Spring Model Research Articles

Design, Modeling, and Characteristics Analysis of Halbach Permanent Magnetic Spring

Magnetic springs, which can be used to replace traditional mechanical springs, have many advantages, such as necessitating no physical contact, generating no friction, no vibration or noise, and having a long lifespan. Nevertheless, their strong nonlinearity limits their widespread application. In this study, we developed a novel permanent magnet spring to address this issue: a Halbach permanent magnetic spring, with a large levitation force and an approximately linear force characteristic curve. First, we introduce the structure and the parameters of the Halbach permanent magnetic spring. Second, we describe the levitation force performance and the stiffness performance of the Halbach permanent magnetic spring using finite element analysis. Third, we analyze the trends through which different parameters influence the levitation force performance and stiffness performance. Finally, we provide recommendations for the future design of and improvement in the Halbach permanent magnetic spring.

Stiffness model for pneumatic spring with air-diaphragm coupling effect

Stiffness model for pneumatic spring with air-diaphragm coupling effect

Comprehensive analysis of mode-I cracking in ice: Exploring full-range rate dependency

Ice, as a crystalline material, demonstrates a unique response in its mode-I fracture toughness to variations in loading rates, characterized by an initial decrease and subsequent increase in toughness. This phenomenon has been explored in limited studies. In this work, an extensive numerical analysis of mode-I ice cracking is conducted by using the distinct lattice spring model (DLSM). Norton-Bailey Drucker-Prager DLSM (NB-DP-DLSM) is initially employed to investigate the classical explanations of creep and stress relaxation for the anomaly observed in ice’s fracture toughness at lower loading rates, but this approach does not successfully replicate the experimentally observed strain rate dependency. Then, two rate-dependent constitutive models are introduced to further examine the mode-I fracture mechanics of ice. Our numerical simulations of three-point bending tests show that the relationship between fracture toughness and strain rate at lower levels is more accurately captured by rate-dependent models. For higher strain rates, our numerical modeling of notched semi-circular bending tests indicates that a rate-independent constitutive model can replicate the loading rate dependency of ice’s mode-I fracture toughness. In conclusion, these observations suggest that a reverse-stage-like dynamic constitutive model for DLSM can potentially capture the full range of loading rate dependencies observed in ice.

The failure of edge-cracked hard roof in underground mining: An analytical study

Hard roof is the primary concern of strata control in underground mining. Various techniques have been utilized to fracture the hard roof and control the failure of strata. Understanding the impact of cracks on strata behaviour is vital for optimizing strata control strategies. In this study, the hard roof was regarded as a beam structure with different loading, support, and boundary conditions. The equivalent spring model was adopted to represent the edge-cracked section of the hard roof, which allows additional rotation at the crack location. A piecewise-defined function was developed for solving equations of hard roof in the vicinity of the crack section. By combining the hard roof beam model and the equivalent spring model, the impact of a crack on the hard roof can be measured. A case study was carried out to explore the impacts of crack location and crack depth on the mechanical state of the hard roof. Results showcase the failure of the hard roof controlled by the crack depth and greatly influenced by the crack location. From the perspective of coal burst prevention, roof fracturing should be implemented at the high-stress area of strata, whereas it has been challenging in practice to determine such a location precisely. To address this challenge, it was suggested that hard roof fracturing should be carried out before coal seam de-stressing, increasing the likelihood of a crack occurring in a high-stress area. By adopting the proposed method, the mechanical state of the edge-cracked hard roof can be quantified.

The effect of the floor-wall interaction on the post-elastic rocking behaviour of segmented CLT shear walls

This paper investigates the impact of floor-wall interaction on the lateral rocking behaviour of two-panel segmented Cross-Laminated Timber (CLT) shear walls through experimental testing and numerical analyses. The study extends a simplified equivalent spring modelling approach to the post-elastic domain to accurately simulate the effects of floor-wall interaction. Experimental withdrawal tests on four types of floor-to-wall connections, utilizing Fully Threaded Screws (FTSs) and Partially Threaded Screws (PTSs) at varying inclinations, revealed that FTSs exhibited higher stiffness and load-carrying capacity but lower deformation capacity compared to PTSs. The maximum force of FTS connections was about twice that of PTS connections, while PTS connections achieved ultimate displacements up to seven times greater. Numerical analyses using a parametric framework showed that floor-wall interaction increases the stiffness and lateral load-carrying capacity of segmented shear walls, with a significant reduction in deformation capacity. These analyses also demonstrated that the equivalent spring can effectively replicate the influence of floor-wall interaction on shear wall lateral response. Expressions for calculating the strength and ductility of this equivalent spring were developed, considering a limited set of geometric and mechanical parameters. These expressions provide a simplified yet accurate method for estimating the effects of floor-wall interaction. The findings suggest that understanding the interaction between floor and wall elements is crucial for optimizing the performance of segmented CLT shear walls in seismic and lateral load scenarios.

Equivalent spring model and efficient numerical strategy for simulating mechanical performance of seam-clip connections in aluminum alloy high-vertical standing seam metal cladding systems

The equivalent spring model has been validated as effective in simulating the mechanical performance of seam-clip connections in typical steel high-vertical standing seam metal cladding systems (SSMCSs). To enhance its applicability, this study further refines the equivalent spring model by developing thickness-related stiffness calibration models and an efficient numerical strategy for seam-clip connections in the aluminum alloy SSMCSs. Tensile tests are initially conducted to analyze the material properties of specimens with three commonly used thicknesses. Subsequently, the mechanical performance of the aluminum alloy seam-clip connections with different thicknesses is systematically studied, followed by the development of thickness-related stiffness calibration models. The tensile test results reveal notable nonlinearities and variability in the mechanical performance. To simulate this mechanical performance equivalently in the finite element (FE) model using data from tensile tests, an efficient numerical strategy is developed. This strategy involves methods for implementing simplified connections to establish the equivalent spring model and techniques for efficiently creating these simplified connections. It is found that both connectors and spring connections with calibrated parameters from tensile tests can effectively replicate the mechanical performance of seam-clip connections. To further validate the effectiveness of the equivalent spring model and the efficient numerical simulation strategy, a comparison of structural responses of the double-span sheet module (DSSM) obtained from tests and FE analysis is performed. The results reveal that all simulated structural responses on the DSSM with three common sheet widths align well with the corresponding experimental results. This highlights the feasibility and efficacy of the equivalent spring model and efficient numerical strategy in simulating the mechanical performance of seam-clip connections and delving deeper into the structural responses of the aluminum alloy high-vertical SSMCSs.

The Method of the Natural Frequency of the Offshore Wind Turbine System Considering Pile–Soil Interaction

Accurately and efficiently evaluating the influence of pile–soil interaction on the overall natural frequency of wind turbines is one of the difficulties in current offshore wind power design. To improve the structural safety and reliability of the offshore wind turbine (OWT) systems, a new closed-form solution method of the overall natural frequency of OWTs considering pile–soil interactions with highly effective calculations is established. In this method, Hamilton’s principle and the equivalent coupled spring model (ECS model) were firstly combined. In Hamilton’s theory, the Timoshenko beam assumption and continuum element theory considering the three-dimensional displacement field of soil were used to simulate the large-diameter monopile–soil interaction under lateral load in multilayer soil. Case studies were used to validate the proposed method’s correctness and efficiency. The results show that when compared with the data of 13 offshore wind projects reported in existing research papers, the difference between the overall natural frequency calculated by the proposed method and that reported in this study is within ±10%. This calculation method achieves the goal of convenient, fast and accurate prediction of the overall natural frequency of offshore wind systems.

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.

Implementation of the Zienkiewicz–Pande Model into a Four‐Dimensional Lattice Spring Model for Plasticity and Fracture

ABSTRACTPlasticity and fracture problems have always been hot topics in numerical methods. In this work, a universal implementation procedure for the elasto‐plastic constitutive model is developed in the four‐dimensional lattice spring model (4D‐LSM), in which the Jaumann stress rate is incorporated to exclude the influence of the rigid rotation in the particle stress, expanding the ability of 4D‐LSM to deal with large elastic deformation problems by its own to large plastic deformation problems. As an example, the Zienkiewicz–Pande (ZP) constitutive model is implemented. Several numerical examples are carried out to check the performance of the implemented model. Through a comparison with analytical solutions, available experimental data, and other numerical results, the stability of the developed plastic framework and the correctness of the stress calculation scheme are verified. Meanwhile, numerical results show that the developed code is capable of solving elasto‐plastic large deformation problems. With the advantage of 4D‐LSM in handling fracture problems, the ability of the embedded model to solve plastic fracture problems is verified with a simple maximum deformation failure criterion.

Development of Johnson-Cook-Distinct Lattice Spring Model and its application in projectile penetration into metal targets

Projectile penetration into metal targets plays an important role in modern protective structure engineering, damage assessment in terrorist attacks, and car clash accident analysis, etc. The penetration process of metal target has been characterized by large deformation, high temperature, high pressure, and dynamic damage, which can be classified as a continuous-discontinuous dynamical process. To capture the dynamic responses of metal materials subjected to high-speed penetration, Johnson-Cook model has been implemented in a continuum-discrete model, namely, Distinct Lattice Spring Model (DLSM). This is achieved by redefining the constitutive model in DLSM, with plasticity hardening (plastic strain effects), strain rate, temperature, damage evolution, equation of state, etc., being taken into account, thus developing a new numerical model called JC-DLSM model. This model is validated through studying Taylor rod high-speed impact experiments, thin metal plate penetration experiments, and projectile impact experiments on titanium alloy targets. Good agreement between numerical modelling and experimental data has been achieved for all cases considered, thereby demonstrating the capability of JC-DLSM to reproduce the damage characteristics and ballistic limits of metals under high-speed impact/penetration. Then, the impacts of projectile shape, metal target surface configuration on the characteristics of penetration damage patterns have been investigated. This contributes to a better understanding of the damage mechanisms of metal targets upon penetration or impact, facilitating the computational means for analysing and optimizing the penetration resistance of protective engineering structures.

Study of Composite Light Weight Leaf Spring

The fundamental Objective of this project is to represent plan and exploratory examination of composite leaf spring made of glass fiber reinforced polymer. The intention is to look at the load carrying capacity, weight and stiffness effective of composite leaf spring with that of steel leaf spring. The design imperatives are stresses and deflections. The measurements of a current ordinary steel leaf spring of a light business vehicle are taken. Static investigation of leaf spring is likew ise performed utilizing analysis software and compared with experimental results. Limited component investigation with full load on 3D model of composite multi leaf spring is done utilizing analysis software and the analytical results are compared with experimentalresult

Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping

In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap d: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient k>0. The oscillation, under the influence of AC–DC voltage V(t)=vdc+vaccos2πTt, is ruled by the following singular differential equation my′′+[A(d−y)3+Ad−y]y′+ky=θ0A2V2(t)(d−y)2.Here, y is the vertical displacement of the moving plate (y is always assumed to be less than d), m>0 is its mass, A>0 is the electrode area, and θ0>0 is the absolute dielectric constant of vacuum. Taking d as the parameter, we show the existence of saddle–node bifurcation of T-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of T-periodic solutions: as d tends to +∞, one of them uniformly tends to +∞ at the rate of d, while the minimum values of the second class tend to, or cross, 0.

Prediction of flanking sound transmission through cross-laminated timber junctions with resilient interlayers

While the low weight and high bending stiffness of cross-laminated timber (CLT) are key to its popularity, these properties also contribute to poor acoustic performance. Notably flanking sound transmission is a critical factor, driving the need for vibration reduction solutions such as resilient interlayers in the junction design. However, due to the complex material behavior of CLT and resilient interlayers, the improvement related to these solutions is difficult to predict. In this research, an analytical model with low computational cost is developed to evaluate the vibration reduction index Kij for CLT junctions with resilient interlayers. The CLT panels are considered as thin orthotropic plates with homogenized material properties. Three potential material models are proposed for the interlayer: it is considered as a thin plate, a thick flexible layer with out-of-plane motion governed by shear or distributed springs. The prediction model is experimentally validated for junctions consisting of CLT panels, with and without resilient interlayers. For junctions without interlayers, the predicted and experimentally determined vibration reduction index Kij correspond most closely when the junction is realized with stiff connectors. In this case, the predictions are moderately accurate with deviations below 5 dB in 1/3 octave bands up to approximately 2000 Hz for both corner and coplanar transmission paths. For junctions with resilient interlayers, the shear interlayer model exhibits the best performance with deviations of less than 5 dB in most 1/3 octave bands. For frequencies below 1000 Hz, the accuracy of the simplified spring model is comparable to that of the thick layer model. Simulations with equivalent isotropic material parameters yield slightly inferior predictions than for orthotropic parameters if the degree of orthotropy of the panels is high.

A novel method for shape pre-optimization of fir-tree couplings in blade-disk connections

This paper presents an innovative approach to the pre-optimization of the blade-disk fir-tree coupling, with the intent of advancing the design and analysis of such connections in turbomachinery. These couplings are subjected to extreme mechanical loads, primarily from centrifugal forces during turbine operation, compounded by aerodynamic and vibrational forces. These conditions pose substantial challenges to ensuring the structural integrity and long-term reliability of the connections. Failure of the blade-disk attachment can lead to catastrophic consequences, making the optimization of these joints crucial.The proposed pre-optimization method addresses this by optimizing the distribution of material between the blade-root, disk-post, and engaging teeth, ensuring uniform load distribution across all teeth. A novel one-dimensional radial spring model is introduced, which simplifies the representation of the joint. The method achieves accurate load-sharing without requiring knowledge of the spring stiffness values, instead relying on stiffness ratios, which in turn are approximated equal to ratios between the areas of throat cross-sections. This approach effectively balances tensile stresses, contact pressures, and, proportionally, local stress peaks, all while streamlining the design process.Validation against finite element models demonstrates the model’s accuracy across different joint configurations with varying numbers of teeth. This method serves as a foundation for more detailed analysis and future optimization efforts, including those targeting specific failure mechanisms such as fretting fatigue and peak stress mitigation. Overall, this work offers a practical and efficient approach to enhancing the performance and reliability of blade-disk connections in turbomachinery.

A Spring Model for Pullout Behavior of Curved, Flexible Structures Embedded in Soil

ABSTRACTThe shape and flexibility of embedded structures, such as tree roots, piles, and anchors, have important impacts on the pullout behavior. However, the rate and manner of mobilization of soil resistances along such structures has not been rigorously explored across a wide range of shapes and structural properties. A spring model for computing compatible displacements of the structure and soil for curved, flexible structures is defined, validated against commonly used methods for computing pile pullout behavior, and then parametrically explored to demonstrate how resistances are mobilized along the length of such structures. The present model allows description of combined axial and transverse loading of these nonlinear structures. The simulation results for the case of normally consolidated clay show that the curvature of a structure causes the distribution of bearing resistance to extend further along the structure than for linear cases. The requirement of equilibrium of the structure produces a coupling between the mobilized bearing and tensile resistance in terms of rate of development and magnitude. Thus, the choices of structure shape impact the magnitude and distribution of mobilized resistance of embedded flexible structures. Implications for anchorage of tree root structures and principles of bioinspired design of anchorage systems are discussed.

Analysis of electrothermal shape memory alloy and disc-type magnetorheological fluid combined transmission

In this article, a disc-type magnetorheological fluid and electrothermal shape memory alloy spring-loaded slider friction combined transmission device is proposed. A heating model of the electrothermal shape memory alloy spring is used to calculate the relationship between current and spring temperature. Experiments are used to investigate the link between spring temperature and restoring force, as well as magnetorheological fluid temperature, magnetic flux density, and shear yield stress. The torque characteristics of the combined device were evaluated utilizing the coupled magnetic field and thermal field finite element method for the characteristics of the two materials. The results showed that the magnetorheological fluid showed an approximately linear decrease in shear yield stress at high temperatures, up to 37.94 %. The heating rate of the electrothermal shape memory alloy spring was influenced by the surrounding temperature, and the heating rate was too fast to easily cause the time lag of the martensite phase transformation, resulting in the spring restoring force becoming smaller, and its restoring force shows a non-linear increase when the temperature rises from 22 °C to 100 °C. By applying the two materials together, the torque of the combined transmission has been significantly increased, by 91.58 % compared to the same size magnetorheological transmission.

Discrete element method simulation of high-speed vehicle collisions with road barrier systems

AbstractThe behavior of road or perimeter protection barriers under vehicle impact are usually investigated based on crash tests and finite element (FE) numerical approaches, which are ether expensive or time-consuming. Several studies have proposed to reduce the computation time of the numerical analysis by substituting the complex FE models of vehicles using simplified mass–spring–damper system models. However, these models have drawbacks since consideration of different vehicle impact angles is difficult and they are unable to correctly simulate the risk of high-speed vehicle collision running over the barrier. In this paper, a new approach is proposed to simulate the collision of vehicles with barriers based on the discrete element method (DEM). Here, to save computation time only a handful of 3D non-spherical particles are used to represent the barrier and vehicle. These particles are generated based on the super-quadric function, which is capable of generating a variety of shapes needed for the model. The contact detection and evaluation are carried out based on discrete function representation of the particles with uniform sampling. The bond between two discrete elements is defined using a nonlinear cohesive beam model since the distance between the elements is relatively large. The simulation results obtained based on this approach are more accurate and complete than the simplified mass–spring models and computationally more efficient than the FE model.

Nanoscale dynamics of the cadherin–catenin complex bound to vinculin revealed by neutron spin echo spectroscopy

We report a neutron spin echo (NSE) study of the nanoscale dynamics of the cell-cell adhesion cadherin-catenin complex bound to vinculin. Our measurements and theoretical physics analyses of the NSE data reveal that the dynamics of full-length α-catenin, β-catenin, and vinculin residing in the cadherin-catenin-vinculin complex become activated, involving nanoscale motions in this complex. The cadherin-catenin complex is the central component of the cell-cell adherens junction (AJ) and is fundamental to embryogenesis, tissue wound healing, neuronal plasticity, cancer metastasis, and cardiovascular health and disease. A highly dynamic cadherin-catenin-vinculin complex provides the molecular dynamics basis for the flexibility and elasticity that are necessary for the AJs to function as force transducers. Our theoretical physics analysis provides a way to elucidate these driving nanoscale motions within the complex without requiring large-scale numerical simulations, providing insights not accessible by other techniques. We propose a three-way "motorman" entropic spring model for the dynamic cadherin-catenin-vinculin complex, which allows the complex to function as a flexible and elastic force transducer.

Hybrid modular construction system “INNO3DJOINTS”: Experimental behaviour and numerical modelling of isolated sub-frames

With the growing demand for sustainable, cost-effective and overall-efficient building solutions, the need for dependable modular construction systems is steadily on the rise. In the present paper a novel hybrid modular construction system named INNO3DJOINTS is introduced, employing cold-formed welded steel tubular columns, fabricated according to EN 10219, and cold-formed steel thin-wall section based truss-girders, joined by the innovative plug-and-play (P&P) connector, designed to provide ease-of-assembly and -disassembly. The experimental investigation conducted on isolated sub-frame configurations of the novel system is presented, where 6 full-scale specimens were subjected to horizontal and vertical loading. The test configurations differed in the P&P joint socket thickness and the absence/presence of the light steel framing (LSF) wall, encased with oriented strand board (OSB). In addition, a numerical model for predicting the system's global behaviour is proposed, developed in SAP2000. Initially, the behaviour of the employed P&P joint configurations, categorized as partial-strength, is characterized using experimental and validated ABAQUS finite element model (FEM) data, resulting in a spring model implemented into the global FEM. Finally, the numerical and experimental results are compared and discussed, leading to conclusions regarding the system's 2D structural performance, identified behaviour governing phenomena, P&P joint influence, LSF wall and OSB contribution, as well as the capabilities of the developed FEM.

Interface stiffness identification of rough and weak bonded interface using developed ultrasonic reflection phase derivative spectrum

The thickness and interface roughness of coatings both affect the interface bonded quality. Existed ultrasonic testing methods based on traditional phase screen approximation or spring model assumption are difficult to simultaneously identify the interface roughness and stiffness of coating. This paper, a new method for integrated identifying coating thickness, interface roughness, and interface stiffness using developed ultrasonic reflection phase derivative spectrum (URPDS) is proposed. A phase-screen-approximated spring-model (PSASM) for ultrasound vertically propagating into rough and weak bonded interface is constructed. On basis of PSASM, a URPDS of coating/substrate structure is developed for identifying the interface stiffness and other parameters of coated parts. Cross-correlation analysis is used to eliminate the phase deviation of URPDS introduced by reference signal and initial phase of tested signal. Sensitivity analysis is used to determine the high-sensitivity regions of URPDS to interface roughness and interface stiffness. Genetic algorithm optimization is used to achieve integrated identification of coating thickness, interface roughness, and interface stiffness. The rationality of PSASM is verified through numerical simulation using a series of coating/substrate models with rough and weak bonded interface, and the relationship between the high-sensitivity regions and the high-precision measurement ranges of interface roughness Rq and interface stiffness Kn is clarified. Ultrasonic experiments are implemented on Nickel-coating samples and coated parts using plane wave probe. The coating thickness, interface roughness, and interface stiffness could be identified accurately, which shows that the proposed URPDS method can identify the interface stiffness of rough contacted dissimilar media or coated parts with rough interface.