ydx2−Hd2ydx2=mgcosθ.Similar to the discretization of the vibration mode equation, the abovementioned equation can also be written in the following matrix form:(31)Ksy=mgcosθinwhich,yT=y1,y2,…,yn;mT=m1,m2,…,mn,where Ks is the static stiffness matrix. This is basically the same as for K1 in equation (14) but with the expressions for elements Di and Ui in the matrix modified as follows:(32)Di=1a42EIi+1−6EIi−Hia2;Ui=1a4−6EIi+2EIi−1−Hia2.According to equation (31), the static profile can be calculated for solving the vibration mode equation.


### 3.5. Multiparameter Identification Algorithm

To accurately identify the cable tension utilizing the measured frequencies in situ, it is necessary to identify some other parameters in the cable vibration system besides the cable tension. The deformation mechanism of a beam and that of a cable are not exactly the same, thus, it is not always trustworthy to use the formulae for a beam to compute the axial stiffness and bending stiffness of a cable. Hence, the mean value of the chord-wise component of the cable tensionHm, the axial stiffness EA, the bending stiffness EI, the rotational constraint stiffness, Kr1 and Kr2, at the two ends of the cable, and the lateral support stiffness, Ks1 and Ks2, at the two ends of the cable, were selected to be the physical parameters that need to be identified in the cable vibration system. A frequency-based sensitivity-updating algorithm was applied for the multiparameter identification of the cable using multiorder frequencies, as shown in Figure 5.Figure 5
Block diagram of the sensitivity-updating algorithm for system parameter identification.The physical parameters to be identified in the cable vibration system constitute the parameter vectorX of the order of p × 1:(33)X=HmEIEAKr1Kr2Ks1Ks2T.Assuming that for thenth-order vibration eigenvalue λn = ωn2, then, the variation of λn is as follows:(34)δλn=λnX+∆X−λnX.Performing Taylor expansion on the first term of the right side of equation (34), one can obtain the following:(35)δλn=∑k=1p∂λn∂xk∆xk=∑k=1p∂λn∂xkxk∆xkxk,where xk is the kth element of the parameter vector X.IfN frequencies are used for the identification of system parameters, from equation (35) the following can be obtained:(36)αF=δλ,in which,(37)F=∆x1/x1eva∆x2/x2eva…∆xp/xpevaT,δλ=δλ1δλ2…δλNT,α=∂λ1∂x1x1eva∂λ1∂x2x2eva⋯∂λ1∂xpxpeva∂λ2∂x1x1eva∂λ2∂x2x2eva⋯∂λ2∂xpxpeva⋮⋮⋮⋮∂λN∂x1x1eva∂λN∂x2x2eva⋯∂λN∂xpxpeva,where xkeva is the initial evaluated value for the kth element of X.The coefficient matrixα in equation (36) is a Jacobian matrix combined with the initial evaluated values of cable parameters. The parameter vector X becomes a dimensionless vector F when the initial evaluated values of cable parameters are introduced into α. In this case, the sensitivity of vibration eigenvalues to the cable parameters is truly reflected in the coefficient matrix α, regardless of the absolute value of each element of X. Introducing the initial values into α is also an effective technique to decrease the condition number of α and improve the stability of solving the linear equations numerically.An iterative solution of equation (36) yields an accurate value for each parameter in the parameter vector X, which comprises the following nine steps:Step 1.
Select the appropriate initial value of each parameter. CalculateHmeva according to the practical formula [3]. Calculate EIeva and EAeva according to the formulae in material mechanics for calculating the bending stiffness and the axial stiffness. The dimension of Kr is the same as the dimension of EI/a, and the dimension of Ks is the same as the dimension of EI/a3. When the ratio of the Kr to EI/a exceeds 4.0, the frequencies of the cable are close to converging. Therefore, the search range of Kr1eva and Kr2eva can be set to be [0.0, 4.0]EI/a. Similarly, according to the result of the parameter sensitivity analysis, when the ratio of the Ks to EI/a3 exceeds 20.0, the frequencies of the cable are close to converging. Therefore, the search range of Ks1eva and Ks2eva can be set to be [0.0, 20.0]EI/a3. In practical engineering, it is essential to conduct a trial of convergence based on experience.Step 2.
Calculate the linear stiffness matrixK1 of the cable.Step 3.
Calculate the static profile of the cable.Step 4.
Calculate the nonlinear stiffness matrixK2 and obtain the overall stiffness matrix K.Step 5.
Perform eigenvalue analysis and obtain the calculated eigenvaluesλtn of the cable.Step 6.
Calculateδλ according to the following formula:(38)δλ=λm1−λt1λm2−λt2…λmn−λtn…λmN−λtNT,where λmn is the measured value (i.e., accurate value) of the nth-order eigenvalue and λtn is the nth-order calculated eigenvalue after t iterations.Step 7.
Calculate the matrixα. Calculation of the vibration eigenvalue derivatives ∂λn/∂xk is the key aspect of the calculation of matrix α. The analytical method for calculating the eigenvalue derivatives of the cable vibration will be described in detail in the Appendix.Step 8.
CalculateF according to equation (39). To obtain a definite solution for equation (39), the number of measured frequencies should be equal to or greater than the number of parameters to be estimated. When the number of measured frequencies is equal to the number of parameters to be identified, i.e., N = p, F can be solved by the following:(39)F=α−1δλ.
When the number of measured frequencies is greater than the number of parameters to be identified, i.e.,N > p, equation (36) becomes an overdetermined equation and can be solved by the least squares technique. The same method applies when N = p. The objective function of the least squares technique is the sum of squares of δλn = (λmn − λtn):(40)E=∑n=1Nδλn2=δλTδλ.
According to the principle of the least squares technique, ∂E/∂F = 0, the following expression can be obtained:(41)F=αTα−1αTδλFk,min<Fk<Fk,maxk=1,2,…,p,where Fk, min and Fk, max are employed to constrain the search range of xk. It is an efficient way to constrain the search ranges of parameters to enhance the robustness of the least squares technique. Based on the actual situation and engineering experience, the search range of xk can be predetermined approximately. For instance, in the cable tension calculation, the error of Hmeva with the calculation result using the practical formula [3] is generally within 20%. Therefore, the variation range of FHm can be constrained to within [–0.2, +0.2].Step 9.
The following equation is used to obtain the values of corrected system parameters, which will be used in the (t + 1)th iteration:(42)xkt+1=1+Fktxkt,where xkt is the kth identification parameter of Xt after the tth iteration. Fkt is the kth element of Ft after the tth iteration.
The abovementioned steps are repeated untilF approaches zero.


## 3.1. Free Vibration Analysis

### 3.1.1. Free Vibration Equation

Figure2 shows the schematic of an inclined cable model along with the coordinate system. Point A is taken as the coordinate origin. Compared with the previous study [6–8], the proposed model gives systematic consideration to the inclination angle, bending stiffness, sag-extensibility, and geometric nonlinearity of the cable. Especially, the complex boundary constraints at the cable ends were first comprehensively modeled, which would be introduced in detail in Section 3.3.Figure 2
Schematic of an inclined cable.Only the in-plane vibration of the cable is considered. Assume that it is uncoupled between the axial vibration and the transverse in-plane vibration of the cable. According to the D’Alembert principle, the equations of the chord-wise and transverse in-plane motions of the inclined cable are derived from the following equations:(1)∂∂sT+τdxds+∂u∂s+mgsinθ=m∂2u∂t2+cu′∂u∂t,(2)∂∂sT+τdyds+∂v∂s−∂2∂s2EI∂2η∂x2+mgcosθ=m∂2v∂t2+cv′∂v∂t,where T(x) denotes the initial cable tension; τx,t denotes the additional tension caused by vibration; s is the arc-length coordinate; yx denotes the static equilibrium profile; ux,t and vx,t represent the displacement in the x-direction and y-direction caused by vibration, respectively; ηx,t=y+v denotes the sum of the displacements in the y-direction; g denotes the gravitational acceleration; EI(x) denotes the bending stiffness; m(x) denotes the mass per unit length; θ denotes the inclination angle; cu′x and cv′x represent the axial and transverse in-plane viscous damping coefficient per unit length, respectively. In addition, Hm denotes the mean value of the chord-wise component of T(x); EA(x) denotes the axial stiffness; and L denotes the cable length.It is generally known thatT+τ=H+hds/dx, where H and h denote the chord-wise component of T and τ, respectively. Furthermore, the axial vibration of the cable can be neglected. Then, equations (1) and (2) become(3)∂∂sH+h+mgsinθ=0.(4)∂∂sH+hdydx+∂v∂x−∂2∂s2EI∂2η∂x2+mgcosθ=m∂2v∂t2+cv′∂v∂t.If the sag-to-span ratio of the cable is no more than 1/10, ∂s can be replaced by ∂x. In this case, we can find that ∂h/∂x = 0 according to equation (3), therefore, h(t) becomes a variable varying with time alone. Substituting the static equilibrium differential equation into equation (4) and neglecting the second-order terms results in(5)Hx∂2v∂x2+H′x∂v∂x+hd2ydx2−∂2∂x2EI∂2v∂x2=m∂2v∂t2+cv′∂v∂t.Equation (5) is the free vibration equation of an inclined cable, and the following steps are taken to transform it into an eigenvalue calculation.


### 3.1.2. Vibration Mode Equation

The vibration of a cable is the superposition of several modes, by using the mode separation method; the variablevx,t can be expressed as follows:(6)vx,t=wxqt,where wx denotes the mode shapes independent of time, and q(t) denotes the generalized coordinates dependent on time alone. Considering the damped free vibration of a cable, q is generally written as qt=ept, where p=−ζω±iωD, in which ζ denotes the damping ratio; ω and ωD denote the nondamping and damped circular frequencies of the free vibration of the cable, respectively.With the separation of the variablevx,t, equation (5) becomes(7)d2dx2EId2wdx2q−Hd2wdx2q−H′dwdxq−hd2ydx2+cv′pwq+mp2wq=0.Expressing the additional cable tension componenth as a linear correlation expression of the generalized coordinates q, then, the generalized coordinates q can be factored out as a common factor. Based on the relationship between the additional cable tension (τ) and the cable elongation, Irvine [1] derived the following expression:(8)∫0Lhds/dx3EAdx=∫0L∂u∂xdx+∫0Ldydx∂v∂xdx.Due to the assumption that chord-wise displacementu equals zero, the first term of the right side of equation (8) is zero. As h is a variable varying with time alone, h can be written as follows:(9)h=∫0Ldy/dx∂v/∂xdx∫0Lds/dx3/EAdx=−∫0Ld2y/dx2wdx∫0Ldy/dx2+13/2/EAdxq.Substituting equation (9) into (7), eliminating the common factor q, and expanding the first term of equation (7) results in(10)d2EIdx2d2wdx2+2dEIdxd3wdx3+EId4wdx4−Hd2wdx2−H′dwdx+∫0Ld2y/dx2wdx∫0Ldy/dx2+13/2/EAdxd2ydx2+cv′pw+mp2w=0.Equation (10) is the nonlinear vibration mode equation of an inclined cable. The first five terms of the left side are the linear stiffness part of the cable while the sixth term denotes the nonlinear stiffness part. By using the finite difference method to discretize this fourth-order nonlinear differential equation, the eigenvalue problem will be solved by a numerical technique below.


## 3.1.1. Free Vibration Equation

Figure2 shows the schematic of an inclined cable model along with the coordinate system. Point A is taken as the coordinate origin. Compared with the previous study [6–8], the proposed model gives systematic consideration to the inclination angle, bending stiffness, sag-extensibility, and geometric nonlinearity of the cable. Especially, the complex boundary constraints at the cable ends were first comprehensively modeled, which would be introduced in detail in Section 3.3.Figure 2
Schematic of an inclined cable.Only the in-plane vibration of the cable is considered. Assume that it is uncoupled between the axial vibration and the transverse in-plane vibration of the cable. According to the D’Alembert principle, the equations of the chord-wise and transverse in-plane motions of the inclined cable are derived from the following equations:(1)∂∂sT+τdxds+∂u∂s+mgsinθ=m∂2u∂t2+cu′∂u∂t,(2)∂∂sT+τdyds+∂v∂s−∂2∂s2EI∂2η∂x2+mgcosθ=m∂2v∂t2+cv′∂v∂t,where T(x) denotes the initial cable tension; τx,t denotes the additional tension caused by vibration; s is the arc-length coordinate; yx denotes the static equilibrium profile; ux,t and vx,t represent the displacement in the x-direction and y-direction caused by vibration, respectively; ηx,t=y+v denotes the sum of the displacements in the y-direction; g denotes the gravitational acceleration; EI(x) denotes the bending stiffness; m(x) denotes the mass per unit length; θ denotes the inclination angle; cu′x and cv′x represent the axial and transverse in-plane viscous damping coefficient per unit length, respectively. In addition, Hm denotes the mean value of the chord-wise component of T(x); EA(x) denotes the axial stiffness; and L denotes the cable length.It is generally known thatT+τ=H+hds/dx, where H and h denote the chord-wise component of T and τ, respectively. Furthermore, the axial vibration of the cable can be neglected. Then, equations (1) and (2) become(3)∂∂sH+h+mgsinθ=0.(4)∂∂sH+hdydx+∂v∂x−∂2∂s2EI∂2η∂x2+mgcosθ=m∂2v∂t2+cv′∂v∂t.If the sag-to-span ratio of the cable is no more than 1/10, ∂s can be replaced by ∂x. In this case, we can find that ∂h/∂x = 0 according to equation (3), therefore, h(t) becomes a variable varying with time alone. Substituting the static equilibrium differential equation into equation (4) and neglecting the second-order terms results in(5)Hx∂2v∂x2+H′x∂v∂x+hd2ydx2−∂2∂x2EI∂2v∂x2=m∂2v∂t2+cv′∂v∂t.Equation (5) is the free vibration equation of an inclined cable, and the following steps are taken to transform it into an eigenvalue calculation.


## 3.1.2. Vibration Mode Equation

The vibration of a cable is the superposition of several modes, by using the mode separation method; the variablevx,t can be expressed as follows:(6)vx,t=wxqt,where wx denotes the mode shapes independent of time, and q(t) denotes the generalized coordinates dependent on time alone. Considering the damped free vibration of a cable, q is generally written as qt=ept, where p=−ζω±iωD, in which ζ denotes the damping ratio; ω and ωD denote the nondamping and damped circular frequencies of the free vibration of the cable, respectively.With the separation of the variablevx,t, equation (5) becomes(7)d2dx2EId2wdx2q−Hd2wdx2q−H′dwdxq−hd2ydx2+cv′pwq+mp2wq=0.Expressing the additional cable tension componenth as a linear correlation expression of the generalized coordinates q, then, the generalized coordinates q can be factored out as a common factor. Based on the relationship between the additional cable tension (τ) and the cable elongation, Irvine [1] derived the following expression:(8)∫0Lhds/dx3EAdx=∫0L∂u∂xdx+∫0Ldydx∂v∂xdx.Due to the assumption that chord-wise displacementu equals zero, the first term of the right side of equation (8) is zero. As h is a variable varying with time alone, h can be written as follows:(9)h=∫0Ldy/dx∂v/∂xdx∫0Lds/dx3/EAdx=−∫0Ld2y/dx2wdx∫0Ldy/dx2+13/2/EAdxq.Substituting equation (9) into (7), eliminating the common factor q, and expanding the first term of equation (7) results in(10)d2EIdx2d2wdx2+2dEIdxd3wdx3+EId4wdx4−Hd2wdx2−H′dwdx+∫0Ld2y/dx2wdx∫0Ldy/dx2+13/2/EAdxd2ydx2+cv′pw+mp2w=0.Equation (10) is the nonlinear vibration mode equation of an inclined cable. The first five terms of the left side are the linear stiffness part of the cable while the sixth term denotes the nonlinear stiffness part. By using the finite difference method to discretize this fourth-order nonlinear differential equation, the eigenvalue problem will be solved by a numerical technique below.


## 3.2. Finite Difference Method for Discretizing the Mode Equation

The differential schemes of individual terms in equation (10) are transformed into difference schemes after using the finite difference method for discretization. Figure 3 shows a schematic of a discretized inclined cable with n internal nodes and n + 1 segments. The projection length on the chord line of each cable segment is expressed as a = L/(n + 1).Figure 3
Schematic of a discretized inclined cable.Defining the difference schemes of mode shape functionswx in the following forms:(11)dwxidx=wi+1−wi−12a,d2wxidx2=wi+1−2wi+wi−1a2d3wxidx3=wi+2−2wi+1+2wi−1−wi−22a3,d4wxidx4=wi+2−4wi+1+6wi−4wi−1+wi−2a4.Furthermore, the difference schemes of EI(x), EA(x), and y(x) are the same as wx.Converting the differential scheme into the difference scheme at nodei (i = 1, 2, …, n) by adopting the abovementioned formulae, which then gives the matrix form of the discretized mode equation:(12)Kw+Cpw+Mp2w=0.In which,(13)K=K1+K2,wT=w1,w2,…,wn;M=diagm1,m2,…,mn;C=diagcv1′,cv2′,…,cvn′,where K1 and K2 are the linear stiffness matrix and the nonlinear stiffness matrix, respectively; n represents the number of internal nodes; wi and mi denote the mode shape displacement and mass per unit length at node i (i = 1, 2, …, n), respectively; cvi′=cvi/a, and cvi denotes the transverse in-plane viscous damping coefficient of the damper at node i.The linear stiffness matrixK1 is expressed as follows:(14)K1=QU1W10RS2U2W2V3D3S3U3W3−−−−−ViDiSiUiWi−−−−−Vn−2Dn−2Sn−2Un−2Wn−2Vn−1Dn−1Sn−1P0VnDnTn×n.Each internal node of the discretization model of the cable corresponds to one row ofK1, in which:(15)Vi=−12a4EIi+1−2EIi−EIi−1Wi=12a4EIi+1+2EIi−EIi−1Si=1a4−2EIi+1+10EIi−2EIi−1+2Hia2Di=1a42EIi+1−6EIi−Hia2+Hi+1−Hi−14a2Ui=1a4−6EIi+2EIi−1−Hia2−Hi+1−Hi−14a2,where EIi and Hi are the bending stiffness and the chord-wise component of the cable tensile force at the internal node i, respectively. D and U incorporate the effect associated with the inclined angle of the cable, compared with previous research on horizontal cables [8]. Meanwhile, the effect of the boundary conditions is incorporated into Q, T, R, and P. The expressions for them need to be determined based on the boundary equilibrium equations described below.The nonlinear stiffness matrixK2 is expressed as follows:(16)K2=rsT;rT=r1,r2,…,rn;sT=s1,s2,…,sn,in which,(17)ri=si∑i=1nti3/EAi;si=yi+1−2yi+yi−1a2;ti=yi+1−yi−1/2a2+11/2.It is necessary to solve the static equilibrium profile due to the self-weight of the cable because some of the terms used depend ony as is reflected in the expressions for K2.When the damping ratioζ is very small, the effect of damping can be neglected. At this time, the damped circular frequencies ωD is equal to the undamped circular frequencies ω of the free vibration of the cable, and p2 = –ω2. Then, equation (12) becomes(18)Kw−Mω2w=0,Equation (18) indicates that the modal frequencies and the mode shapes of a cable in free vibration are the generalized eigenvalues and eigenvectors of the mass matrix and stiffness matrix of a cable, respectively.


## 3.3. Boundary Constraint Equilibrium Equations

In order to calculate the elementsQ, T, R, and P in the matrix K1, the boundary conditions of the cable should be taken into account. In practical engineering, the constraints at the cable ends are usually neither simply pinned nor completely fixed. Herein, it is assumed that the boundary conditions of the cable are elastic support and elastic embedding at both ends. Figure 4 shows an inclined cable model under the boundary conditions of elastic support and elastic embedding, in which Kr1 and Kr2 are the rotational stiffness at the cable ends, and Ks1 and Ks2 are the lateral support stiffness at the cable ends.Figure 4
Schematic of a cable under the boundary of elastic embedding and elastic support.According to the equilibrium of the moment and shear force at the two ends of the cable, the boundary constraint equilibrium equations of the cable are derived as follows:(19)Kr1dwdxx=0−EId2wdx2x=0=0,Ks1wx=0−Hdwdxx=0+EId3wdx3x=0=0,(20)Kr2dwdxx=l+EId2wdx2x=l=0,Ks2wx=l+Hdwdxx=l−EId3wdx3x=l=0.After discretizing the calculation model of the cable, equations (19) and (20) become(21)Kr1w1−w−12a−EI0w1−2w0+w−1a2=0,(22)Ks1w0−H0w1−w−12a+Kr1w1−2w0+w−1a2=0,(23)Kr2wn+2−wn2a+EIn+1wn+2−2wn+1+wna2=0,(24)Ks2wn+1+Hn+1wn+2−wn2a+Kr2wn+2−2wn+1+wna2=0.To determine the elementsQ and T, one should replace w−1 and w0 with a factor containing w1 and replace wn+1 and wn+2 with a factor containing wn. Analogously, to determine the elements R and P, w0 should be expressed by a factor containing w1, and wn+1 should be expressed by a factor containing wn. Based on equations (23)–(26), one can obtain the following equations:(25)w0=2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12w1w−1=Kr1a−2EI0Kr1a+2EI0w1+4EI0Kr1a+2EI0⋅2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12w1wn+1=2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22wnwn+2=Kr2a−2EIn+1Kr2a+2EIn+1wn+4EIn+1Kr2a+2EIn+1⋅2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22wn.Then, the elementsQ, T, R, and P in the matrix K1 can be determined as follows:(26)Q=S1+Kr1a−2EI0Kr1a+2EI0V1+2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12D1+4EI0Kr1a+2EI0V1,(27)T=Sn+Kr2a−2EIn+1Kr2a+2EIn+1Wn+2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22Un+4EIn+1Kr2a+2EIn+1Wn,(28)R=D2+2EI0H0−2Kr12Ks1aKr1a+2EI0+2EI0H0−2Kr12V2,(29)P=Un−1+2EIn+1Hn+1−2Kr22Ks2aKr2a+2EIn+1+2EIn+1Hn+1−2Kr22Wn−1,where EI0 and EIn+1 denote the bending stiffness at the two ends, and H0 and Hn+1 denote the chord-wise component of the cable tensile force at the two ends.


## 3.4. Static Profile of the Cable

One of the premises of finding the value of each element in the stiffness matrixK2 is to obtain the static displacement yi (i = 1, 2, …, n) of each node of the cable under the static equilibrium state. That is, it is necessary to calculate the static equilibrium profile due to the self-weight of the cable. The static profile is often presumed to be a parabola [1]. This assumption is appropriate when the cable is an equal-section horizontal cable that does not consider the bending stiffness. However, when the bending stiffness of the cable cannot be neglected, or the sectional characteristic varies along the cable, the true static profile of the cable will deviate significantly from the parabola. Therefore, the finite difference method is employed to calculate the static profile of the cable in this study.The static equilibrium differential equation of an inclined cable subjected to tension is as follows:(30)d2dx2EId2ydx2−Hd2ydx2=mgcosθ.Similar to the discretization of the vibration mode equation, the abovementioned equation can also be written in the following matrix form:(31)Ksy=mgcosθinwhich,yT=y1,y2,…,yn;mT=m1,m2,…,mn,where Ks is the static stiffness matrix. This is basically the same as for K1 in equation (14) but with the expressions for elements Di and Ui in the matrix modified as follows:(32)Di=1a42EIi+1−6EIi−Hia2;Ui=1a4−6EIi+2EIi−1−Hia2.According to equation (31), the static profile can be calculated for solving the vibration mode equation.


## 3.5. Multiparameter Identification Algorithm

To accurately identify the cable tension utilizing the measured frequencies in situ, it is necessary to identify some other parameters in the cable vibration system besides the cable tension. The deformation mechanism of a beam and that of a cable are not exactly the same, thus, it is not always trustworthy to use the formulae for a beam to compute the axial stiffness and bending stiffness of a cable. Hence, the mean value of the chord-wise component of the cable tensionHm, the axial stiffness EA, the bending stiffness EI, the rotational constraint stiffness, Kr1 and Kr2, at the two ends of the cable, and the lateral support stiffness, Ks1 and Ks2, at the two ends of the cable, were selected to be the physical parameters that need to be identified in the cable vibration system. A frequency-based sensitivity-updating algorithm was applied for the multiparameter identification of the cable using multiorder frequencies, as shown in Figure 5.Figure 5
Block diagram of the sensitivity-updating algorithm for system parameter identification.The physical parameters to be identified in the cable vibration system constitute the parameter vectorX of the order of p × 1:(33)X=HmEIEAKr1Kr2Ks1Ks2T.Assuming that for thenth-order vibration eigenvalue λn = ωn2, then, the variation of λn is as follows:(34)δλn=λnX+∆X−λnX.Performing Taylor expansion on the first term of the right side of equation (34), one can obtain the following:(35)δλn=∑k=1p∂λn∂xk∆xk=∑k=1p∂λn∂xkxk∆xkxk,where xk is the kth element of the parameter vector X.IfN frequencies are used for the identification of system parameters, from equation (35) the following can be obtained:(36)αF=δλ,in which,(37)F=∆x1/x1eva∆x2/x2eva…∆xp/xpevaT,δλ=δλ1δλ2…δλNT,α=∂λ1∂x1x1eva∂λ1∂x2x2eva⋯∂λ1∂xpxpeva∂λ2∂x1x1eva∂λ2∂x2x2eva⋯∂λ2∂xpxpeva⋮⋮⋮⋮∂λN∂x1x1eva∂λN∂x2x2eva⋯∂λN∂xpxpeva,where xkeva is the initial evaluated value for the kth element of X.The coefficient matrixα in equation (36) is a Jacobian matrix combined with the initial evaluated values of cable parameters. The parameter vector X becomes a dimensionless vector F when the initial evaluated values of cable parameters are introduced into α. In this case, the sensitivity of vibration eigenvalues to the cable parameters is truly reflected in the coefficient matrix α, regardless of the absolute value of each element of X. Introducing the initial values into α is also an effective technique to decrease the condition number of α and improve the stability of solving the linear equations numerically.An iterative solution of equation (36) yields an accurate value for each parameter in the parameter vector X, which comprises the following nine steps:Step 1.
Select the appropriate initial value of each parameter. CalculateHmeva according to the practical formula [3]. Calculate EIeva and EAeva according to the formulae in material mechanics for calculating the bending stiffness and the axial stiffness. The dimension of Kr is the same as the dimension of EI/a, and the dimension of Ks is the same as the dimension of EI/a3. When the ratio of the Kr to EI/a exceeds 4.0, the frequencies of the cable are close to converging. Therefore, the search range of Kr1eva and Kr2eva can be set to be [0.0, 4.0]EI/a. Similarly, according to the result of the parameter sensitivity analysis, when the ratio of the Ks to EI/a3 exceeds 20.0, the frequencies of the cable are close to converging. Therefore, the search range of Ks1eva and Ks2eva can be set to be [0.0, 20.0]EI/a3. In practical engineering, it is essential to conduct a trial of convergence based on experience.Step 2.
Calculate the linear stiffness matrixK1 of the cable.Step 3.
Calculate the static profile of the cable.Step 4.
Calculate the nonlinear stiffness matrixK2 and obtain the overall stiffness matrix K.Step 5.
Perform eigenvalue analysis and obtain the calculated eigenvaluesλtn of the cable.Step 6.
Calculateδλ according to the following formula:(38)δλ=λm1−λt1λm2−λt2…λmn−λtn…λmN−λtNT,where λmn is the measured value (i.e., accurate value) of the nth-order eigenvalue and λtn is the nth-order calculated eigenvalue after t iterations.Step 7.
Calculate the matrixα. Calculation of the vibration eigenvalue derivatives ∂λn/∂xk is the key aspect of the calculation of matrix α. The analytical method for calculating the eigenvalue derivatives of the cable vibration will be described in detail in the Appendix.Step 8.
CalculateF according to equation (39). To obtain a definite solution for equation (39), the number of measured frequencies should be equal to or greater than the number of parameters to be estimated. When the number of measured frequencies is equal to the number of parameters to be identified, i.e., N = p, F can be solved by the following:(39)F=α−1δλ.
When the number of measured frequencies is greater than the number of parameters to be identified, i.e.,N > p, equation (36) becomes an overdetermined equation and can be solved by the least squares technique. The same method applies when N = p. The objective function of the least squares technique is the sum of squares of δλn = (λmn − λtn):(40)E=∑n=1Nδλn2=δλTδλ.
According to the principle of the least squares technique, ∂E/∂F = 0, the following expression can be obtained:(41)F=αTα−1αTδλFk,min<Fk<Fk,maxk=1,2,…,p,where Fk, min and Fk, max are employed to constrain the search range of xk. It is an efficient way to constrain the search ranges of parameters to enhance the robustness of the least squares technique. Based on the actual situation and engineering experience, the search range of xk can be predetermined approximately. For instance, in the cable tension calculation, the error of Hmeva with the calculation result using the practical formula [3] is generally within 20%. Therefore, the variation range of FHm can be constrained to within [–0.2, +0.2].Step 9.
The following equation is used to obtain the values of corrected system parameters, which will be used in the (t + 1)th iteration:(42)xkt+1=1+Fktxkt,where xkt is the kth identification parameter of Xt after the tth iteration. Fkt is the kth element of Ft after the tth iteration.
The abovementioned steps are repeated untilF approaches zero.


## 4. Case Study

### 4.1. Introduction to Numerical Cables

Four numerical cables with significantly different characteristics used in the previous studies [8, 11, 15] were adopted to evaluate the method of the present paper. Their main physical parameters are listed in Table 1. Except for those, they have the same length (L = 100 m) and mass per unit length (m = 400 kg/m). Moreover, the cables’ inclination angles θ were all set to 30° and the damping of the cables was ignored. In the discretized model, the cable was divided into 100 segments along the x-direction, with a total of 99 internal nodes (n = 99 and a = 1 m).Table 1
Physical parameters of four numerical cables.
NoHm (106N)E (Pa)A (m2)I (m4)λ2ξCable 12.90361.5988E + 107.8507E − 034.9535E − 060.79605.5Cable 20.72591.7186E + 107.611E − 034.6097E − 0650.70302.7Cable 326.132542.0826E + 137.8633E − 034.9204E − 061.4150.5Cable 40.72594.7834E + 80.7345E − 015.9506E − 0350.7050.5Note. The nondimensional parameters λ2 [3] and ξ [1] characterize the flexural stiffness and sag-extensibility of the cable, respectively. The cable sag-extensibility increases with λ2, and the flexural stiffness decreases with ξ.It can be found in Table1 that Cable 1 has small sag-extensibility and small flexural stiffness; Cable 2 has large sag-extensibility and small flexural stiffness; Cable 3 has small sag-extensibility and large flexural stiffness; and Cable 4 has large sag-extensibility and large flexural stiffness. They were believed to be able to cover a variety of cable features in practical engineering.


### 4.2. Influence of Boundary on Structural Behaviors

#### 4.2.1. Influence on Cable’s Static Profile

The static displacementyi of the cable participated in the formation of matrix K2. In other words, the change of the cable’s static profile caused by the variation of boundary conditions will affect the cable’s nonlinear stiffness matrix. First, to investigate the effect of the rotational stiffnesses at the cable ends on the cable’s static profile, taking Cable 3 as an example, assuming the boundary rotational stiffnesses Kr at two ends were the same and the boundary lateral support stiffnesses Ks were infinity, the static profiles of the cables with different Kr (i.e., EI/a, 2 EI/a, 4 EI/a, and ∞) were calculated and compared, as shown in Figure 6(a). It could be seen that the static deformation of the cable declined with the increase of Kr. However, generally speaking, Kr exhibited a slight effect on the cable’s static profile.Figure 6
Influence of (a) rotational constraint stiffnessKr and (b) lateral support constraint stiffness Ks at the cable ends on cable’s static profile.
(a)(b)Next, as depicted in Figure6(b), the effect of the lateral support stiffnesses at the cable ends on the cable’s static profile was analyzed, taking Cable 2 as an example. Analogously, the boundary lateral support stiffnesses Ks were assumed to be the same at the two ends, and the boundary rotational stiffnesses Kr at the two ends were both set as EI/a. Five conditions were analyzed, namely, Ks = EI/a3, 2 EI/a3, 4 EI/a3, 8 EI/a3, and ∞, respectively. One can find that with the decrease of Ks, the cable’s static profile presented an approximate overall translation, in addition to the increase of the static deformation of the cable. Moreover, when Ks became larger, the sensitivity of the cable’s static profile to Ks decreased.


#### 4.2.2. Influence on Cable’s Vibration Frequencies

As shown in Figure7, the first two frequencies of the vibration of each cable under different boundary conditions were calculated and compared, to learn the influence of the boundary on the cable’s vibration characteristics. Noteworthy is that the boundary constraints at the two cable ends were assumed to be the same during the analysis.Figure 7
Influence of boundary constraint stiffnesses on cable’s first two vibration frequencies.First, keeping the lateral support stiffnessKs being infinity, the first two frequencies of the cables with different boundary rotational stiffnesses Kr (i.e., 0, EI/a, 2 EI/a, and ∞) were compared. One can find in Figure 7 that Kr had a significant influence on Cables 3 and 4 with large flexural stiffnesses, which indicated that the sensitivity of the cable vibration frequency to Kr is affected by the flexural stiffness of the cable. However, larger sag would weaken the influence of Kr. For example, the frequencies of Cable 4 with large sag showed weaker sensitivity to Kr compared to Cable 3 with small sag. These results proved that Kr mainly affects the dynamic characteristics of the cables with large flexural stiffness but small sag.Then, keeping the rotational stiffnessKr at boundary being EI/a, the first two frequencies of the cables with different lateral support stiffnesses Ks (i.e., EI/a3, 2 EI/a3, 10 EI/a3, and ∞) were also compared. It could be found in Figure 7 that the Ks presented a complex influence on the cables’ vibration frequencies. With the increase of Ks: the frequencies of Cables 1 and 2 increased significantly; the frequencies of Cable 3 decreased slightly; and the frequencies of Cable 4 did not change much. The reason could be found in the expressions of the elements Q, T, R, and P within the stiffness matrix K1 shown in equations (29)–(32). One can find that there is a common term EI⋅H−Kr2 in the numerators and denominators of these equations and the term Ks appears only in the denominators. Therefore, the size relationship between Kr2 and EI⋅H will determine the influence of Ks on the elements Q, T, R, and P within the stiffness matrix K1 and further the cables’ vibration frequencies.


### 4.3. Validation of Eigenvalue Derivative Calculation Results

As stated before, the calculation of the vibration eigenvalue derivatives ∂λn/∂xk is the key aspect of the calculation of matrix α. To verify the derived equations for computing the eigenvalue derivatives, the eigenvalue derivatives were first transformed into frequency derivatives using the following equation derived from λ = ω2 = (2πf)2,(43)∂f∂xk=18π2f∂λ∂xk.Then, the calculated frequency derivatives with respect to each cable parameter were presented visually in the form of tangents at several selected points on the frequency-parameter curve (i.e.,f-xk curve). Taking Cable 2 as an example, the derivatives of the first-order frequency with respect to its multiple system parameters were calculated and illustrated in Figure 8 (due to the limited space, the derivatives of the frequencies of the other orders were not shown here). It could be seen that the calculated tangents (blue short solid lines) had good agreement with the actual curve slopes on each frequency-parameter curve, proving the accuracy of the analytical method for calculating ∂λn/∂xk derived in this article.Figure 8
Illustration of calculated derivatives of the first eigenvalue (frequency) of cable 2 with respect to its system parameters: (a)Hm; (b) EI; (c) EA; (d) Kr; and (e) Ks.
(a)(b)(c)(d)(e)Noteworthy is that, when obtaining the curves that frequency varies withHm, EI, and EA, respectively, in Figure 8, the boundary conditions of the cable were assumed to be fixed at the two ends; when obtaining the curve that frequency varies with Kr, it was assumed that the boundary constraints at the two cable ends were the same and Ks were infinity; and when obtaining the curve that frequency varies with Ks, it was assumed that the boundary constraints at the two cable ends were the same and Kr = EI/a.


### 4.4. Multiparameter Identification of the Cable

#### 4.4.1. Model Setup

The four numerical cables in Table1 were adopted to illustrate the effectiveness of the frequency-based sensitivity-updating algorithm for identifying multiple system parameters using the multiorder frequencies. To consider the effects of elastic embedding and elastic support at the cable boundary, for all four cables, assume that the accurate values of the rotational constraint stiffnesses (Kr1 and Kr2) at the two ends of the cables were 2 EI/a and 4 EI/a, respectively, and that the accurate values of the lateral support stiffness (Ks1 and Ks2) at the two ends of the cables were 5 EI/a3 and 10 EI/a3, respectively. Moreover, in the first iteration, it was assumed that the errors of the initial iteration values of Hmeva, EIeva, and EAeva of each cable were all 20%, and the errors of the initial iteration values of Kr1eva, Kr2eva, Ks1eva, and Ks2eva of each cable were all 50%, as shown in Table 2. The accurate frequencies of the cables, fmn, which should be measured in the field in practical engineering, were calculated from the accurate system parameters in Table 2 in this paper, as listed in Table 3.Table 2
Accurate and initial values of system parameters of the four cables.
NoItemsHm (N)EI (Pa∙m4)EA (Pa∙m2)Kr1 (N∙m)Kr2 (N∙m)Ks1 (N/m)Ks2 (N/m)Cable 1Accurate value2.9036E67.9197E41.2552E81.5839E53.1679E53.9599E57.9197E5Initial value2.3229E66.3358E41.0042E87.9197E41.5839E51.9799E53.9599E5Initial error (%)20202050505050Cable 2Accurate value7.2590E057.9222E041.3080E081.5844E053.1689E53.9611E57.9222E5Initial value5.8072E056.3378E041.0464E087.9222E41.5844E51.9806E53.9611E5Initial error (%)20202050505050Cable 3Accurate value2.6133E71.0244E81.6371E112.0488E84.0976E85.1220E81.0244E9Initial value2.0906E78.1952E71.3097E111.0244E82.0488E82.5610E85.1220E8Initial error (%)20202050505050Cable 4Accurate value7.2590E52.8464E61.3080E85.6928E61.1386E71.4232E72.8464E7Initial value5.8072E52.2771E61.0464E82.8464E65.6928E67.1160E61.4232E7Initial error (%)20202050505050Table 3
Accurate modal frequencies of the four cables (Hz).
No1st-order2nd-order3rd-order4th-order5th-order6th-order7th-orderCable 10.4200.8211.2321.6442.0572.4702.885Cable 20.4100.4310.6480.8501.0631.2751.488Cable 31.3742.6714.0445.4596.9328.47210.092Cable 40.4140.4450.6780.9091.1541.4111.682


#### 4.4.2. Results

Using the method described in Section3.5, the identification of the multiple system parameters of the four numerical cables was complemented based on their true model frequencies listed in Table 3. In order to enhance the robustness of the iteration, the variation ranges of FHm, FEI, and FEA were constrained to within [–0.3, +0.3], and the variation ranges of FKr1, FKr1, FKs1, and FKs2 were constrained to within [–0.5, +0.5].Figure9 shows the convergence process of the relative errors of cable system parameters with the iteration times. It can be observed that the cable system parameters obtained after the first iteration still had relatively large errors. After a total of 10 iterations, the relative errors of cable system parameters converged to zero for all four cables. At this time, the system parameters tended to be stable and to converge to the accurate values. The abovementioned results indicated that the frequency-based sensitivity-updating algorithm could be adopted to identify the multiple system parameters quickly and effectively for different kinds of cables.Figure 9
Convergence of the relative errors of cable system parameters with iteration times: (a) Cable 1; (b) Cable 2; (c) Cable 3; and (d) Cable 4.
(a)(b)(c)(d)To give more details, taking Cable 1 as an example, the convergence of the correction coefficients and system parameters of Cable 1 is shown in Figure10. It could be seen that, among the system parameters of the cable, Hm and EA converged faster. After 3 iterations, their identification errors were less than 0.5%. In contrast, the other system parameters converged relatively slowly. After 8 iterations, the identification errors of all system parameters were less than 0.5%. Figure 11 depicts the comparison between the frequencies calculated from the initial iterative values, final iterative values, and accurate values of the cable parameters, respectively. It can be observed that the errors of the frequencies calculated from the initial iterative values of the system parameters were very large. After repeated iterations, the frequencies calculated from the final iterative values showed good agreement with the accurate frequencies. This also illustrated the effectiveness of the frequency-based sensitivity-updating algorithm.Figure 10
Convergence of (a) correction coefficientsFk, (b) Hm, (c) EI, (d) EA, (e) Kr1, (f) Kr2, (g) Ks1, and (h) Ks2 of cable 1 during the identification process.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 11
Comparison of iteration results with accurate values of modal frequencies of cable 1.


## 4.1. Introduction to Numerical Cables

Four numerical cables with significantly different characteristics used in the previous studies [8, 11, 15] were adopted to evaluate the method of the present paper. Their main physical parameters are listed in Table 1. Except for those, they have the same length (L = 100 m) and mass per unit length (m = 400 kg/m). Moreover, the cables’ inclination angles θ were all set to 30° and the damping of the cables was ignored. In the discretized model, the cable was divided into 100 segments along the x-direction, with a total of 99 internal nodes (n = 99 and a = 1 m).Table 1
Physical parameters of four numerical cables.
NoHm (106N)E (Pa)A (m2)I (m4)λ2ξCable 12.90361.5988E + 107.8507E − 034.9535E − 060.79605.5Cable 20.72591.7186E + 107.611E − 034.6097E − 0650.70302.7Cable 326.132542.0826E + 137.8633E − 034.9204E − 061.4150.5Cable 40.72594.7834E + 80.7345E − 015.9506E − 0350.7050.5Note. The nondimensional parameters λ2 [3] and ξ [1] characterize the flexural stiffness and sag-extensibility of the cable, respectively. The cable sag-extensibility increases with λ2, and the flexural stiffness decreases with ξ.It can be found in Table1 that Cable 1 has small sag-extensibility and small flexural stiffness; Cable 2 has large sag-extensibility and small flexural stiffness; Cable 3 has small sag-extensibility and large flexural stiffness; and Cable 4 has large sag-extensibility and large flexural stiffness. They were believed to be able to cover a variety of cable features in practical engineering.


## 4.2. Influence of Boundary on Structural Behaviors

### 4.2.1. Influence on Cable’s Static Profile

The static displacementyi of the cable participated in the formation of matrix K2. In other words, the change of the cable’s static profile caused by the variation of boundary conditions will affect the cable’s nonlinear stiffness matrix. First, to investigate the effect of the rotational stiffnesses at the cable ends on the cable’s static profile, taking Cable 3 as an example, assuming the boundary rotational stiffnesses Kr at two ends were the same and the boundary lateral support stiffnesses Ks were infinity, the static profiles of the cables with different Kr (i.e., EI/a, 2 EI/a, 4 EI/a, and ∞) were calculated and compared, as shown in Figure 6(a). It could be seen that the static deformation of the cable declined with the increase of Kr. However, generally speaking, Kr exhibited a slight effect on the cable’s static profile.Figure 6
Influence of (a) rotational constraint stiffnessKr and (b) lateral support constraint stiffness Ks at the cable ends on cable’s static profile.
(a)(b)Next, as depicted in Figure6(b), the effect of the lateral support stiffnesses at the cable ends on the cable’s static profile was analyzed, taking Cable 2 as an example. Analogously, the boundary lateral support stiffnesses Ks were assumed to be the same at the two ends, and the boundary rotational stiffnesses Kr at the two ends were both set as EI/a. Five conditions were analyzed, namely, Ks = EI/a3, 2 EI/a3, 4 EI/a3, 8 EI/a3, and ∞, respectively. One can find that with the decrease of Ks, the cable’s static profile presented an approximate overall translation, in addition to the increase of the static deformation of the cable. Moreover, when Ks became larger, the sensitivity of the cable’s static profile to Ks decreased.


### 4.2.2. Influence on Cable’s Vibration Frequencies

As shown in Figure7, the first two frequencies of the vibration of each cable under different boundary conditions were calculated and compared, to learn the influence of the boundary on the cable’s vibration characteristics. Noteworthy is that the boundary constraints at the two cable ends were assumed to be the same during the analysis.Figure 7
Influence of boundary constraint stiffnesses on cable’s first two vibration frequencies.First, keeping the lateral support stiffnessKs being infinity, the first two frequencies of the cables with different boundary rotational stiffnesses Kr (i.e., 0, EI/a, 2 EI/a, and ∞) were compared. One can find in Figure 7 that Kr had a significant influence on Cables 3 and 4 with large flexural stiffnesses, which indicated that the sensitivity of the cable vibration frequency to Kr is affected by the flexural stiffness of the cable. However, larger sag would weaken the influence of Kr. For example, the frequencies of Cable 4 with large sag showed weaker sensitivity to Kr compared to Cable 3 with small sag. These results proved that Kr mainly affects the dynamic characteristics of the cables with large flexural stiffness but small sag.Then, keeping the rotational stiffnessKr at boundary being EI/a, the first two frequencies of the cables with different lateral support stiffnesses Ks (i.e., EI/a3, 2 EI/a3, 10 EI/a3, and ∞) were also compared. It could be found in Figure 7 that the Ks presented a complex influence on the cables’ vibration frequencies. With the increase of Ks: the frequencies of Cables 1 and 2 increased significantly; the frequencies of Cable 3 decreased slightly; and the frequencies of Cable 4 did not change much. The reason could be found in the expressions of the elements Q, T, R, and P within the stiffness matrix K1 shown in equations (29)–(32). One can find that there is a common term EI⋅H−Kr2 in the numerators and denominators of these equations and the term Ks appears only in the denominators. Therefore, the size relationship between Kr2 and EI⋅H will determine the influence of Ks on the elements Q, T, R, and P within the stiffness matrix K1 and further the cables’ vibration frequencies.


## 4.2.1. Influence on Cable’s Static Profile

The static displacementyi of the cable participated in the formation of matrix K2. In other words, the change of the cable’s static profile caused by the variation of boundary conditions will affect the cable’s nonlinear stiffness matrix. First, to investigate the effect of the rotational stiffnesses at the cable ends on the cable’s static profile, taking Cable 3 as an example, assuming the boundary rotational stiffnesses Kr at two ends were the same and the boundary lateral support stiffnesses Ks were infinity, the static profiles of the cables with different Kr (i.e., EI/a, 2 EI/a, 4 EI/a, and ∞) were calculated and compared, as shown in Figure 6(a). It could be seen that the static deformation of the cable declined with the increase of Kr. However, generally speaking, Kr exhibited a slight effect on the cable’s static profile.Figure 6
Influence of (a) rotational constraint stiffnessKr and (b) lateral support constraint stiffness Ks at the cable ends on cable’s static profile.
(a)(b)Next, as depicted in Figure6(b), the effect of the lateral support stiffnesses at the cable ends on the cable’s static profile was analyzed, taking Cable 2 as an example. Analogously, the boundary lateral support stiffnesses Ks were assumed to be the same at the two ends, and the boundary rotational stiffnesses Kr at the two ends were both set as EI/a. Five conditions were analyzed, namely, Ks = EI/a3, 2 EI/a3, 4 EI/a3, 8 EI/a3, and ∞, respectively. One can find that with the decrease of Ks, the cable’s static profile presented an approximate overall translation, in addition to the increase of the static deformation of the cable. Moreover, when Ks became larger, the sensitivity of the cable’s static profile to Ks decreased.


## 4.2.2. Influence on Cable’s Vibration Frequencies

As shown in Figure7, the first two frequencies of the vibration of each cable under different boundary conditions were calculated and compared, to learn the influence of the boundary on the cable’s vibration characteristics. Noteworthy is that the boundary constraints at the two cable ends were assumed to be the same during the analysis.Figure 7
Influence of boundary constraint stiffnesses on cable’s first two vibration frequencies.First, keeping the lateral support stiffnessKs being infinity, the first two frequencies of the cables with different boundary rotational stiffnesses Kr (i.e., 0, EI/a, 2 EI/a, and ∞) were compared. One can find in Figure 7 that Kr had a significant influence on Cables 3 and 4 with large flexural stiffnesses, which indicated that the sensitivity of the cable vibration frequency to Kr is affected by the flexural stiffness of the cable. However, larger sag would weaken the influence of Kr. For example, the frequencies of Cable 4 with large sag showed weaker sensitivity to Kr compared to Cable 3 with small sag. These results proved that Kr mainly affects the dynamic characteristics of the cables with large flexural stiffness but small sag.Then, keeping the rotational stiffnessKr at boundary being EI/a, the first two frequencies of the cables with different lateral support stiffnesses Ks (i.e., EI/a3, 2 EI/a3, 10 EI/a3, and ∞) were also compared. It could be found in Figure 7 that the Ks presented a complex influence on the cables’ vibration frequencies. With the increase of Ks: the frequencies of Cables 1 and 2 increased significantly; the frequencies of Cable 3 decreased slightly; and the frequencies of Cable 4 did not change much. The reason could be found in the expressions of the elements Q, T, R, and P within the stiffness matrix K1 shown in equations (29)–(32). One can find that there is a common term EI⋅H−Kr2 in the numerators and denominators of these equations and the term Ks appears only in the denominators. Therefore, the size relationship between Kr2 and EI⋅H will determine the influence of Ks on the elements Q, T, R, and P within the stiffness matrix K1 and further the cables’ vibration frequencies.


## 4.3. Validation of Eigenvalue Derivative Calculation Results

As stated before, the calculation of the vibration eigenvalue derivatives ∂λn/∂xk is the key aspect of the calculation of matrix α. To verify the derived equations for computing the eigenvalue derivatives, the eigenvalue derivatives were first transformed into frequency derivatives using the following equation derived from λ = ω2 = (2πf)2,(43)∂f∂xk=18π2f∂λ∂xk.Then, the calculated frequency derivatives with respect to each cable parameter were presented visually in the form of tangents at several selected points on the frequency-parameter curve (i.e.,f-xk curve). Taking Cable 2 as an example, the derivatives of the first-order frequency with respect to its multiple system parameters were calculated and illustrated in Figure 8 (due to the limited space, the derivatives of the frequencies of the other orders were not shown here). It could be seen that the calculated tangents (blue short solid lines) had good agreement with the actual curve slopes on each frequency-parameter curve, proving the accuracy of the analytical method for calculating ∂λn/∂xk derived in this article.Figure 8
Illustration of calculated derivatives of the first eigenvalue (frequency) of cable 2 with respect to its system parameters: (a)Hm; (b) EI; (c) EA; (d) Kr; and (e) Ks.
(a)(b)(c)(d)(e)Noteworthy is that, when obtaining the curves that frequency varies withHm, EI, and EA, respectively, in Figure 8, the boundary conditions of the cable were assumed to be fixed at the two ends; when obtaining the curve that frequency varies with Kr, it was assumed that the boundary constraints at the two cable ends were the same and Ks were infinity; and when obtaining the curve that frequency varies with Ks, it was assumed that the boundary constraints at the two cable ends were the same and Kr = EI/a.


## 4.4. Multiparameter Identification of the Cable

### 4.4.1. Model Setup

The four numerical cables in Table1 were adopted to illustrate the effectiveness of the frequency-based sensitivity-updating algorithm for identifying multiple system parameters using the multiorder frequencies. To consider the effects of elastic embedding and elastic support at the cable boundary, for all four cables, assume that the accurate values of the rotational constraint stiffnesses (Kr1 and Kr2) at the two ends of the cables were 2 EI/a and 4 EI/a, respectively, and that the accurate values of the lateral support stiffness (Ks1 and Ks2) at the two ends of the cables were 5 EI/a3 and 10 EI/a3, respectively. Moreover, in the first iteration, it was assumed that the errors of the initial iteration values of Hmeva, EIeva, and EAeva of each cable were all 20%, and the errors of the initial iteration values of Kr1eva, Kr2eva, Ks1eva, and Ks2eva of each cable were all 50%, as shown in Table 2. The accurate frequencies of the cables, fmn, which should be measured in the field in practical engineering, were calculated from the accurate system parameters in Table 2 in this paper, as listed in Table 3.Table 2
Accurate and initial values of system parameters of the four cables.
NoItemsHm (N)EI (Pa∙m4)EA (Pa∙m2)Kr1 (N∙m)Kr2 (N∙m)Ks1 (N/m)Ks2 (N/m)Cable 1Accurate value2.9036E67.9197E41.2552E81.5839E53.1679E53.9599E57.9197E5Initial value2.3229E66.3358E41.0042E87.9197E41.5839E51.9799E53.9599E5Initial error (%)20202050505050Cable 2Accurate value7.2590E057.9222E041.3080E081.5844E053.1689E53.9611E57.9222E5Initial value5.8072E056.3378E041.0464E087.9222E41.5844E51.9806E53.9611E5Initial error (%)20202050505050Cable 3Accurate value2.6133E71.0244E81.6371E112.0488E84.0976E85.1220E81.0244E9Initial value2.0906E78.1952E71.3097E111.0244E82.0488E82.5610E85.1220E8Initial error (%)20202050505050Cable 4Accurate value7.2590E52.8464E61.3080E85.6928E61.1386E71.4232E72.8464E7Initial value5.8072E52.2771E61.0464E82.8464E65.6928E67.1160E61.4232E7Initial error (%)20202050505050Table 3
Accurate modal frequencies of the four cables (Hz).
No1st-order2nd-order3rd-order4th-order5th-order6th-order7th-orderCable 10.4200.8211.2321.6442.0572.4702.885Cable 20.4100.4310.6480.8501.0631.2751.488Cable 31.3742.6714.0445.4596.9328.47210.092Cable 40.4140.4450.6780.9091.1541.4111.682


### 4.4.2. Results

Using the method described in Section3.5, the identification of the multiple system parameters of the four numerical cables was complemented based on their true model frequencies listed in Table 3. In order to enhance the robustness of the iteration, the variation ranges of FHm, FEI, and FEA were constrained to within [–0.3, +0.3], and the variation ranges of FKr1, FKr1, FKs1, and FKs2 were constrained to within [–0.5, +0.5].Figure9 shows the convergence process of the relative errors of cable system parameters with the iteration times. It can be observed that the cable system parameters obtained after the first iteration still had relatively large errors. After a total of 10 iterations, the relative errors of cable system parameters converged to zero for all four cables. At this time, the system parameters tended to be stable and to converge to the accurate values. The abovementioned results indicated that the frequency-based sensitivity-updating algorithm could be adopted to identify the multiple system parameters quickly and effectively for different kinds of cables.Figure 9
Convergence of the relative errors of cable system parameters with iteration times: (a) Cable 1; (b) Cable 2; (c) Cable 3; and (d) Cable 4.
(a)(b)(c)(d)To give more details, taking Cable 1 as an example, the convergence of the correction coefficients and system parameters of Cable 1 is shown in Figure10. It could be seen that, among the system parameters of the cable, Hm and EA converged faster. After 3 iterations, their identification errors were less than 0.5%. In contrast, the other system parameters converged relatively slowly. After 8 iterations, the identification errors of all system parameters were less than 0.5%. Figure 11 depicts the comparison between the frequencies calculated from the initial iterative values, final iterative values, and accurate values of the cable parameters, respectively. It can be observed that the errors of the frequencies calculated from the initial iterative values of the system parameters were very large. After repeated iterations, the frequencies calculated from the final iterative values showed good agreement with the accurate frequencies. This also illustrated the effectiveness of the frequency-based sensitivity-updating algorithm.Figure 10
Convergence of (a) correction coefficientsFk, (b) Hm, (c) EI, (d) EA, (e) Kr1, (f) Kr2, (g) Ks1, and (h) Ks2 of cable 1 during the identification process.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 11
Comparison of iteration results with accurate values of modal frequencies of cable 1.


## 4.4.1. Model Setup

The four numerical cables in Table1 were adopted to illustrate the effectiveness of the frequency-based sensitivity-updating algorithm for identifying multiple system parameters using the multiorder frequencies. To consider the effects of elastic embedding and elastic support at the cable boundary, for all four cables, assume that the accurate values of the rotational constraint stiffnesses (Kr1 and Kr2) at the two ends of the cables were 2 EI/a and 4 EI/a, respectively, and that the accurate values of the lateral support stiffness (Ks1 and Ks2) at the two ends of the cables were 5 EI/a3 and 10 EI/a3, respectively. Moreover, in the first iteration, it was assumed that the errors of the initial iteration values of Hmeva, EIeva, and EAeva of each cable were all 20%, and the errors of the initial iteration values of Kr1eva, Kr2eva, Ks1eva, and Ks2eva of each cable were all 50%, as shown in Table 2. The accurate frequencies of the cables, fmn, which should be measured in the field in practical engineering, were calculated from the accurate system parameters in Table 2 in this paper, as listed in Table 3.Table 2
Accurate and initial values of system parameters of the four cables.
NoItemsHm (N)EI (Pa∙m4)EA (Pa∙m2)Kr1 (N∙m)Kr2 (N∙m)Ks1 (N/m)Ks2 (N/m)Cable 1Accurate value2.9036E67.9197E41.2552E81.5839E53.1679E53.9599E57.9197E5Initial value2.3229E66.3358E41.0042E87.9197E41.5839E51.9799E53.9599E5Initial error (%)20202050505050Cable 2Accurate value7.2590E057.9222E041.3080E081.5844E053.1689E53.9611E57.9222E5Initial value5.8072E056.3378E041.0464E087.9222E41.5844E51.9806E53.9611E5Initial error (%)20202050505050Cable 3Accurate value2.6133E71.0244E81.6371E112.0488E84.0976E85.1220E81.0244E9Initial value2.0906E78.1952E71.3097E111.0244E82.0488E82.5610E85.1220E8Initial error (%)20202050505050Cable 4Accurate value7.2590E52.8464E61.3080E85.6928E61.1386E71.4232E72.8464E7Initial value5.8072E52.2771E61.0464E82.8464E65.6928E67.1160E61.4232E7Initial error (%)20202050505050Table 3
Accurate modal frequencies of the four cables (Hz).
No1st-order2nd-order3rd-order4th-order5th-order6th-order7th-orderCable 10.4200.8211.2321.6442.0572.4702.885Cable 20.4100.4310.6480.8501.0631.2751.488Cable 31.3742.6714.0445.4596.9328.47210.092Cable 40.4140.4450.6780.9091.1541.4111.682


## 4.4.2. Results

Using the method described in Section3.5, the identification of the multiple system parameters of the four numerical cables was complemented based on their true model frequencies listed in Table 3. In order to enhance the robustness of the iteration, the variation ranges of FHm, FEI, and FEA were constrained to within [–0.3, +0.3], and the variation ranges of FKr1, FKr1, FKs1, and FKs2 were constrained to within [–0.5, +0.5].Figure9 shows the convergence process of the relative errors of cable system parameters with the iteration times. It can be observed that the cable system parameters obtained after the first iteration still had relatively large errors. After a total of 10 iterations, the relative errors of cable system parameters converged to zero for all four cables. At this time, the system parameters tended to be stable and to converge to the accurate values. The abovementioned results indicated that the frequency-based sensitivity-updating algorithm could be adopted to identify the multiple system parameters quickly and effectively for different kinds of cables.Figure 9
Convergence of the relative errors of cable system parameters with iteration times: (a) Cable 1; (b) Cable 2; (c) Cable 3; and (d) Cable 4.
(a)(b)(c)(d)To give more details, taking Cable 1 as an example, the convergence of the correction coefficients and system parameters of Cable 1 is shown in Figure10. It could be seen that, among the system parameters of the cable, Hm and EA converged faster. After 3 iterations, their identification errors were less than 0.5%. In contrast, the other system parameters converged relatively slowly. After 8 iterations, the identification errors of all system parameters were less than 0.5%. Figure 11 depicts the comparison between the frequencies calculated from the initial iterative values, final iterative values, and accurate values of the cable parameters, respectively. It can be observed that the errors of the frequencies calculated from the initial iterative values of the system parameters were very large. After repeated iterations, the frequencies calculated from the final iterative values showed good agreement with the accurate frequencies. This also illustrated the effectiveness of the frequency-based sensitivity-updating algorithm.Figure 10
Convergence of (a) correction coefficientsFk, (b) Hm, (c) EI, (d) EA, (e) Kr1, (f) Kr2, (g) Ks1, and (h) Ks2 of cable 1 during the identification process.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 11
Comparison of iteration results with accurate values of modal frequencies of cable 1.


## 5. Concluding Remarks

A nonlinear mathematical model of the vibration of the cable was established in this study with a systematical and comprehensive consideration of the influence of the inclination, bending stiffness, and sag-extensibility of the cable, particularly of the rotational constraint stiffness and the lateral support stiffness at the cable ends. The finite difference method was used to discretize the cable model, and the expressions of the linear stiffness matrix and the nonlinear stiffness matrix of the cable were strictly derived. Finally, a frequency-based sensitivity-updating algorithm was applied to identify multiple system parameters of the cable according to multiple frequencies. The results obtained made it possible to draw the following conclusions:(1)
The frequency-based sensitivity-updating algorithm can simultaneously and accurately identify the system parameters for different kinds of cables utilizing multiple frequencies;(2)
Calculation of the derivatives of the eigenvalues of the cable vibration is a key to obtaining the system sensitivity matrix. The developed calculation formulae were proved to be able to calculate accurately the derivatives of the vibration eigenvalues of an inclined cable;(3)
To enhance the robustness of the sensitivity-updating algorithm during the iteration, the variation range of the cable parameter could be constrained to a proper interval;(4)
The cable’s boundary conditions have a complex influence on the static and dynamic characteristics of the cable: the rotational stiffnessKr exhibits a slight effect on the cable’s static profile but the variation of lateral support stiffnesses Ks will cause an approximate overall static profile translation; Kr mainly affects the frequencies of the cables with large flexural stiffness but small sag; and the size relationship between Kr2 and EI⋅H will determine the influence of Ks on frequencies. It is necessary to take into account the effects of boundary conditions of the cable for the estimation of the cable tension.



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*Source: 7795452-2023-02-11.xml*