Nonlinear Stiffness Research Articles

Experimental bifurcation analysis of a clamped beam with designed mechanical nonlinearity

We use control-based continuation (CBC) to perform an experimental bifurcation study of a periodically forced dual-beam. The nonlinearity is of geometric nature, provided by a thin, clamped beam. The overall system exhibits hysteresis and bistability in its open-loop frequency response due to a hardening, Duffing-like nonlinear stiffness, which can be designed or adjusted by choosing the properties of the thin beam. We employ local stabilising feedback control to implement CBC and track stable periodic solutions past the fold points. Thus obtained continuous solution branches are used to generate the solution surface over the plane of excitation amplitude and frequency. This surface features two curves of fold bifurcations that meet at a cusp point, and they delimit the experimentally observed bistability range of this nonlinear beam.

A piezoelectric nonlinear energy sink shunt for vibration damping

The theoretical study and experimental validation of a nonlinear shunt circuit for piezoelectric vibration damping is investigated here. The circuit consists of a resistor, an inductor and a nonlinear cubic voltage source. The shunt acts as an electrical analog to the mechanical nonlinear energy sinks (NESs). These mechanical NESs are passive vibration absorbers that typically have a cubic nonlinear stiffness. They have attractive properties such as saturation of the host system’s vibration amplitude and strongly modulated response. This increases its operational frequency bandwidth and robustness against variations in the properties of host systems compared to linear vibration absorbers. However, the nonlinear nature may induce isolated responses in the host system that induce high vibration amplitudes. This paper investigates if these attractive properties also occur in the electric nonlinear energy sink shunt. An analytical expression for the frequency response is derived through the complexification-averaging method. Bifurcations in the frequency response reveal the occurrence of a quasi-periodic vibration energy exchange between the host system and the voltage over the electrodes of piezoelectric material. This is the main mechanism behind the amplitude saturation of the host system. Other bifurcations also reveal the existence of isolated responses. The nonlinear shunt is then realized with analog multipliers and a synthetic inductor and its performance in vibration damping is experimentally verified for a cantilever beam.

Nonlinear energy harvesting system with multiple stability

The nonlinear energy harvesting systems of the forced vibration with an electron-mechanical coupling are widely used to capture ambient vibration energy and convert mechanical energy into electrical energy. However, the nonlinear response mechanism of the friction induced vibration (FIV) energy harvesting system with multiple stability and stick-slip motion is still unclear. In the current paper, a novel nonlinear energy harvesting model with multiple stability of single-, double- and triple-well potential is proposed based on V-shaped structure spring and the belt conveying system. The dynamic equations for the energy harvesting system with multiple stability and self-excited friction are established by using Euler-Lagrangian equations. Secondly, the static characteristics of the nonlinear restoring force, the friction force, and the potential energy surfaces are obtained to show the nonlinear stiffness, multiple equilibrium points, discontinuous behaviors and multiple well responses. Then, the equilibrium surface of bifurcation sets for the autonomous system is given to show the third-order quasi zero stiffness (QZS3), fifth-order quasi zero stiffness (QZS5), double well (DW) and triple well (TW). The co-dimension bifurcation sets of the self-excited vibration system are analyzed and the corresponding phase portraits for the coexistent of multiple limit cycles are obtained. Furthermore, the analytical formula of amplitude frequency response of the approximated system are obtained by the complex harmonic method. The response amplitudes of charge, current, voltage and power of the forced electron-mechanical coupled vibration system for QZS3, QZS5, DW and TW are analyzed by using the numerically solution. Finally, a prototype of FIV energy harvesting system is manufactured and the experimental system is setup. The experimental works of static restoring forces, damping forcse and the electrical outputs are well agreeable with the numerical results, which testified the proposed FIV energy harvesting model.

Switchable microscale stress response of actin-vimentin composites emerges from scale-dependent interactions.

The mechanical properties of the mammalian cell regulate many cellular functions and are largely dictated by the cytoskeleton, a composite network of protein filaments, including actin, microtubules, and intermediate filaments. Interactions between these distinct filaments give rise to emergent mechanical properties that are difficult to generate synthetically, and recent studies have made great strides in advancing our understanding of the mechanical interplay between actin and microtubule filaments. While intermediate filaments play critical roles in the stress response of cells, their effect on the rheological properties of the composite cytoskeleton remains poorly understood. Here, we use optical tweezers microrheology to measure the linear viscoelastic properties and nonlinear stress response of composites of actin and vimentin with varying molar ratios of actin to vimentin. We reveal a surprising, nearly opposite effect of actin-vimentin network mechanics compared to single-component networks in the linear versus nonlinear regimes. Namely, the linear elastic plateau modulus and zero-shear viscosity are markedly reduced in composites compared to single-component networks of actin or vimentin, whereas the initial response force and stiffness are maximized in composites versus single-component networks in the nonlinear regime. While these emergent trends are indicative of distinct interactions between actin and vimentin, nonlinear stiffening and longtime stress response appear to both be dictated primarily by actin, at odds with previous bulk rheology studies. We demonstrate that these complex, scale-dependent effects arise from the varied contributions of network density, filament stiffness, non-specific interactions, and poroelasticity to the mechanical response at different spatiotemporal scales. Cells may harness this complex behavior to facilitate distinct stress responses at different scales and in response to different stimuli to allow for their hallmark multifunctionality.

Robot base position and spacecraft cabin angle optimization via homogeneous stiffness domain index with nonlinear stiffness characteristics

The use of mobile robots for machining large components has received considerable research interest for the application of industrial robots in the machinery manufacturing sector. However, the low structural stiffness of industrial robots can result in poor machining quality under the action of cutting forces. Therefore, this paper proposes a simultaneous optimization method the mobile robot base position and cabin angle using homogeneous stiffness domain (HSD) index for large spacecraft cabins. First, a nonlinear joint stiffness model that considers the gravity compensator mechanism is established to describe the stiffness characteristics of heavy-duty robots more accurately. Subsequently, a HSD index is proposed to evaluate the overall stiffness values and stiffness fluctuation for all robot postures in the machining program. An optimization model is then established based on the HSD under the constraints of machining accessibility, joint angle limitation and singularity. The optimal base position and cabin angle are determined simultaneously using the sparrow search algorithm. Finally, simulation and milling experiments are used to demonstrate that the optimization method proposed in this paper can effectively improve the machining quality.

An equivalent modeling method for coupling stiffness of metal braided sleeve based on hyper-elastic material function

Abstract Metal hoses are important connecting components in pipelines, used for compensation in length and vibration reduction in system. Due to the complex structure of the metal braided sleeve on the outer layer of the metal hose and the nonlinear stiffness caused by coupling with the metal corrugated pipe during the combination process, modeling and calculation are difficult. In this paper, an equivalent modeling method for the coupling stiffness of metal braided sleeve based on Hyper-elastic material function is proposed. The parameters of Hyper-elastic material function are obtained through experiments. Experiments and simulations show that this method has a high accuracy, which propose an efficient way to solve the problems in theoretical analysis and simulation modeling of metal hose.

A vertical track nonlinear energy sink

Eliminating the effects of gravity and designing nonlinear energy sinks (NESs) that suppress vibration in the vertical direction is a challenging task with numerous damping requirements. In this paper, the dynamic design of a vertical track nonlinear energy sink (VTNES) with zero linear stiffness in the vertical direction is proposed and realized for the first time. The motion differential equations of the VTNES coupled with a linear oscillator (LO) are established. With the strong nonlinearity considered of the VTNES, the steady-state response of the system is analyzed with the harmonic balance method (HBM), and the accuracy of the HBM is verified numerically. On this basis, the VTNES prototype is manufactured, and its nonlinear stiffness is identified. The damping effect and dynamic characteristics of the VTNES are studied theoretically and experimentally. The results show that the VTNES has better damping effects when strong modulation responses (SMRs) occur. Moreover, even for small-amplitude vibration, the VTNES also has a good vibration suppression effect. To sum up, in order to suppress the vertical vibration, an NES is designed and developed, which can suppress the vertical vibration within certain ranges of the resonance frequency and the vibration intensity.

Backstepping control based on adaptive neural network and disturbance observer for reconfigurable variable stiffness actuator

Reconfigurable variable stiffness actuator (RVSA) has attracted increasing attention in robotics due to its safety, compliance, and robustness. However, the control of the RVSA is challenging due to nonlinear factors such as high-order nonlinear dynamic, model uncertainties, time-varying model parameters, and disturbances. In this paper, firstly, a lightweight RVSA structure with both passive and active nonlinear variable stiffness characteristic is developed. Secondly, a dynamic surface backstepping control method based on a radial basis neural network and disturbance observer (DSBC-RBFNN-DOB) is proposed to achieve position control of the lightweight RVSA with matched and unmatched uncertainties. To address solve the “complexity explosion” and noise problems in traditional backstepping control, the dynamic surface backstepping control (DSBC) method is used to design the controller. Then, a method based on radial basis neural network (RBFNN) and disturbance observer (DOB) are used to compensate for the matched and unmatched uncertainties in the link and motor. In this method, the matched uncertainties are compensated using RBFNN, and the DOB is integrated to compensate RBFNN approximation errors and unmatched uncertainties. Through Lyapunov stability analysis, the semi-global boundedness of the controller is proven. Finally, the proposed method is simulated and actually implemented, verifying the effectiveness of the method. Simulation and experimental results show that the root mean square error (RMSE) of the proposed method is only 0.97277° and 0.6418°, respectively. Compared with PID, DSBC, and DSBC-RBFNN, the error reduction percentages in simulation (experiment) are 85.6 % (88.9 %), 49.4 % (88.4 %) and 36.1 % (80.0 %) respectively.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Nonlinear characteristics of torsional stiffness of a coilable mast with triangular section

With numerous applications coilable masts in high-precision astronomical observations, such as X-ray source observations, it is important to investigate mast stiffness. To date, there have been many studies on the bending stiffness of coilable masts, but few studies on their torsional stiffness, especially regarding the nonlinear characteristics of torsional stiffness of coilable masts under large torsional deformation. In this paper, a nonlinear analysis method is presented to examine the torsional stiffness of coilable masts with triangular sections. Based on the second-order bending buckling hypothesis of battens under large torsion deformation, the nonlinear relationship between torsional torque and torsional angle is obtained by analyzing torsional deformation and force of coilable masts. This method is used to analyze the torsional stiffness nonlinearity of a certain type of coilable mast which will be used in a practical application in the future and the results are verified by simulation and testing. The comparison results show that the error is within the acceptable range, which proves the effectiveness of the proposed method.

KINEMATIC RESPONSES AND SEGMENTAL STIFFNESS OF NORMAL AND DEGENERATIVE L4/L5 MOTION SEGMENTS: FINITE ELEMENT STUDY

In this study, finite element (FE) models of an anatomical normal and a degenerated disc of L4–L5 motion segment, developed using different methodologies of volume mesh constructions, and ways of defining the spinal components’ material properties, were exercised and analyzed. The geometrics details were obtained using digitizing technique and segmentation of computed tomography (CT) scans; with material and structural properties of the spinal components defined based on the literature and greyscale values of CT images for the normal and degenerated L4–L5 motion segments, respectively. The predicted kinematic responses under five different physiological loadings in three anatomical reference planes for both models together with published data were analyzed. The predicted results correlated well within the range and in similar trends with published experimental results. The overall predicted responses showed different volume mesh constructions and the representation of spinal components’ material properties were accurate enough to predict the kinematic responses, although experimental results exhibit considerable divergence due to different experimental techniques. Both models exhibited different rotational and translation segmental stiffness in each corresponding loading configuration. The normal L4–L5 segment exhibited greatest axial torsional stiffness, and least extensional stiffness. The degenerated L4–L5 segment was least flexible (greatest stiffness) in extension and most flexible (least stiffness) in flexion. For compression, the axial stiffness of segment of normal segment was about 10% larger than that of the degenerated segment. The en-bloc efforts of all spinal components have contributed to the different nonlinear kinematic characteristics and segmental stiffness of these different L4–L5 motion segments.

Study of vibration suppression and energy harvesting for a Vibration-based Piezoelectric–Electromagnetic energy harvester with nonlinear energy sink

Vibration-based energy harvesting devices are prone to excessive displacement response. The effective vibration suppression is critical to the longevity and safety of energy harvesting systems. In this work, the vibration-based Piezoelectric–Electromagnetic energy harvester with nonlinear energy sink (NES-VBPEEH) is proposed to achieve the dual efficacy goals of vibration suppression and vibration energy harvesting. The electromechanical-coupled distributed governing equations are derived through Hamilton’s principle, Euler’s Bernoulli theory, Gauss’s law and Faraday’s law of electromagnetic induction. The output performances of the degradation model of the NES-VBPEEH are validated by previous work. Parameter analysis indicates that the external resistance of the piezoelectric energy harvester significantly affects the electrical damping of the system and ultimately affects the output response of the NES-VBPEEH. The external resistance and coil turns of the electromagnetic vibration energy harvester can significantly affect the output response of the NES-VBPEEH. The output power of both types of energy harvesters is maximized with respect to the resistance and the coil turns. The output response of the NES-VBPEEH exhibits certain nonlinear characteristics as the external excitation amplitude increases to a certain value. The damping and nonlinear stiffness of the NES can significantly affect the output power of the NES-VBPEEH, and appropriate parameter selection can help to achieve the dual efficacy goals of vibration suppression and vibration energy harvesting for the energy harvester.

Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation

Quasi-zero stiffness (QZS) metastructures with simple structures exhibit a high low-frequency vibration isolation performance, but the bearing capacity and working regions are limited due to the linear positive stiffness elements. In this study, a novel hyperbolic QZS metastructure coupled with nonlinear stiffness is designed by combining a pair of cosine-shaped beams and arc-shaped beams for low-frequency vibration isolation. Finite element models (FEM) and experimental analysis are utilized to examine the static characteristics of the metastructure, revealing a wide QZS region and significant load-bearing capacity. The metastructure can achieve effective vibration isolation properties through reasonably designing the unit cells. The influence of geometrical parameters on the QZS region is investigated to optimize and realize desirable QZS features. The dynamic equation and response of the vibration isolation system are determined using the harmonic balance method. Moreover, the corresponding displacement transmissibility is confirmed by numerical simulation. The results of the dynamic sweep frequency experiment further verify the validity of the theoretical analysis model. The hyperbolic QZS metastructure system possesses good vibration isolation performance due to the introduction and compensation of a nonlinear positive stiffness arc beam element. The proposed hyperbolic QZS metastructure provides a significant route for the stiffness compensation of metastructures.

Exploring nonlinearity in quarter car models with an experimental approach to formulating fractional order form and its dynamic analysis.

This study explores the inherent nonlinearity of quarter car models by employing an experimental and numerical approach. The dynamics of vehicular suspension systems are pivotal for ensuring passenger comfort, vehicle stability, and overall ride quality. In this paper we assessed the impact of various parameters and components on suspension performance, enabled the optimization of ride comfort, stability, and handling characteristics. Firstly, experimental analysis allowed for the investigation of factors that are challenging to model theoretically, such as stiffness nonlinearity and damping characteristics, which may vary under different operating conditions. Time domain and frequency response diagram of the model has been obtained. Secondly, a quarter-car with single degree-of-freedom presented and investigated in fractional order form. Fractional order dynamics emphasize nonlinearities in quarter car models, capturing real-world dynamics effectively. The proposed fractional-order nonlinear quarter car model employed Caputo derivative. For numerical analysis of fractional order system, the Adam-Bashforth-Moulton method is used and the disturbance of road assumed to be stochastic. Results show that the dynamic response of the vehicle can be chaotic. Influence of road roughness amplitude and frequency on vehicle vibration is investigated.

Finite element modelling of pipe piles driven in low-to-medium density chalk under monotonic axial loading

Percussive driving in low-to-medium density chalk creates a thin annulus of fully de-structured ‘putty’ chalk around pile shafts and an outer annular zone where fracturing is more intense than in the natural chalk. This damage and the related generation, and subsequent equalisation, of excess pore pressures impacts the piles’ time-dependent axial loading behaviour, as do other ageing processes. This paper presents finite element analyses of open steel tubular piles driven for the recent ALPACA and ALPACA Plus research projects under monotonic axial loading to failure after extended ageing periods. A nonlinear elastic stiffness model with a nonlocal deviatoric strain-based Mohr-Coulomb failure criterion was employed, with different sets of properties to represent the de-structured, fractured chalk and intact chalk. It is shown that the piles’ axial responses are mainly controlled by the puttified chalk annuli. A simplified but efficient means is adopted to impose pre-loading chalk effective stress conditions that explicitly capture the effects of installation damage and subsequent ageing. The potential for strain-softening in the brittle chalk is examined and the effective stress paths developed in representative chalk elements throughout the loading are considered in conjunction with the mobilisation of accumulated deviatoric strains. The simulations indicate 264% higher shaft capacity in compression than in tension, which is mainly attributed to the internal chalk plug.

Dynamic modeling and verification of rotating compressor blade with crack based on beam element

Aero-engine blades may crack under harsh operating environments. This reduces the reliability of the aero-engine and causes catastrophic accidents. Although dynamic modeling and vibration characteristics of simplified blades with cracks have been analyzed extensively, there are few studies on the actual compressor blade with varying sections and twisted shapes. Based on Timoshenko beam theory, a new modeling approach for the compressor blade is proposed. Next, according to the strain energy release rate and Castingliano theory, the dynamic model of cracked compressor blades is established considering the stiffness nonlinearity induced by the breathing crack. The modeling approach is validated by the measured natural frequencies of an intact blade. Then, the dynamic model of the cracked blade is verified by comparing the modal characteristics and vibration characteristics using ANSYS software. Finally, some results show that the natural frequencies fn1, fn2, and fn4 increase as the pre-twisted angle and stagger angle increase. Due to the breathing effect, some integer frequencies appear in the spectrum including 0fe and 2fe, and the variations in the amplitude of frequencies 0fr and 2fr are monotonic under different aerodynamic amplitudes. The research can provide some help for the modeling of the actual blade and fault diagnosis of cracked blades.

On the displacement control performance of base-isolated structures incorporating nonlinear negative stiffness device

Low-frequency dominant far-field harmonic-like long period ground motions (FGMs) excitations have the potential to amplify the seismic response of base-isolated structures (BIS) and may result in isolator displacements exceeding their maximum limits. The application of negative stiffness devices (NSD) in isolated structures primarily focuses on SDOF systems, which fails to capture the displacement response of the superstructure on the isolation story. To address this issue, this study proposes the application of NSD to a 2DOF system in a BIS, with root mean square value as the metric for displacement transmissibility. Firstly, the displacement transmissibility of base-isolated structures with negative stiffness devices (BIS-NSD) and BIS is derived using the triple-harmonic balance (T-HB) method and the single-harmonic balance (S-HB) method, respectively. Subsequently, the applicability of the T-HB method and the S-HB method in obtaining analytical solution for the displacement transmissibility of BIS-NSD and BIS is revealed and validated through numerical solution. Parameters are selected based on fixed-point theorem for comparative analysis of the displacement and acceleration transmissibility between BIS-NSD and BIS. Finally, a comparative analysis between BIS-NSD and BIS under FGMs confirms the application advantages of NSD. The results indicate that compared to BIS, BIS-NSD significantly reduces the displacement transmissibility of the superstructure and the isolation story in the low-frequency range. As the primary resonance frequency decreases, NSD widens the vibration bandwidth of BIS. BIS-NSD effectively mitigates the increase in displacement caused by low-frequency ground motions. The acceleration transmissibility demonstrates that, without sacrificing the acceleration of the superstructure, it effectively controls the displacement of the isolation story.

An interpenetrating-network theory of the cytoskeletal networks in living cells

Cells undergo dramatic deformations during many physiological and pathological processes such as division and migration, and their mechanical integrity is supported by cytoskeletal networks (i.e. intermediate filaments, F-actin, and microtubules). Recent observations of cytoplasmic microstructure indicate interpenetration among different cytoskeletal networks, and micromechanical experiments have shown evidence of complex characteristics in the mechanical response of the interpenetrating cytoskeletal networks of living cells including viscoelastic, nonlinear stiffening, microdamage, and healing characteristics (Hu et al., 2019). However, a theoretical framework describing such a response is missing, and thus it is not clear how different cytoskeletal components with distinct mechanical properties come together to build the overall complex mechanical features of the cytoskeletal networks. In this work, we address this gap by developing a finite-deformation continuum-mechanical theory with a multi-branch visco-hyperelastic constitutive relation coupled with phase-field damage and healing. The proposed interpenetrating-network model elucidates the coupling among interpenetrating cytoskeletal components, and the roles of finite elasticity, viscoelastic relaxation, damage, and healing in the experimentally-observed mechanical response of interpenetrating cytoskeletal networks in living eukaryotic cells.

Mathematical model of dry stack structural elements with geometric imperfections under a cyclic bending moment

Dry stack structural elements are characterized by nonlinear stiffness that arises from geometric imperfections of their components and the absence of any bonding between them. Moreover, such elements dissipate energy under cyclic loading because of their internal structure. The authors considered dry stack structural elements loaded with a bending moment to propose a relatively simple mathematical model of dry stacks composed of only three elements. The model consists of a linear spring, a nonlinear spring, and a spring with hysteresis in series. In this model, the first element describes the idealized properties of a dry stack element, while the second and third elements correspond to the influence of geometric imperfections and the behaviour of dry joints. Furthermore, the authors described a procedure for determining the parameters of the model based on test results. The proposed solution was verified via experimental studies of temporary support structures consisting of a stack of cuboid elements and a hydraulic jack typically used in the process of building rectification. This study showed that the proposed model adequately describes both the nonlinearity and the energy dissipation under a cyclic bending moment.

Dynamic analysis and performance evaluation of inerter fractional nonlinear quasi-zero stiffness isolator on a multi-span bridge under moving load

In this paper we propose a new vibration isolator system, which combines the advantages of a quasi-zero stiffness (QZS) isolator, a damper exhibiting fractional properties and an inerter, to reduce the vibrations of a multi-span continuous beam bridge excited by moving loads. The inerter produces a fictitious mass amplification effect to improve the controller performance and the fractional order takes into account the previous state of the viscoelastic material. After formulating the dynamics equation using beam theory, the amplitude response is determined analytically using the averaging method. The results obtained from the analytical study are validated using the direct numerical simulation method (Newton–Leipnik method). By comparing the isolation performance of the FQZS (fractional quasi-zero stiffness) isolator and the IFQZS (inerter fractional quasi-zero stiffness) isolator, it is shown that the addition of inertance can significantly suppress the tendency of the curve to slope to the right, allowing us to have a wider isolation frequency range on force transmissibility while improving the efficiency of the isolator. One also shows that increasing the fractional parameter contributes to a decrease in the vibration amplitude of the structure, the amplitude of the force transmitted and the area in which unstable solutions appear. However, beyond a certain value of the fractional parameter, we observe an increase in the latter. In order to further extend the study, bifurcation diagram, phase portrait, time history and power spectral density are explored.