Linear Stiffness Research Articles

Symmetries and symmetry-generated averages of elastic constants up to the sixth order of nonlinearity for all crystal classes, isotropy and transverse isotropy.

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non-crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low-symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank-deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database.

Stochastic optimization of a mass-in-mass cell with piecewise hybrid nonlinear–linear restoring force

This article investigates the optimization under uncertainty of a mass-in-mass meta-cell for its potential use within a metamaterial. The specificity of the proposed mass-in-mass system stems from the hybrid nonlinear–linear stiffness at the inner level. It is well known that these systems are highly sensitivity to small perturbations in loading conditions or design parameters. In fact, the sensitivity is such that the system can exhibit discontinuous behaviors. Therefore the proposed optimization approach not only accounts for sources of uncertainties but also can handle discontinuous responses. The objective of the stochastic optimization is to find the stiffness properties of the mass-in-mass system which minimize the expected value of a specific efficiency metric. In order to better understand the system’s dynamic behavior and the origins of the discontinuities, slow invariant manifolds and frequency response curves are provided. The efficiency of the optimized system with hybrid stiffness is compared with that of a similar optimized system featuring pure cubic nonlinearity.

Theoretical and experimental investigation of flexible air spring stiffness in a tuned liquid column gas damper for vertical vibration control

This study investigates the effectiveness of equivalent linear air spring stiffness in a Vertical Tuned Liquid Column Gas Damper (VTLCGD) for reducing vertical structural vibrations, particularly considering different sealing conditions. The VTLCGD system, designed with a single sealed vertical air column, allows flexible frequency tuning by adjusting the air spring stiffness. It aims to be used in low-frequency structures where conventional VTLCGD designs with two sealed ends are less efficient. The geometric configuration and working principle of the VTLCGD are first described. The equation of motion is derived using liquid dynamic equilibrium and the pressure-volume relationship, with expressions for natural frequency and control force validated through shaking table tests conducted with varying VTLCGD lengths. Experimental investigations are conducted to examine the parameters affecting damper performance using pressure data. The system's vibration reduction performance is then numerically evaluated on a cantilevered floor subjected to human walking. The numerical results, based on the equation of motion for liquid displacement and natural frequency, show good agreement with experimental data, which confirms the effectiveness of using linearized air spring stiffness for frequency tuning. The effects of total liquid length and height difference on control force are further investigated, and a polytropic index of 1.2 is determined for the VTLCGD frequency formula. The VTLCGD system achieves a maximum vibration reduction of 52.63 % in acceleration response on the cantilevered floor compared to the uncontrolled case. Moreover, the variation in stiffness due to changes in the sealing condition is presented to validate the damper's adaptability over a wide frequency range.

Nonlinear normal modes and response to random inputs of systems with bilinear stiffness

This work seeks to provide a comprehensive review of the effects that stiffness bilinearity can have on the nonlinear modes of a system, its response in a random vibration environment, and the connection between the two. Stiffness bilinearity here refers to a continuous piecewise linear force vs displacement function that is composed of two linear regions. This work focuses on a bilinear stiffness function that is regularized so there is a smooth transition at the point where the two linear regions meet. A single-degree-of-freedom system (SDOF) and a two-degree-of-freedom (2DOF) system are explored and several interesting behaviors are shown. The SDOF bilinear spring model is characterized by four parameters: the low amplitude frequency, the ratio of the linear stiffnesses on either side of the transition, the displacement at which the transition occurs, and the rate or sharpness of the transition. The effect of each parameter on the shape of the NNM is described. In a 2DOF system, these parameters have similar effects, but modal coupling is found to play a significant role. When a bilinear system is subjected to random excitation, many harmonics appear in the response for both the SDOF and 2DOF cases. The root-mean-square (RMS) response of the bilinear system can be larger or smaller than the corresponding linear case depending on the values of the parameters and the type of forcing (broadband, bandlimited, in the shape of a vibration mode, etc.). However, many cases were observed in which the RMS response of the bilinear system was almost the same as that of a linear system, and hence the response could be predicted well using linear analysis. It is hoped that the results presented herein can assist engineers in helping to determine when linear analysis would be adequate to predict the failure of a system, when a rigorous nonlinear analysis is required, and what phenomena are likely to be observed in the latter case.

A noninvasive measurement technique for the initial stiffness of the radial artery

Arterial stiffness carries significant implications for cardiovascular disease. Monitoring changes in arterial stiffness is integral to proactive health management, however, current noninvasive methods of quantifying stiffness in vivo rely primarily on linear tangent stiffness, making the measurements vulnerable to the variability of blood pressure and thereby affecting the accuracy in portraying the health status of the arteries. This study proposed a novel methodology for evaluating arterial stiffness that is unaffected by changes in blood pressure. Ultrasound detection techniques are applied to accurately chronicle changes in radial artery diameters across varied blood pressures. Incorporating blood pressure measurements, the initial diameter at cuff blockade, and vessel diameters at systolic and diastolic pressures enables inverse determination of the unstressed initial radial artery stiffness. This method accurately mirrors the results of in vitro experiments employing porcine blood vessels at physiological pressures. The results underscore the technique's ability to quantify arterial mechanical properties precisely. This study offers a groundbreaking strategy for fostering the early detection of atherosclerosis, and aiding artery health regulation.

Rapid design method for composite emergency steel piers with thin-walled steel tube framework and discussion on "temporary-permanent" transition

To address the conflict between the reserve of repair materials for existing bridge piers and the demands for emergency repairs, as well as the challenge of converting temporary emergency structures into permanent ones without significant disassembly, a novel composite emergency steel pier has been developed. This pier utilizes commonly available thin-walled steel tube sections as the framework, I-beams as the connecting elements, and incorporates a "temporary-permanent" transition function. Taking the emergency steel pier for medium-height bridge piers in high-speed rail systems as an example, this study introduces three evaluation metrics—security coefficient (n), stability coefficient k), and variation coefficient (V)—analyzing the factors influencing the design and application thresholds of the emergency steel piers from perspectives of strength, stability, and load distribution uniformity. Based on this, a rapid design method for composite emergency steel piers with thin-walled steel tube frameworks that does not require numerical simulation has been developed and experimentally validated. Research findings indicate that when used for the emergency repair of medium-height or lower bridge piers in high-speed railways, the column line stiffness of the thin-walled steel tube framework composite emergency steel piers should range between 0.34 × 10 exp7 kN/mm to 4.15 × 10 exp7 kN/mm. The transverse and longitudinal spacing of the columns should be 1 m to 3 m and 1.5 m to 3 m respectively. The line stiffness of the plinth beams I and II should be greater than 0.69 kN/mm and 1.01 kN/mm respectively. Within these parameter ranges, the emergency steel pier will not experience strength failure or instability, and the load distribution among the columns remains uniformly favorable. In applications to other scenarios, the rapid generation of composite emergency steel piers with thin-walled steel tube frameworks can be achieved by following three steps: parameter selection and structural configuration determination, calculation of the pier-beam linear stiffness ratio, and criterion-based result verification. The rapid design and implementation of composite emergency steel piers help alleviate the contradiction between material stockpiling and repair demands. Furthermore, the pre-designed "temporary-permanent" transition functionality facilitates integrated operations for emergency repairs and permanent restorations.

Static analysis of violin bow behavior under playing loads.

After discussing the relationship between taper and camber typically used by bowmakers, energy methods are then employed to develop nonlinear boundary value problems describing both the hair tightening problem and static deflection analysis of the violin bow under playing loads, including an analysis of possible hair tension asymmetry by employing a linear springs in series model of the hair. A simple algebraic condition for the distance the frog travels in reaching maximum tension is also presented. Boundary value problems are then applied using tools in scilab, comparing the idealized Tourte taper exhibiting a linear stiffness profile and an alternative design with a significantly different profile, revealing small differences in their behavior under static playing loads and validating previously observed significant increase in hair tension when approaching the tip. While the difference in vertical hair deflection as a function of bow position is extremely small, the Tourte design exhibits somewhat more compliance in the stick when approaching the tip, possibly desirable by players given increasing hair tension. An interesting small loss of hair tension in the lower section of the bow stroke is also discussed, representing additional slight differences in the initial compliance felt by the player when engaging the strings.

Configuration and Parameter Optimization Design of a Novel RBR-2RRR Spherical Hybrid Bionic Shoulder Joint

To improve the workspace, linear displacement stiffness, and driving torque utilization of humanoid robot shoulder joint mechanisms, an offset-designed RBR-2RRR (R represents the revolute pair, and B represents the ball cage joint) spherical hybrid bionic shoulder joint configuration (SHBSJC) is proposed and its structural parameters are optimized. Firstly, the shoulder joint’s physiological structure is biomimetically designed, a prototype mechanism of RBR-2RRR SHBSJC is proposed, and its kinematics are solved. The deformation response of RBR-2RRR and 3-RRR under the same load is compared to verify the obtained configuration can improve the linear displacement stiffness. Considering the workspace and singularity, using the GCI and GDCI as optimization functions, the recommended and adopted values of structural parameters are obtained. The distribution diagrams of the LCI and LDCI demonstrate that the configuration meets performance expectations. To further increase the prototype mechanism’s workspace and match the human shoulder joint’s motion range, an offset-designed RBR-2RRR SHBSJC is proposed, and the offset angle, installation posture angle, and spatial mapping relationship of the mechanism are determined. The results of workspace comparison and virtual model machine action simulation indicate that the final configuration meets the workspace expectations. This work enriches the shoulder joint configuration types and has engineering application value.

A variable-thickness-arch based nonlinear vibration absorber for rotor-bearing systems

There are complex nonlinear vibrations in rotating machinery and their suppression is essential. This study aims to design a nonlinear vibration absorber to suppress the nonlinear vibration of rotor-bearing systems. The nonlinear vibration absorber is composed of a mass ring and circumferentially arranged multiple variable thickness arches mounted on the rotor and rotated synchronously. The absorber provides the nonlinear stiffness through the bending deformation of the arches under concentrated loads and the nonlinear stiffness is adjustable by designing variable thickness in the arches. The dynamic model of a rotor-bearing system coupled with the nonlinear vibration absorber is established. An efficient Galerkin averaging incremental harmonic balance (EGA-IHB) method is developed to analyze the vibration reduction performance of the absorber. The results indicate that the centrifugal force produced by the rotation of the absorber provides negative stiffness depends on the mass and the rotational speed of the absorber. When the absorber has a linear stiffness near the negative stiffness, the absorber demonstrates outstanding capability in suppressing the nonlinear vibrations of rotor-bearing systems. The absorber effectively suppresses both the primary resonance and harmonic resonance of the rotor system. The analysis demonstrated detailed the effects of the absorber parameters on vibration reduction. Increasing absorber damping can decrease both the primary resonance and harmonic resonance of the rotor system slightly. Appropriately increasing the absorber mass and reducing the cubic nonlinear stiffness can enhance the absorber performance in reducing primary resonance vibrations without affecting the vibration characteristics of the rotor system.

Investigation on local mooring stresses of floating offshore wind turbines considering mooring chain geometrical and material nonlinearity

As the renewable energy industry moves towards deep water, mooring chains play an irreplaceable role in position keeping. The geometrical and material nonlinearity of the mooring chains significantly affects moored platform responses and mooring responses. To investigate local mooring stress evolution under wave loads, a simulation method was proposed, combining hydrodynamic loads derived from fully coupled simulations of floating offshore wind turbines (FOWTs). The simulation method, embedding the lumped mass mooring model, was validated by comparing the experimental data from OC4 semi-submersible wind turbines. Their reasonable segment numbers and the mooring structural damping ratio were also discussed. Based on the validated simulation technology, a coupled 3D-mooring analysis was performed, replacing linear mooring stiffness suggested in FAST tutorials with nonlinear mooring stiffness calculated by axial stretching the real-sized 3D chains. The nonlinear mooring stiffness is assumed, taking into account detailed geometry, nonlinear material properties, and contact behavior between chains. The nominal diameter of the real-sized 3D-chains was determined, so that the chains had the same volume per unit length as the validated case, ensuring a similar Morison's force on the mooring line. Then, the local mooring stresses were investigated by loading the derived nonlinear fairlead tension onto both sides of real-sized 3D chains in an explicit finite element software. The evolution of local mooring stresses under design wave loads with different wave headings of 0, 30, 60° was compared and discussed. As wave direction increases, this study indicated that the amplitude of local maximum principal stress decreases and its phase leads. Additionally, its magnitude is mainly influenced by axial tension force and contact behavior.

Study on characterization of moderate energy impact response of elastic-plastic plate structure

This paper presents a study on the characterization of the moderate energy impact response between the elastic-plastic plate and the elastic-plastic sphere. An equivalent contact stiffness is proposed to substitute the linear contact stiffness, and a universal analytical expression is presented according to the intensive FE simulations for different material combinations under moderate impact energy. Then, a modified characterization diagram is suggested using the analytical expression. The modified diagram is validated by the FE simulations and experimental results. The results show the modified diagram can predict the maximum impact force and the response type in a fast and simple way.

Study on quasi-static tensile mechanical properties of biomimetic helicoidal composite laminates

Abstract Numerous biomaterials exhibit intricate helicoidal structures renowned for their exceptional toughness. This study drew inspiration from the collagen fiber structure found in the scales of coelacanths to design composite laminates featuring double helicoidal layup configurations with varying pitch angles. A damage evolution model with damage initiation and linear stiffness degradation based on the Hashin failure criterion was used to describe the damage behavior of the materials. The impact of varying pitch angles on the quasi-static tensile properties of double helicoidal composite laminates is investigated. The results show that the magnitude of the pitch angle has a significant effect on the tensile properties of the laminates, and the tensile modulus of double composite laminates is highest when the pitch angle is 90°.

Gyroscopic rotor dynamics simulation with anisotropy of elastic and damping characteristics of the support

In the work, the differential equations of motion of the gyroscopic rotor, built taking into account the anisotropy of stiffness and damping of the flexible support, are solved analytically, by the method of harmonic balance, convenient for obtaining separately amplitude-frequency and phase-frequency characteristics in the direction of oscillations. The equations of the non-stationary process are obtained by the method of changing amplitudes. It has been found that when the linear stiffness of the elastic support is different in two orthogonal directions, two critical velocities and the corresponding resonant regions arise. In the area of each critical speed, there are two amplitude-frequency curves of oscillations of the main direction and the direction perpendicular to it, respectively. The geometric nonlinearity of damping suppresses the elevations of these amplitude-frequency curves more significantly than linear damping. If only one of the two directions has a damping nonlinearity, then its effect is on the amplitude-frequency curves of the corresponding critical speed. It is more efficient to control resonant amplitudes for smooth resonant transitions by enhancing the linear damping with geometrically nonlinear damping. The results of the analytical solution of the equations of motion agree well with the results of direct modeling and experimental studies.

Tunable High-Static-Low-Dynamic Stiffness Isolator under Harmonic and Seismic Loads

High-Static-Low-Dynamic Stiffness (HSLDS) mechanisms exploit nonlinear kinematics to improve the effectiveness of isolators, preserving controlled static deflections while maintaining low natural frequencies. Although extensively studied under harmonic base excitation, there are still few applications considering real seismic signals and little experimental evidence of real-world performance. This study experimentally demonstrates the beneficial effects of HSLDS isolators over linear ones in reducing the vibrations transmitted to the suspended mass under near-fault earthquakes. A tripod mechanism isolator is presented, and a lumped parameter model is formulated considering a piecewise nonlinear–linear stiffness, with dissipation taken into account through viscous and dry friction forces. Experimental shake table tests are conducted considering harmonic base motion to evaluate the isolator transmissibility in the vertical direction. Excellent agreement is observed when comparing the model to the experimental measurements. Finally, the behavior of the isolator is investigated under earthquake inputs, and results are presented using vertical acceleration time histories and spectra, demonstrating the vibration reduction provided by the nonlinear isolator.

Parameter identification of overhead conductor rail support and its influence on the contact forces

Overhead conductor rail (OCR) serves as a crucial device employed in tunnels to power electric locomotives. The OCR is hung by supports mounted on the tunnel ceiling. Previously, the support was usually considered as a linear spring when modelling the OCR. The linear stiffness and mass adopted to the support model were imprecise, and damping was absent. It may be insufficient to reproduce the pantograph-OCR system (POCR) dynamic behaviour. This paper is devoted to improving the accuracy of POCR dynamic model and makes it more realistic. Field tests are conducted in order to identify the support parameters, including stiffness, damping and mass, for the first time. Then, a novel nonlinear stiffness-damping-mass support model is developed utilising the identified parameters and verified by comparing simulation results with measurement data. Finally, in pursuit of improving current collection quality, the effects of support damping and train speeds on contact forces are investigated. The results show that the support damping can mitigate the OCR’s vibration. The increase of support damping within a specific range, 0–200 Ns/m, has a positive effect on the current collection quality.

Optimal Design of Assembling Robot Considering Different Limb Topologies and Layouts

Abstract A class of 5 degree-of-freedom (DoF) hybrid robots consisting of one translational and two rotational (1T2R) parallel module and 2 T serial module is presented for assembling in the aircraft cabin. The 1T2R parallel modules are with three limb topologies (PRS, RRS, and internal closed loop) and two layouts (symmetrical and “T” shape). Herein, P, R, and S denote prismatic, rotational, and spherical joints. This article presents the multi-objective optimization of the hybrid robot regarding different limb topology, limb layout, and corresponding dimensions as design variables. Considering demands from the in-cabin assembling, kinematic, linear stiffness along z-axis and total mass of the robot are the objectives. The Pareto fronts and cooperative equilibrium point (CEP) indicate that the robot with 3-PRS and 3-RRS parallel module have better performances than the one with internal closed loop. The overall performances of robot with symmetrical layout are superior than the one with “T” shape layout. In addition, two optimization methods are compared. One is to separately optimize six robots with specific topology and layout. The other is to optimize all design variables in a model. It is found that six robots have their own performance zones. Therefore, final optimums of two methods are close to each other. But optimization in one model is able to eliminate unfeasible topologies in the early stage of searching and thus is more efficient. Optimal module is 3-PRS with symmetrical layout. Experiments on the physical prototype validate performances of the optimal robot.

Simulation of soil-structure interaction using inelasticity-separated finite element method

The soil–structure interaction (SSI) effect is nonnegligible during the nonlinear seismic response analysis of large-scale or underground structures. However, the computation of nonlinear response of a structure considering SSI effect usually is a time-consuming process because the numerical model should involve the structure itself, a wide-range soil domain and the soil–structure interface, especially for large-scale structure. This study aims to incorporate the inelasticity-separated finite element method (IS-FEM), which is an algorithm recently developed for efficient structural nonlinear analysis, to improve the efficiency of the soil-structure interaction system. To this end, an inelasticity-separated contact element model for modeling the nonlinear behavior of soil and structure interface is developed in this study. To derive the inelasticity-separated governing equation of this model, a reference elastic stiffness is defined so that the normal and shear deformation of any point pair in the presented contact element is decomposed into reference linear and reference nonlinear parts according to this reference elastic stiffness. As a result, the problem of inconsistency between the requirement of IS-FEM for constant initial linear elastic stiffness and the existence of inflection point for the stress versus relative displacement relationship of contact behavior can be solved. Then, the Goodman element formulation for depicting the nonlinear contact behavior is introduced and an interpolation scheme to establish the reference nonlinear deformation field in an element is incorporated. By combining the proposed model with existing inelasticity-separated elements, a global analytical model of the soil–structure interaction system can be established within the framework of IS-FEM. Because the nonlinearity usually occur in some local regions of the global model and the use of the IS-FEM only require performing these operations for a small scale matrix representing local nonlinearity, the computational efficiency of nonlinear structural response analysis considering SSI effect can be improved significantly. The applicability and efficiency of the proposed method is finally verified by two numerical examples.

Discrete mechanical model for nonlinear dynamical analysis of planar guyed towers considering the unilateral contact of cables

Towers are widely used for power line transmission, wind power plants, TV and radio broadcasting, and telecommunications. To enhance their stability, cables are often employed to anchor these towers to the ground. In this study, we investigate the nonlinear static and dynamic responses of a planar guyed tower in which the unilateral constraints on the cables are considered. A representative discrete mechanical model with two degrees of freedom is developed to simulate the central mast of the tower, and the cables are modeled as unilateral springs with linear stiffness. The nonlinear equilibrium equations are derived using an energy approach that incorporates the dissipative forces, total potential, and kinetic energies into the Euler-Lagrange equations. Unilateral cable contact is directly included in the nonlinear equilibrium equation for the guyed tower, allowing for numerical analysis without the need to evaluate the contact point at each time or load step. Several numerical strategies are employed to obtain nonlinear static equilibrium paths, bifurcation diagrams, phase portraits, and Poincaré sections. Our analyses provide novel results for the influence of unilateral cable contact in nonlinear static and dynamic analysis, evaluating the effects of unilateral contact and prestressing on the results. A parametric analysis reveals that cable contact affects nonlinear oscillations, bifurcation, and stability. Our numerical results indicate that unilateral cable contact introduces less structural stiffness compared to bilateral contact, thereby significantly affecting the static and dynamic stability of a planar guyed tower. This is evidenced by a decrease in the static limit load and alterations in the bifurcation diagrams, where unilateral contact destroys the trivial solutions, leading to periodic and quasi-periodic solutions at low levels of vertical load.

A novel optimization framework using frequency-based substructuring for estimation of linear bolted joint stiffness and damping

Estimating the stiffness and damping coming from a joint is a major challenge. This work proposes a novel framework to estimate the unknown parameters using frequency-based substructuring and a maximum a posteriori optimization approach. The idea is to minimize the discrepancy between the measured responses of an assembled system and the responses obtained by coupling the measured responses of the individual substructures with a joint model. The system’s dynamics are assumed to be linear. However, the FRFs of the coupled system (which drive the objective function) change nonlinearly with the linear joint parameters, making it a nonlinear optimization problem. Therefore, the joint parameters are estimated iteratively using the Gauss minimization method, a well-established linearization scheme. The framework is numerically validated on simple mdof systems, and the method’s effectiveness is tested by adding noise to the generated data. To demonstrate the framework’s applicability on real structures, joint stiffness and damping of a known benchmark structure are estimated based on real measurements. Two different underlying joint models are applied. The first assumes just flexibility at the connection. The second assumes the joint is a separate substructure. The algorithm can find realistic stiffness parameters and a range for the damping parameters providing a good compromise between the measurements in both cases.

Shear resistance design of prefabricated corrugated steel plate shear walls with reinforced boundary elements

This paper investigates the buckling behavior and shear-bearing capacity of prefabricated corrugated steel plate shear walls (PCSPSWs) with reinforced vertical boundary elements (VBEs). The PCSPSWs are solely connected to the frame beams, and therefore, they can be positioned flexibly within the frame. However, there is currently no available shear-bearing capacity design method for such PCSPSWs. Firstly, this paper conducts a numerical investigation into the elastic buckling behaviors of PCSPSWs using finite element analysis (FEA). The prediction formulas for elastic buckling loads are fitted considering the influence of VBEs’ stiffness on the corrugated steel plates (CSPs). Then, the elastic-plastic buckling performance of PCSPSWs is analyzed numerically. The stability factor (φ) is proposed to predict the shear-bearing capacity of PCSPSWs by introducing the normalized slenderness ratio (λn). Subsequently, this paper explores the effects of PCSPSWs’ position within the frame and the influence of linear stiffness ratios between the frame beams and columns on the shear-bearing capacity of PCSPSWs through numerical analysis. Finally, the lower stiffness values of VBEs are discussed to provide some guidance for structural designers.