Linear Stiffness Research Articles

Bioinspired multilayer braided structure with controllable nonlinear mechanical properties for artificial ligaments

Inspired by “crimped” collagen fiber, which induce a low-level load region within the hierarchical structures of ligaments and tendons, this study introduces a braiding technology aiming at characterizing nonlinear mechanical behavior. Samples of the developed artificial ligament with a multilayer braided structure were fabricated and tested. Alterations in the number of braiding layers, wales, and courses significantly affect the mechanical properties of the artificial ligaments. Specifically, the toe length exhibited an increase of up to 104.58 %, while the linear stiffness exhibited variations concomitant with changes in these parameters. Additionally, a mathematical model to describe the relationship between the toe length and linear stiffness was developed based on the braiding parameters and material properties of the fibers. Alternative fibers were employed and tested to demonstrate a consistency with the mathematical model. The correlation coefficients were calculated as 0.9644 for the toe length and 0.9203 for the linear stiffness. Lastly, this study mimics the mechanical behavior of the human medial collateral ligament (MCL) by integrating the artificial ligament into a bio-fidelity knee model with a joint stiffness of 1.77 Nm/deg, resulting in a high degree of similarity with human knee in valgus-varus, further affirming the utility and applicability of this study.

Stress-driven generative design and numerical assessment of customized additive manufactured lattice structures

The rise of additive manufacturing (AM) has positioned lattice infilling as a pivotal strategy for creating lightweight, customized engineering components. This study presents a generative method that enables the conformal design and stiffness prediction of complex gradient strut-node lattice structures. A stress-driven Multi-Agent System (MAS) is introduced for the parametric optimization of lattice material distribution, incorporating geometric limitations, stress factors, and AM constraints. A beam element model simplifies the numerical analysis of the structure linear stiffness. By applying the Response Surface Method (RSM), a numerical model is established, not only conducting a quantitative analysis on the sensitivity of MAS design variables but predicting mechanical performance. This method is validated by designing a supporting component, demonstrating that the optimized lattice design can achieve a linear stiffness 1.4 times greater than that of conventional uniform lattice infills for the same mass. This research provides a comprehensive framework for the efficient design and analysis of irregular lattice structures at a macroscopic scale.

Research on Dynamic Characteristic Coefficients of Integral Squeeze Film Damper

Integral squeeze film damper (ISFD) is a new type of structure that appeared on the basis of traditional squeeze film damper (SFD). Since the oil film in ISFD is a segmented structure without annular flow, the nonlinearity of the oil film force has been improved to a great extent. The dynamic characteristic coefficients of ISFD have a close relationship with its damping performance. This work investigates and studies the dynamic characteristic parameters of ISFD by means of numerical analysis and experimental validation techniques in order to examine the dynamic features and unveil the damping mechanism. The ISFD solid and fluid analysis models are created, and the computational fluid dynamics (CFD) and mechanical performance analyses are completed. The force acting on the ISFD’s S-type elastomer under excitation conditions is revealed in the mechanical property analysis, and the flow characteristics of the internal oil film are investigated in the CFD analysis. It is discovered that the ISFD has good linear damping and stiffness characteristics, and numerical analytical values for the ISFD’s damping and stiffness coefficients are obtained. By constructing a bi-directional excitation test rig, the experimental values of the ISFD stiffness coefficient and damping coefficient are determined. These values are in close agreement with the results of the numerical analysis, confirming the accuracy of the ISFD’s numerical analysis conclusions.

A study of controlling the transverse vibration of a beam-plate system by utilizing a nonlinear coupling oscillator

In the engineering field, prolonged exposure of complex structures to high-strength vibration environments can lead to various engineering challenges. This underscores the importance of effectively managing the vibration of complex structures. Given the extensive utilization of beams and plates in the construction of intricate structures, this study incorporates a nonlinear coupling oscillator into the beam-plate system and formulates a vibration analysis model for this system. The governing equations of the system are derived theoretically and solved numerically using the Galerkin truncation method. This study focuses on the operational modes of the nonlinear coupling oscillator based on accurate numerical results. Meanwhile, the influence of parameters belonging to the nonlinear coupling oscillator on the transverse vibration responses of the beam-plate system is systemically studied. Upon meticulous examination of the numerical results, the operational states of the nonlinear coupling oscillator encompass the normal vibration suppression mode and the quasi-periodic vibration suppression mode. The alteration in the linear elastic coupling stiffness of the beam-plate system impacts the effectiveness of vibration suppression in the nonlinear coupling oscillator. This influence stems from the fact that changes in the linear elastic coupling stiffness directly affect the nonlinear forces exerted on the beam-plate system within a nonlinear coupling oscillator. Within a feasible range, the nonlinear stiffness and viscous damping can be chosen as the adjustable parameters for the nonlinear coupling oscillator to regulate the vibration of the beam-plate system by adjusting the motion mass of the nonlinear coupling oscillator, which presents a challenge. The appropriate utilization of the nonlinear coupling oscillator demonstrates favorable control effectiveness in managing the transverse vibration of the beam-plate system. The study presented in this work offers a method to simultaneously control the vibration of each component within beam-plate coupling systems.

Stiffness Mediated-Mechanosensation of Airway Smooth Muscle Cells on Linear Stiffness Gradient Hydrogels.

In obstructive airway diseases such as asthma and chronic obstructive pulmonary disease (COPD), the extracellular matrix (ECM) protein amount and composition of the airway smooth muscle (ASM) is often remodelled, likely altering tissue stiffness. The underlying mechanism of how human ASM cell (hASMC) mechanosenses the aberrant microenvironment is not well understood. Physiological stiffnesses of the ASM were measured by uniaxial compression tester using porcine ASM layers under 0, 5 and 10% longitudinal stretch above in situ length. Linear stiffness gradient hydrogels (230 kPa range) were fabricated and functionalized with ECM proteins, collagen I (ColI), fibronectin (Fn) and laminin (Ln), to recapitulate the above-measured range of stiffnesses. Overall, hASMC mechanosensation exhibited a clear correlation with the underlying hydrogel stiffness. Cell size, nuclear size and contractile marker alpha-smooth muscle actin (αSMA) expression showed a strong correlation to substrate stiffness. Mechanosensation, assessed by Lamin-A intensity and nuc/cyto YAP, exhibited stiffness-mediated behaviour only on ColI and Fn-coated hydrogels. Inhibition studies using blebbistatin or Y27632 attenuated most mechanotransduction-derived cell morphological responses, αSMA and Lamin-A expression and nuc/cyto YAP (blebbistatin only). This study highlights the interplay and complexities between stiffness and ECM protein type on hASMC mechanosensation, relevant to airway remodelling in obstructive airway diseases.

Vibrational Analysis of Axially Loaded Rayleigh Beams on Variable Winkler Foundations with Implications for Structural Design

In this study, we explore the complex vibration problem of a Rayleigh beam under axial loads, positioned over a variable Winkler foundation with elastic properties changing along its length. The investigation delves into how factors like rotational inertia, axial forces, and foundation properties affect the beam’s dimensionless eigenvalues and eigenmodes. Employing the Differential Transform Method (DTM), we tackle the governing equations across various foundation conditions, including constant, linear, and parabolic variations. The results align well with the previous research, affirming their precision between scenarios considering or neglecting the elastic foundation’s effects. Notably, we emphasize the influence of linear spring stiffness on vibration characteristics, overshadowing rotary spring stiffness. Thus, the important novelty of the study lies in its comprehensive investigation into the vibration dynamics of a Rayleigh beam under axial loads on a variable Winkler foundation with elastic properties varying along its length. The findings illuminate a previously unexplored dynamic: the intricate interplay between linear and rotary spring stiffness, showcasing the dominant impact of linear stiffness on vibration characteristics. Of significant note is the profound effect of axial forces on fundamental eigenvalues, underscoring their critical role in structural behavior. These developments possess substantial practical implications, providing invaluable guidance for the accurate modeling of beams supported by elastic foundation, thereby enhancing structural design and performance.

An Investigation on the Seismic Behavior of the End-Plate Connection between a Steel Beam and the Weak-Axis of an H-Shaped Column Using a U-Shaped Connector

The seismic behavior of the end-plate connections between a steel beam and the weak axis of the H-shaped steel column using a U-shaped connector was investigated using numerical analysis. Finite element (FE) models were established using ABAQUS 6.14 software, and the applicability of the modeling approach was verified by comparing the numerical results with the relevant experimental results. This parametric study of the joint was carried out to analyze the effects of the thickness of the U-shaped connectors, the thickness of the end-plates, the axial compression ratio of the columns, and the linear stiffness ratio of the beam to the column. The results show that the U-shaped connector set in the weak axis of the H-shaped column can form a box-shaped panel zone with the column flange and web. The volume of the panel zone and its resistance to shear deformation are increased through this connection. Finally, the recommended reasonable ranges for the thickness of the U-shaped connector, the thickness of the end-plates, the axial compression ratio of the columns, and the linear stiffness ratio of the beam to the column are proposed in this paper.

Numerical simulation on the load-bearing capacity of CFRP-strengthened concrete arches

To investigate the bearing capacity of concrete structures strengthened by carbon fiber reinforced composite (CFRP), this study conducted numerical simulation research on seven CFRP-strengthened concrete arch structures with five different strengthening schemes. The finite element model was established by HYPERMESH, the simulation results were compared with the experimental results to verify the accuracy of the model. Based on this, the interfacial bonding performance of concrete and CFRP under different strengthening schemes was analyzed, and the static damage mode of the reinforced concrete arch structure was revealed. The results indicate that the application of CFRP to the surface of a concrete arch will increase the linear segmental stiffness, cracking load, and ultimate load of the structure. The five strengthening schemes in this paper provide enhancements ranging from 2.9 % to 57.6 % for concrete arch cracking loads, from 3.3 % to 14.7 % for ultimate loads, and from 1.1 % to 17.6 % for stiffness. If both the inner and outer sides are strengthened with CFRP, the ultimate bearing capacity of the arch structure can be significantly enhanced. Circumferential wrapping of CFRP can delay the concrete cracking and further enhance the bearing capacity of the arch, and the strengthening effect can be maximized. This study can provide a valuable reference for the strengthening of old concrete arch structures and the construction of newly assembled arch tunnels.

Development of novel two-level SMA-based self-centring steel columns for seismic resilience

This paper presents an innovative two-level shape memory alloy (SMA)-based self-centring (SC) steel column, referred to as TSSSC. The TSSSC is interconnected by two series of SMA bolts, differing in diameters and lengths, to achieve multi-yielding. This is intended to fulfil multiple performance-based design objectives and augment the structural ductility of SMA elements. Utilizing the distinctive material property of phase transformation inherent in SMA, the two-level bolts can accomplish multi-level yielding in a straightforward yet effective manner. This is achieved by employing SMA elements of varying diameters in series and sequentially activating them via the material's inherent stiffness hardening. The behaviours of the TSSSC are meticulously analysed through refined finite element models (FEMs), which are corroborated by previous test results. The TSSSC demonstrates commendable multi-yielding behaviour and SC capacity during cyclic behaviour studies. In comparison to conventional SMA-based SC structures, the TSSSC exhibits markedly superior ductility, exceeding 10, whereas traditional structures typically range between 4 and 6. This enhanced ductility ensures that the TSSSC can comprehensively meet the multiple design objectives stipulated in the current performance-based seismic design. The SC and energy dissipation (ED) capacity of the TSSSC can be activated during frequent seismic events, thereby providing a safety buffer for severe earthquakes due to seismic uncertainty. Furthermore, parametric studies are conducted to identify critical design parameters, including the vertical design load, prestress in SMA bolts, and the linear stiffness ratio between the short and long bolts of the two-level SMA bolts. Design recommendations are subsequently proposed based on the findings of these parametric studies.

Vibration Suppression of Piecewise-Linear Stiffness Nes System Under Random Excitation

Nonlinear energy sinks (NESs) are critically important for structural vibration suppression. They can absorb vibrational energy across a broad frequency spectrum, possess strong robustness, and have a relatively small mass. This study addresses the vibration suppression in a piecewise linear stiffness NES system under random excitation. Initially, a theoretical model of the piecewise linear stiffness NES system is developed. The piecewise linear stiffness function is approximated using Legendre polynomial approximation. Following this, the steady-state Fokker–Planck–Kolmogorov (FPK) equation of the system is formulated via the Generalized Harmonic Function Method. The FPK equation is solved using the fourth-order central difference method, and the effectiveness of this approach is validated by comparing the FDM results with numerical simulations. Lastly, the influence of varying system parameters on the stability of the piecewise linear stiffness NES system is analyzed.

Rich phenomenology of the solutions in a fractional Duffing equation

In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to 0 or α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

Coupled behavior and seismic performance assessment of square-over-round RCFDST column with reinforced ribs to steel beam joints under cyclic loading

The cyclic loading test study and finite element analysis (FEA) were conducted to investigate the seismic performance of five square-over-round recycled concrete-filled double skin steel tubular (RCFDST) columns with reinforced ribs to steel beam joints varying parameters such as the length of ribs, hollow ratio, axial compression ratio, the ratio of beam-column flexural linear stiffness and strength of the recycled concrete. It was found that a larger the displacement resulted in a lower the strength degradation. Moreover, the structures with two types of concrete showed similarities in strength degradation. In the experiment, compared to the cases without reinforced ribs, the specimens with reinforced ribs exhibited a larger value of stiffness and related rigidity degeneration primarily, and then showed a slow trend in rigidity degeneration laterally. The ductility of the cases with recycled concrete is twice that of the cases with normal concrete. In addition, the FEA model with optimal β of 0.5 agreed with the experimental results in both their damage patterns and horizontal load-displacement hysteresis curves. According the ratio of performance to cost, the optimal ratio of beam-column flexural line stiffness (km) is found in a range from 1.0 to 1.5. In a whole, the joints with reinforced ribs show generally excellent seismic performance.

Exact periodic solution of the undamped Helmholtz oscillator subject to a constant force and arbitrary initial conditions

The Helmholtz oscillator is a nonlinear mixed‐parity oscillator that models the asymmetric vibrations of many engineering and scientific systems. This paper investigated a general and completely integrable form of the Helmholtz oscillator and derived its exact periodic solution from the first integral of the governing differential equation. The considered Helmholtz oscillator has general linear and nonlinear stiffness constants, is subject to a constant force, and has arbitrary initial conditions. Its exact time period was derived in terms of the complete elliptic integral of the first kind, while the exact displacement was derived in terms of the Jacobi elliptic sine function. The validity of the exact solutions was verified for various combinations of the system parameters and initial conditions by comparing with numerical solutions. The exact solutions and numerical solutions were found to match perfectly. Furthermore, the exact solution was applied to analyze the vibration response of real‐world systems that could be modeled as Helmholtz oscillators. The present exact solutions provide benchmark solutions that could be used to determine the accuracy of new and existing approximate solutions for the Helmholtz oscillator.

Linear and nonlinear stiffness compensation for low-frequency vibration isolation: a comparative study

Linear and nonlinear stiffness compensation for low-frequency vibration isolation: a comparative study

Adaptive MPC-Based Lateral Path-Tracking Control for Automatic Vehicles

Adaptive MPC-Based Lateral Path-Tracking Control for Automatic Vehicles

The spring stiffness profile within a passive, full-leg exoskeleton affects lower-limb joint mechanics while hopping.

Passive, full-leg exoskeletons that act in parallel with the legs can reduce the metabolic power of bouncing gaits like hopping. However, the magnitude of metabolic power reduction depends on the spring stiffness profile of the exoskeleton and is presumably affected by how users adapt their lower-limb joint mechanics. We determined the effects of using a passive, full-leg exoskeleton with degressive (DG), linear (LN) and progressive (PG) stiffness springs on lower-limb joint kinematics and kinetics during stationary, bilateral hopping at 2.4 Hz. We found that the use of a passive, full-leg exoskeleton primarily reduced the muscle-tendon units (MTUs) contribution to overall joint moment and power at the ankle, followed by the knee, due to the average exoskeleton moment arm around each joint. The greatest reductions occurred with DG springs, followed by LN and PG stiffness springs, probably due to differences in elastic energy return. Moreover, the relative distribution of positive joint power remained unchanged when using a passive, full-leg exoskeleton compared with unassisted hopping. Passive, full-leg exoskeletons simultaneously assist multiple lower-limb joints and future assistive devices should consider the effects of spring stiffness profile in their design.

Effects of adipose-derived cell supplementation on tendon-bone healing in a rat model of chronic rotator cuff tear with suprascapular nerve injury.

To investigate the effect of adipose-derived cells (ADCs) on tendon-bone healing in a rat model of chronic rotator cuff tear (RCT) with suprascapular nerve (SN) injury. Adult rats underwent right shoulder surgery whereby the supraspinatus was detached, and SN injury was induced. ADCs were cultured from the animals' abdominal fat. At 6 weeks post-surgery, the animals underwent surgical tendon repair; the ADC (+ve) group (n = 18) received an ADC injection, and the ADC (-ve) group (n = 18) received a saline injection. Shoulders were harvested at 10, 14, and 18 weeks and underwent histological, fluorescent, and biomechanical analyses. In the ADC (+ve) group, a firm enthesis, including dense mature fibrocartilage and well-aligned cells, were observed in the bone-tendon junction and fatty infiltration was less than in the ADC (-ve) group. Mean maximum stress and linear stiffness was greater in the ADC (+ve) compared with the ADC (-ve) group at 18 weeks. ADC supplementation showed a positive effect on tendon-bone healing in a rat model of chronic RCT with accompanying SN injury. Therefore, ADC injection may possibly accelerate recovery in massive RCT injuries.

The effect of trapeziometacarpal joint passive stiffness on mechanical loadings of cartilages

Hypermobility of the trapeziometacarpal joint is commonly considered to be a potential risk factor for osteoarthritis. Nevertheless, the results remain controversial due to a lack of quantitative validation. The objective of this study was to evaluate the effect of joint laxity on the mechanical loadings of cartilage.A patient-specific finite element model of trapeziometacarpal joint passive stiffness was developed. The joint passive stiffness was modeled by creating linear springs all around the joint. The linear spring stiffness was determined by using an optimization process to fit force–displacement data measured during laxity tests performed on eight healthy volunteers. The estimated passive stiffness parameters were then included in a full thumb finite element simulation of a pinch grip task driven by muscle forces to evaluate the effect on trapeziometacarpal loading. The correlation between stiffness and the loading of cartilage in terms of joint contact pressure and maximum shear strain was analyzed.A significant negative correlation was found between the trapeziometacarpal joint passive stiffness and the contact pressure on trapezium cartilage during the simulated pinch grip task.These results therefore suggest that the hypermobility of the trapeziometacarpal joint could affect the contact pressure on trapezium cartilage and support the existence of an increased risk associated with hypermobility.

Nonlinear Vibration and Stiffness Characteristics Analysis of Maglev Train Based on Cubic Displacement Control

A control strategy combining cubic displacement feedback nonlinear control and proportional-differential (PD) linear control is used to control the vibration performance of the maglev system. The maglev system is divided into positive stiffness maglev system, quasi-zero stiffness maglev system and negative stiffness maglev system according to the linear stiffness value of maglev system. Firstly, an improved multi-scale method is used to analyze the vibration characteristics of suspension in the positive stiffness state of the maglev system. Secondly, the influence of control parameters on train vibration amplitude and vibration center displacement under quasi-zero stiffness is studied. Finally, the vibration characteristics of the train when the maglev system is in negative stiffness are analyzed by numerical simulation. The maglev system exhibits the worst vibration performance under negative stiffness compared with positive stiffness and quasi-zero stiffness. The suspension frame is easy to enter the chaotic motion state, and its vibration center is easy to deviate from the equilibrium position and produce large displacement when the maglev system is in the negative stiffness state. The control results show that the control strategy combining the cubic displacement feedback nonlinear control with the PD linear control can make the maglev system exhibit better vibration characteristics under positive and quasi-zero stiffness.