Linear Stiffness Research Articles

Impact of Cross-Tie Material Nonlinearity on the Dynamic Behavior of Shallow Flexible Cable Networks

Cross-ties have proven their efficacy in mitigating vibrations in bridge stay cables. Several factors, such as cross-tie malfunctions due to slackening or snapping, as well as the utilization of high-energy dissipative materials, can introduce nonlinear restoring forces in the cross-ties. While previous studies have investigated the influence of the former on cable network dynamics, the evaluation of the impact of nonlinear cross-tie materials remains unexplored. In this current research, an existing analytical model of a two-shallow-flexible-cable network has been extended to incorporate the cross-tie material nonlinearity in the formulation. The harmonic balance method (HBM) is employed to determine the equivalent linear stiffness of the cross-ties. The dynamic response of a cable network containing nonlinear cross-ties is approximated by comparing it to an equivalent linear system. Additionally, the study delves into the effects of the cable vibration amplitude, cross-tie material properties, installation location, and the length ratio between constituent cables on both the fundamental frequency of the cable network and the equivalent linear stiffness of the cross-ties. The findings reveal that the presence of cross-tie nonlinearity significantly influences the in-plane modal response of the cable network. Not only the frequencies of all the modes are reduced, but the formation of local modes is delayed to a high order. In contrast to an earlier finding based on a linear cross-tie assumption, with nonlinearity present, moving a cross-tie towards the mid-span of a cable would not enhance the in-plane stiffness of the network. Moreover, the impact of the length ratio on the network in-plane stiffness and frequency is contingent on its combined effect on the cross-tie axial stiffness and the lateral stiffness of neighboring cables.

Novel operator-splitting methods for real-time dynamic substructure testing

Improved and corrected Operator-Splitting (OS) methods are proposed for Real-Time Dynamic Substructure Testing (RTDST) in this paper. Firstly, the theory and the stability of two proposed OS methods are investigated. Furthermore, the effectiveness of two proposed OS methods is validated by RTDSTs with one pure mass specimen and one mass-stiffness specimen. By using the predictor velocity and acceleration as well as the correction force approximations, the improved OS method and the corrected OS method are explicit methods for RTDST. It is shown from the analytical and experimental results that the improved OS method is unconditionally stable for linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping system if the mass ratio of the experimental substructure to the numerical substructure is less than 1. In contrast, the improved OS method is unsuitable for RTDST if the mass ratio is larger than 1. The corrected OS method is unconditionally stable for the linear type and softening type stiffness and damping systems and conditionally stable for hardening type stiffness and damping systems. The stability of the corrected OS method is better than the improved OS method, which indicates the advantage of the corrected OS method over the improved OS method.

Programmable hardening and softening cubic inductive shunts for piezoelectric structures: Harmonic balance analysis and experiments

It has been well explored in the literature that intentionally introduced (designed) nonlinearities offer significant advantages in vibration absorption and isolation problems. In the context of using piezoelectric shunt damping for vibration attenuation in flexible structures, synthetic impedance circuits with digital control enable precise tuning capabilities and eliminate the need for bulky analog circuit components when targeting low frequencies, and they have been studied mostly for linear circuits. In this work, we explore nonlinear synthetic impedance-based Duffing shunts with hardening and softening nonlinearities, connected to a cantilever under base excitation in its geometrically and materially linear regime. We demonstrate a shunt of nonlinear impedance equivalent to a positive or negative cubic inductor connected in parallel to a linear inductor and a linear resistor. This shunt, along with the inherent piezoelectric capacitance, provides digital analog of a hardening- or softening-type nonlinear stiffness, linear stiffness, linear damping, and mass, as a synthetic Duffing circuit. Electromechanical modeling of a piezoelectric cantilevered beam shunted to the Duffing oscillator with hardening or softening inductance is performed. Numerical simulations for single-term and multi-term harmonic balance solutions are performed for periodic response. These approximate analytical predictions are compared against time-domain numerical simulations and are experimentally validated for various mechanical excitation amplitudes and nonlinear coefficients. Overall, a very good agreement is obtained between the model simulations and experimental results, in addition to demonstrating the unique digital programming capabilities of this nonlinear shunt.

Nonlinear dynamics of pipes composed of Mooney-Rivlin hyperelastic materials conveying unsteady fluid flow

In the present study, a mathematical model for the nonlinear dynamics of a Mooney-Rivlin type hyperelastic pipe conveying unsteady fluid flow is established. The internal fluid velocity is assumed to comprise an average component superimposed on a harmonically varying component. Subsequently, various numerical methods are employed to investigate the nonlinear responses of the pipe through the forms of amplitude-response, disturbance-response, global bifurcation diagrams, phase trajectories, Poincaré maps, and power spectral densities (PSD). Additionally, the contributions of the Coriolis force and geometric nonlinearity are examined. The results reveal that, in comparison to linearelastic pipes, the combined effect of nonlinear geometric relation and nonlinear constitutive relation introduces specific nonlinear terms, leading to distinct dynamical phenomena. With small disturbed coefficient, the asymmetric period-1 and symmetric period-2 oscillations are respectively observed in the primary and sub-critical regimes. Furthermore, the hyperelasticity and fluid-structure interactions (FSI) play significant, unequal, and competitive roles in influencing the dynamic responses of the pipe by affecting either linear or nonlinear stiffness. Additionally, abundant dynamical phenomena are captured within the system versus system parameters with large disturbed coefficient. The study also demonstrates that the combined effect of nonlinear geometric relation and nonlinear constitutive relation ultimately results in weaker hardening behaviors for Mooney-Rivlin type pipes compared to linearelastic ones.

Insights into Bending Stiffness Modeling of Elementary Flexure Hinges

Flexure hinges are widely used in mechanical devices, especially for micro- or even nano-scale applications, where conventional joints based on conjugate surfaces prove unsuitable. However, to achieve accurate motion of devices whose joints are flexure hinges, knowledge of stiffness models that correlate applied forces or bending moments with the resulting displacements is required. Nonlinear bending models are typically too complex and difficult to implement. Therefore, it is preferred to use linear models, which admit analytical solutions. The purpose of this paper is to show what is lost in terms of accuracy in reducing a nonlinear bending stiffness model associated with a flexure hinge when simplifications are made to obtain an analytical solution. An analysis of the simplification process leading to a linear stiffness model and its analytical solution is presented. From this study arises an increased awareness of flexure joints in terms of the accuracy obtained with their stiffness models, suggesting important information to the user in choosing the level of complexity required to model them in a specific application. A comparison between analytical and numerical results is provided in the form of maps and tables so as to make that choice as clear as possible.

Novel metamaterial structures with negative thermal expansion and tunable mechanical properties

The thermal expansion and stiffness characteristics are significant in determining the responses of lightweight mechanical metamaterials in a thermal environment. This paper proposed a novel trapezoid unit with negative thermal expansion (NTE) by three conventional materials with positive thermal expansion, and constructed a series of lightweight metamaterials by sharing edges and sharing vertexes. The effective linear coefficients of thermal expansion (CTE) of these metamaterial structures are theoretically explored by considering axial and bending deformations. Based on Euler beam theory, the effective linear elastic stiffnesses of these metamaterials are given accurately by the stiffness matrix method. The accuracy of the theoretical effective CTE and stiffness is well verified by numerical modeling. The effective linear CTE, elastic stiffness, and Poisson's ratio of these metamaterials can be programmable by controlling geometrical parameters and material properties. The NTE effect can be achieved in ET, EQ, EH, and VT metamaterials due to the presence of re-entrant structures, in which the ET metamaterial can fulfill the maximum magnitude NTE. The NTE and negative Poisson's ratio can simultaneously be realized in EQ metamaterial. Significantly, a broader range of CTE from negative to positive values can be fulfilled in these metamaterials compared with the conventional triangle unit. Furthermore, the high specific stiffness with extremely low thermal stress can be obtained in EQ, VT, and VH metamaterials, which benefits the stability controlling and enhancing natural frequencies of structures in a thermal environment.

Tensile performance of aluminum alloy roof panel-member clamp connection

In aluminum alloy reticulated shell structures with gusset joints, the roof panel is effectively connected with the structural members through a clamp connection, enabling the roof panels to participate in the force resisting system. The performance of the connection between the roof panels and the members is the foundation for studying the collaborative action of the roof panels and the members. Relevant studies have shown that when the roof panels participate in the force resisting system, there is mainly a tensile force in the roof panels perpendicular to the axis of the members. Therefore, this paper experimentally studied the mechanical performance of the clamp connection between the roof panels and the members under the tensile force in the roof panels, and employed two-dimensional plane digital image correlation (DIC) to measure the deformation of the specimen section during the entire loading process, thus exploring the force transmission mechanism of the connection. The tensile capacity of the connection can be divided into four stages: roof panel sliding, roof panel tension–shear, lower limb bending of the strip, and roof panel bending stages. The mechanical index of the connection is primarily determined by the tension–shear stage of the roof panel, and the ultimate strength and stiffness can be determined as the peak load and linear segment stiffness of the connection at this stage. The refined finite element model was further established, and it was verified that the model could efficiently simulate the mechanical properties of the connection through comparison with the test. Parametric analysis shows that with an increase in the friction coefficient between materials, roof panel thickness, angle between roof panel and flange, and bolt length adjustment, the restraint on the roof panel increases, and the failure mode of the clamp connection changes. Based on the parametric analysis, the recommended values of the friction coefficient between the materials, minimum panel thickness, minimum bolt screw depth, and maximum bolt spacing are provided. Finally, combined with the theoretical analysis, calculation formulas for the tensile strength and stiffness of the connection were proposed. A comparison with the test results indicates that the prediction accuracy of the formula proposed in this study is good.

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

Effect of linear and nonlinear stiffness support on the dynamic response of high-speed railway bridges

In this paper, the effect of linear and nonlinear support vertical stiffness of elastic support on the dynamic response of the railway bridge is investigated. The bridge is modeled by a Euler-Bernoulli beam supported by linear and nonlinear supports stiffness. A constant moving load model HSLM-A is utilized to simulate the moving real train, and based on Hamilton’s principle, the governing partial differential equation of motion of the studied system is derived. A method to deal with the nonlinear boundary conditions is proposed, the fourth-order Runge-Kutta method in combination with the Galerkin method is used to obtain the dynamic response of the bridge. The finite difference method (FDM) is used to validate the proposed solution. The obtained results have shown that the dynamics of the system are governed essentially by a Duffing-like oscillator and proved that the elastic bearings and the cubic nonlinearities of supports have a significant effect on the global behavior of the railway bridge particularly at resonance.

Analysis of point-contact models of the bounce of a hard spinning ball on a compliant frictional surface

Abstract Inspired by the turf–ball interaction in golf, this paper seeks to understand the bounce of a ball that can be modelled as a rigid sphere and the surface as supplying a viscoelastic contact force in addition to Coulomb friction. A general formulation is proposed that models the finite time interval of bounce from touch-down to lift-off. Key to the analysis is understanding transitions between slip and roll during the bounce. Starting from the rigid-body limit with an energetic or Poisson coefficient of restitution, it is shown that slip reversal during the contact phase cannot be captured in this case, which generalizes to the case of pure normal compliance. Yet, the introduction of linear tangential stiffness and damping does enable slip reversal. This result is extended to general weakly nonlinear normal and tangential compliance. An analysis using the Filippov theory of piecewise-smooth systems leads to an argument in a natural limit that lift-off while rolling is non-generic and that almost all trajectories that lift off do so under slip conditions. Moreover, there is a codimension-one surface in the space of incoming velocity and spin which divides balls that lift off with backspin from those that lift off with topspin. The results are compared with recent experimental measurements on golf ball bounce and the theory is shown to capture the main features of the data.

Modeling and analysis of the magnetic force and stiffness of a low-frequency vibration isolator with axial magnetized magnetic rings.

The vibration isolator is a key part of many ultra-precision machines and measuring apparatus. Magnetic suspension vibration isolators (MSVIs) will have excellent application prospects in these instruments to restrain external oscillations. So this paper firstly proposes a new basic configuration of MSVI. Then, in order to study the mechanical characteristics of the MSVI, an analytical expression of the magnetic force is established. The effectiveness of which is demonstrated by the experiment and finite element analysis (FEA). The stiffness of the MSVI is obtained by the derivative of the established analytical magnetic force. Both the axial magnetic force and stiffness appear strong nonlinearity when the inner ring moves at both ends of the fixed outer ring. While the inner ring travels in the middle of the fixed outer one, the axial magnetic force and stiffness indicate approximate linearity with enough bearing capacity. Furthermore, parametric analysis, based on the created magnetic force and stiffness, is performed. The analytical results show that the axial magnetic stiffness may achieve a zero or even negative stiffness value in this range at some size dimensions. The MSVI appears to have a negative stiffness characteristic. More importantly, if a linear and nonlinear positive stiffness spring is combined with the MSVI, it can increase the load capacity of the MSVI. As an example study, the vibration isolation performance of the MSVI is analyzed. The vibration isolation calculation and experiment with the zero stiffness MSVI will be the further focus of the paper.

Softening/hardening dynamics of beam resting on generalized nonlinear foundation with a linear stiffening effect

An asymptotic study of softening/hardening dynamics of beam on a generalized nonlinear foundation is presented, by focusing on its linear stiffening effect on quadratic/cubic nonlinearities. It is found that, though linear stiffening simultaneously weakens both quadratic and cubic nonlinearities, they decrease at different rates, leading to complex competing dynamics in sequel, such as enhanced hardening/weakened softening, weakened hardening, and quenched nonlinearity in the limit, among which softening–hardening transition dynamics is of particular interest when quadratic and cubic nonlinearities ‘balance’ with each other (i.e., a vanishing effective nonlinear coefficient). Close to transition, standard third-order perturbation fails and the predicted responses turn invalid in mediate/high amplitude ranges, which critically depends on transition inclination when transversally crossing the line of vanishing effective nonlinear coefficient. In parallel, linear stiffening effects on softening/hardening dynamics are also investigated from a methodology viewpoint, i.e., it reduces differences between discretization and direct perturbations, with the former becoming more reliable by an increased linear stiffness.

Mechanical Properties of Uncured Thermoset Tow Prepreg: Experiment and Finite Element Analysis

This paper presents an experimental analysis of the tensile behavior of unidirectional carbon/epoxy prepreg, focusing on the nonlinearity observed at the beginning of the stress–strain curve. Due to the material’s high viscosity, securely holding specimens during testing was challenging, prompting modifications in the gripping method to ensure reliable data. By using a longer gauge length, the slippage impact on elastic modulus measurement was minimized, resulting in good repeatability among the test samples. Experimental findings highlighted the significant interaction between fiber waviness and the viscous matrix, leading to stiffness reduction. The linear stiffness of the samples closely matched that of the fibers and remained unaffected by temperature variations. However, at higher temperatures, the epoxy matrix’s decreased viscosity caused an upward shift in the stiffness plot within the non-linear region. To support the experimental findings, a micromechanical model of prepreg tow with fiber waviness was proposed. An RVE model of periodically distributed unidirectional waved cylindrical fibers embedded within the matrix was developed to predict effective material stiffness parameters. The simulation outcomes aligned well with the uniaxial tensile test of the prepreg tow, demonstrating the proposed RVE model’s capability to accurately predict elastic properties, considering factors like fiber arrangement, waviness, and temperature.

Shimmy Suppression of an Aircraft Nose Landing Gear Using Torsional Nonlinear Energy Sink

Abstract Shimmy motion may occur for an aircraft nose landing gear (NLG) during its takeoff, landing, and taxiing procedure, which is an undesirable motion and should be suppressed. Here, a novel torsional nonlinear energy sink (NES) based on the cam-roller mechanism is proposed and applied in the NLG to improve its shimmy performance. The torsional NES is connected with the upper and lower struts of the NLG and its mechanical characteristic is studied. A seven-dimensional dynamic model of the NLG coupled with NES is built, which considers the NLG torsional and lateral motions, and the nonlinear tire model. The numerical continuation method is applied to analyze its shimmy performance, the stable area and limit cycle oscillation (LCO) amplitude are acquired, further compared with those of the original NLG without using the NES. The results show that as the NES is used in the NLG, the torsional shimmy dominated unstable area reduces significantly and the lateral shimmy dominated unstable area decreases slightly, which results in the increase of the stable area of the NLG; the maximum LCO amplitudes for the NLG torsional and lateral shimmy become smaller, the forward speed ranges as the torsional and lateral shimmy occur become narrower, which indicates that the stable forward speed ranges of the NLG become wider; the NLG shimmy performance improves as the NES has larger torsional inertia and damping, and smaller torsional linear stiffness. Thus, the NES can suppress the NLG shimmy motion and enhance its shimmy performance effectively.

Biomechanical Comparison of Two Different Sutures for the Tensile Strength of the Pullout Repair of Posterior Meniscal Root Tear.

Background Medial meniscal posterior root tear has garnered attention because it causes increased joint contact pressure and loss of load distribution function. Currently, the common treatment for posterior meniscal root tears is transtibial pullout repair, and loop suture remains the gold standard procedure. Purpose The aim of this study is to validate whether the meniscus-suture complex exhibits better structural properties after loop stitch using a suture tape versus the conventional thread. Study design and methods This is a controlled laboratory study. In total, 20 menisci harvested from castrated male pigs were prepared and randomly assigned to two suturing techniques: loop stitch using ULTRATAPE (n = 10, tape group) or loop stitch using 2-0 UltraBraid (n = 10, control group). The single-loop stitch meniscus-suture technique was used. A single suture using ULTRATAPE or 2-0 UltraBraid was placed through the meniscus 5 mm medial from the resected edge of the posterior meniscus horn. The meniscus-suture complex specimens were mounted on a tensile tester to apply tensile load on the meniscus-suture complex parallel to the long axis of the stitched suture materials. Each specimen was stretched to failure at a crosshead speed of 50 mm/minute. The structural properties of the meniscus-suture complex (maximum load, linear stiffness, and elongation at failure) were determined. Results In the tape group, all the repaired complexes were torn at the ligature; conversely, in the control group, the threads were torn in three of 10 specimens, and the repaired complexes in the remaining seven specimens were torn at the ligature. No significant between-group differences were observed in linear stiffness (9.0 ± 4.1 vs 7.1 ± 2.9 N/mm) and elongation at failure (15.8 ± 6.1 vs 19.6 ± 20.1 mm); however, the tape group exhibited a significantly higher average maximum load than the control group (104.6 ± 41.2 vs 62.6 ± 32.1 N; P = 0.020). Conclusion Suture tape can be a potential alternative to the conventional thread for Medial meniscal posterior root tear repair. Clinical relevance Medial meniscal posterior root tear repair using tape rather than the conventional thread may be beneficial in obtaining better structural properties.

Effect of negative stiffness nonlinearity on the vibration control effectiveness of tuned negative stiffness inerter damper

Recently, tuned negative stiffness inerter damper (TNSID) integrating the advantages of both negative stiffness and inerter elements has been proposed and verified to be effective for structural vibration control. In previous studies, the stiffness of the negative stiffness element was normally regarded as a constant for simplicity. However, it is well-known that negative stiffness is generally accompanied by nonlinearity, which may lead to undesirable nonlinear vibrations and unstable responses. Therefore, in this paper, the influence of the nonlinearity of negative stiffness, including weakening and strengthening nonlinear negative stiffness, on the performances of two representative TNSIDs with different configurations is investigated. The working mechanisms and optimal parameters of the TNSIDs with simplified linear negative stiffness are introduced firstly. Following that, based on the harmonic balance method, the dynamic responses of the two TNSIDs with different nonlinear negative stiffness are attained. The unstable problems and performance degradation caused by the nonlinearity of negative stiffness are analysed. Parametric studies are further carried out to evaluate the effects of the nonlinear coefficient, excitation amplitude, internal frequency ratio, and damping ratio on the performances of the two nonlinear TNSIDs. Moreover, to eliminate the impact of the negative stiffness nonlinearity, a new structural parameter optimization is conducted by minimizing the maximum displacement amplitude-frequency response of the TNSIDs. The results indicate that the nonlinearity of negative stiffness could lead to the bend and jump of the displacement amplitude-frequency response curves of TNSIDs, as well as superharmonic resonances, especially when the performance of TNSIDs is sensitive to the variation of negative stiffness. Besides, the unstable areas and the maximum displacement amplitude-frequency responses of TNSIDs are dependent on the excitation amplitude and nonlinear coefficient. To avoid the unstable vibration phenomenon and maintain effective vibration control performance, TNSID-P2 is recommended to be equipped with a weakening negative stiffness under both moderate and strong excitations, while TNSID-P1 should be designed with a weakening negative stiffness and work under a moderate excitation due to its performance being highly sensitive to the variation of negative stiffness.

Periodic solutions and frequency lock-in of vortex-induced vibration energy harvesters with nonlinear stiffness

The phenomenon of amplitude jump usually means the presence of multi-solution ranges and unstable periodic solutions, which has not been fully explored in the field of vortex-induced vibration energy harvesting. In this study, semi-analytical periodic solutions of vortex-induced vibration energy harvesters (VIVEHs) are obtained using the incremental harmonic balance method, and their stabilities are determined via the Floquet theory. It is found that the VIVEH with linear stiffness and those with three-order nonlinear stiffness have four saddle-node bifurcation points and two multi-solution ranges, which leads to the occurrence of amplitude jumps. The hardening effect can shift frequency lock-in regions to higher wind speeds, thus increasing the aerodynamic forces, and realizing wide wind speed bandwidths and high voltage outputs. The wind tunnel experiments verify the accuracy of theoretical calculations, the occurrence of amplitude jump phenomena, and the existence of multi-solution ranges. The VIVEH with nonlinear stiffness experimentally reached a maximum voltage of 9.87 V, and a frequency lock-in region of 2.27–5.36 m/s.

Comparative study on failure mechanism of multi-storey planar composite frames with RC beams to CFST columns subjected to compartment fire

This paper presents a theoretical study of the failure mechanism of planar composite frames composed of reinforced concrete (RC) beams to concrete-filled steel tubular (CFST) columns subjected to compartment fire. The numerical analyses on the planar composite frames with multiple stories and spans are meant to reproduce the conditions of framed members in multi-storey composite structures in real buildings. A refined finite element model is established and calibrated against the thermal and mechanical results from physical testing and shows good reliability in general. Using this model, this study proposes comparative analyses on the failure mechanisms of three typical composite frames, and discusses the effects of two significant parameters: the beam-to-column linear stiffness ratio and the fire protection thickness of the composite column. The responses of the heated composite frames, including the rotational deformation and stiffness degradation of framed joints, the internal force redistribution within framed members, and the failure mode of framed structures are investigated and discussed. The results indicate that there are two important parameters should be considered in the design of composite frames in fire.

Optimal Design and Analysis of Nonlinear Tuned Mass Damper System

The tuned mass damper (TMD), as a representative passive control device, has been widely used in various fields such as mechanical vibration and civil engineering. In practical engineering applications, nonlinear characteristics of the TMD inevitably occur due to the installation of limiting devices or large displacements. Neglecting such nonlinearity may adversely affect the control performance of the TMD. Based on this situation, the parameters of the nonlinear tuned mass damper (NTMD) were analyzed, considering the linear and cubic nonlinear stiffness of the TMD, in order to improve its control performance. In this paper, a numerical model of a single-degree-of-freedom controlled structure using an NTMD was established, and the complex variable averaging method was employed to derive the response expression of the structure under the 1:1 resonance condition. Subsequently, the nonlinear design equation for the NTMD frequency was obtained. Based on this nonlinear design formula, the design accuracy was analyzed under 1:1 resonance, undamped and softened stiffness conditions. The results indicate that the NTMD based on a nonlinear design exhibits good control performance.

The X-shaped structure with nonlinear positive stiffness compensation for low-frequency vibration isolation

X-shaped structure with quasi-zero stiffness (QZS) properties are suitable for low-frequency vibration isolation because of its simple structure. The effective working range of classic X-shaped structure (CXS) is narrow for the existence of negative stiffness region, and using positive stiffness compensation to overcome this deficiency is a common way. This paper proposes a novel nonlinear positive stiffness compensation method (NPSC) to improve the load capacity and QZS range of CXS. Different from the previous studies that utilize linear or hardening stiffness compensation, the NPSC exhibits the characteristics of softening stiffness first and then hardening stiffness and is provided by torsion springs simply. The static properties of X-shaped structure with NPSC are investigated. The result indicates the NPSC possesses dual significant advantages. Firstly, the X-shaped structure with NPSC can achieve larger load capacity and wider QZS range than that with linear or hardening stiffness compensation in optimal parameter region. Secondly, the X-shaped structure with NPSC can conveniently overlap the equilibrium position with QZS point to obtain ultra-low frequency vibration isolation effect compared to that with linear stiffness compensation. The motion equation of the proposed system is derived, and the dynamic response is solved through harmonic balance method (HBM). The influence of different parameters on displacement transmissibility is investigated and the result is verified by numerical simulation. The transmissibility comparison between the proposed model and the CXS with linear compensation shows that the proposed model possesses better vibration isolation performance. Besides, the static compression test and dynamic sweep frequency experiment are implemented, and the experiment results verify the validity of proposed model. The proposed NPSC is an important inspiration for the stiffness compensation research of X-shaped structures.