Nonlinear Stiffness Characteristics Research Articles

Skyhook Control Law Extension for Suspension with Nonlinear Spring Characteristics

This work aimed to improve the vehicle body stability and the ride comfort of the tracked military vehicle crew. For this purpose, magnetorheological fluid dampers were used. This process has made the theoretical model of the tracked platform a semi-active suspension system. This modification allows for the application of different control laws to these systems. The usage of the continuous skyhook control law assumes the influence of three fictitious viscous dampers. Their force in this conceptual model is replicated by the magnetorheological dampers of the suspension in the real system. However, the continuous skyhook control law does not take into consideration the nonlinear stiffness characteristics. In this paper, the continuous skyhook control law has been appropriately modified. The modification takes into consideration the nonlinearity of the stiffness characteristics. Applying the modified continuous skyhook control law improves the stability of the vehicle body and the vehicle crew’s ride comfort. All these goals had to be introduced due to the modernization of the tracked military vehicle suspension by replacing the torsion bars with spiral spring packages with nonlinear characteristics.

Experimental analysis of nonlinear characteristics of absorbers with wire rope isolators

Abstract Four different configurations of a mass supporting system consisting of a few prototypes of wire rope isolators are considered in the article. These are named “triangle,” “star,” “parallel,” and “serial” configurations. Reasonable models of a fixture assembly that could be utilized as models for cargo transportation are tested statically and dynamically in the laboratory. The main aim of this study is to investigate the impact of the configuration of damper settings on the effectiveness of vibration and shock isolation (in three directions) and to develop a suitable method to determine nonlinear properties and stiffness characteristics of the tested system. The purpose is also to investigate the sensitivity of the tested systems on the eccentricity of vertical loads and its influence on identification results. Experiments are carried out on the specially designed stand for the model parameters identification.

Quasi-zero stiffness isolator based on bistable structures with variable cross-section

Simple and light-weighted quasi-zero stiffness (QZS) isolators can be designed based on nonlinear and negative stiffness generated by the snap-through effect of bistable structures. Traditionally, the snap-through force of the bistable structure is limited which makes the weight which can be isolated based on this mechanism very low. This paper investigates increasing loading capacity of this kind of isolator by using an optimized and varying sectional profile. Numerical models were derived for the bistable structures with variable sectional distributions. Optimized sections’ alignment of the bistable beam was derived based on the numerical model which was consequently validated by experimental results. Influences of the bistable beams with a variable section on nonlinear stiffness characteristics and performance of the isolator were at last investigated with the harmonic balance method.

A variable positive-negative stiffness joint with low frequency vibration isolation performance

A new vibration isolation joint with combined positive and negative stiffness is proposed, which also supports multi-directional stiffness adjustment. Two sets of opposed buckling Euler beams are used as the variable stiffness structure. The effective length of the beam is changed by moving the slider, which in turn changes the structural stiffness. In the horizontal direction the structure provides adjustable positive stiffness. In the vertical direction it can provide adjustable negative or positive stiffness. A disk spring with nonlinear stiffness characteristics is used as the load-bearing spring and is connected in parallel with the variable stiffness structure. The static and dynamic characteristics show that the structure has good passive low-frequency vibration isolation performance. By actively moving the slider to change the structural stiffness, the equilibrium position of the system can be changed, active actuation can be performed, and active control of ultra-low frequency vibration isolation can be achieved.

Quantification and Modeling of Ankle Stiffness During Standing Balance.

This study investigates the factors contributing to the modulation of ankle stiffness during standing balance and evaluates the reliability of linear stiffness models. A dual-axis robotic platform and a visual feedback display were used to quantify ankle stiffness in both the sagittal and frontal planes while subjects controlled different levels of ankle muscle co-contraction, center-of-pressure (CoP), and loading on the ankle. Results of 40 subjects demonstrated that ankle stiffness in the sagittal plane linearly increased with the increasing level of these three factors. The linear model relating the change in these factors from the baseline measurements during quiet standing to the change in weight normalized ankle stiffness resulted in high reliability (R2 = 0.83). Ankle stiffness in the frontal plane increased with the increasing ankle muscle co-contraction and ankle loading, but the linearity was less obvious. It also exhibited a clear nonlinear trend when CoP was shifted mediolaterally. Consequently, the reliability of the linear model was low for ankle stiffness in the frontal plane (R2 = 0.37). During standing balance, ankle stiffness in the sagittal plane could be well explained by a linear model if ankle muscle activation, CoP, and ankle loading were collectively considered. However, the linear model cannot capture highly variable and nonlinear ankle stiffness characteristics in the frontal plane. The outcomes of this study could benefit the development of lower-extremity robots and their controllers. Furthermore, the ankle stiffness models could be used as a baseline in developing patient-specific ankle rehabilitation protocols.

Nonlinear dynamic force transmissibility of a flywheel rotor supported by angular contact ball bearings

The dynamic force transmissibility (DFT) of aerospace flywheel rotor system (FRS) supported by angular contact ball bearings (ACBBs) is examined in this paper. The influence of combined loads and contact angle variation is considered in the Sjovall formula to accurately solve the load distribution and nonlinear stiffness of ACBB. Subsequently, the lateral vibration model of FRS is established by considering the nonlinear stiffness characteristics of ACBB. The DFT of the system is solved via harmonic balance method and arc length continuation, and the stability of the results is determined. Numerical integration and dynamic tests are utilized to verify the accuracy of harmonic balance results. Based on the proposed model, the effects of rotor unbalance excitation, axial preload, and rotor damping on the DFT of the system are discussed. The soft-stiff transition phenomenon is observed in terms of the varying supporting stiffness of ACBB wherein deformation is measured under axial preload. The value of rotor unbalanced mass determines the nonlinear characteristics of FRS. The results provide an important reference for dynamic performance evaluation and vibration isolation device design of aerospace FRS.

A mapping method of dynamic response and stiffness characteristics for realizing a customized nonlinear oscillator

With nonlinear systems extensively studied for applications in vibration isolation, energy harvesting, etc., most of the studies focused on the response of the nonlinear oscillator defined by some explicit equations, namely the typical oscillator. Rare studies or approaches are devoted to the non-typical nonlinear oscillator for which the stiffness profile is possible defined by an arbitrary curve without an explicit function. This paper proposes a novel design method for both typical and non-typical monostable hardening oscillators, especially for the latter one in order to achieve some arbitrarily specified response. The key idea is to construct a series of linear equivalent oscillators to reproduce the response of the nonlinear oscillator. A trial stiffness characteristic expression of the linear equivalent oscillators has been given as a function of several particular variables. It has been verified that the linear equivalent oscillators can obtain the response for some typical nonlinear oscillators. Moreover, the method can be also used in a reversed way to identify the stiffness characteristics from a given response for either typical or non-typical systems. In particular, this parameter identification feature can be utilized to design a customized oscillator with arbitrarily specified hardening response. By constructing the linear equivalent oscillators to reproduce the desired response, the linear equivalent stiffness expression can be determined, while the customized nonlinear stiffness characteristics can be obtained accordingly. Experiments on an elaborated example have validated effectiveness of the proposed method. It shows that the mapping method can be used to determine the proper stiffness characteristics for customized hardening response beyond the scope of typical nonlinear oscillators.

Dynamic crack propagation in specimens with a surface irregularity

Initiation of cracks and their propagation in prescribed orientations on tubular specimens has been made possible on a dynamic fatigue testing rig developed at the University of Aberdeen. This rig was originally designed to perform experiments on single edge notched beams (SENB) [1,2]. Modifications have recently been made so as to accommodate experimental tests on tubular specimens with a range of sizes and other cross sections. Crack initiation at grooves on such specimens has been followed by crack growth. At the same time, lateral oscillations of cracked specimen have been measured, as well as accelerations of base excitation, masses above and below cracked specimen. Forces on these two masses have been observed by two load cells attached at positions close to specimen. These load cells facilitated the measurement of stresses in experiment. Crack length time histories have also been constructed by applying an alternating current potential difference (ACPD) method. Fatigue cracks were initiated at the pre-cut grooves in aluminium tubular specimens. Three specimens with different groove sizes were tested in fifteen individual experiments. A three-dimensional Finite Element model was established for each specimen so as to calculate the stress concentration factor (SCF). This formed the basis of determining the amplitude of forcing input made possible by an electromechanical shaker. The phase shifts of acquired time histories has provided some indication of energy transfer mechanism during fatigue and system dynamic response. Observations of cracked specimen displacement during experiments was compared against calculated displacement from elastic theory. Nonlinear responses were observed, suggesting nonlinear stiffness characteristics of the specimen due to discontinuities introduced by crack growth and plasticity effects. Comparison of damage combinations in experimental observation was made with predictions from BS EN 1999-1-3. All experimental observations of total damage combination were higher than predicted values by the latter code.

A spatial closed-form nonlinear stiffness model for sheet flexures based on a mixed variational principle including third-order effects

The sheet flexure is commonly used to provide support stiffness in flexure mechanisms for precision applications. While the sheet flexure is often analyzed in a simplified form, e.g. by assuming planar deformation or linearized stiffness, the deformation in practice is spatial and sufficiently large that nonlinear effects due to the geometric stiffness are significant.This paper presents a compact analytical model for the nonlinear stiffness characteristics of spatially deforming sheet flexures under general 3-D load conditions at moderate deformations. This model provides closed-form expressions in a mixed stiffness and compliance matrix format that is tailored to flexure mechanism analysis. The effects of bending, shear, elongation, torsion and warping deformation are taken into account, so that the stiffness in all directions, including the in-plane lateral support direction, is modeled accurately. The model is verified numerically against beam and shell-based finite elements. The approach for deriving closed-form solutions in a nonlinear context is detailed in this paper. The Hellinger–Reissner variational principle with a specific physically motivated set of low-order interpolation functions is shown to be well-suited to the geometrically nonlinear analysis of flexures.An extension of the derivation approach to the nonlinear closed-form analysis of general flexure mechanisms consisting of multiple sheet flexures connected in parallel is presented. This is demonstrated with the case of a spatially deforming parallelogram flexure mechanism and a cross-hinge flexure mechanism.

Acoustical characteristics of segmented plates with contact interfaces

The possibility of shifting sound energy from lower to higher frequency bands is investigated. The system configuration considered is a segmented structure having non-linear stiffness characteristics. It is proposed here that such a frequency-shifting mechanism could complement metamaterial concepts for mass-efficient sound barriers. The acoustical behavior of the material system was studied through a representative two-dimensional model consisting of a segmented plate with a contact interface. Multiple harmonic peaks were observed in response to a purely single frequency excitation, and the strength of the response was found to depend on the degree of non-linearity introduced. The lower and closer an excitation frequency was to the characteristic resonance frequencies of the base system, the stronger was the predicted higher harmonic response. The broadband sound transmission loss of these systems has also been calculated and the low frequency sound transmission loss was found to increase as the level of the broadband incident sound field increased. The present findings support the feasibility of designing material systems that transfer energy from lower frequency bands, where a sound barrier is less efficient, to higher bands where energy is more readily dissipated.

A nonlinear stiffness and nonlinear inertial vibration isolator

A nonlinear stiffness and nonlinear inertial vibration isolator is proposed in this article. It consists of an inerter, a damper, and spring elements. The nonlinear stiffness characteristic is achieved through a negative stiffness structure based on the diamond-shaped structure that generates nonlinear displacement terms. The nonlinear inertial characteristic is implemented via a geometrical nonlinear inerter that produces a nonlinear acceleration term and a nonlinear quadratic velocity term. An averaging method is developed to analyze approximately the dynamic response. The isolation performance is evaluated using four performance criteria: dynamic displacement peak, force transmissibility peak, isolation frequency band, and high-frequency force transmissibility. The effects of the nonlinear inertial characteristic on the dynamic response and isolation performance are examined. The results show that the backbone curve of the nonlinear stiffness and nonlinear inertial vibration isolator has two changing trends: bending first to the right and then to the left and bending directly to the left. The corresponding frequency response curve displays linear, hardening, and softening characteristics. The isolation performance of the nonlinear stiffness and nonlinear inertial vibration isolator is compared with that of the nonlinear stiffness and nonlinear stiffness and linear inertial ones. It could achieve a smaller force transmissibility peak, lower resonant frequency, and larger isolation frequency band than the nonlinear stiffness one and could have smaller dynamic displacement peak and smaller high-frequency force transmissibility than the nonlinear stiffness and linear inertial one. The proposed nonlinear stiffness and nonlinear inertial vibration isolator achieves a better integrated isolation performance among the three vibration isolators.

An improved quasi-zero stiffness vibration isolation system utilizing dry friction damping

Quasi-zero stiffness (QZS) isolators, a nonlinear vibration isolation technique, enhance the isolation performance, by lowering the natural frequencies of an isolation system while providing higher static load bearing capacity compared to similarly performing linear isolators. Despite its performance improvement, the challenge of implementing QZS isolation systems is due to their highly nonlinear stiffness characteristics as a result of cubic like behavior of stiffness elements used. Although increasing linear damping in the isolation alleviates the input dependency of the QZS isolation systems by reducing the resonance amplitudes, it results in increased transmissibility in the isolation region, which is an adverse effect. Therefore, in this study, in order to overcome this, dry friction damping is implemented on the QZS isolation system. Hysteresis loop for the new QZS dry friction element is obtained and a mathematical model is introduced. For the nonlinear isolation system, harmonic balance method is used to transform the nonlinear differential equations into a set of nonlinear algebraic equations. For single harmonic motion, analytical expressions of Fourier coefficients are obtained in terms of elliptic integrals. Numerical solution of the resulting set of nonlinear algebraic equations is obtained via Newton’s method with arc-length continuation. Performance of the isolation system under periodic base excitation is studied for different base excitation levels and the stability of the periodic steady-state solutions is investigated.

An Output Force Control for Robotic Manipulator by Changing the Spring Stiffness

This paper presents an output control for a manipulator by changing the spring stiffness. Through the modeling and analysis of the nonlinear stiffness characteristics of the crank-rocker mechanism, and using the zero stiffness domain search method to select the appropriate spring stiffness, using different spring stiffness to establish different mechanism models, the robot can finally control the output of ideal constant force, and at the same time, the analysis results are applied to the improved design of the tire grabbing manipulator. Through this method, the tire grabbing manipulator becomes a constant grabbing force mechanism, and the mechanism is transformed from a rigid-body mechanism to a pseudo-rigid-body mechanism. The accuracy and stability of the whole system are greatly improved. In this study, the method of adding spring to each joint of the linkage mechanism is applied to the improvement design of the linkage mechanism, and the four-bar constant force mechanism is designed for the first time, which expands the application field of the nonlinear stiffness characteristics of the linkage mechanism, and has great application value to the improvement design of the mechanical system with the linkage mechanism and the control of the output force.

Large-scale testing of a hydraulic non-linear mooring system for floating offshore wind turbines

The mooring system has been recognised as a key area of expense that needs to be addressed to improve the cost competitiveness of floating offshore wind turbines. The devices installed to date have generally adopted designs from the oil and gas industry using heavy mooring materials, providing the required safety margins but with a significant degree of conservatism. Recent interest in the usage of lighter and more compliant mooring materials has shown that they have the potential to reduce peak line loads, which would in-turn reduce costs. However, the lack of operational experience with such materials has limited their adoption in a risk averse industry. This paper reports on the large-scale physical testing of a hydraulic-based mooring component with non-linear stiffness characteristics. The performance of the device is characterised in a laboratory both statically and dynamically, as well as in conditions representative of operating in a sea state using a combined physical and numerical modelling approach. The results show that the dynamic stiffness of the component is a function of load history and hydraulic pre-charge pressure, while the inclusion of the device as part of the OC4 semi-submersible floating wind platform can reduce the peak mooring line loads by up to 9%. Beyond the physical test results, the calculations suggest that the peak load reduction in the modelled scenarios could be as much as 40% if the device can be scaled further. The paper supports the adoption of innovative mooring systems through dedicated component and performance testing.

Learning to Walk a Tripod Mobile Robot Using Nonlinear Soft Vibration Actuators With Entropy Adaptive Reinforcement Learning

Soft mobile robots have shown great potential in unstructured and confined environments by taking advantage of their excellent adaptability and high dexterity. However, there are several issues to be addressed, such as actuating speeds and controllability, in soft robots. In this letter, a new vibration actuator is proposed using the nonlinear stiffness characteristic of a hyperelastic material, which creates continuous vibration of the actuator. By integrating three proposed actuators, we also present an advanced soft mobile robot with high degrees of freedom of movement. However, since the dynamic model of the soft mobile robot is generally hard to obtain(intractable), it is difficult to design a controller for the robot. In this regard, we present a method to train a controller, using a novel reinforcement learning (RL) algorithm called adaptive soft actor-critic (ASAC). ASAC gradually reduces a parameter called an entropy temperature, which regulates the entropy of the control policy. In this way, the proposed method can narrow down the search space during training, and reduce the duration of demanding data collection processes in real-world experiments. For the verification of the robustness and the controllability of our robot and the RL algorithm, experiments for zig-zagging path tracking and obstacle avoidance were conducted, and the robot successfully finished the missions with only an hour of training time.

Predicting Nonlinear Stiffness, Motion Range, and Load-Bearing Capability of Leaf-Type Isosceles-Trapezoidal Flexural Pivot Using Comprehensive Elliptic Integral Solution

A leaf-type isosceles-trapezoidal flexural (LITF) pivot consists of two leaf springs that are situated in the same plane and intersect at a virtual center of motion outside the pivot. The LITF pivot offers many advantages, including large rotation range and monolithic structure. Each leaf spring of a LITF pivot subject to end loads is deflected into an S-shaped configuration carrying one or two inflection points, which is quite difficult to model. The kinetostatic characteristics of the LITF pivot are precisely modeled using the comprehensive elliptic integral solution for the large-deflection problem derived in our previous work, and the strength-checking method is further presented. Two cases are employed to verify the accuracy of the model. The deflected shapes and nonlinear stiffness characteristics within the range of the yield strength are discussed. The load-bearing capability and motion range of the pivot are proposed. The nonlinear finite element results validate the effectiveness and accuracy of the proposed model for LITF pivots.

Spring Effects on Workspace and Stiffness of a Symmetrical Cable-Driven Hybrid Joint

This paper aims to investigate how to determine the basic parameters of the helical compression spring which supports a symmetrical cable-driven hybrid joint (CDHJ) towards the elbow joint of wheelchair-mounted robotic manipulator. The joint design of wheelchair-mounted robotic manipulator needs to consider lightweight but robust, workspace requirements, and variable stiffness elements, so we propose a CDHJ which becomes a variable stiffness joint due the spring under bending and compression provides nonlinear stiffness characteristics. Intuitively, different springs will make the workspace and stiffness of CDHJ different, so we focus on studying the spring effects on workspace and stiffness of CDHJ for its preliminary design. The key to workspace and stiffness analysis of CDHJ is the cable tension, the key to calculate the cable tension is the lateral bending and compression spring model. The spring model is based on Castigliano’s theorem to obtain the relationship between spring force and displacement. The simulation results verify the correctness of the proposed spring model, and show that the spring, with properly chosen parameters, can increase the workspace of CDHJ whose stiffness also can be adjusted to meet the specified design requirements. Then, the modelling method can be extended to other cable-driven mechanism with a flexible compression spring.

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.

Dynamic and Stability Analysis of Multibolt Plane Joints under Normal Forces

In this paper, a stiffness model of contact surfaces based on a modified three-dimensional fractal contact model is built, which is in accordance with the experiment results. Additionally, the static, dynamic, and stable behaviors of the bolt joint between the spindle box and the machine bed are analyzed. The mathematical relationship between fractal parameters of the surface topography and the stiffness of the system was established to accurately study its static behaviors. Asymmetric curves are observed from the load–deflection results and the nonlinear stiffness characteristic is also presented. It is shown that both the stress and the stiffness increase with the increase of the displacement near the static equilibrium position. Meanwhile, a simplified model without the consideration of roughness is compared with joint interfaces composed from milling, scraping, and grinding surfaces. Numerical calculation was employed to investigate effects of design parameters on the system under harmonic excitation when the processing method, excitation force, bolt pre-tightening force, topography parameters, and other structural parameters, i.e., nominal contact area, joint thickness and bolt number, are eventually regarded as the control parameters. The aim of the article is to analysis the influence of these parameters, including surface morphology, on nonlinear characteristics of the bolt interface with fractal contact surfaces andto provide some references to improve the characteristics.

Nonlinear Static and Dynamic Stiffness Characteristics of Support Hydraulic System of TBM

Full-face hard rock tunnel boring machines (TBM) are essential equipment in highway and railway tunnel engineering construction. During the tunneling process, TBM have serious vibrations, which can damage some of its key components. The support system,an important part of TBM, is one path through which vibrational energy from the cutter head is transmitted. To reduce the vibration of support systems of TBM during the excavation process, based on the structural features of the support hydraulic system, a nonlinear dynamical model of support hydraulic systems of TBM is established. The influences of the component structure parameters and operating conditions parameters on the stiffness characteristics of the support hydraulic system are analyzed. The analysis results indicate that the static stiffness of the support hydraulic system consists of an increase stage, stable stage and decrease stage. The static stiffness value increases with an increase in the clearances. The pre-compression length of the spring in the relief valve affects the range of the stable stage of the static stiffness, and it does not affect the static stiffness value. The dynamic stiffness of the support hydraulic system consists of a U-shape and reverse U-shape. The bottom value of the U-shape increases with the amplitude and frequency of the external force acting on the cylinder body, however, the top value of the reverse U-shape remains constant. This study instructs how to design the support hydraulic system of TBM.