Dynamic mechanisms of the cross-tied cable network, especially its nonlinear dynamics, are not yet fully understood. Current studies on the cable network assume an idealized boundary condition, whereas in practical engineering structures, beam's or tower's motion will provide a nonideal excitation. Therefore, a mechanical model, namely, the elastically connected two-cable system (TCS) coupled with the axial moving support, is developed in this paper. By utilizing the extended Hamilton's principle, the corresponding dynamic model is established, and is then discretized into a set of ordinary differential equations (ODEs) by means of Galerkin's method. To solve the ODEs, the method of multiple time scales is employed. In this way, one-to-one internal resonance between the TCS and the support is investigated when primary resonance occurs in the support and/or the first-order mode of the TCS. The frequency-response curves are presented to explore the effects of key parameters on nonlinear behaviors of the system. It is found that in some cases, increasing the excitation amplitude may lead to a reduction in the response. In other words, the two primary resonant excitations are mutually constraining when one of them is relatively small. As the studied parameters change, the left and right peak values of the response curves show an opposite variation trend.

This paper presents a comparative study of the Homotopy Analysis Method (HAM) and Incremental Harmonic Balance Method (IHBM) for solving nonlinear periodic responses in dynamical systems. Both methods are evaluated in terms of accuracy, computational efficiency, convergence behavior, and applicability to strongly nonlinear regimes. Numerical results on Duffing and van der Pol oscillators show that IHBM achieves high accuracy and rapid convergence for steady-state solutions, making it well-suited for engineering simulations when adequate harmonic content and good initial guesses are provided. In contrast, HAM offers analytical flexibility and controllable convergence through auxiliary parameters, enabling solution construction even without prior knowledge of periodicity. However, higher-order approximations in HAM incur significant computational cost. The results highlight the complementary strengths of the two methods: IHBM excels in efficiency for regular periodic responses, whereas HAM provides greater analytical insight into complex nonlinear dynamics. These findings offer practical guidance for selecting appropriate semi-analytical tools based on problem characteristics.

In the field of vibration control on cable-stayed bridges, the countermeasure using cross-ties to suppress unfavorable vibrations of cables is receiving widespread attention. By adding cross-ties to connect different cables to form a cable network system, it can not only effectively improve the geometric configurations of cables but also enhance the in-plane stiffness of the overall structure. Currently, existing studies face the following two issues: (1) focus on the network composed of cables and cross-ties, but the effects of bridge deck vibrations are commonly overlooked; (2) even when bridge deck vibrations are considered, the modeling incorporates only a single cable. Therefore, this paper establishes a multi-aligned-cross-tie multi-cable-beam model to enable the description of vibrations for both bridge deck and multiple cables. Based on the governing differential equations of the beam and cables, by introducing the transfer matrix method (TMM), a theoretical modeling and derivation pattern suitable for linear dynamic problems of the cable-beam-cross-tie coupled system is formulated. Four numerical examples are analyzed, namely one-aligned-cross-tie double/three-cable-beam model and double/three-aligned-cross-tie three-cable-beam model. Meanwhile, the corresponding finite element models (FEMs) are also established, and the frequencies and mode shapes are compared with those obtained by the present method. The results show that the solving strategy in this paper is credible and feasible.

This study systematically elucidates the intrinsic mechanisms by which a nonlinear energy sink (NES) suppresses multimodal coupled vibrations in cable structures. First, a spatial model of the cable-NES system and a simplified model considering only the first three in-plane modes were established. Using the Galerkin method, the strongly nonlinear governing equations of the system were derived, and high-precision analytical solutions were obtained via the homotopy analysis method, with their validity verified by Runge–Kutta numerical integration. Comparative analysis of the two models demonstrates that neglecting out-of-plane modes significantly underestimates the system's vibrational energy, while incorporating out-of-plane degrees of freedom is essential for accurately characterizing the targeted energy transfer path. Furthermore, the vibration suppression performance of the NES was quantified using amplitude-frequency response curves, and the effects of the NES damping ratio, nonlinear stiffness, and installation location on the suppression bandwidth were systematically investigated. The findings reveal that out-of-plane modes play a significant role in coupled vibrations; the NES achieves targeted energy transfer by activating nonlinear energy channels, thereby effectively suppressing multimodal coupled vibrations. Parametric analysis shows that optimizing the NES damping characteristics, nonlinear stiffness, and placement can substantially broaden the suppression bandwidth, enabling sustained and broadband vibration attenuation. This research provides theoretical foundations and design references for the nonlinear vibration control design of engineering cable structures.

Floating structures are composite systems comprising multiple anchor cables and a floating body. The dynamic coupling among these components can induce complex nonlinear responses, which cause the system's large amplitude vibration. Currently, the research focus is limited to the dynamic model with a single cable, while neglecting the indirect coupling between multiple cables such as the modal resonance. Aiming to investigate the coupling mechanism between the components, this study established a dual-cable floating-body model with consideration of the 1:1:1 internal resonance among two anchor cables and the floating body, incorporating primary resonance induced by external excitation applied to either the floating body or one of the cables. The anchor cable and floating body are coupled through the mechanical and displacement boundary conditions at the junction. Galerkin modal truncation is utilized to obtain the reduced ordinary differential equations. Asymptotic perturbation solutions are obtained by the method of multiple scales and verified by the Runge-Kutta numerical method. The results show that the vibrational energy between the cables is transmitted through the floating body; the frequency response curve of the cable exhibits multi-peak characteristics due to the modal resonance mechanism; the symmetry of the system and key structural parameters, such as the cable inclination angle and mass ratio, significantly influence the dynamic response of the cable, while having a comparatively minor effect on the floating body.

Conventional passive control measures for cable vibrations often lack adaptability to varying excitation frequencies and exhibit a narrow control bandwidth. This study aims to suppress cable vibrations using micro-fiber composite (MFC) piezoelectric plates, overcoming the limitations of passive control measures. The control strategy employs the Positive Position Feedback (PPF) method integrated with a resonance mechanism to achieve effective vibration suppression. The in-plane coupled equations of the MFC-cable system are derived via the extended Hamilton principle and solved analytically using the method of multiple scales (MMS) to obtain modulation equations. A comprehensive parametric analysis is conducted. The feedback signal gain λ and control signal gain γ of the PPF controller exhibit a contrasting effect: while both can broaden the control bandwidth and enhance system stability, a larger λ increases the controller’s circuit voltage and energy consumption, whereas a larger γ reduces it. More importantly, for optimal vibration suppression, the PPF controller’s natural frequency should be tuned to align with the excitation frequency rather than the cable’s inherent resonant frequency. The proposed MFC-based active control strategy with PPF controller is demonstrated to be effective. The findings provide critical design guidance, specifically on balancing control performance with energy consumption through gain selection and on the optimal tuning of the controller frequency.

Nonlinear Energy Sink (NES), as an innovative passive vibration control technology, has attracted attention for broadband vibration suppression owing to its high efficiency. This paper investigates the parametric resonance and bifurcation behavior of a spatial cable coupled with an NES under parametric excitation. An accurate nonlinear dynamic model of the cable-NES system is first formulated, capturing the essential nonlinear coupling mechanisms and energy transfer pathways. Theoretical analysis reveals complex nonlinear dynamics, including bifurcations induced by system parameters. The dynamic responses under primary parametric resonance are systematically examined using the Homotopy Analysis Method (HAM), which highlights the significant influence of subtle NES parameter variations on the cables vibration. Furthermore, the fourth subharmonic resonance of the cable under parametric excitation is explored. Results demonstrate that optimal placement of the NES at the cable end yields superior vibration mitigation compared to mid-span attachment. Moreover, appropriate tuning of NES damping enables the cable response to settle into a stable steady state. This study elucidates the internal mechanism of parametric resonance in the cable-NES system, providing a theoretical basis and practical guidance for vibration control in bridge engineering.

Connecting the target (or susceptible) cable with adjacent ones using cross-ties to form a cable network is recognized as an effective countermeasure for suppressing large-amplitude vibrations of cables, and has been implemented in real-world scenarios. Although substantial research has been conducted on the dynamic characteristics of such systems, the underlying mechanisms, particularly regarding nonlinear behaviors, remain unclear. In this paper, a nonlinear global dynamic model for a two-cable network with multiple flexible cross-ties is established. Based on this model, the nonlinear dynamic behaviors of the system under primary resonance are investigated. Modal functions accounting for dynamic axial forces in the cables are first determined to derive the discrete model, i.e., the ordinary differential equation (ODE). The multiple time scales method is employed to solve the ODE and obtain the steady-state solutions, which are then compared and validated against numerical solutions obtained using the Runge–Kutta method. The effects of some key parameters, such as the cross-tie stiffness, the cross-tie position, the initial tension ratio, and the length ratio, on the frequency–response curves are systematically explored. In this way, the influence mechanism of the cross-tie on primary resonance responses of the system’s first three modes is revealed. It is found that parameter variations enhance the hardening spring characteristic by altering the values of the cubic nonlinear coefficients, thereby suppressing the primary resonance responses and achieving vibration control.

Vortex-induced vibration (VIV) and support motion-induced vibration (SMIV), two critical phenomena frequently encountered in engineering, can cause large-amplitude responses in cables, yet their interaction mechanisms remain insufficiently understood. To this end, the vortex-induced resonance of the cable combined with the primary resonance induced by support motion is focused on in this paper for the first time. A refined fluid-structure interaction model for an inclined cable excited by wind load and support motion simultaneously is developed, aiming to investigate the resulting nonlinear coupled dynamics. Firstly, governing differential equation for the cable's cross-flow motion is established, with the fluid force modeled using van der Pol wake oscillator. Then, the continuous dynamic equations and boundary disturbance are attacked directly by method of multiple scales (MMS), based on the concept of boundary resonant modulation, leading to the derivation of modulation equations. Newton-Raphson algorithm is employed to obtain the steady-state solution, from which numerical continuation is initiated, and frequency response curves of the coupled system are plotted using pseudo-arclength algorithm. Furthermore, the proposed fluid-structure interaction dynamic model and results are validated against experimental tests and numerical solutions obtained from Galerkin-truncated model, respectively. Finally, parametric analysis reveals that variations in the amplitude and frequency of support motion significantly affect the system's nonlinear behaviors, potentially changing the amplitudes quantitatively and/or influencing the nonlinear dynamic characteristics (hardening and softening) qualitatively.

In the context of the global push for renewable energy, wind turbines are playing an increasingly pivotal role, and the dynamic characteristics of their blades are crucial for safe operation. To ensure the safety of wind turbine blades, Vestas and Tongji University proposed a new type of blade structure known as the cable-stayed rotor. This paper models the structure as a cable-stayed laminated composite beam (CSLCB), and studies the mechanisms by which material parameters affect the nonlinear response of the system. Unlike existing studies on composite/cable-stayed beams, this work integrates both effects to reveal material-dependent nonlinear behaviors. The governing equations are derived via Hamilton’s principle, with full consideration of the effects of fiber volume fractions and orientation angles. In this way, nonlinear behaviors of the structure are systematically investigated. Steady-state response analysis indicates that the fiber volume fraction qualitatively determines whether the structural response exhibits softening or hardening spring characteristics, while the fiber orientation angle quantitatively affects the response amplitude. It is concluded that adjusting the fiber volume fraction and orientation angle may achieve vibration control.

In large-span cable-stayed bridges, vortex-induced vibration (VIV) of flexible cables is significantly influenced by their geometric nonlinearity and wind profile. This paper presents an integrated analysis of these effects by establishing a framework for analyzing cable VIV. Within this framework, the cable's cross-flow vibration equation is coupled with a van der Pol equation that describes aerodynamic force. Subsequently, the total governing equations are discretized employing a second-order central finite difference method (FDM) and solved numerically. The reliability of fluid-structure coupling solver is confirmed through comparison with existing experimental and numerical data. The results show that an increase in wind profile exponent transforms the cable's VIV response from a single-mode standing wave into a multimodal hybrid wave. Meanwhile, the cable's motion changes from periodic to quasi-periodic. Exponential shear flow induces bidirectional non-uniform energy transfer along the span, revealing the driving mechanism behind multimodal coupling. Additionally, the frequency lock-in mechanism under primary resonance is explored by adopting incremental harmonic balance (IHB) technique. Within the lock-in regime, the system exhibits hysteresis and bistability, with jumps occurring exactly at the critical points where the system enters and exits the lock-in region. These findings provide theoretical insights for design and vibration control of flexible cables.

Experimental results reveal that the dynamic response of a stay cable under forced excitation exhibits significant nonlinear characteristics, manifested by the presence of higher harmonics in addition to the excitation frequency component. To investigate the underlying mechanism, this study establishes multi-mode uncoupled and coupled systems for the first three symmetric inplane modes based on the governing equations of modal motion. The equations are discretized into ordinary differential equations using the Galerkin method, and the homotopy analysis method (HAM) is employed for solution. Response curves are constructed using the NewtonRaphson method combined with the pseudo-arclength algorithm. A systematic investigation is conducted into the effects of key parameters, including excitation amplitude, damping ratio, and excitation under different modal conditions. The results indicate that higher harmonics originate from strong nonlinear internal resonance mechanisms. When the natural frequencies satisfy omega 3 approximate to 2 omega 1, a 2:1 internal resonance pathway facilitates efficient energy transfer from the excited higherorder mode to the lower-order mode. Even when external excitation is applied only to higherorder modes such as the third or fifth, a 3:1 internal resonance near Omega approximate to omega 3 can excite double resonance peaks in all three modes, thereby breaking the one-to-one correspondence between excitation and response observed in linear systems. Under high-amplitude excitation, parameter variations further induce typical nonlinear phenomena such as response curve bifurcation, jumping, and phase drift. This study reveals the nonlinear vibration mechanism of stay cables under the combined action of internal resonance and external excitation resonance, providing a theoretical basis for the nonlinear vibration control of cable structures.

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.

A nonlinear sagged cable, due to its initial curvature, leads to various challenges of empirical mode truncation used by routine Galerkin method when constructing reduced-order model. It is recently elucidated that ( Guo and Rega, 2023a ), the key for refined mode truncation (and thus for correct nonlinear dynamics prediction) is to first eliminate low-order nonlinear terms of spatial continuous structures. This paper focuses on refined truncation of nonlinear sagged cable by leveraging the recent low-order elimination perspective, which is realized by a normal form development. Further comparative studies for both primary resonant and two-to-one internally resonant dynamics of the sagged cable, including nonlinear frequency responses, backbone curves, and Poincaré mapping, demonstrate notable differences between the two different types of models built by either routine or refined truncation, which confirms necessity of the refined mode truncation used for geometrically nonlinear structures like sagged cables.

The nonlinear energy harvesting system enhanced by a resonance mechanism is proposed, which consists of a piezoelectric cantilever beam supported by a nonlinear boundary oscillator. The nonlinear boundary oscillator comprises three springs and a lumped mass, which introduces tunable nonlinear stiffness and modifies the overall dynamic characteristics. The dynamic equations for the electromechanically coupled system are derived using the extended Hamilton principle. The direct multiple-scale method is employed to derive the modulation equations governing the slow dynamics of amplitude and phase. The perturbation solutions are validated against numerical simulations via the Runge–Kutta method. At primary resonance, the coupled system exhibits nonlinear behaviors, including hardening spring behavior, jump phenomena, and multiple solutions. Parametric studies indicate that the mass ratio between the boundary oscillator and the cantilever beam significantly influences the frequency response, enabling enhancement of the electrical output and adjustment of the effective harvesting bandwidth. The nonlinear stiffness parameter primarily affects the extent of hardening nonlinearity, while the load resistance largely determines the electrical output characteristics. Compared with a conventional single piezoelectric cantilever beam, the coupled system yields larger vibration amplitudes and higher electrical output, demonstrating its potential for efficient broadband vibration energy harvesting.

By applying a moving load with the uniform velocity, this paper investigates nonlinear behaviors of a shallow arch model allowing for describing the effects of the initial configuration of the structure. Two different cases are taken into account, namely, simultaneous primary resonances of the first and third modes; simultaneous resonances of the first mode for two-term excitations. First, nonlinear ordinary differential equations (ODEs) of the shallow arch are derived by using Galerkin method. Through introducing different time scales, the ODEs are solved based on the widely utilized method of multiple time scales (MMTS). In this way, the modulation equations for the two cases are derived, on the basis of which the steady state frequency- and force-response curves are obtained. Meanwhile, phase portraits, power spectra and two-parameter bifurcation diagram are also given to assist in analysis on nonlinear behaviors. The results show that the large vibration of the shallow arch under the moving load may occur and primary resonance peaks of the different modes are located in distinct positions.

This paper combines functionally graded materials (FGMs) with the cable-stayed beam to propose a cable-stayed functionally graded beam (CSFGB) model. The influences of temperatures on nonlinear behaviors of the model are analyzed when one-to-two internal resonance of the global mode (FGB) and the local mode (cable) is triggered. First, the governing equations, which consider thermal effects, are derived through the extended Hamilton's principle. Thereafter, the modal functions are obtained based on the boundary conditions. On this basis, the governing equations are discretized by employing Galerkin discretization, resulting in a set of ordinary differential equations (ODEs). The method of multiple time scales is then applied to solve these ODEs and derive the modulation equations. Finally, nonlinear dynamic behaviors of the CSFGB at three different temperatures are analyzed. The results show that as the temperature decreases, the response increases, making chaotic motion more likely to occur. Moreover, the distribution of FGMs has a significant impact on nonlinear responses of the system.

Nonlinear energy sinks (NESs) have received widespread attention due to their broadband vibration absorption ability. This study investigates the vibration suppression of a double-cable beam structure by NES. Firstly, a mechanical model of the double cable-beam-NES structure was established, and the Hamilton principle was used to derive the motion partial differential equation of the double cable-beam-NES structure. The incremental harmonic balance method (IHBM) was employed to analyze the nonlinear dynamic characteristics of the coupled model subjected to external load excitation, and the impact of NES on reducing vibrations in the composite structure was investigated. In addition, the effect of cables' interaction on the response amplitudes of both the cables and the beam was investigated. The results show that in the absence of NES attachment to cable 2, the NES attached to cable 1 does not effectively suppress the response amplitude of cable 2. Compared with the single cable-beam-NES composite structure, the interaction between the cables significantly inhibits the response amplitude of the cable.

The suppression of oscillations in cable-stayed bridges has been a formidable challenge in engineering. Although various types of dampers have been employed in the vibration suppression of cable-stayed bridges, obvious oscillations of cable-stayed bridges still can be observed. The possible reason is dynamic mechanisms of coupling system remain unclear, and the current dynamic theoretical models only consider the damper coupling with a single component (i.e., beam or cable) of cable-stayed bridges. To address this problem, a novel double coupled cable-damper-beam dynamic theoretical model of cable-stayed bridges and its internal resonance analysis is investigated in this paper. There are three innovation points. Firstly, a double coupled cable-damper-beam dynamic theoretical model, involves the direct coupling between the cable and beam ends and the indirect coupling between the cable and beam through dampers, is proposed, which is closer to the real case compared with the non-double coupled model. The geometric nonlinearity of cable and beam is considered in the modeling. Secondly, an important mechanism is revealed, i.e., the stiffness of damper affects the Hopf bifurcation of the model. The increase in damper stiffness enhances the coupling between cable and beam, which lead to the occurrence of the Hopf bifurcation and further induce the large vibration of cables. Thirdly, a very interesting phenomenon is discovered, i.e., reduction in the sag of the cable has almost no effect on its dynamic response, but it leads to a significant increase in the dynamic response of the beam. The phenomenon indicates that the sag of cable has a significant effect on the dynamic mechanism of cable-stayed bridges. Moreover, the proposed model can explore the mechanism of cable-damper-beam more clearly due to its double coupled characteristic, which lays a theoretical foundation for effective vibration suppression and provides the convenience for the optimization design of dampers.

In this paper, we develop an inextensible cantilever beam model coupled with moving foundation to study two-way dynamic coupling behaviors with the consideration of nonlinear inertia and stiffness. Based on the concept of boundary resonant modulation, moving foundation-induced vibration of the beam is asymptotically perturbed into its slow dynamics using direct multi-scale method. Boundary resonant modulation term and effective nonlinearity coefficients characterizing the nonlinear inertia and stiffness are cleared out. Parametric influence of key factors on coupling dynamics, including inertia effect of top mass, mass ratio of top mass to total mass of beam, and mass ratio of foundation to total mass of beam, are discussed. Results show the hardening spring in first mode frequency response dominating by nonlinear stiffness, and a transition to softening spring with the consideration of inertia effect of top mass; the motion of moving foundation can suppress vibration of cantilever beam in the primary resonance domain to some extent.