The damping influence in monitoring the tension of cable using the vibration method

Cable vibration regularly occurs in operating structures and negatively impacts the performance and long-term maintenance of cable-supported structures. Therefore, it is necessary to develop solutions to control and predict the condition of structures, especially cable structures such as stayed cables of cable-stayed bridges, suspenders of suspension bridges, and hangers of arch bridges. Pre-stressed or stayed cables typically consist of steel wire bundles (nearlinear elasticity), a corrosion-resistant high-density polyethylene (HDPE) jacket, protective grease or cement mortar, anchorage assemblies, and sometimes viscoelastic damping. Therefore, the observed damping of vibrations is not only due to the steel but also to the polymer components and the contact surfaces. In terms of material mechanics, this can be modeled in various ways. Steel has low but non-zero internal friction; in addition, fiber friction, anchor slippage, and mortar micro-cracking contribute to the structural damping, along with polymers (HDPE, sealants, and anchor pads). When inspecting cable structures, estimating the tension acting on the cable based on the vibration technique plays a significant role, due to the damping effect of cables. Vibration measurement using accelerometers is more commonly employed in buildings due to its flexible implementation at a low cost, while ensuring good accuracy and meeting construction safety standards. Therefore, practical formulas are proposed to improve the accuracy of determining cable tension for different cable structures, without considering damping. For basic cable systems, damping is typically negligible and may be ignored, allowing accurate tension estimation from natural frequencies alone. In contrast, special cable types such as stay cables often incorporate multiple layers of steel wire bundles (exhibiting near-linear elasticity), as well as corrosion-resistant HDPE jackets. The presence of polymer components and friction at contact surfaces introduces significant damping behavior, which must be considered for reliable cable tension estimation. Owing to the damping of cable material, estimating the tension based on the vibration technique plays a significant role.
To enhance the accuracy of methods for determining cable strength using the vibration method, numerous scientists have researched and refined their techniques for evaluating cable tensile force. Shimada et al. propose a practical workflow: preferably, measure multiple natural frequencies; shape modes and correct for boundary conditions; estimate tensile force with optional adjustment via multi-mode regression; and cross-validate across modes and time [1]. The main advantages include higher accuracy compared to using the first mode, reduced reliance on direct measurements, and enhanced suitability for structural health monitoring (SHM) of stay cables and suspension cables. However, several critical factors must be addressed to ensure reliable field deployment: mode identification, temperature, cable deflection, boundary condition, and sensor calibration [2]. Tabatabai et al. focused on assessing the condition of stay cables using vibration measurement techniques. Instead of relying solely on visual inspection or traditional load testing methods, this study exploits the vibrational behavior of cables to detect changes in tensile force, stiffness, or potential damage. This method offers advantages of feasibility, low cost, and continuous monitoring, thereby supporting maintenance, ensuring safety, and extending the service life of modern cable-stayed bridges [3]. Cho et al. conducted a field study to compare methods for measuring cable tension in cable-stayed bridges. By applying various techniques, the authors evaluated the accuracy, feasibility, and limitations of each method under real-world conditions. They selected suitable measurement solutions for monitoring and maintenance work, thereby enhancing the reliability and safety of the cable system [4]. Two popular methods
(the lift-off method and the vibration method) were compared and evaluated for determining bridge cable tension [5]. The research results not only contribute to improving the reliability of monitoring and maintenance of cable-stayed bridges but also have practical significance for transport infrastructure. In 2017, Jakiel and Manko developed a method for estimating the tension in cable-stayed pedestrian bridges based on measurements of natural frequencies. Through the analysis of experimental vibrations, the authors have developed a computational model that enables the indirect yet effective determination of cable tension, thereby replacing the complex and expensive direct measurement methods for wide application in structural health monitoring and the maintenance of cable-stayed bridges [6]. Suangga et al. focused on determining the tensile force in cable-stayed bridges through the analysis of natural frequencies. The authors conducted experimental measurements and compared them with theoretical models, thereby demonstrating that exploiting the relationship between frequency and cable tension is an effective, minimally invasive, and applicable method for SHM [7]. In 2023, Syamsi et al. investigated the impact of the location of the first sensor relative to the support on determining the tensile force of a cable with two fixed ends. By using a multi-sensor synchronized vibration measurement system, the authors analyzed how this distance change affects the accuracy of the tension estimation results. The results showed that selecting a reasonable sensor location plays a crucial role in reducing errors and enhancing the reliability of the vibration measurement method [8]. Kosco et al. (2024) focused on the development and application of the vibration method to estimate cable tension in cable-stayed and suspension bridges, thereby contributing to the improvement of safety and longevity for the project [9]. However, the similarity among the above studies lies in modeling the cable as an Euler-Bernoulli beam with an axial force and determining the relationship between the undamped vibration of the beam and the cable tension.
In another aspect, some researchers have expanded the beam theory to incorporate different mechanical characteristics, such as the Euler-Bernoulli, Rayleigh, Timoshenko, and Rayleigh-Love models, highlighting the differences in frequency and mode shape, as well as the range of their applications. Ignoring the role of rotational inertia and shear deformation can cause significant errors, especially for short beams or high-frequency vibrations; thus, advanced models such as Rayleigh or Timoshenko are often necessary to predict the dynamic response more accurately [10] [11]. Nathaniel and Alimi researched the dynamic behavior of a uniform Rayleigh beam subjected to an accelerated mass along its length. Instead of considering only static or moving loads with constant velocity, this study extended the analysis to the case of an accelerating mass, thereby clarifying the influence of rotational inertia and shear deformation on the vibration response [12]. Yang et al. (2018) developed an analytical model to analyze the material damping behavior of carbon fiber reinforced composite (CFRP) cantilever beams in the low-frequency domain. The study focused on describing the influence of composite material structure and energy dissipation mechanisms on vibration response, thereby providing a more accurate prediction tool than traditional models [13]. Zhu and Chung (2019) developed a Rayleigh model for a double-support beam that accounts for both longitudinal and rotational motion. The model was dimensionless and discretized using the Galerkin method with eigenfunctions of sinusoids to obtain the mass, gyro, and stiffness matrices for natural frequencies and responses [14]. Moreover, the damping part is specially studied as viscoelastic damping using Kelvin-Voigt models or the Zener model, and then expanded upon with a fractional derivative. Rossikhin and Shitikova researched the development and application of the fractional Kelvin-Voigt model in describing the viscoelastic behavior of materials. Through theoretical analysis and simulation, the authors demonstrate that the introduction of the fractional derivative helps the model to more accurately reflect the time-dependent degradation and deformation characteristics, which cannot be fully described by the classical model [15]. Xu et al. proposed a new method for monitoring cable tension in cable bridges by using multiple sensors to identify mode shapes and boundary conditions via a Bernoulli-Euler model with linear/rotational springs. Field tests on a three-span arch bridge in China achieved $<3\mathrm{\%}$$<3\mathrm{\%}$< 3%<3 \%$<3\mathrm{\%}$ error, far better than beam theory and comparable to the flux method, confirming accuracy and cost-effectiveness [16]. Khang et al. (2021) investigated the forced transverse vibration of EulerBernoulli beams with fractional viscoelasticity using the modal analysis method. Instead of using the classical linear elastic model, the author introduced the fractional operator to describe the material's complex viscoelastic properties more accurately, thereby constructing an analytical solution for the forced vibration response. The results demonstrate that the fractional model can more accurately reflect the actual dynamic behavior of the beam, particularly in cases where the traditional model is no longer applicable [17]. Łabedzki researched the application of the fractional Kelvin-Voigt model for beam vibration analysis by numerical analysis.
Additionally, the study proposed a method to approximate the fractional model using the classical model in some instances, thereby making it more convenient for practical applications. This study contributes to strengthening the role of fractional analysis in modern structural mechanics and vibration analysis [18]. Combining these studies, scientists have investigated cables modeled as beams, such as in the contemporary landscape of cable tension estimation and vibration control, which is moving toward field-robust tools. Le et al. performed artificial neural network models trained on simulations to infer tension under limited prior data, with full-scale validation [19]. Furukawa (2023) introduces a natural frequency approach that jointly identifies tensile force, bending stiffness, and damper parameters for damped cables; subsequently. Then,
damper-model dependence was removed via a mode-shape-based estimator to study, achieving an experimental error of $\le 4\mathrm{\%}$$\le 4\mathrm{\%}$<= 4%\leq 4 \%$\le 4\mathrm{\%}$ and outperforming design-formula methods [20] [21]. Nguyen (2024) formulates the combined action of viscous and HDR dampers, revealing near-additive damping and the governing role of damper-support flexibility. Overall trends favor multi-mode measurements, machine learning, and shape information to reduce modeling bias and optimize damper layouts for accurate, cost-effective SHM. Remaining gaps include field standardization, sensitivity to complex boundary conditions, and long-term calibration needs [22].
In this research, a cable has been modeled as an axially loaded Rayleigh beam, assuming the fractional derivative KelvinVoigt damping model from viscoelastic properties. A tensile force has been applied to the cable under supported conditions at both ends. Some applied calculations were conducted and compared to enhance the procedure for evaluating the tensile force of cables and cable structures, taking into account the damping material.

Under working conditions, the tensile cables are influenced by the effect of the excited environment, which leads to vibration for the cable structure. Their known reaction is the creation of mechanical phenomena such as tensile force, flexural rigidity, damping effects, and rotatory inertia, as shown in Figure 1.

In this study, the inertia due to the axial displacement of the beam included in the Rayleigh beam, as known rotatory inertia, $\rho J(x)$$\rho J(x)$rho J(x)\rho J(x)$\rho J(x)$, is considered to have a damping effect $\gamma (\mathrm{x})$$\gamma (\mathrm{x})$gamma(x)\gamma(\mathrm{x})$\gamma (\mathrm{x})$. This is because, under the plane cross-section assumption, the axial displacements of points within a given cross section are equivalent to a rigid-body rotation about the $z$$z$zz$z$-axis, associated with the sectional moment of inertia $\rho A(x)$$\rho A(x)$rho A(x)\rho A(x)$\rho A(x)$, flexural rigidity $EJ(x)$$EJ(x)$EJ(x)E J(x)$EJ(x)$, cable length, $L$$L$LL$L$, axial tensile force, $N(x)$$N(x)$N(x)N(x)$N(x)$, and the dynamic excited load of $f(x,t)$$f(x,t)$f(x,t)f(x, t)$f(x,t)$ per unit length [12]. A general partial differential equation (PDE) for a vibrating cable, $u(x,t)$$u(x,t)$u(x,t)u(x, t)$u(x,t)$, is the equation of transverse motion, which has been modeled as a Rayleigh beam with tensile force and damping property of its material [14]. In this study, the fractional derivative Kelvin-Voigt model has been used to assess the damping influence on the tensile force of cables by extending the classical Kelvin-Voigt model with by $a^{\text{th}}$$a^{\text{th }}$a^("th ")a^{\text {th }}$a^{\text{th }}$ order fractional derivative with $0<\alpha <1$$0<\alpha <1$0 < alpha < 10<\alpha<1$0<\alpha <1$, as shown in Eqn. (1) [15].

where: $C_a$$C_a$C_(a)C_{a}$C_a$ is the viscoelastic damping coefficient per unit length (with units of Pa.s ${}^{\alpha }$${}^{\alpha }$^(alpha){ }^{\alpha}${}^{\alpha }$ ), obtained through cross-sectional integration, which depends on the excitation frequency as well as the viscoelastic properties of the material. The parameter $a$$a$aa$a$ is a dimensionless fractional order, and ${\partial }_t^{\alpha }$${\partial }_t^{\alpha }$del_(t)^(alpha)\partial_{t}^{\alpha}${\partial }_t^{\alpha }$ denotes the Riemann-Liouville fractional derivative of order $\alpha$$\alpha$alpha\alpha$\alpha$. [23]. Additionally, the fractional derivatives effectively specialize energy propagation and memory effects of viscoelastic materials [18].

For this study, considering free vibration and homogeneous, uniform cross-sections of cable do not change, which lead to ring down artifact $f(x,t)=0$$f(x,t)=0$f(x,t)=0f(x, t)=0$f(x,t)=0$, properties of material from the governing differential equation for the cable vibration as $E,J,\rho$$E,J,\rho$E,J,rhoE, J, \rho$E,J,\rho$, and $A$$A$AA$A$ are clearly assumed as constants per unit of length as Eqn. (2) [12]:

According to the expansion theorem, any function that satisfies the boundary conditions of the beam denotes a possible transverse displacement of the beam can be expressed as a sum of eigenfunction [17] as:

In the time domain, the viscoelastic behavior of the cable material is described by a fractional Kelvin-Voigt constitutive relation, in which the stress is related to the strain through an elastic component and a Riemann-Liouville fractional derivative of order $\alpha$$\alpha$alpha\alpha$\alpha$ acting on the strain field. To solve the governing equation, the method of separation of variables is applied, and the transverse displacement $u(x,t)$$u(x,t)$u(x,t)u(x, t)$u(x,t)$ is expressed as the product of a spatial mode shape $W(x)$$W(x)$W(x)W(x)$W(x)$ and a timedependent response function $\mathrm{T}(\mathrm{t})$$\mathrm{T}(\mathrm{t})$T(t)\mathrm{T}(\mathrm{t})$\mathrm{T}(\mathrm{t})$. By applying the Fourier transform to the fractional Kelvin-Voigt constitutive relation, the formulation is transferred from the time domain to the frequency domain. In this study, the Riemann-Liouville fractional derivative of order $\alpha$$\alpha$alpha\alpha$\alpha$ acting on the strain field is represented by the factor $(i\omega {)}^{\alpha }[17,24]$$(i\omega {)}^{\alpha }[17,24]$(i omega)^(alpha)[17,24](i \omega)^{\alpha}[17,24]$(i\omega {)}^{\alpha }[17,24]$. With $T(t)=D{e}^{{\pm }_i{\omega }_t}$$T(t)=D{e}^{{\pm }_i{\omega }_t}$T(t)=De^(+-_(i)omega_(t))T(t)=D e^{ \pm_{i} \omega_{t}}$T(t)=D{e}^{{\pm }_i{\omega }_t}$ as given in Eqn. (3), substituting the effective complex modulus of the viscoelastic material $E^{\ast }(\omega )=E+C_a{\partial }_t^{\alpha }=E+C_a(i\omega {)}^a$$E^{\ast }(\omega )=E+C_a{\partial }_t^{\alpha }=E+C_a(i\omega {)}^a$E^(**)(omega)=E+C_(a)del_(t)^(alpha)=E+C_(a)(i omega)^(a)E^{*}(\omega)=E+C_{a} \partial_{t}^{\alpha}=E+C_{a}(i \omega)^{a}$E^{\ast }(\omega )=E+C_a{\partial }_t^{\alpha }=E+C_a(i\omega {)}^a$ into Eqn. (5) yields the spatial differential equation given in Eqn. (6). Consequently, the characteristic equation of the system is obtained in Eqn. (7).

The solutions of Eqn. (7) are given by:

In this study, the cable ends are modeled as elastic support boundary conditions by applying stiffness values (like springs) denoted by $k_1$$k_1$k_(1)k_{1}$k_1$ and $k_2$$k_2$k_(2)k_{2}$k_2$, respectively.

where: $k_n$$k_n$k_(n)k_{n}$k_n$ denotes the spring stiffness, with $k_1$$k_1$k_(1)k_{1}$k_1$ at left end $(x=0)$$(x=0)$(x=0)(x=0)$(x=0)$ and $k_2$$k_2$k_(2)k_{2}$k_2$ at the right one $(x=L)$$(x=L)$(x=L)(x=L)$(x=L)$, see Figure 1.

The boundary condition of both cable ends is simply supported [20]:

Complex modulus shall be rewritten [13] [14] as:

The storage modulus ( $E$$E$EE$E$ ), the real part of Eqn. (12), is directly measured from vibration signal, and the dissipative modulus ( $E$$E$EE$E$ '' which is the imaginary part, is indirectly determined as shown in Eqns. (13) and (14) [15].

And the loss factor from cable material is also a function of frequency is derived:

The spatial modes remain unchanged because the damping component only affects the frequency domain. For a nontrivial solution of $C_2$$C_2$C_(2)C_{2}$C_2$ and $C_4$$C_4$C_(4)C_{4}$C_4$ in boundary conditions, the determinant of their coefficient matrix is set equal to zero. Consequently, Eqn. (16) is obtained as follows:

where: ${\beta }_n$${\beta }_n$beta_(n)\beta_{n}${\beta }_n$ is the parameter depends on boundary conditions, $k_1$$k_1$k_(1)k_{1}$k_1$ and $k_2$$k_2$k_(2)k_{2}$k_2$ are the stiffness of spring, $n$$n$nn$n$ is the number of modes. In small damping case, approximation for expressions ${\mathrm{C}}_{\alpha }{\omega }^{\alpha }\ll \mathrm{E}$${\mathrm{C}}_{\alpha }{\omega }^{\alpha }\ll \mathrm{E}$C_(alpha)omega^(alpha)≪E\mathrm{C}_{\alpha} \omega^{\alpha} \ll \mathrm{E}${\mathrm{C}}_{\alpha }{\omega }^{\alpha }\ll \mathrm{E}$ and $\eta \approx ({\mathrm{C}}_{\alpha }/\mathrm{E}{\omega }^{\alpha }\sin (\alpha \pi /2)$$\eta \approx \left({\mathrm{C}}_{\alpha }/\mathrm{E}{\omega }^{\alpha }\sin (\alpha \pi /2)\right.$eta~~(C_(alpha)//Eomega^(alpha)sin(alpha pi//2):}\eta \approx\left(\mathrm{C}_{\alpha} / \mathrm{E} \omega^{\alpha} \sin (\alpha \pi / 2)\right.$\eta \approx ({\mathrm{C}}_{\alpha }/\mathrm{E}{\omega }^{\alpha }\sin (\alpha \pi /2)$ [15]. The fractional exponent $\alpha$$\alpha$alpha\alpha$\alpha$ with range from zero to one governs the nature of the viscoelastic response. When $\alpha$$\alpha$alpha\alpha$\alpha$ is equal one, the model reduces to the classical Kelvin-Voigt formulation, in which the damping term is linear in velocity and the effective loss factor is given by $\eta =\omega C/\mathrm{E}$$\eta =\omega C/\mathrm{E}$eta=omega C//E\eta=\omega C / \mathrm{E}$\eta =\omega C/\mathrm{E}$, where $\omega$$\omega$omega\omega$\omega$ is the angular frequency, $C$$C$CC$C$ is the fractional damping coefficient, and E is the elastic modulus. Conversely, when $\alpha$$\alpha$alpha\alpha$\alpha$ is equal zero, the damping term vanishes and the material tends toward purely elastic behavior without energy dissipation. This illustrates the essential role of $\alpha$$\alpha$alpha\alpha$\alpha$ in modulating energy dissipation and intrinsic material damping.
In this study, the hierarchical Bayesian model [25] is employed to estimate the viscoelastic parameter vector of each cable, $\mathrm{X}=({\mathrm{E}}^{\prime },\mathrm{C}\alpha ,\mathrm{f},\eta )$$\mathrm{X}=\left({\mathrm{E}}^{\prime },\mathrm{C}\alpha ,\mathrm{f},\eta \right)$X=(E^('),Calpha,f,eta)\mathrm{X}=\left(\mathrm{E}^{\prime}, \mathrm{C} \alpha, \mathrm{f}, \eta\right)$\mathrm{X}=({\mathrm{E}}^{\prime },\mathrm{C}\alpha ,\mathrm{f},\eta )$, which governs the fractional Kelvin-Voigt behavior. The physical constitutive relation is expressed through the complex modulus in Eqn. (12), from which the storage and loss components are derived, providing direct insight into the elastic and dissipative characteristics of the cable material. To represent variability across different cables, hierarchical
priors are assigned to the parameters $E^{\prime },C_a,f,\eta$$E^{\prime },C_a,f,\eta$E^('),C_(a),f,etaE^{\prime}, C_{a}, f, \eta$E^{\prime },C_a,f,\eta$ enabling the model to separate true material variability from segment-level estimation noise.
At the segment level, the parameters (a, b) or their corresponding empirical forms are characterized using statistical descriptors such as means $\mu \mathrm{a},\mu \mathrm{b}$$\mu \mathrm{a},\mu \mathrm{b}$mua,mub\mu \mathrm{a}, \mu \mathrm{b}$\mu \mathrm{a},\mu \mathrm{b}$ and their associated standard deviations ${\mu }_{sd}^a,{\mu }_{sd}^b$${\mu }_{sd}^a,{\mu }_{sd}^b$mu_(sd)^(a),mu_(sd)^(b)\mu_{s d}^{a}, \mu_{s d}^{b}${\mu }_{sd}^a,{\mu }_{sd}^b$. For each segment, the parameters ( a , b) are obtained via a local optimization step, and the Hessian matrix at the optimum is used to quantify identification confidence, expressed through ${\mu }_{\mathrm{sd}}^{\mathrm{a}}$${\mu }_{\mathrm{sd}}^{\mathrm{a}}$mu_(sd)^(a)\mu_{\mathrm{sd}}^{\mathrm{a}}${\mu }_{\mathrm{sd}}^{\mathrm{a}}$ and ${\mu }_{sd}^b$${\mu }_{sd}^b$mu_(sd)^(b)\mu_{s d}^{b}${\mu }_{sd}^b$. These segment-wise parameter distributions are then fused within the hierarchical structure, yielding global hyper-parameter estimates and corresponding 99% credible intervals (CI99), such as ${}_a{\mu }_{\text{CI99}}^{\text{low}}-{}_a{\mu }_{\text{CI99}}^{\text{high}}$${}_a{\mu }_{\text{CI99 }}^{\text{low }}-{}_a{\mu }_{\text{CI99 }}^{\text{high }}$_(a)mu_("CI99 ")^("low ")-_(a)mu_("CI99 ")^("high "){ }_{a} \mu_{\text {CI99 }}^{\text {low }}-{ }_{a} \mu_{\text {CI99 }}^{\text {high }}${}_a{\mu }_{\text{CI99 }}^{\text{low }}-{}_a{\mu }_{\text{CI99 }}^{\text{high }}$ and ${}_b{\mu }_{\text{CI99}}^{\text{low}}-{}_b{\mu }_{\text{CI99}}^{\text{high}}$${}_b{\mu }_{\text{CI99 }}^{\text{low }}-{}_b{\mu }_{\text{CI99 }}^{\text{high }}$_(b)mu_("CI99 ")^("low ")-_(b)mu_("CI99 ")^("high "){ }_{b} \mu_{\text {CI99 }}^{\text {low }}-{ }_{b} \mu_{\text {CI99 }}^{\text {high }}${}_b{\mu }_{\text{CI99 }}^{\text{low }}-{}_b{\mu }_{\text{CI99 }}^{\text{high }}$ [25]. The modeling or measurement uncertainty is described by the mean $\sigma$$\sigma$sigma\sigma$\sigma$, its standard deviation $\sigma$$\sigma$sigma\sigma$\sigma$ sd, and the $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ credible bounds ${}_{\sigma }{\mu }_{\text{CI99}}^{\text{low}}-{}_{\sigma }{\mu }_{\text{CI99}}^{\text{high}}$${}_{\sigma }{\mu }_{\text{CI99 }}^{\text{low }}-{}_{\sigma }{\mu }_{\text{CI99 }}^{\text{high }}$_(sigma)mu_("CI99 ")^("low ")-_(sigma)mu_("CI99 ")^("high "){ }_{\sigma} \mu_{\text {CI99 }}^{\text {low }}-{ }_{\sigma} \mu_{\text {CI99 }}^{\text {high }}${}_{\sigma }{\mu }_{\text{CI99 }}^{\text{low }}-{}_{\sigma }{\mu }_{\text{CI99 }}^{\text{high }}$, while ${\mathrm{N}}_{\text{obs}}$${\mathrm{N}}_{\text{obs }}$N_("obs ")\mathrm{N}_{\text {obs }}${\mathrm{N}}_{\text{obs }}$, reports the number of data points supporting each model. Local parameter estimation relies on numerical optimization, and the Hessian at the optimum provides a Gaussian approximation for quantifying local uncertainty.
At the hierarchical level, information from all cables and segments is fused to infer the hyper-parameters, which encode the population-level distribution and inter-cable variability. The joint posterior is explored using Markov Chain Monte Carlo (MCMC), enabling full posterior characterization, convergence diagnostics, and uncertainty propagation for cable-tension prediction. Parameter identifiability such as parameter collinearity or non-uniqueness under certain excitation spectra are examined, and prior sensitivity analyses are conducted to ensure robustness of the inferences.
This hierarchical Bayesian framework provides reliable viscoelastic parameter estimates for each cable and enables probabilistic prediction of cable tension, explicitly distinguishing physical variability from modeling and measurement uncertainty.
In the next section, the vibration data measured at cables of the Hwamung bridge (Busan city) [4] during construction were used to validate the damped vibration model of this study. Then, cable vibration of Phu My Cable-Stayed Bridge (Ho Chi Minh city) were used to investigate tension considering damping with proposed model.

In this research, the proposed method is applied and validated by Choi's data [4], which were collected from the Hwamung bridge, Korea. The cable properties and geometry for BLC02 cable of the Hwamung bridge are shown in Tab. 1, the first six frequencies are shown in Tab. 2, and Tab. 3 will show the comparison of other methods with this study's method.

The proposed model with linear spring at each end is well suitable, as the error is less than $4.71\mathrm{\%}$$4.71\mathrm{\%}$4.71%4.71 \%$4.71\mathrm{\%}$ with $\alpha \approx 0.001$$\alpha \approx 0.001$alpha~~0.001\alpha \approx 0.001$\alpha \approx 0.001$. In the next subsection, calculations for a real measurement were conducted for collected data at Phu My bridge.

In 2017, some measuring trips with ten sessions for each measuring trip was performed for some cables in the Phu My bridge, a cable-stayed bridge crossing the Sai Gon River (Fig. 2). Tab. 4 shows cable properties and geometry for measured cables labeled C2102N, C2207N, C2212N, C2215N, and C2115N. The measured data which shall be processed by the Fourier transform (Fig. 3) to determine the tensile force, have collected about 270 frequency data with some missing data due to noise or error.

Figure 3: Cable vibration Spectrums of Phu My Bridge in 2017.

In this calculation, Hierarchical Bayes model is adopted since the vibration data of cables are naturally grouped by each cable, measurement session, shape mode, and the sample sizes are sparse and unbalanced across groups. Multiple physical factors act together, such as tension, bending stiffness, boundary, and viscoelastic effects, and often include missing data due to noise or error, additional, a framework is needed to allow group-specific behavior while still sharing information across groups. The hierarchical Bayesian approach allows each group to have its own baseline and sensitivity while still sharing information across groups through partial pooling, which gives more stable estimates and full posterior uncertainty for every parameter, which single regression cannot capture. This statistics model also lets us embed physics in two complementary ways: a phenomenological regression that relates tension to explanatory variables such as the reciprocal of storage modulus or a viscoelastic factor, and a physics-informed likelihood that predicts frequencies from tension and section properties to infer tension directly. Results are presented with shrinkage plots, trend lines with credible bands, caterpillar plots of group parameters, posterior predictive checks by mode, and fit-metric tables. Another advantage is that it can easily integrate mechanical principles into the statistical framework, linking observed frequencies to underlying physical quantities such as tension and stiffness.
The data analysis followed a clear workflow, with the first step, vibration measurements were organized into a long-format table including cable name, session, measured frequency, and related covariates. Secondly, derived variables such as effective length or viscoelastic factor were calculated, and modes with reliable signals were selected. Third, Bayesian hierarchical models were built with weakly informative priors and fitted using MCMC sampling. Convergence diagnostics ensured the reliability of the posterior estimates. Fourth, posterior summaries and predictive checks were conducted to evaluate model fit. Finally, graphical outputs were generated: shrinkage plots showing per-cable tension estimates, trend plots with 95 to $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ credible bands illustrating global relationships, caterpillar plots for group parameters, and posterior predictive. These visualizations link the statistical inference directly to mechanical interpretation, highlighting the trend and uncertainty in each cable's estimated tension.
Bayesian hierarchical modeling was applied to estimate the posterior distributions of the model parameters and produce cable-specific predictions with credible intervals. Tensile force is taken as the real part of Eqn. (16). Tab. 5 for the model in Eqn. (18) with $E^{\prime }$$E^{\prime }$E^(')E^{\prime}$E^{\prime }$ and $N$$N$NN$N$ across five cables such as: C2102N, C2207N, C2212N, C2215N, and C2115N. The CI99 for the slope $b$$b$bb$b$ excludes zero, indicating a statistically significant effect at the cable level. Since the predictor is $X=E^{\prime }$$X=E^{\prime }$X=E^(')X=E^{\prime}$X=E^{\prime }$, a positive $b$$b$bb$b$ means $N$$N$NN$N$ increases as $1/E^{\prime }$$1/E^{\prime }$1//E^(')1 / E^{\prime}$1/E^{\prime }$ increases equivalently, $N$$N$NN$N$ decreases as $E^{\prime }$$E^{\prime }$E^(')E^{\prime}$E^{\prime }$ increases. While intercepts a differ substantially by cable, the slopes, are nearly identical, showing that Eqn. (18) captures a common trend shared by all cables: $1/E^{\prime }$$1/E^{\prime }$1//E^(')1 / E^{\prime}$1/E^{\prime }$ has the same influence across cables, and between cable differences are primarily offset rather than differences in sensitivity.

The Fig. 4 shows the relationship between tensile force, $N$$N$NN$N$, and the storage modulus, $E^{\prime }$$E^{\prime }$E^(')E^{\prime}$E^{\prime }$, with medians and $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ prediction ranges for the data sets C2102N, C2207N, C2212N, C2215N, and C2115N. Tensile force ranges from $6\times {10}^7\mathrm{N}$$6\times {10}^7\mathrm{N}$6xx10^(7)N6 \times 10^{7} \mathrm{~N}$6\times {10}^7\mathrm{N}$ to $13\times {10}^7$$13\times {10}^7$13 xx10^(7)13 \times 10^{7}$13\times {10}^7$ N as $E^{\prime }$$E^{\prime }$E^(')E^{\prime}$E^{\prime }$ varies from $2\times {10}^{11}{10}^{11}\mathrm{Pa}$$2\times {10}^{11}{10}^{11}\mathrm{Pa}$2xx10^(11)10^(11)Pa2 \times 10^{11} 10^{11} \mathrm{~Pa}$2\times {10}^{11}{10}^{11}\mathrm{Pa}$ to $12\times {10}^{11}\mathrm{Pa}$$12\times {10}^{11}\mathrm{Pa}$12 xx10^(11)Pa12 \times 10^{11} \mathrm{~Pa}$12\times {10}^{11}\mathrm{Pa}$, while the $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ prediction range shows some uncertainty, especially at $E^{\prime }\approx 2\times {10}^{11}\mathrm{Pa}$$E^{\prime }\approx 2\times {10}^{11}\mathrm{Pa}$E^(')~~2xx10^(11)PaE^{\prime} \approx 2 \times 10^{11} \mathrm{~Pa}$E^{\prime }\approx 2\times {10}^{11}\mathrm{Pa}$ and above, with some data points falling outside the range. The data sets tend to be similar at $E^{\prime }\approx 6\times {10}^{11}\mathrm{Pa}$$E^{\prime }\approx 6\times {10}^{11}\mathrm{Pa}$E^(')~~6xx10^(11)PaE^{\prime} \approx 6 \times 10^{11} \mathrm{~Pa}$E^{\prime }\approx 6\times {10}^{11}\mathrm{Pa}$ and $8\times {10}^{11}\mathrm{Pa}$$8\times {10}^{11}\mathrm{Pa}$8xx10^(11)Pa8 \times 10^{11} \mathrm{~Pa}$8\times {10}^{11}\mathrm{Pa}$, but C 2102 N and C 2115 N have higher tension at some points. Some outliers need to be further considered to improve the accuracy of the prediction model.

In Fig. 5, the posterior distribution of ${\mu }_a$${\mu }_a$mu_(a)\mu_{a}${\mu }_a$, the global mean of cable-level intercepts, in the model $N=a+b/E^{\prime }$$N=a+b/E^{\prime }$N=a+b//E^(')N=a+b / E^{\prime}$N=a+b/E^{\prime }$ was shown with the highest frequency $(400;500)$$(400;500)$(400;500)(400 ; 500)$(400;500)$ samples centered around zero, symmetrical and decreasing from -4000 to 4000 . The distribution is dense at -2000 to 2000 , low at the extremes. From Tab. 5, $a_{\text{mean}}$$a_{\text{mean }}$a_("mean ")a_{\text {mean }}$a_{\text{mean }}$ ranges from 88325831 N to 95032006 N between cables, $a_{\text{sd}}$$a_{\text{sd }}$a_("sd ")a_{\text {sd }}$a_{\text{sd }}$ from 8724 N to $73420\mathrm{N},b_{\text{mean}}$$73420\mathrm{N},b_{\text{mean }}$73420N,b_("mean ")73420 \mathrm{~N}, b_{\text {mean }}$73420\mathrm{N},b_{\text{mean }}$ is stable $652.69\mathrm{N}\cdot \mathrm{Pa}$$652.69\mathrm{N}\cdot \mathrm{Pa}$652.69N*Pa652.69 \mathrm{~N} \cdot \mathrm{~Pa}$652.69\mathrm{N}\cdot \mathrm{Pa}$ (CI99: 476.38 to 932.44). Extreme values ( -4000 ; 4000) are rare exceptions. The model is stable around ${\mu }_a=0$${\mu }_a=0$mu_(a)=0\mu_{a}=0${\mu }_a=0$, but $a_{sd}$$a_{sd}$a_(sd)a_{s d}$a_{sd}$ variation like needs further study to increase reliability.

The graph (Fig. 6) shows the relationship between N and the variable $C_a$$C_a$C_(a)C_{a}$C_a$, with medians and $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ prediction ranges for the data sets C2102N, C2207N, C2212N, C2215N, and C2115N. Tensile force ranges from $6\times {10}^7\mathrm{N}$$6\times {10}^7\mathrm{N}$6xx10^(7)N6 \times 10^{7} \mathrm{~N}$6\times {10}^7\mathrm{N}$ to $13\times {10}^7\mathrm{N}$$13\times {10}^7\mathrm{N}$13 xx10^(7)N13 \times 10^{7} \mathrm{~N}$13\times {10}^7\mathrm{N}$ as $C_a$$C_a$C_(a)C_{a}$C_a$ varies from $5.2\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$$5.2\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$5.2 xx10^(11)Pa*s^(alpha)5.2 \times 10^{11} \mathrm{~Pa} \cdot \mathrm{~s}^{\alpha}$5.2\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$ to $8.3\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$$8.3\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$8.3 xx10^(11)Pa*s^(alpha)8.3 \times 10^{11} \mathrm{~Pa} \cdot \mathrm{~s}^{\alpha}$8.3\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$. The $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ prediction range (light color) shows large variation at $C_a>7\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$$C_a>7\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$C_(a) > 7xx10^(11)Pa*s^(alpha)C_{a}>7 \times 10^{11} \mathrm{~Pa} \cdot \mathrm{~s}^{\alpha}$C_a>7\times {10}^{11}\mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }$, with some data points falling outside the range, possibly outliers. The medians show similarities between the sets, but C2115N tends to be higher at large $C_a$$C_a$C_(a)C_{a}$C_a$. Each colored band represents the $99\mathrm{\%}$$99\mathrm{\%}$99%99 \%$99\mathrm{\%}$ posterior predictive distribution for one cable, including both parameter uncertainty, and observation-level uncertainty. Because these uncertainties are substantial relative to the differences between cables, bands overlap almost entirely in the central region. Overlap therefore represents: shared predictive ranges, absence of strong cable-to-cable separation, and convergence of all posterior predictions to a common tension interval.

Figure 6: Median and 99% predictive band for damping coefficient ( $C_a$$C_a$C_(a)C_{a}$C_a$ ) and tensile force ( $N$$N$NN$N$ ).

The posterior distribution of the global mean of cable-level intercepts ( ${\mu }_a$${\mu }_a$mu_(a)\mu_{a}${\mu }_a$ ) was shown (Fig. 7), the global mean intercept across cables; its center and width summarize the baseline level and its uncertainty, the posterior under $N=a+b/C_a$$N=a+b/C_a$N=a+b//C_(a)N=a+b / C_{a}$N=a+b/C_a$ is unimodal, roughly normal, centered near zero with slight left skew, indicating a stable global baseline and no multimodality; the width conveys uncertainty.
The histogram from Fig. 7 shows the posterior distribution of the global mean of cable-level intercepts, ${\mu }_a$${\mu }_a$mu_(a)\mu_{a}${\mu }_a$, in this model, with the highest frequency from 400 to 500 centered around zero, symmetrical and decreasing from -4000 to 4000 . The distribution is dense at -2000 to 2000 , low at the extremes. Tab. 6, ${\mathrm{a}}_{\text{mean}}$${\mathrm{a}}_{\text{mean }}$a_("mean ")\mathrm{a}_{\text {mean }}${\mathrm{a}}_{\text{mean }}$ ranges from 86523172 N (C2102N) to 94988594 N $(\mathrm{C}2212\mathrm{N}),a_{sd}$$(\mathrm{C}2212\mathrm{N}),a_{sd}$(C2212N),a_(sd)(\mathrm{C} 2212 \mathrm{~N}), a_{s d}$(\mathrm{C}2212\mathrm{N}),a_{sd}$ from $11973\mathrm{N}(\mathrm{C}2102\mathrm{N})$$11973\mathrm{N}(\mathrm{C}2102\mathrm{N})$11973N(C2102N)11973 \mathrm{~N}(\mathrm{C} 2102 \mathrm{~N})$11973\mathrm{N}(\mathrm{C}2102\mathrm{N})$ to $105965\mathrm{N}(\mathrm{C}2215\mathrm{N}),b_{\text{mean}}$$105965\mathrm{N}(\mathrm{C}2215\mathrm{N}),b_{\text{mean }}$105965N(C2215N),b_("mean ")105965 \mathrm{~N}(\mathrm{C} 2215 \mathrm{~N}), b_{\text {mean }}$105965\mathrm{N}(\mathrm{C}2215\mathrm{N}),b_{\text{mean }}$ is stable at $1429.65\mathrm{N}\cdot \mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }\cdot (\mathrm{CI}99:1284.89;1687.51)$$1429.65\mathrm{N}\cdot \mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }\cdot (\mathrm{CI}99:1284.89;1687.51)$1429.65N*Pa*s^(alpha)*(CI 99:1284.89;1687.51)1429.65 \mathrm{~N} \cdot \mathrm{~Pa} \cdot \mathrm{~s}^{\alpha} \cdot(\mathrm{CI99:} 1284.89 ; 1687.51)$1429.65\mathrm{N}\cdot \mathrm{Pa}\cdot {\mathrm{s}}^{\alpha }\cdot (\mathrm{CI}99:1284.89;1687.51)$. Extreme values (-4000; 4000) are rare exceptions.