This paper proposes a slenderness indicator (SI)-based algorithm for identifying cable force in bridge engineering. First, the cable dynamic equation is reduced through time-space coordinate transformation in the dimensionless system, and the fractional power function relationship of the dimensionless frequency-ratio and the slenderness indicator is established through the Taylor series expansion method. On this basis, a practicable algorithm is constructed, which calculates the cable force for each order frequency, and a clear engineering application scope of the algorithm for each order mode is established. The accuracy of the SI-CFI (slenderness indicator-based cable force identification) algorithm is verified through the finite element examples. The feasibility of the SI-CFI algorithm is demonstrated based on the frequencies of the bridge cables using the radar interferometry technology. The overall average error of cables is within 0.9%. It can be seen that the proposed method excels in both error control and identification stability. The algorithm is simple, intuitive, and theoretically supported, which can cover the applicable range of slenderness indicators for engineering cables, providing engineers with a standardized and practicable cable force identification solution.

A multiscale continuum framework is developed to couple the nonlinear Poisson-Boltzmann electrochemical field with the entropic elasticity of polymer brushes, establishing the electrochemo-mechanical constitutive behavior of DNA-functionalized microcantilevers. This constitutive relation is incorporated into the macroscopic beam dynamics through an effective stiffness that accounts for surface-induced nonlinearity. The analysis shows that the effective flexural rigidity exhibits a non-monotonic dependence on ionic conditions, resulting from the competition between electrostatic screening and entropic repulsion. This dependence modifies the nonlinear restoring force and alters the structure of the dynamic response, leading to a transition from weakly nonlinear oscillations to bistable behavior associated with saddle-node bifurcations of periodic solutions. Based on this mechanism, a bifurcation-based sensing scheme is examined, in which DNA hybridization acts as a slowly varying parameter that drives the system across stability boundaries. The resulting transition between coexisting periodic attractors produces a pronounced change in oscillation amplitude. The dynamic response is computed using a hybrid harmonic balance and pseudo-arc-length continuation (HBM-PALC) method, which captures both stable and unstable solution branches. In addition, the global response is analyzed in the excitation amplitude-frequency parameter space. The results indicate that increasing excitation amplitude leads to a sequence of period-doubling bifurcations and eventual transition to aperiodic motion, thereby defining a bounded region of stable periodic operation. The influence of stochastic perturbations near bifurcation points is also discussed in terms of noise-induced transitions between attractors. These results provide a consistent dynamical description of the coupled electro-chemo-mechanical system and clarify the role of bifurcation behavior in sensing applications.

This paper presents a comparative investigation of the dependence of eigen-solutions on damper positioning, using a dimensionless taut-string model with a point viscous damper. Building on the known closed-form eigen-solutions for dampers at the 1/2- and 1/3-span positions, this study newly derives the novel closed-form solution for the 1/4-span configuration. With this newly derived quarter-span solution, the mid-span, one-third-span, and quarter-span cases form a minimal closed-form analytical sequence for examining how damper position organizes the controlled eigenspectrum. For these three canonical positions, the transcendental frequency equation is transformed into an equivalent algebraic polynomial equation through hyperbolic-function identities and a change of variables, with the order of the resulting polynomial determined by the selected position. As the damper position changes from 1/2 to 1/4 span, the characteristic polynomial rises from first to third order, yielding one, two, and three eigenvalue branches, respectively. These branches exhibit different evolutionary features: the mid-span system shows a global critical spectral transition, the one-third-span system undergoes position-specific local branch coalescence and separation, whereas the quarter-span system contains a smoothly evolving conjugate pair of controlled branches among its three branches. The quarter-span solution therefore identifies, within this minimal canonical sequence, a smooth controlled-branch behavior that is structurally related to the critical spectral transitions revealed by the mid-span and one-third-span cases. Through this systematic comparison, the closed-form solutions clarify the position-dependent organization of algebraic order, controlled-branch structure, and spectral-transition behavior. They also provide analytical benchmarks for numerical eigen-solution, branch tracking, optimal damping identification, and subsequent damper-arrangement studies under practical installation constraints.

To resolve the repeated solution of transcendental equations in the free vibration analysis of Rayleigh beams caused by dependence on multiple physical parameters, a general solution approach based on nondimensionalization is proposed in this paper. By introducing specific time and space restoring coefficients, the dynamic equation is fully nondimensionalized and reduced to a most concise form, in which all coefficients are equal to 1 or -1, yielding a universal vibration equation applicable to Rayleigh beams with arbitrary values of physical parameters. Based on this equation, the frequency equations are derived for three typical boundary conditions (clamped-clamped, simply supported, and clamped-free), and dimensionless frequency-beam length curves are obtained to systematically describe the frequency characteristics under different boundary conditions and modal orders. Further investigation indicates that the effect of rotary inertia on natural frequencies is uniformly characterized by the dimensionless beam length. Accordingly, the critical dimensionless beam lengths separating the applicability of the Euler-Bernoulli and Rayleigh beam models are identified. Finally, based on the dimensionless frequency-beam length curves, a normalized method is proposed for frequency computation, and its accuracy is verified through comparisons with finite element results. The study demonstrates that the proposed method is independent of the specific physical parameters of the beam; variations in system parameters only affect the time and space restoring coefficients, while the dimensionless frequency-beam length curves retain their general applicability to arbitrary parameter changes. The method involves only the solution of a single-variable transcendental equation and linear transformations, thereby avoiding nonlinear iterative procedures associated with multi-parameter transcendental equation systems. Consequently, computational efficiency and numerical stability are significantly improved, and repetitive calculations caused by parameter variations are eliminated. The proposed approach provides an efficient and general tool for the vibration analysis of Rayleigh beams.

In practical engineering, the accuracy of vibration-based cable tension identification is significantly affected by multi-factor coupling uncertainties, including sag, bending stiffness, and damping. However, traditional sensitivity analysis methods are insufficient to capture the probabilistic characteristics of cable tension errors and to quantify the global importance of influencing factors. To address this issue, this study proposes a Monte Carlo simulation framework based on Latin Hypercube Sampling (LHS) to quantify the uncertainty of cable tension errors under multi-factor coupling. A cable dynamic model considering sag, bending stiffness, and damping effects is established, and the corresponding piecewise cable tension correction formulas are derived. Subsequently, Monte Carlo simulations are conducted to obtain the probability density distributions of cable tension errors under different working conditions, and global sensitivity analysis is performed using the Standardized Regression Coefficient (SRC) method. The results indicate that bending stiffness is the dominant factor controlling both the magnitude and distribution pattern of cable tension errors, with SRC values exceeding 0.7 in most cases. Sag significantly affects the dispersion and symmetry of the error distribution, and its SRC value can exceed 0.4 under low bending stiffness conditions. In contrast, the effect of damping is negligible, with SRC values consistently below 0.1. This study establishes a probabilistic framework for analyzing cable tension errors under multi-factor coupling and presents a quantitative ranking of influencing factors, thereby offering an effective approach to overcome the limitations of traditional deterministic sensitivity analysis methods and providing a generalizable framework for uncertainty quantification in cable-supported structural systems.

Accurate tension estimation is fundamental to the structural health monitoring of cable-supported bridges, yet it is often impeded by two pervasive issues in real-world structures: non-ideal boundary conditions and significant uncertainty in bending stiffness. These factors collectively undermine the reliability of vibration-based methods, especially for short cables where modeling errors are magnified. To overcome these limitations, this paper first non-dimensionalizes the cable’s dynamic equations through temporal and spatial scaling. Next, by exploiting the property that the non-dimensional frequency equation is controlled solely by the non-dimensional length under any prescribed boundary condition, a boundary-independent fixed-point iterative algorithm is developed to determine this key parameter. Finally, integrating this algorithm with scaling relations yields a practical procedure for accurate tension estimation. The method is rigorously validated through numerical simulations covering three representative boundary types, controlled laboratory experiments, and field testing on an in-service cable-stayed bridge. Results consistently demonstrate that the fixed-point iteration converges stably regardless of support conditions, and the final tension estimates exhibit high accuracy across both short and long cables. This work provides a unified and theoretically consistent framework for cable tension identification under various boundary conditions.

This paper establishes a 1:250 scale model to experimentally investigate the nonlinear dynamic behaviors of a cable-stayed bridge based on the Xiangshangang Bridge. Firstly, the experimental model and some necessary instruments are introduced. Modal analysis is then carried out and the physical parameters of the cables are determined. Subsequently, the nonlinear vibrations of the experimental model are studied by applying a harmonic excitation. In this way, rich out-of-plane and in-plane nonlinear behaviors are uncovered based on a detailed analysis of the forced vibration and superharmonic resonance, especially the superharmonic resonance of the out-of-plane modes. The experimental results reveal the possibility of the occurrence of higher-order superharmonic resonances. Specifically, higher-order superharmonic resonance of the in-plane and out-of-plane modes may be triggered under the external excitation, such as 6:1, 7:1, or even 8:1 superharmonic resonance of the in-plane modes and 2:1, 3:1, or even 4:1 superharmonic resonance of the out-of-plane modes.

In frequency-based cable tension identification, it is common to neglect either bending stiffness or sag to avoid repetitive solutions to transcendental equations or optimization problems for each cable. For rigid cables, bending stiffness cannot be neglected; for long cables using non-contact methods, measuring in-plane frequencies potentially influenced by non-negligible sag is necessary. However, this absence of a clearly delineated theoretical scope for the model’s simplification fosters both empiricism and inconsistency in the methodology. To address these issues, this paper proposes a unified algorithm that is applicable to the parameter identification of both rigid short cables and sagged long cables. A temporal and spatial scaling approach is employed to simplify the dynamic equations of sagged rigid cables, yielding a general frequency equation and corresponding numerical solution for the dimensionless frequency within a parameter space defined by sag and dimensionless cable length. Based on this universal solution and the reciprocal relationship of frequency ratio, a unified algorithm is presented. Using any three measured frequencies, the algorithm can determine the Irvine parameter and any two unknown parameters of cable tension, bending stiffness, and mass density through simple interpolation, thereby eliminating the need for complex calculations such as iterative solutions or equation-solving. Furthermore, a sensitivity analysis is conducted to quantify the robustness of the identified parameters against frequency perturbations. The reliability and accuracy of the proposed method are validated through comparisons with laboratory experiments, real-life bridge cable data from various references, and a field test.

To derive simple, approximate analytical formulas for critical buckling loads of uniform columns under combined end and linearly distributed axial compression for common boundary conditions. The end and distributed compression are perturbed in different orders, followed by time and space coordinate transformations, which leads to the most concise form of dimensionless equation. Buckling loads were determined from the condition where the fundamental frequency approaches zero. The universal pre-solution data ( ω _i-l-g ) were obtained, which avert repeatedly solving frequency transcendental equations for each beam with different physical parameters. Simple, approximate analytical buckling load equations were derived for the four boundary conditions, expressing the critical load as the classical Euler load minus a linear correction term for the distributed load. Good agreement was found with FEM and literature results. Simple, physically meaningful, and practical approximate formulas for buckling loads under combined compression were successfully derived. They extend Euler’s formulas, offer satisfactory accuracy for design, and simplify the calculation compared to existing complex solutions.

The frequency equation for a cable damped at a fraction location, generally being a transcendental equation, was reformulated into an algebraic form. Based on the fundamental theorem of algebra and the characteristics of logarithmic function in the complex domain, the solution structures were revealed and several examples with given damper positions were presented to study the variation of the solutions with the damping coefficient. The results show that: 1) All solutions of the frequency equation of the system could be classified into a finite number of solution branches. 2) The solutions of different orders in the same solution branch share an identical real part and any two adjacent solutions share an identical difference. 3) According to different variations in decay rate and frequency with damping coefficient, all the solution branches could be classified into four categories, each of which showed different trend of variation.

Systems exhibiting Saddle-Node (SN) bifurcations are often characterized by drastic amplitude and phase jumps, representing a crucial state in engineering scenarios. The accurate and efficient prediction of SN points is fundamental for the comprehensive understanding and control of dynamical systems. This paper derives an equation for locating SN points based on the dimensionless governing equation for the in-plane primary resonance of a suspended cable. It reveals that the SN points for the cable are influenced by three key parameters: the cable’s effective nonlinearity Γem, the excitation parameter F2, and damping ratio. Importantly, when the cable is subjected solely to horizontal end excitation, the product of Γem and F2 emerges as a new parameter, Λem. The effects of parameter Λem (Γem), damping, axial, and vertical excitation amplitudes on SN points are investigated. Findings indicate that these key parameters more significantly affect the SN1 (the one near peck point) than SN2, and slight variations in Λem (Γem) or vertical excitation amplitude can lead to substantial alterations in SN1. The effect of Λem (Γem) on SN1 is asymmetric, with the values of σ and a being significantly higher when Λem (Γem) is positive than when negative. The computational results of the SN position equation closely align with experimental observations and the literature, demonstrating good computational efficiency.

Frequency-based algorithms are prevalently used to estimate bridge cable tensions. However, for the non-negligible bending stiffness, the eigenproblem involves a cumbersome procedure of solving transcendental equations, which aroused various treatment skills for simplification or optimization, yielding diverse identification algorithms with different effectiveness, efficiency, and applicability that bring difficulties and confusion in making an appropriate choice for users. Therefore, it is necessary to systematically illustrate these algorithms’ intrinsic relations, differences, and application characteristics. A comprehensive comparative study on five representative algorithms, including analytical and empirical formulas and optimal algorithms, is carried out in this paper by parallelly identifying cable parameters on 24 real-life bridge cables worldwide available in the literature. Results show that: (i) Algorithms of empirical formulas are always based on the modified chord theory or tensioned-simply-supported beam model, resulting in the solution of two simultaneous equations by providing two measured frequencies in case of unknown bending stiffness. (ii) Proper identification procedure is significant in applicating an algorithm. (iii) The FROCPI algorithm is the simplest method with good consistency and broad applicability for long and short cables.

Synchronization and jumping are two distinctive phenomena pervasively present in nonlinear dynamical systems. Cables, as prototypical nonlinear structures, have also been noted to exhibit alternating in-phase and anti-phase synchronous vibrations. In order to investigate the interrelationship between cable jumping and synchronization, an equation for determining the cable’s jump point during in-plane primary resonance has been derived based on a dimensionless nonlinear dynamic model. It has been observed that the jump point and the synchronous vibration of the cable are intrinsically linked to key factors, including the effective nonlinearity coefficient Γem, excitation amplitudes, and damping. Through an analysis of frequency-response and phase-frequency curves, the mechanism governing the alternating patterns of inphase and anti-phase synchronous vibrations in cables with distinct parameters is qualitatively examined. Subsequently, the influences of critical parameters on the jumping points and the regions of in-phase and anti-phase synchronous vibrations are systematically investigated, encompassing both first-order and second-order primary resonances. The results elucidate that in-phase and anti-phase synchronous vibrations between cables fundamentally arise due to disparities in cable-specific parameters or variations in end excitation amplitudes, which consequently lead to a shift in the position of the jump point. Notably, the impact of these key parameters on the jump point position and its amplitude is more pronounced during forward frequency sweeping than during backward frequency sweeping. Furthermore, even minor adjustments in the parameter Γem or the vertical excitation amplitude can markedly alter the position of the jump point when cables are subjected to forward frequency sweeping.

悬索受多个端部激励后可能产生复杂的非线性动力学现象.为研究端部激励相位差对悬索亚谐波响应的影响,建立两端受激励的悬索模型,得到无限维离散运动方程,采用多尺度法研究面内亚谐波共振,并通过数值积分加以验证.文中给出了多激励系统等效激励幅值和响应相位的一般表达式,定性定量分析了面内两两激励的不同组合对响应幅值和相位的影响.结果表明:端部激励相位差对悬索响应幅值的影响以2π为周期变化,关于π对称.端部激励间的相位差可以使响应相位在一个名义单激励系统的响应相位基础上发生移动,当激励相位差逐渐由0转变为π的同时,相频曲线也非同步地位移了 π.激励相位差不改变系统的软硬性质,但会影响响应幅值和共振区宽度,且其共振区宽度关于σ=0对称.

This paper proposes a frequency‐ratio‐offset (FRO)‐based parameter identification algorithm for real bridge cables. By introducing specific time and space scaling factors, the dynamic equation is fully nondimensionalized and reduced to a most concise form, independent of all distributed cable parameters. As a result, the frequency equation has only two variables, which can be easily solved to obtain the universal dimensionless frequency–length curve under specified boundary conditions. Based on these curves, a decoupled identification algorithm is constructed, using the newly defined FRO as a bridge to directly connect the dimensionless beam and its real‐world counterpart. The algorithm enables a simultaneous determination of any two of the three cable parameters using a pair of measured frequencies. Examples involving both laboratory tests and real‐life applications demonstrate the effectiveness and accuracy of the algorithm. Based on a statistical analysis of the fitness of dozens of predicted and measured frequencies, a guideline for frequency selection is suggested, and the influence of measurement error on the identification results is discussed. The proposed algorithm could become a convenient tool for engineers to efficiently identify multiple parameters (e.g., tension and bending stiffness) for a broad class of bridge cables.

Analysis of key parameters is an effective means to understanding modal characteristics of a system. However, because of the large number of components, there are various geomet-ric, material, and physical parameters in a sagged-cable-crosstie structure, which inevitably hinders the reasonable selection of key parameters. Based on the companion (Part 1) pa-per, the effect of four fundamental key parameters, i.e., the Irvine parameter, position and stiffness of crosstie(s), and wave speed ratio, on the modal characteristics of three rep-resentative models: double-cable-single-crosstie, three-cable-single-crosstie, and double-cable-double-crosstie, are investigated by mechanical modeling-based parametric analysis. The simultaneous existence of cross-over and veering phenomena that commonly wouldn't co-exist in single classical cables is found in sagged-cable-crosstie structures. Generally, the frequency curves of all the in-phase modes, out-of-phase modes with symmetrically arranged crossties, and specific out-of-phase modes with crossties just at modal nodes show cross-over phenomena, while that of out-of-phase modes with non-symmetrically arranged crossties show veering phenomena. Setting one or two crossties can only signif-icantly increase the out-of-phase modal frequencies of specific orders, and the increment limits of dimensionless frequency for systems with identical sagged cables are found to be 1 and 2 respectively, no matter how the crosstie stiffness and position are adjusted and how many cables are connected. However, the wave speed difference between sagged cables can enhance the effect of crosstie(s) to further increase modal frequencies, espe-cially for high-order ones, and hence break through the above increment limits. Moreover, the more cables with wave speed differences connected by crossties and the greater the difference in wave speed, the more the system frequency increases. (c) 2023 Elsevier Inc. All rights reserved.

The phase-frequency characteristic is a fundamental feature of cables closely related to the synchronization phenomena. The response phase of a nonlinear vibrating cable under a specific excitation frequency is commonly believed to be the constant value in the linear solution, while higher-order terms (HOTs) are commonly omitted. However, as the variation of cable parameters, the HOTs would significantly contribute to the response and thus change the phase instantaneously. In order to ascertain the instantaneous phase difference between cables with the consideration of the HOTs, the instantaneous phase-frequency characteristics of two parametric-excited nonidentical suspended cables are investigated. The dimensionless dynamic equations of a two-cable system were derived, and the discrete model was obtained using the Galerkin method and then solved by the Multiple Scales Method (MSM). The MSM solution was verified simultaneously using the Runge–Kutta method (R-K) and the Finite Element Method (FEM). Results show that the HOTs’ influence on the instantaneous phase is non-negligible in some frequency ranges. The origination of the instantaneous phase difference between the two distinct cables comes from two aspects: (i) the difference in phase shift values (PSVs) of linear terms; and (ii) the proportion difference of drift terms (DTs) in HOTs.

为了研究动态风对覆冰输电线非线性舞动特征的影响,在原有稳定风作用下覆冰输电线舞动控制方程中添加周期激励载荷,并建立了新的受迫-自激振动控制方程,该控制方程也适用于描述相邻档导线对舞动档导线运动特征的影响。运用多尺度法分别对弱激励和强激励下的受迫-自激振动求解,得到主共振和谐波共振的幅频响应函数,分析了|受迫-自激系统的主共振、超谐波和亚谐波共振。研究表明:弱激励下的主共振,当调谐参数大于0时,风速或激励幅值的增加会使得响应幅值出现跳跃、多值等不稳定的非线性动力学行为,并呈现硬弹簧特征;强激励下的自激系统，当激励频率接近固有频率的整数倍和分数倍时,更容易出现2次超谐波共振和1/2次亚谐波共振;当发生1/2次亚谐波共振时,随着激励幅值的增大,响应幅值也不断增大,共振峰值对应的调谐参数趋向于正轴方向,呈现硬弹簧特征,风速的增加会增强系统的非线性和硬弹簧特征。

Though crosstie has become a promising approach for vibration mitigation of long ca -bles, its mechanism is yet to be fully understood, restricting the set of engineering design theory. Scholars worldwide have continuously proposed various mechanical models to in-vestigate modal characteristics of cable-crosstie structures. However, first, dynamic equa-tions were dimensional or not fully dimensionalized, hindering the definition of essen-tial independent key parameters that govern the modal characteristics of the system; sec -ond, no general expression of the coefficient matrix in characteristic equations was given. This paper proposes a general sagged-cable-crosstie model applicable to structures with a generic number of cables and crossties. A universal dimensionless dynamic equation is de-rived by thorough dimensionless treatment, and the universal expression of a 2N(M + 1) -order coefficient matrix is obtained by introducing boundary, equilibrium, and continuity conditions. A minimal set of dimensionless parameters that govern modal characteristics of any sagged-cable-crosstie system is found, i.e., crosstie positions ej,p, Irvine parameters of cables lambda 2j , dimensionless wave speed parameters alpha j, and dimensionless crosstie stiff-nesses kj,p. Subsequently, the general model degenerates into three representative models: double-cable-single-crosstie, three-cable-single-crosstie, and double-cable-double crosstie. Frequencies of each degenerated model are compared with that of literature and FEM, and good consistency is obtained. (c) 2023 Elsevier Inc. All rights reserved.