Nonlinear Spring Research Articles

Analysis of Displacement Transmissibility and Bifurcation Behavior in Nonlinear Systems with Friction and Nonlinear Spring

Analysis of Displacement Transmissibility and Bifurcation Behavior in Nonlinear Systems with Friction and Nonlinear Spring

Impacts of Transition Piece Designs on the Resilience of Large Offshore Wind Turbines Subject to Combined Earthquake, Wind and Wave Loads and Soil‐Structure Interaction

ABSTRACTThe urgent global drive to mitigate greenhouse gas emissions has significantly boosted renewable energy production, notably expanding offshore wind energy across the globe. With the technological evolution enabling higher‐capacity turbines on larger foundations, these installations are increasingly situated in earthquake‐prone areas, underscoring the critical need to ensure their seismic resilience as they become a pivotal component of the global energy infrastructure. This study scrutinises the dynamic behaviour of a 15 MW offshore wind turbine (OWT) under concurrent earthquake, wind and wave loads, focusing on the performance of the ultra‐high‐strength cementitious grout that bonds the monopile to the transition piece. Employing LS DYNA for numerical simulations, we explored the seismic responses of four OWT designs with diverse transition piece cone angles, incorporating nonlinear soil springs to model soil‐structure interactions (SSIs) and conducting a site response analysis (SRA) to account for local site effects on ground motion amplification. Our findings reveal that transition pieces with larger cone angles exhibit substantially enhanced stress distribution and resistance to grout damage, evidenced by decreased ovalisation in the coned sections of the transition piece and monopile, and improved bending flexibility. The observed disparities in damage across different cone angles highlight shortcomings in current design guidelines pertaining to the prediction of grout stresses in conical transition piece designs, with the current code‐specified calculations predicting higher stresses for transition piece designs with larger cone angles. This study also highlights the code's limitations when accounting for grout damage induced by stress concentrations in the grouted connections under seismic dynamic loading conditions. The results of the study demonstrate the need for refinement of these guidelines to improve the seismic robustness of OWTs, thereby contributing to the resilience of renewable energy infrastructure against earthquake‐induced disruptions.

Tapered curved-beam hinges for electret-based vibration energy harvesting devices

Abstract Interests on vibration energy harvesting have been growing recently for various applications. One of the major development goals for vibration energy harvesters has been improvement in energy conversion efficiency. To pursue that goal, one of the main approaches has been to broaden the spectra of harvesters. Employment of nonlinear springs, such as curved-beam hinges, has proven to be effective for that purpose. The main contribution of the current study is to introduce a lateral taper to the curved beam so as to further optimize the harvester performances. Via numerical analysis by using stochastic differential equations, the study shows that at 0.05g of vibration strength, tapered curved-beam hinges can result in higher electric power output than the non-tapered ones. Deformation-induced stress was taken into consideration as well, in reference to the fracture strength of the material (single-crystal silicon). At lower vibration strength (0.02g), spring nonlinearity becomes weaker, and as a result, the narrowest curved-beam hinge produces the highest output power. Overall, the current study demonstrates that tapering of the curved beam can be a useful addition in the vibration energy harvester design.

Seismic Fragility Analysis of Offshore Wind Turbines Considering Site-Specific Ground Responses

This study investigated the seismic performance and assessed the seismic fragility of an existing pentapod suction-bucket-supported offshore wind turbine, focusing on the amplification of earthquake ground motions. A simplified suction bucket–soil interaction model with nonlinear spring elements was employed within a finite element framework, linking the suction bucket and soil to hypothetical points on the OWT structures at the mudline. Unlike conventional approaches using bedrock earthquake records, this study utilized free-field surface motions as input, derived from bedrock ground motions through one-dimensional wave theory propagation to estimate soil-layer-induced amplification effects. The validity of the simplified model was confirmed, enabling effective assessment of seismic vulnerability through fragility curves. These curves revealed that the amplification effect increases the vulnerability of the OWT system, raising the probability of exceeding damage limit states such as horizontal displacement of the tower top, tower stress, and horizontal displacement at the mudline during small to moderate earthquakes, while decreasing this likelihood during strong earthquakes. Comparisons between the Full Model and the simplified Spring Model reveal that the simplified model reduces computational time by approximately 75%, with similar seismic response accuracy, making it a valuable tool for rapid seismic assessments. This research contributes to enhancing seismic design practices for suction-bucket-supported offshore wind turbines by employing a minimalist finite element model approach.

Examining the behavior of a two-connected-spring pendulum according to non-resonance and resonance conditions

This paper focuses on the planar forced oscillations of a particle linked to its support point through a system comprising two nonlinear springs arranged in series with two viscous dampers. The motion of this point is restricted to being on an elliptic route. Three degrees-of-freedom (DOF) characterize the system’s motion, explained by two differential equations (DEs) and another algebraic equation derived using Lagrange’s equations. The traditional method of multiple-time scales (MMTS) is applied in the context of the time realm. The dynamics of forced and damped oscillations is explored in this work, highlighting non-resonant conditions along with two distinct external resonances. Furthermore, stationary periodic states influenced by external resonances are examined, and the stability of each case is evaluated. The asymptotic solutions of the controlling equations are solved up to the third approximation. The classification of resonance cases is based on the second and third order of the approximate solutions. Therefore, the solvability criteria and the modulation equations (ME) are obtained. The Routh–Hurwitz conditions (RHCs) serve as the basis for assessing the stability and instability criteria. A discussion and graphical representations of the time histories, frequency response curves (FRC), and stability regions are provided to demonstrate the beneficial effects of different physical parameter inputs on the system’s performance. The obtained outcomes can improve the performance of spring-based mechanical systems like vibrational isolation devices, shock absorbers, and suspension mechanisms. Additionally, they have the potential to enhance stability and control in robotic limbs and aircraft landing gear systems, ensuring better stress absorption and stability.

Analytical solutions for large-deflection bending of variable-thickness inhomogeneous functional graded composite circular plates with parameterized boundaries under hygro-thermo-mechanical loads

A brand-new hygro-thermo-mechanical bending model of the inhomogeneous concave and convex composite circular plates having varying thickness undergoing large deformation resting on nonlinear tri-parameter spring and shear elastic layers is proposed. Three-dimensional analytical hygrothermal field of circular plate is demonstrated with non-uniform thermal and moisture diffusion coefficients in axial variation, while thickness and elastic module of material are modeled in quadratically change and sufficiently summarized in normalization. The circled boundaries with arbitrary translational and rotational constraints are parameterized with feasible region of elastic parameters highlighted in rectangular domain by Linear Programming method. Two reduced-order integro-differential governing equations for the composite circular plates under extreme load have been derived, while analytical bending solutions are obtained by an employed homotopy-based analytical scheme with accuracy verified and convergence accelerated by truncation and iteration. Whether thickness or elastic module of composite material is variable, the final outcome on plate structural strength is the discrepancy of bending stiffness with different load capacity, with the former revealing more sensitive than the latter under the condition of same values. Once amplitude and distribution of hygrothermal load are determined, influence on large-deflection bending of circular plates exists at a limited range of nonlinearity, while non-uniform distribution of thermal expansion and moisture concentration aggravates bending effects only within this range.

New implicit time integration schemes for structural dynamics combining high frequency damping and high second order accuracy

The time integration schemes, GA-23 and GA-234, recently proposed by the authors for first order problems, are extended to solve second-order problems in structural dynamics. The resulting methods maintain unconditional stability and user-controlled high-frequency damping. They offer improved accuracy and exhibit less numerical damping in the low-frequency regime, outperforming the well-known generalised-α method. When the high-frequency damping is maximised the new schemes can be cast in the format of backward difference formulae, offering more accurate alternatives to the standard second order formula. The effectiveness of the new time integration schemes is validated through a number of numerical examples, including a linear elastic cantilever beam, a nonlinear spring pendulum, and wave propagation on a string.

Case study of adjacent integral bridges supported on continuous secant piled walls

Integral bridges are often the preferred way of construction for short and medium span bridges in the UK. However, the behaviour of integral bridges is influenced by complex soil-structure interaction behaviour that may be cumbersome to predict and analyse when dealing with unusual structural arrangements. This paper presents a case study for the design of two single span bridges carrying a roundabout over a dual carriageway, supported on continuous secant piled walls. One of the bridges is fully integral while the adjacent structure is articulated at one end due to its skew. An alternative detail for the articulated end is proposed, to eliminate soil pressures and settlements at the back of the abutments as well as reduce the risk of water ingress to the bearings. The complex nature of the structural arrangement presented analytical challenges, that were solved by developing an advanced modelling approach, exploring two proposals. The first is to evaluate the geotechnical model for a range of imposed top of abutment displacements from the onset. The second, using unique non-linear springs and derived P-Y curves describes the more realistic variable nature of soil-structure interaction over the length of the walls leading to reduced soil pressures enhancing design feasibility.

Influence of Specimen Width on Crack Propagation Process in Lightly Reinforced Concrete Beams.

Models used to study the fracture process of concrete are often considered 2D, ignoring the influence of specimen width. However, during the fracture process in pre-cracked concrete beams, the crack length varies along the thickness direction, especially in reinforced concrete. To study the influence of specimen width on reinforced concrete fracture behavior, a 3D numerical method was used to simulate the crack propagation processes of lightly reinforced concrete beams based on Fracture Mechanics. Nonlinear spring elements with different stress-displacement constitutive laws were employed to characterize the softening behavior of concrete and the bond-slip behavior between the steel bars and concrete, respectively. It is assumed that the crack begins to propagate when the maximum stress intensity factor at the crack tip along the beam width reaches the initial fracture toughness of concrete. To verify the validity of the proposed method, the completed crack propagation processes of lightly reinforced concrete three-point bending notched beams were simulated, and the calculated load-crack mouth opening displacement curves showed a reasonable agreement with the experimental data. Moreover, the impact of the 2D reinforced concrete beam model on the crack propagation process was analyzed. The results indicate that at the initial loading stage, the external load P obtained from the 2D model is significantly larger than the result from the presented 3D model.

Stability and bifurcation analysis of a 2DOF dynamical system with piezoelectric device and feedback control

This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

Optimised rotational-spring component-based modelling strategy for seismic resistant steel-concrete composite joints and frames with continuous or isolated slab

Joints and frames in steel-concrete composite systems represent complex mechanical assemblies that require specific calculation procedures to optimise their detailing and structural capacity, particularly under seismic loads. To this aim, component-based modelling approaches should be able to account for the most relevant mechanisms and resistance/stiffness behaviours of individual members, and their mutual interaction. In this paper, two different simplified non-linear approaches are considered for steel-concrete composite beam-to-column joints, and are specifically applied to a seismic resistant case-study frame with X-concentric bracings. Both beam-to-column joints with or continuous (“JA” joint) or fully isolated (“JB” joint) slab are examined. First, non-linear axial springs are assembled and calibrated on the base of a previous study (“Type 1″ model (“T1″)), according to force-displacement relationships proposed in the DPC-ReLUIS Italian guidelines. Successively, a novel modelling approach based on non-linear rotational springs is presented (“Type 2″ model (“T2″)), to further simplify the computational cost of T1 strategy, and allow to efficiently account for the moment-rotation behaviour of the examined joints. The preliminary numerical validation is carried out based on past literature experiments. Moreover, the optimized T2 approach is used to explore the in-plane lateral, seismic performance of a 2D steel-concrete composite frame, which is specifically designed with X-concentric bracings. The seismic capacity of the frame (and the associated interaction of components, especially the joint zone with the bracing system) is addressed on the base of pushover analyses.

Wave dispersion in a damped beam supported by cubic nonlinear springs: A multiscale freewave approach.

Researchers are drawn to exploring wave dispersion in nonlinear systems because of the amplitude-dependent tunability of the band gap. This paper investigates the amplitude-dependent wave dispersion in continuous beam structures supported periodically by nonlinear springs. Additionally, it examines the influence of inherent beam damping on wave dispersion. The analytical framework consists of homogenization of the unit cell and the method of multiple scales with two distinct time scales to derive the wave dispersion equation. The proposed analytical approach for nonlinear wave propagation is validated through numerical finite element simulations. It is observed that the frequency shift is positive for hardening and negative for softening supports. Following this, the dispersion shift over time in the damped systems is examined by considering viscous and strain rate-dependent damping. The sensitivity of strain rate damping to propagation constant and the independence of viscous damping from propagation constant are thoroughly investigated. In a damped system, the frequency shift diminishes over time as the amplitude decreases reducing the effect of nonlinearity. This study opens up avenues for controlling or filtering vibrations through the tunable band gap of continuous nonlinear metamaterials.

Analysing Flexural Response in RC Beams: A Closed-Form Solution Designer Perspective from Detailed to Simplified Modelling

This paper presents a detailed analytical approach for the bending analysis of reinforced concrete beams, integrating both structural mechanics principles and Eurocode 2 provisions. The general analytical expressions derived for the curvature were applied for the transverse displacement analysis of a simply supported reinforced concrete beam under four-point loading, focusing on key limit states: the initiation of cracking, the yielding of tensile reinforcement and the compressive failure of concrete. The displacement’s results were validated through experimental testing, showing a high degree of accuracy in the elastic and crack propagation phases. Deviations in the yielding phase were attributed to the conservative material assumptions within the Eurocode 2 framework, though the analytical model remained reliable overall. To streamline the computational process for more complex structures, a simplified model utilising a non-linear rotational spring was further developed. This model effectively captures the influence of cracking with significantly reduced computational effort, making it suitable for serviceability limit state analyses in complex loading scenarios, such as seismic impacts. The results demonstrate that combining detailed analytical methods with this simplified model provides an efficient and practical solution for the analysis of reinforced concrete beams, balancing precision with computational efficiency.

Dynamic characteristics and damage identification with interface damage of slabs in movable point crossing

As the operating density of China’s high-speed railway increases, the phenomenon of interlayer debonding in movable point crossing (MPC) is becoming more prevalent, which can lead to failure of the frog’s slab track. Meanwhile, vehicle impacts and interlayer debonding can exacerbate abnormal vibration of the vehicle and even lead to derailment. Therefore, based on the theory of rigid-flexible coupling dynamics, a vehicle-turnout-slab coupling dynamics model is established for the first time. In this model, the crossing rails with variable cross sections is considered and the interlayer debonding between turnout slabs is modelled by nonlinear springs. Then, the characteristics of the debonding is identified by applying a wavelet transform to the dynamic response of the vehicle. Finally, different types and sizes of debonding between the layers are considered. The results show that different debonding can lead to two types of contact, partial contact and complete voiding, which can further lead to detrimental effects on vehicle safety. And the limit of the defect size that causes the track slab to void is 1 mm. On this basis, it was found that when MPC’s slab tracks were voided, longer debonding can cause high frequency vibration acceleration of approximately 700 Hz in the vehicle axle box. The new insights obtained by this study could provide guidance for the maintenance period and avoid the development of the damage. This study not only obtained a mechanism for the effect of interlayer debonding on the dynamic response of the vehicle-track system, but also provides a theoretical basis of detecting interlayer debonding for future study.

Aeroelasticity of an aircraft wing with nonlinear energy sink

In this paper, the effectiveness of nonlinear energy sinks on enhancing aeroelastic stability and post-instability response of aircraft wings is investigated. The wing has two degrees of freedom in bending and torsion, and is modelled using an extended Euler-Bernoulli beam theory with hardening nonlinearity. A Nonlinear Energy Sink (NES) absorber is embedded inside the wing distributed along the wing span. The wing attached unsteady aerodynamic loads are simulated using the Wagner's indicial lift model. The structural dynamics of the wing are derived using Extended Hamilton's Principle and it is discretised using Galerkin's method. The NES mass is connected to the wing spar through a linear damper and nonlinear spring with a cubic stiffness nonlinearity. The coupled aeroelastic equations are then transformed to state space. Then, integrated numerically to resolve the bending and torsional response of the wing to study the impact of spanwise and chordwise positions of the embedded NES on flutter suppression and instability response enhancement. The results demonstrate that the NES is most efficient and is most sensitive to changes in the stiffness when placed at the wingtip. For a given chordwise location, it is found that there is a range of flow speeds over which the NES is most effective and reducing the chordwise offset lowers the speed of the peak efficiency range and moves it closer to the flutter speed. In addition, increasing the stiffness coefficient of the NES improves the efficiency of the device in the immediate post-flutter region. Two near optimum NES devices are proposed with a mass ratios of 1% (located at the wingtip) and 2.5% (located at 75% span). Both of these improve the flutter speed by 5%, and reduce the post-flutter response by 64.5% and 59.2%, respectively.

Equivalent spring-like system for two nonlinear springs in series: application in metastructure units design

The paper deals with the problem of design of unit in auxetic metastructure. The unit is modeled as a two-part spring-like system where each part is with individual stiffness. To overcome the problem of analyzing of each of parts separately, the equivalent spring is suggested to be introduced. In the paper, a method for obtaining the equivalent elastic force of the unit is developed. The method is the generalization of the procedure suggested for substitution of a hard and a soft spring in series with an equivalent one. The nonlinearity of original springs is of quadratic order. As a results, it is obtained that the equivalent elastic force for two equal springs remains of the same type as of the original springs (soft or hard). For two opposite type springs in series with equal coefficients, the equivalent force is soft. The method is applicable for any hard and soft nonlinear springs or spring-like systems. Thus the hexagonal auxetic unit which contains a soft and a hard part in series is analyzed. In the paper, a new analytic method for determination of the frequency of vibration for the unit under action of a constant compression force acting along the unit axis is introduced. The method is applied for units which contain two parts: hard–hard, soft–soft, hard–linear, soft–linear and opposite. The obtained approximate vibration results are compared with numerically obtained ones and show good agreement. The advantage of the method is its simplicity as it does not require the nonlinear equation of motion to be solved.

Corrosion failure analysis of interfacial bond performance in circular seawater sea-sand concrete encased weathering steel structures

This paper investigated the bond-slip behavior of circular seawater sea-sand concrete encased weathering steel (CSSCEWS) structures after chloride-induced corrosion. Electrochemical corrosion tests and push-out tests were conducted on fifteen designed CSSCEWS specimens. The results revealed that the corrosion ratio and steel section type distinctly influenced the interfacial bond performance. Both the ultimate and residual bond strengths decreased with the increase of the corrosion ratio, and the decline rate slowed down gradually. Compared to the uncorroded specimens, the average damage ratio of the ultimate strength of the specimens with a corrosion ratio of 8% is 25.9%, while the average damage ratio of the residual strength is 49.1%. The bond toughness continuously decreased with prolonged corrosion, while the bond stiffness and energy dissipation index first increased and then decreased. Additionally, formulas for calculating the corroded bond strengths of CSSCEWS specimens were proposed. Considering the strength damage due to corrosion and the stiffness damage due to slip, a three-stage bond-slip constitutive model was proposed. Finally, this model was applied to the nonlinear spring elements in finite element modeling to effectively simulate the interfacial bond behavior during the loading of corroded CSSCEWS specimens.

Pyrolysis modelling of insulation material in coupled fire-structure simulations

This paper presents a modelling approach to predict the thermodynamical and thermomechanical behaviour of structures with a layer of insulation material under fire, which takes into account the pyrolysis of the insulation and its effects on the structure. First, an existing 1D pyrolysis model is implemented and verified by theoretical and validated by experimental results. Then the model’s 1D setup is integrated into 3D Heat Transfer (HT) analyses of structures. The obtained thermodynamical results, i.e. temperatures as a function of time and the pyrolysis process, are transferred to a thermomechanical Structural Response (SR) analysis. Mechanical results are then obtained via temperature and pyrolysis-dependent material properties. The resulting HT and SR analyses are demonstrated in fire-structure simulations of facades made of sandwich panels, including their supporting frames with steel sections, and bolt and screw connections, modelled by non-linear spring elements. It is shown that in a short time window of 100 s, pyrolysis is limited to certain zones of the panel and for limited depths. Nevertheless, due to the endothermic process, it reduces expansion and bending of the panels, and consequently results in smaller displacements, and delayed failure of the connections. For longer periods, with connection failures neglected, significant pyrolysis takes place, which influences the temperature distribution in the complete interior of the sandwich panel. However, this has only a marginal effect on the structural behaviour. In conclusion, pyrolysis effects are relevant, can be modelled, and may somewhat reduce fire risks in structures. Future research can combine pyrolysis with advanced modelling of bolt and screw connections, using a two-scale method. As such, all relevant details of structures can be modelled and investigated for different fire scenarios, including fire-structure effects, all still to be validated by experiments.

High-speed nonlinear dynamic analysis of dual springs system considering internal structural impact

The significant dynamic responses of helical springs are widely reported as the main reason causing pre-failure in various high-speed conditions. Though the concept of nonlinear springs is proposed to reduce their own dynamic responses, they are found to induce unexpected spike forces during operations. This study presents a dual springs system used within high-speed sports car engines, which aims to mitigate the dynamic responses and spike forces. Analytical models are first developed to calculate the fundamental frequencies of the outer and the inner springs of the system. Then, a transient finite element (FE) model is developed to accurately simulate the dual springs system working at various high engine speeds and validated experimentally by the results of the engine head tests. The simulations reveal that the use of dual springs with different natural frequencies can mitigate the dynamic responses of the whole system to a certain extent. Nevertheless, it is still a compromise as coil clash occurs on each isolated spring though the effects are reduced. The findings of this study indicate that the establishment of direct connections between every component spring would be a promising way to better mitigate the dynamic responses including the effects of coil clash.

A rate-dependent crack bridging model for dynamic fracture of CNT-reinforced polymers

Carbon nanotubes (CNTs) improve the fracture toughness of polymer-based matrix composites by bridging the crack growth path. This research presents a finite element (FE) rate-dependent crack bridging model of Mode-I dynamic crack growth in CNT-reinforced polymers accounting for rate of loading and rapid crack growth. Two distinct CNT bridging and matrix cracking damage mechanisms are taken into account in the fracture process zone (FPZ) accounting for the dissipation of fracture energy. A viscoelastic-viscoplastic material model is adapted for the matrix phase to capture the strain rate effects. The crack in the matrix phase is modeled using cohesive zone elements characterized by experimental data. A rate-dependent traction-separation law obtained from CNT pull-out simulation at relevant crack opening speeds is used to simulate the CNT bridging in the FPZ. Considering the given traction-separation law as a constitutive equation, the CNTs are replaced with non-linear spring elements to facilitate the FE simulation of crack bridging with numerous CNTs in the FPZ. The proposed rate-dependent FE model for crack bridging enables the study of the effect of key CNT factors such as length, orientation, waviness, volume fraction, and agglomeration on fracture energy dissipation at various crack speeds. The developed model can simultaneously consider the interactive effects of all CNT parameters which is useful for considering all processing-induced uncertainties in analyzing the effects of CNTs on the dynamic fracture toughness of nanocomposites.