Mass-spring Model Research Articles

Uphill dynamics: A spring-mass model analysis of sloped walking

This study explores the biomechanics of uphill running by adjusting and analyzing the spring- mass model. Modifications specific to slope conditions were made to the equations of motion. Despite facing challenges in code execution, an in-depth investigation of the models physical characteristics was conducted through both static and dynamic state analyses. The findings aid in the theoretical understanding of motion on sloped surfaces and provide valuable experience for subsequent research.

Ion crystal size and structure in Paul traps.

This article presents a relationship between the size of an ion crystal in a Paul trap and the radio frequency potential applied on the central ring electrode. Using a simple and elegant derivation it has been shown that the distance of an ion in the crystal from the trap center is proportional to power of the applied radio frequency voltage. The validity of this power law has been demonstrated on ion crystals having up to 13 ions. A spring-mass model has been presented to predict structure of ion crystals in Paul traps operating in the Dehmelt regime. Structures are obtained by minimizing the total potential energy stored in an ion ensemble. The total potential energy is taken to be the sum of the electrostatic potential due to ion-ion interaction and the potential energy stored in the springs. Structures of crystals having up to 13 ions predicted by our model have been verified by comparing them with results obtained by direct numerical simulations.

Study on theoretical model and actual deformation of weft-knitted transfer loop based on particle constraint

Abstract In order to derive the structural properties and deformation behavior of the weft-knitted transfer fabric, a multilayer spring-mass geometric circle model with weft-knitted transfer loop is provided in conjunction with the fabric samples’ image. Eight type-value points were utilized to control the transfer loop’s morphological structure. Connections between the type-value points and the particle system were established. Non-uniform rational B spline curves and texture mapping were utilized to create the three-dimensional impression of the weft-knitted transfer loop. By measuring the offset of the type-value points in conjunction, the actual deformation of the weft-knitted transfer fabric was determined. Utilizing a cylindrical envelope box for collision detection, the possible unreasonable interpenetration phenomenon in the simulation was solved, and thus a stable weft-knitted transfer loop structure was obtained. Using Microsoft Visual Studio and C#, the deformation of the weft-knitted transfer fabric was simulated, the simulated weft-knitted transfer fabric’s deformation pattern accurately depicts the fabric’s three-dimensional structure and deformation behavior while also matching the structural characteristics of the fabric sample. The findings demonstrate that the theoretical structural model of weft-knitted transfer fabric constructed can accurately represent the loop’s structure, and the actual deformation of the simulated weft-knitted transfer fabric is consistent with the deformation characteristics of the fabric sample. Weft-knitted transfer fabric’s deformation behavior may be effectively reflected by the multilayer spring-mass geometric circle model, allowing for the modeling of the fabric’s morphology and 3D structure.

High sound pressure level sound absorption in acoustic resonant metamaterials: adapting a mass-spring model for nonlinear effects

In aerospace, conventional materials as micro-perforated panels backed by cavities are used to absorb sound at high sound pressure levels. These solutions are not optimal at low frequencies compared to resonant acoustic metamaterials, which present an acoustic absorption peak with a wavelength-to-thickness ratio greater than conventional materials, i.e., a ratio>>5. Acoustic resonant metamaterials are mainly studied at low sound pressure levels (<100dB). However, they are sensitive to increasing the amplitude of acoustic excitation. The studied metamaterial is composed of a compact array of thin single-perforation panels spaced by thin cylindrical air cavities. In a previous study, a mass-spring model was developed for low sound pressure levels. The perforations are modeled by equivalent masses, whereas cavities by equivalent springs. In the perforations, high values of the acoustic velocity lead to an increase in the acoustic resistance of the material. This effect is considered by the Forchheimer parameter in the mass-spring model. To determine this parameter and to study how it is impacted by the metamaterial geometry, the computational fluid dynamic method is used. The mass-spring model is adapted with a nonlinear term. The adapted model agrees well with simulations by finite element method at sound pressure levels up to 140dB.

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.

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.

Broadband frequency vibration isolation in pipelines using BCC meta-lattice sandwich sleeves

Pipeline structures usually suffer from serious vibration problems due to pressure pulsation and fluid excitation, potentially compromising their structural stability and service life. In this paper, a lightweight body center cubic (BCC) meta-lattice sandwich sleeve is proposed to isolate vibrations in pipelines within a broadband frequency range. The vibration isolation is based on the concepts of the locally resonant metamaterials. An equivalent mass-spring model is conducted to theoretically predict frequency range and elucidate the generation mechanism of the band gap. In addition, an analytical methodology is also proposed to predict the equivalent stiffness of the BCC meta-lattice sleeves. The influence of geometric parameters and configuration layout of the meta-lattice on the flexural wave band gap is further investigated. Numerical studies and experimental tests are also conducted to validate the vibration isolation effectiveness, considering various pipe lengths and different numbers of meta-lattice units. The proposed design provides fundamental support for the design of lightweight vibration isolation structure for pipelines.

Low-velocity impact response of sandwich plates with corrugation star-shaped honeycomb hybrid core

Compared with traditional lightweight corrugation and honeycomb cores, the novel cellular structure exhibiting a negative Poisson's ratio possesses distinctive mechanical deformation features, making it suitable for modeling lightweight sandwich structures. Therefore, the concept of combining the auxetic honeycomb core with folded corrugations is proposed to construct a new type of corrugation star-shaped honeycomb (SSH) hybrid core for studying the dynamic behavior of sandwich panels subjected to low-velocity impact. Integrate Hertz elasticity theory and first-order shear deformation theory (FSDT) to develop an equivalent analytical model, and derive the equations of motion through Hamilton principle. To model contact force interactions during dynamic processes, a spring-mass model is utilized. Analytical solutions are derived for predicting transverse displacement with Duhamel's principle and Navier's method. Numerical simulations are conducted using the Abaqus commercial software, and the validity of the results is confirmed by comparing them with findings in the existing literature. Based on this, effective strategies for enhancing the sandwich panel's resistance to low-velocity impacts are proposed by examining the influence of different side length ratios, thickness ratios, and cell concave angles. In comparison to the corrugation re-entrant hexagonal honeycomb hybrid core sandwich panel structure, the corrugation SSH hybrid core sandwich panel structure reduces transverse displacement by 33.6 % at the same impact velocity.

An Efficient Progressive Grid Generation Method Considering Internal Quality and Boundary Grid Adjustment

Due to its elegant appearance and high structural efficiency, free-form grid structures are increasingly adopted in architectural design. However, it is a challenging task for engineers to generate a uniform and well-shaped grid on a free-form surface while considering the processing of both interior and boundary of grid, especially for composite surfaces. To generate well-shaped grids with uniform rods, regular cells and smooth visual effects over free-form surfaces, this paper develops an innovative triangular grid generation method capable of automatically managing grid boundary with extending initial surface. In the method, nodes of grid structure are considered to be zero mass particles and are progressively added to the extended surface from geometric center to the boundary of surface. A boundary-processing algorithm is then established to evenly distribute nodes on initial boundary curve, while ensuring that grid cells at the boundary do not exhibit elongation. A mechanical simulation system based on spring-mass model is improved to optimize spatial grid structure, rods of grid are regarded as linear springs, a surface attraction force and an anchoring force are performed on grid nodes. Moreover, the proposed method allows architects change grid direction and rods length easily, which can greatly improve the efficiency of design. Several case studies show that the method can effectively avoid distortion of grids and generate well-shaped grids that can meet aesthetic requirements.

Recreating evolutions of human walking gesture by spring-mass walking model

The evolution of human walking gestures is the evolution to adapt to new environments. Evolutionary adaptations in humans, including longer, straighter legs and increased average BMI, were examined using a spring-mass walking model to analyze their impact on locomotion efficiency. Spring-mass model considers walking problems in mechenics dimension. The simulation seperates variations that affects human walking to four independent parameters: mass, touch-down degree, leg length and leg stiffness. Simulation results demonstrated that model has lighter mass, which walks in smaller touch-down degrees gaits by lengthened, robust legs improved walking efficiency. By comparing CM displacement, research also addressed discrepancies between flat and normal gaits. Flat walking compensates its forward moving efficiency and distance per step to gain smaller vertical CM amplitude, accompanied by larger vertical CM amplitude and slower walking speed. Stronger legs and a curved, large-step gait reduced knee and waist pressure in individuals with higher BMI. The study highlights the adaptive significance of human physiological evolution in relation to locomotion efficiency. The study will provide insights into bipedal explanation for evolution tendency and existing problems.

Uncovering Pattern-Transformable Soft Granular Crystals Induced by Microscopic Instability

Abstract Upon compression, some soft granular crystals undergo pattern transformation. Recent studies have unveiled that the underlying mechanism of this transformation is closely tied to microscopic instability, resulting in symmetry breaking. This intriguing phenomenon gives rise to unconventional mechanical properties in the granular crystals, paving the way for potential metamaterial application. However, no consistent approach has been reported for studying other unexplored transformable granular crystals. In this study, we present a systematic approach to identify a new set of pattern-transformable diatomic granular crystals induced by microscopic instability. After identifying the kinematic constraints for diatomic soft granular crystals, we have generated a list of feasible particle arrangements for instability-induced pattern transformation under compression. Instead of computationally intensive finite element models (FEMs) with continuum elements, we adopt a simplified mass-spring model derived from granular contact networks to efficiently evaluate these feasible particle arrangements for pattern transformation. Our numerical analysis encompasses quasi-static analysis and microscopic/macroscopic instability analyses within the framework of linear perturbation. Subsequently, the pattern transformation of the identified particle arrangements is confirmed through quasi-static analyses employing detailed finite element (FE) simulations with continuum elements. Additional numerical simulations with continuum elements reveal that the pattern transformations of particle arrangements are significantly influenced by the initial void volume and some transformed granular crystals may exhibit strong low-frequency directional phononic band-gaps, which were not observed in the initial granular crystals.

Machine Learning Meets Physics-based Modeling: A Mass-spring System to Predict Protein-ligand Binding Affinity.

Computational assessment of the energetics of protein-ligand complexes is a challenge in the early stages of drug discovery. Previous comparative studies on computational methods to calculate the binding affinity showed that targeted scoring functions outperform universal models. The goal here is to review the application of a simple physics-based model to estimate the binding. The focus is on a mass-spring system developed to predict binding affinity against cyclin-dependent kinase. Publications in PubMed were searched to find mass-spring models to predict binding affinity. Crystal structures of cyclin-dependent kinases found in the protein data bank and two web servers to calculate affinity based on the atomic coordinates were employed. One recent study showed how a simple physics-based scoring function (named Taba) could contribute to the analysis of protein-ligand interactions. Taba methodology outperforms robust physics-based models implemented in docking programs such as AutoDock4 and Molegro Virtual Docker. Predictive metrics of 27 scoring functions and energy terms highlight the superior performance of the Taba scoring function for cyclin- dependent kinase. The recent progress of machine learning methods and the availability of these techniques through free libraries boosted the development of more accurate models to address protein-ligand interactions. Combining a naïve mass-spring system with machine-learning techniques generated a targeted scoring function with superior predictive performance to estimate pKi.

Liquid concentration sensing via weakly coupled point defects in a phononic crystal

A pair of weakly coupled point defects in a liquid–solid phononic crystal are employed for liquid concentration sensing. An exemplary case of methanol in ethanol is investigated. As the physical and chemical properties of the two constituents are very similar, sensing is a rather difficult task by other analytical means. The defects are formed by thin-walled polyethylene tubing filled with either the reference liquid (ethanol) or the mixture in a square phononic crystal composed of cylindrical steel rods in water. Finite-element method simulations for the dual defect system, which has a small form factor comparable to the acoustic wavelength in water, revealed that two sharp transmission peaks corresponding to the even and odd modes arising from symmetry lifting due to weak defect interaction behave in different manners as the methanol concentration is varied. Specifically, the even mode frequency exhibits a linear dependence on the concentration, whereas the odd mode is associated with a quadratic shift. Besides, a simple coupled mass-spring model is employed to analyze the observed peak behavior as a function of methanol concentration in ethanol. Accordingly, when the acoustic properties of the two liquids in a binary mixture vary in a parallel manner with temperature, the ratio of the resonance frequencies depends only on the concentration, thus offering a sensing scheme immune to temperature fluctuations. Furthermore, it is shown for a number of water-miscible liquids that the even and odd mode frequencies are representative of the dominant contents of the reference and the sample cells.

Mass-spring model for elastic wave propagation in multilayered van der Waals metamaterials

Multilayered van der Waals (vdW) materials have attracted increasing interest because of the manipulability of their superior optical, electrical, thermal, and mechanical properties. A mass-spring model (MSM) for elastic wave propagation in multilayered vdW metamaterials is reported in this paper. Molecular dynamics (MD) simulations are adopted to simulate the propagation of elastic waves in multilayered vdW metamaterials. The results show that the graphene/MoS2 metamaterials have an elastic wave bandgap in the terahertz range. The MSM for the multilayered vdW metamaterials is proposed, and the numerical simulation results show that this model can well describe the dispersion and transmission characteristics of the multilayered vdW metamaterials. The MSM can predict elastic wave transmission characteristics in multilayered vdW metamaterials stacked with different two-dimensional (2D) materials. The results presented in this paper offer theoretical help for the vibration reduction of multilayered vdW semiconductors.

Equivalent analytical model for liquid sloshing in a 2-D rectangular container with multiple vertical baffles by subdomain partition approach

An equivalent analytical model of sloshing in a two-dimensional (2-D) rigid rectangular container equipped with multiple vertical baffles is presented. Firstly, according to the subdomain partition approach, the total liquid domain is partitioned into subdomains with the pure interface and boundary conditions. The separation of variables is utilized to achieve the velocity potential for subdomains. Then, sloshing characteristics are solved according to continuity and free surface conditions. According to the mode orthogonality of sloshing, the governing motion equation for sloshing under horizontal excitation is given by introducing generalized time coordinates. Besides, by producing the same hydrodynamic shear and overturning moment as those from the original container-liquid-baffle system, a mass-spring analytical model of the continuous liquid sloshing is established. The equivalent masses and corresponding locations are presented in the model. The feasibility of the present approach is verified by conducting comparative investigations. Finally, by utilizing normalized equivalent model parameters, the sloshing behaviors of the baffled container are investigated regarding baffle positions and heights as well as the liquid height, respectively.

Relation between soft tissue energy dissipation and leg stiffness in running at different step frequencies.

This study aims to investigate the relationship between soft tissue energy dissipation and leg stiffness during running. Eight recreational healthy male runners (age: 22.2 ± 1.0 years; height: 1.84 ± 0.03 m; mass: 73.7 ± 5.7 kg) were asked to run at different speeds and step frequencies. Their soft tissue energy dissipation was estimated by the difference between the total mechanical work of the body, measured as the work done to move the body centre of mass relative to the surroundings plus the work to move the limbs relative to the body centre of mass, and lower-limb joint work. A mass-spring model with an actuator was used to analyse the force-length curve of the bouncing mechanism of running. In this way, the stiffness and damping coefficient were assessed at each speed and step frequency. Pearson's correlations were used to describe the relationship between the deviation from the spring-mass model and soft tissue energy fluctuations. The soft tissue dissipation was found to be significantly influenced by step frequency, with both positive and negative work phases decreasing when step frequency increases. Moreover, deviation from a spring-mass model was positively associated with the amount of soft tissue dissipation (r > 0.6). The findings emphasize the substantial role of soft tissues in dissipating or returning energy during running, behaving in a damped-elastic manner. Also, we introduce a novel approach for evaluating the elastic rebound of the body during running. The insights gained may have broad implications for assessing running mechanics, with potential applications in various contexts.

Investigation on vibration characteristics of rock based on circular plate and cylinder model: dimension, geometric shape and boundary condition

Abstract The declaration on the “natural frequency of rock” exists in many engineering areas, and it has caused many misunderstandings. Different from the mass-spring model usually used, the circular plate model and cylinder model are respectively established to clarify the relationship between the vibration characteristics (including natural frequency and vibration mode) and their influencing factors of rock by modal analysis. The effect of dimension, geometric shape, and boundary condition on the vibration characteristics of rock with plate structure is investigated, in which the semi-analytical solutions agree well with the simulation results. By using the cylinder model based upon the Lamé-Navier Eq., the effect of such influencing factors on the vibration characteristics of the block rock sample is further studied and verified by numerical simulation and experimental results. The results suggest that the natural frequency of “rock” (including the experimental rock sample) is strongly dependent on the dimension, geometric shape, and boundary condition. The resonance frequency observed in the excitation experiment is not only closely associated with the natural frequency of a specific order, but also dependent on the dominance of the particular vibration mode. These findings contribute to a better understanding of the rock-breaking mechanism under dynamic loads with a certain excitation frequency.

Examination of Precast Concrete Movement Subjected to Vibration Employing Mass-Spring Model with Two Convective Masses

This article investigates the movement of concrete subjected to vibration employing mass-spring model. For this purpose, two different prefabricated concrete-formworks, on which experimental work was done before, were analysed analytically. The theoretical modeling of precast formworks employed in experiments has been made by the three-dimensional finite element method employing the SAP2000 program. Modeling of mortar is performed to resolve the interaction between the fresh concrete and formwork using the mass-spring model. Considering the dynamic behavior of the fluid, it is possible to define multiple oscillation (convective) masses with different frequency values in addition to the impulse mass. Thus, one impulse mass and two convective masses were used in the mass-spring model. This study is essentially theoretical, and its accuracy has been strengthened by experimental work. The time-dependent results of concrete movement obtained from the dynamic mass-spring model were compared with the measured ones. The matches indicate that the findings are consistent.

Impact dynamic response of stiffened porous functionally graded materials sandwich doubly-curved shells with Arc-type auxetic core

The Arc-type auxetic core is based on the traditional concave hexagonal core and incorporates a bending configuration, facilitating a smooth transition of the cell elements. This modification proves highly effective in alleviating stress concentration. To extend its use in sandwich structures, the dynamic behavior of stiffened porous functionally graded materials (PFGMs) doubly-curved sandwich shell with Arc-type auxetic core under low-velocity impact is a concern that this research addresses. The material properties of external PFGM panels are controlled by two grading patterns (P-FGM and E-FGM) and has four different porosity distributions. The core layer is made of Arc-type auxetic core. A theoretical model that utilizes Hertzian elasticity theory and first-order shear deformation theory (FSDT) is proposed, and the equations of motions are derived using the Hamilton's principle. In addition, to simulate the contact force interaction in dynamic process, the spring-mass model is employed. The analytical solution is presented by using Duhamel's principle and Navier's method to anticipate the transverse displacement. The effectiveness of the obtained data is verified by comparing the findings with those of the existing literature and the numerical results established by the Abaqus commercial software. On this basis, efficient methods for optimizing the PFGM doubly-curved sandwich shell are presented by analyzing the effect of porosity, porosity distribution, gradient index, grading pattern, auxetic core inclined angle and circular arc radius on the energy absorption capability, wherein the sandwich shell of P-FGM reduces the transverse displacement by 24.5 % over E-FGM, and the corresponding gain effects for different porosity distributions are Type B > Type D > Type C > Type A.

Improving low-frequency sound transmission loss of double panels with a plate-type acoustic metamaterial

The sound insulation performance of double-leaf partitions is crucially affected by the mass-air-mass resonance phenomenon, whose frequency reduction can shift the high sound insulation to lower frequencies. In this work, an alternative strategy for exceeding the low limit of the mass-air-mass resonance frequency of conventional double panels is proposed in terms of the effective surface mass density. An equivalent spring-mass model based on the effective medium theory is implemented to intuitively understand the characteristics. A simple double-panel metamaterial is designed by replacing one of the panels with a single-plate metamaterial attached strip masses to decrease the structural complexity. The extended semi-analytical method is applied to predict efficiently the sound transmission loss in the low-frequency range, which is verified by the coupled vibro-acoustic finite element method with excellent agreement. The results demonstrate that the proposed double-panel metamaterial can reduce the minimum of the resonant frequency without changing the areal mass density and panel spacing. The sound insulation can be significantly improved between the reduced mass-air-mass resonance and the anti-resonance frequencies. To provide a better understanding of the sound transmission loss improvement, parametric studies are conducted. The design strategy is validated through experiments conducted in an impedance tube, and the results corroborate the findings. This study is instrumental in designing double-panel partitions that require sound insulation in specific low-frequency regions.