Nonlinear Stiffness Properties Research Articles

Response analysis of the Duffing oscillator in dielectric elastomer transducers

To elucidate the nonlinear oscillations inherent to dielectric elastomer transducers (DETs), this study delves into the dynamic behaviour exhibited by a one-dimensional asymmetric Duffing oscillator, which may exhibit either hardening or softening stiffness. The investigation utilizes hyperelastic and visco-hyperelastic constitutive models to capture the nonlinear stiffness properties of dielectric elastomers within compliant transducers. The canonical Duffing equation is employed to articulate the oscillator’s motion, complemented by both asymptotic and quantitative analyses of unforced vibrations and harmonic excitation. Our results reveal that the distinctive stiffness characteristics observed in DETs arise from the complex interplay among hyperelasticity, rheology and applied electric fields. Contrary to conventional oscillators, Maxwell stress substantially modifies the nonlinear dynamics of DETs, an aspect that has been quantitatively dissected in this study. Furthermore, the influence of additional system parameters on forced oscillation has also been meticulously evaluated.

Design and Performance Evaluation of an Enclosed Inertial Wave Energy Converter with a Nonlinear Stiffness Mechanism

In order to enhance the power generation efficiency and reliability of wave energy converters (WECs), an enclosed inertial WEC with a magnetic nonlinear stiffness mechanism (nonlinear EIWEC) is proposed in this paper. A mathematical model of the nonlinear EIWEC was established based on the Cummins equation and the equivalent magnetic charge method, and the joint simulations were carried out using MATLAB/Simulink 2020 and AMESim 2020 softwares. The effect of the magnetic nonlinear stiffness mechanism (NSM) on the performance of the EIWEC system was investigated. The results show that the nonlinear negative stiffness property of NSM can significantly improve the motion response and output power of EIWEC under low-frequency waves. Compared to EIWEC without NSM (linear EIWEC), nonlinear EIWEC has a higher generation efficiency and wider frequency bandwidth. Additionally, the effects of linear spring, internal mass body, and hydraulic power take-off (PTO) system parameters on the energy conversion capability of the system were analyzed to provide a reference for the design of nonlinear EIWECs. In general, the proposed nonlinear EIWEC could provide good development potential for the scale utilization of wave energy resources.

Equivalent single layer approach for ultimate strength analysis of box girder under bending load

The objective of this study is to assess the ultimate strength of box girder using equivalent single layer (ESL) approach. Box girders, composed of stiffened panels, are subjected to a four-point bending load for the ultimate strength analysis. The results of ESL were validated using FEM and experiments. Modeling of ESL includes a single plate with the same stiffness as the actual stiffened panel topology, as opposed to traditional FEM modeling and test specimen, where the stiffener is explicitly modeled. For ESL approach, VUGENS subroutine in Abaqus software is used to define the non-linear stiffness properties. Material and geometric non-linearities were considered during simulations. The ESL stiffness matrix was derived from unit cell simulations under six loading conditions. In this case, a unit cell is a periodic structure that represents the entire stiffened panel model. Contact interaction is utilized to simulate the controlled movement of a hollow cylinder contacting a box girder, a precise modeling approach essential for comprehensive analysis. The explicit dynamic analysis is used to allow the large deformation and non-linear material behavior on stiffened panels. The effect of initial imperfection, residual stress, and stiffener slenderness on the ultimate strength was studied. The ultimate strength of a box girder is determined by its maximum bending moment. According to the results, the ESL model achieves results similar to the traditional FEM model and experiment. The accuracy of ESL to FEM in the ultimate strength assessment varies depending on the magnitude of residual stress and initial imperfection applied to the stiffened panel.

Experimental evaluation of dimension-stable synthetic fibre rope under investigation of spooling test on the multilayer winch drum

Synthetic fibre rope has been a promising solution in ocean exploring applications benefiting from its excellent mechanical properties and high-strength lightweight. However, spooling on the multilayer winch with fibre rope was noticed as a severe problem as a consequence of its nonlinear lateral stiffness property. Substantially increased layers of winding packages caused the deformation of rope. Therefore, a modified rope evolved from applications in fisheries with tight dimensional stability and resistance to deformation under compression is seen as an alternative selection. In this research, characteristics of a 24 mm high module polyethylene (HMPE) dimension-stable marine handling rope subjected to various tension are investigated by a series of full-scale spooling experimental tests on a specially designed test winch at a diameter ratio of D/d = 600/24. The results compared with another spooling experiment conducted in 2021 using an HMPE rope without a core but at the same D/d ratio reveal that the modified rope could induce greater loads than the former. Besides, analytical and numerical calculations will be discussed to assess the accuracy of experimental measurements. Further, a novel winding model is proposed to estimate the deformation of rope shape, evaluate the performance of rope winding and adjust rope spooling adaptability.

Temperature- and Frequency-Dependent Nonlinearities of an Integrated Hydro-Pneumatic Suspension with Mixed Gas-Oil Emulsion Flow

Hydro-pneumatic suspension (HPS) systems are increasingly being implemented in commercial vehicles and various industrial equipment, which is mainly attributed to the integration of adaptable nonlinear pneumatic stiffness and hydraulic damping properties. The integrated HPS design with a shared gas-oil chamber, however, leads to gas-oil emulsion flow within the suspension chambers, which intricately affects the internal and external properties of the HPS, especially under variations in temperature and excitation frequency. This study experimentally and analytically investigated the temperature- and frequency-dependent properties of the hydro-pneumatic suspension with the gas-oil emulsion. Laboratory experiments were performed under three different near-constant temperatures (30, 40, and 50 °C) in the 0.5–8 Hz frequency range. An analytical model of the HPS was formulated considering the effects of temperature on internal fluid properties, gas-oil emulsion flow between the coupled chambers, the dynamic seal friction, and polytropic change in the gas state. The internal parameters, including the gas volume fraction, the discharge coefficient of the emulsion, and the dynamic friction components, as well as the external stiffness and damping characteristics, were determined. The relationships between these properties and the system temperature, velocity, and excitation frequency were further investigated. The simulated responses obtained under different excitations showed reasonably good agreement with the experimental results of the HPS. The results suggested that increased temperature yielded greater equivalent stiffness and comparable damping properties of the system. The gas volume fraction, discharge coefficient, and magnitude of seal friction generally tended to increase with increasing temperature. Increased excitation frequency led to greater hysteresis in hydraulic damping force and seal friction, and reduced seal friction magnitude and Stribeck effect.

Computationally Efficient Method for Steel Column Buckling in Fire

The stability of axially loaded steel columns with compact rectangular hollow sections at elevated temperatures is studied in this paper. The current Eurocode model for checking the buckling resistance of columns in fire was developed on a similar basis to that for ambient conditions. Due to the effect of the complex non-linear behaviour of steel in fire, the standard design model is not always fully appropriate, and certain parameter ranges may give unsafe results. In this work, an analytical method to determine the buckling resistance of steel columns at elevated temperatures is proposed, accounting for variable non-linear stiffness properties which have significant effects on the flexural buckling resistance of steel columns in fire. A finite element model was developed, and an extensive numerical study was performed to explore the effects of different parameters on the behaviours of steel columns at elevated temperatures. The proposed method is validated by comparing the performance with the results of the numerical model. Its improved accuracy with respect to the current Eurocode method is verified. The advantage of the new technique is its computational efficiency, which is valuable in reliability evaluations or data-based design procedures demanding numerous calculation cycles. The potential of the method for probability-based analysis is supported by the format, which enables us to explicitly handle the uncertainties of essential parameters. The proposed framework is suitable for extension to incorporate different material models and section types.

Experimental investigations on nonlinear mechanical behaviors of Kevlar tether

A Kevlar tether usually exhibits strong nonlinearity in engineering applications, bringing new challenges to the modeling of tether dynamics. The nonlinear mechanical behaviors, including creep behavior, nonlinear stiffness, hysteresis effect, and dynamical property of a Kevlar tether, are investigated through a series of experiments. The longitudinal loading experiment setup is established, from which the relationships between tether deformation, tension, and time can be obtained. The creep process of Kevlar tethers is divided into three stages, namely, linear creep stage, deceleration creep stage, and long-term creep stage. This paper studied the longitudinal nonlinear stiffness of a Kevlar tether, whose nonlinearity is fitted by the cubic function model. The hysteresis effect under single loading and unloading is fitted well by the Kawabata stress-strain model, which verifies the correctness and validity of Kawabata model. The nonlinear dynamical model, which includes the elastic force, hysteresis force, and damping force, is established to describe the general dynamical property of the Kevlar tether. The experiment results verify the correctness of the dynamical model form. To simplify the analysis, a simplified model is proposed to describe the dynamical property of the Kevlar tether, and the parameters have a good consistency. The works in this paper contribute to the accurate modeling of flexible tether and lay the foundation for the further research of tether dynamics and control.

An asymmetric quasi-zero stiffness vibration isolator with long stroke and large bearing capacity

A novel passive asymmetric quasi-zero stiffness vibration isolator (AQZS-VI) comprising two linear springs acting in parallel with one negative stiffness element (NSE) is proposed, of which the NSE is mainly constructed by the combination of cantilever plate spring and L-shaped lever (CPS-LSL). The static model of the isolator is deduced considering the geometrical nonlinearity of the NSE and the bending deformation of plate spring. The nonlinear stiffness properties of the CPS-LSL and the AQZS-VI, as well as the nonlinear damping properties of the AQZS-VI, are discussed. The absolute displacement transmissibility of the AQZS-VI under base displacement excitation is obtained using harmonic balance method, and the effects of different excitation amplitudes and damping factors on the vibration isolation performance are analyzed. Better than other quasi-zero stiffness vibration isolators (QZS-VI) whose NSEs do not provide supporting force at zero stiffness point, the NSE of the AQZS-VI provides more supporting force than the parallel connected linear springs, which is very beneficial for improving the bearing capacity of the isolator. Compared with a typical symmetric QZS-VI with same damping property, the AQZS-VI has longer stroke with low stiffness and lower peak value of displacement transmissibility. The prototype experiments indicate that the AQZS-VI outperforms the linear counterpart with much smaller starting frequency of vibration isolation and lower displacement transmissibility. The proposed AQZS-VI has great potential for applying in various engineering practices with superior vibration isolation performance.

An innovative X-shaped vibration isolation mount with tunable quasi-zero-stiffness property

Passive vibration isolation is always preferable in many engineering practices. To this aim, an innovative, compact, and passive vibration isolation mount is studied in this paper. The novel mount is adjustable to different payloads due to a special oblique and tunable stiffness mechanism, and of high vibration isolation performance with a wider quasi-zero-stiffness range due to the deliberate employment of negative stiffness of the X-shaped structure. The X-shaped structure has been well studied recently due to its excellent nonlinear stiffness and damping properties. In this study, by using of the negative stiffness property within the X-shaped structure, the X-shaped mount (X-mount) can have an obviously larger vibration displacement range which maintains the quasi-zero-stiffness property. A special oblique spring is thus introduced such that the overall equivalent stiffness can be much easily adjusted. Systematic parametric study is conducted to reveal the critical design parameters and their relationship with vibration isolation performance. A prototype and experimental validations are implemented to validate the theoretical results. It is believed that the X-mount would provide an innovative technical upgrade to many existing vibration isolation mounts in various engineering practices and it could also be the first prototyped mount which can offer adjustable quasi-zero stiffness conveniently.

High-performance and robust bistable point absorber wave energy converter

A novel magnetic bistable device (MBD) is proposed to improve the efficiency of energy harvesting of a point absorber wave energy converter (WEC) by decreasing the potential barrier of the bistable energy absorber. The novel MBD is mainly composed of two pairs of coaxial permanent magnet rings with different sizes. For the permanent magnet rings, one pair has the same magnetic field direction to provide negative stiffness and the other magnet rings have nonlinear positive stiffness property with a reverse magnetic field direction. The nonlinear dynamic model is established by adopting the state space model approach during solving the convolution integral of radiation force. The capture width ratio of the WEC with the novel MBD is studied with the change of design parameters in regular wave. Then the effect of structural parameters for the novel MBD on the wave energy capture ratio is discussed. The results indicate that using the novel MBD can significantly improve the performance of point absorber WEC especially under low wave excitations in comparison with that of the WEC with traditional MBD.

Optimization under Uncertainty of a Chain of Nonlinear Resonators using a Field Representation

Chains of resonators are known to exhibit interesting dynamic phenomena. For instance, band gaps are frequency ranges within which waves do not propagate or are strongly attenuated. Through the introduction of nonlinearities, other interesting behaviors can be achieved to manipulate waves and optimize the dynamic behavior of a chain. This study proposes the design optimization under uncertainty of such a chain of resonators for the purpose of vibration mitigation. The stochastic optimization accounts for the presence of design uncertainties (e.g., nonlinear stiffness properties) as well as aleatory uncertainties (e.g., loading conditions). The algorithm is tailored to account for discontinuities in the chain’s response due to nonlinearities. In addition, a field formulation is used to define the properties of the resonators along the chain and reduce the dimensionality of the optimization problem. It is shown that the combination of the stochastic optimization algorithm and the field representation leads to robust designs that could not be achieved with optimal properties constant over the chain.

A Bioinspired Dynamics-Based Adaptive Fuzzy SMC Method for Half-Car Active Suspension Systems With Input Dead Zones and Saturations

Active suspension systems are widely used in vehicles to improve ride comfort and handling performance. However, existing control strategies may be limited by various factors, including insufficient consideration of different operation conditions, such as changing in vehicle mass, defects in strategy design leading to incapability for guaranteeing finite-time stability, lack of considering input effects of dead zone and saturation, excessive energy cost, etc. Importantly, very few results considered the energy-saving performance of active suspension systems although a well-perceived issue in practice. An adaptive fuzzy SMC method based on a bioinspired reference model is established in this article, which is to purposely address these problems and be able to provide finite-time convergence and energy-saving performance simultaneously. The proposed control method effectively utilizes beneficial nonlinear stiffness and nonlinear damping properties that the bioinspired reference model could provide. Therefore, superior vibration suppression performance with less energy consumption and improved ride comfort can all be obtained readily. By using a fuzzy-logic system (FLS), the proposed method is beneficial in compensating for system parameter uncertainties, external disturbances, input dead zones, and saturations. Furthermore, based on the adaptive PD-SMC method, the tracking errors can converge to zeros in finite time. The stability of the equilibrium point of all the states in active suspension systems is theoretically proven by Lyapunov techniques. Finally, simulation results are provided to verify the correctness and effectiveness of the proposed control scheme.

Analysis and design of a novel and compact X-structured vibration isolation mount (X-Mount) with wider quasi-zero-stiffness range

Passive vibration isolation is always preferable in most engineering practices. To this aim, a novel, compact and passive vibration isolation mount is studied in this paper, which is designed by using the X-shaped structure with a much smaller size. The novel mount is adjustable to different payloads due to a special oblique and tunable spring mechanism, and of high vibration isolation performance with a wider quasi-zero-stiffness range due to the deliberate employment of negative stiffness of the X-shaped structure. The X-shaped structure has been well studied recently due to its excellent nonlinear stiffness and damping properties. In this study, it is for the first time to explore the utilization of the negative stiffness property within the X-shaped structure such that the resulting design, the X-structured mount (X-mount), can have an obviously larger vibration displacement range which maintains the quasi-zero-stiffness property. A special oblique spring is thus introduced such that the overall equivalent stiffness of the X-mount can be much easily adjusted according to different payloads. Systematic parametric study is conducted to reveal the critical design parameters and their relationship with vibration isolation performance. A prototype and experimental validations are implemented to validate the theoretical results. It is believed that the X-mount would provide an innovative technical upgrade to many existing vibration isolation mounts in various engineering practices and it could also be the first prototyped mount which can offer adjustable quasi-zero stiffness and adjustable loading capacity conveniently.

The Design and Analysis of a Novel Passive Quasi-Zero Stiffness Vibration Isolator

This project aims to design a novel passive quasi-zero stiffness vibration isolator (QZS-VI) and analyze the static and dynamic mechanical properties of the QZS-VI. First, a novel combination of V-shaped lever, plate spring and cross-shaped structure (VL-PS-CS) vibration isolation platform is designed, and a nonlinear QZS-VI is built by parallel connecting VL-PS-CS and coil spring. Second, the static and dynamic modeling of QZS-VI are derived considering the geometrical nonlinearity of V-shaped lever and the large deflection of plate spring, and the average method is applied to obtain the displacement transmissibility. Third, the effects of different structural parameters (e.g., the lengths of long arm and short arm of V-shaped lever, the assembly angle between two arms, the thickness of plate spring) on the static mechanical and equivalent nonlinear friction properties of QZS-VI are thoroughly investigated. Finally, the vibration isolation performance of the designed QZS-VI is compared with another QZS-VI of buckled beam mechanism and traditional linear vibration isolator. The QZS-VI with VL-PS-CS fully explores the nonlinear advantages of plate spring and V-shaped lever and can achieve excellent high static and low dynamic stiffness and nonlinear friction properties. The superior static mechanical properties and nonlinear friction of QZS-VI can be tuned with different structural parameters. The designed QZS-VI exhibits much smaller resonant frequency, lower peak value and more stability property at the peak frequency than other isolators due to its special nonlinear friction and stiffness properties. The designed QZS-VI is practical, novel and suitable for low-frequency vibration isolation. The innovative structure provides novel insights into the design of passive vibration isolators and has great potential for application in engineering practice.

A novel bio-inspired multi-joint anti-vibration structure and its nonlinear HSLDS properties

A unique bio-inspired multi-joint leg-like or limb-like vibration isolation structure is studied by mimicking the skeleton and joint structures of animal legs, and its advantage in passive vibration isolation is systematically investigated. This bio-inspired structure uses several small quadrilateral structures composed by short adjustable rods and a transverse spring to simulate the multi-articulation of animal legs and adopts long rods to simulate the skeleton. The equivalent static stiffness property, loading capacity and dynamic vibration isolation performance of the structure are systematically studied. It is shown that the nonlinear stiffness characteristics of the structure can lead to very excellent vibration isolation capability in the low-frequency band. Different from many existing quasi-zero-stiffness (QZS) isolators in the literature, this novel anti-vibration structure can demonstrate superior high static but low dynamic stiffness (HSLDS) property and thus excellent vibration isolation performance, subject to large vibration amplitude. Through adjusting proper structural parameters including the rod-length, assembly angle, spring stiffness and layer number according to the influence of the structural parameters on the nonlinear stiffness property, the new structure can have the optimal vibration isolation performance without sacrificing its loading capacity to meet actual requirements. Especially, the length of long rods which is used to simulate the skeleton can be adjusted to obtain better vibration isolation performance than adjusting other parameters, followed by the adjustment of the parameters of small quadrilateral structures. The analysis of the influence of structural parameters on the vibration isolation performance shows that the symmetry between different layers has an important role in maintaining the characteristic of the HSLDS. The comparison with other existing vibration isolation structures of the QZS property also indicates that, the new vibration isolation structure can have a better vibration isolation performance with relatively a smaller size of the overall structure. Experiments are successfully conducted and validated the beneficial nonlinear properties of the structure. This novelbio-inspired anti-vibration structure would have great advantages in practical engineering applications.

Carbon-nanotube reinforcement of DNA-silica nanocomposites yields programmable and cell-instructive biocoatings

Biomedical applications require substrata that allow for the grafting, colonization and control of eukaryotic cells. Currently available materials are often limited by insufficient possibilities for the integration of biological functions and means for tuning the mechanical properties. We report on tailorable nanocomposite materials in which silica nanoparticles are interwoven with carbon nanotubes by DNA polymerization. The modular, well controllable and scalable synthesis yields materials whose composition can be gradually adjusted to produce synergistic, non-linear mechanical stiffness and viscosity properties. The materials were exploited as substrata that outperform conventional culture surfaces in the ability to control cellular adhesion, proliferation and transmigration through the hydrogel matrix. The composite materials also enable the construction of layered cell architectures, the expansion of embryonic stem cells by simplified cultivation methods and the on-demand release of uniformly sized stem cell spheroids.

Implementation of physiological functional spinal units in a rigid-body model of the thoracolumbar spine

Most of the current rigid-body models of the complete thoracolumbar spine do not properly model the intervertebral joint as the highly nonlinear stiffness is not incorporated comprehensively and the effects of compressive load on stiffness is commonly being neglected. Based on published in vitro data of individual intervertebral joint flexibility, multi-level six degree-of-freedom nonlinear stiffness of functional spinal units was modelled and incorporated in a rigid-body model of the thoracolumbar spine. To estimate physiological in vivo conditions of the entire spine, stiffening effects caused by directly applied compressive loads, and contributions to mono-segmental stiffness from the rib cage as well as multi-segmental interactions in the thoracic spine were analysed and implemented. Forward dynamic simulations were performed to simulate in vitro tests that measured the load-displacement response of the spine under various loading conditions. The predicted kinematic responses of the model were in agreement with in vitro measurements, with correlations between simulated and measured segmental displacements varying between 0.66 and 0.97 (p < 0.05) and average deviations below 1.6°. Coupling relationships were found between lateral bending and axial rotation. Under compressive loads, the model behaved stiffer and showed a decreased range of motion: The flexion/extension response of the full thoracolumbar spine under compressive loads up to 800 N was found to strongly correlate with the literature (r = 0.99, p < 0.0001). The implementation of physiological functional spinal units with nonlinear stiffness properties into rigid-body models can enhance accuracy of biomechanical simulations, and enable detailed analysis of spinal kinematics under complex loading conditions seen in vivo.

Dynamics and performance evaluation of a novel tristable hybrid energy harvester for ultra-low level vibration resources

To improve the efficiency of energy harvesting from external ultra-low level vibration resources, this article presents a novel tristable hybrid vibration energy harvester using geometric nonlinearity technique to tuning the resonant frequency and broadening the bandwidth. Based on a simple snap-through spring-mass system, it is composed of a piezoelectric ceramics, two ring permanent magnets and a central inertial mass. The device was designed as harvesting energy from low and ultra-low level vibration resources by exploring structural benefits and nonlinear characteristics. The governing electromechanical coupling equations of motion are derived, and a transformation is performed to decouple the equations. The presented device has different nonlinear stiffness properties of the hardening spring effect, softening spring effect and hardening-softening spring effect, which are ideal for the system to harvest broadband energy and suitable for energy harvesting for changing excitation. Compared with monostable and bistable energy harvesters, the tristable energy harvesting has a wider bandwidth for resonance than the bistable or monostable one to enhance the energy harvesting performance over a broadband ultra-low level frequency region. For ultra-low level stochastic fluctuation, results show that the tristable characteristic can lead to a large response and give out a high harvested power.

Asymmetric Bimanual Control of Dual-Arm Exoskeletons for Human-Cooperative Manipulations

In this paper, two upper limbs of an exoskeleton robot are operated within a constrained region of the operational space with unidentified intention of the human operator's motion as well as uncertain dynamics including physical limits. The new human-cooperative strategies are developed to detect the human subject's movement efforts in order to make the robot behavior flexible and adaptive. The motion intention extracted from the measurement of the subject's muscular effort in terms of the applied forces/torques can be represented to derive the reference trajectory of his/her limb using a viable impedance model. Then, adaptive online estimation for impedance parameters is employed to deal with the nonlinear and variable stiffness property of the limb model. In order for the robot to follow a specific impedance target, we integrate the motion intention estimation into a barrier Lyapunov function based adaptive impedance control. Experiments have been carried out to verify the effectiveness of the proposed dual-arm coordination control scheme, in terms of desired motion and force tracking.

Characterization of a hydro-pneumatic suspension strut with gas-oil emulsion

Abstract The nonlinear stiffness and damping properties of a simple and low-cost design of a hydro-pneumatic suspension (HPS) strut that permits entrapment of gas into the hydraulic oil are characterized experimentally and analytically. The formulation of gas-oil emulsion is studied in the laboratory, and the variations in the bulk modulus and mass density of the emulsion are formulated as a function of the gas volume fraction. An analytical model of the HPS is formulated considering polytropic change in the gas state, seal friction, and the gas-oil emulsion flows through orifices and valves. The model is formulated considering one and two bleed orifices configurations of the strut. The measured data acquired under a nearly constant temperature are used to identify gas volume fraction of the emulsion, and friction and flow discharge coefficients as functions of the strut velocity and fluid pressure. The results suggested that single orifice configuration, owing to high fluid pressure, causes greater gas entrapment within the oil and thus significantly higher compressibility of the gas-oil emulsion. The model results obtained under different excitations in the 0.1–8 Hz frequency range showed reasonably good agreements with the measured stiffness and damping properties of the HPS strut. The results show that the variations in fluid compressibility and free gas volume cause increase in effective stiffness but considerable reduction in the damping in a highly nonlinear manner. Increasing the gas volume fraction resulted in substantial hysteresis in the force–deflection and force–velocity characteristics of the strut.