Multi-degree-of-freedom origami patterns exhibit rich kinematics but are challenging to analyze because many distinct folding motions may coexist at a given configuration. In this work, we introduce an eigenfolding framework that augments conventional kinematics with energetic accessibility analysis. We first formulate exact rigid-foldability using quaternion loop-closure constraints, and compute the instantaneous null space of admissible infinitesimal motions. We then endow each crease with a torsional spring and pose a generalized eigenvalue problem restricted to this null space. The resulting “eigenfoldings” form a canonical, orthonormal basis of admissible motions ordered by “eigenstiffness”, i.e., effective resistance to folding. Applied to one-orbit hexagonal Resch patterns, we discover that their eigenfoldings cluster into four interpretable families (s, p, d, and f), whose spatial symmetries resemble atomic orbitals. Configuration-space maps show how eigenstiffnesses evolve with the folding states of the Resch patterns. Quasi-static compression experiments on physical prototypes confirm the predicted stiffness-based ordering of the tested prototypes, consistent with eigenfolding analysis. Overall, the eigenfolding analysis provides a simple, physically grounded kinematic basis for understanding and designing multi-DOF origami structures.
Inter-shaft bearings in dual-rotor systems serve as crucial load-carrying components that are frequently vulnerable to localized defects. This study investigates the dynamic characteristics of a dual-rotor system with a localized inter-shaft bearing defect, with particular emphasis on the influence of the relative rotational state of the HP and LP rotors, represented by the speed ratio, under variable engine operating conditions. A dynamic model of the dual-rotor system is constructed via the finite element method, into which an inter-shaft bearing outer-race defect model considering time-varying stiffness, defect-induced displacement excitations, and initial clearance is introduced. The resulting governing equations are solved via the Newmark-β method. The results uncover a frequency modulation evolution law whereby the sideband distribution is strongly influenced by the speed ratio. Crucially, this study delineates a distinctive spectrum purification phenomenon, where complex modulation sidebands merge into rotational harmonics at certain speed ratios, manifesting as a simplified pseudo-healthy spectrum. An analytical method is developed to predict the critical speed ratios at which this phenomenon occurs. Moreover, bifurcation analysis reveals persistent non-periodic motion, the nonlinear intensity of which exhibits periodic variations modulated by the speed ratio. Concurrently, the largest Lyapunov exponent identifies bottleneck regions where the system transitions from disordered chaotic states to relatively ordered quasi-periodic states, which can significantly suppress vibration amplitudes. All numerical simulation results are validated on a custom-built dual-rotor test rig. The insights from this study offer a diagnostic reference for condition monitoring and provide a theoretical foundation for active vibration control through speed matching.
For the first time, this paper presents the application of the variational differential quadrature (VDQ) method to analyze the viscoelastic behavior of two-dimensional (2D) soft engineering solids. This contribution is significant as the finite element method (FEM) has been the most widely adopted solution strategy in this field. The VDQ method is free from locking and instability issues which results to eliminating the need for mixed-interpolation or stabilization techniques required in FEM. Furthermore, VDQ is able to model complex geometries as easily as regular domains through a straightforward coordinate transformation. Present analysis employs the Quasi-Linear Viscoelastic (QLV) theory under a plane stress assumption to investigate the finite viscoelasticity of 2D incompressible materials. The weak-form formulations are derived using the P-F stress-strain conjugates (the first Piola-Kirchhoff stress and deformation gradient tensors). By expressing the governing equations in an equivalent matrix-vector form, the integral and differential operators of the VDQ method can be directly incorporated. Several well-established case studies successfully capture the time-dependent large-deformation responses as well as validate the proposed VDQ-QLV model for viscoelastic incompressible materials.
Traditional cubic nonlinear energy sinks often require a sufficiently large input energy to initiate targeted energy transfer, which limits their effectiveness under low-amplitude stochastic excitation. To address this limitation, a softening–fractional nonlinear absorber is proposed in this study. The restoring force combines a fractional-power term, a cubic term, and a saturated softening term. The fractional-power component strengthens nonlinear coupling in the small-amplitude range, the softening component modifies the frequency–energy matching condition, and the cubic term preserves global confinement at large relative displacements. A dimensionless coupled model is established for a primary oscillator attached to the proposed absorber, and the response is examined under both deterministic and stochastic low-energy excitations. The results show that the softening–fractional nonlinear absorber triggers targeted energy transfer at a lower excitation level than a conventional cubic NES and a fractional–cubic NES. Monte Carlo simulations further show that it provides higher stochastic triggering probability, lower sample-to-sample variability of absorber dissipation, and stronger suppression of the peak response of the primary system. Frequency-response analyses indicate that the proposed absorber maintains effective vibration reduction over a wider frequency range. Parameter-plane studies reveal that its performance is governed by the coordinated choice of the fractional exponent, fractional stiffness, and softening coefficient, rather than by increasing any single nonlinear parameter. The results provide useful design guidance for nonlinear absorbers operating in low-energy stochastic environments.
Conventional viscoelastic dampers (VEDs) are susceptible to stiffness degradation, tearing, and interfacial debonding under severe seismic excitations. Existing metal–viscoelastic composite devices may also suffer from strain incompatibility and localized stress concentrations when the two energy-dissipation components are directly coupled. To overcome these generic bottlenecks, this study proposes a high-performance metal-viscoelastic composite damper (MVECD) featuring conceptually novel decoupled architecture. The device encapsulates a viscoelastic core within an external notched aluminum sleeve, utilizing a precisely engineered preset clearance. This unique configuration activates a displacement-dependent, multi-stage dual dissipation mechanism that entirely eliminates premature mechanical interference and achieves superior elastoplastic performance and mechanical robustness. The dynamic properties tests on a square MVECD (SMVECD) demonstrates definitive superiority over normal VEDs (NVEDs). In the initial-displacement stage (d < 1.5 mm), energy is dissipated solely by the viscoelastic components through shear deformation. In the medium-displacement stage, additional energy dissipation is incorporated via the elastoplastic hysteresis of the metal sleeve when the positive displacement exceeds 1.5 mm. At a medium displacement of 4 mm, the mechanical engagement of the aluminum sleeve bolsters the equivalent stiffness and equivalent damping exceed those of the NVED by maximums of 28.29% and 29.59%, respectively. Finite Element Analysis (FEA) of the SMVECD and a new ellipse metal-viscoelastic composite damper (EMVECD) are performed to further validate the three-displacement-stage working mechanism of MVECD. High-fidelity FEA under extreme large-displacement regimes (15 mm) reveals an unparalleled structural envelope-protection effect. The outer sleeve undergoes intense flexural yielding to maximize structural damping—surpassing NVED equivalent stiffness and damping by 51.97% and 49.96%—while completely shielding the internal viscoelastic core, maintaining its internal stress below a safe threshold of 2.516 MPa. This mechanism effectively mitigates the long-standing debonding risks inherent in traditional composite dampers. Furthermore, a novel mathematical model integrating the Equivalent Fractional Order Micro-Structure Zener (EFMS-Zener) framework of viscoelastic materials (VEMs) and a segmented elastoplastic restoring force model of metal sleeves is established. Numerical analyses indicate that the proposed model can accurately characterize the dynamic behaviors of the MVECD across varying amplitudes and frequencies, with maximum damping force errors between numerical and experimental/simulated results remaining within approximately 10%. The proposed MVECD combines viscoelastic high energy dissipation capacity and metallic superior elastoplastic performance, serving as an advanced, high-performance vibration control solution for engineering structures.
Within the theoretical and applied research on periodic microstructures, increasing attention is being directed towards the nonlinear dynamics of mechanical metamaterials. The present study investigates the nonlinear free and forced dynamics of a minimal two degree-of-freedom model that captures the local mechanical interactions between warp and weft yarns in pretensioned plain-weave textile metamaterials. The analysis incorporates the high-amplitude range of oscillations, including the yarn-to-yarn detachment events. The governing equations of motion are characterized by geometric stiffness induced by self-equilibrated yarn pretension and piecewise nonlinear elastic stiffness arising from unilateral inter-yarn contact. Frequency-response curves under harmonic excitation exhibit a systematic softening behavior, initially driven by the nonlinear constitutive law in the attached configuration and further amplified by the stiffness reduction associated with detachment. Direct numerical integration combined with computational continuation reveals a rich nonlinear response scenario, characterized by coexistence of multistable and unstable solutions, onset of bifurcations and generation of superharmonics. The role of the fundamental mechanical parameters – namely yarn pretension and slenderness – is investigated to clarify their influence on the interplay between weak constitutive nonlinearities and strong non-smooth detachment effects. Pretension is identified as a passive tuning parameter for the local dynamic response, providing promising perspectives towards the spectral design of complex textile metamaterials. Finally, exact solutions are qualitatively and quantitatively compared with solutions resulting from third-order asymptotic approximations of the contact law, highlighting the accuracy and limitations of linearized and cubic approximate models.
Downhole stuck tools are a significant source of downtime during drilling, and jarring tools can remedy such problems. In a typical jarring operation, axial impact releases a stuck tool during drilling. Innovative jarring tool designs generate multiple, high-frequency impacts, but existing models of jarring operations focus on conventional jarring with a single impact. In this work, we developed two low-dimensional models to describe a multi-impact jarring system to investigate nonlinear behaviour during multi-impact jarring. A simplified strongly nonlinear one degree-of-freedom model with a realistic dynamic loading scenario was selected for investigation through a direct numerical simulation. Typical nonlinear dynamics characteristics, such as time–history evolutions, phase portraits, and bifurcation diagrams, are presented. In addition, novel steady-state force equilibrium diagrams are constructed. Results of the numerical simulations indicate the system behaviour is sensitive to some operational parameters, namely the excitation frequency. We present examples of the system dynamic responses ranging from periodic to chaotic solutions, strongly affecting the transmitted force onto a stuck pipe. Our analysis shows that the external excitation frequency has a major effect on system operation and can be used to improve tool performance.
The flexoelectric effect refers to the phenomenon of electric polarization induced by strain gradients, which occurs in both dielectrics and semiconductors. In semiconductors, the flexoelectric effect can modulate the redistribution of carriers. The majority of works on flexoelectric semiconductors are focused on their linear behavior at small deformations, and their nonlinear response under finite deformation remains largely unexplored. In this paper, we present the fundamental equations governing the finite deformation of flexoelectric semiconductors, and we establish a self-consistent, nonlinear constitutive relationship for flexoelectric semiconductors. This constitutive relationship incorporates geometric nonlinearity, electrostrictive nonlinearity, and semiconductor physical nonlinearity. Building on this foundation, we investigate the deformation of bars and tubes in the context of flexoelectric semiconductors, both under finite and small deformations. The results reveal that the influence of nonlinear effects on the coupled multiphysics response is not negligible. To address more general problems involving finite deformation in flexoelectric semiconductors, we develop a nonlinear mixed finite element method and numerically investigate the previous examples as benchmarks. Therefore, this paper advances both the theoretical foundations and numerical treatment of higher-order multiphysics mechanics and provides a basis for the rational design of soft flexoelectric semiconductors.
Dielectric elastomer actuators (DEAs) have attracted considerable attention for applications requiring lightweight, compliant, and large-strain electromechanical transduction. However, the reduction of operating voltage through miniaturization introduces pronounced surface and interfacial effects that can significantly alter the dynamic response and stability characteristics of these systems. In this work, a dynamic electromechanical framework is developed to investigate the nonlinear dynamics of miniaturized dielectric elastomer actuators while explicitly incorporating surface elasticity, intrinsic surface energy, and surface tension. The dielectric elastomer is modeled as an incompressible neo-Hookean ideal dielectric, and the governing nonlinear equation of motion is derived using an Euler-Lagrange formulation under finite deformation. An energy-based approach is further employed to predict the onset of dynamic pull-in instability and the associated critical conditions. The dynamic response under both DC and AC voltage excitations is examined through transient time-history analysis, phase portraits, Poincar & eacute; maps, and frequency-response characteristics. Particular attention is devoted to dynamic pull-in instability and its dependence on surface parameters. The predictions of critical instability parameters are verified through direct numerical integration of the nonlinear governing equation, showing close agreement. A comprehensive parametric study reveals how surface elasticity, intrinsic surface energy, and surface tension modify the effective stiffness, resonance behavior, and critical instability thresholds of miniaturized DEAs. The results demonstrate that surface effects substantially shift dynamic pull-in voltages and alter nonlinear oscillatory characteristics, highlighting their critical role in the design and development of miniaturized dielectric elastomer actuators operating under transient electromechanical loading.
Large-span spatial structures, particularly cable-truss configurations, exhibit marked susceptibility to impulsive loading, which may precipitate abrupt and catastrophic loss of structural integrity. Cable elements, functioning as indispensable load-transfer pathways within the structural system, exhibit pronounced susceptibility to impact-induced dynamic responses. Formulating reliable approaches for structural performance assessment and damage-informed maintenance optimization under impact loading remains a critical challenge within the context of intelligent operation and maintenance (O&M) paradigms. This study integrates fundamental structural parameters with diverse loading scenarios to establish an impact-response-oriented maintenance framework tailored for cable-truss systems, thereby enhancing the generalizability of the proposed methodology. A control matrix is formulated by synthesizing mechanical response metrics with maintenance decision-making indicators to encode the mapping correlation between impact-response performance and maintenance data. A mathematical optimisation framework for impact-response-driven maintenance is developed in this work. Structural load-bearing capacity and configurational requirements are imposed as governing constraints, while maintenance efficiency, carbon footprint, and structural reliability serve as competing objective functions. To identify the optimal maintenance strategy, a hybrid optimisation engine integrating a back-propagation (BP) neural network with an improved non-dominated sorting genetic algorithm (INSGA-II) is proposed. The proposed algorithm enables efficient approximation of highly nonlinear relationships, thereby facilitating the acquisition of superior maintenance solutions and achieving coordinated multi-objective optimisation. The proposed framework is verified and implemented through a case study involving an experimentally tested cable-truss model. The mechanical behaviour of the experimental specimen is examined under representative loading scenarios. The optimal maintenance strategy for mitigating structural impact response is derived. A comparative assessment of structural impact responses in both pre- and post-optimisation states is conducted, thereby substantiating the practical feasibility of the proposed maintenance strategy. Concurrently, the merits of the proposed neural network-based algorithm are substantiated across multiple performance metrics, including convergence behaviour, computational cost-effectiveness, and optimisation quality. The data correlations learned by the neural network are integrated with the SHapley Additive exPlanations (SHAP) framework for interpretable machine learning, enabling a deeper investigation into the dominant factors governing impact resistance. For scenarios involving low impact kinetic energy, the most practical strengthening strategy is identified. The proposed methodology offers a dependable analytical framework and valuable insights for maintaining impact-resilient safety throughout the structural service life.
Multimode mechanical resonators can exhibit strong nonlinear modal interactions that significantly influence response amplitude, stability, and effective operating range. Among these interactions, internal resonance is of particular interest because it can induce energy redistribution, modal activation, and complex bifurcation behaviour. For studies aimed at resolving activation of coupled modes and excitation-direction-dependent interaction, resonator platforms in which the dominant motions of the participating modes remain spatially distinguishable are especially advantageous. In this work, internal resonance is investigated theoretically and experimentally in a beam-based resonator with spatially separated modal motions. The first two modes are predominantly associated with the lateral beams and the cross beam, respectively, enabling the modal frequency ratio to be tuned close to 1:2 while preserving clear physical distinguishability of the modal motions. A continuous model is established and reduced to a two-degree-of-freedom nonlinear model, and a continuation method is used to determine the response branches, stability, and the associated bifurcation structure. Stepped steady-state frequency sweep experiments and single-frequency measurements are then conducted to examine the predicted responses. Under excitation at the first mode, the strongly coupled response is realized on a single-period branch, and quasi-periodic motion is observed experimentally in the central interaction region at higher excitation levels. Under excitation at the second mode, the stable single-period branch loses stability through period doubling, and the strongly coupled response is realized on a doubled-period branch. The experiments show good agreement with the theoretical predictions in terms of the main response structure, hysteretic behaviour, coupled-response region, and phase-space characteristics. These results clarify how the coupled response is realized under different excitation directions in this class of beam-based resonators with spatially separated modal motions.
Voice-coil-motor-actuated fast steering mirrors (VCM-FSMs) are widely used in high-dynamic beam-steering systems owing to their large stroke, smooth actuation, low hysteresis, and good scalability. For large-angle VCM-FSMs, a more rigorous dynamic model is needed to supplement conventional linear descriptions and thereby provide a more realistic representation of the dynamic behavior of the practical system. In this work, a dynamic modeling framework is developed by characterizing the system equivalent stiffness and assessing the angle-dependent nonlinear electromagnetic actuation. The analysis shows that the overall equivalent stiffness of the VCM-FSM consists of the intrinsic stiffness of the flexure hinge and the stiffness correction induced by the center-of-mass (CoM) offset. The resulting load-dependent stiffness behavior is affected by the system parameter configuration and may cause frequency-response shifts under different loading conditions. Electromagnetic actuation mainly enters the governing equation through an angle-dependent nonlinear electromagnetic bending moment, which modifies the effective actuation moment rather than the equivalent stiffness and therefore causes the response amplitude to deviate from linear predictions. Experiments are performed to examine the load-dependent stiffness characteristics and the electromagnetic-actuation-induced amplitude response. The results are consistent with theoretical predictions and confirm the effectiveness of the proposed model. This study provides a useful basis for dynamic modeling, response analysis, structural optimization, and control compensation of large-angle VCM-FSMs.
Friction-induced vibration (FIV) is highly sensitive to the mechanical behaviour of the contacting interface. Consequently, predicting its onset and evolution remains challenging, owing to the limited understanding of which contact parameters most strongly govern system stability. To address this problem for large-scale structures, such as braking systems and turbomachinery, this paper analyses the influence of surface roughness and structural parameters on the onset of FIV by integrating semi-analytical normal and tangential rough-contact models with a two-degree-of-freedom (2DOF) mode-coupling model. The onset of system instability is investigated through complex eigenvalue analysis (CEA) of the linearised system while varying the surface roughness parameters, system parameters, and normal load. The findings reveal that surface roughness has a strong and scale-dependent influence on friction-induced vibration stability, and that the corresponding sensitivity of the stability boundary depends strongly on the structural parameters. The proposed framework provides a mechanistic basis for understanding how surface topography and system stiffness distributions influence friction-induced vibration, and assist the future design and tuning of frictional interfaces in practical engineering systems.
Based on the geometrically exact formulation, this paper develops a planar dynamical model for cantilevered pipes conveying fluid subjected to base excitation in arbitrary directions. The proposed model generalizes and unifies the previously reported special cases involving purely transverse and purely axial base excitation. By systematically reducing the model to these two fundamental cases and comparing the results with existing literature, its validity is confirmed. Compared with conventional small/moderate-deformation models, which usually simplify the displacement-rotation relations and treat transverse and axial base excitations separately, the present geometrically exact model can capture the strong geometric nonlinearity and excitation coupling induced by arbitrary-direction base excitation. More importantly, the novelty of this work lies in two aspects: the development of a unified geometrically exact model for arbitrarily directed base excitation, and the clarification of symmetry evolution in the nonlinear dynamical response together with its mathematical and mechanical origins. The results show that the pipe exhibits symmetric response under purely transverse and purely axial excitation, while asymmetric bifurcation occurs when the excitation direction is inclined, causing the response to lose symmetry with respect to the initial equilibrium configuration. Furthermore, the response undergoes a progressive process of symmetry breaking and subsequent symmetry restoration as the excitation direction varies from transverse to inclined and then to axial.
Dynamic modeling is essential for understanding the vibration mechanisms of rolling bearings with localized defects and for supporting reliable fault diagnosis. However, many existing models still idealize defect interaction as an instantaneous event and simplify interface friction, which limits their ability to reproduce waveform details needed for diagnosis. To address this issue, this study proposes a high-fidelity dynamic model for rolling bearings with localized defects that considers progressive contact behavior and interface nonlinearity. The model explicitly describes the pre-entry, main-contact, and exit stages as a rolling element traverses a defect, and it incorporates LuGre-based rolling–sliding friction to capture velocity-dependent nonlinear effects. A multi-DOFs framework is further established to represent the coupled motions of the inner race, outer race, rolling elements, and cage constraint. Simulated and experimental results for both inner- and outer-race defects show that the proposed model not only preserves the correct characteristic frequencies but also reproduces the precursor step response observed before the main impact. Across the tested cases, the relative errors of the main dimensionless indicators remain below 20%, and the DCRM values remain below 0.2, demonstrating improved fidelity for diagnosis-oriented simulation. The proposed formulation therefore provides a physically grounded basis for high-fidelity fault analysis and simulation-assisted diagnosis of defective rolling bearings.
In this paper, the buckling and post-buckling of a peridynamic elastica column is solved analytically and numerically, in a geometrically exact framework. New solutions are derived for a size-dependent Euler elastica problem, here formulated as a mixed local/peridynamic (MLP) beam model. The governing equation of the mixed local/peridynamic elastica is obtained, using variational arguments based on the peridynamic energy functional. The peridynamic kernel is chosen in an exponential form, built from homogeneous differential operators (Helmholtz kernel). An equivalent fourth-order nonlinear differential eigenvalue problem related to the MLP elastica is formulated. It is shown that, for the considered exponential kernels, the pure peridynamic elastica coincides with the nonlocal elastica in the differential sense (nonlocality in the sense of Eringen), and with the nonlocal curvature-driven elastica. The pure peridynamic elastica problem governed by a nonlinear second-order differential equation is analytically solved, by help of Jacobi elliptic functions together with the elliptic integrals of the first and the third kind. The new solution coincides with Lembo’s solution previously derived by an alternative method for the differential nonlocal elastica. The bifurcation diagram of the peridynamic integro-differential eigenvalue problem is investigated by several FD-based numerical approaches, utilizing the Euler and Newton-Raphson methods. The order of convergence is reported for all proposed methods. Finally, an asymptotic analysis applied to the characterization of the fundamental post-buckling branch is performed for this new peridynamic mechanical problem. The length scale of each equivalent nonlocal model has a softening effect for both the bifurcation loads and in the post-buckling regimes.
