This paper revisits the mode derivative (MD) concept with the aim to bring it into a more consistent footing with other standard nonlinear techniques. Different from classical sensitivity analysis used in the original MD, an asymptotic treatment is proposed to reframe the MD procedure with two distinct types of nonlinear frequency modifications taken separately into account, which leads to a rectified mode derivative (MD) definition, with previous theoretical limitations removed. This is also validated by a rectified application to reduced-order modelling of nonlinear structures.

Electrostatically tunable MEMS shallow arches provide compact platforms for achieving large frequency tuning and high sensitivity. Experimental evidence indicates there is a possible softening-hardening-softening stiffness transition near snap-through. A high-order analytical framework is developed to explore nonlinear stiffness transitions in such systems operating near but not reaching the snap-through threshold. A full dynamic map is established, revealing a softening-hardening-softening sequence governed by the competing voltage-dependent contributions of geometric, electrostatic, and stretching-induced nonlinearities. Within this transition, quasi-linear dynamics responses emerge where third-and fifth-order effects offset across the relevant amplitude range. The analytical outcome reveals the underlying mechanism of the experimental phenomenon. These findings establish a predictive route toward voltage-programmable stiffness modulation, offering practical implications for the design of responsive, instability-free MEMS resonators and filters.

Competing dynamics of an inextensible straight beam with a weakly constrained end (i.e., end mass+end spring/stiffness), involved with both inertial nonlinearity (favoring softening) and geometric nonlinearity (favoring hardening), is fully investigated through the lens of asymptotic analysis. Besides monotonic dynamics (either softening or hardening), the current paper focuses on non-monotonic dynamics i.e., mixed hardening/softening (H/S) dynamics. It turns out that the H/S transition curve (or separatrix) plays a key role in locating the above two distinct nonlinear dynamics, i.e., monotonic or non-monotonic one, with the former found away from H/S transition while the latter located in the vicinity of H/S separatrix. In both scenarios, the inextensible beam’s dominant nonlinear dynamics are found to critically depend on the beam’s end mass and end stiffness.

It is challenging to analyze nonlinear dynamics of infinite-dimensional continuous structures due to mutual interactions among different modes. Single-mode approximation has long been assumed valid to approximate primary resonant dynamics of cubic-only (nonlinear) structures, although it is known to possibly fail for structures with both quadratic and cubic terms. This paper revisits this conclusion by focusing on a cubic-only structure with two competing dynamics, i.e., hardening (H) dynamics with frequency response curve (FRC) bending rightward and softening (S) dynamics with FRC bending leftward. Explicitly, for a cubic-only beam with both geometric and inertial nonlinearities, by comparing single-mode discretization perturbation with direct perturbation (with full-mode effect considered), it finds that: close to hardening-to-softening (H/S) transition, i.e., a critical state where hardening balances softening, the widely used single-mode approximation turns insufficient, i.e., giving enormous nonlinear prediction, while it is valid as commonly assumed when the system is away from H/S transition.

Hardening/softening (H/S) dynamics of continuous structures have long been investigated by referring to an effective nonlinear coefficient associated with modulation/averaged equation (derived by perturbation technique). Close to internal resonance, however, this traditional formulation becomes unreliable: the effective coefficient may diverge or become ill-posed, leading to misleading H/S predictions. Being confined to a weakly damping scenario, the current paper aims to develop a refined perturbation framework that explicitly incorporates internal resonance thereby removing this nominal singularity. By focusing on a typical quadratic–cubic system (a sagged cable), two distinct types of crossing-singularity are revealed. Namely, type-I for a generalized H/S transition in which the response exhibits a hardening-to-softening or softening-to-hardening transition, while type-II for a singularity crossing without H/S transition, being hardening-to-hardening or softening-to-softening. Although the main work is currently developed for a typical cable model, the framework can be meaningfully extended to other quadratic–cubic structures involved with internal resonance.

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.

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.

Competing mechanisms like quadratic/cubic nonlinearities associated with continuous structures lead to complicated dynamics, and it is meaningful to identify the dominant mechanism in a specific parameter domain. However, when the same parameter arises simultaneously in different competing mechanisms, either hardening (H) or softening (S), it implies that a certain constraint between various parameters and thus leads to difficulties for identifying its dominant dynamics with reference to physics parameters. For example, in a nonlinear cable model, its stiffness α, initial sag f, and Irvine parameter λc are all closely related to H/S behaviors, but they are not independent. A novel nonlinear hardening/softening dynamic analysis procedure is asymptotically developed by geometrically interpreting the underlying constraint between parameters as a curved surface located in an auxiliary parameter space. For cables, it consists of stiffness α, initial sag f, and Irvine parameter λc. Further, iso-λc curves on this curved constraint surface is properly defined, which represents a family of cable models with the same Irvine parameter and turn out to be useful for the proposed hardening/softening analysis. Though currently developed for a cable model, the framework can be meaningfully extended to other nonlinear structures with an initial curvature like shallow arches, buckled beams, or imperfect beams, etc.

In recent years, nonlinear energy sink (NES) has received widespread attention from scholars as an efficient passive control means. In this paper, the effect of NES on the nonlinear dynamic response of the cable-beam composite structure is investigated. Firstly, a mechanical model of the cable-beam composite structure with a NES under the external excitation of the beam is established. By using Hamilton principle and Galerkin discretization, the ordinary differential equations (ODEs) of the system are derived. Then, the incremental harmonic balance (IHB) method is employed to obtain the frequency response of the system when the beam is subjected to forced excitation. Finally, three working conditions are considered to discuss the effect of NES on the dynamic characteristics of the composite structure. Moreover, the vibration suppression mechanism of NES attached to the cable on the beam members is also investigated. The results demonstrate that when the natural frequencies of beam and each mode of cable are close, NES has good vibration suppression characteristics for both cable and beam. Furthermore, the vibration mitigation effect of NES on beam members has a great relationship with the degree of cable-beam coupling vibration.

Competing mechanisms due to mechanical and geometrical effects can lead to intricate nonlinear dynamics of curved structures like a shallow arch, e.g., slenderness ratio ξ (mechanical parameter), arch rise b (geometrical parameter). It is thus meaningful to correctly locate dominant dynamics in its parameter space. An auxiliary elasto-geometric parameter λa = ξb is newly introduced for shallow arches as inspired by a sagged cable, following which an asymptotic exploration of hardening/softening (H/S) dynamics of a shallow arch is presented by fully leveraging the so-called iso-λa curves located on the surface λa = ξb geometrically defined in three-dimensional parameter space (ξ–b–λa), i.e., a family of arch models associated with the same elasto-geometric parameter λa (but with different slenderness and rise). A critical iso-λa curve corresponding to separatrix of hardening and softening dynamics is identified. Away from the H/S transition curve, monotonic nonlinear dynamics (either hardening or softening) is found. In contrast, more subtle non-monotonic dynamics can occur in the vicinity of the H/S transition curve (separatrix), wherein a higher-order perturbation is required to capture relevant quintic mechanism besides the commonly treated cubic one. The current contribution leads to a full picture or classification of softening/hardening dynamics of shallow arches, and in particular, a closer look into subtle competing dynamics close to H/S transition.

Combined with a frequency detuning idea, i.e., linearization of nonlinear systems around its nonlinear response frequency rather than traditional linear natural frequency, a detuned multiple scale method (dMSM) is investigated by a full evaluation of its performance when being applied to cubic nonlinear systems with either geometrically cubic stiffness or cubic damping, i.e., a hinged–hinged beam and a generalized Van der Pol oscillator. By detailed comparison with standard MSM and focusing on frequency response and backbone curves, it is found that (detuned) dMSM demonstrates a superior performance in prediction of high-amplitude nonlinear behaviors (with weakly nonlinear assumption still valid), for both a geometrically cubic beam and a cubic damped oscillator.

This paper conducts a nonlinear analysis of cable-beam model of cable-stayed bridges by using the exact mode superposition method (EMSM) and the cable-beam dragging method (CBDM), respectively, comparing and exploring their theoretical foundations and practical implications. The EMSM is based on the global mode function of the cable-beam structure for nonlinear analysis, yet it requires more computational resources. The CBDM is based on the cable-beam dragging equations for nonlinear analysis, which can quickly obtain the static equilibrium state and dynamic response of the cable-beam system, but it requires some simplifying assumptions on the cable-beam connection conditions. Research results demonstrate qualitative and quantitative differences between these two methods through parametric analysis on dynamic behaviors, which provide a significant methodological study and a reference for the design and dynamics of composite structures.

Equipping cross-ties on the cables of cable-stayed bridges to form a cable network is regarded as an effective measure to suppress large cable vibrations and has been realized in practical engineering. However, most of the existing studies on the suspended cable (small sag) network have ignored the contributions of dynamic axial forces to transverse forces of the cables. More importantly, the dimension of the characteristic matrix is relatively large. Therefore, the current study introduces the transfer matrix method (TMM) to solve the in-plane free vibration problem of a two-cable network with dynamic axial forces of the cables considered simultaneously. Compared with the methodology utilized by the previous studies, the dimension of the resulting characteristic matrix is smaller and is independent of the number of cross-ties. Three cases, i.e., a two-cable network with two cross-ties, with three cross-ties and with four cross-ties, are explored in detail. Their frequencies and mode shapes are compared with those obtained by finite element model (FEM) to demonstrate the validity of the results in this paper. Meanwhile, the effects of dynamic axial forces on the first two frequencies and mode shapes are analyzed. The results show that the contributions of dynamic axial forces should be taken into account in transverse forces when calculating the first two frequencies of the cable network with flexible cross-ties.

This paper intends to reveal the dynamic mechanism of the inclined cable under wind-induced vibration based on a multi-mode reduced model. The discrete harmonic superposition method is applied to simulate the broadband wind-induced loads. Galerkin's method and the method of multiple scales are utilized to derive the modulation equations. The asymmetrical profile of inclined cable due to the inclination is discussed, of which results demonstrate the parabola assumption of the cable's initial static profile is rational only when its sag-to-span ratio is small enough. Moreover, the drift phenomena and new resonance peak will appear with the variation of frequency and amplitude of the multi-frequency excitations, which indicates the multi-mode resonance mechanism.

Modal analysis is a widely applied method to study the vibration phenomenon of continuum structures, but there is no clear method to solve the modal truncation problem at present. To determine the contribution of different modes to the whole system, a new mode truncation method based on perturbation theory is proposed in this paper. The modes are subjected to perturbation parameters during discretization, and using norm error analysis on the stiffness matrix in different degrees of freedom (DOFs) systems confirms the model number of the continuum structure system. The results show that the DOF identified by the modal perturbation method is related to the perturbation parameter, and the smaller the perturbation parameter is, the fewer modes need to be considered. When the perturbation parameter is large enough, the response of the system can only be accurately explained by truncation to higher-order modes. Finally, the perturbation parameter is fixed to 1, and the traditional Galerkin method is connected to the modal perturbation, making traditional discretization a unique case for the modal perturbation method. This method can significantly reduce the modal truncation error, which is of great significance to the dynamic analysis of engineering applications.

Non-monotonic dynamics of nonlinear continuous structures, i.e., mixed hardening(H)/softening(S) behavior in the vicinity of H/S transition, is comprehensively investigated by developing a generic asymptotic formulation. Non-monotonic dynamics is due to high-order competition between cubic and quintic mechanisms and thus qualitatively distinct from routine monotonic dynamics (either softening or hardening) associated with Duffing-type cubic mechanism. The general theoretical formulation is applied to both a nonlinear foundation beam model and a nonlinear shallow sagged cable model, with various non-monotonic responses found. In particular, by leveraging frequency response curves (FRCs) and backbone curves (BBCs), reversal of FRCs/BBCs in the mixed softening/hardening dynamics is further connected to zero dispersion phenomenon, with its activation condition also established.

The present study focuses on an inextensible beam and its relevant inertia nonlinearity, which are essentially distinct from the commonly treated extensible beam that is dominated by the geometric nonlinearity. Explicitly, by considering a weakly constrained or free end (in the longitudinal direction), the inextensibility assumption and inertial nonlinearity (with and without an initial curvature) are introduced. For a straight beam, a multi-scale analysis of hardening/softening dynamics reveals the effects of the end stiffness/mass. Extending the straight scenario, a refined inextensible curved beam model is further proposed, accounting for both its inertial nonlinearity and geometric nonlinearity induced by the initial curvature. The numerical results for the frequency responses are also presented to illustrate the dynamic effects of the initial curvature and axial constraint, i.e., the end mass and end stiffness.

Vibration suppression of a simply supported beam under moving loads is investigated by employing a nonlinear energy sink (NES), which is an essential nonlinear (cubic) oscillator with a vanishing linear stiffness. Dynamic equations of the beam-NES coupled system are established and Galerkin method is applied to discretize the equations. By focusing on an energy dissipation index, an optimization procedure is employed and the optimal parameters of the NES are found. The results demonstrate that beam responses with a NES are reduced significantly compared to the case without NES. In particular, the NES can irreversibly transfer and dissipate vibration energy in a wide frequency range, which thus performs better than the classical linear tuned mass damper (TMD) through a further comparison study.

Large amplitude vibration of transmission line seriously affects the structural safety. Due to the low bending stiffness of superhigh transmission tower, vortex-induced excitation combining with support motion induced by the tower will cause significant complex dynamic behaviors of the transmission line. To reveal dynamic behaviors, this paper newly proposes a suspended cable model of the transmission line subjected to the boundary motion and vortex-induced vibration. Dynamic mechanism and vibration energy transfer are focused in the condition of primary and subharmonic resonances. Innovative behaviors of the transmission line under two excitations are revealed. Firstly, vortex-induced excitation is very weak and support motion usually plays a dominate role in the dynamic responses. Large amplitude vibration of transmission line observed in practice should be caused more by tower tip motion and the vortex-induced vibration is the incentive. Secondly, different weak support motion can cause different effect on dynamic responses of transmission line under vortex-induced vibration reflecting by different lock-in phenomenon, which leads us the application of active control measure in engineering.