Equivalent Stiffness Coefficient Research Articles

Multi-level stiffness property and isolating-based design of high damping rubber bearings

This study presents the development and analysis of high damping rubber bearings (HDRBs) with enhanced stiffness properties to improve seismic isolation performance. The proposed HDRBs exhibit displacement-dependent nonlinear stiffness and significant damping effects, especially under large deformations caused by various seismic events. A deformation history integral model, calibrated with experimental data, is employed to accurately simulate the mechanical behavior and stiffness-damping characteristics of the HDRBs. The numerical simulations are validated through experimental tests, providing a solid basis for parameter design and performance assessment. The results show that the equivalent stiffness coefficient of the HDRBs increases with deformation amplitude, effectively limiting extreme deformations. Parametric analyses and case studies across a wide range of earthquake scenarios demonstrate that the enhanced stiffness and high damping effects of HDRBs significantly improve seismic isolation efficiency while controlling isolation layer displacement. The performance-based design methodology developed in this research effectively limits bearing deformation, thereby preventing potential superstructure failures. Moreover, the adaptive characteristics of the HDRBs allow for the adjustment of deformation levels according to seismic intensity, ensuring the structural safety of buildings under varying earthquake conditions.

Linear equivalence for motion amplification devices in earthquake engineering

AbstractMotion amplification devices utilized to amplify the motion of dampers can effectively improve the energy dissipation performance of dampers to reduce seismic responses of engineering structures. This study systematically develops a linear equivalence theory for motion amplification devices based on the proposed equivalent Maxwell model. This equivalent model accurately predicts the supplemental damping effect provided by motion amplification devices without approximation. Also, the equivalent model is capable of quantifying the amplification effect of motion amplification devices by means of deriving the analytical expressions of the equivalent damping and stiffness coefficients, which reveal that motion amplification devices simultaneously enhance the original damping and stiffness coefficients by , where is the proposed rigidity motion amplification factor. The representative value of member stiffness is developed to comprehensively evaluate the supporting stiffness of motion amplification devices. All the achieved results strongly support that the proposed linear equivalence theory provides a generic paradigm to explain, measure and compare different types of motion amplification devices in terms of their supplemental damping effects, and hence helps researchers and engineers gain valuable insight into the dynamic properties of motion amplification devices.

Calculation Method of New Assembled Corrugated Steel Initial Support Structure of Highway Tunnel

The assembled corrugated steel initial support structure is a new prefabricated structure in highway tunnel engineering, achieving a balance between economy and safety. This study proposes a simplified calculation method and elucidates the mechanical mechanisms of assembled corrugated steel initial support structures. Firstly, the stiffness characteristics of corrugated steel plates were studied based on full-scale tests. A general equivalent stiffness coefficient table was established. Numerical simulations of corrugated steel flange joints were conducted to explore their bending mechanical properties. A two-stage rotational stiffness model for corrugated steel flange joints was proposed. Finally, a plane strain-spring simplified calculation method for the assembled corrugated steel initial support structure was developed, and the monitoring data from the Qipanshan Tunnel validated the correctness and reliability of the proposed method. The results demonstrate that (1) the plane strain-spring simplified model consists of the planar strain equivalent calculation method for corrugated steel plates and the two-line stiffness equivalent spring of the corrugated steel flange joint. The simplified model was validated as effective by monitoring data. (2) Corrugated steel plates exhibit two stages under loading, namely gap elimination and elastic stages. The elastic stage stiffness of corrugated steel plates decreases with increasing ratio of depth to pitch (RDP), positively correlating with plate thickness when the RDP exceeds 0.333 and otherwise negatively correlated. (3) Corrugated steel lining flange joints exhibit distinct elastic and plastic stages in their linear moment–rotation curves under loading.

Research on the equivalent stiffness of bucket foundations for offshore wind power

Natural frequencies and foundation displacements are vital for the safe operation of offshore wind turbines because of their dependence on the equivalent stiffness of foundations. To investigate this dependency, a finite element method based on substructures is proposed for wide and shallow bucket foundations (aspect ratios of 0.25–1). The equivalent stiffness coefficient (ki) is defined to analyze the influences of the relative skirt flexibility, foundation size, and soil parameters on the equivalent stiffness. The results highlight the predominant impacts of the relative skirt flexibility and aspect ratio on the equivalent stiffness coefficients of bucket foundations. Conversely, certain factors, such as bucket diameter, skirt thickness, and soil modulus, have minimal influences on the equivalent stiffness coefficients. The significance of the Poisson's ratio of the soil is particularly notable, especially concerning the vertical stiffness coefficients. Based on finite element calculations, an empirical formula for bucket foundation equivalent stiffness is proposed and effectively utilized for predicting natural frequencies and evaluating foundation displacements.

Study on axial response of semi-parallel wire suspender with spherical plain bearings

Semi-parallel wire suspenders with spherical plain bearings have untwisting under axial force, and the collision in the untwist process bring adverse effects on the structure. To investigate the axial mechanical response of semi-parallel wire suspenders with spherical plain bearings, the equilibrium equations of suspender are deduced by assuming the distribution of steel wire after deformation, the forms of boundary conditions are distinguished according to the use of spherical plain bearing, and the linear equation expressed by equivalent stiffness coefficients is deduced. The Costello theory and finite element modeling are used to verify the proposed calculation method in this paper, and the influence of suspender parameters on the stiffness equivalent coefficient is analyzed. The results show that the proposed calculation method is suitable for multi-layer semi-parallel wire suspenders; the twist angle and the total number of layers of wires have the greatest effect on the stiffness equivalence factor; when the twisting angle is close to 0, the torsional stiffness of suspender is 0.010–0.21 of that of the steel rod with the same section; the axial tensile stiffness of the conventional suspender is basically the same as that of the steel rod with the same section, but the torsional stiffness is only 0.012–0.023 of the steel rod with the same section, and the tension-torsion coupling effect cannot be ignored. The research results provide a concise method to calculate the axial response of semi-parallel wire suspenders, which lays a foundation for the engineering treatment of the untwisting problem of semi-parallel wire suspender using spherical plain bearings.

Dynamic behavior analysis of systems with friction and hysteretic effects

This paper investigates the steady-state response of a nonlinear system that incorporates both a hysteretic element and frictional contact under harmonic excitation. While previous studies have separately explored hysteretic oscillators and frictional systems, the simultaneous consideration of both elements has received limited attention. In this study, an analytical solution for the steady-state response is proposed, employing the equivalent linearization approach. This method enables the determination of equivalent stiffness and equivalent damping coefficients for the linearized system, which are directly linked to the amplitude of the steady-state response. To validate the derived analytical solution, a numerical model is utilized, demonstrating the agreement between the analytical and numerical results. Moreover, this paper identifies noteworthy dynamic characteristics of the designed system, notably the occurrence of peak amplitude at excitation frequencies lower than the pre-yield natural frequency. Furthermore, the apex point can be adjusted by manipulating the nonlinear parameters. Building upon these observations, a new mounting system is examined, revealing distinctive behavior.

Mechanical Model and Damping Effect of a Particle-Inertial Damper

Particle dampers (PD) are safe, economical, and effective energy-dissipation devices for structures. However, the additional mass of PD must be sufficiently large to provide a better damping effect, and the initial movement condition of particles has a significant impact on the damping effect of PD. In this study, a particle-inertial damper (PID) is proposed to overcome these problems, and its mechanical model is established with and without considering particle collision. Subsequently, the influence of particle rolling friction and particle collision on the inertial amplification capacity as well as the dynamic response of a single degree of freedom (SDOF) structure with non-collision and collision PID (SDOF-PID) are systematically analysed. Finally, the control effects of a PID and a tuned mass damper (TMD) are compared based on two typical optimisation methods. The results indicate that particle rolling friction has little influence on the inertia amplification effect of a PID and the displacement response of a SDOF-PID. Under harmonic excitation, particle collision significantly affects the damping mechanism of a PID by its equivalent inertia coefficient, equivalent damping coefficient, and equivalent stiffness coefficient. The fixed-point theory and ‘performance-cost’ theory can be used to optimise the PID to a certain extent. The damping effect of a PID on the SDOF under the most severe seismic excitation is better than that of the PID under white noise excitation. With respect to the decreasing ratio of 40~50%, the additional mass of the PID is only one thousandth that of the TMD under the same damping capacity demand.

Buckling Analysis of Grid-stiffened Composite Cylindrical Shell

An energy based smeared stiffener model is developed to obtain equivalent stiffness coefficients of a general grid-stiffened composite cylindrical shell. These equivalent stiffness coefficients are used in the Ritz buckling analysis of the shell. Transverse shear properties are important in a stiffened shell and results are obtained for First order Shear Deformation Theory (FSDT) as well as for Classical Laminated Shell Theory (CLST). Parametric study is carried out to find out the effects of various design parameters on specific buckling load of the shell.

A yaw stability-guaranteed hierarchical coordination control strategy for four-wheel drive electric vehicles using an unscented Kalman filter

A novel hierarchical coordination control strategy (HCCS) is offered to guarantee the stability of four-wheel drive electric vehicles (4WD-EVs) combining the Unscented Kalman filter (UKF). First, a dynamics model of the 4WD-EVs is established, and the UKF-based estimator of sideslip angle and yaw rate is constructed concurrently. Second, the equivalent cornering stiffness coefficients are jointly estimated to consider the impact of vehicle uncertainty parameters on the estimator design. Afterwards, a HCCS with two-level controller is presented. The sideslip angle and yaw rate are controlled by an adaptive backstepping-based yaw moment controller, and the computational burden is relieved by an improved adaptive neural dynamic surface control technology in the upper-level controller. Simultaneously, the optimal torque distribution controller of hub motors is developed to optimize the adhesion utilization ratio of tire in the lower-level controller. Finally, the proposed HCCS has shown effective improvement in the trajectory tracking capability and yaw stability of the 4WD-EVs under various maneuver conditions compared with the traditional Luenberger observer-based and the federal-cubature Kalman filter-based hierarchical controller.

Numerical Study on Vibration Response of Compressor Stator Blade Considering Contact Friction of Holding Ring

There are many kinds of assembly structures in heavy-duty gas turbine compressor stator blades which have significant influence on the complex damping vibration characteristics. At present, compressor design is becoming more and more compact, so it is very meaningful to accurately obtain stator blade vibration characteristics of structures with contact damping. Firstly, a fretting slip friction dynamic model was introduced, and then a vibration analysis model of the compressor stator blade with outer ring structure was established based on the slip friction theory. Then, the vibration response of the compressor stator blade was obtained according to various working conditions, and the main factors affecting the vibration characteristics of the stator blade were revealed. Finally, the vibration response of the blade under a particular exciting force condition was simulated. The results show that the damping vibration characteristics of the compressor stator blades were affected by the excitation force and the normal load of the contact surface. The vibration response curve of the stator blade and the equivalent stiffness coefficient of the contact surface were analyzed, and the friction motion of the contact surface changed with the change of the working condition. The model can simulate the nonlinear vibration characteristics of stator blades. This method provides a reference for the vibration safety analysis of compressor stator blades.

Design and experimental characterization of a bypass magnetorheological damper featuring variable stiffness and damping

Magnetorheological (MR) fluid dampers (MRFDs) with variable stiffness and variable damping capability (VSVD-MRFDs) have demonstrated excellent vibration mitigation performance. However, there are limited studies on the development of bypass VSVD-MRFDs which offer both higher dynamic range and output force, apart from simple maintenance and straightforward assembly. In this study, a novel large-capacity VSVD-MRFD with an annular-radial bypass MR valve, as opposed to the typical practice of implementing the valve within the traveling piston in the hydraulic cylinder of the MRFD, is proposed. The main contribution of the present work includes: (a) providing the conceptional design and experimental dynamic characterization of the proposed VSVD-MRFD; (b) investigating the feasibility of the proposed damper for realizing the VSVD characteristics under wide ranges of loading conditions. A test rig was, thus, designed to perform experimental characterization of the proposed VSVD-MRFD under wide ranges of mechanical loading and magnetic field conditions. A qualitative analysis including force-displacement, and force-velocity characteristics, together with a quantitative analysis including dynamic range, equivalent viscous and stiffness coefficients, were conducted as a function of loading frequency, displacement amplitude, and applied current. Results showed a maximum dynamic range and maximum output force of 4.5 and 7.8 kN, respectively. Also, the maximum relative increase in the equivalent viscous and stiffness coefficients were obtained, respectively, as 425% and 488%, when the applied current is increased from zero to 2 A. The results confirm the potential of the proposed VSVD-MRFD for applications in off-road suspension systems. The externally designed bypass MR valve permits a straightforward design modification for realizing wide scalability of damping force in different applications (e.g., off-road vehicle suspension systems).

Influence of Real Lubricant Density–Pressure Behavior on the Dynamic Response of Elastohydrodynamic Lubricated Conjunctions

Abstract The current work investigates the influence of real lubricant density–pressure behavior on the dynamic response of elastohydrodynamic lubricated conjunctions. Such a response is often based on a nonrealistic universal equation of state, despite longstanding evidence of its lack of support by measurements. A finite element framework is employed to investigate the damping and stiffness characteristics of line contact elastohydrodynamic (EHD) lubricating films, subject to a harmonic loading. Both the equivalent stiffness and damping coefficients of lubricating films are found to increase with the base applied external load and its amplitude of oscillation. They decrease however with increasing mean entrainment speed and load oscillation frequency. That is, they both increase as lubricant films get thinner. By comparison with the real density–pressure response of a highly compressible silicon oil, the universal equation of state is shown to underestimate the lubricant film’s stiffness and damping characteristics. The relative deviations in equivalent damping and stiffness coefficients can reach up to about 12% and 25%, respectively. Therefore, realistic lubricant characteristics should always be considered. In particular, the use of the universal equation of state should not be taken for granted, as is customary in the elastohydrodynamic lubrication (EHL) literature. Lubricant density–pressure response is not of a secondary nature when it comes to predicting the dynamic performance characteristics of EHL conjunctions.

Design and Forced Vibration Experiment Study of the Elastic Beam Structure with an Intelligent Boundary

This paper attaches importance to the design and manufacturing of a type of intelligent boundary for vibration control of the elastic beam structure. Artificially set as 12 working states in this study, the intelligent boundary consists of the executing element and the control element. The equivalent stiffness coefficients of the intelligent boundary under each working state are obtained through the experiment. The study on the influence imposed by the intelligent boundary on the forced vibration responses of the elastic beam structure is carried out through the experiment. Based on this work, the equivalent stiffness of the intelligent boundary can be effectively controlled through the change to its working state. The adjustment to the range for the stiffness coefficients of the intelligent boundary can be conducted by changing the line diameter of the springs. In the effective working region, the change to the working states of the intelligent boundary imposes a significant influence on the vibration responses of the elastic beam structure, where an appropriate working state of the intelligent boundary effectively suppresses the vibration at the boundary of the elastic beam. The intelligent boundary can control the vibration of the elastic beam structure through the change to its boundary condition. The control flow of the intelligent boundary is proposed to improve the engineering acceptance of the intelligent boundary.

A vehicle-bridge interaction vibration model considering bridge deck pavement

The bridge pavement subjected to vehicle loads is the most vulnerable component in the highway bridge and overloaded vehicles can result in serious damage to the bridge pavement more easily. Therefore, it is necessary to consider the pavement in vehicle-bridge interaction (VBI) system. However, the bridge pavement is not considered in the traditional VBI system or the viscoelasticity of bridge pavement is ignored. In this study, a new vehicle-pavement-bridge interaction (VPBI) system is established. The viscoelastic asphalt pavement is simulated with the continuously and uniformly distributed spring-damper. The equivalent stiffness coefficient and equivalent damping coefficient of the pavement are determined through dynamic mechanical analysis (DMA) test and finite element method. The equations of motion for VPBI system can be derived using Lagrange equation and the modal superposition method. Then the crucial parameters such as displacement at mid-span of bridge, acceleration of vehicle body, tire contact force, and pavement deformation are obtained. The effects of vehicle velocity, bridge span, tire contact area, pavement stiffness coefficient, and pavement damping coefficient on VPBI responses are analyzed. The results show that (1) The pavement is compressive during vehicle passing bridge and its dynamic deformation fluctuates around initially static deformation of pavement. (2) Increasing the stiffness of pavement can rapidly reduce the deformation of pavement, while the pavement damping has the opposite effect. (3) The increase of bridge span can conduct exponentially growth of dynamic responses of VPBI system. The pavement deformation will increase by 7.9% if the bridge span is lengthened by 5 m. This study provides a reliability response analysis method for VPBI system.

A state‐dependent switching controller design approach for linear systems under control constraints

In the present work, a state-dependent switching controller design approach for controlled systems under control constraints is proposed. First, the controlled system with control gain parameters is converted into a single-degree-of-freedom vibration system by using an invertible transform. Second, based on the physical significance of the single-degree-of-freedom linear vibration systems, two pairs of gain parameters are properly chosen to obtain two subsystems with sufficiently small equivalent negative-damping coefficients and sufficiently large equivalent stiffness coefficient difference. Third, by introducing an identical invertible transform, two subsystems are converted into the form of vibration systems simultaneously. As such, it is reasonable to compare the two corresponding energy functions directly. Fourth, two switching lines with maximum and minimum energy ratio between two subsystems are derived to construct a state-dependent switching law. Combining the selected control gain parameters, a state-dependent switching controller is then designed for stabilizing the original controlled system with control constraints. In the end, the effectiveness and robustness of the developed control approach are illustrated by three numerical examples.

Dynamic Coefficients of Tilting Pad Bearing by Perturbing the Turbulence Model

Tilting pad bearings are appropriate for the trend of high efficiency and reliability design of rotating machinery due to their high stability. The laminar and turbulent flow states exist in the lubricating oil film of high-speed and heavy-load tilting pad bearings simultaneously. By perturbing the multiple flow state lubrication model with a partial derivative method, together with the pad-pivot structural perturbations, the frequency-dependent stiffness and damping coefficients of tilting pad bearings, embracing the effect of dynamical variations of both turbulence and pressure-viscous, were numerically solved in this research. The importance of each perturbed variable was studied, and the results indicate that the perturbed film thickness included in turbulence coefficients perturbations is significant enough to be taken into account otherwise the equivalent stiffness coefficients will be obviously overestimated. Unlike the perturbed film thickness, the consideration of the perturbed viscosity is optional, because it makes the stiffness and damping coefficients larger at both laminar and turbulent flow states. For a simplified simulation and conservative prediction results, the perturbed viscosity can be neglected.

Stability and Modal Conversion Phenomenon of Pipe-In-Pipe Structures with Arbitrary Boundary Conditions by Means of Green’s Functions

In this paper, a closed-form frequency equation of the pipe-in-pipe (PIP) structure with arbitrary boundaries is obtained. The frequency equation is derived from Green’s function of the transverse forced vibration of the PIP structure and takes into account the effects of internal two-phase flow and axial pressure. The reliability of the method in this paper is proved by comparison with the published literature. In the numerical discussion part, the PIP structures with clamped-clamped, clamped-free, and elastic boundary conditions are used as examples to discuss. The effects of equivalent stiffness coefficient, internal flow velocity, and gas volume fraction on the stability of PIP structure are studied. The results show that the stability of the PIP structure is better than that of the single-pipe structure, and the greater the equivalent stiffness coefficient of the elastic layer, the higher the critical flow velocity of the structure. In addition, a modal conversion phenomenon existing in the PIP structure is discovered. There are different forms of modal conversion for different boundary conditions, and the modal conversion makes the order of instability of the PIP structure different from that of a single-pipe. The conclusion of this paper has positive significance for the dynamic research of PIP structure.

Bamboo-inspired self-powered triboelectric sensor for touch sensing and sitting posture monitoring

Self-powered sensors, as the essential information notes in low power consumption of Internet of healthcare, exhibit a great potential to monitor the state of human health in real time. In this work, a bamboo-inspired self-powered triboelectric sensor (BSTS) using 3D-printing and freeze-drying fabrication processes is presented. The designed triboelectric layer with hollow and bamboo-joint microstructures has a low equivalent stiffness coefficient and a great deformation limit, contributing significantly to the high sensitivity and wide detection range of BSTS. An electromechanical model was proposed to better understand the influence mechanism of inner structures in the dielectric layer on the sensing performance. The sensitivity of the BSTS can reach 3.627kPa−1 in the pressure range of 0–8kPa and still keep 1.264kPa−1 in the range of 8–80kPa. Moreover, the BSTS is able to detect tiny pressure as low as 5 Pa and possesses ultra-fast response time of 40 ms. A touch-controlled lighting system and a sitting posture monitoring system based on the BSTS for healthcare application are demonstrated. This strategy paves a way to fabricate the self-powered triboelectric pressure sensor with high sensitivity in all working ranges.

Tuning of relative spatial distributions of a two-component ion system to improve sympathetic cooling efficiency

Herein, a fine-tuning method is proposed for the spatial distributions of a mixed three-dimensional (3D) ion system in dual radio frequency (RF) linear Paul traps to achieve efficient sympathetic cooling. The dual RF field matching, efficient capture method and transient process of the intrinsic micromotion of the mixed ion system are analyzed quantitatively by numerical simulations. The 3D correlation coupling characteristics between intrinsic micromotion and secular motion of ion system are obtained. It is found that reasonable low-frequency trapping potential can produce ultra-low-frequency pulling effect on ions with low mass-to-charge ratio (M/Q), which is beneficial to the dynamic coupling between ions with large M/Q differences. The effects of equivalent stiffness coefficients [Formula: see text] on the relative spatial configuration and dynamic coupling process of mixed 3D ion crystals with large M/Q differences are discussed. By tuning [Formula: see text], radial distributions of laser-cooled ions (LCIs) and sympathetically cooled ions (SCIs) that do not conform to the rules based on M/Q are realized. The optimum sympathetic-cooling efficiency occurs, where [Formula: see text] is approximately equivalent to [Formula: see text]. These results are applicable to studies such as cold ion clocks, quantum logic manipulation, antimatter synthesis, regulation of cold chemical reaction, and precise spectral measurements based on sympathetic cooling.

Measured and investigated nonlinear dynamics parameters on bolted flange joints of combined rotor

The complex micro-slip phenomenon of the contact interface will lead to the nonlinear stiffness of the connection structure, as well as the structural damping and energy dissipation. As the most important connection structure of the combination rotor, the mechanical properties of bolted flange joint interface are needed in the dynamic analysis of the combined rotor. Therefore, it is urgent to model and test the friction contact interface in the nonlinear dynamic analysis of rotor. In this paper, two sets of mechanical characteristics test system were built to test the dynamic parameters of tangential and bending directions of the bolted flange joint interface. Then, the mechanical behavior and the change regularities of dynamics parameters were studied under different external excitation, bolt distribution and tightening torque. The results show that once the bolt preload is above the rated torque, stiffness softening behavior is not significant; and then the tangential stiffness of the joint interface tends to be stable, with the variation range of 8.08∼8.96e8 N/m; the equivalent bending stiffness coefficient is about 3.38∼3.83e6 N·m/rad. With the decrease of bolt preload, the external excitation and the number of bolts have a significant effect on the stiffness reduction of the joint. Finally, the change interval of the dynamics parameters of the interface obtained by the experiment provide basis for the uncertainty dynamic analysis and optimization of the rotor.