Nonlinear Stiffness Research Articles

Role of inertial nonlinearity and coupling stiffness on a series of coupled harvesters

This study investigates the effect of coupling configuration on a state-of-the-art harvesting system consisting of a series of spring-coupled piezoelectric nonlinear monostable tip-loaded cantilever-based vibration energy harvesters (VEHs). The investigation finds favorable coupling configurations that exploit complex nonlinear interaction for improving system performance. The system model incorporates the beam's geometric nonlinearities in the governing equation using inertial and stiffness nonlinear terms. The study first analytically investigates the effect of grounding stiffness on a single harvester's performance. Then, it numerically investigates the effect of increasing array size and changing stiffness distribution. The study discovers a novel hardening/softening transition phenomenon at a critical grounding stiffness value induced by the inertial nonlinear effect. This critical event also maximizes peak output power near the largest eigenfrequency at an optimum grounding stiffness value. At larger arrays, the peak power shows a complex bifurcation behavior, hinting at the presence of multiple attractors. The numerical study of different stiffness configurations and linearized eigenanalysis reveals the necessity of a multilevel optimization strategy focused on both the stiffness ratio between the coupling springs' and their overall stiffness level. This study bridges the gap between previous studies on spring-coupled linear and bistable arrays, offering new insights into coupled nonlinear interactions and proposing novel optimization strategies.

Torsional vibration suppression of cantilever beams system with the PNESI

Considering the intricate design of the conventional inerter structures, the compressive-torsional coupling effects of chiral metamaterial are used to achieve the torsional movement alongside linear movement, forming a simplified inerter mechanism. This simplified design is then integrated with the nonlinear energy sink (NES). In this study, a novel NES with the chiral metamaterial inerter structure is suggested to mitigate vibrations in the cantilever beam system. The introduction includes both the piecewise linear beam, providing nonlinear stiffness, and the chiral metamaterial inerter structure. These components are combined to form the piecewise nonlinear NES with inerter (PNESI) structure. Subsequently, the PNESI structure is integrated with the cantilever beam system, and a corresponding dynamic model is formulated to evaluate its vibration mitigation capabilities. The theoretical results are verified through experiments. The findings indicate that under steady-state vibration conditions, PNESI achieves peak vibration reductions of 88.6% in simulations and 62.7% in experiments for the cantilever beam system.

Analytical study on hybrid thrust foil bearing, structural characterization and measurement of static performance

This work presents a novel hybrid foil thrust bearing (HFTB) with an outer diameter of 154 mm, along with simulation and test results. The HFTB incorporates a high-pressure air/gas injection to the taper portion of the thrust foil bearing, boosting the hydrodynamic effect on the main bearing surface and hydrostatic load capacity at zero speed. The bearing has high load capacity, low power loss, and no friction/wear during startup and shutdown. Firstly, push-pull tests were conducted on a double-acting HFTB configuration to evaluate the nonlinear structural stiffness of the bump foil structure. The measured nonlinear stiffness model was adopted to predict the overall bearing performance under various speeds and loads. The simulation uses an advanced model which predicts the 2D plate top foil deflection with physical bump locations mapped from the actual hardware. The simulation confirms that the hybrid operation significantly increases the load capacity compared to hydrodynamic bearing due to the boost effect of hydrostatic pressure in the main film. Tests were conducted at 10krpm, 15krpm, and 20krpm to measure load capacity, power loss, and average film thickness. At 20krpm, a maximum axial load of 1711 N was achieved before bearing failure. Ultimately, the novel HFTB shows excellent static performance with potential for use in Organic Rankine Cycle (ORC) generators and other large oil-free turbomachines.

Dynamic characteristics of nonlinear vibration isolator for gas turbine

AbstractIn this study, a theoretical model of the vibration isolation system of the gas turbine is developed and numerically solved. A simplified finite element (FE) model was also established to determine the response under the shock load. The results of the FE model are used to verify the effectiveness of the theoretical model and the numerical solution. The influence of isolator stiffness, vibration isolator damping, and vibration isolator nonlinear stiffness coefficient on the shock response of the vibration isolation system is studied using the controlled‐variable method. These parameters (stiffness, damping, and nonlinear coefficient) enter into the shock resistance design of gas turbine vibration isolators.

An adjustable stiffness vibration isolator implemented by a semicircular ring

Nonlinear vibration isolators with quasi-zero-stiffness characteristics offer broadband vibration isolations, while passive quasi-zero-stiffness isolators are unable to adapt to variable working conditions. To address the issue, a nonlinear semi-active vibration isolator based on a semicircular ring structure is proposed. As an elastic component, the semicircular ring exhibits nonlinear stiffness under a vertical compression at the centre. With a semi-active control of the horizontal distance between the two ends of the semicircular ring, a pre-deformed state is induced, and the stiffness characteristics are adjusted accordingly. A chained beam-constraint model is adopted to characterize the pre-deformed process and the vertical compression process, respectively. The adjustable nonlinear restoring force is accurately calculated and adopted for vibration analysis. A harmonic balance method is used to analyze the frequency responses of the isolator under different working conditions. The semicircular ring exhibits softening stiffness under a vertical compression, and the quasi-zero-stiffness region can be effectively adjusted by the pre-deformed process. With the vertical compression reaching a threshold, negative stiffness is presented along with a snap-through phenomenon due to the bi-stability. With the semi-active pre-deformed, the isolator demonstrates different resonant frequencies and different risks in the occurrence of the snap-through during vibration. To avoid the snap-through-induced impact, the control threshold of the pre-deformed is determined, depending on the payload mass, the excitation amplitude, and the damping coefficient. Experiments are performed to validate the adjustable stiffness and the frequency responses of the isolator, and both the jump phenomenon and the snap-through phenomenon are observed.

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.

Nonlinear wire rope isolator with magnetic negative stiffness

The high-static-low-dynamic-stiffness (HSLDS) isolators have been widely employed to achieve low-frequency vibration isolation. However, traditional HSLDS isolators are prone to nonlinear jump phenomena due to the introduction of nonlinear stiffness characteristics, which will cause the performance deterioration under high excitation levels. To overcome this issue, inspired by the dry friction characteristic of wire ropes, the wire ropes are employed to design HSLDS isolators to suppress nonlinear jump phenomenon by introducing hysteretic damping in this paper. Therefore, a nonlinear wire rope isolator with magnetic negative stiffness is proposed to achieve HSLDS with hysteretic damping. The elastic force of wire rope and the magnetic force are modeled to obtain nonlinear restoring force, and then the dynamic model of the proposed isolator is established to investigate isolation performance. The equivalent beam element method is proposed to model wire ropes for obtaining dynamic characteristics by considering tensile and bending stiffnesses. Then, numerical simulation is performed to investigate the isolation performance of the wire rope isolators with different magnetic negative stiffnesses, and demonstrates that the proposed isolator can work in a higher excitation level than the isolator without dry friction of wire ropes. Moreover, experiments are carried out to show that the onset isolation frequency of the proposed isolator can be reduced by 66.7% compared with the wire rope isolator without magnetic negative stiffness. Therefore, it is demonstrated that the proposed wire rope isolator with magnetic negative stiffness has the preferable performance in low frequency isolation, and can improve the adaptability under a higher excitation level.

Research on Spatial Developable Mechanism Considering Revolute Clearance Joints with Irregular Rough Surfaces

Due to assembling, manufacturing errors, and wear, irregular rough surfaces inevitably exist in joints, which will increase the contact stiffness nonlinearity at the joint of the spatial developable mechanism, and the traditional contact force model is difficult to accurately predict the contact force change of irregular rough surface clearance. Aiming at the difficulty of accurately predicting the dynamic behavior of the spatial developable mechanism caused by rough joint clearance, an improved clearance contact force modeling method based on an uncoordinated contact model is proposed in this paper. The influence of rough peak distribution on the contact area is analyzed. The stiffness of the traditional Hertz model is modified based on the probability distribution density function of the rough peak, and an improved contact force model based on dynamic contact stiffness is established. Based on the Lagrange multiplier method, the dynamics model of spatial developable mechanism is constructed. Based on the model, the dynamic analysis of the spatial expansion mechanism is carried out, and the influence of the roughness of the contact surface and the size of the clearance on the dynamic characteristics of the system are explored; as well, the influence of different clearance parameters on the dynamic characteristics of the development mechanism is revealed. Through the analysis, this paper provides a theoretical basis for predicting the effect of a spatial revolute clearance joint on the dynamic characteristics of the mechanism and lays a good foundation for the manufacture and application of the mechanism.

Research on fastener lateral nonlinear stiffness characteristics and its impact on wheel-rail stick-slip vibration

Research on fastener lateral nonlinear stiffness characteristics and its impact on wheel-rail stick-slip vibration

Advances in elastomeric mount concepts for quasi-zero stiffness isolation

Quasi-zero stiffness (QZS) mounts can exploit geometric nonlinearity to minimize stiffness about an operating point which isolates vibrations from a supported source or receiver. Previous research on elastomeric QZS mounts featured 3D printed prototypes but did not consider unique material properties and fabrication artifacts that may impact the static and dynamic behavior of the mounts. Additionally, multi-axis properties and their influence on common, box-like systems supported by several mounts lacks detailed study. To overcome these limitations and improve the practicality of elastomeric QZS mechanisms, this paper discusses nonlinear stiffness, hysteresis, and fabrication issues with elastomeric materials in this context. Multi-material mount designs that improve strength and stability are considered, and their effectiveness for QZS isolation is experimentally evaluated. System-level dynamics are presented for a box-like load supported by QZS mounts. Comparisons between additive and cast materials in the fabrication of QZS mounts are discussed, and the mounts' dynamic stiffness amplification effect is examined for additive and molded materials.

A ring metastructure vibration isolator with thin beams

Conventional linear vibration isolators are difficult to have the function of high load-bearing capacity and low-frequency isolation, however nonlinear metastructure vibration isolators can effectively achieve this function. In this study, a new ring metastructure vibration isolator with thin beams was designed to have high static and low dynamic stiffnesses. We have accurately derived the static expression of these thin beams, obtained the stiffness characteristic and innovatively analyzed the vibration transmission characteristics of the ring isolator by deriving its dispersion equation. The influences of damping, frequency ratio, mass ratio, stiffness ratio, linear stiffness and nonlinear stiffness on the wave vector were obtained using the dispersion equation. Finally, a ring metastructure isolator sample with the high load-bearing capacity was designed and fabricated, and a vibration experiment was conducted to verify that the isolator has a lower vibration transmissibility in the target frequency band. This study not only demonstrates the feasibility of applying thin beams to nonlinear metastructure isolator design from mathematics, but also reveals profoundly its physical connotation from structural dynamics.

Multitarget adaptive virtual fixture based on task learning for hydraulic manipulator

AbstractHeavy‐duty construction tasks implemented by hydraulic manipulators are highly challenging due to unstructured hazardous environments. Considering many tasks have quasirepetitive features (such as cyclic material handling or excavation), a multitarget adaptive virtual fixture (MAVF) method by teleoperation‐based learning from demonstration is proposed to improve task efficiency and safety, by generating an online variable assistance force on the master. First, the demonstration trajectory of picking scattered materials is learned to extract its distribution and the nominal trajectory is generated. Then, the MAVF is established and adjusted online by a defined nonlinear variable stiffness and position deviation from the nominal trajectory. An energy tank is introduced to regulate the stiffness so that passivity and stability can be ensured. Taking the operation mode without virtual fixture (VF) assistance and with traditional weighted adaptation VF as comparisons, two groups of tests with and without time delay were carried out to validate the proposed method.

Effects of Matrix Cracking-Induced Delamination on Thermal Buckling of Composite Laminate

Abstract Delamination and matrix cracking are common damage modes in composite laminates, significantly impacting their structural integrity and performance. Under elevated temperatures, the collision of anisotropic thermal expansion and nonlinear stiffness responses affect the onset of thermal buckling in flying vehicles. This study implements a multi-scale analysis framework, incorporating finite element modeling and analytical methods to obtain the reduction in mechanical properties of the composite laminate and its stiffness degradation based on an effective stiffness model due to pre-existing matrix cracking in plies. Based on non-linear equations, obtained from the principle of virtual work, in combination with first-order shear deformable theory and von Kármán strain-displacement relations, the thermal buckling temperature, post-buckling behavior of composite laminates are investigated. This work paves the way for developing a model to predict the influence of the laminate stiffness, delamination size, transverse crack location, stacking sequence, and plies thickness on the laminate.

Self-excited vibration analysis of spline rotor systems considering parallel and angular misalignments

To address the instability phenomenon of aerospace splines at supercritical speed, the law of self-excited vibration (SEV) phenomenon of rotor systems considering parallel and angular misalignments is systematically investigated. The finite element model (FEM) of a rotor system is developed, incorporating spline internal friction damping. The spline is modeled by a nonlinear time-varying stiffness theory derived from the slicing method and the penalty function method. The parallel and angular misalignments are considered in the nonlinear time-varying stiffness model. The effects of the misalignment on critical speed (CS) and SEV of systems are analyzed. Moreover, the validity of the proposed spline rotor dynamic model is established through a comparison between simulation and experimental results.

Using statistical linearization in experiment design for identification of robotic manipulators

It is shown how nonlinear joint stiffness in industrial robots can be determined quickly and accurately through a combination of statistical linearization and optimized data acquisition configurations. The statistical linearization is carried out using the histogram of the measured motor torques. The result of this linearization is used in a criterion that is minimized to determine optimal configurations for data collection. The proposed approach is validated using data from both simulations and experiments with a medium-size industrial robot. In both cases, there is a significant improvement in accuracy compared to both using conventional linearization and collecting data in a larger but random set of configurations.

Enhancing seismic resilience of structures through optimally designed nonlinear negative stiffness base isolators: Exact closed-form expressions

Enhancing seismic resilience of structures through optimally designed nonlinear negative stiffness base isolators: Exact closed-form expressions

Improving quasi-zero-stiffness isolator performance: two hybrid isolators employing inerter elements

The pursuit of enhanced low-frequency vibration isolation motivates the development of quasi-zero-stiffness vibration isolators (QZS-VIs), yet with inherent limitations. The nonlinear hardening stiffness may distort the resonance peak rightward, narrowing the isolation bandwidth. Additionally, the high sensitivity to excitation variations can exacerbate the rightward distortion and jumping phenomenon. In this study, two types of hybrid isolators (QZS-IE isolators) are developed by integrating inerter elements (IEs) with traditional QZS-VI to enhance isolation performance. Their distinct mass block configurations yield varied enhancement behaviors. The unified dynamic equation for the two QZS-IE isolators is derived using harmonic balance and arc-length methods, validated by Runge-Kutta. The analyses of stability and periodicity clarify their response characteristics. Subsequently, the study investigates the effects of geometry and excitation parameters, comparing the enhancements of the two QZS-IE isolators. The results manifest that the introduction of inerter elements improves low-frequency vibration isolation, suppressing peak transmissibility and reducing isolation beginning frequency. The type I QZS-IE isolator outperforms the type II QZS-IE and traditional QZS isolators in low-frequency isolation performance and excitation sensitivity. However, the high-frequency isolation performance of the type I isolator is hindered by the ascending transmissibility curve post anti-resonant frequency. The type II QZS-IE isolator rectifies resonance peak distortion while avoiding adverse effects. The different configurations of the inerter elements and the related attenuating strategies offer a guideline for diversified vibration isolation objectives.

Forced vibration analysis of a beam-plate system coupled through multiple nonlinear single-freedom-degree systems

A beam-plate coupling system is widely used in the construction of different frameworks across industries according to engineering application requirements. In this paper, a nonlinear single-degree-of-freedom structure is introduced as a coupling element between the beam and the plate. Therefore, a coupling system connected through a plurality of the nonlinear single-degree-of-freedom structures is proposed with exposure of its operational mechanism and the dynamic behavior analysis as required. The nonlinear coupled governing equations of the beam-plate system with series nonlinear energy sinks are established according to the Hamilton principle and solvable by the Galerkin truncation method. Meanwhile, the nonlinear characteristics of transverse vibration of the coupling system are studied for discussing the effect of nonlinear stiffness on a low-frequency response of the beam-plate system. Further, the transverse vibration of the system may vary with changes in the nonlinear stiffness, the external viscous damping, and the motion mass of the nonlinear single-degree-of-freedom structures. Accordingly, the variations along with these factors are investigated. The results indicate that a reasonable nonlinear single-degree-of-freedom structure can have a significant influence on the transverse vibration characteristics of the beam-plate system.

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.