Long short-term memory-enhanced semi-active control of cable vibrations with a magnetorheological damper

Abstract

The large vibrations of stay cables pose significant challenges to the structural performance and safety of cable-stayed bridges. While magnetorheological dampers (MRDs) have emerged as an effective solution for suppressing these vibrations, establishing accurate forward and inverse mapping models for MRDs to facilitate effective semi-active control of cable vibrations remains a formidable task. To address this issue, the current study proposes an innovative strategy that leverages Long Short-Term Memory (LSTM) neural networks for MRD modeling, thus enhancing semi-active control of stay cable vibrations. A high-fidelity data set accurately capturing the MRD dynamics is first generated by coupling finite element analysis and computational fluid dynamic approach. The obtained data set is then utilized for training LSTM-based forward and inverse mapping models of MRD. These LSTM models are subsequently integrated into dynamic computational models for effectively suppressing the stay cable vibrations, culminating in an innovative semi-active control strategy. The feasibility and superiority of the proposed strategy are demonstrated through comprehensive comparative analyses with existing passive, semi-active and active control methodologies involving sinusoidal load, Gaussian white noise load and rain–wind induced aerodynamic load scenarios, paving the way for novel solutions in semi-active vibration control of large-scale engineered structures.

Graphical abstract

Keywords

stay cable vibration / magnetorheological damper / semi-active control / LSTM / cable-stayed bridge

1 Introduction

Cable-stayed bridges, renowned for their aesthetic appeal and structural efficiency, often grapple with the challenge of cable vibrations induced by environmental factors such as wind/rain, earthquakes and vehicular loads, among others. These vibrations pose a potential threat to the performance and safety of the bridge structure [1,2]. To suppress these cable vibrations, several strategies have been explored, among which the use of magnetorheological dampers (MRDs) has emerged as a promising solution [2,3]. MRDs represent an innovative class of smart adjustable dampers, utilizing the magnetorheological fluids (MRFs) as working medium, enabling semi-active control of structural vibrations. Contrary to traditional passive dampers, MRDs exhibit a unique ability to adjust damping forces in real-time in response to variations in piston speed, displacement and control current input, thereby ensuring optimal damping performance [4]. These dampers maintain reliable operational performance even under extreme conditions, offering numerous advantages, such as rapid response, simple structure and low energy consumption. Consequently, MRDs are regarded as a highly effective technology in the control of cable vibrations in cable-stayed bridges. MRDs were first installed on a 115-m long cable of the Dong-ting Lake Bridge and maintained to operate under a constant current lacking the ability to actively adapt to complex external environmental conditions [5]. Later, they were affixed to the cable of the Yellow River Highway Bridge for the control of cable vibrations by assuming a simple relationship between the peak damping force and the current [6].

In practical applications, the primary challenge in realizing semi-active control of cable vibrations using MRDs lies in providing appropriate damping forces based on the real-time vibrational state of the cable (including speed and displacement) and the magnitude of control current. Although the vibrational state of the cable can be obtained through real-time monitoring, determining the optimal damping force requires reverse calculation for the control current input [7]. Therefore, it is of paramount importance to accurately establish both the forward mapping model (obtaining the damping force through the motion state and control current) and the inverse mapping model (obtaining the control current through the motion state and desired damping force) of the MRD. However, the behavior of the MRD involves multiple physical processes, including fluid dynamics, electromagnetics and their interaction, as well as the complex nonlinearity and typical hysteresis characteristics of MRF under a magnetic field [8]. This complexity results in considerable challenges in accurately describing these phenomena through traditional theoretical/empirical equations.

A considerable body of research has been devoted to the modeling and control techniques of MRDs. As outlined in Ref. [9], the Bingham model, being relatively simple, was easy to implement. However, it may fail to accurately describe the behavior of MRDs when the shear strain rate of the fluid undergoes significant changes. The Bouc-Wen model proposed in Ref. [10] can effectively represent the force-displacement relationship of MRDs. Nonetheless, its complex mathematical form requires extensive computation and may not accurately capture the behavior of the damper under extreme conditions. The hyperbolic parameter model proposed in Ref. [11], despite being simple and easily adjustable, may not truly reflect the performance of MRDs due to the inherent cusps issue. Although computational fluid dynamics (CFD) can provide high-precision simulation and visualization of fluid behavior [12], it requires substantial computational resources and is not applicable for real time applications. While the aforementioned modeling schemes are typically used to describe the forward mapping model of MRDs, it is extremely challenging to derive their inverse versions that are crucial for realizing semi-active control of structural vibrations. The polynomial model for MRDs in Ref. [13], obtained by fitting experimental data, facilitated the derivation of its inverse model and the reverse calculation of control current input. However, it is overly simplistic and susceptible to the severe Runge phenomenon. To this end, both back-propagation neural network [14] and multilayer neural network have been employed to conduct the MRD inverse model. It is noted that the lack of time dimension in these networks poses great challenge of them for time series predictions [15].

To address this issue, especially to let the networks remember inputs for a long time, an explicit memory, the long short-term memory (LSTM), was introduced to augment the network [16,17]. Over recent years, LSTM neural networks have demonstrated their robust potential in addressing complex, nonlinear time-series mapping challenges [18–20]. However, their applications in MRD modeling and control remains largely unexplored. Hence, the primary objective of this study is to develop an innovative approach that employs LSTM neural networks to enhance the modeling and semi-active control of MRDs for mitigating cable vibrations of cable-stayed bridges. Accordingly, Section 2 will formulate the solution processes of the MRD numerical model, resulting in the generation of a high-fidelity data set that accurately depicts the MRD dynamics. In Section 3, the high-precision data set will be harnessed for training the LSTM network, whose architecture is optimized through the Particle Swarm Optimization (PSO) algorithm, leading to the establishment of both LSTM-based forward and inverse mapping models for the MRD. Section 4, set against the engineering context of the A20 stay cable of the Second Nanjing Yangtze River Bridge, will merge the developed LSTM-based models for MRD with the dynamic computation model of the stay cable, thereby providing an LSTM-enhanced semi-active control of stay cable vibrations with a MRD. A thorough comparative analysis of the vibration mitigation of stay cables under four distinct load scenarios, employing passive control, LSTM-enhanced semi-active control, Bang-Bang semi-active control and active control methods will be conducted, underscoring the practicability and superiority of the proposed control strategy. Section 5 will conclude the current study.

2 Training data set generation using numerical model

To generate high-fidelity data set that can reflect the MRD performance under various operational conditions, a one-way coupled numerical model of finite element analysis (FEA) and CFD is utilized. Specifically, the COMSOL 6.5 is employed for FEA of the static magnetic field within the flow domain, and the relationship between the magnetic flux density at the throttle channel and the current is formulated. Thereafter, the apparent viscosity of MRF is defined using the hyperbolic tangent model in relation to the shear strain rate and the magnetic flux density, with Fluent 19.2 used for the CFD modeling of the MRD. A number of different input current operating conditions are selected to determine the damping forces at various speeds and displacements, thereby preparing a comprehensive high-fidelity data set of MRD dynamics.

2.1 Magnetorheological damper overview

In this study, a double-rod shear valve-type MRD is selected as the research object. Its basic structure is shown in Fig.1, where component (1) is insulating shell, (2) is electrifiable coil, (3) is fluid-domain piston head (high magnetic permeability), (4) is piston rod (low magnetic permeability), (5) is MRF-filled inner fluid domain, (6) is air domain, (7) is air-domain piston head, and (8) is fluid-domain throttling channel. The damper utilizes MRF-140 CG by LORD Corporation, with its H–B (magnetic field strength−magnetic flux density) and H–τ_y (magnetic field strength−yield stress) curves detailed in Fig.2(a) and Fig.2(b), respectively. The piston head and cylinder adopt a magnetically conductive carbon steel structure AISI-1045, whereas the rod is of a low magnetic conductance metal. The coil adheres to the AWG-24 copper standard. The selected damper dimensions and material parameters are consolidated in Tab.1 [21].

Comprising minute ferromagnetic particles and a carrier fluid, the MRF displays dynamic magnetic responsiveness [22]. While these particles remain randomly dispersed in a non-magnetic environment, a magnetic field causes them to swiftly align with the force lines, leading to a significant viscosity uptick. This immediate and reversible change is governed by magnetic field intensity that is modulated with the electrifiable coil, adjusting the current to alter the MRF viscosity. Increased current enhances the magnetic field in both the inner fluid domain and the throttle channel, elevating MRF viscosity. This surge in resistance as the MRF traverses the throttle channel amplifies the damping force. Conversely, lowering the current decreases viscosity, subsequently reducing the damping force.

2.2 Numerical simulation of magnetorheological damper

The axisymmetric structure of the selected a double-rod shear valve-type MRD enables a two-dimensional simulation based on the parameters in Tab.1 [21]. The magnetic field distribution within the inner flow domain is governed by Maxwell’s equations [23].

(1)\{\begin{array}{l}\mathrm{\nabla }J=0,\\ \mathrm{\nabla }H=J,\\ B=\mathrm{\nabla }A,\end{array}

(2)B={\mu }_0{\mu }_{\mathrm{r}}H,

where H is the magnetic field strength, J means the current density, A represents the magnetic vector potential, B denotes magnetic flux density, and the terms {\mu }_0 and {\mu }_{\mathrm{r}} refer to the material and vacuum magnetic permeability, respectively.

The electrical insulation parameters and magnetic field strengths at relevant boundaries are specified to accurately capture the physical environment. The unstructured mesh is utilized to accommodate complex geometries and facilitate local refinement in areas demanding very high resolution. A strategy integrating the finite element method (FEM) and the boundary element method (BEM) is employed for the discretization of Maxwell’s equations, where the FEM leverages high-order Lagrangian elements to guarantee the accuracy of numerical solution, while the BEM concentrate on the effective treatment of boundary conditions. Accordingly, the computational domain is partitioned into 35726 triangular mesh elements. The mesh configuration around in the throttling channel area is illustrated in Fig.3. To validate mesh independence, a denser mesh consisting of 87228 triangular elements is also constructed. Fig.4 computes the magnetic flux density distribution curves along the centerline of throttling channel at an input current of 4.0 A using both mesh grids. A Newton-Raphson method-based solver in COMSOL 6.5 is used to solve the discretized equations. The near-identical results suggest that the coarser mesh is sufficient to perform the current simulation. Hence, it is the chosen grid for subsequent analyses.

It is well known that the magnetic flux density B within the internal flow domain linearly correlates with the input current I [24], and this relationship is substantiated here by the FEA results in Fig.5. Accordingly, B is formulated as a linear function of I in subsequent CFD simulations.

(3)

where x denotes the distance from any point to the throttle channel center.

Considering the Reynolds number for the MRD working conditions in this study is small than five, a laminar flow model is selected for the CFD simulation. The governing equations for fluid motion are the continuity equation and Navier-Stokes equation [25].

(4)\{\begin{array}{rl}& \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\left(\rho V\right)=0,\\ & \frac{\mathrm{\partial }\left(\rho V\right)}{\mathrm{\partial }t}+\mathrm{\nabla }\left(\rho VV\right)=f_{\mathrm{b}}-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^{2}V,\end{array}

where t is the time, \rho denotes the fluid density, V represents the fluid velocity, p stands for the pressure difference, f_b refers to the volumetric force, \mu signifies the dynamic viscosity of the fluid. A rheological viscosity model is employed to characterize the viscosity of MRF defined by the hyperbolic tangent function [26,27].

(5)\mu =\frac{{\tau }_{\mathrm{y}}\left(B\right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\zeta \dot{\gamma }\right)}{\sqrt{{\alpha }^2+{\dot{\gamma }}^2}}+{\mu }_{\mathrm{p}},

where \alpha and \zeta represent viscosity growth control factors, with respective values of 0.03 and 0.1, {\mu }_{\mathrm{p}} indicates plastic viscosity, \dot{\gamma } represents shear rate.

The boundary conditions similar to those in the parallel-plate theoretical model are adopted in the CFD simulation. The computational domain is simplified into a semi-structured parallel plate based on symmetry [28,29]. Quadrilateral grids are used for discretizing the computational domain, with a focus on grid refinement at the inner walls of the throttle channel. Dynamic mesh technology is employed in the piston head boundary region for automatic grid updating. As shown in Fig.6, the walls on either side of the piston are defined as rigid moving walls, and the internal flow domain is divided into a total of 98,828 elements. The relationship curve between the yield stress {\tau }_{\mathrm{y}} and the magnetic flux density B is fitted using the Hill sigmoidal function, given as

(6){\tau }_{\mathrm{y}}\left(B\right)=\epsilon \frac{B^k}{{\beta }^k+B^k},

where \epsilon = 117024.8, \beta = 0.96766, and k = 1.69363. The air spring force F_s enabling the compressed damper to reset automatically is described using the polytropic process equation for gas.

(7)F_{\mathrm{s}}=P_0{A}_{\mathrm{a}\mathrm{i}\mathrm{r}}{\left(\frac{W_0}{W_0-A_{\mathrm{a}\mathrm{i}\mathrm{r}}x}\right)}^n,

where P₀ signifies the initial pressure, set at 2.5 MPa, A_air indicates the piston area, W₀ is the initial volume, n stands for the polytropic index of gas, set at 1.3, x represents the displacement of the piston.

Equations (5) and (6) form the fundamental basis for coupling of the FEA of magnetic field and the CFD of fluid flow. A User-Defined Function (UDF) serves as a transformative tool that transposes the distribution of magnetic flux density (B), ascertained from static magnetic field FEA, into the spatiotemporal distribution of dynamic viscosity (\mu) within the fluid domain. This transformation is subsequently incorporated into the CFD simulations, where a transient solver employing a coupled algorithm alongside a second-order upwind discretization scheme is deployed. The temporal resolution for solutions is fixed at 0.0001 s. Fig.7(a) and Fig.7(b) display a comparison between the experimental data of the MRD performance curve [21] and the simulated data obtained through the coupled FEA-CFD method for the case of current I = 4 A and piston velocity v = 0.1 cos(2πt). The numerical simulation results closely align with the experimental outcomes, validating the feasibility and accuracy of the FEA-CFD coupling approach for simulating the dynamic characteristics of MRD. It is noted that the significant data discrepancies around 0 s and 8 s are due essentially to the limit of physical experiment for instantly reaching the specified velocity.

2.3 Training data set generation

Various scenarios covering a wide range of input currents and piston velocities are carefully designed to generate a comprehensive training data set for LSTM neural network. First, an array of distinct randomly generated current fluctuations are input to the MRD, along with a sinusoidal piston velocity function with an amplitude of 0.1 m/s and a frequency of 1 Hz. For each type of randomly generated input current, calculations are performed separately using the coupled FEA-CFD method. Fig.8 displays the time-course curves of the electrical signal under five different scenarios, and the corresponding hysteresis loop curves are shown in Fig.9. It can be observed that the MRD exhibits complex behavioral patterns under random current inputs.

In addition, another scenario where both the piston velocity and input current vary randomly (Scenario 6) is designed to better capture the randomness of actual structural vibrations. Fig.10 shows the realized velocity and current signals used for the training data set generation, and Fig.11 presents the corresponding simulation results.

3 Long short-term memory-based mapping models for magnetorheological damper

By addressing the pervasive vanishing and exploding gradient issues that traditional Recurrent Neural Networks typically encounter when processing extended sequences, the LSTM network proves particularly apt for simulating complex dynamic processes inherent to MRD systems. This distinct network cell predominantly comprises four crucial components, namely the input gate, forget gate, output gate and cell state [16]. During the cell state updating process, the input gate modulates the degree to which input information at the current time step is integrated into the current cell state. Concurrently, the forget gate controls the proportion of the previous cell state maintained within the current cell state. Ultimately, the output gate ascertains the extent to which the current cell state contributes to the output value at the current time step.

Fig.12 presents a schematic of a standard LSTM cell, where x_n denotes the cell input at time step n, h_n represents the cell output, s_n and c_n correspond to the long-term cell state and internal state gate, respectively, f_n, i_n, and o_n signify the forget gate, input gate and output gate, respectively, W_~ and b_~ (with subscripts ‘~’ indicated by ‘f ’, ‘i’, ‘c’, or ‘o’) constitute the weight matrices and biases, and σ(·) and tanh(·) refer to the logistic sigmoid and hyperbolic tangent functions, with value ranges of [0,1] and [−1,1], respectively.

In accordance with Fig.12, the forward algorithms, which represent the process of updating the cell state in LSTM, are expressed as

\begin{array}{}\text{(8a)}& {\mathit{f}}_n=\sigma \text{(}{\mathit{n}\mathit{e}\mathit{t}}_{\mathrm{f},n}\text{)},\end{array}

\begin{array}{}\text{(8b)}& {\mathit{i}}_n=\sigma \text{(}{\mathit{n}\mathit{e}\mathit{t}}_{\mathrm{i},n}\text{)},\end{array}

\begin{array}{}\text{(8c)}& {\mathit{c}}_n=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\text{(}{\mathit{n}\mathit{e}\mathit{t}}_{\mathrm{c},n}\text{)},\end{array}

\begin{array}{}\text{(8d)}& {\mathit{o}}_n=\sigma \text{(}{\mathit{n}\mathit{e}\mathit{t}}_{\mathrm{o},n}\text{)},\end{array}

\begin{array}{}\text{(8e)}& {\mathit{s}}_n={\mathit{f}}_n\odot {\mathit{s}}_{n-1}+{\mathit{i}}_n\odot {\mathit{c}}_n,\end{array}

\begin{array}{}\text{(8f)}& {\mathit{h}}_n={\mathit{o}}_n\odot \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\text{(}{\mathit{s}}_n\text{)},\end{array}

where the symbol ‘\odot’ represents the vector dot product. The input of the corresponding gate, net_~,n can be determined as follows

(9)\begin{array}{rl}{\mathit{n}\mathit{e}\mathit{t}}_{\sim ,n}=& {\mathit{W}}_{\sim }\times \left[{\mathit{h}}_{n-1},x_n\right]+{\mathit{b}}_{\sim }\\ =& \left[{\mathit{W}}_{\sim h},{\mathit{W}}_{\sim x}\right]\times \left[{\mathit{h}}_{n-1},x_n\right]+{\mathit{b}}_{\sim }\\ =& {\mathit{W}}_{\sim }{\mathit{h}}_{n-1}+{\mathit{W}}_{\sim x}{x}_n+{\mathit{b}}_{\sim },\end{array}

where the symbol ‘\times’ signifies matrix-vector multiplication.

In a multilayer LSTM network, as illustrated in Fig.13 where the superscript denotes the layer number and the subscript indicates the step number), the inputs x = [x₁,x₂,...,x_n,...,x_N] are employed exclusively in the first layer, while the cell outputs of each layer serve as the inputs for the subsequent layer. Utilizing the shared weight matrices within the same layer, the network recursively implements the algorithms presented in Eqs. (8) and (9) from the 1st to the lth layer. In the output layer, {\mathit{h}}_{\mathrm{N}}^{\mathrm{l}} is mapped to yield the network prediction as follows

(10){\hat{y}}_{\mathrm{N}}={\mathit{W}}_{\mathrm{y}}\times {\mathit{h}}_{\mathrm{N}}^{\mathrm{l}}+{\mathit{b}}_{\mathrm{y}},

where {\hat{y}}_{\mathrm{N}} denotes the prediction of the network at step N, {\mathit{W}}_{\mathrm{y}} and {\mathit{b}}_{\mathrm{y}} correspond to the weight matrix and bias in the output layer, respectively.

3.1 Design of long short-term memory-based models for magnetorheological damper

It is noted that the performance of a LSTM network is determined by both ‘hyper-parameters’ (consisting of input feature quantity, output signal quantity, structural hyper-parameters such as number of hidden layers, activation function type and time steps, training algorithm hyper-parameters such as optimization algorithm, learning rate, batch size and regularization method, and loss functions) and ‘parameters’ (consisting of weight matrices and bias sequences which are typically adaptively adjusted during the training process via error backpropagation to fit nonlinear data). While the former is usually designed based on either experience or optimization, the latter is determined through the process of training the neural network using a specific data set [30].

The first step of the hyper-parameter design is to select appropriate network inputs and outputs. Specifically, the input parameters for the LSTM-based forward mapping model of the MRD include piston displacement, velocity and control current values, with the output parameter being the damping force. For the LSTM-based inverse mapping model, considering the requirements of semi-active control system design, the input parameters are set as piston displacement, velocity and optimal control force (i.e., damping force), with the output parameter being the control current value. Accordingly, both the forward and inverse mapping models of the MRD form a multi-input single-output LSTM model. In addition, the Mean Square Error (MSE), widely used in regression problems, is chosen as the loss function for network training. Based on trial and error, this study constructs an LSTM network structure with three hidden layers connected to a single-layer fully connected neural network for output feature extraction subsequent to the LSTM network output layer. The tanh function is employed as the activation function for the hidden layers, while the fully connected neural network employs the Relu function as its activation function.

To effectively determine optimal values of cell number of each hidden layer, time steps and batch size, an adaptive PSO algorithm, introduced by Eberhart and Kennedy [31], is utilized. The adaptive characteristics of the PSO algorithm allow the LSTM model to rapidly and accurately ascertain the optimal hyper-parameter configuration based on training data attributes, effectively integrating the LSTM model network structure with data features, and enhancing learning efficiency and reliability. Since the Adam optimizer is utilized in the current LSTM network, the learning rate is automatically adjusted during the training process. Fig.14 delineates the specific computational procedure of the PSO algorithm. Twenty particles are concurrently trained to ascertain the optimal hyper-parameters and the network prediction accuracy serves as the fitness function. Tab.2 presents the hyper-parameter selection process and associated fitting error in each iteration. After the 20th iteration, the fitting accuracy no longer improves, yielding the optimal combination of 76 time steps, a batch size of 46, and 486 cells in the hidden layer.

3.2 Training long short-term memory-based models for magnetorheological damper

Numerical simulation data from Scenarios 1 to 4 and Scenario 6 are partitioned into training and testing data sets. Specifically, the initial 80% of the data serve as the training data set while the remaining 20% function as the testing data set. During the training phase, the three-layer LSTM-based model undergoes iterative updates to minimize the error between the predicted control current value and the actual control current value obtained from the training data set. The training process entails adjusting the network weights and biases through backpropagation and gradient descent methodologies.

The variations in training and prediction relative errors are closely monitored throughout the training process for both forward and inverse models, as shown in Fig.15. At the initial stage, the errors are generally high, as the model has not yet sufficiently learned the patterns within the data. As the training epoch increases, the values of these error metrics gradually decrease. The values of absolute error and relative error tend to approach zero for the training data set or the test data set, while they gradually decrease and stabilize, indicating model convergence. It can be observed that the error values for both the training and test data sets are already quite small with increasing numbers of training iterations, which suggests that the model has been trained adequately.

To further evaluate the effectiveness of the model training, Fig.16 presents a graphical portrayal of the fitting performance related to both the training and prediction segments across the data sets associated with Scenarios 1−4 and Scenarios 6. As suggested by the figure, the prediction performance for both the training and test data sets is remarkably satisfactory. When evaluating the fitting performance of the model, it is essential to ensure not only satisfactory fitting results for the training set but also the manifestation of a certain level of generalization capability by the model. To evaluate the generalization capacity of the trained LSTM model, a total of 20000 data points extracted from Scenario 5 are employed as the comprehensive prediction set and subsequently introduced for forecasting purposes. Fig.17 shows the simulation results of the trained forward and inverse models on the data set derived from Scenario 5, both presenting an impressive prediction performance. To clearly present the performance of LSTM models, the Root Mean Squared Error (RMSE) and Coefficient of Determination (R²) are evaluated for both forward and inverse mapping models under various scenarios and the results are listed in Tab.3 and Tab.4. The RMSE and R² values in these tables suggest that both the forward and inverse LSTM models maintain high predictive accuracy.

4 Long short-term memory-enhanced semi-active control of cable vibrations

Fig.18 depicts the principle behind implementing semi-active control utilizing MRDs. The sensors are tasked with real-time monitoring of the load information and dynamic responses of the stable cable when subjected to external loads. The controller computes the optimal control force via a control algorithm and applies it through the MRD for vibration mitigation. Specifically, the optimal control algorithm linear quadratic regulator (LQR) is employed for the determination of the optimal control force based on known structural dynamics, and the corresponding input current is swiftly calculated using the pre-established LSTM-based inverse mapping model for the MRD to ensure the output damping force closely aligns with the optimal control force. Moreover, the LSTM-based forward mapping model facilitates the conversion from input current to output damping force, enabling timely and effective vibration control of cables. It is noted that the dynamics of stay cable is calculated by numerical model in this study rather than measured by sensors.

4.1 Dynamic model of stay cable

The A20 stay cable of the Nanjing Yangtze River Bridge II equipped with a MRD is used to investigate the LSTM-enhanced semi-active control performance [32]. Based on the D’Alembert Principle, the governing equations for the stay cable motion are expressed as [33]

\begin{array}{}\text{(11a)}& A_1\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+A_2\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+A_3\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+A_4\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+F_{\mathrm{p}\mathrm{r}\mathrm{e},x}\left(x,t\right)=m\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}+c_1\frac{\mathrm{\partial }u}{\mathrm{\partial }t},\end{array}

\begin{array}{}\text{(11b)}& A_5\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+A_2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+A_6\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+A_4\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+F_{\mathrm{p}\mathrm{r}\mathrm{e},y}\left(x,t\right)=m\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{t}^2}+c_1\frac{\mathrm{\partial }v}{\mathrm{\partial }t},\end{array}

4.1.1 where

\begin{array}{}\text{(12a)}& A_1=\frac{H}{\sqrt{1+{\left[f^{\prime }\left(x\right)\right]}^2}}+\frac{EA}{{\left(1+{\left[f^{\prime }\left(x\right)\right]}^2\right)}^2},\end{array}

\begin{array}{}\text{(12b)}& A_2=\frac{EA{f}^{\prime }\left(x\right)}{{\left(1+{\left[f^{\prime }\left(x\right)\right]}^2\right)}^2},\end{array}

\begin{array}{}\text{(12c)}& A_3=-\frac{3EA{f}^{\prime }\left(x\right)}{{\left(1+{\left[f^{\prime }\left(x\right)\right]}^2\right)}^3}{f}^{″}\left(x\right),\end{array}

\begin{array}{}\text{(12d)}& A_4=\frac{EA\left(1-2{\left[f^{\prime }\left(x\right)\right]}^2\right)}{{\left(1+{\left[f^{\prime }\left(x\right)\right]}^2\right)}^3}{f}^{″}\left(x\right),\end{array}

\begin{array}{}\text{(12e)}& A_5=\frac{H}{\sqrt{1+{\left[f^{\prime }\right(x\left)\right]}^2}}+\frac{EA{\left[f^{\prime }\right(x\left)\right]}^2}{{(1+{\left[f^{\prime }\right(x\left)\right]}^2)}^2},\end{array}

\begin{array}{}\text{(12f)}& A_6=\frac{EA\left(2f^{\prime }\right(x)-{\left[f^{\prime }\right(x\left)\right]}^3)}{{(1+{\left[f^{\prime }\right(x\left)\right]}^2)}^3}{f}^{″}\left(x\right),\end{array}

where u and v are in-plane dynamic displacements in the x and y directions, respectively, m is mass per unit length; c₁ is the structural distributed damping coefficients per unit length; E is Young’s modulus of the cable; A is the area of the cable cross section; H is the horizontal component of the initial tension of the cable; f(x) is the catenary profile of the cable under gravity action; and F_pre(x,t) is the damper force from MRD. The detailed cable properties are discussed in Refs. [32,33]. The numerical model established yields a fundamental frequency of 2.72 rad/s, while the value obtained in Ref. [34] is 2.64 rad/s, deviating only by 2.8%.

The semi-active control strategy outlined in Fig.19 aims to precisely calculate and apply the optimal damping force in real time to mitigate cable vibrations. This strategy begins by computing the cable dynamic response, including velocity and displacement, at a specific moment t. The system then utilizes the LQR algorithm to deduce the optimal control force based on the cable dynamic response. Following this, the LSTM-based inverse mapping model is used to ascertain the necessary input current value, enabling the MRD to produce a damping force closely matching the optimal control force. Subsequently, the actual damping force is predicted and retrieved through the LSTM forward mapping model. The system updates the load array of cable at the subsequent time t + 1, incorporating the derived damping force F_pre​ to establish a closed-loop control mechanism.

4.2 Case study

Three different load conditions are comprehensively investigated here to assess the cable vibration mitigation performance of the LSTM-enhanced semi-active control strategy, namely sinusoidal load case, Gaussian white noise load case, and rain–wind induced aerodynamic load case.

4.2.1 Uniformly distributed sinusoidal load (Case 1)

Sinusoidal loads, representing a type of periodic dynamic load commonly encountered in traffic and wind loads, are crucial for evaluating the performance of vibration control strategies. Specifically, the dynamic response of cables under the action of uniformly distributed sinusoidal loads F = sinwt is examined. To simulate the most critical condition, the frequency of the sinusoidal load is set to match the fundamental frequency of the cable.

The vibration responses under uncontrolled, passive control, LQR active control and LSTM semi-active control scenarios are compared, and Tab.5 presents the performance of each damping strategy in terms of mid-span vibration amplitude, Root Mean Square (RMS) of control force, and damping rate. Without any damping measures, the cable vibration amplitude reaches up to 178.45 cm. With passive control introduced, the vibration amplitude drops to 90.32 cm, achieving a damping rate of 49.38%. The LQR active control shows the most significant vibration reduction, with the amplitude decreasing to 8.86 cm and a damping rate of 95.03%. The LSTM-enhanced semi-active control strategy exhibits a damping rate of 90.24%, significantly outperforming passive control and closely approaching the high efficiency of LQR active control. Fig.20 displays the time history of cable mid-span vibration under different control strategies. It is evident that the LSTM-enhanced semi-active control performs excellently in reducing vibration amplitude, with its control effect comparable to LQR active control and significantly better than passive control. This result indicates that the LSTM-enhanced semi-active control strategy is not only effective in handling cable vibrations induced by sinusoidal loads but also has significant advantages in damping efficiency and control precision.

4.2.2 Gaussian white noise random load (Case 2)

Gaussian white noise random load, an ideal random process with a uniform spectral density and Gaussian-distributed amplitude, simulates various complex encountered in the real world. To assess the vibration control effectiveness of the LSTM-enhanced semi-active control strategy against Gaussian white noise loads, the random sequence is uniformly applied to all discrete node of the cable numerical model.

The vibration responses under uncontrolled, passive control, LQR active control and LSTM semi-active control scenarios are compared, and Tab.6 summarizes the performance of each damping strategy in terms of mid-span vibration amplitude, RMS of control force, and damping rate. Without control, the cable vibration amplitude is highest, reaching 26.37 cm. Introducing passive control reduces the vibration amplitude to 15.41 cm, achieving a damping rate of 41.56%. Comparatively, LQR active control demonstrates the best damping effect, reducing the vibration amplitude to 7.39 cm with a damping rate of 71.97%. The LSTM-enhanced semi-active control strategy significantly outperforms passive control and closely approaches the LQR active control with a damping rate of 52.11%. Fig.21 visually presents the comparison of damping effects among different control strategies, where the LSTM control effectiveness is notably superior to passive control and closely matches LQR active control for the majority of the time. This result suggests that the LSTM-enhanced semi-active control strategy exhibits outstanding performance in addressing bridge stay cable vibrations caused by Gaussian white noise loads. In addition, Fig.22(a) and Fig.22(b) present the time history of cable min-span displacement and velocity under Gaussian white noise random load with LSTM-enhanced and Bang-Bang semi-active control strategies, further highlighting the significantly improved damping effects of the former approach. It is noted that the RMS of control force from Bang-Bang control method is 3386.72 N with a damping rate of 14.11%. In contrast to the discrete and extreme output of Bang-Bang control, the LSTM network effectively suppresses vibrations by real-time, accurate adjustments of the MRD damping force, tailored to the cable dynamic response.

4.2.3 Rain–wind induced aerodynamic load (Case 3)

Rain–wind induced vibration of stay cables can lead to large amplitude oscillations of the cables, posing a serious threat to the safety and serviceability of bridges. To assess the vibration control effectiveness of the LSTM-enhanced semi-active control strategy against rain–wind induced aerodynamic load, the movement of water line is assumed to follow a sinusoidal motion pattern and the corresponding wind force is calculated using quasi-steady theory [35]. In this case, the most impactful condition is selected with the motion frequency of water line equal to the fundamental frequency of cable, a wind speed of 10 m/s at the height of the cable mid-span and a wind angle of 35° .

The vibration responses under uncontrolled, passive damping, LQR active control, and LSTM-enhanced semi-active control scenarios are compared, and Tab.7 summarizes the performance of all damping strategies in terms of mid-span vibration amplitude, RMS of control force, and damping rate. In the uncontrolled condition, the maximum vibration amplitude is 20.79 cm. Passive control reduces the vibration amplitude to 12.54 cm, achieving a damping rate of 39.68%. LQR active control shows the most significant damping effect, reducing the vibration amplitude to 2.42 cm with a damping rate of 88.36%. The LSTM-enhanced control exhibits an excellent damping rate of 77.37%, outperforming the passive control and closely approaches the LQR active control. Fig.23 provides a detailed comparison of damping effects among different control strategies. The results suggest the LSTM-enhanced semi-active control strategy exhibits outstanding performance in addressing rain–wind induced vibration of stay cables. In addition, Fig.24(a) and Fig.24(b) present the time history of cable vibration displacement and velocity under rain-wind induced aerodynamic load with LSTM-enhanced and Bang-Bang semi-active control strategies, further highlighting the significantly improved damping effects of former approach. It is noted that the RMS of control force from Bang-Bang control method is 1868.03 N with a damping rate of 49.64%.

5 Concluding remarks

This study has successfully developed an innovative semi-active control strategy based on LSTM networks for vibration mitigation of stay cables using a MRD. Specifically, both LSTM-based forward and inverse mapping models of MRD have been established using high-fidelity data set generated by coupling finite element analysis and computational fluid dynamics approaches. Through comprehensive comparison with passive control, active and Bang-Bang semi-active control strategies under the complex and challenging conditions of sinusoidal load, Gaussian white noise load, and rain–wind induced aerodynamic load, this study effectively demonstrated the superiority and applicability of the LSTM-enhanced semi-active control strategy in the vibration control of bridge stay cables. It was shown that the LSTM-enhanced semi-active control strategy presented clear improved damping effects over passive and Bang-Bang semi-active control strategies, and performed very close to the active control strategy with significantly less energy consumption. Future work will focus on further improving the generalizability and robustness of LSTM modeling of MRDs to explore its applications under a broader range of structural and environmental conditions.

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